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+{"text":"\\section{Introduction}\n\nThe COMPASS experiment has recorded a large data set of the\ndiffractive dissociation reaction\n$\\pi^- + p \\to (3\\pi)^- + p_\\text{recoil}$ using a 190~GeV\/$c$ pion\nbeam on a liquid-hydrogen target.  This reaction is known to exhibit a\nrich spectrum of produced intermediate three-pion states.  In the\npast, several candidates for spin-exotic mesons have been reported in\npion-induced diffraction~\\cite{exotic}.  In diffractive reactions the\nbeam hadron is excited to some intermediate state $X^-$ via\n$t$-channel Reggeon exchange with the target.  At 190~GeV\/$c$ beam\nmomentum, pomeron exchange is dominant.  In the reaction considered\nhere, the $X$ decays into the $\\pi^-\\pi^+\\pi^-$ and $\\pi^-\\pi^0\\pi^0$\nfinal states, which are detected by the spectrometer.  The scattering\nprocess is characterized by two kinematic variables: the squared total\ncenter-of-mass energy s and the squared four-momentum transfer to the\ntarget $t = (p_\\text{beam} - p_{X})^2 < 0$.  It is customary to use\nthe reduced four-momentum transfer $t' \\equiv |t| - |t|_\\text{min}$\ninstead of $t$, where $|t|_\\text{min}$ is the minimum value of $|t|$\nfor a given invariant mass of $X$.  After all cuts, the data sample\nconsists of $46 \\times 10^6$ $\\pi^-\\pi^+\\pi^-$ and $3.5 \\times 10^6$\n$\\pi^-\\pi^0\\pi^0$ exclusive events in the analyzed kinematic region of\nthree-pion mass, $0.5 < m_{3\\pi} < 2.5~\\text{GeV}\/c^2$, and\nfour-momentum transfer squared, $0.1 < t' < 1.0~(\\text{GeV}\/c)^2$.\nFigure~\\ref{fig:mass} shows the $\\pi^-\\pi^+\\pi^-$ mass spectrum as\nwell as that of the $\\pi^+\\pi^-$ subsystem.  The known pattern of\nresonances $a_1(1260)$, $a_2(1320)$, and $\\pi_2(1670)$ is seen in the\n$3\\pi$ system along with $\\rho(770)$, $f_0(980)$, $f_2(1270)$, and\n$\\rho_3(1690)$ in the $\\pi^+\\pi^-$ subsystem.\n\n\\begin{figure}[tb]\n  \\centering\n  \\includegraphics[width=0.45\\textwidth]{massX_final}\n  \\includegraphics[width=0.456\\textwidth]{mass_isobar_final}\n  \\caption{Left: $\\pi^-\\pi^+\\pi^-$ invariant mass spectrum in the\n    analyzed range; Right: $\\pi^+\\pi^-$ mass distribution.}\n  \\label{fig:mass}\n\\end{figure}\n\n\n\\section{Partial-Wave Decomposition}\n\nIn order to disentangle the different contributing intermediate $3\\pi$\nstates $X$, a partial-wave analysis (PWA) was performed.  The PWA is\nbased on the isobar model, which assumes that the $X$ decays first\ninto an intermediate resonance, which is called the isobar, and a\n``bachelor pion''.  In a second step, the isobar decays into\n$\\pi^+\\pi^-$.  In accordance with the $\\pi^+\\pi^-$ invariant mass\nspectrum shown in Fig.~\\ref{fig:mass} and with analyses by previous\nexperiments, we include $[\\pi\\pi]_S$, $\\rho(770)$, $f_0(980)$,\n$f_2(1270)$, $f_0(1500)$, and $\\rho_3(1690)$ as isobars into the fit\nmodel.  Here, $[\\pi\\pi]_S$ represents the broad component of the\n$\\pi\\pi$ $S$-wave.  Based on the six isobars, we have constructed a\nset of partial waves that consists of 88 waves in total, including one\nnon-interfering flat wave representing three uncorrelated pions.  This\nconstitues the largest wave set ever used in an analysis of the $3\\pi$\nfinal state.  The partial-wave decomposition is performed in narrow\nbins of the $3\\pi$ invariant mass and makes no assumptions on the\n$3\\pi$ resonance content of the partial waves.  Each $m_{3\\pi}$ bin is\nfurther subdivided into non-equidistant bins in the four-momentum\ntransfer $t'$.  For the $\\pi^-\\pi^+\\pi^-$ channel 11 bins are used,\nfor the $\\pi^-\\pi^0\\pi^0$ final state 8 bins.  With this additional\nbinning in $t'$, the dependence of the partial-wave amplitudes on the\nfour-momentum transfer can be studied in detail.  The details of the\nanalysis model are described in Ref.~\\cite{long_paper}.\n\nThe partial-wave amplitudes are extracted from the data as a function\nof $m_{3\\pi}$ and $t'$ by fitting the five-dimensional kinematic\ndistributions of the outgoing three pions.  The amplitudes do not only\ncontain information about the partial-wave intensities, but also about\nthe relative phases of the partial waves.  The latter are crucial for\nresonance extraction.\n\nPartial waves are defined by the quantum numbers of the $X$ (spin $J$,\nparity $P$, $C$-parity, spin projection $M$), the naturality\n$\\epsilon = \\pm 1$ of the exchange particle, the isobar, and the\norbital angular momentum $L$ between the isobar and the bachelor pion.\nThese quantities are summarized in the partial-wave notation\n$J^{PC}\\,M^\\epsilon\\,\\text{[isobar]}\\,\\pi\\,L$.  Since at the used beam\nenergies pomeron exchange is dominant, 80 of the 88 partial waves in\nthe model have $\\epsilon = +1$.  The $C$-parity is by convention that\nof the neutral isospin partner of the $X^-$.\n\n\n\\section{The $J^{PC} = 1^{-+}$ Spin-Exotic Wave}\n\nThe 88-wave model contains also spin-exotic waves with $J^{PC}$\nquantum numbers that are forbidden for quark-antiquark states in the\nnon-relativistic limit.  The most interesting of these waves is the\n$1^{-+}\\,1^+\\,\\rho(770)\\,\\pi\\,P$ wave, which contributes less than 1\\%\nto the total intensity.  Previous analyses claimed a resonance, the\n$\\pi_1(1600)$, at about 1.6~GeV\/$c^2$ in this\nchannel~\\cite{bnl_1,compass_pb}.  Figure~\\ref{fig:1mp_intens} shows\nthe intensity of this partial wave for the two final states\n($\\pi^-\\pi^+\\pi^-$ in red, $\\pi^-\\pi^0\\pi^0$ in blue).  The two\ndistributions are scaled to have the same integral.\n\n\\begin{figure}[tb]\n  \\centering\n  \\includegraphics[width=0.45\\textwidth]{pwa_pi0_int_1-+1+rhopiP_sum_ccnorm}\n  \\caption{Intensity of the $1^{-+}\\,1^+\\,\\rho(770)\\,\\pi\\,P$ wave\n    summed over all $t'$ bins for the $\\pi^-\\pi^+\\pi^-$ (red) and the\n    $\\pi^-\\pi^0\\pi^0$ (blue) final state.}\n  \\label{fig:1mp_intens}\n\\end{figure}\n\nBoth decay channels are in fair agreement and exhibit a broad\nenhancement extending from about 1.0 to 1.8~GeV\/$c^2$.  In the 1.0 to\n1.2~GeV\/$c^2$ mass range the intensity depends strongly on the details\nof the fit model.  Peak-like structures in this region are probably\ndue to imperfections of the applied PWA model.  A remarkable change of\nthe shape of the intensity spectrum of the\n$1^{-+}\\,1^+\\,\\rho(770)\\,\\pi\\,P$ wave with $t'$ is observed (see\nFig.~\\ref{fig:1mp_tbins}).  At values of $t'$ below about\n0.3~$(\\text{GeV}\/c)^2$, we observe no indication of a resonance peak\naround $m_{3\\pi} = 1.6$~GeV\/$c^2$, where we would expect the\n$\\pi_1(1600)$.  For the $t'$ bins in the interval from 0.449 to\n$1.000~(\\text{GeV}\/c)^2$, the observed intensities exhibit a very\ndifferent shape as compared to the low-$t'$ region, with a structure\nemerging at about 1.6~GeV\/$c^2$ and the intensity at lower masses\ndisappearing rapidly with increasing $t'$.  This is in contrast to\nclean resonance signals like the $a_2(1320)$ in the\n$2^{++}\\,1^+\\,\\rho(770)\\,\\pi\\,D$ wave, which, as expected, do not\nchange their shape with $t'$ (see Fig.~\\ref{fig:2pp_tbins}).  The\nobserved $t'$ behavior of the $1^{-+}$ wave is therefore a strong\nindication that non-resonant contributions play a dominant role.\n\n\\begin{figure}[tb]\n  \\centering\n  \\includegraphics[width=0.45\\textwidth]{h20_tbin1}\n  \\includegraphics[width=0.45\\textwidth]{h20_tbin11}\n  \\caption{Intensity of the $1^{-+}\\,1^+\\,\\rho(770)\\,\\pi\\,P$ wave in\n    different regions of $t'$ for the $\\pi^-\\pi^+\\pi^-$ final state\n    (dark blue).  The partial-wave projections of Monte-Carlo data\n    generated according to a model of the Deck effect are overlaid in\n    green.}\n  \\label{fig:1mp_tbins}\n\\end{figure}\n\n\\begin{figure}[tb]\n  \\centering\n  \\includegraphics[width=0.45\\textwidth]{pwa_pi0_int_2++1+rhopiD_tbins_ccnorm_1}\n  \\includegraphics[width=0.45\\textwidth]{pwa_pi0_int_2++1+rhopiD_tbins_ccnorm_2}\n  \\caption{Intensity of the $2^{++}\\,1^+\\,\\rho(770)\\,\\pi\\,D$ wave in\n    different regions of $t'$ for the $\\pi^-\\pi^+\\pi^-$ (red) and the\n    $\\pi^-\\pi^0\\pi^0$ (blue) final state.}\n  \\label{fig:2pp_tbins}\n\\end{figure}\n\nIt is believed that the non-resonant contribution in the $1^{-+}$ wave\noriginates predominantly from the Deck effect, in which the incoming\nbeam pion dissociates into the isobar and an off-shell pion that\nscatters off the target proton to become on-shell~\\cite{deck}.  As a\nfirst step towards a better understanding of the non-resonant\ncontribution, Monte-Carlo data were generated that are distributed\naccording to a model of the Deck effect.  The model employed here is\nvery similar to that used in Ref.~\\cite{accmor_deck}.  The\npartial-wave projection of these Monte Carlo data is shown as green\npoints in Fig.~\\ref{fig:1mp_tbins}.  In order to compare the\nintensities of real data and the Deck-model pseudo data, the Monte\nCarlo data are scaled to the $t'$-summed intensity of the $1^{-+}$\nwave as observed in real data.  At values of $t'$ below about\n0.3~$(\\text{GeV}\/c)^2$, the intensity distributions of real data and\nDeck Monte Carlo exhibit strong similarities suggesting that the\nobserved intensity might be saturated by the Deck effect.  Starting\nfrom $t' \\approx 0.4~(\\text{GeV}\/c)^2$, the spectral shapes for Deck\npseudo-data and real data deviate from each other with the differences\nincreasing towards larger values of $t'$.  This leaves room for a\npotential resonance signal.  It should be noted, however, that the\nDeck pseudo data contain no resonant contributions. Therefore,\npotential interference effects between the resonant and non-resonant\namplitudes cannot be assessed in this simple approach.\n\n\n\\section{The $a_1(1420)$}\n\nA surprising find in the COMPASS data was a pronounced narrow peak at\nabout 1.4~GeV\/$c^2$ in the $1^{++}\\,0^+\\,f_0(980)\\,\\pi\\,P$ wave (see\nFig.~\\ref{fig:1pp_f0_intens}).  The peak is observed with similar\nshape in the $\\pi^-\\pi^+\\pi^-$ and $\\pi^-\\pi^0\\pi^0$ data and is\nrobust against variations of the PWA model.  In addition to the peak\nin the partial-wave intensity, rapid phase variations with respect to\nmost waves are observed in the 1.4~GeV\/$c^2$ region (see\nFig.~\\ref{fig:1pp_f0_phase}).  The phase motion as well as the peak\nshape change only little with $t'$.\n\n\\begin{figure}[tb]\n  \\centering\n  \\includegraphics[width=0.45\\textwidth]{pwa_pi0_int_1++0+f0piP_sum_ccnorm}\n  \\includegraphics[width=0.45\\textwidth]{fig_1a}\n  \\caption{Left: Intensity of the $1^{++}\\,0^+\\,f_0(980)\\,\\pi\\,P$ wave\n    summed over all $t'$ bins for the $\\pi^-\\pi^+\\pi^-$ (red) and the\n    $\\pi^-\\pi^0\\pi^0$ (blue) final states.  Right: Result of a\n    resonance-model fit to the $\\pi^-\\pi^+\\pi^-$ data~\\cite{a1_1420}.\n    The data points correspond to the red points in the left figure.}\n  \\label{fig:1pp_f0_intens}\n\\end{figure}\n\n\\begin{figure}[tb]\n  \\centering\n  \\includegraphics[width=0.45\\textwidth]{fig_1d}\n  \\includegraphics[width=0.45\\textwidth]{fig_1e}\n  \\caption{Examples for relative phases of the\n    $1^{++}\\,0^+\\,f_0(980)\\,\\pi\\,P$ wave with respect to the\n    $4^{++}\\,1^+\\,\\rho(770)\\,\\pi\\,G$ (left) and the\n    $1^{++}\\,0^+\\,\\rho(980)\\,\\pi\\,S$ wave (right)~\\cite{a1_1420}.  The\n    phases are shown for three different $t'$ regions indicated by the\n    color.}\n  \\label{fig:1pp_f0_phase}\n\\end{figure}\n\nIn order to test the compatibility of the signal with a Breit-Wigner\nresonance, a resonance-model fit was performed using a novel method,\nwhere the intensities and relative phases of three waves\n($1^{++}\\,0^+\\,f_0(980)\\,\\pi\\,P$, $2^{++}\\,1^+\\,\\rho(770)\\,\\pi\\,D$,\nand $4^{++}\\,1^+\\,\\rho(770)\\,\\pi\\,G$) were fit simultaneously in all\n11 $t'$ bins~\\cite{a1_1420}.  Forcing the resonance parameters to be\nthe same across all $t'$ bins leads to an improved separation of\nresonant and non-resonant contribution as compared to previous\nanalyses that did not incorporate the $t'$ information.  The\nBreit-Wigner model describes the peak in the\n$1^{++}\\,0^+\\,f_0(980)\\,\\pi\\,P$ wave well and yields a mass of\n$m_0 = 1414^{+15}_{-13}$~MeV\/$c^2$ and a width of\n$\\Gamma_0 = (153^{+8}_{-23})$~MeV\/$c^2$ for the $a_1(1420)$.  Due to\nthe high precision of the data, the uncertainties are dominated by\nsystematic effects.\n\nThe $a_1(1420)$ signal is remarkable in many ways.  It appears in a\nmass region that is well studied since decades.  However, previous\nexperiments were unable to see the peak, because it contributes only\n0.25\\% to the total intensity.  The $a_1(1420)$ is very close in mass\nto the $1^{++}$ ground state, the $a_1(1260)$.  But it has a much\nsmaller width than the $a_1(1260)$.  The $a_1(1420)$ peak is seen only\nin the $f_0(980)\\,\\pi$ decay mode of the $1^{++}$ waves and lies\nsuspiciously close to the $K\\, \\bar{K}^*(892)$ threshold.\n\nThe nature of the $a_1(1420)$ is still unclear and several\ninterpretations were proposed.  It could be the isospin partner to the\n$f_1(1420)$.  It was also described as a two-quark-tetraquark mixed\nstate~\\cite{wang} and a tetraquark with mixed flavor\nsymmetry~\\cite{chen}.  Other models do not require an additional\nresonance:\nRef.~\\cite{berger1,berger2} proposes resonant re-scattering\ncorrections in the Deck process as an explanation, whereas\nRef.~\\cite{bonn} suggests a branching point in the triangle\nrescattering diagram for\n$a_1(1260) \\to K\\, \\bar{K}^*(892) \\to K\\, \\bar{K}\\, \\pi \\to f_0(980)\\,\n\\pi$.\nMore detailed studies are needed in order to distinguish between these\nmodels.\n\n\n\n\\section*{Acknowledgments}\nThis work was supported by the BMBF, the MLL and the Cluster of\nExcellence Exc153 ``Origin and Structure of the Universe''.\n\n\n\\section*{References}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe weak decays of $\\Lambda_b$ provide valuable information of the\nCKM parameter $V_{cb}$ and serve as an ideal laboratory to study\nnon-perturbative QCD effects in the heavy baryon system. Recently\nthe DELPHI collaboration reported their measurement on the slope\nparameter $\\rho^2$ in the Isgur-Wise function and the the braching\nratio of the exclusive semi-leptonic process\n$\\Lambda_b\\to\\Lambda_cl\\bar\\nu_l$ \\cite{DELPHI}. This experimental\nmeasurement re-excites great theoretical interests in\nsemi-leptonic decays of $\\Lambda_b$ \\cite{AHN,SLL,GKLLW,\nHJKL,PWC,EFG}. From PDG06 \\cite{PDG06}, signals of several\nnon-leptonic processes, such as $\\Lambda_b \\to \\Lambda_c\\pi,\n\\Lambda_ca_1(1260)$ have been observed. These processes are a good\nprobe to test the factorization hypothesis which has been\nextensively explored in the B meson case \\cite{Factorization}. The\nforthcoming LHCb project is expected to accumulate  large samples\nof the b-flavor hadrons and offer an opportunity to study in\ndetail the $\\Lambda_b$ decays. Thus it probably is the time to\ninvestigate the $\\Lambda_b$ weak decays more systematically. In\nthis work, we will concentrate on the $\\Lambda_b\\to\\Lambda_c$\nsemi-leptonic and non-leptonic decays.\n\nAs in the B meson decays, the key for correctly evaluation on the\nrate of the semi-leptonic decays is how to properly derive the\nhadronic matrix element which is parameterized by the $\\Lambda_b \\to\n\\Lambda_c$ transition form factors. There are various approaches in\nthe market to evaluate these form factors: the QCD sum rules\n\\cite{HJKL}, quark models \\cite{CS,AHN,PWC,EFG}, perturbative QCD\napproach \\cite{SLL,GKLLW} {\\it etc.}. In this work, we will study\nthe heavy baryon form factors in the light-front quark model. The\nlight-front quark model is a relativistic quark model based on the\nlight-front QCD \\cite{light}. The basic ingredient is the hadron\nlight-front wave function which is explicitly Lorentz-invariant. The\nhadron spin is constructed using the Melosh rotation. The\nlight-front approach has been widely applied to calculate various\ndecay constants and form factors for the meson cases\n\\cite{Jaus,meson2,CCH1,CCH2,HW}. Different from the case discussed\nin \\cite{CS} where the light-front quark model was also employed, we\nadopt the diquark picture for baryons. It has been known for a long\ntime that two quarks in a color-antitriplet state attract each other\nand may form a correlated diquark \\cite{DJS}. Indeed, the diquark\npicture of baryons is considered to be appropriate for the low\nmomentum transfer processes \\cite{kroll,wilczek,yu,MQS}. In the\nconventional quark model, the heavy baryon is composed of one heavy\nquark and two light quarks. The light spectator quarks participate\nonly in the soft interactions as $\\Lambda_b $ transits into\n$\\Lambda_c$, hence it is reasonable to employ the diquark picture\nfor heavy baryons where the diquark serves as a whole spectator.\nConcretely, under the diquark approximation, $\\Lambda_b$ and\n$\\Lambda_c$ are of the one-heavy-quark-one-light-diquark(ud)\nstructure which is analogous to the meson case and a considerable\nsimplification in the calculations is expected. In fact, some\nnon-perturbative interactions between the two light quarks can be\neffectively absorbed into the constituent diquark mass.  In this\nphenomenological study, we use the rate of the semi-leptonic process\n$\\Lambda_b\\to\\Lambda_c l\\bar \\nu_l$ which will be accurately\nmeasured at LHCb and future ILC, to constrain the light scalar\ndiquark mass.\n\nFor the non-leptonic two-body decays $\\Lambda_b\\to\\Lambda_c M$ where\n$M$ denote light mesons, the amplitude is factorized to a product of\nthe meson decay constant and $\\Lambda_b\\to\\Lambda_c$ transition form\nfactors by the factorization assumption. Because there only a\ncolor-allowed diagram is involved, the factorization assumption is\nbelieved to be reliable in the B meson case \\cite{Factorization}.\nHowever, the theoretical predictions on the non-leptonic two-body\ndecays vary by a factor of 2-3 for various models. The main\ntheoretical uncertainties arise from the model evaluations of the\nform factors. In order to reduce model dependence and obtain a more\nreliable prediction, we study the semi-leptonic decays and\nnon-leptonic processes simultaneously. The present experimental data\nof the semi-leptonic decays (although the errors are still quite\nsizable) set constraints on the phenomenological parameters in the\nlight-front approach. With these parameters, even though not very\naccurate yet, we evaluate the $\\Lambda_b\\to\\Lambda_c$ form factors\nand make predictions on the widths of the  semi-leptonic decay\n$\\Lambda_b\\to\\Lambda_c+l\\bar\\nu$ and non-leptonic two-body decay\n$\\Lambda_b\\to\\Lambda_c+M$ where $M$ is a meson.\n\nWe organize our paper as follows. In section II, we formulate the\nform factors for the transition $\\Lambda_b\\to\\Lambda_c$ in the\nlight-front approach. We will show that in the heavy quark limit,\nthe resultant form factors are related to one universal Isgur-Wise\nfunction. In section III, the formulations of the decay ratios and\nthe polarizations for the semi-leptonic and non-leptonic two-body\ndecays are given. In section IV, we present our numerical results\nand all relevant input parameters are given explicitly. We then\ncompare our numerical results with the predictions by other\napproaches. Finally, Section V is devoted to conclusion and\ndiscussion.\n\n\n\n\\section{$\\Lambda_b\\to\\Lambda_c$ transition form factors in light-front\n approach}\n\nIn the diquark picture, the heavy baryon $\\Lambda_{b(c)}$ is\ncomposed of one heavy quark $b(c)$ and a light diquark [ud]. In\norder to form a color singlet hadron, the diquark [ud] is in a color\nanti-triplet. Because$\\Lambda_{b(c)}$ is at the ground state, the\ndiqaurk is in a $0^+$ scalar state (s=0, l=0) and the orbital\nangular momentum between the diquark and the heavy quark is also\nzero, i.e. $L=l=0$.\n\n\\subsection{Heavy baryon in the light-front approach}\n\nIn the light-front approach, the heavy baryon $\\Lambda_Q$ with total\nmomentum $P$, spin $S=1\/2$ and scalar diquark can be written as\n\\begin{eqnarray}\\label{eq:lfbaryon}\n |\\Lambda_Q(P,S,S_z)\\rangle&=&\\int\\{d^3p_1\\}\\{d^3p_2\\} \\,\n  2(2\\pi)^3\\delta^3(\\tilde{P}-\\tilde{p_1}-\\tilde{p_2}) \\nonumber\\\\\n &&\\times\\sum_{\\lambda_1}\\Psi^{SS_z}(\\tilde{p}_1,\\tilde{p}_2,\\lambda_1)\n  C^{\\alpha}_{\\beta\\gamma}F^{bc}\\left|\\right.\n  Q_{\\alpha}(p_1,\\lambda_1)[q_{1b}^{\\beta}q_{2c}^{\\gamma}](p_2)\\rangle,\n\\end{eqnarray}\nwhere $Q$ represents $b$ or $c$, $[q_1q_2]$ represents $[ud]$,\n$\\lambda$ denotes helicity, $p_1,~ p_2$ are the on-mass-shell\nlight-front momenta defined by\n\\begin{equation}\n \\tilde{p}=(p^+,p_{\\perp}),\\qquad p_\\perp=(p^1,p^2),\\qquad\n p^-=\\frac{m^2+p_{\\perp}^2}{p^+},\n\\end{equation}\nand\n\\begin{eqnarray}\n&&\\{d^3p\\}\\equiv\\frac{dp^+d^2 p_{\\perp}}{2(2\\pi)^3},\\qquad\n  \\delta^3(\\tilde{p})=\\delta(p^+)\\delta^2(p_{\\perp}),\n  \\nonumber\\\\\n&&\\mid Q(p_1,\\lambda_1)[q_1 q_2](p_2)\\rangle=\n b^{\\dagger}_{\\lambda_1}(p_1)a^{\\dagger}(p_2)| 0\\rangle,\\nonumber\\\\\n&&[a(p'), a^{\\dagger}(p)]=2(2\\pi)^3\\delta^3(\\tilde{p}'-\\tilde{p}),\n  \\nonumber\\\\\n&&\\{d_{\\lambda'}(p'),d_{\\lambda}^{\\dagger}(p)\\}=\n  2(2\\pi)^3\\delta^3(\\tilde{p}'-\\tilde{p})\\delta_{\\lambda'\\lambda},\n\\end{eqnarray}\nThe coefficient $C^{\\alpha}_{\\beta\\gamma}$ is a normalized color\nfactor and $F^{bc}$ is a normalized flavor coefficient,\n \\begin{eqnarray}\n && C^{\\alpha}_{\\beta\\gamma}F^{bc}C^{\\alpha'}_{\\beta'\\gamma'}F^{b'c'}\n  \\langle Q_{\\alpha'}(p'_1,\\lambda'_1)[q_{1b'}^{\\beta'}q_{2c'}^{\\gamma'}](p'_2)|\n  Q_{\\alpha}(p_1,\\lambda_1)[q_{1b}^{\\beta}q_{2c}^{\\gamma}](p_2)\\rangle\n  \\nonumber\\\\\n  &&=2^2(2\\pi)^6\\delta^3(\\tilde{p}_1'-\\tilde{p}_1)\\delta^3\n  (\\tilde{p}_2'-\\tilde{p}_2)\\delta_{\\lambda'_1\\lambda_1}.\n \\end{eqnarray}\n\nIn order to describe the internal motion of the constituents, one\nneeds to introduce intrinsic variables $(x_i, k_{i\\perp})$ with\n$i=1,2$ through\n\\begin{eqnarray}\n&&p^+_1=x_1 P^+, \\qquad\\qquad p^+_2=x_2 P^+,\n \\qquad\\qquad x_1+x_2=1, \\nonumber\\\\\n&&p_{1\\perp}=x_1 P_{\\perp}+k_{1\\perp},\n  ~~~ p_{2\\perp}=x_2 P_{\\perp}+k_{2\\perp},\n  ~~~ k_{\\perp}=-k_{1\\perp}=k_{2\\perp},\n\\end{eqnarray}\nwhere $x_i$ are the light-front momentum fractions satisfing $0<x_1,\nx_2<1$. The variables $(x_i, k_{i\\perp})$ are independent of the\ntotal momentum of the hadron and thus are Lorentz-invariant\nvariables. The invariant mass square $M_0^2$ is defined as\n \\begin{eqnarray} \\label{eq:Mpz}\n  M_0^2=\\frac{k_{1\\perp}^2+m_1^2}{x_1}+\n        \\frac{k_{2\\perp}^2+m_2^2}{x_2}.\n \\end{eqnarray}\nThe invariant mass $M_0$ is in general different from the hadron\nmass $M$ which satisfies the physical mass-shell condition\n$M^2=P^2$. This is due to the fact that the baryon, heavy quark\nand diquark can not be on their mass shells simultaneously. We\ndefine the internal momenta as\n \\begin{eqnarray}\n k_i=(k_i^-,k_i^+,k_{i\\bot})=(e_i-k_{iz},e_i+k_{iz},k_{i\\bot})=\n  (\\frac{m_i^2+k_{i\\bot}^2}{x_iM_0},x_iM_0,k_{i\\bot}).\n \\end{eqnarray}\nIt is easy to obtain\n \\begin{eqnarray}\n  M_0&=&e_1+e_2, \\nonumber\\\\\n  e_i&=&\\frac{x_iM_0}{2}+\\frac{m_i^2+k_{i\\perp}^2}{2x_iM_0}\n      =\\sqrt{m_i^2+k_{i\\bot}^2+k_{iz}^2},\\nonumber\\\\\n k_{iz}&=&\\frac{x_iM_0}{2}-\\frac{m_i^2+k_{i\\perp}^2}{2x_iM_0}.\n \\end{eqnarray}\nwhere $e_i$ denotes the energy of the i-th constituent. The\nmomenta $k_{i\\bot}$ and $k_{iz}$ constitute a momentum vector\n$\\vec k_i=(k_{i\\bot}, k_{iz})$ and correspond to the components in\nthe transverse and $z$ directions, respectively.\n\nThe momentum-space function $\\Psi^{SS_z}$ in Eq.\n(\\ref{eq:lfbaryon}) is expressed as\n\\begin{equation}\n \\Psi^{SS_z}(\\tilde{p}_1,\\tilde{p}_2,\\lambda_1)=\n  \\left\\langle\\lambda_1\\left|\\mathcal{R}^{\\dagger}_M(x_1,k_{1\\perp},m_1)\n   \\right|s_1\\right\\rangle\n  \\left\\langle 00;\\frac{1}{2} s_1\\left|\\frac{1}{2}S_z\\right\\rangle\n   \\phi(x,k_{\\perp})\\right.,\n\\end{equation}\nwhere $\\phi(x,k_{\\perp})$  is the light-front wave function which\ndescribes the momentum distribution of the constituents in the\nbound state with $x=x_2,~k_{\\perp}=k_{2\\perp}$; $\\left\\langle\n00;\\frac{1}{2} s_1\\left|\\frac{1}{2}S_z\\right\\rangle\\right.$ is the\ncorresponding Clebsch-Gordan coefficient with spin $s=s_z=0$ for\nthe scalar diquark;\n$\\left\\langle\\lambda_1\\left|\\mathcal{R}^{\\dagger}_M(x_1,k_{1\\perp},m_1)\n\\right|s_1\\right\\rangle$ is the well-known Melosh transformation\nmatrix element which transforms the the conventional spin states\nin the instant form into the light-front helicity eigenstates,\n \\begin{eqnarray}\n \\left\\langle\\lambda_1\\left|\\mathcal{R}^{\\dagger}_M(x_1,k_{1\\perp},m_1)\n \\right|s_1\\right\\rangle &=& \\frac{\\bar u(k_1,\\lambda_1)u_D(k_1,s_1)}{2m_1}\n \\nonumber\\\\\n  &=&\\frac{(m_1+x_1M_0)\\delta_{\\lambda_1 s_1}+i\\vec{\\sigma}_{\\lambda_1 s_1}\n  \\cdot\\vec k_{1\\perp}\\times\\vec n}\n  {\\sqrt{(m_1+x_1M_0)^2+k_{1\\perp}^2}},\n \\end{eqnarray}\nwhere $u_{(D)}$ denotes a Dirac spinor in the light-front\n(instant) form and $\\vec n=(0,0,1)$ is a unit vector in the $z$\ndirection. In practice, it is more convenient to use the covariant\nform for the Melosh transform matrix \\cite{Jaus,CCH2}\n\\begin{eqnarray}\n \\left\\langle\\lambda_1\\left|\\mathcal{R}^{\\dagger}_M(x_1,k_{1\\perp},m_1)\n   \\right|s_1\\right\\rangle \\left\\langle 00;\\frac{1}{2}s_1\\left|\n   \\frac{1}{2}S_z\\right\\rangle\\right.=\\frac{1}{\\sqrt{2(p_1\\cdot\n   \\bar P+m_1M_0)}}\\bar{u}(p_1,\\lambda_1)\\Gamma u(\\bar {P},S_z)\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\n\\Gamma=1, \\qquad \\qquad \\bar {P}=p_1+p_2.\n\\end{eqnarray}\nfor the scalar diquark. If the diquark is a vector which is\nusually supposed to be the case for the $\\Sigma_{c(b)}$ baryon,\nthe Melosh transform matrix should be modified.\n\nThe heavy baryon state is normalized as\n \\begin{eqnarray}\n \\langle\n \\Lambda(P',S',S'_z)|\\Lambda(P,S,S_z)\\rangle=2(2\\pi)^3P^+\n  \\delta^3(\\tilde{P}'-\\tilde{P})\\delta_{S'S}\\delta_{S'_zS_z}.\n \\end{eqnarray}\nThus, the light-front wave function satisfies the constraint\n \\begin{eqnarray}\n \\int\\frac{dxd^2k_{\\perp}}{2(2\\pi^3)}|\\phi(x,k_{\\perp})|^2=1.\n \\end{eqnarray}\n\nIn principle, the wave functions can be obtained by solving the\nlight-front bound state equations. However, it is too hard to\ncalculate them based on the first principle, so that instead, we\nutilize a phenomenological function, and the Gaussian form would\nbe most preferable one,\n \\begin{eqnarray}\n \\phi(x,k_{\\perp})=N\\sqrt{\\frac{\\partial k_{2z}}{\\partial x_2}}\n  {\\rm exp}\\left( \\frac{-\\vec k^2}{2\\beta^2}\\right).\n \\end{eqnarray}\nwith\n \\begin{eqnarray}\n N=4\\left(\\frac{\\pi}{\\beta^2}\\right)^{3\/4},\\qquad\n \\frac{\\partial k_{2z}}{\\partial x_2}=\\frac{e_1e_2}{x_1x_2M_0}.\n \\end{eqnarray}\nwhere $\\beta$ determines the confinement scale of hadron. The\nphenomenological parameters in the light-front quark model are\nquark masses and the hadron wave function parameter $\\beta$ which\nshould be prior determined before numerical computations can be\ncarried out and we will do the job in the later subsections.\n\n\n\\subsection{$\\Lambda_b\\rightarrow \\Lambda_c$ transition form factors}\n\nThe form factors for the weak transition $\\Lambda_Q\\rightarrow\n\\Lambda_{Q'}$  are defined in the standard way as\n\\begin{eqnarray}\\label{s1}\n&& \\langle \\Lambda_{Q'}(P',S',S_z') \\mid \\bar{Q}'\\gamma_{\\mu}\n (1-\\gamma_{5})Q \\mid \\Lambda_{Q}(P,S,S_z) \\rangle  \\nonumber \\\\\n &=& \\bar{u}_{\\Lambda_{Q'}}(P',S'_z) \\left[ \\gamma_{\\mu} f_{1}(q^{2})\n +i\\sigma_{\\mu \\nu} \\frac{ q^{\\nu}}{M_{\\Lambda_{Q}}}f_{2}(q^{2})\n +\\frac{q_{\\mu}}{M_{\\Lambda_{Q}}} f_{3}(q^{2})\n \\right] u_{\\Lambda_{Q}}(P,S_z) \\nonumber \\\\\n &&-\\bar u_{\\Lambda_{Q'}}(P',S'_z)\\left[\\gamma_{\\mu} g_{1}(q^{2})\n  +i\\sigma_{\\mu \\nu} \\frac{ q^{\\nu}}{M_{\\Lambda_{Q}}}g_{2}(q^{2})+\n  \\frac{q_{\\mu}}{M_{\\Lambda_{Q}}}g_{3}(q^{2})\n \\right]\\gamma_{5} u_{\\Lambda_{Q'}}(P,S_z).\n\\end{eqnarray}\nwhere  $q \\equiv P-P'$, $Q$ and $Q'$ denote $b$ and $c$,\nrespectively. The above formulation is the most general expression\nwith only constraint of keeping the Lorentz invariance and parity\nconservation for strong interactions. There are six form factors\n$f_i,~g_i$ in total for the vector and axial vector current $\\bar\nc\\gamma_{\\mu}(1-\\gamma_5)b$ and all the information on the strong\ninteraction is involved in them. Since $S=S'=1\/2$, we will be able\nto write $\\mid\\Lambda_{Q}(P,S,S'_z)\\rangle$ as\n$\\mid\\Lambda_{Q}(P,S_z)\\rangle$ in the following formulations. A\nparametrization is more convenient for the heavy-to-heavy\ntransitions, which is written in terms of the four-velocities as\n \\begin{eqnarray} \\label{s22}\n &&\\langle \\Lambda_{Q'}(v',S_z') \\mid \\bar{Q}'\\gamma_{\\mu}\n  (1-\\gamma_{5})Q \\mid \\Lambda_{Q}(v,S_z) \\rangle  \\nonumber \\\\\n  &=& \\bar{u}_{\\Lambda_{Q'}}(v',S'_z)\\left[F_1(\\omega)\\gamma_{\\mu}\n  +F_2(\\omega)v_{\\mu}+F_3(\\omega)v_{\\mu}^{\\prime}\\right]\n  u_{\\Lambda_{Q}}(v,S_z)- \\nonumber \\\\\n &&\\bar u_{\\Lambda_{Q'}}(v',S'_z)\\left[G_1(\\omega)\\gamma_{\\mu}\n  +G_2(\\omega)v_{\\mu}+G_3(\\omega)v_{\\mu}^{\\prime}\\right]\n \\gamma_{5} u_{\\Lambda_{Q}}(v,S_z),\n \\end{eqnarray}\nwhere $v=\\frac{P}{M_{\\Lambda_Q}},\n~v'=\\frac{P'}{M_{\\Lambda_{Q'}}}$, and $\\omega=v\\cdot v'$. The\nrelation between the two methods is\n \\begin{eqnarray}\n F_1=f_1-\\frac{M_{\\Lambda_Q}+M_{\\Lambda_{Q'}}}{M_{\\Lambda_Q}}f_2,\n  \\qquad F_2=f_2+f_3, \\qquad\n  F_3=\\frac{M_{\\Lambda_{Q'}}}{M_{\\Lambda_Q}}(f_2-f_3), \\nonumber\\\\\n G_1=g_1+\\frac{M_{\\Lambda_Q}-M_{\\Lambda_{Q'}}}{M_{\\Lambda_Q}}g_2,\n  \\qquad G_2=g_2+g_3, \\qquad\n  G_3=\\frac{M_{\\Lambda_{Q'}}}{M_{\\Lambda_Q}}(g_2-g_3).\n \\end{eqnarray}\n\nThe lowest order Feynman diagram for the $\\Lambda_{b}\\to\\Lambda_{c}$\nweak decay is depicted in Fig. \\ref{t1}.  In \\cite{pentaquark2}, the\nlight-front quark model for heavy pentaquark with one heavy quark\nand two light diquarks is presented. In our case, the heavy baryon\n$\\Lambda_{b(c)}$ is composed of a heavy quark b(c) and one diquark\n[ud]. Thus, most of our formulations are similar to that in\n\\cite{pentaquark2} except there is only one diquark in our case.\n\n\\begin{figure}\n\\begin{center}\n\\scalebox{0.8}{\\includegraphics{lightfront.eps}}\n\\end{center}\n\\caption{Feynman diagram for $\\Lambda_{b}\\to\\Lambda_{c}$\ntransitions, where $\\bigotimes$ denotes $V-A$ current\nvertex.}\\label{t1}\n\\end{figure}\n\n\nNow, we are ready to calculate the weak transition matrix\nelements. Using the the light-front quark model description of\n$\\mid \\Lambda_{Q}(P,S,S_z) \\rangle$, we obtain\n\\begin{eqnarray}\\label{s2}\n&& \\langle \\Lambda_{Q'}(P',S_z') \\mid \\bar{Q}'\n\\gamma^{\\mu} (1-\\gamma_{5}) Q \\mid \\Lambda_{Q}(P,S_z) \\rangle  \\nonumber \\\\\n &=& \\int\\{d^3p_2\\}\\frac{\\phi_{\\Lambda_{Q'}}^*(x',k'_{\\perp})\n  \\phi_{\\Lambda_Q}(x,k_{\\perp})}{2\\sqrt{p^+_1p'^+_1(p_1\\cdot \\bar{P}+m_1M_0)\n (p'_1\\cdot \\bar{P'}+m'_1M'_0)}}\\nonumber \\\\\n  &&\\times \\bar{u}(\\bar{P'},S'_z)\\bar{\\Gamma}'_{Lm}\n  (p_1\\!\\!\\!\\!\\!\\slash'+m'_1)\\gamma^{\\mu}(1-\\gamma_{5})\n  (p_1\\!\\!\\!\\!\\!\\slash+m_1)\\Gamma u(\\bar{P},S_z),\n\\end{eqnarray}\nwhere\n \\begin{eqnarray}\n &&\\bar{\\Gamma}'=\\gamma_0\\Gamma\\gamma_0=\\Gamma=1, \\nonumber \\\\\n &&m_1=m_b, \\qquad m'_1=m_c, \\qquad m_2=m_{[ud]},\n \\end{eqnarray}\nand $P$ and $P'$ denote the momenta of initial and final baryons,\n$p_1,~p'_1$ are the momenta of $b$ and $c$ quarks, respectively.\nBecause the diquark is a scalar, one does not need to deal with\nspinors which make computations more complex. In this framework,\nat each effective vertex, only the three-momentum rather than the\nfour-momentum is conserved, hence\n$\\tilde{p}_1-\\tilde{p}'_1=\\tilde{q}$ and\n$\\tilde{p}_2=\\tilde{p}'_2$. From $\\tilde{p}_2=\\tilde{p}'_2$, we\nhave\n \\begin{eqnarray}\n x'=\\frac{P^+}{P^{'+}}x, \\qquad \\qquad\n k'_{\\perp}=k_{\\perp}+x_2q_{\\perp}.\n \\end{eqnarray}\nwith $x=x_2$, $x'=x'_2$. Thus, Eq. (\\ref{s2}) is rewritten as\n\\begin{eqnarray}\\label{s23}\n &&\\langle \\Lambda_{Q'}(P',S_z') \\mid \\bar{Q}' \\gamma^{\\mu}\n  (1-\\gamma_{5}) Q \\mid \\Lambda_{Q}(P,S_z) \\rangle \\nonumber\\\\\n  &=& \\int\\frac{dxd^2k_{\\perp}}{2(2\\pi)^3}\\frac{\n  \\phi_{\\Lambda_{Q'}}(x',k'_{\\perp})\n  \\phi_{\\Lambda_Q}(x,k_{\\perp})}\n  {2\\sqrt{x_1x'_1(p_1\\cdot \\bar{P}+m_1M_0)\n  (p'_1\\cdot \\bar{P'}+m'_1M'_0)}}\\nonumber \\\\\n &&\\times \\bar{u}(\\bar{P'},S'_z)\n  (p_1\\!\\!\\!\\!\\!\\slash'+m'_1)\\gamma^{\\mu}(1-\\gamma_{5})\n  (p_1\\!\\!\\!\\!\\!\\slash+m_1) u(\\bar{P},S_z).\n\\end{eqnarray}\n\nFollowing \\cite{BD}, the Dirac and Pauli form factors can be derived\nfrom the helicity-conserving and helicity-flip matrix elements of\nthe plus component of the vector current operators in the\nlight-front framework. Analogously, the form factors corresponding\nto axial vector current are obtained by the authors of\n\\cite{Schlumpf}. In this work we choose the transverse frame where\n$q^+=0,~q_{\\perp}\\neq0$ which is similar to the treatment in\n\\cite{pentaquark2}. We then have\n \\begin{eqnarray}\n f_1(q^2)&=&\\frac{\\langle \\Lambda_{Q'}(P',\\uparrow)\\mid V^+\\mid\n  \\Lambda_{Q}(P,\\uparrow)\\rangle}{2\\sqrt{P^+P'^+}}, \\nonumber \\\\\n \\frac{f_2(q^2)}{M_{\\Lambda_Q}}&=&-\\frac{\\langle \\Lambda_{Q'}(P',\\uparrow)\n  \\mid V^+\\mid\\Lambda_{Q}(P,\\downarrow)\\rangle}{2q_{\\perp L}\\sqrt{P^+P'^+}},\n  \\nonumber \\\\\n g_1(q^2)&=&\\frac{\\langle \\Lambda_{Q'}(P',\\uparrow)\\mid A^+\\mid\n  \\Lambda_{Q}(P,\\uparrow)\\rangle}{2\\sqrt{P^+P'^+}},\\nonumber\\\\\n \\frac{g_2(q^2)}{M_{\\Lambda_Q}}&=&-\\frac{\\langle \\Lambda_{Q'}(P',\\uparrow)\n  \\mid A^+\\mid\\Lambda_{Q}(P,\\downarrow)\\rangle}{2q_{\\perp L}\\sqrt{P^+P'^+}},\n \\end{eqnarray}\nwhere $q_{\\perp L}=q^1_{\\perp}-iq^2_{\\perp}$.  The above relations\ncan be written in a compact form as\n\\begin{eqnarray}\\label{s3}\n \\langle \\Lambda_{Q'}(P',S_z') \\mid V^+\\mid \\Lambda_{Q}(P,S_z) \\rangle\n  &=&2\\sqrt{P^+P'^+}\\left[f_1(q^2)\\delta_{S'_zS_z}+\n  \\frac{f_2(q^2)}{M_{\\Lambda_Q}}(\\vec{\\sigma}\\cdot\\vec{q}_{\\perp}\n  \\sigma^3)_{S'_zS_z}\\right], \\nonumber \\\\\n \\langle \\Lambda_{Q'}(P',S_z') \\mid A^+\\mid \\Lambda_{Q}(P,S_z) \\rangle\n  &=&2\\sqrt{P^+P'^+}\\left[g_1(q^2)(\\sigma^3)_{S'_zS_z}+\n  \\frac{g_2(q^2)}{M_{\\Lambda_Q}}(\\vec{\\sigma}\\cdot\n  \\vec{q}_{\\perp})_{S'_zS_z}\\right].\n\\end{eqnarray}\nIt is noted that the form factors $f_3(q^2)$ and $g_3(q^2)$ cannot\nbe extracted in terms of the above method for we have imposed the\ncondition $q^+=0$. However, they do not contribute to the\nsemi-leptonic decays $\\Lambda_b\\to\\Lambda_c l\\bar\\nu_l$ and\nvanish in the heavy quark limit.\n\nIn order to extract $f_{1,2}(q^2)$ and $g_{1,2}(q^2)$ from Eq.\n(\\ref{s23}), the following identities are necessary,\n\\begin{eqnarray}\\label{s4}\n \\frac{1}{2}\\sum_{S_z,S'_z}u(\\bar{P},S_z)\\delta_{S'_zS_z}\\bar{u}\n  (\\bar{P}',S'_z)&=&\\frac{1}{4\\sqrt{P^+P'^+}}\n  (\\bar{P}\\!\\!\\!\\!\\slash+M_0)\\gamma^+(\\bar{P'}\\!\\!\\!\\!\\!\\slash+M'_0),\n  \\nonumber \\\\\n \\frac{1}{2}\\sum_{S_z,S'_z}u(\\bar{P},S_z)\n  (\\sigma^3\\sigma^i_{\\perp})_{S'_zS_z}\\bar{u}\n  (\\bar{P}',S'_z)&=&-\\frac{1}{4\\sqrt{P^+P'^+}}\n  (\\bar{P}\\!\\!\\!\\!\\slash+M_0)\\sigma^{i+}(\\bar{P'}\\!\\!\\!\\!\\!\\slash+\n   M'_0), \\nonumber\\\\\n \\frac{1}{2}\\sum_{S_z,S'_z}u(\\bar{P},S_z)(\\sigma^3)_{S'_zS_z}\\bar{u}\n  (\\bar{P}',S'_z)&=&\\frac{1}{4\\sqrt{P^+P'^+}}\n  (\\bar{P}\\!\\!\\!\\!\\slash+M_0)\\gamma^{+}\\gamma_5(\\bar{P'}\n   \\!\\!\\!\\!\\!\\slash+M'_0), \\nonumber\\\\\n \\frac{1}{2}\\sum_{S_z,S'_z}u(\\bar{P},S_z)(\\sigma^i_{\\perp})_{S'_zS_z}\n  \\bar{u}(\\bar{P}',S'_z)&=&-\\frac{1}{4\\sqrt{P^+P'^+}}\n (\\bar{P}\\!\\!\\!\\!\\slash+M_0)\\sigma^{i+}\\gamma_5(\\bar{P'}\n  \\!\\!\\!\\!\\!\\slash+M'_0).\n\\end{eqnarray}\nIt should be noted that $u(\\bar P,S_z)$ is not equal to\n$u(P,S_z)$, but the relation $\\gamma^+u(\\bar\nP,S_z)=\\gamma^+u(P,S_z)$ is satisfied.\n\nFrom Eqs. (\\ref{s23}), (\\ref{s3}) and (\\ref{s4}), the transition\nform factors are obtained,\n\\begin{eqnarray}\\label{s6}\n f_1(q^2)&=&\\frac{1}{8P^+P'^+}\\int{\\frac{dxd^2k_{\\perp}}{2(2\\pi)^3}}\n  \\frac{\\phi_{\\Lambda_{Q'}}(x',k'_{\\perp})\n  \\phi_{\\Lambda_Q}(x,k_{\\perp})}{2\\sqrt{x_1x'_1\n  (p_1\\cdot\\bar{P}+m_1M_0)(p'_1\\cdot \\bar{P'}+m'_1M'_0)}} \\nonumber\\\\\n &&\\times {\\rm Tr}\\left[(\\bar{P}\\!\\!\\!\\!\\slash+M_0)\\gamma^+\n  (\\bar{P'}\\!\\!\\!\\!\\!\\slash+M'_0)(p_1\\!\\!\\!\\!\\!\\slash'+m'_1)\n  \\gamma^{+}(p_1\\!\\!\\!\\!\\!\\slash+m_1)\\right],\\nonumber \\\\\n g_1(q^2)&=&\\frac{1}{8P^+P'^+}\\int{\\frac{dxd^2k_{\\perp}}{2(2\\pi)^3}}\n  \\frac{\\phi_{\\Lambda_{Q'}}(x',k'_{\\perp})\n  \\phi_{\\Lambda_Q}(x,k_{\\perp})}{2\\sqrt{x_1x'_1\n  (p_1\\cdot\\bar{P}+m_1M_0)(p'_1\\cdot \\bar{P'}+m'_1M'_0)}} \\nonumber\\\\\n &&\\times {\\rm Tr}\\left[(\\bar{P}\\!\\!\\!\\!\\slash+M_0)\\gamma^+\\gamma_5\n  (\\bar{P'}\\!\\!\\!\\!\\!\\slash+M'_0)\n  (p_1\\!\\!\\!\\!\\!\\slash'+m'_1)\\gamma^{+}\\gamma_5\n  (p_1\\!\\!\\!\\!\\!\\slash+m_1)\\right],\\nonumber \\\\\n \\frac{f_2(q^2)}{M_{\\Lambda_Q}}&=&-\\frac{1}{8P^+P'^+q^i_{\\perp}}\n  \\int{\\frac{dxd^2k_{\\perp}}{2(2\\pi)^3}}\n  \\frac{\\phi_{\\Lambda_{Q'}}(x',k'_{\\perp})\n  \\phi_{\\Lambda_Q}(x,k_{\\perp})}{2\\sqrt{x_1x'_1\n  (p_1\\cdot\\bar{P}+m_1M_0)(p'_1\\cdot \\bar{P'}+m'_1M'_0)}} \\nonumber \\\\\n &&\\times{\\rm Tr}\\left[(\\bar{P}\\!\\!\\!\\!\\slash+M_0)\\sigma^{i+}\n  (\\bar{P'}\\!\\!\\!\\!\\!\\slash+M'_0)\n  (p_1\\!\\!\\!\\!\\!\\slash'+m'_1)\\gamma^{+}\n  (p_1\\!\\!\\!\\!\\!\\slash+m_1)\\right],\\nonumber \\\\\n \\frac{g_2(q^2)}{M_{\\Lambda_Q}}&=&\\frac{1}{8P^+P'^+q^i_{\\perp}}\n  \\int{\\frac{dxd^2k_{\\perp}}{2(2\\pi)^3}}\n  \\frac{\\phi_{\\Lambda_{Q'}}(x',k'_{\\perp})\n  \\phi_{\\Lambda_Q}(x,k_{\\perp})}{2\\sqrt{x_1x'_1\n  (p_1\\cdot\\bar{P}+m_1M_0)(p'_1\\cdot \\bar{P'}+m'_1M'_0)}} \\nonumber \\\\\n &&\\times{\\rm Tr}\\left[(\\bar{P}\\!\\!\\!\\!\\slash+M_0)\\sigma^{i+}\\gamma_5\n  (\\bar{P'}\\!\\!\\!\\!\\!\\slash+M'_0)\n  (p_1\\!\\!\\!\\!\\!\\slash'+m'_1)\\gamma^{+}\\gamma_5\n  (p_1\\!\\!\\!\\!\\!\\slash+m_1)\\right],\n\\end{eqnarray}\nwith $i=1,2$. The traces can be worked out straightforwardly\n\\begin{eqnarray}\n &&\\frac{1}{8P^+P'^+}{\\rm Tr}\\left[(\\bar{P}\\!\\!\\!\\!\\slash+M_0)\n   \\gamma^+(\\bar{P'}\\!\\!\\!\\!\\!\\slash+M'_0)\n   (p_1\\!\\!\\!\\!\\!\\slash'+m'_1)\\gamma^{+}\n   (p_1\\!\\!\\!\\!\\!\\slash+m_1)\\right]\\nonumber \\\\\n &&~~~~~=-(p_1-x_1\\bar{P})\\cdot(p'_1-x'_1\\bar{P}')+\n   (x_1M_0+m_1)(x'_1M'_0+m'_1),\\nonumber \\\\\n &&\\frac{1}{8P^+P'^+}{\\rm Tr}\\left[(\\bar{P}\\!\\!\\!\\!\\slash+M_0)\n   \\gamma^+\\gamma_5(\\bar{P'}\\!\\!\\!\\!\\!\\slash+M'_0)\n   (p_1\\!\\!\\!\\!\\!\\slash'+m'_1)\\gamma^{+}\\gamma_5\n   (p_1\\!\\!\\!\\!\\!\\slash+m_1)\\right]\\nonumber \\\\\n &&~~~~~=(p_1-x_1\\bar{P})\\cdot(p'_1-x'_1\\bar{P}')+\n   (x_1M_0+m_1)(x'_1M'_0+m'_1),\\nonumber\n \\end{eqnarray}\nand\n \\begin{eqnarray}\\label{s7}\n &&\\frac{1}{8P^+P'^+}{\\rm Tr}\\left[(\\bar{P}\\!\\!\\!\\!\\slash+M_0)\\sigma^{i+}\n   (\\bar{P'}\\!\\!\\!\\!\\!\\slash+M'_0)(p_1\\!\\!\\!\\!\\!\\slash'+m'_1)\\gamma^{+}\n   (p_1\\!\\!\\!\\!\\!\\slash+m_1)\\right]\\nonumber\\\\\n &&~~~~~=(m'_1+x'_1\\bar{M}'_0)(p^i_{1\\perp}- x_1\\bar{P}^i_{\\perp})-\n   (m_1+x_1M_0)(p^{\\prime i}_{1\\perp}- x'_1\\bar{P}'^i_{\\perp}),\\nonumber\\\\\n &&\\frac{1}{8P^+P'^+}{\\rm Tr}\\left[(\\bar{P}\\!\\!\\!\\!\\slash+M_0)\\sigma^{i+}\\gamma_5\n   (\\bar{P'}\\!\\!\\!\\!\\!\\slash+M'_0)(p_1\\!\\!\\!\\!\\!\\slash'+m'_1)\\gamma^{+}\\gamma_5\n   (p_1\\!\\!\\!\\!\\!\\slash+m_1)\\right]\\nonumber\\\\\n &&~~~~~=(m'_1+x'_1\\bar{M}'_0)(p^i_{1\\perp}-x_1\\bar{P}^i_{\\perp})+(m_1+x_1M_0)\n   (p^{\\prime i}_{1\\perp}-x'_1\\bar{P}'^i_{\\perp}).\n\\end{eqnarray}\nUsing $\\bar P^{(\\prime)}=p_1^{(\\prime)}+p_2^{(\\prime)}$ and other\nmomentum relations, the products of momenta in Eqs. (\\ref{s6}) and\n(\\ref{s7}) are given in terms of the internal variables as\n\\begin{eqnarray}\\label{s8}\n &&p_1\\cdot\\bar{P}=e_1M_0=\n   \\frac{m^2_1+x^2_1M^2_0+k_{1\\perp}^2}{2x_1}, \\nonumber\\\\\n &&p'_1\\cdot\\bar{P}'=e'_1M'_0=\n   \\frac{m'^2_1+x'^2_1M'^2_0+{k'}_{1\\perp}^{2}}{2x'_1},\\nonumber\\\\\n &&p_{1\\perp}^{(\\prime)i}-x_1\\bar{P}_{\\perp}^{(\\prime)i}=\n   k_{1\\perp}^{(\\prime)i}, \\nonumber\\\\\n &&(p_1-x_1\\bar{P})\\cdot(p'_1-x'_1\\bar{P}')=\n   -k_{1\\perp}\\cdot k'_{1\\perp}.\n\\end{eqnarray}\n\nAt last, we obtain the final expressions for the\n$\\Lambda_Q\\to\\Lambda_{Q'}$ weak transition form factors\n\\begin{eqnarray}\\label{s9}\n f_1(q^2)&=&\\int{\\frac{dxd^2k_{\\perp}}{2(2\\pi)^3}}\n   \\frac{\\phi_{\\Lambda_{Q'}}(x',k'_{\\perp})\n  \\phi_{\\Lambda_Q}(x,k_{\\perp})\\left[k_{2\\perp}\n   \\cdot k'_{2\\perp}+\\left(x_1M_0+m_1\\right)\n   \\left(x'_1M'_0+m'_1\\right)\\right]}\n   {\\sqrt{\\left[\\left(m_1+x_1M_0\\right)^2+k_{2\\perp}^2\\right]\n   \\left[\\left(m'_1+x_1M'_0\\right)^2+k_{2\\perp}^{'2}\\right]}}, \\nonumber \\\\\n g_1(q^2)&=&\\int{\\frac{dxd^2k_{\\perp}}{2(2\\pi)^3}}\n   \\frac{\\phi_{\\Lambda_{Q'}}(x',k'_{\\perp})\n  \\phi_{\\Lambda_Q}(x,k_{\\perp})[-k_{2\\perp}\n   \\cdot k'_{2\\perp}+(x_1M_0+m_1)(x'_1M'_0+m'_1)]}\n   {\\sqrt{\\left[\\left(m_1+x_1M_0\\right)^2+k_{2\\perp}^2\\right]\n   \\left[\\left(m'_1+x_1M'_0\\right)^2+k_{2\\perp}^{'2}\\right]}}, \\nonumber \\\\\n \\frac{f_2(q^2)}{M_{\\Lambda_Q}}&=&\\frac{1}{q^i_{\\perp}}\n   \\int{\\frac{dxd^2k_{\\perp}}{2(2\\pi)^3}}\n   \\frac{\\phi_{\\Lambda_{Q'}}(x',k'_{\\perp})\n   \\phi_{\\Lambda_Q}(x,k_{\\perp})\n   [(m_1+x_1M_0)k_{1\\perp}^{\\prime i}-(m'_1+x'_1M'_0)k_{1\\perp}^i]}\n   {\\sqrt{\\left[\\left(m_1+x_1M_0\\right)^2+k_{2\\perp}^2\\right]\n   \\left[\\left(m'_1+x_1M'_0\\right)^2+k_{2\\perp}^{'2}\\right]}}, \\nonumber \\\\\n \\frac{g_2(q^2)}{M_{\\Lambda_Q}}&=&\\frac{1}{q^i_{\\perp}}\n   \\int{\\frac{dxd^2k_{\\perp}}{2(2\\pi)^3}}\n   \\frac{\\phi_{\\Lambda_{Q'}}(x',k'_{\\perp})\n   \\phi_{\\Lambda_Q}(x,k_{\\perp})\n   [(m_1+x_1M_0)k_{1\\perp}^{\\prime i}+ (m'_1+x'_1M'_0)\n   {k}_{1\\perp}^i]}{\\sqrt{\\left[\\left(m_1+x_1M_0\\right)^2+\n   k_{2\\perp}^2\\right]\\left[\\left(m'_1+x_1M'_0\\right)^2+\n   k_{2\\perp}^{'2}\\right]}}.\\nonumber\\\\\n\\end{eqnarray}\n\n\n\n\\subsection{The Form factors in the heavy quark limit}\n\nIt is well known that there is a non-trivial symmetry in QCD: the\nheavy quark symmetry (HQS) in the infinite quark mass limit\n\\cite{HQS}. Since the masses of heavy quarks $b$ and $c$ are much\nlarger than the strong interaction scale $\\Lambda_{\\rm QCD}$, the\nspin of the heavy quark decouples from light quark and gluon\ndegrees of freedoms, and an extra symmetry $SU_f(2)\\otimes\nSU_s(2)$ is expected. This flavor and spin symmetry provides\nseveral model-independent relations for the heavy-to-heavy\nbaryonic form factors: the six form factors $f_i,~g_i$ (i=1,2,3)\nare related to a unique universal Isgur-Wise function\n$\\zeta(v\\cdot v')$. In the heavy quark limit, the heavy quark $Q$\nis described by a two-component spinor $Q_v={\\rm e}^{im_Qv\\cdot\nx}\\frac{(1+\\rlap{\\hspace{-0.02cm}\/}{v})}{2}Q$ where $v$ is the velocity of the heavy\nbaryon. The current $\\bar Q'\\gamma_{\\mu}(1-\\gamma_5)Q$ in the full\ntheory is matched onto the current $\\bar{Q}_{v'}' \\gamma_{\\mu}\n(1-\\gamma_{5}) Q_v$ in the heavy quark effective theory (HQET).\nThe baryon bound state and Dirac spinor field are replaced by\n \\begin{eqnarray}\n \\mid\\Lambda_{Q}(P,S_z)\\rangle &\\to& \\sqrt{M_{\\Lambda_Q}}\n \\mid\\Lambda_{Q}(v,S_z)\\rangle,\n  \\nonumber \\\\\n u(\\bar{P},S_z) &\\to& \\sqrt{m_Q}u(v,S_z).\n \\end{eqnarray}\nThe Isgur-Wise function which appears in the transition amplitude\n$\\Lambda_Q\\to\\Lambda_{Q'}$ is defined as \\cite{IW}\n \\begin{eqnarray}\\label{s10}\n  \\langle \\Lambda_{Q'}(v',S_z')\\mid\\bar{Q}_{v'}' \\gamma^{\\mu}\n  (1-\\gamma_{5}) Q_v \\mid \\Lambda_{Q}(v,S_z)\\rangle =\n  \\zeta(\\omega)\\bar{u}(v',S'_z)\\gamma_{\\mu}(1-\\gamma_{5})u(v,S_z),\n \\end{eqnarray}\nwhere $\\omega\\equiv v\\cdot v'$. The heavy flavor symmetry implies\nthat the Isgur-Wise function is normalized to be 1 at the\nzero-recoil point, $\\zeta(1)=1$. The physical form factors are\nobtained as\n \\begin{eqnarray}\\label{s11}\n  f_1(q^2)=g_1(q^2)=\\zeta(\\omega),\\qquad \\qquad\n  f_2=f_3=g_2=g_3=0,\n \\end{eqnarray}\nwhere $q^2=M_{\\Lambda_Q}^2+M_{\\Lambda_{Q'}}^2-\n2M_{\\Lambda_Q}M_{\\Lambda_{Q'}}\\omega$.\n\nSince the momentum of $\\Lambda_Q$  is dominated by the momentum of\nthe heavy quark $Q$, the momentum of the light spectator diquark\n$x$ is of order $\\Lambda_{\\rm QCD}\/m_Q$. The variable $X\\equiv xm_b$ is of\norder of $\\Lambda_{\\rm QCD}$. In analog to the situation for heavy mesons, the\nwave function of $\\Lambda_Q$ should have a scaling behavior in the\nheavy quark limit \\cite{CZL}\n \\begin{eqnarray}\n  \\phi_{\\Lambda_Q}(x,k_{\\bot})\\to \\sqrt{\\frac{m_Q}{X}}\\Phi(X,k_{\\bot}),\n \\end{eqnarray}\nwhere the factor $\\sqrt{m_Q}$ is deliberately factorized out and\nthe rest  of $\\phi_{\\Lambda_Q}(x,k_{\\bot})$  is independent of\n$m_Q$,  $\\Phi(X,k_{\\bot})$ is normalized as\n \\begin{eqnarray} \\label{s24}\n \\int_0^{\\infty}\\frac{dX}{X}\\int\\frac{d^2k_{\\bot}}{2(2\\pi)^3}\n  |\\Phi(X,k_{\\bot})|^2=1.\n \\end{eqnarray}\nFor the on-shell diquark momentum $p_2$, we have\n$p^-_2=\\frac{(p^2_{2\\perp}+m^2_2)}{p^+_2}$, and\n \\begin{eqnarray}\n v\\cdot p_2=\\frac{m_2^2+k_{\\perp}^2+X^2}{2X}.\n \\end{eqnarray}\nHence the wave function $\\Phi(X,k_{\\bot})$ which only depends on\nthe velocity of the baryon, is the same for $\\Lambda_Q$ and\n$\\Lambda_{Q'}$.\n\nWe now consider the transition form factors obtained in the\nprevious section under the heavy quark limit. For the initial\nbaryon, we have\n\\begin{eqnarray}\\label{s12}\n &&M_{\\Lambda_Q}\\to m_Q, \\qquad ~~ M_0\\to m_Q,\\nonumber\\\\\n &&e_1\\to m_Q, \\qquad\\qquad e_2\\to v\\cdot p_2,\\nonumber\\\\\n &&\\vec k^2\\to (v\\cdot p_2)^2-m_2^2,\\nonumber\\\\\n &&p_1\\!\\!\\!\\!\\!\\slash+m_1\\to m_Q(v\\!\\!\\!\\slash+1)\\nonumber \\\\\n &&\\frac{e_1e_2}{x_1x_2M_0}\\to\\frac{m_Q}{X}(v\\cdot p_2),\n\\end{eqnarray}\nand\n \\begin{eqnarray}\n \\Phi(X,k_{\\bot})=4\\sqrt{v\\cdot p_2}\\left(\\frac{\\pi}{\\beta^2_\\infty}\n  \\right)^{\\frac{3}{4}}{\\rm exp}\\left(-\\frac{(v\\cdot p_2)^2-m^2_2}\n  {2\\beta^2_\\infty}\\right).\n \\end{eqnarray}\nThe subscript in $\\beta_{\\infty}$ represents the case of the heavy\nquark limit. Similar expressions can be obtained for the final\nbaryon where a prime sign ``$'$\" would be attached to each\nvariable.\n\nThe calculation of the Isgur-Wise function in the heavy quark limit\nbecomes much simpler than that for $f_i$ and $g_i$ because they can\nbe evaluated directly in the time-like region by choosing a\nreference frame where $q_\\perp=0$ \\cite{pentaquark2}. The  matrix\nelement $\\Lambda_Q\\to\\Lambda_{Q'}$ is\n\\begin{eqnarray}\\label{sw14}\n &&\\langle\\Lambda_{Q'}(v',S_z') \\mid \\bar{Q}_{v'}'\n   \\gamma^{\\mu}(1-\\gamma_{5})Q_v\\mid\\Lambda_{Q}(v,S_z) \\rangle  \\nonumber\\\\\n &=& \\int{\\frac{dX}{X}\\frac{d^2k_{\\perp}}{2(2\\pi)^3}}\n {\\Phi(X,k_{\\perp})\\Phi(X',k_{\\perp}^{\\prime})}\n \\bar{u}(v',S'_z)\\gamma^\\mu(1-\\gamma_5)u(v,S_z),\n\\end{eqnarray}\nwhere $z\\equiv X'\/X$. By comparing the above equation with Eq.\n(\\ref{s10}), we get\n\\begin{eqnarray}\\label{s15}\n \\zeta(\\omega)= \\int{\\frac{dX}{X}\\frac{d^2k_{\\perp}}{2(2\\pi)^3}}\n {\\Phi(X,k_{\\perp})\\Phi(X',k_{\\perp}^{\\prime})}.\n\\end{eqnarray}\nThe obtained Isgur-Wise function $\\zeta(\\omega)$ is an overlapping\nintegration of the initial and final wave functions and no spin\ninformation is left. The variable $z$ is related to $\\omega$ via\n\\begin{eqnarray}\\label{s16}\n z\\rightarrow z_{\\pm}=\\omega\\pm\\sqrt{\\omega^2-1},\n \\qquad\\qquad z_+=\\frac{1}{z_-}.\n\\end{eqnarray}\nwhere $+(-)$ denotes the final baryon recoiling direction. There\nis a symmetry between $z_+$ and $z_-$. $\\zeta(\\omega)$ does not\nchange when we replace $z_+$ by $z_-$, or vice versa. Eq.\n(\\ref{s15}) shows explicitly that $\\zeta(\\omega)$ depends only on\nthe velocities of the initial and final baryons and independent of\nthe heavy quark masses.\n\nThe Isgur-Wise function can also be obtained from Eq. (\\ref{s2})\nby taking the heavy quark limit. It is not difficult to verify\nthat $f_1(q^2)=g_1(q^2)=\\zeta(\\omega)$, $f_2=g_2=0$ in leading\norder of $\\Lambda_{\\rm QCD}\/m_Q$. The normalization of Isgur-Wise function at\nthe zero-recoil point is guaranteed by our normalization condition\nfor wave functions Eq. (\\ref{s24}). Its consistency with forms in\nthe heavy quark limit implies the correctness of the light-front\napproach.\n\n\\section{Semi-leptonic and Non-leptonic decays of transition\n$\\Lambda_b \\to \\Lambda_c$ }\n\nThe polarization effects in exclusive processes, such as $B\\to\n\\phi K^*$ offers non-trivial information about strong interaction,\nwhich is important to test different theoretical approaches. The\ndecays of $\\Lambda_b\\to\\Lambda_c$ indeed contain complex spin\nstructures. In this section, we obtain formulations for the rates\nof semi-leptonic and non-leptonic processes. In this work, we\nconcern only the exclusive decay modes.\n\n\\subsection{Semi-leptonic decays of\n $\\Lambda_b \\to \\Lambda_c l\\bar\\nu_l$ }\n\nThe transition amplitude of $\\Lambda_b\\to\\Lambda_c$ contains\nseveral independent helicity components. The helicity amplitudes\ninduced by the weak vector and axial vector currents are described\nby $H^{V,A}_{\\lambda',\\lambda_W}$ where $\\lambda'$ and $\\lambda_W$\ndenote the helicities of the final baryon and the virtual\n$W$-boson, respectively. According to the definitions of the form\nfactors for $\\Lambda_b \\to \\Lambda_c$ given in Eq. (\\ref{s22}),\nthe helicity amplitudes are related to these form factors through\nthe following expressions \\cite{KKP}\n \\begin{eqnarray}\n H^V_{\\frac{1}{2},0}&=&\\frac{\\sqrt{Q_-}}{\\sqrt{q^2}}\\left(\n  \\left(M_{\\Lambda_b}+M_{\\Lambda_c}\\right)f_1-\\frac{q^2}{M_{\\Lambda_b}}f_2\\right),\\nonumber\\\\\n H^V_{\\frac{1}{2},1}&=&\\sqrt{2Q_-}\\left(-f_1+\n  \\frac{M_{\\Lambda_b}+M_{\\Lambda_c}}{M_{\\Lambda_b}}f_2\\right),\\nonumber\\\\\n H^A_{\\frac{1}{2},0}&=&\\frac{\\sqrt{Q_+}}{\\sqrt{q^2}}\\left(\n  \\left(M_{\\Lambda_b}-M_{\\Lambda_c}\\right)g_1+\\frac{q^2}{M_{\\Lambda_b}}g_2\\right),\\nonumber\\\\\n H^A_{\\frac{1}{2},1}&=&\\sqrt{2Q_+}\\left(-g_1-\n  \\frac{M_{\\Lambda_b}-M_{\\Lambda_c}}{M_{\\Lambda_b}}g_2\\right).\n \\end{eqnarray}\nwhere $Q_{\\pm}=2(P\\cdot P'\\pm M_{\\Lambda_b}M_{\\Lambda_c})=2M_{\\Lambda_b}M_{\\Lambda_c}(\\omega\\pm 1)$. The\namplitudes for the negative helicities are obtained in terms of\nthe relation\n \\begin{eqnarray}\n H^{V,A}_{-\\lambda'-\\lambda_W}=\\pm H^{V,A}_{\\lambda',\\lambda_W},\n  \\end{eqnarray}\nwhere the upper (lower) sign corresponds to V(A).\n\nBecause of the $V-A$ structure of the weak current, the helicity\namplitudes are obtained as\n \\begin{eqnarray}\n H_{\\lambda',\\lambda_W}=H^V_{\\lambda',\\lambda_W}-\n  H^A_{\\lambda',\\lambda_W}.\n \\end{eqnarray}\nThe helicities of the $W$-boson $\\lambda_W$ can be either $0$ or\n$1$, which correspond to the longitudinal and transverse\npolarizations, respectively. Following the definitions in\nliterature, we decompose the decay width into a sum of the\nlongitudinal and transverse parts according to the helicity states\nof the virtual W-boson. The differential decay rate of $\\Lambda_b\n\\to \\Lambda_c l\\bar\\nu_l$ is\n \\begin{eqnarray}\n \\frac{d\\Gamma}{d\\omega}=\\frac{d\\Gamma_L}{d\\omega}+\n \\frac{d\\Gamma_T}{d\\omega},\n \\end{eqnarray}\nand the longitudinally (L) and transversely (T) polarized rates are\nrespectively\\cite{KKP}\n \\begin{eqnarray}\n \\frac{d\\Gamma_L}{d\\omega}&=&\\frac{G_F^2|V_{cb}|^2}{(2\\pi)^3}~\n  \\frac{q^2~p_c~M_{\\Lambda_c}}{12M_{\\Lambda_b}}\\left[\n  |H_{\\frac{1}{2},0}|^2+|H_{-\\frac{1}{2},0}|^2\\right],\\nonumber\\\\\n \\frac{d\\Gamma_T}{d\\omega}&=&\\frac{G_F^2|V_{cb}|^2}{(2\\pi)^3}~\n  \\frac{q^2~p_c~M_{\\Lambda_c}}{12M_{\\Lambda_b}}\\left[\n  |H_{\\frac{1}{2},1}|^2+|H_{-\\frac{1}{2},-1}|^2\\right].\n \\end{eqnarray}\nwhere $p_c=M_{\\Lambda_c}\\sqrt{\\omega^2-1}$ is the momentum of $\\Lambda_c$ in\nthe reset frame of $\\Lambda_b$. Integrating over the solid angle,\nwe obtain the decay rate\n \\begin{eqnarray}\n \\Gamma=\\int_1^{\\omega_{\\rm max}}d\\omega\\frac{d\\Gamma}{d\\omega},\n \\end{eqnarray}\nwhere the upper bound of the integration $\\omega_{\\rm\nmax}=\\frac{1}{2}\\left(\\frac{M_{\\Lambda_{b}}}\n{M_{\\Lambda_{c}}}+\\frac{M_{\\Lambda_{c}}}{M_{\\Lambda_{b}}}\\right)$\nis the maximal recoil. In order to compare our results with those\nin the literatures, we used the variable $\\omega$ in the\nexpression for the differential decay rate. In the heavy quark\nlimit, the decay rate of $\\Lambda_b \\to \\Lambda_c l\\bar{\\nu}_l$ is\nsimplified into\n\\begin{eqnarray}\n \\frac{d\\Gamma}{d\\omega}=\\frac{G_F^{2}|V_{cb}|^2}\n  {24\\pi^3}M_{\\Lambda_{b}}^5 \\sqrt{\\omega^2-1}~r^3 ( 6\\omega\n  r^2-8\\omega^2r-4r+6\\omega)\\zeta(\\omega)^2,\n\\end{eqnarray}\nwith $r=\\frac{M_{\\Lambda_c}}{M_{\\Lambda_b}}$.\n\nThe polarization of the cascade decay $\\Lambda_b\\to\\Lambda_c(\\to\np\\pi)+W(\\to l\\nu)$ is expressed by various asymmetry\nparameters\\cite{EFG,KKP}. Among them, the integrated longitudinal\nand transverse asymmetries are defined by\n \\begin{eqnarray}\n a_L&=&\\frac{\\int_1^{\\omega_{\\rm max}} d\\omega ~q^2~ p_c\n     \\left[ |H_{\\frac{1}{2},0}|^2-|H_{-\\frac{1}{2},0}|^2\\right]}\n     {\\int_1^{\\omega_{\\rm max}} d\\omega ~q^2~ p_c\n     \\left[|H_{\\frac{1}{2},0}|^2+|H_{-\\frac{1}{2},0}|^2\\right]},\n     \\nonumber\\\\\n a_T&=&\\frac{\\int_1^{\\omega_{\\rm max}} d\\omega ~q^2~ p_c\n     \\left[ |H_{\\frac{1}{2},1}|^2-|H_{-\\frac{1}{2},-1}|^2\\right]}\n     {\\int_1^{\\omega_{\\rm max}} d\\omega ~q^2~ p_c\n     \\left[|H_{\\frac{1}{2},1}|^2+|H_{-\\frac{1}{2},-1}|^2\\right]}.\n \\end{eqnarray}\nThe ratio of the longitudinal to transverse decay rates $R$ is\ndefined by\n \\begin{eqnarray}\n R=\\frac{\\Gamma_L}{\\Gamma_T}=\\frac{\\int_1^{\\omega_{\\rm\n     max}}d\\omega~q^2~p_c\\left[ |H_{\\frac{1}{2},0}|^2+|H_{-\\frac{1}{2},0}|^2\n     \\right]}{\\int_1^{\\omega_{\\rm max}}d\\omega~q^2~p_c\n     \\left[ |H_{\\frac{1}{2},1}|^2+|H_{-\\frac{1}{2},-1}|^2\\right]},\n \\end{eqnarray}\nand the  longitudinal $\\Lambda_c$ polarization asymmetry $P_L$ is\ngiven as\n \\begin{eqnarray}\n P_L&=&\\frac{\\int_1^{\\omega_{\\rm max}} d\\omega ~q^2~ p_c\n     \\left[ |H_{\\frac{1}{2},0}|^2-|H_{-\\frac{1}{2},0}|^2+\n     |H_{\\frac{1}{2},1}|^2-|H_{-\\frac{1}{2},-1}|^2\\right]}\n     {\\int_1^{\\omega_{\\rm max}} d\\omega ~q^2~ p_c\n     \\left[|H_{\\frac{1}{2},0}|^2+|H_{-\\frac{1}{2},0}|^2+\n     |H_{\\frac{1}{2},1}|^2+|H_{-\\frac{1}{2},-1}|^2\\right]}\n  \\nonumber\\\\\n  &=&\\frac{a_T+R\\,a_L}{1+R}.\n \\end{eqnarray}\n\n\n\\subsection{Non-leptonic decay  of $\\Lambda_b \\to \\Lambda_c M$}\n\nSeveral exclusive non-leptonic decays of $\\Lambda_b \\to \\Lambda_c\n+M$ where $M$ is a meson, have been measured in recent experiments\n\\cite{PDG06}. From the theoretical aspects, the non-leptonic\ndecays are much more complicated than the semi-leptonic ones\nbecause of the strong interaction. Generally, the present\ntheoretical framework is based on the factorization assumption,\nwhere  the hadronic matrix element is factorized into a product of\ntwo matrix elements of single currents. One can be written as a\ndecay constant while the other is expressed in terms of a few form\nfactors according to the lorentz structure of the current. For the\nweak decays of mesons, such factorization approach is verified to\nwork very well for the color-allowed processes and the\nnon-factorizable contributions are negligible. We have reason to\nbelieve that this would be valid for the baryon case, especially\nas the diquark picture is employed. The decays $\\Lambda_b^0 \\to\n\\Lambda_c^+ M^-$ belong to this type. Thus, the study on these\nmodes could be not only a test for the factorization hypothesis,\nbut also a check of the consistency of the obtained form factors\nin the heavy bottomed baryon system.\n\nFor the non-leptonic decays $\\Lambda_b^0 \\to \\Lambda_c^+ M^-$, the\neffective interaction at the quark level is $b\\to c\\bar{q_1}q_2$.\nThe relevant Hamiltonian is\n \\begin{eqnarray}\n &&{\\cal H}_W=\\frac{G_F}{\\sqrt 2}V_{cb}V_{q_1q_2}^*(c_1O_1+c_2O_2),\n  \\nonumber\\\\\n &&O_1=(\\bar c b)_{V-A} (\\bar q_2q_1)_{V-A},\\qquad\n O_2=(\\bar q_2b)_{V-A} (\\bar c q_1)_{V-A}.\n \\end{eqnarray}\nwhere $c_i$ denotes the short-distance Wilson coefficient,\n$V_{cb}(V_{q_1q_2})$ is the CKM matrix elements, $q_1$ stands for\n$u$ or $c$ and $q_2$ for $d$ or $s$ in the context. Then one needs\nto evaluate the hadronic matrix elements\n \\begin{eqnarray}\n \\langle \\Lambda_c M | {\\cal H}_W | \\Lambda_b\\rangle=\n \\frac{G_F}{\\sqrt 2}V_{cb}V_{q_1q_2}^*\\sum_{i=1,2}c_i~\n \\langle \\Lambda_c M | O_i | \\Lambda_b\\rangle.\n \\end{eqnarray}\nUnder the factorization approximation, the hadronic matrix element\nis reduced to\n \\begin{eqnarray}\n \\langle\\Lambda_c M | O_i | \\Lambda_b\\rangle\n =\\langle\\Lambda_c | J_\\mu |\\Lambda_b\\rangle\n  \\langle M | J^{\\prime\\mu} | 0\\rangle.\n \\end{eqnarray}\nwhere $J(J')$ is the $V-A$ weak current. The first factor\n$\\langle\\Lambda_c | J_\\mu |\\Lambda_b\\rangle$ is parameterized by six form\nfactors as done in Eq. (\\ref{s1}). The second factor defines the\ndecay constants as follows\n \\begin{eqnarray}\n \\langle P(P)|A_{\\mu}|0\\rangle&=&f_PP_{\\mu}, \\nonumber\\\\\n \\langle S(P)|V_{\\mu}|0\\rangle&=&f_SP_{\\mu}, \\nonumber\\\\\n \\langle V(P,\\epsilon)|V_{\\mu}|0\\rangle&=&f_VM_V\\epsilon^*_{\\mu}, \\nonumber\\\\\n \\langle A(P,\\epsilon)|A_{\\mu}|0\\rangle&=&f_VM_A\\epsilon^*_{\\mu},\n \\end{eqnarray}\nwhere $P(V)$ denotes a pseudoscalar (vector) meson, and $S(A)$\ndenotes a scalar (axial-vector) meson. In the definitions, we omit\na factor $(-i)$ for the  pseudoscalar meson decay constant.\n\nIn general, the transition amplitude of $\\Lambda_b\\to\\Lambda_c M$\ncan be written as\n \\begin{eqnarray}\n {\\cal M}(\\Lambda_b\\to\\Lambda_c P)&=&\\bar\n  u_{\\Lambda_c}(A+B\\gamma_5)u_{\\Lambda_b}, \\nonumber \\\\\n {\\cal M}(\\Lambda_b\\to\\Lambda_c V)&=&\\bar\n  u_{\\Lambda_c}\\epsilon^{*\\mu}\\left[A_1\\gamma_{\\mu}\\gamma_5+\n   A_2(p_{\\Lambda_c})_{\\mu}\\gamma_5+B_1\\gamma_{\\mu}+\n   B_2(p_{\\Lambda_c})_{\\mu}\\right]u_{\\Lambda_b},\n \\end{eqnarray}\nwhere $\\epsilon^{\\mu}$ is the polarization vector of the final\nvector or axial-vector mesons. Including the effective Wilson\ncoefficient $a_1=c_1+c_2\/N_c$, the decay amplitudes in the\nfactorization approximation are \\cite{KK,Cheng}\n \\begin{eqnarray}\n A&=&\\lambda f_P(M_{\\Lambda_b}-M_{\\Lambda_c})f_1(M^2), \\nonumber \\\\\n B&=&\\lambda f_P(M_{\\Lambda_b}+M_{\\Lambda_c})g_1(M^2), \\nonumber\\\\\n A_1&=&-\\lambda f_VM\\left[g_1(M^2)+g_2(M^2)\\frac{M_{\\Lambda_b}-M_{\\Lambda_c}}{M_{\\Lambda_b}}\\right],\n \\nonumber\\\\\n A_2&=&-2\\lambda f_VM\\frac{g_2(M^2)}{M_{\\Lambda_b}},\\nonumber\\\\\n B_1&=&\\lambda f_VM\\left[f_1(M^2)-f_2(M^2)\\frac{M_{\\Lambda_b}+M_{\\Lambda_c}}{M_{\\Lambda_b}}\\right],\n \\nonumber\\\\\n B_2&=&2\\lambda f_VM\\frac{f_2(M^2)}{M_{\\Lambda_b}},\n \\end{eqnarray}\nwhere $\\lambda=\\frac{G_F}{\\sqrt 2}V_{cb}V_{q_1q_2}^*a_1$ and $M$ is\nthe meson mass. Replacing  $P$, $V$ by $S$ and $A$ in the above\nexpressions, one can easily obtain similar expressions for scalar\nand axial-vector mesons .\n\nThe decay rates of $\\Lambda_b\\rightarrow\\Lambda_cP(S)$ and up-down\nasymmetries are \\cite{Cheng}\n \\begin{eqnarray}\n \\Gamma&=&\\frac{p_c}{8\\pi}\\left[\\frac{(M_{\\Lambda_b}+M_{\\Lambda_c})^2-M^2}{M_{\\Lambda_b}^2}|A|^2+\n  \\frac{(M_{\\Lambda_b}-M_{\\Lambda_c})^2-M^2}{M_{\\Lambda_b}^2}|B|^2\\right], \\nonumber\\\\\n \\alpha&=&-\\frac{2\\kappa{\\rm Re}(A^*B)}{|A|^2+\\kappa^2|B|^2},\n \\end{eqnarray}\nwhere $p_c$ is the $\\Lambda_c$ momentum in the rest frame of\n$\\Lambda_b$ and $\\kappa=\\frac{p_c}{E_{\\Lambda_c}+M_{\\Lambda_c}}$. For\n$\\Lambda_b\\rightarrow\\Lambda_c V(A)$ decays, the decay rates and\nup-down asymmetries are\n \\begin{eqnarray}\n \\Gamma&=&\\frac{p_c (E_{\\Lambda_c}+M_{\\Lambda_c})}{8\\piM_{\\Lambda_b}}\\left[\n  2\\left(|S|^2+|P_2|^2\\right)+\\frac{E^2}{M^2}\\left(\n  |S+D|^2+|P_1|^2\\right)\\right], \\nonumber\\\\\n \\alpha&=&\\frac{4M^2{\\rm Re}(S^*P_2)+2E^2{\\rm Re}(S+D)^*P_1}\n  {2M^2\\left(|S|^2+|P_2|^2 \\right)+E^2\\left(|S+D|^2+|P_1|^2\n  \\right) },\n \\end{eqnarray}\nwhere $E$ is energy of the vector (axial vector) meson, and\n \\begin{eqnarray}\n  S&=&-A_1, \\nonumber\\\\\n  P_1&=&-\\frac{p_c}{E}\\left(\\frac{M_{\\Lambda_b}+M_{\\Lambda_c}}\n  {E_{\\Lambda_c}+M_{\\Lambda_c}}B_1+B_2\\right), \\nonumber \\\\\n  P_2&=&\\frac{p_c}{E_{\\Lambda_c}+M_{\\Lambda_c}}B_1,\\nonumber\\\\\n  D&=&-\\frac{p^2_c}{E(E_{\\Lambda_c}+M_{\\Lambda_c})}(A_1- A_2).\n \\end{eqnarray}\n\n\n\\section{Numerical Results}\n\nIn this section, we will present our numerical results of  the\nform factors for the transition $\\Lambda_b\\to\\Lambda_c$. Then use\nthem to predict the rates of the exclusive semi-leptonic\n$\\Lambda_b\\rightarrow \\Lambda_c l\\bar{\\nu}_l$, and two-body\nnon-leptonic processes, such as $\\Lambda_b\\to \\Lambda_c M^-$ where\n$M=\\pi,~ K,~ \\rho,~ K^*,~ a_1$.\n\nAt first, we provide our input parameters in the light-front quark\nmodel. The baryon masses $M_{\\Lambda_b}=5.624$ GeV, $M_{\\Lambda_c}=2.285$ GeV are taken\nfrom \\cite{PDG06}. The quark masses and the hadron wave function\nparameter $\\beta$ need to be specified. For the heavy quark masses,\nwe take $m_b$ and $m_c$ from \\cite{CCH2}. Following\n\\cite{pentaquark2}, the mass of a [ud] diquark is assumed to be\nclose to the constitute strange quark mass. In the literature, the\nmass of the constitutent light scalar diquark $m_{[ud]}$ is rather\narbitrary, for example, it is set as: 400 MeV \\cite{pentaquark2},\n500 MeV \\cite{MQS}, 710 MeV \\cite{EFG} and 650$\\sim$800 MeV\n\\cite{Guo}. A recent result from lattice calculation gives the\nscalar diquark mass varies from 1190 to 696 MeV when a hopping\nparameter $\\kappa$ changes from 0.140 to 0.148 \\cite{HKLW}. To\nreduce error and model-dependence, we use the value of $BR(\\Lambda_b\n\\to \\Lambda_c l\\bar{\\nu}_l)=5.0^{+1.1}_{-0.8}({\\rm\nstat})^{+1.6}_{-1.2}({\\rm syst})\\%$ measured by the DELPHI\ncollaboration \\cite{DELPHI} to fix parameters. The present data\nfavors diquark mass as $m_{[ud]}=500$ MeV. All the input parameters\nare collected in Table \\ref{Tab:t1}.\n\n\n\\begin{table}\n\\caption{Quark mass and the parameter $\\beta$ (in units of\n GeV).}\\label{Tab:t1}\n\\begin{ruledtabular}\n\\begin{tabular}{ccccc}\n  $m_c$  & $m_b$  &$m_{[ud]}$ & $\\beta_{c[ud]}$ & $\\beta_{b[ud]}$ \\\\\n  $1.3$  & $4.4$  & 0.50     & $0.35$         & 0.40\n\\end{tabular}\n\\end{ruledtabular}\n\\end{table}\n\n\\subsection{$\\Lambda_b\\to \\Lambda_c$ form factors and the Isgur-Wise\nfunction}\n\nBecause the calculation of form factors is performed in the frame\n$q^+=0$ with $q^2=-q^2_{\\perp}\\leq 0$, only the values of the form\nfactors in the space-like region can be obtained. The advantage of\nthis choice is that the so-called Z-graph contribution arising\nfrom the non-valence quarks vanishes. In this study, another\nadvantage is that it simplifies the calculation of baryonic matrix\nelements. In order to obtain the physical form factors, an\nanalytical extension from the space-like region to the time-like\nregion is required. The form factors in the space-like region can\nbe parameterized in a three-parameter form as\n \\begin{eqnarray}\\label{s14}\n F(q^2)=\\frac{F(0)}{\\left(1-\\frac{q^2}{M_{\\Lambda_b}^2}\\right)\n  \\left[1-a\\left(\\frac{q^2}{M_{\\Lambda_b}^2}\\right)\n  +b\\left(\\frac{q^2}{M_{\\Lambda_b}^2}\\right)^2\\right]},\n \\end{eqnarray}\nwhere $F$ represents the form factor $f_{1,2}$ and $g_{1,2}$. The\nparameters $a,~b$ and $F(0)$ are fixed by performing a\nthree-parameter fit to the form factors in the space-like region\nwhich were obtained in previous sections. We then use these\nparameters to determine the physical form factors in the time-like\nregion. The fitted values of $a,~b$ and $F(0)$ for different form\nfactors $f_{1,2}$ and $g_{1,2}$ are given in Table \\ref{Tab:t4}.\nThe $q^2$ dependence of the form factors is plotted in Fig.\n\\ref{f2}.\n\n\\begin{table}\n\\caption{The $\\Lambda_b\\to \\Lambda_c$ form factors given in the\n  three-parameter form.}\\label{Tab:t2}\n\\begin{ruledtabular}\n\\begin{tabular}{cccc}\n  $F$    &  $F(0)$ &  $a$  &  $b$ \\\\\n  $f_1$  &   0.50568     &   1.00    & 0.75   \\\\\n  $f_2$  &   -0.09943      &   1.50    &  1.43  \\\\\n  $g_1$  &    0.50087     &     1.00  &  0.70  \\\\\n  $g_2$  &      -0.00889   &     1.50  & 1.45\n\\end{tabular}\n\\end{ruledtabular}\n\\end{table}\n\n\\begin{figure}\n\\begin{center}\n\\scalebox{0.8}{\\includegraphics{formfactor.eps}}\n\\end{center}\n\\caption{(a) Form factors  $ f_1$ and $g_1$ \\,\\,\\,  (b)\n Form factors  $f_2$ and $g_2$}\\label{f2}\n\\end{figure}\n\nFrom Table \\ref{Tab:t2} and Fig. \\ref{f2}, we find that the form\nfactors $f_1$ and $g_1$ are nearly equal. At small recoil, i.e.\nlarge $q^2$ region, there is only a tiny difference between the two\nfunctions. Even at the maximal recoil point $q^2=0$, their\ndifference is less than 3\\%. This can be understood by Eq.\n(\\ref{s9}) where the difference between $f_2$ and $g_2$ is at order\nof $\\Lambda_{\\rm QCD}^2\/(M_{\\Lambda_b}M_{\\Lambda_c})$, a next-to-next-to-leading order in the\n$1\/m_Q$ expansion. The form factor $f_2$ and $g_2$ are small\ncomparing to $f_1$ and $g_1$. In practice, $g_2$ is approximately\nzero, and $f_2$ is about $20-30$\\% of $f_1$ and $g_1$. These\nconclusions are consistent with the results of \\cite{GKLLW}. From\nTable \\ref{Tab:t2}, the parameter $a$ for various form factors is\nclose to 1 and the parameter $b$ is small. The results suggest that\nthe $q^2$-dependence of $f_i$ and $g_i$ approximately exhibits a\ndipole behavior $F(q^2)=\\frac{F(0)}{(1-q^2\/M_{\\Lambda_b})^n}$ with $n=2$.\n\nIn the heavy quark limit, the heavy baryons $\\Lambda_b$ and\n$\\Lambda_c$ have the same scale parameter $\\beta^\\infty$ in their\nwave functions. We choose $\\beta^\\infty=0.40$ GeV which is equal to\nthe parameter in the $\\Lambda_b$ wave function. The Isgur-Wise\nfunction is usually parameterized by\n \\begin{eqnarray}\n  \\zeta(\\omega)=1-\\rho^2(\\omega-1)+\\frac{\\sigma^2}{2}(\\omega-1)^2+...,\n \\end{eqnarray}\nwhere $\\rho^2\\equiv-\\frac{d\\zeta(\\omega)}{d\\omega}|_{\\omega=1}$ is\nthe slope parameter and\n$\\sigma^2\\equiv\\frac{d^2\\zeta(\\omega)}{d\\omega^2}|_{\\omega=1}$ is\nthe curvature of the Isgur-Wise function. Our fitted values are\n \\begin{eqnarray}\n \\rho^2&=&1.47, \\\\\n \\sigma^2&=&1.90.\n \\end{eqnarray}\nThe DELPHI collaboration reported their measurement on the slope\nof the Isgur-Wise function $\\zeta(\\omega)=1-\\rho^2(\\omega-1)$ as\n$\\rho^2=2.03\\pm 0.46({\\rm stat})^{+0.72}_{-1.00}({\\rm syst})$ in\nthe semi-leptonic decay $\\Lambda_b\\rightarrow \\Lambda_c l\n\\bar{\\nu}_l$ \\cite{DELPHI}.  The recent theoretical calculations\non the slope parameter $\\rho^2$ are: $\\rho^2=1.35\\pm 0.12$ in QCD\nsum rules \\cite{HJKL}; $\\rho^2=1.51$ in a relativistic quark model\n\\cite{EFG}.  All the results are in agreement with the experiment\ndata within the theoretical and experimental errors. The\nIsgur-Wise function in the total $\\omega$ space is depicted in\nFig. \\ref{f3}. The errors in the parameter $\\beta^\\infty$ has only\na minor effect which is consistent with the B meson case\n\\cite{LWW}.\n\n\n\\begin{figure}[hhh]\n\\begin{center}\n\\scalebox{0.8}{\\includegraphics{IW.eps}}\n\\end{center}\n\\caption{The $\\Lambda_{b} \\rightarrow \\Lambda_{c}$ Isgur-Wise\nfunction $\\zeta(\\omega)$ with diquark mass $m_{[ud]}=500$ MeV.\n}\\label{f3}\n\\end{figure}\n\n\\subsection{Semi-leptonic decay of $\\Lambda_b \\to\n\\Lambda_c +l\\bar{\\nu}_l$}\n\n\nWith the form factors and the Isgur-Wise function given in the\nabove subsection, we are able to calculate the branching ratios\nand various asymmetries of $\\Lambda_b \\to \\Lambda_c l\\bar{\\nu}_l$\ndecay. Table \\ref{Tab:t4} provides our numerical predictions. The\nresults are presented for two cases: with taking the heavy quark\nlimit and without taking the heavy quark limit. The ratio of\nlongitudinal to transverse rates $R>1$ implies that the\nlongitudinal polarization dominates.\n\nThe significant difference for the transverse polarization asymmetry\n$a_T$ in the two cases(with or without taking the heavy quark limit)\nimplies that $a_T$ is sensitive to the heavy quark symmetry breaking\neffects. Thus, measurement of $a_T$ may be an ideal probe to test\nhow well the heavy quark symmetry works in the weak decays of heavy\nbaryons, not only for the rate estimate, but also other relevant\nmeasurable quantities such as $a_T$. Indeed, for the branching ratio\nand the $\\Lambda_c$ polarization asymmetry $P_L$, the deviation in\nthe two cases is at the level of a few percents, thus the heavy\nquark limit provides a good approximation.\n\nWe also compare our results with the predictions by the relativistic\nquark model \\cite{EFG}. The two models result in nearly equal\npredictions for the longitudinal asymmetry $a_L$ and the\n{$\\Lambda_c$} polarization asymmetry $P_L$. This confirms the\nobservation of \\cite{CS} that these quantities are less model\ndependent.\n\n\n\n\\begin{table}\n\\caption{The branching ratios and polarization assymetries of\n$\\Lambda_b\\to \\Lambda_c l\\bar{\\nu}_l$ .}\\label{Tab:t4}\n\\begin{ruledtabular}\n\\begin{tabular}{c|ccccc}\n    &  $Br$ &  $a_L$  &  $a_T$   & $R$    &  $P_L$ \\\\\\hline\n  within the heavy quark limit (this work)\n    &    $6.2\\%$      &   -0.926 & -0.483 & 1.539 & -0.751 \\\\\\hline\n  without the heavy quark limit (this work)\n    &  $6.3\\%$        &   -0.932 & -0.601 & 1.466 & -0.798 \\\\\\hline\n  within the heavy quark limit (in \\cite{EFG})\n    &  $6.2\\%$        &  -0.928  & -0.483 & 1.59  & -0.756 \\\\\\hline\n  with $1\/m_Q$ corrections (in \\cite{EFG})\n    &   $6.9\\%$       &   -0.940 & -0.600 & 1.61  & -0.810\n\\end{tabular}\n\\end{ruledtabular}\n\\end{table}\n\n\n\n\\subsection{Non-leptonic decays of $\\Lambda_b\\to\\Lambda_c+ M$}\n\nThe non-leptonic decays $\\Lambda_b\\to\\Lambda_c +M$ in the\nfactorization approach have been studied in the previous section.\nNow, we present our numerical predictions on the decay rates and\nrelevant measurable quantities. The CKM matrix elements take\nvalues \\cite{PDG06}\n \\begin{eqnarray}\n && V_{ud}=0.9738, \\qquad  V_{us}=0.2257, \\qquad V_{cd}=0.230,  \\qquad\n     \\nonumber\\\\\n && V_{cs}=0.957,~~\\qquad  V_{cb}=0.0416, \\qquad\n \\end{eqnarray}\nand and the effective Wilson coefficient $a_1= 1$. The meson decay\nconstants are shown in Table \\ref{Tab:t5}.\n\n\\begin{table}\n\\caption{Meson decay constants $f$ (in units of\n MeV) \\cite{CCH2}.}\\label{Tab:t5}\n\\begin{ruledtabular}\n\\begin{tabular}{cccccccccc}\n  meson & $\\pi$ & $\\rho$ & $K$ & $K^*$ & $D$ & $D^*$ & $D_s$ & $D_s^*$ & $a_1$ \\\\\\hline\n  $f$   & 131   & 216    & 160 & 210   & 200 & 220   & 230   & 230     & 203\n\\end{tabular}\n\\end{ruledtabular}\n\\end{table}\n\nThe predictions for the branching ratios and up-down asymmetries are\nprovided in Table \\ref{Tab:t6}. The Tables \\ref{Tab:t7} and\n\\ref{Tab:t8} demonstrate comparisons of our result with that in\nother approaches. Some arguments are made in orders:\n\n(1) For the processes with mesons $\\pi,\\rho,D_s,D_s^*,a_1$ being in\nthe final states, the corresponding sub-processes are $b\\to c\\bar u\nd$ or $b\\to c \\bar c s$, which are the Cabibbo-favored processes.\nThe decay ratios fall in the region $4\\times 10^{-3}$ to $1\\times\n10^{-2}$. They are the dominant decay modes which will be measured\nin near future. For the processes with mesons $K,K^*,D,D^*$ in the\nfinal states, the sub-processes are $b\\to c\\bar u s$ or $b\\to c \\bar\nc d$ which are the Cabibbo-suppressed processes. Their decay ratios\nare of order  $(3-5)\\times 10^{-4}$.\n\n(2) In the scheme adopted in this work, we obtain the ratio\n$\\frac{BR(\\Lambda_b^0\\to\\Lambda_c^+ l^-\\bar\\nu_l)}{BR(\\Lambda_b^0\n\\to\\Lambda_c^+\\pi^-)}$ to be $16.8$,  and this theoretical\nprediction is consistent with the experimental measurement (a\npreliminary result) $\\frac{BR(\\Lambda_b^0\\to\\Lambda_c^+\nl^-\\bar\\nu_l)}{BR(\\Lambda_b^0 \\to\\Lambda_c^+\\pi^-)}=20.0\\pm3.0({\\rm\nstat})\\pm 1.2({\\rm syst})$ \\cite{exp2}.\n\n(3) From Table \\ref{Tab:t7}, it is noted that the differences\namong the predictions on the branching ratios for non-leptonic\ndecays by various theoretical approaches are obvious. It is hard\nto decide which model is closer to reality at present, because\nmore precise data are still lacking. It may be more appropriate to\nemploy the experimental data about the semi-leptonic decay as\ninputs to reduce the model dependence of the\n$\\Lambda_b\\to\\Lambda_c$ transition form factors as we did in this\nwork.\n\n\n\n(4) All the up-down asymmetries $\\alpha$ are negative, this result\nreflects the $V-A$ nature of the weak currents.  Table\n\\ref{Tab:t8} shows that the numerical values of the up-down\nasymmetries $\\alpha$ predicted by different approaches are nearly\nthe same except for the process $\\Lambda_b^0\n\\to\\Lambda_c^+D_s^{*-}$ where the difference is about 10\\%.\n\n\n\n\n\n\n\\begin{table}\n\\caption{Branching ratios and up-down asymmetries of non-leptonic\ndecays $\\Lambda_b\\to\\Lambda_c M$ with the light diquark mass\n$m_{[ud]}=500$ MeV.}\\label{Tab:t6}\n\\begin{ruledtabular}\n\\begin{tabular}{c|cc|cc|}\n  & \\multicolumn{2}{c|}{within the heavy quark limit~~~~~~}\n  & \\multicolumn{2}{c|}{without the heavy quark limit~~~~} \\\\\\hline\n  & $Br$                & $\\alpha$ & $Br$ & $\\alpha$          \\\\\\hline\n  $\\Lambda_b^0\\to\\Lambda_c^+ \\pi^-$  & $4.22\\times 10^{-3}$  & $-1$\n                        & $3.75\\times 10^{-3}$  & $-1$      \\\\\\hline\n  $\\Lambda_b^0\\to\\Lambda_c^+ \\rho^-$ & $6.07\\times 10^{-3}$  & $-0.897$\n                        & $6.73\\times 10^{-3}$  & $-0.885$  \\\\\\hline\n  $\\Lambda_b^0\\to\\Lambda_c^+ K^-$    & $3.41\\times 10^{-4}$  & $-1$\n                        & $3.02\\times 10^{-4}$  & $-1$      \\\\\\hline\n  $\\Lambda_b^0\\to\\Lambda_c^+ K^{*-}$  & $3.15\\times 10^{-4}$  & $-0.865$\n                        & $3.50\\times 10^{-4}$  & $-0.857$  \\\\\\hline\n  $\\Lambda_b^0\\to\\Lambda_c^+ a_1^-$  & $5.84\\times 10^{-3}$  & $-0.758$\n                        & $6.49\\times 10^{-3}$  & $-0.760$  \\\\\\hline\n  $\\Lambda_b^0\\to\\Lambda_c^+ D_s^-$  & $1.18\\times 10^{-2}$ & $-0.984$\n                        & $1.14\\times 10^{-2}$ & $-0.982$  \\\\\\hline\n  $\\Lambda_b^0\\to\\Lambda_c^+ D^{*-}_s$& $8.88\\times 10^{-3}$ & $-0.419$\n                        & $9.96\\times 10^{-3}$  & $-0.442$  \\\\\\hline\n  $\\Lambda_b^0\\to\\Lambda_c^+ D^-$  & $5.23\\times 10^{-4}$ & $-0.987$\n                        & $5.01\\times 10^{-4}$  & $-0.986$  \\\\\\hline\n  $\\Lambda_b^0\\to\\Lambda_c^+ {D^*}^-$& $4.61\\times 10^{-4}$& $-0.459$\n                        & $5.12\\times 10^{-4}$  & $-0.481$\n\\end{tabular}\n\\end{ruledtabular}\n\\end{table}\n\n\n\\begin{table}\n\\caption{Branching ratios for non-leptonic decays\n$\\Lambda_b\\to\\Lambda_c +M$ within different theoretical approaches\n(in units of $10^{-2}$).}\\label{Tab:t7}\n\\begin{ruledtabular}\n\\begin{tabular}{ccccccc}\n  & This work & \\cite{Cheng} & \\cite{MGKIIO} & \\cite{FR} & \\cite{GMM} & \\cite{LLS} \\\\\\hline\n  $\\Lambda_b^0\\to\\Lambda_c^+ \\pi^-$  & 0.375 & 0.38 & 0.175 & -    & 0.391 & 0.503 \\\\\\hline\n  $\\Lambda_b^0\\to\\Lambda_c^+ \\rho^-$ & 0.673 & 0.54 & 0.491 & -    & 1.082 & 0.723 \\\\\\hline\n  $\\Lambda_b^0\\to\\Lambda_c^+ K^-$    & 0.030 &  -   & 0.013 &      & -     & 0.037 \\\\\\hline\n  $\\Lambda_b^0\\to\\Lambda_c^+ K^{*-}$ & 0.035 &  -   & 0.027 & -    & -     & 0.037 \\\\\\hline\n  $\\Lambda_b^0\\to\\Lambda_c^+ a_1^-$  & 0.649 &  -   & 0.532 & -    & -     & -     \\\\\\hline\n  $\\Lambda_b^0\\to\\Lambda_c^+ D_s^-$  & 1.140 & 1.1  & 0.770 & 2.23 & 1.291 & -     \\\\\\hline\n  $\\Lambda_b^0\\to\\Lambda_c^+D^{*-}_s$& 0.996 & 0.91 & 1.414 & 3.26 & 1.983 & -     \\\\\\hline\n  $\\Lambda_b^0\\to\\Lambda_c^+ D^-$    & 0.050 &  -   & 0.030 & -     & -     \\\\\\hline\n  $\\Lambda_b^0\\to\\Lambda_c^+ {D^*}^-$& 0.051 &  -   & 0.049 & -     & -\n\\end{tabular}\n\\end{ruledtabular}\n\\end{table}\n\n\\begin{table}[!h]\n\\caption{Up-down asymmetries for non-leptonic decays\n$\\Lambda_b\\to\\Lambda_c M$ within different theoretical\napproaches.}\\label{Tab:t8}\n\\begin{ruledtabular}\n\\begin{tabular}{cccccc}\n  & This work     & \\cite{Cheng} & \\cite{MGKIIO}  & \\cite{FR}    & \\cite{LLS}    \\\\\\hline\n  $\\Lambda_b^0\\to\\Lambda_c^+ \\pi^-$  & -1     & -0.99 & -0.999 & -     &  -1     \\\\\\hline\n  $\\Lambda_b^0\\to\\Lambda_c^+ \\rho^-$ & -0.885 & -0.88 & -0.897 & -     &  -0.885 \\\\\\hline\n  $\\Lambda_b^0\\to\\Lambda_c^+ K^-$    & -1     &  -    & -1     &       &  -1     \\\\\\hline\n  $\\Lambda_b^0\\to\\Lambda_c^+ K^{*-}$ & -0.857 &  -    & -0.865 & -     &  0.885  \\\\\\hline\n  $\\Lambda_b^0\\to\\Lambda_c^+ a_1^-$  & -0.760 &  -    & -0.758 & -     &  -      \\\\\\hline\n  $\\Lambda_b^0\\to\\Lambda_c^+ D_s^-$  & -0.982 & -0.99 & -0.984 & -0.98 &  -      \\\\\\hline\n  $\\Lambda_b^0\\to\\Lambda_c^+D^{*-}_s$& -0.442 & -0.36 & -0.419 & -0.40 &  -      \\\\\\hline\n  $\\Lambda_b^0\\to\\Lambda_c^+ D^-$    & -0.986 &  -    & -0.987 & -     &  -      \\\\\\hline\n  $\\Lambda_b^0\\to\\Lambda_c^+ {D^*}^-$& -0.481 &  -    & -0.459 & -     &  -\n\\end{tabular}\n\\end{ruledtabular}\n\\end{table}\n\n\\section{Conclusions}\n\nIn this work, we investigate extensively the $\\Lambda_b\\to\\Lambda_c$\ntransition form factors in the light-front approach and make\npredictions on the rates for the semi-leptonic decay\n$\\Lambda_b\\to\\Lambda_c l\\bar\\nu_l$ and non-leptonic two-body decays\n$\\Lambda_b\\to\\Lambda_c+ M$. In the light-front quark model, we adopt\nthe diquark picture for the heavy baryons. It is believed that for\nthe the heavy baryons which contain at least one heavy quark, the\nquark-diquark picture seems to work well, therefore one can employ\nit for evaluating the hadronic matrix elements of\n$\\Lambda_b\\to\\Lambda_c$ transitions which are dominated by\nnon-perturbative QCD effects. The light scalar diquark mass\ndetermined from the data on the semi-leptonic decay is about $500$\nMeV. Our numerical results show that the $q^2$-dependence of the\nmomentum transfer of different form factors has a dipole-like\nbehavior. The slope parameter of the universal Isgur-Wise function\nis found to be consistent with that obtained by fitting experimental\ndata. The small difference for the branching ratio of the\nsemi-leptonic decay with and without the heavy quark limit implies\nthat the heavy quark symmetry is good in the heavy bottom baryon\nsystem. However on the other aspect, the transverse polarization\nasymmetry is shown to be sensitive to the heavy quark symmetry\nbreaking, and it is worth further and more accurate studies. Our\nresults for the exclusive non-leptonic two-body decays\n$\\Lambda_b\\to\\Lambda_c+ M$ is modest among the predictions by other\napproaches. The semi-leptonic to non-leptonic $\\Lambda_c^+\\pi^-$\ndecay ratio is well in accord with the experimental measurements.\nThe non-leptonic decays, so far have not been accurately measured,\nand there are only upper bounds for some channels, so that it is\nstill hard to judge the closeness of the present models to the\nphysical reality yet. Fortunately the LHCb will run and a remarkable\namount of data on $\\Lambda_b$ production and decay will be\naccumulated in the future LHCb, then one may be able to verify the\ndifferent models.\n\n\n\n\\section*{Acknowledgement}\n\nThis work was supported in part by NNSFC under contract Nos.\n10475042, 10745002 and 10705015.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe violation of the charge conjugation ($\\mathcal{C}$), spatial reflection ($\\mathcal{P}$), and time reversal ($\\mathcal{T}$) symmetries is the striking feature of the Standard model (SM) \\cite{khriplovich2012cp,schwartz2014quantum,particle2020review}. The sources of the charge-parity ($\\mathcal{CP}$) nonconservation in the SM are Cabibbo-Kobayashi-Maskawa (CKM) \\cite{Cabibbo1963,KobayashiMaskawa1973} and Pontecorvo\u2013Maki\u2013Nakagawa\u2013Sakata (PMNS) matrices \\cite{Pontecorvo1957,MNS1962}, and, possibly, the $\\theta$ term of the strong interaction \\cite{Cheng1988,KimCarosi2010}. One of the $\\mathcal{CPT}$ theorem consequences is that nonconservation of $\\mathcal{CP}$ is equal to the violation of $\\mathcal{T}$-symmetry.\n\nOne of the possible manifestation of the $\\mathcal{CP}$-nonconservation is the electron electric dipole moment (eEDM). In the Standard model the eEDM appears only in the multiloop processes with a high order of the weak coupling constant and, thus, the predicted value is very small. On the other hand, some models of the physics beyond the Standard Model (SM) forecast new $\\mathcal{CP}$-violation sources that can lead to the significant increase of the eEDM \\cite{Fukuyama2012,PospelovRitz2014,YamaguchiYamanaka2020,YamaguchiYamanaka2021}. The presence of the particle superpartners in the Supersymmetry theory (SUSY) would provide new $\\mathcal{P}$, $\\mathcal{T}$-violation sources. Besides, the fluctuations of the $\\theta$ parameter, the axion, in the Peccei\u2013Quinn theory may result in the  $\\mathcal{CP}$-violating processes. Furthermore, the matter-antimatter ratio in the observable universe  \\cite{DineKusenko2003, Sakharov1967} may imply new sources of the charge-parity violation.\n\nThe high precision molecular experiments provide a powerful way to investigate the $\\mathcal{CP}$-violating physics \\cite{baron2014order,ACME:18}.\n\nAs for now, the best experimental bound on the eEDM was obtained for diatomic molecules with closely spaced $\\Omega$-doublets such as ThO \\cite{ACME:18,DeMille:2001,Petrov:14,Vutha:2010,Petrov:15,Petrov:17} and  HfF$^{+}$ \\cite{Cornell:2017,Petrov:18}. These experiments also put constraints on the scalar-pseudoscalar nucleon-electron interaction \\cite{ginges2004violations,PospelovRitz2014,ChubukovLabzowsky2016}.\nExperiments for searching other $\\mathcal{P}$, $\\mathcal{T}$ odd effects, including nuclear magnetic quadrupole moment \\cite{FDK14, Petrov:17b, Kurchavov:20, maison2019theoretical}  and axion mediated interactions \\cite{maison2020study, maison2021axion} are planed. \n\n\nThe vibrational modes of the polyatomic molecules create unique spectral characteristics not possessed by diatomic molecules. For instance, the triatomic species can simultaneously allow  laser-cooling \\cite{Isaev_2017} and possess levels with opposite parity, the so-called $l$-doublets \\cite{Kozyryev:17,hutzler2020polyatomic}. \n\n\n\nThe levels of opposite parities constituting the $l$-doublet are mixed when the external electric field applied so that the molecule becomes polarized. The $\\mathcal{P}$, $\\mathcal{T}$-violation is manifested in the energy splitting, $\\Delta E_{\\mathcal{P},\\mathcal{T}}$, between  the levels with opposite values $\\pm M$ of  total angular momentum projection on the electric field axis.\n\n\nIf the electron has EDM $d_e$ and is affected by the scalar-pseudoscalar interaction with nuclei characterized by the coupling constant $k_s$, these $\\mathcal{P}$, $\\mathcal{T}$-odd effects can be estimated from the  maximum splitting between levels with opposite values of $M$ given by,\n\\begin{equation}\n\\Delta E_{\\mathcal{P},\\mathcal{T}}=2E_{\\rm eff}  d_e + 2E_{\\rm s} k_s,\n\\label{split}\n\\end{equation}\nThe parameters $E_{\\rm eff}$ and $E_{\\rm s}$ are determined by the molecular electronic structure \\cite{KozlovLabzowsky1995, titov2006d, Safronova2017}.\n\n\n\n\n\nThe laser-cooling in one dimension was achieved for alkaline earth metal monohydroxides such as SrOH \\cite{kozyryev2017sisyphus} and, recently, the YbOH \\cite{steimle2019field,augenbraun2020laser}. The latter is considered a promising candidate for the future experiments searching eEDM \\cite{Kozyryev:17}.\n\nThe sensitivity of the \nYbOH molecule to the eEDM was previously computed in \\cite{denis2019enhancement} within the relativistic coupled cluster method. However, the vibrational motion, including the bending modes, of excited vibrational states may influence the value of this parameter. In \\cite{prasannaa2019enhanced} $E_\\mathrm{eff}$ was studied for different nonlinear configurations and strong dependence on the bending\nangle already at Dirac-Hartree-Fock (DHF) level was stressed. This claim is inconsistent with the results of \\cite{gaul2020ab} that used the complex generalized Hartree-Fock and Kohn-Sham methods within zeroth-order regular approximation and also has given the harmonic estimate for $E_\\mathrm{eff}$ at the $v=1$ vibrational level.\n\nPreviously we obtained the rovibrational\nwavefunctions for the molecule RaOH \\cite{ourRaOH}. This allowed us not only to compute the $E_\\mathrm{eff}$ and $E_\\mathrm{s}$ parameters for the first vibrational levels but also to obtain the value of the $l$-doubling that determines the external electric field required for the complete polarization of the molecule. In this paper, we apply the techniques we developed to perform a similar analysis for the YbOH molecule.\n\n\n\n\n\n\\section{Methods}\n\nWe assume that the wavefunction of the molecule can approximately be factorized into the nuclear and electronic parts,\n\\begin{equation}\n\\Psi_{\\rm total}\\simeq\\Psi_{\\rm nuc}(Q)\\psi_{\\rm elec}(Q|q),\n\\label{totalWF}\n\\end{equation}\nwhere $Q$ denotes generalized coordinates of the nuclei and $q$ - generalized coordinates of the electrons. Within the Born-Oppenheimer approximation the electronic part $\\psi_{\\rm elec}(Q|q)$ is the solution of the Dirac-Coulomb equation for the electrons in the field of the nuclei fixed at coordinates $Q$. To describe the configuration of the triatomic molecule we choose $Q$ as the Jacobi coordinates represented in Fig.~\\ref{Jacob}: $\\hat{r}$ and $\\hat{R}$ are unit \nvectors directed along the OH axis and Yb - OH center of mass (c.m.) axis respectively, $\\theta$ is the angle between above axes, $R$ is the distance between Yb and the c.m. of OH. As the frequency of OH vibrational mode is about one order of magnitude larger than other vibrational frequencies in YbOH, we fix OH ligand stretch at the equilibrium distance $r=1.832 \\,a.u.$ \\cite{Huber:1979}.\n\nThe nuclear part of the wavefunction $\\Psi_{\\rm nuc}$ satisfies the Schr\\\"{o}dinger equation,\n\\begin{equation}\n\\hat{H}_{\\rm nuc}\\Psi_{\\rm nuc}(R, \\hat{R}, \\hat{r}) = E \\Psi_{\\rm nuc}(R, \\hat{R}, \\hat{r}).\n\\label{Shreq}\n\\end{equation}\nThe \nnuclear\nHamiltonian takes the form,\n\\begin{equation}\n\\hat{H}_{\\rm nuc}=-\\frac{1}{2\\mu}\\frac{\\partial^2}{\\partial R^2}+\\frac{\\hat{L}^2}{2\\mu R^2}+\\frac{\\hat{j}^2}{2\\mu_{OH}r^2}+V(R,\\theta),\n\\end{equation}\nwhere $\\mu$ is the the $Yb-OH$ reduced mass, $\\mu_{OH}$ is the the OH ligand reduced mass, $\\hat{L}$  is the angular momentum of the Yb-OH system rotation \naround its c.m., $\\hat{j}$ is the ligand angular momentum, and $V(R,\\theta)$ is the effective adiabatic potential obtained from the electronic structure calculations.\n\n\\begin{figure}[h]\n\\centering\n  \\includegraphics[width=0.25\\textwidth]{YbOH.png}\n  \\caption{Jacobi coordinates}\n  \\label{Jacob}\n\\end{figure}\n\n\nThe sensitivity of the spectrum to the $\\mathcal{P}$, $\\mathcal{T}$-odd interactions for the fixed configurations can be described by the parameters,\n\\begin{equation}\nE_{\\rm eff}(R,\\theta)=\\frac{\\langle\\psi_{\\rm elec}(R,\\theta)| \\hat{H}_d|\\psi_{\\rm elec}(R,\\theta)\\rangle}{d_e{\\rm sign}(\\Omega)},\n\\end{equation}\n\\begin{equation}\nE_{\\rm s}(R,\\theta)=\\frac{\\langle\\psi_{\\rm elec}(R,\\theta)| \\hat{H}_s|\\psi_{\\rm elec}(R,\\theta)\\rangle}{k_s{\\rm sign}(\\Omega)},\n\\end{equation}\nthat can be understood as the expectation values for the eEDM and scalar-pseudoscalar nucleon-electron interaction terms in the $\\mathcal{P}$, $\\mathcal{T}$-odd interaction Hamiltonian Hamiltonian\n\\begin{equation}\n\\hat{H}_{\\cancel{\\mathcal{PT}}}=\\hat{H}_d+\\hat{H}_s,\n\\end{equation}\n\\begin{equation}\n\\hat{H_d}=  2d_e\\sum_{i}\n  \\left(\\begin{array}{cc}\n  0 & 0 \\\\\n  0 & \\bf{\\sigma_i E_i} \\\\\n  \\end{array}\\right)\\ \n \\label{Hd},\n\\end{equation}\n\\begin{equation}\n\\hat{H_s}=ik_s\\frac{G_F}{\\sqrt2}\\sum_{j=1}^{N_{\\rm elec}}\\sum_{I=1}^{N_{\\rm nuc}}{\\rho_I\\left(\\vec{r_j}\\right)Z_I}\\gamma^0\\gamma^5,\n\\label{Hs}\n\\end{equation}\nwhere  $\\rho_I$ is the normalized charge density of the  $I$-th nucleon, $G_F$ is Fermi constant, $\\bf{\\sigma}$ are Pauli matrices, $\\bf{E_i}$ is the internal molecular electric field that acts on ith electron.\n\nFor the total molecular wavefunction (\\ref{totalWF}) these parameters should be averaged over the nuclear wavefunction,\n\\begin{equation}\n\\label{Eeffaver}\nE_{\\rm eff,s}=\\int dR d\\hat{R} d\\hat{r} |\\Psi_{\\rm nuc}(R, \\hat{R}, \\hat{r})|^2 E_{\\rm eff,s}(R,\\theta).\n\\end{equation}\n\nThe Ytterbium atom was described by a 28-electron generalized relativistic effective core potential (GRECP) \\cite{titov1999generalized,mosyagin2010shape,mosyagin2016generalized} and a 42-valence electron basis set developed by the PNPI Quantum Chemistry Laboratory \\cite{QCPNPI:Basis}. The cc-pVTZ basis was used for H and O atoms. The calculations were performed on a grid of Jacobi coordinates. The $R$ coordinate ranges from $2.6\\, a.u.$ to $4.3\\,a.u.$ with step $0.1\\, a.u.$ The $\\theta$ angle values are $0^\\circ$, $5^\\circ$, $10^\\circ$, $15^\\circ$, $20^\\circ$, $25^\\circ$, $57^\\circ$, $90^\\circ$, $122^\\circ$, $155^\\circ$ and $180^\\circ$. Extra points near the equilibrium were added to better describe the region most relevant for the lowest vibrational levels.\n\n\nThe molecular two-component pseudospinors were obtained using the Hartree-Fock self-consistent field (SCF) method implemented in the Dirac 19 software \\cite{DIRAC19}. The pseudospinors\nare smoothed in the inner core region, so that the electronic\ndensity in this region is not correct. The operators\nin eqs. (\\ref{Hd},\\ref{Hs}) are heavily concentrated near the\nnucleus and are therefore strongly affected by the wave\nfunction in the inner region. The four-component molecular\nspinors must therefore be restored in the inner region\nof Yb.\nThe MOLGEP program was used to apply the method of one-center restoration of the correct four-component spinors in the core region with help of the equivalent basis sets \\cite{Petrov:02,titov2006d,skripnikov2015theoretical}. The matrix elements of $\\hat{H}_\\mathrm{d}$ and $\\hat{H}_\\mathrm{s}$ were computed in the basis of the restored spinors $\\psi_i$.\n\nRestoration of the basis begins with the creation of an equivalent basis set of atomic\n(one-center) four-component spinors:\n  \\begin{equation}\n  \\left\\{\\left( \\begin{array}{c}\n  f_{nlj}(r)\\chi_{ljm} \\\\\n  g_{nlj}(r)\\chi_{l'jm}\n  \\end{array} \\right) \\right\\},\n  \\end{equation}\nand two-component pseudospinors $\\left\\{ \\tilde{f}_{nlj}(r)\\chi_{ljm}\\right\\}$. Here $f$ - large component, $g$ - small\ncomponent, $\\chi$ - spin-angular part, $n$ - principal quantum number, $j$ and $m$ - total\nelectronic moment and his projection in internuclear axis, $l$ and $l'$ - orbital moment, and $l'=2j-l$.\n\nFor the numerical four-component and two-component atom calculations, the HFD and\nHFJ\/GRECP programs were used to create two equivalent basis sets for\nreconstruction. Molecular pseudo-orbitals then decompose in the basis of\ntwo-component single-center atomic pseudospinors,\n\n\\begin{equation}\n  \\tilde{\\phi}_i(\\mathrm{r})\\approx \\sum_{l=0}^{L_{\\text{max}}}\\sum_{j=|l-1\/2|}^{j=|l+1\/2|}\\sum_{nm}c_{nljm}^i\\tilde{f}_{nlj}(\\mathrm{r})\\chi_{ljm}.\n\\end{equation}\nThen two-component pseudospinors are replaced by equivalent four-component spinors:\n\\begin{equation}\n  \\label{eq4}\n  \\phi_i(\\mathrm{r})\\approx \\sum_{l=0}^{L_{\\text{max}}}\\sum_{j=|l-1\/2|}^{j=|l+1\/2|}\\sum_{nm}c_{nljm}^i\\left( \\begin{array}{c}\n  f_{nlj}(r)\\chi_{ljm} \\\\\n  g_{nlj}(r)\\chi_{l'jm}\n\\end{array} \\right).\n\\end{equation}\nMolecular four-component spinors constructed in this way are orthogonal to the core\nspinor Yb, since atomic basis functions in the equation were calculated for frozen inner\ncore electrons.\nFor the current calculation we put $L_{\\text{max}}=3$, what is enough for accurate calculation of $E_{\\rm eff}$ and $E_\\mathrm{s}$ \\cite{Petrov:02}.\n\n\nFor the correlation computations we have chosen the active space with 30 frozen electrons and 21 active ones. The relativistic coupled cluster method with single, double and perturbative triple excitations (CCSD (T)) implemented in Dirac 19 RELCCSD module was used to compute the points of the potential surface on the grid defined above. Then the function $V(R,\\theta)$ was constructed by the bicubic interpolation Fig.~\\ref{Surface}.\n\n\\begin{figure}[h]\n\\centering\n  \\includegraphics[width=0.5\\textwidth]{potential_surf.png}\n  \\caption{Potential surface $V(R,\\theta)$}\n  \\label{Surface}\n\\end{figure}\n\nThe obtained adiabatic potential was used for numerical computations of the nuclear wavefunctions based on the close-coupled equations \\cite{mcguire1974quantum} obtained as a decomposition of (\\ref{Shreq}) in terms of the eigenfunctions of the molecular and ligand angular momenta. For the details of our approach we refer the reader to our previous work \\cite{ourRaOH}. This way we take into account the anharmonicities of the potential and interaction between the rotational and vibrational degrees of freedom without resorting to the perturbative techniques.\n\nFor the property calculations of the linear configuration we get the one-electron density matrix $\\rho^{(1)}_{ij}$ at CCSD level with help of the MRCC program suite \\cite{MRCC2020}. The density matrix is then contracted with $E_{\\rm eff}$ and $E_{\\rm s}$ matrix elements to obtain the values of the correlation\ncorrections for the linear configurations,\n\\begin{equation}\nE_{\\rm eff,s}(R,\\theta)=\\frac{1}{N_\\mathrm{{elec}}}\\sum_{i,j=1}^{N_{\\rm orb}}\\rho^{(1)}_{ij}\\frac{\\langle \\psi_i|\\hat{H}_{\\rm d,s}|\\psi_j\\rangle}{d_e{\\rm sign}(\\Omega)}.\n\\end{equation}\nRegretfully, the Dirac-MRCC interface works only for the symmetry groups with real representations (such as $C_{2v}$) but not with  ones with complex representations (such as $C_s$) of the nonlinear molecules. Therefore the CCSD correction was obtained only for the linear configurations and are depicted on Fig. \\ref{PTLinear}. Since these corrections constitute only about $1\\%$ of the SCF values near minimum reaching $6\\%$ far from equilibrium point, we assumed that, as a first approximation, it is reasonable to approximate the CCSD correction for the nonlinear molecule by the result computed for the linear molecule:\n\\begin{align}\n&E_{{\\rm eff},{\\rm s}}^{({\\rm total},I)}(R,\\theta)=E_{{\\rm eff},{\\rm s}}^{({\\rm ccsd})}(R,0^\\circ)+ \\nonumber\\\\\n&+\\Big( E_{{\\rm eff},{\\rm s}}^{({\\rm scf})}(R,\\theta)- E_{{\\rm eff},{\\rm s}}^{({\\rm scf})}(R,0^\\circ)\\Big).\\label{EtotalI}\n\\end{align}\n\nIn other words, the dependence of $E_{\\rm eff,s}(R,\\theta)$ on $\\theta$ was calculated at SCF level. To test the validity of our approximation, we made a finite field computation of\n$E_{\\rm eff,s}$\nby CCSD method for the near equilibrium value of $R=3.9\\,\\mathrm{a.u.}$ and different angles. Because this computation is very expensive we were able to obtain the values only for the single value of $R$.\n\n\n\n\n\\section{Results and discussion}\n\nThe spectrum of the nuclear wavefunctions $\\Psi_{\\rm nuc}$ is characterized by the parameters we present in the Table~\\ref{tab:Spectrum}. \nThe agreement between the computed and the experimental values of frequencies is rather good.\n\nOur $l$-doubling value is consistent with an estimate\\cite{HerzbergBook},\n\\begin{equation}\nq \\simeq \n    \\frac{B^2}{\\nu_2}\\Big(1+4\\frac{\\zeta_{21}^2\\nu_{2}^2}{\\nu_{1}^2-\\nu_{2}^2}\\Big)(v+1).\n\\end{equation} Comparing with our results, we can find Coriolis coefficient, $\\zeta_{21}=0.265$. The value of $l$-doubling for YbOH is greater than our result for RaOH molecule, $\\Delta E_{J=1}=2q=14.5$ \\rm MHz\\cite{ourRaOH} as expected from the smaller momentum of inertia  of the  YbOH molecule.\n\n\\begin{table}\n\\caption{\\label{tab:Spectrum} Rovibrational spectrum parameters}\n\\begin{ruledtabular}\n\\begin{tabular}{ccc}\n& Computation & Experiment\\footnote{The number in parenthesis denotes $2\\sigma$ deviation} \\\\\n\\hline\nStretching mode $\\nu_1$\n  & $550 {\\rm cm}^{-1}$ & $529.341(1) {\\rm cm}^{-1}$,\\cite{melville2001visible} \\\\\nBending mode $\\nu_2$\n  & $319 {\\rm cm}^{-1}$ & $339(5) {\\rm cm}^{-1},$\\cite{melville2001visible} \\\\\nRotational constant $B$  & $0.2461 {\\rm cm}^{-1}$ & $0.245434(13) {\\rm cm}^{-1}$,\\cite{melville2001visible}\\\\\n&& $0.2451163(10) {\\rm cm}^{-1}$,\\cite{nakhate2019pure}\\\\\n$l$-doubling $\\Delta E_{J=1}=2q$  & $26 {\\rm MHz}$ & \\\\\n\\end{tabular}\n\\end{ruledtabular}\n\\end{table}\n\n\\begin{figure}[h!]\n  \\includegraphics[width=0.49\\textwidth]{Eeff_Es.png}\n  \\caption{SCF $\\mathcal{P}$,$\\mathcal{T}$-odd parameters for\nnonlinear configurations of YbOH}\n  \\label{PT3d}\n\\end{figure}\n\n\\begin{figure*}[h!]\n\\centering\n  \\includegraphics[width=0.49\\textwidth]{Eeff_linear.png}\n  \\includegraphics[width=0.49\\textwidth]{Es_linear.png}\n  \\caption{SCF (dashed) and CCSD (solid) $\\mathcal{P}$,$\\mathcal{T}$-odd parameters for linear configurations of YbOH}\n  \\label{PTLinear}\n\\end{figure*}\n\nThe dependence of the $E_{{\\rm eff},{\\rm s}}$ on the bending and stretching is depicted on the Fig.~\\ref{PT3d}. As with RaOH \\cite{ourRaOH} we do not confirm the oscillatory behavior claimed in \\cite{prasannaa2019enhanced}.\n\n\nAs described in previous section, the full dependence of $E_{\\rm eff,s}(R,\\theta)$ on $\\theta$ was calculated at SCF level. The finite field computation was performed only for the single value of $R=3.9\\,\\mathrm{a.u.}$. In the Table~\\ref{tab:SCF_FF_Compare} we compare the deviations from the equilibrium value of $E_{\\rm eff}(\\theta)-E_{\\rm eff}(0^\\circ)$ for the SCF and correlation corrections results. The changes of the correlation correction happen to be of the same magnitude as the changes of the SCF values. The deviation becomes significant for large bending angles and, while this does not affect the average values on the rovibrational levels considered, it may become important for the higher excited levels. The similar analysis was made for the $E_{\\rm s}$ values in the Table \\ref{tab:SCF_FF_Compare2}. Unlike SCF values the correlation correction for $E_{\\rm s}$ have a different angular dependence from $E_{\\rm eff}$.\n\nTo take the angular dependence of the correlation correction $\\Delta E^{(corr)}_{{\\rm eff},{\\rm s}}$ into account using the available data we use the following approximation,\n\\begin{align}\n&E_{{\\rm eff},{\\rm s}}^{({\\rm total},II)}(R,\\theta)=E_{{\\rm eff},{\\rm s}}^{({\\rm ccsd})}(R,0^\\circ)+ \\nonumber\\\\\n&+\\Big( E_{{\\rm eff},{\\rm s}}^{({\\rm scf})}(R,\\theta)- E_{{\\rm eff},{\\rm s}}^{({\\rm scf})}(R,0^\\circ)\\Big)\n\\nonumber\\\\\n&+\\Big( E_{{\\rm eff},{\\rm s}}^{({\\rm ccsd})}(3.9\\,\\mathrm{a.u.},\\theta)- E_{{\\rm eff},{\\rm s}}^{({\\rm scf })}(3.9\\,\\mathrm{a.u.},\\theta)\\Big)\n\\nonumber\\\\\n&-\\Big( E_{{\\rm eff},{\\rm s}}^{({\\rm ccsd})}(3.9\\,\\mathrm{a.u.},0^\\circ)- E_{{\\rm eff},{\\rm s}}^{({\\rm scf })}(3.9\\,\\mathrm{a.u.},0^\\circ)\\Big).\n\\label{Etotal}\n\\end{align}\n\n\n\n\\begin{table}\n\\caption{\\label{tab:SCF_FF_Compare} The deviations of $E_{\\rm eff}$ for $R=3.9\\,\\mathrm{a.u.}$ from the equilibrium values in the finite field approach}\n\\begin{ruledtabular}\n\\begin{tabular}{ccc}\nAngle & SCF, GV\/cm & CCSD correction, GV\/cm \\\\\n\\hline\n$5^\\circ$ & -0.006 & \\,\\,0.003 \\\\\n$10^\\circ$ & -0.025 & -0.024 \\\\\n$15^\\circ$ & -0.055 & -0.066 \\\\\n$20^\\circ$ & -0.096 & -0.117\\\\\n$25^\\circ$ & -0.149 & -0.173\\\\\n$57^\\circ$ & -0.629 & -0.209\\\\\n$90^\\circ$ & -0.923 & -0.722\\\\\n$122^\\circ$ & -0.773 & -5.138\n\\end{tabular}\n\\end{ruledtabular}\n\\end{table}\n\n\\begin{table}\n\\caption{\\label{tab:SCF_FF_Compare2} The deviations of $E_{\\rm s}$ for $R=3.9\\,\\mathrm{a.u.}$ from the equilibrium values in the finite field approach}\n\\begin{ruledtabular}\n\\begin{tabular}{ccc}\nAngle & SCF, GV\/cm & CCSD correction, GV\/cm \\\\\n\\hline\n$5^\\circ$ & -0.006 & \\,\\,0.001 \\\\\n$10^\\circ$ & -0.022 & \\,\\,0.002 \\\\\n$15^\\circ$ & -0.049 & \\,\\,0.005 \\\\\n$20^\\circ$ & -0.086 & \\,\\,0.008\\\\\n$25^\\circ$ & -0.132 & \\,\\,0.010\\\\\n$57^\\circ$ & -0.557 & -0.030\\\\\n$90^\\circ$ & -0.816 & -0.297\\\\\n$122^\\circ$ & -0.685 & -0.658\n\\end{tabular}\n\\end{ruledtabular}\n\\end{table}\n\nWe summarize the results for the $\\mathcal{P}$, $\\mathcal{T}$-odd parameters and confront them with the preceding work in the Table~\\ref{tab:EeffEs}. The sensitivities to $\\mathcal{P}$,$\\mathcal{T}$-odd effects is more than two times smaller than for the RaOH molecule\\cite{ourRaOH}. Our results are in concordance with the previous estimates for the fixed geometries. While for the lower levels vibrations do not strongly affect the $E_\\mathrm{eff}$ and $E_\\mathrm{s}$ parameters, for higher levels it may become significant.\n\n\\begin{table}\n\\caption{\\label{tab:EeffEs} Sensitivities to the $\\mathcal{P}$, $\\mathcal{T}$-odd effects for YbOH}\n\\begin{ruledtabular}\n\\begin{tabular}{ccc}\n& $E_{\\rm eff},\\,\\mathrm{GV}\/\\mathrm{cm}$ & $E_{\\rm s}, kHz$ \\\\\n\\hline\nEquilibrium geometry\n  & $23.875$ & $20.659$ \\\\\n \\hline\n\\multicolumn{3}{l}{Angular dependence as in (\\ref{EtotalI})}\\\\\n$v=0$ state\n  & $23.810$ & $20.602$ \\\\\n$v=1$ state\n  & $23.740$ & $20.540$ \\\\\n\\hline\n\\multicolumn{3}{l}{Angular dependence  as in (\\ref{Etotal})}\\\\\n$v=0$ state\n  & $23.716$ & $20.608$ \\\\\n$v=1$ state\n  & $23.576$ & $20.548$ \\\\\n\\hline\nRef.~\\onlinecite{denis2019enhancement} FSCC+Gaunt\\,\\,\n  & $23.37$ &  \\\\\nRef.~\\onlinecite{prasannaa2019enhanced} QZ CCSD\\,\\,\n  & $23.80$ &  \\\\\nRef.~\\onlinecite{gaul2020ab} cGHF\\,\\,\n  & $23.57$ & $20.60$ \\\\\nRef.~\\onlinecite{gaul2020ab} cGKS\\footnote{For the value of $\\Omega=0.495$.}\n  & $17.48$ & $15.25$ \\\\\n\\end{tabular}\n\\end{ruledtabular}\n\\end{table}\n\n\n\n\\begin{acknowledgments}\nThe work is supported by the Russian Science Foundation grant No. 18-12-00227.\n\\end{acknowledgments}\n\\section*{Author Declarations}\n\n\\subsection*{Conflict of interest}\nThe authors have no conflicts to disclose.\n\n\\section*{Availability of data}\nThe data that support the findings of this study are available from the corresponding author upon reasonable request.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nIn DFD, relative defocus blur between images is exploited as a cue for depth \\cite{n1}. However, accurately estimating blur is a difficult problem. After early works \\cite{n1,n2}, there are now many methods for estimating blur addressing major issues such as noise \\cite{n3}, space-variance \\cite{n3,n5} and the blur model \\cite{n4}. \n\nWhen images are captured with different camera settings to induce a relative blur, a relative magnification is also induced between them causing a shift between the corresponding pixels in the images. Since conventional DFD does not factor in such a shift, the blur estimation can be erroneous in presence of magnification (scaling). Approaches to handle scaling can be classified as optical \\cite{n6} or warping based \\cite{n7,n8}. Nayar and Watanabe \\cite{n6} perform an optical correction by using an additional aperture at an analytically calculated position. But this adds to hardware and cost. Ghita et al. \\cite{n7} consider magnification correction in an active DFD scheme by interpolation that exploits an active illumination pattern. An interpolation based approach proposed by Darell and Wohn \\cite{n8} for shape from focus computes the warping by observing the motion of a special pattern. Scaling in DFD depends only on camera parameters and since it is required that camera parameters be known, the scale factor is available apriori. Warping based methods may be preferable due to simplicity but estimation could be compromised due to interpolation effects \\cite{n6}. However, a formal analysis of the magnification effect in DFD is totally lacking in literature. \n\nIn this paper, we formally analyze the magnification effect in DFD theoretically and experimentally. Section $2$ points out important implications of magnification on the relative blur computation. Section $3$ provides a statistical analysis of scaling on blur estimation. Since the analysis implicitly assumes an image warping solution, section $4$ discusses effects of interpolation errors. Section $5$ provides experimental analysis regarding blur estimation accuracy and we conclude in section $6$. We limit ourselves to space-invariant blur since our goal is not to propose a new method but to analyze magnification as an important consideration in DFD. \n\n\\section{Magnification in DFD}\nFor a thin lens model, the blur radius $R$ and depth $D$ can be related as  \n\\begin{equation}\nR = rV\\left(\\frac{1}{F} -\\frac{1}{V} -\\frac{1}{D}\\right)\n\\end{equation}\nwhere $r$ is the aperture radius, $V$ is the distance between the lens and the image plane, $F$ is the focal length. Varying $r$ and $F$ keeping $V$ constant will cause relative blurring without inducing magnification. However, since $F$ is a function of the physical parameters of lens, changing $F$ means physically changing the lens. As mentioned in \\cite{n6}, the sensitivity of blur due to change in $r$ is low. Hence a better way to induce relative blur is to vary $V$. However, this introduces a relative magnification between the two images. The scaling factor can be derived as $s =\\frac{V_1}{V_2}$, where $V_1$ and $V_2$ are two instances of $V$ while capturing the two images (Fig 1). \n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=150pt]{c.eps}\n\\caption{Magnification and blurring with variation in camera parameters}\n\\end{figure}\nAlthough the above scaling factor is defined for a thin lens, our intention is to note that scale is a function of known camera parameters. Hence, even in compound lenses, the scale factor can be known but with more involved calculations. Thus the amount of magnification can be known apriori. We now discuss two important implications of relative scaling on estimating blur. \n\\subsection{Order of scaling and blurring}\nDue to magnification, two types of transformations are involved that relate the two images viz. scaling and blurring. The order of the transformations is vital when estimating the blur. Typically, the image formation is expressed as,   \n\\begin{equation}\ng(x) = h(x)\\ast f(x) + \\eta(x)\n\\end{equation}\nIn frequency domain, we have \n\\begin{equation}\nG(\\omega) = H(\\omega)F(\\omega) + N(\\omega)\n\\end{equation}\nIn DFD, the Gaussian point spread function (PSF) is popular as a blur model because the effects of blurring, diffraction and sampling can be expressed as a Gaussian in view of the central limit theorem \\cite{n1}. The standard deviation $\\sigma$ of the Gaussian PSF referred to as `blur parameter' is related to $R$ in equation (1) as $\\sigma = \\rho R$ where $\\rho$ is a constant. Thus $H(\\omega) = \\exp^{-\\frac{\\sigma^2\\omega^2}{2}}$. \n\nConsidering the order of blurring followed by scaling, equation (3) becomes, \n\\begin{equation}\nG(\\omega) = \\frac{1}{s}H\\left(\\frac{\\omega}{s}\\right)F\\left(\\frac{\\omega}{s}\\right) + N(\\omega)\n\\end{equation}\nHence, $H(\\frac{\\omega}{s}) = \\exp^{-\\frac{\\sigma^2\\omega^2}{2s^2}}$. This effectively means that scaling of the image $f$ is followed by blurring with blur parameter $\\frac{\\sigma}{s}$. However referring to Figure 1 and its related discussion, we see that the shift in the pixels is due to scaling $s$ and the blur around these corresponding pixel is given by equation (1). According to the above thin lens model, these shifted points are spread by a blur having parameter $\\sigma$ rather than $\\frac{\\sigma}{s}$. Thus we conclude that the order of scaling followed by blurring is more valid than vice-versa. Thus the image formation when considering scaling should be\n\\begin{equation}\nG(\\omega) = \\frac{1}{s}H(\\omega)F\\left(\\frac{\\omega}{s}\\right) + N(\\omega)\n\\end{equation}\n\n\\subsection{Effect of scaling on relative blur computation}\nGenerally, given two blurred images $g_1$ and $g_2$ a model that is often used \\cite{n9} is\n\\begin{equation}\ng_2(x) = h_r(x)\\ast g_1(x) + \\eta(x)\n\\end{equation}\nwhere $\\ast$ denotes convolution. This can be derived as follows \n\\begin{eqnarray}\nG_i(\\omega) &=& H_i(\\omega)F(\\omega)\\hspace{1cm} i = 1,2\\\\\n\\Rightarrow\\frac{G_2(\\omega)}{G_1(\\omega)} &=& \\frac{H_2(\\omega)}{H_1(\\omega)} = H_r(\\omega)\\nonumber\\\\\n\\Rightarrow G_2(\\omega) &=& H_r(\\omega)G_1(\\omega)\\nonumber\n\\end{eqnarray}\nIn spatial domain adding noise this gives equation (6). The blur parameter of the relative blur $h_r(x)$ (or $H_r(\\omega)$) is $\\sqrt{\\sigma_2^2 - \\sigma_1^2}$ where $\\sigma_1$ and $\\sigma_2$ are the blur parameters of $H_1(\\omega)$ and $H_2(\\omega)$. This can be deduced from the Gaussian PSF of the blur and the fact that $H_r(\\omega) = \\frac{H_2(\\omega)}{H_1(\\omega)}$. Estimating blur from equation (6) actually means solving for $\\sqrt{\\sigma_2^2 - \\sigma_1^2}$. This relationship along with $\\sigma_2 = \\alpha\\sigma_1 + \\beta$, can be used to solve for $\\sigma_1$ or $\\sigma_2$. Here $\\alpha$ and $\\beta$ are dependent on camera parameters as $\\alpha = \\frac{r_1V_1}{r_2V_2}$ and $\\beta = \\rho r_1V_1\\left(\\frac{1}{F_1} -\\frac{1}{V_1} -\\frac{1}{F_2} + \\frac{1}{V_2}\\right)$ in a general case of varying all the camera parameters \\cite{n2}. Considering scale in the frequency domain version of equation (6) results in a simple extension of (5) as \n\\begin{equation}\nG_2(\\omega) = \\frac{1}{s}H_r(\\omega)G_1\\left(\\frac{\\omega}{s}\\right) + N(\\omega)\n\\end{equation}\nApparently, this should resulting in the estimation of $\\sqrt{\\sigma_2^2 - \\sigma_1^2}$, the blur parameter of $H_r(\\omega)$. However such an extension of (5) is incorrect as shown below. Actually, the scale consideration should be in the basic image formation i.e. equation (7),  \n\\begin{eqnarray}\nG_i(\\omega) &=& \\frac{1}{s_i}H_i(\\omega)F\\left(\\frac{\\omega}{s_i}\\right)\\hspace{1cm}i = 1,2\\\\\n\\Rightarrow G_2(\\omega) &=& \\frac{s_1}{s_2}\\frac{H_2(\\omega)}{H_1\\left(\\frac{s_1}{s_2}\\omega\\right)}{G_1\\left(\\frac{s_1}{s_2}\\omega\\right)}\\nonumber\\\\ \n\\end{eqnarray}\nHere, $s_1$ and $s_2$ are the scaling factors when the focused image  $f$ is transformed into $g_1$ and $g_2$ respectively. We note that since we do not have the focused image, we also do not have $s_1$ and $s_2$. However, we only need the relative scale $s = \\frac{s_2}{s_1}$ which is known apriori from the camera parameters. We now write the modified version of equation (6) as,\n\\begin{equation}\ng_2(x) = h_r(x)\\ast g_2(sx) + \\eta(x)\n\\end{equation}\nHere, the correct relative blur parameter of $h_r$ is $\\sqrt{\\sigma_2^2 - s^2\\sigma_1^2}$ rather than $\\sqrt{\\sigma_2^2 - \\sigma_1^2}$. Thus when accounting for scaling one must consider this modification in the relative blur expression to solve for $\\sigma_1$ or $\\sigma_2$.\n\n\\section{Analysis of the scaling effect}\nIn this section we answer the following question. `Given two images with relative scaling and blurring, how important is scale consideration for blur estimation ?' We show that the Least-Squares (LS) estimator that does not account for scaling is both biased and inefficient. For simplicity, we assume that one of the two images is the focused image.  \n\\subsection{Bias in blur estimation}\nFrom the discussion in the previous section, the true image formation model can be written as\n\\begin{equation}\ng(x) = h(x)\\ast f(sx) + \\eta(x) \\hspace{0.3cm} \\mbox{or} \\hspace{0.3cm}\\underline{g} = F_s\\underline{h} + \\underline{\\eta}\n\\end{equation}\nwhere $F_s$ is a block Toeplitz image matrix corresponding to the focused image $f(sx)$, $\\underline{g}$ is the observed image, $\\underline{h}$ is a blur vector and $\\underline{\\eta}$ is a zero mean additive white Gaussian noise (AWGN) vector. A least-squares solution of the blur estimate $\\hat{h}$ will be\n\\begin{equation}\n\\underline{\\hat{h}} = (F_s^{T}F_s)^{-1}F_s^{T}\\underline{g}\n\\end{equation}\nThe bias in this estimate is then\\\\ \n\\begin{eqnarray}\n\\underline{b_s} = E(\\underline{\\hat{h}})-\\underline{h} & = & E((F_s^{T}F_s)^{-1}F_s^{T}\\underline{g}) - \\underline{h} = (F_s^{T}F_s)^{-1}F_s^{T}F_s\\underline{h} - \\underline{h} = 0\n\\end{eqnarray}\nHowever, if scaling is not taken into account during estimation, then we obtain the estimator\n\\begin{equation}\n\\underline{\\hat{h}} = (F^{T}F)^{-1}F^{T}\\underline{g}\n\\end{equation}\nSince in actuality, $E(\\underline{g}) = F_s\\underline{h}$, this results in a non zero bias if one does not account for the scale factor $s$.\n\\begin{eqnarray}\n\\underline{b} = E(\\underline{\\hat{h}})-\\underline{h} & = & E((F^{T}F)^{-1}F^{T}\\underline{g}) - \\underline{h} = (F^{T}F)^{-1}F^{T}F_s\\underline{h} - \\underline{h} \\neq 0\n\\end{eqnarray}\nThe above bias can be explained as follows. The matrix $F$ consists of entries from the focused image $f(\\textbf{x})$. The matrix $F_s$ consists of entries from $f(s\\textbf{x})$, the scaled version of the focused image. To account for scale, one must convert $f(\\textbf{x})$ to $f(s\\textbf{x})$. Doing this will ideally make the bias to be $(F_s^{T}F_s)^{-1}F_s^{T}F_s\\underline{h} - \\underline{h} = 0$. Not accounting for scale will mean that we are using the focused image $f(\\textbf{x})$ for estimating the blur when the actual focused image is $f(s\\textbf{x})$, the scaled version of the $f(\\textbf{x})$, thus inducing a bias.  \n\n\\subsection{Efficiency of the estimator}\nHere we comment on the efficiency of the estimator that does and does not consider scaling. The efficiency is in the sense of the Cramer-Rao lower bound (CRLB). We do not provide the details due to space constraints. Considering the true image formation (equation (12)), the CRLB for the blur estimate $\\hat{\\underline{h}}$ can be shown to be\n\\begin{equation}\nCov(\\hat{\\underline{h}}) \\geq  \\sigma_v^2(F_s^TF_s)^{-1}\n\\end{equation}      \nwhere $Cov(\\hat{\\underline{h}})$ is the covariance matrix of $\\hat{\\underline{h}}$ and $\\sigma_v^2$ is the noise variance. The LS estimator $\\underline{\\hat{h}} = (F_s^{T}F_s)^{-1}F_s^{T}\\underline{g}$ is an efficient estimator that meets the CRLB. The estimator that does not use the scaled image is $\\underline{\\hat{h}} = (F^{T}F)^{-1}F^{T}\\underline{g}$. As shown above, this estimator incurs a bias $\\underline{b}$. Suppose a new estimator which is formed by subtracting the bias $\\underline{b}$ from $\\underline{\\hat{h}} = (F^{T}F)^{-1}F^{T}\\underline{g}$ i.e. $\\underline{\\hat{h}_{new}} =  (F^{T}F)^{-1}F^{T}\\underline{g}$ - $\\underline{b}$. This new estimator will be unbiased for obvious reasons. However, covariance matrix of this estimator turns out to be, \n\\begin{equation}\nCov(\\underline{\\hat{h}_{new}}) \\geq  \\sigma_v^2(F^TF)^{-1}\n\\end{equation}      \nSince this is not equal to the CRLB, it is an inefficient estimator. Ideally, to achieve an unbiased and efficient estimate of blur parameter $\\sigma$, we must utilize the scale factor $s$ to transform the observed focused image $f(x)$ to its scaled version $f(sx)$ and then compute $\\hat{\\sigma}$ by solving equation (12) in the least-squares sense. However, in practice, the transformation of $f(x)$ to $f(sx)$ involves image interpolation. In the next section we comment on the effects of interpolation errors on the blur estimate.\n\n\\section{Interpolation errors}\nTill now, we implicitly assumed ideal image warping to alleviate the magnification effect. i.e. the images are exactly aligned after warping. However, due to interpolation this will not be so. The difference between the ideal scaled image $f_{is}(x)$ and interpolated version $f_s(x)$ is what we call `interpolation noise',\n\\begin{equation}\nf_s(x) = f_{is}(x) + \\eta_i(x)\n\\end{equation}      \nThe samples of $\\eta_i(x)$ are correlated as explained next. In Figure 2 pixels $P_i$s and $M_i$s belong to the reference image and warped image, respectively. Considering bilinear interpolation as an example, the pixels $P_2, P_5$ contribute to both $M_1$ and $M_2$. Such common contributions during interpolation induce correlation in $\\eta_i(x)$. Also, from the histograms in Figure 3, we can empirically deduce that the pdf of $\\eta(x)$ is heavy tailed and may not be well approximated as Gaussian. Thus interpolation errors induces correlation and non-Gaussianity. From equations (12), (19) and considering ideal scaling in equation (12) \n\\begin{equation}\ng(x) = h(x)\\ast f_{is}(x) + \\eta(x) = h(x)\\ast f_s(x) - h(x)\\ast\\eta_i(x) + \\eta(x)\n\\end{equation}      \nwhere $h(x)\\ast\\eta_i(x)$ is the correlated, non-Gaussian component of the distribution. The LS estimator considered in the previous section is efficient for AWGN \\cite{n10}, and not when the distribution possesses non-Gaussianity and uncorrelatedness \\cite{n11,n12}. However a class of robust M-estimators are shown to be asymptotically efficient even under such pathologies \\cite{n12,n13}. We have explored the performance of some M-estimators for blur estimation. In the next section, we carry out experiments and analyze the blur estimation accuracy.\n\\begin{figure}[h]\n\\centering\n\\includegraphics[width=60pt]{interp.eps}\n\\caption{Pixel positions in the reference image ($P_i$s) and interpolated image ($M_i$s)} \n\\end{figure}\n\\vspace{-1cm}\n\\begin{figure}[h]\n\\centering\n\\begin{tabular}{c c c}\n\\includegraphics[width=80pt]{hist1.eps} & \\includegraphics[width=80pt]{hist2.eps} & \\includegraphics[width=80pt]{hist3.eps}\n\\end{tabular}\n\\caption{Histograms of $\\eta_i(x)$ for three images}\n\\end{figure}\n\n\\section{Results}\nWe experimentally analyzed blur estimation accuracy versus variation in scale, blur and noise.  We include two discontinuity adaptive functions and an absolute difference as robust estimators in our experiments. Discontinuity adaptive functions can serve as robust estimators due to the analogy between discontinuities and outliers \\cite{n14}. Thus the estimators that we experiment with are the least squares estimator (LSE), absolute difference estimator (ABS) and discontinuity adaptive functions (DA1 and DA2)\n\\begin{eqnarray}\n\\mbox{LSE:}\\hspace{0.1cm}\\arg\\min_{\\sigma}E(x) &=& \\arg\\min_{\\sigma}(g_1(x) - h_r(x)\\ast g_2(sx))^2\\\\\n\\mbox{ABS:}\\hspace{0.1cm}\\arg\\min_{\\sigma}E(x) &=& \\arg\\min_{\\sigma}|g_1(x) - h_r(x)\\ast g_2(sx)|\\nonumber\\\\\n\\mbox{DA1:}\\hspace{0.1cm}\\arg\\min_{\\sigma}E(x) &=& \\arg\\min_{\\sigma}(1-\\exp^{-(g_1(x) - h_r(x)\\ast g_2(sx))^2})\\nonumber\\\\\n\\mbox{DA2:}\\hspace{0.1cm}\\arg\\min_{\\sigma}E(x) &=& \\arg\\min_{\\sigma}\\left(1-\\frac{1}{1+(g_1(x) - h_r(x)\\ast g_2(sx))^2}\\right)\\nonumber\n\\end{eqnarray}      \n\\textbf{Variation in scale factor:}\nWe vary the scale factor from 0.7 to 0.95 keeping the blur constant. The blur estimates were computed for various constants values of $\\sigma_1$ and $\\sigma_2$. Figure 4(a) shows the results for $\\sigma_1=0.7$ and $\\sigma_2=1.2$. We observe that there is no proper behavior of the estimated blur with scale variation. This is because the interpolation errors do not depend on the magnitude of the scale but rather on the shifts of individual pixels. We observe that ABS and DA estimators are very accurate as compared to the Least squares.\\\\\\\\\n\\textbf{Variation in blur:} This experiment involves variation of blur with a constant scale factor. In figure 4(b) we show the results for scale factor of 0.9 with $\\sigma_1$ constant at 0.7 and $\\sigma_2$ varying from 0.7 to 1.5. It is quite clear that generally the inaccuracy increases with blurring. However, again the inaccuracy is quite negligible for ABS, DA1 and DA2 estimators.\\\\\\\\\n\\textbf{Variation in noise:} We experimented with noise variances from 1 to 25 with constant scale and blur. Fig 4(c) shows results for $s = 0.9$, $\\sigma_1 = 1$, $\\sigma_2 = 1.5$.  For the LSE, the error is fairly constant but large. For the ABS estimator, the error increases with noise but it is quite small. The DA estimators incur negligible error. \n\\begin{figure}[h]\n\\centering\n\\begin{tabular}{c c c}\n\\hspace{-1cm}\n\\includegraphics[width=130pt]{scale.eps} & \\includegraphics[width=130pt]{blur.eps} & \\includegraphics[width=130pt]{noise.eps}\\\\(a) & (b) & (c)\n\\end{tabular}\n\\caption{Blur estimation results (a) Scale variation (b) Blur variation (c) Noise variation}\n\\end{figure}\n\nThus, estimation accuracy in warping based solution depends lot on the estimator. The LS estimator is inaccurate as interpolation errors cause violation of the underlying assumptions concerning the pdf. However, robust estimators are very accurate despite interpolation effects.\n\n\\section{Conclusion}\nThis work analyzed the inherent magnification effect in DFD. Important issues such as the order of scaling - blurring and the effect of scaling on relative blur computation were discussed. We then carried out statistical analysis and concluded that the estimator that does not handle scaling is both biased and inefficient. Since an image warping solution was inherently assumed, we scrutinized the effect of interpolation on blur estimation accuracy. We conclude that blur estimation using robust estimators performs very well.  \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{sect:intro}\n\nThe current star formation occurs in magnetic rotating cores of molecular clouds, i.e., protostellar\nclouds (hereafter, PSCs). During the gravitational collapse of the PSC, a protostar is formed at its center,\nwhich is observed as an infrared source~\\citep[see][]{Zhang2020}.\nThe young protostar immersed in an extended envelope\nis observed in the submillimeter range as a Class 0 young stellar object (hereafter, YSO). Characteristic\nfeatures of Class 0 YSOs are outflows~\\citep{andre1993}. In Class 0 YSOs, flattened envelopes with radii of$200-10000$~au are observed, as well as small probably Keplerian disks with radius of $5-50$~au~\\citep{ohashi97, wiseman2001}. In YSO envelopes, a large-scale magnetic field with hourglass geometry is detected; within disks, a pinch magnetic field and indications of a toroidal magnetic field are observed~\\citep[see][]{lee2019}. \nThe angular momentum distribution changes in going from the disk to the envelope~\\citep{caselli02}.\n\nThe first simulations of the PSC collapse showed that the collapse is inhomogeneous with the formation of the first hydrostatic core at the cloud center~\\citep{larson1969}. \nDuring the collapse, the magnetic rotating PSC takes a shape oblate along magnetic field lines and\/or the rotation axis~\\citep{scott1980}. The PSC magnetic flux evolution and properties of formed stars are substantially controlled by ionization, recombination, and diffusion MHD effects, in particular, ambipolar diffusion~\\citep{DudSaz1987}.\n\nThe basic questions in the star formation theory are the problems of the angular momentum and catastrophic magnetic braking~\\citep{scott1980, DudSaz1982}. The modern numerical simulations are mostly devoted to the accretion stage of the solar-mass PSC collapse~\\citep[see][]{HennFromang2008, Zhao2020}.\nTo solve the angular momentum problem, it is important to comprehensively study\ninitial PSC collapse stages, when magnetic braking is the most efficient.\n\nPreviously, ~\\cite{paper1} studied the isothermal collapse of magnetic PSCs with masses of $1$ and $10\\,M_{\\odot}$. Simulations showed that a hierarchical PSC structure is formed during isothermal collapse, which consists of a geometrically thick and optically thin envelope, with a geometrically and optically thin quiasi-magnetostatic primary disk (hereafter PD) inside.\n\nThe PD boundary is characterized by a sharp jump in velocity profiles, when almost free fall of the gas from\nthe cloud envelope to its center transforms to slow almost radial motion. At the PD boundary, a fast\nshock magnetohydrodynamic (hereafter MHD) wave is formed, which moves to the cloud periphery.\n\nIn the present study, the approach of~\\cite{paper1} is developed. The collapse of the magnetic rotating PSC with mass of $M_{\\odot}$ is numerically simulated taking into account the formation of the first core. The PD evolution is studied, its mass, size, angular momentum, magnetic flux, and lifetime are\ndetermined. Possible observational manifestations of PDs are discussed.\n\n\\section{Problem statement and numerical method}\n\\label{sect:problem}\n\nA homogeneous spherically symmetric rotating PSC with mass of 10 $M_{\\odot}$ and a temperature of 20 K, in a uniform magnetic field is considered. The initial cloud density is $4\\cdot 10^4$~cm$^{-3}$, the initial cloud radius is $0.1$~pc. The main parameters determining the collapse dynamics are the ratios of the thermal $\\varepsilon_{\\rm t}$, magnetic $\\varepsilon_{\\rm m}$ and rotational $\\varepsilon_{\\rm w}$ energies to the modulus of its gravitational energy. In this paper, we consider the simulation\nwith $\\varepsilon_{\\rm t}= 0.3$, $\\varepsilon_{\\rm m}=0.2$ and $\\varepsilon_{\\rm w}=0.01$. \n\nThe PSC collapse is studied using gravitational MHD equations. Numerical simulation is performed\nusing the two-dimensional MHD code `Enlil'~\\citep{Dud1999, zhilkin09}. The PSC thermal evolution is simulated using the gas law with a density-dependent effective adiabatic index, $\\gamma_{\\rm eff}$~\\citep{MasunInuts2000}. For the isothermal collapse $\\gamma_{\\rm eff}=1.001$ is taken. At the density $\\rho\\geq 10^{-13}$~g~cm$^{-3}$, when the first hydrostatic core is formed~\\citep{larson1969}, $\\gamma_{\\rm eff}=5\/3$. This approach allows us to model the PSC collapse taking into account the first core formation.\n\n\\section{Evolution of the primary disk during the protostellar cloud collapse}\n\\label{sect:fiduc_m}\n\nThe performed simulations confirm the conclusions by~\\cite{paper1}. At the isothermal\ncollapse stage, the cloud gains a hierarchical structure: the envelope takes a shape oblate along magnetic\nfield lines and the rotation axis; a quasi-magnetostatic PD is formed within it. Let us consider the general\npicture of the PD evolution with the emphasis on the angular momentum distribution.\n\nFigure~\\ref{fig:J_PD} shows the quarter of the central part of the collapsing PSC at different time points. The time $t$ is measured in the units of the characteristic collapse time taking into account the effect of electromagnetic and centrifugal forces: $t_{{\\rm fmw}} = t_{{\\rm ff}}(1-\\varepsilon_{\\rm m}-\\varepsilon_{\\rm w})^{-1\/2}$, where $t_{{\\rm ff}}$ is the free fall time~\\cite{DudSaz1982}. At the PD formation\ntime, $t=0.9081$ $t_{{\\rm fmw}}$ (Fig.~\\ref{fig:J_PD}a), its radius is $R_{\\rm pd}\\approx 0.07\\,R_0\\approx 1500$~au, and the ratio of its maximum half-thickness to the radius is $Z_{\\rm pd}\/R_{{\\rm pd}}=0.039$. \nAt the time point $t=0.9268$ $t_{{\\rm fmw}}$ (Fig.~\\ref{fig:J_PD}b), the PD radius is $\\approx 0.22\\,R_0\\approx 4500$~au. At the PD boundary, a fast MHD shock wave is formed, which propagates into the\ncloud envelope~\\citep[see][]{paper1}.\nThe first core is formed at time $t=0.9645$ $t_{{\\rm fmw}}$ (Fig.~\\ref{fig:J_PD}c). Near the core,\nin the region $r<0.04\\,R_0\\approx 800$~au, the PD thickness abruptly decreases, i.e., quasi-magnetostatic equilibrium is broken. Then, gas in this region begins to move from the cloud core to the periphery in parallel to the rotation axis, i.e., the outflow is formed (Fig.~\\ref{fig:J_PD}d). The PD radius continues to increase and\nby the time of $t=0.9966$~$t_{{\\rm fmw}}$ (Fig.~\\ref{fig:J_PD}f) reaches $R_{\\rm pd}\\approx 0.36\\,R_0\\approx 7400$~au. During evolution, the PD becomes\ngeometrically thinner $Z_{\\rm pd}\/R_{{\\rm pd}}=0.028$, and the outflow region sizes increase.\n\n\\begin{figure*}\n   \\centering\n  \\includegraphics[width=0.79\\textwidth]{fig1_eng.png}\n   \\caption{Distribution of the angular momentum (color filling), velocity field (arrows), and magnetic field (white lines) near\nthe primary disk at following time moments: a) $t=0.9081$ $t_{\\rm fmw}$; b) $t=0.9268$ $t_{\\rm fmw}$; c) $t=0.9645$ $t_{\\rm fmw}$; d) $t=0.9762$ $t_{\\rm fmw}$; e) $t=0.9896$ $t_{\\rm fmw}$; f) $t=0.996$ $t_{\\rm fmw}$. The green line is the primary disk boundary.} \n  \n   \\label{fig:J_PD}\n\\end{figure*}\n\nAt the PD formation time (Fig.~\\ref{fig:J_PD}a) the magnetic field is quasi-radial, $B_r\\sim B_z$,in the envelope and\nquasi-uniform, $B_r\\ll B_z$, within the PD. Further the magnetic field takes a toroidal geometry, $B_{\\varphi}\\sim(B_r, B_z)$, behind the front of the fast shock MHD wave and in the outflow region.\n\nAfter the PD formation, the specific angular momentum is accumulated at its boundary (Fig.~\\ref{fig:J_PD}b), and it is further transferred from the PD boundary to the PSC envelope by the fast shock MHD wave\n(Fig.~\\ref{fig:J_PD}c). The angular momentum of the first core is removed by the outflow (Figs~\\ref{fig:J_PD}d--f).\n\n\nAn analysis of the simulation shows that the PD mass increases with time from $0.3\\,M_{\\odot}$ to $5.2\\,M_{\\odot}$. On the contrary, the PSC envelope mass decreases from the time of the PD formation. The total angular\nmomentum of the PSC decreases by 15~\\% relative to the initial value~$J_0$. \n\n\n\n\n\\section{Conclusions and discussion}\n\\label{sect:end}\n\nThe present paper is the development of the study by~\\cite{paper1}, in which the isothermal stage of the PSC collapse was modelled.\nTwo-dimensional numerical MHD simulation of the collapse of the magnetic rotating PSC with mass of 10~$M_{\\odot}$ was performed taking into account the first hydrostatic core formation.\n\nThe simulations show that the hierarchical structure of  the PSC formed at the isothermal collapse stage~\\cite{paper1}, is retained during further evolution. The first hydrostatic core is formed at the center of the quasi-magnetostatic PD. Near the core, $r<0.04\\,R_0\\approx 800$~au, the quasi-magnetostatic equilibrium is broken, and, afterwards,\noutflow arises. The PD size and mass increase from 1500~au to 7400~au and from 0.3~$M_{\\odot}$ to 5.2~$M_{\\odot}$, respectively.\nThese values are close to characteristics of observed Class 0 YSO envelopes~\\citep{ohashi97, wiseman2001}. Therefore, it can\nbe assumed that the observed large-scale oblate envelopes of Class 0 YSOs are PDs.\n\nOur results confirm the conclusions by ~\\cite{paper1} on the PD lifetime and\nmagnetic field geometry in the collapsing PSC. The PD is a long-living structure which continues to\nevolve after the first core formation. The magnetic field geometry is different across the hierarchy. The magnetic field is quasi-radial within the envelope; it is toroidal behind the front of the fast MHD wave\nemerging from the PD soon after its formation and in the outflow region; the\nmagnetic field is quasi-uniform within the PD. The PD plays an important role in the evolution of the specific angular momentum\nin the cloud. The angular momentum is transferred by the fast shock MHD wave travelling\nfrom the PD boundary to the cloud envelope, as well as by the outflow formed near the first core. The total\nangular momentum of the cloud, decreased by 15~\\% relative to the initial value by the time $t=0.9645$ $t_{{\\rm fmw}}$ $=0.1936$ Myrs, when a typical hierarchical PSC structure with outflows.\n\nBased on our results, it can be assumed that the hierarchical structure of collapsing PSCs can\nbe revealed in observations in terms of the distribution of the magnetic field geometry and the angular momentum.\nTo determine the magnetic field geometry at various hierarchy levels, polarization maps of Class 0 YSOs\nshould be constructed in the submillimeter range with high spatial resolution. \n\nThe further study will be directed to the determination of the mass, size, total angular momentum, and magnetic flux of the PD, as well as of the first hydrostatic core during the collapse of magnetic rotating PSCs with various initial cloud parameters and taking into account magnetic field diffusion.\n\n{\\bf Acknowledgments}\nThis study was supported by the Russian Science Foundation, project no. 19-72-10012. The authors are grateful to reviewer E.O. Vasil'ev for helpful comments.\n\n\\bibliographystyle{mnras}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}