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+{"text":"\\section{Introduction}\n\nThe recently reported \\cite{ATLAS\/CMS} 750 GeV diphoton resonance \nby ATLAS and CMS, if confirmed during Run III at the LHC, would \nhave far reaching ramifications in our \nquest for new physics beyond the Standard Model (SM). These \npreliminary results have triggered, not surprisingly perhaps, \na flurry of theoretical papers \\cite{theory} offering a large \nvariety of \nplausible extensions of the SM in order to explain the reported \ndiphoton excess.\n\nIn this paper we propose an\nextension \nof the Minimal Supersymmetric Standard Model (MSSM) which \nnaturally yields resonance states in the TeV range. A simple \nimplementation of this framework is realized in a local $U(1)_{B-L}$ \nextension of the MSSM gauge symmetry. In contrast to radiative \nelectroweak breaking implemented in the MSSM, the additional \nlocal symmetry in our case is spontaneously broken at tree level \nwith a superpotential $W$ whose form is uniquely determined by \na combination of the underlying gauge symmetry and a $U(1)$ global \nR-symmetry. The construction of $W$ utilizes an appropriate pair \nof Higgs superfields ($\\Phi$, $\\bar{\\Phi}$) as well as a gauge singlet \nsuperfield $S$. The resonance states  \narise from the scalar components of the $S-\\Phi-\\bar{\\Phi}$ system, \nand their mass is determined, in the supersymmetric (SUSY) limit, by a \ndimensionless parameter in W which can be much smaller, \nif needed, compared to the typical order unity or so gauge \ncoupling constant. Thus, if required, the resonance states can be \nsignificantly lighter than the mass of the $Z'$ gauge boson \nassociated with $B-L$. Note that with global $U(1)_{B-L}$ any \nconstraint arising from $Z'$ goes away. This is a plausible \nalternative to local $U(1)_{B-L}$ considered here.\n\nThe spontaneous breaking of $U(1)_{B-L}$ leaves \nSUSY unbroken. The symmetry breaking scale $M$ may be \nmuch larger than the TeV SUSY breaking scale. Superpotentials \nof this type have previously \nbeen employed by Dvali and Shafi \\cite{Dvali:1994wj} in their \nconstruction of SUSY trinification models based on \n$SU(3)_c\\times SU(3)_L\\times SU(3)_R$, and later in the \nconstruction of SUSY hybrid inflation models \n\\cite{Dvali:1994ms,Copeland:1994vg}.\n\nThe scalar component of $S$ acquires a non-zero vacuum expectation \nvalue (VEV) proportional to \n$m_{3\/2}$ after SUSY breaking \\cite{Dvali:1994wj,Dvali:1997uq}, and \nthis has been utilized in the past \\cite{Dvali:1997uq,King:1997ia} \nto resolve the MSSM \n$\\mu$ problem. The R-symmetry also protects $S$ from acquiring \narbitrarily large masses. In the present scheme we also use \nthis $\\sim{\\rm TeV}$ VEV of $S$ to provide masses to suitable \nvector-like fields including colored fields which play a role \nin the production and subsequent decay of the scalar \nresonance(s).\n\n\n\nThe renormalizable superpotential of the MSSM with R-parity possesses \nthree global symmetries, namely \nthe baryon number $U(1)_B$, lepton number $U(1)_L$ and a R-symmetry \n$U(1)_R$, where, for simplicity, we ignore the tiny non-perturbative \nviolation of $B$ and $L$ by the $SU(2)_L$ instantons. The new local \n$U(1)$ symmetry, which we identify as $U(1)_{B-L}$, is to be \nspontaneously broken at some scale $M$, and we prefer to implement this \nbreaking by a SUSY generalization of the\nHiggs mechanism. \nMotivated by the MSSM example, we \nrequire that the new superpotential $W$ respects the global $U(1)_B$ \nand $U(1)_L$ symmetries as well as a global $U(1)$ R-symmetry. \n\nThe full renormalizable  superpotential is \n\\bea\nW &=&y_uH_{u}qu^c + y_dH_{d}qd^c + y_{\\nu}H_{u}l\\nu^c +y_{e}H_{d}le^c\n\\nonumber\n\\\\\n& & +\\kappa S (\\Phi\\bar{\\Phi}-M^2) +\\lambda_{\\mu} SH_{u}H_{d}+ \n\\lambda_{\\nu^c}\\bar\\Phi \\nu^c\\nu^c \n\\nonumber\n\\\\ \n& & +\\lambda_{D} S D\\bar{D}+ \\lambda_{q}Dqq + \\lambda_{q^c}\\bar{D}u^cd^c, \n\\label{W}\n\\eea\nwhere $y_u$, $y_d$, $y_{\\nu}$, $y_e$ are the Yukawa coupling constants \nand the family indices  are generally suppressed for simplicity. Here \n$q$, $u^c$, $d^c$, $l$, $\\nu^c$, $e^c$ are the usual quark and lepton \nsuperfields of MSSM including the right handed neutrinos $\\nu^c$, and \n$H_{u}$, $H_{d}$ are the standard electroweak Higgs superfields. The \ngauge singlet $S$ has necessarily the same R-charge as $W$, which we \ntake to be 2. Consequently, $H_{u}$, $H_{d}$ have opposite R-charges, \nwhich can be brought to zero by an appropriate hypercharge ($Y$) \ntransformation. The R-charges of $u^c$ ($\\nu^c$) and $d^c$ ($e^c$) are \nequal and, thus, $B$ and $L$ transformations can make the R-charges of \n$q$, $u^c$, $d^c$, $l$, $\\nu^c$, $e^c$ all equal to unity. \n\nIn order to determine the R-charges and $B-L$ quantum numbers of the \nSM singlets $\\Phi$, $\\bar{\\Phi}$, we introduce the coupling \n$\\bar{\\Phi}\\nu^c\\nu^c$, which implies that their $B-L$ charge is \n$2$, $-2$ respectively, and their R-charges are zero. This \ncoupling generates masses for the right handed neutrinos after the \nbreaking of $U(1)_{B-L}$ to its $Z_2$ subgroup by the VEVs of \n$\\Phi$, $\\bar{\\Phi}$.\n\nWe also introduced the coupling $SD\\bar{D}$, where $D$ ($\\bar{D}$) are\ncolor triplet (antitriplet) and $SU(2)_L$ singlet superfields. (Color \ntriplet vector-like superfields $D$, $\\bar{D}$ are perhaps best \nmotivated in the framework of GUT symmetry $E_6$.) To determine \nthe charges of these fields we need at least one additional coupling. \nTaking the coupling $Dqq$, which is a color and $SU(2)_L$ singlet, the \nR-charges of $D$, $\\bar{D}$ vanish. Also, the $Y$ charge of $D$ becomes -1\/3 \nand consequently, the hypercharge of $\\bar{D}$ is 1\/3. Finally, the $B-L$ \ncharge of $D$ is -2\/3 and that of $\\bar{D}$ is 2\/3 with their lepton numbers \nvanishing. Note that the coupling $\\bar{D}u^cd^c$ is also present since it \nrespects all the symmetries of the model. It is also worth mentioning that \nthe $Z_2$ subgroups of both $U(1)_R$ and $U(1)_{B-L}$ coincide with the \n$Z_2$ matter parity under which all the ordinary (anti)quark and \n(anti)lepton superfields are odd, with the rest of the superfields being\neven. This symmetry remains unbroken by all the soft SUSY \nbreaking terms and all the VEVs. (See Ref.~\\cite{Kibble:1982ae} for a more \ngeneral discussion of unbroken $Z_2$ from $SO(10)$.)  In \nTable~\\ref{tab:fields}, we summarize all the superfields of \nthe model together with their transformation properties under the SM gauge \ngroup ($G_{SM}=SU(3)_c\\times SU(2)_L\\times U(1)_Y$) and their charges under \nthe global symmetries $U(1)_B$, $U(1)_L$, and $U(1)_R$.\n\n\\begin{table}[!t]\n\\caption{Superfield content of the model.}\n\\begin{tabular}{c@{\\hspace{.8cm}}\nc@{\\hspace{.8cm}} c@{\\hspace{.8cm}} c@{\\hspace{.8cm}}c}\n\\toprule\n{Superfields}&{Representions}&\\multicolumn{3}{c}{Global Symmetries}\n\\\\\n{}&{under\n$G_{SM}$}&{$B$} &{$L$} &{$R$}\n\\\\\\colrule\n\\multicolumn{5}{c}{Matter Superfields}\\\\\\colrule\n{$q$} &{$({\\bf 3, 2}, 1\/6)$}&$1\/3$& $0$& $1$\n\\\\\n{$u^c$} & {$({\\bf \\bar 3, 1},-2\/3)$}&$-1\/3$&{$0$}&$1$\n\\\\\n{$d^c$} & {$({\\bf \\bar 3, 1},1\/3)$}  &$-1\/3$&{$0$}&$1$\n\\\\\n{$l$} &{$({\\bf 1, 2}, -1\/2)$} &$0$& $1$&$1$\n \\\\\n{$\\nu^c$} & {$({\\bf 1, 1}, 0)$}&{$0$}&{$-1$} &$1$\n\\\\\n{$e^c$} & {$({\\bf 1, 1}, 1)$}&{$0$}&{$-1$}&$1$ \n\\\\\n\\colrule\n\\multicolumn{5}{c}{Higgs Superfields}\n\\\\\\colrule\n{$H_u$} & {$({\\bf 1, 2},1\/2)$} &$0$&$0$&$0$\n\\\\\n{$H_d$} & {$({\\bf 1, 2},-1\/2)$} &$0$&$0$&$0$\n\\\\\n\\colrule\n{$S$} & {$({\\bf 1, 1},0)$}  &$0$&$0$&$2$ \n\\\\ \n{$\\Phi$} &{$({\\bf 1, 1},0)$} & {$0$}&{$-2$}&{$0$} \n\\\\\n{$\\bar\\Phi$}&{$({\\bf 1, 1},0)$}&{$0$}&{$2$}&{$0$} \n\\\\\n\\colrule\n\\multicolumn{5}{c}{Vector-like Diquark Superfields}\n\\\\\\colrule\n$D$&{$({\\bf 3, 1},-1\/3)$} &$-2\/3$& $0$ &$0$\n\\\\\n$\\bar{D}$&{$({\\bf \\bar 3, 1},1\/3)$}  &$2\/3$&$0$& $0$\n\\\\\n\\botrule\n\\end{tabular}\\label{tab:fields}\n\\end{table}\n\nThe superpotential in Eq.~(\\ref{W}) is\nthe most general renormalizable superpotential which obeys the SM gauge\nsymmetry and the global symmetries $U(1)_B$, $U(1)_L$, and $U(1)_R$. Had \nwe removed the separate baryon and lepton number symmetries and kept only \nthe gauge $U(1)_{B-L}$ symmetry, the renormalizable superpotential terms\n$\\bar{D}ql$, $Du^ce^c$, and $Dd^c\\nu^c$ would be present leading to fast \nproton decay and other baryon and lepton number violating effects \n\\cite{Lazarides:1998iq}. The spontaneous breaking of $U(1)_{B-L}$ to its \n$Z_2$ subgroup will generate a network of local cosmic strings. Their \nstring tension, which is determined by the scale $M$, is relatively small \nand certainly satisfies the most stringent relevant upper bound from pulsar \ntiming arrays \\cite{pulsar}.     \n\nThe `bare' MSSM $\\mu$ term is now replaced by a term $SH_{u}H_{d}$, so that \nthe $\\mu$ term is generated after $S$ acquires a non-zero VEV of order TeV \nfrom soft SUSY breaking \\cite{Dvali:1997uq}. (In the SUSY limit the VEV of \n$S$ is zero.) The VEV of $S$ also plays an essential role, as we will see, \nin the generation of masses for the vector-like fields $D$, $\\bar{D}$ that \nare crucial in the production and decay of the diphoton resonance(s).\n\nThe spontaneous breaking of $U(1)_{B-L}$ implemented with the fields \n$S$, $\\Phi$, $\\bar{\\Phi}$ delivers, in the exact SUSY limit, four spin \nzero particles all with the same mass given by $\\sqrt{2}\\kappa M$.\nThis mass, even for $M\\gg 1~{\\rm TeV}$, can be of \norder TeV (more precisely $\\simeq 750~{\\rm GeV}$ in the present case) by \nselecting an appropriate value for $\\kappa$. We should point out though \nthat depending on the SUSY breaking mechanism, the four resonance states \nmay end up with significantly different masses. The VEV of $S$, with \nsuitable choice of the gauge and R-charges, yields \nmasses for the appropriate fields that are vector-like under the MSSM \ngauge symmetry. This is in addition to possibly additional \nsuch fields that acquire masses from their coupling to the Higgs fields \nwith VEV $M$ that break the $U(1)_{B-L}$ gauge symmetry. The diquarks \nassociated with the vector-like fields may be found \\cite{Gogoladze:2010xd} \nat the LHC.\n\n\nThe spontaneous breaking of $U(1)_{B-L}$ is achieved via the first term \nin the second line of Eq.~(\\ref{W}), which gives the following potential\nfor unbroken SUSY\n\\beq\nV=\\kappa^2|\\Phi\\bar{\\Phi}-M^2|+\\kappa^2|S|^2(|\\Phi|^2+|\\bar{\\Phi}|^2)+\n{\\rm D-terms}.\n\\label{V}\n\\eeq\nHere we assumed that the mass parameter $M$ and the dimensionless coupling \nconstant $\\kappa$ are made real and positive by field rephasing, and the \nscalar components of the superfields are denoted by the same symbol.\nVanishing of the D-terms implies that $|\\Phi|=|\\bar{\\Phi}|$, which yields\n$\\bar{\\Phi}^*=e^{i\\varphi}\\Phi$, while the F-terms vanish for $S=0$ and \n$\\Phi\\bar{\\Phi}=M^2$, which requires that $\\varphi=0$. So, after rotating \n$\\Phi$ and $\\bar{\\Phi}$ to the positive real axis by a $B-L$ transformation, \nwe find that the SUSY vacuum lies at\n\\beq\nS=0 \\quad {\\rm and}\\quad \\Phi=\\bar{\\Phi}=M.  \n\\eeq\nThe mass spectrum of the scalar $S-\\Phi-\\bar{\\Phi}$ system is constructed\nby writing $\\Phi=M+\\delta\\Phi$ and $\\bar{\\Phi}=M+\\delta\\bar{\\Phi}$. In the \nunbroken SUSY limit, we find two complex scalar fields $S$ and \n$\\theta=(\\delta\\Phi+\\delta\\bar{\\Phi})\/\\sqrt{2}$ with equal masses \n$m_{S}=m_{\\theta}=\\sqrt{2}\\kappa M$. Soft SUSY breaking can, of course, \nmix these fields and generate a mass splitting. For example, the trilinear\nsoft term $A\\kappa S \\Phi\\bar{\\Phi}$ yields a mass squared splitting \n$\\pm\\sqrt{2}\\kappa M A$ with the mass eigenstates now being \n$(S+\\theta^*)\/\\sqrt{2}$ and $(S-\\theta^*)\/\\sqrt{2}$. This splitting is \nsmall for $A\\ll\\sqrt{2}\\kappa M$. Let us assume that the mixing \nis in general sub-dominant and ignore it. This simplifies our analysis.\n\nThe soft SUSY breaking terms \n\\beq\nV_1=A\\kappa S \\Phi\\bar{\\Phi}-(A-2m_{3\/2})\\kappa M^2 S\n\\label{V1}\n\\eeq\nin the potential with $m_{3\/2}$ being the gravitino mass and \n$A\\sim m_{3\/2}$, which arise from the first term in the second line \nof Eq.~(\\ref{W}), play an important role in our scheme. Here we \nassume minimal supergravity so that the coefficients of the trilinear \nand linear soft terms are related as shown. Substituting \n$\\Phi=\\bar{\\Phi}=M$ in Eq.~(\\ref{V1}), we obtain a linear term in $S$ \nwhich, together with the mass term $2\\kappa^2 M^2 |S|^2$ of $S$, \ngenerates \\cite{Dvali:1994wj} a VEV for $S$:\n\\beq\n\\vev{S}=-\\frac{m_{3\/2}}{\\kappa}.\n\\label{vev}\n\\eeq \nSubstituting this VEV of $S$ in the superpotential\nterm $\\lambda_{\\mu}SH_{u}H_{d}$, we obtain \\cite{Dvali:1997uq,King:1997ia} \nthe MSSM $\\mu$ term with $\\mu=-\\lambda_{\\mu}m_{3\/2}\/\\kappa$. The crucial \npoint here is that the same VEV generates mass terms \n$-\\lambda_{D}m_{3\/2}D\\bar{D}\/\\kappa$ for the vector-like superfields $D$, \n$\\bar{D}$ via the superpotential terms $\\lambda_{D}SD\\bar{D}$. The \ntrilinear terms corresponding to these superpotential terms\nwill produce mixing between the scalar components of $D$, $\\bar{D}$.\nHowever, we will assume here that this mixing is sub-dominant. \n\nTo preserve gauge coupling unification one should introduce \nvector-like color singlet, $SU(2)_L$ doublet superfields $L$, $\\bar{L}$ \nequal in number to the color triplets $D$, $\\bar{D}$. (With $D$, \n$\\bar{D}$ and $L$, $\\bar{L}$ masses $\\sim {\\rm TeV}$, the gauge \ncouplings stay in the perturbative domain for up to four such \npairs.) These fields with a superpotential coupling \n$\\lambda_L S L\\bar{L}$ can enhance the branching ratio of \nthe decay of the spin zero fields $S$ and $\\theta$ to photons and, in \naddition, allow the decay into $W^{\\pm}$. \nIntroducing the superpotential coupling $Lle^c$, the \nhypercharge of $L$ ($\\bar{L}$) is -1\/2 (1\/2). Their baryon, lepton \nnumbers, and R-charges are all zero. These quantum numbers allow the \nsuperpotential couplings $SLH_u$, $SH_d\\bar{L}$, $Lqd^c$, $\\bar{L}qu^c$, \nand $\\bar{L}l\\nu^c$. Substituting $\\vev{S}$\nin $\\lambda_L S L\\bar{L}$, the superfields $L$, \n$\\bar{L}$ acquire a mass $m_L=-\\lambda_L m_{3\/2}\/\\kappa$. \n\n\n\\begin{figure}[t]\n\\centerline{\\epsfig{file=so10_Stophotons.eps,width=8.7cm}}\n\\caption\nProduction of the bosonic component \nof $S$ at the LHC by gluon ($g$) fusion and its subsequent decay \ninto photons ($\\gamma$). Solid (dashed) lines represent the \nfermionic (bosonic) component of the indicated superfields. The \narrows depict the chirality of the superfields and the crosses \nare mass insertions which must be inserted in each of the lines\nin the loops.}\n\\label{fig1}\n\\end{figure}\n\nThe real scalar $S_1$ and real pseudoscalar $S_2$ components \nof $S~(=(S_1+iS_2)\/\\sqrt{2})$ with \nmass $m_S=\\sqrt{2}\\kappa M$ in the exact SUSY limit can be \nproduced at the LHC by gluon fusion via a \nfermionic $D$, $\\bar{D}$ loop as indicated in Fig.~\\ref{fig1}.\nIn the absence of the vector-like $L$, $\\bar{L}$ superfields,\nthey can decay into gluons, photons, or $Z$ gauge bosons via \nthe same loop diagram, but not to $W^{\\pm}$ bosons since the \n$D$, $\\bar{D}$ are $SU(2)_L$ singlets. The most \npromising decay channel to search for these resonances is\ninto two photons with the relevant diagram \nalso shown in Fig.~\\ref{fig1}. \n\nApplying the results of S.F.~King and R.~Nevzorov in \nRef.~\\cite{theory}, the cross section of the diphoton \nexcess is \n\\beq\n\\label{sigma}\n\\sigma(pp\\rightarrow S_i\\rightarrow\\gamma\\gamma)\\simeq \n\\frac{C_{gg}}{m_{S}s\\Gamma_{S_i}}\n\\Gamma(S_i\\rightarrow gg)\\Gamma(S_i\\rightarrow \\gamma\\gamma),\n\\eeq     \nwhere $i=1,2$, $C_{gg}\\simeq 3163$, $\\sqrt{s}\\simeq \n13~{\\rm TeV}$, $\\Gamma_{S_i}$ is the total decay width of \n$S_i$, and the decay widths of $S_i$ to two gluons ($g$) or \ntwo photons ($\\gamma$) are given by \n\\bea\n\\label{Sg}\n\\Gamma(S_i\\rightarrow gg)=\\frac{n^2\\alpha_s^2m_S^3}\n{256\\,\\pi^3\\vev{S}^2}\\, A_i^2(x),\\\\\n\\label{Sgamma}\n\\Gamma(S_i\\rightarrow \\gamma\\gamma)=\\frac{n^2\\alpha_Y^2m_S^3\n\\cos^4\\theta_W}{4608\\,\\pi^3\\vev{S}^2}\\, A_i^2(x).\n\\eea \nHere $n$ is the number of $D$, $\\bar{D}$ pairs, which are \ntaken, for simplicity, to have a common coupling constant $\\lambda_D$ \nto $S$, $A_1(x)=2x[1+(1-x)\\arcsin^2(1\/\\sqrt{x})]$, \n$A_2(x)=2x\\arcsin^2(1\/\\sqrt{x})$, $x=4m_D^2\/m_S^2> 1$, and \n$\\alpha_s$, $\\alpha_Y$ are the strong and hypercharge fine-structure \nconstants. If the $L$, $\\bar{L}$ superfields are present, they will\nalso contribute to the decay width of $S$ to photons via loop \ndiagrams similar to the ones in the right part of Fig.~\\ref{fig1}, and \nEq.~(\\ref{Sgamma}) will be replaced by\n\\bea\n\\label{SgammaL}\n\\Gamma(S_i\\rightarrow \\gamma\\gamma)&=&\\frac{n^2m_S^3\\alpha_Y^2\\cos^4\\theta_W}\n{4608\\,\\pi^3\\vev{S}^2}A_i^2(x)\n\\nonumber\\\\\n& &\\left[1+\\frac{3A_i(y)}{2A_i(x)}\\left(1+\\frac{\\alpha_2\\tan^2\\theta_W}\n{\\alpha_Y}\\right)\\right]^2,\n\\eea \nwhere $\\alpha_2$ is the $SU(2)_L$ fine-structure constant and \n$y=4m_L^2\/m_S^2> 1$.\n\nThe cross section in Eq.~(\\ref{sigma}) simplifies under the \nassumption that the spin zero fields $S_i$ decay predominantly into \ngluons, namely, $\\Gamma_{S_i}\\simeq\\Gamma(S_i\\rightarrow gg)$. In \nthis case, as pointed by R.~Franceschini et al. \nin Ref.~\\cite{theory}, one obtains $\\sigma(pp\\rightarrow S_i\n\\rightarrow\\gamma\\gamma)\\simeq 8~{\\rm fb}$ if\n\\beq\n\\label{condition}\n\\frac{\\Gamma(S_i\\rightarrow \\gamma\\gamma)}{m_{S}}\n\\simeq 1.1 \\times 10^{-6}.\n\\eeq\nFor $x$ and $y$ just above unity, which guarantees that the \ndecay of $S_i$ to $D$, $\\bar{D}$ and $L$, $\\bar{L}$ pairs is \nkinematically blocked, $A_1(x)$ and $A_2(y)$ are \nmaximized with values $A_1\\simeq 2$ and $A_2\\simeq \\pi^2\/2$.\nNote that $x$ close to unity means $m_D\\simeq 375~{\\rm GeV}$. \nHowever, one should work with somewhat larger $m_D$ as indicated \nby ATLAS and CMS. So we take $m_D=700~{\\rm GeV}$. It is more \nbeneficial to consider the decay of the pseudoscalar $S_2$ since \n$A_2(x)>A_1(x)$ for all $x>1$. Using Eq.~(\\ref{SgammaL}), we \nthen find that the condition in Eq.~(\\ref{condition}) is \nsatisfied for $n m_S\/|\\vev{S}|\\simeq 2.97$. Therefore, for $n=3$, \nwe require that $m_S\/|\\vev{S}|\\simeq 0.99$, which, for $m_S\\simeq \n750~{\\rm GeV}$, implies that $|\\vev{S}|\\simeq 758~{\\rm GeV}$. \nIn this case, $\\lambda_D\\simeq 0.92$ and $\\lambda_L$ is just \nabove 0.49. Comparing Eqs.~(\\ref{Sgamma}) and (\\ref{SgammaL}), \nwe find that the inclusion of the vector-like $L$, $\\bar{L}$ \nsuperfields enhances the decay width of $S_2$ to photons by \nabout a factor 58.5. \n\n\nIn the exact SUSY limit, the complex scalar field $S$ could decay \ninto MSSM Higgsinos (potential dark matter candidate) via the \nsuperpotential term \n$\\lambda_{\\mu}SH_{u}H_{d}$ if this is kinematically allowed.\n$S$ also could decay into right handed \nsneutrinos via the F-term $F_{\\bar{\\Phi}}$ between the superpotential \nterms $\\kappa S\\Phi\\bar{\\Phi}$ and $\\bar{\\Phi}\\nu^{c}\\nu^{c}$ after \nsubstituting the VEV of $\\Phi$.\nThe decay \nwidths in the two cases are\n\\beq\n\\Gamma^S_H=\\frac{\\lambda_{\\mu}^2}{8\\pi}m_{S}, \n\\quad \n\\Gamma^S_{\\nu^c}=\\frac{\\lambda_{\\nu^c}^2}{8\\pi}m_{S},\n\\label{decay}\n\\eeq \nrespectively, where we assumed that the masses of the Higgsinos and \nthe relevant right handed sneutrinos are much smaller than $m_S$.\nDepending on the kinematics the total decay width of the \nresonance could easily lie in the multi-GeV range.\nThe diphoton, dijet, and diboson decay modes in this case\nwould be sub-dominant. \n\nOur estimate of $m_S\/|\\vev{S}|$ \nafter Eq.~(\\ref{condition}) requires that the decay widths of\n$S$ into MSSM Higgsinos and right handed sneutrinos are sub-dominant \nor kinematically blocked. The latter is achieved for $|\\mu|=\n\\lambda_{\\mu}|\\vev{S}|>m_S\/2\\simeq 375~{\\rm GeV}$ (or \n$\\lambda_\\mu\\gtrsim 0.49$) and $\\lambda_{\\nu^c}M>m_S\/2$. \nDemanding that the mass of the $B-L$ gauge boson $m_{Z'}=\n\\sqrt{6}g_{B-L}M>3~{\\rm TeV}$ \\cite{Zprime,okada-okada}, say, we \nfind that $g_{B-L}M\\gtrsim 1225~{\\rm GeV}$ ($g_{B-L}$ is the GUT \nnormalized $B-L$ gauge coupling constant). From \nEq.~(\\ref{vev}) and setting, say, \n$m_{3\/2}=50~{\\rm GeV}$, we obtain\n$\\kappa\\simeq 0.066$, $M\\simeq 8040~{\\rm GeV}$, \n$\\lambda_{\\nu^c}\\gtrsim 0.047$, and $g_{B-L}\\gtrsim 0.15$. \nA gravitino in this mass range is a plausible cold matter \ncandidate -- for a recent discussion and references, see \nRef.~\\cite{Okada:2015vka}. Finally, we have checked that, in this \nexample, $g_{B-L}\\lesssim 0.25$, $\\lambda_{D}$, and $\\lambda_{L}$ \nremain perturbative up to the GUT scale ($M_{\\rm GUT}$). In \nparticular, if $g_{B-L}\\simeq 0.24$, it unifies with the MSSM \ngauge coupling constants at $M_{\\rm GUT}$. So the requirements \nfor a viable diphoton resonance are met.   \n\n\\begin{figure}[t]\n\\centerline{\\epsfig{file=so10_thetatophotons.eps,width=8.7cm}}\n\\caption\nProduction of the bosonic component \nof $\\theta$ at the LHC by gluon ($g$) fusion and its subsequent \ndecay into photons ($\\gamma$). The notation is the same as in \nFig.~\\ref{fig1} with the crosses indicating mass squared \ninsertions.\n}\n\\label{fig3}\n\\end{figure}\n\nThe complex spin zero field $\\theta=(\\delta\\Phi+\\delta\\bar{\\Phi})\/\n\\sqrt{2}=(\\theta_1+i\\theta_2)\/\\sqrt{2}$, which consists of a \nreal scalar ($\\theta_1$) and a real pseudoscalar ($\\theta_2$) field\nand has mass $m_{\\theta}=\\sqrt{2}\\kappa M$ in the SUSY limit, \ncouples to the scalar vector-like fields $D$, $\\bar{D}$ via\nthe F-term $F_S$ between the superpotential terms \n$\\kappa S\\Phi\\bar{\\Phi}$ and $\\lambda_D S D\\bar{D}$. The \ncoupling constant is $\\lambda_D m_{\\theta}$. It also can be produced\nat the LHC by gluon fusion via scalar $D$, $\\bar{D}$ loops as shown \nin Fig.~\\ref{fig3}(a), and decay into two photons via the diagrams in \nFig.~\\ref{fig3}(b). (In the presence of $L$, $\\bar{L}$ superfields \nsimilar diagrams with scalar $L$, $\\bar{L}$ loops also contribute \nto the decay of $\\theta$ into photons.) The important point here is \nthat the mass squared \ninsertions in all the diagrams of Fig.~\\ref{fig3} arise from the soft SUSY \nbreaking trilinear term $A^\\prime\\lambda_D S D\\bar{D}$ and are thus \nequal to $A^\\prime m_{D}$, where $m_{D}=-\\lambda_{D}m_{3\/2}\/\\kappa$ is \nthe mass of the $D$, $\\bar{D}$ superfields generated by the VEV of $S$\nin Eq.~(\\ref{vev}). Consequently, for $A^\\prime\\ll m_{D}$, the cross \nsections for the diphoton excess are suppressed by a factor\n$(A^\\prime\/m_{D})^4$ relative to the ones for the spin zero field $S$.\nLarger soft SUSY breaking trilinear terms will\nenhance the diagrams in Fig.~\\ref{fig3} and also cause larger mixing \nbetween $S$ and $\\theta$. In this case all four spin zero states can \ncontribute to the diphoton excess. \n          \nThe field $\\theta$ can decay, in the exact SUSY\nlimit, into MSSM Higgs fields via the F-term $F_S$ between the \nsuperpotential terms $\\kappa S \\Phi\\bar{\\Phi}$ and $\\lambda_{\\mu} S \nH_{u}H_{d}$ if this is kinematically allowed.\nThe relevant coupling constant is $\\lambda_{\\mu}m_{\\theta}$, \nand thus the decay width is the same as $\\Gamma^S_H$ in \nEq.~(\\ref{decay}) provided that the masses of the Higgs fields are much \nsmaller than $m_{\\theta}$ (for $m_S=m_{\\theta}$). $\\theta$ \nalso could decay into right handed neutrinos via the superpotential term \n$\\lambda_{\\nu^c}\\bar{\\Phi}\\nu^c\\nu^c$\nwith a decay \nwidth equal to $\\Gamma^S_{\\nu^c}$ in Eq.~(\\ref{decay}) under the same \nassumption.\n\n\n\nIn conclusion, we have presented a realistic SUSY model based on a \n$U(1)_{B-L}$ extension \nof the MSSM that contains resonances observable at the LHC and\/or future \ncolliders. The underlying gauge and R-symmetries are such that the \nMSSM $\\mu$ parameter and the masses of vector-like \nsuperfields and a gauge singlet superfield cannot \nbe arbitrarily large. A resonance system consisting of four spin \nzero states arises from a \ngauge singlet scalar and a pair of conjugate Higgs superfields \nresponsible \nfor the $B-L$ breaking. These states are degenerate in mass in the SUSY \nlimit, and depending on the details of SUSY breaking, one or more of \nthese states could explain the observed 750 GeV diphoton excess. Their \ntotal decay widths can lie in \nthe multi-GeV range, depending on the kinematics, in which \ncase the diphoton, diboson and dijet events will be sub-dominant. \n\n\\acknowledgments{Q.S. is supported in part by the DOE grant \nDOE-SC0013880. We thank George Leontaris for discussions and Aditya \nHebbar for help with the figures.}\n\n\\def\\ijmp#1#2#3{{Int. Jour. Mod. Phys.}\n{\\bf #1},~#3~(#2)}\n\\def\\plb#1#2#3{{Phys. Lett. B }{\\bf #1},~#3~(#2)}\n\\def\\zpc#1#2#3{{Z. Phys. C }{\\bf #1},~#3~(#2)}\n\\def\\prl#1#2#3{{Phys. Rev. Lett.}\n{\\bf #1},~#3~(#2)}\n\\def\\rmp#1#2#3{{Rev. Mod. Phys.}\n{\\bf #1},~#3~(#2)}\n\\def\\prep#1#2#3{{Phys. Rep. }{\\bf #1},~#3~(#2)}\n\\def\\prd#1#2#3{{Phys. Rev. D }{\\bf #1},~#3~(#2)}\n\\def\\npb#1#2#3{{Nucl. Phys. }{\\bf B#1},~#3~(#2)}\n\\def\\np#1#2#3{{Nucl. Phys. B }{\\bf #1},~#3~(#2)}\n\\def\\npps#1#2#3{{Nucl. Phys. B (Proc. Sup.)}\n{\\bf #1},~#3~(#2)}\n\\def\\mpl#1#2#3{{Mod. Phys. Lett.}\n{\\bf #1},~#3~(#2)}\n\\def\\arnps#1#2#3{{Annu. Rev. Nucl. Part. Sci.}\n{\\bf #1},~#3~(#2)}\n\\def\\sjnp#1#2#3{{Sov. J. Nucl. Phys.}\n{\\bf #1},~#3~(#2)}\n\\def\\jetp#1#2#3{{JETP Lett. }{\\bf #1},~#3~(#2)}\n\\def\\app#1#2#3{{Acta Phys. Polon.}\n{\\bf #1},~#3~(#2)}\n\\def\\rnc#1#2#3{{Riv. Nuovo Cim.}\n{\\bf #1},~#3~(#2)}\n\\def\\ap#1#2#3{{Ann. Phys. }{\\bf #1},~#3~(#2)}\n\\def\\ptp#1#2#3{{Prog. Theor. Phys.}\n{\\bf #1},~#3~(#2)}\n\\def\\apjl#1#2#3{{Astrophys. J. Lett.}\n{\\bf #1},~#3~(#2)}\n\\def\\apjs#1#2#3{{Astrophys. J. Suppl.}\n{\\bf #1},~#3~(#2)}\n\\def\\n#1#2#3{{Nature }{\\bf #1},~#3~(#2)}\n\\def\\apj#1#2#3{{Astrophys. J.}\n{\\bf #1},~#3~(#2)}\n\\def\\anj#1#2#3{{Astron. J. }{\\bf #1},~#3~(#2)}\n\\def\\mnras#1#2#3{{MNRAS }{\\bf #1},~#3~(#2)}\n\\def\\grg#1#2#3{{Gen. Rel. Grav.}\n{\\bf #1},~#3~(#2)}\n\\def\\s#1#2#3{{Science }{\\bf #1},~#3~(#2)}\n\\def\\baas#1#2#3{{Bull. Am. Astron. Soc.}\n{\\bf #1},~#3~(#2)}\n\\def\\ibid#1#2#3{{\\it ibid. }{\\bf #1},~#3~(#2)}\n\\def\\cpc#1#2#3{{Comput. Phys. Commun.}\n{\\bf #1},~#3~(#2)}\n\\def\\astp#1#2#3{{Astropart. Phys.}\n{\\bf #1},~#3~(#2)}\n\\def\\epjc#1#2#3{{Eur. Phys. J. C}\n{\\bf #1},~#3~(#2)}\n\\def\\nima#1#2#3{{Nucl. Instrum. Meth. A}\n{\\bf #1},~#3~(#2)}\n\\def\\jhep#1#2#3{{J. High Energy Phys.}\n{\\bf #1},~#3~(#2)}\n\\def\\jcap#1#2#3{{J. Cosmol. Astropart. Phys.}\n{\\bf #1},~#3~(#2)}\n\\def\\lnp#1#2#3{{Lect. Notes Phys.}\n{\\bf #1},~#3~(#2)}\n\\def\\jpcs#1#2#3{{J. Phys. Conf. Ser.}\n{\\bf #1},~#3~(#2)}\n\\def\\aap#1#2#3{{Astron. Astrophys.}\n{\\bf #1},~#3~(#2)}\n\\def\\mpla#1#2#3{{Mod. Phys. Lett. A}\n{\\bf #1},~#3~(#2)}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nThe conventional wave-function renormalization prescription extracts WRC by expanding the particle's propagator around its physical mass point in the LSZ reduction formula \\cite{c1}. For scalar boson it is \\cite{c2,c3,c5}\n\\begin{eqnarray}\n  \\frac{i}{p^2-m^2-\\delta m^2+\\Sigma(p^2)}&\\sim&\\frac{i}\n  {(p^2-m^2)(1+Re\\Sigma^{\\prime}(m^2))+Re\\Sigma(m^2)-\\delta m^2+i\\,Im\\Sigma(p^2)} \\nonumber \\\\\n  &=&\\frac{i\\,(1+Re\\Sigma^{\\prime}(m^2))^{-1}}{p^2-m^2+i\\,\\epsilon}\\,, \\nonumber\\\\\n  {\\bf or}\\hspace{3mm}&\\sim&\\frac{i}{(p^2-m^2)(1+\\Sigma^{\\prime}(m^2))\n  +\\Sigma(m^2)-\\delta m^2}\\,=\\,\\frac{i\\,(1+\\Sigma^{\\prime}(m^2))^{-1}}{p^2-m^2+i\\,\\epsilon^{\\prime}}\\,,\n\\end{eqnarray}\nwhere $\\Sigma^{\\prime}(m^2)\\hspace{-1mm}=\\hspace{-1mm}\\partial\\Sigma(m^2)\/\\partial p^2$, $\\epsilon$ and $\\epsilon^{\\prime}$ are small quantities. But not long ago people propose that only the mass definition of the pole of the particle's propagator is gauge invariant \\cite{c6} and physical results are only gauge invariant under the pole mass renormalization prescription \\cite{c9}, so WRC must also be defined on the pole of the particle's propagator. Considering the fact that unstable particle's WRC must contain imaginary part \\cite{c3,c5,cin,c4}, the new wave-function renormalization prescription for boson must be\n\\begin{equation}\n  \\frac{i}{p^2-m^2-\\delta m^2+\\Sigma(p^2)}\\,\\sim\\,\\frac{i}{(p^2-\\bar{s})\n  (1+\\Sigma^{\\prime}(\\bar{s}))}\\,=\\,\\frac{i\\,(1+\\Sigma^{\\prime}(\\bar{s}))^{-1}}{p^2-\\bar{s}}\\,,\n\\end{equation}\nwhere $\\bar{s}=m^2-i m\\Gamma$ is the pole of the boson's propagator \\cite{c6}. Note that the pole mass renormalization prescription has been used in Eq.(2).\n\nFor fermion the new wave-function renormalization prescription is a little complex. The fermion inverse propagator can be written as\n\\begin{equation}\n  i S^{-1}(\\xslash p)\\,=\\,\\xslash p-m-\\delta m+\\Sigma(\\xslash p)\\,\\equiv\\,\n  \\xslash p(a\\gamma_L+b\\gamma_R)+c\\gamma_L+d\\gamma_R\\,,\n\\end{equation}\nwhere $\\gamma_L$ and $\\gamma_R$ are the left- and right- handed helicity operators, and the diagonal fermion self energy is\n\\begin{equation}\n  \\Sigma(\\xslash p)\\,=\\,\\xslash p\\gamma_L\\,\\Sigma^L(p^2)+\\xslash p\\gamma_R\\,\\Sigma^R(p^2)\n  +m(\\gamma_L\\Sigma^{S,L}(p^2)+\\gamma_R\\Sigma^{S,R}(p^2))\\,.\n\\end{equation}\nExpanding the fermion propagator around its pole we get \\cite{cin,c4}\n\\begin{eqnarray}\n  S(\\xslash p)&=&\\frac{i(\\xslash p(a\\gamma_L+b\\gamma_R)-d\\gamma_L-c\\gamma_R)}\n  {p^2 a b - c d} \\nonumber \\\\\n  &\\sim&\\frac{i(m+\\delta m+\\xslash p\\gamma_L(1+\\Sigma^L(\\bar{s}))\n  +\\xslash p\\gamma_R(1+\\Sigma^R(\\bar{s}))-m\\gamma_L\\Sigma^{S,R}(\\bar{s})\n  -m\\gamma_R\\Sigma^{S,L}(\\bar{s}))}{(p^2-m^2+i\\,m\\,\\Gamma)A}\\,,\n\\end{eqnarray}\nwhere $\\bar{s}=m^2-i m\\Gamma$ is the pole of the fermion propagator, and\n\\begin{eqnarray}\n  A&=&(1+\\Sigma^L(\\bar{s}))(1+\\Sigma^R(\\bar{s}))+\\bar{s}(\n  \\Sigma^{L\\prime}(\\bar{s})+\\Sigma^{R\\prime}(\\bar{s})\n  +\\Sigma^{L\\prime}(\\bar{s})\\Sigma^R(\\bar{s})\n  +\\Sigma^{L}(\\bar{s})\\Sigma^{R\\prime}(\\bar{s}) ) \\nonumber \\\\\n  &+&m\\Sigma^{S,L\\prime}(\\bar{s})(m+\\delta m-m\\Sigma^{S,R}(\\bar{s}))\n  +m\\Sigma^{S,R\\prime}(\\bar{s})(m+\\delta m-m\\Sigma^{S,L}(\\bar{s}))\\,.\n\\end{eqnarray}\n\nFrom Eq.(2) and Eq.(5) we can extract boson and fermion's WRC. In section 2 we will do this work. In section 3 we will evaluate the difference of unstable particle's WRC between the new and the conventional wave-function renormalization prescription and discuss the influence of the difference on physical results. Lastly we give our conclusion.\n\n\\section{Determination of wave-function renormalization constants}\n\nIn the LSZ reduction formula one needs to introduce two sets of WRC: the incoming WRC and the outgoing WRC \\cite{c5,cin,c4}. For boson the incoming and outgoing WRC are introduced as follows \\cite{c5}\n\\begin{equation}\n  Z^{\\frac{1}{2}}\\,=\\,<\\hspace{-1mm}\\Omega|\\phi(0)|\\lambda\\hspace{-1mm}>\\,, \\hspace{10mm}\n  \\bar{Z}^{\\frac{1}{2}}\\,=\\,<\\hspace{-1mm}\\lambda|\\phi^{\\dagger}(0)|\\Omega\\hspace{-1mm}>\n  \\,,\n\\end{equation}\nwhere $\\Omega$ is the interaction vacuum, $\\phi$ is the boson's Heisenberg field, and $\\lambda$ is the incoming or outgoing state of S-matrix element. According to the LSZ reduction formula we have from Eq.(2)\n\\begin{equation}\n  Z^{\\frac{1}{2}}\\bar{Z}^{\\frac{1}{2}}\\,=\\,(1+\\Sigma^{\\prime}(m^2-i\\,m\\,\\Gamma))^{-1}\\,.\n\\end{equation}\nAnother condition that boson's WRC must satisfy is \\cite{c5}\n\\begin{equation}\n  \\bar{Z}\\,=\\,Z\\,.\n\\end{equation}\nTherefore we get\n\\begin{equation}\n  \\bar{Z}\\,=\\,Z\\,=\\,(1+\\Sigma^{\\prime}(m^2-i\\,m\\,\\Gamma))^{-1}\\,.\n\\end{equation}\n\nFor fermion the incoming and outgoing WRC are introduced as follows \\cite{c5}:\n\\begin{equation}\n  <\\hspace{-1mm}\\Omega|\\psi(0)|\\lambda\\hspace{-1mm}>\\,=\\,Z^{\\frac{1}{2}}u\\,, \\hspace{10mm}\n  <\\hspace{-1mm}\\lambda|\\bar{\\psi}(0)|\\Omega\\hspace{-1mm}>\\,=\\,\\bar{u}\\bar{Z}^{\\frac{1}{2}}\\,,\n\\end{equation}\nwhere $\\psi$ is the fermion's Heisenberg field and\n\\begin{equation}\n  Z^{\\frac{1}{2}}\\,=\\,Z^{L\\frac{1}{2}}\\gamma_L+Z^{R\\frac{1}{2}}\\gamma_R\\,,\\hspace{10mm}\n  \\bar{Z}^{\\frac{1}{2}}\\,=\\,\\bar{Z}^{L\\frac{1}{2}}\\gamma_R+\\bar{Z}^{R\\frac{1}{2}}\\gamma_{L}\\,.\n\\end{equation}\nThe fermion propagator at resonant region can be expressed as \\cite{c5,cin}\n\\begin{equation}\n  S(\\xslash p)\\,\\sim\\,\\frac{i\\,Z^{\\frac{1}{2}}(\\xslash p+m+i x)\\bar{Z}^{\\frac{1}{2}}}{p^2-m^2+i\\,m\\,\\Gamma}\\,,\n\\end{equation}\nwhere $x$ is a small quantity. Considering Eqs.(5,13) must be placed in the middle of on-shell spinors $\\bar{u}$ and $u$ (or $\\bar{\\nu}$ and $\\nu$) and the fact $\\bar{u}\\gamma_L u=\\bar{u}\\gamma_R u$ (or $\\bar{\\nu}\\gamma_L \\nu=\\bar{\\nu}\\gamma_R \\nu$), we obtain\n\\begin{eqnarray}\n  Z^{L\\frac{1}{2}}\\bar{Z}^{L\\frac{1}{2}}&=&(1+\\Sigma^R(\\bar{s}))\/A\\,, \\nonumber \\\\\n  Z^{R\\frac{1}{2}}\\bar{Z}^{R\\frac{1}{2}}&=&(1+\\Sigma^L(\\bar{s}))\/A\\,, \\nonumber \\\\\n  Z^{L\\frac{1}{2}}\\bar{Z}^{R\\frac{1}{2}}+Z^{R\\frac{1}{2}}\\bar{Z}^{L\\frac{1}{2}}&=&\n  (2m+2\\delta m-m\\Sigma^{S,L}(\\bar{s})-m\\Sigma^{S,R}(\\bar{s}))\/(A(m+i\\,x))\\,.\n\\end{eqnarray}\nAnother condition that fermion's WRC must satisfy is \\cite{c5}\n\\begin{equation}\n  \\bar{Z}^L\\,=\\,Z^L\\,, \\hspace{10mm} \\bar{Z}^R\\,=\\,Z^R\\,.\n\\end{equation}\nTherefore we get\n\\begin{eqnarray}\n  \\bar{Z}^L&=&Z^{L}\\,=\\,(1+\\Sigma^R(\\bar{s}))\/A\\,, \\nonumber \\\\\n  \\bar{Z}^R&=&Z^{R}\\,=\\,(1+\\Sigma^L(\\bar{s}))\/A\\,.\n\\end{eqnarray}\nSince the quantity $x$ is undefined in Eq.(13), the third equation of Eqs.(14) can be used to define $x$. At one-loop level we get \\cite{cin}\n\\begin{equation}\n  x\\,=\\,-\\Gamma\/2\n\\end{equation}\nwhere $\\Gamma$ is the fermion's decay width.\n\nNow we have finished the definition of diagonal WRC. The off-diagonal WRC are out of our consideration, because they are different from the diagonal WRC under the meaning of the LSZ reduction formula.\n\n\\section{Gauge dependence of physical results under the conventional wave-function renormalization prescription}\n\nSince unstable particle's WRC must contain imaginary part \\cite{c3,c5,cin,c4}, the conventional wave-function renormalization prescription must be the second prescription of Eq.(1) for boson, i.e.  \\cite{c5} (see Eq.(9))\n\\begin{equation}\n  \\bar{Z}_o\\,=\\,Z_o\\,=\\,(1+\\Sigma^{\\prime}(m^2))^{-1}\\,,\n\\end{equation}\nwhere the subscript $o$ represents the conventional wave-function renormalization prescription. Comparing with Eqs.(10) we find at two-loop level\n\\begin{equation}\n  Z_o-Z\\,=\\,-i\\,m\\,\\Gamma\\,\\Sigma^{\\prime\\prime}(m^2)\\,.\n\\end{equation}\nFor unstable boson the difference is gauge-parameter dependent. For example for gauge boson W we obtain (see Fig.1)\n\\begin{eqnarray}\n  &&Re[Z_{oW}-Z_W]_{\\xi_W} \\nonumber \\\\\n  =&&\\frac{\\alpha^2}{288 s_w^2}\\bigl{[} \\sum_{i=e,\\mu,\\tau}\n  (1-x_i)^2(2+x_i)+3\\sum_{i=u,c}\\sum_{j=d,s,b}|V_{ij}|^2\\sqrt{1-2(x_i+x_j)+(x_i-x_j)^2} \\nonumber \\\\\n  \\times&&(2-(x_i+x_j)-(x_i-x_j)^2) \\bigr{]}(2\\xi_W^3-3\\xi_W^2-6\\xi_W-5)\\theta[1-\\xi_W]\\,,\n\\end{eqnarray}\nwhere $Re$ takes the real part of the quantity, $\\xi_W$ is the gauge parameter of W, the subscript $\\xi_W$ denotes the $\\xi_W$-dependent part of the quantity, $\\alpha$ is the fine structure constant, $s_w$ is the sine of the weak mixing angle, $x_i=m_i^2\/m_W^2$  and $x_j=m_j^2\/m_W^2$ with $m_W$ the mass of W, $V_{ij}$ is the CKM matrix element \\cite{c11}, and $\\theta$ is the Heaviside function. Note that in the calculations we have used the program packages {\\em FeynArts} and {\\em FeynCalc} \\cite{c10}.\n\\begin{figure}[htbp]\n\\begin{center}\n  \\epsfig{file=WW.ps, width=10.5cm} \\\\\n  \\caption{W one-loop self-energy diagrams containing imaginary part.}\n\\end{center}\n\\end{figure}\n\nFor fermion the conventional wave-function renormalization prescription must be \\cite{c5}\n\\begin{eqnarray}\n  \\bar{Z}_o^L&=&Z_o^{L}\\,=\\,(1+\\Sigma^R(m^2))\/A_1\\,, \\nonumber \\\\\n  \\bar{Z}_o^R&=&Z_o^{R}\\,=\\,(1+\\Sigma^L(m^2))\/A_1\\,,\n\\end{eqnarray}\nwhere\n\\begin{eqnarray}\n  A_1&=&(1+\\Sigma^L(m^2))(1+\\Sigma^R(m^2)) \\nonumber \\\\\n  &+&m^2(\\Sigma^{L\\prime}(m^2)\n  +\\Sigma^{R\\prime}(m^2)+\\Sigma^{L\\prime}(m^2)\\Sigma^R(m^2)\n  +\\Sigma^L(m^2)\\Sigma^{R\\prime}(m^2) ) \\nonumber \\\\\n  &+&m\\Sigma^{S,L\\prime}(m^2)(m+\\delta m-m\\Sigma^{S,R}(m^2)) \\nonumber \\\\\n  &+&m\\Sigma^{S,R\\prime}(m^2)(m+\\delta m-m\\Sigma^{S,L}(m^2))\\,.\n\\end{eqnarray}\nComparing with Eqs.(16) we find\n\\begin{eqnarray}\n  Z_o^L-Z^L&=&-i\\,m\\,\\Gamma(2\\Sigma^{L\\prime}+\\Sigma^{R\\prime}\n  +m^2(\\Sigma^{L\\prime\\prime}+\\Sigma^{R\\prime\\prime}+\\Sigma^{S,L\\prime\\prime}\n  +\\Sigma^{S,R\\prime\\prime}))\\,, \\nonumber \\\\\n  Z_o^R-Z^R&=&-i\\,m\\,\\Gamma(\\Sigma^{L\\prime}+2\\Sigma^{R\\prime}\n  +m^2(\\Sigma^{L\\prime\\prime}+\\Sigma^{R\\prime\\prime}+\\Sigma^{S,L\\prime\\prime}\n  +\\Sigma^{S,R\\prime\\prime}))\\,.\n\\end{eqnarray}\nFor unstable fermion the difference is also gauge-parameter dependent. For example for top quark we obtain (see Fig.2)\n\\begin{eqnarray}\n  &&Re[Z_{ot}^L-Z_t^L]_{\\xi_W} \\nonumber \\\\\n  =&&-\\frac{\\alpha^2}{128 s_w^4 x_t^3}\n  \\sum_{i=d,s,b}|V_{ti}|^2 B_i(x_t^2+(1-2 x_i)x_t+x_i^2+x_i-2) \\nonumber \\\\\n  \\times&&\\sum_{j=d,s,b}\\frac{|V_{tj}|^2}{C_j}(x_t^3-x_j x_t^2-(3\\xi_W^2+3 x_j\\xi_W+2x_j^2)x_t \\nonumber \\\\\n  +&&2(\\xi_W-x_j)^2(\\xi_W+x_j))\\theta[m_t-\\sqrt{\\xi_W}m_W-m_j]\\,, \\nonumber \\\\\n  &&Re[Z_{ot}^R-Z_t^R]_{\\xi_W} \\nonumber \\\\\n  =&&-\\frac{\\alpha^2}{256 s_w^4 x_t^3}\n  \\sum_{i=d,s,b}|V_{ti}|^2 B_i(x_t^2+(1-2 x_i)x_t+x_i^2+x_i-2) \\nonumber \\\\\n  \\times&&\\sum_{j=d,s,b}\\frac{|V_{tj}|^2}{C_j}\n  (x_t^3-(\\xi_W+x_j)x_t^2-(\\xi_W^2+4 x_j\\xi_W+3 x_j^2)x_t \\nonumber \\\\\n  +&&(\\xi_W-x_j)^2(\\xi_W+3 x_j))\\theta[m_t-\\sqrt{\\xi_W}m_W-m_j]\\,,\n\\end{eqnarray}\nwhere $m_t$ is the top quark's mass and $x_t=m_t^2\/m_W^2$, and\n\\begin{equation}\n  B_i\\,=\\,\\sqrt{x_t^2-2(x_i+1)x_t+(x_i-1)^2}\\,, \\hspace{10mm}\n  C_j\\,=\\,\\sqrt{x_t^2-2(\\xi_W+x_j)x_t+(\\xi_W-x_j)^2}\\,.\n\\end{equation}\n\\begin{figure}[htbp]\n\\begin{center}\n  \\epsfig{file=tt.ps, width=7cm} \\\\\n  \\caption{Top quark's one-loop self-energy diagrams containing imaginary part.}\n\\end{center}\n\\end{figure}\n\nThe gauge dependence of Eqs.(20,24) will lead to the decay widths\nof some physical processes gauge-parameter dependent under the\nconventional wave-function renormalization prescription.\nConsidering the physical process $W^{+}\\rightarrow e^{+}\\nu_e$,\nsince positive electron and electronic neutrino are stable\nparticles, their's WRC are same under the new and the conventional\nwave-function renormalization prescription, therefore we only need\nto consider the effect of $Z_{oW}$ on the gauge dependence of the\ndecay width. The result is shown in Fig.3 (the data is cited from\nRef.\\cite{c12}).\n\\begin{figure}[htbp]\n\\begin{center}\n  \\epsfig{file=Wen.eps, width=8cm} \\\\\n  \\caption{Gauge dependence of two-loop $|\\cal{M}$$(W^{+}\\rightarrow e^{+}\\nu_e)|^2$ under\n  the conventional wave-function renormalization prescription, where\n  $\\cal{M}$$_t(W^{+}\\rightarrow e^{+}\\nu_e)$ is the tree-level amplitude.}\n\\end{center}\n\\end{figure}\nConsidering another physical process $t\\rightarrow W^{+}b$, since the decay width of bottom quark is zero at one-loop level, the bottom quark's WRC are same at two-loop level under the two wave-function renormalization prescriptions (see Eqs.(23)), so we only need to consider the effect of $Z_{ot}$ and $Z_{oW}$ on the gauge dependence of the decay width. The result is shown in Fig.4.\n\\begin{figure}[htbp]\n\\begin{center}\n  \\epsfig{file=tWb.eps, width=8cm} \\\\\n  \\caption{Gauge dependence of two-loop $|\\cal{M}$$(t\\rightarrow W^{+}b)|^2$ under\n  the conventional wave-function renormalization prescription, where\n  $\\cal{M}$$_t(t\\rightarrow W^{+}b)$ is the tree-level amplitude.}\n\\end{center}\n\\end{figure}\nFig.3 and Fig.4 show the gauge dependence of the two physical results under the conventional wave-function renormalization prescription is order of $O(1)$ at two-loop level, so the conventional wave-function renormalization prescription will affect the veracity of physical results beyond one-loop level.\n\n\\section{Conclusion}\n\nThe new wave-function renormalization prescription proposed here is based on the pole mass renormalization prescription, in which WRC is extracted by expanding the particle's propagator around its pole rather than its physical mass point as convention. The difference of the new WRC and the conventional WRC is gauge dependent for unstable particles beyond one-loop level. This will lead to some physical results gauge dependent under the conventional wave-function renormalization prescription beyond one-loop level.\n\n\\vspace{5mm} {\\bf \\Large Acknowledgments} \\vspace{2mm}\n\nThe author thanks Prof. Cai-dian Lu for his devoted help.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nEpilepsy is one of the most common chronic neurological disorders characterized by spontaneous recurrent seizures and affects around 70 million patients worldwide\\cite{ref1}. Over 30\\% of the epilepsy patients have refractory seizures which may carry risks of structural damage to the brain and nervous system, comorbidities, and increased mortality\\cite{ref2}. Deep Brain Stimulation (DBS) is a recently FDA-approved neurostimulation therapy that can effectively reduce the occurrences of refractory seizures by delivering electric impulses to a deep brain structure called the anterior nucleus of the thalamus (ANT). Accurate localization of the ANT target is however difficult because of the well documented variability in the ANT size and shape and thalamic atrophy caused by persistent epileptic seizures \\cite{ref3}. Currently, the standard approach to automate this process is the atlas-based technique. While popular, atlas-based methods are known to lack robustness when anatomic differences between atlases and subjects are large. This is particularly acute for ANT-DBS targets that are close to the ventricles, which can be severely enlarged in some patients. \n\nOver the past decade, DL-based techniques such as convolutional neural networks (CNN) have emerged as powerful tools and have achieved unprecedented performances in many medical imaging tasks. However, to train sufficiently robust and accurate models, deep learning methods typically require large amounts of labeled data, which is expensive to collect, especially in the medical domain. In the case of data scarcity and noisy labels, insufficiently trained models may fail catastrophically without any indication. Hence, it is extremely desirable for deep learning models to estimate the uncertainties regarding their outputs in these scenarios. The predictive uncertainty of neural networks can be categorized into two types: \\textit{epistemic} uncertainty and \\textit{aleatoric} uncertainty. Epistemic uncertainty, also known as model uncertainty, accounts for the uncertainty in the model and can be explained away by observing more training data. On the other hand, the aleatoric uncertainty is the input-dependent uncertainty that captures the noise and randomness inherent in observations. Recently, uncertainty estimation has also received increasing attention in medical image analysis. Ayhan et al.\\cite{ref4} proposed to estimate the heteroscedastic aleatoric uncertainty using TTA for classification task. Nair et al.\\cite{ref5} explored the uncertainty estimation for lesion detection and segmentation tasks based on MCDO. Wang et al.\\cite{ref6} proposed a theoretical formulation of TTA and demonstrated its effectiveness in uncertainty estimation for segmentation task. Nevertheless, uncertainty estimation for localization tasks has not been well studied.\n\nIn this work, we developed a novel two-stage deep learning framework aiming at robustly localizing the ANT targets. To the best of our knowledge, this is the first work to develop a learning-based approach for this task. To overcome the problem of data scarcity, we train the models with the pseudo labels which are created based on the available gold-standard annotations using multi-atlas registration. Moreover, we validate two sampling-based uncertainty estimation techniques to assess the localization performance of the developed method. We also propose a novel metric called MAD for sampling-based uncertainty estimation methods in localization tasks. Our experimental results show that the proposed method achieved more robust localization performance than the traditional multi-atlas method and TTA could further improve the robustness. Lastly, we show that the epistemic and hybrid uncertainty estimated by MAD can be used to detect the unreliable localizations and the magnitude of MAD can reflect the degree of unreliability when the predictions are rejected.\n\n\\section{Materials and Methods}\n\\subsection{Data}\nOur own dataset consists of 230 T1-weighted MRI scans from a database of patients who underwent a DBS implantation for movement disorders, i.e., Parkinson Disease or Essential Tremor at Vanderbilt University. The resolution of the images varies from 0.4356$\\times$0.4356$\\times$1 mm\\textsuperscript{3} to 1$\\times$1$\\times$6 mm\\textsuperscript{3}. The ground truth was manually annotated on a different dataset collected by an experienced neurosurgeon. In this dataset, the 3D coordinates of eight ANT targets (one on each side) on four MRI scans were identified and the thalamus mask on one of these volumes was delineated. With the available annotations, we generated the pseudo labels for the ANT targets and for the thalamus masks using multi and single-atlas registration\\cite{ref7}. In this study, 200 MRI scans were randomly selected for training and validation and the remaining 30 images were used for testing. For preprocessing, we use trilinear interpolation to resample all the images to isotropic voxel sizes of 1$\\times$1$\\times$1 mm\\textsuperscript{3} and rescale the image intensities to $[0, 1]$.\n\n\\subsection{Proposed Method}\nTypically, an MRI scan with original resolution cannot be fed to a 3D CNN directly due to the limitation of computational resource. A common approach to solve this problem, i.e., using downsampled images, is not appropriate here because the downsampling operation unavoidably leads to a loss in image resolution. This is a concern in our application because even a few-voxel shift in the deep brain can lead to target predictions that are unacceptable for clinical use. To address this issue, we propose a two-stage framework where the first stage coarsely identifies and crops the thalamus regions from the whole brain MRI and the second stage performs per-voxel regression on the cropped volume to localize the targets at the finest resolution scale. \n\n\\begin{figure}\n\\includegraphics[width=1\\columnwidth]{figure1.png}\n\\centering\n\\caption{The workflow of the proposed two-stage framework and the 3D U-net architecture we used. The segmentation and localization network share the same architecture. The number below each encode\/decoder unit is the channel number of the 3$\\times$3$\\times$3 convolution kernels.} \\label{fig1}\n\\end{figure}\n\nThe workflow of the proposed framework is shown in Figure 1. In this first stage, we train a 3D U-net \\cite{ref8} using the downsampled 80$\\times$80$\\times$80 MRI scans to coarsely segment the thalamus. The output layer of this network has three channels corresponding to background, left thalamus and right thalamus respectively. Once we obtain the segmentation results, we post-process the binary segmentation from each foreground channel by isolating the largest connected component and resample the results back to the original resolution. Thereafter, we compute the bounding box of each isolated component and crop a 64$\\times$64$\\times$64 mm\\textsuperscript{3} volume\naround its center. The cropped volume fully encloses the entire left or right thalamus as well as its contextual information. Before passing the volumes to the second stage, we flip the left-thalamus volumes in the left-right direction so that the inputs of the second stage have consistent orientations. In the second stage, we employ another 3D U-net with the same architecture to localize the ANT target by performing per-voxel regression on the cropped volumes. Since the cropped volumes have the same resolution as the original MRI, there is no performance degradation in localization due to loss in resolution. To allow volume-to-volume mapping, we design the ground truth map to be a 3D Gaussian function centered at the pseudo label position with a standard deviation of 1.5 mm. The maximum value is scaled to 1 and any value below 0.05 is set to 0. During the testing phase, the left-thalamus localization maps are flipped back to the original orientation and the voxel with the \\textbf{maximum} activation in each localization map is taken as the final prediction. \n\n\\subsection{Uncertainty Estimation}\n\\subsubsection{Epistemic Uncertainty}\nWe estimate the epistemic uncertainty of the localization task using the dropout variational inference. Specifically, we train the model with dropout (same as the baseline method) and during the testing phase we perform T stochastic forward passes with dropout to generate Monte Carlo samples from the approximate posterior. Let $y = f(x)$ be the network mapping from input $x$ to output $y$. Let $T$ be the number of Monte Carlo samples and $\\hat{W}_{t}$ be the sampled model weights from MCDO. For regression tasks, the final prediction and the epistemic uncertainty can be estimated by calculating the predictive mean $E(y)\\approx\\frac{1}{T}\\sum_{t=1}^{T}f^{\\hat{W}_{t}}(x)$ and variance $Var(y)\\approx\\frac{1}{T}\\sum_{t=1}^{T}f^{\\hat{W}_{t}}(x)^{T} f^{\\hat{W}_{t}}(x)-E(y)^{T}E(y)$ from these samples. \n\n\\subsubsection{Aleatoric Uncertainty}\nTo estimate the aleatoric uncertainty, we use the TTA technique which is a simple yet effective approach to study locality of testing samples. Recently, Wang et al.\\cite{ref6} provided a theoretical formulation for using TTA to estimate a distribution of prediction by Monte Carlo simulation with prior distributions of image transformation and noise parameters in an image acquisition model. In our image acquisition model, we extend this idea by incorporating both spatial transformations $T_{s}$ and intensity transformation ${T_{i}}$  to simulate the variations of spatial orientations and image brightness and contrast respectively. Our image acquisition model can thus be expressed as: $x = T_s(T_i(x_0))$, where $x$ is our observed testing image and $x_0$ is the image without transformations in latent space. During the testing phase, we aim to reduce the bias caused by transformations in $x$ by leveraging the latent variable $x_0$. Given the prior distributions of the transformation parameters in $T_{s}$ and $T_{i}$, we can estimate $y$ by generating N Monte Carlo samples and the $n_{th}$ Monte Carlo sample can be inferred as: $y_n=T_{s_n}(y_{0_n})=T_{s_n}(f(x_{0_n}))=T_{s_n}(f(T_{i_n}^{-1}(T_{s_n}^{-1}(x_n))))$. The final prediction and aleatoric uncertainty can be obtained by computing the mean and variance from the Monte Carlo samples. \n\n\\subsubsection{Maximum Activation Dispersion}\nFor regression tasks, the uncertainty maps are typically obtained by computing the voxel-wise variance from the Monte Carlo samples. However, this approach fails to generate useful uncertainty maps in our application. In our ground truth maps, the non-zero elements, i.e., foreground voxels within the Gaussian ball, are very sparse compared to the zero elements, and thus more difficult to localize and more likely to produce larger predictive variance. As a result, such uncertainty maps would display higher uncertainty at the predicted targets even if the targets are correctly localized (Figure 2), and thus are not effective for uncertainty estimation regarding the localization performance. To address this issue, we propose a novel metric called Maximum Activation Dispersion (\\textbf{MAD}) which can be directly applied to any sampling-based uncertainty estimation technique. This metric measures the consistency of the maximum activation positions of the Monte Carlo samples and ignores the activation variance at the same position. Note that MAD aims at estimating the image-wise uncertainty regarding the overall localization performance instead of voxel-wise uncertainty produced by uncertainty maps. Let N be the number of Monte Carlo samples and $\\mathbf{p}_n = (x_{n}, y_{n}, z_{n})$ be the maximum activation position of the $n_{th}$ Monte Carlo sample. The maximum activation dispersion is computed as $\\frac{1}{N}\\sum_{n=1}^{N}\\|{\\mathbf{p}_n-\\bar{\\mathbf{p}}\\|}$, where $\\|\\cdot\\|$ is the $L_2$ norm and $\\bar{\\mathbf{p}}= \\frac{1}{N}\\sum_{n=1}^{N}\\mathbf{p_n}$ is the geometric center of all maximum activation positions.\n\n\\begin{figure}\n\\includegraphics[width=1\\columnwidth]{figure2.png}\n\\centering\n\\caption{Visualization of some Monte Carlo samples (A-L) by TTA, their mean (localization map) and variance (uncertainty map). The uncertainty map displays higher uncertainties at the correctly localized position and thus is not effective for localization uncertainty estimation.} \\label{fig2}\n\\end{figure} \n\n\\subsection{Implementation Details}\nTwo five-level 3D U-nets with the same architecture were used in the proposed two-stage framework (Figure 1). In the first stage (segmentation), optimization was performed using the Adam optimizer with a learning rate of \\SI{5e-4}, with Dice loss as the loss function, a batch size of 3, and early stopping based on validation loss with patience of 10 epochs. In the second stage (localization), dropout layers were added to allow MCDO. As suggested by Kendall et al.\\cite{ref9}, applying dropout layers to all the encoders and decoders is too strong a regularizer. To avoid poor training fit, we followed the best dropout configuration in \\cite{ref9} by adding dropout layers with a dropout rate of $p=0.5$ only at the deepest half of encoders and decoders. During training, optimization was performed using the Adam optimizer with a learning rate of \\SI{2e-4}, with a batch size of 6, and early stopping based on validation loss with patience of 5 epochs. A weight decay of \\SI{5e-4} was used. The weighted mean squared error (WMSE) was used as the loss function to alleviate the class imbalance issue by assigning higher weights to the sparse non-zero entries. The models with the smallest validation losses were selected for final evaluation. \n\nDuring the testing phase, we forward passed the testing image once to the deterministic network with dropout turned off (\\textbf{baseline}). With a given prior distribution of transformation parameters in image acquisition model, we forward passed the stochastically transformed testing image $N=100$ times to the deterministic network with dropout turned off and transformed the predictions back to the original orientation. The mean and variance (aleatoric uncertainty) of the Monte Carlo samples were obtained (\\textbf{baseline + TTA}). In the image acquisition model, the spatial transformations were modeled by translation and rotation along arbitrary axis. The intensity transformation was modeled by a smooth and monotonous function called B\u00e9zier Curve, which is generated using two end points $P_0$ and $P_3$ and two control points $P_1$ and $P_2$: $B(t) = (1-t)^3P_0 + 3(1-t)^2tP_1+3(1-t)t^2P_2+t^3P_3, t\\in[0,1]$. In particular, we set $P_0 = (0, 0)$ and $P_3 = (1, 1)$ to obtain an increasing function to avoid invalid transformations.\nThe prior distributions of the spatial and intensity transformation parameters were modeled by uniform distribution $U$ as $s \\sim U(s_0, s_1)$, $r \\sim U(r_0, r_1)$ and $t \\sim U(t_0, t_1)$, where $s$, $r$ and $t$ represent translation (voxels), rotation angle (degrees) and the fractional value for B\u00e9zier Curve. In our experiment, we set $s_0 = -10$, $s_1 = 10$, $r_0 = -20$, $r_1 = 20$, $t_0 = 0$ and $t_1 = 1$. Lastly, we forward passed the same testing image to the stochastic network $T=100$ times with dropout turned on with a rate of $p=0.5$ and obtained the mean and variance (epistemic uncertainty) of the Monte Carlo samples (\\textbf{baseline + MCDO}).\n\nIn our experiment, we evaluated the localization performances of the multi-atlas, baseline, baseline + TTA and baseline + MCDO methods on 30 testing images (60 targets). The axial, sagittal and coronal views centered at the targets predicted by each method were provided to an experienced neurosurgeon to evaluate whether the predicted targets are acceptable for clinical use (the order was shuffled and the evaluator was blind to the method used to predict the target). When the predictions were evaluated as rejected, the evaluator was asked to provide the reasons for rejection. Furthermore, we analyze the aleatoric, epistemic and hybrid (aleatoric + epistemic) uncertainties estimated by MAD on the baseline rejected cases.\n\n\\section{Experimental Results}\nOur results show that among a total number of 60 targets, 53, 55, 57 and 55 targets were evaluated as acceptable for the multi-atlas, baseline, baseline + TTA and baseline + MCDO respectively. In Figure 3, we show the boxplots of aleatoric, epistemic and hybrid uncertainties estimated by MAD. It can be observed that when the rejected predictions are far away from the acceptable positions (red and blue), their estimated uncertainty correspond to the outliers above the upper whisker in the boxplots of epistemic and hybrid uncertainty. On the other hand, when the rejected predictions are close to the acceptable positions (cyan, green and magenta), their uncertainties fall in the range of upper quartile and the upper whisker (cyan and green) and the range of lower quartile and median (magenta), corresponding to their degree of unreliability. \n\n\\begin{figure}\n\\includegraphics[width=1\\columnwidth]{figure3.png}\n\\centering\n\\caption{Boxplots of aleatoric, epistemic and hybrid uncertainties estimated by MAD on the testing set (60 targets). The rejected cases of the baseline method (5 cases) are shown in color. The axial, sagittal and coronal views of the rejected targets are shown with the reasons for rejection provided by the evaluator.} \\label{fig3}\n\\end{figure}\n\nIt can be observed that the epistemic and hybrid uncertainty estimated by MAD could be used to detect unreliable localizations, i.e., the ones that not even in thalamus. Moreover, the magnitudes of MAD could reflect the degree of unreliability when the predictions were rejected. We also observe that even though the MCDO did not improve the localization robustness compared to the baseline method, the epistemic uncertainty obtained by this technique has great value for detecting the unreliable localizations, i.e., the outliers in the boxplot.\n\n\\section{Conclusion}\nIn this study, we present a two-stage deep learning framework to robustly localize the ANT-DBS targets in MRI scans. Results show that the proposed method achieved more robust localization performance than the traditional multi-atlas method and TTA-based aleatoric uncertainty estimation can further improve the localization robustness. We also show that the proposed MAD is a more effective uncertainty estimation metric for localization tasks.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{\\label{sec:level1} Introduction}\n\nOne of the central themes in recent condensed matter physics is searching for new topological states, e.g. the topological insulators (TIs) \\cite{M.Z.Hasan,X.L.Qi}, which are insulating in the bulk but support topologically protected metallic states at the boundary.  This area of studies has becoming more interesting thanks to the introduction of periodical light to the existing materials.  Specifically, by applying periodic, short laser pulse to topological, or even trivial, materials, the driven systems are expected to undergo topological phase transitions, leading to the so-called Floquet TIs \\cite{N.H.Lindner,B.Dora}.  Floquet TIs are intrinsically different from the traditional TIs since they are in non-equilibrium rather than in equilibrium.  Experimentally, the Floquet states have been reported to be observed on the surface of Bi$_2$Se$_3$ \\cite{Y.H.Wang} as well as in a hexagonal lattice made in a photonic crystal \\cite{M.C.Rechtsman}.\n\nSince the application of periodic light to the electron system breaks the time-reversal symmetry (TRS), the resulted Floquet TI belongs to the quantum anomalous Hall (QAH) class, according to the ten-fold classification of topological matter by dimensionality and certain discrete symmetries \\cite{A.P.Schnyder}.  However, in contrast to the static TI characterized by Chern number, which, according to the bulk-boundary correspondence, is directly related to the total number of edge modes in the gap \\cite{Y.Hatsugai}, for the Floquet TI, the topological states are intrinsically dynamical and the correspondence between the bulk and edge states is more subtle.  For example, for a two-band two dimensional system under driving light, in addition to the energy gap located near zero energy, another inequivalent gap would appear at the boundary of first Brillouin zone (BZ) in frequency domain, and in some cases, it would support chiral edge modes despite the fact that the Chern numbers associated with both bands are zero.  Therefore, the Chern number characterization in static system  is insufficient to describe its topological structure. One approach to this problem is proposed by Rudner \\emph{et. al} \\cite{M.S.Rudner}, in which a new topological invariant is constructed that yields the correct edge state structure in the driven case.\n\nFor graphene in particular, it has been proposed that the application of polarized light can turn the trivial equilibrium bands into the nontrivial Floquet TI and different dynamical generalizations of static topological phases can be achieved \\cite{T.Kitagawa,J.I.Inoue,A.Kundu,Z.H.Gu,A.G.Leon,G.Usaj,P.Titum,L.E.F.Torres,A.Quelle,M.A.Sentef, Perez2014, Perez2015}. However, a global phase diagram that can provide us with the information about the specific phase for certain driving protocol has not yet been reported. In Ref. \\cite{A.G.Leon}, a phase diagram in the parameter space spanned  by  the field amplitude and phase polarization has been reported for the modulated honeycomb lattices. Since the driving frequency plays a more important role in determining the topological property of the system, here we try to derive the phase diagram containing the driving frequency by using the Floquet theorem.  The phase diagram will be calculated by analyzing the edge states and the topological structures.  We will mainly focus on the drivings of intermediate frequency that is comparable to the bandwidth of the system.\n\nAs the band overlap may occur in the whole momentum region, compared with many previous studies which focus on the low-energy approximation, here the exact diagonalization calculation is more reliable.  Our work shows that phase diagram of the driven graphene ribbon contains up to ten different topological phases, in the parameter space spanned by the driving frequency and light strength.  Among them, exist the phases with high-Chern number (Chern number larger than 1)  as well as the phases with  counter-propagating edge modes in the gap.  In each phase, the topological structure and its physical origin are analyzed.  We also examine the robustness of the edge states to the inevitable disorder potential.  Finally, the possible experimental proposals of the driven graphene results and the detection method are discussed.\n\n\n\\section{\\label{sec:level1} Model and Method}\n\nWe start from the noninteracting electron system defined on a honeycomb lattice and coupled to a circularly polarized light.  For the polarized light, only the effect of its electric field part is considered, while that of the magnetic field part is neglected.  The electric field is assumed to be in-plane and spatially homogeneous.  We can describe the polarized light by the vector potential ${\\bf A}(\\tau)=A_0(\\text{cos}\\omega\\tau,\\text{sin}\\omega\\tau)$, where $A_0$ is the amplitude and $\\omega=\\frac{2\\pi}{T}$ is the frequency.  In general, the interaction between the light and electrons can be included in the hopping integrals as $t_{ij}(\\tau)=te^{iA_{ij}(\\tau)}$ where $t$ is the nearest-neighboring hopping strength and $A_{ij}(\\tau)=\\frac{e}{\\hbar}{\\bf A}(\\tau)\\cdot({\\bf r}_i-{\\bf r}_j)$ is the phase factor which strongly depends on the hopping direction.\n\nAs the whole system is time-periodic, the powerful Floquet theorem \\cite{H.Sambe,J.H.Shirley} can be applied.  The two-component wavefunction $\\psi_{\\bf k}(\\tau)$ encoding the time-dependent amplitudes on each sublattice can be written as $\\psi_{\\bf k}(\\tau)=e^{-i\\epsilon_{\\bf k}\\tau\/\\hbar}u_{\\bf k}(\\tau)$, where $\\epsilon_{\\bf k}$ is the quasienergy and $u_{\\bf k}(\\tau)=u_{\\bf k}(\\tau+T)$ is the periodic function.  Substituting $\\psi_{\\bf k}(\\tau)$ into the time-dependent Schr\\\"{o}dinger equation $i\\hbar\\partial_\\tau\\psi_{\\bf k}(\\tau)=H(\\tau)\\psi_{\\bf k}(\\tau)$, the latter will be reduced to the following eigenvalue problem:\n\\begin{eqnarray}\nH_{F}({\\bf k},\\tau)u_{\\bf k}(\\tau)=\\epsilon_{\\bf k}u_{\\bf k}(\\tau),\n\\end{eqnarray}\nwhere $H_F({\\bf k},\\tau)=H({\\bf k})-i\\hbar\\frac{\\partial}{\\partial\\tau}$ is the Floquet Hamiltonian.  We can expand $u_{\\bf k}(\\tau)$ into the Fourier series as $u_{\\bf k}(\\tau)=\\sum_{n=-\\infty}^{\\infty}u_{n{\\bf k}}e^{in\\omega \\tau}$, in which $u_{n{\\bf k}}=(u_{n{\\bf k}A},u_{n{\\bf k}B})^T$ denotes the $n$th Floquet mode corresponding to the quasienergy $\\epsilon_{\\bf k}$.  Using the formula of $e^{iz\\text{cos}\\omega\\tau}=\\sum_{n=-\\infty}^{\\infty}J_n(z)e^{in(\\omega \\tau+\\frac{\\pi}{2})}$, with $J_n(x)$ the n-th order Bessel function, Eq. (1) can be written as the time-independent form:\n\\begin{eqnarray}\n&&n\\omega u_{n{\\bf k}A}-t\\sum_m J_m(\\alpha)f_{m{\\bf k}} u_{n-m{\\bf k}B}=\\epsilon_{\\bf k}\nu_{n{\\bf k}A},  \\\\\n&&-t\\sum_m J_m(\\alpha)g_{m{\\bf k}}  u_{n+m{\\bf k}A}+n\\omega u_{n{\\bf k}B}=\\epsilon_{\\bf k} u_{n{\\bf k}B},\n\\end{eqnarray}\nwhere $t$ is the nearest-neighbor (NN) hopping integral and $\\alpha=\\frac{eA_0a_0}{c}$ is the dimensionless strength parameter for light.  The functions $f_{m{\\bf k}}=e^{i{\\bf k}\\cdot{\\bf a}_1}+e^{i({\\bf k}\\cdot{\\bf a}_2+\\frac{4m\\pi}{3})}+e^{i({\\bf k}\\cdot{\\bf a}_3+\\frac{2m\\pi}{3})}$ and $g_{m{\\bf k}}=e^{-i({\\bf k}\\cdot{\\bf a}_1+m\\pi)}\n+e^{-i({\\bf k}\\cdot{\\bf a}_2-\\frac{m\\pi}{3})}+e^{-i({\\bf k}\\cdot{\\bf a}_3+\\frac{m\\pi}{3})}$, where ${\\bf a}_l=a_0(\\text{cos}\\beta_l,\\text{sin}\\beta_l)$ are the NN vectors for a honeycomb lattice and $\\beta_l=\\frac{(4l-1)\\pi}{6}$.  Note that $(f_{m{\\bf k}})^*\\neq g_{m{\\bf k}}$.  Therefore, one has the infinite dimensional eigenvalue problem in the direct product Floquet space of ${\\mathcal R}\\bigotimes{\\mathcal T}$ \\cite{H.Sambe}, with $\\mathcal R$ being the usual Hilbert space and ${\\mathcal T}$ the space of periodic functions spanned by the function of $\\text{exp}(in\\omega t)$.  To solve it effectively, we follow the standard procedure to truncate the Floquet Hamiltonian $H_F$ into a finite dimensional matrix with $-M\\leq n\\leq M$ \\cite{M.S.Rudner}.  This is justified because the $n-$th Floquet component $u_{n{\\bf k}}$ decays rapidly with $|n|$ beyond a finite range in the frequency space. In actual calculations, when choosing $M=5$ the accurate results can be obtained already.\n\nIn a periodically driven system, the temporal periodicity gives rise to the periodicity in frequency space,  that is, the quasienergy $\\epsilon_{\\bf k}$ is equivalent to $\\epsilon_{\\bf k}+p\\omega$ for any integer $p$. Therefore, we often set the quasienergy to live within the quasienergy Brillouin zone (QBZ) as $\\epsilon_{\\bf k}\\in[-\\frac{\\omega}{2},\\frac{\\omega}{2}]$. In the system, we should distinguish two kinds of gaps \\cite{A.G.Leon}: the zero-energy gap $\\Delta_0$ between the conduction and valence bands within $n=0$ Floquet bands as well as the $\\pi-$gap $\\Delta_\\pi$ separating the conduction and valence bands of the neighboring Floquet bands. Physically, the zero-energy gap is caused by the first-order perturbation contribution of virtual processes involving the photon absorption and emission in which the two processes do not commute and thus impart an effective mass to the low-energy Floquet states, while the $\\pi-$gap is produced by the avoided crossings between the $n=0$ and $n=\\pm1$ Floquet bands \\cite{T.Oka,L.E.F.Torres}.\n\nAs there are two inequivalent gaps which will not close simultaneously, there may be topologically distinguished states but with the same Chern number.  When the edges are present in the system, we consider the upper edge of the system and denote the number of chiral edge modes propagating in the $\\hat x(-\\hat x)$ direction by $n_u^+(n_u^-)$, where the index $u=0,\\pi$ characterizes the zero-energy gap and the $\\pi-$gap, respectively. Rudner \\textit{et al.} showed the net chiralities of the edge modes as $C_0=n_0^+-n_0^-$ and $C_{\\pi}=n_{\\pi}^+-n_{\\pi}^-$.  Further the Chern number of the conduction band in the Floquet system can be obtained from the net chiralities as $C=C_0-C_\\pi$ \\cite{M.S.Rudner}.\n\nIn the numerical calculations, the number of $n_u^+$ and $n_u^-$ can be obtained exactly by counting the edge states from the ribbon dispersion according to the bulk-edge correspondence \\cite{Wang2, M.Ezawa}.  In the quasienergy spectrum,  besides the bulk bands, there exist  edge states, which always appear in pairs.  Each pair of chiral edge states contributes one unit to $n_u^{+(-)}$ in the corresponding gap.  More precisely, to evaluate $n_u^{+(-)}$, we need to take into account their locations (upper or bottom edges), which can be determined by analyzing their wave functions, as well as their directions (right or left) of propagation, which can be obtained from the sign of the group velocity $\\frac{d\\epsilon}{dk_x}$.\n\n\n\\section{\\label{sec:level1} Main Results}\n\n\\textit{Phase diagram. --} The graphene system under the modulation of light can exhibit diverse topological edge states, whose behavior strongly depends on the driving frequency $\\omega$ and the light strength parameter $\\alpha$.  In the high-frequency regime $\\omega\\gg W$ where $W=6t$ is the bandwidth, the Floquet Hamiltonian in Eqs. (2) and (3) is approximately block diagonal.  The neighboring Floquet bands are decoupled and well separated by a shift of $\\omega$.  When the frequency decreases and becomes comparable to the bandwidth $\\omega\\sim W$, a hierarchy of band-crossings will appear,  giving rise to energy gaps that may hold edge states.   As a result, the number of edge states will change and the topological phase transition happens.  Physically, the band-crossings can be understood as one-photon or multi-photon resonance. When the frequency further decreases and is much smaller than the bandwidth $\\omega\\ll W$, such band-crossings become serious. The gap between the neighboring Floquet bands, as well as the edge states within it, thus become obscure \\cite{A.G.Leon}.  Because of this reason, we mainly focus on the case of intermediate-frequency $\\omega\\sim W$, even though it is expected that graphene under low-frequency driving would host new states that are absent in the high-frequency region.  As for the parameter $\\alpha$, it renormalizes the hopping integrals, and thus leads to the change of the bandwidth $W$ and correspondingly to the change of the topological structure of the system.\n\n\\begin{figure}\n\\includegraphics[width=9.2cm]{Layout1.pdf}\n\\caption{(Color online) Phase diagram of periodically driven graphene in the parameter space spanned by $\\alpha$ and $\\omega$.  Each phase ($A$ to $J$) are characterized by $(C_0,C_\\pi)$.  We have labeled the phase boundary as solid lines when zero-energy gap is closed and as dotted lines when $\\pi-$gap is closed.}\n\\end{figure}\n\n\\begin{table}\n\\begin{tabular}{c|c|c|c|c|c|c|c|c|c|c}\\hline\n& \\emph{A}& \\emph{B}& \\emph{C}& \\emph{D}& \\emph{E}& \\emph{F}& \\emph{G}& \\emph{H}& \\emph{I}& \\emph{J} \\\\ \\hline\n$(n_0^+,n_0^-)$& (0,1)& (0,1)& (1,1)& (2,2)& (0,1)& (1,1)& (0,1)& (1,1) &(2,2) &(2,2)  \\\\ \\hline\n$(n_\\pi^+,n_\\pi^-)$& (0,0)& (2,0)& (2,0)& (2,0)& (0,4)& (0,4)& (0,2)& (0,2)& (0,4)& (0,2)  \\\\ \\hline \\hline\n$C_0$, $C_\\pi$& -1,0& -1,2& 0,2& 0,2& -1,-4& 0,-4& -1,-2& 0,-2& 0,-4& 0,-2  \\\\ \\hline \\hline\n$C$& -1& -3& -2& -2& 3& 4& 1& 2& 4& 2  \\\\ \\hline\n\\end{tabular}\n\\caption{The topological characteristics of each phase from $A$ to $J$, where ${(n_u^+,n_u^-)}$ give the number of chiral modes within different gaps, $C_u=n_u^+-n_u^-$ denotes the net chirality and $C=C_0-C_\\pi$ is the Chern number of the Floquet system.}\n\\end{table}\n\nThe distinct topological phases hosted by periodically driven graphene can be classified according to the net chiralities of the bulk bands $(C_0,C_\\pi)$ and the characteristics of the edge states in the quasienergy spectrum.  The resulting phase diagram in terms of the driving parameters $\\alpha$ and $\\omega$ is shown in Fig. 1.   Similar diagrams are presented  in Ref.~\\cite{Perez2015}, where the phase transitions in $0$ gap and $\\pi$ gap are separately discussed. We can see that the system undergoes versatile phase transitions as a function of $\\alpha$ and $\\omega$.  For the range of parameters shown, there are five possible phase transition boundaries which  divide the parameter space into ten phases labeled as $A$ to $J$.  All phases have well-defined quasienergy bands: both the zero-energy gap and $\\pi-$gap are well formed.  Here the neighboring phases are always separated by a transition boundary, where the band gap closes at certain value of $k_x$ and the band inversion occurs.  At the phase boundaries, either the zero-energy gap or $\\pi-$gap is closed, which are denoted, respectively, by solid lines or dashed lines.\n\nTo better illustrate these phases, in Table I, the topological characteristics of each phase are summarized, including the number of chiral modes ${(n_u^+,n_u^-)}$, the net chirality $C_u$ within different gaps and the Chern number $C$ of the Floquet system.  It shows that in these phases except phase $A$, there are more than one pair of edge states in the zero-energy gap or $\\pi-$gap, indicating the high-Chern number behavior is very common in the Floquet system.  For the chiralities of these edge states, we find that the edge states in the $\\pi-$gap always have the same chiralities while in the zero-energy gap, the chiralities may be different (see, phase $C$, $F$ and $H$).  Across the phase transition boundaries along which the zero-energy gap or $\\pi$-gap close, the Chern number $|C|$ is changed, respectively, by 1 and 2.\n\nIt should be noted that for phases $C$ and $D$, although they own the same net chirality and the same Chern number value, they have different topological structures as the number of chiral modes in the zero-energy gap are different, which can be seen in Fig. 3(c) and (d) below.  The arguments are the same for phases $F$ and $I$ as well as $H$ and $J$.  This suggests the Chern number alone is not enough to describe the number of edge states.  We need to take into account of the symmetry and the topological structure of the Floquet Hamiltonian to understand the topological properties of the driven system.\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=18cm]{Layout2.pdf}\n\\caption{(Color online) Quasienergy spectra of a finite ribbon (in the $y$ direction) of the periodically driven graphene system.  The eight panels correspond to phase $A$-$H$, as shown in Fig. 1, respectively.  From (a) to (h), the parameters of the driving $(\\alpha,\\omega)$ are taken as $(0.7,6)$, $(0.7,4)$, $(0.7,2.25)$, $(0.7,1.52)$, $(1.3,2.3)$, $(1.3,1.8)$, $(1.45,1.55)$, $(1.45,1.65)$.  Edge states localized on the upper and bottom edge are shown in red and blue, respectively.  }\n\\end{figure*}\n\n\\textit{Ribbon dispersion.--} Next in Fig. 2, the representative spectra of phases $A$ to $H$ are shown for a ribbon structure with 200 lattice in the $y$ direction and infinitely long in the $x$ direction.  Due to the periodicity of the Floquet system, we only plot the quasienergy in the range of $(-0.65\\omega,0.65\\omega)$.  It shows clearly that in all these phases, there exist a pair of chiral edge states crossing at the Dirac point $k_x=\\pi$ in the zero-energy gap.  Such edge states are the same for all different values of photon frequencies and light strength. They come from the linearity of the graphene bands around the Dirac point and have no relevance from the photon resonances.\n\nFor the ribbon structure, the quasi-energy spectra evidently exhibit several nice symmetries of the Floquet Hamiltonian.  First, the Floquet system has the symmetry of spatial inversion and is invariant under the operation of $k_x\\rightarrow-k_x$ and $y\\rightarrow L_y-y$.  Thus an edge state solution $\\epsilon(k_x)$ implies the existence of another edge state solution $\\epsilon(-k_x)$ (or $\\epsilon(2\\pi-k_x)$) with the same quasienergy but localized at the opposite edge.  Therefore, the edge states at $k_x\\neq0,\\pi$ always come in pairs.  Second, the Floquet system exhibits the particle-hole symmetry (PHS): if $\\epsilon(k_x)$ is a quasienergy eigenvalue, another solution of $-\\epsilon(k_x)$ must exist correspondingly. In addition, the PHS combined with the periodicity of the Floquet Hamiltonian lead to the symmetry around $\\epsilon=\\frac{\\omega}{2}$.  In the following, we will discuss these phases one by one.\n\n(1) In phase $A$, i.e. in the high-frequency regime, the electron cannot directly absorb photons due to the off-resonant light.  In $\\pi-$gap, as there exist no edge states, it is topologically trivial.  The effective Hamiltonian can take the same form as the static Hamiltonian, only with the hopping integrals being replaced by the effective ones $t_{ij}\\rightarrow tJ_0(\\alpha)$.  Here the renormalized hopping integrals to the lowest-order do not depend on the hopping directions.  Therefore the driven system can be well compared with the static one except for the renormalized hopping integrals. Early studies about the Floquet TI were mainly focused on this phase \\cite{T.Kitagawa,J.I.Inoue,Y.X.Wang}.\n\n(2) In phase $B$, there are two pairs of edge states in the $\\pi-$gap.  We can see the conduction band of $n=0$ Floquet band overlap with the valence band of $n=1$ Floquet replica, leading to the band inversion around the $\\pi-$gap.  At $\\epsilon=\\frac{\\omega}{2}$, the energy difference between the valence and conduction bands is precisely $\\omega$ \\cite{A.Quelleb}, which allows for the one-photon resonance.  Note the chiralities of edge states here are different from those in the zero-energy gap.  The transport properties about the dc conductance and quantum Hall response have been investigated in this phase \\cite{L.E.F.Torres}, where the transport signatures were suggested to be dominated by the time-averaged density of states.\n\n(3) In phase $C$, inside the zero-energy gap, it shows that besides the edge states at $k_x=\\pi$, there appear additional ones at $k_x=0$.  More importantly, the chiralities of these edge states are opposite with those at $k_x=\\pi$, i.e. they propagate along the opposite directions at the same edge. Therefore, the states at $k_x=0$ are called the counter-propagating edge modes.  The corresponding topological numbers give $n_0^+=n_0^-=1$.  This is in sharp contrast with the static TIs or the QH effect where the edge states in the same gap must own the same chirality.  Compared with the previous studies, where the counter-propagating edge modes are predicted to appear in the $\\pi-$gap of the periodically driven QH system (either the periodically kicked \\cite{M.Lababidi} or harmonically driven \\cite{Z.Y.Zhou} Hofstadter model ), here our study shows that such modes can also appear in the zero-energy gap of the periodically driven graphene.\n\n\\begin{figure*}\n\\centering\n\\includegraphics[width=18cm]{Layout3.pdf}\n\\caption{(Color online) (a) The quasienergy eigenvalues of a periodically driven graphene on a finite $40\\times20$ lattice for $\\alpha=1.3$ and $\\omega=1.8$, corresponding to phase $F$ in Fig. 2.  The horizontal axis is the index of the eigenvalues. The curve I is the spectrum without perturbation, while II and III are for case (1) and (2), respectively. They have been shifted vertically from each other for clarity.  (b),(c) Contour plot of the wavefunction in the Floquet system when disorder exist.  The parameters used for (b) and (c) correspond to the crosses in line II and III in (a), respectively. }\n\\end{figure*}\n\nEvidently in Fig. 2(c), to avoid the band-crossings between the valence band of $n=1$ Floquet band and the conduction band of $n=-1$ Floquet band, the band inversion happens around $\\epsilon=0$ which creates the counter-propagating edge modes.  At $\\epsilon=0$, the energy difference between the conduction band and the valence band is $2\\omega$ \\cite{A.Quelleb}, so the edge states can be regarded as being induced by the two-photon resonances.  In the framework of the low-energy effective theory, Quelle \\textit{et al.} recongnized the significance of the band inversions and referred the edge states appearing at $k_x=0$ as a realization of the Bernevig-Hughes-Zhang model of the HgTe\/CdTe quantum well \\cite{A.Quelle}.\n\n(4) Phase $D$ can be obtained from phase C by decreasing the frequency.  Intrinsically it has same topological structure as phase C. However, within the zero-energy gap,   the original linear structure of the edge states in phase C evolves into a helical structure, that is, the modes around $k_x=0$ cross each other for three times.  This means that the propagation direction of the original edge states at a given edge changes. Correspondingly the topological numbers give $n_0^+=2$, $n_0^-=2$.  However, the net chiralities remain the same as in phase $C$.  The helical structure in the low-frequency region can be attributed to the nonlinearity of the bands near $k_x=\\pi$.\n\n(5) Phase $F$ can be treated as one obtained from phase by increasing the driving magnitude. The difference with phase C is the appearance of four, instead of two, pairs of edge states within the $\\pi-$gap. Moreover, these four pair of edge states have the same chirality, but opposite to those in phase $C$. The topological number $n_\\pi^+$ vanishes while $n_\\pi^-=4$.  This topological phase transition is accompanied by the change of Chern number of conduction band from $-2$ to $4$.\n\n(6) Phase $H$ is obtained from phase $F$ by further increase of the driving magnitude. Within the $\\pi-$gap, exist two pairs of edge states with the same chirality as in phase $F$. Correspondingly, the Chern number of the conduction band changes from $4$ to $2$.\nFrom the phase diagram in Fig. 1, it shows only when the light strength $\\alpha$ is large that the topological structures that are similar to phase $F$ and $H$ may happen in the $\\pi-$gap.  This is because the structures of the Floquet bands are changed (not only becoming flat) due to the strong light field.  As a result, the Chern number will change in each Floquet band and the edge states with different number and chirality will appear.\n\nHere we have studied different topological phases of the periodically driven graphene and found several new phases, from $D$ through $H$, which exhibit novel topological structure that have not been discussed before.  Our analysis suggest that the one-photon resonances can create the edge states in the $\\pi-$ gap, while the two-photon resonances will induce the counter-propagating edge states in the zero-energy gap.\n\n\\textit{Robustness of the edge states.--} As the driven topological phases belong to the QAH class \\cite{A.P.Schnyder}, the edge states should not be affected by the disorder potential inevitably presenting in real materials.  Here we try to numerically check the robustness of the edge states by solving the quasienergy of periodically driven graphene.  The size of the system is set as $L_x\\times L_y$.  We choose the periodic boundary condition in the $x-$direction and open boundary condition in the $y-$direction.  Two kinds of static perturbations are considered: (1) random onsite potential $U_i\\in[-\\frac{t}{2},\\frac{t}{2}]$; (2) exceedingly strong onsite potential $U\\sim10^3 t$ on the edge (at position $(20,1)$).  In Ref.~\\cite{Perez2014}, the robustness was also tested against vacancies included randomly with a very low density.\n\nIn Fig. 3(a), we plot the quasienergy spectra of the driven system with the above perturbations included.  For comparison, the quasienergy spectra without disorder is also plotted as curve I.  For curve II, it shows due to the scattering of the impurity potentials, more bulk states are scattered into the zero-energy gap and $\\pi-$gap, so the curve becomes smooth.  While for curve III, the strong onsite potential shows minor influence on the low-energy spectra.   The results demonstrate the robustness of the edge states, similar to the static ones. Further in Fig. 3(b) and (c), we plot the contour of the wavefunction $\\psi(x,y,t=0)$ corresponding to quasienergy $\\epsilon=0$ and $\\epsilon=\\frac{\\omega}{2}$.  We note the main contribution to the wavefunction comes from the $n=0$ Floquet component, indicating the correctness of the cutoff of the photon number in the calculation.  For the given quasienergies, the wavefunctions are both localized on the upper edge.  Compared with the unperturbed case, the presence of disorder distorts the wavefunction but does not break the property of edge states.\n\n\n\\section{\\label{sec:level1} Discussions and Summaries}\n\nWe discuss the feasibility of our results in experiment. The periodically driven graphene system exhibiting the topological nontrivial phases have been proven to be realized in different physical setups.  In real graphene material the relevant hopping parameter is $t=2.7$eV and the NN bond length $a_0=1.4{\\AA}$, which requires rather strong driven light field with the frequency in the THz region and the amplitude $A_0\\sim 10^{-2}$V$\\cdot$s\/m.  There are also proposals with artificial honeycomb structure where the bands can also be described by the tight-binding model, such as the photonic crystals \\cite{M.C.Rechtsman} with extended helical wave guides in the third spatial dimension as well as the shaken optical lattices \\cite{G.Jotzu,P.Hauke}.  In addition, the microwave honeycomb crystals \\cite{M.Bellec} have also been put forward, where the hopping parameter is of the order of a few MHz \\cite{M.Bellec} while the driving frequency can be varied up to GHz \\cite{S.Gehler}, which makes the experiment more easily to operate.\n\nThe Floquet spectrum and the gaps due to the photon resonance in the driven system can be detected by using the ARPES technique, as recently demonstrated on the surface of 3D TI \\cite{Y.H.Wang}.  The edge modes in the nontrivial gap can be detected by measuring  the spectral function $\\rho(k_x,\\omega)$ \\cite{F.Li, F.Li2}, which is well captured by momentum-resolved radio-frequency spectroscopy \\cite{J.T.Stewart}.  On the other hand, the transport properties of electrons \\cite{L.E.F.Torres} may provide the direct signal of the nontrivial phase in the driven system. However, there are less studies in this topic \\cite{L.E.F.Torres} and further investigations are needed in the future.\n\nIn summary, we have studied the driven graphene under the circularly polarized light and  obtained a global phase diagram as a function of the frequency and strength of the light.  We analyze the topological structures of different phases in detail, especially the the high-Chern number behavior and the counter-propagating edge modes.  We suggest such counter-propagating modes may exist only in the driven system.  In addition, we demonstrate the robustness of the edge modes.  Our work may deepen the understanding of the driven non-equilibrium system and help us to search the new topological states of matter.\n\n\n\\section{\\label{sec:level1} Acknowledgements}\n\nThis work is supported by Natural Science Foundation of Jiangsu Province, China under grant No. BK20140129.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nConvolutional neural network (CNN) based object detectors \\cite{girshick2015fast,he2017mask,ren2015faster,liu2016ssd,lin2017focal} have achieved state-of-the-art performance due to the strong discriminative power and generalization ability. However, the CNN based detection methods require massive computation and storage resources to achieve ideal performance, which limits their deployment on mobile devices. Therefore, it is desirable to develop detectors with lightweight architectures and few parameters.\n\nTo reduce the complexity of deep neural networks, several model compression methods have been proposed including pruning \\cite{molchanov2019importance,zhao2019variational,he2017channel}, low-rank decomposition \\cite{lin2018holistic,peng2018extreme,kim2019efficient}, quantization \\cite{wang2019learning,li2019fully,gong2019differentiable}, knowledge distillation \\cite{wei2018quantization,wang2019private,chen2017learning}, architecture design \\cite{sandler2018mobilenetv2,zhang2018shufflenet,qin2019thundernet} and architecture search \\cite{wu2019fbnet,tan2019mnasnet}. Among these methods, network quantization reduces the bitwidth of the network parameters and activations for efficient inference. In the extreme cases, binarizing weights and activations of neural networks decreases the storage and computation cost by $32\\times$ and $64\\times$ respectively. However, deploying binary neural networks with constrained representational capacity in object detection causes numerous false positives due to the information redundancy in the networks.\n\nIn this paper, we present a BiDet method to learn binarized neural networks including the backbone part and the detection part for efficient object detection. Unlike existing methods which directly binarize the weights and activations in one-stage or two-stage detectors, our method fully utilizes the representational capacity of the binary neural networks for object detection via redundancy removal, so that the detection precision is enhanced with false positive elimination. More specifically, we impose the information bottleneck (IB) principle on binarized object detector learning, where we simultaneously limit the amount of information in high-level feature maps and maximize the mutual information between object detection and the learned feature maps. Meanwhile, the learned sparse object priors are utilized in IB, so that the posteriors are enforced to be concentrated on informative prediction and the uninformative false positives are eliminated. Figure \\ref{fp} (a) and (b) show an example of predicted positives obtained by Xnor-Net \\cite{rastegari2016xnor} and our BiDet respectively, where the false positives are significantly reduced in the latter. Figure \\ref{fp} (c) and (d) depict the information plane dynamics for the training and test sets respectively, where our BiDet removes the information redundancy and fully utilizes the representational power of the networks. Extensive experiments on the PASCAL VOC \\cite{everingham2010pascal} and COCO \\cite{lin2014microsoft} datasets show that our BiDet outperforms the state-of-the-art binary neural networks in object detection across various architectures. Moreover, BiDet can be integrated with other compact object detectors to acquire faster speedup and less storage. Our contributions include:\n\\begin{itemize}[leftmargin=*]\n\t\\item To the best of our knowledge, we propose the first binarized networks containing the backbone and detection parts for efficient object detection.\n\t\\item We employ the IB principle for redundancy removal to fully utilize the capacity of binary neural networks and learn the sparse object priors to concentrate posteriors on informative detection prediction, so that the detection accuracy is enhanced with false positive elimination.\n\t\\item We evaluate the proposed BiDet on the PASCAL VOC and the large scale COCO datasets for comprehensive comparison with state-of-the-art binary neural networks in object detection.\n\\end{itemize}\n\n\\section{Related Work}\n\\textbf{Network Quantization: }Network quantization has been widely studied in recent years due to its efficiency in storage and computation. Existing methods can be divided into two categories: neural networks with weights and activations in one bit or multiple bits. Binary neural networks reduce the model complexity significantly due to the extremely high compression ratio. Hubara \\etal \\cite{hubara2016binarized} and Rastegari \\etal \\cite{rastegari2016xnor} binarized both weights and activations in neural networks and replaced the multiply-accumulation with xnor and bitcount operations, where straight-through estimators were applied to relax the non-differentiable sign function for back-propagation. Liu \\etal \\cite{liu2018bi} added extra shortcut between consecutive convolutional blocks to strengthen the representational capacity of the network. They also used custom gradients to optimize the non-differentiable networks. Binary neural networks perform poorly on difficult tasks such as object detection due to the low representational capacity, multi-bit quantization strategies have been proposed with wider bitwidth. Jacob \\etal \\cite{jacob2018quantization} presented an 8-bit quantized model for inference in object detection and their method can be integrated with efficient architectures. Wei \\etal \\cite{wei2018quantization} applied the knowledge distillation to learn 8-bit neural networks in small size from large full-precision models. Li \\etal \\cite{li2019fully} proposed fully quantized neural networks in four bits with hardware-friendly implementation. Meanwhile, the instabilities during training were overcome by the presented techniques. Nevertheless, multi-bit neural networks still suffer from heavy storage and computation cost. Directly applying binary neural networks with constrained representational power in object detection leads to numerous false positives and significantly degrades the performance due to the information redundancy in the networks.\n\n\\iffalse\nYang \\etal \\cite{yang2019quantization} leveraged the soft quantization strategy by approximating the rigid sign function with the sigmoid layer, where the discrepancy between the optimization objective and the gradient was minimized.\n\\fi\n\n\\textbf{Object Detection: }Object detection has aroused comprehensive interest in computer vision due to its wide application. Modern CNN based detectors are categorized into two-stage and one-stage detectors. In the former, R-CNN \\cite{girshick2014rich} was among the earliest CNN-based detectors with the pipeline of bounding box regression and classification. Progressive improvements were proposed for better efficiency and effectiveness. Fast R-CNN \\cite{girshick2015fast} presented the ROIpooling in the detection framework to achieve better accuracy and faster inference. Faster R-CNN \\cite{ren2015faster} proposed the Region Proposal Networks to effectively generate region proposals instead of hand-crafted ones. FPN \\cite{lin2017feature} introduced top-down architectures with lateral connections and the multi-scale features to integrate low-level and high-level features. In the latter regard, SSD \\cite{liu2016ssd} and YOLO \\cite{redmon2016you} directly predicted the bounding box and the class without region proposal generation, so that real-time inference was achieved on GPUs with competitive accuracy. RetinaNet \\cite{lin2017focal} proposed the focal loss to solve the problem of foreground-background class imbalance. However, CNN based detectors suffer from heavy storage and computational cost so that their deployment is limited.\n\n\\begin{figure*}[t]\n\t\\centering\n\t\\includegraphics[height=5cm, width=14.5cm]{BiDet_pipeline.pdf}\n\t\\caption{The pipeline of the information bottleneck based detectors, which consist of the backbone part and the detection part. The solid line represents the forward propagation in the network, while the dashed line means sampling from a parameterized distribution $\\Phi$. The high-level feature map $F$ is sampled from the distribution parameterized by the backbone network. The one-stage and two-stage detector framework can be both employed in the detection part of our BiDet. For the one-stage detectors, the head network parameterizes the distribution of object classes and location. For two-stage detectors, Region Proposal Networks (RPN) parameterize the prior distribution of location and the posteriors are parameterized by the refining networks. (best viewed in color).}\n\t\\vspace{-0.3cm}\n\t\\label{pipeline}\n\\end{figure*}\n\n\\textbf{Information Bottleneck: }The information bottleneck (IB) principle was first proposed by \\cite{tishby2000information} with the goal of extracting relevant information of the input with respect to the task, so that the IB principle are widely applied in compression. The IB principle enforces the mutual information between the input and learned features to be minimized while simultaneously maximizing the mutual information between the features and groundtruth of the tasks. Louizos \\etal \\cite{louizos2017bayesian} and Ullrich \\etal \\cite{ullrich2017soft} utilized the Minimal Description Length (MDL) principle that is equivalent to IB to stochastically quantize deep neural networks. Moreover, they used the sparse horseshoe and Gaussian mixture priors for weight learning in order to reduce the quantization errors. Dai \\etal \\cite{dai2018compressing} pruned individual neurons via variational IB so that redundancy between adjacent layers was minimized by aggregating useful information in a subset of neurons. Despite the network compression, IB is also utilized in compact feature learning. Amjad \\etal \\cite{amjad2019learning} proposed stochastic deep neural networks where IB could be utilized to learn efficient representations for classification. Shen \\etal \\cite{shen2019embarrassingly} imposed IB on existing hash models to generate effective binary representations so that the data semantics were fully utilized. In this paper, we extend the IB principle to squeeze the redundancy in binary detection networks, so that the false positives are alleviated and the detection precision is significantly enhanced.\n\n\n\\section{Approach}\nIn this section, we first extend the IB principle that removes the information redundancy to object detection. Then we present the details of learning the sparse object priors for object detection, which concentrate posteriors on informative prediction with false positive elimination. Finally, we propose the efficient binarized object detectors.\n\\subsection{Information Bottleneck for Object Detection}\nThe information bottleneck (IB) principle directly relates to compression with the best hypothesis that the data misfit and the model complexity should simultaneously be minimized, so that the redundant information irrelevant to the task is exclusive in the compressed model and the capacity of the lightweight model is fully utilized. The task of object detection can be regarded as a Markov process with the following Markov chain:\n\\begin{align}\n\tX\\rightarrow F\\rightarrow L,C\n\\end{align}\nwhere $X$ means the input images and $F$ stands for the high-level feature maps output by the backbone part. $C$ and $L$ represent the predicted classes and location of the objects respectively. According to the Markov chain, the objective of the IB principle is written as follows:\n\\begin{align}\n\t\\min\\limits_{\\phi_b,\\phi_d} ~~ I(X;F)-\\beta I(F;C,L)\n\t\\label{IB}\n\\end{align} where $\\phi_b$ and $\\phi_d$ are the parameters of the backbone and the detection part respectively. $I(X;Y)$ means the mutual information between two random variables $X$ and $Y$. Minimizing the mutual information between the images and the high-level feature maps constrains the amount of information that the detector extracts, and maximizing the mutual information between the high-level feature maps and object detection enforces the detector to preserve more information related to the task. As a result, the redundant information irrelevant to object detection is removed. Figure \\ref{pipeline} shows the pipeline for information bottleneck based detectors, the IB principle can be imposed on the conventional one-stage and two-stage detectors. We rewrite the first term of (\\ref{IB}) according to the definition of mutual information:\n\\begin{align}\n\tI(X;F)=\\mathbb{E}_{\\bm{x}\\sim p(\\bm{x})}\\mathbb{E}_{\\bm{f}\\sim p(\\bm{f}|\\bm{x})}\\log\\frac{ p(\\bm{f}|\\bm{x})}{p(\\bm{f})}\n\t\\label{feature_map}\n\\end{align}where $\\bm{x}$ and $\\bm{f}$ are the specific input images and the corresponding high-level feature maps. $p(\\bm{x})$ and $p(\\bm{f})$ are the prior distribution of $\\bm{x}$ and $\\bm{f}$ respectively, and $\\mathbb{E}$ represents the expectation.  $p(\\bm{f}|\\bm{x})$ is the posterior distribution of the high-level feature map conditioned on the input. We parameterize $p(\\bm{f}|\\bm{x})$ by the backbone due to its intractability, where evidence-lower-bound (ELBO) minimization is applied for relaxation. To estimate $I(X;F)$, we sample the training set to obtain the image $\\bm{x}$ and sample the distribution parameterized by the backbone to acquire the corresponding high-level feature map $\\bm{f}$.\n\n \t \\begin{figure}[t]\n\t\t\\centering\n\t\t\\includegraphics[height=6.5cm, width=8cm]{BiDet_priors.pdf}\n\t\\caption{The detected objects and the corresponding confidence score (a) before and (b) after optimizing (\\ref{alter_sparse}). The contrast of confidence score among different detected objects is significantly enlarged by minimizing alternate objective. As the NMS eliminates the positives with confidence score lower than the threshold, the sparse object priors are acquired and the posteriors are enforced to be concentrated on informative prediction. (best viewed in color).}\n\t\\vspace{-0.3cm}\n\t\\label{priors}\n\\end{figure}\n\n\nThe location and classification of objects based on the high-level feature map are independent, as the bounding box location and the classification probability are obtained via different network branches in the detection part. The mutual information in the second term of (\\ref{IB}) is factorized:\n\\begin{align}\n\tI(F;C,L)=I(F;C)+I(F;L)\n\\end{align}Similar to (\\ref{feature_map}), we rewrite the mutual information between the high-level feature maps and the classes as follows:\n\\begin{align}\n\tI(F;C)=\\mathbb{E}_{\\bm{f}\\sim p(\\bm{f}|\\bm{x})}\\mathbb{E}_{\\bm{c}\\sim p(\\bm{c}|\\bm{f})}\\log\\frac{p(\\bm{c}|\\bm{f})}{p(\\bm{c})}\n\t\\label{mi_class}\n\\end{align}where $\\bm{c}$ is the object class labels including the background class. $p(\\bm{c})$ and $p(\\bm{c}|\\bm{f})$ are the prior class distribution and posterior class distribution when given the feature maps respectively. Same as the calculation of (\\ref{feature_map}), we employ the classification branch networks in the detection part to parameterize the distribution. Meanwhile, we divide the images to blocks for multiple object detection. For one-stage detectors such as SSD \\cite{liu2016ssd}, we project the high-level feature map cells to the raw image to obtain the block partition. For two-stage detectors such as Faster R-CNN \\cite{ren2015faster}, we scale the ROI to the original image for block split. $\\bm{c}\\in \\mathbb{Z}^{1\\times b}$ represents the object class in $b$ blocks of the image. We define $c_{i}$ as the $i_{th}$ element of $\\bm{c}$, which demonstrates the class of the object whose center is in the $i_{th}$ block of the image. The class of a block is assigned to background if the block does not contain the center of any groundtruth objects.\n\nAs the localization contains shift parameters and scale parameters for anchors, we rewrite the mutual information between the object location and high-level feature maps:\n\n\\vspace{-0.3cm}\n\\begin{small}\n\\begin{align*}\n\tI(F;L)=\\mathbb{E}_{\\bm{f}\\sim p(\\bm{f}|\\bm{x})}\\mathbb{E}_{\\bm{l}_1\\sim p(\\bm{l}_1|\\bm{f})}\\mathbb{E}_{\\bm{l}_2\\sim p(\\bm{l}_2|\\bm{f})}\\log\\frac{p(\\bm{l}_1|\\bm{f})p(\\bm{l}_2|\\bm{f})}{p(\\bm{l}_1)p(\\bm{l}_2)}\n\\end{align*}\n\\end{small}where $\\bm{l}_1 \\in \\mathbb{R}^{2\\times b}$ represents the horizontal and vertical shift offset of the anchors in $b$ blocks of the image, and $\\bm{l}_2 \\in \\mathbb{R}^{2\\times b}$ means the height and width scale offset of the anchors. For the anchor whose center  $(x,y)$ is in the $j_{th}$ block with height $h$ and width $w$, the offset changes the bounding box in the following way: $(x,y)\\rightarrow (x,y)+\\bm{l}_{1,j}$ and $(h,w)\\rightarrow (h,w)\\cdot exp(\\bm{l}_{2,j})$, where $\\bm{l}_{1,j}$ and $\\bm{l}_{2,j}$ represent the $j_{th}$ column of $\\bm{l}_1$ and $\\bm{l}_2$. The priors and the posteriors of shift offset conditioned on the feature maps  are denoted as $p(\\bm{l}_1)$ and $p(\\bm{l}_1|\\bm{f})$ respectively. Similarly, the scaling offset has the prior and the posteriors given feature maps $p(\\bm{l}_2)$ and $p(\\bm{l}_2|\\bm{f})$.\nWe leverage the localization branch networks in the detection part for distribution parameterization.\n\n\\subsection{Learning Sparse Object Priors}\nSince the feature maps are binarized in BiDet, we utilize the binomial distribution with equal probability as the priors for each element of the high-level feature map $\\bm{f}$. We assign the priors for object localization in the following form: $p(\\bm{l}_{1,j})= N(\\bm{\\mu}_{1,j}^0,\\bm{\\Sigma}_{1,j}^0)$ and $p(\\bm{l}_{2,j})= N(\\bm{\\mu}_{2,j}^0,\\bm{\\Sigma}_{2,j}^0)$, where $N(\\bm{\\mu}, \\bm{\\Sigma})$ means the Gaussian distribution with mean $\\bm{\\mu}$ and covariance matrix $\\bm{\\Sigma}$. For one-stage detectors, the object localization priors $p(\\bm{l}_{1,j})$ and $p(\\bm{l}_{2,j})$ are hypothesized to be the two-dimensional standard normal distribution. For two-stage detectors, Region Proposal Networks (RPN) output the parameters of the Gaussian priors.\n\nAs numerous false positives emerge in the binary detection networks, learning sparse object priors for detection part enforces the posteriors to be concentrated on informative detection prediction with false positive elimination. The priors for object classification is defined as follows:\n\\begin{align*}\n\tp(c_i)=\\mathbb{I}_{M_i}\\cdot cat(\\frac{1}{n+1}\\cdot\\bm{1}^{n+1})+(1-\\mathbb{I}_{M_i})\\cdot cat([1,\\bm{0}^{n}])\n\\end{align*}where $\\mathbb{I}_x$ is the indicator function with $\\mathbb{I}_1=1$ and $\\mathbb{I}_0=0$,  and $M_i$ is the $i_{th}$ element of the block mask $\\bm{M}\\in\\{0,1\\}^{1\\times b}$. $cat(\\bm{K})$ means the categorical distribution with the parameter $\\bm{K}$. $\\bm{1}^{n}$ and $\\bm{0}^{n}$ are the all-one and zero vectors in $n$ dimensions respectively, where $n$ is the number of class. The multinomial distribution with equal probability is utilized for the class prior in the $i_{th}$ block if $M_i$ equals to one.  Otherwise, the categorical distribution with the probability $1$ for background and zero probability for other classes is leveraged for the prior class distribution. When $M_i$ equals to zero, the detection part definitely predicts the background for object classification in the $i_{th}$ block according to (\\ref{mi_class}). In order to obtain sparse priors for object classification with fewer predicted positives, we minimize the $L_1$ norm of the block mask $\\bm{M}$. We propose an alternative way to optimize $\\bm{M}$ due to the non-differentiability, where the objective is written as follows:\n\\begin{align}\n\t\\min\\limits_{s_i} -\\frac{1}{m}\\sum_{i=1}^{m}s_i \\log s_i\n\t\\label{alter_sparse}\n\\end{align}where $m=||\\bm{M}||_1$ represents the number of detected foreground objects in the image, and $s_i$ is the normalized confidence score for the $i_{th}$ predicted foreground object with $\\sum_{i=1}^{m}s_i=1$. As shown in Figure \\ref{priors}, minimizing (\\ref{alter_sparse}) increases the contrast of confidence score among different predicted objects, and predicted objects with low confidence score are assigned to be negative by the non-maximum suppression (NMS) algorithms. Therefore, the block mask becomes sparser with fewer predicted objects, and the posteriors are concentrated on informative prediction with uninformative false positive elimination.\n\n\n\n\\subsection{Efficient Binarized Object Detectors}\nIn this section, we first briefly introduce neural networks with binary weights and activations, and then detail the learning objectives of our BiDet.  Let $\\bm{W^l_r}$ be the real-valued weights and $\\bm{A^l_r}$ be the full-precision activations of the $l_{th}$ layer in a given L-layer detection model. During the forward propagation, the weights and activations are binarized via the sign function: $\\bm{W^l_b}=sign(\\bm{W^l_r})$ and $\\bm{A^l_b}=sign(\\bm{W^l_r}\\odot\\bm{A^l_b})$. $sign$ means the element-wise sign function which maps the number larger than zero to one and otherwise to minus one, and $\\odot$ indicates the element-wise binary product consisting of xnor and bitcount operations. Due to the non-differentiability of the sign function, straight-through estimator (STE) is employed to calculate the approximate gradients and update the real-valued weights in the back-propagation stage. The learning objectives for the proposed BiDet is written as follows:\n\\vspace{-0.5cm}\n\n\\begin{small}\n\\begin{align}\n&\\min J = J_1+J_2\\notag\\\\\n&=(\\sum_{t,s}\\log \\frac{ p(f_{st}|\\bm{x})}{p(f_{st})}-\\beta\\sum_{i=1}^{b}\\log\\frac{p(c_i|\\bm{f})p(\\bm{l}_{1,i}|\\bm{f})p(\\bm{l}_{2,i}|\\bm{f})}{p(c_i)p(\\bm{l}_{1,i})p(\\bm{l}_{2,i})})\\notag\\\\\n& \\quad-\\gamma\\cdot\\frac{1}{m}\\sum_{i=1}^{m}s_i \\log s_i\n\\label{objective}\n\\end{align}\n\\end{small}where $\\gamma$ is a hyperparameter that balances the importance of false positive elimination. The posterior distribution $p(c_i|\\bm{f})$ is hypothesized to be the categorical distribution $cat(\\bm{K}_i)$, where $\\bm{K}_i\\in \\mathbb{R}^{1\\times(n+1)}$ is the parameter and $n$ is the number of classes. We assume the posterior of the shift and scale offset follows the Gaussian distribution: $p(\\bm{l}_{1,j}|\\bm{f})= N(\\bm{\\mu}_{1,j},\\bm{\\Sigma}_{1,j})$ and $p(\\bm{l}_{2,j}|\\bm{f})= N(\\bm{\\mu}_{2,j},\\bm{\\Sigma}_{2,j})$. The posteriors of the element in the $s_{th}$ row and $t_{th}$ column of binary high-level feature maps $p(f_{st}|\\bm{x})$ is assigned to binomial distribution $cat([p_{ts},1-p_{ts}])$, where $p_{ts}$ is the probability for $f_{st}$ to be one. All the posterior distribution is parameterized by the neural networks. $J_1$ represents for the information bottleneck employed in object detection, which aims to remove information redundancy and fully utilize the representational power of the binary neural networks. The goal of $J_2$ is to enforce the object priors to be sparse so that the posteriors are encouraged to be concentrated on informative prediction with false positive elimination.\n\nIn the learning objective, $p(f_{st})$ in the binomial distribution is a constant. Meanwhile, the sparse object classification priors are imposed via $J_2$ so that $p(c_i)$ is also regarded as a constant. For one-stage detectors, constant $p(\\bm{l}_{1,i})$ and $p(\\bm{l}_{2,i})$ follows standard normal distribution. For two-stage detectors, $p(\\bm{l}_{1,i})$ and $p(\\bm{l}_{2,i})$ are parameterized by RPN, which is learned by the objective function. The last layer of the backbone that outputs the parameters of the binary high-level feature maps is real-valued in training for Monte-Carlo sampling and is binarzed with the sign function during inference. Meanwhile, the layers that output the parameters for object class and location distribution remain real-valued for accurate detection. During inference, we drop the network branch of covariance matrix for location offset, and assign all location prediction with the mean value to accelerate computation. Moreover, the prediction of object classes is set to that with the maximum probability to avoid time-consuming stochastic sampling in inference.\n\n\n\\section{Experiments}\nIn this section, we conducted comprehensive experiments to evaluate our proposed method on two datasets for object detection: PASCAL VOC \\cite{everingham2010pascal} and COCO \\cite{lin2014microsoft}. We first describe the implementation details of our BiDet, and then we validate the effectiveness of IB and sparse object priors for binarized object detectors by ablation study. Finally, we compare our method with state-of-the-art binary neural networks in the task of object detection to demonstrate superiority of the proposed BiDet.\n\n\\subsection{Datasets and Implementation Details}\nWe first introduce the datasets that we carried out experiments on and data preprocessing techniques:\n\n\\textbf{PASCAL VOC: }The PASCAL VOC dataset contains natural images from 20 different classes. We trained our model on the VOC 2007 and VOC 2012 trainval sets which consist of around 16k images, and we evaluated our method on VOC 2007 test set including about 5k images. Following \\cite{everingham2010pascal}, we used the mean average precision (mAP) as the evaluation criterion.\n\n\\textbf{COCO: }The COCO dataset consists of images from 80 different categories. We conducted experiments on the 2014 COCO object detection track. We trained our model with the combination of 80k images from the training set and 35k images sampled from validation set (trainval35k \\cite{bell2016inside}) and tested our method on the remaining 5k images in the validation set (minival \\cite{bell2016inside}). Following the standard COCO evaluation metric \\cite{lin2014microsoft},  we report the average precision (AP) for IoU $\\in  \\left[0.5 : 0.05 : 0.95\\right]$ denoted as mAP@$\\left[.5, .95\\right]$. We also report AP$_{50}$, AP$_{75}$ as well as AP$_{s}$, AP$_{m}$ and AP$_{l}$ to further analyze our method.\n\nWe trained our BiDet with the SSD300 \\cite{liu2016ssd} and Faster R-CNN \\cite{ren2015faster} detection framework whose backbone were VGG16 \\cite{simonyan2014very} and ResNet-18 \\cite{he2016deep} respectively. Following the implementation of binary neural networks in \\cite{hubara2016binarized}, we remained the first and last layer in the detection networks real-valued. We used the data augmentation techniques in \\cite{liu2016ssd} and \\cite{ren2015faster} when training our BiDet with SSD300 and Faster R-CNN detection frameworks respectively.\n\nIn most cases, the backbone network was pre-trained on ImageNet \\cite{russakovsky2015imagenet} in the task of image classification. Then we jointly finetuned the backbone part and trained the detection part for the object detection task. The batchsize was assigned to be $32$, and the Adam optimizer \\cite{kingma2014adam} was applied. The learning rate started from 0.001 and decayed twice by multiplying $0.1$ at the $6_{th}$ and $10_{th}$ epoch out of $12$ epochs. Hyperparamters $\\beta$ and $\\gamma$ were set as $10$ and $0.2$ respectively.\n\n \t \\begin{figure}[t]\n\t\t\\centering\n\t\t\\includegraphics[height=7.5cm, width=8.5cm]{BiDet_ablation.pdf}\n\t\\caption{Ablation study w.r.t. hyperparameters $\\beta$ and $\\gamma$, where the variety of (a) mAP, (b) the mutual information between high-level feature maps and the object detection $I(F;L,C)$ , (c) the number of false positives and (d) the number of false negatives are demonstrated. (best viewed in color).}\n\t\\vspace{-0.3cm}\n\t\\label{ablation}\n\\end{figure}\n\n\n\n\\begin{table*}[t]\n\t\\footnotesize\n\t\\caption{Comparison of parameter size, FLOPs and mAP (\\%) with the state-of-the-art binary neural networks in both one-stage and two-stage detection frameworks on PASCAL VOC. The detector with the real-valued and multi-bit backbone is given for reference. BiDet (SC) means the proposed method with extra shortcut for the architectures. }\n\t\\label{MAPVOC}\n\t\\centering\n\t\\vspace{0.1cm}\n\\renewcommand\\arraystretch{1.2}\n\\begin{tabular}{|c|c|c|c|c|c|c|c|}\n\t\\hline\n\tFramework & Input & Backbone & Quantization & W\/A (bit) & \\#Params & MFLOPs & mAP\\\\\n\t\\hline\n\t\\multirow{10}{*}{SSD300} & \\multirow{10}{*}{$300\\times300$} & VGG16 & \\multirow{2}{*}{$-$} & \\multirow{2}{*}{$32\/32$} & $100.28$MB & $31,750$ & $72.4$\\\\\n\n\t& & MobileNetV1 & & & $30.07$MB & $1,150$ & $68.0$\\\\\n\t\\cline{3-8}\n\t& & \\multirow{7}{*}{VGG16} & TWN & $2\/32$ & $24.54$MB & $8,531$ & $67.8$\\\\\n\n\t& & & DoReFa-Net & $4\/4$ & $29.58$MB & $4,661$ & $69.2$\\\\\n\t\\cline{4-8}\n\t& & & BNN & \\multirow{3}{*}{$1\/1$} & $22.06$MB & $1,275$ & $42.0$\\\\\n\n\t& & & Xnor-Net & & $22.16$MB & $1,279$ & $50.2$\\\\\n\t& & & BiDet & & $22.06$MB & $1,275$ & $\\bm{52.4}$\\\\\n\t\\cline{4-8}\n\n\t& & & Bi-Real-Net & \\multirow{2}{*}{$1\/1$}  & $21.88$MB & $1,277$ & $63.8$\\\\\n\n\t\n\n\t& & & BiDet (SC) & & $21.88$MB & $1,277$ & $\\bm{66.0}$\\\\\n\t\\cline{3-8}\n\t& & \\multirow{2}{*}{MobileNetV1} & Xnor-Net & \\multirow{2}{*}{$1\/1$}  & $22.48$MB & $836$ & $48.9$\\\\\n\t& &  & BiDet & & $22.48$MB & $836$ & $\\bm{51.2}$\\\\\n\t\\hline\n\t\\multirow{8}{*}{Faster R-CNN} & \\multirow{8}{*}{$600\\times1000$} & \\multirow{8}{*}{ResNet-18} & $-$ & $32\/32$ & $47.35$MB & $36,013$ & $74.5$\\\\\n\t\\cline{4-8}\n\t& & & TWN & $2\/32$ & $3.83$MB & $9,196$ & $69.9$\\\\\n\n\t& & & DoReFa-Net & $4\/4$ & $6.73$MB & $4,694$ & $71.0$\\\\\n\t\\cline{4-8}\n\t& & & BNN & \\multirow{3}{*}{$1\/1$} & $2.38$MB & $779$ & $35.6$\\\\\n\n\t& & & Xnor-Net & & $2.48$MB & $783$ & $48.4$\\\\\n\t& & & BiDet  & & $2.38$MB & $779$ & $\\bm{50.0}$\\\\\n\t\\cline{4-8}\n\n\t& & & Bi-Real-Net & \\multirow{2}{*}{$1\/1$} & $2.39$MB & $781$ & $58.2$\\\\\n\n\n\t& & & BiDet (SC) & & $2.39$MB & $781$ & $\\bm{59.5}$\\\\\n\t\\hline\n\\end{tabular}\n\t\\vspace{-0.2cm}\n\\end{table*}\n\n\\begin{table*}[t]\n\t\\footnotesize\n\t\\caption{Comparison of mAP@$\\left[.5, .95\\right]$ (\\%), AP with different IOU threshold and AP for objects in various sizes with state-of-the-art binarized object detectors in SSD300 and Faster R-CNN detection framework on COCO, where the performance of real-valued and multi-bit detectors is reported for reference. BiDet (SC) means the proposed method with extra shortcut for the architectures.}\n\t\\label{MAPCOCO}\n\t\\centering\n\t\\vspace{0.1cm}\n\t\\renewcommand\\arraystretch{1.2}\n\\begin{tabular}{|c|c|c|c|c|cc|ccc|}\n\t\\hline\n\tFramework & Input & Backbone & Quantization & mAP@$\\left[.5, .95\\right]$ & AP$_{50}$ & AP$_{75}$ (\\%) & AP$_{s}$ & AP$_{m}$ & AP$_{l}$\\\\\n\t\\hline\n\t\\multirow{8}{*}{SSD300} & \\multirow{8}{*}{$300\\times300$} & \\multirow{8}{*}{VGG16} & $-$ & $23.2$ & $41.2$ & $23.4$ & $5.3$ & $23.2$ & $39.6$\\\\\n\t\\cline{4-10}\n\t& &  & TWN & $16.9$ & $33.0$ & $15.8$ & $5.0$ & $16.9$ & $27.2$\\\\\n\n\t& & & DoReFa-Net & $19.5$ & $35.0$ & $19.6$ & $5.1$ & $20.5$ & $32.8$\\\\\n\t\\cline{4-10}\n\t& &  & BNN & $6.2$ & $15.9$ & $3.8$ & $2.4$ & $10.0$ & $9.9$\\\\\n\n\t& & & Xnor-Net & $8.1$ & $19.5$ & $5.6$ & $2.6$ & $8.3$ & $13.3$\\\\\n\t& & & BiDet  & $\\bm{9.8}$ & $\\bm{22.5}$ & $\\bm{7.2}$ & $\\bm{3.1}$ & $\\bm{10.8}$ & $\\bm{16.1}$\\\\\n\t\\cline{4-10}\n\n\t& & & Bi-Real-Net & $11.2$ & $26.0$ & $8.3$ & $3.1$ & $12.0$ & $18.3$\\\\\n\n\n\t& & & BiDet (SC)& $\\bm{13.2}$ & $\\bm{28.3}$ & $\\bm{10.5}$ & $\\bm{5.1}$ & $\\bm{14.3}$ & $\\bm{20.5}$\\\\\n\t\\hline\n\t\\multirow{8}{*}{Faster R-CNN} & \\multirow{8}{*}{$600\\times1000$} & \\multirow{8}{*}{ResNet-18} & $-$ & $26.0$ & $44.8$ & $27.2$ & $10.0$ & $28.9$ & $39.7$\\\\\n\t\\cline{4-10}\n\t& & & TWN & $19.7$ & $35.3$ & $19.7$ & $5.1$ & $20.7$ & $33.3$\\\\\n\n\t& & & DoReFa-Net & $22.9$ & $38.6$ & $23.7$ & $8.0$ & $24.9$ & $36.3$\\\\\n\t\\cline{4-10}\n\t& & & BNN & $5.6$ & $14.3$ & $2.6$ & $2.0$ & $8.5$ & $9.3$\\\\\n\n\t& & & Xnor-Net & $10.4$ & $21.6$ & $8.8$ & $2.7$ & $11.8$ & $15.9$\\\\\n\t& & & BiDet & $\\bm{12.1}$ & $\\bm{24.8}$ & $\\bm{10.1}$ & $\\bm{4.1}$ & $\\bm{13.5}$ & $\\bm{17.7}$\\\\\n\t\\cline{4-10}\n\n\t& & & Bi-Real-Net & $14.4$ & $29.0$ & $13.4$ & $3.7$ & $15.4$ & $24.1$\\\\\n\n\t\n\n\t& & & BiDet (SC) & $\\bm{15.7}$ & $\\bm{31.0}$ & $\\bm{14.4}$ & $\\bm{4.9}$ & $\\bm{16.7}$ & $\\bm{25.4}$\\\\\n\t\\hline\n\\end{tabular}\n\t\\vspace{-0.2cm}\n\\end{table*}\n\n\n\n\\subsection{Ablation Study}\nSince the IB principle removes the redundant information in binarized object detectors and the learned sparse object priors concentrate the posteriors on informative prediction with false positive alleviation, the detection accuracy is enhanced significantly. To verify the effectiveness of the IB principle and the learned sparse priors, we conducted the ablation study to evaluate our BiDet w.r.t. the hyperparameter $\\beta$ and $\\gamma$ in the objective function. We adopted the SSD detection framework with VGG16 backbone for our BiDet on the PASCAL VOC dataset. We report the mAP, the mutual information between high-level feature maps and the object detection $I(F;L,C)$, the number of false positives and the number of false negatives with respect to $\\beta$ and $\\gamma$ in Figure \\ref{ablation} (a), (b), (c) and (d) respectively. Based on the results, we observe the influence of the IB principle and the learned sparse object priors as follows.\n\nBy observing Figure \\ref{ablation} (a) and (b), we conclude that mAP and $I(F;L,C)$ are positively correlated as they demonstrate the detection performance and the amount of related information respectively. Medium $\\beta$ provides the optimal trade-off between the amount of extracted information and the related information so that the representational capacity of the binary object detectors is fully utilized with redundancy removal. Small $\\beta$ fails to leverage the representational power of the networks as the amount of extracted information is limited by regularizing the high-level feature maps, while large $\\beta$ enforces the networks to learn redundant information which leads to significant over-fitting. Meanwhile, medium $\\gamma$ offers optimal sparse object priors that enforces the posteriors to concentrate on most informative prediction. Small $\\gamma$ is not capable of sparsifying the predicted objects, and large $\\gamma$ disables the posteriors to represent informative objects with excessive sparsity.\n\nBy comparing the variety of false positives and false negatives w.r.t. $\\beta$ and $\\gamma$, we know that medium $\\beta$ decreases false positives most significantly and changing $\\beta$ does not varies the number of false negatives notably, which means that the redundancy removal only alleviates the uninformative false positives while remains the informative true positives unchanged. Meanwhile, small $\\gamma$ fails to constrain the false positives and large $\\gamma$ clearly increases the false negatives, which both degrades the performance significantly.\n\n\n\n\\begin{figure*}[t]\n\t\\centering\n\t\\includegraphics[height=5cm, width=17.5cm]{BiDet_visualization.pdf}\n\t\\caption{Qualitative results on PASCAL VOC. Images in the top row shows the object predicted by Xnor-Net, while the images with the objects detected by our BiDet are displayed in the bottom row. The proposed BiDet removes the information redundancy to fully utilize the network capacity, and learns the sparse object priors to eliminate false positives (best viewed in color).}\n\t\\label{visualization}\n\\end{figure*}\n\n\\subsection{Comparison with the State-of-the-art Methods}\nIn this section, we compare the proposed BiDet with the state-of-the-art binary neural networks including BNN \\cite{courbariaux2015binaryconnect}, Xnor-Net \\cite{rastegari2016xnor} and Bi-Real-Net \\cite{liu2018bi} in the task of object detection on the PASCAL VOC and COCO datasets. For reference, we report the detection performance of the multi-bit quantized networks containing DoReFa-Net \\cite{zhou2016dorefa} and TWN \\cite{li2016ternary} and the lightweight networks MobileNetV1 \\cite{howard2017mobilenets}.\n\n\\textbf{Results on PASCAL VOC: }Table \\ref{MAPVOC} illustrates the comparison of computation complexity, storage cost and the mAP across different quantization methods and detection frameworks.  Our BiDet significantly accelerates the computation and saves the storage by $24.90\\times$ and $4.55\\times$ with the SSD300 detector and $46.23\\times$ and $19.81\\times$ with the Faster R-CNN detector. The efficiency is enhanced more notably in the Faster R-CNN detector, as there are multiple real-valued output layers of the head networks in SSD300 for multi-scale feature extraction.\n\nCompared with the state-of-the-art binary neural networks, the proposed BiDet improves the mAP of Xnor-Net by $2.2\\%$ and $1.6\\%$ with SSD300 and Faster R-CNN frameworks respectively with fewer FLOPs and the number of parameters than Xnor-Net. As demonstrated in \\cite{liu2018bi}, adding extra shortcut between consecutive convolutional layers can further enhance the representational power of the binary neural networks, we also employ architecture with additional skip connection to evaluate our BiDet in networks with stronger capacity. Due to the information redundancy, the performance of Bi-Real-Net with constrained network capacity is degraded significantly compared with their full-precision counterparts in both one-stage and two-stage detection frameworks. On the contrary, our BiDet imposes the IB principle on learning binary neural networks for object detection and fully utilizes the network capacity with redundancy removal. As a result, the proposed BiDet increases the mAP of Bi-Real-Net by $2.2\\%$ and $1.3\\%$ in SSD300 and Faster R-CNN detectors respectively without additional computational and storage cost. Figure \\ref{visualization} shows the qualitative results of Xnor-Net and our BiDet in the SSD300 detection framework with VGG16, where the proposed BiDet significantly alleviates the false positives.\n\nDue to the different pipelines in one-stage and two-stage detectors, the mAP gained from the proposed BiDet with Faster R-CNN is less than SSD300. As analyzed in \\cite{lin2017focal}, one-stage detectors face the severe positive-negative class imbalance problem which two-stage detectors are free of, so that one-stage detectors are usually more vulnerable to false positives. Therefore, one-stage object detection framework obtains more benefits from the proposed BiDet, which learns the sparse object priors to concentrate the posteriors on informative prediction with false positive elimination.\n\nMoreover, our BiDet can be integrated with other efficient networks in object detection for further computation speedup and storage saving. We employ our BiDet as a plug-and-play module in SSD detector with the MobileNetV1 network and saves the computational and storage cost by $1.47\\times$ and $1.38\\times$ respectively. Compared with the detectors that directly binarize weights and activations in MobileNetV1 with Xnor-Net, BiDet improves the mAP by a sizable margin, which depicts the effectiveness of redundancy removal for networks with extremely low capacity.\n\n\n\n\\textbf{Results on COCO: }The COCO dataset is much more challenging for object detection than PASCAL VOC due to the high diversity and large scale. Table \\ref{MAPCOCO} demonstrates mAP, AP with different IOU threshold and AP of objects in various sizes. Compared with the state-of-the-art binary neural networks Xnor-Net, our BiDet improves the mAP by $1.7\\%$ and $1.7\\%$ in SSD300 and Faster R-CNN detection framework respectively due to the information redundancy removal. Moreover, the proposed BiDet also enhances the binary one-stage and two-stage detectors with extra shortcut by $2.0\\%$ and $1.3\\%$ on mAP. Comparing with the baseline methods of network quantization, our method achieves better performance in the AP with different IOU threshold and AP for objects in different sizes, which demonstrates the universality in different application settings.\n\n\\iffalse\nIn short, our BiDet method removes the redundant information to fully utilize the representational power of binary neural networks and learns the sparse object priors to concentrate posteriors on informative prediction with false positive elimination, so that the performance on object detection is significantly enhanced.\n\\fi\n\n\n\\section{Conclusion}\nIn this paper, we have proposed a binarized neural network learning method called BiDet for efficient object detection. The presented BiDet removes the redundant information via information bottleneck principle to fully utilize the representational capacity of the networks and enforces the posteriors to be concentrated on informative prediction for false positive elimination, through which the detection precision is significantly enhanced. Extensive experiments depict the superiority of BiDet in object detection compared with the state-of-the-art binary neural networks.\n\n\\section*{Acknowledgement}\nThis work was supported in part by the National Key Research and Development Program of China under Grant 2017YFA0700802, in part by the National Natural Science Foundation of China under Grant 61822603, Grant U1813218, Grant U1713214, and Grant 61672306, in part by the Shenzhen Fundamental Research Fund (Subject Arrangement) under Grant JCYJ20170412170602564, and in part by Tsinghua University Initiative Scientific Research Program.\n\n{\\small\n\\bibliographystyle{ieee_fullname}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}