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+{"text":"\\section{Introduction \\label{sec:intro}}\nAt very beginning of enrichment history of the Galactic halo,\ninterstellar medium (ISM) was certainly chemically inhomogeneous, \nbecause star forming regions at that time were locally confined \nand stellar ejecta did not spread uniformly throughout the halo.\nOnly a limited number of Type II supernovae (SN\\,II) exploded\nand a mean spatial separation of SN\\,II was much larger than \na typical radius of single supernova remnant (Audouze \\& \nSilk 1995). Thus, the ISM was enriched locally. \nA trace of such inhomogeneous ISM should be frozen \nin extremely metal-deficient \nstars which we observe today (e.g. McWilliam 1997).\n\nNevertheless, no clear evidence which suggests  \nthat the halo was inhomogeneous in the past has so far been found. \nMcWilliam et al. (1995), Audouze \\& Silk (1995), \nand McWilliam (1997) claimed that large scatters of {\\it heavy metal}  \nabundances (e.g., Sr, Y, Zr, Ba, and Eu) \nof extremely metal-poor halo stars (Beers, Preston, \\& Shectman 1992; \nMcWilliam et al. 1995; \nRyan, Norris, \\& Beers 1996) imply that the early halo was inhomogeneous. \nHowever, these scatters may \nsimply reflect various lifetime of stars producing heavy metals, \nwhose origins are poorly understood yet. \nThe abundance of each heavy metal should sharply rise at a time when \nbulk of massive stars that produce it start to explode. Thus stars formed \nduring such epoch should show a large scatter in relative abundance of \nheavy metals with \nrespect to iron while they have nearly identical iron content. \nTherefore large scatters of heavy metals at certain values of [Fe\/H] do \nnot necessarily imply an inhomogeneous ISM in an early halo. \n\nOn the other hand, even if relative abundance of $\\alpha$-elements \n(e.g., O, Mg, Si, Ca, etc) with respect to iron do show little scatter \nat any value of [Fe\/H], it does not necessarily mean that an early \nhalo was homogeneous. Even if a star formation history of each region \nin the halo is different from each other, evolutionary paths on the \n[$\\alpha$\/Fe]-[Fe\/H] plane should be nearly identical in an early \nevolutionary stage during which SN\\,II were dominant sources of nuclear \nenrichment. The evolutionary path depends on a star formation rate (SFR) \nvery weak. \n\nAlthough little is known about an early halo, there seems to be a clear \nevidence for the inhomogeneous ISM in a solar neighbourhood disc; \nlong-lived dwarfs (e.g., Edvardsson et al 1993) show a spread of $\\sim1$\\,dex \nin iron abundance along poorly defined age-metallicity relation. \nBy relaxing a usual assumption of well mixing, Copi (1997) estimated \nchemical evolution of the solar neighbourhood in a Monte Carlo fashion. \nThe solar neighbourhood is modeled by 1000 independent \nregions. The evolutionary history of one particular region is determined \nrandomly based on the SFR and the initial mass function (IMF). \nAlthough he successfully reproduced the observed spreads of  elemental ratios, \nhe failed to explain the scatter appeared in the age-metallicity relation. \nBateman \\& Larson (1993) suggested that random walk processes of atomic \nand molecular gas clouds are dominant mixing mechanisms of iron in the \npresent day solar neighbourhood disc. Wielen, Fuchs, \\& Dettbarn (1996) \nsuggested that stellar orbital diffusion in combination with radial abundance \ngradients can induce inhomogeneity in the ISM. However, van den Hoek \\& \nde Jong (1997) showed that the stellar orbital diffusion  \ndoes not explain the abundance variation sufficiently. They suggested instead \nthat a sequential star formation and an infall of metal-deficient gas \nplay an important role in preventing the ISM from being mixed. \nWilmes \\& K\\\"oppen (1995) studied chemical evolution of isolated \nindividual ISM parcels and showed that mixing is inefficient \nin the galactic disc. Pilyugin (1996) suggested that major galaxy mergers\nform multiple stellar populations of different metallicities in the disc. \nWhile these mechanisms can well explain the scatters along the age-metallicity \nrelation, one should also keep in mind that stellar abundances may not reflect \nabundances of ISM from which stars formed, \nbecause chemical condensation processes \nsuch as grain formation in circumstellar envelope (e.g. Henning \\& \nG\\\"urler 1986) and\/or thermal diffusion in stellar \natmosphere (e.g. Bahcall \\& Pinsonneault 1996) might work efficiently.\n\nIf the early Galactic halo was chemically inhomogeneous, the metallicity \nspread should appear among halo stars of the same age. Unfortunately, \nthe present day stellar isochrone fitting cannot distinguish age of \nstars born in a young halo. Therefore, it is not clear at all if the \nmetallicity spread exists among coeval halo stars. Instead, we show \nin this paper that one can use an observed cumulative metallicity \ndistribution function of halo stars to constrain stochastic chemical \nevolution models for an early halo and indicate that the Galactic halo \nhas never been chemically homogeneous in its history. \n\nIn section 2, we give prescriptions for our stochastic chemical evolution \nmodels, and in section 3 we describe behaviours of theoretical metallicity \ndistributions. We confront model results with observational data in section 4 \nand give discussions and conclusions in section 5 and 6, respectively. \n\n\\section{Model \\label{model}}\n\\subsection{Outline} \nWe have built up stochastic chemical evolution models for the Galactic halo. \nWe assume that the halo is spheroidal and that the gas of primordial \nabundance distributed uniformly at the beginning. We divide the halo into \nmany cubic blocks having a volume of $l_{\\rm b}^3=4 \\pi R_{\\rm m}^3\/3$, \nwhere $R_{\\rm m}$ \nis a radius of supershell caused by SN\\,II that contributed dominantly \nto an early stage of chemical evolution. Progenitor of SN\\,II are massive \nstars which tend to form in star clusters and associations \n(Blaauw 1964; Humphreys 1978; Heiles 1987).      \nSupernova remnants in OB associations and star clusters are suggested \nto produce surrounding shell structures, or supershells \n(e.g., Tenorio-Tangle \\& Bodenheimer 1988). \nCash et al. (1980) predicted that the supershell is produced after a series \nof SN\\,II explosions.  A radius of supershell can roughly be given as \na maximum size of the area enriched by synthesised heavy elements. \nThus we assume that a volume of the block is equal to that of the supershell. \nThe ISM in the supershell should be well-mixed, because \nstellar winds and SN\\,II explosions induce \nthe Rayleigh-Taylor and the Kelvin-Helmholtz instabilities\n(Tenorio-Tagle \\& Bodenheimer 1988; Allen \\& Burton 1993). \nThe time scale of mixing is less than 0.0015 Gyr in a case of isonised gas \n(Roy \\& Kunth 1995), thus is much shorter than a typical lifetime \n($\\sim 0.01$\\,Gyr) of the OB association. Once the mixing took place, \nthe ISM in a block uniformly enriched by newly synthesised elements. \nTherefore, individual regions can be regarded as independent one-zones and \na standard chemical evolution model can be applied for each of them. \n\nBefore an onset of star formation, all blocks have identical physical \nproperties. We therefore follow chemical enrichment histories of \n1000 neighbouring blocks instead of the all. Unless a number of blocks \nconsidered is too small, the resulting behaviour of each block is \nsufficiently stable. By simulating stochastic chemical evolution for 1000 \nblocks simultaneously, we calculate time variations of abundances in \nindividual blocks and study chemical inhomogeneity in the halo. \n\n\\subsection{Block Size}\nIn an analogy to an expansion of single supernova, \nwe roughly estimate a radius of the supershell as follows, \nalthough more detailed models for the supershells were \npublished by several authors \n(Bruhweiler et al. 1980; McCray \\& Kafatos 1987; Mac Low \\& McCray 1988). \nCioffi, McKee, \\& Bertschinger (1988) gave an analytical \nestimate for a radius $r_{\\rm m}$ of a supernova remnant (SNR): \n\\begin{equation}\nr_{\\rm m}=69~E_{51}^{0.32}~n^{-0.41}~v_{10}^{-0.43}~(Z\/Z_\\odot)^{-0.05}~~{\\rm{pc}}, \n\t\\label{eqn:Rm}\n\\end{equation}\nwhere $E_{51}$ is an initial kinetic energy of the SNR \nin units of $10^{51}$ erg, $n$ is the number density of surrounding ISM, \n$v_{10}$ is an ISM velocity dispersion in units of 10 km s$^{-1}$, \nand $Z\/Z_\\odot$ is the ISM heavy element abundance. We calculate a radius \n$R_{\\rm m}$ of the supershell by\nreplacing $E_{51}$ with $n_{\\rm{II}}\\,E_{51}$, where $n_{\\rm{II}}$ is the number of SN\\,II in an OB association. \nHereafter we adopt $E_{51}=1$ (Shigeyama, Nomoto, \\& Hashimoto 1988). \nWe assume that the velocity dispersion of ISM is equal to the sound velocity,  \nand adopt $v \\simeq 10$~km\\,s$^{-1}$ or $v_{10} \\simeq 1$, since  \nthe temperature should be near $10^4$~K (Hoyle 1953; Silk 1977) in the \ninitial halo. \nSince we are interested in chemical evolution in the halo, \nwe fix $Z\/Z_\\odot=0.06$ which corresponds to \n[O\/Fe]$\\simeq 0.4$ (Barbuy 1988; Nissen et al. 1994) and    \n[Fe\/H]$=-1.6$; i.e., a peak iron abundance \nof the metallicity distribution function \nobtained for long-lived stars in the halo (Laird et al. 1988). \n\nStars having $M> 8 M_\\odot$, corresponding to main-sequence spectral type \nB3, eventually become SN\\,II. The number of supernovae $n_{\\rm{II}}$ \nis the number of such stars per OB association. We adopt $n_{\\rm{II}}=40$ \naccording to Heiles (1987) who derived the average number of stars \nwith $M> 8 M_\\odot$ per clusters from actual counting of O stars in clusters \nin the solar neighbourhood together with the IMF derived from clusters in the \nsolar vicinity. The number Heiles (1987) got was $\\sim 28$, which was then \ncorrected for runaway O stars that will deposit energy within the supershell. \nSignificant deviations from the average value of $n_{\\rm{II}}$ are \napparently rare although $n_{\\rm{II}}$ may cover a wide range of values \n(Tenorio-Tangle \\& Bodenheimer 1988). \n\nThe number density of the ISM is given as \n$n=M_{\\rm h}\/(\\mu m_{\\rm H} V_{\\rm h})$, where $m_{\\rm H}$ is the proton mass, \n $M_{\\rm h}$ is the mass, $V_{\\rm h}=\\frac{4}{3} \\pi R_{\\rm h}^3$ is \nthe halo volume, and $\\mu=1.3$ is a mean molecular weight corresponding \nto the primordial compositions.\n\nSaito (1979) derived an empirical relation between   \nthe binding energy $\\Omega_{\\rm G}$ and the mass $M_{\\rm G}$ \nby analysing surface brightness distributions\nand line-of-sight velocity dispersions of spheroidal systems:   \n\\begin{equation}\n\\Omega_{\\rm G}=1.66 \\times 10^{60} \\left[\\frac{M_{\\rm G}}\n{10^{12}M_\\odot} \\right]^{1.45}~~~{\\rm{erg}}.  \n\\end{equation}\nUnder the assumption of spherical geometry, \nthe radius $R(M_{\\rm G})$ of a galaxy is given as \n\\begin{equation}   \nR(M_{\\rm G})\n=26.1 \\left[\\frac{M_{\\rm G}}{10^{12}~M_\\odot} \\right]^{0.55}~~\\rm{kpc}.\n\t\\label{eqn:Rh}\n\\end{equation}\nWe assume that the early Galactic halo follows \nthe same mass-radius relation given by equation (\\ref{eqn:Rh}). \nThe virial theorem tells that a system with initially no kinetic energy \nattains virial equilibrium by reducing a radius \nto half the initial value.  Since we study chemical evolution \nat the very beginning, the radius of halo $R_{\\rm h}$ \nshould be taken as $2R(M_{\\rm h})$. \n\nIf we assume the halo was $M_{\\rm h}= 4 \\cdot 10^{11}~M_\\odot$ \n(Fich \\& Tremaine 1991), then we obtain $R_{\\rm h}=2R(M_{\\rm h})=31.5$~kpc, \nand $n=0.1$cm$^{-3}$. According to equation (1), the supershell radius \n$R_{\\rm m}$ depends weakly on $n$. Since the amount of gas condensed into \nstars in the early stage of halo evolution is at most 25 percent, \n$R_{\\rm m}$ changes only by 12 percent. We therefore assume $n$ constant \nin time. Putting $n_{\\rm{II}}E_{51}=40$, $n=0.1$cm$^{-3}$, \n$v_{10} \\simeq 1$, and $Z\/Z_\\odot=0.06$ into equation (1), we obtain \n$R_{\\rm m} \\sim 660$~pc and $l_{\\rm b} \\equiv (4\\pi R_m^3\/3)^{1\/3}=1~$kpc \nfor our standard model. \n\nA time $t_{\\rm m}$ when a SNR merges with the surrounding ISM\nis given by (Cioffi et al. 1988), \n\\begin{equation}\nt_{\\rm m}=2 \\cdot 10^6E_{51}^{0.32}~n^{-0.37}~v_{10}^{-1.43}~(Z\/Z_\\odot)^{-0.05}~~{\\rm{yr}},\n\t\\label{eqn:Tm}\n\\end{equation}\nand a cooling time  $t_{\\rm c}$ inside a SNR (Cox 1972) as, \n\\begin{equation}\nt_{\\rm c}=5.7 \\cdot 10^4~E_{51}^{4\/17}~n^{-9\/17}~~{\\rm{yr}}.  \n\t\\label{eqn:tc}\n\\end{equation}\nA lifetime of supershell $T_{\\rm m}$ and a cooling time $T_{\\rm c}$ \ninside the supershell can be derived by replacing $E_{51}$ in equations (4) \nand (5) with $n_{\\rm II}E_{51}$, respectively. Assuming the same parameters \ndiscussed above, we obtain $T_{\\rm m} \\simeq 1.7 \\cdot 10^{-2}$~Gyr and \n$T_{\\rm c} \\simeq 4.6 \\cdot 10^{-4}$\\,Gyr. We note that $T_{\\rm m}$ \nis much longer than $T_{\\rm c}$. \n\n\\subsection{Star Formation Probability}\nConsidering an idealised situation that the OB associations distribute \nuniformly in the halo, we assume that a star formation probability \n$P_{\\rm{SF}}(t)$ in a block for a time interval $t$ and $t+\\Delta t$ \nis given as,  \n\\begin{eqnarray}\nP_{\\rm{SF}}(t) & = & \\frac{l_{\\rm b}^3}{V_{\\rm h}} \\frac{1}{n_{\\rm{II}}}\n\\int^{t+\\Delta t}_{t}N_{\\rm{II}}(t') dt' \\nonumber \\\\\n\t& & \\nonumber \\\\\n       & = & \\frac{3}{4 \\pi} \\left(\\frac{l_{\\rm b}}{R_{\\rm h}}\n \\right)^3 \\frac{1}{n_{\\rm{II}}} \\int^{t+\\Delta t}_t N_{\\rm{II}}(t') dt', \n\\end{eqnarray}\nwhere $N_{\\rm{II}}(t)$ is the total number of progenitor \nof SN\\,II in the halo and is given as,  \n\\begin{equation}\nN_{\\rm{II}}(t)\n=M_{\\rm h} C(t) f_{\\rm{II}},  \n\t\\label{eqn:nsfb}\n\\end{equation}\nwhere $C(t)$ is the SFR per unit mass and $f_{II}$ is the number fraction \nof SN\\,II defined as, \n\\begin{equation}\nf_{\\rm{II}}= \\int_{m_{\\rm{II}}}^{m_u} \\phi(m) m^{-1}dm,\n\\end{equation}\nwith $m_l=0.1~M_\\odot$, $m_u=50~M_\\odot$, and $m_{\\rm{II}}=10~M_\\odot$  \nas the lower and the upper mass limits, \nand the lower SN\\,II mass limit, respectively. The IMF adopted here has \nmass spectrum independent of time:\n\\begin{equation}\n\\phi(m)=\\frac{(x-1)~m_l^{x-1}}{1-(m_l\/m_u)^{x-1}}~m^{-x},~~(m_l\\le m \\le m_u). \n\t\\label{eqn:imf}\n\\end{equation}\nWe adopt the Salpeter IMF (Salpeter 1955) which has a slope $x=1.35$ \nin this definition. For $C(t)$, we adopt the Schmidt law (Schmidt 1959):   \n\\begin{equation}\nC(t)=\\omega^{-1}~G(t)^{\\,p}~~~~~(p=1),\n\\end{equation}\nwhere $\\omega$ and $G(t)$ are time scale of star formation \nand a gas fraction in the halo, respectively.  \n $P_{\\rm{SF}}(t)$ is thus finally given as:\n\\begin{eqnarray}\n\\lefteqn{P_{\\rm{SF}}(t) = \\frac{3}{4 \\pi}\\,\\left(\\frac{l_{\\rm b}}{R_{\\rm h}}\n\\right)^3 n_{\\rm{II}}^{-1} f_{\\rm{II}}\n\\,M_{\\rm h}\\, \\omega^{-1} G(t)^p\\, \\Delta t}\\nonumber \\\\\n&& \\hspace{4cm}(p=1). \n\t\\label{eqn:psf} \n\\end{eqnarray}\n\nWe adopt $m_{\\rm II}=10M_\\odot$ in equation (8), which is slightly larger than \n$m_{\\rm II}=8M_\\odot$ which Heiles (1987) adopted.  \nThus, equation (11) may underestimate $P_{\\rm{SF}}(t)$. \nA difference corresponding to the adopted $m_{\\rm II}$ is \n$\\sim 12$ percent. \nHowever, an exact value of $m_{\\rm II}$ is still uncertain and, therefore, \nwe adopt $m_{\\rm II}=10M_\\odot$ for our standard model and change it later \nas a free parameter. \n\n\\subsection{Chemical Evolution}\nFor the blocks in which stars are born (hereafter the star forming blocks),  \nwe apply a chemical evolution model developed by \nPagel \\& Tautvai$\\check{{\\rm s}}$ien$\\dot{{\\rm e}}$ \n(1995; hereafter PT95). We adopt the iron as a tracer of chemical evolution. \nIn PT95, the iron abundance is calculated \nby using an instantaneous recycling approximation \nand a delayed production approximation  \nwhich assumes that the iron is additionally produced  \nby Type Ia supernovae (SNIa) after a fixed time lag $\\tau_1$.\nWe assume that the SFR $c(t)$ in a block is proportional \nto the gas fraction $g(t)$. The iron abundance $Z_{\\rm{Fe}}(t)$ is given by, \n\\begin{footnotesize}\n\\begin{eqnarray} \n\t\\label{eqn:delay}\n\\lefteqn{\\frac{d(gZ_{\\rm{Fe}})}{dt}}\\nonumber  \\\\\n&&= \\left\\{\n   \\begin{array}{ll}\n\t-Z_{\\rm{Fe}}(t)c(t)+p_0c(t), & (t < \\tau_1),\\\\  \n        -Z_{\\rm{Fe}}(t)c(t)+p_0c(t)+p_1c(t-\\tau_1), & (t \\ge \\tau_1),\n    \\end{array}\n \\right. \n\\end{eqnarray}\n\\end{footnotesize}\nwhere $p_0$ and $p_1$ are yields of iron corresponding to the instantaneous \nrecycling approximation and the delayed production approximation, \nrespectively.  \nThe first term on the right hand side (r.h.s.) represents \nthe net amount of iron locked into newly formed stars. \nThe second and the third terms are \ncorresponding to the instantaneous recycling approximation\nand the delayed production approximation, respectively. \nWe adopt $p_0=0.28$, $p_1=0.42$, and $\\tau_1=1.3$~Gyr   \naccording to PT95. \nSince the theoretical yields of the iron are uncertain due to difficulties \nin modeling the explosion mechanism (Timmes, Woosley, \\& Weaver 1995), \nwe use the yields of PT95 which are calibrated empirically. \n  \nThe SFR in each block $c(t)$ is determined by an assumed number of SN\\,II \nexploded in a block during a time interval $t$ and $t + \\Delta t$:   \n\\begin{equation}\n\\int^{t+ \\Delta t}_{t}m_{\\rm b} f_{\\rm{II}} c(t')~dt'   \n=n_{\\rm{II}},   \n\\end{equation}\nwhere $m_{\\rm b}= M_{\\rm h} l_{\\rm b}^3\/V_{\\rm h} \n\\simeq 3 \\cdot 10^6\\,M_\\odot$ is the initial gas mass of each block. \n\n\\subsection{Standard Models}\nTable\\,1 gives a list of parameters adopted for our standard models\\,S1, \nS2, and S3. In model\\,S1 we assume that the OB associations are formed \nrandomly at every $\\Delta t \\sim \\frac{1}{3}T_{\\rm m}=0.006$\\,Gyr. \nIn model S2 we introduce an effect of periodic mixing due to the \nturbulent diffusion. A particle with a sound velocity $v=10 {\\rm km\\,s^{-1}}$ \nwould across a block of size $l_{\\rm b}=1$\\,kpc in $T_{\\rm d}=0.1$\\,Gyr. \nThus, we assume in this model that every 0.1\\,Gyr the turbulent diffusion \nmixes the 27 blocks surrounding the star forming block periodically. \n\n\\begin{table}[t]\n\\small\n\\begin{center}\nTable~1. \\hspace{4pt}Parameters of standard model.\\\\\n\\end{center}\n\\vspace{6pt}\n\\begin{tabular*}{\\columnwidth}{@{\\hspace{\\tabcolsep}\n\\extracolsep{\\fill}}lll}\n\\hline\\hline\\\\[-6pt]\n$M_{\\rm h}$ & $4~10^{11}~M_\\odot$ & mass of the halo\\\\\n$R_{\\rm h}$ & 31.5~kpc & radius of the halo\\\\\n$\\omega$ & 5~Gyr & time scale of star formation in the halo \\\\\n$l_{\\rm b}$ & 1~kpc & size of a block\\\\\n$v_{10}$ & 1 & velocity dispersion of ISM \\\\\n & & \\hspace{2cm} in units of 10 km~s$^{-1}$ \\\\\n$m_l$ & 0.1$M_\\odot$ & IMF lower mass limit \\\\\n$m${\\tiny II} & 10$M_\\odot$ & lower mass limit for SN\\,II\\\\\n$n_{\\rm{II}}$ & 40 & number of SNe\\,II in a block\\\\\n$\\Delta t$ & $6 \\cdot 10^6$~yr & duration of star formation\\\\\n\\hline\n\\end{tabular*}\n\\end{table}\n\n The supershell structures are often mentioned as a trigger of massive star \nformation. The evidences have been discussed by many authors  \n(Blaauw 1964; Elmegreen \\& Lada 1977; \nLada, Blitz, and Elmegreen 1979; Elmegreen 1982; 1985a, b).\nIf OB associations form in this way, \nthey should be born only in the surrounding area of the \nstar forming regions and chemical enrichment should be locally confined \nthere. The effects of this stimulated star formation are studied in model~S3. \nWe first distribute OB associations randomly and then assign randomly \na star forming block for the next generation among 27 blocks adjoining to \neach star forming block of the first generation and so on. In this way, \nthe star formation propagates from one block to another. \n\nThe time step adopted for each simulation is 0.001\\,Gyr. Calculations are \nstopped at 1\\,Gyr, since our main interest is in early chemical evolution \nof the halo.  \n\nThe time scale of star formation $\\omega$ in the halo \nis quite uncertain. On the contrary, it is quite often assumed that\n$\\omega$ in elliptical galaxies is of the order\nof free-fall time $t_{\\rm ff}=(3 \\pi \/ 32G \\bar{\\rho})^{1\/2}$\n(Larson 1974). One could assume that $\\omega$ in the halo is the same\nas that in giant ellipticals. However, mean stellar metallicities\nof giant ellipticals are nearly solar (Arimoto et al. 1997), \nwhile that of Galactic halo stars  \nis $\\sim$ 1\/40 solar (Ryan \\& Norris 1991; Carney et al. 1996).\nBoth in giant ellipticals and in the galactic halo, Mg seems to\nbe enhanced with respect to Fe by at least a factor of 2\n(McWilliam et al 1995; Ryan et al 1996). \nThis implies that star formation in both systems is  \ncharacterised by the same IMF and that the star formation stopped\nbefore the onset of SNIa explosions ($\\sim 1$ Gyr).\nIf this is the case, $\\omega$ in the Galactic halo \nshould be 40 times longer than that of giant ellipticals.\nThus we obtain $\\omega \\simeq 40 t_{\\rm ff} \\sim 6$\\,Gyr \nfor our halo model with $M_{\\rm h}= 4 \\cdot 10^{11}~M_\\odot$ and \n$R_{\\rm h}=31.5$~kpc.\nSince $\\omega \\sim 6$\\,Gyr is very close to $\\omega  \n\\sim 5$\\,Gyr derived for the solar neighbourhood disc (Arimoto, \nYoshii, \\& Takahara 1992), we shall adopt $\\omega=5$\\,Gyr in our standard \nmodels. In other words, we adopt the same SFR fo the Galactic\nhalo and the solar neighbourhood disc.\n  \n\\section{Metallicity Distribution \\label{subsubsec:sk}}\n\nFigure~1 shows the number fraction of blocks \nwhich at least once have experienced the chemical enrichment. Model\\,S1 \npredicts  that all the blocks have been enriched before $\\sim 0.2$~Gyr. \nIn model\\,S2, this happens earlier than model\\,S1,  because the iron \nspreads beyond the supershells due to the turbulent diffusion, \nthus much wider area are enriched even if the SFRs \nare the same as in model\\,S1. On the other hand, \nnearly 8\\% \nof the block in model\\,S3 has never been enriched till 1~Gyr, \nthis is because the stimulated star formation \ntends to localise the chemical enrichment. \n\\begin{figure}\n\\epsfxsize=150pt\n\\epsfbox{fig01.epsi}\n\\caption[]{The evolution of the number fraction of blocks \nwhich at least once have experienced chemical enrichment \nin the standard models\\,S1, S2, and S3. \n}\n\\end{figure}\n\n\nFigures\\,2a~and~2b show the evolution of mean and median iron abundances, \nrespectively. \nHereafter we discuss statistical properties \nexcept for the first $0.04$\\,Gyr, during which the number of \nenriched blocks is too small to define the statistical properties.   \nFigures\\,2a~and~2b indicate that the median metallicities of \nmodels\\,S1~and~S2 are nearly the same as the mean metallicities. \nThis suggests that the metallicity distributions of the two models \nhave symmetric shapes. The median of model\\,S3 is always higher \nthan the mean, indicating that the metallicity distribution has a tail \ntoward lower metallicity.\n\n\\begin{figure}[h]\n\\begin{center}\n\\epsfxsize=9cm\n\\epsfbox{fig02.epsi} \n\\caption[]{\n(a)~~The evolution of the mean iron abundance \nin the standard models\\,S1, S2, and S3. \n(b)~~\nThe same as figure\\,2(a), but for the median iron abundance. \n}\n\\end{center}\n\\end{figure}\n\nFigure~3 illustrates a frequency distribution of iron abundance  \nof the 1000 blocks, $N$([Fe\/H]), of model\\,S1 at 0.01, 0.05, 0.1, 0.2, 0.5, \nand 1\\,Gyr. For an illustrating purpose, $N$($-5 <$[Fe\/H]$\\le -4.8$) \nis artificially assigned for \na fraction of metal-free blocks. Figure~3 shows that the metallicity \ndistribution of model\\,S1 has roughly 1.3 \\,dex spread of the iron \nabundance, suggesting that the ISM in the halo was inhomogeneous till 1\\,Gyr. \n\n\\begin{figure*}[t]\n\\begin{center}\n\\epsfxsize=9cm\n\\epsfbox{fig3.epsi}\n\\caption[]{\nThe frequency distribution of iron abundance of the 1000 blocks in model\\,S1 \nat 0.01, 0.05, 0.1, 0.2, 0.5, and 1\\,Gyr. \n}\n\\end{center}\n\\end{figure*}\n\\begin{figure}[h]\n\\epsfxsize=5cm\n\\epsfbox{fig4.epsi}\n\\caption[]{\nThe evolution of standard deviation of metallicity distribution \nin the standard models\\,S1, S2, and S3. \n}\n\\end{figure}\n\nFigure\\,4 gives the evolution of standard deviation of the metallicity \ndistribution obtained by the standard models. In model\\,S1, the standard \ndeviation keeps nearly constant at $s(t) \\simeq 0.24$ after $\\sim 0.2$\\,Gyr, \nwhich corresponds to roughly $1.3$\\,dex dispersion of the iron \nabundance. Model\\,S3 suggests much stronger abundance spread than model\\,S1. \nThe standard deviation of model\\,S3 increases with time and exceeds \n$0.5$ at 1\\,Gyr, which is roughly equivalent to 2~dex dispersion of the iron \nabundance among all the enriched blocks. \nOn the other hand, the standard deviation of model\\,S2 decreases  \nquickly to nearly zero, showing that the ISM was homogeneous \nfrom the very beginning of its chemical evolution.\n\n\\begin{center}\n\\begin{figure}\n\\epsfysize=10cm\n\\epsfbox{fig5.epsi}\n\\caption[]{\n(a)~~The evolution of skewness of metallicity distribution \nof the standard models\\,S1, S2\\, and S3. \n(b)~~The same as figure\\,5a, but for the kurtosis \nof metallicity distribution. \n}\n\\end{figure}\n\\end{center}\nNow we focus on the shape of the metallicity distribution. \nFigures~5\\,a~and~5\\,b illustrate the skewness and the kurtosis of the \nmetallicity distributions of the standard models, respectively \n(e.g. Stuart \\& Ord 1987). The skewness characterises a degree of asymmetry \nof a distribution around its mean. \nIt characterises only a shape of the distribution. The definition is \n\\begin{equation}\n{\\rm {Skew}}(x_1,~x_2,....,~x_N)= \\frac{1}{N} \\sum_{i=1}^{N}(\\frac{x_i-\\bar{x}}{s})^3,\n\t\\label{eqn:skew}\n\\end{equation}\nwhere $\\bar{x}$ and $s$ are the mean and the standard deviation \nof the measured value $x_1,~x_2,...,~x_N$. A positive value of skewness \nsignifies a distribution with an asymmetric tail extending out towards \nlarger $x$, while a negative value signifies a distribution whose tail \nextends out towards smaller $x$. \nThe definition of the kurtosis is\n\\begin{equation}\n{\\rm{Kurt}}(x_1,~x_2,....,~x_N)= \\{ \\frac{1}{N}\n \\sum_{i=1}^{N}(\\frac{x_i-\\bar{x}}{s})^4 \\} -3,  \n\t\\label{eqn:kurt}\n\\end{equation}\nwhere the first term of r.h.s. becomes 3 for a Gaussian distribution. \nThe kurtosis measures the relative peakedness \nor flatness of a distribution relative to the Gaussian distribution. \nThe distribution with positive kurtosis has the outline of the Matterhorn \nfor example. The distribution with negative kurtosis is outlined \nof a lump of meat-loaf. \nFigure \\,5a gives the evolution of skewness of the standard models. \nThe skewness of model\\,S1 is nearly zero, which means a symmetric \nshape of the metallicity distribution. The skewness of model\\,S2 behaves \nirregularly due to nearly null standard deviation.\nThe skewness of model\\,S3 is always negative and decreases\nas times goes on, \nshowing a tail of the metallicity distribution extending toward lower \nmetallicity at later stages. This is due to the localised chemical enrichment \ncaused by the \nstimulated star formation. Figure\\,5b shows the evolution of kurtosis. \nThe kurtosis of model\\,S1 is nearly equal to zero, thus \nthe metallicity distribution has a Gaussian like shape.\nOn the contrary, the kurtosis of \nmodel\\,S2 is very large. This model suggests the homogeneous ISM, \nsince the kurtosis should be infinitely large if samples have the same value. \nIn model\\,S3, the kurtosis gradually increases up to $\\sim$ 2, \nwhich means that \nthe metallicity distribution has a sharp peak. Thus the skewness and \nthe kurtosis of model\\,S3 indicate that the metallicity distribution \nhas a peaked shape with a tail extending to the lower metallicity. \n\nAs a summary, only model\\,S2 shows that the ISM was chemically \nhomogeneous in the early halo, while both models\\,S1~and~S3 show \nthat the ISM in the halo was not well-mixed till at least 1\\,Gyr. Indeed, \nthe scatters of the iron abundance are roughly\n1.4 dex in model\\,S1 and 2 dex in model\\,S3, respectively. \nWe have repeated the simulations several times  and confirm \nthat the model results show always the same tendencies. \n\n\\section{Observational Constraints \\label{subsubsec:gp}}\n\nLong-lived stars in the halo should keep the original metallicities\nof the ISM from which they formed. It is true that little observational\ninformation is available for constraining chemical evolution models\nof the halo, but we will show that a cumulative metallicity \ndistribution function of the\nlong-lived halo stars can potentially give a powerful clue to obtain the\nbest model prescriptions. \n\n\\subsection{Comparison with the Standard Models}\n\nFigures\\,6~and~7 present generalised metallicity \ndistribution functions (GMDFs), $P$([Fe\/H]), of the standard models\nS1 -- S3 together with the empirical GMDFs of long-lived halo stars\ntaken from Ryan \\& Norris (1991) and Carney et al. (1996). \n\\begin{figure*}[t]\n\\begin{center}\n\\epsfxsize=9cm\n\\epsfbox{fig6.epsi}\n\\caption[]\n{\nThe generalised metallicity distribution functions (GMDFs) of\nmodel\\,S1 at 0.01, 0.05, 0.1, 0.2, 0.5, and 1\\,Gyr (thick solid line).  \nThin solid and dotted lines illustrate the empirical halo GMDFs taken from \nRyan \\& Norris (1991) and Carney et al. (1996), respectively.\n}\n\\end{center}\n\\end{figure*}\nThe definition of GMDF is given by Laird et al. (1988):  \n\\begin{equation}\nP(x)=\\frac{1}{N \\sigma \\sqrt{2 \\pi}} \n\\sum_{i=1}^{N} \\exp\\left [-\\frac{(x-x_i)^2}{2 \\sigma^2} \\right ], \n\\end{equation}\nwhere $\\sigma$ reads as a typical error in the observed values\n$x_1$, $x_2$, ..., $x_N$. \n\nQuite often, the empirical metallicity distribution function is presented \nin the form of a histogram. However, Searle \\& Zinn (1978) suggested that \nthe bins of a conventional histogram should be replaced by a continuous \ndistribution function such as a Gaussian to obtain a better approximation \nto the actual abundance distribution function, since the binning distorts \nthe data. Therefore, we convolve the observed histograms of stellar\nmetallicity distribution to include the uncertainty of observational data \nproperly (Laird et al 1988) and \nconvolve the theoretical GMDFs by using equation (16) \nwith $\\sigma=0.15$ (Ryan \\& Norris 1991; Carney et al. 1996).\nWe note that the GMDFs of Ryan \\& Norris (1991)\nand Carney et al. (1996) are almost identical, except that the latter shows\na small hump at [Fe\/H] $ \\simeq -2.7$.\n\nFigure\\,6 shows that model\\,S1 gives good fits to the observed GMDFs   \nat 0.5~Gyr and 1~Gyr. The star formation in the \nhalo must have virtually stopped at around 0.5 Gyr, otherwise the resulting\nGMDF gives too high peak metallicity. This would happen if the gas escapes\nfrom the halo and accretes on to the bulge and disc dissipationally.\nA precise value of the epoch when the star formation terminated is \nof course model dependent. However, the GMDFs of\nhalo stars strongly suggest that the star formation in the halo did not\nlast longer than, say, 0.5-1 Gyr.\n\n\nUpper and lower panels in figure\\,7 illustrate the GMDFs of models\\,S2~and~S3, \nrespectively. Clearly, model\\,S2 is inconsistent with the observations. \nModel\\,S2 gives too sharp GMDFs to reproduce the observations.\nWe have also studied several alternative cases in which we assume much \nlonger time interval for the periodic turbulent mixing, much smaller\nsize of the smoothed out area, and lower or higher probability \nof star formation, and have confirmed that these models give\nmuch sharper GMDFs than the empirical ones; thus inconsistent \nwith the observations. On the contrary, the GMDF of \nmodel\\,S3 is consistent with those observed. \nEquivalent models to model\\,S3, but with higher or lower SFRs,\nare also calculated. These models predict similar inhomogeneous ISM \nenrichment. Their GMDFs are also consistent with the empirical ones \nat 0.1\\,Gyr (higher SFR) and 0.5\\,Gyr (lower SFR).  \nThe standard deviations of metallicity distribution function of these models \nincrease with time, showing the same trends as those of model\\,S3. \nWithout additional informations, \nit is rather difficult to conclude which star formation\nmechanism, the spontaneous star formation (model\\,S1) \nor the stimulated one (model\\,S3),\nis responsible for the early enrichment of the Galactic halo.\n\n\\begin{figure*}[t]\n\\epsfxsize=9cm\n\\epsfbox{fig7.epsi}\n\\caption[]\n{\nThe GMDFs of model\\,S2 (upper three panels) and \nmodel\\,S3 (lower three panels) at 0.1, 0.5, and 1\\,Gyr.  \nThick solid lines show the theoretical GMDFs. \nThin solid and dotted lines have the same meaning as in Fig.\\, 6. \n}\n\\end{figure*}\n\nTo judge a goodness of the model fit to the empirical GMDFs,    \nwe have performed the $\\chi^2$ statistics. \nLet $x_i$ and $y_i$ be the theoretical and the observed  $P$([Fe\/H]$_i$),\nrespectively. Then, the $\\chi^2$ value is defined as, \n\\begin{equation}\n\\chi^2=\\sum_i \\frac{(x_i-y_i)^2}{x_i+y_i}. \n\\end{equation}\nIn columns (8)-(10) of table 2, we give the $\\chi^2$ values of\nthe standard models at 0.1, 0.5, and 1\\,Gyr, respectively. The best fit\nis realised by model S1 at 0.5 Gyr, but model S1 at 1 Gyr and model S3\nat 0.5 Gyr also give a reasonable fit.\n\n\\begin{table*}\n\\small\n\\begin{center}\nTable~2.\\hspace{4pt}Models and results.\\\\\n\\vspace{6pt}\n\\end{center}\n\\vspace{6pt}\n\\begin{tabular*}{\\textwidth}{@{\\hspace{\\tabcolsep}\n\\extracolsep{\\fill}}lrrlcrrrrrr}\n\\hline\\hline\\\\[-6pt]\nModel & $l_{\\rm b}\/R_{\\rm h}$ & {\\scriptsize $\\omega^{-1} \\Delta t$} & & & \\multicolumn{3}{c}{$s(t)$} & \\multicolumn{3}{c}{$\\chi^2$}  \\\\\n\\multicolumn{3}{c}{} &  & & 0.1 &0.5 &1 & 0.1 &0.5 &1 \\\\  \n\\hline\nS1 & 0.03 & 1.2 &~~& & 0.27 & 0.24 & 0.24 &~~ 75.11 & 11.01 & 11.85 \\\\\nS2 & 0.03 & 1.2 &  & & 0.27 & 0.01 & 0.01 & 76.82 & 26.78 & 22.63 \\\\\nS3 & 0.03 & 1.2 &  &~~& 0.33 & 0.45 & 0.54 & ~~68.16 & 11.89 & 20.36 \\\\\n& & \\\\\nA & 0.03 & 0.6 &   &~~& 0.34 & 0.34 & 0.29 & ~~53.52 & 4.68 & 15.82 \\\\\nB & 0.03 & 3 &  &~~& 0.18 & 0.10 & 0.10 & ~~95.12 & 24.40  & 16.79 \\\\\nC & 0.01 & 1.2 &  &~~& 0.32 & 0.44 & 0.42 & ~~9.25 &  11.68 & 19.73 \\\\\nD & 0.03 & 1.2 & $n_{\\rm{II}}=20$  &~~& 0.15 & 0.11 & 0.11 & ~~92.31 & 21.39 & 17.19 \\\\\nE & 0.03 & 1.2 & $m_{\\rm{II}}$=8$M_\\odot$  &~~& 0.24 & 0.16 & 0.17 & ~~83.33 & 16.70 & 14.02 \\\\\nF & 0.03 & 1.2 & $m_l=0.05M_\\odot$  &~~& 0.30 & 0.33 & 0.30 & ~~62.80 & 6.19 & 11.77 \\\\\nG & 0.03 & 1.2 &  $M_{\\rm h}=2 \\cdot 10^{11}M_\\odot$ & & 0.30 &  0.42 & 0.37 & ~~49.38 & 4.00 &  13.06 \\\\\n\\hline\n\\end{tabular*}\n\n\\vspace{6pt}\n\n\\noindent\nModel\\,S2 assumes propagation of star formation. Model\\,S3 assumes  \nthe mixing of ISM by turbulent diffusion whose priod is taken \nto be $10^8$~yr. See text in detail.\n\n\\end{table*}\n\nIn addition to the standard models, we show additional 7 models A - G \nto demonstrate the parameter dependence of our results. \nAll the 7 models are equivalent to model\\,S1, but with different parameters, \ni.e., these models assume spontaneous star formation and no turbulent mixing. \nThe parameters of these 7 models are summarised in table 2.\nIn this table, column (1) gives an identification of the model, columns \n(2) and (3) give the adopted values of \n$l_{\\rm b}\/R_{\\rm h}$ and $\\omega^{-1} \\Delta t$, respectively.\nColumn (4) gives the parameter that is different from the standard models. \nColumns (5) - (7) indicate the standard deviation $s(t)$ at\n0.1, 0.5, and 1\\,Gyr, respectively.\nThe $\\chi^2$ values for the best fit epochs of the standard models \nare 11.01 (model\\,S1) and 11.89 (model\\,S3) at 0.5\\,Gyr. \nTherefore, we consider a good fit is realised if the model gives the \n$\\chi^2$ less than 12 in columns (8) - (10). \n\nIn models A - C,  we study how the evolution of $s(t)$ and $\\chi^2$\ndepend on $l_{\\rm b}\/R_{\\rm h}$ and $\\omega^{-1} \\Delta t$.\nTwo cases of $l_{\\rm b}\/R_{\\rm h}=$ 0.01 and 0.03 are studied.    \nFor $\\omega^{-1} \\Delta t$, we consider three cases, \ni.e. $\\omega^{-1} \\Delta t=0.6 \\cdot 10^{-3}$, \n$1.2 \\cdot 10^{-3}$, and $3 \\cdot 10^{-3}$.    \nThe standard deviation of model\\,A is always larger than\nthat of model\\,S1, which suggests that the ISM is always \nchemically inhomogeneous. In model\\,B, higher SFRs lessen  \nthe differences of chemical enrichment among the blocks. \nThe standard deviation \nof model\\,B is nearly equal to $0.1$ at 1\\,Gyr, predicting relatively \nwell-mixed ISM with respect to model\\,S1. However, the $\\chi^2$  \nof model\\,B indicates that this model is inconsistent with the observations.  \nIn model\\,C, given smaller $l_{\\rm b}\/R_{\\rm h}$, the best fit epoch \nto the observations comes earlier than model\\,S1. \nThe gas in the star forming blocks of model\\,C is converted into stars more \nefficiently than model\\,S1, since  \nequation (14) indicates that the SFR in a block is proportional to \n$n_{\\rm{II}}\/m_{\\rm b}$, i.e., $ (l_{\\rm b}\/R_{\\rm h})^{-3}$. \nIn model\\,D, we reduce the number of SN\\,II progenitor \nin a single OB association and assume $n_{\\rm{II}}=20$  \ninstead of 40 (Blaauw 1964). This \nincreases the probability of star formation (see equation (11)) and \nhomogenises the ISM quickly. \nThe $\\chi^2$ of model\\,D indicates poor fits to the observations.  \nModel\\,E assumes $m_{\\rm{II}}=8M_\\odot$. \nThe number fraction of SN\\,II progenitor in model\\,E is increased \nand the SFR becomes twice of model\\,S1. \nThus, both the evolutionary behaviour of standard deviation \nand the $\\chi^2$ fits are very similar to those of model\\,B. \nWe can reject this model, since the $\\chi^2$ gives very poor fits.\nModels\\,F~and~G assume $m_l=0.05M_\\odot$ \nand $M_{\\rm h}=2 \\cdot 10^{11}\\,M_\\odot$, respectively. Equation (11) \nindicates that the probability of star formation $P(t)$ in a block, \nnamely the SFR in the halo, is proportional to the number \nfraction $f_{\\rm{II}}$ of SN\\,II progenitor \nand the mass of the halo. Thus \nin models\\,F~and~G with the lower SFR in the halo, \nthe standard deviation of metallicity distribution \nfunction becomes slightly larger than that of model\\,S1. \nThe $\\chi^2$ values of models\\,F~and~G show that these models \ngive a good fit to the observed GMDFs in the halo at 0.5\\,Gyr. \n\nIn summary, the different parameters change the SFR in the halo.  \nIf the SFR is low, the standard deviation of metallicity \ndistribution becomes large, i.e. the ISM is inhomogeneous, and \ngood fits to the observed GMDFs are resulted. \nWhile if the SFR in the halo is high, \nthe standard deviation becomes small (well-mixed ISM) and  \nthe resulting GMDFs show poor fits to the observations, since  \nstars born in the similarly enriched ISM dominate \nthe cumulative metallicity distribution function at later phase. \nThus the predicted GMDF becomes too narrow to be consistent with \nthe observations. \n\n\\subsection{Comparison with Mass-Loss Models}\nIn previous sections, we assume that the halo was a closed system. \nHartwick (1976) showed that a simple model predicts \ntoo many metal-poor stars and is inconsistent with \nthe cumulative distribution of the iron abundances \nfor metal-poor ([Fe\/H]$\\le -0.52$) globular clusters in the Galactic halo. \nHe modified the simple model \nand proposed a model which allows for gas outflow from the halo. \nWe therefore consider effects of mass loss from the halo in this subsection. \nThe gas will eventually either escape to intergalacitc space or to accrete \nonto the galactic plane. \n\nSince the gas removal from the halo could be related to energy injection \nfrom hot OB stars and SN\\,II (Hartwick 1976), we assume that the mass loss \nrate $dD\/dt$ is proportional to the SFR $C(t)$:\n\\begin{equation}\n\\frac{dD}{dt}=b\\,C(t). \n\\end{equation}\nThe mass loss rate $b$ and the SFR are taken from \nHartwick (1976), i.e., $b=10$ and $\\omega^{-1}=0.3$, respectively. \nWe assume the same gas outflow rate for all the blocks in \nthe halo. In model ML1, the parameters are the same as those of model S1 \nexcept for $b$ and $\\omega^{-1}$. We assume that OB associations are born \nrandomly in space and do not consider the ISM mixing by turbulent \ndiffusion. While in model ML2, we study the effects of turbulent mixing. \nWe assume periodic ISM mixing, similarly to model S2. The period is \ntaken as 0.1\\,Gyr, the same as that of model S2. Other prescriptions \nfor model ML2 are same as those of model ML1. \n\nFigures\\,8a~and~8b show the evolution of mean iron abundance and \nthat of standard deviation of the mass loss models, respectively.\nFigure~9  shows the theoretical GMDFs plotted with the observed GMDFs.   \nThese figures show that the GMDFs obtained by model\\,ML1 agree with \nthe observed GMDFs and the ISM is always \ninhomogeneous in this model. On the other hand, \nthe GMDFs of model\\,ML2 are too sharp and \nare inconsistent with the observations, \nbecause the ISM in this model is well-mixed. \nIn model ML1, the best fit is achieved at an epoch later than model S1. \nIn the case of one-zone model, a shift of the peak abundance can be \nexplained by a decrease of effective yield caused by the mass-loss \n(Hartwick 1976; Pagel 1992). Although our model is not one-zone, \nthe mass-loss gives a similar effect to the peak abundance of \nthe cumulative metallicity distribution function.  \n \n\\begin{figure}[t]\n\\epsfxsize=5cm\n\\epsfbox{fig8a.epsi}\n\\epsfxsize=5cm\n\\epsfbox{fig8b.epsi}\n\\caption[]{\n(a) ~~\nThe same as figure\\,2(a), but for mass loss models ML1 and ML2. \nThe SFR $\\omega^{-1}=0.3$ and the mass-loss rate $b=10$ are the same as those \nof Hartwick (1976). \n(b) ~~\nThe same as figure\\,4, but for mass-loss models\\,ML1 and ML2.\n}\n\\end{figure}\n\\begin{figure*}[ht]\n\\epsfxsize=9cm\n\\epsfbox{fig9.epsi}\n\\caption[]{\nThe GMDFs of model\\,ML1 (upper three panels) and \nmodel\\,ML2 (lower three panels) at 0.1, 0.5, and 1\\,Gyr.  \nThick solid lines show the theoretical GMDFs. \nThin solid and dotted lines have the same meaning as in Fig.\\, 6. \n}\n\\end{figure*}\n\nThe GMDFs obtained by assuming $b=10$ and $\\omega^{-1}=0.2$ \n(the same SFR as that of model~S1) also fit to the observed GMDFs and \nthe best fit is realised at around 1\\,Gyr. \nThe chemical enrichment in this model \nis also inhomogeneous.  \nA model adopting $b=5$ also gives consistent GMDFs with \nthe observations, although \nthe best fit epoch appears earlier than model\\,ML1 due to a larger effective  \nyield. Gas outflow may occur in the vicinity of star forming regions, \nif major energy sources are massive \nstars. When we take this into account and consider a model \nin which gas is expelled only from the star forming blocks, \nwe obtain the same results on a fitting goodness and \ninhomogeneity of the ISM as those obtained by models\\,ML1 and ML2.\n\n\nAs a conclusion, whether or not mass loss is taken into account, \nthe ISM in the halo at the beginning must be \ninhomogeneous. \n\n\\section{Discussions \\label{sec:discussion}}\n\nObservations show neither clear age-metallicity relation nor clear \nevidence for metallicity gradient in the halo. \nBecause of these observational results,\nGalactic halo formation scenario favoured recently \n(e.g Carney et al. 1996) is  \nthe one proposed by Searle \\& Zinn (1978), i.e. \naccretion of small proto-galactic fragments contributed to the \nhalo population. However, in the poorly mixed halo, \nthe age-metallicity relation and the metallicity gradient \ncannot be expected even if the Galaxy formed from one massive cloud. \nThus it is not necessary to introduce the accretion \nof proto-galactic fragments \nto interpret the lack of the age-metallicity relation \nand the metallicity gradient in the halo. \n\nThe scatters observed in the relative abundances of neutron capture elements \nto the iron, particularly [Sr\/Fe], are often claimed \nas evidences for the inhomogeneous enrichment of the ISM \nin the halo at the very beginning. \nIn figure\\,10, Sr abundances of halo stars taken from \nGratton \\& Sneden (1988; 1994), Magain (1989), \nMcWilliam et al. (1995), and Ryan et al. (1996) are plotted. \nThe observations show that a majority of the metal-poor stars \nwith [Fe\/H]\\,$\\le -2.5$ distribute at $-1.5 \\le$\\,[Sr\/Fe]\\,$\\le 0.6$. \nThe relative abundance of Sr tends to decrease clearly with an increase of \nthe iron abundance. \n\nWe show two theoretical evolutionary paths of Sr on figure\\,10. \nThe models are calculated by adopting  \nthe same parameters as those of models\\,S1~(left path)~and~C (right path). \nThe SFR of model\\,S1 is nearly one tenth of model\\,C.   \nCo-existence of regions with different SFR is a view of \nthe chemical evolution in the halo considered here. We have simplified \nthe situation and assumed the same SFR in each block. \nHowever, different SFR in each star forming region is more realistic. \nThese models are consistent with the observed behaviour of [Sr\/Fe] \nand the empirical metallicity distribution functions. \n\n\nFollowing PT95 and Pagel \\& Tautvai$\\check{\\rm{s}}$ien$\\dot{\\rm{e}}$ \n(1997; hereafter PT97)),  \nwe calculate the chemical evolution of Sr: \n\\begin{equation}\n\\frac{d}{dt}(gZ_{\\rm{Sr}})=-Z_{\\rm{Sr}}(t)c(t)\n+p_0c(t)+ \\sum_{i=1}^{3} p_ic(t-\\tau_i), \n\\end{equation}\nwhere $g(t)$, $c(t)$, and $Z_{\\rm{Sr}}(t)$ are the gas fraction, \nthe SFR,  and abundance of Sr in a block, respectively. \nThe first and the second terms \non the r.h.s. have the same meaning as those in equation\\,(12). \nThe third term comes from the delayed production,\nwhere $p_i$ is the yield corresponding to \nthe fixed time lag $\\tau_i$. Here $p_0=0.01$, $p_1=0.08$, \n$p_2=0.39$, $p_3=0.23$, $\\tau_1=0.023$\\,Gyr, \n$\\tau_2=0.025$\\,Gyr, and $\\tau_3=2.7$\\,Gyr are assumed (PT97).      \nThe evolution of the iron abundance is calculated by equation\\,(12). \n\nThe theoretical paths in figure\\,10 show that [Sr\/Fe] begins \nto increase rapidly \nat [Fe\/H]\\,$\\sim -3.4$ in model\\,S1 and [Fe\/H]\\,$\\sim -2.3$ in model\\,C,  \ndepending on the SFR. These paths roughly outline the dispersion of \nobserved [Sr\/Fe] at [Fe\/H]\\,$\\le -2.5$. \nThe models are calculated with a time step of $0.001$\\,Gyr, \nwhich is shown by filled pentagons on the theoretical paths in figure\\,10. \nThe chemical evolution from [Sr\/Fe]\\,$\\sim -1.3$ to [Sr\/Fe]\\,$\\sim 0$ \nshould take place within a short time interval ($\\sim 0.012$\\,Gyr).\nAfter the rapid increase, \nthe paths show temporal increase and decrease, and then converge. \nThe declines of theoretical [Sr\/Fe] are not because of \ndelayed production of the iron (in other words, \ncontribution from SNIa which produces \nthe bulk of iron in the solar neighbourhood), since  \nwe assume that the time delay of iron $\\tau$ \nin equation (\\ref{eqn:delay}) is equal to $1.3$\\,Gyr (see section 2.3) \nand consider chemical evolution before $1$\\,Gyr. \nThe declines of [Sr\/Fe] are due to \nonsets of the next formation of OB associations.  \nThus different SFR in each star forming region can explain \nthe trends defined by a majority of observed stars \non [Sr\/Fe] vs [Fe\/H] diagram.  \n\n\\begin{figure}[t]\n\\epsfxsize=6cm\n\\epsfbox{fig10.epsi}\n\\caption[]{\nChemical evolution of strontium.  Observational data are taken  \nfrom Gratton \\& Sneden (1988; 1994), Magain (1989), \nMcWilliam et al. (1995), and Ryan et al. (1996). \nTwo theoretical paths correspond to different SFR in a block. \nThe left and right paths are obtained by adopting the same parameters \nas those of models\\,S1~and~C, respectively. \nEach filled pentagon on the paths indicates \na time step $0.001$~Gyr. \n}\n\\end{figure}\n\nApparently the models are inconsistent with the observational \ndata on figure\\,10. \nMcWillam et al (1995), Sneden et al (1996), and McWillam (1998) \nhave shown that the heavy element abundances in the super-solar \n[Sr\/Fe] stars are dominated by the r-process abundance patterns.  \nThe r-process elements can only be produced by Type II supernova events. \nTherefore one may argue that the \ntime-delay model for Sr is inadequate to explain the observations. \nIf the strontium is produced in all progenitors of SN\\,II, however, \n[Sr\/Fe] values observed in metal-poor stars should be always super-solar. \nOn the contrary, as we mentioned before, figure 10 shows that [Sr\/Fe] \ntends to decrease at lower iron abundance ([Fe\/H]\\,$\\le -2.5$). \nThis drop suggests that the strontium was formed slightly later \nthan the iron (see Mathews, Bazan, \\& Cowan 1992). \n\nIf the large scatter of [Sr\/Fe] observed for stars of iron abundance in \nthe range of $-3.6 \\le$[Fe\/H]$\\le -2.6$ is indeed due to a sharp increase \nof the Sr production rate, the Sr abundances of four extremely metal-dificient \nstars with [Fe\/H]$\\sim -4$ seem to show some evidence against it. However, \nthe iron abundance measurement of such very low metal stars is \nextremely difficult and these iron abundances could well be as large as \n[Fe\/H]$\\stackrel{>}{_\\sim}-3.6$ (M. Spite, private communications). \n\nThe theoretical [Sr\/Fe] is too low to fit to the observed \nextremely high [Sr\/Fe] value. \nHowever, the abundance of these stars might not reflect \nthe composition of ISM from which they formed (e.g. McClure 1984). \nWe should reject CH stars when we discuss the Galactic chemical evolution, \nbecause atomosphers of CH stars are thought to have been enriched \nby elements transfered from evolved companion AGB stars (McClure 1984). \nMcWillam et al (1995) and McWilliam (1998) reported that \nCS22898-027 ([Fe\/H]$=-2.36$) and CS22947-187 ([Fe\/H]$=-2.5$) are \nprobably CH stars. \nObservational errors might also reflect the [Sr\/Fe]-[Fe\/H] diagram.   \nFor a star CS22891-209 ([Fe\/H]$\\sim -3.2$),  \nPrimas, Molaro, \\& Castelli (1994) and McWilliam et al (1995) reported [Sr\/Fe]$=0.92$ and \n[Sr\/Fe]$=-0.07$, respectively, although \nthey used the same lines and had the same S\/N.  \n  \n\n\n\\section{Conclusion}  \n\nA stochastic chemical evolution model has been built to study an early \nhistory of metal enrichment in the Galactic halo. The metallicity \ndistribution function of long-lived halo stars is found to be a \nclue to obtain the best model prescriptions. \nWe find that the star formation in the halo virtually terminated by \n$\\sim 1$ Gyr and that the halo has never been chemically homogeneous \nin its star formation history. The star formation in the halo \ncould either be spontaneous or stimulated, which keep the halo always \ninhomogeneous and the turbulent mixing is found to be inefficient. \nThis conclusion does not depend whether the mass loss from the halo is taken \ninto account or not. The observed ratios of the $\\alpha$-elements with \nrespect to the iron do not show scatters on the [$\\alpha$\/Fe]-[Fe\/H] \nplane, but this does not imply that the ISM in the halo was homogeneous \nbecause the chemical evolution path on this diagram is degenerate in the \nSFR. On the other hand, the apparent spread of [Sr\/Fe] ratio among \nmetal-poor halo stars does not reflect an inhomogeneous metal enrichment, \ninstead it is due to a sharp increase in the production rate of \nstrontium that is probably synthesised in slightly less massive stars \nthan the progenitor of iron-producing SN\\,II. \n\n\n\\vspace{1cm}\n\nWe are grateful to an anonymous referee for a careful reading \nof the manuscript and for useful comments.\nC.I. thanks to the Japan Society for Promotion of \nScience for a financial support. \nThis work was financially supported in part \nby a Grant-in-Aid for the Scientific Research (No. 0940311) \nby the Japanese Ministry of Education, Culture, Sports\nand Science. \n\n\n\\section*{References}\n\\small\n\n\\re\nAllen D.A., Burton, M.G. 1993, Nature 363, 54\n\\re Arimoto N., Matsushita, K., Ishimaru, Y., Ohashi, T., Renzini A. 1997, ApJ 477, 128\n\\re Arimoto N., Yoshii, Y., Takahara, F. 1992, A\\&A 253, 21\n\\re  Audouze J., Silk J. 1995, ApJ 451, L49 \n\\re Bahcall J.N., Pinsonneault M.H. 1996, AAS 189, 5601\n\\re  Bateman N.P.T., Larson R. 1993, ApJ 407, 634\n\\re Barbuy B. 1988, A\\&A 191, 121\n\\re  Beers T.C., Preston G.W., Shectman S.A. 1992, AJ 103, 1987\n\\re  Blaauw A. 1964,  ARA\\&A 2, 213\n\\re  Bruhweiler F.C., Gull, T., Kafatos M.,  Sofia S. 1980, ApJ 238, L27\n\\re Cash W., Charles P., Bowyer S., Walter F., Garmire G., Riegler G. 1980, ApJ 238, L71\n\\re Carraro G.,  Chiosi C. 1994, A\\&A 281, 35\n\\re Carney S.G., Laird J.B., Latham D.W., Aguilar L.A. 1996, A\\&A 112, 668 \n\\re  Cioffi D.F., McKee C.F., Bertschinger E. 1988, ApJ 334, 252\n\\re Copi C.J. 1997, ApJ 487, 704\n\\re  Cox D. P. 1972, ApJ 178, 159\n\\re  Edvardsson B., Andersen J., Gustafsson B., Lmbert D.L., Tomkin J. 1993, A\\&A 275, 101\n\\re  Elmegreen B.G. 1982, in Submillimeter Wave Astronomy, \ned. J.E. Beckman, J.P. Phillips (Cambridge University Press) p5\n\\re  -----. 1985a, in Birth and Infancy of Stars, \ned. R. Lucas, A. Omont,  R. Stora (Amsterdam: Elsevier) p215 \n\\re  -----. 1985b, in Brith and Evolution of Massive Stars \nand Stellar Collapse, ed W. Boland, \nH. van Woerden (Dordrecht: Reidel) p227\n\\re  Elmegreen B.G., Lada, C.J. 1977, ApJ 214, 725\n\\re  Fich M., Tremaine, S. 1991, ARA\\&A 29, 409\n\\re  Gratton R.G., Sneden C. 1988,  A\\&A 204, 193\n\\re  Gratton R.G., Sneden C. 1994,  A\\&A 287, 927\n\\re  Hartwick, F.D.A., 1976, ApJ 209, 418\n\\re  Henning T., G$\\ddot{\\rm u}$rler J. 1986, Ap\\&SS 128, 199\n\\re  Heiles C. 1987, ApJ 315, 555\n\\re  van den Hoek L.B., de Jong T. 1997, A\\&A 318, 231 \n\\re  Hoyle F., 1953, ApJ 118, 513\n\\re Humphreys, R.M. 1978, ApJS, 38, 309\n\\re  Lada C.J., Blitz L., Elmegreen B. 1979, in Protostars and Planets, ed T. Gehrels (Tucson: University of Arizona Press) p368 \n\\re  Laird J.B., Rupen M.P., Carney B.W., Latham D.W. 1988, AJ 96, 1908\n\\re Larson R.B. 1974 MNRAS, 169, 229\n\\re  Mac Low M.-M., McCray R. 1988, ApJ 324, 776\n\\re  Magain P. 1989, A\\&A  209, 211\n\\re Mathews, G.J., Bazan, G., Cowan, J.J. 1992, ApJ 391, 719\n\\re  McCray R., Kafatos M. 1987, 317, 190\n\\re McClure R.D. 1984, PASP, 96, 117 \n\\re  McWilliam A 1997, ARA\\&A 35, 503\n\\re  McWilliam A 1998, AJ 115, 1640\n\\re  McWilliam A., Preston G. W., Sneden C., Shectman S. 1995, AJ 109, 2757\n\\re Nissen P.E., Gustafsson B., Edvardsson B., Gilmore G. 1994, A\\&A 285, 440\n\\re  Pagel B.E.J., 1992, in The Stellar Populations of Galaxies, IAU Symposium No. 149, edited by B. Barbuy \\& Renzini (Kluwer, Dordrecht), p. 133 \n\\re  Pagel B.E.J., Tautvai$\\check{{\\mbox s}}$ien$\\dot{{\\mbox e}}$ G. 1995, MNRAS 276, 505 (PT95)\n\\re  Pagel B.E.J., Tautvai$\\check{{\\mbox s}}$ien$\\dot{{\\mbox e}}$ G. 1997, MNRAS 288, 108 (PT97)\n\\re  Pilyugin L.S. 1996, A\\&A 313, 803\n\\re Primas F., Molaro, P., Castelli, F. 1994, A\\&A 290, 885\n\\re  Roy J.-R., Kunth D. 1995, A\\&A 294, 432\n\\re  Ryan S.G., Norris J.E. 1991, AJ 101, 1865\n\\re  Ryan S.G., Norris J.E., Beers T.C. 1996, ApJ 471, 254\n\\re Saito M. 1979, PASJ 31, 181\n\\re  Salpeter E.E. 1955, ApJ 121, 161\n\\re Searle L., Zinn R. 1978, ApJ 225, 357\n\\re  Schmidt M 1959, ApJ 129, 243\n\\re  Seal L., Zinn R. 1978, ApJ 225, 357\n\\re  Silk J. 1977, ApJ 214, 718\n\\re  Shigeyama T., Nomoto K., Hashimoto M. 1988, A\\&A 196, 141\n\\re Sneden, C., McWilliam, A., Preston, G.W., Cowan, J.J., Burris, D.I., \nArmosky, B.J., 1996, ApJ 467, 819\n\\re Stuart A., Ord, J.K. 1987, in Kendall's Advanced Theory \nof Statistics, 5th ed. (London: Griffin and Co.)\n\\re  Tenorio-Tagle G., Bodenheimer P. 1988, ARA\\&A 26, 145\n\\re  Timmes F. X.,  Woosley  S. E., Weaver T. A. 1995, ApJS 98, 617\n\\re  Tinsley B. M. 1980, Fund. Cosmic Phys. 5, 287\n\\re Wielen R., Fuchs B., Dettbarn C. 1996, A\\&A 314, 438\n\\re Wilmes M., K$\\ddot{\\rm o}$ppen J. 1995, A\\&A 294, 47\n\n\\end{document}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nStudy of the electromagnetic properties of baryons constitutes very important \nsource of information in exploring their internal structure, and can\nprovide valuable insight in understanding the nonperturbative aspect of QCD.\nIn recent years, study of the properties of heavy baryons has become a\nsubject of growing interest due to the experimental observation of many\nheavy baryons. All ground state baryons containing single charm and bottom\nquarks, except $\\Omega_b^\\ast$ baryon, are observed and their masses are\nmeasured (for a review, see \\cite{Rfrd01}). Moreover, a number of negative\nparity baryons have also been observed.\n\nThese exciting experimental results have stimulated the theoretical studies\nalong these lines. The mass and magnetic moments of the heavy baryons can\nserve useful information about their inner structure. Experimentally the\nmagnetic moments of all members of the octet $J^P={1^+\\over 2}$ baryons\n(except $\\Sigma^0$ baryon), and two members of the decuplet\n$J^P={3^+\\over 2}$ baryons are measured \\cite{Rfrd01}.\n\nThe magnetic moments of the $J^P={1^+\\over 2}$ light and heavy baryons have\nextensively been calculated in numerous theoretical approaches. The\napproaches based on naive quark model in \\cite{Rfrd02,Rfrd03},\nrelativistic quark model\\cite{Rfrd04}, nonrelativistic quark\nmodel\\cite{Rfrd05}, chiral quark model \\cite{Rfrd06},\nchiral perturbation theory \\cite{Rfrd07}, hypercentral model \\cite{Rfrd08},\nsoliton model \\cite{Rfrd09}\ntraditional \\cite{Rfrd10} and light version of the QCD sum rules method\n(LCSR) \\cite{Rfrd11,Rfrd12,Rfrd13,Rfrd14} have\nbeen employed in studying the masses and magnetic\nmoments of the heavy baryons. The magnetic moments of the negative parity\nheavy baryons has recently been considered within the framework of the LCSR\nmethod in \\cite{Rfrd15}.\n\nIn this work, we extend our analysis to determine the magnetic\nmoments for the $\\Sigma_Q \\to \\Lambda_Q$ and $\\Xi_Q^\\prime \\to \\Xi_Q$\ntransitions between the negative parity baryons.\n\nIn the following section we derive the light cone sum rules for the\nmagnetic moments of the aforementioned transitions. Section 3 is devoted to\nthe numerical analysis of the obtained sum rules for the transition\nmagnetic moments. A comparison of our predictions with the results from\nother approaches is given also.\n\n\\section{Theoretical framework}\nFollowing the philosophy of the QCD sum rules,\nthe magnetic moments of the $\\Sigma_Q \\to \\Lambda_Q$ and $\\Xi_Q^\\prime \\to\n\\Xi_Q$ transitions of the baryons with negative parity can be obtained in\nLCSR by matching two representations of the relevant correlation function\nwritten in terms of the hadronic and quark-gluon languages. For this purpose\nwe use the correlation function\n\\begin{eqnarray}\n\\label{efrd01}\n\\Pi (p,q) = i \\int d^4x e^{ipx} \\left< 0 \\left| \\mbox{\\rm T} \\{\n\\eta_Q (x) \\bar{\\eta}_Q(0) \\}\\right| 0 \\right>_\\gamma ~,\n\\end{eqnarray}\nwhere $\\gamma$ is the external electromagnetic field, $\\eta_Q$ is the\ninterpolating current of the heavy baryon with single heavy quark.\nThis correlation function can be calculated at hadronic level by inserting a\ncomplete set of hadrons carrying the same quantum numbers of the correlation\nfunction, and isolating the contributions arising from the ground states\nwhich have poles in $p^2$ and $(p+q)^2$. The interpolating current can\ninteract with both negative and positive parity baryons, therefore it can be\nwritten in the following form,\n\\begin{eqnarray}\n\\label{efrd02}\n\\Pi(p,q) \\!\\!\\! &=& \\!\\!\\!\n{\\langle 0 \\vert \\eta \\vert B_2^{(+)}(p,s) \\rangle \\over p^2-m_{B_2^{(+)}}^2} \\langle B_2^{(+)}(p,s)\n\\gamma(q)\\vert\nB_1^{(+)}(p+q,s) \\rangle {\\langle B_1^{(+)}(p+q,s) \\vert \\bar{\\eta}(0) \\rangle \\over\n(p+q)^2-m_{B_1^{(+)}}^2} \\nonumber \\\\\n&+& \\!\\!\\! {\\langle 0 \\vert \\eta \\vert B_2^{(-)}(p,s) \\rangle \\over p^2-m_{B_2^{(-)}}^2} \\langle B_2^{(-)}(p,s)\n\\gamma(q) \\vert\nB_1^{(-)}(p+q,s) \\rangle {\\langle B_1^{(-)}(p+q,s) \\vert \\bar{\\eta}(0) \\rangle \\over\n(p+q)^2-m_{B_1^{(-)}}^2} \\nonumber \\\\\n&+& \\!\\!\\! {\\langle 0 \\vert \\eta \\vert B_2^{(+)}(p,s) \\rangle \\over p^2-m_{B_2^{(+)}}^2} \\langle B_2^{(+)}(p,s)\n\\gamma(q)\\vert\nB_1^{(-)}(p+q,s) \\rangle {\\langle B_1^{(-)}(p+q,s) \\vert \\bar{\\eta}(0) \\rangle \\over\n(p+q)^2-m_{B_1^{(-)}}^2} \\nonumber \\\\\n&+& \\!\\!\\! {\\langle 0 \\vert \\eta \\vert B_2^{(-)}(p,s) \\rangle \\over p^2-m_{B_2^{(-)}}^2} \\langle B_2^{(-)}(p,s)\n\\gamma(q)\\vert\nB_1^{(+)}(p+q,s) \\rangle {\\langle B_1^{(+)}(p+q,s) \\vert \\bar{\\eta}(0) \\rangle \\over\n(p+q)^2-m_{B_1^{(+)}}^2} + \\cdots~,\n\\end{eqnarray}\nwhere $B^{(\\pm)}$ and $m_{B^{(\\pm)}}$ correspond to positive (negative)\nparity baryons and their masses, respectively; $q$ is the photon momentum;\nand dots correspond to the higher states contributions.\n\nThe matrix elements in Eq. (\\ref{efrd02}) are defined as,\n\\begin{eqnarray}\n\\label{efrd03}\n\\langle 0 \\vert \\eta \\vert B^{(+)}(p)\\rangle \\!\\!\\! &=& \\!\\!\\!  \\lambda_{B^{(+)}} u^{(+)} (p)~,\\nonumber \\\\\n\\langle 0 \\vert \\eta \\vert B^{(-)}(p)\\rangle \\!\\!\\! &=& \\!\\!\\!  \\lambda_{B^{(-)}} \\gamma_5 u^{(-)} (p)~,\\nonumber \\\\\n\\langle B_2^{(+)}(p) \\gamma(q) \\vert \\eta \\vert B_1^{(+)}(p+q)\\rangle \\!\\!\\! &=& \\!\\!\\! e \\varepsilon^\\mu\n\\bar{u}^{(+)}(p)\n\\left[\\gamma_\\mu f_1 - {i \\sigma_{\\mu\\nu} q^\\nu \\over  m_{B_1^{(+)}}\n+ m_{B_2^{(+)}} } f_2 \\right] u^{(+)}(p+q) \\nonumber \\\\\n\\!\\!\\! &=& \\!\\!\\! e \\varepsilon^\\mu \\bar{u}^{(+)}(p) \\left[ (f_1+f_2) \\gamma_\\mu - {(2p+q)_\\mu\n\\over m_{B_1^{(+)}} + m_{B_2^{(+)}} } f_2 \\right] u^{(+)}(p+q)~, \\nonumber \\\\\n\\langle B_2^{(-)}(p) \\gamma(q) \\vert \\eta \\vert B_1^{(+)}(p+q)\\rangle \\!\\!\\! &=& \\!\\!\\! e \\varepsilon^\\mu\n\\bar{u}^{(-)}(p)\n\\left[\\gamma_\\mu f_1^T - {i \\sigma_{\\mu\\nu} q^\\nu \\over \nm_{B_1^{(+)}} + m_{B_2^{(-)}} } f_2^T\n\\right] \\gamma_5 u^{(+)}(p+q) \\nonumber \\\\\n\\!\\!\\! &=& \\!\\!\\! e \\varepsilon^\\mu \\bar{u}^{(-)}(p)\n\\left[\\left( f_1^T - {m_{B_1^{(+)}} - m_{B_2^{(-)}} \\over \nm_{B_1^{(+)}} + m_{B_2^{(-)}} } f_2^T \\right) \\gamma_\\mu \\right. \\nonumber \\\\\n&-& \\!\\!\\! \\left. {(2p+q)_\\mu \\over m_{B_1^{(+)}} + m_{B_2^{(-)}} } f_2^T \\right] \\gamma_5\nu^{(+)}(p+q)~, \\nonumber \\\\ \n\\langle B_2^{(-)}(p) \\gamma(q) \\vert \\eta \\vert B_1^{(-)}(p+q)\\rangle \\!\\!\\! &=& \\!\\!\\! e \\varepsilon^\\mu\n\\bar{u}^{(-)}(p)\n\\left[\\gamma_\\mu f_1^\\ast - {i \\sigma_{\\mu\\nu} q^\\nu \\over \nm_{B_1^{(-)}} + m_{B_2^{(-)}} } f_2^\\ast\n\\right] u^{(-)}(p+q) \\nonumber \\\\\n\\!\\!\\! &=& \\!\\!\\! e \\varepsilon^\\mu \\bar{u}^{(-)}(p) \\left[\\left(f_1^\\ast + f_2^\\ast\n\\right) \\gamma_\\mu - {(2p+q)_\\mu \\over m_{B_1^{(-)}} + m_{B_2^{(-)}} }\nf_2^\\ast \\right] u^{(-)}(p+q)~,\n\\end{eqnarray}    \nwhere $\\varepsilon^\\mu$ is the four-polarization vector.\n\nPerforming summation over spins of the\nheavy baryons, for the correlation function from the hadronic side we get,\n\\begin{eqnarray}\n\\label{efrd04} \n\\Pi (p,q) \\!\\!\\! &=& \\!\\!\\! A ({\\not\\!{p}}_2 +  m_{B_2^{(+)}}) \n\\not\\!{\\varepsilon} ({\\not\\!{p}}_1 +  m_{B_1^{(+)}})~\\nonumber \\\\\n&+& \\!\\!\\! B ({\\not\\!{p}}_2 -  m_{B_2^{(-)}})       \n\\not\\!{\\varepsilon} ({\\not\\!{p}}_1 -  m_{B_1^{(-)}})~\\nonumber \\\\\n&+& \\!\\!\\! C  ({\\not\\!{p}}_2 -  m_{B_2^{(-)}})       \n\\not\\!{\\varepsilon} \\gamma_5({\\not\\!{p}}_1 +  m_{B_1^{(+)}})~\\nonumber\\\\\n&+& \\!\\!\\! D  ({\\not\\!{p}}_2 +  m_{B_2^{(+)}})       \n\\not\\!{\\varepsilon} \\gamma_5({\\not\\!{p}}_1 -  m_{B_1^{(-)}})~,\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\n\\label{efrd05}\nA\\!\\!\\! &=& \\!\\!\\! { \\lambda_{B_1^{(+)}}  \\lambda_{B_2^{(+)}} (f_1+f_2) \\over\n( m_{B_1^{(+)}}^2-p_1^2)( m_{B_2^{(+)}}^2-p_2^2)} \\nonumber \\\\\nB\\!\\!\\! &=& \\!\\!\\! { \\lambda_{B_1^{(-)}}  \\lambda_{B_2^{(-)}} (f_1^\\ast+f_2^\\ast) \\over\n( m_{B_1^{(-)}}^2-p_1^2)( m_{B_2^{(-)}}^2-p_2^2)} \\nonumber \\\\\nC\\!\\!\\! &=& \\!\\!\\! { \\lambda_{B_1^{(-)}} \\lambda_{B_2^{(+)}} \\over\n( m_{B_1^{(-)}}^2-p_1^2)( m_{B_2^{(+)}}^2-p_2^2)}\n\\left[f_1^T + { m_{B_1^{(-)}} -  m_{B_2^{(+)}} \\over m_{B_1^{(-)}} + m_{B_2^{(+)}}} f_2^T \\right]\\nonumber \\\\\nD\\!\\!\\! &=& \\!\\!\\! { \\lambda_{B_1^{(+)}} \\lambda_{B_2^{(-)}} \\over\n( m_{B_1^{(+)}}^2-p_1^2)( m_{B_2^{(-)}}^2-p_2^2)}\n\\left[f_1^T - { m_{B_1^{(+)}} - m_{B_2^{(-)}} \\over   m_{B_1^{(+)}} + m_{B_2^{(-)}} } f_2^T \\right]~,\n\\end{eqnarray}\nwhere $p_1 = p+q$ and $p_2=p$.\n\nAmong the terms in Eq. (\\ref{efrd04})\n\\begin{eqnarray}\n\\label{nolabel01}\nf_1+f_2~,~~(f_1^\\ast+f_2^\\ast)~,~~f_1^T + { m_{B_1^{(-)}} - m_{B_2^{(+)}}\n\\over  m_{B_1^{(-)}} + m_{B_2^{(+)}} } f_2^T~,~~f_1^T - { m_{B_1^{(+)}} -\nm_{B_2^{(-)}} \\over m_{B_1^{(+)}} + m_{B_2^{(-)}} } f_2^T~,\\nonumber\n\\end{eqnarray}\nthat are proportional to $\\gamma_\\mu$, the first two correspond\nto the magnetic moments of the positive to\npositive, negative to negative transitions, respectively;\nand the third and the fourth ones describe the transition magnetic moments between positive and\nnegative parity baryons at $q^2=0$.\nOur aim in the present work\nis to calculate the transition magnetic moment between the negative parity\nbaryons, and therefore we should find a way to remove the other three contributions.\n\nIn order to determine the transition magnetic moments between negative\nparity baryons four equations are needed, for which we choose the following\nfour Lorentz structures, $(\\varepsilon\\!\\cdot\\! p) I$,\n$(\\varepsilon\\!\\cdot\\! p) \\rlap\/{p}$, $\\rlap\/{p}\\rlap\/{\\varepsilon}$ and\n$\\rlap\/{\\varepsilon}$. Solving finally these four coupled equations, we\nobtain the unknown coefficient $B$ which describes the negative to negative\nparity transition.\n\nIt follows from Eq. (\\ref{efrd01}) that interpolating currents are needed\nin order to calculate the correlation function in terms of quarks and\ngluons. Here, it should be remembered that hadrons containing single heavy\nquark belong to either sextet or anti-triplet representations of $SU(3)$.\nSextet (anti-triplet) representation is symmetric (antisymmetric) with\nrespect to the exchange of light quarks. In constructing the interpolating\ncurrents belonging to sextet and anti-triplet representations we will use\nthis fact, whose explicit forms are given as (see \\cite{Rfrd16}),\n\\begin{eqnarray}\n\\label{efrd06}\n\\eta^{(s)} \\!\\!\\! &=& \\!\\!\\! -{1\\over \\sqrt{2}} \\varepsilon^{abc} \\Big\\{\n(q_1^{aT} C Q^b) \\gamma_5 q_2^c +\nt (q_1^{aT} C \\gamma_5 Q^b) q_2^c +\n(q_2^{aT} C Q^b) \\gamma_5 q_1^c + (q_2^{aT} C\n\\gamma_5 Q^b) q_1^c\\Big\\}~, \\nonumber \\\\\n\\eta^{(a)} \\!\\!\\! &=& \\!\\!\\! -{1\\over \\sqrt{6}} \\varepsilon^{abc} \\Big\\{\n2 (q_1^{aT} C q_2^b) \\gamma_5 Q^c +\n2 t (q_1^{aT} C \\gamma_5 q_2^b) Q^c +\n(q_1^{aT} C Q^b) \\gamma_5 q_2^c \\nonumber \\\\\n&+& \\!\\!\\! t (q_1^{aT} C \\gamma_5 Q^b) q_2^c -\n(q_2^{aT} C Q^b) \\gamma_5 q_1^c -\nt (q_2^{aT} C \\gamma_5 Q^b) q_1^c\\Big\\}~.\n\\end{eqnarray}\nIn this expression $a,b,c$ are the color indices; $C$ is the\ncharge conjugation operator; and $t$ is a free parameter (the choice $t=-1$\ncorrespond to the Ioffe current). The light quark contents of the heavy\n$\\Sigma_Q$, $\\Xi_Q^\\prime$, $\\Lambda_Q$ and $\\Xi_Q$ baryons are presented in\nTable (1).\n\n\n\\begin{table}[h]\n\n\\renewcommand{\\arraystretch}{1.3}\n\\addtolength{\\arraycolsep}{-0.5pt}\n\\small\n$$\n\\begin{array}{|c|c|c|c|c|c|c|c|c|}\n\\hline \\hline   \n & \\Sigma_{c(b)}^{+(0)} & \\Sigma_{c(b)}^{0(-)} &\n             \\Xi_{c(b)}^{\\prime 0(-)}  & \\Xi_{c(b)}^{\\prime +(0)}  &\n \\Lambda_{c(b)}^{+(0)} & \n\\Xi_{c(b)}^{0(-)}  & \\Xi_{c(b)}^{+(0)} \\\\   \n\\hline \\hline\nq_1   & u & d & d & u & u & d & u \\\\\nq_2   & d & d & s & s & d & s & s \\\\\n\\hline \\hline \n\\end{array}\n$$\n\\caption{Light quark contents of the heavy\n$\\Sigma_Q$, $\\Xi_Q^\\prime$, $\\Lambda_Q$ and $\\Xi_Q$ baryons.}\n\\renewcommand{\\arraystretch}{1}\n\\addtolength{\\arraycolsep}{-1.0pt}\n\\end{table}      \n\n\nHaving the interpolating currents containing single heavy baryon \nat hand, the correlation function can easily be calculated. The correlation\nfunctions describing the sextet to anti-triplet transition magnetic moments\nin the light cone version of the sum rules contain the following\ncontributions. The photon interacts with light or heavy quarks\nperturbatively. This contribution can be obtained by replacing one of the\nfree quark operators by,\n\n\\begin{eqnarray}\n\\label{efrd07}\nS^{free} (x) \\to -{1\\over 2}\\int d^4y S^{free} (x-y) \\gamma^\\mu S^{free} (y)\ny^\\nu {\\cal F}_{\\mu\\nu}~,\n\\end{eqnarray}\nand the other two propagators are taken as the free quark operator. In Eq.\n(\\ref{efrd07}) the Fock-Schwinger gauge, i.e., $A_\\mu = {1\\over 2}\n{\\cal F}_{\\mu\\nu} y^\\nu$ has been used.\nThe other type of contribution can be calculated by replacing one of the\npropagators in the same manner as is given in Eq. (\\ref{efrd07}), and replacing\nthe other one (or both) by the ``full\" light quark operator.\n\nLast type of contribution is the nonperturbative one, that can be obtained\nby replacing one of the light quark operators with,\n\\begin{eqnarray}                                                                       \n\\label{efrd08}\nS_{\\mu\\nu}^{ab} \\to -{1\\over 4} \\left( q^a \\Gamma_i q^b \\right)\n\\left(\\Gamma_i \\right)_{\\mu\\nu},\n\\end{eqnarray} \nwhere $\\Gamma_i$ are the full set of Dirac matrices; and the remaining two\nother propagators are taken as the full quark propagators. In calculating\nthese contributions, the expressions of the light and heavy quark\npropagators in external field are needed. The light cone expansion of the\npropagator in external field is performed in \\cite{Rfrd17}, and it is found\nthat the contributions of the three-particle $\\bar{q}G q$, and the four-particle\n$\\bar{q}G^2 q$, $\\bar{q} q \\bar{q}q$ nonlocal operators are small. Keeping\nthis fact in mind, the expressions of the light and heavy quark propagators\nin external field are given by,\n\\begin{eqnarray}\n\\label{efrd09}\nS_q(x) \\!\\!\\! &=& \\!\\!\\! {i \\rlap\/x\\over 2\\pi^2 x^4} - {m_q\\over 4 \\pi^2 x^2} -\n{\\left< \\bar q q \\right>\\over 12} \\left(1 - i {m_q\\over 4} \\rlap\/x \\right) -\n{x^2\\over 192} m_0^2 \\left< \\bar q q \\right>  \\left( 1 -\ni {m_q\\over 6}\\rlap\/x \\right) \\nonumber \\\\\n&-& \\!\\!\\! i g_s \\int_0^1 du \\Bigg[{\\rlap\/x\\over 16 \\pi^2 x^2} G_{\\mu \\nu} (ux)\n\\sigma_{\\mu \\nu} - {i\\over 4 \\pi^2 x^2} u x^\\mu G_{\\mu \\nu} (ux) \\gamma^\\nu \\nonumber \\\\\n&-& \\!\\!\\! i {m_q\\over 32 \\pi^2} G_{\\mu \\nu} \\sigma^{\\mu\n \\nu} \\left( \\ln {-x^2 \\Lambda^2\\over 4}  +\n 2 \\gamma_E \\right) \\Bigg]~, \\nonumber \\\\ \\nonumber \\\\\nS_Q(x) \\!\\!\\! &=& \\!\\!\\! {m_Q^2 \\over 4 \\pi^2} \\Bigg\\{ {K_1(m_Q\\sqrt{-x^2}) \\over \\sqrt{-x^2}} +\ni {\\rlap\/{x} \\over (\\sqrt{-x^2})^2} K_2(m_Q\\sqrt{-x^2}) \\Bigg\\} \\nonumber \\\\ \n&-& \\!\\!\\! {g_s \\over 16 \\pi^2} \\int_0^1 du\nG_{\\mu\\nu}(ux) \\left[ \\left(\\sigma^{\\mu\\nu} \\rlap\/x + \\rlap\/x\n\\sigma^{\\mu\\nu}\\right) {K_1 (m_Q\\sqrt{-x^2})\\over \\sqrt{-x^2}} +\n2 \\sigma^{\\mu\\nu} K_0(m_Q\\sqrt{-x^2})\\right]~.\n\\end{eqnarray}\nHere $\\Lambda$ is the cut-off energy separating the perturbative and\nnonperturbative regions, $K_i$ are the modified Bessel functions.\n\nIt should be noted here that, in the expression (\\ref{efrd08}) which is used\nto calculate nonperturbative contributions, there appears the matrix\nelements of the nonlocal operators between vacuum and one photon states of\nthe form $\\langle \\gamma(q) \\vert \\bar{q} \\Gamma_i q \\vert 0 \\rangle$, in which all\nnonperturbative effects are encoded. These matrix elements are given in\nterms of the photon distribution amplitudes as \\cite{Rfrd18},\n\n\\begin{eqnarray}\n\\label{efrd10}\n&&\\langle \\gamma(q) \\vert  \\bar q(x) \\sigma_{\\mu \\nu} q(0) \\vert  0\n\\rangle  = -i e_q \\bar q q (\\varepsilon_\\mu q_\\nu - \\varepsilon_\\nu\nq_\\mu) \\int_0^1 du e^{i \\bar u qx} \\left(\\chi \\varphi_\\gamma(u) +\n\\frac{x^2}{16} \\mathbb{A}  (u) \\right) \\nonumber \\\\ &&\n-\\frac{i}{2(qx)}  e_q \\langle \\bar q q \\rangle \\left[x_\\nu \\left(\\varepsilon_\\mu - q_\\mu\n\\frac{\\varepsilon x}{qx}\\right) - x_\\mu \\left(\\varepsilon_\\nu -\nq_\\nu \\frac{\\varepsilon x}{q x}\\right) \\right] \\int_0^1 du e^{i \\bar\nu q x} h_\\gamma(u)\n\\nonumber \\\\\n&&\\langle \\gamma(q) \\vert  \\bar q(x) \\gamma_\\mu q(0) \\vert 0 \\rangle\n= e_q f_{3 \\gamma} \\left(\\varepsilon_\\mu - q_\\mu \\frac{\\varepsilon\nx}{q x} \\right) \\int_0^1 du e^{i \\bar u q x} \\psi^v(u)\n\\nonumber \\\\\n&&\\langle \\gamma(q) \\vert \\bar q(x) \\gamma_\\mu \\gamma_5 q(0) \\vert 0\n\\rangle  = - \\frac{1}{4} e_q f_{3 \\gamma} \\epsilon_{\\mu \\nu \\alpha\n\\beta } \\varepsilon^\\nu q^\\alpha x^\\beta \\int_0^1 du e^{i \\bar u q\nx} \\psi^a(u)\n\\nonumber \\\\\n&&\\langle \\gamma(q) | \\bar q(x) g_s G_{\\mu \\nu} (v x) q(0) \\vert 0\n\\rangle = -i e_q \\langle \\bar q q \\rangle \\left(\\varepsilon_\\mu q_\\nu - \\varepsilon_\\nu\nq_\\mu \\right) \\int {\\cal D}\\alpha_i e^{i (\\alpha_{\\bar q} + v\n\\alpha_g) q x} {\\cal S}(\\alpha_i)\n\\nonumber \\\\\n&&\\langle \\gamma(q) | \\bar q(x) g_s \\tilde G_{\\mu \\nu} i \\gamma_5 (v\nx) q(0) \\vert 0 \\rangle = -i e_q \\langle \\bar q q \\rangle \\left(\\varepsilon_\\mu q_\\nu -\n\\varepsilon_\\nu q_\\mu \\right) \\int {\\cal D}\\alpha_i e^{i\n(\\alpha_{\\bar q} + v \\alpha_g) q x} \\tilde {\\cal S}(\\alpha_i)\n\\nonumber \\\\\n&&\\langle \\gamma(q) \\vert \\bar q(x) g_s \\tilde G_{\\mu \\nu}(v x)\n\\gamma_\\alpha \\gamma_5 q(0) \\vert 0 \\rangle = e_q f_{3 \\gamma}\nq_\\alpha (\\varepsilon_\\mu q_\\nu - \\varepsilon_\\nu q_\\mu) \\int {\\cal\nD}\\alpha_i e^{i (\\alpha_{\\bar q} + v \\alpha_g) q x} {\\cal\nA}(\\alpha_i)\n\\nonumber \\\\\n&&\\langle \\gamma(q) \\vert \\bar q(x) g_s G_{\\mu \\nu}(v x) i\n\\gamma_\\alpha q(0) \\vert 0 \\rangle = e_q f_{3 \\gamma} q_\\alpha\n(\\varepsilon_\\mu q_\\nu - \\varepsilon_\\nu q_\\mu) \\int {\\cal\nD}\\alpha_i e^{i (\\alpha_{\\bar q} + v \\alpha_g) q x} {\\cal\nV}(\\alpha_i) \\nonumber \\\\ && \\langle \\gamma(q) \\vert \\bar q(x)\n\\sigma_{\\alpha \\beta} g_s G_{\\mu \\nu}(v x) q(0) \\vert 0 \\rangle  =\ne_q \\langle \\bar q q \\rangle \\left\\{\n        \\left[\\left(\\varepsilon_\\mu - q_\\mu \\frac{\\varepsilon x}{q x}\\right)\\left(g_{\\alpha \\nu} -\n        \\frac{1}{qx} (q_\\alpha x_\\nu + q_\\nu x_\\alpha)\\right) \\right. \\right. q_\\beta\n\\nonumber \\\\ && -\n         \\left(\\varepsilon_\\mu - q_\\mu \\frac{\\varepsilon x}{q x}\\right)\\left(g_{\\beta \\nu} -\n        \\frac{1}{qx} (q_\\beta x_\\nu + q_\\nu x_\\beta)\\right) q_\\alpha\n\\nonumber \\\\ && -\n         \\left(\\varepsilon_\\nu - q_\\nu \\frac{\\varepsilon x}{q x}\\right)\\left(g_{\\alpha \\mu} -\n        \\frac{1}{qx} (q_\\alpha x_\\mu + q_\\mu x_\\alpha)\\right) q_\\beta\n\\nonumber \\\\ &&+\n         \\left. \\left(\\varepsilon_\\nu - q_\\nu \\frac{\\varepsilon x}{q.x}\\right)\\left( g_{\\beta \\mu} -\n        \\frac{1}{qx} (q_\\beta x_\\mu + q_\\mu x_\\beta)\\right) q_\\alpha \\right]\n   \\int {\\cal D}\\alpha_i e^{i (\\alpha_{\\bar q} + v \\alpha_g) qx} {\\cal T}_1(\\alpha_i)\n\\nonumber \\\\ &&+\n        \\left[\\left(\\varepsilon_\\alpha - q_\\alpha \\frac{\\varepsilon x}{qx}\\right)\n        \\left(g_{\\mu \\beta} - \\frac{1}{qx}(q_\\mu x_\\beta + q_\\beta x_\\mu)\\right) \\right. q_\\nu\n\\nonumber \\\\ &&-\n         \\left(\\varepsilon_\\alpha - q_\\alpha \\frac{\\varepsilon x}{qx}\\right)\n        \\left(g_{\\nu \\beta} - \\frac{1}{qx}(q_\\nu x_\\beta + q_\\beta x_\\nu)\\right)  q_\\mu\n\\nonumber \\\\ && -\n         \\left(\\varepsilon_\\beta - q_\\beta \\frac{\\varepsilon x}{qx}\\right)\n        \\left(g_{\\mu \\alpha} - \\frac{1}{qx}(q_\\mu x_\\alpha + q_\\alpha x_\\mu)\\right) q_\\nu\n\\nonumber \\\\ &&+\n         \\left. \\left(\\varepsilon_\\beta - q_\\beta \\frac{\\varepsilon x}{qx}\\right)\n        \\left(g_{\\nu \\alpha} - \\frac{1}{qx}(q_\\nu x_\\alpha + q_\\alpha x_\\nu) \\right) q_\\mu\n        \\right]\n    \\int {\\cal D} \\alpha_i e^{i (\\alpha_{\\bar q} + v \\alpha_g) qx} {\\cal T}_2(\\alpha_i)\n\\nonumber \\\\ &&+\n        \\frac{1}{qx} (q_\\mu x_\\nu - q_\\nu x_\\mu)\n        (\\varepsilon_\\alpha q_\\beta - \\varepsilon_\\beta q_\\alpha)\n    \\int {\\cal D} \\alpha_i e^{i (\\alpha_{\\bar q} + v \\alpha_g) qx} {\\cal T}_3(\\alpha_i)\n\\nonumber \\\\ &&+\n        \\left. \\frac{1}{qx} (q_\\alpha x_\\beta - q_\\beta x_\\alpha)\n        (\\varepsilon_\\mu q_\\nu - \\varepsilon_\\nu q_\\mu)\n    \\int {\\cal D} \\alpha_i e^{i (\\alpha_{\\bar q} + v \\alpha_g) qx} {\\cal T}_4(\\alpha_i)\n                        \\right\\}~,\n\\end{eqnarray}\nwhere $\\varphi_\\gamma(u)$ is the leading twist-2, $\\psi^v(u)$,\n$\\psi^a(u)$, ${\\cal A}$ and ${\\cal V}$ are the twist-3, and\n$h_\\gamma(u)$, $\\mathbb{A}$, ${\\cal T}_i$ ($i=1,~2,~3,~4$) are the\ntwist-4 photon DAs, and $\\chi$ is the magnetic susceptibility.\nThe measure ${\\cal D} \\alpha_i$ is defined as\n\\begin{eqnarray}\n\\label{nolabel05}\n\\int {\\cal D} \\alpha_i = \\int_0^1 d \\alpha_{\\bar q} \\int_0^1 d\n\\alpha_q \\int_0^1 d \\alpha_g \\delta(1-\\alpha_{\\bar\nq}-\\alpha_q-\\alpha_g)~.\\nonumber\n\\end{eqnarray}\n\nAs has already been noted, in determination of the magnetic moment\nresponsible for the negative to negative parity transition, four equations\nare needed, and for this purpose we choose the coefficients of the\nstructures $(\\varepsilon\\!\\cdot\\! p) I$,\n$(\\varepsilon\\!\\cdot\\! p) \\rlap\/{p}$, $\\rlap\/{p}\\rlap\/{\\varepsilon}$ and  \n$\\rlap\/{\\varepsilon}$. The sum rules for the negative to negative parity\ntransition magnetic moments can be obtained by choosing the coefficients of\nthe aforementioned Lorentz structures $\\Pi_i$, and equate them to the\ncorresponding coefficients in hadronic part. Solving then the linear\nequations for the coefficients describing the negative to negative\ntransition magnetic moments, and performing Borel transformation over the\nvariables $-p^2$ and $-(p+q)^2$ in order to suppress higher states and\ncontinuum contribution, we finally obtain the magnetic moment for the negative\nto negative parity baryon transitions as is given below,\n\\begin{eqnarray}                                                                        \n\\label{efrd11}\n\\mu \\!\\!\\! &=& \\!\\!\\! {e^{m_{B_1^{(-)}}^2\/2M^2} e^{m_{B_2^{(-)}}^2\/2M^2}  \\over\n2 \\lambda_{B_1^{(-)}}  \\lambda_{B_2^{(-)}}\n\\left( m_{B_1^{(+)}} +  m_{B_1^{(-)}}\\right)\n\\left( m_{B_2^{(+)}} +  m_{B_2^{(-)}}\\right)}\n\\Big\\{ \\left( m_{B_1^{(+)}} + m_{B_2^{(-)}}\\right)\n\\left(\\Pi_1^{B} - m_{B_2^{(+)}} \\Pi_2^{B}\\right) \\nonumber \\\\\n&+& \\!\\!\\! 2 m_{B_2^{(+)}} \\Pi_3^{B} - 2 \\Pi_4^{B}\\Big\\}~.\n\\end{eqnarray}\nIn this expression we take $M_1^2=M_2^2=2 M^2$, since the masses of the\n$\\Sigma_Q$, $\\Lambda_Q$, $\\Xi_Q^\\prime$ and $\\Xi_Q$ baryons are very close\nto each other. The expressions of $\\Pi_i^B$ are presented in Appendix A.\n\nIt follows from Eq. (\\ref{efrd11}) that in determination of the magnetic\nmoments of the $\\Sigma_Q \\to \\Lambda_Q$ and $\\Xi_Q^\\prime \\to \\Xi_Q$\ntransitions, the residues of the negative parity heavy baryons are\nnecessary. These residues can be determined from the analysis of the\ntwo-point correlation function\n\\begin{eqnarray}\n\\label{nolabel06} \n\\Pi(q^2)= i \\int d^4x e^{iqx} \\langle 0 \\vert \\mbox{\\rm T} \\left\\{ \\eta_Q(x)\n\\bar{\\eta}_Q(0) \\right\\} \\vert 0 \\rangle~, \\nonumber\n\\end{eqnarray}\nwhere $\\eta_Q$ is the interpolating current for the corresponding heavy\nbaryon given by Eq. (\\ref{efrd06}). This interpolating current interacts\nwith both positive and negative parity heavy baryons. Saturating this\ncorrelation function with the ground states of positive and negative parity\nbaryons we have,\n\\begin{eqnarray}\n\\label{nolabel07}\n\\Pi (q^2) = {\\vert \\lambda_{B^{(-)}} \\vert^2 (\\not\\!p - m_{B^{(-)}}) \\over\nm_{B^{(-)}}^2-p^2} +\n{\\vert \\lambda_{B^{(+)}} \\vert^2 (\\not\\!p + m_{B^{(+)}}) \\over m_{B^{(+)}}^2-p^2}~\\nonumber .\n\\end{eqnarray}\nEliminating the contributions coming from the positive parity baryons, the\nfollowing sum rules for the residue and mass of the negative parity baryons\nare obtained,\n\\begin{eqnarray}\n\\label{nolabel08}\n\\vert \\lambda_{B^{(-)}} \\vert^2 \\!\\!\\! &=& \\!\\!\\! {1\\over \\pi} {e^{m_{B^{(-)}}^2\/M^2} \\over m_{B^{(+)}}+m_{B^{(-)}}} \n\\int ds e^{-s\/M^2} \\left[ m_{B^{(+)}} \\mbox{\\rm Im} \\Pi_1^M(s) - \\mbox{\\rm Im}\n\\Pi_2^M(s) \\right]~,\\nonumber \\\\\n\\label{nolabel09}\nm_{B^{(-)}}^2 \\!\\!\\! &=& \\!\\!\\! {\\int_{m_Q^2}^{s_0} s ds e^{-s\/M^2} \\left[  m_{B^{(+)}} \\mbox{\\rm Im}\n\\Pi_1^M(s) - \\mbox{\\rm Im} \\Pi_2^M(s) \\right] \\over\n\\int_{m_Q^2}^{s_0} ds e^{-s\/M^2} \\left[  m_{B^{(+)}} \\mbox{\\rm Im}     \n\\Pi_1^M(s) - \\mbox{\\rm Im} \\Pi_2^M(s) \\right]}~ \\nonumber.\n\\end{eqnarray}\nHere $\\Pi_1^M$ and $\\Pi_2^M$ are the invariant functions corresponding to the\nstructures $\\not\\!\\!p$ and $I$, respectively. The expressions of $\\Pi_1^M$ and\n$\\Pi_2^M$ for the $\\Sigma_Q^0$ baryon are presented in Appendix B.\n\n\\section{Numerical analysis}\n\nIn this section we present our numerical results for the magnetic moments of\nthe $\\Sigma_Q \\to \\Lambda_Q$ and $\\Xi_Q^\\prime \\to \\Xi_Q$ transitions for\nnegative parity baryons derived from the LCSR. In this numerical analysis\nthe values of the the values of the relevant input parameters entering to\nthe LCSR are needed. The main nonperturbative input of LCSR is the DAs which\nare all calculated in \\cite{Rfrd18}, and for completeness we present\ntheir expressions in Appendix C. The other input parameters needed in the\nnumerical analysis are, quark condensate $\\langle \\bar q q \\rangle$, $m_0^2$, magnetic\nsusceptibility $\\chi$ of quarks, etc. In further numerical calculations\nwe use $\\left[ \\langle \\bar u u \\rangle = \\langle \\bar d d \\rangle \\right]_{\\mu=1~GeV} = -(0.243)^3~GeV^3$\n\\cite{Rfrd19}, $\\langle \\bar s s \\rangle \\vert_{\\mu=1~GeV} = 0.8 \\langle \\bar u u \\rangle\\vert_{\\mu=1~GeV}$,\n$m_0^2=(0.8\\pm 0.2)~GeV^2$ \\cite{Rfrd20}. The magnetic susceptibility was\ndetermined within the QCD sum rules in \\cite{Rfrd21,Rfrd22,Rfrd23}).\n\nHaving all necessary ingredients at hand, we are now ready to perform\nthe numerical analysis for the transition magnetic moments of the negative\nparity baryons. Sum rules contain also three auxiliary parameters in the\ninterpolating current other than\nthose input parameters given above: Borel mass parameter $M^2$, continuum\nthreshold $s_0$, and the arbitrary parameter $t$. We demand that the\nmagnetic moment should be independent on these auxiliary parameters.\nTherefore we shall look for the ``working regions\" of these parameters,\nwhere magnetic moments exhibit good stability with respect to their variations\nin respective domains. It should be remembered  that the continuum  threshold\n$s_0$ is not arbitrary but related to the first excited states. The\ndifference $\\sqrt{s_0}-m_{ground}$ is the energy needed to transfer the\nbaryon to its first excited state. Usually this difference varies in the\nrange $0.3~GeV \\le \\sqrt{s_0}-m_{ground} \\le 0.8~GeV$, and in our analysis\nwe choose the average value $\\sqrt{s_0}-m_{ground}=0.5~GeV$.\n\nHaving determined the value of $s_0$, next we try to find the ``working\nregions\" of the Borel parameter $M^2$. The upper bound of $M^2$ is obtained\nby demanding that contributions of higher states and continuum constitute\nabout 40\\% of the perturbative part.The lower bound is determined from the\ncondition that higher twist contributions are less than the leading twist\ncontributions. Our analysis shows that the working regions of $M^2$ where\nboth conditions are satisfied are\n\\begin{eqnarray}\n\\label{nolabel09}\n2.5~GeV^2 \\le M^2 \\le 4.0~GeV^2,~~\\mbox{for $\\Sigma_c$, $\\Xi_c^\\prime$,\n$\\Lambda_c$, $\\Xi_c$}~, \\nonumber \\\\\n4.5~GeV^2 \\le M^2 \\le 7.0~GeV^2,~~\\mbox{for $\\Sigma_b$, $\\Xi_b^\\prime$,\n$\\Lambda_b$, $\\Xi_b$}~. \\nonumber\n\\end{eqnarray}\nAs an example, in Figs. (1) and (2) we present the dependence of\n$\\mu_{\\Sigma_b^0 \\to \\Lambda_b^0}$ on $M^2$ at several fixed values of the\narbitrary parameter $t$, at $s_0=40.0~GeV^2$ and  $s_0=42.5~GeV^2$. We\nobserve from these figures that transition magnetic moment exhibits good\nstability when $M^2$ varies in the region $3.0~GeV^2 \\le M^2 \\le 4.0~GeV^2$.   \nIn Figs. (3) and (4) we present the dependence of\n$\\mu_{\\Sigma_b^0 \\to \\Lambda_b^0}$ on $\\cos\\theta$ (where $t=\\tan\\theta$ at\ntwo fixed values of $M^2$ and at $s_0=40.0~GeV^2$ and  $s_0=42.5~GeV^2$,\nrespectively. We see from these figures that, when $\\cos\\theta$ varies in\nthe domain $-1.0 \\le \\cos\\theta \\le -0.7$, the magnetic moment demonstrates\ngood stability with respect to the variation in $\\cos\\theta$. Our final\nresult for the transition magnetic moment is $\\mu_{\\Sigma_b^0 \\to\n\\Lambda_b^0}=(-0.3 \\pm 0.05) \\mu_N$.\n\nThe analysis of the sum rules for the other transition magnetic moments are\nalso calculated, whose values can be summarized as,\n\\begin{eqnarray}\n\\label{nolabel10}\n\\mu_{\\Sigma_c^+ \\to \\Lambda_c^+} \\!\\!\\! &=& \\!\\!\\! (0.25 \\pm 0.05) \\mu_N~, \\nonumber \\\\\n\\mu_{\\Xi_c^{\\prime 0} \\to \\Xi_c^{0}} \\!\\!\\! &=& \\!\\!\\! (0.08 \\pm 0.01) \\mu_N~, \\nonumber \\\\\n\\mu_{\\Xi_c^{\\prime +} \\to \\Xi_c^{+}} \\!\\!\\! &=& \\!\\!\\! (0.20 \\pm 0.05) \\mu_N~, \\nonumber \\\\\n\\mu_{\\Xi_b^{\\prime 0} \\to \\Xi_b^{0}} \\!\\!\\! &=& \\!\\!\\! (-0.008 \\pm 0.001) \\mu_N~, \\nonumber \\\\\n\\mu_{\\Xi_b^{\\prime -} \\to \\Xi_b^{-}} \\!\\!\\! &=& \\!\\!\\! (0.10 \\pm 0.01) \\mu_N~, \\nonumber\n\\end{eqnarray}\nwhere upper signs correspond to the electric charge of the corresponding\nnegative parity baryons. \nIt can easily be seen from these results that the transition magnetic moments between\nthe neutral $\\Xi^\\prime$ and $\\Xi$ baryons are very close to zero. The\nmagnetic moments for the $\\Sigma_c^+ \\to \\Lambda_c^+$ and $\\Xi_b^{\\prime +}\n\\to \\Xi_b^+$ transitions are very close to each other which follows from\n$SU(3)$ symmetry arguments.\n\nIn conclusion, the transition magnetic moments of the negative parity,\nspin-1\/2 heavy baryons are estimated within the QCD sum rules. The\ncontributions coming from the positive to positive, as well as positive to\nnegative parity transitions are eliminated by constructing various sum\nrules. It is obtained that the magnetic moments between neutral, negative\nparity heavy $\\Xi_Q^{\\prime 0}$ and $\\Xi_Q^0$ baryons are very small.\nMoreover, it is found that the magnetic moments for the $\\Sigma_Q \\to\n\\Lambda_Q$ and $\\Xi_Q^{\\prime \\pm} \\to \\Xi_Q^\\pm$ transitions of the\nnegative parity heavy baryons are quite large and can be measured in future\nexperiments.\n\n\\newpage\n\n\n\n\\section*{Appendix A}\nIn this Appendix we present the expressions of the invariant functions\n$\\Pi_i^B$ appearing in\nthe sum rules for the magnetic moment of \n$\\Xi_b^{\\prime 0} \\to \\Xi_b^0$ transition. Here in this appendix, and in\nappendix B the masses of the light quarks are\nneglected.\\\\\\\\\n\n\\setcounter{equation}{0}\n\\setcounter{section}{0}\n\n\n\n{\\bf 1) Coefficient of the $(\\varepsilon\\!\\cdot\\!p) I$ structure}\n\n\\begin{eqnarray}\n\\Pi_1^B \\!\\!\\! &=& \\!\\!\\!\n- {1 \\over 32 \\sqrt{3} \\pi^2}\n(-1 + t) m_b^2 M^4 \\Big\\{4 (2 + t) m_b^2 \\left(e_u \\langle \\bar s s \\rangle - e_s \\langle \\bar u u \\rangle\\right) {\\cal I}_3 \\nonumber \\\\\n&+& \\!\\!\\! e_b \\left(\\langle \\bar s s \\rangle - \\langle \\bar u u \\rangle\\right) \\Big[(7 + 3 t) {\\cal I}_2 - 2 (3 + t) \nm_b^2 {\\cal I}_3\\Big]\\Big\\} \\nonumber \\\\\n&+& \\!\\!\\! {\\sqrt{3} \\over 8 \\pi^2}\n(-1 + t) m_b^4 M^4 \\left(e_s \\langle \\bar s s \\rangle - e_u \\langle \\bar u u \\rangle\\right) \n   {\\cal I}_3 \\widetilde{j}(h_\\gamma) \\nonumber \\\\\n&+& \\!\\!\\! {1 \\over 16 \\sqrt{3} \\pi^2} \n (-1 + t) (3 + t) (e_s - e_u) f_{3\\gamma} m_b^3 M^4 \n   \\left({\\cal I}_2 - m_b^2 {\\cal I}_3\\right) \\psi^v(u_0) \\nonumber \\\\\n&+& \\!\\!\\! {e^{-m_b^2\/M^2}\\over 768 \\sqrt{3} \\pi^2}\n(-1 + t) M^2 \\Big\\{12 m_0^2 \\left(e_u \\langle \\bar s s \\rangle - e_s \\langle \\bar u u \\rangle\\right) \n     \\Big[4 + t \\left(2 + m_b^2 e^{m_b^2\/M^2} {\\cal I}_2\\right)\\Big] \\nonumber \\\\\n&+& \\!\\!\\! e_b \\left(\\langle \\bar s s \\rangle - \\langle \\bar u u \\rangle\\right) \n     \\Big[ 24 (7 + 3 t) m_b^2 e^{m_b^2\/M^2} \\left(-{\\cal I}_1 + m_b^2 {\\cal I}_2\\right) + \n      m_0^2 \\Big(7 (1 + t) + (29 + 17 t) m_b^2 e^{m_b^2\/M^2} {\\cal I}_2\\Big)\\Big]\\Big\\} \\nonumber \\\\\n&+& \\!\\!\\! {e^{-m_b^2\/M^2}\\over 1152 \\sqrt{3} m_b \\pi^2}\n(-1 + t) f_{3\\gamma} M^2 \n\\Big\\{e_u \\Big[-96 (1 + t) m_b \\pi^2 \\langle \\bar s s \\rangle - (3 + t) \\langle g_s^2 G^2 \\rangle \n       \\left(-1 + 3 m_b^2 e^{m_b^2\/M^2} {\\cal I}_2\\right)\\Big] \\nonumber \\\\\n&+& \\!\\!\\! e_s \\Big[96 (1 + t) m_b \\pi^2 \\langle \\bar u u \\rangle + \n      (3 + t) \\langle g_s^2 G^2 \\rangle \\left(-1 + 3 m_b^2 e^{m_b^2\/M^2} {\\cal I}_2\\right)\\Big]\\Big\\} \\psi^v(u_0) \\nonumber \\\\\n&+& \\!\\!\\! {e^{-m_b^2\/M^2}\\over 48 \\sqrt{3} M^2}\n(-1 + t^2) f_{3\\gamma} m_0^2 m_b^2 \\left(e_u \\langle \\bar s s \\rangle - e_s \\langle \\bar u u \\rangle\\right) \\psi^v(u_0) \\nonumber \\\\\n&+& \\!\\!\\! {e^{-m_b^2\/M^2}\\over 6912 \\sqrt{3} M^4 \\pi^2}\n(-1 + t) \\langle g_s^2 G^2 \\rangle m_b^2 \\left(e_u \\langle \\bar s s \\rangle - e_s \\langle \\bar u u \\rangle\\right) \\Big[-3 (2 + t) m_0^2 + \n   8 (1 + t) f_{3\\gamma} \\pi^2 \\psi^v(u_0)\\Big] \\nonumber \\\\\n&+& \\!\\!\\! {e^{-m_b^2\/M^2}\\over 1728 \\sqrt{3} M^6}\n(-1 + t^2) f_{3\\gamma} \\langle g_s^2 G^2 \\rangle m_0^2 m_b^2 \\left(e_u \\langle \\bar s s \\rangle - e_s \\langle \\bar u u \\rangle\\right) \\psi^v(u_0) \\nonumber \\\\\n&-& \\!\\!\\! {e^{-m_b^2\/M^2}\\over 3456 \\sqrt{3} M^8}\n(-1 + t^2) f_{3\\gamma} \\langle g_s^2 G^2 \\rangle m_0^2 m_b^4 \\left(e_u \\langle \\bar s s \\rangle - e_s \\langle \\bar u u \\rangle\\right) \\psi^v(u_0) \\nonumber \\\\\n&+& \\!\\!\\! {e^{-m_b^2\/M^2}\\over 2304 \\sqrt{3} \\pi^2} \n(-1 + t) \\Big[4 (2 + t) \\langle g_s^2 G^2 \\rangle \\left(e_u \\langle \\bar s s \\rangle - e_s \\langle \\bar u u \\rangle\\right) + \n    3 (29 + 17 t) e_b m_0^2 m_b^2 e^{m_b^2\/M^2} \\left(\\langle \\bar s s \\rangle - \\langle \\bar u u \\rangle\\right) {\\cal I}_1\\Big] \\nonumber \\\\\n&-& \\!\\!\\! {e^{-m_b^2\/M^2}\\over 192 \\sqrt{3} \\pi^2}\n(-1 + t) \\langle g_s^2 G^2 \\rangle \\left(e_s \\langle \\bar s s \\rangle - e_u \\langle \\bar u u \\rangle\\right) \n   \\widetilde{j}(h_\\gamma) \\nonumber \\\\\n&-& \\!\\!\\! {e^{-m_b^2\/M^2}\\over 288 \\sqrt{3}}\n(-1 + t^2) f_{3\\gamma} m_0^2 \\left(e_u \\langle \\bar s s \\rangle - e_s \\langle \\bar u u \\rangle\\right) \\psi^v(u_0)~. \\nonumber\n\\end{eqnarray}\n\\\\\\\\\\\\\n\n{\\bf 2) Coefficient of the $(\\varepsilon \\! \\cdot\\! p)\\!\\not\\!p$ structure}\n\n\\begin{eqnarray}\n\\Pi_2^B \\!\\!\\! &=& \\!\\!\\!\n{1\\over 8 \\sqrt{3} \\pi^2}\n(-2 + t + t^2) m_b^3 M^2 \\Big[(e_b + e_u) \\langle \\bar s s \\rangle - (e_b + e_s) \\langle \\bar u u \\rangle\\Big] \n  \\left({\\cal I}_2 - m_b^2 {\\cal I}_3\\right) \\nonumber \\\\\n&-& \\!\\!\\! {e^{-m_b^2\/M^2}\\over 1152 \\sqrt{3} m_b M^2 \\pi^2}\n(-2 + t + t^2) \\langle g_s^2 G^2 \\rangle m_0^2 \\left(e_u \\langle \\bar s s \\rangle - e_s \\langle \\bar u u \\rangle\\right) \\nonumber \\\\\n&+& \\!\\!\\! {e^{-m_b^2\/M^2}\\over 2304 \\sqrt{3} M^4 \\pi^2}\n(-2 + t + t^2) \\langle g_s^2 G^2 \\rangle m_0^2 m_b \\left(e_u \\langle \\bar s s \\rangle - e_s \\langle \\bar u u \\rangle\\right) \\nonumber \\\\\n&-& \\!\\!\\! {e^{-m_b^2\/M^2}\\over 576 \\sqrt{3} m_b \\pi^2}\n(-1 + t) (2 + t) \\langle g_s^2 G^2 \\rangle \\left(e_u \\langle \\bar s s \\rangle - e_s \\langle \\bar u u \\rangle\\right) \\nonumber \\\\\n&+& \\!\\!\\! {\\sqrt{3}\\over 64 \\pi^2}\n(-1 + t^2) m_0^2 m_b \\left(e_u \\langle \\bar s s \\rangle - e_s \\langle \\bar u u \\rangle\\right) {\\cal I}_1 \\nonumber \\\\\n&+& \\!\\!\\! {1\\over 192 \\sqrt{3} \\pi^2}\n (-1 + t) m_b \\Big\\{\\Big[(2 + t) e_u \\langle g_s^2 G^2 \\rangle - \n      3 (3 + 2 t) e_b m_0^2 m_b^2 -3 (7 + 5 t) e_u m_0^2 m_b^2 \\Big] \\langle \\bar s s \\rangle \\nonumber \\\\\n&+& \\!\\!\\! \\Big[-(2 + t) e_s \\langle g_s^2 G^2 \\rangle + 3 (3 + 2 t) e_b m_0^2 m_b^2 + 3 (7 + 5 t) e_s m_0^2 \n       m_b^2\\Big] \\langle \\bar u u \\rangle\\Big\\} {\\cal I}_2~. \\nonumber\n\\end{eqnarray}\n\\\\\\\\\n\n\n\n{\\bf 3) Coefficient of the $\\not\\!p\\!\\!\\not\\!\\varepsilon$ structure}\n\n\\begin{eqnarray}\n\\Pi_3^B \\!\\!\\! &=& \\!\\!\\!\n{1 \\over 256 \\sqrt{3} \\pi^4}\n(-1 + t) m_b^3 M^6 \\Big[ -3 (3 + t) (e_s - e_u) \n    \\left({\\cal I}_2 - 2 m_b^2 {\\cal I}_3 + m_b^4 {\\cal I}_4\\right) \\nonumber \\\\\n&+& \\!\\!\\! 8 (-1 + t) m_b \\pi^2 \n    \\left(e_s \\langle \\bar s s \\rangle - e_u \\langle \\bar u u \\rangle\\right) \\chi \\left({\\cal I}_3 - m_b^2 {\\cal\nI}_4\\right) \\varphi_\\gamma^\\prime (u_0)\\Big] \\nonumber \\\\\n&-& \\!\\!\\! {1 \\over 768 \\sqrt{3} \\pi^4}\n(-1 + t) (3 + t) m_b^3 M^4 \\Big[ \\langle g_s^2 G^2 \\rangle (e_u - e_s) + \n    24 (e_b - e_u) m_b \\pi^2 \\langle \\bar s s \\rangle \\nonumber \\\\\n&+& \\!\\!\\!  24 (-e_b + e_s) m_b \\pi^2 \\langle \\bar u u \\rangle \\Big]\n   {\\cal I}_3 \\nonumber \\\\\n&+& \\!\\!\\! {1 \\over 1024 \\sqrt{3} \\pi^4}\n(e_s - e_u) m_b M^4 \\Big\\{-(-1 + t) (3 + t) \\langle g_s^2 G^2 \\rangle {\\cal I}_2 + \n   16 f_{3\\gamma} m_b^2 \\pi^2 \\left(-{\\cal I}_2\n+ m_b^2 {\\cal I}_3\\right) \\nonumber \\\\ \n&\\times& \\!\\!\\!    \\Big[2 (-1 + t) (3 + t) \\psi^v(u_0) - (-1 + t^2) \n      \\psi^{a\\prime}(u_0)\\Big] \\Big\\} \\nonumber \\\\\n&+& \\!\\!\\! {1 \\over 128 \\sqrt{3} \\pi^2}\n(-1 + t) m_b^2 M^4 \\left(e_s \\langle \\bar s s \\rangle - e_u \\langle \\bar u u \\rangle\\right) {\\cal I}_2 \n   \\Big\\{(5 + t) i_1({\\cal S},1) + (1 + 5 t) i_1(\\widetilde{\\cal S},1) \\nonumber \\\\\n&+& \\!\\!\\! 2 i_1({\\cal T}_1,1) + i_1({\\cal T}_2,1) + \n    2 i_1({\\cal T}_3,1) - 5 i_1({\\cal T}_4,1) - 6 i_1({\\cal S},v)\n - 2 i_1(\\widetilde{\\cal S},v) \\nonumber \\\\\n&-& \\!\\!\\! t \\Big[2 i_1({\\cal T}_1,1) - 5 i_1({\\cal T}_2,1) + 2 i_1({\\cal T}_3,1) + i_1({\\cal T}_4,1) + 2 i_1({\\cal S},v) + \n      6 i_1(\\widetilde{\\cal S},v) \\nonumber \\\\\n&+& \\!\\!\\! 4 i_1({\\cal T}_2,v)- 4 i_1({\\cal T}_3,v)\\Big] - \n    4 i_1({\\cal T}_3,v) + 4 i_1({\\cal T}_4,v)\\Big\\} \\nonumber \\\\\n&-& \\!\\!\\! {1 \\over 128 \\sqrt{3} \\pi^2} \n(-1 + t) m_b^4 M^4 \\left(e_s \\langle \\bar s s \\rangle - e_u \\langle \\bar u u \\rangle\\right) {\\cal I}_3 \n   \\Big\\{4 (2 + t) i_1({\\cal S},1) + (4 + 8 t) i_1(\\widetilde{\\cal S},1) \\nonumber \\\\\n&-& \\!\\!\\! 4 \\Big[(-1 + t) i_1({\\cal T}_1,1) - i_1({\\cal T}_2,1) + 2 i_1({\\cal T}_4,1) + 3 i_1({\\cal S},v) + \n      i_1(\\widetilde{\\cal S},v) + i_1({\\cal T}_2,v) \\nonumber \\\\\n&+& \\!\\!\\! t \\Big(-2 i_1({\\cal T}_2,1) + i_1({\\cal T}_4,1) + \n        i_1({\\cal S},v) + 3 i_1(\\widetilde{\\cal S},v)\n+ i_1({\\cal T}_2,v)\\Big)\\Big] \\nonumber \\\\\n&+& \\!\\!\\! 4 (1 + t) i_1({\\cal T}_4,v) + 8 (2 + t) \\widetilde{j}(h_\\gamma) + \n    (-1 + t) \\mathbb{A}^\\prime (u_0)\\Big\\} \\nonumber \\\\\n&+& \\!\\!\\! {e^{-m_b^2\/M^2}\\over 768 \\sqrt{3} \\pi^2}\n(-1 + t) M^2 \\Big\\{m_0^2 \\left(e_u \\langle \\bar s s \\rangle - e_s \\langle \\bar u u \\rangle\\right) \\Big[-6 (3 + t) + \n      (7 + t) m_b^2 e^{m_b^2\/M^2} {\\cal I}_2\\Big] \\nonumber \\\\\n&+& \\!\\!\\! e_b \\left(\\langle \\bar s s \\rangle - \\langle \\bar u u \\rangle\\right)\n\\Big[(11 + 5 t) m_0^2 - 24 (3 + t) m_b^2 e^{m_b^2\/M^2} \\left(-{\\cal I}_1 +\nm_b^2 {\\cal I}_2\\right)\\Big]\\Big\\} \\nonumber \\\\\n&-& \\!\\!\\! {e^{-m_b^2\/M^2}\\over 2304 \\sqrt{3} m_b \\pi^2}\n(-1 + t) (3 + t) f_{3\\gamma} M^2 \n   \\Big[-(e_s - e_u) \\langle g_s^2 G^2 \\rangle - 96 m_b \\pi^2 \\left(e_u \\langle \\bar s s \\rangle \n- e_s \\langle \\bar u u \\rangle \\right) \\nonumber \\\\\n&+& \\!\\!\\! 3 (e_s - e_u) \\langle g_s^2 G^2 \\rangle m_b^2 e^{m_b^2\/M^2} {\\cal I}_2\\Big] \\psi^v(u_0) \\nonumber \\\\\n&-& \\!\\!\\! {1 \\over 2304 \\sqrt{3} \\pi^2}\n(-1 + t)^2 \\langle g_s^2 G^2 \\rangle m_b^2 M^2 \\left(e_s \\langle \\bar s s \\rangle - e_u \\langle \\bar u u \\rangle\\right) \\chi {\\cal I}_2 \n   \\varphi_\\gamma^\\prime (u_0) \\nonumber \\\\\n&+& \\!\\!\\! {e^{-m_b^2\/M^2}\\over 4608 \\sqrt{3} m_b \\pi^2}\nf_{3\\gamma} M^2 \\Big[96 t (-1 + t) m_b \\pi^2 \\left(e_u \\langle \\bar s s \\rangle -\ne_s \\langle \\bar u u \\rangle\\right) \\nonumber \\\\\n&+& \\!\\!\\! (-1 + t^2) (e_s - e_u) \\langle g_s^2 G^2 \\rangle \\left(-1 + 3 m_b^2 e^{m_b^2\/M^2} {\\cal I}_2\\right)\\Big] \n  \\psi^{a\\prime}(u_0) \\nonumber \\\\\n&-& \\!\\!\\! {e^{-m_b^2\/M^2}\\over 192 \\sqrt{3} M^2}\n(-1 + t) f_{3\\gamma} m_0^2 m_b^2 \\left(e_u \\langle \\bar s s \\rangle - e_s \\langle \\bar u u \\rangle\\right) \n   \\Big[2 (3 + t) \\psi^v(u_0) + t \\psi^{a\\prime}(u_0)\\Big] \\nonumber \\\\\n&+& \\!\\!\\! {e^{-m_b^2\/M^2}\\over 27648 \\sqrt{3} M^4 \\pi^2}\n(-1 + t) \\langle g_s^2 G^2 \\rangle m_b^2 \\left(e_u \\langle \\bar s s \\rangle - e_s \\langle \\bar u u \\rangle\\right) \\Big\\{3 (3 + t) m_0^2 \\nonumber \\\\\n&-& \\!\\!\\! 8 f_{3\\gamma} \\pi^2 \\Big[2 (3 + t) \\psi^v(u_0) + \n     t \\psi^{a\\prime}(u_0)\\Big]\\Big\\} \\nonumber \\\\\n&-& \\!\\!\\! {e^{-m_b^2\/M^2}\\over 6912 \\sqrt{3} M^6}\n(-1 + t) f_{3\\gamma} \\langle g_s^2 G^2 \\rangle m_0^2 m_b^2 \\left(e_u \\langle \\bar s s \\rangle - e_s \\langle \\bar u u \\rangle\\right) \n   \\Big[2 (3 + t) \\psi^v(u_0) + t \\psi^{a\\prime}(u_0)\\Big] \\nonumber \\\\\n&+& \\!\\!\\! {e^{-m_b^2\/M^2}\\over 13824 \\sqrt{3} M^8}\n(-1 + t) f_{3\\gamma} \\langle g_s^2 G^2 \\rangle m_0^2 m_b^4 \\left(e_u \\langle \\bar s s \\rangle - e_s \\langle \\bar u u \\rangle\\right) \n  \\Big[2 (3 + t) \\psi^v(u_0) + t \\psi^{a\\prime}(u_0)\\Big] \\nonumber \\\\\n&+& \\!\\!\\! {e^{-m_b^2\/M^2}\\over 9216 \\sqrt{3} \\pi^2}\n(-1 + t) \\langle g_s^2 G^2 \\rangle \\left(e_s \\langle \\bar s s \\rangle - e_u \\langle \\bar u u \\rangle\\right) \\Big\\{3 (1 + t) i_1({\\cal S},1) + \n   3 (1 + t) i_1(\\widetilde{\\cal S},1) \\nonumber \\\\\n&+& \\!\\!\\! 2 i_1({\\cal T}_1,1) + 3 i_1({\\cal T}_2,1) - 2 i_1({\\cal T}_3,1) - \n   3 i_1({\\cal T}_4,1) - 6 i_1({\\cal S},v) - 2 i_1(\\widetilde{\\cal S},v) - 4 i_1({\\cal T}_2,v) \\nonumber \\\\\n&+& \\!\\!\\! 4 i_1({\\cal T}_3,v) + 16 \\widetilde{j}(h_\\gamma) - \\mathbb{A}^\\prime (u_0) + \n   t \\Big[-2 i_1({\\cal T}_1,1) + 3 i_1({\\cal T}_2,1) + 2 i_1({\\cal T}_3,1) - 3 i_1({\\cal T}_4,1) \\nonumber \\\\\n&-& \\!\\!\\! 2 i_1({\\cal S},v) - 6 i_1(\\widetilde{\\cal S},v) - 4 i_1({\\cal T}_3,v) + 4 i_1({\\cal T}_4,v) + \n     8 \\widetilde{j}(h_\\gamma) + \\mathbb{A}^\\prime (u_0)\\Big]\\Big\\} \\nonumber \\\\\n&-& \\!\\!\\! {e^{-m_b^2\/M^2}\\over 2304 \\sqrt{3} \\pi^2}\n(-1 + t) \\Big\\{(3 + t) \\langle g_s^2 G^2 \\rangle \\left(e_u \\langle \\bar s s \\rangle - e_s \\langle \\bar u u \\rangle\\right) + \n    3 (11 + 5 t) e_b m_0^2 m_b^2 e^{m_b^2\/M^2} \\left(\\langle \\bar s s \\rangle - \\langle \\bar u u \\rangle\\right) {\\cal I}_1 \\nonumber \\\\\n&+& \\!\\!\\! 2 f_{3\\gamma} m_0^2 \\pi^2 \\left(e_u \\langle \\bar s s \\rangle - e_s \\langle \\bar u u \\rangle\\right) \\Big[2 (11 + 5 t) \\psi^v(u_0) + \n      (2 + 5 t) \\psi^{a\\prime}(u_0)\\Big]\\Big\\}~. \\nonumber\n\\end{eqnarray}\n\\\\\\\\\\\\\n\n\n\n\n\n{\\bf 4) Coefficient of the $\\not\\!\\varepsilon$ structure}\n\n\\begin{eqnarray}\n\\Pi_4^B \\!\\!\\! &=& \\!\\!\\!\n{\\sqrt{3}\\over 32 \\pi^4}\n(1 + t + t^2) (e_s - e_u) m_b^4 M^8 \n  \\left(-{\\cal I}_3 + m_b^2 {\\cal I}_4\\right) \\nonumber \\\\\n&+& \\!\\!\\! {1\\over 128 \\sqrt{3} \\pi^2}\n(e_s - e_u) f_{3\\gamma} m_b^2 M^6 \\left[-3 (1 + t)^2 {\\cal I}_2 + \n    4 (1 + t + t^2) m_b^2 {\\cal I}_3\\right] i_2({\\cal A},v) \\nonumber \\\\\n&+& \\!\\!\\! {1\\over 128 \\sqrt{3} \\pi^2}\n(e_s - e_u) f_{3\\gamma} m_b^2 M^6 \\left[-3 (1 + t)^2 {\\cal I}_2 + \n    2 (1 + 4 t + t^2) m_b^2 {\\cal I}_3\\right] i_2({\\cal V},v) \\nonumber \\\\\n&-& \\!\\!\\! {1\\over 32 \\sqrt{3} \\pi^2}\n(1 + t + t^2) (e_s - e_u) f_{3\\gamma} m_b^4 M^6 {\\cal I}_3\n\\left[4 \\psi^v(u_0)-\\psi^{a\\prime}(u_0)\\right] \\nonumber \\\\\n&+& \\!\\!\\! {\\sqrt{3}\\over 64 \\pi^2}\n(-1 + t^2) m_b^3 M^6 \\left (e_s \\langle \\bar s s \\rangle - e_u \\langle \\bar u u \\rangle\\right) \\chi \n   \\left({\\cal I}_2 - m_b^2 {\\cal I}_3\\right) \\varphi_\\gamma^\\prime (u_0) \\nonumber \\\\\n&+& \\!\\!\\! {e^{-m_b^2\/M^2}\\over 1536 \\sqrt{3} m_b \\pi^2}\n(-1 + t^2) \\langle g_s^2 G^2 \\rangle M^4 \\left(e_s \\langle \\bar s s \\rangle - e_u \\langle \\bar u u \\rangle\\right)\n\\chi \\left(-1 + 3 m_b^2 e^{m_b^2\/M^2} {\\cal I}_2\\right)\n\\varphi_\\gamma^\\prime (u_0) \\nonumber \\\\\n&-& \\!\\!\\! {1 \\over 16 \\sqrt{3} \\pi^2}\n(-2 + t + t^2) m_b^5 M^4 \\left[(e_b + e_u) \\langle \\bar s s \\rangle - (e_b + e_s)\n\\langle \\bar u u \\rangle\\right] {\\cal I}_3 \\nonumber \\\\\n&-& \\!\\!\\! {\\sqrt{3} \\over 128 \\pi^2}\n(-1 + t^2) m_b M^4 \\left(e_s \\langle \\bar s s \\rangle - e_u \\langle \\bar u u \\rangle\\right) {\\cal I}_1 \n    \\left[i_1({\\cal T}_1,1) + i_1({\\cal T}_3,1)\\right] \\nonumber \\\\\n&-& \\!\\!\\! {1 \\over 128 \\sqrt{3} \\pi^2}\n(-1 + t) m_b M^4 \\left(e_s \\langle \\bar s s \\rangle - e_u \\langle \\bar u u \\rangle\\right) {\\cal I}_1 \n   \\Big\\{(5 + t) i_1({\\cal S},1) \\nonumber \\\\\n&-& \\!\\!\\! (1 + 5 t) \\left[i_1(\\widetilde{\\cal S},1) + i_1({\\cal T}_2,1)\\right] - \n    (5 + t) i_1({\\cal T}_4,1)\\Big\\} \\nonumber \\\\\n&-& \\!\\!\\! {1 \\over 768 \\sqrt{3} \\pi^4}\nm_b^2 M^4 {\\cal I}_2 \\Big\\{-(1 + t + t^2) e_s \\langle g_s^2 G^2 \\rangle + \n    (1 + t + t^2) e_u \\langle g_s^2 G^2 \\rangle \\nonumber \\\\\n&-& \\!\\!\\! 48 (-2 + t + t^2) e_b m_b \\pi^2 \n     \\left(\\langle \\bar s s \\rangle - \\langle \\bar u u \\rangle\\right) + 3 (-1 + t) m_b \\pi^2 \\left(e_s \\langle \\bar s s \\rangle - e_u \\langle \\bar u u \\rangle\\right) \\nonumber \\\\\n&\\times& \\!\\!\\! \\Big[2 (-1 + 7 t) \\left(i_1(\\widetilde{\\cal S},1) + i_1({\\cal T}_2,1)\\right) + \n      2 (-7 + t) \\left(i_1({\\cal S},1) - i_1({\\cal T}_4,1)\\right) \\nonumber\\\\\n&+& \\!\\!\\! 8 (2 + t) i_1({\\cal S},v) - \n      4 (-1 + t) i_1(\\widetilde{\\cal S},v) + 8 t i_1({\\cal T}_4,v) + \n      4 (3 + t) \\left(i_1({\\cal T}_2,v) - 2 \\widetilde{j}(h_\\gamma)\\right) \\nonumber \\\\\n&+& \\!\\!\\! 3 (1 + t) \\left(-4 (i_1({\\cal T}_1,1) + i_1({\\cal T}_3,v)) + \\mathbb{A}^\\prime\n      (u_0)\\right)\\Big]\\Big\\} \\nonumber \\\\\n&-& \\!\\!\\! {e^{-m_b^2\/M^2}\\over 9216 \\sqrt{3} \\pi^2}\n(-1 + t)^2 (e_s - e_u) f_{3\\gamma} \\langle g_s^2 G^2 \\rangle M^2 \\left[i_2({\\cal A},v) -\ni_2({\\cal V},v)\\right] \\nonumber \\\\\n&-& \\!\\!\\! {e^{-m_b^2\/M^2}\\over 9216 \\sqrt{3} m_b \\pi^2}\n(-1 + t) \\langle g_s^2 G^2 \\rangle M^2 \\left(e_s \\langle \\bar s s \\rangle - e_u \\langle \\bar u u \\rangle\\right) \\Big\\{4 (-1 + t) i_1({\\cal S},1) \\nonumber \\\\\n&+& \\!\\!\\! 2 \\Big[2 (-1 + t) i_1(\\widetilde{\\cal S},1) - 3 (1 + t) i_1({\\cal T}_1,1) - 2 i_1({\\cal T}_2,1) + \n      3 i_1({\\cal T}_3,1) \\nonumber \\\\\n&+& \\!\\!\\!  2 \\Big(i_1({\\cal T}_4,1) + 4 i_1({\\cal S},v) + i_1(\\widetilde{\\cal S},v) + \n        3 i_1({\\cal T}_2,v) - 3 i_1({\\cal T}_3,v) - 6 \\widetilde{j}(h_\\gamma) \\Big) \\nonumber \\\\\n&+& \\!\\!\\! t \\Big(2 i_1({\\cal T}_2,1) + 3 i_1({\\cal T}_3,1) - 2 i_1({\\cal T}_4,1)\n+ 4 i_1({\\cal S},v) - 2 i_1(\\widetilde{\\cal S},v) + 2 i_1({\\cal T}_2,v) \\nonumber \\\\\n&-& \\!\\!\\! 6 i_1({\\cal T}_3,v) + 4  i_1({\\cal T}_4,v) - \n          4 \\widetilde{j}(h_\\gamma)\\Big)\\Big] + 3 (1 + t) \\mathbb{A}^\\prime (u_0)\\Big\\} \\nonumber \\\\\n&-& \\!\\!\\! {1 \\over 128 \\sqrt{3} \\pi^2}\n(-1 + t) (7 + 5 t) m_0^2 m_b^3 M^2 \\left(e_u \\langle \\bar s s \\rangle - e_s \\langle \\bar u u \\rangle\\right) {\\cal I}_2 \\nonumber \\\\\n&-& \\!\\!\\! {e^{-m_b^2\/M^2}\\over 384 \\sqrt{3} m_b \\pi^2}\n(-2 + t + t^2) M^2 \\left(e_u \\langle \\bar s s \\rangle - e_s \\langle \\bar u u \\rangle\\right) \\Big[\\langle g_s^2 G^2 \\rangle \\left(1 -\ne^{m_b^2\/M^2} m_b^2 {\\cal I}_2\\right) \\nonumber \\\\\n&-& \\!\\!\\! 6 m_0^2 m_b^2 + 32 f_{3\\gamma} m_b^2 \\pi^2 \\psi^v(u_0)\\Big] \\nonumber \\\\\n&+& \\!\\!\\! {e^{-m_b^2\/M^2}\\over 2304 \\sqrt{3} \\pi^2}\n(1 + t + t^2) (e_s - e_u) f_{3\\gamma} \\langle g_s^2 G^2 \\rangle M^2 \n\\left[4 \\psi^v(u_0) - \\psi^{a\\prime}(u_0)\\right] \\nonumber \\\\\n&+& \\!\\!\\! {e^{-m_b^2\/M^2}\\over 48 \\sqrt{3}}\n(1 + t - 2 t^2) e_s f_{3\\gamma} m_b M^2 \\langle \\bar u u \\rangle \\psi^{a\\prime}(u_0) \\nonumber \\\\\n&+& \\!\\!\\! {e^{-m_b^2\/M^2}\\over 384 \\sqrt{3} \\pi^2} \n(-1 + t) m_b M^2 \\Big[- e^{m_b^2\/M^2}3 (3 + 2 t) e_b m_0^2 m_b^2 \n\\left(\\langle \\bar s s \\rangle - \\langle \\bar u u \\rangle\\right) {\\cal I}_2 \\nonumber \\\\\n&+& \\!\\!\\! 8 (1 + 2 t) e_u f_{3\\gamma} \\pi^2 \\langle \\bar s s \\rangle \\psi^{a\\prime}(u_0)\\Big] \\nonumber \\\\\n&+& \\!\\!\\! {e^{-m_b^2\/M^2}\\over 1152 \\sqrt{3} M^2 \\pi^2}\n(-1 + t) m_b \\left(e_u \\langle \\bar s s \\rangle - e_s \\langle \\bar u u \\rangle\\right) \\Big\\{(2 + t) \\langle g_s^2 G^2 \\rangle m_0^2 \\nonumber \\\\\n&+& \\!\\!\\! f_{3\\gamma} \\left(\\langle g_s^2 G^2 \\rangle - 6 m_0^2 m_b^2\\right) \\pi^2 \\Big[-4 (2 + t) \\psi^v(u_0) + \n     (1 + 2 t) \\psi^{a\\prime}(u_0)\\Big]\\Big\\} \\nonumber \\\\\n&-& \\!\\!\\! {e^{-m_b^2\/M^2}\\over 13824 \\sqrt{3} M^4 \\pi^2}\n(-1 + t) \\langle g_s^2 G^2 \\rangle m_b \\left(e_u \\langle \\bar s s \\rangle - e_s \\langle \\bar u u \\rangle\\right) \\Big\\{3 (2 + t) m_0^2 m_b^2 \\nonumber \\\\\n&+& \\!\\!\\!    2 f_{3\\gamma} (3 m_0^2 - 2 m_b^2) \\pi^2 \\Big[4 (2 + t) \\psi^v(u_0) - \n      (1 + 2 t) \\psi^{a\\prime}(u_0)\\Big]\\Big\\} \\nonumber \\\\\n&+& \\!\\!\\! {e^{-m_b^2\/M^2}\\over 2304 \\sqrt{3} M^6}\n(-1 + t) f_{3\\gamma} \\langle g_s^2 G^2 \\rangle m_0^2 m_b^3 \\left(e_u \\langle \\bar s s \\rangle - e_s \\langle \\bar u u \\rangle\\right) \n  \\Big[4 (2 + t) \\psi^v(u_0) - (1 + 2 t) \\psi^{a\\prime}(u_0)\\Big] \\nonumber \\\\\n&-& \\!\\!\\! {e^{-m_b^2\/M^2}\\over 13824 \\sqrt{3} M^8}\n(-1 + t) f_{3\\gamma} \\langle g_s^2 G^2 \\rangle m_0^2 m_b^5 \\left(e_u \\langle \\bar s s \\rangle - e_s \\langle \\bar u u \\rangle\\right) \n   \\Big[4 (2 + t) \\psi^v(u_0) - (1 + 2 t) \\psi^{a\\prime}(u_0)\\Big] \\nonumber \\\\\n&+& \\!\\!\\! {e^{-m_b^2\/M^2}\\over 18432 \\sqrt{3} \\pi^2}\n(-1 + t) \\langle g_s^2 G^2 \\rangle m_b \\left(e_s \\langle \\bar s s \\rangle - e_u \\langle \\bar u u \\rangle\\right) \\Big\\{4 (-1 + t) i_1({\\cal S},1) \\nonumber \\\\\n&+& \\!\\!\\! 2 \\Big[2 (-1 + t) i_1(\\widetilde{\\cal S},1) - 3 (1 + t) i_1({\\cal T}_1,1)\n- 2 i_1({\\cal T}_2,1) + 3 i_1({\\cal T}_3,1) + 2 i_1({\\cal T}_4,1) +\n8 i_1({\\cal S},v) \\nonumber \\\\\n&+& \\!\\!\\! 2 i_1(\\widetilde{\\cal S},v) + 6 i_1({\\cal T}_2,v) - 6 i_1({\\cal T}_3,v)\n- 12 \\widetilde{j}(h_\\gamma) + \nt \\Big(2 i_1({\\cal T}_2,1) + 3 i_1({\\cal T}_3,1) - 2 i_1({\\cal T}_4,1)\\nonumber \\\\\n&+& \\!\\!\\! 4 i_1({\\cal S},v) - 2 i_1(\\widetilde{\\cal S},v) + 2 i_1({\\cal T}_2,v)\n- 6 i_1({\\cal T}_3,v) + 4 i_1({\\cal T}_4,v) - 4 \\widetilde{j}(h_\\gamma)\\Big)\\Big]\n+ 3 (1 + t) \\mathbb{A}^\\prime (u_0)\\Big\\} \\nonumber \\\\\n&+& \\!\\!\\! {e^{-m_b^2\/M^2}\\over 2304 \\sqrt{3} m_b \\pi^2} \n\\left(e_u \\langle \\bar s s \\rangle - e_s \\langle \\bar u u \\rangle\\right) \\Big\\{-(-2 + t + t^2) \\langle g_s^2 G^2 \\rangle (m_0^2 - 2 m_b^2) \\nonumber \\\\\n&+& \\!\\!\\! 18 (-1 + t^2) f_{3\\gamma} m_0^2 m_b^2 \\pi^2 \\Big[-4 \\psi^v(u_0) + \n      \\psi^{a\\prime}(u_0)\\Big]\\Big\\}~. \\nonumber\n\\end{eqnarray}\n\n\nThe functions $i_n~(n=1,2)$, and $\\widetilde{j}_1(f(u))$\nare defined as:\n\\begin{eqnarray}\n\\label{nolabel}\ni_1(\\phi,f(v)) \\!\\!\\! &=& \\!\\!\\! \\int {\\cal D}\\alpha_i \\int_0^1 dv\n\\phi(\\alpha_{\\bar{q}},\\alpha_q,\\alpha_g) f(v) \\delta^\\prime(k-u_0)~, \\nonumber \\\\\ni_2(\\phi,f(v)) \\!\\!\\! &=& \\!\\!\\! \\int {\\cal D}\\alpha_i \\int_0^1 dv\n\\phi(\\alpha_{\\bar{q}},\\alpha_q,\\alpha_g) f(v) \\delta^{\\prime\\prime}(k-u_0)~, \\nonumber \\\\\n\\widetilde{j}(f(u)) \\!\\!\\! &=& \\!\\!\\! \\int_{u_0}^1 du f(u)~, \\nonumber \\\\\n{\\cal I}_n \\!\\!\\! &=& \\!\\!\\! \\int_{m_b^2}^{\\infty} ds\\, {e^{-s\/M^2} \\over s^n}~,\\nonumber\n\\end{eqnarray}\nwhere \n\\begin{eqnarray}\nk = \\alpha_q + \\alpha_g \\bar{v}~,~~~~~u_0={M_1^2 \\over M_1^2\n+M_2^2}~,~~~~~M^2={M_1^2 M_2^2 \\over M_1^2 +M_2^2}~.\\nonumber\n\\end{eqnarray}\n\n\n\n\n\n\\newpage\n\n\\section*{Appendix B}\n\n\nExpressions of the invariant amplitudes $\\Pi_1^M$ and\n$\\Pi_2^M$ entering into the mass sum rule for the negative parity\nheavy $\\Xi_b^{\\prime 0}$ baryon. \\\\\\\\\n\n\\setcounter{equation}{0}\n\\setcounter{section}{0}\n\n\n\n{\\bf 1) Coefficient of the $\\not\\!p$ structure}\n\n\\begin{eqnarray}\n&&\\Pi_1^M =\n{3\\over 256 \\pi^4} \\Big\\{ - m_b^4  M^6  [5 + t (2 + 5 t)] \\left[\nm_b^4 {\\cal I}_5 -\n2 m_b^2 {\\cal I}_4 +\n{\\cal I}_3\\right] \\Big\\} \\nonumber \\\\\n&+& \\!\\!\\! {1\\over 192 \\pi^4} m_b^4 M^2 \\left[\\langle g_s^2 G^2 \\rangle  (1 + t + t^2 ) - 18 m_b \\pi^2 (-1+t^2)\n\\left(\\langle \\bar s s \\rangle+\\langle \\bar u u \\rangle\\right)\\right] {\\cal I}_3 \\nonumber \\\\\n&+& \\!\\!\\! {1\\over 3072 \\pi^4} m_b^2 M^2 \\left[-\\langle g_s^2 G^2 \\rangle  (13 + 10 t + 13 t^2 ) + 288 m_b \\pi^2\n(-1+t^2) \\left(\\langle \\bar s s \\rangle+\\langle \\bar u u \\rangle\\right)\\right] {\\cal I}_2 \\nonumber \\\\\n&+& \\!\\!\\! {e^{-m_b^2\/M^2} \\over 73728 m_b M^2 \\pi^4} \\Big\\{ - \\langle g_s^2 G^2 \\rangle^2 m_b (1 + t)^2 +\n768 m_b m_0^2 \\langle \\bar s s \\rangle \\langle \\bar u u \\rangle \\pi^4 (-1 + t)^2 \\nonumber \\\\\n&-& \\!\\!\\! 56 \\langle g_s^2 G^2 \\rangle m_0^2 \\pi^2 (-1 + t^2) \\left(\\langle \\bar s s \\rangle + \\langle \\bar u u \\rangle\\right) \\Big\\} \\nonumber \\\\\n&+& \\!\\!\\! {1 \\over 768 M^2 \\pi^2} \\langle g_s^2 G^2 \\rangle m_b (-1 + t^2) (\\langle \\bar s s \\rangle + \\langle \\bar u u \\rangle) {\\cal  E}_1 \\nonumber \\\\\n&+& \\!\\!\\! {e^{-m_b^2\/M^2} \\over 18432 M^4  \\pi^2} m_b m_0^2\n\\left[\\langle g_s^2 G^2 \\rangle \\left(\\langle \\bar u u \\rangle+\\langle \\bar s s \\rangle\\right) (-1+t^2) +\n384 m_b \\langle \\bar s s \\rangle \\langle \\bar u u \\rangle \\pi^2 (-1+t)^2 \\right] \\nonumber \\\\\n&+& \\!\\!\\! {e^{-m_b^2\/M^2} \\over 1728 M^6} m_b^2 \\langle g_s^2 G^2 \\rangle (-1 + t)^2 \\langle \\bar s s \\rangle \\langle \\bar u u \\rangle \\nonumber \\\\\n&+& \\!\\!\\! {e^{-m_b^2\/M^2} \\over 1728 M^8} m_b^2 m_0^2 \\langle g_s^2 G^2 \\rangle (-1 + t)^2 \\langle \\bar s s \\rangle \\langle \\bar u u \\rangle \\nonumber \\\\\n&-& \\!\\!\\! {e^{-m_b^2\/M^2} \\over 3456 M^{10}} m_b^4 m_0^2 \\langle g_s^2 G^2 \\rangle (-1 + t)^2 \\langle \\bar s s \\rangle \\langle \\bar u u \\rangle \\nonumber \\\\\n&-& \\!\\!\\! {e^{-m_b^2\/M^2} \\over 768 m_b \\pi^2} \\left[ \\langle g_s^2 G^2 \\rangle \\left(\\langle \\bar u u \\rangle+\\langle \\bar s s \\rangle\\right)\n(-1+t^2) + 32 m_b \\langle \\bar s s \\rangle \\langle \\bar u u \\rangle \\pi^2 (-1+t)^2 \\right] \\nonumber \\\\\n&+& \\!\\!\\! {1\\over 256 \\pi^2} m_b \\left(\\langle \\bar u u \\rangle+\\langle \\bar s s \\rangle\\right) (-1+t^2) \\left[\n\\left(\\langle g_s^2 G^2 \\rangle - 13 m_b^2 m_0^2\\right) {\\cal I}_2 + 6 m_0^2 {\\cal I}_1\n\\right]~,\\nonumber\n\\end{eqnarray}\n\\\\\\\\\n{\\bf 2) Coefficient of the $I$ structure}\n\\begin{eqnarray}\n&&\\Pi_2^M =\n-{3\\over 256 \\pi^4} \\Big\\{ - m_b^3  M^6 (-1+t)^2 \\left[          \nm_b^4 {\\cal I}_4 -                          \n2 m_b^2 {\\cal I}_3 +                               \n{\\cal I}_2\\right] \\Big\\} \\nonumber \\\\       \n&+& \\!\\!\\! {1\\over 3072 \\pi^4} m_b M^4 \\Big\\{4 m_b^2 \\left[\\langle g_s^2 G^2 \\rangle (-1+t)^2 + 72 m_b\n\\left(\\langle \\bar u u \\rangle+\\langle \\bar s s \\rangle\\right) \\pi^2 (-1+t^2) \\right] {\\cal I}_3\n- 3 \\langle g_s^2 G^2 \\rangle (-1+t)^2 {\\cal I}_2 \\Big\\} \\nonumber \\\\\n&-& \\!\\!\\! {7 e^{-m_b^2\/M^2} \\over 256 \\pi^2} m_0^2 M^2\n\\left(\\langle \\bar u u \\rangle+\\langle \\bar s s \\rangle\\right) (-1 + t^2) \\nonumber \\\\\n&+& \\!\\!\\! {1\\over 1024 \\pi^4} m_b M^2 \\Big\\{\nm_b \\left[ 3 m_b \\langle g_s^2 G^2 \\rangle (-1+t)^2 + 4 m_0^2 \\left(\\langle \\bar u u \\rangle+\\langle \\bar s s \\rangle\\right) \\pi^2 (-1+t^2)\n\\right] {\\cal I}_2 -\n2 \\langle g_s^2 G^2 \\rangle (-1+t)^2 {\\cal I}_1 \\Big\\} \\nonumber \\\\\n&-& \\!\\!\\! {e^{-m_b^2\/M^2} \\over 73728 M^2 \\pi^4} m_b \n\\left[ \\langle g_s^2 G^2 \\rangle^2 (-1 + t)^2 +\n1536 m_0^2 \\langle \\bar s s \\rangle \\langle \\bar u u \\rangle \\pi^4 (3 + 2 t + 3 t^2) \\right] \\nonumber \\\\\n&+& \\!\\!\\! {e^{-m_b^2\/M^2} \\over 18432 M^4  \\pi^2} m_b \n\\left[ -11 m_b m_0^2 \\langle g_s^2 G^2 \\rangle \\left(\\langle \\bar u u \\rangle+\\langle \\bar s s \\rangle\\right) (-1+t^2)\\right. \\nonumber \\\\\n&-& \\!\\!\\! \\left. 32 \\left(\\langle g_s^2 G^2 \\rangle - 12 m_0^2 m_b^2 \\right) \\langle \\bar s s \\rangle \\langle \\bar u u \\rangle (5 + 2 t + 5 t^2) \n\\right] \\nonumber \\\\\n&+& \\!\\!\\! {e^{-m_b^2\/M^2} \\over 1728 M^6} m_b \\left( m_b^2 - 3 m_0^2 \\right)\n\\langle g_s^2 G^2 \\rangle \\langle \\bar s s \\rangle \\langle \\bar u u \\rangle (5 + 2 t + 5 t^2) \\nonumber \\\\\n&+& \\!\\!\\! {e^{-m_b^2\/M^2} \\over 576 M^8} m_b^3 m_0^2 \\langle g_s^2 G^2 \\rangle \\langle \\bar s s \\rangle \\langle \\bar u u \\rangle (5 + 2 t + 5 t^2)\n\\nonumber \\\\\n&-& \\!\\!\\! {e^{-m_b^2\/M^2} \\over 3456 M^{10}} m_b^5 m_0^2 \\langle g_s^2 G^2 \\rangle \\langle \\bar s s \\rangle \\langle \\bar u u \\rangle (5 + 2 t + 5\nt^2) \\nonumber \\\\\n&+& \\!\\!\\! {e^{-m_b^2\/M^2} \\over 36864 m_b \\pi^4} \\left[ \\langle g_s^2 G^2 \\rangle^2 (-1+t)^2\n- 1536 m_b^2 \\langle \\bar s s \\rangle \\langle \\bar u u \\rangle \\pi^4 (5 + 2 t + 5 t^2) \\right. \\nonumber \\\\\n&+& \\!\\!\\! \\left.96 m_b \\langle g_s^2 G^2 \\rangle \\left(\\langle \\bar u u \\rangle+\\langle \\bar s s \\rangle\\right) \\pi^2 (-1+t^2) \\right]~. \\nonumber\n\\end{eqnarray}\n\\\\\\\\\n\n\nExpressions of the invariant amplitudes $\\Pi_1^M$ and\n$\\Pi_2^M$ entering into the mass sum rule for the negative parity\nheavy $\\Xi_b^0$ baryon.\\\\\\\\\n\n{\\bf 3) Coefficient of the $\\not\\!p$ structure}\n\n\\begin{eqnarray}\n&&\\Pi_1^M =\n- {1\\over 256 \\pi^4}\n(-3 m_b^4 M^6 (5 + 2 t + 5 t^2) \\left({\\cal I}_3 - 2 m_b^2 {\\cal I}_4 + m_b^4 {\\cal I}_5\\right) \\nonumber \\\\\n&+& \\!\\!\\! {1\\over 3072 \\pi^4}\nm_b^2 M^2 \\Big[3 \\langle g_s^2 G^2 \\rangle (1 + t)^2 {\\cal I}_2 - 16 \\langle g_s^2 G^2 \\rangle m_b^2 (1 + t + t^2) {\\cal I}_3 \\nonumber \\\\\n&-& \\!\\!\\! 32 m_b \\pi^2 (-1 + t) (1 + 5 t) \\left(\\langle \\bar s s \\rangle + \\langle \\bar u u \\rangle\\right) \n\\left(-{\\cal I}_2 + m_b^2 {\\cal I}_3\\right)\\Big] \\nonumber \\\\\n&+& \\!\\!\\! {e^{-m_b^2\/M^2} \\over 221184 m_b M^2 \\pi^4} \n\\langle g_s^2 G^2 \\rangle^2 m_b (13 + 10 t + 13 t^2) + 768 m_0^2 m_b \\pi^4 (-1 + t) (25 + 23 t) \\langle \\bar s s \\rangle \\langle \\bar u u \\rangle \\nonumber \\\\\n&-& \\!\\!\\! 8 \\langle g_s^2 G^2 \\rangle \\pi^2 (-1 + t) (\\langle \\bar s s \\rangle + \\langle \\bar u u \\rangle) \\left[m_0^2 (1 + 5 t) + \n    12 m_b^2 e^{m_b^2\/M^2} (5 + t) {\\cal I}_1\\right] \\nonumber \\\\\n&+& \\!\\!\\! {e^{-m_b^2\/M^2} \\over 55296 M^4 \\pi^2}\nm_0^2 m_b (-1 + t) \\left[384 m_b \\pi^2 (13 + 11 t) \\langle \\bar s s \\rangle \\langle \\bar u u \\rangle + \n   \\langle g_s^2 G^2 \\rangle (31 + 11 t) \\left(\\langle \\bar s s \\rangle + \\langle \\bar u u \\rangle\\right)\\right] \\nonumber \\\\\n&+& \\!\\!\\! {e^{-m_b^2\/M^2} \\over 5184 M^6}\n\\langle g_s^2 G^2 \\rangle m_b^2 (-1 + t) (13 + 11 t) \\langle \\bar s s \\rangle \\langle \\bar u u \\rangle \\nonumber \\\\\n&+& \\!\\!\\! {e^{-m_b^2\/M^2} \\over 5184 M^8}\n\\langle g_s^2 G^2 \\rangle m_0^2 m_b^2 (-1 + t) (13 + 11 t) \\langle \\bar s s \\rangle \\langle \\bar u u \\rangle \\nonumber \\\\\n&-& \\!\\!\\! {e^{-m_b^2\/M^2} \\over 10368 M^{10}}\n\\langle g_s^2 G^2 \\rangle m_0^2 m_b^4 (-1 + t) (13 + 11 t) \\langle \\bar s s \\rangle \\langle \\bar u u \\rangle \\nonumber \\\\\n&+& \\!\\!\\! {e^{-m_b^2\/M^2} \\over 6912 m_b \\pi^2}\n(-1 + t) \\Big\\{\\langle g_s^2 G^2 \\rangle (1 + 5 t) \\left(\\langle \\bar s s \\rangle + \\langle \\bar u u \\rangle\\right) \n\\left(-1 + 3 m_b^2 e^{m_b^2\/M^2} {\\cal I}_2\\right) \\nonumber \\\\\n&-& \\!\\!\\! 3 m_b \\Big[32 \\pi^2 (13 + 11 t) \\langle \\bar s s \\rangle \\langle \\bar u u \\rangle + 3 m_0^2 m_b e^{m_b^2\/M^2} \\left(\\langle \\bar s s \\rangle + \\langle \\bar u u \\rangle\\right) \n      [-6 (1 + t) {\\cal I}_1 + m_b^2 (7 + 11 t) \n{\\cal I}_2]\\Big]\\Big\\}~.\\nonumber\n\\end{eqnarray}\n\n{\\bf 4) Coefficient of the $I$ structure}\n\\begin{eqnarray}\n&&\\Pi_2^M =\n- {1\\over 256 \\pi^4}\nm_b^3 M^6 (-1 + t) (13 + 11 t) \\left({\\cal I}_2 - 2 m_b^2 {\\cal I}_3 +\nm_b^4 {\\cal I}_4\\right) \\nonumber \\\\\n&-& \\!\\!\\! {1\\over 9216 \\pi^4}\nm_b M^4 (-1 + t) \\Big[-96 m_b^3 \\pi^2 (1 + 5 t) (\\langle \\bar s s \\rangle + \\langle \\bar u u \\rangle) {\\cal I}_3 + \n    \\langle g_s^2 G^2 \\rangle (13 + 11 t) \\left(3 {\\cal I}_2 - 4 m_b^2 {\\cal I}_3\\right)\\Big] \\nonumber \\\\\n&-& \\!\\!\\! {e^{-m_b^2\/M^2} \\over 3072 \\pi^4}\nM^2 (-1 + t) \\Big\\{4 m_0^2 \\pi^2 \\left(\\langle \\bar s s \\rangle + \\langle \\bar u u \\rangle\\right) \\left[1 + 5 t +\nm_b^2 e^{m_b^2\/M^2} (5 + t) {\\cal I}_2\\right] \\nonumber \\\\\n&+& \\!\\!\\!  \\langle g_s^2 G^2 \\rangle m_b e^{m_b^2\/M^2} \\left[-2 (-1 + t) {\\cal I}_1 +\n3 m_b^2 (3 + 5 t) {\\cal I}_2\\right]\\Big\\} \\nonumber \\\\\n&+& \\!\\!\\! {e^{-m_b^2\/M^2} \\over 221184 M^2 \\pi^4}\nm_b (-1 + t) \\Big[\\langle g_s^2 G^2 \\rangle^2 (11 + 13 t) - 1536 m_0^2 \\pi^4 (-1 + t) \\langle \\bar s s \\rangle \\langle \\bar u u \\rangle \\Big] \\nonumber \\\\\n&+& \\!\\!\\! {e^{-m_b^2\/M^2} \\over 55296 M^4 \\pi^2}\nm_b \\Big\\{1152 m_0^2 m_b^2 \\pi^2 (5 + 2 t + 5 t^2) \\langle \\bar s s \\rangle \\langle \\bar u u \\rangle \\nonumber \\\\\n&-& \\!\\!\\! \\langle g_s^2 G^2 \\rangle \\Big[96 \\pi^2 (5 + 2 t + 5 t^2) \\langle \\bar s s \\rangle \\langle \\bar u u \\rangle + m_0^2 m_b (-1 + t) (29 + t) \n      \\left(\\langle \\bar s s \\rangle + \\langle \\bar u u \\rangle\\right)\\Big]\\Big\\} \\nonumber \\\\\n&+& \\!\\!\\! {e^{-m_b^2\/M^2} \\over 1728 M^6}\n\\langle g_s^2 G^2 \\rangle m_b (-3 m_0^2 + m_b^2) (5 + 2 t + 5 t^2) \\langle \\bar s s \\rangle \\langle \\bar u u \\rangle \\nonumber \\\\\n&+& \\!\\!\\! {e^{-m_b^2\/M^2} \\over 576 M^8}\n\\langle g_s^2 G^2 \\rangle m_0^2 m_b^3 (5 + 2 t + 5 t^2) \\langle \\bar s s \\rangle \\langle \\bar u u \\rangle \\nonumber \\\\\n&-& \\!\\!\\! {e^{-m_b^2\/M^2} \\over 3456 M^{10}}\n\\langle g_s^2 G^2 \\rangle m_0^2 m_b^5 (5 + 2 t + 5 t^2) \\langle \\bar s s \\rangle \\langle \\bar u u \\rangle \\nonumber \\\\\n&+& \\!\\!\\! {e^{-m_b^2\/M^2} \\over 110592 m_b \\pi^4}\n\\Big[\\langle g_s^2 G^2 \\rangle^2 (11 + 2 t - 13 t^2) - 4608 m_b^2 \\pi^4 (5 + 2 t + 5 t^2) \\langle \\bar s s \\rangle \\langle \\bar u u \\rangle \\nonumber \\\\\n&-& \\!\\!\\! 32 \\langle g_s^2 G^2 \\rangle m_b \\pi^2 (-7 + t) (-1 + t) \\left(\\langle \\bar s s \\rangle + \\langle \\bar u u \\rangle\\right)\\Big]~.\\nonumber\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\n{\\cal I}_n = \\int_{m_b^2}^{\\infty} ds\\, {e^{-s\/M^2} \\over s^n}~.\\nonumber\n\\end{eqnarray}\n\n\n\n\n\n\\newpage\n\n\\section*{Appendix C: Photon distribution amplitudes}\n\\setcounter{equation}{0}\n\\setcounter{section}{0}\n\n\nExplicit forms of the photon DAs\n\\cite{Rfrd18}.\n\n\\begin{eqnarray}\n\\label{nolabel27}\n\\varphi_\\gamma(u) \\!\\!\\! &=& \\!\\!\\! 6 u \\bar u \\Big[ 1 + \\varphi_2(\\mu)\nC_2^{\\frac{3}{2}}(u - \\bar u) \\Big]~,\n\\nonumber \\\\\n\\psi^v(u) \\!\\!\\! &=& \\!\\!\\! 3 [3 (2 u - 1)^2 -1 ]+\\frac{3}{64} (15\nw^V_\\gamma - 5 w^A_\\gamma)\n                        [3 - 30 (2 u - 1)^2 + 35 (2 u -1)^4]~,\n\\nonumber \\\\\n\\psi^a(u) \\!\\!\\! &=& \\!\\!\\! [1- (2 u -1)^2] [ 5 (2 u -1)^2 -1 ]\n\\frac{5}{2}\n    \\Bigg(1 + \\frac{9}{16} w^V_\\gamma - \\frac{3}{16} w^A_\\gamma\n    \\Bigg)~,\n\\nonumber \\\\\n{\\cal A}(\\alpha_i) \\!\\!\\! &=& \\!\\!\\! 360 \\alpha_q \\alpha_{\\bar q} \\alpha_g^2\n        \\Bigg[ 1 + w^A_\\gamma \\frac{1}{2} (7 \\alpha_g - 3)\\Bigg]~,\n\\nonumber \\\\\n{\\cal V}(\\alpha_i) \\!\\!\\! &=& \\!\\!\\! 540 w^V_\\gamma (\\alpha_q - \\alpha_{\\bar q})\n\\alpha_q \\alpha_{\\bar q}\n                \\alpha_g^2~,\n\\nonumber \\\\\nh_\\gamma(u) \\!\\!\\! &=& \\!\\!\\! - 10 (1 + 2 \\kappa^+ ) C_2^{\\frac{1}{2}}(u\n- \\bar u)~,\n\\nonumber \\\\\n\\mathbb{A}(u) \\!\\!\\! &=& \\!\\!\\! 40 u^2 \\bar u^2 (3 \\kappa - \\kappa^+ +1 ) +\n        8 (\\zeta_2^+ - 3 \\zeta_2) [u \\bar u (2 + 13 u \\bar u) + \n                2 u^3 (10 -15 u + 6 u^2) \\ln(u) \\nonumber \\\\ \n&+& \\!\\!\\! 2 \\bar u^3 (10 - 15 \\bar u + 6 \\bar u^2)\n        \\ln(\\bar u) ]~,\n\\nonumber \\\\\n{\\cal T}_1(\\alpha_i) \\!\\!\\! &=& \\!\\!\\! -120 (3 \\zeta_2 + \\zeta_2^+)(\\alpha_{\\bar\nq} - \\alpha_q)\n        \\alpha_{\\bar q} \\alpha_q \\alpha_g~,\n\\nonumber \\\\\n{\\cal T}_2(\\alpha_i) \\!\\!\\! &=& \\!\\!\\! 30 \\alpha_g^2 (\\alpha_{\\bar q} - \\alpha_q)\n    [(\\kappa - \\kappa^+) + (\\zeta_1 - \\zeta_1^+)(1 - 2\\alpha_g) +\n    \\zeta_2 (3 - 4 \\alpha_g)]~,\n\\nonumber \\\\\n{\\cal T}_3(\\alpha_i) \\!\\!\\! &=& \\!\\!\\! - 120 (3 \\zeta_2 - \\zeta_2^+)(\\alpha_{\\bar\nq} -\\alpha_q)\n        \\alpha_{\\bar q} \\alpha_q \\alpha_g~,\n\\nonumber \\\\\n{\\cal T}_4(\\alpha_i) \\!\\!\\! &=& \\!\\!\\! 30 \\alpha_g^2 (\\alpha_{\\bar q} - \\alpha_q)\n    [(\\kappa + \\kappa^+) + (\\zeta_1 + \\zeta_1^+)(1 - 2\\alpha_g) +\n    \\zeta_2 (3 - 4 \\alpha_g)]~,\\nonumber \\\\\n{\\cal S}(\\alpha_i) \\!\\!\\! &=& \\!\\!\\! 30\\alpha_g^2\\{(\\kappa +\n\\kappa^+)(1-\\alpha_g)+(\\zeta_1 + \\zeta_1^+)(1 - \\alpha_g)(1 -\n2\\alpha_g)\\nonumber \\\\ \n&+& \\!\\!\\!\\zeta_2\n[3 (\\alpha_{\\bar q} - \\alpha_q)^2-\\alpha_g(1 - \\alpha_g)]\\}~,\\nonumber \\\\\n\\widetilde {\\cal S}(\\alpha_i) \\!\\!\\! &=& \\!\\!\\!-30\\alpha_g^2\\{(\\kappa -\n\\kappa^+)(1-\\alpha_g)+(\\zeta_1 - \\zeta_1^+)(1 - \\alpha_g)(1 -\n2\\alpha_g)\\nonumber \\\\ \n&+& \\!\\!\\!\\zeta_2 [3 (\\alpha_{\\bar q} -\n\\alpha_q)^2-\\alpha_g(1 - \\alpha_g)]\\}. \\nonumber\n\\end{eqnarray}\nThe parameters entering  the above DA's are borrowed from\n\\cite{Rfrd18} whose values are $\\varphi_2(1~GeV) = 0$, \n$w^V_\\gamma = 3.8 \\pm 1.8$, $w^A_\\gamma = -2.1 \\pm 1.0$, \n$\\kappa = 0.2$, $\\kappa^+ = 0$, $\\zeta_1 = 0.4$, $\\zeta_2 = 0.3$, \n$\\zeta_1^+ = 0$, and $\\zeta_2^+ = 0$.\n\n\n\\newpage\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nIn hadron physics, quark model has become a well-established tool for the classification of various hadronic states.  Most predictions of the quark model have already been experimentally confirmed, but the quest for  doubly heavy baryons, baryonic states made of two heavy charm\/bottom quarks,  has  been conducted  for a long time. These baryonic states  had never been observed  in experiments until 2017, when the LHCb collaboration announced the discovery of $\\Xi_{cc}^{++}$ via $\\Xi_{cc}^{++} \\to \\Lambda_c^{+} K^- \\pi^+ \\pi^+$~\\cite{Aaij:2017ueg} with the decay mode  suggested in Ref.~\\cite{Yu:2017zst}. This discovery is subsequently confirmed  in 2018 in the $\\Xi_{cc}^{++} \\to \\Xi_c^+ \\pi^+$ decay~\\cite{Aaij:2018gfl}, and  meanwhile triggered  a series of  experimental investigations~\\cite{Aaij:2018wzf,Aaij:2019jfq,Aaij:2019dsx,Aaij:2019uaz}.  Now  studies of doubly-heavy baryons now open a  window to study the hadron spectroscopy and strong interactions  in a baryonic system in the presence of two heavy constituent quarks.\n\nAmong various properties  on doubly heavy baryons,     weak decays are of special importance. In the experimental searches for new type of particles the firstly-discovered ones are usually the ground states, which can only be reconstructed via weak decay final states.  Thus theoretical analysis of  their weak decays can greatly help  to optimize the experimental resources. Meanwhile, there exists rich dynamics in weak decay processes and currently only few theoretical approaches are available, which makes them a wonderland full of challenges and opportunities.\n\nOn the theoretical side an ingredient in the weak decay is the transition matrix element of the parent particle to a daughter particle, which can be parameterized as form factors. Fortunately,  there are various available   methods for this part of transition on the market. Thus the  decays of a doubly heavy baryon to a singly heavy baryon transition are studied intensively~\\cite{Yu:2017zst,Wang:2017mqp,Hu:2017dzi,Gutsche:2017hux,Sharma:2017txj,Zhao:2018zcb,Zhao:2018mrg,Yu:2019lxw,Gutsche:2019iac,Onishchenko:2000yp,Ebert:2004ck,Ebert:2005ip,Albertus:2006wb,Albertus:2012jt,Dhir:2018twm,Xing:2018lre,Zhang:2018llc,Jiang:2018oak,Ke:2019lcf,Shi:2019hbf,Shi:2019fph,Hu:2019bqj,Gerasimov:2019jwp}. In particular, some of  the form factors are studied  under the light front quark model~\\cite{Wang:2017mqp,Zhao:2018mrg,Xing:2018lre,Ke:2019lcf}, QCD sum rules~\\cite{Shi:2019hbf} and light cone sum rules~\\cite{Shi:2019fph,Hu:2019bqj}.\n\n\n\nThe light-front quark model is originally developed in meson decays~ \\cite{Jaus:1999zv,Jaus:1989au,Jaus:1991cy,Cheng:1996if,Cheng:2003sm,Cheng:2004yj,Ke:2009ed,Ke:2009mn,Cheng:2009ms,Lu:2007sg,Wang:2007sxa,Wang:2008xt,Wang:2008ci,Wang:2009mi,Chen:2009qk,Li:2010bb,Verma:2011yw,Shi:2016gqt,Chang:2019mmh,Chang:2019obq}. During the last decade it was applied to baryon decays with the help of quark-diquark picture~\\cite{Ke:2007tg,Wei:2009np,Ke:2012wa,Zhu:2018jet,Ke:2017eqo}, and it   is also interesting to notice that Ref.~\\cite{Ke:2019lcf} has adopted the three-quark transition for the form factors.  Under the quark-diquark  picture,  the two spectator quarks play the role of the antiquark in a mesonic system and are treated as a system of spin-$0$ or $1$. In  the calculation the vertex functions are associated with the couplings of a baryon to its quark and diquark constituents.  Following a recent work~\\cite{Chua:2018lfa} we revise the vertex function concerned with the spin-$1$ diquark system in this paper, and consequently we update  the form factors of a doubly heavy baryon to a singly heavy baryon transitions under the light front quark model. As argued in a previous work \\cite{Zhao:2018mrg}, both spin-$1\/2$ to spin-$1\/2$ and spin-$1\/2$ to spin-$3\/2$ transitions are important to the potential discovery channels, and thus both transitions are studied  in this work. Meanwhile, we investigate the charged current induced transitions as well as the flavor changing neutral current (FCNC) induced ones. To be more specific, we will explore  the following transitions:\n\\begin{enumerate}\n\t\\item  the spin-${1}\/{2}$ to spin-${1}\/{2}$ transition  with the charged current~\\footnote{In the following, we will abbreviate  the spin-$S_1$ to spin-$S_2$ transition as the $S_1\\to S_2$ transition. If there is no special note, spin-$1\/2$ and spin-$3\/2$ are all with positive parity. We will omit the positive sign of positive parity in the following.},\n\t\\begin{itemize}\n\t\\item $c\\to d,s$ process,\n\t\\begin{align*}\n\t&\\left.\n            \\begin{array}{lcl}\n\t\t\\Xi_{cc}^{++}(ccu) & \\to&\\Lambda_{c}^{+}(dcu)\/\\Sigma_{c}^{+}(dcu)\/\\Xi_{c}^{(\\prime)+}(scu),\\\\\n\t\t\\Xi_{cc}^{+}(ccd) & \\to&\\Sigma_{c}^{0}(dcd)\/\\Xi_{c}^{0}(scd)\/\\Xi_{c}^{\\prime0}(scd),\\\\\n\t\t\\Omega_{cc}^{+}(ccs) & \\to&\\Xi_{c}^{0}(dcs)\/\\Xi_{c}^{\\prime0}(dcs)\/\\Omega_{c}^{0}(scs),\n\t\t\\end{array}\n       \\right.\n       \\left.\n            \\begin{array}{lcl}\n            \\Xi_{bc}^{+}\/\\Xi_{bc}^{\\prime+}(cbu) & \\to&\\Lambda_{b}^{0}(dbu)\/\\Sigma_{b}^{0}(dbu)\/\\Xi_{b}^{(\\prime)0}(sbu),\\\\\n\t\\Xi_{bc}^{0}\/\\Xi_{bc}^{\\prime0}(cbd) & \\to&\\Sigma_{b}^{-}(dbd)\/\\Xi_{b}^{-}(sbd)\/\\Xi_{b}^{\\prime-}(sbd),\\\\\n\t\\Omega_{bc}^{0}\/\\Omega_{bc}^{\\prime0}(cbs) & \\to&\\Xi_{b}^{-}(dbs)\/\\Xi_{b}^{\\prime-}(dbs)\/\\Omega_{b}^{-}(sbs);\\end{array}\n       \\right.\n\t\\end{align*}\n\t\\item $b\\to u,c$ process,\n\\begin{align*}\n&\\left.\n            \\begin{array}{lcl}\n\t\\Xi_{bb}^{0}(bbu) & \\to&\\Sigma_{b}^{+}(ubu)\/\\Xi_{bc}^{+}(cbu)\/\\Xi_{bc}^{\\prime+}(cbu),\\\\\n\t\\Xi_{bb}^{-}(bbd) & \\to&\\Lambda_{b}^{0}(ubd)\/\\Sigma_{b}^{0}(ubd)\/\\Xi_{bc}^{0}(cbd)\/\\Xi_{bc}^{\\prime0}(cbd),\\\\\n\t\\Omega_{bb}^{-}(bbs) & \\to&\\Xi_{b}^{0}(ubs)\/\\Xi_{b}^{\\prime0}(ubs)\/\\Omega_{bc}^{0}(cbs)\/\\Omega_{bc}^{\\prime0}(cbs),\n\\end{array}\n       \\right.\\quad\n       \\left.\n            \\begin{array}{lcl}\n\t\\Xi_{bc}^{+}\/\\Xi_{bc}^{\\prime+}(bcu) & \\to&\\Sigma_{c}^{++}(ucu)\/\\Xi_{cc}^{++}(ccu),\\\\\n\t\\Xi_{bc}^{0}\/\\Xi_{bc}^{\\prime0}(bcd) & \\to&\\Lambda_{c}^{+}(ucd)\/\\Sigma_{c}^{+}(ucd)\/\\Xi_{cc}^{+}(ccd),\\\\\n\t\\Omega_{bc}^{0}\/\\Omega_{bc}^{\\prime0}(bcs) & \\to&\\Xi_{c}^{+}(ucs)\/\\Xi_{c}^{\\prime+}(ucs)\/\\Omega_{cc}^{+}(ccs);\n\t\\end{array}\n       \\right.\n\\end{align*}\n\\end{itemize}\n\t\\item the   ${1}\/{2}\\to {1}\/{2}$ transition   with FCNC,\n\n\\begin{itemize}\n\t\t\\item $c\\to u$ process,\n\t\t\\begin{align*}\n       \\left.\n            \\begin{array}{lcl}\n            \\Xi_{cc}^{++}(ccu) & \\to&\\Sigma_{c}^{++}(ucu),\\\\\n\t\t\\Xi_{cc}^{+}(ccd) & \\to&\\Lambda_{c}^{+}(ucd)\/\\Sigma_{c}^{+}(ucd),\\\\\n\t\t\\Omega_{cc}^{+}(ccs) & \\to&\\Xi_{c}^{+}(ucs)\/\\Xi_{c}^{\\prime+}(ucs),\n            \\end{array}\n       \\right.\\quad\\quad\n       \\left.\n            \\begin{array}{lcl}\n           \\Xi_{cb}^{+}\/\\Xi_{cb}^{\\prime+}(cbu) & \\to& \\Sigma_{b}^{+}(ubu),\\\\\n\t\t\\Xi_{cb}^{0}\/\\Xi_{cb}^{\\prime0}(cbd) & \\to& \\Lambda_{b}^{0}(ubd)\/\\Sigma_{b}^{0}(ubd),\\\\\n\t\t\\Omega_{cb}^{0}\/\\Omega_{cb}^{\\prime0}(cbs) & \\to&\\Xi_{b}^{0}(ubs)\/\\Xi_{b}^{\\prime0}(ubs);\n            \\end{array}\n       \\right.\n        \\end {align*}\n\\end{itemize}\n\\begin{itemize}\n\t\t\\item $b\\to d,s$ process,\n\t\t\\begin{align*}\n       \\left.\n            \\begin{array}{lcl}\n            \\Xi_{bb}^{0}(bbu) & \\to&\\Lambda_{b}^{0}(dbu)\/\\Sigma_{b}^{0}(dbu)\/\\Xi_{b}^{(\\prime)0}(sbu),\\\\\n\t\t\\Xi_{bb}^{-}(bbd) & \\to&\\Sigma_{b}^{-}(dbd)\/\\Xi_{b}^{-}(sbd)\/\\Xi_{b}^{\\prime-}(sbd),\\\\\n\t\t\\Omega_{bb}^{-}(bbs) & \\to&\\Xi_{b}^{-}(dbs)\/\\Xi_{b}^{\\prime-}(dbs)\/\\Omega_{b}^{-}(sbs),\n            \\end{array}\n       \\right.\\quad\\quad\n       \\left.\n            \\begin{array}{lcl}\n           \\Xi_{bc}^{+}\/\\Xi_{bc}^{\\prime+}(bcu) & \\to&\\Lambda_{c}^{+}(dcu)\/\\Sigma_{c}^{+}(dcu)\/\\Xi_{c}^{(\\prime)+}(scu),\\\\\n\t\t\\Xi_{bc}^{0}\/\\Xi_{bc}^{\\prime0}(bcd) & \\to&\\Sigma_{c}^{0}(dcd)\/\\Xi_{c}^{0}(scd)\/\\Xi_{c}^{\\prime0}(scd),\\\\\n\t\t\\Omega_{bc}^{0}\/\\Omega_{bc}^{\\prime0}(bcs) & \\to&\\Xi_{c}^{0}(dcs)\/\\Xi_{c}^{\\prime0}(dcs)\/\\Omega_{c}^{0}(scs);\n            \\end{array}\n       \\right.\n        \\end {align*}\n\\end{itemize}\n\\item the  ${1}\/{2}\\to {3}\/{2}$ transition   induced by the charged current,\n\\begin{itemize}\n\t\\item $c\\to d,s$ process,\n\t\\begin{align*}\n\t\\left.\n\\begin{array}{lcl}\n\t\t\\Xi_{cc}^{++}(ccu) & \\to&\\Sigma_{c}^{*+}(dcu)\/\\Xi_{c}^{\\prime*+}(scu),\\\\\n\t\t\\Xi_{cc}^{+}(ccd) & \\to&\\Sigma_{c}^{*0}(dcd)\/\\Xi_{c}^{\\prime*0}(scd),\\\\\n\t\t\\Omega_{cc}^{+}(ccs) & \\to&\\Xi_{c}^{\\prime*0}(dcs)\/\\Omega_{c}^{*0}(scs),\n\t\t\\end{array}\n       \\right.\\quad\\quad\n       \\left.\n            \\begin{array}{lcl}\n            \\Xi_{bc}^{+}\/\\Xi_{bc}^{\\prime+}(cbu) & \\to&\\Sigma_{b}^{*0}(dbu)\/\\Xi_{b}^{\\prime*0}(sbu),\\\\\n\t\t\\Xi_{bc}^{0}\/\\Xi_{bc}^{\\prime0}(cbd) & \\to&\\Sigma_{b}^{*-}(dbd)\/\\Xi_{b}^{\\prime*-}(sbd),\\\\\n\t\t\\Omega_{bc}^{0}\/\\Omega_{bc}^{\\prime0}(cbs) & \\to&\\Xi_{b}^{\\prime*-}(dbs)\/\\Omega_{b}^{*-}(sbs).\n\t\t\\end{array}\n       \\right.\n\t\\end{align*}\n    \\item  $b\\to u,c$ process,\n\t\\begin{align*}\n\t\\left.\n\\begin{array}{lcl}\n\t\t\\Xi_{bb}^{0}(bbu) & \\to&\\Sigma_{b}^{*+}(ubu)\/\\Xi_{bc}^{*+}(cbu),\\\\\n\t\t\\Xi_{bb}^{-}(bbd) & \\to&\\Sigma_{b}^{*0}(ubd)\/\\Xi_{bc}^{*0}(cbd),\\\\\n\t\t\\Omega_{bb}^{-}(bbs) & \\to&\\Xi_{b}^{\\prime*0}(ubs)\/\\Omega_{bc}^{*0}(cbs),\n\t\t\\end{array}\n       \\right.\\quad\\quad\n       \\left.\n            \\begin{array}{lcl}\n            \\Xi_{bc}^{+}\/\\Xi_{bc}^{\\prime+}(bcu) & \\to&\\Sigma_{c}^{*++}(ucu)\/\\Xi_{cc}^{*++}(ccu),\\\\\n\t\t\\Xi_{bc}^{0}\/\\Xi_{bc}^{\\prime0}(bcd) & \\to&\\Sigma_{c}^{*+}(ucd)\/\\Xi_{cc}^{*+}(ccd),\\\\\n\t\t\\Omega_{bc}^{0}\/\\Omega_{bc}^{\\prime0}(bcs) & \\to&\\Xi_{c}^{\\prime*+}(ucs)\/\\Omega_{cc}^{*+}(ccs);\n\t\t\\end{array}\n       \\right.\n\t\\end{align*}\n\\end{itemize}\n\\item the  ${1}\/{2}\\to {3}\/{2}$ transition   with FCNC,\n\\begin{itemize}\n\t\t\\item $c\\to u$ process,\n\t\t\\begin{align*}\n       \\left.\n            \\begin{array}{lcl}\n            \\Xi_{cc}^{++}(ccu) & \\to&\\Sigma_{c}^{*++}(ucu),\\\\\n\t\t\\Xi_{cc}^{+}(ccd) & \\to&\\Sigma_{c}^{*+}(ucd),\\\\\n\t\t\\Omega_{cc}^{+}(ccs) & \\to&\\Xi_{c}^{\\prime*+}(ucs),\n            \\end{array}\n       \\right.\\quad\\quad\n       \\left.\n            \\begin{array}{lcl}\n           \\Xi_{cb}^{+}\/\\Xi_{cb}^{\\prime+}(cbu) & \\to& \\Sigma_{b}^{*+}(ubu),\\\\\n\t\t\\Xi_{cb}^{0}\/\\Xi_{cb}^{\\prime0}(cbd) & \\to& \\Sigma_{b}^{*0}(ubd),\\\\\n\t\t\\Omega_{cb}^{0}\/\\Omega_{cb}^{\\prime0}(cbs) & \\to&\\Xi_{b}^{\\prime*0}(ubs);\n            \\end{array}\n       \\right.\n        \\end {align*}\n\\end{itemize}\n\\begin{itemize}\n\\item $b\\to d,s$ process,\n\\begin{align*}\n\\left.\n\\begin{array}{lcl}\n\\Xi_{bb}^{0}(bbu) & \\to\\Sigma_{b}^{*0}(dbu)\/\\Xi_{b}^{\\prime*0}(sbu),\\\\\n\\Xi_{bb}^{-}(bbd) & \\to\\Sigma_{b}^{*-}(dbd)\/\\Xi_{b}^{\\prime*-}(sbd),\\\\\n\\Omega_{bb}^{-}(bbs) & \\to\\Xi_{b}^{\\prime*-}(dbs)\/\\Omega_{b}^{*-}(sbs),\n\\end{array}\n       \\right.\\quad\\quad\n       \\left.\n            \\begin{array}{lcl}\n\\Xi_{bc}^{+}\/\\Xi_{bc}^{\\prime+}(bcu) & \\to\\Sigma_{c}^{*+}(dcu)\/\\Xi_{c}^{\\prime*+}(scu),\\\\\n\\Xi_{bc}^{0}\/\\Xi_{bc}^{\\prime0}(bcd) & \\to\\Sigma_{c}^{*0}(dcd)\/\\Xi_{c}^{\\prime*0}(scd),\\\\\n\\Omega_{bc}^{0}\/\\Omega_{bc}^{\\prime0}(bcs) & \\to\\Xi_{c}^{\\prime*0}(dcs)\/\\Omega_{c}^{*0}(scs);\n\\end{array}\n       \\right.\n\\end{align*}\n\\end{itemize}\n\\end{enumerate}\nIn the above, the quark components have been explicitly given  in the brackets,\nin which the first quarks denote the quarks participating in the  weak decays.\nThe initial baryons are all doubly heavy baryons.\nThe spin-parity $J^{P}$ quantum numbers of the doubly heavy baryons has been listed in Tab.~\\ref{tab:JPC}.\n\n\n\\begin{table}[!htb]\n\\footnotesize\n\\caption{The spin-parity $J^{P}$ quantum numbers and quark composition for doubly heavy baryons. The symbol $S_{h}^{\\pi}$ indicates the spin-parity of the system consisting of two heavy quarks. The light quark $q$ represents $u,d$ quark.}\\label{tab:JPC}\n\\begin{center}\n\\begin{tabular}{cccc|cccccc} \\hline \\hline\nBaryon      & Quark Content  &  $S_h^\\pi$  &$J^P$   & Baryon & Quark Content &   $S_h^\\pi$  &$J^P$   \\\\ \\hline\n$\\Xi_{cc}$ & $\\{cc\\}q$  & $1^+$ & $1\/2^+$ &   $\\Xi_{bb}$ & $\\{bb\\}q$  & $1^+$ & $1\/2^+$ & \\\\\n$\\Xi_{cc}^*$ & $\\{cc\\}q$  & $1^+$ & $3\/2^+$ &   $\\Xi_{bb}^*$ & $\\{bb\\}q$  & $1^+$ & $3\/2^+$ & \\\\ \\hline\n$\\Omega_{cc}$ & $\\{cc\\}s$  & $1^+$ & $1\/2^+$ &   $\\Omega_{bb}$ & $\\{bb\\}s$  & $1^+$ & $1\/2^+$ & \\\\\n$\\Omega_{cc}^*$ & $\\{cc\\}s$  & $1^+$ & $3\/2^+$ &   $\\Omega_{bb}^*$ & $\\{bb\\}s$  & $1^+$ & $3\/2^+$ &  \\\\ \\hline\n$\\Xi_{bc}'$ & $[bc]q$  & $0^+$ & $1\/2^+$ &   $\\Omega_{bc}'$ & $[bc]s$  & $0^+$ & $1\/2^+$ & \\\\\n$\\Xi_{bc}$ & $\\{bc\\}q$  & $1^+$ & $1\/2^+$ &   $\\Omega_{bc}$ & $\\{bc\\}s$  & $1^+$ & $1\/2^+$ & \\\\\n$\\Xi_{bc}^*$ & $\\{bc\\}q$  & $1^+$ & $3\/2^+$ &   $\\Omega_{bc}^*$ & $\\{bc\\}s$  & $1^+$ & $3\/2^+$ &\n \\\\ \\hline \\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\nThe lowest-lying  doubly heavy baryons with $J^{P}=1\/2^{+}$ for example the doubly charm SU(3) triplets $\\Xi_{cc}^{++}$ ($ccu$), $\\Xi_{cc}^{+}$ ($ccd$), and $\\Omega^{+}_{cc}$($ccs$) shown in Fig.~\\ref{fig:doubly_heavy_baron},  can only weak decay.\nThree doubly bottom baryons $\\Xi_{bb}^{0}$ ($bbu$), $\\Xi_{bb}^{-}$ ($bbd$), and $\\Omega^{-}_{bb}$($bbs$) can also constitute one  SU(3) triplet  similar to Fig.~\\ref{fig:doubly_heavy_baron} with the replacement $c\\to b$. While the bottom-charm baryons could form two sets of SU(3) triplets, ($\\Xi_{bc},\\Omega_{bc}$) and ($\\Xi_{bc}^{\\prime},\\Omega_{bc}^{\\prime}$). The difference between the two sets is the different total spin of $bc$ system as shown in Tab.~\\ref{tab:JPC}, In fact there could be   mixing between them. However, the detailed mixing scheme between the two triplets is still unclear, the initial baryons include two triplets, ($\\Xi_{bc},\\Omega_{bc}$) and ($\\Xi_{bc}^{\\prime},\\Omega_{bc}^{\\prime}$) in this work. The doubly heavy baryons with $J^{P}=3\/2^{+}$ can decay into the lowest-lying ones radiatively if the mass splitting is small, or decay into the lowest-lying ones with the emission of a light pion when they are   heavy enough. The final baryons include doubly heavy baryons and singly heavy baryons. The singly heavy baryons can compose one SU(3) anti-triplets $\\boldsymbol{\\bar{3}}$ and one SU(3) sextet $\\boldsymbol{6}$ as shown in Fig.~\\ref{fig:singly_heavy}.\nTaking the transition ${\\cal B}_{bc}\\to {\\cal B}_{c}$ with $b\\to s$ as an example, the final baryons $\\Xi_{c}^{\\prime+}$, $\\Xi_{c}^{\\prime0}$ and $\\Omega_{c}^{0}$ belong to the presentation of $\\boldsymbol{6}$, while $\\Xi_{c}^{+}$ and $\\Xi_{c}^{0}$ are included in the $\\boldsymbol{\\bar{3}}$, as can be seen from Fig.~\\ref{fig:singly_heavy}.\n\n\n\\begin{figure}[!]\n\\includegraphics[width=0.3\\columnwidth]{doubly_heavy_baron}\n\\caption{Spin-$1\/2$ doubly charmed baryons. It is similar for the doubly bottom baryons and the bottom-charm baryons.}\n\\label{fig:doubly_heavy_baron}\n\\end{figure}\n\n\\begin{figure}[!]\n\\includegraphics[width=0.6\\columnwidth]{singly_heavy_baryon}\n\\caption{Spin-$1\/2$ singly charmed baryons. Here (a) represents SU(3) anti-triplets $\\boldsymbol{\\bar{3}}$ and (b) represents SU(3) sextets $\\boldsymbol{6}$. The spin-3\/2 singly charmed baryons only have SU(3) sextets $\\boldsymbol{6}$ as shown by panel (b) just with the replacement ``${\\cal B}_{c}\\to {\\cal B}_{c}^{*}$\".\nFor spin-$1\/2$ and spin-$3\/2$ singly bottomed baryons, a replacement $c\\to b$ is needed.}\n\\label{fig:singly_heavy}\n\\end{figure}\n\nThis paper is organized as follows. In Sec.~II, we will present the framework of the light-front approach under the diquark picture,  and then the flavor-spin wave functions will be discussed. In the appendix, we will provide a new approach to derive the flavor-spin factors.  Numerical results of various transition  form factors are shown in this section. In Sec.~III,   phenomenological applications of the doubly heavy baryon decays will be carried out, including numerical results of the decay widths, branching ratios and $\\Gamma_{L}\/\\Gamma_{T}$s of the semileptonic weak decays of doubly heavy baryons. The SU(3) symmetry breaking effect and error estimations will be also discussed in Sec.~III. A brief summary is given in the last section. The appendix  also contains some brief description of the flavor-spin wave functions, and  helicity amplitudes.\n\n\\section{Theoretical framework}\n\n\nThe theoretical framework for the charged current and FCNC induced baryonic   transitions    will be briefly introduced in this section, including the definitions of the states for spin-$1\/2$ and $3\/2$ baryons, and the extraction of the transition form factors. More details can be found in Refs.~\\cite{Ke:2017eqo,Ke:2007tg}.\nFlavor-spin wave functions will be given in the second subsection, while a new derivation is given in the appendices.\n\n\\subsection{Light-front quark model}\\label{subsec_lightfrontquarkmodel}\n\n\n\n\\begin{figure}[htp]\n\\includegraphics[width=0.4\\columnwidth]{decay} \\caption{Feynman diagram for doubly heavy baryons $B$ into a spin-1\/2 and spin-3\/2 ground-state baryons $B^{\\prime}$ with two spectator quarks as a diquark. Here $P$ and $P^{\\prime}$ are the momentum of the initial and final baryons, respectively. In quark level, the transition is one heavy quark $Q_{1}$ with momentum $p_{1}$ decays into a lighter quark $q_{1}$ with momentum $p_{1}^{\\prime}$, and the diquark with momentum $p_{2}$. The black ball means the weak interaction vertex.}\n\\label{fig:decay}\n\\end{figure}\n\n\nFor the $J^{P}=1\/2^{+}$ baryon states, their wave functions in light-front quark model can be written as\n\\begin{eqnarray}\n\t|{\\cal B}(P,S,S_{z})\\rangle & = & \\int\\{d^{3}p_{1}\\}\\{d^{3}p_{2}\\}2(2\\pi)^{3}\\delta^{3}(\\tilde{P}-\\tilde{p}_{1}-\\tilde{p}_{2})\\nonumber \\\\\n\t&  & \\times\\sum_{\\lambda_{1},\\lambda_{2}}\\Psi^{SS_{z}}(\\tilde{p}_{1},\\tilde{p}_{2},\\lambda_{1},\\lambda_{2})|Q_{1}(p_{1},\\lambda_{1})({\\rm{di}})(p_{2},\\lambda_{2})\\rangle,\\label{eq:state_vector}\n\t\\end{eqnarray}\nhere $Q_{1}=b,c$ is initial heavy quark, and ``$({\\rm{di}})$\" presents the diquark shown in Fig.~\\ref{fig:decay}. $\\Psi^{SS_{z}}$ is the momentum-space wave function and can be shown with the following equation,\n\t\\begin{equation}\n\t\\Psi^{SS_{z}}(\\tilde{p}_{1},\\tilde{p}_{2},\\lambda_{1},\\lambda_{2})=\\frac{1}{\\sqrt{2(p_{1}\\cdot\\bar{P}+m_{1}M_{0})}}\\bar{u}(p_{1},\\lambda_{1})\\Gamma_{S(A)} u(\\bar{P},S_{z})\\phi(x,k_{\\perp}),\\label{eq:momentum_wave_function_1\/2}\n\t\\end{equation}\nhere $\\Gamma$ is the coupling vertex of the decay quark $Q_{1}$ and the diquark in the baryon state, and when the diquark is a scalar diquark, the coupling vertex is defined as $\\Gamma_{S}=1$.\nIn Ref.~\\cite{Chua:2018lfa}, when an axial-vector diquark is involved,\nthe vertex should be\n\\begin{align}\n\t\\Gamma_{A} & =\\frac{\\gamma_{5}}{\\sqrt{3}}\\left(\\slashed\\epsilon^{*}(p_{2},\\lambda_{2})-\\frac{M_0+m_1+m_2}{\\bar{P}\\cdot p_2+m_2M_0}\\epsilon^{*}(p_{2},\\lambda_{2})\\cdot\\bar{P}\\right).\\label{eq:momentum_wave_function_1\/2gamma}\n\t\\end{align}\nIn Eq.~(\\ref{eq:momentum_wave_function_1\/2}), $\\phi$ is a Gaussian-type function:\n\t\\begin{equation}\n\t\\phi=4\\left(\\frac{\\pi}{\\beta^{2}}\\right)^{3\/4}\\sqrt{\\frac{e_{1}e_{2}}{x_{1}x_{2}M_{0}}}\\exp\\left(\\frac{-\\vec{k}^{2}}{2\\beta^{2}}\\right).\\label{eq:Gauss}\n\t\\end{equation}\n\t\n\nIn analogy to the $1\/2^{+}$ baryon case, a  $3\/2^{+}$\nbaryon state has a similar expression to Eq. (\\ref{eq:state_vector}) except a different coupling vertex:\n\\begin{equation}\n\\Psi^{SS_{z}}(\\tilde{p}_{1},\\tilde{p}_{2},\\lambda_{1},\\lambda_{2})=\n\\frac{1}{\\sqrt{2(p_{1}\\cdot\\bar{P}+m_{1}M_{0})}}\\bar{u}(p_{1},\\lambda_{1})\\Gamma^{\\alpha}_{A}(p_{2},\\lambda_{2})u_{\\alpha}(\\bar{P},S_{z})\\phi(x,k_{\\perp}),\\label{eq:momentum_wave_fuction_3\/2}\n\\end{equation}\nwhere\n\\begin{equation}\n\t\\Gamma_{A}^{\\alpha}=-\\left(\\epsilon^{*\\alpha}(p_{2},\\lambda_{2})-\\frac{p_2^{\\alpha}}{\\bar{P}\\cdot p_2+m_2M_0}\\epsilon^{*}(p_{2},\\lambda_{2})\\cdot\\bar{P}\\right).\n\\end{equation}\nWith the help of Eqs. (\\ref{eq:state_vector}) and (\\ref{eq:momentum_wave_function_1\/2}),  the spin-$1\/2$ to spin-$1\/2$ transition matrix element with $\\rm{(V-A)}$ and tensor current can be derived as\n\\begin{eqnarray}\n&  & \\langle{\\cal B}^{\\prime}_{f}(P^{\\prime},S^{\\prime}=\\frac{1}{2}, S_{z}^{\\prime})|\\bar{q}_{1}\\gamma_{\\mu}(1-\\gamma_{5})Q_{1}|{\\cal B}_{i}(P, S=\\frac{1}{2}, S_{z})\\rangle\\nonumber \\\\\n& = & \\int\\{d^{3}p_{2}\\}\\frac{\\phi^{\\prime}(x^{\\prime},k_{\\perp}^{\\prime})\\phi(x,k_{\\perp})}{2\\sqrt{p_{1}^{+}p_{1}^{\\prime+}(p_{1}\\cdot\\bar{P}+m_{1}M_{0})(p_{1}^{\\prime}\\cdot\\bar{P}^{\\prime}+m_{1}^{\\prime}M_{0}^{\\prime})}}\\nonumber \\\\\n&  & \\times\\sum_{\\lambda_{2}}\\bar{u}(\\bar{P}^{\\prime},S_{z}^{\\prime})\\bar{\\Gamma}^{\\prime}_{S(A)}(\\slashed p_{1}^{\\prime}+m_{1}^{\\prime})\\gamma_{\\mu}(1-\\gamma_{5})(\\slashed p_{1}+m_{1})\\Gamma_{S(A)} u(\\bar{P},S_{z}),\\label{eq:matrix_element_half}\\\\\n&  & \\langle{\\cal B}^{\\prime}_{f}(P^{\\prime},S^{\\prime}=\\frac{1}{2}, S_{z}^{\\prime})|\\bar{q}_{1}i\\sigma_{\\mu\\nu}\\frac{q^{\\nu}}{M}(1+\\gamma_{5})Q_{1}|{\\cal B}_{i}(P, S=\\frac{1}{2}, S_{z})\\rangle\\nonumber \\\\\n& = & \\int\\{d^{3}p_{2}\\}\\frac{\\phi^{\\prime}(x^{\\prime},k_{\\perp}^{\\prime})\\phi(x,k_{\\perp})}{2\\sqrt{p_{1}^{+}p_{1}^{\\prime+}(p_{1}\\cdot\\bar{P}+m_{1}M_{0})(p_{1}^{\\prime}\\cdot\\bar{P}^{\\prime}+m_{1}^{\\prime}M_{0}^{\\prime})}}\\nonumber \\\\\n&  & \\times\\sum_{\\lambda_{2}}\\bar{u}(\\bar{P}^{\\prime},S_{z}^{\\prime})\\bar{\\Gamma}^{\\prime}_{S(A)}(\\slashed p_{1}^{\\prime}+m_{1}^{\\prime})i\\sigma_{\\mu\\nu}\\frac{q^{\\nu}}{M}(1+\\gamma_{5})(\\slashed p_{1}+m_{1})\\Gamma_{S(A)} u(\\bar{P},S_{z}).\\label{eq:matrix_element_half_tensor}\n\\end{eqnarray}\nWith the help of Eqs. (\\ref{eq:state_vector}), (\\ref{eq:momentum_wave_function_1\/2})\nand (\\ref{eq:momentum_wave_fuction_3\/2}), the spin-1\/2 to spin-$3\/2$ transition matrix element with $\\rm{(V-A)}$ and tensor current can be derived as\n\\begin{eqnarray}\n&  & \\langle{\\cal B}^{\\prime*}_{f}(P^{\\prime},S^{\\prime}=\\frac{3}{2},S_{z}^{\\prime})|\\bar{q}_{1}\\gamma^{\\mu}(1-\\gamma_{5})Q_{1}|{\\cal B}_{i}(P,S=\\frac{1}{2},S_{z})\\rangle\\nonumber \\\\\n& = & \\int\\{d^{3}p_{2}\\}\\frac{\\phi^{\\prime}(x^{\\prime},k_{\\perp}^{\\prime})\\phi(x,k_{\\perp})}{2\\sqrt{p_{1}^{+}p_{1}^{\\prime+}(p_{1}\\cdot\\bar{P}+m_{1}M_{0})(p_{1}^{\\prime}\\cdot\\bar{P}^{\\prime}+m_{1}^{\\prime}M_{0}^{\\prime})}}\\nonumber \\\\\n&  & \\times\\sum_{\\lambda_{2}}\\bar{u}_{\\alpha}(\\bar{P}^{\\prime},S_{z}^{\\prime})\\left[\\bar{\\Gamma}^{\\prime\\alpha}_{A}(\\slashed p_{1}^{\\prime}+m_{1}^{\\prime})\\gamma^{\\mu}(1-\\gamma_{5})(\\slashed p_{1}+m_{1})\\Gamma_{A}\\right]u(\\bar{P},S_{z}),\\label{eq:matrix_element_onehalf}\\\\\n&  & \\langle{\\cal B}^{\\prime*}_{f}(P^{\\prime},S^{\\prime}=\\frac{3}{2},S_{z}^{\\prime})|\\bar{q}_{1}i\\sigma_{\\mu\\nu}\\frac{q^{\\nu}}{M}(1+\\gamma_{5})Q_{1}|{\\cal B}_{i}(P,S=\\frac{1}{2},S_{z})\\rangle\\nonumber \\\\\n& = & \\int\\{d^{3}p_{2}\\}\\frac{\\phi^{\\prime}(x^{\\prime},k_{\\perp}^{\\prime})\\phi(x,k_{\\perp})}{2\\sqrt{p_{1}^{+}p_{1}^{\\prime+}(p_{1}\\cdot\\bar{P}+m_{1}M_{0})(p_{1}^{\\prime}\\cdot\\bar{P}^{\\prime}+m_{1}^{\\prime}M_{0}^{\\prime})}}\\nonumber \\\\\n&  & \\times\\sum_{\\lambda_{2}}\\bar{u}_{\\alpha}(\\bar{P}^{\\prime},S_{z}^{\\prime})\\left[\\bar{\\Gamma}^{\\prime\\alpha}_{A}(\\slashed p_{1}^{\\prime}+m_{1}^{\\prime})i\\sigma_{\\mu\\nu}\\frac{q^{\\nu}}{M}(1+\\gamma_{5})(\\slashed p_{1}+m_{1})\\Gamma_{A}\\right]u(\\bar{P},S_{z}).\\label{eq:matrix_element_onehalf_tensor}\n\\end{eqnarray}\nIn Eqs.~(\\ref{eq:momentum_wave_function_1\/2})-(\\ref{eq:momentum_wave_function_1\/2gamma})\nand (\\ref{eq:momentum_wave_fuction_3\/2})-(\\ref{eq:matrix_element_onehalf_tensor}),\n\\begin{equation}\nm_{1}=m_{Q},\\quad m_{1}^{\\prime}=m_{q},\\quad m_{2}=m_{(di)},\n\\end{equation}\nand $p_{1}$ and $p_{1}^{\\prime}$ are the four-momentum of the initial and final quark, respectively. $P$ and $P^{\\prime}$ are\nthe four-momentum of the initial baryons ${\\cal B}$ and final baryon states ${\\cal B}^{\\prime}$, respectively.\n$q_{1}=u,d,s,c$ means the lighter quark in the final states shown in Fig.~\\ref{fig:decay}. When the diquark is a scalar diquark, the coupling vertex is defined as,\n\\begin{equation}\n\\Gamma_{S}=\\bar{\\Gamma}^{\\prime}_{S}=1, \\label{scalar diquark}\n\\end{equation}\nand when an axial-vector diquark is involved, the vertex should be\n\\begin{align}\n\\bar{\\Gamma}^{\\prime}_{A} & =\\frac{1}{\\sqrt{3}}\\left(-\\slashed\\epsilon(p_{2},\\lambda_{2})+\\frac{M_0^{\\prime}+m_1^{\\prime}+m_2}{\\bar{P}^{\\prime}\\cdot p_2+m_2M_0^{\\prime}}\\epsilon(p_{2},\\lambda_{2})\\cdot\\bar{P}^{\\prime}\\right)\\gamma_{5},\\label{axial-vector diquarkprime}\n\\end{align}\nand\n\\begin{equation}\n\t\\bar{\\Gamma}_{A}^{\\prime\\alpha}=-\\left(\\epsilon^{\\alpha}(p_{2},\\lambda_{2})-\\frac{p_2^{\\alpha}}{\\bar{P}^{\\prime}\\cdot p_2+m_2M_0^{\\prime}}\\epsilon(p_{2},\\lambda_{2})\\cdot\\bar{P}^{\\prime}\\right).\n\\end{equation}\nThe $1\/2\\to1\/2$ transition matrix elements can be parameterized as\n\\begin{eqnarray}\n&&\\langle{\\cal B}^{\\prime}_{f}(P^{\\prime}, S^{\\prime}=\\frac{1}{2},S_{z}^{\\prime})|\\bar{q}_{1}\\gamma_{\\mu}(1-\\gamma_{5})Q_{1}|{\\cal B}_{i}(P,S=\\frac{1}{2},S_{z})\\rangle\\nonumber \\\\\n& &=  \\bar{u}(P^{\\prime},S_{z}^{\\prime})\n\\Big[\\gamma_{\\mu}f_{1,S(A)}(q^{2})\n+i\\sigma_{\\mu\\nu}\\frac{q^{\\nu}}{M}f_{2,S(A)}(q^{2})\n+\\frac{q_{\\mu}}{M}f_{3,S(A)}(q^{2})\\Big]u(P,S_{z})\\nonumber \\\\\n && \\quad-  \\bar{u}(P^{\\prime},S_{z}^{\\prime})\n\\Big[\\gamma_{\\mu}g_{1,S(A)}(q^{2})\n+i\\sigma_{\\mu\\nu}\\frac{q^{\\nu}}{M}g_{2,S(A)}(q^{2})\n+\\frac{q_{\\mu}}{M}g_{3,S(A)}(q^{2})\\Big]\\gamma_{5}u(P,S_{z}),\\label{eq:matrix_element_2}\n\\end{eqnarray}\n\\begin{eqnarray}\n&&\\langle{\\cal B}^{\\prime}_{f}(P^{\\prime},S^{\\prime}=\\frac{1}{2},S_{z}^{\\prime})\n|\\bar{q}_{1}i\\sigma_{\\mu\\nu}\\frac{q^{\\nu}}{M}(1+\\gamma_{5})Q_{1}|{\\cal B}_{i}(P,S=\\frac{1}{2},S_{z})\\rangle \\nonumber \\\\\n&&=\\bar{u}(P^{\\prime},S_{z}^{\\prime})\n\\Big[\\frac{f_{1,S(A)}^{T}(q^2)}{M(M^{\\prime}-M)}(q^2\\gamma_{\\mu}-{\\slashed q} q_{\\mu})\n+i\\sigma_{\\mu\\nu}\\frac{q^{\\nu}}{M}f_{2,S(A)}^{T}(q^{2})\\Big]u(P,S_{z})\\nonumber \\\\\n& &\\quad+ \\bar{u}(P^{\\prime},S_{z}^{\\prime})\n\\Big[\\frac{g_{1,S(A)}^{T}(q^2)}{M(M+M^{\\prime})}(q^2\\gamma_{\\mu}-{\\slashed q} q_{\\mu})+i\\sigma_{\\mu\\nu}\\frac{q^{\\nu}}{M}g_{2,S(A)}^{T}(q^{2})\n\\Big]\\gamma_{5}u(P,S_{z}).\\label{eq:matrix_element_2p}\n\\end{eqnarray}\n\t\nThen the extraction of these form factors $f_{1,2,3,S(A)}$ can be performed as Ref.~\\cite{Ke:2017eqo}.\nMultiply $\\bar{u}(\\bar{P},S_{z})(\\bar{\\Gamma}^{\\mu})_{i}u(\\bar{P}^{\\prime},S_{z}^{\\prime})$\nand $\\bar{u}(P,S_{z})(\\Gamma^{\\mu})_{i}u(P^{\\prime},S_{z}^{\\prime})$\non the $\\langle{\\cal B}^{\\prime}_{f}(P^{\\prime},S^{\\prime}=\\frac{1}{2},S_{z}^{\\prime})\n|\\bar{q}_{1}\\gamma_{\\mu}Q_{1}|{\\cal B}_{i}(P,S=\\frac{1}{2},S_{z})\\rangle$\npart of Eq.~(\\ref{eq:matrix_element_half}) and Eq.~(\\ref{eq:matrix_element_2}), respectively.\nAt the same time, the approximation $P^{(\\prime)}\\to\\bar{P}^{(\\prime)}$ need to be taken for the integral.\nAfter summing the polarizations of the initial and final baryon states up, we can get the three linear equations as follows,\n\\begin{eqnarray}\n&  & {\\rm Tr}\\Big\\{(\\Gamma^{\\mu})_{i}(\\slashed P^{\\prime}+M^{\\prime})\\Big[\\gamma_{\\mu}f_{1,S(A)}(q^2)\n+i\\sigma_{\\mu\\nu}\\frac{q^{\\nu}}{M}f_{2,S(A)}(q^2)\n+\\frac{q_{\\mu}}{M}f_{3,S(A)}(q^2)\\Big](\\slashed P+M)\\Big\\}\\nonumber \\\\\n& = & \\int\\{d^{3}p_{2}\\}\n\\frac{\\phi^{\\prime}(x^{\\prime},k_{\\perp}^{\\prime})\\phi(x,k_{\\perp})}\n{2\\sqrt{p_{1}^{+}p_{1}^{\\prime+}\n(p_{1}\\cdot\\bar{P}+m_{1}M_{0})(p_{1}^{\\prime}\\cdot\\bar{P}^{\\prime}\n+m_{1}^{\\prime}M_{0}^{\\prime})}}\\nonumber \\\\\n&  & \\times\\sum_{\\lambda_{2}}{\\rm Tr}\\left[(\\bar{\\Gamma}^{\\mu})_{i}(\\bar{\\slashed P}^{\\prime}+M_{0}^{\\prime})\\bar{\\Gamma}^{\\prime}_{S(A)}(\\slashed p_{1}^{\\prime}+m_{1}^{\\prime})\\gamma_{\\mu}(\\slashed p_{1}+m_{1})\\Gamma_{S(A)}(\\bar{\\slashed P}+M_{0})\\right],\n\\label{eq:solve_fi}\n\\end{eqnarray}\nwith $(\\bar{\\Gamma}^{\\mu})_{i}=\\{\\gamma^{\\mu},\\bar{P}^{\\mu},\\bar{P}^{\\prime\\mu}\\}$ and $(\\Gamma^{\\mu})_{i}=\\{\\gamma^{\\mu},P^{\\mu},P^{\\prime\\mu}\\}$.\nUsing the above Eq.~(\\ref{eq:solve_fi}), we can get the specific expression of the form factors $f_{1,2,3,S(A)}$ as follows:\n\\begin{eqnarray}\nf_1&=&\\frac{q^2 [B_1 (M+M')^2-2 B_2 (2 M+M')-2 B_3 (M+2\n   M')]-B_1 (q^2)^2-2 (B_2-B_3) (M-M') (M+M')^2}{4\n [(M-M')^2-q^2] [(M+M')^2-q^2]^2},\\notag\\\\\nf_2&=&\\frac{M (M+M') [B_1 (M+M')^2+2 B_2 (M'-2 M)+2 B_3 (M-2 M')]-M\n   q^2 [B_1 (M+M')+2 B_2+2 B_3]}{4\n [(M-M')^2-q^2] [(M+M')^2-q^2]^2},\\notag\\\\\nf_3&=&\\frac{M \\{B_1 (M-M') [q^2-(M+M')^2]+2 B_2 \\left(4 M^2-M\n   M'+M'^2-q^2\\right)+2 B_3 \\left(-M^2+M M'-4 M'^2+q^2\\right)\\}}{4\n [(M-M')^2-q^2] [(M+M')^2-q^2]^2},\\notag\n\\end{eqnarray}\nwith\n\\begin{eqnarray}\nB_i&=&\\int\\{d^{3}p_{2}\\}\n\\frac{\\phi^{\\prime}(x^{\\prime},k_{\\perp}^{\\prime})\\phi(x,k_{\\perp})}\n{2\\sqrt{p_{1}^{+}p_{1}^{\\prime+}\n(p_{1}\\cdot\\bar{P}+m_{1}M_{0})(p_{1}^{\\prime}\\cdot\\bar{P}^{\\prime}\n+m_{1}^{\\prime}M_{0}^{\\prime})}}\\nonumber \\\\\n&  &\\times\\sum_{\\lambda_{2}}{\\rm Tr}\\left[(\\bar{\\Gamma}^{\\mu})_{i}(\\bar{\\slashed P}^{\\prime}+M_{0}^{\\prime})\\bar{\\Gamma}^{\\prime}_{S(A)}(\\slashed p_{1}^{\\prime}+m_{1}^{\\prime})\\gamma_{\\mu}(\\slashed p_{1}+m_{1})\\Gamma_{S(A)}(\\bar{\\slashed P}+M_{0})\\right].\n\\end{eqnarray}\nThe form factors $g_{1,2,3,S(A)}$ can be calculated using the similar process,\n\\begin{eqnarray}\ng_1&=&\\frac{q^2 [R_1 (M-M')^2+2 R_2 (M'-2 M)-2 R_3 (M-2\n   M')]-R_1 (q^2)^2-2 (R_2-R_3) (M+M') (M-M')^2}{4\n   [(M-M')^2-q^2]^2 [(M+M')^2-q^2]},\\notag\\\\\ng_2&=&\\frac{M q^2 [R_1 (M-M')+2 R_2+2 R_3]-M (M-M') [R_1 (M-M')^2-2\n   R_2 (2 M+M')+2 R_3 (M+2 M')]}{4\n   [(M-M')^2-q^2]^2 [(M+M')^2-q^2]},\\notag\\\\\ng_3&=&\\frac{M \\{R_1 (M+M') [(M-M')^2-q^2]-2 R_2 \\left(4 M^2+M\n   M'+M'^2-q^2\\right)+2 R_3 \\left(M^2+M M'+4 M'^2-q^2\\right)\\}}{4\n   [(M-M')^2-q^2]^2 [(M+M')^2-q^2]},\\notag\n \\end{eqnarray}\nwith\n\\begin{eqnarray}\nR_i&=&\\int\\{d^{3}p_{2}\\}\n\\frac{\\phi^{\\prime}(x^{\\prime},k_{\\perp}^{\\prime})\\phi(x,k_{\\perp})}\n{2\\sqrt{p_{1}^{+}p_{1}^{\\prime+}\n(p_{1}\\cdot\\bar{P}+m_{1}M_{0})(p_{1}^{\\prime}\\cdot\\bar{P}^{\\prime}\n+m_{1}^{\\prime}M_{0}^{\\prime})}}\\nonumber \\\\\n& &\\times\\sum_{\\lambda_{2}}{\\rm Tr}\\left[(\\bar{\\Gamma}^{\\mu})_{i}(\\bar{\\slashed P}^{\\prime}+M_{0}^{\\prime})\\bar{\\Gamma}^{\\prime}_{S(A)}(\\slashed p_{1}^{\\prime}+m_{1}^{\\prime})\\gamma_{\\mu}\\gamma_{5}(\\slashed p_{1}+m_{1})\\Gamma_{S(A)}(\\bar{\\slashed P}+M_{0})\\right].\n\\end{eqnarray}\nThen tensor form factors $f_{1,2,S(A)}^{T}$ or $g_{1,2,S(A)}^{T}$ defined by Eq.~(\\ref{eq:matrix_element_2p}) can also be extracted in the similar way with the form factors $f_{1,2,3,S(A)}$ and $g_{1,2,3,S(A)}$, the differences are only $(\\Gamma^{\\mu})_{i}=\\{\\gamma^{\\mu},P^{\\mu}\\}$ and $(\\bar{\\Gamma}^{\\mu})_{i}=\\{\\gamma^{\\mu},\\bar{P}^{\\mu}\\}$,\n\\begin{eqnarray}\nf_1^T&=&\\frac{M (M-M') \\{B^T_1 [q^2-(M+M')^2]+6 B^T_2 (M+M')\\}}{4\n [(M-M')^2-q^2] [(M+M')^2-q^2]^2},\\notag\\\\\nf_2^T&=&\\frac{M (M+M')^2 [B^T_1 (M+M')-2 B^T_2]-M q^2 [B^T_1 (M+M')+4 B^T_2]}{4\n [(M-M')^2-q^2] [(M+M')^2-q^2]^2},\\notag\\\\\ng_1^T&=&\\frac{M (M+M') \\{R^T_1 [(M-M')^2-q^2]+6 R^T_2 (M'-M)\\}}{4\n   [(M-M')^2-q^2]^2 [(M+M')^2-q^2]},\\notag\\\\\ng_2^T&=&\\frac{M q^2 [R^T_1 (M-M')+4 R^T_2]+M (M-M')^2 [R^T_1 (M'-M)+2 R^T_2]}{4\n   [(M-M')^2-q^2]^2 [(M+M')^2-q^2]},\n\\end{eqnarray}\nwith\n\\begin{eqnarray}\nB_i^T&=&\\int\\{d^{3}p_{2}\\}\n\\frac{\\phi^{\\prime}(x^{\\prime},k_{\\perp}^{\\prime})\\phi(x,k_{\\perp})}\n{2\\sqrt{p_{1}^{+}p_{1}^{\\prime+}\n(p_{1}\\cdot\\bar{P}+m_{1}M_{0})(p_{1}^{\\prime}\\cdot\\bar{P}^{\\prime}\n+m_{1}^{\\prime}M_{0}^{\\prime})}}\\nonumber \\\\\n&  &\\times\\sum_{\\lambda_{2}}{\\rm Tr}\\left[(\\bar{\\Gamma}^{\\mu})_{i}(\\bar{\\slashed P}^{\\prime}+M_{0}^{\\prime})\\bar{\\Gamma}^{\\prime}_{S(A)}(\\slashed p_{1}^{\\prime}+m_{1}^{\\prime})i\\sigma_{\\mu\\nu}\\frac{q^{\\nu}}{M}(\\slashed p_{1}+m_{1})\\Gamma_{S(A)}(\\bar{\\slashed P}+M_{0})\\right],\\notag\\\\\nR_i^T&=&\\int\\{d^{3}p_{2}\\}\n\\frac{\\phi^{\\prime}(x^{\\prime},k_{\\perp}^{\\prime})\\phi(x,k_{\\perp})}\n{2\\sqrt{p_{1}^{+}p_{1}^{\\prime+}\n(p_{1}\\cdot\\bar{P}+m_{1}M_{0})(p_{1}^{\\prime}\\cdot\\bar{P}^{\\prime}\n+m_{1}^{\\prime}M_{0}^{\\prime})}}\\nonumber \\\\\n& & \\times\\sum_{\\lambda_{2}}{\\rm Tr}\\left[(\\bar{\\Gamma}^{\\mu})_{i}(\\bar{\\slashed P}^{\\prime}+M_{0}^{\\prime})\\bar{\\Gamma}^{\\prime}_{S(A)}(\\slashed p_{1}^{\\prime}+m_{1}^{\\prime})i\\sigma_{\\mu\\nu}\\frac{q^{\\nu}}{M}\\gamma_{5}(\\slashed p_{1}+m_{1})\\Gamma_{S(A)}(\\bar{\\slashed P}+M_{0})\\right].\n\\end{eqnarray}\n\n\nThe $1\/2\\to3\/2$ transition matrix elements can be parameterized in a similar form as follows.\n\\begin{eqnarray}\n&&\t\\langle{\\cal B}^{\\prime*}_{f}(P^{\\prime},S^{\\prime}=\\frac{3}{2},S_{z}^{\\prime})|\\bar{q}_{1}\\gamma^{\\mu}(1-\\gamma_{5})Q_{1}|{\\cal B}_{i}(P,S=\\frac{1}{2},S_{z})\\rangle \\nonumber\\\\\n&& =  \\bar{u}_{\\alpha}(P^{\\prime},S_{z}^{\\prime})\\Big[\\mathtt{f}_{1}(q^2)\\frac{P^{\\alpha}}{M}(\\gamma^{\\mu}-\\frac{\\slashed q}{q^2}q^{\\mu})+\\mathtt{f}_{2}(q^2)\\frac{P^{\\alpha}}{M^2}(\\frac{M^2-M^{\\prime 2}}{q^2}q^{\\mu}-{\\cal P}^{\\mu})\\nonumber\\\\\n&&\\quad\\quad\\quad\\quad\\quad\\quad\n+\\mathtt{f}_{3}(q^2)\\frac{P^{\\alpha}}{M^2}\\frac{M^2-M^{\\prime 2}}{q^2}q^{\\mu}+\\mathtt{f}_{4}(q^{2})(g^{\\alpha\\mu}-\\frac{q^{\\alpha}q^{\\mu}}{q^2})\\Big]\\gamma_{5}u(P,S_{z})\\nonumber\\\\\n& &\\quad- \\bar{u}_{\\alpha}(P^{\\prime},S_{z}^{\\prime})\\Big[\\mathtt{g}_{1}(q^2)P^{\\alpha}(\\gamma^{\\mu}-\\frac{\\slashed q}{q^2}q^{\\mu})+\\mathtt{g}_{2}(q^2)\\frac{P^{\\alpha}}{M^2}(\\frac{M^2-M^{\\prime 2}}{q^2}q^{\\mu}-{\\cal P}^{\\mu})\\nonumber\\\\\n&&\\quad\\quad\\quad\\quad\\quad\\quad\n+\\mathtt{g}_{3}(q^2)\\frac{P^{\\alpha}}{M^2}\\frac{M^2-M^{\\prime 2}}{q^2}q^{\\mu}+\\mathtt{g}_{4}(q^{2})(g^{\\alpha\\mu}-\\frac{q^{\\alpha}q^{\\mu}}{q^2})\\Big]u(P,S_{z}),\\label{eq:matrix_element_32nVA}\n\\end{eqnarray}\n\\begin{eqnarray}\n&&\t\\langle{\\cal B}_{f}^{\\prime*}(P^{\\prime},S^{\\prime}=\\frac{3}{2},S_{z}^{\\prime})|\\bar{q}_{1}i\\sigma_{\\mu\\nu}\\frac{q^{\\nu}}{M}(1+\\gamma_{5})Q_{1}|{\\cal B}_{i}(P,S=\\frac{1}{2},S_{z})\\rangle\\nonumber\\\\\n&& =  \\bar{u}_{\\alpha}(P^{\\prime},S_{z}^{\\prime})\n\\Big[\\mathtt{f}_{1}^{T}(q^2)\\frac{P^{\\alpha}}{M}(\\gamma^{\\mu}-\\frac{\\slashed q}{q^2}q^{\\mu})+\\mathtt{f}_{2}^{T}(q^2)\\frac{P^{\\alpha}}{M^2}(\\frac{M^2-M^{\\prime 2}}{q^2}q^{\\mu}-{\\cal P}^{\\mu}) \\nonumber\\\\\n&&\\quad\\quad\\quad\\quad\\quad\\quad +\\mathtt{f}_{3}^{T}(q^2)\\frac{P^{\\alpha}}{M^2}\\frac{M^2-M^{\\prime 2}}{q^2}q^{\\mu}+\\mathtt{f}_{4}^{T}(q^{2})(g^{\\alpha\\mu}-\\frac{q^{\\alpha}q^{\\mu}}{q^2})\\Big]\\gamma_{5}u(P,S_{z})\\nonumber\\\\\n& &\\quad + \\bar{u}_{\\alpha}(P^{\\prime},S_{z}^{\\prime})\\Big[\\mathtt{g}_{1}^{T}(q^2)\\frac{P^{\\alpha}}{M}(\\gamma^{\\mu}-\\frac{\\slashed q}{q^2}q^{\\mu})+\\mathtt{g}_{2}^{T}(q^2)\\frac{P^{\\alpha}}{M^2}(\\frac{M^2-M^{\\prime 2}}{q^2}q^{\\mu}-{\\cal P}^{\\mu})\\nonumber\\\\\n&&\\quad\\quad\\quad\\quad\\quad\\quad\n+\\mathtt{g}_{3}^{T}(q^2)\\frac{P^{\\alpha}}{M^2}\\frac{M^2-M^{\\prime 2}}{q^2}q^{\\mu}+\\mathtt{g}_{4}^{T}(q^{2})(g^{\\alpha\\mu}-\\frac{q^{\\alpha}q^{\\mu}}{q^2})\\Big]u(P,S_{z}).\\label{eq:matrix_element_32nT}\n\\end{eqnarray}\nHere $q^{\\mu}=P^{\\mu}-P^{\\prime\\mu}$ and ${\\cal P}^{\\mu}=P^{\\mu}+P^{\\prime\\mu}$.\nIn the previous work~\\cite{Zhao:2018mrg}, the $1\/2\\to3\/2$ transition matrix elements have been parameterized with form factors $\\mathtt{f}_{1,2,3,4}^{\\prime(T)}$ and $\\mathtt{g}_{1,2,3,4}^{\\prime(T)}$ in following form,\n\\begin{eqnarray}\n&&\t\\langle{\\cal B}_{f}^{\\prime*}(P^{\\prime},S^{\\prime}=\\frac{3}{2},S_{z}^{\\prime})|\\bar{q}_{1}\\gamma^{\\mu}(1-\\gamma_{5})Q_{1}|{\\cal B}_{i}(P,S=\\frac{1}{2},S_{z})\\rangle \\nonumber\\\\\n&& =  \\bar{u}_{\\alpha}(P^{\\prime},S_{z}^{\\prime})\\Big[\\gamma^{\\mu}P^{\\alpha}\\frac{\\mathtt{f}_{1}^{\\prime}(q^{2})}{M}+\\frac{\\mathtt{f}_{2}^{\\prime}(q^{2})}{M^{2}}P^{\\alpha}P^{\\mu} +\\frac{\\mathtt{f}_{3}^{\\prime}(q^{2})}{MM^{\\prime}}P^{\\alpha}P^{\\prime\\mu}+\\mathtt{f}_{4}^{\\prime}(q^{2})g^{\\alpha\\mu}\\Big]\\gamma_{5}u(P,S_{z})\\nonumber\\\\\n& &\\quad-\\bar{u}_{\\alpha}(P^{\\prime},S_{z}^{\\prime})\\Big[\\gamma^{\\mu}P^{\\alpha}\\frac{\\mathtt{g}_{1}^{\\prime}(q^{2})}{M}+\\frac{\\mathtt{g}_{2}^{\\prime}(q^{2})}{M^{2}}P^{\\alpha}P^{\\mu} +\\frac{\\mathtt{g}_{3}^{\\prime}(q^{2})}{MM^{\\prime}}P^{\\alpha}P^{\\prime\\mu}+\\mathtt{g}_{4}^{\\prime}(q^{2})g^{\\alpha\\mu}\\Big]u(P,S_{z}),\\label{eq:matrix_element_32VA} \\\\\n&&\\langle{\\cal B}^{\\prime*}_{f}(P^{\\prime},S^{\\prime}=\\frac{3}{2},S_{z}^{\\prime})|\\bar{q}_{1}i\\sigma_{\\mu\\nu}\\frac{q^{\\nu}}{M}(1+\\gamma_{5})Q_{1}|{\\cal B}_{i}(P,S=\\frac{1}{2},S_{z})\\rangle \\nonumber\\\\& &=  \\bar{u}_{\\alpha}(P^{\\prime},S_{z}^{\\prime})\\Big[\\gamma^{\\mu}P^{\\alpha}\\frac{\\mathtt{f}_{1}^{\\prime T}(q^{2})}{M}+\\frac{\\mathtt{f}_{2}^{\\prime T}(q^{2})}{M^{2}}P^{\\alpha}P^{\\mu} +\\frac{\\mathtt{f}_{3}^{\\prime T}(q^{2})}{MM^{\\prime}}P^{\\alpha}P^{\\prime\\mu}+\\mathtt{f}_{4}^{\\prime T}(q^{2})g^{\\alpha\\mu}\\Big]\\gamma_{5}u(P,S_{z})\\nonumber\\\\\n& &\\quad + \\bar{u}_{\\alpha}(P^{\\prime},S_{z}^{\\prime})\\Big[\\gamma^{\\mu}P^{\\alpha}\\frac{\\mathtt{g}_{1}^{\\prime T}(q^{2})}{M}+\\frac{\\mathtt{g}_{2}^{\\prime T}(q^{2})}{M^{2}}P^{\\alpha}P^{\\mu} +\\frac{\\mathtt{g}_{3}^{\\prime T}(q^{2})}{MM^{\\prime}}P^{\\alpha}P^{\\prime\\mu}+\\mathtt{g}_{4}^{\\prime T}(q^{2})g^{\\alpha\\mu}\\Big]u(P,S_{z}).\\label{eq:matrix_element_32t}\n\\end{eqnarray}\nThen the form factors $\\mathtt{f}_{1,2,3,4}^{(T)}$ and $\\mathtt{g}_{1,2,3,4}^{(T)}$ defined by Eqs.~(\\ref{eq:matrix_element_32nVA})-(\\ref{eq:matrix_element_32nT}) can be related  with $\\mathtt{f}_{1,2,3,4}^{\\prime(T)}$ and $\\mathtt{g}_{1,2,3,4}^{\\prime(T)}$ defined by Eqs.~(\\ref{eq:matrix_element_32VA})-(\\ref{eq:matrix_element_32t}) by the following formulas:\n\\begin{align}\n&\\mathtt{f}_{1}^{(T)}(q^2)=\\mathtt{f}_{1}^{\\prime(T)}(q^2),\\quad\n\\mathtt{f}_{2}^{(T)}(q^2)=-\\frac{1}{2}\\Big[\\mathtt{f}_{2}^{\\prime(T)}(q^2)\n+\\frac{M}{M^{\\prime}}\\mathtt{f}_{3}^{\\prime(T)}(q^2)\\Big],\\quad\n\\mathtt{f}_{4}^{(T)}(q^2)=\\mathtt{f}_{4}^{\\prime(T)}(q^2),\\label{eq:ff23f124t}\\\\\n&\\mathtt{f}_{3}^{(T)}(q^2)=\\frac{M^{2}}{M^2-M^{\\prime2}}\n\\Big[\\mathtt{f}_{1}^{\\prime(T)}(q^2)\\frac{-M-M^{\\prime}}{M}+\\mathtt{f}_{4}^{\\prime(T)}(q^2)\\Big]\n+\\frac{1}{2}\\Big[\\mathtt{f}_{2}^{\\prime(T)}(q^2)\n+\\frac{M}{M^{\\prime}}\\mathtt{f}_{3}^{\\prime(T)}(q^2)\\Big]\\nonumber\\\\\n&\\qquad\\qquad\\quad+\\frac{1}{2}\\frac{q^{2}}{M^2-M^{\\prime2}}\n\\Big[\\mathtt{f}_{2}^{\\prime(T)}(q^2)\n-\\frac{M}{M^{\\prime}}\\mathtt{f}_{3}^{\\prime(T)}(q^2)\\Big],\\label{eq:ff23f3t}\\\\\n&\\mathtt{g}_{1}^{(T)}(q^2)=\\mathtt{g}_{1}^{\\prime(T)}(q^2),\\quad\n\\mathtt{g}_{2}^{(T)}(q^2)=-\\frac{1}{2}\n\\Big[\\mathtt{g}_{2}^{\\prime(T)}(q^2)\n+\\frac{M}{M^{\\prime}}\\mathtt{g}_{3}^{\\prime(T)}(q^2)\\Big],\\quad\n\\mathtt{g}_{4}(q^2)=\\mathtt{g}_4^{\\prime(T)}(q^2),\\label{eq:ff23g124t}\\\\\n&\\mathtt{g}_{3}^{(T)}(q^2)=\n\\frac{M^{2}}{M^2-M^{\\prime2}}\n\\Big[\\mathtt{g}_{1}^{\\prime(T)}(q^2)\\frac{M-M^{\\prime}}{M}+\\mathtt{g}_{4}^{\\prime(T)}(q^2)\\Big]\n+\\frac{1}{2}\\Big[\\mathtt{g}_{2}^{\\prime(T)}(q^2)\n+\\frac{M}{M^{\\prime}}\\mathtt{g}_{3}^{\\prime(T)}(q^2)\\Big]\\nonumber\\\\\n&\\qquad\\qquad\\quad+\\frac{1}{2}\\frac{q^{2}}{M^2-M^{\\prime2}}\n\\Big[\\mathtt{g}_{2}^{\\prime(T)}(q^2)\n-\\frac{M}{M^{\\prime}}\\mathtt{g}_{3}^{\\prime(T)}(q^2)\\Big].\\label{eq:ff23g3t}\n\\end{align}\nMultiplying Eq.~(\\ref{eq:matrix_element_32nT}) by $q^{\\mu}$ will  yield\n\\begin{eqnarray}\n\\bar{u}_{\\alpha}(P^{\\prime},S_{z}^{\\prime})\n\\Big[\\mathtt{f}_{3}^{ T}(q^2)P^{\\alpha}\\frac{M^2-M^{\\prime 2}}{M^2}\\Big]\\gamma_{5}u(P,S_{z})&=& 0,\\nonumber \\\\\n\\bar{u}_{\\alpha}(P^{\\prime},S_{z}^{\\prime})\n\\Big[\\mathtt{g}_{3}^{ T}(q^2)P^{\\alpha}\\frac{M^2-M^{\\prime 2}}{M^2}\\Big]u(P,S_{z})&=& 0,\\label{eq:matrix_element_32f3g3}\n\\end{eqnarray}\nand one obtains $\\mathtt{f}_{3}^{T}(q^2)=\\mathtt{g}_{3}^{T}(q^2)=0$.\nThen using Eqs.~(\\ref{eq:ff23f124t} )- (\\ref{eq:ff23g3t}), one could get\n\\begin{align}\n&\\mathtt{f}_{1}^{T}(q^2)=\\frac{M}{M^{\\prime}+M}\n\\Big\\{\\mathtt{f}_{4}^{\\prime T}(q^2)\n+\\frac{M^2-M^{\\prime2}}{M^{2}}\n\\Big[\\frac{1}{2}\\Big(\\mathtt{f}_{2}^{\\prime T}(q^2)\n+\\frac{M}{M^{\\prime}}\\mathtt{f}_{3}^{\\prime T}(q^2)\\Big)\n+\\frac{1}{2}\\frac{q^{2}}{M^2-M^{\\prime2}}\n\\Big(\\mathtt{f}_{2}^{\\prime T}(q^2)\n-\\frac{M}{M^{\\prime}}\\mathtt{f}_{3}^{\\prime T}(q^2)\\Big)\\Big]\\Big\\},\\\\\n&\\mathtt{g}_{1}^{T}(q^2)=\\frac{M}{M^{\\prime}-M}\n\\Big\\{\\mathtt{g}_{4}^{\\prime T}(q^2)\n+\\frac{M^2-M^{\\prime2}}{M^{2}}\n\\Big[\\frac{1}{2}\\Big(\\mathtt{g}_{2}^{\\prime T}(q^2)\n+\\frac{M}{M^{\\prime}}\\mathtt{g}_{3}^{\\prime T}(q^2)\\Big)\n+\\frac{1}{2}\\frac{q^{2}}{M^2-M^{\\prime2}}\n\\Big(\\mathtt{g}_{2}^{\\prime T}(q^2)-\\frac{M}{M^{\\prime}}\\mathtt{g}_{3}^{\\prime T}(q^2)\n\\Big)\\Big]\\Big\\}.\n\\end{align}\n\nThese form factors $\\mathtt{f}_{1,2,3,4}^{\\prime}$ and $\\mathtt{g}_{1,2,3,4}^{\\prime}$ can be extracted in the following way~\\cite{Ke:2017eqo}.\nMultiply $\\bar{u}(P,S_{z})(\\bar{\\Gamma}_{5}^{\\mu\\beta})_{i}u_{\\beta}(P^{\\prime},S_{z}^{\\prime})$\nand $\\bar{u}(P,S_{z})(\\Gamma_{5}^{\\mu\\beta})_{i}u_{\\beta}(P^{\\prime},S_{z}^{\\prime})$\non the ``$\\langle{\\cal B}^{\\prime*}_{f}(P^{\\prime},S^{\\prime}=\\frac{3}{2},S_{z}^{\\prime})|\\bar{q}_{1}\\gamma^{\\mu}Q_{1}|{\\cal B}_{i}(P,S=\\frac{1}{2},S_{z})\\rangle$\"\npart of Eq.~(\\ref{eq:matrix_element_onehalf}) and Eq.~(\\ref{eq:matrix_element_32VA}), respectively.\nAt the same time, the approximation $P^{(\\prime)}\\to\\bar{P}^{(\\prime)}$ need to be taken for the integral.\nAfter summing the polarizations of the initial and final baryon states up, we can get the four equations as follows,\n\\begin{eqnarray}\n&&{\\rm Tr}\\Big\\{ u_{\\beta}(P^{\\prime},S_{z}^{\\prime})\\bar{u}_{\\alpha}(P^{\\prime},S_{z}^{\\prime})\n\\Big[\\gamma^{\\mu}P^{\\alpha}\\frac{\\mathtt{f}_{1}^{\\prime}(q^{2})}{M}\n+\\frac{\\mathtt{f}_{2}^{\\prime}(q^{2})}{M^{2}}P^{\\alpha}P^{\\mu}\n+\\frac{\\mathtt{f}_{3}^{\\prime}(q^{2})}{MM^{\\prime}}P^{\\alpha}P^{\\prime\\mu}\n+\\mathtt{f}_{4}^{\\prime}(q^{2})g^{\\alpha\\mu}\\Big]\\gamma_{5} (\\slashed P+M)(\\Gamma_{5}^{\\mu\\beta})_{i}\\Big\\}\\nonumber\\\\\n &&= \\int\\{d^{3}p_{2}\\}\\frac{\\phi^{\\prime}(x^{\\prime},k_{\\perp}^{\\prime})\\phi(x,k_{\\perp})}{2\\sqrt{p_{1}^{+}p_{1}^{\\prime+}(p_{1}\\cdot\\bar{P}+m_{1}M_{0})(p_{1}^{\\prime}\\cdot\\bar{P}^{\\prime}+m_{1}^{\\prime}M_{0}^{\\prime})}}\\nonumber \\\\\n&  &\\quad \\times\\sum_{S_{z}^{\\prime}\\lambda_{2}}{\\rm Tr}\\Big\\{ u_{\\beta}(\\bar{P}^{\\prime},S_{z}^{\\prime})\\bar{u}_{\\alpha}(\\bar{P}^{\\prime},S_{z}^{\\prime})\\bar{\\Gamma}^{\\prime\\alpha}_{A}(\\slashed p_{1}^{\\prime}+m_{1}^{\\prime})\\gamma_{\\mu}(\\slashed p_{1}+m_{1})\\Gamma_{A}(\\bar{\\slashed P}+M_{0})(\\bar{\\Gamma}_{5}^{\\mu\\beta})_{i}\\Big\\},\\label{eq:Fisolving}\n\\end{eqnarray}\nwith $(\\Gamma_{5}^{\\mu\\beta})_{i}=\\{\\gamma^{\\mu}P^{\\beta},P^{\\prime\\mu}P^{\\beta},P^{\\mu}P^{\\beta},g^{\\mu\\beta}\\}\\gamma_{5}$ and $(\\bar{\\Gamma}_{5}^{\\mu\\beta})_{i}=\\{\\gamma^{\\mu}\\bar{P}^{\\beta},\\bar{P}^{\\prime\\mu}\\bar{P}^{\\beta},\\bar{P}^{\\mu}\\bar{P}^{\\beta},g^{\\mu\\beta}\\}\\gamma_{5}$.\n\nThe analytic expression of form factors $\\mathtt{f}_{1,2,3,4}^{\\prime}$ can be got by solving the above four equations,\n\\begin{eqnarray}\n{\\mathtt{f}_{1}^{\\prime}}(q^2)&=&\\frac{{M}{M^{\\prime}}}{2\\big[{M}^4-2{M}^2\n({M^{\\prime 2}}+{q^2})+({M^{\\prime 2}}-{q^2})^2\n\\big]^2}\n\\big\\{-4{M^{\\prime}}\\big[{H_1}\n\\big(({M}-{M^{\\prime}})^2-{q^2}\\big)+{H_3}\n{M^{\\prime}}\\big]\\nonumber\\\\\n&&-2{H_2}\\big({M}^2-4{M}\n{M^{\\prime}}+{M^{\\prime 2}}-{q^2}\\big)+{H_4}\\big[{M}^4-2\n{M}^2\n\\big({M^{\\prime 2}}+{q^2}\\big)+\\big({M^{\\prime 2}}-{q^2}\\big)^2\n\\big]\\big\\},\n\\\\\n{\\mathtt{f}_{2}^{\\prime}}(q^2)&=&\\frac{{M}^2{M^{\\prime 2}}}{\\big[({M}-{M^{\\prime}})^2-{q^2}\\big]^3\n\\big[({M}+{M^{\\prime}})^2-{q^2}\\big]^2}\n\\big\\{2{M^{\\prime}}\n\\big[{H_1}\\big(({M}-{M^{\\prime}})^2-{q^2}\\big)+10\n{H_3}{M^{\\prime}}\\big]\\nonumber\\\\\n&&-4{H_2}\\big(2{M}^2+{M}\n{M^{\\prime}}+2{M^{\\prime 2}}-2{q^2}\\big)\n+{H_4}\\big[{M}^4-2\n{M}^2\n\\big({M^{\\prime 2}}+{q^2}\\big)+\\big({M^{\\prime 2}}-{q^2}\\big)^2\n\\big]\\big\\},\n\\end{eqnarray}\n\\begin{eqnarray}\n{\\mathtt{f}_{3}^{\\prime}}(q^2)&=&\\frac{{M}{M^{\\prime}}}{\\left[({M}-{M^{\\prime}})^2-{q^2}\\right]^3\n\\left[({M}+{M^{\\prime}})^2-{q^2}\\right]^2}\\times\\nonumber\\\\\n&&\\big\\{{M^{\\prime}}\\left[{H_1}\n\\left(({M}-{M^{\\prime}})^2-{q^2}\\right)\\left({M}^2-4\n{M}{M^{\\prime}}+{M^{\\prime 2}}-{q^2}\\right)\n-4{H_3}{M^{\\prime}}\n\\left(2{M}^2+{M}{M^{\\prime}}+2{M^{\\prime 2}}-2\n{q^2}\\right)\\right]\\nonumber\\\\\n&&+2{H_2}\\big[{M}^4-2{M}^3\n{M^{\\prime}}+2{M}^2\\left(6{M^{\\prime 2}}-{q^2}\\right)+2{M}\n{M^{\\prime}}\n\\left({q^2}-{M^{\\prime 2}}\\right)+\\left({M^{\\prime 2}}-{q^2}\\right)^2\n\\big]\\nonumber\\\\\n&&-{H_4}\\left[({M}-{M^{\\prime}})^2-{q^2}\\right]\n\\left({M}^2-{M}{M^{\\prime}}+{M^{\\prime 2}}-{q^2}\\right)\n\\left[({M}+{M^{\\prime}})^2-{q^2}\\right]\\big\\},\n\\\\\n{\\mathtt{f}_{4}^{\\prime}}(q^2)&=&\\frac{1}{2\\left[({M}-{M^{\\prime}})^2-{q^2}\\right]^2\n\\left[({M}+{M^{\\prime}})^2-{q^2}\\right]}\n\\big\\{{M^{\\prime}}\\left[{H_1}\n\\left({q^2}-({M}-{M^{\\prime}})^2\\right)+2{H_3}\n{M^{\\prime}}\\right]\\nonumber\\\\\n&&-2{H_2}\\left({M}^2-{M}\n{M^{\\prime}}+{M^{\\prime 2}}-{q^2}\\right)\n+{H_4}\\big[{M}^4-2\n{M}^2\n\\left({M^{\\prime 2}}+{q^2}\\right)+\\left({M^{\\prime 2}}-{q^2}\\right)^2\n\\big]\\big\\},\n\\end{eqnarray}\nwhere $H_i$ is defined as follows,\n\\begin{eqnarray}\n\tH_{i} & = & \\int\\{d^{3}p_{2}\\}\\frac{\\phi^{\\prime}(x^{\\prime},k_{\\perp}^{\\prime})\\phi(x,k_{\\perp})}{2\\sqrt{p_{1}^{+}p_{1}^{\\prime+}(p_{1}\\cdot\\bar{P}+m_{1}M_{0})(p_{1}^{\\prime}\\cdot\\bar{P}^{\\prime}+m_{1}^{\\prime}M_{0}^{\\prime})}}\\nonumber \\\\\n\t&  & \\quad\\times\\sum_{S_{z}^{\\prime}\\lambda_{2}}{\\rm Tr}\\Big\\{ u_{\\beta}(\\bar{P}^{\\prime},S_{z}^{\\prime})\\bar{u}_{\\alpha}(\\bar{P}^{\\prime},S_{z}^{\\prime})\\bar{\\Gamma}^{\\prime\\alpha}_{A}(\\slashed p_{1}^{\\prime}+m_{1}^{\\prime})\\gamma_{\\mu}(\\slashed p_{1}+m_{1})\\Gamma_{A}(\\bar{\\slashed P}+M_{0})(\\bar{\\Gamma}_{5}^{\\mu\\beta})_{i}\\Big\\}.\\label{eq:Fi_L}\n\\end{eqnarray}\nWith the same method, one can obtain the form\nfactors $\\mathtt{g}_{1,2,3,4}^{\\prime}$. $\\mathtt{g}_{1}^{\\prime}$ and $\\mathtt{g}_{3}^{\\prime}$ are similar to $\\mathtt{f}_{1}^{\\prime}$ and $\\mathtt{f}_{3}^{\\prime}$ respectively except for $M^{\\prime}\\to -M^{\\prime}$ and $H_{i}\\to K_{i}$, and $\\mathtt{g}_{2}^{\\prime}$ and $\\mathtt{g}_{4}^{\\prime}$ are similar to $\\mathtt{f}_{2}^{\\prime}$ and $\\mathtt{f}_{4}^{\\prime}$ respectively except for $M^{\\prime}\\to -M^{\\prime}$ and $H_{i}\\to -K_{i}$. For example, we have\n\\begin{eqnarray}\n{\\mathtt{g}_{1}^{\\prime}}(q^2)&=&-\\frac{{M}{M^{\\prime}}}{2\\left[{M}^4-2{M}^2\n\\left({M^{\\prime 2}}+{q^2}\\right)+\\left({M^{\\prime 2}}-{q^2}\\right)^2\n\\right]^2}\n\\Big\\{4{M^{\\prime}}\\left[{K_1}\n\\left(({M}+{M^{\\prime}})^2-{q^2}\\right)-{K_3}\n{M^{\\prime}}\\right]\\nonumber\\\\\n&&-2{K_2}\\left({M}^2+4{M}\n{M^{\\prime}}+{M^{\\prime 2}}-{q^2}\\right)\n+{K_4}\\left[{M}^4-2\n{M}^2\n\\left({M^{\\prime 2}}+{q^2}\\right)+\\left({M^{\\prime 2}}-{q^2}\\right)^2\n\\right]\\Big\\},\n\\end{eqnarray}\nwhere $K_i$ is defined by the following Eq.~(\\ref{eq:Gi_1}),\n\\begin{eqnarray}\nK_{i} & = & \\int\\{d^{3}p_{2}\\}\\frac{\\phi^{\\prime}(x^{\\prime},k_{\\perp}^{\\prime})\\phi(x,k_{\\perp})}{2\\sqrt{p_{1}^{+}p_{1}^{\\prime+}(p_{1}\\cdot\\bar{P}+m_{1}M_{0})(p_{1}^{\\prime}\\cdot\\bar{P}^{\\prime}+m_{1}^{\\prime}M_{0}^{\\prime})}}\\nonumber \\\\\n&  & \\times\\sum_{S_{z}^{\\prime}\\lambda_{2}}{\\rm Tr}\\Big\\{ u_{\\beta}(\\bar{P}^{\\prime},S_{z}^{\\prime})\\bar{u}_{\\alpha}(\\bar{P}^{\\prime},S_{z}^{\\prime})\\bar{\\Gamma}^{\\prime\\alpha}_{A}(\\slashed p_{1}^{\\prime}+m_{1}^{\\prime})\\gamma_{\\mu}\\gamma_{5}(\\slashed p_{1}+m_{1})\\Gamma_{A}(\\bar{\\slashed P}+M_{0})(\\bar{\\Gamma}^{\\mu\\beta})_{i}\\Big\\},\\label{eq:Gi_1}\n\\end{eqnarray}\nwith $(\\bar{\\Gamma}^{\\mu\\beta})_{i}=\\{\\gamma^{\\mu}\\bar{P}^{\\beta},\\bar{P}^{\\prime\\mu}\\bar{P}^{\\beta},\\bar{P}^{\\mu}\\bar{P}^{\\beta},g^{\\mu\\beta}\\}$.\nNote that $\\mathtt{f}_{1,2,3,4}^{\\prime T}$ and $\\mathtt{g}_{1,2,3,4}^{\\prime T}$ should not be independent. Multiplying the Eq.~(\\ref{eq:matrix_element_32t}) by $q^{\\mu}$ leads to\n\\begin{eqnarray}\n\\bar{u}_{\\alpha}(P^{\\prime},S_{z}^{\\prime})\\Big[(-M-M^{\\prime})P^{\\alpha}\\frac{\\mathtt{f}_{1}^{\\prime T}(q^{2})}{M}+\\frac{\\mathtt{f}_{2}^{\\prime T}(q^{2})}{M^{2}}P^{\\alpha}P\\cdot q +\\frac{\\mathtt{f}_{3}^{\\prime T}(q^{2})}{MM^{\\prime}}P^{\\alpha}P^{\\prime}\\cdot q+\\mathtt{f}_{4}^{\\prime T}(q^{2})q^{\\alpha}\\Big]\\gamma_{5}u(P,S_{z})& = &0 ,\\nonumber \\\\\n \\bar{u}_{\\alpha}(P^{\\prime},S_{z}^{\\prime})\\Big[(M-M^{\\prime})P^{\\alpha}\\frac{\\mathtt{g}_{1}^{\\prime T}(q^{2})}{M}+\\frac{\\mathtt{g}_{2}^{\\prime T}(q^{2})}{M^{2}}P^{\\alpha}P\\cdot q +\\frac{\\mathtt{g}_{3}^{\\prime T}(q^{2})}{MM^{\\prime}}P^{\\alpha}P^{\\prime}\\cdot q+\\mathtt{g}_{4}^{\\prime T}(q^{2})q^{\\alpha}\\Big]u(P,S_{z})&  =&0,\\label{eq:matrix_element_32t1}\n\\end{eqnarray}\nand the following two relations can be arrived,\n\\begin{equation}\n\\frac{P^{\\alpha}}{M}\\mathtt{f}_{1}^{\\prime T}(q^2)=\\frac{1}{(M+M^{\\prime})}\\Big[\\frac{\\mathtt{f}_{2}^{\\prime T}(q^{2})}{M^{2}}P^{\\alpha}P\\cdot q +\\frac{\\mathtt{f}_{3}^{\\prime T}(q^{2})}{MM^{\\prime}}P^{\\alpha}P^{\\prime}\\cdot q+\\mathtt{f}_{4}^{\\prime T}(q^{2})q^{\\alpha}\\Big],\\label{eq:f1pT}\n\\end{equation}\n\\begin{equation}\n\\frac{P^{\\alpha}}{M}\\mathtt{g}_{1}^{\\prime T}(q^2)=-\\frac{1}{(M-M^{\\prime})}\\Big[\\frac{\\mathtt{g}_{2}^{\\prime T}(q^{2})}{M^{2}}P^{\\alpha}P\\cdot q +\\frac{\\mathtt{f}_{3}^{\\prime T}(q^{2})}{MM^{\\prime}}P^{\\alpha}P^{\\prime}\\cdot q+\\mathtt{g}_{4}^{\\prime T}(q^{2})q^{\\alpha}\\Big].\\label{eq:g1pT}\n\\end{equation}\nWith the above relations, $\\langle{\\cal B}^{\\prime}(P^{\\prime},S_{z}^{\\prime})|\\bar{q}_{1}i\\sigma_{\\mu\\nu}\\frac{q^{\\nu}}{M}(1+\\gamma_{5})Q_{1}|{\\cal B}(P,S_{z})\\rangle$\ncan be parameterized with the form factors $\\mathtt{f}_{2,3,4}^{\\prime T}$ and $\\mathtt{g}_{2,3,4}^{\\prime T}$.\nThese form factors $\\mathtt{f}_{2,3,4}^{\\prime T}$ and $\\mathtt{g}_{2,3,4}^{\\prime T}$ can be extracted in the same way as we have conducted on the form factors $\\mathtt{f}_{1,2,3,4}^{\\prime}$ and $\\mathtt{g}_{1,2,3,4}^{\\prime}$~\\cite{Ke:2017eqo}.\nMultiply $\\bar{u}(P,S_{z})(\\bar{\\Gamma}_{5}^{\\mu\\beta})_{i}u_{\\beta}(P^{\\prime},S_{z}^{\\prime})$\nand $\\bar{u}(P,S_{z})(\\Gamma_{5}^{\\mu\\beta})_{i}u_{\\beta}(P^{\\prime},S_{z}^{\\prime})$\non the ``$\\langle{\\cal B}^{\\prime}_{f}(P^{\\prime},S^{\\prime}=\\frac{1}{2},S_{z}^{\\prime})\n|\\bar{q}_{1}i\\sigma_{\\mu\\nu}\\frac{q^{\\nu}}{M}Q_{1}|{\\cal B}_{i}(P,S=\\frac{1}{2},S_{z})\\rangle$\"\npart of Eq.~(\\ref{eq:matrix_element_onehalf_tensor}) and Eq.~(\\ref{eq:matrix_element_32t}), respectively.\nAt the same time, the approximation $P^{(\\prime)}\\to\\bar{P}^{(\\prime)}$ need to be taken for the integral.\nAfter summing the polarizations of the initial and final baryon states up, we can get the three equations as follows,\n\\begin{eqnarray}\n&&\\int\\{d^{3}p_{2}\\}\\frac{\\phi^{\\prime}(x^{\\prime},k_{\\perp}^{\\prime})\\phi(x,k_{\\perp})}{2\\sqrt{p_{1}^{+}p_{1}^{\\prime+}(p_{1}\\cdot\\bar{P}+m_{1}M_{0})(p_{1}^{\\prime}\\cdot\\bar{P}^{\\prime}+m_{1}^{\\prime}M_{0}^{\\prime})}}\\nonumber \\\\\n&&\\qquad\\times\\sum_{S_{z}^{\\prime}\\lambda_{2}}{\\rm Tr}\\Big\\{ u_{\\beta}(\\bar{P}^{\\prime},S_{z}^{\\prime})\\bar{u}_{\\alpha}(\\bar{P}^{\\prime},S_{z}^{\\prime})\\bar{\\Gamma}^{\\prime\\alpha}_{A}(\\slashed p_{1}^{\\prime}+m_{1}^{\\prime})i\\sigma_{\\mu\\nu}\\frac{q^{\\nu}}{M}(\\slashed p_{1}+m_{1})\\Gamma_{A}(\\bar{\\slashed P}+M_{0})(\\bar{\\Gamma}_{5}^{\\mu\\beta})_{i}\\Big\\}\\nonumber\\\\\n&&= {\\rm Tr}\\Big\\{ u_{\\beta}(P^{\\prime},S_{z}^{\\prime})\\bar{u}_{\\alpha}(P^{\\prime},S_{z}^{\\prime})\n\\Big[\\frac{\\gamma^{\\mu}}{M+M^{\\prime}}\\Big(\\frac{\\mathtt{f}_{2,A}^{\\prime T}(q^{2})}{M^{2}}P^{\\alpha}P\\cdot q\n+\\frac{\\mathtt{f}_{3,A}^{\\prime T}(q^{2})}{MM^{\\prime}}P^{\\alpha}P^{\\prime}\\cdot q\n+\\mathtt{f}_{4,A}^{\\prime T}(q^{2})q^{\\alpha}\\Big)\\nonumber\\\\\n&&\\qquad\\qquad+\\frac{\\mathtt{f}_{2,A}^{\\prime T}(q^{2})}{M^{2}}P^{\\alpha}P^{\\mu}\n+\\frac{\\mathtt{f}_{3,A}^{\\prime T}(q^{2})}{MM^{\\prime}}P^{\\alpha}P^{\\prime\\mu}\n+\\mathtt{f}_{4,A}^{\\prime T}(q^{2})g^{\\alpha\\mu}\\Big]\n\\gamma_{5} (\\slashed P+M)(\\Gamma_{5}^{\\mu\\beta})_{i}\\Big\\}\n,\\label{eq:FiT_solving}\n\\end{eqnarray}\nwith $(\\bar{\\Gamma}_{5}^{\\mu\\beta})_{i}=\\{\\gamma^{\\mu}\\bar{P}^{\\beta},\\bar{P}^{\\prime\\mu}\\bar{P}^{\\beta},g^{\\mu\\beta}\\}\\gamma_{5}$ and $(\\Gamma_{5}^{\\mu\\beta})_{i}=\\{\\gamma^{\\mu}P^{\\beta},P^{\\prime\\mu}P^{\\beta},g^{\\mu\\beta}\\}\\gamma_{5}$.\n\nThen the expressions of form factors $\\mathtt{f}_{2,3,4}^{\\prime T}$\ncould be got by solving the above three equations and are shown with the following formulas:\n\\begin{eqnarray}\n{\\mathtt{f}_{2}^{\\prime T}}(q^2)&=&\\frac{{M}^2{M^{\\prime 2}}\n\\left\\{\\left[({M}-{M^{\\prime}})^2-{q^2}\\right]\\left[2{H^T_1}\n{M^{\\prime}}+{H^T_3}\n\\left(({M}+{M^{\\prime}})^2-{q^2}\\right)\\right]-4{H^T_2}\n\\left(2{M}^2+{M}{M^{\\prime}}-3{M^{\\prime 2}}-2\n{q^2}\\right)\\right\\}}{\\left[({M}-{M^{\\prime}})^2-{q^2}\\right]^3\\left[({M}+{M^{\\prime}})^2-{q^2}\\right]^2},\n\\\\\n{\\mathtt{f}_{3}^{\\prime T}}(q^2)&=&\\frac{{M}{M^{\\prime}}}{\\left[({M}-\n{M^{\\prime}})^2-{q^2}\\right]^3\n\\left[({M}+{M^{\\prime}})^2-{q^2}\\right]^2}\\times\\\\\n&&\\Big\\{\\left[({M}-{M^{\\prime}})^2-{q^2}\\right]\\left[{H^T_1}{M^{\\prime}}\n\\left({M}^2-4{M}\n{M^{\\prime}}+{M^{\\prime 2}}-{q^2}\\right)-{H^T_3}\\left({M}^2-{M}\n{M^{\\prime}}+{M^{\\prime 2}}-{q^2}\\right)\n\\left(({M}+{M^{\\prime}})^2-{q^2}\\right)\\right]\\nonumber\\\\\n&&2{H^T_2}\n\\left[{M}^4-2{M}^3{M^{\\prime}}+{M}^2\\left(8\n{M^{\\prime 2}}-2{q^2}\\right)+2{M}{M^{\\prime}}\\left({q^2}-2\n{M^{\\prime 2}}\\right)-3{M^{\\prime}}^4+2{M^{\\prime 2}}\n{q^2}+{q^4}\\right]\\Big\\},\\nonumber\n\\\\\n{\\mathtt{f}_{4}^{\\prime T}}(q^2)&=&\\frac{\\left[({M}-{M^{\\prime}})^2-{q^2}\\right]\n\\left[{H^T_3}\n\\left(({M}+{M^{\\prime}})^2-{q^2}\\right)-{H^T_1}\n{M^{\\prime}}\\right]+2{H^T_2}[{M}\n({M^{\\prime}}-{M})+{q^2}]}{2\n\\left[({M}-{M^{\\prime}})^2-{q^2}\\right]^2\n\\left[({M}+{M^{\\prime}})^2-{q^2}\\right]},\n\\end{eqnarray}\nwhere $H_i^{T}$ is defined in Eq.~(\\ref{eq:FiT_L}),\n\\begin{eqnarray}\nH_{i}^{T} & = & \\int\\{d^{3}p_{2}\\}\\frac{\\phi^{\\prime}(x^{\\prime},k_{\\perp}^{\\prime})\\phi(x,k_{\\perp})}{2\\sqrt{p_{1}^{+}p_{1}^{\\prime+}(p_{1}\\cdot\\bar{P}+m_{1}M_{0})(p_{1}^{\\prime}\\cdot\\bar{P}^{\\prime}+m_{1}^{\\prime}M_{0}^{\\prime})}}\\nonumber \\\\\n&  & \\times\\sum_{S_{z}^{\\prime}\\lambda_{2}}{\\rm Tr}\\Big\\{ u_{\\beta}(\\bar{P}^{\\prime},S_{z}^{\\prime})\\bar{u}_{\\alpha}(\\bar{P}^{\\prime},S_{z}^{\\prime})\\bar{\\Gamma}^{\\prime\\alpha}_{A}(\\slashed p_{1}^{\\prime}+m_{1}^{\\prime})i\\sigma_{\\mu\\nu}\\frac{q^{\\nu}}{M}(\\slashed p_{1}+m_{1})\\Gamma_{A}(\\bar{\\slashed P}+M_{0})(\\bar{\\Gamma}_{5}^{\\mu\\beta})_{i}\\Big\\}.\\label{eq:FiT_L}\n\\end{eqnarray}\nWith the same method, one can obtain the form\nfactors $g_{2,3,4}^{\\prime T}$. $\\mathtt{g}_{2}^{\\prime T}$ and $\\mathtt{g}_{4}^{\\prime T}$ are similar to $\\mathtt{f}_{2}^{\\prime T}$ and $\\mathtt{f}_{4}^{\\prime T}$ respectively except for $M^{\\prime}\\to -M^{\\prime}$ and $H_{i}^{T}\\to -K_{i}^{T}$. $\\mathtt{g}_{3}^{\\prime T}$ is similar to $\\mathtt{f}_{3}^{\\prime T}$ except for $M^{\\prime}\\to -M^{\\prime}$ and $H_{i}^{T}\\to K_{i}^{T}$. For example, we have\n\\begin{eqnarray}\n{\\mathtt{g}_{2}^{\\prime T}}(q^2)&=&-\\frac{{M}^2{M^{\\prime 2}}\n}{\\left[({M}-{M^{\\prime}})^2-{q^2}\\right]^2\\left[({M}+{M^{\\prime}})^2-{q^2}\\right]^3}\\times\n\\nonumber\\\\\n&&\\left\\{\\left[({M}+{M^{\\prime}})^2-{q^2}\\right]\\left[{K^T_3}\n\\left(({M}-{M^{\\prime}})^2-{q^2}\\right)-2{K^T_1}\n{M^{\\prime}}\\right]+4{K^T_2}\\left(-2{M}^2+{M}{M^{\\prime}}+3\n{M^{\\prime 2}}+2{q^2}\\right)\\right\\},\n\\end{eqnarray}\nwhere $K_i^{T}$ is defined by the following Eq.~(\\ref{eq:GiT_L}),\n\\begin{eqnarray}\nK_{i}^{T} & = & \\int\\{d^{3}p_{2}\\}\\frac{\\phi^{\\prime}(x^{\\prime},k_{\\perp}^{\\prime})\\phi(x,k_{\\perp})}{2\\sqrt{p_{1}^{+}p_{1}^{\\prime+}(p_{1}\\cdot\\bar{P}+m_{1}M_{0})(p_{1}^{\\prime}\\cdot\\bar{P}^{\\prime}+m_{1}^{\\prime}M_{0}^{\\prime})}}\\nonumber \\\\\n&  & \\times\\sum_{S_{z}^{\\prime}\\lambda_{2}}{\\rm Tr}\\Big\\{ u_{\\beta}(\\bar{P}^{\\prime},S_{z}^{\\prime})\\bar{u}_{\\alpha}(\\bar{P}^{\\prime},S_{z}^{\\prime})\\bar{\\Gamma}^{\\prime\\alpha}_{A}(\\slashed p_{1}^{\\prime}+m_{1}^{\\prime})i\\sigma_{\\mu\\nu}\\frac{q^{\\nu}}{M}\\gamma_5(\\slashed p_{1}+m_{1})\\Gamma_{A}(\\bar{\\slashed P}+M_{0})(\\bar{\\Gamma}^{\\mu\\beta})_{i}\\Big\\},\\label{eq:GiT_L}\n\\end{eqnarray}\nwith $(\\bar{\\Gamma}^{\\mu\\beta})_{i}=\\{\\gamma^{\\mu}\\bar{P}^{\\beta},\\bar{P}^{\\prime\\mu}\\bar{P}^{\\beta},g^{\\mu\\beta}\\}$.\n\n\\subsection{Flavor-spin wave functions}\n\\label{sec:flavorspin}\nIn the above subsection, we have presented the explicit expressions of form factors. However, a physical form factor should be a linear combination of the transition form factors with a scalar and an axial-vector diquark spectator.\n\\begin{equation}\n{\\rm [form~factor]}^{\\rm physical}(q^2)=\n c_{S}\\times {\\rm [form~factor]}_{S}^{\\rm in~Subsec.A}+c_{A}\\times {\\rm [form~factor]}_{A}^{\\rm in~Subsec.A},\\label{eq:physical_ff}\n\\end{equation}\nand here $c_{S}$ and $c_{A}$ are the overlapping factors which are derived from the flavor spin wave functions of the initial and final baryon states with $S$ and $A$ corresponding to  the scalar and the axial vector diquark spectator of these doubly heavy baryons decays. The  hadronic  matrix elements can be written as\n\\begin{equation}\n\\langle B^{\\prime}|\\Gamma_{\\mu}|B\\rangle=c_{S}\\langle q_{1}[Q_{2}q]_{S}|\\Gamma_{\\mu}|Q_{1}[Q_{2}q]_{S}\\rangle+c_{A}\\langle q_{1}\\{Q_{2}q\\}_{A}|\\Gamma_{\\mu}|Q_{1}\\{Q_{2}q\\}_{A}\\rangle,\\label{eq:csca}\n\\end{equation}\nand the form factors $f_{i,S}$  and $f_{i,A}$ extracted from  Eq.~(\\ref{eq:solve_fi}) are involved with the transition matrix elements $\\langle q_{1}[Q_{2}q]_{S}|\\Gamma_{\\mu}|Q_{1}[Q_{2}q]_{S}\\rangle$\nand $\\langle q_{1}\\{Q_{2}q\\}_{A}|\\Gamma_{\\mu}|Q_{1}\\{Q_{2}q\\}_{A}\\rangle$, respectively.\nHere the current $\\Gamma_{\\mu}$ is  $\\Gamma_{\\mu}=\\bar q_1\\gamma_{\\mu}(1-\\gamma_5)Q_1$ or $\\bar q_1\\sigma_{\\mu\\nu}\\frac{q^{\\nu}}{M}(1+\\gamma_5)Q_1$.\nEqs.~(\\ref{eq:physical_ff}) and (\\ref{eq:csca}) are for the  $1\/2\\to1\/2$ transitions. For the transition $1\/2\\to 3\/2$, the diquark in the final state baryons can not be a scalar state, so the transition matrix element $\\langle q_{1}[Q_{2}q]_{S}|\\Gamma_{\\mu}|Q_{1}[Q_{2}q]_{S}\\rangle$ is zero and the physical form factor is given as\n\\begin{equation}\n{\\rm [form~factor]}^{\\rm physical}(q^2)=c_{A}\\times {\\rm [form~factor]}_{A}^{\\rm in~Subsec.A}.\\label{eq:physical_ff3}\n\\end{equation}\nIn this subsection, via performing the inner\nproduct of the flavor-spin wave functions of the initial and final states, the overlapping factors $c_{S}$ and $c_{A}$\nin Eqs.~(\\ref{eq:physical_ff}) and (\\ref{eq:physical_ff3}) can be calculated easily.\nFor shortage of the paper, the detail calculation of the wave functions for\nthe initial and final baryons is arranged in the Appendix~\\ref{app:wave_functions}.\nThe flavor spin wave functions for the doubly charmed SU(3) triplets $\\Xi_{cc}^{++}$, $\\Xi_{cc}^{+}$ and $\\Omega_{cc}^{+}$ are\\begin{align}\n\\mathcal{B}_{cc}=\\frac{1}{\\sqrt{2}}\\Big[\\Big(-\\frac{\\sqrt{3}}{2}c^{1}(c^{2}q)_{S}+\\frac{1}{2}c^{1}(c^{2}q)_{A}\\Big)+(c^{1}\\leftrightarrow c^{2})\\Big],\n\\end{align}\nhere the two charm quarks noted by $c^{1}$ and $c^{2}$ are symmetric.\nThe flavor spin wave functions of the doubly bottomed SU(3) triplets $\\Xi_{bb}^{0}$, $\\Xi_{bb}^{-}$ and $\\Omega_{bb}^{-}$\ncan be obtained through the replacement $c\\to b$. While the bottom-charm baryons could form two sets of SU(3) triplets, ($\\Xi_{bc},\\Omega_{bc}$) and ($\\Xi_{bc}^{\\prime},\\Omega_{bc}^{\\prime}$).\nThe flavor spin wave functions of bottom-charm baryons ($\\Xi_{bc},\\Omega_{bc}$) can be given as\n\\begin{align}\n\\mathcal{B}_{bc} & =  -\\frac{\\sqrt{3}}{2}b(cq)_{S}+\\frac{1}{2}b(cq)_{A}  =  -\\frac{\\sqrt{3}}{2}c(bq)_{S}+\\frac{1}{2}c(bq)_{A},\n,\\quad q=u,~d,~s,\n\\end{align}\nwhile the flavor spin wave functions of bottom-charm baryons ($\\Xi_{bc}^{\\prime},\\Omega_{bc}^{\\prime}$) are given as\n\\begin{align}\n\\mathcal{B}_{bc}^{\\prime} & =  -\\frac{1}{2}b(cq)_{S}-\\frac{\\sqrt{3}}{2}b(cq)_{A}  = \\frac{1}{2}c(bq)_{S}+\\frac{\\sqrt{3}}{2}c(bq)_{A},\\quad q=u,~d,~s.\n\\end{align}\nThe flavor-spin wave functions of the anti-triplet singly charmed baryons can be shown as follows,\n\\begin{align}\n\\Lambda_{c}^{+} & = -\\frac{1}{2}d(cu)_{S}+\\frac{\\sqrt{3}}{2}d(cu)_{A}  = \\frac{1}{2}u(cd)_{S}-\\frac{\\sqrt{3}}{2}u(cd)_{A},\\nonumber \\\\\n\\Xi_{c}^{+} & =  -\\frac{1}{2}s(cu)_{S}+\\frac{\\sqrt{3}}{2}s(cu)_{A}   = \\frac{1}{2}u(cs)_{S}-\\frac{\\sqrt{3}}{2}u(cs)_{A},\\nonumber\\\\\n\\Xi_{c}^{0} & =  -\\frac{1}{2}s(cd)_{S}+\\frac{\\sqrt{3}}{2}s(cd)_{A}  =  \\frac{1}{2}d(cs)_{S}-\\frac{\\sqrt{3}}{2}d(cs)_{A}.\\label{eq:flavor_spin_anti-triplet}\n\\end{align}\nwhile the flavor spin wave functions of the sextet of singly charmed baryons are demonstrated as\n\\begin{align}\n\\Sigma_{c}^{++} & =\\frac{1}{\\sqrt{2}}\\Big[\\frac{\\sqrt{3}}{2}u^{1}(cu^{2})_{S}+\\frac{1}{2}u^{1}(cu^{2})_{A}+(u^{1}\\leftrightarrow u^{2})\\Big],\\nonumber \\\\\n\\Sigma_{c}^{+} & =  \\frac{\\sqrt{3}}{2}d(cu)_{S}+\\frac{1}{2}d(cu)_{A}  =   \\frac{\\sqrt{3}}{2}u(cd)_{S}+\\frac{1}{2}u(cd)_{A},\\nonumber\\\\\n\\Sigma_{c}^{0} & =\\frac{1}{\\sqrt{2}}\\Big[\\frac{\\sqrt{3}}{2}d^{1}(cd^{2})_{S}+\\frac{1}{2}d^{1}(cd^{2})_{A}+(d^{1}\\leftrightarrow d^{2})\\Big],\\nonumber \\\\\n\\Xi_{c}^{\\prime+} & = \\frac{\\sqrt{3}}{2}s(cu)_{S}+\\frac{1}{2}s(cu)_{A}  = \\frac{\\sqrt{3}}{2}u(cs)_{S}+\\frac{1}{2}u(cs)_{A},\\nonumber \\\\\n\\Xi_{c}^{\\prime0} & = \\frac{\\sqrt{3}}{2}s(cd)_{S}+\\frac{1}{2}s(cd)_{A}   = \\frac{\\sqrt{3}}{2}d(cs)_{S}+\\frac{1}{2}d(cs)_{A},\\nonumber \\\\\n\\Omega_{c}^{0} & =\\frac{1}{\\sqrt{2}}\\Big[\\frac{\\sqrt{3}}{2}s^{2}(cs^{1})_{S}+\\frac{1}{2}s^{2}(cs^{1})_{A}+(s^{1}\\leftrightarrow s^{2})\\Big].\\label{eq:flavor_spin_sextet}\n\\end{align}\nThen we can get the wave functions of the singly bottom baryons by changing $c$ in Eqs.~(\\ref{eq:flavor_spin_anti-triplet})-(\\ref{eq:flavor_spin_sextet}) to $b$.\nWhile for the baryons ${\\cal B}^{*}$ with spin-$3\/2$ in the final states,\ntheir flavor spin wave function are given as follows,\n\\begin{eqnarray}\n&&{\\cal B}_{Qqq^{\\prime}}^{*}=q(Qq^{\\prime})_{A}=q^{\\prime}(Qq)_{A},~{\\cal B}_{Qqq}^{*}=\\sqrt{2}q(Qq)_{A},\\\\\n&&{\\cal B}_{QQ^{\\prime}q}^{*}=Q(Q^{\\prime}q)_{A}=Q^{\\prime}(Qq)_{A},~{\\cal B}_{QQq}^{*}=\\sqrt{2}Q(Qq)_{A},\n\\end{eqnarray}\nwith $q^{(\\prime)}=u,d,s$, and $Q^{(\\prime)}=c,b$.\n\nWith the above wave functions of doubly heavy baryons and singly heavy baryons,\nthe overlapping factors $c_{S,A}$ for each transition can be got.\nThe corresponding results of the overlapping factors $c_{S,A}$ for\nthe $1\/2\\to1\/2$ transitions induced by the charged current\nand the FCNC in Eq.~(\\ref{eq:csca})\nare collected in Tab. \\ref{Tab:overlapping_factors_22}.\nFor the $1\/2\\to3\/2$ transitions induced by the charged current and the FCNC,\nthe numerical results of the overlapping factors $c_{A}$ are listed in Tab.~\\ref{Tab:overlapping_factors_23}.\nUnder SU(3) symmetry the doubly heavy baryons can be formed into triplets and the singly heavy baryons can  be formed into an anti-triplet and a sextet. The overlapping factors $c_{S,A}$ can be calculated with SU(3) approach, and the detail calculation can be found in the Appendix~\\ref{su3approach}. Using the SU(3) approach, one gets the same numerical results of $c_{S(A)}$ as those listed in Tabs. \\ref{Tab:overlapping_factors_22} and \\ref{Tab:overlapping_factors_23}.\nThen for a spin $1\/2$ finial state with a scalar and an axial-vector diquark, the physical form factors are then obtained by\n\\begin{equation}\n\tf^{\\frac{1}{2}\\to\\frac{1}{2}}_{i}=c_{S}f_{i,S}+c_{A}f_{i,A},\\quad\n\tg^{\\frac{1}{2}\\to\\frac{1}{2}}_{i}=c_{S}g_{i,S}+c_{A}g_{i,A},\\quad f^{\\frac{1}{2}\\to\\frac{1}{2},T}_{i}=c_{S}f_{i,S}^{T}+c_{A}f_{i,A}^{T},\\quad g^{\\frac{1}{2}\\to\\frac{1}{2},T}_{i}=c_{S}g_{i,S}^{T}+c_{A}g_{i,A}^{T},\\label{eq:physical_ff22}\n\\end{equation}\nwhere these form factors $f_{i,S(A)}$, $g_{i,S(A)}$, $f_{i,S(A)}^{T}$\nor $g_{i,S(A)}^{T}$ are defined by Eqs.~(\\ref{eq:matrix_element_2}) and (\\ref{eq:matrix_element_2p}).\nHowever, for a spin $3\/2$ finial state with only an axial-vector diquark, the physical form factors are then obtained by\n\\begin{equation}\n\tf^{\\frac{1}{2}\\to\\frac{3}{2}}_{i}=c_{A}\\mathtt{f}_{i},\\quad\n\tg^{\\frac{1}{2}\\to\\frac{3}{2}}_{i}=c_{A}\\mathtt{g}_{i},\\quad f^{\\frac{1}{2}\\to\\frac{3}{2},T}_{i}=c_{A}\\mathtt{f}_{i}^{T},\\quad g^{\\frac{1}{2}\\to\\frac{3}{2},T}_{i}=c_{A}\\mathtt{g}_{i}^{T},\\label{eq:physical_ff23}\n\\end{equation}\nwhere these form factors $\\mathtt{f}_{i}$, $\\mathtt{g}_{i}$, ${\\mathtt{f}}_{i}^{T}$\nor $\\mathtt{g}_{i}^{T}$ are defined in Eqs.~(\\ref{eq:matrix_element_32nVA})-(\\ref{eq:matrix_element_32nT}).\n\n\n\n\n\\begin{table}\n\\caption{Numerical results of the overlapping factors for the $1\/2\\to1\/2$ transitions induced by $c\\to d,s$, $b\\to u,c$ and $c\\to u$, $b\\to d,s$. For example, the physical form factor of transition $\\Xi_{cc}^{++}\\to \\Lambda_{c}^{+}$, $f^{\\frac{1}{2}\\to\\frac{1}{2}}_{1}=c_{S}f_{1,S}+c_{A}f_{1,A}$ can be calculated with $c_{S}=\\sqrt{6}\/4$ and $c_{A}=\\sqrt{6}\/4$.}\n\\label{Tab:overlapping_factors_22}\n\\begin{tabular}{c|c|c|c|c|c|c|c|c}\n\\hline \\hline\ntransitions& $c_{S}$ & $c_{A}$\t&transitions& $c_{S}$ & $c_{A}$& transitions&$c_{S}$ & $c_{A}$\\tabularnewline\\hline\n$\\Xi_{cc}^{++}(ccu)\\to\\Lambda_{c}^{+}(dcu)$ & $\\frac{\\sqrt{6}}{4}$ & $\\frac{\\sqrt{6}}{4}$&$\\Xi_{bc}^{+}(cbu)\\to\\Lambda_{b}^{0}(dbu)$ & $\\frac{\\sqrt{3}}{4}$ & $\\frac{\\sqrt{3}}{4}$&$\\Xi_{bc}^{\\prime+}(cbu)\\to\\Lambda_{b}^{0}(dbu)$ & $-\\frac{1}{4}$ & $\\frac{3}{4}$\\tabularnewline\\hline\n$\\Xi_{cc}^{++}(ccu)\\to\\Sigma_{c}^{+}(dcu)$ & $-\\frac{3\\sqrt{2}}{4}$ & $\\frac{\\sqrt{2}}{4}$&$\\Xi_{bc}^{+}(cbu)\\to\\Sigma_{b}^{0}(dbu)$ & $-\\frac{3}{4}$ & $\\frac{1}{4}$&\t$\\Xi_{bc}^{\\prime+}(cbu)\\to\\Sigma_{b}^{0}(dbu)$ & $\\frac{\\sqrt{3}}{4}$ & $\\frac{\\sqrt{3}}{4}$\\tabularnewline\\hline\n$\\Xi_{cc}^{++}(ccu)\\to\\Xi_{c}^{+}(scu)$ & $\\frac{\\sqrt{6}}{4}$ & $\\frac{\\sqrt{6}}{4}$&$\\Xi_{bc}^{+}(cbu)\\to\\Xi_{b}^{0}(sbu)$ & $\\frac{\\sqrt{3}}{4}$ & $\\frac{\\sqrt{3}}{4}$&$\\Xi_{bc}^{\\prime+}(cbu)\\to\\Xi_{b}^{0}(sbu)$ & $-\\frac{1}{4}$ & $\\frac{3}{4}$\\tabularnewline\\hline\n$\\Xi_{cc}^{++}(ccu)\\to\\Xi_{c}^{\\prime+}(scu)$ & $-\\frac{3\\sqrt{2}}{4}$ & $\\frac{\\sqrt{2}}{4}$&\t$\\Xi_{bc}^{+}(cbu)\\to\\Xi_{b}^{\\prime0}(sbu)$ & $-\\frac{3}{4}$ & $\\frac{1}{4}$&$\\Xi_{bc}^{\\prime+}(cbu)\\to\\Xi_{b}^{\\prime0}(sbu)$ & $\\frac{\\sqrt{3}}{4}$ & $\\frac{\\sqrt{3}}{4}$\\tabularnewline\\hline\n$\\Xi_{cc}^{+}(ccd)\\to\\Sigma_{c}^{0}(dcd)$ & $-\\frac{3}{2}$ & $\\frac{1}{2}$&\t$\\Xi_{bc}^{0}(cbd)\\to\\Sigma_{b}^{-}(dbd)$ & $-\\frac{3\\sqrt{2}}{4}$ & $\\frac{\\sqrt{2}}{4}$&$\\Xi_{bc}^{\\prime0}(cbd)\\to\\Sigma_{b}^{-}(dbd)$ & $\\frac{\\sqrt{6}}{4}$ & $\\frac{\\sqrt{6}}{4}$\\tabularnewline\\hline\n$\\Xi_{cc}^{+}(ccd)\\to\\Xi_{c}^{0}(scd)$ & $\\frac{\\sqrt{6}}{4}$ & $\\frac{\\sqrt{6}}{4}$&$\\Xi_{bc}^{0}(cbd)\\to\\Xi_{b}^{-}(sbd)$ & $\\frac{\\sqrt{3}}{4}$ & $\\frac{\\sqrt{3}}{4}$&$\\Xi_{bc}^{\\prime0}(cbd)\\to\\Xi_{b}^{-}(sbd)$ & $-\\frac{1}{4}$ & $\\frac{3}{4}$\\tabularnewline\\hline\n$\\Xi_{cc}^{+}(ccd)\\to\\Xi_{c}^{\\prime0}(scd)$ & $-\\frac{3\\sqrt{2}}{4}$ & $\\frac{\\sqrt{2}}{4}$&$\\Xi_{bc}^{0}(cbd)\\to\\Xi_{b}^{\\prime-}(sbd)$ & $-\\frac{3}{4}$ & $\\frac{1}{4}$&$\\Xi_{bc}^{\\prime0}(cbd)\\to\\Xi_{b}^{\\prime-}(sbd)$ & $\\frac{\\sqrt{3}}{4}$ & $\\frac{\\sqrt{3}}{4}$\\tabularnewline\\hline\n$\\Omega_{cc}^{+}(ccs)\\to\\Xi_{c}^{0}(dcs)$ & $-\\frac{\\sqrt{6}}{4}$ & $-\\frac{\\sqrt{6}}{4}$&$\\Omega_{bc}^{0}(cbs)\\to\\Xi_{b}^{-}(dbs)$ & $-\\frac{\\sqrt{3}}{4}$ & $-\\frac{\\sqrt{3}}{4}$&\t$\\Omega_{bc}^{\\prime0}(cbs)\\to\\Xi_{b}^{-}(dbs)$ & $\\frac{1}{4}$ & $-\\frac{3}{4}$\\tabularnewline\\hline\n$\\Omega_{cc}^{+}(ccs)\\to\\Xi_{c}^{\\prime0}(dcs)$ & $-\\frac{3\\sqrt{2}}{4}$ & $\\frac{\\sqrt{2}}{4}$&$\\Omega_{bc}^{0}(cbs)\\to\\Xi_{b}^{\\prime-}(dbs)$ & $-\\frac{3}{4}$ & $\\frac{1}{4}$&$\\Omega_{bc}^{\\prime0}(cbs)\\to\\Xi_{b}^{\\prime-}(dbs)$ & $\\frac{\\sqrt{3}}{4}$ & $\\frac{\\sqrt{3}}{4}$\\tabularnewline\\hline\n$\\Omega_{cc}^{+}(ccs)\\to\\Omega_{c}^{0}(scs)$ & $-\\frac{3}{2}$ & $\\frac{1}{2}$&$\\Omega_{bc}^{0}(cbs)\\to\\Omega_{b}^{-}(sbs)$ & $-\\frac{3\\sqrt{2}}{4}$ & $\\frac{\\sqrt{2}}{4}$&\t$\\Omega_{bc}^{\\prime0}(cbs)\\to\\Omega_{b}^{-}(sbs)$ & $\\frac{\\sqrt{6}}{4}$ & $\\frac{\\sqrt{6}}{4}$\\tabularnewline\n\\hline \\hline\n$\\Xi_{bb}^{0}(bbu)\\to\\Sigma_{b}^{+}(ubu)$ & $-\\frac{3}{2}$ & $\\frac{1}{2}$&$\\Xi_{bc}^{+}(bcu)\\to\\Sigma_{c}^{++}(ucu)$ & $-\\frac{3\\sqrt{2}}{4}$ & $\\frac{\\sqrt{2}}{4}$&$\\Xi_{bc}^{\\prime+}(bcu)\\to\\Sigma_{c}^{++}(ucu)$ & $-\\frac{\\sqrt{6}}{4}$ & $-\\frac{\\sqrt{6}}{4}$\\tabularnewline\\hline\n$\\Xi_{bb}^{0}(bbu)\\to\\Xi_{bc}^{+}(cbu)$ & $\\frac{3\\sqrt{2}}{4}$ & $\\frac{\\sqrt{2}}{4}$&$\\Xi_{bc}^{+}(bcu)\\to\\Xi_{cc}^{++}(ccu)$ & $\\frac{3\\sqrt{2}}{4}$ & $\\frac{\\sqrt{2}}{4}$&$\\Xi_{bc}^{\\prime+}(bcu)\\to\\Xi_{cc}^{++}(ccu)$ & $\\frac{\\sqrt{6}}{4}$ & $-\\frac{\\sqrt{6}}{4}$\\tabularnewline\\hline\n$\\Xi_{bb}^{0}(bbu)\\to\\Xi_{bc}^{\\prime+}(cbu)$ & $-\\frac{\\sqrt{6}}{4}$ & $\\frac{\\sqrt{6}}{4}$&$\\Xi_{bc}^{0}(bcd)\\to\\Lambda_{c}^{+}(ucd)$ & $-\\frac{\\sqrt{3}}{4}$ & $-\\frac{\\sqrt{3}}{4}$&$\\Xi_{bc}^{\\prime0}(bcd)\\to\\Lambda_{c}^{+}(ucd)$ & $-\\frac{1}{4}$ & $\\frac{3}{4}$\\tabularnewline\\hline\n$\\Xi_{bb}^{-}(bbd)\\to\\Lambda_{b}^{0}(ubd)$ & $-\\frac{\\sqrt{6}}{4}$ & $-\\frac{\\sqrt{6}}{4}$&\t$\\Xi_{bc}^{0}(bcd)\\to\\Sigma_{c}^{+}(ucd)$ & $-\\frac{3}{4}$ & $\\frac{1}{4}$&$\\Xi_{bc}^{\\prime0}(bcd)\\to\\Sigma_{c}^{+}(ucd)$ & $-\\frac{\\sqrt{3}}{4}$ & $-\\frac{\\sqrt{3}}{4}$\\tabularnewline\\hline\n$\\Xi_{bb}^{-}(bbd)\\to\\Sigma_{b}^{0}(ubd)$ & $-\\frac{3\\sqrt{2}}{4}$ & $\\frac{\\sqrt{2}}{4}$&\t$\\Xi_{bc}^{0}(bcd)\\to\\Xi_{cc}^{+}(ccd)$ & $\\frac{3\\sqrt{2}}{4}$ & $\\frac{\\sqrt{2}}{4}$&$\\Xi_{bc}^{\\prime0}(bcd)\\to\\Xi_{cc}^{+}(ccd)$ & $\\frac{\\sqrt{6}}{4}$ & $-\\frac{\\sqrt{6}}{4}$\\tabularnewline\\hline\n$\\Xi_{bb}^{-}(bbd)\\to\\Xi_{bc}^{0}(cbd)$ & $\\frac{3\\sqrt{2}}{4}$ & $\\frac{\\sqrt{2}}{4}$&$\\Omega_{bc}^{0}(bcs)\\to\\Xi_{c}^{+}(ucs)$ & $-\\frac{\\sqrt{3}}{4}$ & $-\\frac{\\sqrt{3}}{4}$&$\\Omega_{bc}^{\\prime0}(bcs)\\to\\Xi_{c}^{+}(ucs)$ & $-\\frac{1}{4}$ & $\\frac{3}{4}$\\tabularnewline\\hline\n$\\Xi_{bb}^{-}(bbd)\\to\\Xi_{bc}^{\\prime0}(cbd)$ & $-\\frac{\\sqrt{6}}{4}$ & $\\frac{\\sqrt{6}}{4}$&$\\Omega_{bc}^{0}(bcs)\\to\\Xi_{c}^{\\prime+}(ucs)$ & $-\\frac{3}{4}$ & $\\frac{1}{4}$&$\\Omega_{bc}^{\\prime0}(bcs)\\to\\Xi_{c}^{\\prime+}(ucs)$ & $-\\frac{\\sqrt{3}}{4}$ & $-\\frac{\\sqrt{3}}{4}$\\tabularnewline\\hline\n$\\Omega_{bb}^{-}(bbs)\\to\\Xi_{b}^{0}(ubs)$ & $-\\frac{\\sqrt{6}}{4}$ & $-\\frac{\\sqrt{6}}{4}$&\t$\\Omega_{bc}^{0}(bcs)\\to\\Omega_{cc}^{+}(ccs)$ & $\\frac{3\\sqrt{2}}{4}$ & $\\frac{\\sqrt{2}}{4}$&$\\Omega_{bc}^{\\prime0}(bcs)\\to\\Omega_{cc}^{+}(ccs)$ & $\\frac{\\sqrt{6}}{4}$ & $-\\frac{\\sqrt{6}}{4}$\\tabularnewline\\hline\n$\\Omega_{bb}^{-}(bbs)\\to\\Xi_{b}^{\\prime0}(ubs)$ & $-\\frac{3\\sqrt{2}}{4}$ & $\\frac{\\sqrt{2}}{4}$&$\\Omega_{bb}^{-}(bbs)\\to\\Omega_{bc}^{0}(cbs)$ & $\\frac{3\\sqrt{2}}{4}$ & $\\frac{\\sqrt{2}}{4}$&$\\Omega_{bb}^{-}(bbs)\\to\\Omega_{bc}^{\\prime0}(cbs)$ & $-\\frac{\\sqrt{6}}{4}$ & $\\frac{\\sqrt{6}}{4}$\\tabularnewline\n\\hline\t\\hline\n$\\Xi_{cc}^{++}(ccu)\\to\\Sigma_{c}^{++}(ucu)$  & $-\\frac{3}{2}$  & $\\frac{1}{2}$  &$\\Xi_{bc}^{+}(cbu)\\to\\Sigma_{b}^{+}(ubu)$  & $-\\frac{3\\sqrt{2}}{4}$  & $\\frac{\\sqrt{2}}{4}$\n&$\\Xi_{bc}^{\\prime+}(cbu)\\to\\Sigma_{b}^{+}(ubu)$  & $\\frac{\\sqrt{6}}{4}$  &$\\frac{\\sqrt{6}}{4} $\\tabularnewline\n\\hline\n$\\Xi_{cc}^{+}(ccd)\\to\\Lambda_{c}^{+}(ucd)$  & $-\\frac{\\sqrt{6}}{4}$  & $-\\frac{\\sqrt{6}}{4}$  &$\\Xi_{bc}^{0}(cbd)\\to\\Lambda_{b}^{+}(ubd)$  & $-\\frac{\\sqrt{3}}{4}$  & $-\\frac{\\sqrt{3}}{4}$\n&$\\Xi_{bc}^{\\prime0}(cbd)\\to\\Lambda_{b}^{0}(ubd)$  & $\\frac{1}{4}$  &$-\\frac{3}{4} $\\tabularnewline\n\\hline\n$\\Xi_{cc}^{+}(ccd)\\to\\Sigma_{c}^{+}(ucd)$  & $-\\frac{3\\sqrt{2}}{4}$  & $\\frac{\\sqrt{2}}{4}$  &$\\Xi_{bc}^{0}(cbd)\\to\\Sigma_{b}^{0}(ubd)$  & $-\\frac{3}{4}$  & $\\frac{1}{4}$\n&$\\Xi_{bc}^{\\prime0}(cbd)\\to\\Sigma_{b}^{0}(ubd)$  & $\\frac{\\sqrt{3}}{4}$  &$\\frac{\\sqrt{3}}{4} $\\tabularnewline\n\\hline\n$\\Omega_{cc}^{+}(ccs)\\to\\Xi_{c}^{+}(ucs)$  & $-\\frac{\\sqrt{6}}{4}$  & $\\frac{\\sqrt{6}}{4}$  &$\\Omega_{bc}^{0}(cbs)\\to\\Xi_{b}^{0}(ubs)$  & $-\\frac{\\sqrt{3}}{4}$  & $-\\frac{\\sqrt{3}}{4}$\n&$\\Omega_{bc}^{\\prime0}(cbs)\\to\\Xi_{b}^{0}(ubs)$  & $\\frac{1}{4}$  &$-\\frac{3}{4} $\\tabularnewline\n\\hline\n$\\Omega_{cc}^{+}(ccs)\\to\\Xi_{c}^{\\prime+}(ucs)$  & $-\\frac{3\\sqrt{2}}{4}$  & $\\frac{\\sqrt{2}}{4}$  &$\\Omega_{bc}^{0}(cbs)\\to\\Xi_{b}^{\\prime0}(ubs)$  & $-\\frac{3}{4}$  & $\\frac{1}{4}$\n&$\\Omega_{bc}^{\\prime0}(cbs)\\to\\Xi_{b}^{\\prime0}(ubs)$  & $\\frac{\\sqrt{3}}{4}$  &$\\frac{\\sqrt{3}}{4} $\\tabularnewline\n\\hline\\hline\n$\\Xi_{bb}^{0}(bbu)\\to\\Xi_{b}^{0}(sbu)$  & $\\frac{\\sqrt{6}}{4}$  & $\\frac{\\sqrt{6}}{4}$  &$\\Xi_{bc}^{+}(bcu)\\to\\Xi_{c}^{+}(scu)$  & $\\frac{\\sqrt{3}}{4}$  & $\\frac{\\sqrt{3}}{4}$\n&$\\Xi_{bc}^{\\prime+}(bcu)\\to\\Xi_{c}^{+}(scu)$  & $\\frac{1}{4}$  &$-\\frac{3}{4} $\\tabularnewline\\hline\n$\\Xi_{bb}^{-}(bbd)\\to\\Xi_{b}^{-}(sbd)$ & $\\frac{\\sqrt{6}}{4}$  & $\\frac{\\sqrt{6}}{4}$& $\\Xi_{bc}^{0}(bcd)\\to\\Xi_{c}^{0}(scd)$  & $\\frac{\\sqrt{3}}{4}$  & $\\frac{\\sqrt{3}}{4}$\n&$\\Xi_{bc}^{\\prime0}(bcd)\\to\\Xi_{c}^{0}(scd)$  &  $\\frac{1}{4}$  & $-\\frac{3}{4} $\\tabularnewline\\hline\n$\\Xi_{bb}^{0}(bbu)\\to\\Xi_{b}^{\\prime0}(sbu)$  &  $-\\frac{3\\sqrt{2}}{4}$  & $\\frac{\\sqrt{2}}{4}$\n& $\\Xi_{bc}^{+}(bcu)\\to\\Xi_{c}^{\\prime+}(scu)$  & $-\\frac{3}{4}$& $\\frac{1}{4}$\n&$\\Xi_{bc}^{\\prime+}(bcu)\\to\\Xi_{c}^{\\prime+}(scu)$  & $-\\frac{\\sqrt{3}}{4}$ & $-\\frac{\\sqrt{3}}{4}$\\tabularnewline\\hline\n$\\Xi_{bb}^{-}(bbd)\\to\\Xi_{b}^{\\prime-}(sbd)$ & $-\\frac{3\\sqrt{2}}{4}$ & $\\frac{\\sqrt{2}}{4}$\n&$\\Xi_{bc}^{0}(bcd)\\to\\Xi_{c}^{\\prime0}(scd)$  &  $-\\frac{3}{4}$  & $\\frac{1}{4}$\n&$\\Xi_{bc}^{\\prime0}(bcd)\\to\\Xi_{c}^{\\prime0}(scd)$  &  $-\\frac{\\sqrt{3}}{4}$  & $-\\frac{\\sqrt{3}}{4}$\\tabularnewline\\hline\n$\\Omega_{bb}^{-}(bbs)\\to\\Omega_{b}^{-}(sbs)$  & $-\\frac{3}{2}$  & $\\frac{1}{2}$ & $\\Omega_{bc}^{0}(bcs)\\to\\Omega_{c}^{0}(scs)$  & $-\\frac{3\\sqrt{2}}{4}$  & $\\frac{\\sqrt{2}}{4}$ &$\\Omega_{bc}^{\\prime0}(bcs)\\to\\Omega_{c}^{0}(scs)$  & $-\\frac{\\sqrt{6}}{4}$& $-\\frac{\\sqrt{6}}{4}$ \\tabularnewline\\hline\n$\\Xi_{bb}^{0}(bbu)\\to\\Lambda_{b}^{0}(dbu)$  & $\\frac{\\sqrt{6}}{4}$  & $\\frac{\\sqrt{6}}{4}$\n & $\\Xi_{bc}^{+}(bcu)\\to\\Lambda_{c}^{+}(dcu)$  & $\\frac{\\sqrt{3}}{4}$  & $\\frac{\\sqrt{3}}{4}$& $\\Xi_{bc}^{\\prime+}(bcu)\\to\\Lambda_{c}^{+}(dcu)$  & $\\frac{1}{4}$  & $-\\frac{3}{4}$\\tabularnewline\\hline\n$\\Omega_{bb}^{-}(bbs)\\to\\Xi_{b}^{-}(dbs)$  & $-\\frac{\\sqrt{6}}{4}$  & $-\\frac{\\sqrt{6}}{4}$\n&$\\Omega_{bc}^{0}(bcs)\\to\\Xi_{c}^{0}(dcs)$  & $-\\frac{\\sqrt{3}}{4}$  & $-\\frac{\\sqrt{3}}{4}$ & $\\Omega_{bc}^{\\prime0}(bcs)\\to\\Xi_{c}^{0}(dcs)$  & $-\\frac{1}{4}$  & $\\frac{3}{4}$\\tabularnewline\\hline\n$\\Xi_{bb}^{0}(bbu)\\to\\Sigma_{b}^{0}(dbu)$  & $-\\frac{3\\sqrt{2}}{4}$  & $\\frac{\\sqrt{2}}{4}$&$\\Xi_{bc}^{+}(bcu)\\to\\Sigma_{c}^{+}(dcu)$  & $-\\frac{3}{4}$  & $\\frac{1}{4}$& $\\Xi_{bc}^{\\prime+}(bcu)\\to\\Sigma_{c}^{+}(dcu)$  & $-\\frac{\\sqrt{3}}{4}$  & $-\\frac{\\sqrt{3}}{4}$\\tabularnewline\\hline\n $\\Xi_{bb}^{-}(bbd)\\to\\Sigma_{b}^{-}(dbd)$  & $-\\frac{3}{2}$  & $\\frac{1}{2}$&$\\Xi_{bc}^{0}(bcd)\\to\\Sigma_{c}^{0}(dcd)$  & $-\\frac{3\\sqrt{2}}{4}$  & $\\frac{\\sqrt{2}}{4}$& $\\Xi_{bc}^{\\prime0}(bcd)\\to\\Sigma_{c}^{0}(dcd)$  & $-\\frac{\\sqrt{6}}{4}$  & $-\\frac{\\sqrt{6}}{4}$\\tabularnewline\\hline\n$\\Omega_{bb}^{-}(bbs)\\to\\Xi_{b}^{\\prime-}(dbs)$  & $-\\frac{3\\sqrt{2}}{4}$  & $\\frac{\\sqrt{2}}{4}$&$\\Omega_{bc}^{0}(bcs)\\to\\Xi_{c}^{\\prime0}(dcs)$  & $-\\frac{3}{4}$  & $\\frac{1}{4}$& $\\Omega_{bc}^{\\prime0}(bcs)\\to\\Xi_{c}^{\\prime0}(dcs)$  & $-\\frac{\\sqrt{3}}{4}$  & $-\\frac{\\sqrt{3}}{4}$\\tabularnewline\\hline\t\\hline\n\\end{tabular}\n\\end{table}\t\n\n\\begin{table}\n\\caption{Numerical results of the overlapping factors for the $1\/2\\to3\/2$ transitions induced by $c\\to d,s$, $b\\to u,c$ and $c\\to u$, $b\\to d,s$. For example, the physical form factor of transition $\\Xi_{cc}^{++}\\to \\Sigma_{c}^{*+}$, $f^{\\frac{1}{2}\\to\\frac{3}{2}}_{1}=c_{A}\\mathtt{f}_{1}$ can be calculated with $c_{A}=1\/\\sqrt{2}$.}\n\\label{Tab:overlapping_factors_23}\n\\begin{tabular}{c|c|c|c|c|c}\n\\hline\\hline\ntransitions  & $c_{A}$  & transitions  & $c_{A}$  & transitions  & $c_{A}$ \\tabularnewline\n\\hline\n$\\Xi_{cc}^{++}(ccu)\\to\\Sigma_{c}^{*+}(dcu)$  & $\\frac{1}{\\sqrt{2}}$  & $\\Xi_{bc}^{+}(cbu)\\to\\Sigma_{b}^{*0}(dbu)$  & $\\frac{1}{2}$  & $\\Xi_{bc}^{\\prime+}(cbu)\\to\\Sigma_{b}^{*0}(dbu)$  & $\\frac{\\sqrt{3}}{2}$ \\tabularnewline\n\\hline\n$\\Xi_{cc}^{+}(ccd)\\to\\Sigma_{c}^{*0}(dcd)$  & $1$  & $\\Xi_{bc}^{0}(cbd)\\to\\Sigma_{b}^{*-}(dbd)$  & $\\frac{\\sqrt{2}}{2}$  & $\\Xi_{bc}^{\\prime0}(cbd)\\to\\Sigma_{b}^{*-}(dbd)$  & $\\frac{\\sqrt{6}}{2}$ \\tabularnewline\n\\hline\n$\\Omega_{cc}^{+}(ccs)\\to\\Xi_{c}^{\\prime*0}(dcs)$  & $\\frac{1}{\\sqrt{2}}$  & $\\Omega_{bc}^{0}(cbs)\\to\\Xi_{b}^{\\prime*-}(dbs)$  & $\\frac{1}{2}$  & $\\Omega_{bc}^{\\prime0}(cbs)\\to\\Xi_{b}^{\\prime*-}(dbs)$  & $\\frac{\\sqrt{3}}{2}$ \\tabularnewline\n\\hline\n$\\Xi_{cc}^{++}(ccu)\\to\\Xi_{c}^{\\prime*+}(scu)$  & $\\frac{1}{\\sqrt{2}}$  & $\\Xi_{bc}^{+}(cbu)\\to\\Xi_{b}^{\\prime*0}(sbu)$  & $\\frac{1}{2}$  & $\\Xi_{bc}^{\\prime+}(cbu)\\to\\Xi_{b}^{\\prime*0}(sbu)$  & $\\frac{\\sqrt{3}}{2}$ \\tabularnewline\n\\hline\n$\\Xi_{cc}^{+}(ccd)\\to\\Xi_{c}^{\\prime*0}(scd)$  & $\\quad\\frac{1}{\\sqrt{2}}$  & $\\Xi_{bc}^{0}(cbd)\\to\\Xi_{b}^{\\prime*-}(sbd)$  & $\\frac{1}{2}$  & $\\Xi_{bc}^{\\prime0}(cbd)\\to\\Xi_{b}^{\\prime*-}(sbd)$  & $\\frac{\\sqrt{3}}{2}$ \\tabularnewline\n\\hline\n$\\Omega_{cc}^{+}(ccs)\\to\\Omega_{c}^{*0}(scs)$  & $1$  & $\\Omega_{bc}^{0}(cbs)\\to\\Omega_{b}^{*-}(sbs)$  & $\\frac{\\sqrt{2}}{2}$  & $\\Omega_{bc}^{\\prime0}(cbs)\\to\\Omega_{b}^{*-}(sbs)$  & $\\frac{\\sqrt{6}}{2}$ \\tabularnewline\n\\hline \\hline\n$\\Xi_{bb}^{0}(bbu)\\to\\Sigma_{b}^{*+}(ubu)$  & $1$ & $\\Xi_{bc}^{+}(bcu)\\to\\Sigma_{c}^{*++}(ucu)$  & $\\frac{\\sqrt{2}}{2}$ & $\\Xi_{bc}^{\\prime+}(bcu)\\to\\Sigma_{c}^{*++}(ucu)$  & $-\\frac{\\sqrt{6}}{2}$\\tabularnewline\n\\hline\n$\\Xi_{bb}^{-}(bbd)\\to\\Sigma_{b}^{*0}(ubd)$  & $\\frac{1}{\\sqrt{2}}$ & $\\Xi_{bc}^{0}(bcd)\\to\\Sigma_{c}^{*+}(ucd)$  & $\\frac{1}{2}$ & $\\Xi_{bc}^{\\prime0}(bcd)\\to\\Sigma_{c}^{*+}(ucd)$  & $-\\frac{\\sqrt{3}}{2}$\\tabularnewline\n\\hline\n$\\Omega_{bb}^{-}(bbs)\\to\\Xi_{b}^{\\prime*0}(ubs)$  & $\\frac{1}{\\sqrt{2}}$ & $\\Omega_{bc}^{0}(bcs)\\to\\Xi_{c}^{\\prime*+}(ucs)$  & $\\frac{1}{2}$ & $\\Omega_{bc}^{\\prime0}(bcs)\\to\\Xi_{c}^{\\prime*+}(ucs)$  & $-\\frac{\\sqrt{3}}{2}$\\tabularnewline\n\\hline\n$\\Xi_{bb}^{0}(bbu)\\to\\Xi_{bc}^{*+}(cbu)$  & $\\quad\\frac{1}{\\sqrt{2}}$ & $\\Xi_{bc}^{+}(bcu)\\to\\Xi_{cc}^{*++}(ccu)$  & $\\frac{\\sqrt{2}}{2}$ & $\\Xi_{bc}^{\\prime+}(bcu)\\to\\Xi_{cc}^{*++}(ccu)$  & $-\\frac{\\sqrt{6}}{2}$\\tabularnewline\n\\hline\n$\\Xi_{bb}^{-}(bbd)\\to\\Xi_{bc}^{*0}(cbd)$  & $\\frac{1}{\\sqrt{2}}$ & $\\Xi_{bc}^{0}(bcd)\\to\\Xi_{cc}^{*+}(ccd)$  & $\\frac{\\sqrt{2}}{2}$ & $\\Xi_{bc}^{\\prime0}(bcd)\\to\\Xi_{cc}^{*+}(ccd)$  & $-\\frac{\\sqrt{6}}{2}$\\tabularnewline\n\\hline\n$\\Omega_{bb}^{-}(bbs)\\to\\Omega_{bc}^{*0}(cbs)$  & $\\frac{1}{\\sqrt{2}}$ & $\\Omega_{bc}^{0}(bcs)\\to\\Omega_{cc}^{*+}(ccs)$  & $\\frac{\\sqrt{2}}{2}$ & $\\Omega_{bc}^{\\prime0}(bcs)\\to\\Omega_{cc}^{*+}(ccs)$  & $-\\frac{\\sqrt{6}}{2}$\\tabularnewline\n\t\t\\hline\t\\hline\n\t\t$\\Xi_{cc}^{++}(ccu)\\to\\Sigma_{c}^{*++}(ucu)$ & $1$ & $\\Xi_{bc}^{+}(cbu)\\to\\Sigma_{b}^{*+}(ubu)$ & $\\frac{1}{\\sqrt{2}}$ &$\\Xi_{bc}^{\\prime+}(cbu)\\to\\Sigma_{b}^{*+}(ubu)$ & $\\frac{\\sqrt{6}}{2}$ \\tabularnewline\n\t\t\\hline\n\t\t$\\Xi_{cc}^{+}(ccd)\\to\\Sigma_{c}^{*+}(ucd)$ & $\\frac{1}{\\sqrt{2}}$ & $\\Xi_{bc}^{0}(cbd)\\to\\Sigma_{b}^{*0}(ubd)$ & $\\frac{1}{2}$ &$\\Xi_{bc}^{\\prime0}(cbd)\\to\\Sigma_{b}^{*0}(ubd)$ & $\\frac{\\sqrt{3}}{2}$ \\tabularnewline\n\t\t\\hline\n\t\t$\\Omega_{cc}^{+}(ccs)\\to\\Xi_{c}^{\\prime*+}(ucs)$ & $\\frac{1}{\\sqrt{2}}$ & $\\Omega_{bc}^{0}(cbs)\\to\\Xi_{b}^{\\prime*0}(ubs)$ & $\\frac{1}{2}$ &$\\Omega_{bc}^{\\prime0}(cbs)\\to\\Xi_{b}^{\\prime*0}(ubs)$ & $\\frac{\\sqrt{3}}{2}$ \\tabularnewline\n\t\t\\hline\\hline\n\t\t$\\Xi_{bb}^{0}(bbu)\\to\\Xi_{b}^{\\prime*0}(sbu)$ & $\\frac{1}{\\sqrt{2}}$ & $\\Xi_{bc}^{+}(bcu)\\to\\Xi_{c}^{\\prime*+}(scu)$ & $\\frac{1}{2}$ &$\\Xi_{bc}^{\\prime+}(bcu)\\to\\Xi_{c}^{\\prime*+}(scu)$ & $-\\frac{\\sqrt{3}}{2}$ \\tabularnewline\n\t\t\\hline\n\t\t$\\Xi_{bb}^{-}(bbd)\\to\\Xi_{b}^{\\prime*-}(sbd)$ & $\\frac{1}{\\sqrt{2}}$ &$\\Xi_{bc}^{0}(bcd)\\to\\Xi_{c}^{\\prime*0}(scd)$ & $\\frac{1}{2}$ & $\\Xi_{bc}^{\\prime0}(bcd)\\to\\Xi_{c}^{\\prime*0}(scd)$ & $-\\frac{\\sqrt{3}}{2}$ \\tabularnewline\n\t\t\\hline\n\t\t$\\Omega_{bb}^{-}(bbs)\\to\\Omega_{b}^{*-}(sbs)$ & $1$ &$\\Omega_{bc}^{0}(bcs)\\to\\Omega_{c}^{*0}(scs)$ & $\\frac{\\sqrt{2}}{2}$ &$\\Omega_{bc}^{\\prime0}(bcs)\\to\\Omega_{c}^{*0}(scs)$ & $-\\frac{\\sqrt{6}}{2}$  \\tabularnewline\n\t\t\\hline\n         $\\Xi_{bb}^{0}(bbu)\\to\\Sigma_{b}^{*0}(dbu)$ & $\\frac{1}{\\sqrt{2}}$&\n\t\t $\\Xi_{bc}^{+}(bcu)\\to\\Sigma_{c}^{*+}(dcu)$ & $\\frac{1}{2}$& $\\Xi_{bc}^{\\prime+}(bcu)\\to\\Sigma_{c}^{*+}(dcu)$ & $-\\frac{\\sqrt{3}}{2}$\\tabularnewline\n\t\t\\hline\n\t\t $\\Xi_{bb}^{-}(bbd)\\to\\Sigma_{b}^{*-}(dbd)$ & $1$&$\\Xi_{bc}^{0}(bcd)\\to\\Sigma_{c}^{*0}(dcd)$ & $\\frac{\\sqrt{2}}{2}$& $\\Xi_{bc}^{\\prime0}(bcd)\\to\\Sigma_{c}^{*0}(dcd)$ & $-\\frac{\\sqrt{6}}{2}$\\tabularnewline\n\t\t\\hline$\\Omega_{bb}^{-}(bbs)\\to\\Xi_{b}^{\\prime*-}(dbs)$ & $\\frac{1}{\\sqrt{2}}$& $\\Omega_{bc}^{0}(bcs)\\to\\Xi_{c}^{\\prime*+}(dcs)$ & $\\frac{1}{2}$\n\t\t& $\\Omega_{bc}^{\\prime0}(bcs)\\to\\Xi_{c}^{\\prime*+}(dcs)$ & $-\\frac{\\sqrt{3}}{2}$\\tabularnewline\n\t\t\\hline\t\\hline\n\t\\end{tabular}\n\\end{table}\n\n\n\n\\section{Numerical results of form factors}\nThe masses of quarks are taken from Refs.~\\cite{Lu:2007sg,Wang:2007sxa,Wang:2008xt,\nWang:2008ci,Wang:2009mi,Chen:2009qk,Li:2010bb,\nVerma:2011yw,Shi:2016gqt},\n\\begin{eqnarray}\n m_u=m_d= 0.25~{\\rm GeV}, \\;\\; m_s=0.37~{\\rm GeV}, \\;\\; m_c=1.4~{\\rm GeV}, \\;\\; m_b=4.8~{\\rm GeV}.\\label{eq:mass_quark}\n\\end{eqnarray}\nThe masses of diquark are approximatively taken as,\n\\begin{eqnarray}\nm_{[cq]}=m_{c}+m_{q}~\\text{and}~m_{[bq]}=m_{b}+m_{q}~\\text{with}~q=u,d,s.\\label{eq:mass_diquark}\n\\end{eqnarray}\nThe masses of all baryons, lifetime of parent baryons and shape parameters $\\beta$ in Eq.~(\\ref{eq:Gauss}) are collected in Tab.~\\ref{Tab:para_doubly_heavy}\n~\\cite{1707.01621,Brown:2014ena,Aaij:2018wzf,Cheng:2018mwu,Karliner:2014gca,Kiselev:2001fw,Olive:2016xmw}.\n\\begin{table}[!htb]\n\\caption{Masses of all baryons (in unit of GeV), lifetimes (in unit of fs) of parent baryons and the shape parameters $\\beta$'s in the Gaussian-type wave functions Eq.~(\\ref{eq:Gauss})~\\cite{1707.01621,Brown:2014ena,Aaij:2018wzf,Cheng:2018mwu,Karliner:2014gca,Kiselev:2001fw,Olive:2016xmw}.}\n\\label{Tab:para_doubly_heavy} %\n\\begin{tabular}{c|c|c|c|c|c|c|c|c|c}\n\\hline \\hline\nbaryons  & $\\Xi_{cc}^{++}$  & $\\Xi_{cc}^{+}$  & $\\Omega_{cc}^{+}$  & $\\Xi_{bc}^{+}$  & $\\Xi_{bc}^{0}$  & $\\Omega_{bc}^{0}$  & $\\Xi_{bb}^{0}$  & $\\Xi_{bb}^{-}$  & $\\Omega_{bb}^{-}$ \\tabularnewline\n\\hline\nmasses  & $3.621$  & $3.621$   & $3.738$   & $6.943$   & $6.943$   & $6.998$   & $10.143$  & $10.143$   & $10.273$\\tabularnewline\n\\hline\nlifetimes  & $256$  & $45$  & $180$   & $244$   & $93$   & $220$  & $370$   & $370$  & $800$\\tabularnewline\n\\hline \\hline\nbaryons&~$\\Lambda_{c}^{+}$~  & ~$\\Sigma_{c}^{++}$~  & ~$\\Sigma_{c}^{+}$~  & ~$\\Sigma_{c}^{0}$~  & ~$\\Xi_{c}^{+}$~  & ~$\\Xi_{c}^{\\prime+}$~  & ~$\\Xi_{c}^{0}$~  & ~$\\Xi_{c}^{\\prime0}$~  & ~$\\Omega_{c}^{0}$~ \\tabularnewline\\hline\nmasses&$2.286$  & $2.454$  & $2.453$  & $2.454$  & $2.468$  & $2.576$  & $2.471$  & $2.578$  & $2.695$ \\tabularnewline\\hline\nbaryons&$\\Lambda_{b}^{0}$  & $\\Sigma_{b}^{+}$  & $\\Sigma_{b}^{0}$  & $\\Sigma_{b}^{-}$  & $\\Xi_{b}^{0}$ & $\\Xi_{b}^{\\prime0}$ & $\\Xi_{b}^{-}$  & $\\Xi_{b}^{\\prime-}$  & $\\Omega_{b}^{-}$\\tabularnewline\\hline\nmasses&$5.620$  & $5.811$  & $5.814$  & $5.816$  & $5.793$ & $5.935$ & $5.795$  & $5.935$  & $6.046$\\tabularnewline\n\\hline\\hline\nbaryons&$\\Sigma_{c}^{*++}$ & $\\Sigma_{c}^{*+}$  & $\\Sigma_{c}^{*0}$ & $\\Xi_{c}^{\\prime*+}$ & $\\Xi_{c}^{\\prime*0}$  & $\\Omega_{c}^{*0}$ & $\\Xi_{cc}^{*++}$ & $\\Xi_{cc}^{*+}$  & $\\Omega_{cc}^{*+}$\\tabularnewline\\hline\nmasses&$2.518$  & $2.518$  & $2.518$  & $2.646$  & $2.646$  & $2.766$  & $3.692$  & $3.692$  & $3.822$ \\tabularnewline\\hline\nbaryons&$\\Sigma_{b}^{*+}$ & $\\Sigma_{b}^{*0}$  & $\\Sigma_{b}^{*-}$ & $\\Xi_{b}^{\\prime*0}$ & $\\Xi_{b}^{\\prime*-}$  & $\\Omega_{b}^{*-}$ & $\\Xi_{bc}^{*+}$ & $\\Xi_{bc}^{*0}$  & $\\Omega_{bc}^{*0}$\\tabularnewline\\hline\nmasses&$5.832$  & $5.833$  & $5.835$  & $5.949$  & $5.955$  & $6.085$  & $6.985$  & $6.985$  & $7.059$ \\tabularnewline\n\\hline\\hline\n~~~$\\beta_{u[cq]}$ & $\\beta_{d[cq]}$ & $\\beta_{s[cq]}$ & $\\beta_{c[cq]}$ & $\\beta_{b[cq]}$& $\\beta_{u[bq]}$ & $\\beta_{d[bq]}$ & $\\beta_{s[bq]}$ & $\\beta_{c[bq]}$ & $\\beta_{b[bq]}$~~~\\tabularnewline\n\\hline\n$0.470$ & $0.470$ & $0.535$ & $0.753$ & $0.886$ &$0.562$ & $0.562$ & $0.623$ & $0.886$ & $1.472$\\tabularnewline\n\\hline \\hline\n\\end{tabular}\n\\end{table}\nUsing these analytical expression of form factors shown in Subsec.~\\ref{subsec_lightfrontquarkmodel} and the input parameters listed in Eqs.~(\\ref{eq:mass_quark})-(\\ref{eq:mass_diquark}) and Tab.~\\ref{Tab:para_doubly_heavy}, one can calculate the form factors with the scalar or axial vector diquarks in Eqs.~(\\ref{scalar diquark}), (\\ref{eq:momentum_wave_function_1\/2gamma}) and (\\ref{axial-vector diquarkprime}). While for each form factors are functions of $q^2$, in order to obtain the dependence of form factors on the momentum $q^2$, we take the following parametrization scheme for $b\\to u,d,s,c$ processes,\n\\begin{align}\n&F(q^{2})=\\frac{F(0)}{1-\\frac{q^{2}}{m_{{\\rm fit}}^{2}}+\\delta\\left(\\frac{q^{2}}{m_{{\\rm fit}}^{2}}\\right)^{2}}, \\label{eq:main_fit_formula_bdecay}\n\\end{align}\nhere $F(0)$ is the numerical result of form factor at $q^2=0$, $m_{\\rm fit}$ and $\\delta$ are two parameters waiting for fitting from numerical result of form factor at different $q^2$ values.\nWhen the fitting result of $m_{\\rm fit}$ is an imaginary result using the above parametrization scheme, we need to take the modified parametrization scheme as follows,\n\\begin{align}\n&F(q^{2})=\\frac{F(0)}{1+\\frac{q^{2}}{m_{{\\rm fit}}^{2}}+\\delta\\left(\\frac{q^{2}}{m_{{\\rm fit}}^{2}}\\right)^{2}}, \\label{eq:auxiliary_fit_formula_bdecay}\n\\end{align}\nIn the tables, we mark the imaginary results with superscripts  ``$*$\".  While for c quark decay process, the single pole structure is assumed\n\\begin{align}\n&F(q^{2})=\\frac{F(0)}{1-\\frac{q^{2}}{m_{{\\rm pole}}^{2}}},\\label{eq:main_fit_formula_cdecay}\n\\end{align}\nfor $c\\to u,~d,~s$ decays, $m_{{\\rm pole}}$s are respectively $1.87$, $1.87$, $1.97$ GeV.\n\\begin{itemize}\n\\item\nFor the charged current transition ${1}\/{2}\\to{1}\/{2}$, the results for  form factors with a scalar diquark or an axial-vector diquark spectator are shown in Tabs. \\ref{Tab:ff_ccc}, \\ref{Tab:ff_bbb} and \\ref{Tab:ff_bcc}.\nAs shown in Eq.~(\\ref{eq:matrix_element_2}), the numerical results of the form factors can be used to calculate the physical hadronic transition matrix elements.\nWe take $\\Xi_{bb}^{0}\\to\\Sigma_{b}^{+}$ as an example to show the $q^2$-dependence of form factors in Fig.~\\ref{fig:Fxibbsigmab}. There is no singular point for the form factors $f_{1,2,3}$ and $g_{1,2,3}$ in the integration interval shown in Fig.~\\ref{fig:Fxibbsigmab}.\n\\begin{figure}\n\\includegraphics[width=0.8\\columnwidth]{formfactorbbtob22.eps}\n\\caption{$q^2$ dependence of the form factors for the transition $\\Xi_{bb}^{0}\\to\\Sigma_{b}^{+}$. The two graphs in the first line correspond to form factors with scalar diquarks, the two graphs in the second correspond to form factors with axial-vector diquarks. The numerical result of $F(0)$, $\\delta$ and $m_{fit}$ are shown in Tab.~\\ref{Tab:ff_bbb}.}\n\\label{fig:Fxibbsigmab}\n\\end{figure}\n\\item  For the FCNC transition ${1}\/{2}\\to{1}\/{2}$, the results for  form factors with a scalar diquark or an axial-vector diquark spectator are shown in Tabs. \\ref{Tab:ff_cucc_axial},~\\ref{Tab:ff_bsbb_axial} and \\ref{Tab:ff_bdbb_axial}. With the help of the results of the form factors and Eqs.~(\\ref{eq:matrix_element_2})-(\\ref{eq:matrix_element_2p}), one can calculate the physical hadronic transition matrix elements.  $\\Xi_{bb}^{0}\\to\\Lambda_{b}^{0}$ is taken as an example to show the $q^2$-dependence of these form factors which are depicted in Fig.~\\ref{fig:Fxibblambdab}.\nAs one can see, these form factors are stable and no divergence exists in the integration interval.\n\\begin{figure}\n\\includegraphics[width=0.8\\columnwidth]{xibblambda.eps}\n\\caption{$q^2$ dependence of the form factors for $\\Xi_{bb}^{0}\\to\\Lambda_{b}^{0}$ . The first four graphs correspond to form factors with scalar diquark, the last four graphs correspond to form factors with axial-vector diquark. Here the numerical results of $F(0)$, $\\delta$ and $m_{fit}$ are shown in Tab.~\\ref{Tab:ff_bsbb_axial}.}\n\\label{fig:Fxibblambdab}\n\\end{figure}\n\\item  For the charged current transition ${1}\/{2}\\to{3}\/{2}$, the results for  form factors with an axial-vector diquark spectator are shown in Tabs. \\ref{Tab:ff32_cdscc} and \\ref{Tab:ff32_bucbb}. As shown in Eq.~(\\ref{eq:matrix_element_32nVA}), the numerical results of the form factors can be used to calculate the physical hadronic transition matrix elements. In Fig.~\\ref{fig:Fxibbsigmastar} we use $\\Xi_{bb}^{0}\\to\\Sigma_{b}^{*0}$ as an example to show the $q^2$-dependence of form factors. As shown in Fig.~\\ref{fig:Fxibbsigmastar} these form factors are stable, which indicates our fitting result in the Tabs. \\ref{Tab:ff32_cdscc} and \\ref{Tab:ff32_bucbb} are reliable.\n\\begin{figure}\n\\includegraphics[width=0.8\\columnwidth]{xibbsigmastar.eps}\n\\caption{$q^2$ dependence of the transition $\\Xi_{bb}\\to\\Sigma_{b}^{*}$ form factors. The numerical result of the parameters $F(0)$, $\\delta$ and $m_{fit}$ are shown in Tab.~\\ref{Tab:ff32_bucbb}.}\n\\label{fig:Fxibbsigmastar}\n\\end{figure}\n\\item For the FCNC transition ${1}\/{2}\\to{3}\/{2}$, the results for  form factors with an axial-vector diquark are shown in Tabs. \\ref{Tab:fcnc32_cu}, \\ref{Tab:fcnc32_bd} and~\\ref{Tab:fcnc32_bs}. As shown in Eqs.~(\\ref{eq:matrix_element_32nVA}) and (\\ref{eq:matrix_element_32nT}), the numerical results of the form factors can be used to calculate the physical hadronic transition matrix elements. To describe the dependence of form factors on $q^2$, we take the transition $\\Omega_{bb}\\to\\Xi_{b}^{\\prime*-}$  as an example shown in Fig.~\\ref{fig:FOmegabbXibS}. The curves are all approaching to zero at large $q^2$ stably.\n\\begin{figure}\n\\includegraphics[width=0.8\\columnwidth]{xibbsigmastarfcnc.eps}\n\\caption{$q^2$ dependence of the transition $\\Omega_{bb}^{-}\\to\\Xi_{b}^{\\prime*-}$ form factors. The numerical result of the parameters $F(0)$, $\\delta$ and $m_{fit}$ are shown in Tab.~\\ref{Tab:fcnc32_bd}.}\n\\label{fig:FOmegabbXibS}\n\\end{figure}\n\\end{itemize}\n\\begin{table}\\scriptsize\n \\caption{\n Numerical results for the transition $1\/2\\to 1\/2$ form factors $f_{i,S(A)}$ and $g_{i,S(A)}$ at $q^2=0$ of $c\\to d,s$ processes. The parametrization scheme in Eq.~(\\ref{eq:main_fit_formula_cdecay}) is introduced for these form factors,\nand the values of the singly pole $m_{\\rm pole}$s are taken as $1.87, ~1.97~{\\rm GeV}$ for $c\\to d,s$, respectively.}\\label{Tab:ff_ccc}\n\\begin{tabular}{c|c|c|c|c|c|c|c}\n\\hline\\hline\n$F$ &$F(0)$ &$F$ &$F(0)$ &$F$ &$F(0)$ &$F$ &$F(0)$\\\\ \\hline\n$f_{1,S}^{\\Xi_{cc}^{++}\\to\\Lambda_{c}^{+}}$ &$0.495$ &\n$g_{1,S}^{\\Xi_{cc}^{++}\\to\\Lambda_{c}^{+}}$ &$0.332$ &\n$f_{1,A}^{\\Xi_{cc}^{++}\\to\\Lambda_{c}^{+}}$ &$0.489$ &\n$g_{1,A}^{\\Xi_{cc}^{++}\\to\\Lambda_{c}^{+}}$ &$-0.111$ \\\\\n$f_{2,S}^{\\Xi_{cc}^{++}\\to\\Lambda_{c}^{+}}$ &$-0.621$ &\n$g_{2,S}^{\\Xi_{cc}^{++}\\to\\Lambda_{c}^{+}}$ &$1.004$ &\n$f_{2,A}^{\\Xi_{cc}^{++}\\to\\Lambda_{c}^{+}}$ &$0.290$ &\n$g_{2,A}^{\\Xi_{cc}^{++}\\to\\Lambda_{c}^{+}}$ &$-0.325$ \\\\\n$f_{3,S}^{\\Xi_{cc}^{++}\\to\\Lambda_{c}^{+}}$ &$0.832$ &\n$g_{3,S}^{\\Xi_{cc}^{++}\\to\\Lambda_{c}^{+}}$ &$-2.957$ &\n$f_{3,A}^{\\Xi_{cc}^{++}\\to\\Lambda_{c}^{+}}$ &$0.648$ &\n$g_{3,A}^{\\Xi_{cc}^{++}\\to\\Lambda_{c}^{+}}$ &$0.943$ \\\\ \\hline\n$f_{1,S}^{\\Xi_{cc}\\to\\Sigma_{c}}$ &$0.536$ &\n$g_{1,S}^{\\Xi_{cc}\\to\\Sigma_{c}}$ &$0.422$ &\n$f_{1,A}^{\\Xi_{cc}\\to\\Sigma_{c}}$ &$0.529$ &\n$g_{1,A}^{\\Xi_{cc}\\to\\Sigma_{c}}$ &$-0.141$ \\\\\n$f_{2,S}^{\\Xi_{cc}\\to\\Sigma_{c}}$ &$-0.732$ &\n$g_{2,S}^{\\Xi_{cc}\\to\\Sigma_{c}}$ &$0.561$ &\n$f_{2,A}^{\\Xi_{cc}\\to\\Sigma_{c}}$ &$0.427$ &\n$g_{2,A}^{\\Xi_{cc}\\to\\Sigma_{c}}$ &$-0.177$ \\\\\n$f_{3,S}^{\\Xi_{cc}\\to\\Sigma_{c}}$ &$0.620$ &\n$g_{3,S}^{\\Xi_{cc}\\to\\Sigma_{c}}$ &$-0.808$ &\n$f_{3,A}^{\\Xi_{cc}\\to\\Sigma_{c}}$ &$0.423$ &\n$g_{3,A}^{\\Xi_{cc}\\to\\Sigma_{c}}$ &$0.215$ \\\\ \\hline\n$f_{1,S}^{\\Xi_{cc}\\to\\Xi_{c}}$ &$0.588$ &\n$g_{1,S}^{\\Xi_{cc}\\to\\Xi_{c}}$ &$0.424$ &\n$f_{1,A}^{\\Xi_{cc}\\to\\Xi_{c}}$ &$0.582$ &\n$g_{1,A}^{\\Xi_{cc}\\to\\Xi_{c}}$ &$-0.141$ \\\\\n$f_{2,S}^{\\Xi_{cc}\\to\\Xi_{c}}$ &$-0.817$ &\n$g_{2,S}^{\\Xi_{cc}\\to\\Xi_{c}}$ &$1.105$ &\n$f_{2,A}^{\\Xi_{cc}\\to\\Xi_{c}}$ &$0.270$ &\n$g_{2,A}^{\\Xi_{cc}\\to\\Xi_{c}}$ &$-0.358$ \\\\\n$f_{3,S}^{\\Xi_{cc}\\to\\Xi_{c}}$ &$1.056$ &\n$g_{3,S}^{\\Xi_{cc}\\to\\Xi_{c}}$ &$-2.936$ &\n$f_{3,A}^{\\Xi_{cc}\\to\\Xi_{c}}$ &$0.873$ &\n$g_{3,A}^{\\Xi_{cc}\\to\\Xi_{c}}$ &$0.927$ \\\\ \\hline\n$f_{1,S}^{\\Xi_{cc}\\to\\Xi_{c}^{\\prime}}$ &$0.626$ &\n$g_{1,S}^{\\Xi_{cc}\\to\\Xi_{c}^{\\prime}}$ &$0.507$ &\n$f_{1,A}^{\\Xi_{cc}\\to\\Xi_{c}^{\\prime}}$ &$0.620$ &\n$g_{1,A}^{\\Xi_{cc}\\to\\Xi_{c}^{\\prime}}$ &$-0.169$ \\\\\n$f_{2,S}^{\\Xi_{cc}\\to\\Xi_{c}^{\\prime}}$ &$-0.904$ &\n$g_{2,S}^{\\Xi_{cc}\\to\\Xi_{c}^{\\prime}}$ &$0.641$ &\n$f_{2,A}^{\\Xi_{cc}\\to\\Xi_{c}^{\\prime}}$ &$0.397$ &\n$g_{2,A}^{\\Xi_{cc}\\to\\Xi_{c}^{\\prime}}$ &$-0.203$ \\\\\n$f_{3,S}^{\\Xi_{cc}\\to\\Xi_{c}^{\\prime}}$ &$0.858$ &\n$g_{3,S}^{\\Xi_{cc}\\to\\Xi_{c}^{\\prime}}$ &$-0.014$ &\n$f_{3,A}^{\\Xi_{cc}\\to\\Xi_{c}^{\\prime}}$ &$0.665$ &\n$g_{3,A}^{\\Xi_{cc}\\to\\Xi_{c}^{\\prime}}$ &$-0.057$ \\\\ \\hline\n$f_{1,S}^{\\Omega_{cc}^{+}\\to\\Xi_{c}^{0}}$ &$0.501$ &\n$g_{1,S}^{\\Omega_{cc}^{+}\\to\\Xi_{c}^{0}}$ &$0.357$ &\n$f_{1,A}^{\\Omega_{cc}^{+}\\to\\Xi_{c}^{0}}$ &$0.496$ &\n$g_{1,A}^{\\Omega_{cc}^{+}\\to\\Xi_{c}^{0}}$ &$-0.119$ \\\\\n$f_{2,S}^{\\Omega_{cc}^{+}\\to\\Xi_{c}^{0}}$ &$-0.666$ &\n$g_{2,S}^{\\Omega_{cc}^{+}\\to\\Xi_{c}^{0}}$ &$0.875$ &\n$f_{2,A}^{\\Omega_{cc}^{+}\\to\\Xi_{c}^{0}}$ &$0.351$ &\n$g_{2,A}^{\\Omega_{cc}^{+}\\to\\Xi_{c}^{0}}$ &$-0.283$ \\\\\n$f_{3,S}^{\\Omega_{cc}^{+}\\to\\Xi_{c}^{0}}$ &$0.741$ &\n$g_{3,S}^{\\Omega_{cc}^{+}\\to\\Xi_{c}^{0}}$ &$-2.588$ &\n$f_{3,A}^{\\Omega_{cc}^{+}\\to\\Xi_{c}^{0}}$ &$0.555$ &\n$g_{3,A}^{\\Omega_{cc}^{+}\\to\\Xi_{c}^{0}}$ &$0.819$ \\\\ \\hline\n$f_{1,S}^{\\Omega_{cc}^{+}\\to\\Xi_{c}^{\\prime0}}$ &$0.529$ &\n$g_{1,S}^{\\Omega_{cc}^{+}\\to\\Xi_{c}^{\\prime0}}$ &$0.419$ &\n$f_{1,A}^{\\Omega_{cc}^{+}\\to\\Xi_{c}^{\\prime0}}$ &$0.523$ &\n$g_{1,A}^{\\Omega_{cc}^{+}\\to\\Xi_{c}^{\\prime0}}$ &$-0.140$ \\\\\n$f_{2,S}^{\\Omega_{cc}^{+}\\to\\Xi_{c}^{\\prime0}}$ &$-0.741$ &\n$g_{2,S}^{\\Omega_{cc}^{+}\\to\\Xi_{c}^{\\prime0}}$ &$0.553$ &\n$f_{2,A}^{\\Omega_{cc}^{+}\\to\\Xi_{c}^{\\prime0}}$ &$0.453$ &\n$g_{2,A}^{\\Omega_{cc}^{+}\\to\\Xi_{c}^{\\prime0}}$ &$-0.175$ \\\\\n$f_{3,S}^{\\Omega_{cc}^{+}\\to\\Xi_{c}^{\\prime0}}$ &$0.593$ &\n$g_{3,S}^{\\Omega_{cc}^{+}\\to\\Xi_{c}^{\\prime0}}$ &$-0.857$ &\n$f_{3,A}^{\\Omega_{cc}^{+}\\to\\Xi_{c}^{\\prime0}}$ &$0.398$ &\n$g_{3,A}^{\\Omega_{cc}^{+}\\to\\Xi_{c}^{\\prime0}}$ &$0.234$ \\\\ \\hline\n$f_{1,S}^{\\Omega_{cc}^{+}\\to\\Omega_{c}^{0}}$ &$0.618$ &\n$g_{1,S}^{\\Omega_{cc}^{+}\\to\\Omega_{c}^{0}}$ &$0.500$ &\n$f_{1,A}^{\\Omega_{cc}^{+}\\to\\Omega_{c}^{0}}$ &$0.613$ &\n$g_{1,A}^{\\Omega_{cc}^{+}\\to\\Omega_{c}^{0}}$ &$-0.167$ \\\\\n$f_{2,S}^{\\Omega_{cc}^{+}\\to\\Omega_{c}^{0}}$ &$-0.901$ &\n$g_{2,S}^{\\Omega_{cc}^{+}\\to\\Omega_{c}^{0}}$ &$0.653$ &\n$f_{2,A}^{\\Omega_{cc}^{+}\\to\\Omega_{c}^{0}}$ &$0.430$ &\n$g_{2,A}^{\\Omega_{cc}^{+}\\to\\Omega_{c}^{0}}$ &$-0.208$ \\\\\n$f_{3,S}^{\\Omega_{cc}^{+}\\to\\Omega_{c}^{0}}$ &$0.837$ &\n$g_{3,S}^{\\Omega_{cc}^{+}\\to\\Omega_{c}^{0}}$ &$-0.159$ &\n$f_{3,A}^{\\Omega_{cc}^{+}\\to\\Omega_{c}^{0}}$ &$0.645$ &\n$g_{3,A}^{\\Omega_{cc}^{+}\\to\\Omega_{c}^{0}}$ &$-0.005$ \\\\ \\hline\n\\hline\n$f_{1,S}^{\\Xi_{bc}^{(\\prime)+}\\to\\Lambda_{b}^{0}}$ &$0.455$ &\n$g_{1,S}^{\\Xi_{bc}^{(\\prime)+}\\to\\Lambda_{b}^{0}}$ &$0.274$ &\n$f_{1,A}^{\\Xi_{bc}^{(\\prime)+}\\to\\Lambda_{b}^{0}}$ &$0.454$ &\n$g_{1,A}^{\\Xi_{bc}^{(\\prime)+}\\to\\Lambda_{b}^{0}}$ &$-0.091$ \\\\\n$f_{2,S}^{\\Xi_{bc}^{(\\prime)+}\\to\\Lambda_{b}^{0}}$ &$-1.471$ &\n$g_{2,S}^{\\Xi_{bc}^{(\\prime)+}\\to\\Lambda_{b}^{0}}$ &$2.114$ &\n$f_{2,A}^{\\Xi_{bc}^{(\\prime)+}\\to\\Lambda_{b}^{0}}$ &$0.023$ &\n$g_{2,A}^{\\Xi_{bc}^{(\\prime)+}\\to\\Lambda_{b}^{0}}$ &$-0.702$ \\\\\n$f_{3,S}^{\\Xi_{bc}^{(\\prime)+}\\to\\Lambda_{b}^{0}}$ &$1.469$ &\n$g_{3,S}^{\\Xi_{bc}^{(\\prime)+}\\to\\Lambda_{b}^{0}}$ &$-15.140$ &\n$f_{3,A}^{\\Xi_{bc}^{(\\prime)+}\\to\\Lambda_{b}^{0}}$ &$1.319$ &\n$g_{3,A}^{\\Xi_{bc}^{(\\prime)+}\\to\\Lambda_{b}^{0}}$ &$5.021$ \\\\ \\hline\n$f_{1,S}^{\\Xi_{bc}^{(\\prime)}\\to\\Sigma_{b}}$ &$0.517$ &\n$g_{1,S}^{\\Xi_{bc}^{(\\prime)}\\to\\Sigma_{b}}$ &$0.370$ &\n$f_{1,A}^{\\Xi_{bc}^{(\\prime)}\\to\\Sigma_{b}}$ &$0.516$ &\n$g_{1,A}^{\\Xi_{bc}^{(\\prime)}\\to\\Sigma_{b}}$ &$-0.123$ \\\\\n$f_{2,S}^{\\Xi_{bc}^{(\\prime)}\\to\\Sigma_{b}}$ &$-1.716$ &\n$g_{2,S}^{\\Xi_{bc}^{(\\prime)}\\to\\Sigma_{b}}$ &$1.388$ &\n$f_{2,A}^{\\Xi_{bc}^{(\\prime)}\\to\\Sigma_{b}}$ &$0.305$ &\n$g_{2,A}^{\\Xi_{bc}^{(\\prime)}\\to\\Sigma_{b}}$ &$-0.460$ \\\\\n$f_{3,S}^{\\Xi_{bc}^{(\\prime)}\\to\\Sigma_{b}}$ &$1.115$ &\n$g_{3,S}^{\\Xi_{bc}^{(\\prime)}\\to\\Sigma_{b}}$ &$-7.892$ &\n$f_{3,A}^{\\Xi_{bc}^{(\\prime)}\\to\\Sigma_{b}}$ &$0.946$ &\n$g_{3,A}^{\\Xi_{bc}^{(\\prime)}\\to\\Sigma_{b}}$ &$2.598$ \\\\ \\hline\n$f_{1,S}^{\\Xi_{bc}^{(\\prime)}\\to\\Xi_{b}}$ &$0.537$ &\n$g_{1,S}^{\\Xi_{bc}^{(\\prime)}\\to\\Xi_{b}}$ &$0.353$ &\n$f_{1,A}^{\\Xi_{bc}^{(\\prime)}\\to\\Xi_{b}}$ &$0.536$ &\n$g_{1,A}^{\\Xi_{bc}^{(\\prime)}\\to\\Xi_{b}}$ &$-0.118$ \\\\\n$f_{2,S}^{\\Xi_{bc}^{(\\prime)}\\to\\Xi_{b}}$ &$-1.851$ &\n$g_{2,S}^{\\Xi_{bc}^{(\\prime)}\\to\\Xi_{b}}$ &$2.362$ &\n$f_{2,A}^{\\Xi_{bc}^{(\\prime)}\\to\\Xi_{b}}$ &$-0.055$ &\n$g_{2,A}^{\\Xi_{bc}^{(\\prime)}\\to\\Xi_{b}}$ &$-0.785$ \\\\\n$f_{3,S}^{\\Xi_{bc}^{(\\prime)}\\to\\Xi_{b}}$ &$1.854$ &\n$g_{3,S}^{\\Xi_{bc}^{(\\prime)}\\to\\Xi_{b}}$ &$-16.090$ &\n$f_{3,A}^{\\Xi_{bc}^{(\\prime)}\\to\\Xi_{b}}$ &$1.700$ &\n$g_{3,A}^{\\Xi_{bc}^{(\\prime)}\\to\\Xi_{b}}$ &$5.333$ \\\\ \\hline\n$f_{1,S}^{\\Xi_{bc}^{(\\prime)}\\to\\Xi_{b}^{\\prime}}$ &$0.599$ &\n$g_{1,S}^{\\Xi_{bc}^{(\\prime)}\\to\\Xi_{b}^{\\prime}}$ &$0.454$ &\n$f_{1,A}^{\\Xi_{bc}^{(\\prime)}\\to\\Xi_{b}^{\\prime}}$ &$0.599$ &\n$g_{1,A}^{\\Xi_{bc}^{(\\prime)}\\to\\Xi_{b}^{\\prime}}$ &$-0.151$ \\\\\n$f_{2,S}^{\\Xi_{bc}^{(\\prime)}\\to\\Xi_{b}^{\\prime}}$ &$-2.076$ &\n$g_{2,S}^{\\Xi_{bc}^{(\\prime)}\\to\\Xi_{b}^{\\prime}}$ &$1.439$ &\n$f_{2,A}^{\\Xi_{bc}^{(\\prime)}\\to\\Xi_{b}^{\\prime}}$ &$0.237$ &\n$g_{2,A}^{\\Xi_{bc}^{(\\prime)}\\to\\Xi_{b}^{\\prime}}$ &$-0.477$ \\\\\n$f_{3,S}^{\\Xi_{bc}^{(\\prime)}\\to\\Xi_{b}^{\\prime}}$ &$1.474$ &\n$g_{3,S}^{\\Xi_{bc}^{(\\prime)}\\to\\Xi_{b}^{\\prime}}$ &$-3.628$ &\n$f_{3,A}^{\\Xi_{bc}^{(\\prime)}\\to\\Xi_{b}^{\\prime}}$ &$1.303$ &\n$g_{3,A}^{\\Xi_{bc}^{(\\prime)}\\to\\Xi_{b}^{\\prime}}$ &$1.172$ \\\\ \\hline\n$f_{1,S}^{\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{b}^{-}}$ &$0.499$ &\n$g_{1,S}^{\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{b}^{-}}$ &$0.330$ &\n$f_{1,A}^{\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{b}^{-}}$ &$0.498$ &\n$g_{1,A}^{\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{b}^{-}}$ &$-0.110$ \\\\\n$f_{2,S}^{\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{b}^{-}}$ &$-1.841$ &\n$g_{2,S}^{\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{b}^{-}}$ &$1.818$ &\n$f_{2,A}^{\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{b}^{-}}$ &$-0.025$ &\n$g_{2,A}^{\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{b}^{-}}$ &$-0.603$ \\\\\n$f_{3,S}^{\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{b}^{-}}$ &$1.357$ &\n$g_{3,S}^{\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{b}^{-}}$ &$-12.730$ &\n$f_{3,A}^{\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{b}^{-}}$ &$1.195$ &\n$g_{3,A}^{\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{b}^{-}}$ &$4.214$ \\\\ \\hline\n$f_{1,S}^{\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{b}^{\\prime-}}$ &$0.552$ &\n$g_{1,S}^{\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{b}^{\\prime-}}$ &$0.418$ &\n$f_{1,A}^{\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{b}^{\\prime-}}$ &$0.551$ &\n$g_{1,A}^{\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{b}^{\\prime-}}$ &$-0.139$ \\\\\n$f_{2,S}^{\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{b}^{\\prime-}}$ &$-2.104$ &\n$g_{2,S}^{\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{b}^{\\prime-}}$ &$1.030$ &\n$f_{2,A}^{\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{b}^{\\prime-}}$ &$0.196$ &\n$g_{2,A}^{\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{b}^{\\prime-}}$ &$-0.340$ \\\\\n$f_{3,S}^{\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{b}^{\\prime-}}$ &$1.013$ &\n$g_{3,S}^{\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{b}^{\\prime-}}$ &$-3.102$ &\n$f_{3,A}^{\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{b}^{\\prime-}}$ &$0.834$ &\n$g_{3,A}^{\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{b}^{\\prime-}}$ &$0.998$ \\\\ \\hline\n$f_{1,S}^{\\Omega_{bc}^{(\\prime)0}\\to\\Omega_{b}^{-}}$ &$0.640$ &\n$g_{1,S}^{\\Omega_{bc}^{(\\prime)0}\\to\\Omega_{b}^{-}}$ &$0.510$ &\n$f_{1,A}^{\\Omega_{bc}^{(\\prime)0}\\to\\Omega_{b}^{-}}$ &$0.639$ &\n$g_{1,A}^{\\Omega_{bc}^{(\\prime)0}\\to\\Omega_{b}^{-}}$ &$-0.170$ \\\\\n$f_{2,S}^{\\Omega_{bc}^{(\\prime)0}\\to\\Omega_{b}^{-}}$ &$-2.590$ &\n$g_{2,S}^{\\Omega_{bc}^{(\\prime)0}\\to\\Omega_{b}^{-}}$ &$0.944$ &\n$f_{2,A}^{\\Omega_{bc}^{(\\prime)0}\\to\\Omega_{b}^{-}}$ &$0.028$ &\n$g_{2,A}^{\\Omega_{bc}^{(\\prime)0}\\to\\Omega_{b}^{-}}$ &$-0.312$ \\\\\n$f_{3,S}^{\\Omega_{bc}^{(\\prime)0}\\to\\Omega_{b}^{-}}$ &$1.363$ &\n$g_{3,S}^{\\Omega_{bc}^{(\\prime)0}\\to\\Omega_{b}^{-}}$ &$4.871$ &\n$f_{3,A}^{\\Omega_{bc}^{(\\prime)0}\\to\\Omega_{b}^{-}}$ &$1.182$ &\n$g_{3,A}^{\\Omega_{bc}^{(\\prime)0}\\to\\Omega_{b}^{-}}$ &$-1.665$ \\\\ \\hline\n\\hline\n\\end{tabular}\n\\end{table}\n\\begin{table}\\footnotesize \\label{Tab:ff_bbb}\n \\caption{Numerical results for the transition $1\/2\\to 1\/2$ form factors $f^{(T)}_{i,S(A)}$ and $g^{(T)}_{i,S(A)}$ of doubly bottom baryon ${\\cal B}_{bb}$ decay with $b\\to u,c$ processes. The parametrization scheme in Eq.~(\\ref{eq:auxiliary_fit_formula_bdecay}) is introduced for these form factors with asterisk, and Eq.~(\\ref{eq:main_fit_formula_bdecay}) for all the other ones.}\n\\begin{tabular}{c|c|c|c|c|c|c|c|c|c|c|c}\n\\hline\\hline\n$F$ &$F(0)$ &$m_{\\rm{fit}}$ &$\\delta$ &$F$ &$F(0)$ &$m_{\\rm{fit}}$ &$\\delta$\n&$F$ &$F(0)$ &$m_{\\rm{fit}}$ &$\\delta$\\\\ \\hline\n$f_{1,S}^{\\Xi_{bb}\\to\\Sigma_{b}}$ &$\n0.102$ &$3.35$ &$0.84$ &\n$f_{2,S}^{\\Xi_{bb}\\to\\Sigma_{b}}$ &$\n-0.149$ &$3.06$ &$0.92$ &\n$f_{3,S}^{\\Xi_{bb}\\to\\Sigma_{b}}$ &$\n-0.004^{*}$ &$3.13^{*}$ &$0.99^{*}$ \\\\\n$g_{1,S}^{\\Xi_{bb}\\to\\Sigma_{b}}$ &$\n0.094$ &$3.36$ &$0.76$ &\n$g_{2,S}^{\\Xi_{bb}\\to\\Sigma_{b}}$ &$\n0.013$ &$3.36$ &$1.11$ &\n$g_{3,S}^{\\Xi_{bb}\\to\\Sigma_{b}}$ &$\n-0.223$ &$3.35$ &$1.14$ \\\\\n$f_{1,A}^{\\Xi_{bb}\\to\\Sigma_{b}}$ &$\n0.102$ &$3.18$ &$0.94$ &\n$f_{2,A}^{\\Xi_{bb}\\to\\Sigma_{b}}$ &$\n0.096$ &$3.28$ &$0.95$ &\n$f_{3,A}^{\\Xi_{bb}\\to\\Sigma_{b}}$ &$\n-0.068$ &$3.44$ &$1.13$ \\\\\n$g_{1,A}^{\\Xi_{bb}\\to\\Sigma_{b}}$ &$\n-0.031$ &$3.35$ &$0.76$ &\n$g_{2,A}^{\\Xi_{bb}\\to\\Sigma_{b}}$ &$\n-0.003$ &$3.46$ &$1.21$ &\n$g_{3,A}^{\\Xi_{bb}\\to\\Sigma_{b}}$ &$\n0.071$ &$3.37$ &$1.16$ \\\\ \\hline\n$f_{1,S}^{\\Xi_{bb}\\to\\Xi_{bc}^{(\\prime)}}$ &$\n0.471$ &$4.23$ &$0.76$ &\n$f_{2,S}^{\\Xi_{bb}\\to\\Xi_{bc}^{(\\prime)}}$ &$\n-0.659$ &$3.15$ &$0.46$ &\n$f_{3,S}^{\\Xi_{bb}\\to\\Xi_{bc}^{(\\prime)}}$ &$\n0.096$ &$3.36$ &$6.76$ \\\\\n$g_{1,S}^{\\Xi_{bb}\\to\\Xi_{bc}^{(\\prime)}}$ &$\n0.458$ &$3.54$ &$0.19$ &\n$g_{2,S}^{\\Xi_{bb}\\to\\Xi_{bc}^{(\\prime)}}$ &$\n-0.087^{*}$ &$0.97^{*}$ &$0.06^{*}$&\n$g_{3,S}^{\\Xi_{bb}\\to\\Xi_{bc}^{(\\prime)}}$ &$\n0.573$ &$1.34$ &$-0.33$ \\\\\n$f_{1,A}^{\\Xi_{bb}\\to\\Xi_{bc}^{(\\prime)}}$ &$\n0.469$ &$3.88$ &$0.74$ &\n$f_{2,A}^{\\Xi_{bb}\\to\\Xi_{bc}^{(\\prime)}}$ &$\n0.318$ &$5.18$ &$2.26$ &\n$f_{3,A}^{\\Xi_{bb}\\to\\Xi_{bc}^{(\\prime)}}$ &$\n-0.079^{*}$ &$5.06^{*}$ &$3.69^{*}$ \\\\\n$g_{1,A}^{\\Xi_{bb}\\to\\Xi_{bc}^{(\\prime)}}$ &$\n-0.153$ &$3.53$ &$0.19$ &\n$g_{2,A}^{\\Xi_{bb}\\to\\Xi_{bc}^{(\\prime)}}$ &$\n0.032^{*}$ &$0.61^{*}$ &$0.03^{*}$ &\n$g_{3,A}^{\\Xi_{bb}\\to\\Xi_{bc}^{(\\prime)}}$ &$\n-0.208^{*}$ &$1.62^{*}$ &$-0.00^{*}$ \\\\ \\hline\n$f_{1,S}^{\\Xi_{bb}^{-}\\to\\Lambda_{b}^{0}}$ &$\n0.100$ &$3.40$ &$0.86$ &\n$f_{2,S}^{\\Xi_{bb}^{-}\\to\\Lambda_{b}^{0}}$ &$\n-0.136$ &$3.10$ &$0.93$ &\n$f_{3,S}^{\\Xi_{bb}^{-}\\to\\Lambda_{b}^{0}}$ &$\n0.008$ &$0.32$ &$-0.01$ \\\\\n$g_{1,S}^{\\Xi_{bb}^{-}\\to\\Lambda_{b}^{0}}$ &$\n0.087$ &$3.57$ &$0.91$ &\n$g_{2,S}^{\\Xi_{bb}^{-}\\to\\Lambda_{b}^{0}}$ &$\n0.041$ &$2.70$ &$0.89$ &\n$g_{3,S}^{\\Xi_{bb}^{-}\\to\\Lambda_{b}^{0}}$ &$\n-0.298$ &$2.99$ &$0.89$ \\\\\n$f_{1,A}^{\\Xi_{bb}^{-}\\to\\Lambda_{b}^{0}}$ &$\n0.100$ &$3.22$ &$0.96$ &\n$f_{2,A}^{\\Xi_{bb}^{-}\\to\\Lambda_{b}^{0}}$ &$\n0.092$ &$3.36$ &$0.99$ &\n$f_{3,A}^{\\Xi_{bb}^{-}\\to\\Lambda_{b}^{0}}$ &$\n-0.055$ &$3.79$ &$1.45$ \\\\\n$g_{1,A}^{\\Xi_{bb}^{-}\\to\\Lambda_{b}^{0}}$ &$\n-0.029$ &$3.56$ &$0.91$ &\n$g_{2,A}^{\\Xi_{bb}^{-}\\to\\Lambda_{b}^{0}}$ &$\n-0.013$ &$2.68$ &$0.92$ &\n$g_{3,A}^{\\Xi_{bb}^{-}\\to\\Lambda_{b}^{0}}$ &$\n0.096$ &$2.98$ &$0.89$ \\\\ \\hline\n$f_{1,S}^{\\Omega_{bb}^{-}\\to\\Xi_{b}^{0}}$ &$\n0.098$ &$3.36$ &$0.86$ &\n$f_{2,S}^{\\Omega_{bb}^{-}\\to\\Xi_{b}^{0}}$ &$\n-0.137$ &$3.09$ &$0.95$ &\n$f_{3,S}^{\\Omega_{bb}^{-}\\to\\Xi_{b}^{0}}$ &$\n0.004^{*}$ &$0.88^{*}$ &$0.06^{*}$ \\\\\n$g_{1,S}^{\\Omega_{bb}^{-}\\to\\Xi_{b}^{0}}$ &$\n0.086$ &$3.50$ &$0.89$ &\n$g_{2,S}^{\\Omega_{bb}^{-}\\to\\Xi_{b}^{0}}$ &$\n0.034$ &$2.70$ &$0.89$ &\n$g_{3,S}^{\\Omega_{bb}^{-}\\to\\Xi_{b}^{0}}$ &$\n-0.283$ &$3.01$ &$0.93$ \\\\\n$f_{1,A}^{\\Omega_{bb}^{-}\\to\\Xi_{b}^{0}}$ &$\n0.097$ &$3.19$ &$0.97$ &\n$f_{2,A}^{\\Omega_{bb}^{-}\\to\\Xi_{b}^{0}}$ &$\n0.090$ &$3.32$ &$0.99$ &\n$f_{3,A}^{\\Omega_{bb}^{-}\\to\\Xi_{b}^{0}}$ &$\n-0.057$ &$3.65$ &$1.35$ \\\\\n$g_{1,A}^{\\Omega_{bb}^{-}\\to\\Xi_{b}^{0}}$ &$\n-0.029$ &$3.49$ &$0.89$ &\n$g_{2,A}^{\\Omega_{bb}^{-}\\to\\Xi_{b}^{0}}$ &$\n-0.010$ &$2.67$ &$0.91$ &\n$g_{3,A}^{\\Omega_{bb}^{-}\\to\\Xi_{b}^{0}}$ &$\n0.091$ &$3.01$ &$0.92$ \\\\ \\hline\n$f_{1,S}^{\\Omega_{bb}^{-}\\to\\Xi_{b}^{\\prime0}}$ &$\n0.099$ &$3.33$ &$0.85$ &\n$f_{2,S}^{\\Omega_{bb}^{-}\\to\\Xi_{b}^{\\prime0}}$ &$\n-0.147$ &$3.06$ &$0.94$ &\n$f_{3,S}^{\\Omega_{bb}^{-}\\to\\Xi_{b}^{\\prime0}}$ &$\n-0.005^{*}$ &$3.63^{*}$ &$1.60^{*}$ \\\\\n$g_{1,S}^{\\Omega_{bb}^{-}\\to\\Xi_{b}^{\\prime0}}$ &$\n0.091$ &$3.35$ &$0.79$ &\n$g_{2,S}^{\\Omega_{bb}^{-}\\to\\Xi_{b}^{\\prime0}}$ &$\n0.013$ &$3.19$ &$0.99$ &\n$g_{3,S}^{\\Omega_{bb}^{-}\\to\\Xi_{b}^{\\prime0}}$ &$\n-0.226$ &$3.29$ &$1.13$ \\\\\n$f_{1,A}^{\\Omega_{bb}^{-}\\to\\Xi_{b}^{\\prime0}}$ &$\n0.098$ &$3.16$ &$0.96$ &\n$f_{2,A}^{\\Omega_{bb}^{-}\\to\\Xi_{b}^{\\prime0}}$ &$\n0.094$ &$3.26$ &$0.96$ &\n$f_{3,A}^{\\Omega_{bb}^{-}\\to\\Xi_{b}^{\\prime0}}$ &$\n-0.066$ &$3.42$ &$1.15$ \\\\\n$g_{1,A}^{\\Omega_{bb}^{-}\\to\\Xi_{b}^{\\prime0}}$ &$\n-0.030$ &$3.34$ &$0.79$ &\n$g_{2,A}^{\\Omega_{bb}^{-}\\to\\Xi_{b}^{\\prime0}}$ &$\n-0.003$ &$3.23$ &$1.00$ &\n$g_{3,A}^{\\Omega_{bb}^{-}\\to\\Xi_{b}^{\\prime0}}$ &$\n0.072$ &$3.31$ &$1.14$ \\\\ \\hline\n$f_{1,S}^{\\Omega_{bb}^{-}\\to\\Omega_{bc}^{(\\prime)}}$ &$\n0.457$ &$4.27$ &$0.81$ &\n$f_{2,S}^{\\Omega_{bb}^{-}\\to\\Omega_{bc}^{(\\prime)}}$ &$\n-0.633$ &$3.20$ &$0.49$ &\n$f_{3,S}^{\\Omega_{bb}^{-}\\to\\Omega_{bc}^{(\\prime)}}$ &$\n0.128$ &$2.70$ &$1.76$ \\\\\n$g_{1,S}^{\\Omega_{bb}^{-}\\to\\Omega_{bc}^{(\\prime)}}$ &$\n0.432$ &$3.76$ &$0.31$ &\n$g_{2,S}^{\\Omega_{bb}^{-}\\to\\Omega_{bc}^{(\\prime)}}$ &$\n0.019^{*}$ &$2.06^{*}$ &$0.34^{*}$&\n$g_{3,S}^{\\Omega_{bb}^{-}\\to\\Omega_{bc}^{(\\prime)}}$ &$\n-0.004^{*}$ &$1.68^{*}$ &$0.21^{*}$ \\\\\n$f_{1,A}^{\\Omega_{bb}^{-}\\to\\Omega_{bc}^{(\\prime)}}$ &$\n0.455$ &$3.92$ &$0.79$ &\n$f_{2,A}^{\\Omega_{bb}^{-}\\to\\Omega_{bc}^{(\\prime)}}$ &$\n0.302$ &$5.44$ &$2.83$ &\n$f_{3,A}^{\\Omega_{bb}^{-}\\to\\Omega_{bc}^{(\\prime)}}$ &$\n-0.041^{*}$ &$2.82^{*}$ &$0.64^{*}$ \\\\\n$g_{1,A}^{\\Omega_{bb}^{-}\\to\\Omega_{bc}^{(\\prime)}}$ &$\n-0.144$ &$3.75$ &$0.31$ &\n$g_{2,A}^{\\Omega_{bb}^{-}\\to\\Omega_{bc}^{(\\prime)}}$ &$\n-0.003^{*}$ &$1.88^{*}$ &$0.27^{*}$ &\n$g_{3,A}^{\\Omega_{bb}^{-}\\to\\Omega_{bc}^{(\\prime)}}$ &$\n-0.014^{*}$ &$1.57^{*}$ &$0.19^{*}$ \\\\ \\hline\n\\hline\n\n\\end{tabular}\n\\end{table}\n\\begin{table}\\footnotesize\\label{Tab:ff_bcc}\n \\caption{Same with Tab.~\\ref{Tab:ff_bbb} expect for bottom charm  baryon ${\\cal B}_{bc^{(\\prime)}}$ decay.}\n\\begin{tabular}{c|c|c|c|c|c|c|c|c|c|c|c}\n\\hline\\hline\n$F$ &$F(0)$ &$m_{\\rm{fit}}$ &$\\delta$ &$F$ &$F(0)$ &$m_{\\rm{fit}}$ &$\\delta$\n&$F$ &$F(0)$ &$m_{\\rm{fit}}$ &$\\delta$\\\\ \\hline\n$f_{1,S}^{\\Xi_{bc}^{(\\prime)}\\to\\Sigma_{c}}$ &$\n0.143$ &$3.76$ &$0.66$ &\n$f_{2,S}^{\\Xi_{bc}^{(\\prime)}\\to\\Sigma_{c}}$ &$\n-0.067$ &$3.23$ &$0.72$ &\n$f_{3,S}^{\\Xi_{bc}^{(\\prime)}\\to\\Sigma_{c}}$ &$\n0.001^{*}$ &$1.12^{*}$ &$0.11^{*}$ \\\\\n$g_{1,S}^{\\Xi_{bc}^{(\\prime)}\\to\\Sigma_{c}}$ &$\n0.123$ &$4.17$ &$0.85$ &\n$g_{2,S}^{\\Xi_{bc}^{(\\prime)}\\to\\Sigma_{c}}$ &$\n0.046$ &$2.81$ &$0.76$ &\n$g_{3,S}^{\\Xi_{bc}^{(\\prime)}\\to\\Sigma_{c}}$ &$\n-0.197$ &$3.07$ &$0.68$ \\\\\n$f_{1,A}^{\\Xi_{bc}^{(\\prime)}\\to\\Sigma_{c}}$ &$\n0.138$ &$3.34$ &$0.77$ &\n$f_{2,A}^{\\Xi_{bc}^{(\\prime)}\\to\\Sigma_{c}}$ &$\n0.147$ &$3.42$ &$0.71$ &\n$f_{3,A}^{\\Xi_{bc}^{(\\prime)}\\to\\Sigma_{c}}$ &$\n-0.095$ &$3.54$ &$0.78$ \\\\\n$g_{1,A}^{\\Xi_{bc}^{(\\prime)}\\to\\Sigma_{c}}$ &$\n-0.041$ &$4.07$ &$0.81$ &\n$g_{2,A}^{\\Xi_{bc}^{(\\prime)}\\to\\Sigma_{c}}$ &$\n-0.011$ &$2.75$ &$0.95$ &\n$g_{3,A}^{\\Xi_{bc}^{(\\prime)}\\to\\Sigma_{c}}$ &$\n0.057$ &$3.04$ &$0.68$ \\\\ \\hline\n$f_{1,S}^{\\Xi_{bc}^{(\\prime)}\\to\\Xi_{cc}}$ &$\n0.546$ &$5.01$ &$0.64$ &\n$f_{2,S}^{\\Xi_{bc}^{(\\prime)}\\to\\Xi_{cc}}$ &$\n-0.245$ &$3.63$ &$0.45$ &\n$f_{3,S}^{\\Xi_{bc}^{(\\prime)}\\to\\Xi_{cc}}$ &$\n0.049$ &$2.84$ &$1.49$ \\\\\n$g_{1,S}^{\\Xi_{bc}^{(\\prime)}\\to\\Xi_{cc}}$ &$\n0.512$ &$5.12$ &$0.51$ &\n$g_{2,S}^{\\Xi_{bc}^{(\\prime)}\\to\\Xi_{cc}}$ &$\n0.086$ &$4.19$ &$2.20$ &\n$g_{3,S}^{\\Xi_{bc}^{(\\prime)}\\to\\Xi_{cc}}$ &$\n-0.492$ &$6.16$ &$5.80$ \\\\\n$f_{1,A}^{\\Xi_{bc}^{(\\prime)}\\to\\Xi_{cc}}$ &$\n0.536$ &$4.17$ &$0.61$ &\n$f_{2,A}^{\\Xi_{bc}^{(\\prime)}\\to\\Xi_{cc}}$ &$\n0.488$ &$4.33$ &$0.59$ &\n$f_{3,A}^{\\Xi_{bc}^{(\\prime)}\\to\\Xi_{cc}}$ &$\n-0.167$ &$5.52$ &$1.81$ \\\\\n$g_{1,A}^{\\Xi_{bc}^{(\\prime)}\\to\\Xi_{cc}}$ &$\n-0.171$ &$4.96$ &$0.49$ &\n$g_{2,A}^{\\Xi_{bc}^{(\\prime)}\\to\\Xi_{cc}}$ &$\n-0.019$ &$4.63$ &$5.41$ &\n$g_{3,A}^{\\Xi_{bc}^{(\\prime)}\\to\\Xi_{cc}}$ &$\n0.133$ &$7.25$ &$12.30$ \\\\ \\hline\n$f_{1,S}^{\\Xi_{bc}^{(\\prime)0}\\to\\Lambda_{c}^{+}}$ &$\n0.143$ &$3.79$ &$0.68$ &\n$f_{2,S}^{\\Xi_{bc}^{(\\prime)0}\\to\\Lambda_{c}^{+}}$ &$\n-0.055$ &$3.27$ &$0.73$ &\n$f_{3,S}^{\\Xi_{bc}^{(\\prime)0}\\to\\Lambda_{c}^{+}}$ &$\n0.009^{*}$ &$0.52^{*}$ &$0.06^{*}$ \\\\\n$g_{1,S}^{\\Xi_{bc}^{(\\prime)0}\\to\\Lambda_{c}^{+}}$ &$\n0.117$ &$4.51$ &$1.16$ &\n$g_{2,S}^{\\Xi_{bc}^{(\\prime)0}\\to\\Lambda_{c}^{+}}$ &$\n0.070$ &$2.80$ &$0.77$ &\n$g_{3,S}^{\\Xi_{bc}^{(\\prime)0}\\to\\Lambda_{c}^{+}}$ &$\n-0.224$ &$2.99$ &$0.68$ \\\\\n$f_{1,A}^{\\Xi_{bc}^{(\\prime)0}\\to\\Lambda_{c}^{+}}$ &$\n0.138$ &$3.37$ &$0.80$ &\n$f_{2,A}^{\\Xi_{bc}^{(\\prime)0}\\to\\Lambda_{c}^{+}}$ &$\n0.147$ &$3.47$ &$0.74$ &\n$f_{3,A}^{\\Xi_{bc}^{(\\prime)0}\\to\\Lambda_{c}^{+}}$ &$\n-0.087$ &$3.71$ &$0.88$ \\\\\n$g_{1,A}^{\\Xi_{bc}^{(\\prime)0}\\to\\Lambda_{c}^{+}}$ &$\n-0.039$ &$4.38$ &$1.08$ &\n$g_{2,A}^{\\Xi_{bc}^{(\\prime)0}\\to\\Lambda_{c}^{+}}$ &$\n-0.019$ &$2.75$ &$0.89$ &\n$g_{3,A}^{\\Xi_{bc}^{(\\prime)0}\\to\\Lambda_{c}^{+}}$ &$\n0.067$ &$2.96$ &$0.68$ \\\\ \\hline\n$f_{1,S}^{\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{c}^{+}}$ &$\n0.133$ &$3.66$ &$0.70$ &\n$f_{2,S}^{\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{c}^{+}}$ &$\n-0.060$ &$3.17$ &$0.77$ &\n$f_{3,S}^{\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{c}^{+}}$ &$\n0.004$ &$0.59$ &$-0.04$ \\\\\n$g_{1,S}^{\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{c}^{+}}$ &$\n0.111$ &$4.15$ &$0.97$ &\n$g_{2,S}^{\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{c}^{+}}$ &$\n0.053$ &$2.77$ &$0.84$ &\n$g_{3,S}^{\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{c}^{+}}$ &$\n-0.204$ &$2.98$ &$0.73$ \\\\\n$f_{1,A}^{\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{c}^{+}}$ &$\n0.129$ &$3.29$ &$0.82$ &\n$f_{2,A}^{\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{c}^{+}}$ &$\n0.135$ &$3.37$ &$0.77$ &\n$f_{3,A}^{\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{c}^{+}}$ &$\n-0.084$ &$3.52$ &$0.86$ \\\\\n$g_{1,A}^{\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{c}^{+}}$ &$\n-0.037$ &$4.07$ &$0.93$ &\n$g_{2,A}^{\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{c}^{+}}$ &$\n-0.014$ &$2.73$ &$0.98$ &\n$g_{3,A}^{\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{c}^{+}}$ &$\n0.061$ &$2.95$ &$0.73$ \\\\ \\hline\n$f_{1,S}^{\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{c}^{\\prime+}}$ &$\n0.133$ &$3.64$ &$0.69$ &\n$f_{2,S}^{\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{c}^{\\prime+}}$ &$\n-0.067$ &$3.14$ &$0.76$ &\n$f_{3,S}^{\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{c}^{\\prime+}}$ &$\n-0.001^{*}$ &$2.30^{*}$ &$0.40^{*}$ \\\\\n$g_{1,S}^{\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{c}^{\\prime+}}$ &$\n0.115$ &$3.97$ &$0.82$ &\n$g_{2,S}^{\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{c}^{\\prime+}}$ &$\n0.038$ &$2.81$ &$0.84$ &\n$g_{3,S}^{\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{c}^{\\prime+}}$ &$\n-0.185$ &$3.04$ &$0.74$ \\\\\n$f_{1,A}^{\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{c}^{\\prime+}}$ &$\n0.129$ &$3.27$ &$0.81$ &\n$f_{2,A}^{\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{c}^{\\prime+}}$ &$\n0.136$ &$3.34$ &$0.75$ &\n$f_{3,A}^{\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{c}^{\\prime+}}$ &$\n-0.089$ &$3.42$ &$0.80$ \\\\\n$g_{1,A}^{\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{c}^{\\prime+}}$ &$\n-0.038$ &$3.89$ &$0.79$ &\n$g_{2,A}^{\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{c}^{\\prime+}}$ &$\n-0.010$ &$2.76$ &$1.04$ &\n$g_{3,A}^{\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{c}^{\\prime+}}$ &$\n0.055$ &$3.03$ &$0.74$ \\\\ \\hline\n$f_{1,S}^{\\Omega_{bc}^{(\\prime)0}\\to\\Omega_{cc}^{+}}$ &$\n0.540$ &$4.79$ &$0.61$ &\n$f_{2,S}^{\\Omega_{bc}^{(\\prime)0}\\to\\Omega_{cc}^{+}}$ &$\n-0.267$ &$3.42$ &$0.42$ &\n$f_{3,S}^{\\Omega_{bc}^{(\\prime)0}\\to\\Omega_{cc}^{+}}$ &$\n0.034$ &$5.44$ &$48.00$ \\\\\n$g_{1,S}^{\\Omega_{bc}^{(\\prime)0}\\to\\Omega_{cc}^{+}}$ &$\n0.513$ &$4.64$ &$0.31$ &\n$g_{2,S}^{\\Omega_{bc}^{(\\prime)0}\\to\\Omega_{cc}^{+}}$ &$\n0.039^{*}$ &$4.04^{*}$ &$2.72^{*}$ &\n$g_{3,S}^{\\Omega_{bc}^{(\\prime)0}\\to\\Omega_{cc}^{+}}$ &$\n-0.379^{*}$ &$6.47^{*}$ &$10.50^{*}$ \\\\\n$f_{1,A}^{\\Omega_{bc}^{(\\prime)0}\\to\\Omega_{cc}^{+}}$ &$\n0.532$ &$4.05$ &$0.60$ &\n$f_{2,A}^{\\Omega_{bc}^{(\\prime)0}\\to\\Omega_{cc}^{+}}$ &$\n0.476$ &$4.22$ &$0.59$ &\n$f_{3,A}^{\\Omega_{bc}^{(\\prime)0}\\to\\Omega_{cc}^{+}}$ &$\n-0.178$ &$4.72$ &$0.99$ \\\\\n$g_{1,A}^{\\Omega_{bc}^{(\\prime)0}\\to\\Omega_{cc}^{+}}$ &$\n-0.171$ &$4.53$ &$0.31$ &\n$g_{2,A}^{\\Omega_{bc}^{(\\prime)0}\\to\\Omega_{cc}^{+}}$ &$\n-0.004^{*}$ &$2.41^{*}$ &$0.57^{*}$ &\n$g_{3,A}^{\\Omega_{bc}^{(\\prime)0}\\to\\Omega_{cc}^{+}}$ &$\n0.098^{*}$ &$4.79^{*}$ &$3.69^{*}$ \\\\ \\hline\n\\hline\n\\end{tabular}\n\\end{table}\n\n\\begin{table}\\footnotesize\\label{Tab:ff_cucc_axial}\n\\caption{Numerical results for the transition $1\/2\\to 1\/2$ form factors $f_{i,S(A)}$ and $g_{i,S(A)}$ at $q^2=0$ of $c\\to u$ processes. The parametrization scheme in Eq.~(\\ref{eq:main_fit_formula_cdecay}) is introduced for these form factors,\nand the value of the singly pole $m_{\\rm pole}$ is taken as $1.87~{\\rm GeV}$ for $c\\to u$.}\n\\begin{tabular}{c|c|c|c|c|c|c|c}\n\\hline\\hline\n$F$ &$F(0)$ &$F$ &$F(0)$ &$F$ &$F(0)$ &$F$ &$F(0)$\\\\ \\hline\n$f_{1,S}^{\\Xi_{cc}\\to\\Sigma_{c}}$ &$0.536$ &\n$g_{1,S}^{\\Xi_{cc}\\to\\Sigma_{c}}$ &$0.422$ &\n$f_{1,A}^{\\Xi_{cc}\\to\\Sigma_{c}}$ &$0.529$ &\n$g_{1,A}^{\\Xi_{cc}\\to\\Sigma_{c}}$ &$-0.141$ \\\\\n$f_{2,S}^{\\Xi_{cc}\\to\\Sigma_{c}}$ &$-0.732$ &\n$g_{2,S}^{\\Xi_{cc}\\to\\Sigma_{c}}$ &$0.558$ &\n$f_{2,A}^{\\Xi_{cc}\\to\\Sigma_{c}}$ &$0.428$ &\n$g_{2,A}^{\\Xi_{cc}\\to\\Sigma_{c}}$ &$-0.176$ \\\\\n$f_{3,S}^{\\Xi_{cc}\\to\\Sigma_{c}}$ &$0.619$ &\n$g_{3,S}^{\\Xi_{cc}\\to\\Sigma_{c}}$ &$-0.788$ &\n$f_{3,A}^{\\Xi_{cc}\\to\\Sigma_{c}}$ &$0.422$ &\n$g_{3,A}^{\\Xi_{cc}\\to\\Sigma_{c}}$ &$0.208$ \\\\\n$f_{1,S}^{T\\Xi_{cc}\\to\\Sigma_{c}}$ &$0.181$ &\n$g_{1,S}^{T\\Xi_{cc}\\to\\Sigma_{c}}$ &$1.958$ &\n$f_{1,A}^{T\\Xi_{cc}\\to\\Sigma_{c}}$ &$0.282$ &\n$g_{1,A}^{T\\Xi_{cc}\\to\\Sigma_{c}}$ &$-0.670$ \\\\\n$f_{2,S}^{T\\Xi_{cc}\\to\\Sigma_{c}}$ &$0.482$ &\n$g_{2,S}^{T\\Xi_{cc}\\to\\Sigma_{c}}$ &$0.415$ &\n$f_{2,A}^{T\\Xi_{cc}\\to\\Sigma_{c}}$ &$-0.150$ &\n$g_{2,A}^{T\\Xi_{cc}\\to\\Sigma_{c}}$ &$-0.135$ \\\\ \\hline\n$f_{1,S}^{\\Xi_{cc}^{+}\\to\\Lambda_{c}^{+}}$ &$0.495$ &\n$g_{1,S}^{\\Xi_{cc}^{+}\\to\\Lambda_{c}^{+}}$ &$0.332$ &\n$f_{1,A}^{\\Xi_{cc}^{+}\\to\\Lambda_{c}^{+}}$ &$0.489$ &\n$g_{1,A}^{\\Xi_{cc}^{+}\\to\\Lambda_{c}^{+}}$ &$-0.111$ \\\\\n$f_{2,S}^{\\Xi_{cc}^{+}\\to\\Lambda_{c}^{+}}$ &$-0.621$ &\n$g_{2,S}^{\\Xi_{cc}^{+}\\to\\Lambda_{c}^{+}}$ &$1.004$ &\n$f_{2,A}^{\\Xi_{cc}^{+}\\to\\Lambda_{c}^{+}}$ &$0.290$ &\n$g_{2,A}^{\\Xi_{cc}^{+}\\to\\Lambda_{c}^{+}}$ &$-0.325$ \\\\\n$f_{3,S}^{\\Xi_{cc}^{+}\\to\\Lambda_{c}^{+}}$ &$0.832$ &\n$g_{3,S}^{\\Xi_{cc}^{+}\\to\\Lambda_{c}^{+}}$ &$-2.957$ &\n$f_{3,A}^{\\Xi_{cc}^{+}\\to\\Lambda_{c}^{+}}$ &$0.648$ &\n$g_{3,A}^{\\Xi_{cc}^{+}\\to\\Lambda_{c}^{+}}$ &$0.943$ \\\\\n$f_{1,S}^{T\\Xi_{cc}^{+}\\to\\Lambda_{c}^{+}}$ &$0.178$ &\n$g_{1,S}^{T\\Xi_{cc}^{+}\\to\\Lambda_{c}^{+}}$ &$0.928$ &\n$f_{1,A}^{T\\Xi_{cc}^{+}\\to\\Lambda_{c}^{+}}$ &$0.271$ &\n$g_{1,A}^{T\\Xi_{cc}^{+}\\to\\Lambda_{c}^{+}}$ &$-0.323$ \\\\\n$f_{2,S}^{T\\Xi_{cc}^{+}\\to\\Lambda_{c}^{+}}$ &$0.363$ &\n$g_{2,S}^{T\\Xi_{cc}^{+}\\to\\Lambda_{c}^{+}}$ &$0.430$ &\n$f_{2,A}^{T\\Xi_{cc}^{+}\\to\\Lambda_{c}^{+}}$ &$-0.133$ &\n$g_{2,A}^{T\\Xi_{cc}^{+}\\to\\Lambda_{c}^{+}}$ &$-0.140$ \\\\ \\hline\n$f_{1,S}^{\\Omega_{cc}^{+}\\to\\Xi_{c}^{+}}$ &$0.501$ &\n$g_{1,S}^{\\Omega_{cc}^{+}\\to\\Xi_{c}^{+}}$ &$0.356$ &\n$f_{1,A}^{\\Omega_{cc}^{+}\\to\\Xi_{c}^{+}}$ &$0.495$ &\n$g_{1,A}^{\\Omega_{cc}^{+}\\to\\Xi_{c}^{+}}$ &$-0.119$ \\\\\n$f_{2,S}^{\\Omega_{cc}^{+}\\to\\Xi_{c}^{+}}$ &$-0.664$ &\n$g_{2,S}^{\\Omega_{cc}^{+}\\to\\Xi_{c}^{+}}$ &$0.883$ &\n$f_{2,A}^{\\Omega_{cc}^{+}\\to\\Xi_{c}^{+}}$ &$0.349$ &\n$g_{2,A}^{\\Omega_{cc}^{+}\\to\\Xi_{c}^{+}}$ &$-0.285$ \\\\\n$f_{3,S}^{\\Omega_{cc}^{+}\\to\\Xi_{c}^{+}}$ &$0.744$ &\n$g_{3,S}^{\\Omega_{cc}^{+}\\to\\Xi_{c}^{+}}$ &$-2.622$ &\n$f_{3,A}^{\\Omega_{cc}^{+}\\to\\Xi_{c}^{+}}$ &$0.559$ &\n$g_{3,A}^{\\Omega_{cc}^{+}\\to\\Xi_{c}^{+}}$ &$0.830$ \\\\\n$f_{1,S}^{T\\Omega_{cc}^{+}\\to\\Xi_{c}^{+}}$ &$0.178$ &\n$g_{1,S}^{T\\Omega_{cc}^{+}\\to\\Xi_{c}^{+}}$ &$1.214$ &\n$f_{1,A}^{T\\Omega_{cc}^{+}\\to\\Xi_{c}^{+}}$ &$0.269$ &\n$g_{1,A}^{T\\Omega_{cc}^{+}\\to\\Xi_{c}^{+}}$ &$-0.418$ \\\\\n$f_{2,S}^{T\\Omega_{cc}^{+}\\to\\Xi_{c}^{+}}$ &$0.397$ &\n$g_{2,S}^{T\\Omega_{cc}^{+}\\to\\Xi_{c}^{+}}$ &$0.421$ &\n$f_{2,A}^{T\\Omega_{cc}^{+}\\to\\Xi_{c}^{+}}$ &$-0.135$ &\n$g_{2,A}^{T\\Omega_{cc}^{+}\\to\\Xi_{c}^{+}}$ &$-0.138$ \\\\ \\hline\n$f_{1,S}^{\\Omega_{cc}^{+}\\to\\Xi_{c}^{\\prime+}}$ &$0.529$ &\n$g_{1,S}^{\\Omega_{cc}^{+}\\to\\Xi_{c}^{\\prime+}}$ &$0.418$ &\n$f_{1,A}^{\\Omega_{cc}^{+}\\to\\Xi_{c}^{\\prime+}}$ &$0.523$ &\n$g_{1,A}^{\\Omega_{cc}^{+}\\to\\Xi_{c}^{\\prime+}}$ &$-0.139$ \\\\\n$f_{2,S}^{\\Omega_{cc}^{+}\\to\\Xi_{c}^{\\prime+}}$ &$-0.739$ &\n$g_{2,S}^{\\Omega_{cc}^{+}\\to\\Xi_{c}^{\\prime+}}$ &$0.560$ &\n$f_{2,A}^{\\Omega_{cc}^{+}\\to\\Xi_{c}^{\\prime+}}$ &$0.451$ &\n$g_{2,A}^{\\Omega_{cc}^{+}\\to\\Xi_{c}^{\\prime+}}$ &$-0.177$ \\\\\n$f_{3,S}^{\\Omega_{cc}^{+}\\to\\Xi_{c}^{\\prime+}}$ &$0.596$ &\n$g_{3,S}^{\\Omega_{cc}^{+}\\to\\Xi_{c}^{\\prime+}}$ &$-0.901$ &\n$f_{3,A}^{\\Omega_{cc}^{+}\\to\\Xi_{c}^{\\prime+}}$ &$0.402$ &\n$g_{3,A}^{\\Omega_{cc}^{+}\\to\\Xi_{c}^{\\prime+}}$ &$0.249$ \\\\\n$f_{1,S}^{T\\Omega_{cc}^{+}\\to\\Xi_{c}^{\\prime+}}$ &$0.181$ &\n$g_{1,S}^{T\\Omega_{cc}^{+}\\to\\Xi_{c}^{\\prime+}}$ &$1.961$ &\n$f_{1,A}^{T\\Omega_{cc}^{+}\\to\\Xi_{c}^{\\prime+}}$ &$0.278$ &\n$g_{1,A}^{T\\Omega_{cc}^{+}\\to\\Xi_{c}^{\\prime+}}$ &$-0.669$ \\\\\n$f_{2,S}^{T\\Omega_{cc}^{+}\\to\\Xi_{c}^{\\prime+}}$ &$0.478$ &\n$g_{2,S}^{T\\Omega_{cc}^{+}\\to\\Xi_{c}^{\\prime+}}$ &$0.411$ &\n$f_{2,A}^{T\\Omega_{cc}^{+}\\to\\Xi_{c}^{\\prime+}}$ &$-0.147$ &\n$g_{2,A}^{T\\Omega_{cc}^{+}\\to\\Xi_{c}^{\\prime+}}$ &$-0.134$ \\\\ \\hline\n$f_{1,S}^{\\Xi_{bc}^{(\\prime)}\\to\\Sigma_{b}}$ &$0.146$ &\n$g_{1,S}^{\\Xi_{bc}^{(\\prime)}\\to\\Sigma_{b}}$ &$0.227$ &\n$f_{1,A}^{\\Xi_{bc}^{(\\prime)}\\to\\Sigma_{b}}$ &$0.144$ &\n$g_{1,A}^{\\Xi_{bc}^{(\\prime)}\\to\\Sigma_{b}}$ &$-0.076$ \\\\\n$f_{2,S}^{\\Xi_{bc}^{(\\prime)}\\to\\Sigma_{b}}$ &$2.315$ &\n$g_{2,S}^{\\Xi_{bc}^{(\\prime)}\\to\\Sigma_{b}}$ &$-0.795$ &\n$f_{2,A}^{\\Xi_{bc}^{(\\prime)}\\to\\Sigma_{b}}$ &$3.504$ &\n$g_{2,A}^{\\Xi_{bc}^{(\\prime)}\\to\\Sigma_{b}}$ &$0.272$ \\\\\n$f_{3,S}^{\\Xi_{bc}^{(\\prime)}\\to\\Sigma_{b}}$ &$-0.468$ &\n$g_{3,S}^{\\Xi_{bc}^{(\\prime)}\\to\\Sigma_{b}}$ &$13.260$ &\n$f_{3,A}^{\\Xi_{bc}^{(\\prime)}\\to\\Sigma_{b}}$ &$-0.559$ &\n$g_{3,A}^{\\Xi_{bc}^{(\\prime)}\\to\\Sigma_{b}}$ &$-4.494$ \\\\\n$f_{1,S}^{T\\Xi_{bc}^{(\\prime)}\\to\\Sigma_{b}}$ &$-0.007$ &\n$g_{1,S}^{T\\Xi_{bc}^{(\\prime)}\\to\\Sigma_{b}}$ &$1.817$ &\n$f_{1,A}^{T\\Xi_{bc}^{(\\prime)}\\to\\Sigma_{b}}$ &$0.023$ &\n$g_{1,A}^{T\\Xi_{bc}^{(\\prime)}\\to\\Sigma_{b}}$ &$-0.618$ \\\\\n$f_{2,S}^{T\\Xi_{bc}^{(\\prime)}\\to\\Sigma_{b}}$ &$0.465$ &\n$g_{2,S}^{T\\Xi_{bc}^{(\\prime)}\\to\\Sigma_{b}}$ &$0.095$ &\n$f_{2,A}^{T\\Xi_{bc}^{(\\prime)}\\to\\Sigma_{b}}$ &$0.093$ &\n$g_{2,A}^{T\\Xi_{bc}^{(\\prime)}\\to\\Sigma_{b}}$ &$-0.031$ \\\\ \\hline\n$f_{1,S}^{\\Xi_{bc}^{(\\prime)0}\\to\\Lambda_{b}^{0}}$ &$0.129$ &\n$g_{1,S}^{\\Xi_{bc}^{(\\prime)0}\\to\\Lambda_{b}^{0}}$ &$0.169$ &\n$f_{1,A}^{\\Xi_{bc}^{(\\prime)0}\\to\\Lambda_{b}^{0}}$ &$0.127$ &\n$g_{1,A}^{\\Xi_{bc}^{(\\prime)0}\\to\\Lambda_{b}^{0}}$ &$-0.056$ \\\\\n$f_{2,S}^{\\Xi_{bc}^{(\\prime)0}\\to\\Lambda_{b}^{0}}$ &$1.699$ &\n$g_{2,S}^{\\Xi_{bc}^{(\\prime)0}\\to\\Lambda_{b}^{0}}$ &$-0.164$ &\n$f_{2,A}^{\\Xi_{bc}^{(\\prime)0}\\to\\Lambda_{b}^{0}}$ &$2.582$ &\n$g_{2,A}^{\\Xi_{bc}^{(\\prime)0}\\to\\Lambda_{b}^{0}}$ &$0.060$ \\\\\n$f_{3,S}^{\\Xi_{bc}^{(\\prime)0}\\to\\Lambda_{b}^{0}}$ &$-0.288$ &\n$g_{3,S}^{\\Xi_{bc}^{(\\prime)0}\\to\\Lambda_{b}^{0}}$ &$4.308$ &\n$f_{3,A}^{\\Xi_{bc}^{(\\prime)0}\\to\\Lambda_{b}^{0}}$ &$-0.370$ &\n$g_{3,A}^{\\Xi_{bc}^{(\\prime)0}\\to\\Lambda_{b}^{0}}$ &$-1.491$ \\\\\n$f_{1,S}^{T\\Xi_{bc}^{(\\prime)0}\\to\\Lambda_{b}^{0}}$ &$-0.005$ &\n$g_{1,S}^{T\\Xi_{bc}^{(\\prime)0}\\to\\Lambda_{b}^{0}}$ &$0.885$ &\n$f_{1,A}^{T\\Xi_{bc}^{(\\prime)0}\\to\\Lambda_{b}^{0}}$ &$0.021$ &\n$g_{1,A}^{T\\Xi_{bc}^{(\\prime)0}\\to\\Lambda_{b}^{0}}$ &$-0.304$ \\\\\n$f_{2,S}^{T\\Xi_{bc}^{(\\prime)0}\\to\\Lambda_{b}^{0}}$ &$0.344$ &\n$g_{2,S}^{T\\Xi_{bc}^{(\\prime)0}\\to\\Lambda_{b}^{0}}$ &$0.100$ &\n$f_{2,A}^{T\\Xi_{bc}^{(\\prime)0}\\to\\Lambda_{b}^{0}}$ &$0.067$ &\n$g_{2,A}^{T\\Xi_{bc}^{(\\prime)0}\\to\\Lambda_{b}^{0}}$ &$-0.032$ \\\\ \\hline\n$f_{1,S}^{\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{b}^{0}}$ &$0.145$ &\n$g_{1,S}^{\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{b}^{0}}$ &$0.204$ &\n$f_{1,A}^{\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{b}^{0}}$ &$0.143$ &\n$g_{1,A}^{\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{b}^{0}}$ &$-0.068$ \\\\\n$f_{2,S}^{\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{b}^{0}}$ &$2.119$ &\n$g_{2,S}^{\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{b}^{0}}$ &$-0.471$ &\n$f_{2,A}^{\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{b}^{0}}$ &$3.200$ &\n$g_{2,A}^{\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{b}^{0}}$ &$0.163$ \\\\\n$f_{3,S}^{\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{b}^{0}}$ &$-0.393$ &\n$g_{3,S}^{\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{b}^{0}}$ &$8.680$ &\n$f_{3,A}^{\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{b}^{0}}$ &$-0.482$ &\n$g_{3,A}^{\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{b}^{0}}$ &$-2.955$ \\\\\n$f_{1,S}^{T\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{b}^{0}}$ &$-0.004$ &\n$g_{1,S}^{T\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{b}^{0}}$ &$1.399$ &\n$f_{1,A}^{T\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{b}^{0}}$ &$0.024$ &\n$g_{1,A}^{T\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{b}^{0}}$ &$-0.476$ \\\\\n$f_{2,S}^{T\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{b}^{0}}$ &$0.419$ &\n$g_{2,S}^{T\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{b}^{0}}$ &$0.103$ &\n$f_{2,A}^{T\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{b}^{0}}$ &$0.086$ &\n$g_{2,A}^{T\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{b}^{0}}$ &$-0.033$ \\\\ \\hline\n$f_{1,S}^{\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{b}^{\\prime0}}$ &$0.160$ &\n$g_{1,S}^{\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{b}^{\\prime0}}$ &$0.259$ &\n$f_{1,A}^{\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{b}^{\\prime0}}$ &$0.158$ &\n$g_{1,A}^{\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{b}^{\\prime0}}$ &$-0.086$ \\\\\n$f_{2,S}^{\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{b}^{\\prime0}}$ &$2.720$ &\n$g_{2,S}^{\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{b}^{\\prime0}}$ &$-1.137$ &\n$f_{2,A}^{\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{b}^{\\prime0}}$ &$4.095$ &\n$g_{2,A}^{\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{b}^{\\prime0}}$ &$0.385$ \\\\\n$f_{3,S}^{\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{b}^{\\prime0}}$ &$-0.571$ &\n$g_{3,S}^{\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{b}^{\\prime0}}$ &$19.110$ &\n$f_{3,A}^{\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{b}^{\\prime0}}$ &$-0.668$ &\n$g_{3,A}^{\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{b}^{\\prime0}}$ &$-6.448$ \\\\\n$f_{1,S}^{T\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{b}^{\\prime0}}$ &$-0.005$ &\n$g_{1,S}^{T\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{b}^{\\prime0}}$ &$2.412$ &\n$f_{1,A}^{T\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{b}^{\\prime0}}$ &$0.026$ &\n$g_{1,A}^{T\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{b}^{\\prime0}}$ &$-0.817$ \\\\\n$f_{2,S}^{T\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{b}^{\\prime0}}$ &$0.535$ &\n$g_{2,S}^{T\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{b}^{\\prime0}}$ &$0.097$ &\n$f_{2,A}^{T\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{b}^{\\prime0}}$ &$0.112$ &\n$g_{2,A}^{T\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{b}^{\\prime0}}$ &$-0.031$ \\\\ \\hline\n\\hline\n\\end{tabular}\n\\end{table}\n\n\\begin{table}\\footnotesize\\label{Tab:ff_bdbb_axial}\n\\caption{Numerical results for the transition $1\/2\\to 1\/2$ form factors $f^{(T)}_{i,S(A)}$ and $g^{(T)}_{i,S(A)}$ of $b\\to d$ processes. The parametrization scheme in Eq.~(\\ref{eq:auxiliary_fit_formula_bdecay}) is introduced for these form factors with asterisk, and Eq.~(\\ref{eq:main_fit_formula_bdecay}) for all the other ones.}\n\\setlength{\\tabcolsep}{0.4mm}{\n\\begin{tabular}{c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c}\n\\hline\\hline\n$F$ &$F(0)$ &$m_{\\rm{fit}}$ &$\\delta$ &$F$ &$F(0)$ &$m_{\\rm{fit}}$ &$\\delta$&\n$F$ &$F(0)$ &$m_{\\rm{fit}}$ &$\\delta$&$F$ &$F(0)$ &$m_{\\rm{fit}}$ &$\\delta$\n\\\\ \\hline\n$f_{1,S}^{\\Xi_{bb}^{0}\\to\\Lambda_{b}^{0}}$ &$\n0.100$ &$3.40$ &$0.86$ &\n$g_{1,S}^{\\Xi_{bb}^{0}\\to\\Lambda_{b}^{0}}$ &$\n0.087$ &$3.57$ &$0.91$ &\n$f_{1,A}^{\\Xi_{bb}^{0}\\to\\Lambda_{b}^{0}}$ &$\n0.100$ &$3.22$ &$0.96$ &\n$g_{1,A}^{\\Xi_{bb}^{0}\\to\\Lambda_{b}^{0}}$ &$\n-0.029$ &$3.56$ &$0.91$ \\\\\n$f_{2,S}^{\\Xi_{bb}^{0}\\to\\Lambda_{b}^{0}}$ &$\n-0.136$ &$3.10$ &$0.93$ &\n$g_{2,S}^{\\Xi_{bb}^{0}\\to\\Lambda_{b}^{0}}$ &$\n0.041$ &$2.70$ &$0.89$ &\n$f_{2,A}^{\\Xi_{bb}^{0}\\to\\Lambda_{b}^{0}}$ &$\n0.092$ &$3.36$ &$0.99$ &\n$g_{2,A}^{\\Xi_{bb}^{0}\\to\\Lambda_{b}^{0}}$ &$\n-0.013$ &$2.68$ &$0.92$ \\\\\n$f_{3,S}^{\\Xi_{bb}^{0}\\to\\Lambda_{b}^{0}}$ &$\n0.008$ &$0.32$ &$-0.01$ &\n$g_{3,S}^{\\Xi_{bb}^{0}\\to\\Lambda_{b}^{0}}$ &$\n-0.298$ &$2.99$ &$0.89$ &\n$f_{3,A}^{\\Xi_{bb}^{0}\\to\\Lambda_{b}^{0}}$ &$\n-0.055$ &$3.79$ &$1.45$ &\n$g_{3,A}^{\\Xi_{bb}^{0}\\to\\Lambda_{b}^{0}}$ &$\n0.096$ &$2.98$ &$0.89$ \\\\\n$f_{1,S}^{T\\Xi_{bb}^{0}\\to\\Lambda_{b}^{0}}$ &$\n0.072$ &$3.32$ &$0.89$ &\n$g_{1,S}^{T\\Xi_{bb}^{0}\\to\\Lambda_{b}^{0}}$ &$\n0.114$ &$2.93$ &$1.12$ &\n$f_{1,A}^{T\\Xi_{bb}^{0}\\to\\Lambda_{b}^{0}}$ &$\n0.104$ &$3.27$ &$0.90$ &\n$g_{1,A}^{T\\Xi_{bb}^{0}\\to\\Lambda_{b}^{0}}$ &$\n-0.040$ &$2.94$ &$1.10$ \\\\\n$f_{2,S}^{T\\Xi_{bb}^{0}\\to\\Lambda_{b}^{0}}$ &$\n0.075$ &$3.30$ &$0.76$ &\n$g_{2,S}^{T\\Xi_{bb}^{0}\\to\\Lambda_{b}^{0}}$ &$\n0.091$ &$3.73$ &$0.93$ &\n$f_{2,A}^{T\\Xi_{bb}^{0}\\to\\Lambda_{b}^{0}}$ &$\n-0.045$ &$3.20$ &$1.74$ &\n$g_{2,A}^{T\\Xi_{bb}^{0}\\to\\Lambda_{b}^{0}}$ &$\n-0.030$ &$3.74$ &$0.93$ \\\\\n\\hline\n$f_{1,S}^{\\Xi_{bb}\\to\\Sigma_{b}}$ &$\n0.102$ &$3.35$ &$0.83$ &\n$g_{1,S}^{\\Xi_{bb}\\to\\Sigma_{b}}$ &$\n0.094$ &$3.36$ &$0.76$ &\n$f_{1,A}^{\\Xi_{bb}\\to\\Sigma_{b}}$ &$\n0.102$ &$3.18$ &$0.94$ &\n$g_{1,A}^{\\Xi_{bb}\\to\\Sigma_{b}}$ &$\n-0.031$ &$3.35$ &$0.76$ \\\\\n$f_{2,S}^{\\Xi_{bb}\\to\\Sigma_{b}}$ &$\n-0.150$ &$3.06$ &$0.92$ &\n$g_{2,S}^{\\Xi_{bb}\\to\\Sigma_{b}}$ &$\n0.012$ &$3.42$ &$1.17$ &\n$f_{2,A}^{\\Xi_{bb}\\to\\Sigma_{b}}$ &$\n0.096$ &$3.28$ &$0.95$ &\n$g_{2,A}^{\\Xi_{bb}\\to\\Sigma_{b}}$ &$\n-0.003$ &$3.57$ &$1.32$ \\\\\n$f_{3,S}^{\\Xi_{bb}\\to\\Sigma_{b}}$ &$\n-0.004^{*}$ &$3.24^{*}$ &$1.09^{*}$ &\n$g_{3,S}^{\\Xi_{bb}\\to\\Sigma_{b}}$ &$\n-0.222$ &$3.36$ &$1.16$ &\n$f_{3,A}^{\\Xi_{bb}\\to\\Sigma_{b}}$ &$\n-0.068$ &$3.44$ &$1.13$ &\n$g_{3,A}^{\\Xi_{bb}\\to\\Sigma_{b}}$ &$\n0.070$ &$3.38$ &$1.17$ \\\\\n$f_{1,S}^{T\\Xi_{bb}\\to\\Sigma_{b}}$ &$\n0.072$ &$3.25$ &$0.87$ &\n$g_{1,S}^{T\\Xi_{bb}\\to\\Sigma_{b}}$ &$\n0.154$ &$2.75$ &$1.17$ &\n$f_{1,A}^{T\\Xi_{bb}\\to\\Sigma_{b}}$ &$\n0.105$ &$3.21$ &$0.88$ &\n$g_{1,A}^{T\\Xi_{bb}\\to\\Sigma_{b}}$ &$\n-0.053$ &$2.75$ &$1.14$ \\\\\n$f_{2,S}^{T\\Xi_{bb}\\to\\Sigma_{b}}$ &$\n0.083$ &$3.22$ &$0.74$ &\n$g_{2,S}^{T\\Xi_{bb}\\to\\Sigma_{b}}$ &$\n0.090$ &$4.00$ &$1.25$ &\n$f_{2,A}^{T\\Xi_{bb}to\\Sigma_{b}}$ &$\n-0.047$ &$3.16$ &$1.71$ &\n$g_{2,A}^{T\\Xi_{bb}\\to\\Sigma_{b}}$ &$\n-0.029$ &$4.02$ &$1.27$\n\\\\\\hline\n$f_{1,S}^{\\Omega_{bb}^{-}\\to\\Xi_{b}^{-}}$ &$\n0.098$ &$3.36$ &$0.86$ &\n$g_{1,S}^{\\Omega_{bb}^{-}\\to\\Xi_{b}^{-}}$ &$\n0.086$ &$3.50$ &$0.89$ &\n$f_{1,A}^{\\Omega_{bb}^{-}\\to\\Xi_{b}^{-}}$ &$\n0.097$ &$3.19$ &$0.97$ &\n$g_{1,A}^{\\Omega_{bb}^{-}\\to\\Xi_{b}^{-}}$ &$\n-0.029$ &$3.49$ &$0.89$ \\\\\n$f_{2,S}^{\\Omega_{bb}^{-}\\to\\Xi_{b}^{-}}$ &$\n-0.137$ &$3.09$ &$0.95$ &\n$g_{2,S}^{\\Omega_{bb}^{-}\\to\\Xi_{b}^{-}}$ &$\n0.034$ &$2.70$ &$0.88$ &\n$f_{2,A}^{\\Omega_{bb}^{-}\\to\\Xi_{b}^{-}}$ &$\n0.090$ &$3.32$ &$0.99$ &\n$g_{2,A}^{\\Omega_{bb}^{-}\\to\\Xi_{b}^{-}}$ &$\n-0.010$ &$2.68$ &$0.90$ \\\\\n$f_{3,S}^{\\Omega_{bb}^{-}\\to\\Xi_{b}^{-}}$ &$\n0.004^{*}$ &$0.96^{*}$ &$0.06^{*}$ &\n$g_{3,S}^{\\Omega_{bb}^{-}\\to\\Xi_{b}^{-}}$ &$\n-0.282$ &$3.01$ &$0.93$ &\n$f_{3,A}^{\\Omega_{bb}^{-}\\to\\Xi_{b}^{-}}$ &$\n-0.057$ &$3.64$ &$1.34$ &\n$g_{3,A}^{\\Omega_{bb}^{-}\\to\\Xi_{b}^{-}}$ &$\n0.091$ &$3.01$ &$0.93$ \\\\\n$f_{1,S}^{T\\Omega_{bb}^{-}\\to\\Xi_{b}^{-}}$ &$\n0.069$ &$3.28$ &$0.90$ &\n$g_{1,S}^{T\\Omega_{bb}^{-}\\to\\Xi_{b}^{-}}$ &$\n0.119$ &$2.88$ &$1.14$ &\n$f_{1,A}^{T\\Omega_{bb}^{-}\\to\\Xi_{b}^{-}}$ &$\n0.101$ &$3.24$ &$0.91$ &\n$g_{1,A}^{T\\Omega_{bb}^{-}\\to\\Xi_{b}^{-}}$ &$\n-0.041$ &$2.89$ &$1.13$ \\\\\n$f_{2,S}^{T\\Omega_{bb}^{-}\\to\\Xi_{b}^{-}}$ &$\n0.074$ &$3.26$ &$0.77$ &\n$g_{2,S}^{T\\Omega_{bb}^{-}\\to\\Xi_{b}^{-}}$ &$\n0.088$ &$3.74$ &$0.98$ &\n$f_{2,A}^{T\\Omega_{bb}^{-}\\to\\Xi_{b}^{-}}$ &$\n-0.044$ &$3.18$ &$1.73$ &\n$g_{2,A}^{T\\Omega_{bb}^{-}\\to\\Xi_{b}^{-}}$ &$\n-0.029$ &$3.75$ &$0.99$ \\\\\\hline\n$f_{1,S}^{\\Omega_{bb}^{-}\\to\\Xi_{b}^{\\prime-}}$ &$\n0.099$ &$3.33$ &$0.85$ &\n$g_{1,S}^{\\Omega_{bb}^{-}\\to\\Xi_{b}^{\\prime-}}$ &$\n0.091$ &$3.35$ &$0.79$ &\n$f_{1,A}^{\\Omega_{bb}^{-}\\to\\Xi_{b}^{\\prime-}}$ &$\n0.098$ &$3.16$ &$0.96$ &\n$g_{1,A}^{\\Omega_{bb}^{-}\\to\\Xi_{b}^{\\prime-}}$ &$\n-0.030$ &$3.34$ &$0.79$ \\\\\n$f_{2,S}^{\\Omega_{bb}^{-}\\to\\Xi_{b}^{\\prime-}}$ &$\n-0.147$ &$3.06$ &$0.94$ &\n$g_{2,S}^{\\Omega_{bb}^{-}\\to\\Xi_{b}^{\\prime-}}$ &$\n0.013$ &$3.19$ &$0.99$ &\n$f_{2,A}^{\\Omega_{bb}^{-}\\to\\Xi_{b}^{\\prime-}}$ &$\n0.094$ &$3.26$ &$0.96$ &\n$g_{2,A}^{\\Omega_{bb}^{-}\\to\\Xi_{b}^{\\prime-}}$ &$\n-0.003$ &$3.23$ &$1.00$ \\\\\n$f_{3,S}^{\\Omega_{bb}^{-}\\to\\Xi_{b}^{\\prime-}}$ &$\n-0.005^{*}$ &$3.63^{*}$ &$1.60^{*}$ &\n$g_{3,S}^{\\Omega_{bb}^{-}\\to\\Xi_{b}^{\\prime-}}$ &$\n-0.226$ &$3.29$ &$1.13$ &\n$f_{3,A}^{\\Omega_{bb}^{-}\\to\\Xi_{b}^{\\prime-}}$ &$\n-0.066$ &$3.42$ &$1.15$ &\n$g_{3,A}^{\\Omega_{bb}^{-}\\to\\Xi_{b}^{\\prime-}}$ &$\n0.072$ &$3.31$ &$1.14$ \\\\\n$f_{1,S}^{T\\Omega_{bb}^{-}\\to\\Xi_{b}^{\\prime-}}$ &$\n0.069$ &$3.24$ &$0.88$ &\n$g_{1,S}^{T\\Omega_{bb}^{-}\\to\\Xi_{b}^{\\prime-}}$ &$\n0.148$ &$2.76$ &$1.19$ &\n$f_{1,A}^{T\\Omega_{bb}^{-}\\to\\Xi_{b}^{\\prime-}}$ &$\n0.101$ &$3.20$ &$0.89$ &\n$g_{1,A}^{T\\Omega_{bb}^{-}\\to\\Xi_{b}^{\\prime-}}$ &$\n-0.051$ &$2.76$ &$1.17$ \\\\\n$f_{2,S}^{T\\Omega_{bb}^{-}\\to\\Xi_{b}^{\\prime-}}$ &$\n0.080$ &$3.20$ &$0.76$ &\n$g_{2,S}^{T\\Omega_{bb}^{-}\\to\\Xi_{b}^{\\prime-}}$ &$\n0.087$ &$3.93$ &$1.21$ &\n$f_{2,A}^{T\\Omega_{bb}^{-}\\to\\Xi_{b}^{\\prime-}}$ &$\n-0.045$ &$3.15$ &$1.71$ &\n$g_{2,A}^{T\\Omega_{bb}^{-}\\to\\Xi_{b}^{\\prime-}}$ &$\n-0.029$ &$3.95$ &$1.23$ \\\\\n\\hline\n\\hline\n$f_{1,S}^{\\Xi_{bc}^{(\\prime)+}\\to\\Lambda_{c}^{+}}$ &$\n0.143$ &$3.74$ &$0.56$ &\n$g_{1,S}^{\\Xi_{bc}^{(\\prime)+}\\to\\Lambda_{c}^{+}}$ &$\n0.117$ &$4.48$ &$1.06$ &\n$f_{1,A}^{\\Xi_{bc}^{(\\prime)+}\\to\\Lambda_{c}^{+}}$ &$\n0.138$ &$3.28$ &$0.59$ &\n$g_{1,A}^{\\Xi_{bc}^{(\\prime)+}\\to\\Lambda_{c}^{+}}$ &$\n-0.039$ &$4.35$ &$0.97$ \\\\\n$f_{2,S}^{\\Xi_{bc}^{(\\prime)+}\\to\\Lambda_{c}^{+}}$ &$\n-0.055$ &$3.19$ &$0.55$ &\n$g_{2,S}^{\\Xi_{bc}^{(\\prime)+}\\to\\Lambda_{c}^{+}}$ &$\n0.070$ &$2.67$ &$0.52$ &\n$f_{2,A}^{\\Xi_{bc}^{(\\prime)+}\\to\\Lambda_{c}^{+}}$ &$\n0.147$ &$3.39$ &$0.57$ &\n$g_{2,A}^{\\Xi_{bc}^{(\\prime)+}\\to\\Lambda_{c}^{+}}$ &$\n-0.019$ &$2.60$ &$0.56$ \\\\\n$f_{3,S}^{\\Xi_{bc}^{(\\prime)+}\\to\\Lambda_{c}^{+}}$ &$\n0.009$ &$2.60$ &$3.48$ &\n$g_{3,S}^{\\Xi_{bc}^{(\\prime)+}\\to\\Lambda_{c}^{+}}$ &$\n-0.224$ &$2.90$ &$0.50$ &\n$f_{3,A}^{\\Xi_{bc}^{(\\prime)+}\\to\\Lambda_{c}^{+}}$ &$\n-0.087$ &$3.63$ &$0.70$ &\n$g_{3,A}^{\\Xi_{bc}^{(\\prime)+}\\to\\Lambda_{c}^{+}}$ &$\n0.067$ &$2.87$ &$0.50$ \\\\\n$f_{1,S}^{T\\Xi_{bc}^{(\\prime)+}\\to\\Lambda_{c}^{+}}$ &$\n0.068$ &$3.54$ &$0.61$ &\n$g_{1,S}^{T\\Xi_{bc}^{(\\prime)+}\\to\\Lambda_{c}^{+}}$ &$\n-0.010^{*}$ &$0.80^{*}$ &$0.34^{*}$ &\n$f_{1,A}^{T\\Xi_{bc}^{(\\prime)+}\\to\\Lambda_{c}^{+}}$ &$\n0.138$ &$3.42$ &$0.58$ &\n$g_{1,A}^{T\\Xi_{bc}^{(\\prime)+}\\to\\Lambda_{c}^{+}}$ &$\n-0.002^{*}$ &$2.31^{*}$ &$0.68^{*}$ \\\\\n$f_{2,S}^{T\\Xi_{bc}^{(\\prime)+}\\to\\Lambda_{c}^{+}}$ &$\n0.110$ &$3.85$ &$0.51$ &\n$g_{2,S}^{T\\Xi_{bc}^{(\\prime)+}\\to\\Lambda_{c}^{+}}$ &$\n0.142$ &$3.59$ &$0.43$ &\n$f_{2,A}^{T\\Xi_{bc}^{(\\prime)+}\\to\\Lambda_{c}^{+}}$ &$\n-0.046$ &$2.79$ &$1.24$ &\n$g_{2,A}^{T\\Xi_{bc}^{(\\prime)+}\\to\\Lambda_{c}^{+}}$ &$\n-0.045$ &$3.62$ &$0.42$ \\\\\n\\hline\n$f_{1,S}^{\\Xi_{bc}^{(\\prime)}\\to\\Sigma_{c}}$ &$\n0.143$ &$3.71$ &$0.55$ &\n$g_{1,S}^{\\Xi_{bc}^{(\\prime)}\\to\\Sigma_{c}}$ &$\n0.123$ &$4.13$ &$0.75$ &\n$f_{1,A}^{\\Xi_{bc}^{(\\prime)}\\to\\Sigma_{c}}$ &$\n0.138$ &$3.24$ &$0.57$ &\n$g_{1,A}^{\\Xi_{bc}^{(\\prime)}\\to\\Sigma_{c}}$ &$\n-0.041$ &$4.03$ &$0.70$ \\\\\n$f_{2,S}^{\\Xi_{bc}^{(\\prime)}\\to\\Sigma_{c}}$ &$\n-0.067$ &$3.15$ &$0.54$ &\n$g_{2,S}^{\\Xi_{bc}^{(\\prime)}\\to\\Sigma_{c}}$ &$\n0.046$ &$2.70$ &$0.52$ &\n$f_{2,A}^{\\Xi_{bc}^{(\\prime)}\\to\\Sigma_{c}}$ &$\n0.147$ &$3.34$ &$0.55$ &\n$g_{2,A}^{\\Xi_{bc}^{(\\prime)}\\to\\Sigma_{c}}$ &$\n-0.012$ &$2.58$ &$0.57$ \\\\\n$f_{3,S}^{\\Xi_{bc}^{(\\prime)}\\to\\Sigma_{c}}$ &$\n0.001^{*}$ &$1.22^{*}$ &$0.11^{*}$ &\n$g_{3,S}^{\\Xi_{bc}^{(\\prime)}\\to\\Sigma_{c}}$ &$\n-0.197$ &$2.98$ &$0.51$ &\n$f_{3,A}^{\\Xi_{bc}^{(\\prime)}\\to\\Sigma_{c}}$ &$\n-0.095$ &$3.46$ &$0.61$ &\n$g_{3,A}^{\\Xi_{bc}^{(\\prime)}\\to\\Sigma_{c}}$ &$\n0.057$ &$2.95$ &$0.50$ \\\\\n$f_{1,S}^{T\\Xi_{bc}^{(\\prime)}\\to\\Sigma_{c}}$ &$\n0.064$ &$3.51$ &$0.59$ &\n$g_{1,S}^{T\\Xi_{bc}^{(\\prime)}\\to\\Sigma_{c}}$ &$\n0.006$ &$7.53$ &$17.40$ &\n$f_{1,A}^{T\\Xi_{bc}^{(\\prime)}\\to\\Sigma_{c}}$ &$\n0.135$ &$3.37$ &$0.56$ &\n$g_{1,A}^{T\\Xi_{bc}^{(\\prime)}\\to\\Sigma_{c}}$ &$\n-0.007$ &$3.65$ &$0.79$ \\\\\n$f_{2,S}^{T\\Xi_{bc}^{(\\prime)}\\to\\Sigma_{c}}$ &$\n0.119$ &$3.71$ &$0.47$ &\n$g_{2,S}^{T\\Xi_{bc}^{(\\prime)}\\to\\Sigma_{c}}$ &$\n0.140$ &$3.70$ &$0.49$ &\n$f_{2,A}^{T\\Xi_{bc}^{(\\prime)}\\to\\Sigma_{c}}$ &$\n-0.047$ &$2.77$ &$1.19$ &\n$g_{2,A}^{T\\Xi_{bc}^{(\\prime)}\\to\\Sigma_{c}}$ &$\n-0.044$ &$3.74$ &$0.49$ \\\\\n\\hline\n$f_{1,S}^{\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{c}^{0}}$ &$\n0.133$ &$3.60$ &$0.57$ &\n$g_{1,S}^{\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{c}^{0}}$ &$\n0.111$ &$4.09$ &$0.83$ &\n$f_{1,A}^{\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{c}^{0}}$ &$\n0.129$ &$3.18$ &$0.59$ &\n$g_{1,A}^{\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{c}^{0}}$ &$\n-0.037$ &$4.01$ &$0.79$ \\\\\n$f_{2,S}^{\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{c}^{0}}$ &$\n-0.060$ &$3.07$ &$0.55$ &\n$g_{2,S}^{\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{c}^{0}}$ &$\n0.053$ &$2.63$ &$0.54$ &\n$f_{2,A}^{\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{c}^{0}}$ &$\n0.135$ &$3.28$ &$0.58$ &\n$g_{2,A}^{\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{c}^{0}}$ &$\n-0.014$ &$2.56$ &$0.58$ \\\\\n$f_{3,S}^{\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{c}^{0}}$ &$\n0.003$ &$0.35$ &$-0.02$ &\n$g_{3,S}^{\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{c}^{0}}$ &$\n-0.204$ &$2.88$ &$0.52$ &\n$f_{3,A}^{\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{c}^{0}}$ &$\n-0.084$ &$3.43$ &$0.65$ &\n$g_{3,A}^{\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{c}^{0}}$ &$\n0.061$ &$2.85$ &$0.52$ \\\\\n$f_{1,S}^{T\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{c}^{0}}$ &$\n0.064$ &$3.40$ &$0.60$ &\n$g_{1,S}^{T\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{c}^{0}}$ &$\n-0.001^{*}$ &$0.92^{*}$ &$0.14^{*}$ &\n$f_{1,A}^{T\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{c}^{0}}$ &$\n0.128$ &$3.30$ &$0.58$ &\n$g_{1,A}^{T\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{c}^{0}}$ &$\n-0.004$ &$7.27$ &$16.40$ \\\\\n$f_{2,S}^{T\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{c}^{0}}$ &$\n0.105$ &$3.64$ &$0.50$ &\n$g_{2,S}^{T\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{c}^{0}}$ &$\n0.131$ &$3.54$ &$0.48$ &\n$f_{2,A}^{T\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{c}^{0}}$ &$\n-0.044$ &$2.73$ &$1.19$ &\n$g_{2,A}^{T\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{c}^{0}}$ &$\n-0.041$ &$3.57$ &$0.48$ \\\\\n\\hline\n$f_{1,S}^{\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{c}^{\\prime0}}$ &$\n0.133$ &$3.58$ &$0.55$ &\n$g_{1,S}^{\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{c}^{\\prime0}}$ &$\n0.116$ &$3.90$ &$0.68$ &\n$f_{1,A}^{\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{c}^{\\prime0}}$ &$\n0.129$ &$3.16$ &$0.58$ &\n$g_{1,A}^{\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{c}^{\\prime0}}$ &$\n-0.039$ &$3.83$ &$0.65$ \\\\\n$f_{2,S}^{\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{c}^{\\prime0}}$ &$\n-0.067$ &$3.04$ &$0.55$ &\n$g_{2,S}^{\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{c}^{\\prime0}}$ &$\n0.038$ &$2.67$ &$0.55$ &\n$f_{2,A}^{\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{c}^{\\prime0}}$ &$\n0.136$ &$3.25$ &$0.56$ &\n$g_{2,A}^{\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{c}^{\\prime0}}$ &$\n-0.009$ &$2.57$ &$0.60$ \\\\\n$f_{3,S}^{\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{c}^{\\prime0}}$ &$\n-0.001^{*}$ &$1.89^{*}$ &$0.46^{*}$ &\n$g_{3,S}^{\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{c}^{\\prime0}}$ &$\n-0.185$ &$2.95$ &$0.53$ &\n$f_{3,A}^{\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{c}^{\\prime0}}$ &$\n-0.089$ &$3.33$ &$0.61$ &\n$g_{3,A}^{\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{c}^{\\prime0}}$ &$\n0.055$ &$2.93$ &$0.53$ \\\\\n$f_{1,S}^{T\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{c}^{\\prime0}}$ &$\n0.062$ &$3.38$ &$0.58$ &\n$g_{1,S}^{T\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{c}^{\\prime0}}$ &$\n0.010$ &$2.96$ &$0.46$ &\n$f_{1,A}^{T\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{c}^{\\prime0}}$ &$\n0.126$ &$3.26$ &$0.56$ &\n$g_{1,A}^{T\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{c}^{\\prime0}}$ &$\n-0.008$ &$3.05$ &$0.51$ \\\\\n$f_{2,S}^{T\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{c}^{\\prime0}}$ &$\n0.110$ &$3.56$ &$0.48$ &\n$g_{2,S}^{T\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{c}^{\\prime0}}$ &$\n0.129$ &$3.62$ &$0.52$ &\n$f_{2,A}^{T\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{c}^{\\prime0}}$ &$\n-0.045$ &$2.72$ &$1.16$ &\n$g_{2,A}^{T\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{c}^{\\prime0}}$ &$\n-0.041$ &$3.65$ &$0.53$ \\\\\n\\hline\n\\hline\n\\end{tabular}}\n\\end{table}\n\\begin{table}\\footnotesize\\label{Tab:ff_bsbb_axial}\n\\caption{Same with Tab.~\\ref{Tab:ff_bdbb_axial} except for $b\\to s$ process.}\n\\setlength{\\tabcolsep}{0.4mm}{\n\\begin{tabular}{c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c}\n\\hline\\hline\n$F$ &$F(0)$ &$m_{\\rm{fit}}$ &$\\delta$ &$F$ &$F(0)$ &$m_{\\rm{fit}}$ &$\\delta$&\n$F$ &$F(0)$ &$m_{\\rm{fit}}$ &$\\delta$&$F$ &$F(0)$ &$m_{\\rm{fit}}$ &$\\delta$\n\\\\ \\hline\n$f_{1,S}^{\\Xi_{bb}\\to\\Xi_{b}}$ &$\n0.141$ &$3.56$ &$0.81$ &\n$g_{1,S}^{\\Xi_{bb}\\to\\Xi_{b}}$ &$\n0.122$ &$3.73$ &$0.85$ &\n$f_{1,A}^{\\Xi_{bb}\\to\\Xi_{b}}$ &$\n0.140$ &$3.35$ &$0.89$ &\n$g_{1,A}^{\\Xi_{bb}\\to\\Xi_{b}}$ &$\n-0.041$ &$3.71$ &$0.84$ \\\\\n$f_{2,S}^{\\Xi_{bb}\\to\\Xi_{b}}$ &$\n-0.189$ &$3.16$ &$0.83$ &\n$g_{2,S}^{\\Xi_{bb}\\to\\Xi_{b}}$ &$\n0.056$ &$2.79$ &$0.82$ &\n$f_{2,A}^{\\Xi_{bb}\\to\\Xi_{b}}$ &$\n0.123$ &$3.55$ &$0.96$ &\n$g_{2,A}^{\\Xi_{bb}\\to\\Xi_{b}}$ &$\n-0.017$ &$2.77$ &$0.85$ \\\\\n$f_{3,S}^{\\Xi_{bb}\\to\\Xi_{b}}$ &$\n0.016$ &$0.34$ &$-0.02$ &\n$g_{3,S}^{\\Xi_{bb}\\to\\Xi_{b}}$ &$\n-0.406$ &$3.14$ &$0.86$ &\n$f_{3,A}^{\\Xi_{bb}\\to\\Xi_{b}}$ &$\n-0.066$ &$4.27$ &$1.85$ &\n$g_{3,A}^{\\Xi_{bb}\\to\\Xi_{b}}$ &$\n0.130$ &$3.13$ &$0.86$ \\\\\n$f_{1,S}^{T\\Xi_{bb}\\to\\Xi_{b}}$ &$\n0.091$ &$3.43$ &$0.82$ &\n$g_{1,S}^{T\\Xi_{bb}\\to\\Xi_{b}}$ &$\n0.156$ &$2.84$ &$0.95$ &\n$f_{1,A}^{T\\Xi_{bb}\\to\\Xi_{b}}$ &$\n0.134$ &$3.38$ &$0.83$ &\n$g_{1,A}^{T\\Xi_{bb}\\to\\Xi_{b}}$ &$\n-0.054$ &$2.85$ &$0.93$ \\\\\n$f_{2,S}^{T\\Xi_{bb}\\to\\Xi_{b}}$ &$\n0.108$ &$3.42$ &$0.70$ &\n$g_{2,S}^{T\\Xi_{bb}\\to\\Xi_{b}}$ &$\n0.128$ &$3.96$ &$0.96$ &\n$f_{2,A}^{T\\Xi_{bb}\\to\\Xi_{b}}$ &$\n-0.061$ &$3.25$ &$1.54$ &\n$g_{2,A}^{T\\Xi_{bb}\\to\\Xi_{b}}$ &$\n-0.042$ &$3.97$ &$0.97$ \\\\\\hline\n$f_{1,S}^{\\Xi_{bb}\\to\\Xi_{b}^{\\prime}}$ &$\n0.143$ &$3.52$ &$0.79$ &\n$g_{1,S}^{\\Xi_{bb}\\to\\Xi_{b}^{\\prime}}$ &$\n0.130$ &$3.53$ &$0.70$ &\n$f_{1,A}^{\\Xi_{bb}\\to\\Xi_{b}^{\\prime}}$ &$\n0.142$ &$3.31$ &$0.87$ &\n$g_{1,A}^{\\Xi_{bb}\\to\\Xi_{b}^{\\prime}}$ &$\n-0.043$ &$3.51$ &$0.70$ \\\\\n$f_{2,S}^{\\Xi_{bb}\\to\\Xi_{b}^{\\prime}}$ &$\n-0.202$ &$3.13$ &$0.81$ &\n$g_{2,S}^{\\Xi_{bb}\\to\\Xi_{b}^{\\prime}}$ &$\n0.024$ &$3.45$ &$1.24$ &\n$f_{2,A}^{\\Xi_{bb}\\to\\Xi_{b}^{\\prime}}$ &$\n0.129$ &$3.46$ &$0.91$ &\n$g_{2,A}^{\\Xi_{bb}\\to\\Xi_{b}^{\\prime}}$ &$\n-0.007$ &$3.51$ &$1.35$ \\\\\n$f_{3,S}^{\\Xi_{bb}\\to\\Xi_{b}^{\\prime}}$ &$\n0.003^{*}$ &$1.05^{*}$ &$0.10^{*}$ &\n$g_{3,S}^{\\Xi_{bb}\\to\\Xi_{b}^{\\prime}}$ &$\n-0.316$ &$3.58$ &$1.24$ &\n$f_{3,A}^{\\Xi_{bb}\\to\\Xi_{b}^{\\prime}}$ &$\n-0.080$ &$3.77$ &$1.24$ &\n$g_{3,A}^{\\Xi_{bb}\\to\\Xi_{b}^{\\prime}}$ &$\n0.100$ &$3.61$ &$1.27$ \\\\\n$f_{1,S}^{T\\Xi_{bb}\\to\\Xi_{b}^{\\prime}}$ &$\n0.091$ &$3.37$ &$0.79$ &\n$g_{1,S}^{T\\Xi_{bb}\\to\\Xi_{b}^{\\prime}}$ &$\n0.198$ &$2.70$ &$1.00$ &\n$f_{1,A}^{T\\Xi_{bb}\\to\\Xi_{b}^{\\prime}}$ &$\n0.135$ &$3.32$ &$0.80$ &\n$g_{1,A}^{T\\Xi_{bb}\\to\\Xi_{b}^{\\prime}}$ &$\n-0.068$ &$2.70$ &$0.97$ \\\\\n$f_{2,S}^{T\\Xi_{bb}\\to\\Xi_{b}^{\\prime}}$ &$\n0.117$ &$3.35$ &$0.68$ &\n$g_{2,S}^{T\\Xi_{bb}\\to\\Xi_{b}^{\\prime}}$ &$\n0.127$ &$4.19$ &$1.26$ &\n$f_{2,A}^{T\\Xi_{bb}\\to\\Xi_{b}^{\\prime}}$ &$\n-0.063$ &$3.20$ &$1.51$ &\n$g_{2,A}^{T\\Xi_{bb}\\to\\Xi_{b}^{\\prime}}$ &$\n-0.042$ &$4.22$ &$1.28$ \\\\\\hline\n$f_{1,S}^{\\Omega_{bb}^{-}\\to\\Omega_{b}^{-}}$ &$\n0.139$ &$3.49$ &$0.80$ &\n$g_{1,S}^{\\Omega_{bb}^{-}\\to\\Omega_{b}^{-}}$ &$\n0.125$ &$3.53$ &$0.74$ &\n$f_{1,A}^{\\Omega_{bb}^{-}\\to\\Omega_{b}^{-}}$ &$\n0.137$ &$3.29$ &$0.88$ &\n$g_{1,A}^{\\Omega_{bb}^{-}\\to\\Omega_{b}^{-}}$ &$\n-0.042$ &$3.52$ &$0.73$ \\\\\n$f_{2,S}^{\\Omega_{bb}^{-}\\to\\Omega_{b}^{-}}$ &$\n-0.198$ &$3.13$ &$0.83$ &\n$g_{2,S}^{\\Omega_{bb}^{-}\\to\\Omega_{b}^{-}}$ &$\n0.028$ &$3.16$ &$1.00$ &\n$f_{2,A}^{\\Omega_{bb}^{-}\\to\\Omega_{b}^{-}}$ &$\n0.125$ &$3.44$ &$0.92$ &\n$g_{2,A}^{\\Omega_{bb}^{-}\\to\\Omega_{b}^{-}}$ &$\n-0.008$ &$3.16$ &$1.03$ \\\\\n$f_{3,S}^{\\Omega_{bb}^{-}\\to\\Omega_{b}^{-}}$ &$\n0.003^{*}$ &$1.00^{*}$ &$0.09^{*}$ &\n$g_{3,S}^{\\Omega_{bb}^{-}\\to\\Omega_{b}^{-}}$ &$\n-0.332$ &$3.44$ &$1.12$ &\n$f_{3,A}^{\\Omega_{bb}^{-}\\to\\Omega_{b}^{-}}$ &$\n-0.077$ &$3.77$ &$1.28$ &\n$g_{3,A}^{\\Omega_{bb}^{-}\\to\\Omega_{b}^{-}}$ &$\n0.106$ &$3.46$ &$1.14$ \\\\\n$f_{1,S}^{T\\Omega_{bb}^{-}\\to\\Omega_{b}^{-}}$ &$\n0.088$ &$3.36$ &$0.81$ &\n$g_{1,S}^{T\\Omega_{bb}^{-}\\to\\Omega_{b}^{-}}$ &$\n0.186$ &$2.71$ &$1.01$ &\n$f_{1,A}^{T\\Omega_{bb}^{-}\\to\\Omega_{b}^{-}}$ &$\n0.130$ &$3.31$ &$0.82$ &\n$g_{1,A}^{T\\Omega_{bb}^{-}\\to\\Omega_{b}^{-}}$ &$\n-0.064$ &$2.72$ &$0.98$ \\\\\n$f_{2,S}^{T\\Omega_{bb}^{-}\\to\\Omega_{b}^{-}}$ &$\n0.112$ &$3.33$ &$0.70$ &\n$g_{2,S}^{T\\Omega_{bb}^{-}\\to\\Omega_{b}^{-}}$ &$\n0.123$ &$4.10$ &$1.19$ &\n$f_{2,A}^{T\\Omega_{bb}^{-}\\to\\Omega_{b}^{-}}$ &$\n-0.060$ &$3.20$ &$1.51$ &\n$g_{2,A}^{T\\Omega_{bb}^{-}\\to\\Omega_{b}^{-}}$ &$\n-0.041$ &$4.12$ &$1.21$ \\\\\n\\hline\n\\hline\n$f_{1,S}^{\\Xi_{bc}^{(\\prime)}\\to\\Xi_{c}}$ &$\n0.203$ &$4.07$ &$0.66$ &\n$g_{1,S}^{\\Xi_{bc}^{(\\prime)}\\to\\Xi_{c}}$ &$\n0.167$ &$4.99$ &$1.32$ &\n$f_{1,A}^{\\Xi_{bc}^{(\\prime)}\\to\\Xi_{c}}$ &$\n0.196$ &$3.56$ &$0.74$ &\n$g_{1,A}^{\\Xi_{bc}^{(\\prime)}\\to\\Xi_{c}}$ &$\n-0.056$ &$4.81$ &$1.19$ \\\\\n$f_{2,S}^{\\Xi_{bc}^{(\\prime)}\\to\\Xi_{c}}$ &$\n-0.079$ &$3.37$ &$0.65$ &\n$g_{2,S}^{\\Xi_{bc}^{(\\prime)}\\to\\Xi_{c}}$ &$\n0.097$ &$2.84$ &$0.70$ &\n$f_{2,A}^{\\Xi_{bc}^{(\\prime)}\\to\\Xi_{c}}$ &$\n0.203$ &$3.68$ &$0.69$ &\n$g_{2,A}^{\\Xi_{bc}^{(\\prime)}\\to\\Xi_{c}}$ &$\n-0.027$ &$2.78$ &$0.83$ \\\\\n$f_{3,S}^{\\Xi_{bc}^{(\\prime)}\\to\\Xi_{c}}$ &$\n0.015^{*}$ &$1.44^{*}$ &$0.74^{*}$ &\n$g_{3,S}^{\\Xi_{bc}^{(\\prime)}\\to\\Xi_{c}}$ &$\n-0.329$ &$3.08$ &$0.60$ &\n$f_{3,A}^{\\Xi_{bc}^{(\\prime)}\\to\\Xi_{c}}$ &$\n-0.110$ &$4.05$ &$0.92$ &\n$g_{3,A}^{\\Xi_{bc}^{(\\prime)}\\to\\Xi_{c}}$ &$\n0.098$ &$3.04$ &$0.60$ \\\\\n$f_{1,S}^{T\\Xi_{bc}^{(\\prime)}\\to\\Xi_{c}}$ &$\n0.085$ &$3.85$ &$0.74$ &\n$g_{1,S}^{T\\Xi_{bc}^{(\\prime)}\\to\\Xi_{c}}$ &$\n-0.021^{*}$ &$0.92^{*}$ &$0.23^{*}$ &\n$f_{1,A}^{T\\Xi_{bc}^{(\\prime)}\\to\\Xi_{c}}$ &$\n0.180$ &$3.69$ &$0.69$ &\n$g_{1,A}^{T\\Xi_{bc}^{(\\prime)}\\to\\Xi_{c}}$ &$\n-0.001^{*}$ &$1.90^{*}$ &$0.27^{*}$ \\\\\n$f_{2,S}^{T\\Xi_{bc}^{(\\prime)}\\to\\Xi_{c}}$ &$\n0.160$ &$4.13$ &$0.54$ &\n$g_{2,S}^{T\\Xi_{bc}^{(\\prime)}\\to\\Xi_{c}}$ &$\n0.202$ &$3.86$ &$0.47$ &\n$f_{2,A}^{T\\Xi_{bc}^{(\\prime)}\\to\\Xi_{c}}$ &$\n-0.064$ &$5.43$ &$29.40$ &\n$g_{2,A}^{T\\Xi_{bc}^{(\\prime)}\\to\\Xi_{c}}$ &$\n-0.064$ &$3.88$ &$0.46$ \\\\\n\\hline\n$f_{1,S}^{\\Xi_{bc}^{(\\prime)}\\to\\Xi_{c}^{\\prime}}$ &$\n0.204$ &$4.04$ &$0.64$ &\n$g_{1,S}^{\\Xi_{bc}^{(\\prime)}\\to\\Xi_{c}^{\\prime}}$ &$\n0.174$ &$4.66$ &$0.99$ &\n$f_{1,A}^{\\Xi_{bc}^{(\\prime)}\\to\\Xi_{c}^{\\prime}}$ &$\n0.197$ &$3.53$ &$0.72$ &\n$g_{1,A}^{\\Xi_{bc}^{(\\prime)}\\to\\Xi_{c}^{\\prime}}$ &$\n-0.058$ &$4.52$ &$0.91$ \\\\\n$f_{2,S}^{\\Xi_{bc}^{(\\prime)}\\to\\Xi_{c}^{\\prime}}$ &$\n-0.090$ &$3.35$ &$0.64$ &\n$g_{2,S}^{\\Xi_{bc}^{(\\prime)}\\to\\Xi_{c}^{\\prime}}$ &$\n0.074$ &$2.86$ &$0.70$ &\n$f_{2,A}^{\\Xi_{bc}^{(\\prime)}\\to\\Xi_{c}^{\\prime}}$ &$\n0.204$ &$3.63$ &$0.67$ &\n$g_{2,A}^{\\Xi_{bc}^{(\\prime)}\\to\\Xi_{c}^{\\prime}}$ &$\n-0.019$ &$2.80$ &$0.89$ \\\\\n$f_{3,S}^{\\Xi_{bc}^{(\\prime)}\\to\\Xi_{c}^{\\prime}}$ &$\n0.007$ &$0.07$ &$-0.00$ &\n$g_{3,S}^{\\Xi_{bc}^{(\\prime)}\\to\\Xi_{c}^{\\prime}}$ &$\n-0.300$ &$3.15$ &$0.61$ &\n$f_{3,A}^{\\Xi_{bc}^{(\\prime)}\\to\\Xi_{c}^{\\prime}}$ &$\n-0.118$ &$3.86$ &$0.80$ &\n$g_{3,A}^{\\Xi_{bc}^{(\\prime)}\\to\\Xi_{c}^{\\prime}}$ &$\n0.088$ &$3.12$ &$0.62$ \\\\\n$f_{1,S}^{T\\Xi_{bc}^{(\\prime)}\\to\\Xi_{c}^{\\prime}}$ &$\n0.083$ &$3.82$ &$0.71$ &\n$g_{1,S}^{T\\Xi_{bc}^{(\\prime)}\\to\\Xi_{c}^{\\prime}}$ &$\n-0.006$ &$0.50$ &$-0.03$ &\n$f_{1,A}^{T\\Xi_{bc}^{(\\prime)}\\to\\Xi_{c}^{\\prime}}$ &$\n0.177$ &$3.65$ &$0.67$ &\n$g_{1,A}^{T\\Xi_{bc}^{(\\prime)}\\to\\Xi_{c}^{\\prime}}$ &$\n-0.006$ &$6.30$ &$5.21$ \\\\\n$f_{2,S}^{T\\Xi_{bc}^{(\\prime)}\\to\\Xi_{c}^{\\prime}}$ &$\n0.169$ &$4.01$ &$0.51$ &\n$g_{2,S}^{T\\Xi_{bc}^{(\\prime)}\\to\\Xi_{c}^{\\prime}}$ &$\n0.200$ &$3.95$ &$0.52$ &\n$f_{2,A}^{T\\Xi_{bc}^{(\\prime)}\\to\\Xi_{c}^{\\prime}}$ &$\n-0.065$ &$5.16$ &$23.60$ &\n$g_{2,A}^{T\\Xi_{bc}^{(\\prime)}\\to\\Xi_{c}^{\\prime}}$ &$\n-0.063$ &$3.98$ &$0.52$ \\\\\n\\hline\n$f_{1,S}^{\\Omega_{bc}^{(\\prime)0}\\to\\Omega_{c}^{0}}$ &$\n0.192$ &$3.91$ &$0.66$ &\n$g_{1,S}^{\\Omega_{bc}^{(\\prime)0}\\to\\Omega_{c}^{0}}$ &$\n0.165$ &$4.40$ &$0.90$ &\n$f_{1,A}^{\\Omega_{bc}^{(\\prime)0}\\to\\Omega_{c}^{0}}$ &$\n0.187$ &$3.45$ &$0.74$ &\n$g_{1,A}^{\\Omega_{bc}^{(\\prime)0}\\to\\Omega_{c}^{0}}$ &$\n-0.055$ &$4.29$ &$0.85$ \\\\\n$f_{2,S}^{\\Omega_{bc}^{(\\prime)0}\\to\\Omega_{c}^{0}}$ &$\n-0.091$ &$3.25$ &$0.67$ &\n$g_{2,S}^{\\Omega_{bc}^{(\\prime)0}\\to\\Omega_{c}^{0}}$ &$\n0.064$ &$2.86$ &$0.77$ &\n$f_{2,A}^{\\Omega_{bc}^{(\\prime)0}\\to\\Omega_{c}^{0}}$ &$\n0.191$ &$3.55$ &$0.70$ &\n$g_{2,A}^{\\Omega_{bc}^{(\\prime)0}\\to\\Omega_{c}^{0}}$ &$\n-0.017$ &$2.81$ &$0.96$ \\\\\n$f_{3,S}^{\\Omega_{bc}^{(\\prime)0}\\to\\Omega_{c}^{0}}$ &$\n0.004^{*}$ &$0.98^{*}$ &$0.07^{*}$ &\n$g_{3,S}^{\\Omega_{bc}^{(\\prime)0}\\to\\Omega_{c}^{0}}$ &$\n-0.288$ &$3.13$ &$0.66$ &\n$f_{3,A}^{\\Omega_{bc}^{(\\prime)0}\\to\\Omega_{c}^{0}}$ &$\n-0.114$ &$3.72$ &$0.80$ &\n$g_{3,A}^{\\Omega_{bc}^{(\\prime)0}\\to\\Omega_{c}^{0}}$ &$\n0.085$ &$3.11$ &$0.67$ \\\\\n$f_{1,S}^{T\\Omega_{bc}^{(\\prime)0}\\to\\Omega_{c}^{0}}$ &$\n0.081$ &$3.68$ &$0.72$ &\n$g_{1,S}^{T\\Omega_{bc}^{(\\prime)0}\\to\\Omega_{c}^{0}}$ &$\n-0.001^{*}$ &$0.90^{*}$ &$0.07^{*}$ &\n$f_{1,A}^{T\\Omega_{bc}^{(\\prime)0}\\to\\Omega_{c}^{0}}$ &$\n0.169$ &$3.54$ &$0.68$ &\n$g_{1,A}^{T\\Omega_{bc}^{(\\prime)0}\\to\\Omega_{c}^{0}}$ &$\n-0.006$ &$3.65$ &$0.58$ \\\\\n$f_{2,S}^{T\\Omega_{bc}^{(\\prime)0}\\to\\Omega_{c}^{0}}$ &$\n0.159$ &$3.86$ &$0.53$ &\n$g_{2,S}^{T\\Omega_{bc}^{(\\prime)0}\\to\\Omega_{c}^{0}}$ &$\n0.188$ &$3.86$ &$0.57$ &\n$f_{2,A}^{T\\Omega_{bc}^{(\\prime)0}\\to\\Omega_{c}^{0}}$ &$\n-0.063$ &$4.90$ &$19.50$ &\n$g_{2,A}^{T\\Omega_{bc}^{(\\prime)0}\\to\\Omega_{c}^{0}}$ &$\n-0.060$ &$3.90$ &$0.57$ \\\\\n\\hline\n\\hline\n\\end{tabular}}\n\\end{table}\n\\begin{table}\\footnotesize\\label{Tab:ff32_cdscc}\n\\caption{Numerical results for the transition $1\/2\\to 3\/2$ form factors $\\mathtt{f}_{i}$ and $\\mathtt{g}_{i}$ at $q^2=0$ of $c\\to d,s$ processes. The parametrization scheme in Eq.~(\\ref{eq:main_fit_formula_cdecay}) is introduced for these form factors,\nand the value of the singly pole $m_{\\rm pole}$s are taken as $1.87, ~1.97~{\\rm GeV}$ for $c\\to d,s$, respectively.}\n\\begin{tabular}{c|c|c|c|c|c|c|c}\n\\hline\\hline\n$F$ &$F(0)$&$F$ &$F(0)$ &$F$ &$F(0)$&$F$ &$F(0)$ \\\\ \\hline\n$\\mathtt{f}_{1}^{\\Xi_{cc}^{++}\\to\\Sigma_{c}^{*+}}$ &$-0.979$ &\n$\\mathtt{f}_{2}^{\\Xi_{cc}^{++}\\to\\Sigma_{c}^{*+}}$ &$-0.645$ &\n$\\mathtt{f}_{3}^{\\Xi_{cc}^{++}\\to\\Sigma_{c}^{*+}}$ &$0.047$ &\n$\\mathtt{f}_{4}^{\\Xi_{cc}^{++}\\to\\Sigma_{c}^{*+}}$ &$-1.969$ \\\\\n$\\mathtt{g}_{1}^{\\Xi_{cc}^{++}\\to\\Sigma_{c}^{*+}}$ &$-5.792$ &\n$\\mathtt{g}_{2}^{\\Xi_{cc}^{++}\\to\\Sigma_{c}^{*+}}$ &$-3.602$ &\n$\\mathtt{g}_{3}^{\\Xi_{cc}^{++}\\to\\Sigma_{c}^{*+}}$ &$0.947$ &\n$\\mathtt{g}_{4}^{\\Xi_{cc}^{++}\\to\\Sigma_{c}^{*+}}$ &$0.393$ \\\\\\hline\n$\\mathtt{f}_{1}^{\\Xi_{cc}^{+}\\to\\Sigma_{c}^{*0}}$ &$-0.979$ &\n$\\mathtt{f}_{2}^{\\Xi_{cc}^{+}\\to\\Sigma_{c}^{*0}}$ &$-0.645$ &\n$\\mathtt{f}_{3}^{\\Xi_{cc}^{+}\\to\\Sigma_{c}^{*0}}$ &$0.047$ &\n$\\mathtt{f}_{4}^{\\Xi_{cc}^{+}\\to\\Sigma_{c}^{*0}}$ &$-1.969$ \\\\\n$\\mathtt{g}_{1}^{\\Xi_{cc}^{+}\\to\\Sigma_{c}^{*0}}$ &$-5.792$ &\n$\\mathtt{g}_{2}^{\\Xi_{cc}^{+}\\to\\Sigma_{c}^{*0}}$ &$-3.602$ &\n$\\mathtt{g}_{3}^{\\Xi_{cc}^{+}\\to\\Sigma_{c}^{*0}}$ &$0.947$ &\n$\\mathtt{g}_{4}^{\\Xi_{cc}^{+}\\to\\Sigma_{c}^{*0}}$ &$0.393$ \\\\\\hline\n$\\mathtt{f}_{1}^{\\Omega_{cc}^{+}\\to\\Xi_{c}^{\\prime*0}}$ &$-1.017$ &\n$\\mathtt{f}_{2}^{\\Omega_{cc}^{+}\\to\\Xi_{c}^{\\prime*0}}$ &$-0.665$ &\n$\\mathtt{f}_{3}^{\\Omega_{cc}^{+}\\to\\Xi_{c}^{\\prime*0}}$ &$0.046$ &\n$\\mathtt{f}_{4}^{\\Omega_{cc}^{+}\\to\\Xi_{c}^{\\prime*0}}$ &$-2.045$ \\\\\n$\\mathtt{g}_{1}^{\\Omega_{cc}^{+}\\to\\Xi_{c}^{\\prime*0}}$ &$-6.244$ &\n$\\mathtt{g}_{2}^{\\Omega_{cc}^{+}\\to\\Xi_{c}^{\\prime*0}}$ &$-3.856$ &\n$\\mathtt{g}_{3}^{\\Omega_{cc}^{+}\\to\\Xi_{c}^{\\prime*0}}$ &$0.980$ &\n$\\mathtt{g}_{4}^{\\Omega_{cc}^{+}\\to\\Xi_{c}^{\\prime*0}}$ &$0.389$ \\\\\\hline\n$\\mathtt{f}_{1}^{\\Xi_{cc}^{++}\\to\\Xi_{c}^{\\prime*+}}$ &$-1.148$ &\n$\\mathtt{f}_{2}^{\\Xi_{cc}^{++}\\to\\Xi_{c}^{\\prime*+}}$ &$-0.714$ &\n$\\mathtt{f}_{3}^{\\Xi_{cc}^{++}\\to\\Xi_{c}^{\\prime*+}}$ &$0.049$ &\n$\\mathtt{f}_{4}^{\\Xi_{cc}^{++}\\to\\Xi_{c}^{\\prime*+}}$ &$-2.297$ \\\\\n$\\mathtt{g}_{1}^{\\Xi_{cc}^{++}\\to\\Xi_{c}^{\\prime*+}}$ &$-10.350$ &\n$\\mathtt{g}_{2}^{\\Xi_{cc}^{++}\\to\\Xi_{c}^{\\prime*+}}$ &$-6.428$ &\n$\\mathtt{g}_{3}^{\\Xi_{cc}^{++}\\to\\Xi_{c}^{\\prime*+}}$ &$1.315$ &\n$\\mathtt{g}_{4}^{\\Xi_{cc}^{++}\\to\\Xi_{c}^{\\prime*+}}$ &$0.404$ \\\\\\hline\n$\\mathtt{f}_{1}^{\\Xi_{cc}^{+}\\to\\Xi_{c}^{\\prime*0}}$ &$-1.148$ &\n$\\mathtt{f}_{2}^{\\Xi_{cc}^{+}\\to\\Xi_{c}^{\\prime*0}}$ &$-0.714$ &\n$\\mathtt{f}_{3}^{\\Xi_{cc}^{+}\\to\\Xi_{c}^{\\prime*0}}$ &$0.049$ &\n$\\mathtt{f}_{4}^{\\Xi_{cc}^{+}\\to\\Xi_{c}^{\\prime*0}}$ &$-2.297$ \\\\\n$\\mathtt{g}_{1}^{\\Xi_{cc}^{+}\\to\\Xi_{c}^{\\prime*0}}$ &$-10.350$ &\n$\\mathtt{g}_{2}^{\\Xi_{cc}^{+}\\to\\Xi_{c}^{\\prime*0}}$ &$-6.428$ &\n$\\mathtt{g}_{3}^{\\Xi_{cc}^{+}\\to\\Xi_{c}^{\\prime*0}}$ &$1.315$ &\n$\\mathtt{g}_{4}^{\\Xi_{cc}^{+}\\to\\Xi_{c}^{\\prime*0}}$ &$0.404$ \\\\\\hline\n$\\mathtt{f}_{1}^{\\Omega_{cc}^{+}\\to\\Omega_{c}^{*0}}$ &$-1.178$ &\n$\\mathtt{f}_{2}^{\\Omega_{cc}^{+}\\to\\Omega_{c}^{*0}}$ &$-0.732$ &\n$\\mathtt{f}_{3}^{\\Omega_{cc}^{+}\\to\\Omega_{c}^{*0}}$ &$0.049$ &\n$\\mathtt{f}_{4}^{\\Omega_{cc}^{+}\\to\\Omega_{c}^{*0}}$ &$-2.359$ \\\\\n$\\mathtt{g}_{1}^{\\Omega_{cc}^{+}\\to\\Omega_{c}^{*0}}$ &$-10.670$ &\n$\\mathtt{g}_{2}^{\\Omega_{cc}^{+}\\to\\Omega_{c}^{*0}}$ &$-6.551$ &\n$\\mathtt{g}_{3}^{\\Omega_{cc}^{+}\\to\\Omega_{c}^{*0}}$ &$1.337$ &\n$\\mathtt{g}_{4}^{\\Omega_{cc}^{+}\\to\\Omega_{c}^{*0}}$ &$0.415$ \\\\\\hline\n\\hline\n$\\mathtt{f}_{1}^{\\Xi_{bc}^{(\\prime)+}\\to\\Sigma_{b}^{*0}}$ &$-1.684$ &\n$\\mathtt{f}_{2}^{\\Xi_{bc}^{(\\prime)+}\\to\\Sigma_{b}^{*0}}$ &$-0.928$ &\n$\\mathtt{f}_{3}^{\\Xi_{bc}^{(\\prime)+}\\to\\Sigma_{b}^{*0}}$ &$0.026$ &\n$\\mathtt{f}_{4}^{\\Xi_{bc}^{(\\prime)+}\\to\\Sigma_{b}^{*0}}$ &$-3.365$ \\\\\n$\\mathtt{g}_{1}^{\\Xi_{bc}^{(\\prime)+}\\to\\Sigma_{b}^{*0}}$ &$-12.570$ &\n$\\mathtt{g}_{2}^{\\Xi_{bc}^{(\\prime)+}\\to\\Sigma_{b}^{*0}}$ &$-6.337$ &\n$\\mathtt{g}_{3}^{\\Xi_{bc}^{(\\prime)+}\\to\\Sigma_{b}^{*0}}$ &$1.412$ &\n$\\mathtt{g}_{4}^{\\Xi_{bc}^{(\\prime)+}\\to\\Sigma_{b}^{*0}}$ &$0.561$ \\\\\\hline\n$\\mathtt{f}_{1}^{\\Xi_{bc}^{(\\prime)0}\\to\\Sigma_{b}^{*-}}$ &$-1.690$ &\n$\\mathtt{f}_{2}^{\\Xi_{bc}^{(\\prime)0}\\to\\Sigma_{b}^{*-}}$ &$-0.931$ &\n$\\mathtt{f}_{3}^{\\Xi_{bc}^{(\\prime)0}\\to\\Sigma_{b}^{*-}}$ &$0.026$ &\n$\\mathtt{f}_{4}^{\\Xi_{bc}^{(\\prime)0}\\to\\Sigma_{b}^{*-}}$ &$-3.377$ \\\\\n$\\mathtt{g}_{1}^{\\Xi_{bc}^{(\\prime)0}\\to\\Sigma_{b}^{*-}}$ &$-12.810$ &\n$\\mathtt{g}_{2}^{\\Xi_{bc}^{(\\prime)0}\\to\\Sigma_{b}^{*-}}$ &$-6.479$ &\n$\\mathtt{g}_{3}^{\\Xi_{bc}^{(\\prime)0}\\to\\Sigma_{b}^{*-}}$ &$1.419$ &\n$\\mathtt{g}_{4}^{\\Xi_{bc}^{(\\prime)0}\\to\\Sigma_{b}^{*-}}$ &$0.559$ \\\\\\hline\n$\\mathtt{f}_{1}^{\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{b}^{\\prime*-}}$ &$-1.949$ &\n$\\mathtt{f}_{2}^{\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{b}^{\\prime*-}}$ &$-1.019$ &\n$\\mathtt{f}_{3}^{\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{b}^{\\prime*-}}$ &$0.034$ &\n$\\mathtt{f}_{4}^{\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{b}^{\\prime*-}}$ &$-3.878$ \\\\\n$\\mathtt{g}_{1}^{\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{b}^{\\prime*-}}$ &$-23.010$ &\n$\\mathtt{g}_{2}^{\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{b}^{\\prime*-}}$ &$-12.440$ &\n$\\mathtt{g}_{3}^{\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{b}^{\\prime*-}}$ &$1.732$ &\n$\\mathtt{g}_{4}^{\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{b}^{\\prime*-}}$ &$0.475$ \\\\\\hline\n$\\mathtt{f}_{1}^{\\Xi_{bc}^{(\\prime)+}\\to\\Xi_{b}^{\\prime*0}}$ &$-1.936$ &\n$\\mathtt{f}_{2}^{\\Xi_{bc}^{(\\prime)+}\\to\\Xi_{b}^{\\prime*0}}$ &$-1.034$ &\n$\\mathtt{f}_{3}^{\\Xi_{bc}^{(\\prime)+}\\to\\Xi_{b}^{\\prime*0}}$ &$0.030$ &\n$\\mathtt{f}_{4}^{\\Xi_{bc}^{(\\prime)+}\\to\\Xi_{b}^{\\prime*0}}$ &$-3.863$ \\\\\n$\\mathtt{g}_{1}^{\\Xi_{bc}^{(\\prime)+}\\to\\Xi_{b}^{\\prime*0}}$ &$-24.640$ &\n$\\mathtt{g}_{2}^{\\Xi_{bc}^{(\\prime)+}\\to\\Xi_{b}^{\\prime*0}}$ &$-13.000$ &\n$\\mathtt{g}_{3}^{\\Xi_{bc}^{(\\prime)+}\\to\\Xi_{b}^{\\prime*0}}$ &$1.926$ &\n$\\mathtt{g}_{4}^{\\Xi_{bc}^{(\\prime)+}\\to\\Xi_{b}^{\\prime*0}}$ &$0.583$ \\\\\\hline\n$\\mathtt{f}_{1}^{\\Xi_{bc}^{(\\prime)0}\\to\\Xi_{b}^{\\prime*-}}$ &$-1.960$ &\n$\\mathtt{f}_{2}^{\\Xi_{bc}^{(\\prime)0}\\to\\Xi_{b}^{\\prime*-}}$ &$-1.045$ &\n$\\mathtt{f}_{3}^{\\Xi_{bc}^{(\\prime)0}\\to\\Xi_{b}^{\\prime*-}}$ &$0.031$ &\n$\\mathtt{f}_{4}^{\\Xi_{bc}^{(\\prime)0}\\to\\Xi_{b}^{\\prime*-}}$ &$-3.910$ \\\\\n$\\mathtt{g}_{1}^{\\Xi_{bc}^{(\\prime)0}\\to\\Xi_{b}^{\\prime*-}}$ &$-25.950$ &\n$\\mathtt{g}_{2}^{\\Xi_{bc}^{(\\prime)0}\\to\\Xi_{b}^{\\prime*-}}$ &$-13.770$ &\n$\\mathtt{g}_{3}^{\\Xi_{bc}^{(\\prime)0}\\to\\Xi_{b}^{\\prime*-}}$ &$1.961$ &\n$\\mathtt{g}_{4}^{\\Xi_{bc}^{(\\prime)0}\\to\\Xi_{b}^{\\prime*-}}$ &$0.572$ \\\\\\hline\n$\\mathtt{f}_{1}^{\\Omega_{bc}^{(\\prime)0}\\to\\Omega_{b}^{*-}}$ &$-2.345$ &\n$\\mathtt{f}_{2}^{\\Omega_{bc}^{(\\prime)0}\\to\\Omega_{b}^{*-}}$ &$-1.177$ &\n$\\mathtt{f}_{3}^{\\Omega_{bc}^{(\\prime)0}\\to\\Omega_{b}^{*-}}$ &$0.046$ &\n$\\mathtt{f}_{4}^{\\Omega_{bc}^{(\\prime)0}\\to\\Omega_{b}^{*-}}$ &$-4.661$ \\\\\n$\\mathtt{g}_{1}^{\\Omega_{bc}^{(\\prime)0}\\to\\Omega_{b}^{*-}}$ &$-49.590$ &\n$\\mathtt{g}_{2}^{\\Omega_{bc}^{(\\prime)0}\\to\\Omega_{b}^{*-}}$ &$-27.300$ &\n$\\mathtt{g}_{3}^{\\Omega_{bc}^{(\\prime)0}\\to\\Omega_{b}^{*-}}$ &$2.529$ &\n$\\mathtt{g}_{4}^{\\Omega_{bc}^{(\\prime)0}\\to\\Omega_{b}^{*-}}$ &$0.428$ \\\\\\hline\n\\hline\n\\end{tabular}\n\\end{table}\n\\begin{table}\\footnotesize\\label{Tab:ff32_bucbb}\n\\caption{Numerical results for the transition $1\/2\\to 3\/2$ form factors $\\mathtt{f}_{i}$ and $\\mathtt{g}_{i}$ of $b\\to u,c$ processes. The parametrization scheme shown with Eq.~(\\ref{eq:auxiliary_fit_formula_bdecay}) is introduced for these form factors with asterisk, and Eq.~(\\ref{eq:main_fit_formula_bdecay}) for all the other ones.}\n\\begin{tabular}{c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c}\n\\hline \\hline\n$F$  & $F(0)$  & $m_{{\\rm {fit}}}$  & $\\delta$  & $F$  & $F(0)$  & $m_{{\\rm {fit}}}$  & $\\delta$ & $F$  & $F(0)$  & $m_{{\\rm {fit}}}$  & $\\delta$  & $F$  & $F(0)$  & $m_{{\\rm {fit}}}$  & $\\delta$\\tabularnewline\n\\hline\n$\\mathtt{f}_{1}^{\\Xi_{bb}\\to\\Sigma_{b}^{*}}$  & $-0.172$  & $3.15$  & $0.89$  & $\\mathtt{g}_{1}^{\\Xi_{bb}\\to\\Sigma_{b}^{*}}$  & $-0.108$  & $2.70$  & $1.84$  & $\\mathtt{f}_{1}^{\\Xi_{bc}^{(\\prime)}\\to\\Sigma_{c}^{*}}$  & $-0.108$  & $3.24$  & $0.57$  & $\\mathtt{g}_{1}^{\\Xi_{bc}^{(\\prime)}\\to\\Sigma_{c}^{*}}$  & $0.021$  & $3.86$  & $1.02$ \\tabularnewline\n$\\mathtt{f}_{2}^{\\Xi_{bb}\\to\\Sigma_{b}^{*}}$  & $-0.129$  & $3.17$  & $0.90$  & $\\mathtt{g}_{2}^{\\Xi_{bb}\\to\\Sigma_{b}^{*}}$  & $-0.056^{*}$  & $0.51^{*}$  & $0.06^{*}$  & $\\mathtt{f}_{2}^{\\Xi_{bc}^{(\\prime)}\\to\\Sigma_{c}^{*}}$  & $-0.145$  & $3.26$  & $0.57$  & $\\mathtt{g}_{2}^{\\Xi_{bc}^{(\\prime)}\\to\\Sigma_{c}^{*}}$  & $0.120$  & $3.45$  & $0.60$ \\tabularnewline\n$\\mathtt{f}_{3}^{\\Xi_{bb}\\to\\Sigma_{b}^{*}}$  & $0.002$  & $2.95$  & $0.91$  & $\\mathtt{g}_{3}^{\\Xi_{bb}\\to\\Sigma_{b}^{*}}$  & $0.136$  & $3.15$  & $0.89$  & $\\mathtt{f}_{3}^{\\Xi_{bc}^{(\\prime)}\\to\\Sigma_{c}^{*}}$  & $0.003$  & $3.07$  & $0.55$  & $\\mathtt{g}_{3}^{\\Xi_{bc}^{(\\prime)}\\to\\Sigma_{c}^{*}}$  & $0.096$  & $3.23$  & $0.57$ \\tabularnewline\n$\\mathtt{f}_{4}^{\\Xi_{bb}\\to\\Sigma_{b}^{*}}$  & $-0.355$  & $3.18$  & $0.86$  & $\\mathtt{g}_{4}^{\\Xi_{bb}\\to\\Sigma_{b}^{*}}$  & $0.099$  & $3.46$  & $0.82$  & $\\mathtt{f}_{4}^{\\Xi_{bc}^{(\\prime)}\\to\\Sigma_{c}^{*}}$  & $-0.270$  & $3.45$  & $0.53$  & $\\mathtt{g}_{4}^{\\Xi_{bc}^{(\\prime)}\\to\\Sigma_{c}^{*}}$  & $0.174$  & $3.76$  & $0.54$ \\tabularnewline\n\\hline\n$\\mathtt{f}_{1}^{\\Omega_{bb}^{-}\\to\\Xi_{b}^{\\prime*0}}$  & $-0.168$  & $3.14$  & $0.91$  & $\\mathtt{g}_{1}^{\\Omega_{bb}^{-}\\to\\Xi_{b}^{\\prime*0}}$  & $-0.099$  & $2.71$  & $1.87$ & $\\mathtt{f}_{1}^{\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{c}^{\\prime*+}}$  & $-0.106$  & $3.15$  & $0.58$  & $\\mathtt{g}_{1}^{\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{c}^{\\prime*+}}$  & $0.016$  & $5.08$  & $3.70$ \\tabularnewline\n$\\mathtt{f}_{2}^{\\Omega_{bb}^{-}\\to\\Xi_{b}^{\\prime*0}}$  & $-0.125$  & $3.16$  & $0.92$  & $\\mathtt{g}_{2}^{\\Omega_{bb}^{-}\\to\\Xi_{b}^{\\prime*0}}$  & $-0.049^{*}$  & $0.27^{*}$  & $0.01^{*}$  & $\\mathtt{f}_{2}^{\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{c}^{\\prime*+}}$  & $-0.135$  & $3.17$  & $0.58$  & $\\mathtt{g}_{2}^{\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{c}^{\\prime*+}}$  & $0.100$  & $3.54$  & $0.74$ \\tabularnewline\n$\\mathtt{f}_{3}^{\\Omega_{bb}^{-}\\to\\Xi_{b}^{\\prime*0}}$  & $0.002$  & $2.95$  & $0.93$  & $\\mathtt{g}_{3}^{\\Omega_{bb}^{-}\\to\\Xi_{b}^{\\prime*0}}$  & $0.132$  & $3.14$  & $0.91$  & $\\mathtt{f}_{3}^{\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{c}^{\\prime*+}}$  & $0.003$  & $2.98$  & $0.56$  & $\\mathtt{g}_{3}^{\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{c}^{\\prime*+}}$  & $0.095$  & $3.14$  & $0.59$ \\tabularnewline\n$\\mathtt{f}_{4}^{\\Omega_{bb}^{-}\\to\\Xi_{b}^{\\prime*0}}$  & $-0.347$  & $3.17$  & $0.88$  & $\\mathtt{g}_{4}^{\\Omega_{bb}^{-}\\to\\Xi_{b}^{\\prime*0}}$  & $0.097$  & $3.42$  & $0.82$  & $\\mathtt{f}_{4}^{\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{c}^{\\prime*+}}$  & $-0.259$  & $3.32$  & $0.54$  & $\\mathtt{g}_{4}^{\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{c}^{\\prime*+}}$  & $0.157$  & $3.67$  & $0.57$ \\tabularnewline\n\\hline\n$\\mathtt{f}_{1}^{\\Xi_{bb}\\to\\Xi_{bc}^{*}}$  & $-0.752$  & $3.52$  & $0.53$  & $\\mathtt{g}_{1}^{\\Xi_{bb}\\to\\Xi_{bc}^{*}}$  & $-2.306$  & $2.41$  & $1.77$  & $\\mathtt{f}_{1}^{\\Xi_{bc}^{(\\prime)}\\to\\Xi_{cc}^{*}}$  & $-0.479$  & $3.88$  & $0.41$  & $\\mathtt{g}_{1}^{\\Xi_{bc}^{(\\prime)}\\to\\Xi_{cc}^{*}}$  & $-0.219^{*}$  & $1.02^{*}$  & $0.54^{*}$ \\tabularnewline\n$\\mathtt{f}_{2}^{\\Xi_{bb}^{0}\\to\\Xi_{bc}^{*}}$  & $-0.473$  & $3.69$  & $0.62$  & $\\mathtt{g}_{2}^{\\Xi_{bb}\\to\\Xi_{bc}^{*}}$  & $-1.572^{*}$  & $3.12^{*}$  & $12.40^{*}$  & $\\mathtt{f}_{2}^{\\Xi_{bc}^{(\\prime)}\\to\\Xi_{cc}^{*}}$  & $-0.443$  & $3.96$  & $0.42$  & $\\mathtt{g}_{2}^{\\Xi_{bc}^{(\\prime)}\\to\\Xi_{cc}^{*}}$  & $0.027^{*}$  & $1.55^{*}$  & $0.28^{*}$ \\tabularnewline\n$\\mathtt{f}_{3}^{\\Xi_{bb}\\to\\Xi_{bc}^{*}}$  & $0.011$  & $3.09$  & $0.40$  & $\\mathtt{g}_{3}^{\\Xi_{bb}\\to\\Xi_{bc}^{*}}$  & $0.920$  & $3.52$  & $0.53$  & $\\mathtt{f}_{3}^{\\Xi_{bc}^{(\\prime)}\\to\\Xi_{cc}^{*}}$  & $0.019$  & $3.79$  & $0.32$  & $\\mathtt{g}_{3}^{\\Xi_{bc}^{(\\prime)}\\to\\Xi_{cc}^{*}}$  & $0.629$  & $3.87$  & $0.41$ \\tabularnewline\n$\\mathtt{f}_{4}^{\\Xi_{bb}\\to\\Xi_{bc}^{*}}$  & $-1.513$  & $3.57$  & $0.50$  & $\\mathtt{g}_{4}^{\\Xi_{bb}\\to\\Xi_{bc}^{*}}$  & $0.376$  & $6.07$  & $3.37$  & $\\mathtt{f}_{4}^{\\Xi_{bc}^{(\\prime)}\\to\\Xi_{cc}^{*}}$  & $-1.038$  & $4.18$  & $0.35$  & $\\mathtt{g}_{4}^{\\Xi_{bc}^{(\\prime)}\\to\\Xi_{cc}^{*++}}$  & $0.573$  & $5.76$  & $1.35$ \\tabularnewline\n\\hline\n$\\mathtt{f}_{1}^{\\Omega_{bb}^{-}\\to\\Omega_{bc}^{*0}}$  & $-0.724$  & $3.57$  & $0.57$  & $\\mathtt{g}_{1}^{\\Omega_{bb}^{-}\\to\\Omega_{bc}^{*0}}$  & $-1.877$  & $2.49$  & $2.09$  & $\\mathtt{f}_{1}^{\\Omega_{bc}^{(\\prime)0}\\to\\Omega_{cc}^{*+}}$  & $-0.500$  & $3.70$  & $0.37$  & $\\mathtt{g}_{1}^{\\Omega_{bc}^{(\\prime)0}\\to\\Omega_{cc}^{*+}}$  & $-0.368$  & $6.29$  & $257.00$ \\tabularnewline\n$\\mathtt{f}_{2}^{\\Omega_{bb}^{-}\\to\\Omega_{bc}^{*0}}$  & $-0.459$  & $3.73$  & $0.66$  & $\\mathtt{g}_{2}^{\\Omega_{bb}^{-}\\to\\Omega_{bc}^{*0}}$  & $-1.229^{*}$  & $1.70^{*}$  & $1.53^{*}$  & $\\mathtt{f}_{2}^{\\Omega_{bc}^{(\\prime)0}\\to\\Omega_{cc}^{*+}}$  & $-0.445$  & $3.77$  & $0.38$  & $\\mathtt{g}_{2}^{\\Omega_{bc}^{(\\prime)0}\\to\\Omega_{cc}^{*+}}$  & $-0.149^{*}$  & $0.38^{*}$  & $0.03^{*}$ \\tabularnewline\n$\\mathtt{f}_{3}^{\\Omega_{bb}^{-}\\to\\Omega_{bc}^{*0}}$  & $0.010$  & $3.19$  & $0.44$  & $\\mathtt{g}_{3}^{\\Omega_{bb}^{-}\\to\\Omega_{bc}^{*0}}$  & $0.869$  & $3.57$  & $0.57$  & $\\mathtt{f}_{3}^{\\Omega_{bc}^{(\\prime)0}\\to\\Omega_{cc}^{*+}}$  & $0.019$  & $3.48$  & $0.24$  & $\\mathtt{g}_{3}^{\\Omega_{bc}^{(\\prime)0}\\to\\Omega_{cc}^{*+}}$  & $0.665$  & $3.69$  & $0.37$ \\tabularnewline\n$\\mathtt{f}_{4}^{\\Omega_{bb}^{-}\\to\\Omega_{bc}^{*0}}$  & $-1.458$  & $3.62$  & $0.55$  & $\\mathtt{g}_{4}^{\\Omega_{bb}^{-}\\to\\Omega_{bc}^{*0}}$  & $0.398$  & $5.28$  & $1.86$  & $\\mathtt{f}_{4}^{\\Omega_{bc}^{(\\prime)0}\\to\\Omega_{cc}^{*+}}$  & $-1.071$  & $3.94$  & $0.31$  & $\\mathtt{g}_{4}^{\\Omega_{bc}^{(\\prime)0}\\to\\Omega_{cc}^{*+}}$  & $0.530$  & $6.13$  & $2.23$ \\tabularnewline\n\\hline\\hline\n\\end{tabular}\n\\end{table}\n\n\\begin{table}\\footnotesize\\label{Tab:fcnc32_cu}\n\n\\caption{\nNumerical results for the transition $1\/2\\to 3\/2$ form factors $\\mathtt{f}_{i}$ and $\\mathtt{g}_{i}$ at $q^2=0$ of $c\\to u$ processes. The parametrization scheme in Eq.~(\\ref{eq:main_fit_formula_cdecay}) is introduced for these form factors,\nand the value of the singly pole $m_{\\rm pole}$ is $1.87~{\\rm GeV}$.}\n\\begin{tabular}{c|c|c|c|c|c|c|c}\n\\hline\\hline\n$F$  & $F(0)$  & $F$  & $F(0)$  & $F$  & $F(0)$  & $F$  & $F(0)$ \\tabularnewline\n\\hline\n$\\mathtt{f}_{1}^{\\Xi_{cc}\\to\\Sigma_{c}^{*}}$  & $-0.979$  & $\\mathtt{g}_{1}^{\\Xi_{cc}\\to\\Sigma_{c}^{*}}$  & $-5.792$  & $\\mathtt{f}_{1}^{\\Xi_{bc}^{(\\prime)}\\to\\Sigma_{b}^{*}}$  & $-0.733$  & $\\mathtt{g}_{1}^{\\Xi_{bc}^{(\\prime)}\\to\\Sigma_{b}^{*}}$  & $-2.193$ \\tabularnewline\n$\\mathtt{f}_{2}^{\\Xi_{cc}\\to\\Sigma_{c}^{*}}$  & $-0.645$  & $\\mathtt{g}_{2}^{\\Xi_{cc}\\to\\Sigma_{c}^{*}}$  & $-3.602$  & $\\mathtt{f}_{2}^{\\Xi_{bc}^{(\\prime)}\\to\\Sigma_{b}^{*}}$  & $-1.380$  & $\\mathtt{g}_{2}^{\\Xi_{bc}^{(\\prime)}\\to\\Sigma_{b}^{*}}$  & $-1.195$ \\tabularnewline\n$\\mathtt{f}_{3}^{\\Xi_{cc}\\to\\Sigma_{c}^{*}}$  & $0.047$  & $\\mathtt{g}_{3}^{\\Xi_{cc}\\to\\Sigma_{c}^{*}}$  & $0.947$  & $\\mathtt{f}_{3}^{\\Xi_{bc}^{(\\prime)}\\to\\Sigma_{b}^{*}}$  & $0.095$  & $\\mathtt{g}_{3}^{\\Xi_{bc}^{(\\prime)}\\to\\Sigma_{b}^{*}}$  & $0.537$ \\tabularnewline\n$\\mathtt{f}_{4}^{\\Xi_{cc}\\to\\Sigma_{c}^{*}}$  & $-1.969$  & $\\mathtt{g}_{4}^{\\Xi_{cc}\\to\\Sigma_{c}^{*}}$  & $0.393$  & $\\mathtt{f}_{4}^{\\Xi_{bc}^{(\\prime)}\\to\\Sigma_{b}^{*}}$  & $-1.726$  & $\\mathtt{g}_{4}^{\\Xi_{bc}^{(\\prime)}\\to\\Sigma_{b}^{*}}$  & $0.157$ \\tabularnewline\n$\\mathtt{f}_{1}^{T\\Xi_{cc}\\to\\Sigma_{c}^{*}}$  & $0.534$  & $\\mathtt{g}_{1}^{T\\Xi_{cc}\\to\\Sigma_{c}^{*}}$  & $-2.469$  & $\\mathtt{f}_{1}^{T\\Xi_{bc}^{(\\prime)}\\to\\Sigma_{b}^{*}}$  & $0.221$  & $\\mathtt{g}_{1}^{T\\Xi_{bc}^{(\\prime)}\\to\\Sigma_{b}^{*}}$  & $12.050$ \\tabularnewline\n$\\mathtt{f}_{2}^{T\\Xi_{cc}\\to\\Sigma_{c}^{*}}$  & $0.330$  & $\\mathtt{g}_{2}^{T\\Xi_{cc}\\to\\Sigma_{c}^{*}}$  & $-1.394$  & $\\mathtt{f}_{2}^{T\\Xi_{bc}^{(\\prime)}\\to\\Sigma_{b}^{*}}$  & $0.379$  & $\\mathtt{g}_{2}^{T\\Xi_{bc}^{(\\prime)}\\to\\Sigma_{b}^{*}}$  & $6.959$ \\tabularnewline\n$\\mathtt{f}_{4}^{T\\Xi_{cc}\\to\\Sigma_{c}^{*}}$  & $1.076$  & $\\mathtt{g}_{4}^{T\\Xi_{cc}\\to\\Sigma_{c}^{*}}$  & $0.032$  & $\\mathtt{f}_{4}^{T\\Xi_{bc}^{(\\prime)}\\to\\Sigma_{b}^{*}}$  & $0.518$  & $\\mathtt{g}_{4}^{T\\Xi_{bc}^{(\\prime)}\\to\\Sigma_{b}^{*}}$  & $0.122$ \\tabularnewline\n\\hline\n$\\mathtt{f}_{1}^{\\Omega_{cc}^{+}\\to\\Xi_{c}^{\\prime*+}}$  & $-1.269$  & $\\mathtt{g}_{1}^{\\Omega_{cc}^{+}\\to\\Xi_{c}^{\\prime*+}}$  & $-13.390$  & $\\mathtt{f}_{1}^{\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{b}^{\\prime*0}}$  & $-0.916$  & $\\mathtt{g}_{1}^{\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{b}^{\\prime*0}}$  & $-7.008$ \\tabularnewline\n$\\mathtt{f}_{2}^{\\Omega_{cc}^{+}\\to\\Xi_{c}^{\\prime*+}}$  & $-0.811$  & $\\mathtt{g}_{2}^{\\Omega_{cc}^{+}\\to\\Xi_{c}^{\\prime*+}}$  & $-8.823$  & $\\mathtt{f}_{2}^{\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{b}^{\\prime*0}}$  & $-1.902$  & $\\mathtt{g}_{2}^{\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{b}^{\\prime*0}}$  & $-4.130$ \\tabularnewline\n$\\mathtt{f}_{3}^{\\Omega_{cc}^{+}\\to\\Xi_{c}^{\\prime*+}}$  & $0.090$  & $\\mathtt{g}_{3}^{\\Omega_{cc}^{+}\\to\\Xi_{c}^{\\prime*+}}$  & $1.366$  & $\\mathtt{f}_{3}^{\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{b}^{\\prime*0}}$  & $0.167$  & $\\mathtt{g}_{3}^{\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{b}^{\\prime*0}}$  & $0.746$ \\tabularnewline\n$\\mathtt{f}_{4}^{\\Omega_{cc}^{+}\\to\\Xi_{c}^{\\prime*+}}$  & $-2.532$  & $\\mathtt{g}_{4}^{\\Omega_{cc}^{+}\\to\\Xi_{c}^{\\prime*+}}$  & $0.129$  & $\\mathtt{f}_{4}^{\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{b}^{\\prime*0}}$  & $-2.162$  & $\\mathtt{g}_{4}^{\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{b}^{\\prime*0}}$  & $0.104$ \\tabularnewline\n$\\mathtt{f}_{1}^{T\\Omega_{cc}^{+}\\to\\Xi_{c}^{\\prime*+}}$  & $0.693$  & $\\mathtt{g}_{1}^{T\\Omega_{cc}^{+}\\to\\Xi_{c}^{\\prime*+}}$  & $-4.387$  & $\\mathtt{f}_{1}^{T\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{b}^{\\prime*0}}$  & $0.277$  & $\\mathtt{g}_{1}^{T\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{b}^{\\prime*0}}$  & $18.900$ \\tabularnewline\n$\\mathtt{f}_{2}^{T\\Omega_{cc}^{+}\\to\\Xi_{c}^{\\prime*+}}$  & $0.395$  & $\\mathtt{g}_{2}^{T\\Omega_{cc}^{+}\\to\\Xi_{c}^{\\prime*+}}$  & $-2.582$  & $\\mathtt{f}_{2}^{T\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{b}^{\\prime*0}}$  & $0.505$  & $\\mathtt{g}_{2}^{T\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{b}^{\\prime*0}}$  & $10.710$ \\tabularnewline\n$\\mathtt{f}_{4}^{T\\Omega_{cc}^{+}\\to\\Xi_{c}^{\\prime*+}}$  & $1.383$  & $\\mathtt{g}_{4}^{T\\Omega_{cc}^{+}\\to\\Xi_{c}^{\\prime*+}}$  & $-0.022$  & $\\mathtt{f}_{4}^{T\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{b}^{\\prime*0}}$  & $0.648$  & $\\mathtt{g}_{4}^{T\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{b}^{\\prime*0}}$  & $0.140$ \\tabularnewline\n\\hline \\hline\n\\end{tabular}\n\\end{table}\n\n\\begin{table}\\footnotesize\n\\caption{Numerical results for the transition $1\/2\\to 3\/2$ form factors $\\mathtt{f}_{i}$ and $\\mathtt{g}_{i}$ of $b\\to d$ processes. The parametrization scheme shown with Eq.~(\\ref{eq:auxiliary_fit_formula_bdecay}) is introduced for these form factors with asterisk, and Eq.~(\\ref{eq:main_fit_formula_bdecay}) for all the other ones.}\\label{Tab:fcnc32_bd}\n\\begin{tabular}{c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c}\n\\hline\\hline\n$F$  & $F(0)$  & $m_{{\\rm {fit}}}$  & $\\delta$  & $F$  & $F(0)$  & $m_{{\\rm {fit}}}$  & $\\delta$ & $F$  & $F(0)$  & $m_{{\\rm {fit}}}$  & $\\delta$  & $F$  & $F(0)$  & $m_{{\\rm {fit}}}$  & $\\delta$\\tabularnewline\n\\hline\n$\\mathtt{f}_{1}^{\\Xi_{bb}\\to\\Sigma_{b}^{*}}$  & $-0.172$  & $3.15$  & $0.89$  & $\\mathtt{g}_{1}^{\\Xi_{bb}\\to\\Sigma_{b}^{*}}$  & $-0.108$  & $2.70$  & $1.84$  & $\\mathtt{f}_{1}^{\\Xi_{bc}^{(\\prime)}\\to\\Sigma_{c}^{*}}$  & $-0.108$  & $3.23$  & $0.54$  & $\\mathtt{g}_{1}^{\\Xi_{bc}^{(\\prime)}\\to\\Sigma_{c}^{*}}$  & $0.021$  & $3.86$  & $1.01$\\tabularnewline\n$\\mathtt{f}_{2}^{\\Xi_{bb}\\to\\Sigma_{b}^{*}}$  & $-0.129$  & $3.17$  & $0.90$  & $\\mathtt{g}_{2}^{\\Xi_{bb}\\to\\Sigma_{b}^{*}}$  & $-0.057^{*}$  & $0.54^{*}$  & $0.06^{*}$  & $\\mathtt{f}_{2}^{\\Xi_{bc}^{(\\prime)}\\to\\Sigma_{c}^{*}}$  & $-0.145$  & $3.25$  & $0.54$  & $\\mathtt{g}_{2}^{\\Xi_{bc}^{(\\prime)}\\to\\Sigma_{c}^{*}}$  & $0.120$  & $3.44$  & $0.58$ \\tabularnewline\n$\\mathtt{f}_{3}^{\\Xi_{bb}\\to\\Sigma_{b}^{*}}$  & $0.002$  & $2.95$  & $0.91$  & $\\mathtt{g}_{3}^{\\Xi_{bb}\\to\\Sigma_{b}^{*}}$  & $0.136$  & $3.15$  & $0.89$  & $\\mathtt{f}_{3}^{\\Xi_{bc}^{(\\prime)}\\to\\Sigma_{c}^{*}}$  & $0.003$  & $3.06$  & $0.51$  & $\\mathtt{g}_{3}^{\\Xi_{bc}^{(\\prime)}\\to\\Sigma_{c}^{*}}$  & $0.096$  & $3.21$  & $0.54$ \\tabularnewline\n$\\mathtt{f}_{4}^{\\Xi_{bb}\\to\\Sigma_{b}^{*}}$  & $-0.355$  & $3.18$  & $0.86$  & $\\mathtt{g}_{4}^{\\Xi_{bb}\\to\\Sigma_{b}^{*}}$  & $0.099$  & $3.46$  & $0.82$  & $\\mathtt{f}_{4}^{\\Xi_{bc}^{(\\prime)}\\to\\Sigma_{c}^{*}}$  & $-0.270$  & $3.44$  & $0.51$  & $\\mathtt{g}_{4}^{\\Xi_{bc}^{(\\prime)}\\to\\Sigma_{c}^{*}}$  & $0.174$  & $3.75$  & $0.52$ \\tabularnewline\n$\\mathtt{f}_{1}^{T,\\Xi_{bb}\\to\\Sigma_{b}^{*}}$  & $0.091$  & $3.14$  & $0.89$  & $\\mathtt{g}_{1}^{T,\\Xi_{bb}\\to\\Sigma_{b}^{*}}$  & $-0.215$  & $2.82$  & $0.82$  & $\\mathtt{f}_{1}^{T,\\Xi_{bc}^{(\\prime)}\\to\\Sigma_{c}^{*}}$  & $0.082$  & $3.22$  & $0.54$  & $\\mathtt{g}_{1}^{T,\\Xi_{bc}^{(\\prime)}\\to\\Sigma_{c}^{*}}$  & $-0.025$  & $2.94$  & $0.44$ \\tabularnewline\n$\\mathtt{f}_{2}^{T,\\Xi_{bb}\\to\\Sigma_{b}^{*}}$  & $0.067$  & $3.16$  & $0.90$  & $\\mathtt{g}_{2}^{T,\\Xi_{bb}\\to\\Sigma_{b}^{*}}$  & $-0.107$  & $3.05$  & $3.24$  & $\\mathtt{f}_{2}^{T,\\Xi_{bc}^{(\\prime)}\\to\\Sigma_{c}^{*}}$  & $0.109$  & $3.24$  & $0.54$  & $\\mathtt{g}_{2}^{T,\\Xi_{bc}^{(\\prime)}\\to\\Sigma_{c}^{*}}$  & $0.080$  & $3.82$  & $0.79$ \\tabularnewline\n$\\mathtt{f}_{4}^{T,\\Xi_{bb}\\to\\Sigma_{b}^{*}}$  & $0.188$  & $3.17$  & $0.86$  & $\\mathtt{g}_{4}^{T,\\Xi_{bb}\\to\\Sigma_{b}^{*}}$  & $0.019$  & $3.34$  & $1.14$  & $\\mathtt{f}_{4}^{T,\\Xi_{bc}^{(\\prime)}\\to\\Sigma_{c}^{*}}$  & $0.207$  & $3.43$  & $0.51$  & $\\mathtt{g}_{4}^{T,\\Xi_{bc}^{(\\prime)}\\to\\Sigma_{c}^{*}}$  & $0.086$  & $3.59$  & $0.55$ \\tabularnewline\n\\hline\n$\\mathtt{f}_{1}^{\\Omega_{bb}^{-}\\to\\Xi_{b}^{\\prime*-}}$  & $-0.168$  & $3.14$  & $0.91$  & $\\mathtt{g}_{1}^{\\Omega_{bb}^{-}\\to\\Xi_{b}^{\\prime*-}}$  & $-0.103$  & $2.70$  & $1.86$  & $\\mathtt{f}_{1}^{\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{c}^{\\prime*0}}$  & $-0.106$  & $3.13$  & $0.55$  & $\\mathtt{g}_{1}^{\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{c}^{\\prime*0}}$  & $0.016$  & $5.13$  & $3.95$ \\tabularnewline\n$\\mathtt{f}_{2}^{\\Omega_{bb}^{-}\\to\\Xi_{b}^{\\prime*-}}$  & $-0.126$  & $3.16$  & $0.92$  & $\\mathtt{g}_{2}^{\\Omega_{bb}^{-}\\to\\Xi_{b}^{\\prime*-}}$  & $-0.052^{*}$  & $0.46^{*}$  & $0.04^{*}$  & $\\mathtt{f}_{2}^{\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{c}^{\\prime*0}}$  & $-0.135$  & $3.16$  & $0.55$  & $\\mathtt{g}_{2}^{\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{c}^{\\prime*0}}$  & $0.100$  & $3.53$  & $0.71$ \\tabularnewline\n$\\mathtt{f}_{3}^{\\Omega_{bb}^{-}\\to\\Xi_{b}^{\\prime*-}}$  & $0.002$  & $2.95$  & $0.93$  & $\\mathtt{g}_{3}^{\\Omega_{bb}^{-}\\to\\Xi_{b}^{\\prime*-}}$  & $0.132$  & $3.14$  & $0.91$  & $\\mathtt{f}_{3}^{\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{c}^{\\prime*0}}$  & $0.003$  & $2.96$  & $0.52$  & $\\mathtt{g}_{3}^{\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{c}^{\\prime*0}}$  & $0.095$  & $3.12$  & $0.55$ \\tabularnewline\n$\\mathtt{f}_{4}^{\\Omega_{bb}^{-}\\to\\Xi_{b}^{\\prime*-}}$  & $-0.348$  & $3.16$  & $0.88$  & $\\mathtt{g}_{4}^{\\Omega_{bb}^{-}\\to\\Xi_{b}^{\\prime*-}}$  & $0.097$  & $3.42$  & $0.83$  & $\\mathtt{f}_{4}^{\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{c}^{\\prime*0}}$  & $-0.259$  & $3.31$  & $0.52$  & $\\mathtt{g}_{4}^{\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{c}^{\\prime*0}}$  & $0.157$  & $3.66$  & $0.55$ \\tabularnewline\n$\\mathtt{f}_{1}^{T,\\Omega_{bb}^{-}\\to\\Xi_{b}^{\\prime*-}}$  & $0.088$  & $3.13$  & $0.91$  & $\\mathtt{g}_{1}^{T,\\Omega_{bb}^{-}\\to\\Xi_{b}^{\\prime*-}}$  & $-0.206$  & $2.82$  & $0.83$  & $\\mathtt{f}_{1}^{T,\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{c}^{\\prime*0}}$  & $0.080$  & $3.13$  & $0.55$  & $\\mathtt{g}_{1}^{T,\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{c}^{\\prime*0}}$  & $-0.032$  & $2.77$  & $0.45$ \\tabularnewline\n$\\mathtt{f}_{2}^{T,\\Omega_{bb}^{-}\\to\\Xi_{b}^{\\prime*-}}$  & $0.065$  & $3.15$  & $0.92$  & $\\mathtt{g}_{2}^{T,\\Omega_{bb}^{-}\\to\\Xi_{b}^{\\prime*-}}$  & $-0.101$  & $3.05$  & $3.24$  & $\\mathtt{f}_{2}^{T,\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{c}^{\\prime*0}}$  & $0.101$  & $3.15$  & $0.55$  & $\\mathtt{g}_{2}^{T,\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{c}^{\\prime*0}}$  & $0.064$  & $4.16$  & $1.26$ \\tabularnewline\n$\\mathtt{f}_{4}^{T,\\Omega_{bb}^{-}\\to\\Xi_{b}^{\\prime*-}}$  & $0.181$  & $3.16$  & $0.88$  & $\\mathtt{g}_{4}^{T,\\Omega_{bb}^{-}\\to\\Xi_{b}^{\\prime*-}}$  & $0.019$  & $3.30$  & $1.14$  & $\\mathtt{f}_{4}^{T,\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{c}^{\\prime*0}}$  & $0.197$  & $3.31$  & $0.52$  & $\\mathtt{g}_{4}^{T,\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{c}^{\\prime*0}}$  & $0.075$  & $3.52$  & $0.58$ \\tabularnewline\n\\hline \\hline\n\\end{tabular}\n\\end{table}\n\n\\begin{table}\\footnotesize\n\\caption{Same with Tab.~\\ref{Tab:fcnc32_bd} except for $b\\to s$ process.}\\label{Tab:fcnc32_bs}\n\\begin{tabular}{c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c}\n\\hline \\hline\n$F$  & $F(0)$  & $m_{{\\rm {fit}}}$  & $\\delta$  & $F$  & $F(0)$  & $m_{{\\rm {fit}}}$  & $\\delta$ & $F$  & $F(0)$  & $m_{{\\rm {fit}}}$  & $\\delta$  & $F$  & $F(0)$  & $m_{{\\rm {fit}}}$  & $\\delta$\\tabularnewline\n\\hline\n$\\mathtt{f}_{1}^{\\Xi_{bb}\\to\\Xi_{b}^{\\prime*}}$  & $-0.232$  & $3.16$  & $0.62$  & $\\mathtt{g}_{1}^{\\Xi_{bb}\\to\\Xi_{b}^{\\prime*}}$  & $-0.142$  & $2.30$  & $0.87$  & $\\mathtt{f}_{1}^{\\Xi_{bc}^{(\\prime)}\\to\\Xi_{c}^{\\prime*}}$  & $-0.153$  & $3.42$  & $0.54$  & $\\mathtt{g}_{1}^{\\Xi_{bc}^{(\\prime)}\\to\\Xi_{c}^{\\prime*}}$  & $0.040$  & $4.00$  & $1.07$ \\tabularnewline\n$\\mathtt{f}_{2}^{\\Xi_{bb}\\to\\Xi_{b}^{\\prime*}}$  & $-0.172$  & $3.20$  & $0.64$  & $\\mathtt{g}_{2}^{\\Xi_{bb}\\to\\Xi_{b}^{\\prime*}}$  & $-0.065^{*}$  & $0.81^{*}$  & $0.28^{*}$  & $\\mathtt{f}_{2}^{\\Xi_{bc}^{(\\prime)}\\to\\Xi_{c}^{\\prime*}}$  & $-0.196$  & $3.45$  & $0.55$  & $\\mathtt{g}_{2}^{\\Xi_{bc}^{(\\prime)}\\to\\Xi_{c}^{\\prime*}}$  & $0.182$  & $3.63$  & $0.61$ \\tabularnewline\n$\\mathtt{f}_{3}^{\\Xi_{bb}\\to\\Xi_{b}^{\\prime*}}$  & $0.003$  & $2.91$  & $0.57$  & $\\mathtt{g}_{3}^{\\Xi_{bb}\\to\\Xi_{b}^{\\prime*}}$  & $0.191$  & $3.17$  & $0.62$  & $\\mathtt{f}_{3}^{\\Xi_{bc}^{(\\prime)}\\to\\Xi_{c}^{\\prime*}}$  & $0.005$  & $3.25$  & $0.48$  & $\\mathtt{g}_{3}^{\\Xi_{bc}^{(\\prime)}\\to\\Xi_{c}^{\\prime*}}$  & $0.141$  & $3.40$  & $0.54$ \\tabularnewline\n$\\mathtt{f}_{4}^{\\Xi_{bb}\\to\\Xi_{b}^{\\prime*}}$  & $-0.479$  & $3.21$  & $0.61$  & $\\mathtt{g}_{4}^{\\Xi_{bb}\\to\\Xi_{b}^{\\prime*}}$  & $0.142$  & $3.59$  & $0.66$  & $\\mathtt{f}_{4}^{\\Xi_{bc}^{(\\prime)}\\to\\Xi_{c}^{\\prime*}}$  & $-0.375$  & $3.66$  & $0.51$  & $\\mathtt{g}_{4}^{\\Xi_{bc}^{(\\prime)}\\to\\Xi_{c}^{\\prime*}}$  & $0.251$  & $4.05$  & $0.54$ \\tabularnewline\n$\\mathtt{f}_{1}^{T,\\Xi_{bb}\\to\\Xi_{b}^{\\prime*}}$  & $0.126$  & $3.16$  & $0.62$  & $\\mathtt{g}_{1}^{T,\\Xi_{bb}\\to\\Xi_{b}^{\\prime*}}$  & $-0.283$  & $2.70$  & $0.51$  & $\\mathtt{f}_{1}^{T,\\Xi_{bc}^{(\\prime)}\\to\\Xi_{c}^{\\prime*}}$  & $0.120$  & $3.41$  & $0.54$  & $\\mathtt{g}_{1}^{T,\\Xi_{bc}^{(\\prime)}\\to\\Xi_{c}^{\\prime*}}$  & $-0.028$  & $2.91$  & $0.36$ \\tabularnewline\n$\\mathtt{f}_{2}^{T,\\Xi_{bb}\\to\\Xi_{b}^{\\prime*}}$  & $0.092$  & $3.19$  & $0.63$  & $\\mathtt{g}_{2}^{T,\\Xi_{bb}\\to\\Xi_{b}^{\\prime*}}$  & $-0.131$  & $2.43$  & $1.08$  & $\\mathtt{f}_{2,A}^{T,\\Xi_{bc}^{(\\prime)}\\to\\Xi_{c}^{\\prime*}}$  & $0.152$  & $3.44$  & $0.54$  & $\\mathtt{g}_{2}^{T,\\Xi_{bc}^{(\\prime)}\\to\\Xi_{c}^{\\prime*}}$  & $0.125$  & $4.07$  & $0.86$ \\tabularnewline\n$\\mathtt{f}_{4}^{T,\\Xi_{bb}\\to\\Xi_{b}^{\\prime*}}$  & $0.260$  & $3.20$  & $0.61$  & $\\mathtt{g}_{4}^{T,\\Xi_{bb}\\to\\Xi_{b}^{\\prime*}}$  & $0.031$  & $3.45$  & $0.78$  & $\\mathtt{f}_{4}^{T,\\Xi_{bc}^{(\\prime)}\\to\\Xi_{c}^{\\prime*}}$  & $0.295$  & $3.66$  & $0.50$  & $\\mathtt{g}_{4}^{T,\\Xi_{bc}^{(\\prime)}\\to\\Xi_{c}^{\\prime*}}$  & $0.124$  & $3.89$  & $0.57$ \\tabularnewline\n\\hline\n$\\mathtt{f}_{1}^{\\Omega_{bb}^{-}\\to\\Omega_{b}^{*-}}$  & $-0.230$  & $3.14$  & $0.63$  & $\\mathtt{g}_{1}^{\\Omega_{bb}^{-}\\to\\Omega_{b}^{*-}}$  & $-0.145$  & $2.29$  & $0.86$  & $\\mathtt{f}_{1}^{\\Omega_{bc}^{(\\prime)}\\to\\Omega_{c}^{*}}$  & $-0.152$  & $3.32$  & $0.55$  & $\\mathtt{g}_{1}^{\\Omega_{bc}^{(\\prime)}\\to\\Omega_{c}^{*}}$  & $0.035$  & $4.65$  & $2.31$ \\tabularnewline\n$\\mathtt{f}_{2}^{\\Omega_{bb}^{-}\\to\\Omega_{b}^{*-}}$  & $-0.169$  & $3.18$  & $0.64$  & $\\mathtt{g}_{2}^{\\Omega_{bb}^{-}\\to\\Omega_{b}^{*-}}$  & $-0.068^{*}$  & $1.21^{*}$  & $0.84^{*}$  & $\\mathtt{f}_{2}^{\\Omega_{bc}^{(\\prime)0}\\to\\Omega_{c}^{*0}}$  & $-0.186$  & $3.36$  & $0.55$  & $\\mathtt{g}_{2}^{\\Omega_{bc}^{(\\prime)0}\\to\\Omega_{c}^{*0}}$  & $0.158$  & $3.72$  & $0.75$ \\tabularnewline\n$\\mathtt{f}_{3}^{\\Omega_{bb}^{-}\\to\\Omega_{b}^{*-}}$  & $0.003$  & $2.89$  & $0.58$  & $\\mathtt{g}_{3}^{\\Omega_{bb}^{-}\\to\\Omega_{b}^{*-}}$  & $0.189$  & $3.15$  & $0.62$  & $\\mathtt{f}_{3}^{\\Omega_{bc}^{(\\prime)0}\\to\\Omega_{c}^{*0}}$  & $0.004$  & $3.15$  & $0.49$  & $\\mathtt{g}_{3}^{\\Omega_{bc}^{(\\prime)0}\\to\\Omega_{c}^{*0}}$  & $0.141$  & $3.30$  & $0.55$ \\tabularnewline\n$\\mathtt{f}_{4}^{\\Omega_{bb}^{-}\\to\\Omega_{b}^{*-}}$  & $-0.474$  & $3.18$  & $0.61$  & $\\mathtt{g}_{4}^{\\Omega_{bb}^{-}\\to\\Omega_{b}^{*-}}$  & $0.137$  & $3.56$  & $0.66$  & $\\mathtt{f}_{4}^{\\Omega_{bc}^{(\\prime)0}\\to\\Omega_{c}^{*0}}$  & $-0.365$  & $3.53$  & $0.51$  & $\\mathtt{g}_{4}^{\\Omega_{bc}^{(\\prime)0}\\to\\Omega_{c}^{*0}}$  & $0.231$  & $3.96$  & $0.57$ \\tabularnewline\n$\\mathtt{f}_{1}^{T,\\Omega_{bb}^{-}\\to\\Omega_{b}^{*-}}$  & $0.123$  & $3.14$  & $0.63$  & $\\mathtt{g}_{1}^{T,\\Omega_{bb}^{-}\\to\\Omega_{b}^{*-}}$  & $-0.277$  & $2.69$  & $0.51$  & $\\mathtt{f}_{1}^{T,\\Omega_{bc}^{(\\prime)0}\\to\\Omega_{c}^{*0}}$  & $0.118$  & $3.32$  & $0.55$  & $\\mathtt{g}_{1}^{T,\\Omega_{bc}^{(\\prime)0}\\to\\Omega_{c}^{*0}}$  & $-0.037$  & $2.75$  & $0.39$ \\tabularnewline\n$\\mathtt{f}_{2}^{T,\\Omega_{bb}^{-}\\to\\Omega_{b}^{*-}}$  & $0.089$  & $3.17$  & $0.64$  & $\\mathtt{g}_{2}^{T,\\Omega_{bb}^{-}\\to\\Omega_{b}^{*-}}$  & $-0.128$  & $2.43$  & $1.07$  & $\\mathtt{f}_{2}^{T,\\Omega_{bc}^{(\\prime)0}\\to\\Omega_{c}^{*0}}$  & $0.143$  & $3.35$  & $0.55$  & $\\mathtt{g}_{2}^{T,\\Omega_{bc}^{(\\prime)0}\\to\\Omega_{c}^{*0}}$  & $0.105$  & $4.37$  & $1.31$ \\tabularnewline\n$\\mathtt{f}_{4}^{T,\\Omega_{bb}^{-}\\to\\Omega_{b}^{*-}}$  & $0.254$  & $3.18$  & $0.61$  & $\\mathtt{g}_{4}^{T,\\Omega_{bb}^{-}\\to\\Omega_{b}^{*-}}$  & $0.030$  & $3.42$  & $0.78$  & $\\mathtt{f}_{4}^{T,\\Omega_{bc}^{(\\prime)0}\\to\\Omega_{c}^{*0}}$  & $0.284$  & $3.53$  & $0.51$  & $\\mathtt{g}_{4}^{T,\\Omega_{bc}^{(\\prime)0}\\to\\Omega_{c}^{*0}}$  & $0.111$  & $3.81$  & $0.60$ \\tabularnewline\n\\hline \\hline\n\\end{tabular}\n\\end{table}\n\n\n\\section{Semi-leptonic weak decays}\nFor the charged current process $c\\to d,s~l^{+}\\nu_{l}$, the effective Hamiltonian is\n\\begin{equation}\n{\\cal H}_{{\\rm eff}}(c\\to d,s~l^{+}\\nu_{l})=\n\\frac{G_{F}}{\\sqrt{2}}\\Big(V_{cd}^{*}[\\bar{d}\\gamma_{\\mu}(1-\\gamma_5)c][\\bar{\\nu}_{l}\\gamma_{\\mu}(1-\\gamma_5)l]+V_{cs}^{*}[\\bar{s}\\gamma_{\\mu}(1-\\gamma_5)c][\\bar{\\nu}_{l}\\gamma_{\\mu}(1-\\gamma_5)l]\\Big)\\label{eq:efhvc},\n\\end{equation}\nand for $b\\to u,c~l^{-}\\bar{\\nu}_{l}$, the effective Hamiltonian has been given as\n\\begin{equation}\n{\\cal H}_{{\\rm eff}}(b\\to u,c~l^{-}\\bar{\\nu}_{l})=\n\\frac{G_{F}}{\\sqrt{2}}\\Big(V_{ub}[\\bar{u}\\gamma_{\\mu}(1-\\gamma_5)b][\\bar{l}\\gamma_{\\mu}(1-\\gamma_5)\\nu_{l}]+V_{cb}[\\bar{c}\\gamma_{\\mu}(1-\\gamma_5)b][\\bar{l}\\gamma_{\\mu}(1-\\gamma_5)\\nu_{l}]\\Big)\\label{eq:efhvb}.\n\t\\end{equation}\nWhile for the FCNC process $b\\to sl^{+}l^{-}$, the effective Hamiltonian can be given as\n\\begin{equation}\n{\\cal H}_{{\\rm eff}}(b\\to sl^{+}l^{-})=-\\frac{G_{F}}{\\sqrt{2}}V_{tb}V_{ts}^{*}\\sum_{i=1}^{10}C_{i}(\\mu)O_{i}(\\mu)\\label{eq:efhfcnc}.\n\\end{equation}\nIn Eqs.~(\\ref{eq:efhvc}), (\\ref{eq:efhvb}) and (\\ref{eq:efhfcnc}), the Fermi constant $G_F$ and the CKM matrix elements are taken from Ref.~\\cite{Tanabashi:2018oca}:\n\\begin{eqnarray}\n&&G_F=1.166\\times 10^{-5}{\\rm GeV}^{-2},\\quad|V_{cd}|=0.218,\\quad|V_{cs}|=0.997,\\quad|V_{cb}|=0.0422,\\nonumber\\\\\n&&~|V_{ub}|=0.00394,\\quad|V_{ts}|=0.0394,\\quad|V_{td}|=0.0081,\\quad|V_{tb}|=1.019.\n\\end{eqnarray}\nThe reader interested in the explicit forms of operators $O_{i}$ in Eq.~(\\ref{eq:efhfcnc}) can consult Ref.~\\cite{Buchalla:1995vs}. Wilson coefficients $C_{i}$ for each operators $O_{i}$ are calculated in the leading logarithmic approximation, with $m_{W}=80.4$~GeV and $\\mu=m_{b,{\\rm pole}}$~\\cite{Buchalla:1995vs} and can be given as follows,\n\\begin{eqnarray}\n&&C_{1}=1.107,\\quad C_{2}=-0.248,\\quad C_{3}=-0.011,\\quad C_{4}=-0.026, \\nonumber\\\\\n&&C_{5}=-0.007,\\quad C_{6}=-0.031,\\quad C_{7}^{{\\rm eff}}=-0.313,\\quad C_{9}=4.344,\\quad C_{10}=-4.669,\n\\end{eqnarray}\nFor the FCNC process ${\\cal B}_{b}\\to{\\cal B}_{s}^{\\prime}l^{+}l^{-}$, the amplitude can be obtained in following form,\n\\begin{eqnarray}\n{\\cal M}({\\cal B}\\to{\\cal B}^{\\prime}l^{+}l^{-}) & = & -\\frac{G_{F}}{\\sqrt{2}}V_{tb}V_{ts}^{*}\\frac{\\alpha_{{\\rm em}}}{2\\pi}\\Big\\{\\Big(C_{9}^{{\\rm eff}}(q^{2})\\langle{\\cal B}^{\\prime}|\\bar{s}\\gamma_{\\mu}(1-\\gamma_{5})b|{\\cal B}\\rangle-2m_{b}C_{7}^{{\\rm eff}}\\langle{\\cal B}^{\\prime}|\\bar{s}i\\sigma_{\\mu\\nu}\\frac{q^{\\nu}}{q^{2}}(1+\\gamma_{5})b|{\\cal B}\\rangle\\Big)\\bar{l}\\gamma^{\\mu}l\\nonumber \\\\\n&  & \\qquad\\qquad\\qquad\\quad+\\,C_{10}\\langle{\\cal B}^{\\prime}|\\bar{s}\\gamma_{\\mu}(1-\\gamma_{5})b|{\\cal B}\\rangle\\bar{l}\\gamma^{\\mu}\\gamma_{5}l\\Big\\}.\\label{eq:the_amplitude}\n\\end{eqnarray}\nIn Refs.~\\cite{Li:2009rc,Lu:2011jm,Giri:2005mt}, the signs before $C_{7}^{{\\rm eff}}$ are different. In this paper, we take the same sign with the ones in Refs.~\\cite{Li:2009rc,Lu:2011jm}, which is different from the one in Ref.~\\cite{Giri:2005mt}. In the above Eq.~(\\ref{eq:the_amplitude}), $C_{7}^{{\\rm eff}}$ and $C_{9}^{{\\rm eff}}$ are obtained as ~\\cite{Buras:1994dj}\n\\begin{eqnarray}\nC_{7}^{{\\rm eff}} & = & C_{7}-C_{5}\/3-C_{6},\\nonumber \\\\\nC_{9}^{{\\rm eff}}(q^{2}) & = & C_{9}(\\mu)+h(\\hat{m}_{c},\\hat{s})C_{0}-\\frac{1}{2}h(1,\\hat{s})(4C_{3}+4C_{4}+3C_{5}+C_{6})\\nonumber \\\\\n&  & -\\frac{1}{2}h(0,\\hat{s})(C_{3}+3C_{4})+\\frac{2}{9}(3C_{3}+C_{4}+3C_{5}+C_{6}),\n\\end{eqnarray}\nwhere $\\hat{s}=q^{2}\/m_{b}^{2}$, $C_{0}=C_{1}+3C_{2}+3C_{3}+C_{4}+3C_{5}+C_{6}$,\nand $\\hat{m}_{c}=m_{c}\/m_{b}$.\nThe auxiliary functions $h$ have been given as\n\\begin{eqnarray}\nh(z,\\hat{s}) & = & -\\frac{8}{9}\\ln\\frac{m_{b}}{\\mu}-\\frac{8}{9}\\ln z+\\frac{8}{27}+\\frac{4}{9}x-\\frac{2}{9}(2+x)|1-x|^{1\/2}\\times\\begin{cases}\n\\left(\\ln\\left|\\frac{\\sqrt{1-x}+1}{\\sqrt{1-x}-1}\\right|-i\\pi\\right), & x\\equiv\\frac{4z^{2}}{\\hat{s}}<1\\\\\n2\\arctan\\frac{1}{\\sqrt{x-1}}, & x\\equiv\\frac{4z^{2}}{\\hat{s}}>1\n\\end{cases},\\nonumber \\\\\nh(0,\\hat{s}) & = & -\\frac{8}{9}\\ln\\frac{m_{b}}{\\mu}-\\frac{4}{9}\\ln\\hat{s}+\\frac{8}{27}+\\frac{4}{9}i\\pi.\n\\end{eqnarray}\n\nWhile for the FCNC process $b\\to d l^+l^-$, the corresponding effective Hamiltonian and amplitude can be got by taking a  replacement $s\\to d$ similarly.\n\n\t\n\n\n\\subsection{Decay widths}\n\\subsubsection{the charged current transition}\n\n\n\\begin{figure}\n\\includegraphics[width=0.45\\columnwidth]{dynamic.eps}\n\\caption{ Kinematics for the charged current induced decay mode. }\n\\label{fig:dynamic}\n\\end{figure}\n\nFor the charged current induced transition, the kinematics are shown in Fig.~\\ref{fig:dynamic}, and the helicity amplitudes are defined by\n\\begin{equation}\nHV_{\\lambda^{\\prime},\\lambda_{W}}^{\\lambda}\\equiv\\langle{\\cal B}_{f}^{(*)}(\\lambda^{\\prime})|\\bar{q}\\gamma^{\\mu}Q|{\\cal B}_{i}(\\lambda)\\rangle\\epsilon_{W\\mu}^{*}(\\lambda_{W})\\quad\n{\\rm and}\\quad HA_{\\lambda^{\\prime},\\lambda_{W}}^{\\lambda}\\equiv\\langle{\\cal B}_{f}^{(*)}(\\lambda^{\\prime})|\\bar{q}\\gamma^{\\mu}\\gamma_{5}Q|{\\cal B}_{i}(\\lambda)\\rangle\\epsilon_{W\\mu}^{*}(\\lambda_{W}),\n\\end{equation}\nhere $\\epsilon_{\\mu}$ and $q_{\\mu}$ are the polarization vector and four-momentum of the virtual propagator W, and $\\lambda_{W}$ means the polarization of the virtual propagator W. $\\lambda$ and $\\lambda^{\\prime}$ are the helicities of the baryon in the initial and final baryon states, respectively. The detail derivation process of helicity amplitudes can be found in Appendix~\\ref{appendix:helicity}.\nThese helicity amplitudes are related to the form factors by the following\nexpressions.\n\\begin{itemize}\n\\item The transition $B_{i}(\\lambda)\\to B_{f}(\\lambda^{\\prime})$ matrix elements are parameterized as shown in Eq.~\\eqref{eq:matrix_element_2},\nand the helicity amplitudes of $1\/2\\to1\/2$ charged current transition can be expressed with following equations,\n\\begin{align}\nHV_{\\frac{1}{2},0}^{-\\frac{1}{2}} & =-i\\frac{\\sqrt{Q_{-}}}{\\sqrt{q^{2}}}\\left((M+M^{\\prime})f_{1}^{\\frac{1}{2}\\to\\frac{1}{2}}-\\frac{q^{2}}{M}f_{2}^{\\frac{1}{2}\\to\\frac{1}{2}}\\right),\\quad\nHV_{\\frac{1}{2},1}^{\\frac{1}{2}}  =i\\sqrt{2Q_{-}}\\left(-f_{1}^{\\frac{1}{2}\\to\\frac{1}{2}}+\\frac{M+M^{\\prime}}{M}f_{2}^{\\frac{1}{2}\\to\\frac{1}{2}}\\right),\\nonumber\\\\\nHA_{\\frac{1}{2},0}^{-\\frac{1}{2}} & =-i\\frac{\\sqrt{Q_{+}}}{\\sqrt{q^{2}}}\\left((M-M^{\\prime})g_{1}^{\\frac{1}{2}\\to\\frac{1}{2}}+\\frac{q^{2}}{M}g_{2}^{\\frac{1}{2}\\to\\frac{1}{2}}\\right),\\quad\nHA_{\\frac{1}{2},1}^{\\frac{1}{2}}  =i\\sqrt{2Q_{+}}\\left(-g_{1}^{\\frac{1}{2}\\to\\frac{1}{2}}-\\frac{M-M^{\\prime}}{M}g_{2}^{\\frac{1}{2}\\to\\frac{1}{2}}\\right),\\label{eq:helicty22v}\n\\end{align}\nwith $Q_{\\pm}=2(P\\cdot P^{\\prime}\\pm MM^{\\prime})=(M\\pm M^{\\prime})^{2}-q^{2}$. Here $f_i^{\\frac{1}{2}\\to\\frac{1}{2}}$ and $g_{i}^{\\frac{1}{2}\\to\\frac{1}{2}}$ are the physical form factors which are defined by Eq.~(\\ref{eq:physical_ff22}). $M$ and $M^{\\prime}$ are the  masses for the initial and final baryon. While the negative helicity amplitudes have the following relations with the corresponding positive ones,\n\\begin{equation}\nHV_{-\\lambda^{\\prime},-\\lambda_{W}}^{-\\lambda}=HV_{\\lambda^{\\prime},\\lambda_{W}}^{\\lambda}\\quad\\text{and}\\quad HA_{-\\lambda^{\\prime},-\\lambda_{W}}^{-\\lambda}=-HA_{\\lambda^{\\prime},\\lambda_{W}}^{\\lambda}.\n\\end{equation}\nThen the total helicity amplitudes for (V-A) current can be shown as follows,\n\\begin{equation}\nH_{\\lambda^{\\prime},\\lambda_{W}}^{\\lambda}=HV_{\\lambda^{\\prime},\\lambda_{W}}^{\\lambda}-HA_{\\lambda^{\\prime},\\lambda_{W}}^{\\lambda}.\n\\end{equation}\nThe polarized differential decay widths can be given as\n\\begin{align}\n\\frac{d\\Gamma_{L}}{dq^{2}} & =\\frac{G_{F}^{2}|V_{CKM}|^{2}}{(2\\pi)^{3}}\\frac{q^{2}|\\vec{P}^{\\prime}|}{24M^{2}}\n(|H_{\\frac{1}{2},0}^{-\\frac{1}{2}}|^{2}+|H_{-\\frac{1}{2},0}^{\\frac{1}{2}}|^{2}),\\label{eq:longi-1}\\\\\n\\frac{d\\Gamma_{T}}{dq^{2}} & =\\frac{G_{F}^{2}|V_{CKM}|^{2}}{(2\\pi)^{3}}\\frac{q^{2}|\\vec{P}^{\\prime}|}{24M^{2}}\n(|H_{\\frac{1}{2},1}^{\\frac{1}{2}}|^{2}+|H_{-\\frac{1}{2},-1}^{-\\frac{1}{2}}|^{2}),\\label{eq:trans-1}\n\\end{align}\nwith $|\\vec{P}^{\\prime}|=\\sqrt{Q_{+}Q_{-}}\/2M$.\n\\item The $B_{i}(\\lambda)\\to B_{f}^{*}(\\lambda^{\\prime})$ transition matrix element are parameterized with Eq.~\\eqref{eq:matrix_element_32nVA}, and the helicity amplitudes of the transitions ${1}\/{2}\\to{3}\/{2}$ induced by charged current can be expressed with following equations,\n\\begin{eqnarray}\nHV_{3\/2,1}^{-1\/2} & = & - i\\sqrt{Q_{-}}f_{4}^{\\frac{1}{2}\\to\\frac{3}{2}},\\quad\nHV_{1\/2,1}^{1\/2}  =  i\\sqrt{\\frac{Q_{-}}{3}}\\left[f_{4}^{\\frac{1}{2}\\to\\frac{3}{2}}-\\frac{Q_{+}}{MM^{\\prime}}f_{1}^{\\frac{1}{2}\\to\\frac{3}{2}}\\right],\\\\\nHV_{1\/2,0}^{-1\/2} & = &\n i\\sqrt{\\frac{2}{3}}\\frac{\\sqrt{Q_{-}}}{\\sqrt{q^{2}}}\n\\Big[\\frac{M^2-M^{\\prime2}-q^2}{2M^{\\prime}}f_{4}^{\\frac{1}{2}\\to\\frac{3}{2}}-\n\\frac{M- M^{\\prime}}{2MM^{\\prime}}Q_{+}f_{1}^{\\frac{1}{2}\\to\\frac{3}{2}}\n-\\frac{Q_{+}Q_{-}}{2M^{2}M^{\\prime}}f_{2}^{\\frac{1}{2}\\to\\frac{3}{2}}\\Big],\\label{eq:helicty23v}\\\\\nHA_{3\/2,1}^{-1\/2} & = &  i\\sqrt{Q_{+}}f_{4}^{\\frac{1}{2}\\to\\frac{3}{2}},\\quad\nHA_{1\/2,1}^{1\/2}  =  i\\sqrt{\\frac{Q_{+}}{3}}\\left[g_{4}^{\\frac{1}{2}\\to\\frac{3}{2}}-\\frac{Q_{-}}{MM^{\\prime}}g_{1}^{\\frac{1}{2}\\to\\frac{3}{2}}\\right],\\\\\nHA_{1\/2,0}^{-1\/2} & = &\n- i\\sqrt{\\frac{2}{3}}\\frac{\\sqrt{Q_{+}}}{\\sqrt{q^{2}}}\n\\Big[\\frac{M^2-M^{\\prime2}-q^2}{2M^{\\prime}}g_{4}^{\\frac{1}{2}\\to\\frac{3}{2}}+\n\\frac{M+ M^{\\prime}}{2MM^{\\prime}}Q_{-}g_{1}^{\\frac{1}{2}\\to\\frac{3}{2}}\n-\\frac{Q_{+}Q_{-}}{2M^{2}M^{\\prime}}g_{2}^{\\frac{1}{2}\\to\\frac{3}{2}}\\Big],\\label{eq:helicty23a}\n\\end{eqnarray}\nhere $f_i^{\\frac{1}{2}\\to\\frac{3}{2}}$ and $g_{i}^{\\frac{1}{2}\\to\\frac{3}{2}}$ are physics form factors introduced with Eq.~(\\ref{eq:physical_ff23}).\n$M$ and $M^{\\prime}$ are the masses of the initial and final baryon states, respectively. While the negative helicity amplitudes have the following relations with the corresponding positive ones,\n\\begin{equation}\nHV_{-\\lambda^{\\prime},-\\lambda_{W}}^{-\\lambda}=- HV_{\\lambda^{\\prime},\\lambda_{W}}^{\\lambda}\\quad \\text{and} \\quad HA_{-\\lambda^{\\prime},-\\lambda_{W}}^{-\\lambda}= HA_{\\lambda^{\\prime},\\lambda_{W}}^{\\lambda}.\n\\end{equation}\nThen we can get the total helicity amplitudes,\n\\begin{equation}\nH_{\\lambda^{\\prime},\\lambda_{W}}^{\\lambda}=\nHV_{\\lambda^{\\prime},\\lambda_{W}}^{\\lambda}-HA_{\\lambda^{\\prime},\\lambda_{W}}^{\\lambda}.\n\\end{equation}\nThe polarized differential decay widths can be given as\n\\begin{eqnarray}\n\\frac{d\\Gamma_{L}}{dq^2} & = & \\frac{G_{F}^{2}}{(2\\pi)^{3}}|V_{{\\rm CKM}}|^{2}\\frac{q^{2}|\\vec{P}^{\\prime}|}{24M^2}[|H_{1\/2,0}^{-1\/2}|^{2}+|H_{-1\/2,0}^{1\/2}|^{2}],\\label{eq:longi-2}\\\\\n\\frac{d\\Gamma_{T}}{dq^2} & = & \\frac{G_{F}^{2}}{(2\\pi)^{3}}|V_{{\\rm CKM}}|^{2}\\frac{q^{2}|\\vec{P}^{\\prime}|}{24M^2}[|H_{1\/2,1}^{1\/2}|^{2}\n+|H_{-1\/2,-1}^{-1\/2}|^{2}+|H_{3\/2,1}^{-1\/2}|^{2}+|H_{-3\/2,-1}^{1\/2}|^{2}].\\label{eq:trans-2}\n\\end{eqnarray}\n\\end{itemize}\n\\subsubsection{the FCNC transition}\nFor the FCNC induced transition, we adopt the helicity amplitudes as follows,\n\\begin{eqnarray}\nH_{\\lambda^{\\prime},\\lambda_{V}}^{{\\cal V}_{l},\\lambda} & \\equiv & \\Big(C_{9}^{{\\rm eff}}(q^{2})\\langle{\\cal B^{(*)}}(\\lambda^{\\prime})|\\bar{s}\\gamma^{\\mu}(1-\\gamma_{5})b|{\\cal B}(\\lambda)\\rangle-C_{7}^{{\\rm eff}}2m_{b}\\langle{\\cal B}^{(*)}(\\lambda^{\\prime})|\\bar{s}i\\sigma^{\\mu\\nu}\\frac{q_{\\nu}}{q^{2}}(1+\\gamma_{5})b|{\\cal B}(\\lambda)\\rangle\\Big)\\epsilon_{\\mu}^{*}(\\lambda_{V}),\\nonumber \\\\\nH_{\\lambda^{\\prime},t}^{{\\cal V}_{l},\\lambda} & \\equiv & \\Big(C_{9}^{{\\rm eff}}(q^{2})\\langle{\\cal B}^{(*)}(\\lambda^{\\prime})|\\bar{s}\\gamma^{\\mu}(1-\\gamma_{5})b|{\\cal B}(\\lambda)\\rangle\\Big)\\frac{q_{\\mu}}{\\sqrt{q^{2}}},\n\\label{eq:HV2}\n\\end{eqnarray}\nand\n\\begin{eqnarray}\nH_{\\lambda^{\\prime},\\lambda_{V}}^{{\\cal A}_{l},\\lambda} & \\equiv & \\Big(C_{10}\\langle{\\cal B^{(*)}}(\\lambda^{\\prime})|\\bar{s}\\gamma^{\\mu}(1-\\gamma_{5})b|{\\cal B}(\\lambda)\\rangle\\Big)\\epsilon_{\\mu}^{*}(\\lambda_{V}),\\nonumber \\\\\nH_{\\lambda^{\\prime},t}^{{\\cal A}_{l},\\lambda} & \\equiv & \\Big(C_{10}\\langle{\\cal B^{(*)}}(\\lambda^{\\prime})|\\bar{s}\\gamma^{\\mu}(1-\\gamma_{5})b|{\\cal B}(\\lambda)\\rangle\\Big)\\frac{q_{\\mu}}{\\sqrt{q^{2}}},\n\\end{eqnarray}\nhere $\\epsilon_{\\mu}$ and $q_{\\mu}$ are the polarization vector and four-momentum of the virtual vector propagator V, and $\\lambda_{V}$ means the polarization of the virtual vector propagator V. $\\lambda$ and $\\lambda^{\\prime}$ are the helicities of the baryon in the initial and final baryon states, respectively.\nIn the following, the superscripts ``${\\cal V}_{l}$\" and ``${\\cal A}_{l}$\" denote the\ncorresponding leptonic counterparts $\\bar{l}\\gamma^{\\mu}l$ and $\\bar{l}\\gamma^{\\mu}\\gamma_{5}l$, respectively.\n\\begin{itemize}\n\\item  The transition $B_{i}(\\lambda)\\to B_{f}(\\lambda^{\\prime})$ matrix elements are parameterized with   Eqs.~\\eqref{eq:matrix_element_2}-\\eqref{eq:matrix_element_2p},\n    and the helicity amplitudes of $1\/2\\to1\/2$ induced by FCNC transition can be obtained with following expressions,\n\\begin{align}\n\tHV_{\\frac{1}{2},0}^{{\\cal V}_{l},-\\frac{1}{2}} & =-i\\frac{\\sqrt{Q_{-}}}{\\sqrt{q^{2}}}\\left((M+M^{\\prime})F_{1}^{{\\cal V}_{l}}-\\frac{q^{2}}{M}F_{2}^{{\\cal V}_{l}}\\right),\\quad\n\tHV_{\\frac{1}{2},1}^{{\\cal V}_{l},\\frac{1}{2}}  =i\\sqrt{2Q_{-}}\\left(-F_{1}^{{\\cal V}_{l}}+\\frac{M+M^{\\prime}}{M}F_{2}^{{\\cal V}_{l}}\\right),\\nonumber \\\\\n\tHA_{\\frac{1}{2},0}^{{\\cal V}_{l},-\\frac{1}{2}} & =-i\\frac{\\sqrt{Q_{+}}}{\\sqrt{q^{2}}}\\left((M-M^{\\prime})G_{1}^{{\\cal V}_{l}}+\\frac{q^{2}}{M}G_{2}^{{\\cal V}_{l}}\\right),\\quad\n\tHA_{\\frac{1}{2},1}^{{\\cal V}_{l},\\frac{1}{2}}  =i\\sqrt{2Q_{+}}\\left(-G_{1}^{{\\cal V}_{l}}-\\frac{M-M^{\\prime}}{M}G_{2}^{{\\cal V}_{l}}\\right),\\label{eq:hv22}\n\t\\end{align}\n\tand\n\t\\begin{eqnarray}\n\tHV_{-\\lambda^{\\prime},-\\lambda_{V}}^{{\\cal V}_{l},-\\lambda} & = & HV_{\\lambda^{\\prime},\\lambda_{V}}^{{\\cal V}_{l},\\lambda},\n\t\\quad\n\tHA_{-\\lambda^{\\prime},-\\lambda_{V}}^{{\\cal V}_{l},-\\lambda}  = -HA_{\\lambda^{\\prime},\\lambda_{V}}^{{\\cal V}_{l},\\lambda}.\n\t\\end{eqnarray}\nwhere the ``HV\" and ``HA\" are corresponding to the $\\Gamma^{\\mu}$ and $\\Gamma^{\\mu}\\gamma_{5}$ parts in Eq.~(\\ref{eq:HV2}), respectively.\n\tThe total helicity amplitude can be given as\n\t\\begin{equation}\n\tH_{\\lambda^{\\prime},\\lambda_{V}}^{{\\cal V}_{l},\\lambda}=HV_{\\lambda^{\\prime},\\lambda_{V}}^{{\\cal V}_{l},\\lambda}-HA_{\\lambda^{\\prime},\\lambda_{V}}^{{\\cal V}_{l},\\lambda}.\n\t\\end{equation}\nThe specific expressions of $H_{\\lambda^{\\prime},\\lambda_{V}}^{{\\cal A}_{l},\\lambda}$ are similar with the ones of $H_{\\lambda^{\\prime},\\lambda_{V}}^{V,\\lambda}$, except\n\t\\begin{eqnarray}\n\tF_{i}^{{\\cal V}_{l}}  \\to  F_{i}^{{\\cal A}_{l}} \\qquad \\rm{and} \\qquad\n\tG_{i}^{{\\cal V}_{l}}  \\to  G_{i}^{{\\cal A}_{l}}.\n\t\\end{eqnarray}\n\tFurthermore, the timelike polarizations of the virtual vector propagator V\nfor the helicity amplitudes, $H^{{\\cal A}_{l}}_{t}$ are necessary for FCNC induced transitions,\n\t\\begin{align}\n\tHV_{-\\frac{1}{2},t}^{{\\cal A}_{l},\\frac{1}{2}}=HV_{\\frac{1}{2},t}^{{\\cal A}_{l},-\\frac{1}{2}}=-i\\frac{\\sqrt{Q_{+}}}{\\sqrt{q^{2}}}\\left((M-M^{\\prime})F_{1}^{{\\cal A}_{l}}+\\frac{q^{2}}{M}F_{3}^{{\\cal A}_{l}}\\right),\\nonumber \\\\\n\t-HA_{-\\frac{1}{2},t}^{{\\cal A}_{l},\\frac{1}{2}}=HA_{\\frac{1}{2},t}^{{\\cal A}_{l},-\\frac{1}{2}}=-i\\frac{\\sqrt{Q_{-}}}{\\sqrt{q^{2}}}\\left((M+M^{\\prime})G_{1}^{{\\cal A}_{l}}-\\frac{q^{2}}{M}G_{3}^{{\\cal A}_{l}}\\right)\\label{eq:helicty22fcnc}\n   \\end{align}\n\tand\n\t\\begin{equation}\n\tH_{\\lambda^{\\prime},t}^{{\\cal A}_{l},\\lambda}=HV_{\\lambda^{\\prime},t}^{{\\cal A}_{l},\\lambda}-HA_{\\lambda^{\\prime},t}^{{\\cal A}_{l},\\lambda}.\t\\label{eq:HV22A}\n\t\\end{equation}\nIn the above Eq.~(\\ref{eq:hv22}-\\ref{eq:HV22A}), the following notations have been introduced:\n\t\\begin{eqnarray}\n\tF_{1}^{{\\cal V}_{l}}(q^{2}) & \\equiv & C_{9}^{{\\rm eff}}(q^{2})f_{1}^{\\frac{1}{2}\\to\\frac{1}{2}}(q^{2})-C_{7}^{{\\rm eff}}\\frac{2m_{b}}{M^{\\prime}-M}f_{1}^{\\frac{1}{2}\\to\\frac{1}{2},T}(q^{2}),\\nonumber \\\\\n    \\quad F_{2}^{{\\cal V}_{l}}(q^{2})  &\\equiv&  C_{9}^{{\\rm eff}}(q^{2})f_{2}^{\\frac{1}{2}\\to\\frac{1}{2}}(q^{2})-C_{7}^{{\\rm eff}}\\frac{2m_{b}M}{q^{2}}f_{2}^{\\frac{1}{2}\\to\\frac{1}{2},T}(q^{2}),\\nonumber \\\\\n\tG_{1}^{{\\cal V}_{l}}(q^{2}) & \\equiv & C_{9}^{{\\rm eff}}(q^{2})g_{1}^{\\frac{1}{2}\\to\\frac{1}{2}}(q^{2})+C_{7}^{{\\rm eff}}\\frac{2m_{b}}{M^{\\prime}+M}g_{1}^{\\frac{1}{2}\\to\\frac{1}{2},T}(q^{2}),\\nonumber \\\\\n\t G_{2}^{{\\cal V}_{l}}(q^{2})  &\\equiv & C_{9}^{{\\rm eff}}(q^{2})g_{2}^{\\frac{1}{2}\\to\\frac{1}{2}}(q^{2})+C_{7}^{{\\rm eff}}\\frac{2m_{b}M}{q^{2}}g_{2}^{\\frac{1}{2}\\to\\frac{1}{2},T}(q^{2}),\n\t\\end{eqnarray}\nand\n\\begin{eqnarray}\n\tF_{i}^{{\\cal A}_{l}}(q^{2}) & \\equiv & C_{10}f_{i}^{\\frac{1}{2}\\to\\frac{1}{2}}(q^{2}),\\quad\n\t G_{i}^{{\\cal A}_{l}}(q^{2})  \\equiv  C_{10}g_{i}^{\\frac{1}{2}\\to\\frac{1}{2}}(q^{2})\\quad(i=1,2,3).\n\\end{eqnarray}\nHere $f_{i}^{\\frac{1}{2}\\to\\frac{1}{2},(T)}(q^2)$ and $g_{i}^{\\frac{1}{2}\\to\\frac{1}{2},(T)}(q^2)$ are the physical form factors illuminated by Eq.~(\\ref{eq:physical_ff22}).\t\nThe longitudinally and transversely polarized differential decay widths read,\n\\begin{eqnarray}\n\\frac{d\\Gamma_{L}}{dq^{2}} & = &\\frac{G_{F}^{2}|V_{\\rm CKM}|^2\\alpha_{em}^{2}|\\vec{P}^{\\prime}||\\vec{p}_{1}|}{24(2\\pi)^{5}M^{2}\\sqrt{q^{2}}}\\Big\\{(q^{2}+2m_{l}^{2})(|H_{-\\frac{1}{2},0}^{{\\cal V}_{l},\\frac{1}{2}}|^{2}+|H_{\\frac{1}{2},0}^{{\\cal V}_{l},-\\frac{1}{2}}|^{2})\\nonumber \\\\\n&  & +(q^{2}-4m_{l}^{2})(|H_{-\\frac{1}{2},0}^{{\\cal A}_{l},\\frac{1}{2}}|^{2}+|H_{\\frac{1}{2},0}^{{\\cal A}_{l},-\\frac{1}{2}}|^{2})+6m_{l}^{2}(|H_{-\\frac{1}{2},t}^{{\\cal A}_{l},\\frac{1}{2}}|^{2}+|H_{\\frac{1}{2},t}^{{\\cal A}_{l},-\\frac{1}{2}}|^{2})\\Big\\},\\label{eq:longfcnc-1}\\\\\n\\frac{d\\Gamma_{T}}{dq^{2}} & = & \\frac{G_{F}^{2}|V_{\\rm CKM}|^2\\alpha_{em}^{2}|\\vec{P}^{\\prime}||\\vec{p}_{1}|}{24(2\\pi)^{5}M^{2}\\sqrt{q^{2}}}\\Big\\{(q^{2}+2m_{l}^{2})(|H_{\\frac{1}{2},1}^{{\\cal V}_{l},\\frac{1}{2}}|^{2}+|H_{-\\frac{1}{2},-1}^{{\\cal V}_{l},-\\frac{1}{2}}|^{2})\\nonumber \\\\\n&  &+(q^{2}-4m_{l}^{2})(|H_{\\frac{1}{2},1}^{{\\cal A}_{l},\\frac{1}{2}}|^{2}+|H_{-\\frac{1}{2},-1}^{{\\cal A}_{l},-\\frac{1}{2}}|^{2})\\Big\\}.\\label{eq:tranfcnc-1}\n\\end{eqnarray}\nwith $V_{\\rm CKM}=V_{tb}V_{ts}^{*}$ for $b\\to s$ processes, $V_{\\rm CKM}=V_{tb}V_{td}^{*}$ for $b\\to d$ processes and\n$|\\vec{p}_{1}|=\\frac{1}{2}\\sqrt{q^2-4m_{l}^2}$.\n\\item The transition ${1}\/{2}\\to{3}\/{2}$ matrix elements are parameterized with Eqs.~\\eqref{eq:matrix_element_32nVA}-\\eqref{eq:matrix_element_32nT},\nand the helicity amplitudes induced by FCNC can be given by the following expressions,\n\\begin{eqnarray}\nHV_{3\/2,1}^{{\\cal V}_{l},-1\/2} & = & - i\\sqrt{Q_{-}}{\\cal F}_{4}^{{\\cal V}_{l}},\\quad HV_{1\/2,1}^{{\\cal V}_{l},1\/2}  =\ni\\sqrt{\\frac{Q_{-}}{3}}\\left[{\\cal F}_{4}^{{\\cal V}_{l}}-\\frac{Q_{+}}{MM^{\\prime}}{\\cal F}_{1}^{{\\cal V}_{l}}\\right],\\label{eq:hv23}\\\\\nHV_{1\/2,0}^{{\\cal V}_{l},-1\/2} & = &  i\\sqrt{\\frac{2}{3}}\\frac{\\sqrt{Q_{-}}}{\\sqrt{q^{2}}}\n\\Big[\\frac{M^2-M^{\\prime2}-q^2}{2M^{\\prime}}{\\cal F}_{4}^{{\\cal V}_{l}}-\\frac{M- M^{\\prime}}{2MM^{\\prime}}Q_{+}{\\cal F}_{1}^{{\\cal V}_{l}}-\\frac{Q_{+}Q_{-}}{2M^{2}M^{\\prime}}{\\cal F}_{2}^{{\\cal V}_{l}}\\Big]\\label{eq:hv232},\n\\\\\nHA_{3\/2,1}^{{\\cal V}_{l},-1\/2} & = &  i\\sqrt{Q_{+}}{\\cal G}_{4}^{{\\cal V}_{l}},\\quad HA_{1\/2,1}^{{\\cal V}_{l},1\/2}  =  i\\sqrt{\\frac{Q_{+}}{3}}\\left[{\\cal G}_{4}^{{\\cal V}_{l}}-\\frac{Q_{-}}{MM^{\\prime}}{\\cal G}_{1}^{{\\cal V}_{l}}\\right],\n\\\\\nHA_{1\/2,0}^{{\\cal V}_{l},-1\/2} & = & - i\\sqrt{\\frac{2}{3}}\\frac{\\sqrt{Q_{+}}}{\\sqrt{q^{2}}}\n\\Big[\\frac{M^2-M^{\\prime2}-q^2}{2M^{\\prime}}{\\cal G}_{4}^{{\\cal V}_{l}}\n+\\frac{M+ M^{\\prime}}{2MM^{\\prime}}Q_{-}{\\cal G}_{1}^{{\\cal V}_{l}}\n-\\frac{Q_{+}Q_{-}}{2M^{2}M^{\\prime}}{\\cal G}_{2}^{{\\cal V}_{l}}\\Big].\\label{eq:helicty23fcnc}\n\\end{eqnarray}\nand\n\\begin{eqnarray}\nHV_{-\\lambda^{\\prime},-\\lambda_{W}}^{{\\cal V}_{l},-\\lambda}&=&- HV_{\\lambda^{\\prime},\\lambda_{W}}^{{\\cal V}_{l},\\lambda},\\quad\nHA_{-\\lambda^{\\prime},-\\lambda_{W}}^{{\\cal V}_{l},-\\lambda}=HA_{\\lambda^{\\prime},\\lambda_{W}}^{{\\cal V}_{l},\\lambda}.\n\\end{eqnarray}\nwhere the ``HV\" and ``HA\" are corresponding to the $\\Gamma^{\\mu}$ and $\\Gamma^{\\mu}\\gamma_{5}$ parts in Eq.~(\\ref{eq:HV2}), respectively.\nThen we can get the total helicity amplitudes,\n\\begin{equation} H_{\\lambda^{\\prime},\\lambda_{V}}^{{\\cal V}_{l},\\lambda}=HV_{\\lambda^{\\prime},\\lambda_{V}}^{{\\cal V}_{l},\\lambda}-HA_{\\lambda^{\\prime},\\lambda_{V}}^{{\\cal V}_{l},\\lambda}.\n\\end{equation}\nThe specific expressions of $H_{\\lambda^{\\prime},\\lambda_{V}}^{{\\cal A}_{l},\\lambda}$ are similar with the ones of $H_{\\lambda^{\\prime},\\lambda_{V}}^{V,\\lambda}$, except\n\\begin{eqnarray}\n{\\cal F}_{i}^{{\\cal V}_{l}}  \\to  {\\cal F}_{i}^{{\\cal A}_{l}}\\qquad \\rm{and}\\qquad\n{\\cal G}_{i}^{{\\cal V}_{l}} \\to  {\\cal G}_{i}^{{\\cal A}_{l}}.\n\\end{eqnarray}\nFurthermore, the timelike polarizations of the virtual vector propagator V\nfor the helicity amplitudes, $H^{{\\cal A}_{l}}_{t}$ are necessary for FCNC induced transitions,\n\\begin{align} &-HV_{-\\frac{1}{2},t}^{{\\cal A}_{l},\\frac{1}{2}}=HV_{\\frac{1}{2},t}^{{\\cal A}_{l},-\\frac{1}{2}}\n=i\\sqrt{\\frac{2}{3}}\\sqrt{Q_{+}}\\frac{Q_{-}}{2MM^{\\prime}}\\frac{M^2-M^{\\prime 2}}{M}{\\cal F}_3^{{\\cal A}_{l}},\\nonumber \\\\ &HA_{-\\frac{1}{2},t}^{{\\cal A}_{l},\\frac{1}{2}}=HA_{\\frac{1}{2},t}^{{\\cal A}_{l},-\\frac{1}{2}}\n=-i\\sqrt{\\frac{2}{3}}\\sqrt{Q_{-}}\\frac{Q_{+}}{2MM^{\\prime}}\\frac{M^2-M^{\\prime 2}}{M}{\\cal G}_3^{{\\cal A}_{l}}\n\\end{align}\nand\n\\begin{equation}\nH_{\\lambda^{\\prime},t}^{{\\cal A}_{l},\\lambda}=HV_{\\lambda^{\\prime},t}^{{\\cal A}_{l},\\lambda}-HA_{\\lambda^{\\prime},t}^{{\\cal A}_{l},\\lambda}.\\label{eq:HV23A}\n\\end{equation}\nIn Eqs.~(\\ref{eq:hv23}-\\ref{eq:HV23A}), the following notations are introduced.\n\\begin{eqnarray}\n{\\cal F}_{i}^{{\\cal V}_{l}}(q^{2}) & \\equiv & C_{9}^{{\\rm eff}}(q^{2})f_{i}^{\\frac{1}{2}\\to\\frac{3}{2}}(q^{2})-C_{7}^{{\\rm eff}}\\frac{2m_{b}M}{q^{2}}f_{i}^{\\frac{1}{2}\\to\\frac{3}{2},T}(q^{2}),\\nonumber\\\\\n {\\cal G}_{i}^{{\\cal V}_{l}}(q^{2}) & \\equiv & C_{9}^{{\\rm eff}}(q^{2})g_{i}^{\\frac{1}{2}\\to\\frac{3}{2}}(q^{2})+C_{7}^{{\\rm eff}}\\frac{2m_{b}M}{q^{2}}g_{i}^{\\frac{1}{2}\\to\\frac{3}{2},T}(q^{2}),\\nonumber\\\\\n{\\cal F}_{i}^{{\\cal A}_{l}}(q^{2}) & \\equiv & C_{10}f_{i}^{\\frac{1}{2}\\to\\frac{3}{2}}(q^{2}),\\quad {\\cal G}_{i}^{{\\cal A}_{l}}(q^{2})  \\equiv  C_{10}g_{i}^{\\frac{1}{2}\\to\\frac{3}{2}}(q^{2}),\\quad(i=1,2,3,4).\n\\end{eqnarray}\nHere $f_{i}^{\\frac{1}{2}\\to\\frac{3}{2},(T)}(q^2)$ and $g_{i}^{\\frac{1}{2}\\to\\frac{3}{2},(T)}(q^2)$ are physics form factors illuminated by Eq.~(\\ref{eq:physical_ff23}).\nThe longitudinally and transversely polarized differential decay widths read\n\\begin{eqnarray}\n\\frac{d\\Gamma_{L}}{dq^{2}} & = &\\frac{G_{F}^{2}|V_{\\rm CKM}|^2\\alpha_{em}^{2}|\\vec{P}^{\\prime}||\\vec{p}_{1}|}{24(2\\pi)^{5}M^{2}\\sqrt{q^{2}}}\n\\Big\\{(q^{2}+2m_{l}^{2})(|H_{-\\frac{1}{2},0}^{{\\cal V}_{l},\\frac{1}{2}}|^{2}+|H_{\\frac{1}{2},0}^{{\\cal V}_{l},-\\frac{1}{2}}|^{2})\n\\nonumber \\\\\n&  & +(q^{2}-4m_{l}^{2})(|H_{-\\frac{1}{2},0}^{{\\cal A}_{l},\\frac{1}{2}}|^{2}+|H_{\\frac{1}{2},0}^{{\\cal A}_{l},-\\frac{1}{2}}|^{2})+6m_{l}^{2}(|H_{-\\frac{1}{2},t}^{{\\cal A}_{l},\\frac{1}{2}}|^{2}+|H_{\\frac{1}{2},t}^{{\\cal A}_{l},-\\frac{1}{2}}|^{2})\\Big\\},\\label{eq:longfcnc-2}\\\\\n\\frac{d\\Gamma_{T}}{dq^{2}} & = & \\frac{G_{F}^{2}|V_{\\rm CKM}|^2\\alpha_{em}^{2}|\\vec{P}^{\\prime}||\\vec{p}_{1}|}{24(2\\pi)^{5}M^{2}\\sqrt{q^{2}}}\n\\Big\\{(q^{2}+2m_{l}^{2})(|H_{\\frac{1}{2},1}^{{\\cal V}_{l},\\frac{1}{2}}|^{2}+|H_{-\\frac{1}{2},-1}^{{\\cal V}_{l},-\\frac{1}{2}}|^{2}\n+|H_{\\frac{3}{2},1}^{{\\cal V}_{l},-\\frac{1}{2}}|^{2}+|H_{-\\frac{3}{2},-1}^{{\\cal V}_{l},\\frac{1}{2}}|^{2})\\nonumber \\\\\n&  &+(q^{2}-4m_{l}^{2})(|H_{\\frac{1}{2},1}^{{\\cal A}_{l},\\frac{1}{2}}|^{2}+|H_{-\\frac{1}{2},-1}^{{\\cal A}_{l},-\\frac{1}{2}}|^{2}\n+|H_{\\frac{3}{2},1}^{{\\cal A}_{l},-\\frac{1}{2}}|^{2}+|H_{-\\frac{3}{2},-1}^{{\\cal A}_{l},\\frac{1}{2}}|^{2})\\Big\\}.\\label{eq:tranfcnc-2}\n\\end{eqnarray}\nwith $V_{\\rm CKM}=V_{tb}V_{ts}^{*}$ for $b\\to s$ processes, $V_{\\rm CKM}=V_{tb}V_{td}^{*}$ for $b\\to d$ processes and\n$|\\vec{p}_{1}|=\\frac{1}{2}\\sqrt{q^2-4m_{l}^2}$.\n\\end{itemize}\nIn the end, the total differential decay width can be written as\n\\begin{eqnarray}\n\\frac{d\\Gamma}{dq^{2}} & = & \\frac{d\\Gamma_{L}}{dq^{2}}+\\frac{d\\Gamma_{T}}{dq^{2}},\n\\end{eqnarray}\nthen we can calculate total width using the following integral,\n\\begin{equation}\n\\Gamma=\\int_{q^2_{\\rm min}}^{(M-M^{\\prime})^{2}}dq^{2}\\frac{d\\Gamma}{dq^{2}},\n\\end{equation}\nwhere $q^2_{\\rm min}= 0$ for these decays with charged current, while $q^2_{\\rm min}=4m_{l}^2$ for other decays with FCNC.\nAt the same time, the ratio of the longitudinal to transverse decay rates $\\Gamma_{L}\/\\Gamma_{T}$ can be calculated.\n\\subsection{Results for semi-leptonic decays}\n\\begin{itemize}\n\\item\nFor the transition ${1}\/{2}\\to{1}\/{2}$ with $\\rm{V-A}$ current, the integrated partial decay widths, the relevant branching ratios and $\\Gamma_{L}\/\\Gamma_{T}$s are shown in Tab.~\\ref{Tab:branchingv22}. The dependence of $q^2$ of the differential decay widths can be shown in Fig.~\\ref{fig:decaywidthbbtob22}.\n\\begin{figure}\n\\includegraphics[width=0.8\\columnwidth]{decaywidthbbtob22.eps}\n\\caption{The differential decay widths $d\\Gamma_{L}\/dq^2$ and $d\\Gamma_{T}\/dq^2$ for the processes ${\\cal B}_{bb}\\to{\\cal B}_{b}({\\cal B}_{bc})l^-\\bar{\\nu}_{l}$ dependence on $q^2$. Blue solid line: $d\\Gamma_{L}\/dq^2$ defined with Eq.~(\\ref{eq:longi-1}), red dashes line: $d\\Gamma_{T}\/dq^2$ defined with Eq.~(\\ref{eq:trans-1}).}\n\\label{fig:decaywidthbbtob22}\n\\end{figure}\n\\item  For the transition ${1}\/{2}\\to{1}\/{2}$ induced by FCNC, the integrated partial decay widths, the relevant branching ratios and $\\Gamma_{L}\/\\Gamma_{T}$s are shown in Tab.~\\ref{Tab:branchingratio22}. The dependence of $q^2$ of the differential decay widths can be shown in Fig.~\\ref{fig:decaywidthbbtobFCNC22}.\n\n\n\\begin{figure}\n\\includegraphics[width=0.9\\columnwidth]{decaywidthbbtobFCNC22.eps}\n\\caption{The differential decay widths $d\\Gamma_{L}\/dq^2$ and $d\\Gamma_{T}\/dq^2$ for the processes ${\\cal B}_{bb}\\to{\\cal B}_{b}l^+l^{-}$ dependence on $q^2$. Blue solid line: $d\\Gamma_{L}\/dq^2$ defined with Eq.~(\\ref{eq:longfcnc-1}), red dashes line: $d\\Gamma_{T}\/dq^2$ defined with Eq.~(\\ref{eq:tranfcnc-1}).}\n\\label{fig:decaywidthbbtobFCNC22}\n\\end{figure}\n\n\n\\item  For the transition ${1}\/{2}\\to{3}\/{2}$ with $\\rm{V-A}$ current , the integrated partial decay widths, the relevant branching ratios and $\\Gamma_{L}\/\\Gamma_{T}$s are shown in Tab.~\\ref{Tab:branchingv23}.\nThe dependence of $q^2$ of the differential decay widths can be shown in Fig.~\\ref{fig:decaywidthbbtob23}.\n\n\n\\begin{figure}\n\\includegraphics[width=0.9\\columnwidth]{decaywidthbbtob23.eps}\n\\caption{The differential decay widths $d\\Gamma_{L}\/dq^2$ and $d\\Gamma_{T}\/dq^2$ for the processes ${\\cal B}_{bb}\\to{\\cal B}_{b}^{*}({\\cal B}_{bc}^{*})l^-\\bar{\\nu}_{l}$ dependence on $q^2$. Blue solid line: $d\\Gamma_{L}\/dq^2$ defined with Eq.~(\\ref{eq:longi-2}), red dashes line: $d\\Gamma_{T}\/dq^2$ defined with Eq.~(\\ref{eq:trans-2}).}\n\\label{fig:decaywidthbbtob23}\n\\end{figure}\n\n\n\\item For the transition ${1}\/{2}\\to{3}\/{2}$ with FCNC, the integrated partial decay widths, the relevant branching ratios and $\\Gamma_{L}\/\\Gamma_{T}$s are shown in Tab.~\\ref{Tab:branching23fcnc}. The dependence of $q^2$ of the differential decay widths can be shown with Fig.~\\ref{fig:decaywidthbbtobfcnc23}.\n\\begin{figure}\n\\includegraphics[width=0.9\\columnwidth]{decaywidthbbtobfcnc23.eps}\n\\caption{The differential decay widths $d\\Gamma_{L}\/dq^2$ and $d\\Gamma_{T}\/dq^2$ for the processes ${\\cal B}_{bb}\\to{\\cal B}_{b}^{*}l^+l^{-}$ dependence on $q^2$. Blue solid line: $d\\Gamma_{L}\/dq^2$ defined with Eq.~(\\ref{eq:longfcnc-2}), red dashes line: $d\\Gamma_{T}\/dq^2$ defined with Eq.~(\\ref{eq:tranfcnc-2}).}\n\\label{fig:decaywidthbbtobfcnc23}\n\\end{figure}\n\\end{itemize}\n\\begin{table}\n\\caption{The decay widths, branching ratios and $\\Gamma_{L}\/\\Gamma_{T}$s for\nthe transition ${1}\/{2}\\to{1}\/{2}$ with the charge current.}\n\\label{Tab:branchingv22}\n\\begin{tabular}{l|c|c|c|l|c|c|c}\n\\hline\\hline\nchannels  & $\\Gamma\/\\text{~GeV}$  & ${\\cal B}$  & $\\Gamma_{L}\/\\Gamma_{T}$  & channels  & $\\Gamma\/\\text{~GeV}$  & ${\\cal B}$  & $\\Gamma_{L}\/\\Gamma_{T}$\\tabularnewline\n\\hline\n$\\Xi_{cc}^{++}\\to\\Lambda_{c}^{+}l^{+}\\nu_{l}$  & $7.97\\times10^{-15}$  & $3.10\\times10^{-3}$  & $2.42$ & $\\Xi_{bb}^{0}\\to\\Sigma_{b}^{+}l^{-}\\bar{\\nu}_{l}$  & $1.06\\times10^{-16}$  & $5.96\\times10^{-5}$  & $1.27$\\tabularnewline\n$\\Xi_{cc}^{++}\\to\\Sigma_{c}^{+}l^{+}\\nu_{l}$  & $1.09\\times10^{-14}$  & $4.25\\times10^{-3}$  & $0.86$ & $\\Xi_{bb}^{0}\\to\\Xi_{bc}^{+}l^{-}\\bar{\\nu}_{l}$  & $6.02\\times10^{-14}$  & $3.38\\times10^{-2}$  & $1.42$\\tabularnewline\n$\\Xi_{cc}^{++}\\to\\Xi_{c}^{+}l^{+}\\nu_{l}$  & $8.74\\times10^{-14}$  & $3.40\\times10^{-2}$  & $3.07$ & $\\Xi_{bb}^{0}\\to\\Xi_{bc}^{\\prime+}l^{-}\\bar{\\nu}_{l}$  & $3.21\\times10^{-14}$  & $1.81\\times10^{-2}$  & $0.84$\\tabularnewline\n$\\Xi_{cc}^{++}\\to\\Xi_{c}^{\\prime+}l^{+}\\nu_{l}$  & $1.43\\times10^{-13}$  & $5.57\\times10^{-2}$  & $0.94$ & $\\Xi_{bb}^{-}\\to\\Lambda_{b}^{0}l^{-}\\bar{\\nu}_{l}$  & $2.39\\times10^{-17}$  & $1.35\\times10^{-5}$  & $5.93$\\tabularnewline\n$\\Xi_{cc}^{+}\\to\\Sigma_{c}^{0}l^{+}\\nu_{l}$  & $2.17\\times10^{-14}$  & $1.48\\times10^{-3}$  & $0.86$ & $\\Xi_{bb}^{-}\\to\\Sigma_{b}^{0}l^{-}\\bar{\\nu}_{l}$  & $5.29\\times10^{-17}$  & $2.98\\times10^{-5}$  & $1.27$\\tabularnewline\n$\\Xi_{cc}^{+}\\to\\Xi_{c}^{0}l^{+}\\nu_{l}$  & $8.63\\times10^{-14}$  & $5.90\\times10^{-3}$  & $3.10$ & $\\Xi_{bb}^{-}\\to\\Xi_{bc}^{0}l^{-}\\bar{\\nu}_{l}$  & $6.02\\times10^{-14}$  & $3.38\\times10^{-2}$  & $1.42$\\tabularnewline\n$\\Xi_{cc}^{+}\\to\\Xi_{c}^{\\prime0}l^{+}\\nu_{l}$  & $1.41\\times10^{-13}$  & $9.67\\times10^{-3}$  & $0.95$ & $\\Xi_{bb}^{-}\\to\\Xi_{bc}^{\\prime0}l^{-}\\bar{\\nu}_{l}$  & $3.21\\times10^{-14}$  & $1.81\\times10^{-2}$  & $0.84$\\tabularnewline\n$\\Omega_{cc}^{+}\\to\\Xi_{c}^{0}l^{+}\\nu_{l}$  & $5.87\\times10^{-15}$  & $1.60\\times10^{-3}$  & $2.94$ & $\\Omega_{bb}^{-}\\to\\Xi_{b}^{0}l^{-}\\bar{\\nu}_{l}$  & $2.18\\times10^{-17}$  & $2.65\\times10^{-5}$  & $5.98$\\tabularnewline\n$\\Omega_{cc}^{+}\\to\\Xi_{c}^{\\prime0}l^{+}\\nu_{l}$  & $1.03\\times10^{-14}$  & $2.83\\times10^{-3}$  & $0.87$ & $\\Omega_{bb}^{-}\\to\\Xi_{b}^{\\prime0}l^{-}\\bar{\\nu}_{l}$  & $4.87\\times10^{-17}$  & $5.92\\times10^{-5}$  & $1.28$\\tabularnewline\n$\\Omega_{cc}^{+}\\to\\Omega_{c}^{0}l^{+}\\nu_{l}$  & $2.80\\times10^{-13}$  & $7.67\\times10^{-2}$  & $0.94$ & $\\Omega_{bb}^{-}\\to\\Omega_{bc}^{0}l^{-}\\bar{\\nu}_{l}$  & $5.24\\times10^{-14}$  & $6.37\\times10^{-2}$  & $1.64$\\tabularnewline\n &  &  &  & $\\Omega_{bb}^{-}\\to\\Omega_{bc}^{\\prime0}l^{-}\\bar{\\nu}_{l}$  & $2.55\\times10^{-14}$  & $3.11\\times10^{-2}$  & $0.89$\\tabularnewline\n\\hline\n$\\Xi_{bc}^{+}\\to\\Lambda_{b}^{0}l^{+}\\nu_{l}$  & $4.62\\times10^{-15}$  & $1.71\\times10^{-3}$  & $2.13$  & $\\Xi_{bc}^{+}\\to\\Sigma_{c}^{++}l^{-}\\bar{\\nu}_{l}$  & $8.00\\times10^{-17}$  & $2.97\\times10^{-5}$  & $1.13$\\tabularnewline\n$\\Xi_{bc}^{+}\\to\\Sigma_{b}^{0}l^{+}\\nu_{l}$  & $5.54\\times10^{-15}$  & $2.06\\times10^{-3}$  & $0.79$  & $\\Xi_{bc}^{+}\\to\\Xi_{cc}^{++}l^{-}\\bar{\\nu}_{l}$  & $4.26\\times10^{-14}$  & $1.58\\times10^{-2}$  & $2.21$\\tabularnewline\n$\\Xi_{bc}^{+}\\to\\Xi_{b}^{0}l^{+}\\nu_{l}$  & $4.89\\times10^{-14}$  & $1.81\\times10^{-2}$  & $2.70$  & $\\Xi_{bc}^{0}\\to\\Lambda_{c}^{+}l^{-}\\bar{\\nu}_{l}$  & $1.76\\times10^{-17}$  & $2.48\\times10^{-6}$  & $6.24$\\tabularnewline\n$\\Xi_{bc}^{+}\\to\\Xi_{b}^{\\prime0}l^{+}\\nu_{l}$  & $6.73\\times10^{-14}$  & $2.50\\times10^{-2}$  & $0.89$  & $\\Xi_{bc}^{0}\\to\\Sigma_{c}^{+}l^{-}\\bar{\\nu}_{l}$  & $4.00\\times10^{-17}$  & $5.65\\times10^{-6}$  & $1.13$\\tabularnewline\n$\\Xi_{bc}^{0}\\to\\Sigma_{b}^{-}l^{+}\\nu_{l}$  & $1.10\\times10^{-14}$  & $1.55\\times10^{-3}$  & $0.79$  & $\\Xi_{bc}^{0}\\to\\Xi_{cc}^{+}l^{-}\\bar{\\nu}_{l}$  & $4.26\\times10^{-14}$  & $6.01\\times10^{-3}$  & $2.21$\\tabularnewline\n$\\Xi_{bc}^{0}\\to\\Xi_{b}^{-}l^{+}\\nu_{l}$  & $4.85\\times10^{-14}$  & $6.85\\times10^{-3}$  & $2.71$  & $\\Omega_{bc}^{0}\\to\\Xi_{c}^{+}l^{-}\\bar{\\nu}_{l}$  & $1.40\\times10^{-17}$  & $4.69\\times10^{-6}$  & $6.21$\\tabularnewline\n$\\Xi_{bc}^{0}\\to\\Xi_{b}^{\\prime-}l^{+}\\nu_{l}$  & $6.73\\times10^{-14}$  & $9.51\\times10^{-3}$  & $0.89$  & $\\Omega_{bc}^{0}\\to\\Xi_{c}^{\\prime+}l^{-}\\bar{\\nu}_{l}$  & $3.27\\times10^{-17}$  & $1.09\\times10^{-5}$  & $1.16$\\tabularnewline\n$\\Omega_{bc}^{0}\\to\\Xi_{b}^{-}l^{+}\\nu_{l}$  & $2.93\\times10^{-15}$  & $9.81\\times10^{-4}$  & $2.73$  & $\\Omega_{bc}^{0}\\to\\Omega_{cc}^{+}l^{-}\\bar{\\nu}_{l}$  & $4.11\\times10^{-14}$  & $1.37\\times10^{-2}$  & $2.15$\\tabularnewline\n$\\Omega_{bc}^{0}\\to\\Xi_{b}^{\\prime-}l^{+}\\nu_{l}$  & $3.96\\times10^{-15}$  & $1.33\\times10^{-3}$  & $0.90$  &  &  &  & \\tabularnewline\n$\\Omega_{bc}^{0}\\to\\Omega_{b}^{-}l^{+}\\nu_{l}$  & $1.01\\times10^{-13}$  & $3.36\\times10^{-2}$  & $1.03$  &  &  &  & \\tabularnewline\n\\hline\n$\\Xi_{bc}^{\\prime+}\\to\\Lambda_{b}^{0}l^{+}\\nu_{l}$  & $6.24\\times10^{-15}$  & $2.31\\times10^{-3}$  & $0.74$  & $\\Xi_{bc}^{\\prime+}\\to\\Sigma_{c}^{++}l^{-}\\bar{\\nu}_{l}$  & $3.31\\times10^{-17}$  & $1.23\\times10^{-5}$  & $5.75$\\tabularnewline\n$\\Xi_{bc}^{\\prime+}\\to\\Sigma_{b}^{0}l^{+}\\nu_{l}$  & $2.02\\times10^{-15}$  & $7.50\\times10^{-4}$  & $3.75$  & $\\Xi_{bc}^{\\prime+}\\to\\Xi_{cc}^{++}l^{-}\\bar{\\nu}_{l}$  & $1.86\\times10^{-14}$  & $6.90\\times10^{-3}$  & $0.95$\\tabularnewline\n$\\Xi_{bc}^{\\prime+}\\to\\Xi_{b}^{0}l^{+}\\nu_{l}$  & $5.91\\times10^{-14}$  & $2.19\\times10^{-2}$  & $0.88$  & $\\Xi_{bc}^{\\prime0}\\to\\Lambda_{c}^{+}l^{-}\\bar{\\nu}_{l}$  & $1.38\\times10^{-17}$  & $1.95\\times10^{-6}$  & $1.21$\\tabularnewline\n$\\Xi_{bc}^{\\prime+}\\to\\Xi_{b}^{\\prime0}l^{+}\\nu_{l}$  & $2.65\\times10^{-14}$  & $9.83\\times10^{-3}$  & $4.33$  & $\\Xi_{bc}^{\\prime0}\\to\\Sigma_{c}^{+}l^{-}\\bar{\\nu}_{l}$  & $1.65\\times10^{-17}$  & $2.34\\times10^{-6}$  & $5.76$\\tabularnewline\n$\\Xi_{bc}^{\\prime0}\\to\\Sigma_{b}^{-}l^{+}\\nu_{l}$  & $4.01\\times10^{-15}$  & $5.67\\times10^{-4}$  & $3.78$  & $\\Xi_{bc}^{\\prime0}\\to\\Xi_{cc}^{+}l^{-}\\bar{\\nu}_{l}$  & $1.86\\times10^{-14}$  & $2.63\\times10^{-3}$  & $0.95$\\tabularnewline\n$\\Xi_{bc}^{\\prime0}\\to\\Xi_{b}^{-}l^{+}\\nu_{l}$  & $5.84\\times10^{-14}$  & $8.26\\times10^{-3}$  & $0.88$  & $\\Omega_{bc}^{\\prime0}\\to\\Xi_{c}^{+}l^{-}\\bar{\\nu}_{l}$  & $1.14\\times10^{-17}$  & $3.81\\times10^{-6}$  & $1.27$\\tabularnewline\n$\\Xi_{bc}^{\\prime0}\\to\\Xi_{b}^{\\prime-}l^{+}\\nu_{l}$  & $2.65\\times10^{-14}$  & $3.75\\times10^{-3}$  & $4.33$  & $\\Omega_{bc}^{\\prime0}\\to\\Xi_{c}^{\\prime+}l^{-}\\bar{\\nu}_{l}$  & $1.35\\times10^{-17}$  & $4.52\\times10^{-6}$  & $5.85$\\tabularnewline\n$\\Omega_{bc}^{\\prime0}\\to\\Xi_{b}^{-}l^{+}\\nu_{l}$  & $3.38\\times10^{-15}$  & $1.13\\times10^{-3}$  & $0.92$  & $\\Omega_{bc}^{\\prime0}\\to\\Omega_{cc}^{+}l^{-}\\bar{\\nu}_{l}$  & $1.85\\times10^{-14}$  & $6.18\\times10^{-3}$  & $0.95$\\tabularnewline\n$\\Omega_{bc}^{\\prime0}\\to\\Xi_{b}^{\\prime-}l^{+}\\nu_{l}$  & $1.62\\times10^{-15}$  & $5.42\\times10^{-4}$  & $4.25$  &  &  &  & \\tabularnewline\n$\\Omega_{bc}^{\\prime0}\\to\\Omega_{b}^{-}l^{+}\\nu_{l}$  & $4.40\\times10^{-14}$  & $1.47\\times10^{-2}$  & $4.76$  &  &  &  & \\tabularnewline\n\\hline\\hline\n\\end{tabular}%\n\\end{table}\n\\begin{table}\n\\caption{The decay widths, branching ratios and $\\Gamma_{L}\/\\Gamma_{T}$s for the  transition ${1}\/{2}\\to{1}\/{2}$ with FCNC.}\\label{Tab:branchingratio22}\n\\begin{tabular}{l|c|c|c|l|c|c|c}\n\\hline \\hline\nchannels  & $\\Gamma\/\\text{~GeV}$  & ${\\cal B}$  & $\\Gamma_{L}\/\\Gamma_{T}$ & channels & $\\Gamma\/\\text{~GeV}$  & ${\\cal B}$  & $\\Gamma_{L}\/\\Gamma_{T}$\\tabularnewline\n\\hline\n$\\Xi_{bb}^{0}\\to\\Lambda_{b}^{0}e^{+}e^{-}$  & $4.15\\times10^{-21}$  & $2.33\\times10^{-9}$  & $5.28$ & $\\Xi_{bb}^{0}\\to\\Xi_{b}^{0}e^{+}e^{-}$  & $1.62\\times10^{-19}$  & $9.13\\times10^{-8}$  & $4.70$\\tabularnewline\n$\\Xi_{bb}^{0}\\to\\Sigma_{b}^{0}e^{+}e^{-}$  & $1.05\\times10^{-20}$  & $5.91\\times10^{-9}$  & $0.90$ & $\\Xi_{bb}^{0}\\to\\Xi_{b}^{\\prime0}e^{+}e^{-}$  & $4.32\\times10^{-19}$  & $2.43\\times10^{-7}$  & $0.85$\\tabularnewline\n$\\Xi_{bb}^{-}\\to\\Sigma_{b}^{-}e^{+}e^{-}$  & $2.10\\times10^{-20}$  & $1.18\\times10^{-8}$  & $0.90$ & $\\Xi_{bb}^{-}\\to\\Xi_{b}^{-}e^{+}e^{-}$  & $1.62\\times10^{-19}$  & $9.12\\times10^{-8}$  & $4.69$\\tabularnewline\n$\\Omega_{bb}^{-}\\to\\Xi_{b}^{-}e^{+}e^{-}$  & $3.79\\times10^{-21}$  & $4.61\\times10^{-9}$  & $5.24$ & $\\Xi_{bb}^{-}\\to\\Xi_{b}^{\\prime-}e^{+}e^{-}$  & $4.32\\times10^{-19}$  & $2.43\\times10^{-7}$  & $0.85$\\tabularnewline\n$\\Omega_{bb}^{-}\\to\\Xi_{b}^{\\prime-}e^{+}e^{-}$  & $9.71\\times10^{-21}$  & $1.18\\times10^{-8}$  & $0.90$ & $\\Omega_{bb}^{-}\\to\\Omega_{b}^{-}e^{+}e^{-}$  & $8.05\\times10^{-19}$  & $9.79\\times10^{-7}$  & $0.85$\\tabularnewline\n$\\Xi_{bb}^{0}\\to\\Lambda_{b}^{0}\\mu^{+}\\mu^{-}$  & $3.98\\times10^{-21}$  & $2.24\\times10^{-9}$  & $6.88$ & $\\Xi_{bb}^{0}\\to\\Xi_{b}^{0}\\mu^{+}\\mu^{-}$  & $1.56\\times10^{-19}$  & $8.75\\times10^{-8}$  & $5.99$\\tabularnewline\n$\\Xi_{bb}^{0}\\to\\Sigma_{b}^{0}\\mu^{+}\\mu^{-}$  & $8.69\\times10^{-21}$  & $4.89\\times10^{-9}$  & $1.33$ & $\\Xi_{bb}^{0}\\to\\Xi_{b}^{\\prime0}\\mu^{+}\\mu^{-}$  & $3.61\\times10^{-19}$  & $2.03\\times10^{-7}$  & $1.20$\\tabularnewline\n$\\Xi_{bb}^{-}\\to\\Sigma_{b}^{-}\\mu^{+}\\mu^{-}$  & $1.74\\times10^{-20}$  & $9.77\\times10^{-9}$  & $1.33$ & $\\Xi_{bb}^{-}\\to\\Xi_{b}^{-}\\mu^{+}\\mu^{-}$  & $1.56\\times10^{-19}$  & $8.75\\times10^{-8}$  & $5.99$\\tabularnewline\n$\\Omega_{bb}^{-}\\to\\Xi_{b}^{-}\\mu^{+}\\mu^{-}$  & $3.63\\times10^{-21}$  & $4.41\\times10^{-9}$  & $6.90$ & $\\Xi_{bb}^{-}\\to\\Xi_{b}^{\\prime-}\\mu^{+}\\mu^{-}$  & $3.61\\times10^{-19}$  & $2.03\\times10^{-7}$  & $1.20$\\tabularnewline\n$\\Omega_{bb}^{-}\\to\\Xi_{b}^{\\prime-}\\mu^{+}\\mu^{-}$  & $7.98\\times10^{-21}$  & $9.71\\times10^{-9}$  & $1.35$ & $\\Omega_{bb}^{-}\\to\\Omega_{b}^{-}\\mu^{+}\\mu^{-}$  & $6.70\\times10^{-19}$  & $8.14\\times10^{-7}$  & $1.21$\\tabularnewline\n$\\Xi_{bb}^{0}\\to\\Lambda_{b}^{0}\\tau^{+}\\tau^{-}$  & $1.51\\times10^{-22}$  & $8.49\\times10^{-11}$  & $5.83$ & $\\Xi_{bb}^{0}\\to\\Xi_{b}^{0}\\tau^{+}\\tau^{-}$  & $6.68\\times10^{-21}$  & $3.76\\times10^{-9}$  & $5.71$\\tabularnewline\n$\\Xi_{bb}^{0}\\to\\Sigma_{b}^{0}\\tau^{+}\\tau^{-}$  & $3.39\\times10^{-22}$  & $1.91\\times10^{-10}$  & $1.16$ & $\\Xi_{bb}^{0}\\to\\Xi_{b}^{\\prime0}\\tau^{+}\\tau^{-}$  & $1.54\\times10^{-20}$  & $8.65\\times10^{-9}$  & $1.05$\\tabularnewline\n$\\Xi_{bb}^{-}\\to\\Sigma_{b}^{-}\\tau^{+}\\tau^{-}$  & $6.76\\times10^{-22}$  & $3.80\\times10^{-10}$  & $1.16$ & $\\Xi_{bb}^{-}\\to\\Xi_{b}^{-}\\tau^{+}\\tau^{-}$  & $6.65\\times10^{-21}$  & $3.74\\times10^{-9}$  & $5.69$\\tabularnewline\n$\\Omega_{bb}^{-}\\to\\Xi_{b}^{-}\\tau^{+}\\tau^{-}$  & $1.22\\times10^{-22}$  & $1.49\\times10^{-10}$  & $5.52$ & $\\Xi_{bb}^{-}\\to\\Xi_{b}^{\\prime-}\\tau^{+}\\tau^{-}$  & $1.54\\times10^{-20}$  & $8.65\\times10^{-9}$  & $1.05$\\tabularnewline\n$\\Omega_{bb}^{-}\\to\\Xi_{b}^{\\prime-}\\tau^{+}\\tau^{-}$  & $2.96\\times10^{-22}$  & $3.60\\times10^{-10}$  & $1.17$ & $\\Omega_{bb}^{-}\\to\\Omega_{b}^{-}\\tau^{+}\\tau^{-}$  & $2.78\\times10^{-20}$  & $3.37\\times10^{-8}$  & $1.08$\\tabularnewline\n\\hline\n$\\Xi_{bc}^{+}\\to\\Lambda_{c}^{+}e^{+}e^{-}$  & $3.71\\times10^{-21}$  & $1.37\\times10^{-9}$  & $5.29$ & $\\Xi_{bc}^{+}\\to\\Xi_{c}^{+}e^{+}e^{-}$  & $1.19\\times10^{-19}$  & $4.43\\times10^{-8}$  & $4.90$\\tabularnewline\n$\\Xi_{bc}^{+}\\to\\Sigma_{c}^{+}e^{+}e^{-}$  & $9.04\\times10^{-21}$  & $3.35\\times10^{-9}$  & $0.81$ & $\\Xi_{bc}^{+}\\to\\Xi_{c}^{\\prime+}e^{+}e^{-}$  & $2.97\\times10^{-19}$  & $1.10\\times10^{-7}$  & $0.77$\\tabularnewline\n$\\Xi_{bc}^{0}\\to\\Sigma_{c}^{0}e^{+}e^{-}$  & $1.81\\times10^{-20}$  & $2.56\\times10^{-9}$  & $0.81$ & $\\Xi_{bc}^{0}\\to\\Xi_{c}^{0}e^{+}e^{-}$  & $1.19\\times10^{-19}$  & $1.69\\times10^{-8}$  & $4.90$\\tabularnewline\n$\\Omega_{bc}^{0}\\to\\Xi_{c}^{0}e^{+}e^{-}$  & $3.03\\times10^{-21}$  & $1.01\\times10^{-9}$  & $5.14$ & $\\Xi_{bc}^{0}\\to\\Xi_{c}^{\\prime0}e^{+}e^{-}$  & $2.97\\times10^{-19}$  & $4.20\\times10^{-8}$  & $0.77$\\tabularnewline\n$\\Omega_{bc}^{0}\\to\\Xi_{c}^{\\prime0}e^{+}e^{-}$  & $7.69\\times10^{-21}$  & $2.57\\times10^{-9}$  & $0.82$ & $\\Omega_{bc}^{0}\\to\\Omega_{c}^{0}e^{+}e^{-}$  & $5.17\\times10^{-19}$  & $1.73\\times10^{-7}$  & $0.78$\\tabularnewline\n$\\Xi_{bc}^{+}\\to\\Lambda_{c}^{+}\\mu^{+}\\mu^{-}$  & $3.54\\times10^{-21}$  & $1.31\\times10^{-9}$  & $7.20$ & $\\Xi_{bc}^{+}\\to\\Xi_{c}^{+}\\mu^{+}\\mu^{-}$  & $1.13\\times10^{-19}$  & $4.18\\times10^{-8}$  & $7.17$\\tabularnewline\n$\\Xi_{bc}^{+}\\to\\Sigma_{c}^{+}\\mu^{+}\\mu^{-}$  & $7.66\\times10^{-21}$  & $2.84\\times10^{-9}$  & $1.11$ & $\\Xi_{bc}^{+}\\to\\Xi_{c}^{\\prime+}\\mu^{+}\\mu^{-}$  & $2.41\\times10^{-19}$  & $8.93\\times10^{-8}$  & $1.15$\\tabularnewline\n$\\Xi_{bc}^{0}\\to\\Sigma_{c}^{0}\\mu^{+}\\mu^{-}$  & $1.53\\times10^{-20}$  & $2.17\\times10^{-9}$  & $1.11$ & $\\Xi_{bc}^{0}\\to\\Xi_{c}^{0}\\mu^{+}\\mu^{-}$  & $1.13\\times10^{-19}$  & $1.59\\times10^{-8}$  & $7.16$\\tabularnewline\n$\\Omega_{bc}^{0}\\to\\Xi_{c}^{0}\\mu^{+}\\mu^{-}$  & $2.89\\times10^{-21}$  & $9.68\\times10^{-10}$  & $6.95$ & $\\Xi_{bc}^{0}\\to\\Xi_{c}^{\\prime0}\\mu^{+}\\mu^{-}$  & $2.41\\times10^{-19}$  & $3.41\\times10^{-8}$  & $1.15$\\tabularnewline\n$\\Omega_{bc}^{0}\\to\\Xi_{c}^{\\prime0}\\mu^{+}\\mu^{-}$  & $6.52\\times10^{-21}$  & $2.18\\times10^{-9}$  & $1.13$ & $\\Omega_{bc}^{0}\\to\\Omega_{c}^{0}\\mu^{+}\\mu^{-}$  & $4.19\\times10^{-19}$  & $1.40\\times10^{-7}$  & $1.17$\\tabularnewline\n$\\Xi_{bc}^{+}\\to\\Lambda_{c}^{+}\\tau^{+}\\tau^{-}$  & $3.28\\times10^{-22}$  & $1.22\\times10^{-10}$  & $12.5$ & $\\Xi_{bc}^{+}\\to\\Xi_{c}^{+}\\tau^{+}\\tau^{-}$  & $8.64\\times10^{-21}$  & $3.21\\times10^{-9}$  & $11.9$\\tabularnewline\n$\\Xi_{bc}^{+}\\to\\Sigma_{c}^{+}\\tau^{+}\\tau^{-}$  & $6.92\\times10^{-22}$  & $2.57\\times10^{-10}$  & $1.68$ & $\\Xi_{bc}^{+}\\to\\Xi_{c}^{\\prime+}\\tau^{+}\\tau^{-}$  & $1.73\\times10^{-20}$  & $6.41\\times10^{-9}$  & $1.72$\\tabularnewline\n$\\Xi_{bc}^{0}\\to\\Sigma_{c}^{0}\\tau^{+}\\tau^{-}$  & $1.39\\times10^{-21}$  & $1.96\\times10^{-10}$  & $1.67$ & $\\Xi_{bc}^{0}\\to\\Xi_{c}^{0}\\tau^{+}\\tau^{-}$  & $8.60\\times10^{-21}$  & $1.22\\times10^{-9}$  & $11.8$\\tabularnewline\n$\\Omega_{bc}^{0}\\to\\Xi_{c}^{0}\\tau^{+}\\tau^{-}$  & $2.12\\times10^{-22}$  & $7.09\\times10^{-11}$  & $9.20$ & $\\Xi_{bc}^{0}\\to\\Xi_{c}^{\\prime0}\\tau^{+}\\tau^{-}$  & $1.73\\times10^{-20}$  & $2.44\\times10^{-9}$  & $1.71$\\tabularnewline\n$\\Omega_{bc}^{0}\\to\\Xi_{c}^{\\prime0}\\tau^{+}\\tau^{-}$  & $5.17\\times10^{-22}$  & $1.73\\times10^{-10}$  & $1.55$ & $\\Omega_{bc}^{0}\\to\\Omega_{c}^{0}\\tau^{+}\\tau^{-}$  & $2.62\\times10^{-20}$  & $8.77\\times10^{-9}$  & $1.60$\\tabularnewline\n\\hline\n$\\Xi_{bc}^{\\prime+}\\to\\Lambda_{c}^{+}e^{+}e^{-}$  & $3.23\\times10^{-21}$  & $1.20\\times10^{-9}$  & $0.84$ & $\\Xi_{bc}^{\\prime+}\\to\\Xi_{c}^{+}e^{+}e^{-}$  & $1.08\\times10^{-19}$  & $4.02\\times10^{-8}$  & $0.82$\\tabularnewline\n$\\Xi_{bc}^{\\prime+}\\to\\Sigma_{c}^{+}e^{+}e^{-}$  & $3.50\\times10^{-21}$  & $1.30\\times10^{-9}$  & $4.76$ & $\\Xi_{bc}^{\\prime+}\\to\\Xi_{c}^{\\prime+}e^{+}e^{-}$  & $1.15\\times10^{-19}$  & $4.25\\times10^{-8}$  & $4.60$\\tabularnewline\n$\\Xi_{bc}^{\\prime0}\\to\\Sigma_{c}^{0}e^{+}e^{-}$  & $7.01\\times10^{-21}$  & $9.90\\times10^{-10}$  & $4.76$ & $\\Xi_{bc}^{\\prime0}\\to\\Xi_{c}^{0}e^{+}e^{-}$  & $1.08\\times10^{-19}$  & $1.53\\times10^{-8}$  & $0.82$\\tabularnewline\n$\\Omega_{bc}^{\\prime0}\\to\\Xi_{c}^{0}e^{+}e^{-}$  & $2.78\\times10^{-21}$  & $9.30\\times10^{-10}$  & $0.87$ & $\\Xi_{bc}^{\\prime0}\\to\\Xi_{c}^{\\prime0}e^{+}e^{-}$  & $1.15\\times10^{-19}$  & $1.62\\times10^{-8}$  & $4.59$\\tabularnewline\n$\\Omega_{bc}^{\\prime0}\\to\\Xi_{c}^{\\prime0}e^{+}e^{-}$  & $2.93\\times10^{-21}$  & $9.80\\times10^{-10}$  & $4.75$ & $\\Omega_{bc}^{\\prime0}\\to\\Omega_{c}^{0}e^{+}e^{-}$  & $1.97\\times10^{-19}$  & $6.58\\times10^{-8}$  & $4.60$\\tabularnewline\n$\\Xi_{bc}^{\\prime+}\\to\\Lambda_{c}^{+}\\mu^{+}\\mu^{-}$  & $2.70\\times10^{-21}$  & $1.00\\times10^{-9}$  & $1.20$ & $\\Xi_{bc}^{\\prime+}\\to\\Xi_{c}^{+}\\mu^{+}\\mu^{-}$  & $8.72\\times10^{-20}$  & $3.23\\times10^{-8}$  & $1.26$\\tabularnewline\n$\\Xi_{bc}^{\\prime+}\\to\\Sigma_{c}^{+}\\mu^{+}\\mu^{-}$  & $3.34\\times10^{-21}$  & $1.24\\times10^{-9}$  & $6.40$ & $\\Xi_{bc}^{\\prime+}\\to\\Xi_{c}^{\\prime+}\\mu^{+}\\mu^{-}$  & $1.08\\times10^{-19}$  & $4.00\\times10^{-8}$  & $6.68$\\tabularnewline\n$\\Xi_{bc}^{\\prime0}\\to\\Sigma_{c}^{0}\\mu^{+}\\mu^{-}$  & $6.68\\times10^{-21}$  & $9.44\\times10^{-10}$  & $6.40$ & $\\Xi_{bc}^{\\prime0}\\to\\Xi_{c}^{0}\\mu^{+}\\mu^{-}$  & $8.72\\times10^{-20}$  & $1.23\\times10^{-8}$  & $1.26$\\tabularnewline\n$\\Omega_{bc}^{\\prime0}\\to\\Xi_{c}^{0}\\mu^{+}\\mu^{-}$  & $2.32\\times10^{-21}$  & $7.77\\times10^{-10}$  & $1.25$ & $\\Xi_{bc}^{\\prime0}\\to\\Xi_{c}^{\\prime0}\\mu^{+}\\mu^{-}$  & $1.08\\times10^{-19}$  & $1.52\\times10^{-8}$  & $6.67$\\tabularnewline\n$\\Omega_{bc}^{\\prime0}\\to\\Xi_{c}^{\\prime0}\\mu^{+}\\mu^{-}$  & $2.79\\times10^{-21}$  & $9.34\\times10^{-10}$  & $6.36$ & $\\Omega_{bc}^{\\prime0}\\to\\Omega_{c}^{0}\\mu^{+}\\mu^{-}$  & $1.85\\times10^{-19}$  & $6.19\\times10^{-8}$  & $6.68$\\tabularnewline\n$\\Xi_{bc}^{\\prime+}\\to\\Lambda_{c}^{+}\\tau^{+}\\tau^{-}$  & $1.60\\times10^{-22}$  & $5.93\\times10^{-11}$  & $0.90$ & $\\Xi_{bc}^{\\prime+}\\to\\Xi_{c}^{+}\\tau^{+}\\tau^{-}$  & $4.26\\times10^{-21}$  & $1.58\\times10^{-9}$  & $0.91$\\tabularnewline\n$\\Xi_{bc}^{\\prime+}\\to\\Sigma_{c}^{+}\\tau^{+}\\tau^{-}$  & $2.70\\times10^{-22}$  & $1.00\\times10^{-10}$  & $8.06$ & $\\Xi_{bc}^{\\prime+}\\to\\Xi_{c}^{\\prime+}\\tau^{+}\\tau^{-}$  & $7.27\\times10^{-21}$  & $2.70\\times10^{-9}$  & $8.91$\\tabularnewline\n$\\Xi_{bc}^{\\prime0}\\to\\Sigma_{c}^{0}\\tau^{+}\\tau^{-}$  & $5.40\\times10^{-22}$  & $7.63\\times10^{-11}$  & $8.04$ & $\\Xi_{bc}^{\\prime0}\\to\\Xi_{c}^{0}\\tau^{+}\\tau^{-}$  & $4.26\\times10^{-21}$  & $6.02\\times10^{-10}$  & $0.91$\\tabularnewline\n$\\Omega_{bc}^{\\prime0}\\to\\Xi_{c}^{0}\\tau^{+}\\tau^{-}$  & $1.27\\times10^{-22}$  & $4.24\\times10^{-11}$  & $0.86$ & $\\Xi_{bc}^{\\prime0}\\to\\Xi_{c}^{\\prime0}\\tau^{+}\\tau^{-}$  & $7.25\\times10^{-21}$  & $1.02\\times10^{-9}$  & $8.86$\\tabularnewline\n$\\Omega_{bc}^{\\prime0}\\to\\Xi_{c}^{\\prime0}\\tau^{+}\\tau^{-}$  & $1.86\\times10^{-22}$  & $6.21\\times10^{-11}$  & $6.86$ & $\\Omega_{bc}^{\\prime0}\\to\\Omega_{c}^{0}\\tau^{+}\\tau^{-}$  & $1.02\\times10^{-20}$  & $3.41\\times10^{-9}$  & $7.60$\\tabularnewline\n\\hline \\hline\n\\end{tabular}%\n\\end{table}\n\\begin{table}\n\\caption{The decay widths, branching ratios and $\\Gamma_{L}\/\\Gamma_{T}$s for the  transition ${1}\/{2}\\to{3}\/{2}$ with the charge current.}\n\\label{Tab:branchingv23}\n\\begin{tabular}{l|c|c|c|l|c|c|c}\n\\hline\\hline\nchannels  & $\\Gamma\/\\text{~GeV}$  & ${\\cal B}$  & $\\Gamma_{L}\/\\Gamma_{T}$  & channels  & $\\Gamma\/\\text{~GeV}$  & ${\\cal B}$  & $\\Gamma_{L}\/\\Gamma_{T}$\\tabularnewline\n\\hline\n$\\Xi_{cc}^{++}\\to\\Sigma_{c}^{*+}l^{+}\\nu_{l}$  & $1.43\\times10^{-15}$  & $5.55\\times10^{-4}$  & $0.92$ & $\\Xi_{bb}^{0}\\to\\Sigma_{b}^{*+}l^{-}\\bar{\\nu}_{l}$  & $2.33\\times10^{-17}$  & $1.31\\times10^{-5}$  & $0.94$\\tabularnewline\n$\\Xi_{cc}^{+}\\to\\Sigma_{c}^{*0}l^{+}\\nu_{l}$  & $2.85\\times10^{-15}$  & $1.95\\times10^{-4}$  & $0.92$ & $\\Xi_{bb}^{-}\\to\\Sigma_{b}^{*0}l^{-}\\bar{\\nu}_{l}$  & $1.16\\times10^{-17}$  & $6.52\\times10^{-6}$  & $0.94$\\tabularnewline\n$\\Omega_{cc}^{+}\\to\\Xi_{c}^{\\prime*0}l^{+}\\nu_{l}$  & $1.35\\times10^{-15}$  & $3.69\\times10^{-4}$  & $0.93$ & $\\Omega_{bb}^{-}\\to\\Xi_{b}^{\\prime*0}l^{-}\\bar{\\nu}_{l}$  & $1.11\\times10^{-17}$  & $1.35\\times10^{-5}$  & $0.97$\\tabularnewline\n$\\Xi_{cc}^{++}\\to\\Xi_{c}^{\\prime*+}l^{+}\\nu_{l}$  & $1.74\\times10^{-14}$  & $6.76\\times10^{-3}$  & $1.08$ & $\\Xi_{bb}^{0}\\to\\Xi_{bc}^{*+}l^{-}\\bar{\\nu}_{l}$  & $3.68\\times10^{-15}$  & $2.07\\times10^{-3}$  & $0.42$\\tabularnewline\n$\\Xi_{cc}^{+}\\to\\Xi_{c}^{\\prime*0}l^{+}\\nu_{l}$  & $1.74\\times10^{-14}$  & $1.19\\times10^{-3}$  & $1.08$ & $\\Xi_{bb}^{-}\\to\\Xi_{bc}^{*0}l^{-}\\bar{\\nu}_{l}$  & $3.68\\times10^{-15}$  & $2.07\\times10^{-3}$  & $0.42$\\tabularnewline\n$\\Omega_{cc}^{+}\\to\\Omega_{c}^{*0}l^{+}\\nu_{l}$  & $3.45\\times10^{-14}$  & $9.45\\times10^{-3}$  & $1.07$ & $\\Omega_{bb}^{-}\\to\\Omega_{bc}^{*0}l^{-}\\bar{\\nu}_{l}$  & $4.57\\times10^{-15}$  & $5.56\\times10^{-3}$  & $0.45$\\tabularnewline\n\\hline\n$\\Xi_{bc}^{+}\\to\\Sigma_{b}^{*0}l^{+}\\nu_{l}$  & $1.16\\times10^{-15}$  & $4.31\\times10^{-4}$  & $0.69$ & $\\Xi_{bc}^{+}\\to\\Sigma_{c}^{*++}l^{-}\\bar{\\nu}_{l}$  & $3.55\\times10^{-17}$  & $1.31\\times10^{-5}$  & $0.89$\\tabularnewline\n$\\Xi_{bc}^{0}\\to\\Sigma_{b}^{*-}l^{+}\\nu_{l}$  & $2.29\\times10^{-15}$  & $3.24\\times10^{-4}$  & $0.69$ & $\\Xi_{bc}^{0}\\to\\Sigma_{c}^{*+}l^{-}\\bar{\\nu}_{l}$  & $1.77\\times10^{-17}$  & $2.51\\times10^{-6}$  & $0.89$\\tabularnewline\n$\\Omega_{bc}^{0}\\to\\Xi_{b}^{\\prime*-}l^{+}\\nu_{l}$  & $7.38\\times10^{-16}$  & $2.47\\times10^{-4}$  & $0.81$ & $\\Omega_{bc}^{0}\\to\\Xi_{c}^{\\prime*+}l^{-}\\bar{\\nu}_{l}$  & $1.37\\times10^{-17}$  & $4.59\\times10^{-6}$  & $0.95$\\tabularnewline\n$\\Xi_{bc}^{+}\\to\\Xi_{b}^{\\prime*0}l^{+}\\nu_{l}$  & $1.36\\times10^{-14}$  & $5.04\\times10^{-3}$  & $0.78$ & $\\Xi_{bc}^{+}\\to\\Xi_{cc}^{*++}l^{-}\\bar{\\nu}_{l}$  & $1.06\\times10^{-14}$  & $3.92\\times10^{-3}$  & $1.46$\\tabularnewline\n$\\Xi_{bc}^{0}\\to\\Xi_{b}^{\\prime*-}l^{+}\\nu_{l}$  & $1.30\\times10^{-14}$  & $1.84\\times10^{-3}$  & $0.79$ & $\\Xi_{bc}^{0}\\to\\Xi_{cc}^{*+}l^{-}\\bar{\\nu}_{l}$  & $1.06\\times10^{-14}$  & $1.49\\times10^{-3}$  & $1.46$\\tabularnewline\n$\\Omega_{bc}^{0}\\to\\Omega_{b}^{*-}l^{+}\\nu_{l}$  & $1.50\\times10^{-14}$  & $5.03\\times10^{-3}$  & $1.00$ & $\\Omega_{bc}^{0}\\to\\Omega_{cc}^{*+}l^{-}\\bar{\\nu}_{l}$  & $7.31\\times10^{-15}$  & $2.44\\times10^{-3}$  & $1.21$\\tabularnewline\n\\hline\n$\\Xi_{bc}^{\\prime+}\\to\\Sigma_{b}^{*0}l^{+}\\nu_{l}$  & $3.48\\times10^{-15}$  & $1.29\\times10^{-3}$  & $0.69$ & $\\Xi_{bc}^{\\prime+}\\to\\Sigma_{c}^{*++}l^{-}\\bar{\\nu}_{l}$  & $1.06\\times10^{-16}$  & $3.94\\times10^{-5}$  & $0.89$\\tabularnewline\n$\\Xi_{bc}^{\\prime0}\\to\\Sigma_{b}^{*-}l^{+}\\nu_{l}$  & $6.87\\times10^{-15}$  & $9.71\\times10^{-4}$  & $0.69$ & $\\Xi_{bc}^{\\prime0}\\to\\Sigma_{c}^{*+}l^{-}\\bar{\\nu}_{l}$  & $5.32\\times10^{-17}$  & $7.52\\times10^{-6}$  & $0.89$\\tabularnewline\n$\\Omega_{bc}^{\\prime0}\\to\\Xi_{b}^{\\prime*-}l^{+}\\nu_{l}$  & $2.21\\times10^{-15}$  & $7.40\\times10^{-4}$  & $0.81$ & $\\Omega_{bc}^{\\prime0}\\to\\Xi_{c}^{\\prime*+}l^{-}\\bar{\\nu}_{l}$  & $4.12\\times10^{-17}$  & $1.38\\times10^{-5}$  & $0.95$\\tabularnewline\n$\\Xi_{bc}^{\\prime+}\\to\\Xi_{b}^{\\prime*0}l^{+}\\nu_{l}$  & $4.08\\times10^{-14}$  & $1.51\\times10^{-2}$  & $0.78$ & $\\Xi_{bc}^{\\prime+}\\to\\Xi_{cc}^{*++}l^{-}\\bar{\\nu}_{l}$  & $3.17\\times10^{-14}$  & $1.17\\times10^{-2}$  & $1.46$\\tabularnewline\n$\\Xi_{bc}^{\\prime0}\\to\\Xi_{b}^{\\prime*-}l^{+}\\nu_{l}$  & $3.90\\times10^{-14}$  & $5.51\\times10^{-3}$  & $0.79$ & $\\Xi_{bc}^{\\prime0}\\to\\Xi_{cc}^{*+}l^{-}\\bar{\\nu}_{l}$  & $3.17\\times10^{-14}$  & $4.48\\times10^{-3}$  & $1.46$\\tabularnewline\n$\\Omega_{bc}^{\\prime0}\\to\\Omega_{b}^{*-}l^{+}\\nu_{l}$  & $4.51\\times10^{-14}$  & $1.51\\times10^{-2}$  & $1.00$ & $\\Omega_{bc}^{\\prime0}\\to\\Omega_{cc}^{*+}l^{-}\\bar{\\nu}_{l}$  & $2.19\\times10^{-14}$  & $7.33\\times10^{-3}$  & $1.21$\\tabularnewline\n\\hline\\hline\n\\end{tabular}\n\\end{table}\n\n\\begin{table}\n\\caption{The decay widths, branching ratios and $\\Gamma_{L}\/\\Gamma_{T}$s for\nthe transition ${1}\/{2}\\to{3}\/{2}$ with FCNC.}\n\\label{Tab:branching23fcnc} %\n\\begin{tabular}{l|c|c|c|l|c|c|c}\n\\hline \\hline\nchannels  & $\\Gamma\/\\text{~GeV}$  & ${\\cal B}$  & $\\Gamma_{L}\/\\Gamma_{T}$ & channels  & $\\Gamma\/\\text{~GeV}$  & ${\\cal B}$  & $\\Gamma_{L}\/\\Gamma_{T}$\\tabularnewline\n\\hline\n$\\Xi_{bb}^{0}\\to\\Sigma_{b}^{*0}e^{+}e^{-}$  & $3.27\\times10^{-21}$  & $1.84\\times10^{-9}$  & $0.80$ & $\\Xi_{bb}^{0}\\to\\Xi_{b}^{\\prime*0}e^{+}e^{-}$  & $1.45\\times10^{-19}$  & $8.15\\times10^{-8}$  & $0.75$\\tabularnewline\n$\\Xi_{bb}^{-}\\to\\Sigma_{b}^{*-}e^{+}e^{-}$  & $6.52\\times10^{-21}$  & $3.67\\times10^{-9}$  & $0.80$ & $\\Xi_{bb}^{-}\\to\\Xi_{b}^{\\prime*-}e^{+}e^{-}$  & $1.43\\times10^{-19}$  & $8.05\\times10^{-8}$  & $0.74$\\tabularnewline\n$\\Omega_{bb}^{-}\\to\\Xi_{b}^{\\prime*-}e^{+}e^{-}$  & $3.06\\times10^{-21}$  & $3.72\\times10^{-9}$  & $0.81$ & $\\Omega_{bb}^{-}\\to\\Omega_{b}^{*-}e^{+}e^{-}$  & $2.71\\times10^{-19}$  & $3.30\\times10^{-7}$  & $0.74$\\tabularnewline\n$\\Xi_{bb}^{0}\\to\\Sigma_{b}^{*0}\\mu^{+}\\mu^{-}$  & $2.56\\times10^{-21}$  & $1.44\\times10^{-9}$  & $1.35$ & $\\Xi_{bb}^{0}\\to\\Xi_{b}^{\\prime*0}\\mu^{+}\\mu^{-}$  & $1.19\\times10^{-19}$  & $6.69\\times10^{-8}$  & $1.12$\\tabularnewline\n$\\Xi_{bb}^{-}\\to\\Sigma_{b}^{*-}\\mu^{+}\\mu^{-}$  & $5.11\\times10^{-21}$  & $2.87\\times10^{-9}$  & $1.35$ & $\\Xi_{bb}^{-}\\to\\Xi_{b}^{\\prime*-}\\mu^{+}\\mu^{-}$  & $1.17\\times10^{-19}$  & $6.58\\times10^{-8}$  & $1.12$\\tabularnewline\n$\\Omega_{bb}^{-}\\to\\Xi_{b}^{\\prime*-}\\mu^{+}\\mu^{-}$  & $2.40\\times10^{-21}$  & $2.92\\times10^{-9}$  & $1.36$ & $\\Omega_{bb}^{-}\\to\\Omega_{b}^{*-}\\mu^{+}\\mu^{-}$  & $2.22\\times10^{-19}$  & $2.70\\times10^{-7}$  & $1.12$\\tabularnewline\n$\\Xi_{bb}^{0}\\to\\Sigma_{b}^{*0}\\tau^{+}\\tau^{-}$  & $1.27\\times10^{-22}$  & $7.13\\times10^{-11}$  & $1.76$ & $\\Xi_{bb}^{0}\\to\\Xi_{b}^{\\prime*0}\\tau^{+}\\tau^{-}$  & $8.18\\times10^{-21}$  & $4.60\\times10^{-9}$  & $1.87$\\tabularnewline\n$\\Xi_{bb}^{-}\\to\\Sigma_{b}^{*-}\\tau^{+}\\tau^{-}$  & $2.52\\times10^{-22}$  & $1.42\\times10^{-10}$  & $1.76$ & $\\Xi_{bb}^{-}\\to\\Xi_{b}^{\\prime*-}\\tau^{+}\\tau^{-}$  & $7.96\\times10^{-21}$  & $4.48\\times10^{-9}$  & $1.88$\\tabularnewline\n$\\Omega_{bb}^{-}\\to\\Xi_{b}^{\\prime*-}\\tau^{+}\\tau^{-}$  & $1.14\\times10^{-22}$  & $1.39\\times10^{-10}$  & $1.75$ & $\\Omega_{bb}^{-}\\to\\Omega_{b}^{*-}\\tau^{+}\\tau^{-}$  & $1.45\\times10^{-20}$  & $1.76\\times10^{-8}$  & $1.85$\\tabularnewline\n\\hline\n$\\Xi_{bc}^{+}\\to\\Sigma_{c}^{*+}e^{+}e^{-}$  & $3.51\\times10^{-21}$  & $1.30\\times10^{-9}$  & $0.71$ & $\\Xi_{bc}^{+}\\to\\Xi_{c}^{\\prime*+}e^{+}e^{-}$  & $1.24\\times10^{-19}$  & $4.61\\times10^{-8}$  & $0.66$\\tabularnewline\n$\\Xi_{bc}^{0}\\to\\Sigma_{c}^{*0}e^{+}e^{-}$  & $7.02\\times10^{-21}$  & $9.92\\times10^{-10}$  & $0.71$ & $\\Xi_{bc}^{0}\\to\\Xi_{c}^{\\prime*0}e^{+}e^{-}$  & $1.24\\times10^{-19}$  & $1.76\\times10^{-8}$  & $0.66$\\tabularnewline\n$\\Omega_{bc}^{0}\\to\\Xi_{c}^{\\prime*0}e^{+}e^{-}$  & $2.77\\times10^{-21}$  & $9.26\\times10^{-10}$  & $0.75$ & $\\Omega_{bc}^{0}\\to\\Omega_{c}^{*0}e^{+}e^{-}$  & $2.07\\times10^{-19}$  & $6.92\\times10^{-8}$  & $0.68$\\tabularnewline\n$\\Xi_{bc}^{+}\\to\\Sigma_{c}^{*+}\\mu^{+}\\mu^{-}$  & $3.09\\times10^{-21}$  & $1.15\\times10^{-9}$  & $0.91$ & $\\Xi_{bc}^{+}\\to\\Xi_{c}^{\\prime*+}\\mu^{+}\\mu^{-}$  & $1.07\\times10^{-19}$  & $3.98\\times10^{-8}$  & $0.86$\\tabularnewline\n$\\Xi_{bc}^{0}\\to\\Sigma_{c}^{*0}\\mu^{+}\\mu^{-}$  & $6.18\\times10^{-21}$  & $8.74\\times10^{-10}$  & $0.91$ & $\\Xi_{bc}^{0}\\to\\Xi_{c}^{\\prime*0}\\mu^{+}\\mu^{-}$  & $1.07\\times10^{-19}$  & $1.52\\times10^{-8}$  & $0.86$\\tabularnewline\n$\\Omega_{bc}^{0}\\to\\Xi_{c}^{\\prime*0}\\mu^{+}\\mu^{-}$  & $2.41\\times10^{-21}$  & $8.07\\times10^{-10}$  & $0.98$ & $\\Omega_{bc}^{0}\\to\\Omega_{c}^{*0}\\mu^{+}\\mu^{-}$  & $1.77\\times10^{-19}$  & $5.91\\times10^{-8}$  & $0.92$\\tabularnewline\n$\\Xi_{bc}^{+}\\to\\Sigma_{c}^{*+}\\tau^{+}\\tau^{-}$  & $4.19\\times10^{-22}$  & $1.55\\times10^{-10}$  & $1.36$ & $\\Xi_{bc}^{+}\\to\\Xi_{c}^{\\prime*+}\\tau^{+}\\tau^{-}$  & $1.33\\times10^{-20}$  & $4.91\\times10^{-9}$  & $1.37$\\tabularnewline\n$\\Xi_{bc}^{0}\\to\\Sigma_{c}^{*0}\\tau^{+}\\tau^{-}$  & $8.38\\times10^{-22}$  & $1.18\\times10^{-10}$  & $1.36$ & $\\Xi_{bc}^{0}\\to\\Xi_{c}^{\\prime*0}\\tau^{+}\\tau^{-}$  & $1.33\\times10^{-20}$  & $1.87\\times10^{-9}$  & $1.37$\\tabularnewline\n$\\Omega_{bc}^{0}\\to\\Xi_{c}^{\\prime*0}\\tau^{+}\\tau^{-}$  & $2.55\\times10^{-22}$  & $8.53\\times10^{-11}$  & $1.37$ & $\\Omega_{bc}^{0}\\to\\Omega_{c}^{*0}\\tau^{+}\\tau^{-}$  & $1.72\\times10^{-20}$  & $5.77\\times10^{-9}$  & $1.37$\\tabularnewline\n\\hline\n$\\Xi_{bc}^{\\prime+}\\to\\Sigma_{c}^{*+}e^{+}e^{-}$  & $1.05\\times10^{-20}$  & $3.90\\times10^{-9}$  & $0.71$ & $\\Xi_{bc}^{\\prime+}\\to\\Xi_{c}^{\\prime*+}e^{+}e^{-}$  & $3.73\\times10^{-19}$  & $1.38\\times10^{-7}$  & $0.66$\\tabularnewline\n$\\Xi_{bc}^{\\prime0}\\to\\Sigma_{c}^{*0}e^{+}e^{-}$  & $2.11\\times10^{-20}$  & $2.98\\times10^{-9}$  & $0.71$ & $\\Xi_{bc}^{\\prime0}\\to\\Xi_{c}^{\\prime*0}e^{+}e^{-}$  & $3.73\\times10^{-19}$  & $5.27\\times10^{-8}$  & $0.66$\\tabularnewline\n$\\Omega_{bc}^{\\prime0}\\to\\Xi_{c}^{\\prime*0}e^{+}e^{-}$  & $8.31\\times10^{-21}$  & $2.78\\times10^{-9}$  & $0.75$ & $\\Omega_{bc}^{\\prime0}\\to\\Omega_{c}^{*0}e^{+}e^{-}$  & $6.21\\times10^{-19}$  & $2.08\\times10^{-7}$  & $0.68$\\tabularnewline\n$\\Xi_{bc}^{\\prime+}\\to\\Sigma_{c}^{*+}\\mu^{+}\\mu^{-}$  & $9.27\\times10^{-21}$  & $3.44\\times10^{-9}$  & $0.91$ & $\\Xi_{bc}^{\\prime+}\\to\\Xi_{c}^{\\prime*+}\\mu^{+}\\mu^{-}$  & $3.22\\times10^{-19}$  & $1.19\\times10^{-7}$  & $0.86$\\tabularnewline\n$\\Xi_{bc}^{\\prime0}\\to\\Sigma_{c}^{*0}\\mu^{+}\\mu^{-}$  & $1.85\\times10^{-20}$  & $2.62\\times10^{-9}$  & $0.91$ & $\\Xi_{bc}^{\\prime0}\\to\\Xi_{c}^{\\prime*0}\\mu^{+}\\mu^{-}$  & $3.22\\times10^{-19}$  & $4.55\\times10^{-8}$  & $0.86$\\tabularnewline\n$\\Omega_{bc}^{\\prime0}\\to\\Xi_{c}^{\\prime*0}\\mu^{+}\\mu^{-}$  & $7.24\\times10^{-21}$  & $2.42\\times10^{-9}$  & $0.98$ & $\\Omega_{bc}^{\\prime0}\\to\\Omega_{c}^{*0}\\mu^{+}\\mu^{-}$  & $5.31\\times10^{-19}$  & $1.77\\times10^{-7}$  & $0.92$\\tabularnewline\n$\\Xi_{bc}^{\\prime+}\\to\\Sigma_{c}^{*+}\\tau^{+}\\tau^{-}$  & $1.26\\times10^{-21}$  & $4.66\\times10^{-10}$  & $1.36$ & $\\Xi_{bc}^{\\prime+}\\to\\Xi_{c}^{\\prime*+}\\tau^{+}\\tau^{-}$  & $3.98\\times10^{-20}$  & $1.47\\times10^{-8}$  & $1.37$\\tabularnewline\n$\\Xi_{bc}^{\\prime0}\\to\\Sigma_{c}^{*0}\\tau^{+}\\tau^{-}$  & $2.51\\times10^{-21}$  & $3.55\\times10^{-10}$  & $1.36$ & $\\Xi_{bc}^{\\prime0}\\to\\Xi_{c}^{\\prime*0}\\tau^{+}\\tau^{-}$  & $3.98\\times10^{-20}$  & $5.62\\times10^{-9}$  & $1.37$\\tabularnewline\n$\\Omega_{bc}^{\\prime0}\\to\\Xi_{c}^{\\prime*0}\\tau^{+}\\tau^{-}$  & $7.65\\times10^{-22}$  & $2.56\\times10^{-10}$  & $1.37$ & $\\Omega_{bc}^{\\prime0}\\to\\Omega_{c}^{*0}\\tau^{+}\\tau^{-}$  & $5.17\\times10^{-20}$  & $1.73\\times10^{-8}$  & $1.37$\\tabularnewline\n\\hline \\hline\n\\end{tabular}\n\\end{table}\n\n\nSome comments on the results for phenomenological observables are given as follows.\n\\begin{itemize}\n\\item It can be seen in Tabs.~\\ref{Tab:branchingv22}-\\ref{Tab:branching23fcnc} that the decay widths for the four cases have the following hierarchical difference.\n\\begin{eqnarray}\n  &&\\Gamma{\\rm(the~transition~1\/2\\to 1\/2~with~charged~current)}>\\Gamma{\\rm(the~transition~1\/2\\to 3\/2~with~charged~current)}\\nonumber\\\\\n  &&>\\Gamma{\\rm(the~transition~1\/2\\to 1\/2~with~FCNC)}>\\Gamma{\\rm(the~transition~1\/2\\to 3\/2~with~FCNC)}.\\nonumber\n\\end{eqnarray}\nIn the transition $1\/2\\to 1\/2$ and $1\/2\\to 3\/2$ with FCNC cases, the decay widths are very close to each other for $l=e\/\\mu$ cases, while it is about one order of magnitude smaller for $l=\\tau$ case. This can be attributed to the much smaller phase space for $l=\\tau$ case.\n\\item A reasonable modification with momentum-space wave function $\\Psi^{SS_{z}}$ in the case of an axial-vector diquark involved is performed in this work in Eqs.~(\\ref{eq:momentum_wave_function_1\/2}) and (\\ref{eq:momentum_wave_function_1\/2gamma}).\nWhile, in Refs.~\\cite{Wang:2017mqp,Xing:2018lre,Zhao:2018mrg}, the momentum-space wave function $\\Psi^{SS_{z}}$ in the case of an axial-vector diquark involved is defined as\n\\begin{eqnarray}\t\n& \\Psi^{SS_{z}}(\\tilde{p}_{1},\\tilde{p}_{2},\\lambda_{1},\\lambda_{2})=\\frac{A}{\\sqrt{2(p_{1}\\cdot\\bar{P}+m_{1}M_{0})}}\\bar{u}(p_{1},\\lambda_{1})\\Gamma u(\\bar{P},S_{z})\\phi(x,k_{\\perp}), \\label{eq:momentum_wave_function_1\/2zhao}\\\\\n&\\Gamma  =-\\frac{1}{\\sqrt{3}}\\gamma_{5}\\left(\\slashed\\epsilon^{*}(p_{2},\\lambda_{2})\\right) \\quad {\\rm with}\\quad A=\\sqrt{\\frac{3(m_{1}M_{0}+p_{1}\\cdot\\bar{P})}{3m_{1}M_{0}+p_{1}\\cdot\\bar{P}+2(p_{1}\\cdot p_{2})(p_{2}\\cdot\\bar{P})\/m_{2}^{2}}}.\\label{eq:momentum_wave_function_1\/2gammazhao}\n\\end{eqnarray}\nIn Ref.~\\cite{Wang:2017mqp} the extraction approach is different from the one used in this work and in Refs.~\\cite{Xing:2018lre,Zhao:2018mrg}.\nIn order to find out the impact on the form factors and decay widths of these two factors: the extraction method of the form factors and the baryon wave function related to the axial vector diquark, we list numerical results of theses form factors of the three decay channels and the corresponding partial decay widths in Tab.~\\ref{difference}. The corresponding numerical results in Refs.~\\cite{Wang:2017mqp,Xing:2018lre,Zhao:2018mrg} are also given in Tab.~\\ref{difference}.\n\n\\begin{table}\n\\caption{The comparison of form factors and decay widths $\\Gamma$ between this work (This) and Zhao's work~\\cite{Wang:2017mqp,Xing:2018lre,Zhao:2018mrg}, and \"SE\" means ``the same extraction method of the form factors as this work\".}\n\\label{difference}\n\\begin{center}\n\\begin{tabular}{c|c|c|c|c|c|c|c|c|c|c|c|c|c}\n\\hline\\hline\nchannel & $f_{1,S}$ & $f_{2,S}$ & $f_{3,S}$ & $g_{1,S}$ & $g_{2,S}$ & $g_{3,S}$ & $f_{1,A}$ & $f_{2,A}$ & $f_{3,A}$ & $g_{1,A}$ & $g_{2,A}$ & $g_{3,A}$&$\\Gamma\/{\\rm GeV}$\\tabularnewline\n\\hline\n$\\Xi_{cc}^{++}\\to\\Lambda_{c}^{+}l^{+}\\nu_{l}$[This] & 0.495 & -0.621 & 0.832 & 0.332 & 1.004 & -2.957 & 0.489 & 0.290 & 0.648 & -0.111 & -0.325 & 0.943&$7.97\\times10^{-15}$\\tabularnewline\n\\hline\n$\\Xi_{cc}^{++}\\to\\Lambda_{c}^{+}l^{+}\\nu_{l}$~[SE]& 0.495 & -0.621 & 0.832 & 0.332 & 1.004 & -2.957 & 0.479 & 0.268 &0.650& -0.111 & -0.307 &1.702&$7.96\\times10^{-15}$\\tabularnewline\n\\hline\n$\\Xi_{cc}^{++}\\to\\Lambda_{c}^{+}l^{+}\\nu_{l}$~\\cite{Wang:2017mqp}& 0.653 & -0.738 &  & 0.533 & -0.053 &  & 0.637 & 0.725 &  & -0.167 & -0.028 & &$1.05\\times10^{-14}$ \\tabularnewline\n\\hline\\hline\nchannel & $f_{1,A}$ & $f_{2,A}$ & $f_{3,A}$ & $g_{1,A}$ & $g_{2,A}$ & $g_{3,A}$ & $f_{1,A}^{T}$ & $f_{2,A}^{T}$ & $g_{1,A}^{T}$ & $g_{2,A}^{T}$ &  & &$\\Gamma\/{\\rm GeV}$\\tabularnewline\n\\hline\n$\\Xi_{bb}^{0}\\to\\Xi_{b}^{0}e^{+}e^{-}$ [This]& 0.140 & 0.123 & -0.066 & -0.041 & -0.017 & 0.130&$0.134$ & -0.061 &$-0.054$ & -0.042 &  & &$1.62\\times10^{-19}$  \\tabularnewline\n\\hline\n$\\Xi_{bb}^{0}\\to\\Xi_{b}^{0}e^{+}e^{-}$~\\cite{Xing:2018lre}& 0.138 & 0.132 & -0.068 & -0.030 & -0.055 & 0.261 &$0.134$& -0.066 & $0.032$ & -0.049  &  & &$1.98\\times10^{-19}$\\tabularnewline\n\\hline\\hline\nchannel & $f_{1,A}$ & $f_{2,A}$ & $f_{3,A}$ & $f_{4,A}$ & $g_{1,A}$ & $g_{2,A}$ & $g_{3,A}$ & $g_{4,A}$ &  &  &  & &$\\Gamma\/{\\rm GeV}$\\tabularnewline\n\\hline\n$\\Xi_{cc}^{++}\\to\\Sigma_{c}^{*+}l^{+}\\nu_{l}$[This] & -0.979 & -0.645 & 0.047 & -1.969 & -5.792 & -3.602 & 0.947 & 0.393 &  &  &  & &$1.43\\times10^{-15}$ \\tabularnewline\n\\hline\n$\\Xi_{cc}^{++}\\to\\Sigma_{c}^{*+}l^{+}\\nu_{l}$~\\cite{Zhao:2018mrg}& -1.121 & 1.845 & -1.703 & -1.827 & -8.292 & -5.262 & 0.942 & 0.295 &  &  &  & &$1.26\\times10^{-15}$ \\tabularnewline\n\\hline\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\nFirstly, comparing each first two lines for $\\Xi_{cc}^{++}\\to\\Lambda_{c}^{+}l^{+}\\nu_{l}$, $\\Xi_{bb}^{0}\\to\\Xi_{b}^{0}e^{+}e^{-}$ and $\\Xi_{cc}^{++}\\to\\Sigma_{c}^{*+}l^{+}\\nu_{l}$, we could find that partial decay width differences coming from the different wave function with axial-vector diquark are small, but there are some differences among the form factors. Secondly, comparing the second line and third line of the channel $\\Xi_{cc}^{++}\\to\\Lambda_{c}^{+}l^{+}\\nu_{l}$, we could find the extracting approach of these form factors will bring in some effect in the form factors and decay widths; So the effect in the form factors and decay widths brought in by the extraction approach is much larger than that of the definition of the wave function $\\Psi^{SS_{z}}$.\n\\item Since there exist uncertainties in the lifetimes of the parent baryons, there may be some small fluctuations in the results for branching ratios. Form Tab.~\\ref{Tab:branchingratio22}, we may find that\n     \\begin{eqnarray}\n     &&{\\cal B}(\\Xi^{++}_{cc}\\to \\Xi^{\\prime +}_{c}l^{+}\\nu_{l})=5.57\\times10^{-2},\\quad\n    {\\cal B}(\\Xi^{++}_{cc}\\to \\Xi^{+}_{c}l^{+}\\nu_{l})=3.40\\times10^{-2},\\\\\n     &&{\\cal B}(\\Omega^{+}_{cc}\\to \\Omega^{0}_{c}l^{+}\\nu_{l})=7.67\\times10^{-2},\\quad\n     {\\cal B}(\\Xi^{+}_{bc}\\to \\Xi^{\\prime 0}_{b}l^{+}\\nu_{l})=2.50\\times10^{-2}.\n     \\end{eqnarray}\n    These channels may be firstly examined at experimental facilities like   LHC or BelleII.\n\\item Take the four processes $\\Xi^{++}_{cc}\\to\\Lambda_{c}^{+}l^{+}\\nu_{l}$, $\\Xi^{++}_{cc}\\to\\Sigma_{c}^{*+}l^{+}\\nu_{l}$, $\\Xi^{0}_{bb}\\to\\Xi_{b}^{0}e^{+}e^{-}$ and $\\Xi^{0}_{bb}\\to\\Xi_{b}^{\\prime*0}e^{+}e^{-}$ as examples. The uncertainties for the partial decay widths caused by the model parameters and the single pole assumption for $c\\to d,s$ channels are listed as\n      \\begin{eqnarray}\n      \\Gamma(\\Xi^{++}_{cc}\\to\\Lambda_{c}^{+}l^{+}\\nu_{l})&=&(7.97\\pm0.65\\pm1.28\\pm1.55\\pm1.65)\\times10^{-15}~{\\rm GeV},\\nonumber\\\\\n      \\Gamma(\\Xi^{++}_{cc}\\to\\Sigma_{c}^{*+}l^{+}\\nu_{l})&=&(1.43\\pm0.23\\pm0.29\\pm0.29\\pm0.16)\\times10^{-15}~{\\rm GeV},\n      \\label{eq:vverrors}\n      \\end{eqnarray}\nwhere these errors come from $\\beta_{i}$, $\\beta_{f}$, $m_{\\rm di}$ and $m_{\\rm pole}$ respectively;\n\\begin{eqnarray}\n      \\Gamma(\\Xi^{0}_{bb}\\to\\Xi_{b}^{0}e^{+}e^{-})&=&(1.62\\pm0.69\\pm0.96\\pm0.17)\\times10^{-19}~{\\rm GeV},\\nonumber\\\\\n      \\Gamma(\\Xi^{0}_{bb}\\to\\Xi_{b}^{\\prime*0}e^{+}e^{-})&=&(1.45\\pm0.19\\pm0.70\\pm0.43)\\times10^{-19}~{\\rm GeV},\n      \\label{eq:fcncerrors}\n\\end{eqnarray}\nwhere these errors come from $\\beta_{i}$, $\\beta_{f}$, $m_{\\rm di}$, respectively.\nTaking $\\Xi_{cc}^{++}\\to\\Lambda^{+}_{c}$ as an example, the error estimates for the form factors can be found in Tab.~\\ref{Tab:errorsformfractors}.\n\\begin{table}\n\\caption{Error estimates for the form factors, taking $\\Xi_{cc}^{++}\\to\\Lambda^{+}_{c}$ as an example. The first number is the central value, and following 3 errors come from $\\beta_{i}=\\beta_{\\Xi_{cc}^{++}}$, $\\beta_{f}=\\beta_{\\Lambda^{+}_{c}}$ and $m_{di}=m_{(cu)}$, respectively. These parameters are all varied by $10\\%$.}\n\\label{Tab:errorsformfractors}\n\\begin{tabular}{c|c|c|c}\n\\hline\\hline\n$F$ & $F(0)$ & $F$ & $F(0)$\\tabularnewline\n\\hline\n$f_{1,S}^{\\Xi_{cc}^{++}\\to\\Lambda_{c}^{+}}$ & $0.495\\pm0.020\\pm0.034\\pm0.042$ & $f_{1,A}^{\\Xi_{cc}^{++}\\to\\Lambda_{c}^{+}}$ & $0.489\\pm0.019\\pm0.034\\pm0.042$\\tabularnewline\n\\hline\n$f_{2,S}^{\\Xi_{cc}^{++}\\to\\Lambda_{c}^{+}}$ & $-0.621\\pm0.119\\pm0.065\\pm0.227$ & $f_{2,A}^{\\Xi_{cc}^{++}\\to\\Lambda_{c}^{+}}$ & $0.290\\pm0.074\\pm0.080\\pm0.199$\\tabularnewline\n\\hline\n$f_{3,S}^{\\Xi_{cc}^{++}\\to\\Lambda_{c}^{+}}$ & $0.832\\pm0.130\\pm0.165\\pm0.202$ & $f_{3,A}^{\\Xi_{cc}^{++}\\to\\Lambda_{c}^{+}}$ & $0.648\\pm0.122\\pm0.170\\pm0.194$\\tabularnewline\n\\hline\n$g_{1,S}^{\\Xi_{cc}^{++}\\to\\Lambda_{c}^{+}}$ & $0.332\\pm0.020\\pm0.004\\pm0.086$ & $g_{1,A}^{\\Xi_{cc}^{++}\\to\\Lambda_{c}^{+}}$ & $-0.111\\pm0.007\\pm0.001\\pm0.003$\\tabularnewline\n\\hline\n$g_{2,S}^{\\Xi_{cc}^{++}\\to\\Lambda_{c}^{+}}$ & $1.004\\pm0.059\\pm0.199\\pm0.170$ & $g_{2,A}^{\\Xi_{cc}^{++}\\to\\Lambda_{c}^{+}}$ & $-0.325\\pm0.021\\pm0.065\\pm0.058$\\tabularnewline\n\\hline\n$g_{3,S}^{\\Xi_{cc}^{++}\\to\\Lambda_{c}^{+}}$ & $-2.957\\pm0.973\\pm0.804\\pm0.731$ & $g_{3,A}^{\\Xi_{cc}^{++}\\to\\Lambda_{c}^{+}}$ & $0.943\\pm0.330\\pm0.264\\pm0.247$\\tabularnewline\n\\hline\\hline\n\\end{tabular}\n\\end{table}\nIt can be seen from Eqs.~(\\ref{eq:vverrors}-\\ref{eq:fcncerrors}) and Tab.~\\ref{Tab:errorsformfractors} that, the uncertainties caused by these parameters may be sizable.\n\\item The ratios $\\Gamma_{L}\/\\Gamma_{T}$s have the following rule:\n\\begin{eqnarray}\n&&c\\to d:~\\Gamma_{L}\/\\Gamma_{T}(\\Xi_{cc}^{++}\\to\\Sigma_{c}^{+}l^{+}\\nu_{l})=\n\\Gamma_{L}\/\\Gamma_{T}(\\Xi_{cc}^{+}\\to\\Sigma_{c}^{0}l^{+}\\nu_{l})=\n\\Gamma_{L}\/\\Gamma_{T}(\\Omega_{cc}^{+}\\to\\Xi_{c}^{\\prime0}l^{+}\\nu_{l}),\\label{eq:cdcc}\\\\\n&&c\\to s:~\\Gamma_{L}\/\\Gamma_{T}(\\Xi_{cc}^{++}\\to\\Xi_{c}^{\\prime+}l^{+}\\nu_{l})=\n\\Gamma_{L}\/\\Gamma_{T}(\\Xi_{cc}^{+}\\to\\Xi_{c}^{\\prime0}l^{+}\\nu_{l})=\n\\Gamma_{L}\/\\Gamma_{T}(\\Omega_{cc}^{+}\\to\\Omega_{c}^{0}l^{+}\\nu_{l}),\\label{eq:cscc}\n\\end{eqnarray}\nfor these decay channels in Eq.~(\\ref{eq:cdcc}) have the same decay in quark level and the final single heavy baryons all in the sextets which leads to the same overlapping factors in the SU(3) sysmetry. In other decay channels the similar relations exist.\n For the transition $1\/2\\to3\/2$, we have the following relations:\n \\begin{eqnarray}\n &&\\Gamma_{L}\/\\Gamma_{T}(B_{bc}^{\\prime}\\to B_{b}^{*}l^{+}\\nu_{l})=\\Gamma_{L}\/\\Gamma_{T}(B_{bc}\\to B_{b}^{*}l^{+}\\nu_{l}), \\label{eq:cdsbc}\\\\\n  &&\\Gamma_{L}\/\\Gamma_{T}(B_{bc}^{\\prime}\\to B_{c}^{*}l^{-}\\bar\\nu_{l})=\\Gamma_{L}\/\\Gamma_{T}(B_{bc}\\to B_{c}^{*}l^{-}\\bar\\nu_{l}),\\label{eq:bucbc}\\\\\n &&\\Gamma_{L}\/\\Gamma_{T}(B_{bc}^{\\prime}\\to B_{c}^{*}l^{+}l^{-})=\\Gamma_{L}\/\\Gamma_{T}(B_{bc}\\to B_{c}^{*}l^{+}l^{-}),\\label{eq:bdsbc}\n \\end{eqnarray}\n for these decay channels have the same decay in quark level and the spin of final single heavy baryons are all $3\/2$, only with the axial-vector diquark spectator which have the same overlapping factors and form factors. As shown in Figs.~\\ref{fig:decaywidthbbtob22}-\\ref{fig:decaywidthbbtobfcnc23} and Tabs.~\\ref{Tab:branchingv22}-~\\ref{Tab:branching23fcnc}, these decay channels with same $\\Gamma_{L}\/\\Gamma_{T}$ have the similar plots of the dependence of $d\\Gamma_{L}\/dq^2$ and $d\\Gamma_{T}\/dq^2$ on $q^2$.\n \\item From Figs.~\\ref{fig:decaywidthbbtobFCNC22} and \\ref{fig:decaywidthbbtobfcnc23}, it can be found that, at small $q^2$, there are some divergence of the $d\\Gamma_{L}\/dq^2$ and $d\\Gamma_{T}\/dq^2$ for the cases\n ${\\cal B}_{bb}\\to{\\cal B}_{b}^{(*)} e^{+}e^{-}\/ \\mu^{+}\\mu^{-}$ and ${\\cal B}_{bb}\\to{\\cal B}_{bc}^{(*)} l^{-}\\bar{\\nu}_{l}$, because $\\frac{1}{\\sqrt{q^2}}$ is included in their helicity amplitudes shown with Eqs.~(\\ref{eq:helicty23v}), (\\ref{eq:hv22}), (\\ref{eq:helicty22fcnc}),\n  (\\ref{eq:hv232}), (\\ref{eq:helicty23fcnc}).\n \\end{itemize}\n\n\\subsection{SU(3) symmetry for semileptonic decays}\n\\renewcommand{\\thetable}{D\\arabic{table}}\nRecently, an analysis of weak decays of doubly-heavy baryons based on flavor symmetry is available in Ref.~\\cite{Wang:2017azm,Shi:2017dto}. In the SU(3) symmetry limit,  there exist the a number of relations\namong these semileptonic decay widths, which we are going to examine in the following.\n\n\\begin{itemize}\n\t\n\\item For $c\\to d,s$ process, we have\n\\begin{align*}\n&\\Gamma(\\Xi_{cc}^{++}\\to\\Lambda_{c}^{+}l^{+}\\nu_{l}) =\\Gamma(\\Omega_{cc}^{+}\\to\\Xi_{c}^{0}l^{+}\\nu_{l}),\\quad\n\\Gamma(\\Xi_{cc}^{++}\\to\\Xi_{c}^{+}l^{+}\\nu_{l})  =\\Gamma(\\Xi_{cc}^{+}\\to\\Xi_{c}^{0}l^{+}\\nu_{l}),\\\\\n&\\Gamma(\\Xi_{cc}^{+}\\to\\Sigma_{c}^{(*)0}l^{+}\\nu_{l})=2\\Gamma(\\Xi_{cc}^{++}\\to\\Sigma_{c}^{(*)+}l^{+}\\nu_{l})  =2\\Gamma(\\Omega_{cc}^{+}\\to\\Xi_{c}^{\\prime(*)0}l^{+}\\nu_{l}),\\\\\n&\\Gamma(\\Omega_{cc}^{+}\\to\\Omega_{c}^{(*)0}l^{+}\\nu_{l}) =2\\Gamma(\\Xi_{cc}^{++}\\to\\Xi_{c}^{\\prime(*)+}l^{+}\\nu_{l}) =2\\Gamma(\\Xi_{cc}^{+}\\to\\Xi_{c}^{\\prime(*)0}l^{+}\\nu_{l}),\\\\\n&\\Gamma(\\Xi_{bc}^{(\\prime)+}\\to\\Lambda_{b}^{0}l^{+}\\nu_{l}) =\\Gamma(\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{b}^{-}l^{+}\\nu_{l}),\\quad\n\\Gamma(\\Xi_{bc}^{(\\prime)+}\\to\\Xi_{b}^{0}l^{+}\\nu_{l})  =2\\Gamma(\\Xi_{bc}^{(\\prime)0}\\to\\Xi_{b}^{-}l^{+}\\nu_{l}),\\\\\n&2\\Gamma(\\Xi_{bc}^{(\\prime)+}\\to\\Sigma_{b}^{(*)0}l^{+}\\nu_{l})  =\\Gamma(\\Xi_{bc}^{(\\prime)0}\\to\\Sigma_{b}^{(*)-}l^{+}\\nu_{l})=2\\Gamma(\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{b}^{\\prime(*)-}l^{+}\\nu_{l}),\\\\\n&2\\Gamma(\\Xi_{bc}^{(\\prime)+}\\to\\Xi_{b}^{\\prime(*)0}l^{+}\\nu_{l}) =2\\Gamma(\\Xi_{bc}^{(\\prime)0}\\to\\Xi_{b}^{\\prime(*)-}l^{+}\\nu_{l})=\\Gamma(\\Omega_{bc}^{(\\prime)0}\\to\\Omega_{b}^{(*)-}l^{+}\\nu_{l}).\n\\end{align*}\n\\item For $b\\to u,c$ process, we have\n\\begin{align*}\n&\\Gamma(\\Xi_{bb}^{-}\\to\\Lambda_{b}^{0}l^{-}\\bar{\\nu}_{l})  =\\Gamma(\\Omega_{bb}^{-}\\to\\Xi_{b}^{0}l^{-}\\bar{\\nu}_{l}),\\\\\n&\\Gamma(\\Xi_{bb}^{0}\\to\\Xi_{bc}^{(*)+}l^{-}\\bar{\\nu}_{l})  =\\Gamma(\\Xi_{bb}^{-}\\to\\Xi_{bc}^{(*)0}l^{-}\\bar{\\nu}_{l})=\\Gamma(\\Omega_{bb}^{-}\\to\\Omega_{bc}^{(*)0}l^{-}\\bar{\\nu}_{l}),\\\\\n&\\Gamma(\\Xi_{bb}^{0}\\to\\Sigma_{b}^{(*)+}l^{-}\\bar{\\nu}_{l})  =2\\Gamma(\\Xi_{bb}^{-}\\to\\Sigma_{b}^{(*)0}l^{-}\\bar{\\nu}_{l})=2\\Gamma(\\Omega_{bb}^{-}\\to\\Xi_{b}^{\\prime(*)0}l^{-}\\bar{\\nu}_{l}),\\\\\n&\\Gamma(\\Xi_{bc}^{(\\prime)+}\\to\\Xi_{cc}^{(*)++}l^{-}\\bar{\\nu}_{l})  =\\Gamma(\\Xi_{bc}^{(\\prime)0}\\to\\Xi_{cc}^{(*)+}l^{-}\\bar{\\nu}_{l})=\\Gamma(\\Omega_{bc}^{(\\prime)0}\\to\\Omega_{cc}^{(*)+}l^{-}\\bar{\\nu}_{l}),\\\\\n&\\Gamma(\\Xi_{bc}^{(\\prime)+}\\to\\Sigma_{c}^{(*)++}l^{-}\\bar{\\nu}_{l}) =2\\Gamma(\\Xi_{bc}^{(\\prime)0}\\to\\Sigma_{c}^{(*)+}l^{-}\\bar{\\nu}_{l})=2\\Gamma(\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{c}^{\\prime(*)+}l^{-}\\bar{\\nu}_{l}).\n\\end{align*}\n\n\\end{itemize}\n\n\nAccording to the flavor SU(3) symmetry, there exist the following\nrelations among these FCNC process. These relations can be readily\nderived using the overlapping factors given in Tab. \\ref{Tab:overlapping_factors_22}.\n\\begin{itemize}\n\\item For $b\\to d$ process, we have\n\t\\begin{eqnarray}\n\t&  & \\Gamma(\\Xi_{bb}^{0}\\to\\Lambda_{b}^{0}l^{+}l^{-})=\\Gamma(\\Omega_{bb}^{-}\\to\\Xi_{b}^{-}l^{+}l^{-}),\\nonumber \\\\\n\t&  & 2\\Gamma(\\Xi_{bb}^{0}\\to\\Sigma_{b}^{(*)0}l^{+}l^{-})=\\Gamma(\\Xi_{bb}^{-}\\to\\Sigma_{b}^{(*)-}l^{+}l^{-})\n=2\\Gamma(\\Omega_{bb}^{-}\\to\\Xi_{b}^{\\prime(*)0}l^{+}l^{-}),\\nonumber\\\\\n\t&  & \\Gamma(\\Xi_{bc}^{(\\prime)+}\\to\\Lambda_{c}^{+}l^{+}l^{-})=\\Gamma(\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{c}^{0}l^{+}l^{-}),\\nonumber \\\\\n\t&  & 2\\Gamma(\\Xi_{bc}^{(\\prime)+}\\to\\Sigma_{c}^{(*)+}l^{+}l^{-})=\\Gamma(\\Xi_{bc}^{(\\prime)0}\\to\\Sigma_{c}^{(*)0}l^{+}l^{-})\n=2\\Gamma(\\Omega_{bc}^{(\\prime)0}\\to\\Xi_{c}^{\\prime(*)0}l^{+}l^{-}).\\nonumber\n\t\\end{eqnarray}\n\n\t\\item For $b\\to s$ process, we have\n\t\\begin{eqnarray}\n\t&  & \\Gamma(\\Xi_{bb}^{0}\\to\\Xi_{b}^{0}l^{+}l^{-})=\\Gamma(\\Xi_{bb}^{-}\\to\\Xi_{b}^{-}l^{+}l^{-}),\\nonumber \\\\\n\t&  & 2\\Gamma(\\Xi_{bb}^{0}\\to\\Xi_{b}^{\\prime(*)0}l^{+}l^{-})=2\\Gamma(\\Xi_{bb}^{-}\\to\\Xi_{b}^{\\prime(*)-}l^{+}l^{-})\n=\\Gamma(\\Omega_{bb}^{-}\\to\\Omega_{b}^{(*)-}l^{+}l^{-}),\\nonumber \\\\\n\t&  & \\Gamma(\\Xi_{bc}^{(\\prime)+}\\to\\Xi_{c}^{+}l^{+}l^{-})=\\Gamma(\\Xi_{bc}^{(\\prime)0}\\to\\Xi_{c}^{0}l^{+}l^{-}),\\nonumber \\\\\n\t&  & 2\\Gamma(\\Xi_{bc}^{(\\prime)+}\\to\\Xi_{c}^{\\prime(*)+}l^{+}l^{-})=2\\Gamma(\\Xi_{bc}^{(\\prime)0}\\to\\Xi_{c}^{\\prime(*)0}l^{+}l^{-})\n=\\Gamma(\\Omega_{bc}^{(\\prime)0}\\to\\Omega_{c}^{(*)0}l^{+}l^{-}).\\nonumber\n\t\\end{eqnarray}\n\t\n\\end{itemize}\t\nComparing the above equations predicted by SU(3) symmetry with the corresponding results in this work, we have the following remarks:\n\\begin{itemize}\n\t\\item  most of our numerical results are respected very well with the SU(3) symmetry relations, except for the following ones\n\t\\begin{align}\n\t\\Gamma(\\Xi_{cc}^{++}\\to\\Lambda_{c}^{+}l^{+}\\nu_{l}) &= \\Gamma(\\Omega_{cc}^{+}\\to\\Xi_{c}^{0}l^{+}\\nu_{l}),\\quad \\Gamma(\\Xi_{bc}^{+}\\to\\Lambda_{b}^{0}l^{+}\\nu_{l})  = \\Gamma(\\Omega_{bc}^{0}\\to\\Xi_{b}^{-}l^{+}\\nu_{l}),\\nonumber\\\\ \\Gamma(\\Xi_{bc}^{+}\\to\\Sigma_{b}^{(*)0}l^{+}\\nu_{l}) &= \\Gamma(\\Omega_{bc}^{0}\\to\\Xi_{b}^{\\prime(*)-}l^{+}\\nu_{l}),\\quad \\Gamma(\\Xi_{bc}^{+}\\to\\Xi_{b}^{\\prime(*)0}l^{+}\\nu_{l}) = \\frac{1}{2}\\Gamma(\\Omega_{bc}^{0}\\to\\Omega_{b}^{(*)-}l^{+}\\nu_{l}),\\nonumber\\\\ \\Gamma(\\Xi_{bc}^{0}\\to\\Sigma_{c}^{(*)+}l^{-}\\bar{\\nu}_{l}) &= \\Gamma(\\Omega_{bc}^{0}\\to\\Xi_{c}^{\\prime(*)+}l^{-}\\bar{\\nu}_{l}).\\label{eq:su3_breaking}\n\\end{align}\nThese five relations  are broken considerably: larger  than 20\\% but still less than 50\\% using the definition of $({\\rm Max}[\\Gamma_{{\\rm LHS}},\\Gamma_{{\\rm RHS}}]-{\\rm Min}[\\Gamma_{{\\rm LHS}},\\Gamma_{{\\rm RHS}}])\/{\\rm Max}[\\Gamma_{{\\rm LHS}},\\Gamma_{{\\rm RHS}}]$. Since the mass difference between the $u$ and $d$ quark has been neglected in this work, the isospin symmetry is well respected. But since the strange quark is much heavier,  the SU(3) relations for the channels involving   $u,d$ quark and $s$ quark can be significantly broken. All relations given in  Eq.~(\\ref{eq:su3_breaking})  are of this type.\n\n\\item The first 4 relations  in Eq.~(\\ref{eq:su3_breaking}) involve the $c$ quark decay but  the last one involves the $b$ quark decay. It indicates that the $c$ quark decay modes tend to break SU(3) symmetry easily. This can be understood since  the phase space of the $c$ quark decay is smaller, and thus the decay amplitude  is more  sensitive to the mass of the initial and final baryons.\n\\item SU(3) symmetry breaking is larger for the $Qs$ diquark involved case than that for the $Qu\/Qd$ diquark involved case with $Q=b,c$. SU(3) symmetry breaking is larger for the $cq$ diquark involved case than that for the $bq$ diquark involved case with $q=u,d,s$.\n\\item SU(3) symmetry breaking is smaller for $l=e\/\\mu$ cases than that for $l=\\tau$ case. This can be attributed to the much smaller phase space for $l=\\tau$ case. Smaller phase space is more sensitive to the variation of the masses of baryons in the initial and final states.\n\t\\end{itemize}\n\n\n\n\\section{Conclusion}\n\nIn this paper, we have  presented  a\nsystematic investigation of  transition form factors of   doubly heavy baryon  decays in the light front approach. Our main results for the form factors with the $q^2$ distributions are collected in Tabs.~\\ref{Tab:ff_ccc}-\\ref{Tab:fcnc32_bd}. The present analysis is the sequel and update of the  previous works on weak decays of doubly heavy baryons, Ref.~\\cite{Wang:2017mqp,Zhao:2018mrg,Xing:2018lre}. It improves upon the  previous work by\n\\begin{itemize}\n\\item adding predictions for the FCNC process with the spin-1\/2 to spin-3\/2 transition;\n\\item using a new extraction of the Lorentz structure for the form factors;\n\\item updating  the wave function for spin-$1\/2$ baryons states with an  axial-vector diquark ;\n\\item updating  the wave function for spin-$3\/2$ baryons states;\n\\item presenting  a new derivation of  the overlapping factors using flavor  SU(3) symmetry approach;\n\\item presenting   the momentum distribution   of form factors.\n\\end{itemize}\nUsing these form factors, we have also preformed the calculation of phenomenological observables of these corresponding semileptonic weak decays of doubly heavy baryons with the results  shown in Tabs.~\\ref{Tab:branchingv22}-\\ref{Tab:branching23fcnc}. The flavor  SU(3) symmetry and sources of symmetry breaking are also discussed in great details.\nWe find that,\n\\begin{itemize}\n\\item most  branching ratios for spin-1\/2 to spin-1\/2  with $c\\to d,s$ processes\nare at the order  $10^{-3}\\sim10^{-2}$,  which might be  examined at experimental facilities at LHC or Belle-II;\n\\item the different extraction approaches could give   sizable differences to form factors;\n\\item the uncertainties of form factors and decay widths caused by model parameters are sizable;\n\\item the ratios $\\Gamma_{L}\/\\Gamma_{T}$s have the rules shown in Eqs.~(\\ref{eq:cdcc})-(\\ref{eq:bdsbc}), for these decay channels have the same decay in quark level and the same overlapping factors and form factors;\n\\item most of our results are comparable to the theoretical results in Refs.~\\cite{Wang:2017mqp,Zhao:2018mrg,Xing:2018lre};\n\\item since the mass difference between the $u$ and $d$ quark has been neglected and the strange quark is much heavier, the SU(3) relations shown in Eq.~(\\ref{eq:su3_breaking}) for the channels involving $u,d$ quark and $s$ quark can be  broken;\n\n\\item the SU(3) symmetry breaking is sizable in the charmed\nbaryon decays, while for the bottomed case the SU(3) symmetry breaking is small.\n\\end{itemize}\n\nThis work completes the study of form factors in the traditional light-front quark model with the quark-diquark constituent viewpoint.   We hope our phenomenology predictions for these semi-leptonic decays could be tested by LHCb and other experiments in the future.\n\n\n\\section*{Acknowledgements}\nWe thank Prof. Wei Wang,  Fu-Sheng Yu and Zhen-Xing Zhao for fruitful discussions.\nThis work is supported in part by National\nNatural Science Foundation of China under Grants No.\n11735010, 11911530088, and 11765012,  Natural Science Foundation of Shanghai under Grants\nNo.~15DZ2272100, and by Key Laboratory for Particle\nPhysics, Astrophysics and Cosmology, Ministry of Education.\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{INTRODUCTION}\n\\label{intro_section}\n\n\nThe performance of consensus-based, \nmulti-agent networks, such as the response to external stimuli, depends on \nrapidly transitioning from one operating point (consensus value) to another, e.g.,  in  flocking of natural systems,~\\cite{Huth_92,Vicsek_95}, as well as engineered systems such as autonomous vehicles, swarms of robots, e.g.,~\\cite{Jadbabaie_03,Ren_Beard_05,Olfati_Saber_06} \nand other  networked systems such as aerospace control~\\cite{Mark_aero_2019} microgrids~\\cite{Cucuzzella_19_Microgrids,Schiffer_16_microgrids}, flexible structures~\\cite{Naiming_19_flex_structures}.\nRapid cohesive transitions, e.g., in the orientation of the agents from one consensus value to another, is seen in flocking behaviour during predator attacks and migration ~\\cite{Ioannou_12, nagy2010hierarchical}. \nThus, there is interest to increase the convergence rate\nto consensus for such networked multi-agent systems. \n\n\n\\vspace{0.1in}\nThere  is a fundamental limit to the achievable rate of convergence  using existing neighbor-based update laws\nfor a given network, e.g., of the form \n\\begin{align}\n\\hat{X}[k+1] &   \n=\n\\hat{X}[k] + u[k]\n=  \\hat{X}[k] -  \\alpha  L \\hat{X}[k],\n\\label{system_eq}\n\\end{align}\nwhere the \ncurrent state is $\\hat{X}[k]$, the updated state is $\\hat{X}[k+1]$,  $\\alpha$ is the update gain, \n$L$ is the graph Laplacian, and $k$ represents the time instants $t_k = k \\delta_t$ with $\\delta_t$ as the sampling time-period. \nThe \nconvergence rate depends on the  eigenvalues of the matrix $P = \\left(\\textbf{I}- \\alpha L\\right)$ \\cite{fagnani2017introduction}, which in turn depends on the \neigenvalues of the graph Laplacian $L$. \nFor example, if the underlying graph  is undirected and connected, it is well known that convergence to consensus can be achieved provided the update gain $\\alpha$ is  \nsufficiently small, e.g.,~\\cite{Olfati_Murray_07}. \nThe update gain can be selected to maximize the convergence rate, and typically, \na larger gain $\\alpha$ tends to increase the convergence rate. \nNevertheless, for a given graph  (i.e., a given graph Laplacian $L$), \nthe range of the acceptable update gain $\\alpha$ \nis limited, which in turn, limits the achievable rate of convergence~\\cite{Devasia_2019_IJC}.\nTypically, the convergence rate tends to be slow if the number of agent inter-connections is small compared to the number of agents, e.g.,~\\cite{Carli_08}. \n{ Faster convergence can be achieved using \nrandomized time-varying connections,} as shown in,   e.g.,~\\cite{Carli_08}. The update sequence  of the agents can also be arranged to improve convergence, e.g.,~\\cite{Fanti_15}.\nThe problem is that the graph connectivity might be fixed and therefore the  Laplacian $L$ cannot be varied over time. In such cases, with a fixed Laplacian $L$, \nthe need to maintain stability limits the range of acceptable update gain $\\alpha$, and therefore, \nlimits the rate of convergence. \nThis convergence-rate limitation motivates ongoing efforts to develop  new approaches to improve the network performance, e.g.,~\\cite{fast_convergence_16_duan}. Furthermore, in addition to convergence-rate, an important consideration is robustness of the approach,  e.g., as studied in~\\cite{Zhenhong_robust_19, Montijano_robust_15}. \n\n\n\n\\vspace{0.1in}\nSince the  neighbor-based update ($u$ in Eq.~\\eqref{system_eq}) can be obtained from the gradient of the \nLaplacian potential \n$\\Phi_{{\\cal{G}}} = \\hat{X}^T L \\hat{X} $ for undirected graphs, i.e., \n$ u  = -(\\alpha \/ 2)\\nabla \\Phi_{{\\cal{G}}}$, \nNestertov-type accelerated approaches, used to speed up  gradient-based optimization~\\cite{Rumelhart_86,QIAN1999145,Nesterov_83,Jakovetic_Moura_2014,VanScoy_Lynch_18},  can be used to  improve the \nconvergence rate.\nPrevious works have considered the\nuse of some parts of the  accelerated gradients (from  optimization theory)  for graph-based multi-agent networks. \nFor example, the addition of a momentum term \n(of the form $ \\hat{X}[k] -  \\hat{X}[k-1],  $ as in, e.g., ~\\cite{Rumelhart_86}) in the update law  has been shown to improve  the response speed under update-bandwidth limits~\\cite{Devasia_2018_JDSMC,Devasia_2019_IJC}. These works have also shown that the use of such reinforcement can lead to  non-diffusive,  wave-like response propagation seen in natural systems such as bird flocks~\\cite{Attanasi_14}.\nSimilarly, the addition of a  Nesterov term without the momentum term, also referred to as an outdated-feedback  (of the form\n $ L(\\hat{X}[k] -  \\hat{X}[k-1]) $,   as in e.g.,~\\cite{Nesterov_83}), has been shown to result in  faster convergence in~\\cite{Cao_ren_2010,Moradian_19}, and to enable a linear rate of convergence using a time-varying gain  in~\\cite{Bu_2018}.   Time-varying gains, however,  require a global resetting\n of each agent's gain at start of each transition, which might not be always feasible because the \n start of a transition might not be known to all agents. \n  { \nThe combination of  both, the momentum term and the outdated-feedback term,  can further improve the  convergence rate of consensus-based networks when compared to the use of either term alone~\\cite{Devasia_ICPS_2019,Devasia_20,tiwari2019cohesive}. \n}\nNote that an advantage of  such  accelerated-gradient-based approach is that the update can be implemented by \nusing an accelerated delayed-self-reinforcement  (A-DSR), where each agent only uses \ncurrent and past information from the network. This use of already existing information is advantageous since the \nconvergence improvement is achieved without the need to change the network connectivity \nand without the need for additional information from the network.\nNevertheless, the update law for more general graphs with non-symmetric Laplacian (e.g., general directed graphs) \ncannot be obtained from the gradient of the \ngraph potential~\\cite{OlfatiSaberIEEETAC_04,Zhang_hui_IJC_2015}.  \nGradients along local agent-wise potential have been considered along with weight-balancing to improve the performance for directed graphs, \ne.g.,~\\cite{Makhdoumi_2015,Khan_Xin_2020}.\nHowever, these approaches rely on the graph being strongly connected, which excludes applications such as platoons where overall information flow between two agents is not bi-directional over the  graph. \nTherefore, the current  Nesterov-based approach and its stability analysis  cannot be directly applied for general  directed graphs (which are not strongly connected),  which are addressed in the current work.\n\n\n\n\n\n\\vspace{0.1in}\nThe main contribution of this article is to  design a \nNesterov-type  accelerated update for  general  graph networks \nusing a local potential function for each agent.\nHowever, since the resulting update law does not necessarily reduce the overall Laplacian potential~\\cite{Zhang_hui_IJC_2015}, the convergence studies from optimization methods cannot be used to establish stability~\\cite{Jakovetic_Moura_2014,VanScoy_Lynch_18}. \nMoreover, while Lyapunov functions can be found to study stability for general directed graphs~\\cite{Zhang_hui_IJC_2015},  the gradient of these Lyapunov functions does not lead to the control update law, and hence accelerated methods cannot be directly applied using these Lyapunov functions.  \nPrior methods that use either the momentum term alone or the outdated-feedback term alone also do not address the stability when both terms are used for general directed  graphs. In this context, a  contribution of this article is to  develop stability conditions for the proposed  generalized  accelerated  approach, with both the momentum and outdated-feedback terms.\nThe current article \nexpands on our  prior work in ~\\cite{Devasia_ICPS_2019}, which used a fixed  ratio between the momentum and outdated-feedback terms,\n{by (i)~proposing  the general  case with varying ratios between the momentum and outdated-feedback terms; (ii)~developing a stability condition for the generalized approach, (iii)~designing the A-DSR to achieve fast response while maximizing structural robustness, (iv) illustrating the importance of momentum term over the outdated-feedback term for graph networks with real spectrum and (v)~presenting experimental results to comparatively evaluate  the performance, with and without A-DSR.}\n\n\\vspace{0.1in}\nThe article begins by presenting the structurally-robust,  convergence-rate improvement problem, along with the limits of  standard consensus-based   update in Section~\\ref{problem_formulation}. \nThe proposed A-DSR based approach is introduced in Section~\\ref{Section_General_ADSR_approach}, and the stability conditions of the A-DSR approach are developed in Section~\\ref{Section_stability_proofs}, followed by the derivation of analytical Robust A-DSR approach for maximizing robustness in Section~\\ref{subsection_robust_convergence_with_ADSR}. \n Section~\\ref{simulation-section}  comparatively evaluates the performance with and without A-DSR through simulations, and Section~\\ref{experimental-section} presents experimental results. Lastly, conclusions from the article are reported in Section~\\ref{conclusion-section}.\n\n \n\n\n\\vspace{0.1in}\n\\section{Problem formulation}\n\\label{problem_formulation}\n\\vspace{-0.01in}\nThis section introduces graph-based consensus dynamics used to model networked systems, and describes the convergence limits with structural robustness achievable due to stability bounds on the update gain in standard neighbor-based consensus dynamics. Finally, the problem statement of the article is stated.\n\n\n\n\\vspace{0.1in}\n\\subsection{Background: graph-based control}\n\\label{Section_graph_based_control}\n\n\nLet the multi-agent network be modeled using a graph representation, where \nthe connectivity of the agents is represented by \na directed graph (digraph) \n${\\cal{G}} = \\left({\\cal{V}}, {\\cal{E}}\\right)$, e.g., as defined in~\\cite{Olfati_Murray_07}. \nHere, the  agents are represented by nodes $ {\\cal{V}}= \\left\\{ 1, 2, \\ldots, {n\\!+\\!1} \\right\\}$, $n>1$ and their connectivity by edges $ {\\cal{E}}   \\subseteq {\\cal{V}} \\times {\\cal{V}} $, where each agent $j$ belonging to the set of \nneighbors $N_i  \\subseteq {\\cal{V}} $  of the agent $i$ satisfies  $ j \\ne i$ and $(j,i) \\in {\\cal{E}} $. \n\n\n\n\\vspace{0.1in}\nThe evolution of the  multi-agent network is defined using the graph ${\\cal{G}}$, as in Eq.~\\eqref{system_eq}.  \nThe elements\n $l_{i,j}$ of the $(n+1)\\times(n+1)$ Laplacian $L$  of the graph ${\\cal{G}}$ are real and given by \n\\begin{eqnarray}\n\\label{eq_Laplacian_defn}\nl_{i,j} & =   \\left\\{ \n\\begin{array}{ll}\n-a_{i,j} < 0, \t& {\\mbox{if}} ~ j \\in N_i \\\\\n\\sum_{m=1}^{n+1} a_{i,m}, & {\\mbox{if}} ~ j = i,  \\\\\n0 & {\\mbox{otherwise,}}\n\\end{array}  \n\\right.\n\\end{eqnarray}\nwhere \nthe weight $a_{i,j}$ is nonzero (and positive) \nif and only if $j$ is in the set of neighbors $N_i  \\subseteq {\\cal{V}} $  of the agent $i$, \neach row of the Laplacian $L$ adds to zero, i.e., from Eq.~\\eqref{eq_Laplacian_defn}, \nthe $(n+1) \\times 1$ vector of ones  ${\\textbf{1}}_{n+1}= [1, \\ldots, 1]^T$  is a right eigenvector of the Laplacian $L$ with eigenvalue $0$,   \n\\begin{eqnarray}\n\\label{eq_first_lambda_eigenvector}\nL{\\textbf{1}}_{n+1} & = 0  {\\textbf{1}}_{n+1} .\n\\end{eqnarray}\n\n\n\n\n\n\\subsubsection{Network dynamics}\nOne of the agents is assumed to be a virtual source agent \\cite{leonard2001virtual}, which can be \nused to specify a desired consensus value $X_s$.  Without loss of \ngenerality, the state $\\hat{X}_{n+1}$ of last $n+1$ node is assumed to be a virtual source agent $X_s$, where $s=n+1$. Moreover, each \nagent in the network should have access to the virtual source agent ${X}_s$  through the network, as formalized below. \nNote that this is a less stringent requirement than the graph without the virtual source being strongly connected. \n\n\\vspace{0.1in}\n\\begin{Assum}[Rooted graph]\n\\label{assum_digraph_properties}\nThe digraph ${\\cal{G}}$ is assumed  to have a directed path from the source node ${n+1}$ to any other  node $i$ in the graph, i.e.,  $ i \\in {\\cal{V}} \\setminus \\!{(n+1)}$. \n\\hfill\\ensuremath{\\square} \n\\end{Assum}\n\n\n\n\n\n\n\\vspace{0.1in}\nSome properties of the  graph  ${\\cal{G}}$ without the source node $s=n+1$, i.e., ${\\cal{G}}\\!\\setminus\\!s$, \nare listed below.  \nIn particular, consider the  $n \\times n$ pinned Laplacian  matrix $K$ associated with ${\\cal{G}}\\!\\setminus\\!s$ \nobtained by removing the row and column associated with the source node $n+1$ through  the partitioning of the Laplacian $L$, i.e., \n\\begin{equation}\nL  = \n\\left[\n\\begin{array}{c}\n\\begin{array}{c|c}\nK  &  -B  \n\\end{array}\n\\\\[0.01in]\n\\hline \n\\vspace{0.001in}\n L_b\\\\\n\n\\end{array}\n\\right]\n\\label{eq_def_pinned_K}\n\\end{equation}\nwhere $L_b$\nis the $1 \\times (n+1)$ size row vector of Laplacian $L$ corresponding the source node $s=n+1$ and\n$B$ is an  $n \\times 1$ vector \n\\begin{equation}\n\\label{eq_B_def}\n\\begin{array}{rl}\nB & = [ a_{1,s}, a_{2,s}, \\ldots, a_{n,s}]^T\n \\\\ &  \n ~= [ B_{1}, B_{2}, \\ldots, B_{n}]^T , \n\\end{array}\n\\end{equation}\nand non-zero value of $B_j$ implies that the agent $j$ is \ndirectly connected to the source $X_s$. The properties of the pinned Laplacian $K$ follow from Assumption~\\ref{assum_digraph_properties}, e.g., see~\\cite{Olfati_Murray_07}.\n\\begin{enumerate}\n\\item \nThe  pinned Laplacian matrix $K$  is invertible, i.e., \n \\begin{equation}\n\\det{(K)} \\ne 0.\n\\label{eq_K_eigenvector}\n \\end{equation}\n \\item \n The eigenvalues of the pinned Laplacian $K$ have strictly-positive, real parts.\n \\item \n The product of the inverse of the pinned Laplacian $K$ with $B$ leads to an  $n \\times 1$ vector of  ones, ${\\textbf{1}}_n$, i.e., \n\\begin{eqnarray}\n\\label{eq_KinvtimesB}\nK ^{-1} B & = {\\textbf{1}}_n.\n\\end{eqnarray}\n\\end{enumerate}\n\n \n\n\n\\noindent \nThe dynamics of the $n$ non-source agents with state vector, \n$X$ represented by the remaining graph ${\\cal{G}}\\!\\setminus\\!s$, can be  given by \n\\begin{equation}\n\\begin{array}{rl}\nX[k+1] &  = X[k]  -\\alpha  K X[k] + \\alpha   B  X_s[k]\n \\\\\n&  = \\left(   {\\bf{I}_n} - \\alpha  K     \\right) X[k]   + \\alpha   B  X_s [k] \\\\\n& = P  X[k] + \\alpha   B  X_s[k]  .\n\\label{system_non_source}\n\\end{array}\n\\end{equation}\nwhere the matrix $P   ~ =  \\textbf{I}_{n}-\\alpha  K$, $ {\\textbf{I}}_{n}$ is the $n\\times n$ identity matrix, \nand $\\alpha$ is the  update gain.\n\n\n\\vspace{0.1in}\n\\subsubsection{Stability conditions}\n\\label{Section_noDSR_stability}\nBounds can be established on the \n update gain $\\alpha$ to ensure stability. \nFor  any  eigenvalue  $\\lambda_{K,m} = a_m + j b_m$ of graph Laplacian $K$, with real part $a_m>0$ (from Assumption~\\ref{assum_digraph_properties}) and imaginary part $b_m$, the corresponding eigenvalue of the matrix $P$ is\ngiven by \n\\begin{equation}\n    \\lambda_{P,m} = 1-\\alpha(a_m + j b_m). \n\\end{equation}\nFor stability of the non-source dynamics in Eq.~\\eqref{system_non_source}, the magnitude of $\\lambda_{P,m}$ needs to be less than one, i.e., \n\\begin{equation}\n    \\left| 1-\\alpha (a_m + j b_m)\\right| <1, \\forall\\; m \\in \\{1,2, \\hdots, n\\}.\n\\end{equation}\nThis condition for stability is met if the update gain $\\alpha$ satisfies~\\cite{Devasia_2019_IJC}\n\\begin{equation}\n    0 < \\alpha < \\min_{1 \\leq m \\leq n} \\frac{2 a_m}{a_m^2 + b_m^2} = \\overline{\\alpha}.\n     \\label{eq_stability_condition_lem_Stability_and_Update_gain}\n\\end{equation}\n\n\n\n\n\n\\vspace{0.1in}\n\\subsubsection{Convergence to consensus}\n\\label{sec_Stable_consensus}\nWith a stabilizing  update gain $\\alpha$ as in Eq.~\\eqref{eq_stability_condition_lem_Stability_and_Update_gain}, the state $X$ of the network (of all non-source agents) converges to a fixed source value $X_s$, e.g., \nfor a step change in the source value $X_s$ from $x_i$ to $x_f$, i.e., $X_s[k] = x_i$, $\\forall\\; k<0$ (initial desired state) and $X_s[k] = x_f$, $\\forall\\; k \\ge 0$.\nSince the eigenvalues $\\lambda_{P,m}$  of the matrix\n$P$ are inside the unit circle, the solution to Eq.~\\eqref{system_non_source} for the step input \nconverges,\n\\begin{equation}\n    \\begin{array}{rcl}\n  X[k+1] - X[k]  &  = & \n  P \\left( X[k] - X[k-1]  \\right) \\\\\n  & = & P^k \\left( X[1] - X[0]  \\right)\n\\rightarrow 0 ,\n\\end{array}\n\\label{Eq_controlled_gen_soln}\n\\end{equation}\nas $ k \\rightarrow \\infty$ because $ P^{k} \\rightarrow 0$. \nThus, $\\lim_{k\\rightarrow\\infty}   X[k+1] =\\lim_{k\\rightarrow\\infty}   X[k] $, and from the first line of Eq.~\\eqref{system_non_source}, \n\\begin{equation}\n  \\lim_{k\\rightarrow\\infty} K X[k] =   B  x_f  .\n\\end{equation}  \nAs a result, from the invertibility of $K$ in Eq.~\\eqref{eq_K_eigenvector}, \nand $K ^{-1} B  = {\\textbf{1}}_n$ from Eq.~\\eqref{eq_KinvtimesB}, the limit for the state $X[k]$ is found to be \n\\begin{eqnarray}\nX[k]  \\rightarrow K^{-1} B  x_f \n= \n\\rightarrow {\\textbf{1}}_{n} x_f ~~\\mbox{as}\\quad k \\rightarrow \\infty.\n\\label{system_non_source_stability}\n\\end{eqnarray}\n\n\\noindent Thus, the control law in Eq.~\\eqref{system_non_source} achieves consensus. \n\n\n\\vspace{0.1in}\n\\subsubsection{Spectral radius and rate of convergence}\nThe rate of convergence to consensus depends on the spectral radius $\\sigma(P)$ of the matrix $P$ given by \n\\begin{equation}\n\\sigma(P) =  \\max_m | \\lambda_{P,m} |  = \\max_m | 1- \\alpha \\lambda_{K,m} | {<1}.\n\\label{Eq_matrix_norm_ineq_0}\n\\end{equation}\nNote that for any $\\epsilon > 0$, say \n\\begin{equation}\n    \\epsilon = \\frac{1-\\sigma(P)}{2} > 0\n\\end{equation}\nthere exists a nonsingular matrix $Q$ such that the \nmodified vector norm $\\| X \\| = \\|Q X \\|_\\infty $  with the corresponding induced matrix norm  $\\| \\cdot \\|$ satisfies, see~\\cite{Ortega_87} (Section 5.3.5), \n\\begin{equation}\n\\| P \\| ~  \\le  ~\\sigma(P) + \\epsilon \n~ = \\frac{1 + \\sigma(P) }{2} ~ < 1.\n\\label{Eq_matrix_norm_ineq}\n\\end{equation}\nHence, from Eq.~\\eqref{Eq_controlled_gen_soln}, \n\\begin{equation}\n\\begin{array}{rcl}\n  \\| X[k+1] - X[k] \\|   &  \\le  &  \\| P \\|^{k+1} \\|   \\textbf{1}_n (x_f- x_i)  \\|   \\\\ \n  &  \\le  & \\left[ \\sigma(P)+\\epsilon \\right]^{k+1} \n  \\|   \\textbf{1}_n ( x_f-   x_i )   \\|.  \n\\end{array}\n\\label{Eq_rate_convergence_rate}\n\\end{equation}\nSince $\\epsilon$ can be chosen to be arbitrarily small, minimizing the spectral radius $\\sigma(P)$ of the matrix $P$ \nresults in faster convergence.\n\n\n\n\\vspace{0.1in}\n\\subsection{ {Convergence with structural robustness}} \n\\label{Limit_of_convergence_rate}\nThe structural  robustness of the network's stability depends on the spectral radius $\\sigma(P)$ of the matrix $P$ \\cite{fagnani2017introduction}. \nFor the network to be stable, the eigenvalues of the matrix $P$ need to be inside the unit circle. Hence,  \nthe smallest distance $d$ of its eigenvalues\n$\\lambda_{P,m} $ from the unit circle is a measure of the network's structural stability, i.e., robustness to perturbations, where \n\\begin{equation}\nd  =   1 - \\sigma(P). \n\\label{eq_optimal_no_dsr_robustness}\n\\end{equation}\nMinimizing the spectral radius $\\sigma(P)$ results in increased structural robustness.  \n{ Therefore, rapid structurally-robust convergence} is achieved during transitions if the spectral radius $\\sigma(P)$ is minimized. \n\\noindent The optimal update gain $\\alpha^*$ for minimum spectral radius $\\sigma^*$, with the standard consensus dynamics (in Eq.~\\eqref{system_non_source}) referred to as no-DSR approach hereon, can be found through a search based method, as\n\n\\begin{equation}\n    \\sigma^* = \\min_{\\alpha} \\sigma(P) = \\min_{0<\\alpha <\\overline{\\alpha}} \\left[   \\max_m | 1- \\alpha \\lambda_{K,m} |\\right].\n\\label{Eq_optimal_sigma_cmplx}\n\\end{equation}\n\n\n\n\\begin{Rem}{[Optimal no-DSR for real spectrum]}\n\\label{alpha_for_min_spectralradius_lemma}\nFor the special case when the graph has real spectrum, i.e., \neigenvalues of the pinned Laplacian $K$ \nare real and satisfy\n     \\begin{equation} \n 0< \\underline\\lambda = \\lambda_{K,1}  \\le \\lambda_{K,2}  \\le \\hdots \\le  \\lambda_{K,n} = \\overline\\lambda = \\sigma(K) , \n \\label{eq_ordering_eigenvalues}\n \\end{equation}\n  the stability condition in Eq.~\\eqref{eq_stability_condition_lem_Stability_and_Update_gain}, becomes \n\\begin{equation}\n    0<\\alpha<  \n     \\frac{2}{ \\left(     \\overline \\lambda \\right)\n      } = \\overline{\\alpha} .\n    \\label{Eq_noDSR_stablitity_limit_real}\n\\end{equation}\n If the extremal eigenvalues are distinct, i.e., \n $\\underline\\lambda \\ne \\overline\\lambda$, then \n the update gain $\\alpha^*$ that minimizes the spectral radius $ \\sigma(P) $ is given by~\\cite{xiao2004fast}\n\\begin{equation}\n    \\alpha^* = \\frac{2}{\\overline \\lambda + \\underline \\lambda} <  \n     \\frac{2}{  \\left(     \\overline \\lambda \\right) }  , \n    \\label{update_gain_for_max_robust}\n\\end{equation}\nand \nthe associated minimum spectral radius  is \n\\begin{equation}\n  \\sigma^* =   \\sigma(P^*) = \\frac{\\overline\\lambda-\\underline\\lambda}{\\overline\\lambda+\\underline\\lambda}.\n    \\label{minimum_spectral_radius}\n\\end{equation}\nIf the extremal eigenvalues are the same, $\\underline\\lambda = \\overline\\lambda$ (e.g., in first-order platoon networks), then \nthe spectral radius of the matrix ($\\sigma(P)$) can be made the ideal value of zero, $ \\sigma^*  =0$,  resulting in maximally fast convergence. \n \\end{Rem}\n\n\n\\subsection{The robust convergence optimization problem}\n\\vspace{-0.01in}\nThe range of acceptable update gain $\\alpha$ in Eq.~\\eqref{eq_stability_condition_lem_Stability_and_Update_gain}, limits the convergence rate. \nThe research problem addressed is to  further reduce the spectral radius of the matrix $P$,  i.e. to improve \nthe structural robustness and convergence rate, \nwhen each agent can modify  its update law \n\\begin{enumerate}\n\\item \nusing only existing information from the network neighbors, { and} \n\\item \nwithout changing the network structure (network connectivity $K$).\n\\end{enumerate}\n\n\n\n\n\n\n\n\\vspace{0.1in}\n\\section{Proposed Solution}\n\\label{proposed_approach}\nThis section introduces the proposed  Accelerated Delayed Self Reinforcement (A-DSR) approach to \nachieve {\nstructurally-robust convergence} and establishes stability conditions. \n\n\n\\subsection{The A-DSR approach} \n\\label{Section_General_ADSR_approach}\n\\vspace{0.1in}\n\\subsubsection{Graph's Laplacian potential}\n\\label{Sec_Graphs_Laplacian_potential}\nFor undirected graphs, the control law $u$ in  Eq.~\\eqref{system_eq} can be considered as a  gradient-based search on the \ngraph's Laplacian potential $\\Phi_{{\\cal{G}}}$\n\\cite{OlfatiSaberIEEETAC_04,Hongwei_lewis_12}\n\\begin{align}\n\\Phi_{{\\cal{G}}}(\\hat{X}) & =   \\frac{1}{2} \\sum_{i,j =1}^{n} a_{i,j}\\left( \\hat{X}_j - \\hat{X}_i \\right)^2\n = \\hat{X}^T L \\hat{X}, \n\\label{system_eq_potential}\n\\end{align}\n\\noindent which results in the standard graph-based update law as in   Eq.~\\eqref{system_eq}, \n\\begin{align}\n\\hat{X}[k+1] & =   \\hat{X}[k]   -   {\\frac{\\alpha}{2}} \\nabla \\Phi_{{\\cal{G}}}(\\hat{X}[k]) \\nonumber \\\\\n& =   \\hat{X}[k] -  \\alpha  L \\hat{X}[k] .\n\\label{system_gradient}\n\\end{align}\n\n\n\n\n\n\\vspace{0.1in}\n\\subsubsection{Nesterov's accelerated-gradient-based update}\nIn general, the convergence of the gradient-based approach as in Eq.~\\eqref{system_gradient} \n can be improved using accelerated methods. In particular, applying  \nthe  Nesterov modification~\\cite{Rumelhart_86,QIAN1999145} of the traditional gradient-based method to  \nEq.~\\eqref{system_gradient}  \nresults in the accelerated-gradient-based modification of the system in Eq.~\\eqref{system_eq}  to \n\\begin{align}\n\\hat{X}[k+1] & = \\hat{X}[k+1]\n+ \\beta \\left(\\hat{X}[k] -\\hat{X}[k-1]  \\right) \n\\nonumber \\\\ \n & \\qquad \n-{\\frac{\\alpha}{2}}\\nabla \\Phi_{{\\cal{G}}}\\left\\{\\hat{X}[k] +\\beta \\left(\\hat{X}[k] -\\hat{X}[k-1]  \\right)  \\right\\}  \\nonumber \\\\ \n& =  \\hat{X}[k] \n+  \\beta\\left( \\hat{X}[k]-\\hat{X}[k-1] \\right)\n\\nonumber \\\\\n& \\qquad\n-  \\hat{\\alpha}  L\\left\\{ \\hat{X}[k] + \\beta\\left(\\hat{X}[k]-\\hat{X}[k-1] \\right) \\right\\}    ,\n\\label{System_Nesterov_gradient_approach}\n\\end{align}\n\\noindent \nwhere $\\beta$ is a scalar gain on the Nesterov-based terms and \n\\begin{equation}\n    \\hat{\\alpha} = \\alpha(1+\\beta).\n    \\label{Eq_alpha_hat_nesterov}\n\\end{equation}\nConsequently,  the dynamics of the non-source agents \n$X$ represented by the remaining graph ${\\cal{G}}\\!\\setminus\\!s$, i.e., Eq.~\\eqref{system_non_source}, becomes\n\\begin{equation}\n\\begin{array}{rl}\nX[k+1] &  = X[k]   -\\hat{\\alpha}  K \\left\\{ X[k]  + \\beta\\left({X}[k]-{X}[k-1]  \\right)\n\\right\\} \\\\\n&    \\qquad + \\beta\\left({X}[k]-{X}[k-1] \\right) \\\\\n& \\qquad+ \\hat{\\alpha}  B \\{X_s[k] + \\beta\\left(X_s[k]-X_s[k-1]\\right) \\}\n\\end{array}\n\\label{accelerated_system_non_source}\n\\end{equation}\n\n \n\\noindent\nThe additional third term $ \\beta \\left({X}[k]-{X}[k-1] \\right)$  \non the right hand side of Eq.~\\eqref{accelerated_system_non_source} is referred to as the momentum term (this term alone forms the Heavy ball method in \\cite{ghadimi2015global}) and  the similar terms \n inside the curly   brackets of the second and fourth terms \nare referred to as the outdated-feedback addition.  \n\n\n\\vspace{0.1in}\n\\subsubsection{Directed graphs}\n\\noindent \nFor general directed graphs, the  potential function $\\Phi_{{\\cal{G}}}$\nin Eq.~\\eqref{system_eq_potential} \ndoes not lead to the standard update equations~\\cite{OlfatiSaberIEEETAC_04,Zhang_hui_IJC_2015}. \nNevertheless, motivated by the gradient-based approach,  for each non-source agent, $1\\le i \\le n$, a modified potential can be considered as\n\n\\begin{align}\n\\Phi_{{\\cal{G}},i} (\\hat{X}) &  =   \\sum_{j =1}^{n+1}  a_{i,j}\\left(\\hat{X}_i -\\hat{X}_j \\right)^2 . \n\\label{system_eq_potential_local}\n\\end{align}\n\n\\noindent \nHere \n$\\Phi_{{\\cal{G}},i}$ is  a localized version of the graph's  Laplacian potential~\\cite{OlfatiSaberIEEETAC_04,Zhang_hui_IJC_2015}, whose gradient with respect to $\\hat{X}_i = X_i$\n\n\\begin{equation}\n    u_i(\\hat{X}) = - \\frac{\\alpha}{2} \\frac{\\partial \\Phi_{{\\cal{G}},i}}{\\partial \\hat{X}_i} = -\\alpha K_i X + \\alpha B_i X_s\n    \\label{single_agent_input}\n\\end{equation}\n\n\\noindent\nwith $K_i$ as the $i^{th}$ row of $K$, $B_i$ the $i^{th}$ row of the source connectivity vector $B$, will lead to the standard update equations for each agent's state $X_i$ in the state vector $X$ of non-source agents, as\n\n\\begin{equation}\n    \\begin{aligned}\n    X_i[k+1] &= X_i[k] - \\alpha K_i X[k] +\\alpha B_i X_s[k].\n    \\end{aligned}\n    \\label{single_agent_gradient}\n\\end{equation}\n\n\\suppressfloats\n\\begin{figure}[!ht]\n\\begin{center}\n\\includegraphics[width=.90\\columnwidth]{images\/Fig_1_DSR_blockdiag_new_iee_tran_edited.eps}\n\\vspace{0.01in}\n\\caption{(Top) Implementation of standard consensus-based approach to multi-agent networks for the  $i^{th}$  agent as in Eq.~\\eqref{system_non_source}. \n(Bottom) Accelerated delayed self reinforcement (A-DSR) approach for the $i^{th}$ agent  in Eq.~\\eqref{accelerated_system_non_source_A_DSR}\nwithout using additional network information.\n}\n\\label{fig_1_control_implementation}\n\\end{center}\n\\end{figure}\n\n\n\n\n\\noindent \nThe application of the  accelerated-gradient approach\nfor local potential $\\Phi_{{\\cal{G}},i} (\\hat{X})$ in Eq.~\\eqref{system_eq_potential_local} (which does not necessarily decrease the graph potential ($\\Phi_{{\\cal{G}}}(\\hat{X})$) in Eq.~\\eqref{system_eq_potential},~\\cite{Rumelhart_86,QIAN1999145}) \nleads to the same Eq.~\\eqref{accelerated_system_non_source}. \n\n\n\\vspace{0.1in}\n\\subsubsection{\n{A-DSR update}}\n\\label{Proposed A-DSR}\nThe  Nesterov-update law in Eq.~\\eqref{accelerated_system_non_source} uses the\nsame  gain $\\beta$\nfor the momentum  and the outdated-feedback terms (Nesterov's accelerated method in \\cite{Nesterov:2014:ILC:2670022}). A generalization of this is to use different gains $\\beta_1, \\beta_2$ for the outdated-feedback and momentum terms (respectively), as used before in optimization theory  \\cite{lessard2016analysis},   \n\\begin{equation}\n\\begin{array}{rl}\nX[k+1] &  = X[k]   -\\hat{\\alpha}  K \\{ X[k]  + \\beta_1\\left({X}[k]-{X}[k-1] \\right) \\}   \\\\\n&    \\qquad + \\beta_2\\left({X}[k]-{X}[k-1] \\right)\\\\\n& \\qquad+ \\hat{\\alpha}   B  \\{X_s[k] + \\beta_1\\left(X_s[k]-X_s[k-1]\\right) \\}    .\n\\end{array}\n\\label{accelerated_system_non_source_A_DSR}\n\\end{equation}\n\\noindent \nwhere, from Eq.~\\eqref{Eq_alpha_hat_nesterov}, $\\hat{\\alpha}=\\alpha(1+\\beta_1)$. The above  accelerated approach, is referred to as the accelerated delayed self reinforcement (A-DSR) in the following, since it does not require  additional information from the network, or having to change the network connectivity. Rather, each agent uses delayed versions of known information to reinforce its own update. \nTo illustrate, for each non-source agent $i$, let $x_i$ be the information obtained from the network, i.e., \n\\begin{equation}\n\\begin{array}{rl}\nx_i[k]  &  =  \\hat{\\alpha} K_i  X[k] ,\n\\end{array}\n\\label{accelerated_system_non_source_2}\n\\end{equation}\nwhere $K_i$ is  the $i^{th}$ row of the pinned Laplacian $K$.\nThen, the update  of agent $X_i$ is,  from Eq.~\\eqref{accelerated_system_non_source_A_DSR}, \n\\begin{equation}\n\\begin{array}{rl}\nX_i[k+1] \n& = X_i[k] - \\{x_i[k]  +\\beta_1(x_i[k] -x_i[k-1]) \\} \\\\\n&  \\qquad + \\beta_2\\left({X}_i[k]-{X}_i[k-1] \\right) \\\\\n&\\qquad+ \\hat{\\alpha}   B_i  \\{X_s[k] + \\beta_1\\left(X_s[k]-X_s[k-1]\\right) \\} ,  \n\\end{array}\n\\label{accelerated_system_non_source_single}\n\\end{equation}\nwhere  $B_i$ is the $i^{th}$ row of the source connectivity vector \n$B$. \nThe delayed self-reinforcement (DSR) approach, however, requires each agent to store delayed versions $X_i[k-1]$ and $ x_i[k-1]$ of its current  state $X_i[k]$ and information $x_i[k]$ from the network, \nas illustrated in Fig.~\\ref{fig_1_control_implementation}.  ~ \\hfill \\hfill\\ensuremath{\\square}\n\n\n\n\\begin{Rem}\n\\label{different_update_methods}\nThe A-DSR method in Eq.~\\eqref{accelerated_system_non_source_A_DSR} without the momentum term ( i.e., $\\beta_2=0$) is referred to as the Outdated-feedback method,  \nwithout the outdated-feedback term ( i.e., $\\beta_1=0$) is referred to as the Momentum method, and with equal parameters \n( i.e., $\\beta_1= \\beta_2 = \\beta$) is referred to as the Nesterov-update method.  \n\\end{Rem}\n\n\n\\begin{Rem}[Stability for directed graphs] The local Laplacian potential in Eq.~\\eqref{system_eq_potential_local} whose gradient is used for deriving each agent's standard update (in Eq.~\\eqref{single_agent_gradient}) and A-DSR approach (in Eq.~\\eqref{accelerated_system_non_source_single}), doesn't reduce the overall Laplacian potential \\cite{Zhang_hui_IJC_2015} of the directed graph. Thus the convergence studies from optimization theory (which require the graphs to be strongly connected, \ne.g.,~\\cite{Makhdoumi_2015,Khan_Xin_2020}) cannot be used to establish stability for general directed graphs.\n\\end{Rem}\n\n\n\n\\subsection{\n{ Stability of A-DSR}}\n\\label{Section_stability_proofs}\nThe stability conditions  for the   general A-DSR approach in Eq.~\\eqref{accelerated_system_non_source_A_DSR}  are presented below.\n\n\\subsubsection{Diagonalizing the pinned Laplacian}\n\\label{section_diagonalization_stability}\nThe network with A-DSR in Eq.~\\eqref{accelerated_system_non_source_A_DSR}  can be  decomposed into subsystems using an \ninvertible transformation matrix $P_K$\nas \n\\begin{equation}\n    \\begin{aligned}\n    X[k] = P_K X_J[k], \n    \\end{aligned}\n    \\label{velocity_transform_J}\n\\end{equation}\nwhere the \ntransformation matrix $P_K$ is selected to diagonalize the  pinned Laplacian $K$ as\n\\begin{equation}\n    \\begin{aligned}\n    K_J = P_K^{-1} K P_K\n    \\end{aligned}\n    \\label{jordan_form_K}\n\\end{equation}\nwhere the diagonal terms of matrix $K_J$ are the eigenvalues $ \\lambda_{K,m}$ for $m= {1,2, ..., n}$, which can be complex and with multiplicity greater than 1. \nSince input doesn't affect stability, setting $X_s[k] = 0, \\; \\forall\\, k$, and pre-multiplying the Eq.~\\eqref{accelerated_system_non_source_A_DSR} with $P_k^{-1}$ results in  \n\n\\begin{equation}\n    \\begin{aligned}\n    &X_J[k+1] - X_J[k]  +\\hat{\\alpha} K_J(X_J[k]\\\\ &+ \\beta_1(X_J[k]-X_J[k-1]))\\\\ \n    & - \\beta_2 (X_J[k]-X_J[k-1]) = 0. \n    \\end{aligned}\n    \\label{general_nesterov_jordan_form}\n\\end{equation}\n\\noindent \nThe stability of network with A-DSR in Eq.~\\eqref{accelerated_system_non_source_A_DSR} is equivalent to the stability of Eq.~\\eqref{general_nesterov_jordan_form} in the transformed coordinate.\n\n\\vspace{0.1in}\n\\subsubsection{Characteristic equations}\nTaking the z-transform of Eq.~\\eqref{general_nesterov_jordan_form} results in \n \\begin{equation}\n     \\begin{aligned}\n     &(z^2 {\\textbf{I}}_{n} - z \\left[ (1+\\beta_2){\\textbf{I}}_{n} - \\hat{\\alpha} (1+\\beta_1) K_J \\right] \n     \\\\ &\n     - \\left(\\hat{\\alpha} \\beta_1 K_J  - \\beta_2 {\\textbf{I}}_{n} \\right) ) X_J(z) = 0. \n     \\end{aligned}\n     \\label{general_z_transform}\n \\end{equation}\nTherefore, the network with A-DSR update in Eq.~\\eqref{accelerated_system_non_source_A_DSR} is stable if and only if, for each eigenvalue $\\lambda_{K,m}$ of the pinned Laplacian $K$,  the roots of the following characteristic equation  \n \\begin{equation}\n     \\begin{aligned}\n     D(z) & = z^2\n     +z \\left[  \\hat{\\alpha} (1+\\beta_1){ \\lambda_{K,m}   -(1+\\beta_2)   }  \\right]  \\\\\n     & \\quad \\quad  + (\\beta_2 -\\hat{\\alpha}\\beta_1 { \\lambda_{K,m}     } )\n     \\quad = 0 \n     \\end{aligned}\n     \\label{general_char_eq}\n \\end{equation} \nhave magnitude less than one. \nFor the case of complex eigenvalue $\\lambda_{K,m} = a_m + j b_m$, the \nreal Jordan form of the z-transform of the diagonalized general A-DSR update equation in Eq.~\\eqref{general_nesterov_jordan_form}, for the block associated with the Laplacian eigenvalue  pair $a_m\\pm jb_m$ is\n\\begin{equation}\n    \\begin{aligned}\n     z^2 {\\textbf{I}}_{2} -&z \\left((1+\\beta_2){\\textbf{I}}_{2}\n     -\\hat{\\alpha}(1+\\beta_1)\\begin{bmatrix}\na_m & b_m \\\\\n-b_m & a_m\n\\end{bmatrix}\\right) \\\\\n-& \\left( \\hat{\\alpha} \\beta_1\\begin{bmatrix}\na_m & b_m \\\\\n-b_m & a_m\n\\end{bmatrix} - \\beta_2 {\\textbf{I}}_{2} \\right), \n    \\end{aligned}\n    \\label{complex_char_matrix_eq}\n\\end{equation}\nwhere ${\\textbf{I}}_{2}$ denotes an identity matrix of size $2 \\times 2$, $a_m$ and $b_m$ are the real and imaginary parts of $\\lambda_{K,m}$. The determinant of Eq.~\\eqref{complex_char_matrix_eq} yields a fourth order equation of the form \n\n\\begin{equation}\n    D(z) =  z^4 + a_3 z^3 + a_2 z^2 + a_1 z + a_0 = 0, \n    \\label{complex_char_eq_complex}\n\\end{equation}\n\n\\noindent where \n\\begin{equation}\n    \\begin{aligned}\n    a_0 &= (a_m \\hat{\\alpha} \\beta_1 - \\beta_2)^2 + \\hat{\\alpha}^2 b_m^2 \\beta_1^2, \\\\\n    a_1 &= -2\\{ (a_m \\hat{\\alpha} \\beta_1 - \\beta_2)^2 + a_m \\hat{\\alpha} \\beta_1(a_m \\hat{\\alpha} - 1) \\\\ &~+ \\hat{\\alpha}  (\\hat{\\alpha} b_m^2 \\beta_1 - {a_m}\\beta_2) + \\hat{\\alpha}^2 b_m^2\\beta_1^2 + \\beta_2  \\}, \\\\\n    a_2 &= (a_m \\hat{\\alpha} \\beta_1 - \\beta_2)^2 + (a_m \\hat{\\alpha} - 1)^2 + 2 \\hat{\\alpha}^2 \\beta_1(a_m^2+b_m^2), \\\\\n    &~ + 4 \\beta_2 -2 a_m \\hat{\\alpha} (2\\beta_1+\\beta_2) + \\hat{\\alpha}^2 b_m^2 (\\beta_1^2+1), \\\\\n    a_3 &= 2 a_m \\hat{\\alpha} (\\beta_1+1) - 2 (\\beta_2+1) .\n    \\end{aligned}\n    \\label{lemma2_a0toa4_coeff_Def}\n\\end{equation}\nHence, for the complex eigenvalue case, the stability of the  A-DSR (in Eq.~\\eqref{accelerated_system_non_source_A_DSR}) can be determined by obtaining conditions for the roots of Eq.~\\eqref{complex_char_eq_complex} to be within unit circle on the complex plane.\n\n\\vspace{0.1in}\n\\subsubsection{Stability conditions}\nStability conditions follow from the Jury test.\n\\vspace{0.1in}\n\\begin{Lem}{[Jury test based stability]}\n\\label{general_ADSR_lemma_complex}\nThe generalized A-DSR in Eq.~\\eqref{accelerated_system_non_source_A_DSR} is stable if and only if the A-DSR gains $\\hat{\\alpha},\\; \\beta_1$ and $\\beta_2$ satisfy the following conditions, for  each eigenvalue $\\lambda_{K,m}$ of the pinned Laplacian $K$.\n\n\\begin{enumerate} \n\n\\item \nIf the eigenvalue $\\lambda_{K,m} = a_m $ is real valued, then \n\\begin{equation}\n    \\begin{aligned}\n     (i)~ &\n    0  < \\hat{\\alpha}  \\\\ \n      (ii)~ &\n    \\left[ \\hat{\\alpha} { a_m     }(\\beta_1 +\\frac{1}{2})-1 \\right] < \\beta_2 < \\left( \\hat{\\alpha} \\beta_1 \n    { a_m\n    }+1 \\right)  \n    \\end{aligned}\n    \\label{general_stability_cond}\n\\end{equation}\n\n\\item \nIf the eigenvalue $\\lambda_{K,m} = a_m + j b_m$ is complex valued (i.e., $b_m \\ne 0$), then \n\\begin{equation}\n    \\begin{aligned}\n    (i)~ & 0 < \\hat{\\alpha}^2 ,   \\\\ \n   (ii)~ &  0 < (2(\\beta_2+1)-\\hat{\\alpha}(2\\beta_1+1)a_m)^2  \\\\\n   & \\qquad \\qquad + \\hat{\\alpha}^2(2\\beta_1+1)^2b_m^2 , \\\\\n    (iii)~ & -1 <  (a_m \\hat{\\alpha} \\beta_1 - \\beta_2)^2 + \\hat{\\alpha}^2 b_m^2 \\beta_1^2 < 1, \\\\ \n    (iv)~ & |a_0a_3-a_1|  < |a_0^2-1|,  \\\\ \n    (v)~ &  |a_2  (a_0^2-1)(a_0-1)  -(a_0a_1-a_3)(a_0a_3-a_1)| \\\\  \n    & ~ \n    < |(a_0^2-1)^2-(a_0a_3-a_1)^2|.  \n    \\end{aligned}\n    \\label{stability_ADSR_complex}\n\\end{equation}\n\n\n\\end{enumerate}\n\\end{Lem}\n\n\n\n\n\n\n\n\n\n\n\\vspace{0.1in}\n\n\n\n{\\bf{Proof}~}\nIf the eigenvalue $\\lambda_{K,m}$ is real valued, \nthen, the Jury test leads to the following three  necessary and sufficient  conditions for\nthe roots of the characteristic equation\nin Eq.~\\eqref{general_char_eq} \nto have magnitude less than one.\n\n\\begin{enumerate}\n    \\item $D(z=1) > 0$\n    \\begin{equation}\n    \\begin{aligned}\n        &1 + \n        \\left[ \\hat{\\alpha}(1+\\beta_1){\\lambda_{K,m}     } -(1+\\beta_2) \\right] +(\\beta_2-\\hat{\\alpha} \\beta_1 { \\lambda_{K,m}     }) >0 \\\\\n        &=> \\hat{\\alpha} > 0, \n    \\end{aligned}\n    \\label{eq_alpha_positive_condition}\n    \\end{equation}\n    which is satisfied due to the first condition in Eq.~\\eqref{general_stability_cond}. \n     \\item $(-1)^2D(z=-1) > 0$\n    \\begin{equation}\n    \\begin{aligned}\n       &1 - \\left[ \\hat{\\alpha}(1+\\beta_1){ \\lambda_{K,m}     } -(1+\\beta_2) \\right] \n        +(\\beta_2-\\hat{\\alpha} \\beta_1 { \\lambda_{K,m}     }) >0  \\\\\n        &=> \\beta_2 > \n        \\hat{\\alpha} { \\lambda_{K,m}     } \\left(\\beta_1 +\\frac{1}{2}\\right)-1 \n    \\end{aligned}\n    \\end{equation}\n    \n    or \n        \\begin{equation}\n     (\\hat{\\alpha} \\beta_1 { \\lambda_{K,m}     }-1) + \\frac{\\hat{\\alpha}  { \\lambda_{K,m}     }}{2} < \\beta_2.\n     \\label{Lemma_1_Proof_cond_2}\n       \\end{equation}\n    \n    \\item \n     $ | D(z=0)|  < 1$ \n    $$|\\beta_2 -\\hat{\\alpha}\\beta_1 { \\lambda_{K,m}     }|<1$$ or \n     \\begin{equation}\n    \\begin{aligned}\n        (\\hat{\\alpha} \\beta_1 { \\lambda_{K,m}     }-1) < \\beta_2 < (1+\\hat{\\alpha} \\beta_1 { \\lambda_{K,m}     }).\n    \\end{aligned}\n         \\label{Lemma_1_Proof_cond_3}\n    \\end{equation}\n\\end{enumerate}\n\n\\noindent \nAs $\\hat{\\alpha}  { \\lambda_{K,m}     } > 0 $ (since $\\hat{\\alpha}>0$ from Eq.~\\eqref{eq_alpha_positive_condition} and $ { \\lambda_{K,m}     }>0$ from Assumption~\\ref{assum_digraph_properties}), \nthe condition in Eq.~\\eqref{Lemma_1_Proof_cond_2}  is more stringent than the lower bound on $\\beta_2$ in Eq.~\\eqref{Lemma_1_Proof_cond_3}, resulting in condition (ii) of Eq.~\\eqref{general_stability_cond}.\n\n\\vspace{0.1in}\n If the eigenvalue $\\lambda_{K,m}=a_m + jb_m$ is complex valued ($b_m \\neq 0$), \nthen, the Jury test leads to the following necessary and sufficient conditions for\nstable roots of the characteristic equation\nin Eq.~\\eqref{complex_char_eq_complex}.\n\n\n\\begin{enumerate}\n    \\item $D(z=1) > 0$\n    \\begin{equation}\n    \\begin{aligned}\n        &1 + a_3 + a_2 + a_1 + a_0 > 0, \\\\\n        &=> \\hat{\\alpha}^2 (a_m^2 + b_m^2 ) > 0,\n    \\end{aligned}\n    \\end{equation}\n    \\noindent which can be simplified further (since $a_m>0$ from Assumption~\\ref{assum_digraph_properties}) as condition (i) in Eq.~\\eqref{stability_ADSR_complex}.\n    %\n    \n    \\item $(-1)^4 D(z=-1)>0$  resulting in condition~(ii) of Eq.~\\eqref{stability_ADSR_complex} since \n      \\begin{equation}\n         \\begin{aligned}\n         & 0 < 1 - a_3 + a_2 - a_1 + a_0  \\\\\n         &= (2(\\beta_2+1)-\\hat{\\alpha}(2\\beta_1+1)a_m)^2  \\\\\n         & \\qquad \\qquad + \\hat{\\alpha}^2(2\\beta_1+1)^2b_m^2 .\n        \\end{aligned}\n    \\end{equation}\n\n    \n    \\item $| D(z=0) | < 1$\n        \\begin{equation}\n            \\begin{aligned}\n             &| (a_m \\hat{\\alpha} \\beta_1 - \\beta_2)^2 + \\hat{\\alpha}^2 b_m^2 \\beta_1^2 |<1.\\\\ \n             &=> -1 <  (a_m \\hat{\\alpha} \\beta_1 - \\beta_2)^2 + \\hat{\\alpha}^2 b_m^2 \\beta_1^2 < 1. \n            \\end{aligned}\n        \\end{equation}    \n \\end{enumerate}\n   In addition to the above three conditions (which are similar to the real eigenvalue case), the complex case has two additional stability conditions (iv) and (v) in Eq.~\\eqref{stability_ADSR_complex} from the Jury test. \n\\hfill\\ensuremath{\\square}\n\n\n\n\n\n\n\n\\vspace{0.1in}\n\\subsubsection{Robust stability with general A-DSR}\nIndependently varying the gains of momentum and outdated-feedback terms gives additional flexibility, which can be used to further improve the robust  convergence when compared to the case without A-DSR. More formally, the general A-DSR approach can be used to  minimize the maximum magnitude $\\bar{z}_m$ of the roots $(z_{\\lambda_{K,m},1}, z_{\\lambda_{K,m},2})$ of the characteristic equation $D(z)=0$ in Eq.~\\eqref{general_char_eq} associated with the eigenvalues  $\\lambda_{K,m}$ of the pinned Laplacian $K$, i.e., \n\\begin{equation}\n    \\sigma^* = \\min_{{\\hat{\\alpha}}, \\beta_1, \\beta_2}  \\left[   \\max_m ( \\bar{z}_m  )  |\\right], \n\\label{Eq_optimal_sigma_cmplx}\n\\end{equation}\nwhere \n $\\bar{z}_m = \\max (|z_{\\lambda_{K,m},1}|, |z_{\\lambda_{K,m},2}|)$. \n \n \n\n\\vspace{0.1in}\n\\subsection{Graphs with  real spectrum}\n\\label{subsection_robust_convergence_with_ADSR}\n\n In general, using different gains for the momentum  and outdated-feedback terms (i.e., different values of $\\beta_1, \\beta_2$) can yield better performance than using the same gains for each term. However, for graphs with real spectrum (which includes all undirected graphs), the momentum term is sufficient to yield fast convergence and balanced robustness, as shown below. \n\n\\vspace{0.1in}\n\n\\begin{Assum}[Real spectrum] In this section, the pinned Laplacian $K$ is assumed to have real eigenvalues, ordered as in Eq.~\\eqref{eq_ordering_eigenvalues}.\n\\end{Assum}\n\n\n\\vspace{0.1in}\n\\subsubsection{Stability given range of Laplacian eigenvalues}\nThe application of \nLemma~\\ref{general_ADSR_lemma_complex}  requires knowledge of all eigenvalues $ {\\lambda_{K,m}     }$  of the pinned Laplacian $K$. The following corollary provides sufficient conditions for stability in terms of the range $[\\underline\\lambda~ \\overline\\lambda]$ of the eigenvalues $ {\\lambda_{K,m}}$ from Eq.~\\eqref{eq_ordering_eigenvalues}. \nTo begin, the  stability condition for general A-DSR update in Eq.~\\eqref{general_stability_cond} is used to  deduce stability for the other (Nesterov-update, momentum and outdated-feedback defined in Remark~\\ref{different_update_methods}) methods for graphs with real spectrum. \n\n\\begin{Cor}\n\\label{corrolary_Acc_methods}\nThe network update as in Eq.~\\eqref{accelerated_system_non_source_A_DSR}, for the following accelerated methods,\nis stable if and only if $\\hat{\\alpha} > 0$, and the gains  satisfy the following {for each eigenvalue  $\\lambda_{K,m}$ of the pinned Laplacian $K$. }\n\\begin{enumerate}\n    \\item Nesterov-update method in Eq.~\\eqref{accelerated_system_non_source}\\cite{Devasia_ICPS_2019} \n    with $\\beta_1= \\beta_2 = \\beta$:\n    \\begin{equation}\n        \\begin{aligned}\n       \\frac{\\hat{\\alpha}{ \\lambda_{K,m}     }}{2}-1 < \\beta (1 - \\hat{\\alpha} { \\lambda_{K,m}     }) < 1.\n        \\end{aligned}\n        \\label{nesterov_stability_1}\n    \\end{equation}\n    \n    \\item Momentum method ($\\beta_1 = 0$):\n    \\begin{equation}\n        \\begin{aligned}\n        \\frac{\\hat{\\alpha} {\\lambda_{K,m}     }}{2}-1 < \\beta_2 < 1.\n        \\end{aligned}\n        \\label{momentum_stability}\n    \\end{equation}\n    \n    \\item Outdated-feedback method ($\\beta_2 = 0$):\n    \\begin{equation}\n        \\begin{aligned}\n        -1 &<  \\hat{\\alpha} { \\lambda_{K,m}     }  \\beta_1 < 1 - \\frac{\\hat{\\alpha} { \\lambda_{K,m}     }}{2}.\n        \\end{aligned}\n        \\label{acc_stability}\n    \\end{equation}\n\\end{enumerate}\n\\end{Cor}\n\\vspace{0.1in}\n{\\bf{Proof}~}\nFor the Nesterov-update method  ($\\beta_1=\\beta_2=\\beta$), the stability condition in Eq.~\\eqref{general_stability_cond} becomes\n\\begin{equation}\n\\begin{aligned}\n     \\left[ \\hat{\\alpha} { \\lambda_{K,m}     }(\\beta +\\frac{1}{2})-1 \\right] &< \\beta < \\left( \\hat{\\alpha} \\beta \n    { \\lambda_{K,m}\n    }+1 \\right), \n\\end{aligned}\n\\label{eq_proof_cor_1_1}\n\\end{equation}\nand  subtracting $\\hat{\\alpha} \\beta\\lambda_{K,m}$ from both sides results in  Eq.~\\eqref{nesterov_stability_1}. \nFor the momentum method, \nEq.~\\eqref{general_stability_cond} becomes  Eq.~\\eqref{momentum_stability} with $\\beta_1=0$.\nFor the outdated-feedback method, with $\\beta_2=0$,  Eq.~\\eqref{general_stability_cond} becomes\n\\begin{equation}\n         \\left[\\hat{\\alpha} { \\lambda_{K,m}     }(\\beta_1 +\\frac{1}{2})-1 \\right] ~< 0 < \\left( \\hat{\\alpha} \\beta_1 \n    { \\lambda_{K,m}\n    }+1 \\right).\n    \\label{corr_1_proof_ac_method_1_1}\n\\end{equation}\nThe left inequality in Eq.~\\eqref{corr_1_proof_ac_method_1_1} can be simplified to\n\\begin{equation}\n    \\hat{\\alpha}\\lambda_{K,m} \\beta_1< 1 - \\frac{\\hat{\\alpha}\\lambda_{K,m}}{2}\n\\end{equation}\nand the right inequality becomes \n\\begin{equation}\n   \\hat{\\alpha} \\lambda_{K,m} \\beta_1 > -1, \n\\end{equation}\nresulting in the stability condition  in Eq.~\\eqref{acc_stability}. \n\\hfill\\ensuremath{\\square}\n\n\n\\vspace{0.1in} \n\\begin{Cor}\n\\label{corrolary_Acc_methods_range}\nThe network update as in Eq.~\\eqref{accelerated_system_non_source_A_DSR}, for the following accelerated methods,\nis stable if and only if $\\hat{\\alpha} > 0$, and the gains satisfy the following, where \n\\begin{equation}\n    \\begin{aligned}\n \\lambda_{*} & =\n \\left\\{ \n \\begin{array}{rcl} \n\\underline{\\lambda} \n& {\\mbox{if}} & \\beta_1 \\le  -\\frac{1}{2}  \\\\\n\\overline{\\lambda} \n& {\\mbox{if}} & \\beta_1 >   -\\frac{1}{2}\n \\end{array}\n \\right. \n \\\\\n %\n  \\lambda^{*} & =\n \\left\\{ \n \\begin{array}{rcl} \n\\overline{\\lambda} \n& {\\mbox{if}} & \\beta_1 \\le  0  \\\\\n\\underline{\\lambda} \n& {\\mbox{if}} & \\beta_1 >   0\n \\end{array}\n \\right.\n %\n    \\end{aligned}\n    \\label{general_stability_cond_eigVal_range_2}\n\\end{equation}\n\\begin{enumerate} \n\\item \nGeneralized A-DSR method: \n\\begin{equation}\n    \\begin{aligned}\n    \\left[ \\hat{\\alpha} { \\lambda_{*}     }(\\beta_1 +\\frac{1}{2})-1 \\right] &< \\beta_2 < \\left( \\hat{\\alpha} \\beta_1 \n    { \\lambda^{*}\n    }+1 \\right) .  \n    \\end{aligned}\n    \\label{general_stability_cond_eigVal_range_1}\n\\end{equation}\n\\item Nesterov-update method  ($\\beta_1=\\beta_2=\\beta$): \n\\begin{equation}\n\\begin{aligned}\n     \\left[ \\hat{\\alpha} { \\lambda_{*}     }(\\beta +\\frac{1}{2})-1 \\right] &< \\beta < \\left( \\hat{\\alpha} \\beta \n    { \\lambda^*\n    }+1 \\right).\n\\end{aligned}\n\\label{eq_proof_cor_3_1}\n\\end{equation}\n    \\item Momentum method ($\\beta_1 = 0$):\n     \\begin{equation}\n        \\begin{aligned}\n        \\frac{\\hat{\\alpha} \\lambda_*}{2}-1 &< \\beta_2 < 1.\n       \\end{aligned}\n        \\label{momentum_stability_extremum}\n    \\end{equation}\n    \\item Outdated-feedback method ($\\beta_2 = 0$):\n    \\begin{equation}\n         \\left[ \\hat{\\alpha} { \\lambda_*     }(\\beta_1 +\\frac{1}{2})-1 \\right] ~< 0 < \\left( \\hat{\\alpha} \\beta_1 \n    { \\lambda^*\n    }+1 \\right).\n    \\label{corr_1_proof_ac_method_3_1}\n\\end{equation}\n\n\\end{enumerate}\n\\end{Cor}\n\\vspace{0.1in}\n\n\n\\vspace{0.1in}\n{\\bf{Proof}~}\nThis follows from Lemma~\\ref{general_ADSR_lemma_complex} and the proof of  Corollary~\\ref{corrolary_Acc_methods} since \n$$\\hat{\\alpha} \\lambda_{K,m} (\\beta_1 + \\frac{1}{2} ) \\le \\hat{\\alpha} \\lambda_{*} \\beta_1 , \\quad  \n \\hat{\\alpha} \\lambda^{*} \\beta_1  \\le \\hat{\\alpha} \\lambda_{K,m} \\beta_1 $$ \n for all eigenvalues $\\lambda_{K,m} $ of the pinned Laplacian $K$. Therefore, the conditions in this corollary are more stringent that the conditions in Lemma~\\ref{general_ADSR_lemma_complex} and   Corollary~\\ref{corrolary_Acc_methods}.\n\n\\hfill\\ensuremath{\\square}\n\n\n\n\\subsubsection{Optimal A-DSR for graphs with real spectrum}\nFast convergence with structural robustness for A-DSR in networks with real spectrum is presented below, which is similar to the \nstructurally-robust convergence without A-DSR in Section~\\ref{Limit_of_convergence_rate}.\nNote that the characteristic equation in  Eq.~\\eqref{general_char_eq} with A-DSR for networks with real spectrum is equivalent to that of a standard  second order system of the form, \n\n\\begin{equation}\n    \\begin{aligned}\n    D(z) & = \n     z^2 + 2\\zeta_{\\lambda_{K,m}} \\omega_{\\lambda_{K,m}} z + \\omega_{\\lambda_{K,m}}^2 =0, \n    \\end{aligned}\n    \\label{general_char_eq_normal_form}\n\\end{equation}\n\n\n\n\n\\noindent\nwhere \n\\begin{equation}\n\\begin{aligned}\n\\omega_{\\lambda_{K,m}}^2 & =  (\\beta_2 -\\hat{\\alpha}\\beta_1 { \\lambda_{K,m}     } ), \\\\ \\zeta_{\\lambda_{K,m}} & = \\frac{ \\hat{\\alpha} (1+\\beta_1) \\lambda_{K,m}   -(1+\\beta_2)     }{2 \\omega_{\\lambda_{K,m}}} , \n\\end{aligned}\n\\label{Eq_def_damping_nat_freq}\n\\end{equation}\n\n\\noindent\nwith two roots\n($ z_{\\lambda_{K,m},i}$, $i \\in \\left\\{1,2\\right\\})$ \nassociated with each  real eigenvalue $\\lambda_{K,m}$ of the pinned Laplacian $K$. As in the case without A-DSR, the goal is to select the roots \n($z_{\\underline\\lambda,i}, z_{\\overline\\lambda,i}, i\\in\\left\\{1,2\\right\\}$) \nof the characteristic equation in Eq.~\\eqref{general_char_eq} for A-DSR,  associated with the \nextremal eigenvalues $\\lambda ={\\underline{\\lambda}, \\overline{\\lambda}}$ of the pinned Laplacian $K$, to be  equidistant from origin (for similar {structural robustness}) \n\\begin{equation}\n| z_{\\underline\\lambda}| = \n| z_{\\underline\\lambda,1}| = \n| z_{\\underline\\lambda,2}| = \n|z_{\\overline\\lambda,1}| =\n|z_{\\overline\\lambda,2}| =\n|z_{\\overline\\lambda}| \n \\label{Eq_ideal_ADSR_1}\n\\end{equation}\nand be farthest away from the unit circle (for fast convergence), i.e., by choosing the A-DSR parameters $\\hat{\\alpha}, \\beta_1, \\beta_2$ to solve the following minimization problem\n\\begin{equation}\n\\min_{\\hat{\\alpha}, \\beta_1, \\beta_2} \n\\left[ \n| z_{\\underline\\lambda}| = \n|z_{\\overline\\lambda}| \n\\right].\n \\label{Eq_ideal_ADSR_2}\n\\end{equation}\nFurthermore,  the roots of  Eq.~\\eqref{general_char_eq} associated with the \ndominant eigenvalue  $\\underline\\lambda$ of the pinned Laplacian are critically damped and positive, i.e., \n\\begin{equation}\n\\zeta_{\\underline\\lambda} = -1, \n\\quad  \\quad \n z_{\\underline\\lambda,1} = z_{\\underline\\lambda,2} > 0, \n \\label{Eq_ideal_ADSR_3}\n\\end{equation}\n as in the case without A-DSR, which can help to reduce  oscillations in the response. \n\n\n\\vspace{0.1in}\n\\begin{Lem}\n{[Parameter selection for Robust A-DSR]}\n\\label{robust_ADSR_lemma}\n{\nLet (i)~the A-DSR parameter be chosen to be positive $\\hat{\\alpha}>0$ to meet the stability condition in Eq.~\\eqref{general_stability_cond}, and (ii) the \npinned Laplacian $K$ have at least two distinct eigenvalues, i.e., $\\overline{\\lambda}\\neq\\underline{\\lambda}$ in \nEq.~\\eqref{eq_ordering_eigenvalues}.\nThen, the A-DSR parameters  ($\\hat{\\alpha}, \\beta_1, \\beta_2$)\n\\begin{equation}\n\\begin{aligned}\n     \\hat{\\alpha} &= \\frac{4}{(\\sqrt{\\overline\\lambda}+\\sqrt{\\underline\\lambda})^2},~~  \n    \\beta_1 = 0, ~~\n    \\beta_2 = \\frac{(\\sqrt{\\overline\\lambda}-\\sqrt{\\underline\\lambda})^2}{(\\sqrt{\\overline\\lambda}+\\sqrt{\\underline\\lambda})^2}\n\\end{aligned}\n    \\label{gains_lemma_for_robust_ADSR}\n\\end{equation}\nresult in \n\\begin{enumerate}\n    \\item\n    balanced robustness of the extremal modes, i.e., satisfies Eq.~\\eqref{Eq_ideal_ADSR_1}, \n    the roots \n    $z_{\\underline\\lambda,i}, z_{\\overline\\lambda,i}, i\\in\\left\\{1,2\\right\\}$ \n    as in Eq.~\\eqref{Eq_ideal_ADSR_1}\n    \\item critical damping of the dominant mode, i.e., $\\zeta_{\\underline{\\lambda}}=-1$ as in Eq.~\\eqref{Eq_ideal_ADSR_3}, and \n    \\item \n    optimal convergence, i.e., achieves the minimization in Eq.~\\eqref{Eq_ideal_ADSR_2}.\n\\end{enumerate}\n}\n\\end{Lem}\n\n\\vspace{0.1in}\n\\noindent\n{\\bf{Proof}~} \nThis is shown below in four  steps. \n\n\\vspace{0.05in}\n\\noindent \n{\\bf{Step 1}} is to show that \nthe roots  of  Eq.~\\eqref{general_char_eq} associated with the \nextremal eigenvalue  $\\overline\\lambda$ of the pinned Laplacian cannot be overdamped. \nNote that if the damping ratio $\\zeta_{\\overline\\lambda}$ of the \n roots $z_{\\overline\\lambda,1}, z_{\\overline\\lambda,2}$ in   Eq.~\\eqref{general_char_eq} associated with the \nextremal eigenvalue  $\\overline\\lambda$ \nis larger than one in magnitude, i.e.,  $| \\zeta_{\\overline\\lambda}| > 1$, then the roots  \n\\begin{equation}\n\\begin{aligned}\nz_{\\overline\\lambda,1} & =-\\left(\\zeta_{\\overline\\lambda} ~ \\omega_{\\overline\\lambda}\n\\right) + \n\\omega_{\\overline\\lambda}\n\\sqrt{\\zeta_{\\overline\\lambda}^2 \n-1} \\\\\nz_{\\overline\\lambda,2} & =\n-\\left(\\zeta_{\\overline\\lambda} ~ \\omega_{\\overline\\lambda}\n\\right) - \n\\omega_{\\overline\\lambda}\n\\sqrt{\\zeta_{\\overline\\lambda}^2 -1}~\n,  \n\\end{aligned}\n\\label{roots_of_2ndorder_Lemma3_1}\n\\end{equation}\nare real and distinct and have \ndifferent magnitudes \n$ |z_{\\overline\\lambda,1}| \\ne \n|z_{\\overline\\lambda,2}| $, which cannot satisfy the lemma's equidistant condition as in Eq.~\\eqref{Eq_ideal_ADSR_1}. \nTherefore, the roots $z_{\\overline\\lambda,1}, z_{\\overline\\lambda,2}$ of  Eq.~\\eqref{general_char_eq} associated with the \nextremal eigenvalue  $\\overline\\lambda$ of the pinned Laplacian cannot be overdamped, i.e., \\begin{equation}\n | \\zeta_{\\overline\\lambda}| \\le 1. \\label{eq_damping_values_Lemma3_2}\n\\end{equation}\n\n\\vspace{0.05in}\n\\noindent \n{\\bf{Step 2}} is to show that \nthe equidistant condition of the lemma, as in Eq.~\\eqref{Eq_ideal_ADSR_1}, leads to a zero outdated-feedback gain, $\\beta_1=0$. \nSince the magnitude of the damping ratio is not more than one,  \n$ | \\zeta_{\\overline\\lambda}| \\le 1$ from Eq.~\\eqref{eq_damping_values_Lemma3_2},  the term $\\zeta_{\\overline\\lambda}^2 \n-1$ becomes non-positive in Eq.~\\eqref{roots_of_2ndorder_Lemma3_1}, and therefore its square root is  either a complex number (when $|\\zeta_{\\overline{\\lambda}}|<1$) or zero (when $|\\zeta_{\\overline{\\lambda}}|=1$), and thus the magnitudes of the roots  become \n\\begin{equation}\n    |z_{\\overline\\lambda,1}| = |z_{\\overline\\lambda,2}| = |z_{\\overline\\lambda}| = \\omega_{\\overline\\lambda}  = \\sqrt{\\beta_2-\\hat{\\alpha}\\beta_1\\overline\\lambda} ~.\n\\end{equation}\nSimilarly, the magnitudes of the roots associated with the extremal value $\\underline\\lambda$\nwith damping ration $\\zeta_{\\underline\\lambda} = -1 $ in Eq.~\\eqref{Eq_ideal_ADSR_3}, are \n\\begin{equation}\n    |z_{\\underline\\lambda,1}| = |z_{\\underline\\lambda,2}| = |z_{\\underline\\lambda}| = \\omega_{\\underline\\lambda}  = \\sqrt{\\beta_2-\\hat{\\alpha}\\beta_1\\underline\\lambda} ~.\n\\end{equation}\nTo satisfy the equidistant condition, \n$$ |z_{\\underline\\lambda}| = \\sqrt{\\beta_2-\\hat{\\alpha}\\beta_1\\underline\\lambda}\n= \n\\sqrt{\\beta_2-\\hat{\\alpha}\\beta_1\\overline\\lambda} = \n|z_{\\overline\\lambda}|,$$\nand since \n$\\hat{\\alpha}>0$ and $\\underline\\lambda \\ne \\overline\\lambda$,  $\\beta_1=0$. Thus,\nthe magnitude of the roots (associated with the extremal eigenvalues) are \n\\begin{equation}\n    |z_{\\overline\\lambda}| = |z_{\\underline\\lambda}| =\n    \\omega_{\\overline\\lambda}  =\n    \\omega_{\\underline\\lambda}  =\\sqrt{\\beta_2}\n    \\label{eq_Lemma_3_proof_mag_beta1_0}\n\\end{equation}\n\n\\vspace{0.05in}\n\\noindent \n{\\bf{Step 3}} is to show that \nthe roots  of  Eq.~\\eqref{general_char_eq} associated with the \nextremal eigenvalue  $\\overline\\lambda$ are critically damped. \nUsing the damping ratio definition for the extremal modes, $\\zeta_{\\overline\\lambda}$ and $\\zeta_{\\underline\\lambda}$ in  Eq.~\\eqref{Eq_def_damping_nat_freq},  with $\\beta_1=0$ and $\\zeta_{\\underline\\lambda}= -1$, and\nsubstituting for $\\omega_{\\overline\\lambda}, \n    \\omega_{\\underline\\lambda}$ from Eq.~\\eqref{eq_Lemma_3_proof_mag_beta1_0}, results in \n\\begin{equation}\n\\begin{aligned}\n    -1 &= \\frac{\\hat{\\alpha}\\underline\\lambda-(1+\\beta_2)}{2\\sqrt{\\beta_2}} \\\\\n    \\zeta_{\\overline\\lambda} &= \\frac{\\hat{\\alpha}\\overline\\lambda-(1+\\beta_2)}{2\\sqrt{\\beta_2}}\n    \\end{aligned}\n    \\label{Eq_extremal_mode_damping_exprsn}\n\\end{equation}\n\n\\noindent Solving the two equations in Eq.~\\eqref{Eq_extremal_mode_damping_exprsn} for the magnitude $\\sqrt{\\beta_2}$ of the extremal roots results in \n\\begin{equation}\n    \\sqrt{\\beta_2} =\\frac{ \\hat{\\alpha} (\\overline\\lambda-\\underline\\lambda)}{2(1+\\zeta_{\\overline\\lambda})} , \\label{Eq_extremal_mode_general_magnitude}\n\\end{equation}\nwhich is minimized over damping ratio $|\\zeta_{\\overline\\lambda}| \\le 1$ by \nselecting \n\\begin{equation}\n\\zeta_{\\overline\\lambda} = 1. \n\\label{Eq_extremal_mode_damping_optimal}\n\\end{equation}\nNote that the magnitude of the roots (associated with the extremal eigenvalues) becomes, from Eqs.~\\eqref{eq_Lemma_3_proof_mag_beta1_0}, and \\eqref{Eq_extremal_mode_general_magnitude},   \n\\begin{equation}\n    |z_{\\overline\\lambda}| = |z_{\\underline\\lambda}| = \\sqrt{\\beta_2}\n    = \n    \\frac{ \\hat{\\alpha}(\\overline\\lambda-\\underline\\lambda)}{4}\n    .\n    \\label{eq_Lemma_3_proof_mag_beta1_02_}\n\\end{equation}\n\n\n\\vspace{0.05in}\n\\noindent \n{\\bf{Step 4}} is to find the optimal A-DSR gains $\\hat{\\alpha}$ and $\\beta_2$. \nSubstituting $\\zeta_{\\overline\\lambda} = 1$ from Eq.~\\eqref{Eq_extremal_mode_damping_optimal} into  Eq.~\\eqref{Eq_extremal_mode_damping_exprsn}, results in\n\\begin{equation}\n\\begin{aligned}\n    \\hat{\\alpha}\\overline\\lambda\n    & = (1+\\beta_2) +2 \\sqrt{\\beta_2}  \\\\\n     \\hat{\\alpha}\\underline\\lambda\n    & = (1+\\beta_2)-2 \\sqrt{\\beta_2}.\n    \\end{aligned}\n    \\label{Eq_extremal_mode_damping_exprsn_2}\n\\end{equation}\nDividing the two equations to eliminate $\\hat{\\alpha}$ yields a quadratic equation for $\\sqrt{\\beta_2}$, the magnitude  of the roots, \n\\begin{equation}\n\\begin{aligned}\n    (\\overline\\lambda - \\underline\\lambda)\n    \\beta_2 \n   -2  \n   (\\overline\\lambda + \\underline\\lambda) \\sqrt{\\beta_2}\n   +(\\overline\\lambda - \\underline\\lambda) & =0,\n       \\end{aligned}\n    \\label{Eq_extremal_mode_damping_exprsn_4}\n\\end{equation}\nwith solutions\n\\begin{equation}\n\\begin{aligned}\n \\sqrt{\\beta_2} & = \n \\frac{(\\overline\\lambda+\\underline\\lambda)\\pm2\\sqrt{\\overline\\lambda\\underline\\lambda}}{(\\overline\\lambda-\\underline\\lambda)}.\n       \\end{aligned}\n    \\label{Eq_extremal_mode_damping_exprsn_5}\n\\end{equation}\n\n\\noindent \nSince $\\overline\\lambda>\\underline\\lambda>0$, the smaller root in Eq.~\\eqref{Eq_extremal_mode_damping_exprsn_5} is chosen for maximizing structural robustness, resulting in \n\\begin{equation}\n    \\begin{aligned}\n    \\sqrt{\\beta_2} &= \\frac{(\\overline\\lambda + \\underline\\lambda) -\n 2\\sqrt{\\overline\\lambda\\underline\\lambda}\n }{(\\overline\\lambda - \\underline\\lambda)} \n %\n ~=\n \\frac{(\\sqrt{\\overline\\lambda}-\\sqrt{\\underline\\lambda})}{(\\sqrt{\\overline\\lambda}+\\sqrt{\\underline\\lambda})}\n    \\end{aligned}\n    \\label{Eq_extremal_mode_damping_exprsn_6}\n\\end{equation}\nand from Eq.~\\eqref{eq_Lemma_3_proof_mag_beta1_02_},  \n\\begin{equation}\n    \\begin{aligned}\n    \\hat{\\alpha} &= \\frac{4}{(\\overline\\lambda-\\underline\\lambda)}\\sqrt{\\beta_2} ~=  \\frac{4}{(\\sqrt{\\overline\\lambda}+\\sqrt{\\underline\\lambda})^2}.\n    \\end{aligned}\n    \\label{Eq_extremal_mode_damping_exprsn_7}\n\\end{equation}\n\\hfill\\ensuremath{\\square} \n\n\n\\begin{Rem}[No outdated-feedback in Robust A-DSR] As $\\beta_1=0$ for Robust A-DSR (from Lemma~\\ref{robust_ADSR_lemma} in Eq.~\\eqref{gains_lemma_for_robust_ADSR}), the outdated-feedback term is zero for maximum robustness in A-DSR based approach for networks with real spectrum. Thus, only momentum term is found to be important for improving both robustness and convergence rate of general networks without loops.\n\n\\end{Rem}\n\n\n\n\n\n\\subsubsection{Stability with momentum term only}\n\\vspace{0.1in}\n\\begin{Lem}\n\\label{stable_robust_ADSR_lemma1}\n{[Stability of Robust A-DSR]}\nLet the  A-DSR parameters $\\hat{\\alpha}$, $\\beta_1$ and $\\beta_2$ be selected as in Eq.~\\eqref{gains_lemma_for_robust_ADSR} from Lemma~\\ref{robust_ADSR_lemma} and let the extremal eigenvalues be distinct, i.e.,\n$\\overline{\\lambda}\\neq\\underline{\\lambda}$. Then, the resulting network with the general A-DSR is stable, i.e.,\nthe \nroots\n($ z_{\\lambda_{K,m},i}$, $i \\in \\left\\{1,2\\right\\})$ \nof characteristic Eq.~\\eqref{general_char_eq_normal_form}  \n(associated with each eigenvalue $\\lambda_{K,m}$ of the pinned Laplacian $K$) have magnitude less than one. \n\n\\end{Lem}\n\n{\\bf{Proof}~} \nWith the optimal parameters in Eq.~\\eqref{gains_lemma_for_robust_ADSR}, \nthe damping ratio \n$\\zeta_{\\lambda_{K,m}}$ of\nthe \nroots\n($ z_{\\lambda_{K,m},i}$, $i \\in \\left\\{1,2\\right\\})$ \nof Eq.~\\eqref{general_char_eq_normal_form} \nassociated with each eigenvalue $\\lambda_{K,m}$ of the pinned Laplacian $K$ is given by \n\\begin{equation}\n    \\begin{aligned}\n    \\zeta_{\\lambda_{K,m}} &= \\frac{\\hat{\\alpha}\\lambda_{K,m}-1-\\beta_2}{2\\sqrt{\\beta_2}} , \n    \\end{aligned} \n    \\label{Eq_lemma_middle_eigvals_1}\n\\end{equation}\n\n\\noindent which makes the damping ratio $\\zeta_{\\lambda_{K,m}}$  linear in the eigenvalue $\\lambda_{K,m}$, and varying between  $\\zeta_{\\underline\\lambda}=-1$ to $\\zeta_{\\overline\\lambda}=1$. This  implies that any eigenvalue between the extremal ones is underdamped, i.e.\n\\begin{equation}\n|\\zeta_{\\lambda_{K,m}}|<1,\\;\\forall\\;\\underline\\lambda<\\lambda_{K,m}<\\overline\\lambda  \n\\label{eq_eigenvalue_distribution}\n\\end{equation}\n\n\\noindent As a result, the magnitude of the roots of the characteristic polynomial for $\\lambda_{K,m}$ is \n\n\\begin{equation}\n    |z_{\\lambda_{K,m},1}| = |z_{\\lambda_{K,m},2}|= \\sqrt{\\beta_2} = \\frac{\\sqrt{\\overline\\lambda} - \\sqrt{\\underline\\lambda}}{\\sqrt{\\overline\\lambda} + \\sqrt{\\underline\\lambda}} < 1,\\; \\forall\\; \\overline \\lambda > \\underline \\lambda > 0, \n    \\label{eq_root_location_robust_ADSR}\n\\end{equation}\n\n\\noindent which shows that the roots are strictly within the unit circle resulting in stability.   \n\n\\hfill\\ensuremath{\\square}\n\n\n\\vspace{0.1in} \n\\begin{Rem}[Balanced structural robustness] From Eq.~\\eqref{eq_root_location_robust_ADSR}, all the roots \nof the characteristic equation in Eq.~\\eqref{general_char_eq_normal_form},  \nassociated with the Robust A-DSR, have the same magnitude and lie on a circle centered at the origin. Therefore, the roots are equally structurally robust, i.e.,  they are \nequidistant from the unit circle.\nThus, the A-DSR with  optimal  parameters, as in Eq.~\\eqref{gains_lemma_for_robust_ADSR} from Lemma~\\ref{robust_ADSR_lemma}, \nleads to balanced structural robustness in networks with real spectrum. \n\\end{Rem}\n\n\\vspace{0.1in}\n\\section{Results and Discussion}\nThis section comparatively evaluates  the Optimal no-DSR and the Robust A-DSR approaches using simulation results for an example network's structural robustness and convergence rate during transition. Additionally, the improvements in\nconvergence rate with the Robust A-DSR are validated with an experimental system.\n\n\n\n\n\\vspace{0.1in}\n\n\n\\subsection{Simulation results}\n\\label{simulation-section}\n\n\n\\vspace{0.1in}\n\n\\subsubsection{Example transition problem}\nThe  network considered here has four  agents ($n=4$) represented by nodes $X_i$, where $1\\le i \\le 4$,  with node  connectivity represented by the graph  in  Figure~\\ref{pathgraph}. Note that the eigenvalues of the given network's Laplacian are real, however the underlying graph is not strongly connected. Moreover, the graph (even without the source $X_s$) is  not balanced.\n\n\n\n\\begin{figure}[!ht]\n   \\centering\n        \\includegraphics[width=.80\\columnwidth]{images\/Fig_2_graph_node_circular_directed.eps} \n    \\caption{ Graph of example network  with four agents ($n=4$).  Non-source agents are $X_i, 1 \\le i\\le 4$, and the source agent is $X_s$. The edge between  agents $X_4$ (the agent with source input) and $X_3$ is undirected, the others are directed.}\n    \\label{pathgraph} \n\\end{figure}\n\n\\noindent \nThe virtual source agent $X_s$ determines the desired consensus value for the network and is connected to the agent  $X_4$, i.e. the leader. The connecting edges are all directed in the non-source graph network, except for the undirected edge between the leader $X_4$ and follower agent $X_3$ which makes the graph Laplacian asymmetric.  \nThe system dynamics with no-DSR for the example network, is given by  Eq.~\\eqref{system_non_source},  with the pinned-Laplacian $K$ and $B$ given as  \n\n\n\\begin{equation}\n    \\begin{aligned}\n  &K =  \\left[\n\\begin{array}{c c c c}\n1  &  -1 & 0 & 0  \\\\\n0 & 1 & -1 & 0 \\\\ \n0 & 0 & 1 & -1 \\\\\n0 & 0 & -1 & 2 \\\\\n\\end{array}\n\\right] \n&B = \\left[\n\\begin{array}{c}\n      0\\\\\n      0\\\\\n      0\\\\\n      1\\\\\n\\end{array}\n\\right].\n    \\end{aligned}\n    \\label{pathgraph_Laplacian}\n\\end{equation}\nAs discussed in Section~\\ref{Sec_Graphs_Laplacian_potential}, the \n gradient of the asymmetric Laplacian potential $\\Phi_{{\\cal{G}}}(\\hat{X}) =\\hat{X}^TL\\hat{X}$ in Eq.~\\eqref{system_eq_potential} does not lead to standard neighbor-based update in Eq.~\\eqref{system_non_source}, where $L$ is the graph Laplacian (from Eq.~\\eqref{eq_def_pinned_K}) and $\\hat{X}$ is the state vector including source agent.\n \n\\vspace{0.1in}\n\n\\subsubsection{Optimal no-DSR for example network}\n\\label{consensus-based-simul-subsection}\n\n\n\n\n\n\nThe optimal update gain $\\alpha^*$ \nfrom Eq.~\\eqref{update_gain_for_max_robust}, \nfor minimum spectral radius $\\sigma(P)=\\sigma(P^*)$, is determined using the extremal eigenvalues $\\overline\\lambda = 2.618$ and $\\underline\\lambda = 0.382$ of the pinned-Laplacian $K$ in  Eq.~\\eqref{pathgraph_Laplacian}, \n{ using Eq.~\\eqref{update_gain_for_max_robust}, } as\n\\begin{equation}\n    \\alpha^* = \\frac{2}{\\overline\\lambda + \\underline\\lambda} = \\frac{2}{2.6180 + 0.3820} = 0.6667. \n    \\label{optimal_updategain_pathgraph}\n\\end{equation}\nThe  measure of structural robustness $d^*$ \nwith Optimal no-DSR is, \n{from Eq.~\\eqref{eq_optimal_no_dsr_robustness}, }\n\\begin{equation}\n    d^* = 1 - \\sigma^* = 0.255, \n\\end{equation}\n\\noindent with the optimal spectral radius $\\sigma^* = 0.745$ (from Eq.~\\eqref{minimum_spectral_radius}), as illustrated in  Figure~\\ref{location_of_roots_robust_consensus}. \n\n\n\\begin{figure}[!t]\n\\begin{center}\n    \\includegraphics[width=0.90\\columnwidth]{images\/Fig_3_pole_location_consensus.eps}\n    \\caption{{ Optimal spectral radius ($\\sigma^*$) for Optimal no-DSR.} \n     Location of the eigenvalues of matrix P, $\\lambda_{P,m}=1-\\alpha^* \\lambda_{K,m}$  with optimal update gain $\\alpha^*$ from Eq.~\\eqref{optimal_updategain_pathgraph}. The  spectral radius  with Optimal no-DSR is  $\\sigma^*= 0.745$, \n     as in Eq.~\\eqref{minimum_spectral_radius} \n     }\n    \\label{location_of_roots_robust_consensus}\n    \\end{center}\n\\end{figure}\n\n\n\\begin{figure}[!ht]\n  \\begin{center}\n    \\includegraphics[width=0.75\\columnwidth]{images\/Fig_4_uncertain_roots_plot_P.eps}\n    \\caption{Perturbation $e$ can lead to loss of real spectrum but stability is still structurally robust,  i.e., stability is maintained for small perturbations $e$. Location of the eigenvalues $\\lambda_{P_{e}} = 1 - \\alpha^* \\lambda_{K_{e}}$  of matrix $P_e=\\textbf{I}_4-\\alpha^* K_e$, where  $K_e$ from Eq.~\\eqref{Eq_Laplacian_with_Error} has a perturbation term $e$, which varies from $10^{(-5)}$ (blue) to $10^{(-1)}$ (red). \n    }\n    \\label{Fig_roots_for_uncertainK}\n    \\end{center}\n\\end{figure}\n\n\\begin{figure}[!t]\n\\begin{center}\n  \n    \\includegraphics[width=0.9\\columnwidth]{images\/Fig_5_pole_location_critically_damped_ADSR.eps}\n    \\caption{\n     Optimal spectral radius ($\\sigma^*$) with Robust A-DSR. \n    Location of the roots of characteristic polynomials with Robust A-DSR as in Lemma~\\ref{robust_ADSR_lemma} for each eigenvalue $\\lambda_{K,m}$ of pinned-Laplacian $K$ in Eq.~\\eqref{pathgraph_Laplacian}. The\n   spectral radius  with Robust A-DSR is  $\\sigma^*=0.447$, an improvement of $40\\%$ compared to Optimal no-DSR. \n   The spectral radius with search-based A-DSR is similar to the Robust A-DSR, with similar root locations.\n    }\n    \\label{location_of_roots_critically_damped_ADSR}\n    \\end{center}\n\\end{figure}\n\n\n\n\n\n\n\n\n\nTo assess the transition response, a simulation was  performed with the virtual agent's state $X_s$ changing from an initial value $X_s[k] = x_i$ for all $k<0$ to a final value $X_s[k] = x_f$ for all\n $0\\le k$. \nIt was assumed that the non-source agents are initially at consensus, i.e., \n$X[0] =  x_i {\\textbf{1}}_n$.\nWith the update gain from Eq.~\\eqref{optimal_updategain_pathgraph}, the \nsimulated response of the Optimal no-DSR method for a change in virtual agent state $X_s$ from $x_i=0$ to $x_f=100$ is shown in Figure~\\ref{ADSR_simul}.\nThe settling time ($T_s$) of the network's response, defined as the time taken for all the agents' states to achieve and remain within $95\\%$ of the desired change $x_{\\Delta}=x_f-x_i = 100$ in the consensus state was found to be $14$ sampling time periods ($k=14$) from the simulated response.\n\n\n\n\n\n\n\n\n\n\\begin{Rem}[Structural robustness with real spectrum]\n\\label{Structural_robustness_real_spectrum}\nThe  addition of edges (even with small edge weights) could lead to loss of the real spectrum property. Nevertheless, the stability will still be structurally robust (with or without A-DSR) since the roots of the any general polynomial (and the characteristic equations in particular) are continuous in its coefficients. \nTo illustrate, the roots of the pinned-Laplacian with a perturbation term $e$ \n\\begin{equation}\nK_e = \\begin{bmatrix}\n1 &-1 &0 &0 \\\\\ne &1-e &-1 &0 \\\\\n0 &0 &1 &-1 \\\\\n0 &0 &-1 &2\n\\end{bmatrix}\n\\label{Eq_Laplacian_with_Error}\n\\end{equation}\nare continuous with respect to $e$.\nThe corresponding location of roots of $P_e = \\textbf{I}_4-\\alpha^* K_e$ with Optimal no-DSR update gain $\\alpha^* = 0.6667$ (obtained from Eq.~\\eqref{optimal_updategain_pathgraph}) are shown in Figure~\\ref{Fig_roots_for_uncertainK} for increasing perturbation $e$. Although, the resulting spectrum is no longer real, the stability is structurally robust, i.e., stability is maintained for small perturbations $e$. \n\\end{Rem}\n\n\n \n\\vspace{0.1in}\n\\subsubsection{A-DSR improves structural robustness} \nThe A-DSR approach in Eq.~\\eqref{accelerated_system_non_source_A_DSR} under  Subsection~\\ref{Proposed A-DSR}  is used to improve the example network's structural robustness. The spectral radius of the network is minimized over the range of A-DSR parameters $\\hat{\\alpha},\\beta_1$ and $\\beta_2$, \n\n\\begin{equation}\n  \\sigma^*  =\\min_{\\hat{\\alpha},\\beta_1,\\beta_2}\\left[ \\max_{m} \\left( \\max_{1\\le i \\le 2}  | z_{\\lambda_{K,m},i}| \\right) \\right], \n   \\label{Eq_spectral_radius_minimization}\n\\end{equation}\n\\noindent where $z_{\\lambda_{K,m},i}$ with $i \\in \\left\\{1,2\\right\\}$ are the  roots \nof the characteristic Eqs.~\\eqref{general_char_eq_normal_form} associated with eigenvalue $\\lambda_{K,m}$ of the pinned-Laplacian $K$,  and the search space is constrained by the stability conditions in Eq.~\\eqref{general_stability_cond}. The optimum parameters for minimum spectral radius, found through a  numerical search, and the resulting performance are tabulated in Table~\\ref{table_simul}.\nWith these optimal parameter selections, the corresponding roots of the characteristic polynomial with A-DSR, in Eq.~\\eqref{accelerated_system_non_source_A_DSR}, for each eigenvalue $\\lambda_{K,m}$, are shown in Figure~\\ref{location_of_roots_critically_damped_ADSR}.  \nThe optimal spectral radius is given by \n $ \\sigma^* = 0.447$, which is a reduction of $40\\%$ when compared to the Optimal no-DSR case for this example network.  \nFor the same state transition from \n$x_i= 0$  to $x_f=100$ in the consensus state, the corresponding $5\\%$\nsettling time is $7$ sampling time periods ($k=7$), which is a $50\\%$ improvement over the Optimal no-DSR case.  Thus, the A-DSR approach improves both the structural robustness and the convergence rate when compared to the Optimal no-DSR case. \n\n\n\n\n\n\n\n\\subsubsection{Robust A-DSR's performance similar to A-DSR}\nInstead of a numerical search to optimize the parameters as in the A-DSR case, the \nRobust A-DSR, proposed in Subsection~\\ref{subsection_robust_convergence_with_ADSR}, yields closed-form expressions for selection of its parameters as in Eq.~\\eqref{gains_lemma_for_robust_ADSR}. With the Robust A-DSR, the corresponding roots of the characteristic polynomials in Eq.~\\eqref{general_char_eq_normal_form}, for each eigenvalue $\\lambda_{K,m}$, are shown in Figure~\\ref{location_of_roots_critically_damped_ADSR}. Note that the  roots corresponding to the extremal eigenvalues $\\underline{\\lambda}, \\overline{\\lambda}$ are real valued and critically damped, as in  Lemma~\\ref{robust_ADSR_lemma}. Furthermore, the other roots of characteristic equation, for intermediate eigenvalues $\\lambda$ satisfying $\\underline\\lambda<\\lambda<\\overline\\lambda$, lie on a circle with radius equal to magnitude of the critically damped extremal modes as shown in Figure~\\ref{location_of_roots_critically_damped_ADSR}, \nwhich follows from Lemma~\\ref{stable_robust_ADSR_lemma1}. \nOverall, the spectral radius $\\sigma^*$ of the example network, with Robust A-DSR, is equal to the magnitude of the roots, i.e., \n$\\sigma^* = \\sqrt{\\beta_2} = 0.447$.  \n\n\n\nThe performance of the Robust A-DSR is similar to the optimized search-based A-DSR (see Table~\\ref{table_simul}).\nIn particular, the spectral radius of $ \\sigma^* = 0.447$ with Robust A-DSR is smaller by $40\\%$ when compared to $\\sigma(P^*)=0.745$ with the Optimal no-DSR method (see Table~\\ref{table_simul}), thus improving the structural robustness.\nAdditionally, the settling time $T_s$  with Robust A-DSR  was found to be $7$ sampling time periods from the simulation result (which corresponds to a $50\\%$ improvement in convergence rate) as shown in Figure~\\ref{ADSR_simul}.  \n\n\n\n{\\tiny\n\\begin{table}[!ht]\n\\caption{{Simulation results for minimizing (min of) Spectral Radius ($\\sigma$) and Settling Time ($T_s$): Comparison of robustness ($\\sigma$) \\& convergence rates ($T_s$) of network responses using Optimal no-DSR (Eq.~\\eqref{system_non_source}), A-DSR (Eq.~\\eqref{accelerated_system_non_source_A_DSR}), Nesterov-update (Eq.~\\eqref{accelerated_system_non_source}), and the Outdated-feedback and the Momentum  methods}\n}\n\\centering\n\\begin{tabular}{c c c c c c c}\n\t\\hline\\hline\n\t\t\\\\    \n\t\tMethod & min  & $\\hat{\\alpha}$ &$\\beta_1$ &$\\beta_2$ & $\\sigma$ &$T_s (k)$  \\\\\n\t\t& of   &    \\\\\n\t\t[0.5ex]\n\t\t\\hline \\hlin\n\t\t\\\\\n\t\\textbf{Robust} & &0.80 &0 &0.20 & 0.4472 &7  \\\\ \n\t \\textbf{A-DSR}& & & & &  &  \\\\ \n\t\t\\hline \n\t\\textbf{A-DSR} &$ \\sigma$ &0.7997 &0.0002 &0.2005 &0.4472 &7  \\\\\n\t\\textbf{}&$ T_s$ &0.6303 &0.2376 &0.3868 &0.6634 &6 \\\\\n\t\t\\hline\n\t\t\t\\textbf{Momentum} &$ \\sigma$ &0.7995 &0 &0.2006 &0.4479 &7 \\\\\n\t&$ T_s$ &0.8388 &0 &0.2347 &0.4845 &6 \\\\\n\t\t\\hline\n\t\\textbf{Nesterov} &$ \\sigma$ &0.4830 &0.3992 &0.3992 &0.5706 &11 \\\\\n\t\\textbf{-update}&$ T_s$ &0.5212&0.4684 &0.4684 &0.7599 &7 \\\\\n\t\t\\hline\n\t\\textbf{Outdated} &$ \\sigma$ &0.9638 &-0.1414 &0 &0.5973 &8 \\\\\n\t\\textbf{-feedback}&$ T_s$ &1.0874 &-0.1881 &0 &0.7318 &6\\\\\n\t\\hline \\\\\n\t\\textbf{Optimal } & &0.6667 &0 &0 &0.745 &14 \\\\  \n\t\t\\textbf{no-DSR}& & & & & &\\\\\n\t\t\\hline\n\t\t\\hline  \n\\end{tabular}\n\\label{table_simul}\n\\end{table}\n}\n\n\n\n\\begin{figure}[!ht]\n   \\hspace{-0.5cm}\n        \\includegraphics[width=.99\\columnwidth]{images\/Fig_6_simul_robustADSRvsOptimalnoDSR.eps}\n    \\caption{Simulated network state responses with Robust A-DSR (in red) and Optimal no-DSR (in blue, with $\\alpha = \\alpha^* = 0.6667$), where the Robust A-DSR parameters are chosen as $\\hat{\\alpha} = 0.80$, $\\beta_1 =0$ and $\\beta_2=0.20$ from Eq.~\\eqref{gains_lemma_for_robust_ADSR} for $\\overline\\lambda = 2.618$ and $\\underline\\lambda = 0.382$, showing the settling time $T_s = 7$ sampling time periods ($50\\%$ improvement w.r.t. Optimal no-DSR method $T_s = 14$ sampling time periods) \n    }\n    \\label{ADSR_simul} \n\\end{figure}\n\n\n\n\\begin{Rem}[Momentum term  $\\beta_2$ and \nsettling time $T_s$] \nFor the Robust A-DSR approach, \nthe settling time $T_s$ can be estimated analytically \nin terms of the momentum term $\\beta_2$. \nSince all the roots of the characteristic equation in Eq.~\\eqref{eq_root_location_robust_ADSR} have the same magnitude,  the dynamics associated with the under-damped roots of the Robust A-DSR converge faster than critically-damped, \nreal-valued roots $\\sqrt{\\beta_2}$.\nThe corresponding real-valued continuous-time roots $s_{cont}$ are at $s_{cont} = (\\ln{\\sqrt{\\beta_2}})\/\\delta_t$, which can be used to predict the $5\\%$ settling time $T_s$ as (in number of sampling time periods) \n\\begin{equation}\n    T_{s} \\approx \\frac{5}{|s_{cont}|\\delta_t} ~\n    ~ \\frac{5}{|\\ln{\\sqrt{\\beta_2}}|} ~= 6.2, \n    \\label{Eq_rem_Ts_1}\n\\end{equation}\nwhich matches the simulation-based value of $7$ sampling time periods. Thus, a larger momentum term $\\beta_2$ results in faster settling.\n\\end{Rem} \n\n\n\n \n \n In summary, the Robust A-DSR approach provides similar improvements as with the general A-DSR approach,  in both the structural robustness and the convergence rate when compared to the\nOptimal no-DSR approach. The advantage of  the Robust A-DSR approach is that it provides an analytical approach for  selecting the control parameters instead of the numerical search  with the general A-DSR. \n\n\n\n\n\n\n\n\n\n\n\n\\subsubsection{Comparison of constrained accelerated approaches}\nAlthough constrained, the Robust A-DSR (with $\\beta_1=0$)  outperforms both the Nesterov-update method (with $\\beta_1=\\beta_2=\\beta$) as well as the Outdated-feedback method (with $\\beta_2=0$). \nOptimal parameters for the \nNesterov-update as well as the Outdated-feedback methods \nwere also found using the same optimization in Eq.~\\eqref{Eq_spectral_radius_minimization}\nwith the additional constraints $\\beta_1=\\beta_2=\\beta$ for Nesterov-update method and $\\beta_2=0$ for Outdated-feedback method. \nThe search space for parameters were constrained as in Corollary~\\ref{corrolary_Acc_methods}. \nThe optimal parameters of Nesterov-update and Outdated-feedback methods and the performance are provided in Table~\\ref{table_simul}. \nWhen compared to the Optimal no-DSR case, \nthe Nesterov-update improves the spectral radius by $23.4\\%$ which is less than the improvement of $40\\%$ with the Robust A-DSR approach. The Outdated-feedback method also improves the\nspectral radius when compared to the no-DSR case, but the improvement ($19.9\\%$) is even smaller than the Nesterov-update case with $23.4\\%$. Similarly, the settling time improvement of $50\\%$ with Robust A-DSR when compared to Optimal no-DSR is larger than the improvement of $21.43\\%$ with the Nesterov-update and $42.9\\%$ improvement with the Outdated-feedback.\nThus,  while the Robust A-DSR is constrained, it still matches the performance of the general optimal A-DSR, and  outperforms both the Nesterov-update method as well as the Outdated-feedback method. \n\n \n\n\n\n\n\\begin{Rem}[Outdated-feedback versus momentum] \nWhen simultaneously improving both the structural robustness and the convergence rate, of the two components of the A-DSR, the momentum component (associated with $\\beta_2$) has more significant impact than the outdated-feedback component (associated with $\\beta_1$). \n\\end{Rem}\n\n\n\\vspace{0.1in}\n\\subsubsection{Convergence improvement without structural robustness} The above results focused on increasing both the structural robustness and convergence rate. However, the parameters of the accelerated update methods can be chosen purely for optimizing the convergence rate (i.e. minimizing the settling time $T_s$). The resulting optimized parameters (found through a numerical search) and the performance are quantified in Table~\\ref{table_simul}. \n\n\nThe accelerated methods achieve smaller settling time $T_s$ when the parameters are optimized for achieving a faster convergence rate. \nFor instance, the settling time $T_s$ with A-DSR (search based) improves to $6$ sampling time periods (see Table~\\ref{table_simul}), which is faster than Robust A-DSR  and Nesterov-update each taking $7$ sampling time periods, and an improvement of $57.1\\%$ over the Optimal no-DSR case. However, this improvement in settling time $T_s$ is accompanied by a decrease in structural robustness of the network. For example, with A-DSR parameters selected for fast convergence, the  spectral radius $\\sigma$ increased to $\\sigma = 0.6237$ from  $\\sigma=\\sigma^*=0.4472$ for the case when the parameters were selected to maximize bot the structural robustness and convergence rate. Among the other accelerated approaches, the Momentum method  also achieves the same settling time of $6$ sampling time periods as the A-DSR case, indicating the importance the momentum term in improving the convergence rate of the given example network. A similar loss in structural robustness is seen with the Momentum and Outdated-feedback approaches when the parameters are optimized purely for faster convergence rate, as seen in Table~\\ref{table_simul}. The loss in structural robustness (for this example) is more with the  Outdated-feedback than with the Momentum method. \n\n\n\nThe simulation results show that the network's convergence-rate alone can be improved with the general A-DSR further than that achieved with Robust A-DSR. However, this increase in convergence-rate alone involves a loss in structural robustness. Moreover, the A-DSR parameters are found using a numerical search method. \n\nIn contrast, the parameters of the Robust A-DSR can be found analytically and it achieves similar convergence rate as the A-DSR optimized for convergence-rate alone. Moreover, the performance improvement with the Robust A-DSR (as well as the A-DSR), in terms of both the \nstructural robustness \nand the rapidity of transition, is better than the  performance with the standard no-DSR consensus method.\n\n\n\\subsection{Experimental results}\n\\label{experimental-section}\n\nA mobile-bot network is used for experimental evaluation of the proposed A-DSR approach. \n\n\\subsubsection{System description}\nThe experimental setup consists of four mobile-bot agents that move in a straight line. The network connectivity is the same as in the simulation example. The bots aim to maintain a spacing of $d_o$ between them, and the state $X_i$ of each bot $i$ is defined as the displacement from the initial equally-spaced configuration, as shown in Figure~\\ref{bot_network}. The virtual source input $X_s$ determines the desired position of the network.  \n\n\n \n\n\\begin{figure}[!th] \n\\begin{center}\n\\includegraphics[width=0.95\\columnwidth]{images\/Fig_7_botsnewfig.eps}\n    \\caption{\n Experimental test bed consisting of four mobile-bot agents moving in a straight line, with the same  connectivity  as  in the example simulation network in Figure~\\ref{pathgraph}. Each $i^{th}$ agent's state is its displacement $X_i$ from its initial position.\n     }\n    \\label{bot_network}\n\\end{center}\n\\end{figure}\n\n\n\n\n\n\\subsubsection{Bot's update computation}\nThe desired displacement $X_i[k+1]$ at the next time step $k+1$ is computed using local relative-distance measurements available at time step $k$ by each bot $i$ using \ndistance sensors (Ultrasonic HC-SR04 to the front, and Infrared GP2Y0A21YK at the back).\nThese  measurements of each bot $i$ include  \n\\begin{equation}\n     X_{f,i}[k]=(X_{i+1}[k]-X_i[k]) + d_0, \n\\end{equation} \nthe relative displacement w.r.t. the front bot  $i+1$ (which is $X_s$ for leader bot $i=4$), and \n\\begin{equation}\nX_{b,i}[k] = d_0 - (X_{i-1}[k]-X_i[k]), \n\\end{equation}\n the relative displacement \nw.r.t. the back bot $i-1$ where $2< i< 4$,  and $d_0$ is the desired offset distance between the bots in the experimental setup. These relative-distance measurements \n ($ X_{f,i}[k], X_{b,i}[k] $) are used to determine the neighbor information needed to evaluate the update law, i.e., to obtain $K_iX[k]$, where $K_i$ is the $i^{th}$ row of the pinned-Laplacian in Eq.~\\eqref{pathgraph_Laplacian}. For example, \n \\begin{equation} \n \\begin{aligned}\n K_i X[k] & =   K_{i,i+1} (X_{i}[k]-X_{i+1}[k])  \\\\\n   & \\qquad + K_{i,i-1}  (X_{i}[k] - X_{i-1}[k])  \\\\\n & =   K_{i,i+1} (d_0 - X_{f,i}[k]) + K_{i,i-1}  (X_{b,i}[k] - d_0). \n  \\end{aligned}\n \\label{eq_relative_positon_measurements}\n \\end{equation}\nThus, the relative-distance  measurements ($ X_{f,i}[k], X_{b,i}[k] $) at time step $k$ \nenable each bot $i$ to compute its update, \ni.e., to find the desired position $X_i[k+1]$ at the next time step  according to Eq.~\\eqref{accelerated_system_non_source_single}, where parameters $\\beta_1$ and $\\beta_2$ are zero for the no-DSR case.\n\n\n\n\\subsubsection{Bot's feedback control}\nEach $i^{th}$ bot's controller aims to match its \nstate (displacement)\n$X_{i}(t)$ to be the desired  state $X_i[k+1]$  by the next time step, i.e., when time $t=t_{k+1}$.\nThis is accomplished using a velocity-feedback inner-loop and a \nposition-feedback outer-loop,  as shown in  Figure~\\ref{SensorsOnBot}, using \nmeasurements of the agent state $X_i(t)$ from magnetic encoders on each bot $i$. \n\n\n\n\n\n\n\\begin{figure}[!ht]\n\\centering\n      \\includegraphics[width=\\columnwidth]{images\/Fig_8_each_bot_controoler_vel_pos.eps} \n\\caption{\n{Each $i^{th}$ bot's control system includes: \na) distance sensors to the front and, b) back , c) micro-controller for on-board computation, and d) wheels with magnetic encoders on motors to estimate each bot's displacement, $X_{i}(t)$. \nTo ensure that the bot achieves $X_i[k+1]$, an inner-loop controller with gain $k_v$ to track desired velocity $V_i[k]$ in Eq.~\\eqref{Eq_desired_Velocity_at_tk} and an outer-loop controller with gain $k_x$ for position error ($\\Tilde{X}_i(t)$ in Eq.~\\eqref{Eq_position_error}) are implemented. \n}\n}\n\\label{SensorsOnBot} \n\\end{figure}\n\n\n\n\n\n\\noindent \nIn particular, the desired velocity for the time period $[t_k, t_{k+1})$ is computed as \n\\begin{equation}\nV_i[k] =  \\frac{X_i[k+1]-X_i[k]}{\\delta_t}, \n\\label{Eq_desired_Velocity_at_tk}\n\\end{equation} \nwhere $\\delta_t$ is the discrete time step (in seconds) for the update method. \nThe desired velocity $V_i[k]$ is then tracked using an inner-loop controller with gain $k_v$ as shown in Figure~\\ref{SensorsOnBot}. Additionally, an outer-loop feedback with gain  $k_x$ is used to correct for position error ($\\Tilde{X}_i(t)$) at any time $t\\in [t_k, t_{k+1})$, determined as\n\n\\begin{equation}\n    \\Tilde{X}_i(t) = (X_i[k]+\\Delta X_i(t)) - X_i(t), \n    \\label{Eq_position_error}\n\\end{equation}\n\n\\noindent where $\\Delta X_{i}(t) = V_i[k] (t-t_k) = V_i[k]\\Delta t$, as  shown in Figure~\\ref{SensorsOnBot}.\n\nThe selection of position transition magnitude for the experiment was based on velocity limits of $20$ cm\/s for the bots. The initial position was $x_i=0$, and the final position was $x_f =100$ cm. Therefore the sampling time period $\\delta_t$ was chosen as $4$ s to ensure that the bots could meet the maximum position transitions of $80$ cm in one sampling-time period $\\delta_t$, seen in simulations in Figure~\\ref{ADSR_simul}, with the bot's feedback gains $k_v =5$ and $k_x = 1$.\n\n\n\n\n\\begin{table}[!t]\n\t\\caption{{ Experimental results. Comparison of convergence rate in position responses with Robust A-DSR and Optimal no-DSR for multi-agent network in Figure~\\ref{bot_network}, using settling time $T_s$.}}\n\t\\centering\n\\begin{tabular}{c c c}\n\t\\hline\\hline\n\t\t\\\\ \n\t\tMethod & Trial  &$T_s (k)$  \\\\%  \n[0.5ex]\n\t\t%\n\t\t\\hline \\hlin\n\t\t\\\\\n\t\\textbf{Robust A-DSR} & Trial 1 &11  \\\\ \n\t       &Trial 2  &11  \\\\ \n\t ($\\hat{\\alpha}=0.80,\\beta_1=0, \\beta_2=0.20$)& Trial 3  &10  \\\\ \n\t &Trial 4  &  11\\\\ \n\t &Trial 5  &  11\\\\ \n\t &Trial 6  &  10\\\\ \n\t &Trial 7  &  9\\\\ \n\t\t\\hline \n\t\t\\bf{Mean Response} & &\\bf{10} \\\\\n\t\t\\hline  \\\\\n\t\\textbf{Optimal no-DSR} &Trial1  &17 \\\\  \n\t&Trial2 & 16\\\\\n\t($\\alpha = \\alpha^*=0.67$)&Trial3 & 16\\\\\n\t&Trial4 & 15\\\\\n\t&Trial5 & 15\\\\\n\t&Trial6 & 18\\\\\n\t&Trial7 & 15\\\\\n\t\n\t\t\\hlin\n\t\t\\bf{Mean Response} & &\\bf{16} \\\\\n\t\t\\hline\n\t\t\\hline\n\\end{tabular}\n\t\\label{table:ExpDSR}\n\\end{table}\n\n\n \\begin{figure}[!t]\n\t\\hspace{-0.5cm}\n\t\\includegraphics[width=0.99\\columnwidth]{images\/Fig_9_exp_positioncontrol_4bots_withmean.eps}\n\t\\caption{Experimental position responses over 7 trials (in lighter shade) and their mean (in dark lines) in the experiments comparing the Optimal no-DSR (in blue) and Robust A-DSR (in red) methods for fast convergence. The experiments on average show an improvement with Robust A-DSR of $37.5\\%$ in $T_s$ (from 16 time steps to 10 time steps).\n\t}\n\t\\label{averageExp_withinput} \n\\end{figure}\n\n\n\\subsubsection{Convergence rate improvement}\nThe improvement in convergence rate of transition response in the example network with Robust A-DSR, over  Optimal no-DSR, is evaluated through the experimental mobile-bot network.\n\nA transition in desired position (defined using virtual source $X_s$) from $x_i=0$ cm to $x_f=100$ cm, similar to simulations, is implemented on the mobile-bot network. Each bot, initially in consensus with position zero, responds as the transition information propagates through the bot network (in Figure~\\ref{bot_network}). This state transition is implemented using Optimal no-DSR and Robust A-DSR, with parameters given in Table~\\ref{table_simul}, and the observations of convergence rates from seven trials (with both the approaches) are tabulated in Table~\\ref{table:ExpDSR}. The position responses of the bots during the transition are plotted in Figure~\\ref{averageExp_withinput}, for each of the seven trials with Optimal no-DSR (in light blue) and Robust A-DSR (in light red). The mean responses for both approaches, obtained from averaging over the seven trials, are also shown in Figure~\\ref{averageExp_withinput}.\n\n\n\n\n\n\\indent \nRobust A-DSR shows  improvement  in convergence rate of the bot network's transition response, improving the settling time (within $5\\%$ of the final position) by $4$ to $9$ time periods ($27$\\% to $50$\\%), when compared with Optimal no-DSR, similar to that observed in simulations. The mean response converges $6$ time periods faster with Robust A-DSR (an improvement of $37.5\\%$) when compared with Optimal no-DSR, see Table~\\ref{table:ExpDSR}.\nThus, the convergence rate improvements observed in simulations with Robust A-DSR, with analytically determined parameters, over Optimal no-DSR are verified with similar results from experimental studies of position transition in the mobile-bot network.\n\n\n\n\n\n\n\n\\section{CONCLUSIONS}\n\\label{conclusion-section}\nThe article introduced an accelerated delayed self reinforcement (A-DSR) approach, based on local potential, for improving the structural robustness and convergence rate beyond the limits of standard consensus-based networks. Of the two terms in the accelerated approach, it was shown that the momentum term has substantially more impact when compared to the outdated-feedback term for improving convergence rate and robustness in networks with real spectrum. A Robust A-DSR approach was developed, with analytical expressions for its parameters, that closely matches the performance of the general A-DSR approach, which alleviates the  need for numerical search when selecting parameters of the general A-DSR. Moreover, experimental results verified the improved convergence rate with Robust A-DSR over Optimal no-DSR. \n\nThe A-DSR approach, presented in this work, assumes scalar gains \nfor the outdated-feedback and momentum terms, which can be extended in future work by using different, possibly nonlinear or time-varying gains for each agent in the network. Further, the proposed Robust A-DSR approach, can be used to accelerate convergence and improve performance of networks with uncertainty, for instance, distributed sensing in presence of communication delays, operation of multi-agent networks with a human-in-the-loop where the human or network model is uncertain, and transporting flexible structures with uncertain stiffness values using mobile bots. Further work is needed to explore the suitability of the Robust A-DSR for these applications. \n\n\n\n\n\n\n\n\n\\bibliographystyle{unsrt}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sec:introduction}\n\nWhen two galaxies merge, a supermassive binary black hole (SMBBH) is expected to form \\citep{Merritt2005}. The potential interaction of the new system with the surrounding gas and the dynamical friction of stars might shrink the binary separation to sub-parsec scales \\citep{Begelman1980,\nMayer2007, Escala2004, Escala2005, Merrit2004, Merritt2006, Dotti2007,\nDotti2009b, Shi2012,SesanaKhan2015,Mirza2017,Khan2019, Tiede+2020, dittmann2022}. At those separations, energy and angular momentum is extracted from the system by gravitational radiation until the BHs merge \\citep{PhysRevLett.95.121101,PhysRevLett.96.111101, PhysRevLett.96.111102}. In the near future, gravitational waves from SMBBH mergers might be observable in the mHZ frequency band by the \\textit{Laser Interferometer Space Antenna} \\citep[LISA,][]{LISA2017} and by \\textit{pulsar timing} techniques in\nthe nHz range \\citep{Nanograv2020, EPTA2016, Reardon2015}. \n\nSince the environment of these systems is most likely gas-rich, a SMBBH could also emit electromagnetic radiation through accretion \\citep{Barnes1992, Barnes1996, Mihos1996,Mayer2007, Dotti2012, Mayer2013, Derdzinski2019}. At subparsec scales, SMBBHs cannot be spatially resolved and they might be hard to distinguish from ordinary active galactic nuclei (AGN). There are many proposed signatures to identify the presence of a SMBBH using electromagnetic waves, for instance: Doppler variations due to the orbital motion \\citep{dorazio2015nature}, binary periodicities in the light curves \\citep{Valtonen2006, Graham2015a,Graham2015b, Liu2019,Saade2020}, interruption of jet emission \\citep{Shoenmakers2000, Liu2003} and \"spin-flips\" of the BH after merger \\citep{Merritt2002}, dual-radio cores \\citep{Rodriguez2006}, profile shifts of broad emission lines \\citep{Dotti2009, Bogdanovic2009}, a ``notch\" in the optical\/IR spectrum \\citep{Roedig2014, Sesana2012}, periodicities in the thermal spectrum due to a short residence time for gas in the disk \\citep{Bowen2019} and X-ray periodicities \\citep{Sesana2012,Roedig2014}. The feasibility of detecting some of these signatures depends strongly on binary properties such as mass-ratio and orbital separation (see also \\cite{Krolik2019} for likely source counts). \n\nBecause the interstellar gas of the merged galaxy would have a considerable amount of angular momentum, a circumbinary disk should form around the binary \\citep{Springel2005, Chapon2013}. For mass-ratios close to one, the system would present a gap of radius $\\sim 2a$ between the binary's semimajor axis $a$ and the edge of the circumbinary disk. Accretion then occurs through two streams from the circumbinary to each black hole. Depending on its angular momentum, the material from the streams can eventually start orbiting the BHs, forming ``mini-disks''. On the other hand, the time-dependent quadrupole potential of the binary system can induce a density concentration in the edge of the circumbinary disk on a small azimuthal range, breaking the axisymmetric accretion \\citep{MM08}. This feature is usually referred as the \\textit{lump} \\citep{Shi12, Noble12, noble2021}. When the lump is formed, it behaves as a coherent $m=1$ density mode orbiting the system at a frequency $\\sim 0.25 \\: \\Omega_{\\rm bin}$, where $\\Omega_{\\rm bin}$ is the binary frequency \\citep{noble2021, armengol2021}. The time-dependence of the gas available for accretion onto the black holes is then dominated by the lump. If the residence time of the gas in the mini-disk is shorter (or comparable to) the modulation period of the lump, then the accretion rate and luminosity of the mini-disks are modulated by the lump  \\citep{Bowen2018, dAscoli2018, Bowen2019}. Whether this modulation happens depends mostly on the orbital separation (see Section \\ref{sec-results})\n\nNumerical simulations are key tools to make accurate models from these highly non-linear systems. Circumbinary accretion has been largely investigated using viscous hydrodynamical models in 2D with Newtonian gravity \\citep{MM08,DOrazio13,Farris2014a,Farris2014b,DOrazio2016,Munoz2016,Miranda2017,\nDerdzinski2019,Munoz2019,Moody2019,Mosta2019,Duffell2020,Zrake2020,Munoz2020a,Munoz2020b,\nTiede2020,Derdzinski2021}. In these simulations, the system can be evolved for $\\mathcal{O}(100-1000)$ binary orbits, when a relaxed stage is reached, and the effects of the initial conditions are suppressed. However, these simulations rely on artificial sink conditions for the black hole and their stress model is not self-consistent, i.e. they adopt an ad-hoc description for the viscosity.\n\nMore realistic simulations using MHD have been performed in Newtonian gravity \\citep{Shi12,Shi2015}, in approximated GR \\citep{Noble12, Zilhao2015, armengol2021, noble2021, Bowen2018, Bowen2019},  and in full numerical relativity \\citep{Gold14, Gold2014b, Farris11, paschalidis2021minidisk,cattorini2021,giacomazzo2012}. MHD simulations are more computationally expensive than two-dimensional $\\alpha-$viscous simulations because they demand three-dimensional domains and fine resolutions. Moreover, GRMHD simulations including both the mini-disks and the circumbinary disk need to resolve different dynamical timescales, evolve the spacetime metric, and thus are usually too expensive to evolve for many orbits.\n\nSince the properties of the mini-disks are necessarily tied to the circumbinary accretion, it is important to perform simulations linking these regimes in order to obtain a proper global description of the system. A viable method to accomplish this was presented in \\cite{Bowen2018}, where a snapshot of an evolved circumbinary disk simulation \\citep{Noble12} was used as initial data for studying mini-disk accretion onto non-spinning black holes. In this way, \\cite{Bowen2019} showed that mini-disks at close binary separations exhibit a quasi-periodic filling and depletion cycle determined by the lump and the short inflow time of the mini-disks.\n\nAn important property of supermassive binary black holes that many of these studies miss is the spin \\citep{Lin1979}. When binary BHs are approaching merger, spins can have an important impact on the spacetime evolution: they can alter the orbital motion of the system \\citep{Campanelli:2006uy,  hemberger2013final,healy2018hangup}, induce precession and nutation \\citep{Campanelli:2006fy}, repeatedly flip their sign \\citep{Lousto:2014ida, Lousto:2015uwa, lousto2016spin, Lousto:2018dgd} and even tilt the orbital orientation \\citep{kesden2014gravitational}. Spins also have a key role in the accretion of matter into BHs \\citep{KHH05} and binary BHs. For instance, accretion rate per unit mass near the circumbinary disk's inner edge depends on the spin, altering the mass profile in the inner part of the disk \\citep{armengol2021}. Moreover, the character of the flow within the mini-disks depends on the ratio between the radius at which they are fed and the radius of the innermost stable circular orbit (ISCO) \\citep{Bowen2018, Gold14, paschalidis2021minidisk}, which is strongly dependent on the spin. \n\nOn the other hand, spinning black holes are expected to launch electromagnetic outflows \\citep{dVH03, HKdVH04, mckinney2004measurement, dVHKH05, KHH05, TchekhovskoyChap}. The generation of a net Poynting flux in a binary BH system approaching merger has been modeled within General Relativistic Force-Free Electrodynamics (GRFFE) \\citep{palenzuela2010magnetospheres, palenzuela2010dual, neilsen2011, moesta2012detectability} and ideal GRMHD \\citep{Giacomazzo12,Kelly2017, Kelly2021,Farris12,Gold:2013APS,Gold14,paschalidis2021minidisk}. In the force-free regime, given a homogeneous plasma threaded by a constant magnetic field, the Blandford-Znajek (BZ) mechanism \\citep{blanford1977} proved to operate efficiently around each BH, leading to a pair of collimated jets in a double-helical structure that coalesces after merger \\citep{palenzuela2010magnetospheres}. The dynamics of binary BHs in a homogeneous medium have also been investigated in the ideal GRMHD regime, where the inertia of the plasma is taken into account; in this case, accretion of the plasma onto the BHs leads to a steeper growth of the initial magnetic field during the inspiral, via compression and magnetic winding, resulting in higher luminosities \\citep{Giacomazzo:2012ApJ}. \nAt the same time, the inertia of the (homogenous) plasma falling onto the BHs interferes with the propagation of the electromagnetic flux, causing less collimated magnetic structures compared with GRFFE  \\citep[see also][]{Kelly2017, Kelly2021}.  Further, \\cite{Farris12,Gold:2013APS,Gold14,paschalidis2021minidisk} performed GRMHD simulations that include the circumbinary disk in the domain and found the development of net Poynting fluxes emitted from the polar regions of each BH that coalesce at larger distances.\n\nThere are only a few GRMHD simulations of circumbinary accretion into spinning SMBBHs. Very recently, \\cite{armengol2021} presented the first long-term circumbinary GRMHD simulation of spinning black holes (with the inner cavity excised), and \\cite{paschalidis2021minidisk} presented the first numerical relativity simulation of mini-disks around spinning BHs, confirming previous expectations that the ISCO plays a key role in the mini-disk mass \\citep{Bowen2018,Bowen2019,Gold2014b}. A natural next step is to consider more realistic simulations where both mini-disks and a properly relaxed circumbinary disk around spinning BBHs are taken into account. This is important to make accurate predictions of the light curves and spectra of these systems.\n\nIn this work, we present the results of a GRMHD simulation of mini-disk accretion around spinning black holes of spins $a = 0.6~M$ aligned with the orbital angular momentum. We evolve the ideal GRMHD equations on top of a binary BH spacetime that is moving on a quasi-circular orbit starting at $20 M$ separation. We use an approximate BBH spacetime that uses Post-Newtonian (PN) trajectories for the BHs in the inspiral regime, but is valid at every space point, including the BH horizons. As initial data for the plasma, we use a steady-state snapshot of a circumbinary disk simulation performed in \\cite{Noble12}. We compare our new simulation with a previous non-spinning simulation \\citep{Bowen2018,Bowen2019} that uses the same initial data. \n\nThe paper is structured as follows: In Section \\ref{sec-simusetup}, we present the simulations set-up, the numerical methods, the spacetime approximation that we use, and the initial data. In Section \\ref{sec-results}, we present our results. First, we give an overview of the main features of the system and the relation with previous simulations. In Section \\ref{sec-mdot} we investigate the accretion rate, inflow time, and mass evolution of the mini-disks, and in Section \\ref{sec-struct} we analyze the mini-disks structure, the specific angular momentum distribution, and the azimuthal density modes. Then, in Section \\ref{sec-emfluxes} we analyze the outflows, the magnetized structure of the system, and the variability of the Poynting flux, comparing spinning and non-spinning results. Finally, in Section \\ref{sec-discussion} we discuss some of the implications of our results and in Section \\ref{sec-conclusions} we summarize the main points and results of the paper.\n\n\\textbf{Notation and conventions.} We use the signature $(-,+,+,+)$ and we follow the Misner-Thorne-Wheeler convention for tensor signs. We use geometrized units, $G=c=1$. We use Latin letters $a,b,c,...=0,1,2,3$ for four dimensional components of tensors, and $i,j,k,...=1,2,3$ for space components.\n\n\\section{Simulation set-up}\n\\label{sec-simusetup}\n\nWe evolve the equations of ideal GRMHD in the spacetime of a binary black hole system using the finite-volume code \\harm. Our goal is to analyze the effects of the black hole spin in the minidisks. For that purpose, we compare two simulations of an equal-mass binary black hole, with and without spins. The non-spinning simulation, denoted as \\texttt{S0}{}, was performed in \\cite{Bowen2019} using an analytical metric built by matching different spacetimes \\citep{Mundim2014}. We perform a new simulation, denoted \\texttt{S06}{}, with BHs having spins of $\\chi=0.6$ aligned with the orbital angular momentum of the binary, using an approximate analytical metric described in \\cite{combi2021} (a concise summary is given in Sec.~\\ref{sec:metric}). For both simulations, we use the same grid and initial data in order to have a faithful comparison (see Section~\\ref{sec:metric} for details). \n\n\\subsection{GRMHD equations of motion}\n\\label{sec:EOM}\n\n\nAssuming that the plasma does not influence the spacetime, we evolve the ideal GRMHD equations in the dynamical binary BH metric described in Section \\ref{sec:metric}. The equations of motion are given by the conservation of the baryon number, the conservation of the energy-momentum tensor, and Maxwell's equations with the ideal MHD condition:\n\\begin{equation*}\n\\nabla_a (\\rho u^a) = 0, \\quad \\nabla_{a} {T^{a}}_{b} = \\mathcal{F}_{b}, \n\\end{equation*}\n\\begin{equation}\n\\nabla_{a} \\:^{*} F^{ab} = 0,  \\quad u_{a} F^{ab} =0,\n\\end{equation}\nwhere $\\rho$ is the rest-mass density, $F^{ab}\/\\sqrt{4 \\pi}$ is the Faraday tensor\\footnote{Following \\cite{Noble09}, we absorb the factor $1\/\\sqrt{4 \\pi}$ in the definition of the tensor $F^{ab}$.}, $u^{a}$ is the four-velocity of the fluid, $\\mathcal{F}_b$ is the radiated energy-momentum per 4-volume unit, and the MHD energy-momentum tensor is\n\\begin{equation}\nT^{ab} := (\\rho h + b^2) u^a u^b + (P + \\frac{1}{2} b^2) g^{ab} - b^a b^b,\n\\end{equation}\nwhere $h:= (1+ \\epsilon +P\/\\rho)$ is the specfic enthalpy, $\\epsilon$ is the specific internal energy, $P$ is the pressure, $b^a:= \\:^{*}F^{ab}u_b:$ is the four-vector magnetic field, and $b^2:=b^ab_a$ is proportional to the magnetic pressure, $p_{\\rm m} := b^2\/2$. We follow \\cite{Noble12,Noble09} and write these coupled equations of motion in manifest conservative form as\n\\begin{equation}\n\\partial_t {\\bf U}\\left({{\\mathbf{P}}}\\right) =                                                                       \n-\\partial_i {\\bf F}^i\\left({{\\mathbf{P}}}\\right) + \\mathbf{S}\\left({{\\mathbf{P}}}\\right),\n\\label{eq:cons-form-mhd}\n\\end{equation}\nwhere ${\\bf P}$ are the \\textit{primitive} variables, ${\\bf U}$ the \\textit{conserved} variables, ${\\bf F}^i$ the \\textit{fluxes}, and $\\mathbf{S}$ the \\textit{source} terms.  These are given explicitly as:\n\\begin{eqnarray}\n\\mathbf{P}: & = & [\\rho, u, \\tilde{u}^j, B^{j}] \\ ,\n\\label{primitive-mhd} \\\\\n{\\bf U}\\left({{\\mathbf{P}}}\\right) & = & \\sqrt{-g} \\left[ \\rho u^t ,\\, {T^t}_t +\n  \\rho u^t ,\\, {T^t}_j ,\\, B^j\\right] \\ , \\label{cons-U-mhd} \\\\\n{\\bf F}^i\\left({{\\mathbf{P}}}\\right) & = & \\sqrt{-g} \\left[ \\rho u^i ,\\, {T^i}_t +\n  \\rho u^i ,\\, {T^i}_j ,\\, \\left(b^i u^j - b^j u^i \\right)\\right], \\label{cons-flux-mhd} \\\\\n\\mathbf{S}\\left({{\\mathbf{P}}}\\right) & = & \\sqrt{-g} \\left[ 0 ,\\,\n  {T^\\kappa}_\\lambda {\\Gamma^\\lambda}_{t \\kappa} - \\mathcal{F}_t ,\\,\n  {T^\\kappa}_\\lambda {\\Gamma^\\lambda}_{j \\kappa} - \\mathcal{F}_j ,\\, 0\n  \\right], \\label{cons-source-mhd}\n\\end{eqnarray}    \nwhere $g$ is the determinant of the metric, ${\\Gamma^\\lambda}_{\\alpha \\beta}$ \nare the Christoffel symbols, $u:=\\rho \\epsilon$ is the internal energy, $\\tilde{u}^j:= u^j -g^{tj}\/g^{tt}$ is the velocity relative to the normal spacelike hypersurface, and $B^{j}:= \\: ^{*} F^{it}$ is the magnetic field, which is both a conserved and a primitive variable \\footnote{We denote the magnetic field in the frame of normal observers (proportional to the constrained-transported field) as $B^i$, while we denote the magnetic field in the frame of the fluid as $b^{a} = \\left(1\/u^t\\right)\\left({\\delta^a}_\\nu + u^a u_b \\right)B^b$.}. We close the system with a $\\Gamma$-law equation of state, $P=(\\Gamma-1) \\rho \\epsilon$, where we set $\\Gamma= 5\/3$.\n\nThe source term in the energy-momentum conservation ensures that part of the dissipated energy caused by MHD turbulence is converted to radiation that escapes from system. We assume radiation is removed from each cell independently of all the others, isotropically in the fluid frame. In this way, we set $\\mathcal{F}_{a} = {\\cal L}_{c}u_\\beta$, where $\\mathcal{L}_c$ is the cooling function. We use the prescription used in \\cite{Noble12} for the rest-frame cooling rate per unit volume:\n\\begin{equation}\n  \\mathcal{L}_{c} = \\frac{\\rho \\epsilon}{t_{\\mathrm{cool}}} \\left( \\frac{\\Delta S}{S_0} + \\left| \\frac{\\Delta S}{S_0}\\right| \\right)^{1\/2} \\ , \n\\end{equation}\nwhere $t_{\\mathrm{cool}}$ is the cooling timescale, where the disk radiates away any local increase in entropy, $\\Delta S := S - S_0$, where $S := p \/ \\rho^{\\Gamma}$ is the local entropy. Our target entropy, $S_0 = 0.01$, is the initial entropy of each accretion disk in the simulation. The timescale $t_{\\mathrm{cool}}$ is determined by the local fluid orbital period, following the prescriptions in \\cite{Bowen17,dAscoli2018}.\n\n\\begin{center}\n\\begin{deluxetable}{c c c}[ht!]\n\\tablewidth{\\columnwidth}\n\\tablecolumns{3}\n\\tablecaption{\\label{tab:grid}}\n\\tablehead\n{\n  \\colhead{Simulation}  & \\colhead{\\texttt{S06}}     & \\colhead{\\texttt{S0}}\n}\n\\startdata\nSpin parameter [$\\chi$]   \t \t\t     &  0.6    &  0.0       \\\\\nBH$_{1,2}$ mass [$M_{\\rm BH}$] \t\t         &    0.5    & \t\t\t\\\\\nMass-ratio [$q$] \t\t         &    1    & \t\t\t\\\\\nFinal time [$t_f$]               & $8000 \\: M$ & $6000 \\: M$ \\\\\nFinal separation [$r_{12}(t_f)$] & $16.6 \\: M$ & $17.8 \\: M$ \\\\\n\\# Orbits \t\t\t\t\t\t & $15$\t   & $12.5$    \\\\\nInit. separation [$r_{12}(0)$]   &  $20 \\: M$      &    \\\\\nInit. total mini-disk mass [$M_0$] & $20$  &   \\\\\nAverage orbital period [$T_{\\mathcal{B}}$] &  & $530\\: M$   \\\\\nLump orbital frequency [$\\Omega_{\\rm lump}$] & $0.28 \\:  \\Omega_{\\mathcal{B}}$ &    \\\\\nISCO radius [$r_{\\rm ISCO}$] & $2.82 \\:  M_{\\rm BH}$ & $5.0 \\:  M_{\\rm BH}$   \\\\\nTruncation radius [$r_{\\rm trunc}$] & $0.4 \\: r_{12}$ &    \\\\\n\n\\hline \\\\\nGrid [$(x^1 \\times x^2 \\times x^3)$] &      & $(600,160,640)$ \\\\\nPhysical Size [($r_{\\rm min},r_{\\rm max}$)]   &  & $(2M,260M)$ \\\\ \n\\enddata\n\\tablecomments{Physical and grid parameters of both non-spinning and spinning simulations.} \n\\end{deluxetable}\n\\end{center}\n\n\n\\subsection{Code details, grid, and boundary conditions}\n\nWe solve equations \\eqref{eq:cons-form-mhd} using the high-resolution, shock-capturing methods implemented in \\harm. Following \\cite{Noble12}, we use a piecewise parabolic reconstruction of the primitive variables for the local Lax-Friedrichs flux at each cell interface and the Flux CT method to maintain the solenoidal constraint \\citep{toth2000}. Once the numerical fluxes are found, the equations are evolved in time using the method of lines with a second-order Runge-Kutta method. The primitive variables are recovered from the evolved conserved variables using the 2D method developed in \\cite{Noble06}.\n\n\\begin{figure}[htb!]\n  \\centering\n  \\includegraphics[width=\\columnwidth]{\\figfolder\/init_data.png}\n  \\caption{Initial data used in the simulation with quasi-equilibrated mini-toris around the black holes. In white thin lines we plot the warped spherical grid every each 50 cells.}\n  \\label{fig-initdata}\n\\end{figure}\n\n\\begin{figure*}[htb!]\n  \\centering\n  \\includegraphics[width=1.0\\columnwidth]{\\figfolder\/spinrho423.png}\n  \\includegraphics[width=\\columnwidth]{\\figfolder\/bh2_spinpol.png}\n  \\caption{ Left Panel: Rest-mass density snapshot of the fluid in the equatorial plane for \\texttt{S06}{}. Right panel: Rest-mass density snapshot of the fluid in the meridional plane corrotating with the (second) black hole. White stream lines represent the comoving magnetic field projected on the meridional plane.}\n  \\label{fig-rhp}\n\\end{figure*}\n\nThe grid and boundary conditions used in this simulation are the same as in \\cite{Bowen2018} and \\cite{Bowen2019}. We use a time-dependent, double fish-eye, warped spherical grid, centered in the center of mass of the binary system, developed in \\cite{WARPED} (see full details of the grid used in \\cite{Bowen2019}). The maximum physical size of the grid is set to $r_{\\rm max}=13 \\: r_{12}(0)$, containing the circumbinary disk of \\texttt{RunSE} in \\cite{Noble12}, that we use as initial data. We use outflow boundary conditions on the radial ($x^1$) boundaries, demanding the physical radial velocity $u^r$ to be oriented out of the domain; if not, we reset the radial velocity to zero and solve for the remaining velocity components. Poloidal coordinates ($x^2$) have reflective, axisymmetric boundary conditions at the polar axis cutout and the azimuthal coordinates ($x^3$) have periodic boundary conditions.\n\nThe resolution is given by $N_{r} \\times N_{\\theta} \\times N_{\\phi} = 600{\\times}160{\\times}640$ cells. The shape of the grid in the circumbinary region matches the grid used in \\cite{Noble12} and is sufficient to resolve the magnetorotational instability (MRI) in the circumbinary disk. Because of our polar grid resolution and off-grid-center location of the BHs in the spherical grid, our configuration does not include a full 32 cells per scale height in the mini-disks on the side farthest from the center-of-mass. \n\n\\subsection{Binary BH spacetime and initial data}\n\\label{sec:metric}\n\nThe spacetime of the binary black hole is approximated by superposing two Kerr spacetimes on a Minkowski background.  We describe it in terms of harmonic coordinates.  The metric can be written schematically as\n\\begin{equation}\ng_{ab} = \\eta_{ab} + M_1 \\mathcal{H}^1_{ab} (\\vec{x}_1,\\vec{v}_1)+M_2 \\mathcal{H}^2_{ab} (\\vec{x}_2,\\vec{v}_2),\n\\label{eq-supmetric}\n\\end{equation}\nwhere $\\eta_{ab}$ is the Cartesian Minkowski metric, $\\mathcal{H}^A_{ab}$ is the boosted black hole term for $A=1,2$, $M_A$ is the mass, and $\\lbrace \\vec{x}_A(t), \\vec{v}_A(t) \\rbrace$ are the position and velocity of the black hole. The trajectories are obtained by solving the Post-Newtonian equations of motion for a spinning binary BH in quasi-circular motion at 3.5 PN order. In \\cite{combi2021}, we showed that this analytical metric constitutes a good approximation to a vacuum solution of Einstein's field equations for a binary approaching merger\\secondrev{, see also \\cite{east2012}, \\cite{varma2018} and \\cite{ma2021extending} for similar approaches in the context of numerical relavity}. The metric \\eqref{eq-supmetric} is computationally efficient, compared with previous approaches, and easy to handle for different parameters. In the non-spinning simulation, the spacetime was represented by an semi-analytical metric built by stitching different approximate solutions of Einstein's equation \\citep{Mundim2014}. \\firstrev{In \\cite{combi2021}, we compared the matching and superposed metrics evolving two GRMHD simulations for the non-spinning case and we found that they are completely equivalent in this context. Moreover, we analyze the spacetime scalars and integrated Hamiltonian constraints for each one and we found that (a) they remain small and well-behaved up to a separation of $10 ~M$, and (b) the constraints remain invariant when we change the BH spin, and thus no pathologies are introduced by the spin}.\n\n\\secondrev{\nThe level at which these constraints are violated is comparable to the very low level achieved in the numerical simulations performed in \\cite{zlochower2016inspiraling}, who used the matching metric as initial data for evolving Einstein's equations. Constraint violations in this\nevolution, which can be damped using the CCZ4 scheme, mainly introduce deviations to the trajectories (eccentricity) and errors in the masses and spins of the BHs. In our analytical metric, however, there are no such dynamical effects since we solve the trajectories using the PN approximation and the BH masses\/spins are fixed. The small constraint violations, relative to the mass of the BHs, might produce small errors in the gas dynamics, but these are washed out by MHD turbulence in our simulation \\citep{Zilhao2015}.}\n\nAlthough the metric uses PN trajectories and thus is only valid in the inspiral regime, it is mathematically well-defined at every point in space, including the horizons of the black holes; hence, no artificial sink terms or large excisions are needed in the evolution. We apply, however, a mask inside the horizon to avoid the singularity of each black hole. In particular, we do not evolve the hydro fluxes inside the masked region and we set to zero the magnetic fluxes. This allows us to evolve the induction equations in the whole domain and preserve the solenoidal constraints.\n\nFor this simulation, we use an equal-mass black hole binary, $M_1=M_2=M\/2$, with an initial separation of $r_{12}(0)=20\\:M$, where relativistic effects are important \\citep{Bowen17} and the orbit is shrinking due to gravitational radiation. In \\texttt{S06}{} we set the spins of the black holes perpendicular to the orbital plane (i.e. no precession) with a moderate value of $\\chi_i \\equiv a_i\/M = 0.6$. Since spin couples with the orbital motion, the trajectories of the holes change with respect to a non-spinning system. In particular, the inspiral is delayed because of the \\textit{hang-up} effect \\citep{Campanelli:2006fg}, and the orbital frequency increases (see Figure \\ref{fig-traj}). \n\n\\begin{figure}\n  \\includegraphics[width=\\columnwidth]{\\figfolder\/trajectories.pdf}\n \n  \\caption{Properties a binary black hole system with aligned spins of $\\chi=0.0$ (blue dashed line), $\\chi=0.6$ (solid orange line), and $\\chi=0.9$ (green dashed line). In the top panel, we plot the radial velocity, $\\dot{r}_{12}$,  normalized with its initial value, in the middle panel the orbital separation $r_{12}$, and in the bottom panel the period of the orbit, $P$.}\n  \\label{fig-traj}\n\\end{figure}\n\n\\firstrev{In order to compare with the previous simulation, we take the same initial data for the matter fields as in \\cite{Bowen2018}}. We start with a snapshot of the circumbinary disk from \\cite{Noble12}, previously evolved for $5 \\times 10^4 M$ ($\\sim 80$ orbits). At this time, the disk is in a turbulent state and accretion into the cavity is dominated by a $m=1$ density mode, the so-called lump, that is orbiting at the inner edge of the circumbinary disk. In \\cite{Noble12}, a zero spin PN metric in harmonic coordinates was used to evolve the system. As shown in \\cite{armengol2021}, the bulk properties of the circumbinary disk are not sensitive to the spin, even for high values, so it is a good initial state for our spinning simulation. We interpolate this data onto our grid and we initialize two mini-disks inside the cavity, see Figure \\ref{fig-initdata}. We then clean magnetic divergences introduced by the interpolation to the new grid using a projection method as explained in \\cite{Bowen2018}.\n\n\\subsection{Diagnostics}\n\nProperties of the circumbinary disk and other global properties of the system are better analyzed in the center of mass coordinates. On the other hand, to analyze mini-disk properties such as fluxes, we compute quantities on the (moving) BH frame. We define the BH frame at a given time slice with a boosted coordinate system centered at each BH (see \\cite{combi2021}); we denote these BH coordinates with a bar, $\\lbrace \\bar{t}, \\bar{r}, \\bar{\\theta}, \\bar{\\phi} \\rbrace$. \\firstrev{We notices that all our diagnostic are written in the harmonic coordinate gauge}. Fluxes and other local properties in this frame are computed in post-process, interpolating the global grid into a spherical grid centered in the BH with the Python package \\texttt{naturalneighbor} \\footnote{\\url{https:\/\/github.com\/innolitics\/natural-neighbor-interpolation}} that implements a fast Discrete Sibson interpolation \\citep{interpolation}. \n\nWeighted surface averages of an MHD quantity $\\mathcal{Q}$ with respect to a quantity $\\sigma$ are defined as\n\\begin{equation}\n\\langle \\mathcal{Q} \\rangle_\\sigma := \\frac{\\int dA \\: \\sigma \\: \\mathcal{Q}}{\\int dA \\:  \\sigma},\n\\end{equation}\nwhere $dA:= d\\theta d\\phi \\sqrt{-g}$. A time-average of a surface-average is defined as\n\\begin{equation}\n\\langle \\langle \\mathcal{Q} \\rangle  \\rangle := \\frac{1}{\\Delta t} \\int^{t_{i}}_{t_{f}} dt \\langle \\mathcal{Q} \\rangle,\n\\end{equation}\nwhere we always sum over a given time interval after the initial transient. \n\n\\section{Results}\n\\label{sec-results}\n\n\\subsection{Overview of the system and previous studies}\n\nIn the steady-state of an equal-mass SMBBH, a lump orbits the edge of the circumbinary disk at an average  frequency $\\langle \\Omega_{\\rm lump} \\rangle = 0.28 ~ \\Omega_{\\mathcal{B}}$ \\citep{Noble12, Shi12, DOrazio2013, DOrazio2016, Farris15a, Farris15b, armengol2021, noble2021} modulating the accretion into the cavity. When one of the BHs passes near the lump, it peels off part of lump's inner edge, forming a stream that feeds the black hole with a beat frequency of $\\Omega_{\\rm beat} :=  \\Omega_{\\mathcal{B}} - \\langle \\Omega_{\\rm lump} \\rangle \\sim 0.72 \\: \\Omega_{\\mathcal{B}}$. This stream is almost ballistic, formed by fluid particles with relatively low angular momentum \\citep{Shi2015}. This material can start orbiting the black hole as it approaches, forming a mini-disk. The maximum size of the mini-disk is determined by the tidal truncation radius of the binary, or Hill's sphere. The residence time of matter in the mini-disks is determined by the ratio of the truncation radius and the radius of the ISCO. At close relativistic separations, such as the ones here, the mini-disks will be out of inflow equilibrium with the circumbinary lump accretion, and thus the masses oscillate quasi periodically in a filling and depletion cycle \\citep{Bowen2018, Bowen2019}.\n\nFor relativistic binaries, the tidal truncation radius is approximately at $r_t \\sim 0.4 \\: r_{12}(t)$ \\citep{Bowen17}, similar to the Newtonian value, estimated to be $\\sim 0.3 \\: r_{12}$ \\citep{Paczynski:1977, Papaloizou:1977a, ArtymLubow94}. Since spin is a second order effect in the effective potential \\citep{armengol2021}, mild values of spin do not change the truncation radius significantly for binary separations greater or close to $20M$. The most relevant difference between our two simulations, \\texttt{S06}{} and \\texttt{S0}{}, is the location of the ISCO: $r_{\\rm ISCO}(\\chi=0.6) = 2.82 \\: M_{\\rm BH_{i}} $ for \\texttt{S06}{}, and $r_{\\rm ISCO}(\\chi=0.0)= 5.0 \\: M_{\\rm BH_{i}} $ for \\texttt{S0}{} (both in given here in harmonic coordinates). The smaller ISCOs of the spinning black holes allow material with lower angular momentum to maintain circular orbits closer to the BH instead of plunging in directly (see Section \\ref{sec-struct}). \n\nIn the following sections, we analyze how the size of the ISCO plays a role in the accretion rate, inflow time, and periodicities, as well as in the structure of the mini-disks. We will also examine how the presence of an ergosphere in the spinning case helps the black holes to produce more Poynting flux. We will also analyze how the variability of the fluxes is connected with the variability of the accretion rate, dominated by the lump. \n\n\\subsection{Mass evolution, accretion rate, and inflow time}\n\\label{sec-mdot}\nWe start the analysis by calculating the \\textbf{integrated rest-mass} of each mini-disk, defined as\n\\begin{equation}\nM := \\int^{r_t(t)}_{r_{\\mathcal{H}}}   dV \\: \\rho u^{t},\n\\label{eq-mass}\n\\end{equation}\nwhere $r_t(t):= 0.4 \\: r_{12}(t)$ is the truncation radius, $r_{\\mathcal{H}}$ is the BH horizon, and $dV:= \\sqrt{-g} d^3x$ is the volume element. In both simulations there is an initial transient due to the initial conditions that lasts approximately $\\sim 3$ orbits for \\texttt{S0}{} and $\\sim 4$ orbits for \\texttt{S06}{} (see Figure \\ref{fig-massminidisk}). Both simulations start with two quasi-equilibrated mini-tori around the holes, with a specific angular momentum distribution adapted specifically for non-spinning black holes, following the prescription described in \\cite{Bowen17}. Since we are also using this initial data for the spinning simulation, the initial tori have an excess of angular momentum, making the transient slightly longer in \\texttt{S06}{}. We analyze each simulation after this transient, marked in the plots as a vertical line. As a time unit, we use $T_{\\mathcal{B}} = 530 \\: M$, the average binary period of the non-spinning simulation.\n\n\\begin{figure}[htb!]\n  \\centering\n  \\includegraphics[width=\\columnwidth]{\\figfolder\/masses_comparison.pdf}\n  \\caption{Upper panel: the mass fraction evolution $M_i$ of each minidisk for \\texttt{S06}{} (solid lines) and \\texttt{S0}{} (dashed lines), where we define $M_{\\rm tot}:= M_1(t)+M_2(t)$. Lower panel: total mass evolution for \\texttt{S06}{} (solid line) and \\texttt{S0}{} (dashed line), where $M_0:= M_1(0)+M_2(0)$. The black line indicates the end of the transient phase.}\n  \\label{fig-massminidisk}\n\\end{figure}\n\nWe find that although the mini-disks of \\texttt{S06}{}, like those in \\texttt{S0}{}, go through a filling-depletion cycle, the mini-disks around spinning BHs in \\texttt{S06}{} are more massive than in \\texttt{S0}{} by a factor of $2$ through most of the evolution (Figure~\\ref{fig-massminidisk}), although they both follow the same decay. When the mass fraction of a mini-disk is more than $50\\%$ of the total mass, we say that the disk is in its \\textit{high state}; otherwise, it is in its \\textit{low state}. The cycle of the mass fraction is similar in both simulations, although marginally smaller in amplitude for \\texttt{S06}{}. The frequency of the cycle is associated with the orbital frequency and thus is higher in the spinning case, as can be seen plainly after $\\sim 8$ orbits (upper panel {color{blue} of Fig.~\\ref{fig-massminidisk}}). On the other hand, at $t=11 \\: T_{\\mathcal{B}}$, we observe a slight increase of mass in the system. Because the lump grows and oscillates radially around the cavity \\citep{armengol2021}, it generates stronger accretion events onto the black holes with a lower frequency rate. \n\nWe also analyze the accretion rate evolution at the horizon in the black hole rest-frame, which is given by\n\\begin{equation}\n\\dot{M} := \\oint_{r_{\\mathcal{H}}}  d\\bar{A} \\: u^{\\bar{r}} \\rho.\n\\label{eq:massflux}\n\\end{equation}\n\nHere, and in the remainder of this paper, overbars indicate (harmonic) coordinates whose origin is the center of one of the black holes. We plot the sum of the accretion rate in each BH, $\\dot{M}_{\\rm Tot}$, for each simulation in Figure \\ref{fig-accrate}. We find that the accretion rate evolution is overall similar after the transient in both \\texttt{S06}{} and \\texttt{S0}{}. Moreover, the time-dependence of the mini-disk mass is, to a first approximation, a smoothed version of the accretion rate's time-dependence, e.g. see Figure \\ref{fig-accratemassbh1} for BH$_1$ in \\texttt{S06}{}.\n\n\\begin{figure}\n  \\includegraphics[width=\\columnwidth]{\\figfolder\/acc_rate_tot.pdf}\n  \\caption{Total accretion rate evolution $\\dot{M}_{\\rm Tot} \\equiv \\dot{M}_{\\rm BH_2} + \\dot{M}_{\\rm BH_2}$ in \\texttt{S06}{} (solid lines) and \\texttt{S0} (dashed lines)}\n  \\label{fig-accrate}\n\\end{figure}\n\nThe differences in the masses per cycle are closely related to the inflow time of particles in the mini-disk. It is useful then to define an (Eulerian) inflow time as the characteristic time for a fluid element to move past a fixed radius $r$ \\citep{KHH05}. On average, this can be defined as\n\\begin{equation}\nt_{\\rm inflow}^{-1} :=  \\frac{1}{\\bar{r}} \\Big \\langle \\langle V^{\\bar{r}} \\rangle_{\\rho} \\Big \\rangle,\n\\end{equation}\nwhere $V^{\\bar{r}}:= u^{\\bar{r}}\/u^{\\bar{t}}$ is the transport velocity. In Figure \\ref{fig-inflow}, we show the inflow time as a function of coordinate radius for BH$_1$, averaged in time for the first part (solid lines) and second part (dashed lines) of both simulations. Inside the truncation radius, the inflow time is consistently longer in \\texttt{S06}{} than in \\texttt{S0}, generally by tens of percent. \n\n\\begin{figure}\n  \\includegraphics[width=\\columnwidth]{\\figfolder\/accratemassbh1.pdf}\n  \\caption{Accretion rate ($\\dot{M}_{\\rm BH_1}$) in red solid lines and mass ($M_{\\rm BH_1}$) in black dot-dashed lines for a mini-disk around BH$_1$ in \\texttt{S06}{}. Both quantities were rescaled for plotting. }\n  \\label{fig-accratemassbh1}\n\\end{figure}\n\nAs the binary shrinks, the inflow time at the truncation radius diminishes from $\\sim 0.41 \\: T_{\\mathcal{B}}$ to $\\sim 0.31 \\: T_{\\mathcal{B}}$ for the spinning simulation. The average inflow time at the truncation radius is well below the beat period, $ T_{\\rm beat}  = 1.3 T_{\\mathcal{B}}$, on which the mini-disk refills. Our mean inflow time is also significantly shorter than typical inflow timescales of Keplerian orbits around single black holes \\citep{KHH05}. This suggests that accretion in our mini-disks is driven by different mechanisms than single BH disks. Moreover, the similar measures of accretion rates at the horizon for \\texttt{S06}{} and \\texttt{S0}{} (see again Figure \\ref{fig-accrate}), seem to imply a shared accretion mechanism between spinning and non-spinning systems (see Sec.~\\ref{sec-struct}).\n\n\\begin{figure}[htb]\n  \\includegraphics[width=\\columnwidth]{\\figfolder\/inflow_vs_r.png}\n  \\caption{Inflow time as a function of radius  in harmonic coordinates for BH$_1$ in \\texttt{S06}{} (maroon) and \\texttt{S0}{} (dark blue). Solid lines are time averages over the first half of the simulation, while dot-dashed lines are time averages of the last half of the simulation. Thin lines show the instantaneous inflow time every $300M$ for the spinning case. The maroon (dark blue) vertical line represents the ISCO of the spinning (non-spinning) black hole. The vertical black line is the initial truncation radius of the binary.}\n  \\label{fig-inflow}\n\\end{figure}\n\n\n\\begin{figure}[ht!]\n  \\includegraphics[width=\\columnwidth]{\\figfolder\/psd_all.pdf}\n  \\caption{Power spectral density of the mini-disk's masses for \\texttt{S06}{} (upper panel) and \\texttt{S0}{} (lower pannel) using a Welch algorithm with a Hamming window size and a frequency of $10M$. The confidence intervals at $3 \\sigma$ are shown as shadowed areas.}\n  \\label{fig-psd}\n\\end{figure}\n\n\nThe quasi-periodic behavior of the system can be described through a power density spectrum (PSD) of the mini-disk's masses. In Figure \\ref{fig-psd} we show the PSD, normalized with the power of the highest peak within the simulation. Most features of the system's quasi-periodicities are shared in both simulations and were described in detail in \\cite{Bowen2019}. We find, still, some interesting differences. The PSDs for $M_1$ and $M_2$ taken individually peak at the beat frequency in both simulations. The PSD of the total mass of the mini-disks, $M_1+M_2$, has a peak at $2\\Omega_{\\rm beat}$ in \\texttt{S0}{}, while the latter is severely damped in \\texttt{S06}{}. Indeed, if the individual masses vary with a characteristic frequency $\\Omega_{\\rm beat}$, and these are out of phase with the same amplitude, we expect their sum to vary with $2\\Omega_{\\rm beat}$. In \\texttt{S06}{}, however, the inflow time of the mini-disks is larger and the depletion period of a mini-disk briefly coexists with the filling period of the other mini-disk, reducing the variability of the total mass. On the other hand, the beat frequency is slightly higher for \\texttt{S06}{} ($\\Omega_{\\rm beat} = 0.71 \\Omega_{\\mathcal{B}} $) than \\texttt{S0}{} ($\\Omega_{\\rm beat}= 0.68 \\Omega_{\\mathcal{B}} $) as the orbital frequency of the spinning BHs is higher. Further, since we evolved the binary for longer, in \\texttt{S06}{} we find a more prominent amplitude at $\\sim 0.20 \\Omega_{\\mathcal{B}}$. We can associate this low-frequency power to the radial oscillations of the lump around the cavity that produces additional accretion events \\citep{armengol2021}. We observe this frequency is closely related but different from the orbital frequency of the lump at $0.28 \\Omega_{\\mathcal{B}}$, which could be related to the orbital frequency increasing during the inspiral (notice we measure the PSD at fixed frequencies).\n\nFinally, because we use a spherical grid with a central cutout, we cannot analyze the effects of the sloshing of matter between mini-disks \\citep{Bowen17}. To estimate how much mass we lose through the cutout, we compute the accretion rate at the inner boundary of the grid. This mass loss constitutes only $5 \\%$ of the total mass accreted by the BHs throughout the simulation, although the instantaneous accretion can be close to $20\\%$ of the accretion onto a single BH.  We do not expect this small mass loss to alter the main conclusions of this work, namely, the differences between mini-disks in spinning and non-spinning BBH.\n \n\\subsection{Structure and orbital motion in mini-disks}\n\\label{sec-struct}\n\nIn this section, we analyze in detail how the structure of the mini-disks in \\texttt{S06}{} compares with mini-disks in \\texttt{S0}{}. We first focus on how the spin changes the surface density distribution and the azimuthal density modes in the mini-disks. We then investigate the angular momentum of the fluid and how it compares with the angular momentum at the ISCO. \n\n\\begin{figure}[ht]\n  \\includegraphics[width=\\columnwidth]{\\figfolder\/double_dens.png}\n  \\caption{Surface density snapshot for \\texttt{S06}{} (upper row) and \\texttt{S0}{} (lower row) at $t=4000M$ and $t=4060M$ respectively, where the phase of the binary is the same in both simulations. White dashed lines indicate the truncation radius and solid white lines indicate the ISCO. The sense of rotation of the binary is counter-clockwise.}\n  \\label{fig-surfden-2D}\n\\end{figure}\n\n\\begin{figure}[ht]\n  \\includegraphics[width=\\columnwidth]{\\figfolder\/surf_density_phiavg.pdf}\n  \\caption{Surface density average in the azimuthal ranges $\\Delta \\phi_1=(\\pi\/4,3\\pi\/4) $ (positive $y_{\\rm BH}$-axis) and $\\Delta \\phi_2=(5\\pi\/4,7\\pi\/4)$ (negative $y_{\\rm BH}$-axis) for BH$_1$ in \\texttt{S06} (upper panel) and \\texttt{S0} (lower panel). Solid lines represent a time average on the high state of the cycle, while dot-dashed lines represent a time average over the low state. For reference, we indicate the direction of the orbital BH velocity. Dashed vertical lines indicate the position of the ISCO.}\n  \\label{fig-surfden}\n\\end{figure}\n\nThe surface density is defined as $\\Sigma(t, r,\\phi) := \\int  d\\theta \\rho \\sqrt{-g}\/\\sqrt{-\\sigma}$, where \\firstrev{we use $\\sqrt{-\\sigma}= \\sqrt{g_{\\phi \\phi} g_{rr}}$ as the surface metric of the equatorial plane.} In Figure \\ref{fig-surfden-2D}, we plot the surface density in both \\texttt{S06}{} and \\texttt{S0}{}, for the same orbital phase at the $7$th orbit. In this plot, the mini-disk around BH$_1$ (right side) is in the peak of the mass cycle. In both simulations we can clearly notice the lump stream plunging directly into the hole. There is also circularized gas orbiting BH$_1$ in both simulations, but there is much more of it in \\texttt{S06}{}. On the other hand, we observe that BH$_2$ (left side), in its low state, has a noticeable disk structure in \\texttt{S06}{}, while the material is already depleted for \\texttt{S0}{}.\n\nWe can quantify these differences in structure by computing the average surface density over two ranges of $\\bar{\\phi}$, as measured in the BH frame, representing the front and back of the mini-disk with respect to the orbital motion. We define\n\\begin{equation}\n\\langle \\Sigma(r,t) \\rangle := \\frac{ \\int_{\\Delta \\bar{\\phi}} d\\bar{l} \\: \\Sigma }{\\int_{\\Delta \\bar{\\phi}} d\\bar{l} },\n\\end{equation}\nwhere $d\\bar{l}:= d\\bar{\\phi} \\sqrt{g_{\\bar{\\phi} \\bar{\\phi}} (\\bar{\\theta}=\\pi\/2)}$.  In Figure \\ref{fig-surfden}, we plot  $\\langle \\langle \\Sigma(r) \\rangle \\rangle $ for $\\Delta \\bar{\\phi}_1=(\\pi\/4,3\\pi\/4) $ and $\\Delta \\bar{\\phi}_2=(5\\pi\/4,7\\pi\/4)$, averaging in time over the high and low state of the mini-disk separately. In the high-state, the mini-disk accumulates more material at the front while the back of the mini-disk is flatter. In the low-state, both simulations show a flatter profile, with a slightly higher density at the front. The asymmetry between front and back arises because of the orbital motion of the black holes, capturing and accumulating the stream material as they orbit. In \\texttt{S06}{}, the density profile is steeper near the ISCO for high and low states. In the outer part of the mini-disk, the slope of the surface density for both \\texttt{S0}{} and \\texttt{S06}{} have a similar profile, indicating a common truncation radius. The density is higher in \\texttt{S06}{} by a factor of $\\sim 2$ in both states.\n\n\\begin{figure}[htb]\n  \\includegraphics[width=\\columnwidth]{\\figfolder\/modes_spin_full.pdf}\n  \\caption{Azimuthal density modes for BH$_1$ in \\texttt{S06}{} (upper panel) and \\texttt{S0}{} (lower panel) normalized with the zero mode $D_0$. The black vertical line represents the end of the transient in \\texttt{S06}.}\n  \\label{fig-modes}\n\\end{figure}\n\nAnother important property of mini-disks in relativistic binaries is the presence of non-trivial azimuthal density modes. When the accreting stream of the lump impacts the mini-disk, it generates a pressure wave, forming strong spiral shocks \\citep{Bowen2018}. This induces an $m=1$ density mode in the mini-disk that competes with the $m=2$ mode excited by the tidal interaction of the companion black hole. The spiral wave patterns can be analyzed decomposing the mini-disk rest-mass  density in azimuthal Fourier modes \\citep{Zurek1986}, $\\rho(\\bar{\\phi})\\equiv \\sum_m D_{m} \\exp{(-im\\bar{\\phi})}$, where\n\\begin{equation}\nD_{m} := \\int^{r_t(t)}_{r_{\\mathcal{H}}} d\\bar{V} \\rho \\exp{(-im\\bar{\\phi})}.\n\\end{equation}\n\nLet us compare these modes in \\texttt{S0}{} and \\texttt{S06}{} for BH$_1$. From Figure \\ref{fig-modes}, we observe that both simulations share common features. In both simulations, the mini-disks are mainly dominated by $m=1$ modes, followed closely by $m=2$ modes. We also observe important $m=3,4$ contributions. The modes are excited in the high state of the cycle, where the mini-disks increase their mass and the stream is accreted onto the black holes. In \\texttt{S0}{}, the amplitudes of the modes are noticeably larger than \\texttt{S06}{}, while in the latter the amplitudes grow as the system evolves. The growth of modes in \\texttt{S06}{} is correlated with the mass decrease of the mini-disks. This behavior could indicate that, as the mini-disks become less massive, the density modes are really representing the single-arm stream of the lump that plunges directly into the hole. Azimuthal modes grow to large amplitudes in the high phase of a mini-disk only when the material orbiting the BH is less or equally massive than the material with low angular momentum that plunges from the lump, occurring around $10 T_{\\mathcal{B}}$ in \\texttt{S06}{} (see Figure \\ref{fig-circmass} and discussion below). This is another consequence of the disk-like structure of the mini-disks surviving for longer time in the spinning case.\n\n\n\\begin{figure}[htb]\n  \\includegraphics[width=\\columnwidth]{\\figfolder\/sam_vs_r.png}\n  \\caption{Specific angular momentum as a function of radius for \\texttt{S06}{} (upper panel) and \\texttt{S0}{} (lower panel) for both BHs. The time-averages are in solid lines, and the individual values are the very thin lines. The Keplerian value is plotted in dashed green lines.}\n  \\label{fig-samvsr}\n\\end{figure}\n\nOur analysis so far indicates that a considerable amount of the matter in the mini-disk region is plunging directly from the lump to the black hole. In order to further analyze the orbital motion of the fluid in the mini-disk, we compute the density-weighted specific angular momentum in the BH frame:\n\\begin{equation}\n\\langle \\bar{\\ell} \\rangle_{\\rho} :=  \\langle -u_{\\bar{\\phi}}\/u_{\\bar{t}} \\rangle_{\\rho}.\n\\end{equation}\n\nIn Figure \\ref{fig-samvsr}, we show the time-average of $\\langle \\bar{\\ell} \\rangle_{\\rho}$ for both BHs and both simulations. In absolute terms, $\\langle \\bar{\\ell} \\rangle_\\rho$ is nearly the same for both the spinning and non-spinning cases, with the spinning case only slightly greater.  This is because the specific angular momentum of the material that falls into the cavity is essentially determined by the stresses at the inner edge of the circumbinary disk. These stresses are determined by binary torques and the plasma Reynolds and magnetic stresses \\citep{Shi12,Noble12}. Indeed, in \\cite{armengol2021} we found that these quantities depend weakly on spin outside the cavity.\nOn the other hand, their relation to their respective Keplerian (circular orbit) values, $\\bar{\\ell}_K(\\bar{r},\\chi)$, is quite different because they depend strongly on the spin. In \\texttt{S06}{}, the distribution of the angular momentum tracks closely the Keplerian value. For \\texttt{S0}{}, the behavior is always sub-Keplerian on average. \n\nLet us assume that the angular momentum distribution of the circumbinary streams is independent of spin for a fixed binary separation and mass-ratio. In that case, our simulation data indicate that the angular momentum with which the streams arrive at the mini-disk would be greater than the ISCO angular momentum when the BH spin is $\\chi> 0.45$. This estimate could serve as a crude criterion for determining whether mini-disks form in relativistic binaries.\n\n\\begin{figure}[htb]\n  \\includegraphics[width=\\columnwidth]{\\figfolder\/mass_circ.pdf}\n  \\caption{Sub-Keplerian and (super-)Keplerian components of the mass for BH$_1$ in \\texttt{S06}{} (upper panel) and \\texttt{S0}{} (lower panel)}\n  \\label{fig-circmass}\n\\end{figure}\n\nWe can also use the specific angular momentum to distinguish the material in the mini-disk with high angular momentum that manages to orbit the black hole from the low angular momentum part that plunges in. To do so, we recompute the mass as in equation~\\eqref{eq-mass}, taking fluid elements with $\\bar{\\ell}< \\bar{\\ell}_K$ and $\\bar{\\ell} \\geq \\bar{\\ell}_K$ separately. In Figure \\ref{fig-circmass} we plot the evolution of the sub-Keplerian and super-Keplerian mass components for BH$_1$ in \\texttt{S06}{} and \\texttt{S0}{}. In \\texttt{S06}{} after the initial transient, a little more than half of the mass comes from relatively high angular momentum fluid. As the system inspirals, however, the truncation radius decreases, and the masses of these two components become nearly equal. In \\texttt{S0}{}, on the other hand, most of the fluid has relatively low angular momentum. This sub-Keplerian component has roughly the same mass in \\texttt{S06}{} and \\texttt{S0}{}, while the mass of the high angular momentum component of the fluid is much greater in \\texttt{S06}{}, as expected.\n\nAlthough a fair amount of the mass in the mini-disk has relatively high angular momentum and manages to orbit the black hole in \\texttt{S06}, the accreted mass onto the BH, in both simulations, is always dominated by the low angular momentum part that plunges directly. To demonstrate this, we compute the average accretion rates for low and high angular momentum particles as we did with the mass. Figure \\ref{fig-mdot-am} shows that the total accretion rate onto the BH has a flat radial profile in both \\texttt{S06}{} and \\texttt{S0}{}, with very similar average values. Accretion by low angular momentum particles dominates at all radii, although the high angular momentum contribution becomes comparable to the low angular momentum one near the ISCO for \\texttt{S06}{}.\n\n\\begin{figure}[htb]\n  \\includegraphics[width=\\columnwidth]{\\figfolder\/mdot_circ.pdf}\n  \\caption{Time averaged accretion rate in BH$_1$ for \\texttt{S06}{} (solid lines) and \\texttt{S0}{} (dashed lines) considering particles with low (blue) and high (red) angular momentum. The vertical dashed blue and dot-dashed red lines mark the ISCO for \\texttt{S0}{} and \\texttt{S06}{}, respectively.}\n  \\label{fig-mdot-am}\n\\end{figure}\n\nWe can also compute the density-weighted specific energy, $E:=\\langle -u_{\\bar{t}} \\rangle_{\\rho}$, the mass-weighted sum of rest-mass, kinetic, and binding energy for individual fluid elements. As can be seen in Figure~\\ref{fig-ut}, on average, fluid in the mini-disks around the spinning black holes is more bound than in the non-spinning case. On the other hand, fluid in both \\texttt{S06}{} and \\texttt{S0}{} is more bound than particles on circular orbits. Near the ISCO, the specific energy drops sharply inward in both cases, as is often found when accretion physics is treated in MHD: stress does not cease at the ISCO when magnetic fields are present.\n\n\\begin{figure}[htb]\n  \\includegraphics[width=\\columnwidth]{\\figfolder\/minusut_vs_r.pdf}\n  \\caption{Density-weighted specific energy $E = - u_{\\bar{t}}$, averaged in time as a function of radius for BH$_1$ in \\texttt{S06}{} and \\texttt{S0}{}. The dashed line represents the specific energy for a geodesic particle in circular motion in each case. Vertical lines indicate the location of the ISCO.}\n  \\label{fig-ut}\n\\end{figure}\n\nWhen the mini-disk is in its high state, the spiral shocks heat up the mini-disks and increase their aspect ratio, $h\/r$. Given our cooling prescription, the entropy is kept close to its original value, regulating the aspect ratio to $h\/r=0.1$. However, the gas scale height increases dramatically where the lump stream impacts the mini-disk. At the peak of the accretion cycle, the aspect ratio rises to $h\/r \\sim 2$, but the gas cools before the next accretion event.\n\n\\subsection{Electromagnetic and hydrodynamical fluxes}\n\\label{sec-emfluxes}\n\nIn this section we analyze the extraction of energy from the system in two forms: outward electromagnetic luminosities, arising from a Poynting flux, and unbound material. In particular, we analyze how these energy fluxes change with spin and how their variability is characterized by the same periodicity as the accretion. \n\nThe electromagnetic luminosity from each mini-disk  evaluated in the BH frame is:\n\\begin{equation}\nL_{\\rm EM} (t,\\bar{r}) = \\oint_{\\bar{r}} d\\bar{A} \\: \\mathcal{S}^{\\bar{r}},\n\\end{equation}\nwhere the Poynting flux is $\\mathcal{S}^{\\bar{i}}:=(T_{\\rm EM})_{\\: \\: \\bar{t}}^{\\bar{i}}$. In Figure~\\ref{fig-poyntvst}, we plot the EM luminosity as a function of time for \\texttt{S06}{} and \\texttt{S0}{}, evaluated on spheres of radius $\\bar{r}=10 \\: M $ that follow each black hole.  These luminosities are normalized to the average accretion rate $\\langle \\dot{M} \\rangle = 0.002$, so they are equivalent to the rest-mass efficiency of the jets.  The most noteworthy element in Figure~\\ref{fig-poyntvst} is that the EM luminosity is an order of magnitude larger in S06 than S0. In \\texttt{S0}{}, there are no clear long-term trends while in \\texttt{S06}{}, there is a secular growth of $L_{\\rm EM}$ until $7.5  T_{\\mathcal{B}}$, when it starts declining. \n\n\n\\begin{figure}[htb]\n  \\includegraphics[width=\\columnwidth]{\\figfolder\/poynting_flux_spin_nospin.pdf}\n  \\caption{EM luminosity evolution in the BH frame from a sphere at $r=10M$ for \\texttt{S06}{} (upper panel) and \\texttt{S0}{} (lower panel).}\n  \\label{fig-poyntvst}\n\\end{figure}\n\nThe instantaneous efficiency  $\\eta = L_{\\rm EM}(t)\/\\dot{M}(t)$ increases during the first several orbits, saturating at $\\approx 0.05$ in the case of \\texttt{S06}{}, but an order of magnitude lower for \\texttt{S0}{},  as can be seen in Figure \\ref{fig-efficiency}. The efficiency in \\texttt{S06}{} with spin parameter 0.6 is rather larger than $0.013$, the value found by \\cite{dVHKH05} for the efficiency of a single BH with specific angular momentum of 0.5 surrounded by a statistically time-steady disk. \\secondrev{Both \\texttt{S0}{} and \\texttt{S06}{} present a similar secular growth of $\\eta(t)$ until $10 T_{\\mathcal{B}}$, where they slightly drop and plateau.}  \\firstrev{Measuring fluxes in the comoving frame, we found that non-spinning black holes produce negligible EM luminosity, which is consistent with the fundamental idea of the Blandford-Znajek mechanism \\citep{blanford1977, komissarov2001} and has been confirmed in many simulations \\citep{McKG04,dVHKH05,HK06,TchekhovskoyChap}. However, because the BHs are orbiting around the center of mass this could power additional EM fluxes \\citep{neilsen2011,palenzuela2010dual}. In order to capture the electromagnetic fluxes from the entire binary, we move to the center of mass frame and calculate the Poynting flux through a sphere of radius $r=100 \\: M$ surrounding the binary system.}\n\n\\firstrev{In Figure \\ref{fig-poyntingcm-t}, we plot this quantity for both \\texttt{S06}{} and \\texttt{S0}{} as a function of retarded time $t-r\/\\langle v \\rangle$, where $\\langle v \\rangle$ is the mean velocity of the outflow. We also plot the sum of the Poynting fluxes around each mini-disk, measured in the BH comoving frame. We notice that the fluxes in \\texttt{S0}{} are in average five times larger in the center of mass frame compared with the BH frame, suggesting that there is a kinematic contribution to $L_{\\rm EM}$ from the orbital motion of the black holes and possibly from the circumbinary disk. The luminosities in \\texttt{S06}{}, on the other hand, differ between frames only by tens of percent, suggesting that the rotation of the black hole dominates the extraction of EM energy here \\footnote{We have also checked the Poynting luminosity as a function of radius and we observed that, in average, it has varations of around $30 \\%$ far from the source, which is expectable as we lose resolution and the outflow interacts with the atmosphere floor of the simulation.}. During our simulations, the speed of the black holes remains fairly constant at $v\\sim 0.1$ but, closer to the merger, the orbital speed might enhance significantly the electromagnetic luminosity, as seen e.g. by \\citep{palenzuela2010dual, Farris2012, kelly2020electromagnetic}.} \n\nIn both \\texttt{S06}{} and \\texttt{S0}{} the EM luminosities are variable, and in both a Fourier power spectrum reveals a periodic modulation at the frequency of the circumbinary disk's inner edge, i.e., the ``lump\" frequency.  However, in \\texttt{S06}{} there is an additional modulation of similar amplitude at twice the beat frequency, proving it is tied closely to accretion, see Figure \\ref{fig-psd_lum}.\n\n\n\\begin{figure}[htb]\n  \\includegraphics[width=\\columnwidth]{\\figfolder\/inst_efficiency_bh.pdf}\n  \\caption{Instantaneous efficiency $\\eta$ of the Poynting flux for \\texttt{S06}{} (solid lines) and \\texttt{S0}{} (dashed lines) in the BH frame measured at $\\bar{r}=10M$.}\n  \\label{fig-efficiency}\n\\end{figure}\n\n\\begin{figure*}[ht!]\n\\begin{center}\n  \\includegraphics[width=\\columnwidth]{\\figfolder\/spin_poynt_meridional_bh1.png}\n  \\includegraphics[width=\\columnwidth]{\\figfolder\/nospin_poynt_meridional_bh1.png}\n  \\caption{Meridional plot of a time average Poynting scalar for BH$_1$ in \\texttt{S06}{} (left) and in \\texttt{S0}{} (right). The black hole is at $x \\sim 10M$ and the center of mass is at $x=0M$. The red lines represent the division between bound and unbound material, while the dot-dashed white lines represent the magnetically dominated material.}\n  \\label{fig-ponynt-pol}\n\\end{center}\n\\end{figure*}\n\n\\begin{figure}[htb]\n  \\includegraphics[width=\\columnwidth]{\\figfolder\/poyntingflows_10100M.pdf}\n  \\caption{Evolution of the total Poynting flux measured in the BH frame (dashed lines) and in the (inertial) center of mass frame at $100 ~M$(solid lines) for both \\texttt{S06}{} and \\texttt{S0}{}. For the center of mass fluxes, we use the retarded time $t-r\/\\langle v \\rangle$ to account for the delay.}\n  \\label{fig-poyntingcm-t}\n\\end{figure}\n\n\nTo further study the spatial distribution of the electromagnetic flux, in Figure \\ref{fig-ponynt-pol}, we plot a meridional slice of the Poynting scalar $\\mathcal{P}:= \\mathcal{S}^{\\bar{i}} \\mathcal{S}_{\\bar{i}}$ for both \\texttt{S06}{} and \\texttt{S0}{},  averaged in time over half an orbit at $4000 ~ M$. The dotted-dashed white lines are defined by $b^2\/\\rho=1$, containing the regions that better approximate the magnetically dominated region, while the red solid lines are defined by the region where $h u_t =-1$, where the fluid becomes unbound. As expected, \\texttt{S06}{} has a much more prominent Poynting jet structure than \\texttt{S0}{}, which has almost zero jet power. The poloidal distribution of $\\mathcal{P}$ in \\texttt{S06}{} around the black hole has a parabolic shape, with most of the flux being emitted at mid-latitudes. Each individual jet shape is similar to those found around single BHs \\citep{nakamura2018}. In both cases, we can also notice the strong (bound) Poynting fluxes generated by magnetic stresses in the disk at the equatorial plane.\n\n\n\\begin{figure}[htb]\n  \\includegraphics[width=\\columnwidth]{\\figfolder\/poynting_sphere.png}\n  \\caption{Time average of Poynting scalar $\\mathcal{P}$ projected on a sphere of radius 60 M for spinning (left sphere) and non-spinning (right sphere) for unbound elements of fluid.}\n  \\label{fig-poyntingcm}\n\\end{figure}\n\n\n\\begin{figure}[htb]\n  \\includegraphics[width=\\columnwidth]{\\figfolder\/hydroflows_10060m.pdf}\n  \\caption{Hydro luminosity as a function of time measured in the center of mass frame for \\texttt{S06}{} (green lines) and \\texttt{S0}{} (blue lines) at different radii.}\n  \\label{fig-koutflows}\n\\end{figure}\n\n\nIn Figure \\ref{fig-poyntingcm}, we plot $\\mathcal{P}$, averaged over half an orbit in the corotating frame of the binary, for a sphere of radius $r=60 ~M$ at the center of mass.  In \\texttt{S06}{} the Poynting flux has a double cone structure that extends to larger polar angles than does the Poynting flux in \\texttt{S0}{}, likely because of interaction between the two jets.  On the other hand, in \\texttt{S0}{}, the Poynting flux is distributed on a uniform ring around the axis of the binary.  Unfortunately, our simulation coordinates require a cutout in the grid covering the polar axis running through the center of mass. This prevents an accurate study of the interaction of the jets. Nonetheless, because this cutout is small ($2 ^{\\circ}$) compared to the angular size of the jets, we are still able to pick out important features.\n\nBesides the electromagnetic fluxes, there are also hydrodynamical fluxes from the system. We define the hydro luminosity as the integral of the energy flux component of the hydrodynamic stress tensor minus the contribution of the rest-mass energy flux at that radius in order to get the `usable' energy flux \\citep{HK06}:\n\\begin{equation}\nL_{\\rm H} (t,r) := \\Big( \\oint_r dA \\:(T_{\\rm H})^r_t \\Big ) - \\dot{M}_{\\rm jet}\n\\label{eq-hydroflux}\n\\end{equation}\n\nIn Figure~\\ref{fig-koutflows}, we plot the hydro luminosity in the center of mass frame as a function of time for both \\texttt{S0}{} and \\texttt{S06}{}, at different radii, normalized by the averaged value of the accretion rate,  so that these luminosities, too, can be described in the language of rest-mass efficiency. In the integral over flux (Eqn.~\\eqref{eq-hydroflux}), we include only fluid elements that are both unbound according to the Bernoulli criterion ($-h u_t > 1$) and moving outward ($u^r>0$) (cf. \\cite{dVHKH05}). \\firstrev{Like the EM luminosity in S06, but not S0, the hydro luminosities are modulated for both spin cases at the ``lump\" frequency and at twice the beat frequency. \\secondrev{Figure~\\ref{fig-psd_lum} shows the Fourier power spectrum for the spinning case. We observe in Figure~\\ref{fig-koutflows} a secular growth of the hydro fluxes in \\texttt{S06}{} while in \\texttt{S0}{} they remain rather constant, with an average efficiency of $\\sim 2 \\%$.  Also like the EM case, the hydro energy flux is considerably greater in S06 than S0, but by a factor $\\sim 5$.  Such a contrast resembles the differences between the hydro efficiencies measured in spinning and non-spinning single BH simulations \\citep{HK06}.}}\n\n\\secondrev{We caution, however, that the luminosities measured at $100~M$ may not be the luminosities received at infinity. Energy can be easily converted from EM to hydro or vice versa.  Here we quote the values at $100~M$ because they are the largest we can measure within our grid.  Nonetheless the comparison between S06 and S0 demonstrates clearly that spinning BBH are much more efficient at creating coherent outflows carrying energy both electromagnetically and hydrodynamically than non-spinning BBH.}\n\n\n\\begin{figure}[htb]\n  \\includegraphics[width=\\columnwidth]{\\figfolder\/psd_all_lums.pdf}\n  \\caption{Power spectral density of the hydro ($L_{\\rm H}$) and EM luminosities ($L_{\\rm EM}$) for \\texttt{S06}{} (thick lines) and \\texttt{S0}{} (dashed lines) at $100~ M$ using a Welch algorithm with a Hamming window size and a frequency of $10~M$. The confidence intervals at $3 \\sigma$ are shown as shadowed areas for \\texttt{S06}{}. The two main peaks are given by twice the beat frequency, $2 \\Omega_{\\rm beat} = 1.4 \\Omega_{\\rm bin}$, and the lump accretion periodicity $\\sim 0.22\\Omega_{\\rm bin}$}\n  \\label{fig-psd_lum}\n\\end{figure}\n\n\\section{Discussion}\n\\label{sec-discussion}\n\n\nAlthough a few candidates have been identified, the existence of supermassive black hole binaries has not been confirmed. The direct detection of their gravitational waves by LISA or pulsar timing arrays (PTA) remains at least a decade into the future. Nevertheless, upcoming wide-field surveys such as the Vera C. Rubin Observatory, SDSS-V, and DESI, may discover many SMBBH candidates through their electromagnetic emission. \n\nIn order to confirm the presence of a SMBBH, we need to build accurate models and predictions of their electromagnetic signatures. Our GRMHD simulations will be useful for this purpose: as a next step, in \\cite{gutierrez2021}, we use these simulations to extract light curves and spectra using ray-tracing techniques \\citep{Noble07,dAscoli2018} with different radiation models and different masses. The results in this paper constitute the foundations to interpret the underlying physics of those predictions.\n\nCircumbinary and mini-disk accretion onto an equal-mass binary system has been largely studied in the past in the context of 2D $\\alpha-$viscous simulations. These simulations are particularly good for analyzing the very long-term behavior of the system, evolving sometimes for $1000$ orbits. Close to the black holes and at close separations, however, the inclusion of 3D MHD and accurate spacetime dynamics becomes necessary in order to describe the proper mechanisms of accretion and outflow. 2D $\\alpha-$disk simulations are not able to include spin effects and most of them do not include GR effects (see, however, \\cite{RyanMacFadyen17}). On the other hand, in this work we analyze the balance of hydro accretion from the circumbinary streams and conventional accretion from the internal stresses of the mini-disk; to properly model the latter, we need MHD. Moreover, the presence of a proper black hole, and its horizon, makes the accretion processes entirely self-consistent without adding adhoc sink conditions as used in Newtonian simulations (see,  however, \\cite{dittmann2021}). Finally, 3D MHD simulations are necessary to model magnetically-dominated regions and jets. The connection of the accretion and the production of electromagnetic luminosity was one of the main motivations of this work, and impossible to analyze in 2D hydro simulations.\n\nRecently, \\cite{paschalidis2021minidisk} presented GRMHD simulations of a system similar to the one analyzed in this paper: equal-mass, spinning binary black holes approaching merger. It is then interesting to compare our results and highlight the differences with their model and analysis. In their paper, they use a slightly higher spin value ($\\chi = 0.75$) and explore different spin configurations, including antialigned and up-down directions with respect to the orbital angular momentum. Their system has different thermodynamics than ours, using an ideal-gas state equation with $\\Gamma=4\/3$ and no cooling. Their focus is on the mass budget of the mini-disk (as in \\cite{Bowen2019}) and the electromagnetic luminosity when spin is included. They report that spinning black holes have more massive mini-disks and the electromagnetic luminosity is higher, with quantitative measures similar to what we find in this paper. \n\n\\firstrev{In our work, we analyze in great detail, for the first time, the accretion mechanisms onto the mini-disk and their connection to the circumbinary disk. We show that the BHs accretes in two different ways: through direct plunging of the stream from the lump's inner edge (that dominates the accretion), and through `conventional' stresses of the circular component orbiting the mini-disk. This is qualitatively different than single BHs disks and a direct consequenece of the short inflow time determined by $r_{\\rm ISCO} \/ r_{\\rm trunc}$; for larger separations and higher spins, we expect mini-disks to behave closer to conventional single BH disks}. Our simulations also differ significantly in the grid setup and initial data. We start our simulations with an evolved circumbinary disk snapshot, taken from \\cite{Noble12}, which is already turbulent and presents a lump (starting the simulation from a quasi-stationary torus, the lump appears after $\\sim 50$ orbits at these seperations, once the inner edge has settled). This is very important to accurately describe the periodicities of the system given by the beat frequency, which is set by the orbital motion of the lump. \\firstrev{These quasi-periodicities might be different if the thermodynamics change, e.g. if there is no cooling, although currently there are no sufficiently long 3D GRMHD simulations of circumbinary disks exploring this.} Interestingly, we found that the Poynting flux is also modulated by the beat frequency. For BBH approaching merger, this constitutes a possible independent observable if this periodicity is translated to jet emission. As expected, for spinning BHs, we also found more powerful Poynting fluxes, in agreement with \\cite{paschalidis2021minidisk}.\n\nWith our careful analysis of the accretion onto the mini-disks, we show that a disk-like structure survives for longer as the binary shrinks when the black holes have spin. Further explorations with higher spins will show how far these structures survive very close to merger.\n\n\\section{Conclusions}\n\\label{sec-conclusions}\n\nWe have performed a GRMHD accretion simulation of an equal-mass binary black hole with aligned spins of $a= 0.6 ~ M_{\\rm BH}$ approaching merger. We have compared this simulation with a previous non-spinning simulation of the same system, analyzing the main differences in mini-disk accretion and the variabilities induced by the circumbinary disk accretion. Our main findings can be summarized as follows:\n\n\\begin{itemize}\n\n\\item Mini-disks in \\texttt{S06}{}, where BHs have aligned spins $\\chi=0.6$, are more massive than in \\texttt{S0}{}, where BHs have zero spins, by a factor of two. The mass and accretion rate of mini-disks have quasi-periodicities determined by the beat frequency in both simulations (see Section \\ref{sec-mdot}).\n\n\\item The material in the mini-disk region can be separated into two components of relatively high and low angular momentum. The low angular momentum component mostly plunges directly from the lump edge, forming a strong single-arm stream. The (supra) Keplerian angular momentum component of the fluid is determined by the size of the ISCO and the truncation radius. We have shown that most of the mini-disk mass in \\texttt{S0}{} is sub-Keplerian while in \\texttt{S06}{} most of the material follows closely the Keplerian value up to the end of the evolution when it becomes comparable to the low angular momentum component. For binary parameters as in this simulation, we have also predicted a critical value of $\\chi \\sim 0.45$ for which most of the mass will have low angular momentum relative to the ISCO.\n\n\\item The accretion rates at the horizon in \\texttt{S06}{} and \\texttt{S0}{} are very similar through the evolution (see Fig. \\ref{fig-accrate}), as they are dominated by the plunging material from the lump stream (see Figure \\ref{fig-mdot-am}).\n\n\\item In \\texttt{S06}{} a jet-like structure is formed self-consistently around each BH (see Figures \\ref{fig-rhp} and \\ref{fig-ponynt-pol}). Due to the black hole spin, there is a well-defined Poynting flux (see Figure \\ref{fig-ponynt-pol}, left panel) with an efficiency of $\\eta \\sim 8 \\%$ (see Figure \\ref{fig-efficiency}).  On the other hand, in \\texttt{S0}{} the efficiency is closer to $\\eta \\sim 4 \\%$, and its Poynting flux is more homogeneous in space (see Figure \\ref{fig-ponynt-pol}, right panel).\n\n\\item The time evolution of the Poynting flux is modulated by the quasi-periodicity of the accretion, determined by the beat frequency. In the spinning case, the fluxes in the comoving frame grow and start decreasing at $7.5 T_{\\mathcal{B}}$, while in the non-spinning these remain fairly constant.\n\n\\end{itemize}\n\n\\section*{Acknowledgments}\n\nWe thank Eduardo Guti\\'errez, Vassilios Mewes, Carlos Lousto, and Alex Dittman, for useful discussions.\nL.~C., F.~L.~A, M.~C., acknowledge support from AST-2009330, AST- 754 1028087, AST-1516150, \nPHY-1707946. \nL.~C also acknowledges support from a CONICET (Argentina) fellowship. \n\nS.~C.~N. was supported by AST-1028087, AST-1515982 and OAC-1515969, \nand by an appointment to the NASA Postdoctoral Program at the Goddard Space Flight Center administrated by USRA through a\ncontract with NASA. J.H.K. was supported by AST-1028111, PHY-1707826, and AST-2009260\nD.~B.~B. is supported by the US Department of Energy through the Los Alamos \nNational Laboratory. Los Alamos National Laboratory is operated by Triad National \nSecurity, LLC, for the National Nuclear Security Administration of U.S. \nDepartment of Energy (Contract No. 89233218CNA000001).\n\nComputational resources were provided by the Blue Waters sustained-petascale computing NSF Projects No. OAC-1811228 and No. OAC-1516125\", \nand replace with \"Computational resources were provided by the TACC's Frontera supercomputer allocation No. PHY-20010 and AST-20021\nAdditional resources were provided by the RIT's BlueSky and Green Pairie Clusters   \nacquired with NSF grants AST-1028087, PHY-0722703, PHY-1229173 and PHY-1726215.\n\nThe views and opinions expressed in this paper are those of the authors \nand not the views of the agencies or US government.\n\n\\hfill \\break\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}