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+{"text":"\\section{Introduction} \\label{intro}\nLattice regularization of chiral gauge theories has remained a long standing\nproblem of nonperturbative investigation of quantum field theory. Lack of\nchiral gauge invariance in $L\\chi GT$ proposals is responsible for\nthe longitudinal gauge degrees of freedom ({\\em dof}) coupling to\nfermionic {\\em dof} and eventually spoiling the chiral nature of the theory.\nThe well-known example is the Smit-Swift proposal of $L\\chi GT$ \\cite{smit}.\nAlthough in a recent development using a Dirac operator that\nsatisfies the Ginsparg-Wilson relation, it was possible to formulate a\n$L\\chi GT$ without violating gauge-invariance or locality \\cite{luscher}, an\nexplicit model for nonperturbative numerical studies is still not \navailable.\n \nIn this paper, we follow the gauge fixing approach to $L\\chi GT$\n\\cite{golter1}. The obvious remedy to control the longitudinal gauge {\\em dof}\nis to gauge fix with a target theory in mind. The Roma proposal \\cite{roma}\ninvolving gauge fixing passed perturbative tests but does not address the\nproblem of gauge fixing of compact gauge fields and the associated problem of\nlattice artifact Gribov copies. The formal problem is that\nfor compact\ngauge fixing a BRST-invariant partition function as well as (unnormalized)\nexpectation values of BRST invariant operators vanish as a consequence of\nlattice Gribov copies \\cite{neuberger}. Shamir and Golterman \\cite{golter1}\nhave proposed to\nkeep the gauge fixing part of the action BRST noninvariant and tune\ncounterterms to recover BRST in the continuum. In their formalism, the\ncontinuum limit is to be taken from within the broken ferromagnetic (FM) phase\napproaching another broken phase which is called ferromagnetic directional\n(FMD) phase, with the mass of the gauge field vanishing at the FM-FMD\ntransition. This was tried out in a $U(1)$ Smit-Swift model and so\nfar the results show that in the pure gauge sector, QED is recovered in\nthe continuum limit \\cite{recent} and in the {\\em reduced} model limit \n(to be defined below) free chiral fermions in the appropriate chiral \nrepresentation are obtained \\cite{bock1}. Tuning with counterterms has \nalso not posed any practical problem, actually very little tuning is \nnecessary. Efforts are currently underway to extend this gauge fixing \nproposal to include nonabelian gauge groups \\cite{nab}. \n      \nWithout gauge fixing the longitudinal gauge {\\em dof}, which are \nradially frozen scalar fields, are rough and nonperturbative even if the \ntransverse gauge coupling may be weak (this is because with the standard \nlattice measure, each point on the gauge orbit has equal weight). \nThe theory in the continuum limit, taken at the transition \nbetween the broken symmetry ferromagnetic (FM) phase and the \nsymmetric paramagnetic (PM) phase, displays undesired nonperturbative \neffects of the scalar-fermion coupling that usually spells disaster for \nthe chiral theory. The job of the gauge fixing is to introduce a new \ncontinuous phase transition, from the FM phase to a new broken symmetry \nphase (FMD), at which the gauge symmetry is recovered and at the same \ntime the gauge fields become smooth.\n  \nThe problem can be cleanly studied in the reduced model as explained in the\nfollowing. When one gauge transforms a gauge non-invariant theory, one picks\nup the longitudinal gauge degrees of freedom (radially frozen scalars)\nexplicitly in the action. The reduced model is then obtained   \nby making the lattice gauge field unity for \nall links, {\\em i.e.}, by switching off the transverse gauge coupling. \nThe action becomes that of a chiral Yukawa theory with interaction \nbetween the fermions and the longitudinal gauge {\\em dof}. \nThe reduced model would have a phase structure similar to the full \ntheory, {\\em e.g.}, the gauge fixed theory in the reduced limit \nwill have a FM-FMD transition in addition to the FM-PM transition. \nNow for the gauge fixing proposal to work, the scalars need \nto decouple from the fermions at the FM-FMD transition leaving the \nfermions free in the appropriate chiral representation. Passing the \nreduced model test is an important first step for any $L\\chi GT$ \nproposal that breaks gauge invariance.  \n\nIn the reduced model derived from the gauge fixed theory the \nscalar fields become smooth and expandable in a perturbative \nseries as $1+{\\cal O}(\\mbox{coupling constant})$ at the FM-FMD \ntransition. If continuum limit can be taken near the point in the \ncoupling parameter space around which this perturbative expansion is \ndefined, the scalar fields will decouple from the theory. \nThe parameterization of the gauge fixing action turns out to be a good \none, because this continuum limit can be taken ($i$) very easily by \napproaching the FM-FMD transition almost perpendicularly by tuning \nessentially one counterterm, and ($ii$) at a point on this transition \nline which is reasonably far away from the expansion point. This has \nbeen possible in \\cite{bock1} and again in the present work.  \n\nA central claim of the gauge fixing proposal \nis that it is universal, {\\em i.e.}, it should work with any lattice \nfermion action that has the correct classical continuum limit. This is \nbecause the central idea as discussed above is independent of the \nparticular lattice fermion regularization. In the \npresent paper we want to confirm the universality claim by applying the \nproposal to domain wall fermions \\cite{kaplan} with $U(1)$ gauge group. \nFor this purpose we have chosen the waveguide formulation \\cite{kaplan2} \nof the domain wall fermion and investigate in the reduced model. This \nmodel was investigated before without gauge fixing and the free domain \nwall spectrum was not obtained in the reduced limit \\cite{golter2}.  \nMirror chiral modes were found at the waveguide boundaries in addition \nto the chiral modes at the domain wall or anti-domain wall.\n \nIn section II we present the gauge-fixed domain wall fermion action for a\n$U(1)$ chiral gauge theory and then go to the so-called reduced\nmodel by switching off the transverse gauge coupling. In section III we\nperform a weak coupling perturbation theory in the reduced model for the\nfermion propagators and mass matrix to 1-loop. However, in sections\n\\ref{ferm_m} and \\ref{ferm_m1} we have used special boundary conditions\n(instead of the actual Kaplan boundary conditions) to arrive at explicit\nexpressions for the overlap of the opposite chiral modes. Our numerical \nresults for \nthe quenched phase diagram and chiral fermion propagators at the domain \nwall and anti-domain wall and at the waveguide boundaries are presented \nand compared with the perturbative results in section IV. We summarize \nin the concluding section V. In Appendix A, we describe the special \nboundary conditions used in sections \\ref{ferm_m} and \\ref{ferm_m1}.\nIn Appendix B we schematically discuss how using Kaplan boundary conditions\none can arrive at the same qualitative conclusion about the 1-loop\noverlap of the opposite chiral modes.\n \n\\section{Gauge-fixed Domain Wall Action} \\label{gfdwa}\n\nKaplan's free domain wall fermion action \\cite{kaplan} on a\n$4+1$-dimensional lattice is given by (lattice constant is taken to be\nunity throughout this paper),\n\\begin{equation}\nS_F = \\sum_{XY} \\overline{\\psi}_X \\left[ \\partial\\!\\!\\!\/_5 - w_5 +\n{\\bf M}\\right]_{XY} \\psi_Y  \\label{dwact}\n\\end{equation}\nwhere $\\overline{\\psi}$ and $\\psi$ are the fermion fields, and  \n$\\partial\\!\\!\\!\/_5$ and $w_5$ are respectively the 5-dimensional\nDirac operator and the Wilson term,\n\\begin{eqnarray}\n(\\partial\\!\\!\\!\/_5)_{XY} &=&\n\\frac{1}{2}\\sum_{\\alpha=1}^5 \\gamma_\\alpha \\left( \n\\delta_{X+\\hat{\\alpha},Y} - \\delta_{X-\\hat{\\alpha},Y} \\right),            \n\\nonumber \\\\ (w_5)_{XY} &=& \\frac{r}{2} \\sum_{\\alpha=1}^5\n\\left( \\delta_{X+\\hat{\\alpha},Y} + \\delta_{X-\\hat{\\alpha},Y} - 2\\delta_{XY}\n\\right),  \\label{par5}\n\\end{eqnarray}\nThe $\\gamma_\\alpha$'s are the five hermitian euclidean gamma matrices,\n$r$ is the Wilson parameter, $X=(x,s),~Y=(y,t)$ label the sites of\nthe $L^4 L_s$ lattice and $L_s$ is the extent of the 5th dimension:\n$0 \\leq s,t \\leq L_s-1$. We are interested in\ntaking the continuum limit in the 4 space-time dimensions only. It is\nconvenient to look at the 5-th dimension as a flavor space.\n\n\n\nWith periodic boundary conditions in the 5th or\n$s$-direction  ($s,t = L_s \\Rightarrow s,t=0$) and the domain wall mass\n${\\bf M}$ taken as\n\\begin{equation}\n{\\bf M}_{XY}=m(s)\\delta_{XY},\\;\\; {\\rm where},\n\\end{equation}\n\\begin{equation}\nm(s) =  \\begin{array}{rl}\n-m_0, &  0 <s< L_s\/2 \\\\\n 0,   &  s = 0, L_s\/2 \\\\\n m_0, &  L_s\/2 <s< L_s\n\\end{array} \\label{dwmass}\n\\end{equation}\nthe model possesses a lefthanded (LH) chiral mode bound to the domain wall\nat $s=0$ and a righthanded (RH) chiral mode bound to the anti-domain wall\nat $s=L_s\/2$. For $m_0 L_s\\gg 1$, these modes have exponentially\nsmall overlap. The chiral modes exist for momenta $p$\nbelow a critical momentum $p_c$, i.e. $\\vert \\hat{p} \\vert < p_c$, where\n$\\hat{p}^2 = 2\\sum_\\mu[1-\\cos(p_\\mu)]$ and $p_c^2 = 4 - 2m_0\/r$. Taking\nthe Wilson parameter $r=1$ the choice of $m_0$ is then restricted to\n$0<m_0<2$.\n\nA 4-dimensional gauge field which is same for all\n$s$-slices can be coupled to fermions only for a restricted number of\n$s$-slices around the anti-domain wall \\cite{golter2} with a view to\ncoupling only to the RH mode at the anti-domain wall. The gauge field\nis thus confined within a {\\em waveguide},\n\\begin{eqnarray}\nWG &=& (s: s_0 < s \\leq s_1) \\nonumber \\\\\n{\\rm with} \\hspace{0.5cm} s_0 &=& \\frac{L_s+2}{4}-1, \\hspace{0.7cm} s_1 =\n\\frac{3L_s+2}{4}-1.\n\\end{eqnarray}\nWith this choice, $(L_s-2)$ has to be a multiple of 4. For convenience,\nthe boundaries at ($s_0,s_0+1$) and ($s_1,s_1+1$) are denoted waveguide\nboundary $I$ and $II$ respectively.\n\nThe gauge transformations on the fermion fields are defined as follows:\n\\begin{eqnarray}\n\\psi_x^s \\rightarrow g_x\\psi_x^s, \\hspace{0.8cm} &\\overline{\\psi}_x^s &\n\\rightarrow \\overline{\\psi}_x^s g_x^\\dagger \\hspace{0.8cm} s \\in WG\n\\nonumber \\\\\n\\psi_x^s \\rightarrow \\psi_x^s, \\hspace{0.8cm} &\\overline{\\psi}_x^s&\n\\rightarrow \\overline{\\psi}_x^s \\hspace{1.15cm} s \\in\\!\\!\\!\\!\\!\/ \\;WG\n\\label{gsym}\n\\end{eqnarray}   \nwhere $g_x\\in G$, the gauge group. Other symmetries of the model remain \nthe same as in \\cite{golter2}.\n\nObviously, the hopping terms from $s_0$ to $s_0+1$ and that\nfrom $s_1$ to $s_1+1$ would break the local gauge invariance of the\naction. This is taken care of by gauge transforming the action and\nthereby picking up the pure gauge {\\em dof} or a radially frozen scalar\nfield $\\varphi$ (St\\\"{u}ckelberg field) at the waveguide boundary,\nleading to the gauge-invariant action (with $\\varphi_x\\rightarrow g_x\n\\varphi_x$ and (\\ref{gsym})) :\n\\begin{eqnarray}\nS_{F} & = & \\sum_{s \\in WG}\n\\overline{\\psi}^s \\left( D\\!\\!\\!\\!\/\\,(U) - W(U) + m(s) \\right) \\psi^s\n+ \\sum_{s\\not\\in WG} \\overline{\\psi}^s \\left( \\partial\\!\\!\\!\/\n- w + m(s) \\right) \\psi^s \\nonumber \\\\\n&+& \\;\\sum_s \\overline{\\psi}^s \\psi^s - \\sum_{s\\neq s_0,s_1}\n\\left( \\overline{\\psi}^s P_L \\psi^{s+1} + \\overline{\\psi}^{s+1} P_R\n\\psi^s \\right) \\nonumber \\\\\n&-& \\;y \\left( \\overline{\\psi}^{s_0} \\varphi^\\dagger P_L \\psi^{s_0+1} +\n\\overline{\\psi}^{s_0+1} \\varphi P_R \\psi^{s_0} \\right)\n- y \\left( \\overline{\\psi}^{s_1} \\varphi P_L\n\\psi^{s_1+1} + \\overline{\\psi}^{s_1+1} \\varphi^\\dagger P_R \\psi^{s_1}\n\\right) \\label{wgact}\n\\end{eqnarray}\nwhere we have taken the Wilson parameter $r=1$\nand have suppressed all indices other than $s$. The projector $P_{L(R)}$\nis $(1\\mp \\gamma_5)\/2$ and $y$ is the Yukawa coupling introduced by hand at\nthe waveguide boundaries. $D\\!\\!\\!\\!\/\\;(U)$ and $W(U)$ are respectively the\ngauge covariant Dirac operator and the Wilson term in 4 space-time dimensions.\n$\\partial\\!\\!\\!\/$ and $w$ are the 4-dimensional versions of (\\ref{par5}).\n\nThe gauge-fixed pure gauge action for $U(1)$, where the ghosts are free\nand decoupled, is:\n\\begin{equation}\nS_B(U) = S_g(U) + S_{gf}(U) + S_{ct}(U) \\label{ggact}\n\\end{equation}\nwhere, $S_g$ is the usual Wilson plaquette action;\nthe gauge fixing term $S_{gf}$ (as proposed by Shamir and Golterman) and \nthe gauge field mass counter term $S_{ct}$ are given by (for a \ndiscussion of relevant counterterms see \\cite{golter1,bock2}),\n\\begin{eqnarray}\nS_{gf}(U) & = & \\tilde{\\kappa} \\left( \\sum_{xyz} \\Box(U)_{xy}\n\\Box (U)_{yz} - \\sum_x B_x^2 \\right), \\label{gfact} \\\\\nS_{ct}(U)  & = &  - \\kappa \\sum_{x\\mu} \\left( U_{\\mu x} +\nU_{\\mu x}^\\dagger \\right),\n\\end{eqnarray}\nwhere $\\Box(U)$ is the covariant lattice laplacian and\n\\begin{equation}\nB_x  = \\sum_\\mu \\left( \\frac{V_{\\mu x-\\hat{\\mu}} +\nV_{\\mu x}}{2} \\right)^2  \\label{bxsq}\n\\end{equation}\nwith\n$V_{\\mu x} = \\frac{1}{2i} \\left( U_{\\mu x} - U_{\\mu x}^\\dagger \\right)$\nand $\\tilde{\\kappa} = 1\/(2\\xi g^2)$.\n\n$S_{gf}$ is not just a naive lattice transcription of the continuum \ncovariant gauge fixing term, it has in addition appropriate irrelevant \nterms. As a result, $S_{gf}$ has a unique absolute minimum at $U_{\\mu \nx}=1$, validating weak coupling perturbation theory (WCPT) around $g=0$ \nor $\\tilde{\\kappa}=\\infty$ and in the naive continuum limit it reduces \nto $\\frac{1}{2\\xi} \\int d^4x (\\partial_\\mu A_\\mu)^2$. \n\nObviously, the action $S_B(U)$ is not gauge invariant. By giving it a gauge\ntransformation the resulting action $S_B(\\varphi^\\dagger_x U_{\\mu x}\n\\varphi_{x+\\hat{\\mu}})$ is gauge-invariant with $U_{\\mu x}\\rightarrow g_x U_{\\mu\nx} g^\\dagger_{x+\\hat{\\mu}}$ and $\\varphi_x \\rightarrow g_x \\varphi_x$, $g_x \\in\nU(1)$. By restricting to the trivial orbit, we arrive at the so-called\n{\\bf reduced model} action\n\\begin{equation}\nS_{reduced} = S_F(U=1) + S_B(\\varphi^\\dagger_x \\;1 \\;\\;\\varphi_{x+\\hat{\\mu}})\n\\label{reduced}\n\\end{equation}\nwhere $S_F(U=1)$ is obtained quite easily from eq.(\\ref{wgact}) and\n\\begin{equation}\nS_B(\\varphi^\\dagger_x \\;1 \\;\\;\\varphi_{x+\\hat{\\mu}})\n= -\\kappa \\sum_x \\varphi^\\dagger_x(\\Box \\varphi)_x + \\tilde{\\kappa}\n\\sum_x \\left[\\varphi^\\dagger_x(\\Box^2 \\varphi)_x - B^2_x \\right] \\label{redB}\n\\end{equation}\nnow is a higher-derivative scalar field theory action. $B_x$ in (\\ref{redB})\nis same as in (\\ref{bxsq}) with\n\\begin{equation}\nV_{\\mu x} = \\frac{1}{2i}\\left(\\varphi^\\dagger_x\\varphi_{x+\\hat{\\mu}} -\n\\varphi^\\dagger_{x+\\hat{\\mu}} \\varphi_x\\right).\n\\end{equation}\n\nIn the following, we investigate the action (\\ref{reduced}) at $y=1$ by\nanalytical and numerical methods. Some numerical results with other values of\n$y$ have been presented in \\cite{npbps}. The waveguide model strictly\nat $y=0$ would give rise to opposite\nchiral modes at the waveguide boundaries as can be seen from fermion current\nconsiderations \\cite{golter2} (and also from numerical simulation) \nand would thereby spoil the chiral nature of the theory. It is an \ninteresting question to investigate the model for $0<y<1$. Analysis of \nthe results for small values of $y (<1)$ is tricky and will be discussed \nin a separate article \\cite{big}.\n\n\\section{Weak Coupling Perturbation Theory in the Reduced Model} \\label{wcpt}\nAt $y=1$, we\ncarry out a WCPT in the coupling $1\/\\tilde{\\kappa}$ for the fermion\npropagators to 1-loop. In order to develop perturbation theory, in reduced\nmodel, we expand,\n\\begin{equation}\n\\varphi_x = \\exp(ib\\theta_x) = 1 +ib \\theta_x  -\\frac{1}{2} b^2 \\theta_x^2\n+ {\\cal O}(b^3)\n\\end{equation}\nwhere,\n$b=1\/\\sqrt{2\\tilde{\\kappa}} $  and $\\theta_x$ is dimensionless, \nleading to\n\\begin{equation}\nS = S_F^{(0)}(\\psi,\\overline{\\psi};y) + S_B^{(0)}(\\theta) + S^{({\\rm\nint})}(\\psi,\\overline{\\psi},\\theta;y)  \\label{Sint}\n\\end{equation}\nwhere $S^{(0)}$'s are free actions and $S^{({\\rm int})}$ is the interaction\npart.\n\n\\subsection{Scalar propagator at tree level} \\label{scal_p}\nFrom $S_B^{(0)}(\\theta)$ one gets the free propagator for the compact scalar\n$\\theta$ \\cite{bock2},\n\\begin{equation}\n{\\cal G}(k) = \\frac{1}{\\hat{k^2}(\\hat{k^2} + \\omega^2)},\n\\hspace{1.0cm} \\omega^2=\\frac{\\kappa}{\\tilde{\\kappa}}\n\\end{equation}\nwhere, $\\hat{k}_\\mu = 2\\sin(k_\\mu\/2)$.\n\n\\subsection{$LL$ and $RR$ fermion propagators at tree level and at 1-loop}\n\\label{ferm_p}\n\\subsubsection{Tree level}\nWith $y=1$, $S_F^{(0)}(\\psi,\\overline{\\psi};y=1)$ is the free domain wall\naction (\\ref{dwact}).\nFree fermion propagators at $y=1$\nare obtained in momentum space for 4-spacetime dimensions while staying\nin the coordinate space for the 5th dimension following \\cite{aoki1}\n(results in \\cite{aoki1,aoki2,blum} cannot be directly used because of\ndifference in implementation of the domain wall (\\ref{dwmass})). The\nfree action is written as,\n\\begin{equation}\nS_F^{(0)}(y=1) = \\sum_{p,s,t} \\overline{\\widetilde{\\psi_p^s}} \\left[\ni\\overline{p\\!\\!\\!\/} \\delta_{s,t} + M_{st}P_L + M^\\dagger_{st} P_R\n\\right] \\widetilde{\\psi_p^t}\n\\end{equation}\nwhere, $M_{st} = F(p)\\delta_{s,t} + (M_0)_{st}$, $(M_0)_{st} =\n[1+m(s)]\\delta_{s,t} - \\delta_{s+1,t}$,\n$F(p) = \\sum_\\mu (1 - \\cos(p_\\mu))$,\n$\\overline{p}_\\mu = \\sin(p_\\mu)$ and $\\overline{p\\!\\!\\!\/} =\n\\gamma_\\mu\\overline{p}_\\mu$.\nThe free fermion propagator can formally be written as,\n\\begin{eqnarray}\n\\Delta(p) &=& \\left[ i\\overline{p\\!\\!\\!\/} + M P_L +\n                                  M^\\dagger P_R \\right]^{-1} \\nonumber \\\\\n &=& ( -i\\overline{p\\!\\!\\!\/} + M^\\dagger ) P_L G_L(p) +\n   ( -i\\overline{p\\!\\!\\!\/} + M ) P_R G_R(p)  \\label{fprop}\n\\end{eqnarray}\nwhere,\n\\begin{eqnarray}\nG_L(p) & = & \\frac{1}{\\sum_\\mu \\overline{p}_\\mu^2 + M M^\\dagger}\n\\label{gl} \\\\\nG_R(p) & = & \\frac{1}{\\sum_\\mu \\overline{p}_\\mu^2 +\nM^\\dagger M}. \\label{gr}\n\\end{eqnarray}\nSolution of $G_L$ is obtained by writing (\\ref{gl}) explicitly:\n\\begin{equation}\n\\left[ \\overline{p}^2+1+B(s)^2 \\right] (G_L)_{s,t} - B(s+1)\n(G_L)_{s+1,t} - B(s) (G_L)_{s-1,t} = \\delta_{s,t} \\label{green}\n\\end{equation}\nand similarly for $G_R$. In (\\ref{green}), $B(s)= F(p)+1+m(s)$.\nWe show only the calculations for obtaining $G_L$ and henceforth drop\nthe subscript $L$.\n\nSetting the notation as follows:\n\\begin{eqnarray}\nG &=& G^-, \\;\\;\\;\\;B(s)=F(p)+1-m_0=a_- \\hspace{0.8cm} {\\rm for}\\;\\;\\;\\;\n0 <s \\leq L_s\/2-1 \\label{nomen1}\\\\\n{\\rm and} \\hspace{0.4cm} G &=& G^+, \\;\\;\\;\\;B(s)=F(p)+1+m_0=a_+\n\\hspace{0.8cm} {\\rm for}\\;\\;\\;\\;L_s\/2 <s\\leq L_s-1, \\label{nomen2}\n\\end{eqnarray}\nthe equations for $G^\\pm$ are given by,\n\\begin{eqnarray}\n\\left( \\overline{p}^2+1+a^2_{-} \\right) G^{-}_{s,t} - a_{-} G^{-}_{s+1,t}\n- a_{-} G^{-}_{s-1,t} &=& \\delta_{s,t}, \\label{lmslice} \\\\\n\\left( \\overline{p}^2+1+a^2_{+} \\right) G^{+}_{s,t} - a_{+} G^{+}_{s+1,t}\n- a_{+} G^{+}_{s-1,t} &=& \\delta_{s,t}. \\label{lpslice}\n\\end{eqnarray}\n\nThe ranges of $s$ in eqs.(\\ref{nomen1}, \\ref{nomen2}) for which $G^-$\nand $G^+$ are defined, are applicable\nonly to the translationally invariant eqs.(\\ref{lmslice},\n\\ref{lpslice}). In general for the translationally noninvariant\neq.(\\ref{green}) we also define $G^-$ and $G^+$ at $s=0$ and $L_s\/2$,\nthe ones excluded by (\\ref{lmslice}, \\ref{lpslice}). The $\\pm$\nsuperscript to $G$ at $s=0,\\;L_s\/2$ is decided by the translationally\ninvariant $s$-sector from which $s=0$ or $L_s\/2$ is approached in\neq.(\\ref{green}). We have used this notation for the boundary conditions\neq.(\\ref{dwbc}) below.\n\nThe solutions of the eqs.(\\ref{lmslice}, \\ref{lpslice}) are expressed\nas sum of homogeneous and inhomogeneous solutions:\n\\begin{equation}\nG^{\\pm}_{s,t}(p) = g_{\\pm}^{(1)}(t)e^{-\\alpha_{\\pm}(p)s} +\n                   g_{\\pm}^{(2)}(t)e^{\\alpha_{\\pm}(p)s}\n+ \\frac{\\cosh[\\alpha_{\\pm}(p)(\\vert s-t \\vert - l\/2)]}\n{2a_{\\pm}\\sinh(\\alpha_{\\pm}(p))\\sinh(\\alpha_{\\pm}(p)l\/2)}, \\label{lmsol}\n\\end{equation}\nwhere, $l=L_s\/2$ and\n\\begin{equation}\n\\cosh(\\alpha_{\\pm}(p))=\\frac{1}{2}\\left(\na_{\\pm}+\\frac{1+\\overline{p}^2} {a_{\\pm}} \\right).\n\\end{equation}\nThe third term in (\\ref{lmsol})\nis the inhomogeneous solution. To avoid singularities in\n$\\alpha_{\\pm}(p)$ when $a_{\\pm}$ is zero further restricts the allowed range\nof $m_0$ to $0<m_0<1$. In this paper we have taken $m_0=0.5$.\n\nIn order to get the complete\nsolution we need to determine the unknown functions $g_{\\pm}^{(1)}(t)$\nand $g_{\\pm}^{(2)}(t)$ in (\\ref{lmsol}), which are obtained by\nconsidering boundary conditions from eqs.(\\ref{lmslice}, \\ref{lpslice})\nat $s=0,\\;1,\\;L_s\/2-1,\\;L_s\/2,\\;L_s\/2+1,\\;L_s-1$,\n\\begin{eqnarray}\na_0 G^{-}_{0,t}(p) &=& a_{+} G^{+}_{L_s,t}(p), \\nonumber \\\\\na_0 G^{+}_{L_s\/2,t}(p) &=& a_{-} G^{-}_{L_s\/2,t}(p), \\nonumber \\\\\n(\\overline{p}^2+1+a_0^2) G^{-}_{0,t}(p) - a_{-}G^{-}_{1,t}(p)\n&=& \\delta_{0,t} + a_0 G^{+}_{L_s-1,t}(p), \\nonumber \\\\\n(\\overline{p}^2+1+a_0^2) G^{+}_{L_s\/2,t}(p) - a_{+}G^{+}_{L_s\/2+1,t}\n&=& \\delta_{L_s\/2,t} + a_0 G^{-}_{L_s\/2-1,t}. \\label{dwbc}\n\\end{eqnarray}\nwith $B(s)=F(p)+1=a_0$ at $s=0,L_s\/2$. It is to be\nnoted that these boundary conditions are significantly different from\nthe ones given in \\cite{aoki1} because of the difference in implementation\nof the domain wall. $G^-_{0,t}$ and $G^+_{L_s\/2,t}$ (and the corresponding\nones from $G_R$) are used to determine\nthe free chiral propagators at the domain wall and anti-domain wall for\ncomparison with numerical data in Fig.\\ref{waw}. We will see later that\nthese chiral propagators do not receive any 1-loop self-energy corrections.\n\nSubstituting\n$\\overline{p}^2+1+a_0^2=F_0$ and $2a_{\\pm}\\sinh(\\alpha_{\\pm}(p))\n\\sinh(\\alpha_{\\pm}(p)l\/2)=X_{\\pm}$ and using the boundary conditions,\neqs.(\\ref{dwbc}), we arrive at an equation of the form,\n\\begin{equation}\n{\\bf A}\\cdot{\\bf g}(t) = {\\bf X}(t), \\label{mateq}\n\\end{equation}\nwhere, ${\\bf g}(t) = (g_{-}^{(1)}\\;\\;\\;g_{-}^{(2)}\\;\\;\\;g_{+}^{(1)}\\;\\;\\;\ng_{+}^{(2)})$ is a 4-component vector, ${\\bf X}(t)$ is another 4-component\nvector and ${\\bf A}$ is a $4\\times 4$ matrix as given below,\n\n\\[ {\\bf A} = \\left( \\begin{array}{cccc}\na_0 & a_0 & -a_{+}e^{-\\alpha_{+}L_s} & -a_{+}e^{\\alpha_{+}L_s} \\\\\na_{-}e^{-\\alpha_{-}L_s\/2} & a_{-}e^{\\alpha_{-}L_s\/2} & -a_0e^{-\\alpha_{+}\nL_s\/2} & -a_0e^{\\alpha_{+}L_s\/2} \\\\\nF_0-a_{-}e^{-\\alpha_{-}} & F_0-a_{-}e^{\\alpha_{-}} & -a_0e^{-\\alpha_{+}\n(L_s-1)} & -a_0e^{\\alpha_{+}(L_s-1)} \\\\\n-a_0e^{-\\alpha_{-}(L_s\/2-1)} & -a_0e^{\\alpha_{-}(L_s\/2-1)} &\n(F_0-a_{+}e^{-\\alpha_{+}})e^{-\\alpha_{+}L_s\/2} &\n(F_0-a_{+}e^{\\alpha_{+}})e^{\\alpha_{+}L_s\/2}\n\\end{array} \\right) \\]\nand\n\\[ {\\bf X}(t) = \\left( \\begin{array}{ll}\na_{+}X_{+}\\cosh[\\alpha_{+}(\\vert L_s-t \\vert -l\/2)] & \\hspace{-0.9cm} -\\,\na_0X_{-}\\cosh[\\alpha_{-}(\\vert -t \\vert -l\/2)] \\\\\n& \\\\\na_0X_{+}\\cosh[\\alpha_{+}(\\vert L_s\/2-t \\vert -l\/2)] & \\hspace{-0.7cm} -\\,\na_{-}X_{-}\\cosh[\\alpha_{-}(\\vert L_s\/2-t \\vert -l\/2)] \\\\\n& \\\\\n\\delta_{0,t} + a_0X_{+}\\cosh[\\alpha_{+}(\\vert L_s-1-t \\vert\\, - &\n\\hspace{-0.3cm} l\/2)] - F_0X_{-}\\cosh[\\alpha_{-}(\\vert -t \\vert -l\/2)] + \\\\\n& \\hspace{3.5cm} a_{-}X_{-}\\cosh[\\alpha_{-}(\\vert 1-t \\vert -l\/2)] \\\\\n& \\\\\n\\delta_{L_s\/2,t} + a_{+}X_{+}\\cosh[\\alpha_{+}(\\vert L_s\/2+1-&\\!\\!t \\vert\n-l\/2)] - F_0X_{+}\\cosh[\\alpha_{+}(\\vert L_s\/2-t \\vert -l\/2)] + \\\\\n& \\hspace{2.4cm} a_0X_{-}\\cosh[\\alpha_{-}(\\vert L_s\/2-1-t \\vert -l\/2)]\n\\end{array} \\right). \\]\n\nThe explicit expressions for ${\\bf A}$ and ${\\bf X}(t)$ are obviously\ndifferent from similar expressions given in \\cite{aoki1,aoki2,blum} because\nof the differences in domain wall implementation as already discussed\nearlier.\n\nThe solution to the eqs.(\\ref{mateq}) is very complicated in general,\nparticularly for finite $L_s$. However, $g_{\\pm}(t)$ can be obtained for\nfinite $L_s$ by solving the above equations numerically for different $t$\nvalues. This way we can easily construct the free fermion propagators at any\ngiven $s$-slice, including the zero mode propagators at $s=t=0,L_s\/2$.\n\nThe solutions for $(G_R)_{s,t}$ and the resulting propagators are \nobtained in exactly the same way. However, in this case the explicit \nforms for (\\ref{green}), (\\ref{lmsol}), (\\ref{dwbc}) and matrices ${\\bf \nA}$ and ${\\bf X}(t)$ in (\\ref{mateq}) are obviously different.\n\n\\subsubsection{1-loop} \\label{ferm_p1}\nNext we calculate the chiral fermion propagators to 1-loop. {\\em Half-circle}\ndiagrams which are diagonal in flavor space contributes to $LL$ and $RR$\npropagator self-energies. However, the\nself-energies are nonzero only at the waveguide boundaries $I$ and $II$. \n\n\nRetaining up to ${\\cal O}(b^2)$ in the interaction term\n$S^{({\\rm int})}(\\psi,\\overline{\\psi},\\theta;y=1)$ in (\\ref{Sint}),\nwe find the vertices necessary to calculate the self-energies to\n1-loop.\n\n\\vspace{-0.8cm}\n\\begin{center}\n\\begin{figure}\n\\begin{picture}(300,85)(0,65)\n\\ArrowLine(75,85)(125,85)\n\\ArrowLine(125,85)(175,85)\n\\ArrowLine(175,85)(225,85)\n\\DashArrowArcn(150,85)(35,180,0){3}\n\\Text(80,92)[]{\\footnotesize{$L$}}\n\\Text(92,76)[]{$p$, $s_0+1$}\n\\Text(125,92)[]{\\footnotesize{$R$}}\n\\Text(175,92)[]{\\footnotesize{$R$}}\n\\Text(150,76)[]{$p-k$, $s_0$}\n\\Text(220,92)[]{\\footnotesize{$L$}}\n\\Text(208,76)[]{$p$, $s_0+1$}\n\\Text(150,110)[]{$k$}\n\\end{picture}\n\\caption{1-loop self-energy contribution to $LL$ propagator at $WG$ boundary\n$I$.} \\label{hcirc}\n\\end{figure}\n\\end{center}\n\nThe $LL$ propagator on the $(s_0+1)$-th slice at the waveguide boundary\n$I$ receives a nonzero  self-energy contribution from the\nhalf-circle diagram,\n\\begin{eqnarray}\n-\\left(\\Sigma_{LL}^I(p)\\right)_{st} &= &\\int_{BZ} \\frac{d^4k}{(2\\pi)^4}\nb^2 \\left[-i\\gamma_\\mu (\\overline{p-k})_\\mu P_L\nG_L(p-k)\\right]_{s_0,s_0} {\\cal G}(k) \\;\\delta_{s,s_0+1}\\delta_{t,s_0+1}\n\\label{fselfl} \\\\\n& \\rightarrow & \\frac{b^2}{L^4} \\sum_k \\left[{\\cal\nS}^{(0)}_{RR}(p-k)\\right]_{s_0,s_0}\\;\\; \\frac{1}{\\hat{k}^2 \n(\\hat{k}^2+\\omega^2)} \\;\\delta_{s,s_0+1}\\delta_{t,s_0+1} \\label{fselfd}\n\\end{eqnarray}\nwhere  the expression in the square bracket in\n(\\ref{fselfl}) is the free $RR$ propagator \n$[{\\cal S}^{(0)}_{RR}]_{s_0,s_0}$ on the $s_0$-slice. \nEq.(\\ref{fselfl}) assumes \ninfinite 4 space-time volume while in eq.(\\ref{fselfd}) a finite \nspace-time volume $L^4$ is considered. \n\nUsing (\\ref{fselfd}) we numerically evaluate the analytic \n1-loop propagator on a given finite lattice, using\n\\begin{equation}\n{\\cal S}_{LL} = {\\cal S}^{(0)}_{LL} + {\\cal S}^{(0)}_{LL}  \n\\left[-\\Sigma^{I,II}_{LL}\\right] {\\cal S}^{(0)}_{LL},\n\\end{equation}\nand in the section $IV$ \ncompare with nonperturbative numerical results. To avoid the infra-red \nproblem in the scalar propagator, we use anti-periodic boundary \ncondition in one of the space-time directions in evaluating \n(\\ref{fselfd}). \n\nIn a similar\nway, 1-loop corrected $RR$ or $LL$ propagators are obtained at all the\n$s$-slices of the waveguide boundaries $I$ and $II$, {\\em i.e.}, at the\nslices $s_0$, $s_0+1$, $s_1$ and $s_1+1$.\n\n\\subsection{Fermion mass matrix at tree level and 1-loop} \\label{ferm_m}\nAnother issue of interest is the spread of the\nwavefunctions of the two chiral zero mode solutions along the discrete\n$s$-direction and their possible overlap. A finite overlap would\nmean an induced Dirac mass. The extra dimension, as already pointed out\nin the discussion following (\\ref{par5}), can be interpreted as a flavor\nspace with one LH chiral fermion, one RH chiral fermion and $(L_s\/2-1)$\nheavy fermions on each sector of $1\\leq s\\leq L_s\/2-1$ and $ L_s\/2+1\\leq\ns\\leq L_s-1$. \n\\subsubsection{Tree level}\nFor the spread of the zero modes at the tree level, \none needs only to solve\n$M_0 u_L =0$ and $M_0^\\dagger u_R=0$ \\cite{kaplan,aoki0} where $M_0=\nM(p=0)$ (keeping the momentum $p$ non-zero is unnecessary in this\ndiscussion). However, for radiative correction on the domain wall mass\n$m(s)$, the heavy mode spreads are also needed. Accordingly we consider\nflavor diagonalization of $M^\\dagger_0 M_0$ ($M_0$ is not hermitian) as\nin \\cite{aoki2,blum}:\n\\begin{eqnarray}\n\\left( M^\\dagger_0 M_0 \\right)_{st} \\left( \\phi^{(0)}_j \\right)_t &=& \\left[\n\\lambda_j^{(0)} \\right]^2 \\left( \\phi^{(0)}_j \\right)_s \\label{eveq1}\\\\\n\\left( M_0 M^\\dagger_0 \\right)_{st} \\left( \\Phi^{(0)}_j \\right)_t &=& \\left[\n\\lambda_j^{(0)} \\right]^2 \\left( \\Phi^{(0)}_j \\right)_s. \\label{eveq2}\n\\end{eqnarray}\n\nThe index $j$ for the eigenvalues and the eigenvectors is basically a\nflavor index, but unlike $s$ and $t$ which vary from $0$ to $L_s-1$, it is\ntaken symmetric around the domain wall $j=0$. Index $j$ varies from\n$-L_s\/2$ to $L_s\/2$. From periodic boundary condition, $L_s\/2$ and \n$-L_s\/2$  are the same point in flavor space corresponding to the \nanti-domain wall. It is to be noted that $j$ appears \nexplicitly in the heavy mode solutions below. However, it may be pointed \nout that the index $j$ need not be chosen this way. One could also \ndefine $j$ in the same way as the flavor indices $s$ and $t$, only in \nthat case the explicit solutions below would have a different \nappearance and to our taste less tractable. \n\nTo solve the above tree level eigenequations (later also at 1-loop), we \nshould ideally use the Kaplan boundary conditions because we used in our\naction the Kaplan way of implementing the domain wall. However, that would\nmake an explicit calculation of the heavy eigenmodes quite complicated and\nalmost intractable. We have hence devised a special set of boundary\nconditions (look at Appendix A for details on the boundary conditions) \nwith the property that\nno information can be passed through the wall and the antiwall. This \nmakes the calculation less cumbersome and explicit expressions can be \nobtained in manageable forms. We stress that these are not \nthe actual boundary conditions in the particular domain wall \nimplementation we have taken. However, we show in Appendix B that using \nthe correct boundary conditions ({\\em i.e.}, the Kaplan boundary \nconditions) also would lead to the same qualitative \nconclusions about the nature of 1-loop corrections to the eigenvalues, \nstability of the zero modes, and particularly the 1-loop overlap of the \nopposite chiral zero modes. \n  \nWith our special boundary conditions (used only in sections \\ref{ferm_m}\nand \\ref{ferm_m1}), the explicit form of the \neigenfunctions are obviously different from \\cite{aoki2,blum} and are \nobtained as, \n\n\\vspace{0.3cm}\n\\leftline{Zero ($L$-handed) mode:}\n\\begin{eqnarray}\n(j=0) \\hspace{2.0cm}\n\\left( \\phi^{(0)}_j \\right)_s &=& {\\cal A}\\;\n\\exp(-\\tilde{\\alpha}s) \\hspace{1.9cm} 0\\leq s\\leq L_s\/2 \\\\\n(j=L_s\/2) \\hspace{2.7cm}\n&=& {\\cal A}\\; \\exp\\left[-\\tilde{\\alpha}(L_s-s)\\right] \n\\hspace{0.8cm} L_s\/2\\leq s\\leq L_s \\\\\n{\\cal A} &=& \\left(\n\\frac{1-\\exp(-2\\tilde{\\alpha})}{1-\\exp(-\\tilde{\\alpha}L_s)}\n\\right)^{1\/2} \\nonumber \\\\\n{\\rm with}, \\hspace{2.0cm} \\exp(-\\tilde{\\alpha}) &=& 1-m_0\n\\end{eqnarray}\n\n\\leftline{Heavy mode $(\\vert j \\vert \\neq 0,\\,L_s\/2)$:}\n\\begin{eqnarray}\n(0<j<L_s\/2) \\hspace{1.0cm} \\left( \\phi^{(0)}_j \\right)_s\n&=& \\sqrt{\\frac{4}{L_s}} \\,\\sin\\beta_j(s-1)\n\\hspace{2.1cm} 0\\leq s\\leq L_s\/2 \\\\\n(-L_s\/2<j<0) \\hspace{2.0cm} &=& \\sqrt{\\frac{4}{L_s}} \n\\,\\sin\\beta_{-j}(L_s\/2-s+1) \\hspace{0.7cm} L_s\/2\\leq s\\leq L_s .\n\\end{eqnarray}\nThe parameters involved are as follows:\n\\begin{eqnarray}\n\\left(\\lambda^{(0)}_j \\right)^2 &= & 1 + \\tilde{a}(s)^2 \n-2\\tilde{a}(s)\\cos \\beta_j, \\label{eigv} \\\\  \n\\tilde{a}(s) & = & 1 + m(s) \\label{apm} \n\\end{eqnarray}\nand for nontrivial heavy mode solutions, \n$\\beta_j = 2\\pi j\/L_s$ with $\\vert j \\vert \\ne 0,\\,L_s\/2$.\n\nSolutions to (\\ref{eveq2}) for all $j$ and $s$ easily follow:\n\\begin{equation}         \n\\left( \\Phi^{(0)}_j \\right)_s = \\left(\n\\phi^{(0)}_j \\right)_{L_s\/2-s}, \\label{modes}\n\\end{equation}\n\nIt is obvious that the\neigenvectors $\\phi^{(0)}_{j=0}$ and $\\Phi^{(0)}_{j=0}$ correspond \nrespectively to the\nLH and RH chiral zero modes at the domain wall and the anti-domain wall\nin the region $0\\leq s\\leq L_s\/2$. Similarly the\neigenvectors $\\phi^{(0)}_{j=L_s\/2}$ and $\\Phi^{(0)}_{j=L_s\/2}$ correspond to\nthe same chiral zero modes at the domain wall and the anti-domain wall \nin the region\n$L_s\/2\\leq s \\leq L_s$. The $\\vert j \\vert \\neq 0,\\,L_s\/2$ eigenvectors \nare for the heavy flavor modes. All solutions are real.\n\nThe overlap of the opposite chiral modes in the region $0\\leq s\\leq L_s\/2$\ndoes not depend explicitly on $s$ and is given at the tree level by,\n\\begin{equation}\n\\Phi^{(0)}_{j=0}\\; \\phi^{(0)}_{j=0} = {\\cal A}^2\\;\n\\exp(-\\tilde{\\alpha}L_s\/2) \n\\end{equation}\nand similarly in the region $L_s\/2\\leq s\\leq L_s$. The exponentially \ndamped mixing of the LH and RH chiral modes does not induce any Dirac \nmass at the tree level for large $L_s$.\n \nIt is also noted that $\\phi^{(0)}$ and \n$\\Phi^{(0)}$ diagonalize $M_0$ and $M_0^\\dagger$ in the following \nmanner: \n\\begin{equation}\n\\left( \\Phi^{(0)}_j\\right)_s \\left( M_0 \\right)_{st}\n\\left(\\phi^{(0)}_{j^\\prime} \\right)_t = \\lambda_j^{(0)} \\delta_{j,j^\\prime}\n= \\left( \\phi^{(0)}_j\\right)_s \\left(M_0^\\dagger\\right)_{st}\n\\left(\\Phi^{(0)}_{j^\\prime} \\right)_t   \\label{diag}\n\\end{equation}\nThe way the formalism is set up, the absolute sign of $\\lambda_j^{(0)}$ \nin eq.(\\ref{diag}) is arbitrary.\n\n\\subsubsection{1-loop}\nFlavor off-diagonal {\\em tadpole} diagrams\nproduce the self-energies for the $LR$ and $RL$ parts of the fermion\npropagator. Again, the\nself-energies are nonzero only at the waveguide boundaries $I$ and $II$.\n\n\\vspace{-0.7cm}\n\\begin{center}\n\\begin{figure}\n\\begin{picture}(300,85)(0,65)\n\\ArrowLine(90,85)(150,85)\n\\ArrowLine(150,85)(210,85)\n\\DashArrowArcn(150,105)(20,180,0){3}\n\\DashCArc(150,105)(20,180,360){3}\n\\Text(95,92)[]{\\footnotesize{$L$}}\n\\Text(112,76)[]{$p$, $s_0+1$}\n\\Text(205,92)[]{\\footnotesize{$R$}}\n\\Text(198,76)[]{$p$, $s_0$}\n\\Text(150,115)[]{$k$}\n\\end{picture}\n\\caption{Tadpole contribution to 1-loop $LR$ propagator at $WG$ \nboundary $I$.} \\label{tadp}\n\\end{figure}\n\\end{center}\n\nFor the $LR$ propagator connecting $s_0$ and $s_0+1$ at the waveguide\nboundary $I$, the self-energy contribution from the tadpole diagram is\ngiven by,\n\\begin{eqnarray}\n-\\left(\\Sigma_{LR}^I(p)\\right)_{st} &=& \\frac{1}{2}b^2P_L \\int_{BZ}\n\\frac{d^4k}{(2\\pi)^4} \\frac{1}{\\hat{k}^2 (\\hat{k}^2+\\omega^2)}\\;\n\\delta_{s,s_0}\\delta_{t,s_0+1} \\\\\n&=& \\frac{1}{2}b^2P_L \\,{\\cal T}\n\\;\\delta_{s,s_0}\\delta_{t,s_0+1}  \\label{tadpld}\n\\end{eqnarray}\nwhere ${\\cal T}\\sim 0.04$ is the tadpole loop integral.\n\nSimilarly the self-energy contribution to the $LR$ propagator at the\nwaveguide boundary $II$ connecting $s_1$ and $s_1+1$ comes from a\ntadpole diagram and is given by,\n\\begin{equation}\n-\\left(\\Sigma_{LR}^{II}(p)\\right)_{st}\n= \\frac{1}{2}b^2P_L \\,{\\cal T}\n\\;\\delta_{s,s_1}\\delta_{t,s_1+1}.  \\label{tadpld2}\n\\end{equation}\n\nThe mass parameter $M_0$ gets modified at 1-loop as:\n\\begin{eqnarray}\n(M_0)_{st}P_L \\rightarrow (\\widetilde{M}_0)_{st}P_L &=&\n(M_0)_{st}P_L+\\left[-(\\Sigma^I_{LR}(0))_{st} \\right] +\n\\left[-(\\Sigma^{II}_{LR}(0))_{st}\\right]   \\nonumber \\\\\n&\\equiv & (M_0)_{st}P_L + b^2 ({\\bf \\Sigma}_{LR})_{st} P_L\\,.\n\\end{eqnarray}\n$\\Sigma^{I,\\,II}_{LL,RR}(0) =0$ identically. $(M_0^\\dagger)_{st}P_R$\ngets modified accordingly. \n\nBecause of our use of the special boundary conditions (Appendix A) which do\nnot allow any communication through the walls, 1-loop correction to \nthe eigenvectors and eigenvalues involves either \n$s_0$ (waveguide boundary $I$) or $s_1$ (waveguide boundary $II$) \ndepending on which sector of $j$ is considered. In Appendix B we show that \nif the actual (Kaplan) boundary conditions are used, one gets contribution \nat 1-loop level also from another diagram, called there the {\\em global\nloop} diagram. However, the conclusions are qualitatively the same.  \n\n\\subsection{Mass matrix diagonalization at 1-loop} \\label{ferm_m1}\nAt 1-loop level we organize the corrections to the eigenvectors and the\neigenvalues of the fermion mass matrix squared as follows:\n\\begin{eqnarray}\n\\phi^{(0)} \\rightarrow \\phi &=& (1 + b^2 \\phi^{(1)})\\phi^{(0)}, \n\\nonumber \\\\\n\\Phi^{(0)} \\rightarrow \\Phi &=& (1 + b^2 \\Phi^{(1)})\\Phi^{(0)}, \n\\nonumber \\\\\n\\left(\\lambda^{(0)}\\right)^2 \\rightarrow \\lambda^2 & = &\n\\left(\\lambda^{(0)} + b^2 \\lambda^{(1)}\\right)^2.\n\\end{eqnarray}\n$\\lambda^{(1)}$  is found to be\n\\begin{equation}\n\\lambda^{(1)}_j = \\left(\\Phi^{(0)}_j\\right)_s ({\\bf\n\\Sigma}_{LR})_{st} \\left(\\phi^{(0)}_j\\right)_t\n\\end{equation}\nThe eq.(\\ref{diag}) gets modified in 1-loop as\n\\begin{equation}\n(\\Phi_j)_s (\\widetilde{M}_0)_{st} (\\phi_{j^\\prime})_t =\n\\lambda_j \\,\\delta_{j,j^\\prime} + {\\cal O}(b^4) =\n(\\phi_j)_s (\\widetilde{M}_0^\\dagger)_{st}\n(\\Phi_{j^\\prime})_t.\n\\end{equation}\n\nThe 1-loop correction to the eigenvalues, \n$(\\delta \\lambda)_j=b^2 \\lambda^{(1)}_j$,  is given by:\n\\begin{eqnarray}\n(\\delta \\lambda)_j &=& \\frac{2b^2}{L_s}\\,{\\cal T} \\sin\\beta_j(L_s\/2-s_0-1)\n\\sin\\beta_js_0 + {\\cal O}(b^4) \\hspace{2.0cm} {\\rm for}\\;\\;1\\leq j \\leq\n(L_s\/2-1),   \\label{delm1}  \\\\\n &=& \\frac{2b^2}{L_s}\\,{\\cal T} \\sin\\beta_j(s_1+1) \n\\sin\\beta_j(L_s\/2-s_1) + {\\cal O}(b^4) \\hspace{1.75cm} {\\rm \nfor}\\;\\;-(L_s\/2-1) \\leq j \\leq -1, \\label{delm2} \\\\  \n&=& \\frac{b^2}{2}\\;\n{\\cal A}^2\\,{\\cal T} \\exp[-\\tilde{\\alpha} (L_s\/2+1)]+ {\\cal O}(b^4) \n\\;\\;\\;\\;\\stackrel{L_s\\rightarrow \\infty} {\\longrightarrow} \\;0 \n\\hspace{1.2cm} {\\rm for}\\;\\;j=0,L_s\/2. \\label{delm3}\n\\end{eqnarray}\nWe notice in \neq.(\\ref{delm3}) that the 1-loop correction to zero mode eigenvalue is \nexponentially damped and the zero modes are hence perturbatively stable.\n \nThe 1-loop expression for the chiral zero mode at the domain wall, in \nthe region $0\\leq s\\leq L_s\/2$, is,\n\\begin{equation}\n(\\phi_{j=0})_s = {\\cal A} \\left[ \\exp(-\\tilde{\\alpha} s) -\n\\frac{2b^2}{L_s}{\\cal T} \\exp(-\\tilde{\\alpha} (s_0+1))\n\\sum_{j^\\prime=1}^{L_s\/2-1} \\frac{1}{\\lambda^{(0)}_{j^\\prime}}\n\\sin\\beta_{j^\\prime}(L_s\/2-s_0-1)\\sin\\beta_{j^\\prime}(s-1) \\right].\n\\end{equation}\nWe obtain a similar 1-loop expression for the chiral mode at the \nanti-domain wall. \n\nThe overlap of the opposite chiral modes in the region $0\\leq \ns\\leq L_s\/2$ at 1-loop is given by,\n\\begin{eqnarray}\n\\Phi_{j=0}\\;\\phi_{j=0} = {\\cal A}^2 \\exp(-\\tilde{\\alpha}L_s\/2)\n&-& \\left(\\frac{2b^2}{L_s}\\right){\\cal A}^2{\\cal T}\\,{\\cal F}_1 \\,\n\\exp[-\\tilde{\\alpha}(s_0+1)]\n\\nonumber\\\\\n&-& \\left(\\frac{2b^2}{L_s}\\right){\\cal A}^2{\\cal T}\\,{\\cal F}_2 \\,\n\\exp[-\\tilde{\\alpha}(L_s\/2-s_0)], \\label{mixing}\n\\end{eqnarray}\nwhere,\n\\begin{eqnarray*}\n{\\cal F}_1 &=& \\sum_{j^\\prime,s} \\frac{1}{\\lambda^{(0)}_{j^\\prime}}\n\\exp[-\\tilde{\\alpha}(L_s\/2-s)]\n\\sin\\beta_{j^\\prime}(s-1) \\sin\\beta_{j^\\prime}(L_s\/2-s_0-1) \\\\\n{\\cal F}_2 &=& \\sum_{j^\\prime,s} \\frac{1}{\\lambda^{(0)}_{j^\\prime}} \n\\exp(-\\tilde{\\alpha}s)\n\\sin\\beta_{j^\\prime}s_0 \\sin\\beta_{j^\\prime}(L_s\/2-s-1).\n\\end{eqnarray*}\nEq.(\\ref{mixing}) clearly shows that 1-loop corrections to the mixing of \nthe LH and RH modes are also exponentially damped. This guards against \nany induced Dirac mass in the domain wall model for large enough $L_s$ \nand the waveguide boundaries $I$ and $II$ chosen approximately \nequidistant from the domain wall and the anti-domain wall.   \nIn this context we point out that in the Smit-Swift model a shift \nsymmetry for the singlet fermion ensures that no fermion mass counter \nterm is needed.\n\nUsing the actual Kaplan boundary conditions would make the \nrelatively nice explicit expressions in eqs.(\\ref{delm1} -- \\ref{mixing})\nmuch less manageable, to say the least. Diagonalization of the fermion mass\nmatrix upto 1-loop  using the Kaplan boundary condition is discussed in\nAppendix B. \n   \n\\section{Numerical Results} \\label{numr}\nIn the quenched approximation, we have first numerically confirmed the\nphase diagram in \\cite{bock3} of the reduced model in\n($\\kappa,\\tilde{\\kappa}$) plane. The phase diagram shown schematically in\nFig.\\ref{phases} has the interesting feature that for large enough\n$\\tilde{\\kappa}$, there is a continuous phase transition between the\nbroken phases FM and FMD. FMD phase is\ncharacterized by loss of rotational invariance and the continuum limit is to\nbe taken from the FM side of the transition. In the full theory with \ngauge fields, the gauge symmetry reappears at this transition and the gauge\nboson mass vanishes, but the longitudinal gauge {\\em dof} remain decoupled.\nIn Fig.\\ref{phases} PM is the symmetric phase and AM is the broken\nanti-ferromagnetic\nphase. The numerical details involved in reconstruction of the phase \ndiagram and the fermionic measurements that follow will be \navailable in \\cite{big}. \n\n\\begin{center}\n\\begin{figure}\n\\hspace{0.1cm}\\psfig{figure=ph1.ps,width=6.5cm,height=5.0cm}\\\\\n\n\\caption{Schematic quenched phase diagram.} \\label{phases}\n\\end{figure}\n\\end{center}\n\\vspace{0.1cm}\n\nFor calculating the fermion propagators, as in \\cite{bock1} we have \nchosen the point $\\kappa =0.05$, $\\tilde{\\kappa}=0.2$ (gray blob in \nFig.\\ref{phases}). Although this point is far away from $\\tilde{\\kappa}=\\infty$, \naround which we did our perturbation theory in the previous section, the \nimportant issue here is to choose a point near the FM-FMD transition and \naway from the FM-PM transition. The results below show that for the \nfermion propagators there is \nexcellent agreement between numerical results obtained at $\\kappa =0.05$, \n$\\tilde{\\kappa}=0.2$ and perturbation theory. \n   \nNumerically  on $4^3 16$ and $6^3 16$ \nlattices with $L_s=22$ and $m_0=0.5$ we look for chiral modes at the \ndomain wall ($s=0$), the anti-domain wall ($s=11$), and at the waveguide \nboundaries ($s=5,6$ and $s=16,17$). Error bars in all the figures are \nsmaller than the symbols. \n\nFig.\\ref{waw} shows the $RR$ propagator $|S_{RR}|$ and the $LL$ propagator\n$|S_{LL}|$ at the domain and anti-domain wall as a function of a component of\nmomentum $p_4$ for both $\\vec{p}=(0,0,0)$ (physical mode) and $(0,0,\\pi)$\n(first doubler mode) at $y=1$. From the figures, it is clear that\nthe doubler does not exist, only the physical $RR$ ($LL$) propagator seems to\nhave a pole at $p=(0,0,0,0)$ at the anti-domain (domain) wall. In all the\nfigures, NS, PT and FF respectively indicate data from numerical simulation,\nfrom perturbation theory and from free fermion propagator by direct inversion\nof the free fermion matrix.\n\n\\begin{figure}\n\\vspace{-1.0cm}\n\\parbox{8cm}{\\psfig{figure=y1s0br1.ps,width=8.6cm,height=10.6cm}} \\ \\\n\\parbox{8cm}{\\psfig{figure=y1s11br1.ps,width=8.6cm,height=10.6cm}}\n\\vspace{-2.7cm}\n\\caption{Chiral propagators at domain wall $s=0$ and at anti-domain\nwall $s=11$ ($L_s=22$; a.p.b.c. in $L_4$, $y=1.0$).} \\label{waw}\n\\end{figure}\n\\vspace{0.5cm}\n\n\\begin{figure}\n\\vspace{-1.5cm}\n\\parbox{8cm}{\\psfig{figure=y1s5br1.ps,width=8.6cm,height=10.6cm}} \\ \\\n\\parbox{8cm}{\\psfig{figure=y1s6br1.ps,width=8.6cm,height=10.6cm}}\n\\vspace{-2.9cm}\n\\caption{$RR$ propagator at waveguide boundary $s=5$ and\n$LL$ propagator at waveguide boundary $s=6$ ($L_s=22$; a.p.b.c. in\n$L_4$, $y=1.0$).} \\label{wgb}\n\\end{figure}\n\nFor Fig.\\ref{waw}, PT also means zeroth order perturbation theory, {\\em i.e.},\nnumerical solution of propagator following eq.(\\ref{mateq}) (as noted\nbefore in subsubsection \\ref{ferm_p1} the self-energy contributions to\nthe $LL$ and $RR$ propagators are nonzero only at the waveguide boundaries,\nthe propagators in Fig.\\ref{waw} do not get any 1-loop correction).\nWe have PT\nresults also for $6^3\\times 16$ lattice but have chosen not to show them\nbecause they fall right on top of the numerical data. Instead PT results are\nshown for $8^3\\times 20$ lattice for which the $p_4$ points are distinct. The\ndotted line in all figures refer to the propagator from PT using a\n$256^3\\times 1024$ lattice. {\\em The curves stay the same irrespective of\nmethods or lattice size}. Based on the above, we can conclude that there are\n{\\em only free RH fermions} at the anti-domain wall, and at the domain wall\nthere are {\\em only free LH fermions}.\n\nFig.\\ref{wgb} show no evidence of a chiral mode at the waveguide boundaries\n$s=5$ and 6 and excellent agreement with 1-loop perturbation\ntheory. Here too doublers do not exist. Actually the agreement with the FF\nmethod (direct inversion of the free domain wall fermion matrix on a given\nfinite lattice) is also excellent, because the 1-loop corrections are almost\ninsignificant. For clarity, in Fig.\\ref{wgb} we have not shown the\n$LL$ propagator on $s=5$ and the $RR$ propagator at $s=6$, but conclusions are\nthe same.\n\nSimilar investigation at the other waveguide boundary $s=16, 17$ also does\nnot show any  chiral modes. Previous investigations of the domain wall \nwaveguide model without gauge fixing \\cite{golter2} have shown that the \nwaveguide boundaries are the most likely places to have the unwanted \nmirror modes. This is why we have mostly concentrated in showing that \nthere are no mirror chiral modes at these boundaries, although we have \nlooked for chiral modes everywhere along the flavor dimension. In fact, \nwe do not see any evidence of a chiral mode anywhere other than at the \ndomain wall and the anti-domain wall. \n\n\\section{Discussion} \\label{disc}\n\nWe have followed the gauge fixing proposal of Shamir and Golterman and \napplied it to domain wall fermions for a $U(1)$ $L\\chi GT$. By switching \noff the transverse gauge {\\em dof}, we arrive at the so-called  \nreduced model. We have determined the quenched phase diagram of the \nmodel and confirmed that there is a continuous phase transition from a \nbroken symmetry ferromagnetic (FM) phase to a broken symmetry \nrotationally noninvariant ferromagnetic directional (FMD) phase with the \nproperties that at this transition the longitudinal gauge {\\em dof}, \n{\\em i.e.}, the $\\varphi$ fields get decoupled. We have come to this \nconclusion by performing a WCPT for the fermion propagators and \ncomparing them to nonperturbative numerical simulations.\n\nLet us now contrast this with the previous attempt \\cite{golter2} of \nlattice regularizing a chiral gauge theory with domain wall fermions. No \ngauge fixing was done and in the reduced model, a combination of \nanalytic and numerical methods showed that mirror chiral modes were \ndynamically generated in the theory.\n\nAs in the Smit-Swift model \\cite{smit}, the mirror chiral modes are a \nreflection of the undesired presence of the longitudinal gauge {\\em \ndof} in the continuum limit. Without gauge fixing, these radially \nfrozen group-valued scalar $\\varphi$ fields are generally \nnonperturbative or rough for any value of the transverse gauge coupling, \neven in the reduced model limit. If the continuum limit is taken at the \nFM to a symmetric paramagnetic (PM) phase transition, the $\\varphi$ \nfields survive with radial modes and physical effects of them coupling \nwith the rest of the theory become manifest in the form of mirror chiral \nmodes etc..\n\nIn the gauge fixing approach of Shamir and Golterman, care has been \nexercised so that the gauge fixing term is not just a naive lattice \ntranscription of the continuum gauge fixing condition. There are \nappropriate additions of irrelevant terms in the covariant gauge fixing \nterm so that a unique perturbative vacuum exists. With this gauge \nfixing in the reduced model limit the $\\varphi$ fields \nbecome smooth and can be perturbatively expanded as $1 + {\\cal \nO}(1\/\\sqrt{\\tilde{\\kappa}}) + \\cdots$ around $\\tilde{\\kappa}=\\infty$. \nWe have found in our investigation that as long as $\\tilde{\\kappa}$ is \ntaken sufficiently large so that a continuum limit exists from the FM \nphase to the FMD phase (not the PM phase), the model is as good as at \nthe perturbative limit $\\tilde{\\kappa}=\\infty$. This seems to hold true \nin our particular implementation, {\\em i.e., the domain wall fermion \ncase}, in a {\\em strong sense}, because the perturbative 1-loop \ncorrections ({\\em i.e.}, the leading nontrivial corrections) to the \nfermion propagators are found to be negligible (Figs. 4 and 5). In gauge \nfixing the Smit-Swift model, however, 1-loop corrections to the fermion \npropagators were small but not negligible \\cite{bock1}. In the present \ninvestigation with domain wall fermions, to the accuracy of our \ncalculations, the reduced model spectrum is that of a free domain wall \nmodel with absolutely no trace of the $\\varphi$ fields.\n\nIn our investigation of the reduced model, only a counterterm with \ncoefficient $\\kappa$ was sufficient to reach the FM-FMD phase transition \nby decreasing $\\kappa$ from the FM-side. A dimension-three fermion mass \ncounterterm was not needed, because domain wall fermions have the robust \nproperty that mixing between the opposite chiral modes at the domain \nwall and the anti-domain wall is exponentially damped and is \nnegligible for large $L_s$. \n\nWe have carried out and presented the \nperturbative results for fermion propagators and mass matrix in \nreasonable detail, because, although the technique employed is not new, \nthe explicit results are available mostly for only the QCD (wall) \nimplementation rather than the wall-antiwall implementation suitable for \na chiral gauge theory. With transverse gauge fields back on (and \nfermions in an anomaly-free representation), the full gauge-fixed domain \nwall model of $L\\chi GT$ is ready for a perturbative treatment with the \nsame techniques as used in this paper if the transverse gauge coupling \nis perturbative. The model is also ready for a nonperturbative treatment \nby numerical simulation for strong transverse gauge coupling. \n\nOur numerical computations have been done in the quenched approximation \nand they agree for the fermion propagators very well with zeroth order \nperturbation theory with the 1-loop corrections almost negligible in \nour case. Effects of fermion loops would enter these calculations at \nleast at the 2-loop level.  Inclusion of dynamical fermions in our \nnonperturbative numerical investigation should not affect our results \nabout the spectrum of the model at all if the relevant part of the phase \ndiagram remains qualitatively the same.\n \nSo far the gauge fixing method is applicable only to the abelian theory. \nExtension to a nonabelian gauge group is nontrivial and  is \nbeing pursued at the current time \\cite{nab}.\n\nAs commented at the end of section II, the reduced gauge fixed \ndomain wall model  with $0<y<1$ is interesting because at $y=0$ the \nmodel is known to have mirror modes at the waveguide boundaries. This \nwill be taken up in a separate publication \\cite{big}. \n\n                                                                         \n\\acknowledgements \n \nThe authors thank Yigal Shamir, Maarten Golterman, Palash Pal and \nKrishnendu Mukherjee for useful discussions. One of the authors (SB) \nwould like to thank the Theory Group of Saha Institute of Nuclear \nPhysics for providing the facilities.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nDark energy has been modelled by a large variety of theories since decades. Among these, many rely on the introduction of additional scalar fields whose dynamics, at the origin of the late-time acceleration of the expansion of the Universe, is determined by arbitrary parametric functions, potentials and\/or coupling (see \\textit{e.g.} \\cite{Brax_2018}). These are the so-called scalar-tensor theories of modified gravity. In particular, the class of Horndeski theories is of great interest as it contains all models of modified gravity with a single additional scalar field leading to second-order equations of motion  \\cite{Horndeski:1974wa,Deffayet:2011gz}. Extensions of Horndeski theories to scalar-tensor theories of one scalar field with equations of motion of higher orders have also been explored \\cite{Langlois:2015skt,Langlois:2018dxi}. Particular Horndeski theories are described by the specification of four arbitrary functions of the scalar field and its kinetic energy, leading to a huge variety of models and phenomenological behaviours.\n\nAmong these wide classes of models, some can be built from first physical principles or arguments of symmetry. For instance, the Galileon model \\cite{Nicolis:2008in} and its covariant extension \\cite{Deffayet:2009wt} was built by imposing a galilean symmetry for the scalar field, leaving only five free numerical parameters. We can also cite, among many others, the pure kinetic gravity theory \\cite{GUBITOSI2011113}, massive gravity in the non-relativistic limit \\cite{deRham:2010ik,deRham:2010kj} and the DBI-Galileon \\cite{deRham:2010eu} which is the main object of this paper. \n\nThe DBI-Galileon is a model that falls into the class of Brane-world scenarios of extra-dimension theories, where the matter fields are confined on a 4D brane while gravity can propagate into the additional spatial dimensions. Of most interest for the DBI-Galileon is the case of a single extra-dimension as it has been shown that theories with more co-dimensions exhibit ghosts either in the flat or self-accelerating de Sitter solution \\cite{Hinterbichler:2010xn}. The action include a volume term for the 4D brane in the 5D bulk which leads to the well-known Dirac-Born-Infeld (DBI) action. This action, and DBI-like extensions, can lead to a self-accelerating solution and has been thoroughly studied as a candidate model in the early Universe cosmic inflation paradigm \\cite{Silverstein:2003hf,Langlois:2008qf}. In addition, the DBI-Galileon model exhibits the Galileon Lagrangians in the non-relativistic limit \\cite{deRham:2010eu} but giving a physical meaning to their free parameters: the Planck  mass in the brane, the Planck mass in the bulk, etc. In particular, the brane tension here plays the role of the cosmological constant which brings a possible interpretation of its nature.\n\nThe original probe brane construction has been revisited in \\cite{Zumalacarregui:2012us} where the matter metric is disformally related to a standard gravitational metric, or in \\cite{Goon:2012dy} in the framework of spontaneous symmetry breaking for the 5D space-time symmetries broken by the presence of the brane, bridging the gap with Brane-world scenarios developed in the context of quantum field theory and an interpretation of the scalar field as a Nambu-Goldstone boson \\cite{Dobado:2000gr,Cembranos:2001my}. The DBI-Galileon model has been studied extensively in special cases of the maximally symmetric bulk geometry \\cite{Burrage:2011bt,Goon:2011qf}. However, to our knowledge, no study of the DBI-Galileon in the late-time cosmology setting as a potential candidate for Dark Energy has been performed so far.\n\nIn this paper we develop the DBI-Galileon theory in the non relativistic limit (Section~\\ref{sec:dbigalileon}) and study its dynamics in the Friedmann-Lema\u00eetre-Robertson-Walker flat metric (Section~\\ref{sec:background}). The perturbation stability is explored in Section~\\ref{sec:perturbations} and then discussed in Section~\\ref{sec:discussion}.\n\n\\section{DBI-Galileon in the non-relativistic limit}\\label{sec:dbigalileon}\n\n\\subsubsection*{DBI-Galileon \\\\}\n\nWe are interested in the description of a four dimensional brane universe embedded in a five dimensional bulk from the cosmological perspective. In this context, it has been shown in \\cite{deRham:2010eu} that the most general action on the brane is given by the 4D Lovelock terms \\cite{Lovelock:1971yv} inside the brane and the boundary terms associated to the 5D Lovelock terms in the bulk:\n\\begin{equation}\n    \\mathcal{S} = \\int dx^{4} \\sqrt{-g} \\left( -\\Lambda - M_{5}^{3}K + \\frac{M_{P}^{2}}{2}R - \\beta\\frac{M_{5}^{3}}{m^{2}}\\mathcal{K}_{GB} + \\mathcal{L}_{m} \\left( \\tilde{q}_{\\mu\\nu}, \\psi_{m} \\right) \\right), \\label{eq:dRT action}\n\\end{equation}\nwhere $g$ is the induced metric on the brane, $\\Lambda$ the brane cosmological constant, $K$ is the extrinsic curvature of the brane, $R$ is the Ricci scalar on the brane, $\\mathcal{K}_{GB}$ is the boundary term on the brane of the Gauss-Bonnet scalar in the bulk and $\\mathcal{L}_{m}$ is the Lagrangian density of matter that lives confined in the brane. In the above action has been defined the 4D Planck mass $M_{P}$, its 5D counterpart $M_{5}$ and their ratio $m=M_{5}^{3}\/M_{P}^{2}$, and $\\beta$ is an arbitrary constant parameter.\n\nBecause the action is defined on the 4-dimensional brane, the above quantities are expressed in terms of the induced metric $g$, as opposed to the 5-dimensional bulk metric $G$. Assuming we have a coordinate system $\\left(x^{\\mu}, y \\right)$ in the bulk, Greek letters being defined to span from 0 to 3, such that the length element in this frame is $ds^{2} = q_{\\mu\\nu} dx^{\\mu}dx^{\\nu} + \\left( dy \\right)^{2}$. The brane position in the bulk is defined by $y = \\pi \\left( x^{\\mu} \\right)$ such that we can express the induced metric from the bulk metric:\n\\begin{equation}\n    g_{\\mu\\nu} = q_{\\mu\\nu} + \\partial_{\\mu} \\pi \\partial_{\\nu} \\pi \\qquad \\mathrm{and} \\qquad g^{\\mu\\nu} = q^{\\mu\\nu} - \\gamma^{2} \\partial^{\\mu} \\pi \\partial^{\\nu} \\pi\n\\end{equation}\nwith the Lorentz factor $\\gamma = \\left( 1+ \\left( \\partial \\pi \\right)^{2} \\right)^{-1\/2}$. From this expression of the induced metric, we can explicitly write the action \\eqref{eq:dRT action} using the bulk metric $q$ and the scalar field $\\pi$. As an illustration, see how the cosmological constant part of the action on the brane can be expressed:\n\\begin{equation}\n    \\mathcal{S}_{\\Lambda} = - \\Lambda \\int d^{4}x \\sqrt{-g} = - \\Lambda \\int d^{4}x \\sqrt{-q} \\sqrt{1 + \\left( \\partial\\pi \\right)^{2}}\n\\end{equation}\nAs was pointed out in the original paper by de Rham and Tolley \\cite{deRham:2010eu}, we recover a DBI term in the action. This leads us to associate the cosmological constant $\\Lambda$ with a brane tension $f$, following $\\Lambda = f^{4}$ for unit convenience.\n\n\\subsubsection*{Non-relativistic limit \\\\}\n\nIf the derivatives of the field vanish, $\\partial_{\\mu} \\pi = 0$, then we recover standard GR. Because we are interested in the cosmological setting, where predictions from the $\\Lambda$CDM model based on GR match precisely a large range of observations, we consider only small corrections to GR. Therefore, we consider the DBI-Galileon in the so-called non-relativistic limit where $\\left( \\partial \\pi \\right)^{2} \\ll 1$. In particular, the DBI part of the action in the non-relativistic limit becomes:\n\\begin{equation}\n    \\mathcal{S}_{f} = -f^{4} \\int d^{4}x \\sqrt{-q} \\left[ 1 + \\frac{\\nabla_{\\mu} \\pi \\nabla^{\\mu} \\pi}{2} - \\frac{\\left( \\nabla_{\\mu} \\pi \\nabla^{\\mu} \\pi \\right)^{2}}{8} + \\ldots \\right]\n\\end{equation}\nWe see that, to have a canonically normalized field, we can proceed to the field redefinition $\\pi \\rightarrow \\varphi= f^{2} \\pi $. Up to degree 5 in $\\partial \\varphi\/f^{2}$, we find the following Lagrangian operators for the DBI-Galileon in the non-relativistic limit:\n\\begin{eqnarray}\n    && \\mathcal{L}_{f} = 1 + \\frac{\\mathcal{L}_{2}}{2f^{4}} - \\frac{X^{2}}{2} + \\ldots \\\\\n    && \\mathcal{L}_{K} = - \\frac{\\mathcal{L}_{3}}{2f^{6}} - \\frac{2X}{f^{6}}\\left[ \\psi \\right] + \\ldots \\\\\n    && \\mathcal{L}_{R} = \\bar{R} + \\frac{1}{f^{4}} \\left( \\left[ \\Phi \\right]^{2} - \\left[ \\Phi^{2} \\right] - \\frac{f^{4}X}{2}\\bar{R} - 2\\bar{R}_{\\mu\\nu}\\nabla^{\\mu}\\varphi \\nabla^{\\nu}\\varphi \\right) + \\frac{\\mathcal{L}_{4}}{4f^{8}} + \\ldots \\\\\n    && \\mathcal{L}_{\\mathcal{K}_{GB}} = \\frac{2}{f^{6}} \\left( \\left( - \\bar{R}_{\\mu\\nu}\\left[ \\Phi \\right] + 2 \\bar{R}_{\\mu\\rho} \\Phi_{\\nu}^{\\rho} + \\bar{R}_{\\mu\\rho\\nu\\lambda}\\Phi^{\\rho\\lambda} \\right) \\nabla^{\\mu}\\varphi \\nabla^{\\nu}\\varphi + \\frac{1}{3}\\left[ \\Phi \\right]^{3} - \\left[ \\Phi \\right]\\left[ \\Phi^{2} \\right] + \\frac{2}{3}\\left[ \\Phi^{3} \\right] \\right. \\nonumber \\\\\n    && \\left.  - \\frac{1}{2}\\bar{R}\\left[ \\psi \\right] \\right) + \\frac{\\mathcal{L}_{5}}{3f^{10}} + \\ldots\n\\end{eqnarray}\nWe defined on the above expressions the tensor $\\Phi_{\\mu\\nu} = \\nabla_{\\mu}\\nabla_{\\nu} \\varphi$, and the three scalars $\\left[ \\Phi \\right] = \\Phi_{\\mu}^{\\mu}$, $\\left[ \\psi \\right] = \\partial^{\\mu} \\varphi \\cdot \\Phi_{\\mu\\nu} \\cdot \\partial^{\\nu} \\varphi$ and $X = -\\left( \\partial \\varphi \\right)^{2}\/2f^{4}$. Furthermore, the $\\mathcal{L}_{2 \\ldots 5}$ Lagrangian operators are the covariant Galileon model Lagrangian operators as defined in \\cite{Nicolis:2008in,Deffayet:2009wt}. Therefore, the theory described here is a generalization of the Galileon model. We can note that the additional terms in $\\mathcal{L}_{R}$ and $\\mathcal{L}_{\\mathcal{K}_{GB}}$ compared to the Galileon Lagrangian operators vanish in the particular case of a flat geometry. We conclude that the DBI-Galileon in the non-relativistic limit is a different coviariantization of the flat Galileon than the covariant Galileon \\cite{Deffayet:2009wt} or dRGT massive gravity \\cite{deRham:2010ik,deRham:2010kj}, with an additional terms in $\\mathcal{L}_{f}$ and another one in $\\mathcal{L}_{K}$. That point noted, we will stay at leading order in $X$ in the following. Being a theory of a scalar field interacting with a metric, with equations of motion of at most second order, it can be described as a Horndeski theory \\cite{Horndeski:1974wa,Deffayet:2011gz} with the following Horndeski functions:\n\\begin{align}\nG_{2} & = \\mathcal{A} \\left( \\varphi \\right) - f^{4} \\left( 1 - X + \\ldots \\right) \\label{eq:Horndeski G2} \\\\\nG_{3} & = \\frac{M_{5}^{3}}{f^{2}} \\left( X + \\ldots \\right) \\\\\nG_{4} & = \\frac{M_{P}^{2}}{2} \\left( 1 - X -  X^2\/2 \\ldots \\right) \\\\\nG_{5} & = -2\\beta\\frac{M_{5}^{3}}{m^{2}f^{2}} \\left( X + X^2 + \\ldots \\right) \\label{eq:Horndeski G5}\n\\end{align}\n\n\n\\section{Cosmological background evolution}\\label{sec:background}\n\nWe expand the dynamics of the fields around a flat FLRW background on the 4D slice of the bulk along $x^{\\mu}$ where the properties of the brane, $g_{\\mu\\nu}$ and $\\varphi$, are defined. On this slice, the length element is:\n\\begin{equation}\n    ds^{2} = q_{\\mu\\nu}dx^{\\mu}dx^{\\nu} = -dt^{2} + a^{2} \\left( t \\right)\\delta_{ij}dx^{i}dx^{j} \\label{eq:FLRW bulk}\n\\end{equation}\n\nIt might appear more natural to expand around a flat FLRW background on the brane with its induced metric $g_{\\mu\\nu}$. However, the two metrics $q_{\\mu\\nu}$ and $g_{\\mu\\nu}$ are related by a disformal transformation involving the scalar field which, at the background level, depends only on the physical time:\n\\begin{equation}\n    \\bar{g}_{\\mu\\nu} = \\bar{q}_{\\mu\\nu} + \\frac{\\left( \\dot{\\bar{\\varphi}} \\left( t \\right) \\right)^{2}}{f^{4}} \\delta_{\\mu 0}\\delta_{\\nu 0}\n\\end{equation}\nwhere barred quantities are taken at the background level. Therefore, the geometry on the brane is also of the FLRW type, but with a different definition for the physical time, leading to a different scale factor and expansion history. Because in our case $\\dot{\\bar{\\varphi}} \\ll f^{2}$, the expansion history on the 4D slice and on the brane will, then, be approximately the same. In addition, equations~\\eqref{eq:Horndeski G2}-\\eqref{eq:Horndeski G5} determine a self-consistent Horndeski theory of gravity with the metric $q_{\\mu\\nu}$ and the scalar field $\\varphi$ without referring to the induced metric $g_{\\mu\\nu}$.\nThus, in the following we apply the well-known techniques used in the context of Horndeski theories.\n\nIn the non-relativistic limit, action~\\eqref{eq:dRT action} reads:\n\\begin{equation}\\label{eq:action_galdbi}\n    \\mathcal{S} = \\int dx^{4} \\sqrt{-q} \\left(\\mathcal{L}_f + \\mathcal{L}_K + \\mathcal{L}_R + \\mathcal{L}_{\\mathcal{K}_{GB}} + \\mathcal{L}_{m} \\left( q_{\\mu\\nu}, \\psi_{m} \\right) \\right).\n\\end{equation}\nWe define $\\Omega_m^0$ and $\\Omega_r^0$ the standard present energy density parameters for pressureless matter and radiation respectively, and $\\bar{H}$ the normalized Hubble rate $H\/H_0$ with $H_0$ the present Hubble constant. Prime symbol denotes the derivative with respect to $\\ln a$. We set: \n\\begin{equation}\n\\tilde{x} = \\frac{\\varphi' H_0}{f^2}, \\quad \\Omega_\\Lambda^0 = \\frac{\\Lambda}{3 H_0^2 M_P^2}= \\frac{f^4}{3 H_0^2 M_P^2}, \\quad \\eta = \\frac{M_5^3}{M_P^2 H_0}, \\quad \\xi = \\frac{\\beta}{\\eta}, \\quad \\kappa = \\frac{M_P H_0}{f^2}.\n\\end{equation}\nThen, the two Friedmann equations derived from action~\\eqref{eq:action_galdbi} are:\n\\begin{eqnarray}\n    && \\bar{H}^2 =  - \\frac{3}{2} \\bar{H}^4 \\tilde{x}^2- \\frac{15}{8} \\bar{H}^6 \\tilde{x}^4  + \\eta \\bar{H}^4 \\tilde{x}^3 - \\frac{10}{3} \\xi \\bar{H}^6 \\tilde{x}^3 - \\frac{14}{3} \\xi \\bar{H}^8 \\tilde{x}^5 \\nonumber \\\\\n    && + \\frac{\\Omega_m^0}{a^3} + \\frac{\\Omega_r^0}{a^4} + \\Omega_\\Lambda^0 \\left( 1 + \\frac{1}{2}\\bar{H}^2 \\tilde{x}^2 \\right) \\\\[5pt]\n    && \\bar{H}^2 +\\frac{2}{3}\\bar{H} \\bar{H}' = - \\frac{2}{3} \\xi\\left( 2\\bar{H}^6 \\tilde{x}^3 + 5\\bar{H}^5 \\tilde{x}^3 \\bar{H}'  + 3\\bar{H}^6 \\tilde{x}^2 \\tilde{x}' + 2\\bar{H}^8 \\tilde{x}^5 + 5\\bar{H}^8 \\tilde{x}^4 \\tilde{x}'  + 7\\bar{H}^7 \\tilde{x}^5 \\bar{H}'\\right) \\nonumber \\\\    \n    && - \\frac{1}{2}\\bar{H}^4 \\tilde{x}^2 - \\bar{H}^3 \\tilde{x}^2 \\bar{H}'  - \\frac{2}{3} \\bar{H}^4 \\tilde{x} \\tilde{x}' + \\frac{1}{3} \\eta \\bar{H}^3 \\tilde{x}^2 (\\bar{H}\\tilde{x})' - \\frac{3}{8} \\bar{H}^6 \\tilde{x}^4 - \\frac{5}{4} \\bar{H}^5 \\tilde{x}^4 \\bar{H}'- \\bar{H}^6 \\tilde{x}^3 \\tilde{x}' \\nonumber \\\\\n    && - \\frac{\\Omega_r^0}{3a^{4}} + \\Omega_\\Lambda^0 \\left( 1 - \\frac{1}{2}\\bar{H}^2 \\tilde{x}^2 \\right)\n\\end{eqnarray}\nWe see that we recover the $\\Lambda$CDM equations when setting $\\tilde{x}$ to zero, but with a physical interpretation of the cosmological constant as the brane tension. The DBI model proposed here is then an extension of the standard model of cosmology, and as such follows a late accelerated expansion but with a physical interpretation of the origin of $\\Lambda$ as a brane tension.\nUsing the same methodology and similar notations as in \\cite{Stephen-Appleby_2012}, we derive the field $\\varphi$ equation of motion from action~\\eqref{eq:action_galdbi}. This leads to a system of two coupled equations\n\\begin{equation}\n    \\left.\\begin{array}{ll}\n\\tilde x' & = - \\tilde x + \\dfrac{\\alpha \\lambda - \\sigma \\gamma}{\\sigma \\beta - \\alpha \\omega}  \\\\\n\\bar H' & = \\dfrac{\\omega \\gamma - \\beta\\lambda}{\\sigma \\beta - \\alpha\\omega}\n\\end{array}\\right.\n\\end{equation}\nwith\n\\begin{align*}\n\t\\beta &= 2\\bar{H}^4+ \\frac{9}{2} \\bar{H}^6 \\tilde{x}^2 - \\Omega_\\Lambda^0 \\bar{H}^2  - 2 \\eta \\bar{H}^4 \\tilde{x} + 4 \\xi \\bar{H}^6 \\tilde{x} + \\frac{40}{3} \\xi \\bar{H}^8 \\tilde{x}^3\n \\\\\n\t\\alpha & = 6 \\bar{H}^3 \\tilde{x} -  \\Omega_\\Lambda^0 \\bar{H} \\tilde{x} + \\frac{15}{2} \\bar{H}^5 \\tilde{x}^3 - 3 \\eta \\bar{H}^3 \\tilde{x}^2 + 10 \\xi \\bar{H}^5 \\tilde{x}^2  + \\frac{70}{3} \\xi \\bar{H}^7 \\tilde{x}^4 \n \\\\\n\t\\gamma & = 4 \\bar{H}^4 \\tilde{x} -2  \\Omega_\\Lambda^0 \\bar{H}^2\\tilde{x}  -  \\eta \\bar{H}^4 \\tilde{x}^2 + 2 \\xi \\bar{H}^6 \\tilde{x}^2 - \\frac{10}{3} \\xi \\bar{H}^8 \\tilde{x}^4 \n \\\\\n\t\\omega & = \\frac{4}{3} \\bar{H}^4 \\tilde{x} + \\bar{H}^6 \\tilde{x}^3 - \\frac{1}{3} \\eta \\bar{H}^4 x^2 + 2 \\xi \\bar{H}^6 \\tilde{x}^2 + \\frac{10}{3} \\xi \\bar{H}^8 \\tilde{x}^4\n \\\\\n\t\\sigma &=  \\frac{2}{3} \\bar{H} + 2 \\bar{H}^3 \\tilde{x}^2 + \\frac{15}{12} \\bar{H}^5 \\tilde{x}^4 - \\frac{1}{3} \\eta \\bar{H}^3 \\tilde{x}^3 + \\frac{10}{3} \\xi \\bar{H}^5 \\tilde{x}^3 + \\frac{14}{3} \\xi \\bar{H}^7 \\tilde{x}^5 \n \\\\\n\t\\lambda &= \\bar{H}^2  + \\frac{\\Omega_r^0 }{3 a^4} - \\Omega_\\Lambda^0 + \\frac{1}{2} \\Omega_\\Lambda^0 \\bar{H}^2 \\tilde{x}^2 - \\frac{1}{3} \\bar{H}^4 \\tilde{x}^2 - \\frac{5}{8} \\bar{H}^6 \\tilde{x}^4 + \\frac{1}{3} \\eta \\bar{H}^4 \\tilde{x}^3  -\\frac{2}{3} \\xi \\bar{H}^6 \\tilde{x}^3  - 2 \\xi \\bar{H}^8 \\tilde{x}^5\n\\end{align*}\nGiven values for the parameters $\\Omega_m^0$, $ \\eta$, $\\xi$ and $\\kappa$, and initial conditions $\\tilde x_0$ and $H_0$, this system can be integrated to compute background cosmology observables like the distance moduli of type Ia supernovae. In Figure~\\ref{fig:hubble_diagram} we illustrate this with a Hubble diagram prediction compared with recent type Ia supernova data \\cite{Scolnic2018}. As an initial condition for $\\tilde{x}$, we chose to set $\\tilde{x}_0 = 6\\times 10^{-8}$ today. This is the maximum value allowed by the constraint on gravitational wave speed (see Section~\\ref{sec:tensorial}) coming from the quasi simultaneous observation of photons and gravitational waves after neutron star merger event GW170817A~\\cite{LIGOScientific:2017zic}.  Nevertheless, before discussing more the cosmological scenarios proposed by the DBI-Galileon model, stability conditions much be computed first to assess the viability of the models for any set of parameters at the perturbation level.\n\n\\begin{figure}\n\\centering\n\\includegraphics[width=0.7\\textwidth]{hubble_diagramme.pdf}\n\\caption{Hubble diagram prediction for the non-relativistic DBI-Galileon model for $\\Omega_\\Lambda^0 = 0.7$, $\\tilde x_0 = 6 \\times 10^{-8}$, $\\eta = \\xi=0$ (blue) compared with binned Pantheon data (black points) \\cite{Scolnic2018}. We used $\\mathcal{M}=23.81$ for the offset magnitude of the diagram. Residuals to the fit are presented in the bottom panel.\n}\\label{fig:hubble_diagram}\n\\end{figure}\n\n\n\\section{Stability conditions}\\label{sec:perturbations}\n\nTo be viable as a description of our Universe, the model has to fulfill stability conditions. These requirements apply to degrees of freedom propagating around the fixed background, \\textit{i.e.} to the cosmological perturbations, that are capable of undermining the stability of the Universe. In the determination of the stability conditions for the DBI-Galileon model in the non-relativistic limit, we use the formalism described in \\cite{DeFelice:2011bh} for Horndeski theories. In our case, these stability conditions are defined from the following quantities derived from the particular Horndeski functions \\eqref{eq:Horndeski G2} to \\eqref{eq:Horndeski G5}:\n\\begin{align}\n    \\omega_{1} & = 1 +  \\frac{1}{2} \\bar{H}^2 \\tilde{x}^2  + \\frac{3}{8} \\bar{H}^4 \\tilde{x}^4  + 2 \\xi \\bar{H}^4 \\tilde{x}^3  + 2\\xi \\bar{H}^6 \\tilde{x}^5 \\\\\n    \\omega_{2} &  = 2\\bar{H} + 3 \\bar{H}^3 \\tilde{x}^2   + \\frac{15}{4} \\bar{H}^5 \\tilde{x}^4  -\\eta \\bar{H}^3 \\tilde{x}^3  + 10 \\xi \\bar{H}^5\\tilde{x}^3  + 14 \\xi \\bar{H}^7 \\tilde{x}^5\\\\\n    \\omega_{3} & = 9\\left( -\\bar{H}^2 + \\frac{1}{2}\\Omega_\\Lambda^0 \\bar{H}^2\\tilde{x}^2 - 3 \\bar{H}^4 \\tilde{x}^2  + 2 \\eta \\bar{H}^4 \\tilde{x}^3 -10 \\xi \\bar{H}^6 \\tilde{x}^3- \\frac{45}{8} \\bar{H}^6 \\tilde{x}^4   -\\frac{56}{3}\\xi \\bar{H}^8 \\tilde{x}^5 \\right)  \\\\\n    \\omega_{4} & = 1 - \\frac{1}{2} \\bar{H}^2 \\tilde{x}^2  - \\frac{1}{8}\\bar{H}^4 \\tilde{x}^4 + 2 \\xi \\bar{H}^3 \\tilde{x}^2 (\\bar{H} \\tilde{x})'  + 2 \\xi \\bar{H}^5 \\tilde{x}^4 (\\bar{H}\\tilde{x})'\n\\end{align}\n\n\n\\subsection*{Tensorial stability conditions}\\label{sec:tensorial}\n\nIn order to avoid ghosts and Laplacian instabilities, we impose the following constraints on the sign of the kinetic term and on the sign of the gravitational waves speed squared:\n\\begin{align}\n    Q_{t} & \\equiv \\frac{\\omega_{1}}{4} =  \\frac{1}{4} + \\frac{1}{8}\\bar{H}^2\\tilde{x}^2   + \\frac{3}{32}\\bar{H}^4\\tilde{x}^4 + \\frac{1}{2} \\xi \\bar{H}^4 \\tilde{x}^3  + \\frac{1}{2}\\xi \\bar{H}^6 \\tilde{x}^5   > 0 \\\\\n    c_{t}^{2} & \\equiv \\frac{\\omega_{4}}{\\omega_{1}}  = \\frac{1 - \\frac{1}{2}\\bar{H}^2\\tilde{x}^2  - \\frac{1}{8}\\bar{H}^4 \\tilde{x}^4 + 2 \\xi \\bar{H}^3 \\tilde{x}^2 (\\bar{H} \\tilde{x})'  + 2 \\xi \\bar{H}^5 \\tilde{x}^4 (\\bar{H}\\tilde{x})'}{1 +  \\frac{1}{2} \\bar{H}^2 \\tilde{x}^2  + \\frac{3}{8} \\bar{H}^4 \\tilde{x}^4  + 2 \\xi \\bar{H}^4 \\tilde{x}^3  + 2\\xi \\bar{H}^6 \\tilde{x}^5} \\geq 0  \n\\end{align}\n\nIn particular, we see that the gravitational wave speed depends on $\\tilde{x}$, and tends to 1 when  $\\tilde{x} \\rightarrow 0$ :\n\\begin{align}\n4 Q_t & \\simeq 1 + 2(\\bar{H} \\tilde{x})^2 \\\\\n    c_{t} & \\simeq 1 - \\frac{1}{2} (\\bar{H} \\tilde{x})^2\n\\end{align}\nGiven the very tight constraint on the speed of gravitational waves, equal to the speed of light up to a $\\sim 10^{-15}$ difference \\cite{LIGOScientific:2017zic,Creminelli:2017sry,Ezquiaga:2017ekz}, this justifies \\textit{a posteriori} the relevance of the non-relativistic limit where $\\tilde{x} \\ll 1$. Moreover, we see that tensorial perturbations are stable in this limit since $Q_t > 0$.\n\n\\subsection*{Scalar stability conditions}\n\nSimilar stability conditions apply to the scalar degrees of freedom, here including the scalar perturbations of matter components:\n\\begin{align}\n    Q_{s} & \\equiv \\frac{\\omega_{1} \\left( 4\\omega_{1}\\omega_{3} + 9\\omega_{2}^{2} \\right)}{3\\omega_{2}^{2}} > 0 \\label{eq:Qs} \\\\\n    c_{s}^{2} & \\equiv \\frac{3 \\left( 2\\omega_{1}^{2}\\omega_{2}H - \\omega_{2}^{2}\\omega_{4} + 4\\omega_{1}\\omega_{2}\\dot{\\omega}_{1} - 2\\omega_{1}^{2}\\dot{\\omega}_{2} \\right) - 6\\omega_{1}^{2} \\sum \\left( 1+w_{i} \\right) \\rho_{i}}{\\omega_{1} \\left( 4\\omega_{1}\\omega_{3} + 9\\omega_{2}^{2} \\right)} \\geq 0\n\\end{align}\nwhere $w_{i}$ and $\\rho_{i}$ are respectively the equation of state parameter and the energy density of the fluid $i$, and the sum runs over all the components of the Universe (here only pressureless matter and radiation). At the lowest order in $\\tilde{x}$, we get:\n\\begin{align}\nQ_s & \\simeq \\frac{3}{2} (\\Omega_\\Lambda^0 - \\bar{H}^2) \\tilde{x}^2 \\\\\nc_{s}^{2} & \\simeq 1 + \\frac{2 \\left( \\eta \\bar{H}^{2}\\tilde{x}' - \\bar{H}\\h' - 2\\xi\\bar{H}^{4}\\tilde{x}' \\right)}{3 \\left( \\Omega_{\\Lambda}^{0} - \\bar{H}^{2} \\right)}\n\\end{align}\n\nWith $\\tilde{x} \\ll 1$, a fit of the DBI-Galileon model to data leads to cosmological parameters close to the standard model ones: $\\Omega_m^0 \\approx 0.3$ and $\\Omega_\\Lambda^0 \\approx 0.7$ \\cite{Planck2018}. Therefore, from the first Friedmann equation, we get $\\Omega_\\Lambda^0 < \\bar{H}^2$ for all relevant models in agreement with cosmological observations. As $Q_s \\leq 0$, the DBI-Galileon model contains scalar instabilities unless it reduces to GR. One way to avoid this would be to add a spatial curvature to the metric, but with a strong energy density (at least $\\sim 0.3$) which is also excluded by observations \\cite{Planck2018}. \n\n\n\\section{Discussion}\\label{sec:discussion}\n\n\\subsection*{Physical interpretation}\n\nFrom the definition \\eqref{eq:Qs}, we see that the dominant terms come $\\G2$ (giving the $\\Omega_\\Lambda^0$ term) and $\\G4$ (giving the $\\bar{H}^2$ term). The competition between the two terms leads to the ghost-like behaviour in a cosmological setting: $Q_s \\leq 0$. In other words, it is the result of the competition between the DBI and the Einstein-Hilbert terms. The DBI action will have the effect of stretching the brane towards an extremal surface, whereas the Einstein-Hilbert term on the brane will tend to make the brane contract on itself from the effect of curvature. However, in the non-relativistic limit of the DBI-Galileon, the Einstein-Hilbert term destabilizes the scalar field perturbations and the stretching effect from the cosmological constant is not strong enough to counterbalance, leading to an instantaneous decay of the vacuum state.\n\nBecause the DBI-Galileon action is the most general one can find of a 4D probe brane in a 5D bulk, we expect this statement to be quite general for all such theories studied in the current context.  Indeed, this is true in a standard cosmological setting which is realised with an FLRW slicing of the bulk space-time (equivalent to an FLRW background on the brane). Therefore, the only way to evade this ghostly behaviour in cosmology is to include the full relativistic dynamics of the theory ($\\tilde{x} \\sim 1$). We have seen that, in this case, we expect significant deviations of the speed of gravitational waves $c_t$ from $c$. This is not a definitive impossibility though if the full DBI-Galileon is viewed as an effective theory valid only at cosmological scales for which the speed of gravitational waves has not been probed \\cite{deRham:2018red,Ezquiaga:2018btd}. Indeed, the constraint on the gravitational speed from the observation of GW170817 in coincidence with GRB170817A \\cite{LIGOScientific:2017zic} is only valid on small scales probed by LIGO and Virgo. A modification of the dispersion relation of gravitational waves at small scales from operators present in the UV complete theory could allow $c_{t} \\neq c$ on cosmological scales while being compatible with current astrophysical observations. Waiting for the next generation of gravitational wave interferometers, in particular LISA, which will be able to probe this relation at larger scales \\cite{Barausse:2020rsu}, this possibility remains open.\n\n\\subsection*{Direct coupling to matter}\n\nIn the context of cosmology, where standard model matter is present, there might be direct coupling to the scalar field. In that case, the metric $\\tilde{q}$ to which matter is sensitive is different than the space-time metric $q$:\n\\begin{equation}\\label{eq:action_galdbi-qtilde}\n    \\mathcal{S} = \\int dx^{4} \\sqrt{-q} \\left(\\mathcal{L}_f + \\mathcal{L}_K + \\mathcal{L}_R + \\mathcal{L}_{\\mathcal{K}_{GB}}\\right) + \\int dx^{4} \\sqrt{-\\tilde q} \\mathcal{L}_{m} \\left(\\tilde q_{\\mu\\nu}, \\psi_{m} \\right).\n\\end{equation}\nIt has been shown in \\cite{Bekenstein:1992pj} that the two metrics are related by a disformal transformation of the following form:\n\\begin{equation}\n    q_{\\mu\\nu} = A \\left( \\varphi, \\tilde{X} \\right) \\tilde{q}_{\\mu\\nu} + B \\left( \\varphi, \\tilde{X} \\right) \\frac{\\partial_{\\mu}\\varphi \\partial_{\\nu}\\varphi}{f^{4}} \\label{eq:disformal transformation}\n\\end{equation}\nwhere $A$ and $B$ are arbitrary functions of the scalar field and $\\tilde{X} = -\\tilde{q}_{\\mu\\nu}\\partial_{\\mu}\\varphi\\partial_{\\nu}\\varphi\/2f^{4}$. For simplicity and following the treatment of the covariant Galileon \\cite{Stephen-Appleby_2012}, we assume that $A$ and $B$ are constant parameters. This can be further justified by the fact that, a dependency on $X$ would introduce, in general, higher order terms which would go beyond the framework of Horndeski theories \\cite{Bettoni:2013diz}, and a dependency on $\\varphi$ would, in general, break the shift symmetry followed by the scalar field $\\varphi$ in the probe brane context. Note that, when $A = -B$, matter is coupled to the induced metric on the brane.\n\nContrary to the covariant Galileon, the DBI-Galileon action is not invariant by such a change of reference frame. However, new terms that can not be absorbed into a redefinition of the parameters arise only at higher order in $\\tilde{X}$. Therefore, the non-relativistic dynamics is not change by the introduction of a direct coupling between the scalar field and matter of the form \\eqref{eq:disformal transformation} with constant parameters. In particular, this does not prevent the perturbations around the FLRW background from showing ghost instabilities.\n\\subsection*{Generalization}\n\nThe DBI-Galileon is a particular example of the more general class of shift-symmetric Horndeski theories.  These are subclass of Horndeski theories which are invariant under a shift symmetry of the scalar field $\\varphi \\rightarrow \\varphi + c$ \\cite{Sotiriou:2013qea,Sotiriou:2014pfa}. In these theories, the arbitrary Horndeski functions are restricted to be functions of $X$ alone. In order to make the non-relativistic limit apparent, we Taylor expand these arbitrary functions around GR:\n\\begin{align}\n    G_{2} & \\equiv \\Lambda + \\sum_{n=1}^{+\\infty} g_{2}^{\\left( n \\right)} X^{n} \\\\\n    G_{3} & \\equiv \\sum_{n=1}^{+\\infty} g_{3}^{\\left( n \\right)} X^{n} \\\\\n    G_{4} & \\equiv \\frac{M_{P}^{2}}{2} + \\sum_{n=1}^{+\\infty} g_{4}^{\\left( n \\right)} X^{n} \\\\\n    G_{5} & \\equiv \\sum_{n=1}^{+\\infty} g_{5}^{\\left( n \\right)} X^{n}\n\\end{align}\n\nThe constant terms in $G_{3}$ and $G_{5}$ do not appear in the expansion as they lead to total derivative terms. Because the Horndeski functions depend only on $X$, the $\\omega$ functions that determine the stability conditions reduce to:\n\\begin{eqnarray}\n    \\omega_{1} & \\equiv & 2 \\G4 - 2X\\left( 2\\G{4,X} + \\dot{\\phi}H\\G{5,X} \\right) \\\\\n    \\omega_{2} & \\equiv & 4 H\\G4 - 2X \\left( \\dot{\\phi}\\G{3,X} + 8 H\\G{4,X} + 5 \\dot{\\phi}H^{2} \\G{5,X} \\right) - 4X^{2}H \\left( 4 \\G{4,XX} + \\dot{\\phi}H\\G{5,XX} \\right) \\\\\n    \\omega_{3} & \\equiv & -18H^{2}\\G4 + 3X \\left( \\G{2,X} + 12 \\dot{\\phi}H\\G{3,X} + 42H^{2} \\G{4,X} + 30 \\dot{\\phi}H^{3} \\G{5,X} \\right) \\nonumber \\\\\n    && + 6X^{2} \\left( \\G{2,XX} + 3\\dot{\\phi}H\\G{3,XX} + 48 H^{2}\\G{4, XX} + 13H^{3}\\dot{\\phi}\\G{5,XX} \\right) \\nonumber \\\\\n    && + 12 X^{3}H^{2} \\left( 6\\G{4,XXX} + H\\dot{\\phi}\\G{5,XXX} \\right) \\\\\n    \\omega_{4} & \\equiv & 2\\G4 - 2X\\ddot{\\phi} \\G{5,X}\n\\end{eqnarray}\n\nFrom these, we can compute the quantity $Q_{s}$ up to first order in $X$:\n\\begin{equation}\n    Q_{s} = \\frac{X}{H^{2}} \\left( \\g2^{\\left( 1 \\right)} + 6H^{2} \\g4^{\\left( 1 \\right)} \\right) + O \\left( X^{\\frac{3}{2}} \\right)\n\\end{equation}\n\nThis leads to a very simple formulation of the no-ghost condition, independent of $X$, in the context of Shift-Symmetric Horndeski theories in the non-relativistic limit:\n\\begin{equation}\n     \\g2^{\\left( 1 \\right)} + 6H^{2} \\g4^{\\left( 1 \\right)} > 0\n\\end{equation}\n\nIn the context of the brane galileon, where $\\g4^{\\left( 1 \\right)} = -M_{P}^{2}\/2$ and $\\g2^{\\left( 1 \\right)} = \\Lambda$, this is equivalent to the inequality which is never fulfilled in flat space:\n\\begin{equation}\n    \\Lambda - 3M_{P}^{2}H^{2} > 0 \\quad \\Leftrightarrow \\quad \\Omega_{\\Lambda}^{0} > \\bar{H}^{2}\n\\end{equation}\n\nOther stability conditions are given by:\n\\begin{align}\n    c_{s}^{2} & \\simeq 1 + \\frac{2\\Ddot{\\phi} \\g3^{\\left( 1 \\right)} + 4\\Dot{H}\\g4^{\\left( 1 \\right)} + 2\\Ddot{\\phi}H^{2}\\g5^{\\left( 1 \\right)}}{\\g2^{\\left( 1 \\right)} + 6H^{2} \\g4^{\\left( 1 \\right)}} > 0\\\\\n    Q_t & \\simeq \\frac{M_{P}^{2}}{4} \\\\\n    c_t^{2} & \\simeq 1\n\\end{align}\nwhere we expressed these quantities at the lowest order. The two tensorial conditions are, thus, automatically satisfied in this context. On the other hand, the stability conditions for scalar perturbations at the lowest order give a simple inequality involving the parameters of the Taylor expansion, that can be easily checked at the background level.\n\n\n\\section{Conclusion}\\label{sec:conclusion}\n\nWe described the DBI-Galileon theory of a four-dimensional brane evolving in a 5D bulk space-time in the non-relativistic limit where its local kinetic energy is small compared to its tension. This model belongs to the class of shift-symmetric Horndeski theories, themselves being a subclass of the more general family of Horndeski theories. From the construction of the DBI-Galileon model, the free parameters of the model acquire a physical meaning. In particular, the interpretation of the cosmological constant is linked to the brane tension energy density. We derived the equations driving the evolution of the late-time Universe around a spatially flat FLRW cosmological background and studied the stability of scalar and tensorial perturbations. This model reduces to an expansion around standard GR, and therefore around standard $\\Lambda$CDM in the cosmological context. As such, it is naturally compatible with data in the non-relativistic limit provided the effect of the scalar field is small enough, even considering the speed of the gravitational waves. However, it revealed fatal ghostly behaviour for scalar perturbations  around the FLRW background. From there, we derived the corresponding stability conditions for shift-symmetric Horndeski theories in the non-relativistic limit in the cosmological context and found very simple formulations for these conditions.\n\n\\section*{Acknowledgements}\n\nWe would like to thank Marc Besan\u00e7on, Arnaud de Mattia and Vanina Ruhlmann-Kleider for their comments on the present paper. We also want to thank David Langlois for useful and interesting comments and suggestions.\n\n\n\\section*{References}\n\\bibliographystyle{iopart-num}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\\label{sec1}\n\nThe supersymmetry technique is a powerful method in random matrix theory and disordered systems. For a long time it was thought to be applicable for Gaussian probability densities only \\cite{Efe83,VerZir85,VWZ85,Efe97}. Due to universality on the local scale of the mean level spacing \\cite{BreZee93a,BreZee93b,HacWei95,GMW98}, this restriction was not a limitation for calculating in quantum chaos and disordered systems. Indeed, results of Gaussian ensembles are identical for large matrix dimension with other invariant matrix ensembles on this scale. In the Wigner--Dyson theory \\cite{Bee97} and its corrections for systems with diffusive dynamics \\cite{Mir00}, Gaussian ensembles are sufficient. Furthermore, universality was found on large scale, too \\cite{AJM90}. This is of paramount importance when investigating matrix models in high--energy physics.\n\nThere are, however, situations in which one can not simply resort to Gaussian random matrix ensembles. The level densities in high--energy physics \\cite{BIPZ78} and finance \\cite{LCBP99} are needed for non-Gaussian ensembles. But these one--point functions strongly depend on the matrix ensemble. Other examples are bound--trace and fixed--trace ensembles \\cite{Meh04}, which are both norm--dependent ensembles \\cite{Guh06}, as well as ensembles derived from a non-extensive entropy principle \\cite{TVT04,BBP04,Abu04}. In all these cases one is interested in the non-universal behavior on special scales.\n\nRecently, the supersymmetry method was extended to general rotation invariant probability densities \\cite{Guh06,Som07,LSZ07,KGG08}. There are two approaches. The first one is the generalized Hubbard--Stratonovich transformation \\cite{Guh06,KGG08}. With help of a proper Dirac--distribution in superspace an integral over rectangular supermatrices was mapped to a supermatrix integral with non-compact domain in the Fermion--Fermion block. The second approach is the superbosonization formula \\cite{Som07,LSZ07} mapping the same integral over rectangular matrices as before to a supermatrix integral with compact domain in the Fermion--Fermion block.\n\nIn this work, we prove the equivalence of the generalized Hubbard--Stratonovich transformation with the superbosonization formula. The proof is based on integral identities between supersymmetric Wishart--matrices and quadratic supermatrices. The orthogonal, unitary and unitary-symplectic classes are dealt with in a unifying way.\n\nThe article is organized as follows. In Sec. \\ref{sec2}, we give a motivation and introduce our notation. In Sec. \\ref{sec3}, we define rectangular supermatrices and the supersymmetric version of Wishart-matrices built up by supervectors. We also give a helpful corollary for the case of arbitrary matrix dimension discussed in Sec. \\ref{sec7}. In Secs. \\ref{sec4} and \\ref{sec5}, we present and further generalize the superbosonization formula and the generalized Hubbard--Stratonovich transformation, respectively. The theorem stating the equivalence of both approaches is given in Sec. \\ref{sec6} including a clarification of their mutual connection. In Sec. \\ref{sec7}, we extend both theorems given in Secs. \\ref{sec4} and \\ref{sec5} to arbitrary matrix dimension. Details of the proofs are given in the appendices.\n\n\\section{Ratios of characteristic polynomials}\\label{sec2}\n\nWe employ the notation defined in Refs.  \\cite{KKG08,KGG08}. ${\\rm Herm\\,}(\\beta,N)$ is either the set of $N\\times N$ real symmetric ($\\beta=1$), $N\\times N$ hermitian ($\\beta=2$) or $2N\\times 2N$ self-dual ($\\beta=4$) matrices, according to the Dyson--index $\\beta$. We use the complex representation of the quaternionic numbers $\\mathbb{H}$. Also, we define\n\\begin{equation}\\label{2.0}\n \\gamma_1=\\left\\{\\begin{array}{ll} 1 & ,\\ \\beta\\in\\{2,4\\} \\\\ 2 & ,\\ \\beta=1 \\end{array}\\right. \\quad ,\\quad \\gamma_2=\\left\\{\\begin{array}{ll} 1 & ,\\ \\beta\\in\\{1,2\\} \\\\ 2 & ,\\ \\beta=4 \\end{array}\\right.\n\\end{equation}\nand $\\tilde{\\gamma}=\\gamma_1\\gamma_2$.\n\nThe central objects in many applications of supersymmetry are averages over ratios of characteristic polynomials \\cite{AkeFyo03,AkePot04,BorStr05}\n\\begin{eqnarray}\n  \\fl Z_{k_1k_2}(E^-) & = & \\int\\limits_{{\\rm Herm\\,}(\\beta,N)}P(H)\\frac{\\prod\\limits_{n=1}^{k_2}\\det\\left(H-(E_{n2}-\\imath\\varepsilon)\\leavevmode\\hbox{\\small1\\kern-3.8pt\\normalsize1}_{\\gamma_2N}\\right)}{\\prod\\limits_{n=1}^{k_1}\\det\\left(H-(E_{n1}-\\imath\\varepsilon)\\leavevmode\\hbox{\\small1\\kern-3.8pt\\normalsize1}_{\\gamma_2N}\\right)}d[H]\\nonumber\\\\\n  & = & \\int\\limits_{{\\rm Herm\\,}(\\beta,N)}P(H){\\rm Sdet\\,}^{-1\/\\tilde{\\gamma}}\\left(H\\otimes \\leavevmode\\hbox{\\small1\\kern-3.8pt\\normalsize1}_{\\tilde{\\gamma}(k_1+k_2)}- \\leavevmode\\hbox{\\small1\\kern-3.8pt\\normalsize1}_{\\gamma_2N}\\otimes E^-\\right)d[H]\\label{2.1}\n\\end{eqnarray}\nwhere $P$ is a sufficiently integrable probability density on the matrix set ${\\rm Herm\\,}(\\beta,N)$ invariant under the group\n\\begin{equation}\\label{2.2}\n {\\rm U\\,}^{(\\beta)}(N)=\\left\\{\\begin{array}{ll} \n              \\Or(N)\t& ,\\ \\beta=1\\\\\n\t      {\\rm U\\,}(N)\t& ,\\ \\beta=2\\\\\n\t      {\\rm USp\\,}(2N)\t& ,\\ \\beta=4\n             \\end{array}\\right. .\n\\end{equation}\nHere, we assume that $P$ is analytic in its real independent variables. We use the same measure for $d[H]$ as in Ref.  \\cite{KKG08} which is the product over all real independent differentials, see also Eq.~\\eref{t1.4}. Also, we define $E={\\rm diag\\,}(E_{11},\\ldots,E_{k_11},E_{12},\\ldots,E_{k_22})\\otimes \\leavevmode\\hbox{\\small1\\kern-3.8pt\\normalsize1}_{\\tilde{\\gamma}}$ and $E^-=E-\\imath\\varepsilon \\leavevmode\\hbox{\\small1\\kern-3.8pt\\normalsize1}_{\\tilde{\\gamma}(k_1+k_2)}$.\n\nThe generating function of the $k$--point correlation function \\cite{BreHik00,Zir06,Guh06,KGG08}\n\\begin{equation}\\label{2.3}\n R_{k}(x)=\\gamma_2^{-k}\\int\\limits_{{\\rm Herm\\,}(\\beta,N)}P(H)\\prod\\limits_{p=1}^k\\tr\\delta(x_p-H)d[H]\n\\end{equation}\nis one application and can be computed starting from the matrix Green function and Eq.~\\eref{2.1} with $k_1=k_2=k$. Another example is the $n$--th moment of the characteristic polynomial \\cite{MehNor01,Fyo02,Zir06}\n\\begin{equation}\\label{2.4}\n \\widehat{Z}_{n}(x,\\mu)=\\int\\limits_{{\\rm Herm\\,}(\\beta,N)}P(H)\\Theta(H){\\det}^n\\left(H-E \\leavevmode\\hbox{\\small1\\kern-3.8pt\\normalsize1}_{\\gamma_2k}\\right)d[H],\n\\end{equation}\nwhere the Heavyside--function for matrices $\\Theta(H)$ is unity if $H$ is positive definite and zero otherwise. \\cite{KGG08} \n\nWith help of Gaussian integrals, we get an integral expression for the determinants in Eq.~\\eref{2.1}. Let $\\Lambda_j$ be the Grassmann space of $j$--forms. We consider a complex Grassmann algebra \\cite{Ber87} $\\Lambda=\\bigoplus\\limits_{j=0}^{2\\gamma_2Nk_2}\\Lambda_j$ with $\\gamma_2Nk_2$ pairs $\\{\\zeta_{jn},\\zeta_{jn}^*\\}$, $1\\leq n\\leq k_2,\\ 1\\leq j\\leq \\gamma_2N$, of Grassmann variables and use the conventions of Ref.  \\cite{KKG08} for integrations over Grassmann variables. Due to the $\\mathbb{Z}_2$--grading, $\\Lambda$ is a direct sum of the set of commuting variables $\\Lambda^0$ and of anticommuting variables $\\Lambda^1$. The body of an element in $\\Lambda$ lies in $\\Lambda_0$ while the Grassmann generators are elements in $\\Lambda_1$.\n\nLet $\\imath$ be the imaginary unit. We take $\\gamma_2Nk_1$ pairs $\\{z_{jn},z_{jn}^*\\}$, $1\\leq n\\leq k_1,\\ 1\\leq j\\leq \\gamma_2N$, of complex numbers and find for Eq.~\\eref{2.1}\n\\begin{equation}\\label{2.5}\n \\fl Z_{k_1k_2}(E^-)= (2\\pi)^{\\gamma_2N(k_2-k_1)}\\imath^{\\gamma_2Nk_2}\\int\\limits_{\\mathfrak{C}} \\mathcal{F}P(K)\\exp\\left(-\\imath{\\rm Str\\,} BE^-\\right)d[\\zeta]d[z]\n\\end{equation}\nwhere $d[z]=\\prod\\limits_{p=1}^{k_1}\\prod\\limits_{j=1}^{\\gamma_2N}dz_{jp}dz_{jp}^*$ , $d[\\zeta]=\\prod\\limits_{p=1}^{k_2}\\prod\\limits_{j=1}^{\\gamma_2N}(d\\zeta_{jp}d\\zeta_{jp}^*)$ and $\\mathfrak{C}=\\mathbb{C}^{\\gamma_2k_1N}\\times\\Lambda_{2\\gamma_2Nk_2}$. The characteristic function appearing in \\eref{2.5} is defined as\n\\begin{equation}\\label{2.6}\n \\mathcal{F}P(K)=\\int\\limits_{{\\rm Herm\\,}(\\beta,N)}P(H)\\exp\\left(\\imath\\tr HK\\right)d[H] .\n\\end{equation}\nThe two matrices\n\\begin{equation}\\label{2.7}\n K = \\frac{1}{\\tilde{\\gamma}}V^\\dagger V\\qquad{\\rm and}\\qquad\n B = \\frac{1}{\\tilde{\\gamma}}VV^\\dagger\n\\end{equation}\nare crucial for the duality between ordinary and superspace. While $K$ is a $\\gamma_2N\\times\\gamma_2N$ ordinary matrix whose entries have nilpotent parts, $B$ is a $\\tilde{\\gamma}(k_1+k_2)\\times\\tilde{\\gamma}(k_1+k_2)$ supermatrix. They are composed of the rectangular $\\gamma_2N\\times\\tilde{\\gamma}(k_1+k_2)$ supermatrix\n\\begin{eqnarray}\n V^\\dagger|_{\\beta\\neq2} & = & (z_1,\\ldots,z_{k_1},Yz_1^*,\\ldots,Yz_{k_1}^*,\\zeta_1,\\ldots,\\zeta_{k_2},Y\\zeta_1^*,\\ldots,Y\\zeta_{k_2}^*) ,\\nonumber\\\\\n V|_{\\beta\\neq2} & = & (z_1^*,\\ldots,z_{k_1}^*,Yz_1,\\ldots,Yz_{k_1},-\\zeta_1^*,\\ldots,-\\zeta_{k_2}^*,Y\\zeta_1,\\ldots,Y\\zeta_{k_2})^T ,\\nonumber\\\\\n V^\\dagger|_{\\beta=2} & = & (z_1,\\ldots,z_{k_1},\\zeta_1,\\ldots,\\zeta_{k_2}) ,\\nonumber\\\\\n V|_{\\beta=2} & = & (z_1^*,\\ldots,z_{k_1}^*,-\\zeta_1^*,\\ldots,-\\zeta_{k_2}^*)^T .\\label{2.9}\n\\end{eqnarray}\nThe transposition ``$T$'' is the ordinary transposition and is not the supersymmetric one. However, the adjoint ``$\\dagger$'' is the complex conjugation with the supersymmetric transposition ``$T_{\\rm S}$''\n\\begin{equation}\\label{2.9b}\n \\sigma^{T_{\\rm S}}=\\left[\\begin{array}{cc} \\sigma_{11} & \\sigma_{12} \\\\ \\sigma_{21} & \\sigma_{22} \\end{array}\\right]^{T_{\\rm S}}=\\left[\\begin{array}{cc} \\sigma_{11}^T & \\sigma_{21}^T \\\\ -\\sigma_{12}^T & \\sigma_{22}^T \\end{array}\\right],\n\\end{equation}\nwhere $\\sigma$ is an arbitrary rectangular supermatrix. We introduce the constant $\\gamma_2N\\times\\gamma_2N$ matrix\n\\begin{equation}\\label{2.10}\n Y=\\left\\{\\begin{array}{ll}\n             \\leavevmode\\hbox{\\small1\\kern-3.8pt\\normalsize1}_N & ,\\ \\beta=1\\\\\n\t     Y_s^T\\otimes \\leavevmode\\hbox{\\small1\\kern-3.8pt\\normalsize1}_N & ,\\ \\beta=4\n   \\end{array}\\right.\\ ,\\ \\ Y_s=\\left[\\begin{array}{cc} 0 & 1 \\\\ -1 & 0 \\end{array}\\right] .\n\\end{equation}\nThe crucial duality relation \\cite{Guh06,KGG08}\n\\begin{equation}\\label{2.11}\n \\tr K^m={\\rm Str\\,} B^m\\ ,\\ \\ m\\in\\mathbb{N} ,\n\\end{equation}\nholds, connecting invariants in ordinary and superspace. As $\\mathcal{F}P$ inherits the rotation invariance of $P$, the duality relation \\eref{2.11} yields\n\\begin{equation}\\label{2.12}\n \\fl Z_{k_1k_2}(E^-)= (2\\pi)^{\\gamma_2N(k_2-k_1)}\\imath^{\\gamma_2Nk_2}\\int\\limits_{\\mathfrak{C}} \\Phi(B)\\exp\\left(-\\imath{\\rm Str\\,} BE^-\\right)d[\\zeta]d[z] .\n\\end{equation}\nHere, $\\Phi$ is a supersymmetric extension of a representation $\\mathcal{F}P_0$ of the characteristic function,\n\\begin{equation}\\label{2.13}\n \\Phi(B)=\\mathcal{F}P_0({\\rm Str\\,} B^m|m\\in\\mathbb{N})=\\mathcal{F}P_0(\\tr K^m|m\\in\\mathbb{N})=\\mathcal{F}P(K) .\n\\end{equation}\nThe representation $\\mathcal{F}P_0$ is not unique \\cite{BEKYZ07}. However, the integral \\eref{2.12} is independent of a particular choice \\cite{KGG08}.\n\nThe supermatrix $B$ fulfills the symmetry\n\\begin{equation}\\label{2.14}\n B^*=\\left\\{\\begin{array}{ll}\n                \\widetilde{Y}B\\widetilde{Y}^T &,\\ \\beta\\in\\{1,4\\}, \\\\\n\t\t\\widetilde{Y}B^*\\widetilde{Y}^T &,\\ \\beta=2\n               \\end{array}\\right.\n\\end{equation}\nwith the supermatrices\n\\begin{equation}\\label{2.15}\n \\fl\\widetilde{Y}|_{\\beta=1}=\\left[\\begin{array}{ccc} 0 &  \\leavevmode\\hbox{\\small1\\kern-3.8pt\\normalsize1}_{k_1} & 0 \\\\  \\leavevmode\\hbox{\\small1\\kern-3.8pt\\normalsize1}_{k_1} & 0 & 0 \\\\ 0 & 0 & Y_s\\otimes \\leavevmode\\hbox{\\small1\\kern-3.8pt\\normalsize1}_{k_2} \\end{array}\\right]\\quad,\\quad\\widetilde{Y}|_{\\beta=4}=\\left[\\begin{array}{ccc} Y_s\\otimes \\leavevmode\\hbox{\\small1\\kern-3.8pt\\normalsize1}_{k_1} & 0 & 0 \\\\ 0 & 0 &  \\leavevmode\\hbox{\\small1\\kern-3.8pt\\normalsize1}_{k_2} \\\\ 0 &  \\leavevmode\\hbox{\\small1\\kern-3.8pt\\normalsize1}_{k_2} & 0 \\end{array}\\right]\n\\end{equation}\nand $\\widetilde{Y}|_{\\beta=2}=  \\leavevmode\\hbox{\\small1\\kern-3.8pt\\normalsize1}_{k_1+k_2}$ and is self-adjoint for every $\\beta$. Using the $\\pi\/4$--rotations\n\\begin{equation}\\label{2.16}\n \\fl U|_{\\beta=1}=\\frac{1}{\\sqrt{2}}\\left[\\begin{array}{ccc}  \\leavevmode\\hbox{\\small1\\kern-3.8pt\\normalsize1}_{k_1} &  \\leavevmode\\hbox{\\small1\\kern-3.8pt\\normalsize1}_{k_1} & 0 \\\\ -\\imath \\leavevmode\\hbox{\\small1\\kern-3.8pt\\normalsize1}_{k_1} & \\imath \\leavevmode\\hbox{\\small1\\kern-3.8pt\\normalsize1}_{k_1} & 0 \\\\ 0 & 0 & \\sqrt{2}\\  \\leavevmode\\hbox{\\small1\\kern-3.8pt\\normalsize1}_{2k_2} \\end{array}\\right],\\ U|_{\\beta=4}=\\frac{1}{\\sqrt{2}}\\left[\\begin{array}{ccc} \\sqrt{2}\\  \\leavevmode\\hbox{\\small1\\kern-3.8pt\\normalsize1}_{2k_1} & 0 & 0 \\\\ 0 &  \\leavevmode\\hbox{\\small1\\kern-3.8pt\\normalsize1}_{k_2} &  \\leavevmode\\hbox{\\small1\\kern-3.8pt\\normalsize1}_{k_2}\\\\ 0 & -\\imath \\leavevmode\\hbox{\\small1\\kern-3.8pt\\normalsize1}_{k_2} & \\imath \\leavevmode\\hbox{\\small1\\kern-3.8pt\\normalsize1}_{k_2} \\end{array}\\right]\n\\end{equation}\nand $U|_{\\beta=2}= \\leavevmode\\hbox{\\small1\\kern-3.8pt\\normalsize1}_{k_1+k_2}$, $\\widehat{B}=UBU^\\dagger$ lies in the well-known symmetric superspaces \\cite{Zir96},\n\\begin{eqnarray}\n \\fl\\widetilde{\\Sigma}_{\\beta,\\gamma_1k_1,\\gamma_2k_2}^{(\\dagger)} & = &\\Biggl\\{\\sigma\\in{\\rm Mat}(\\tilde{\\gamma}k_1\/\\tilde{\\gamma}k_2)\\Biggl|\\sigma^\\dagger=\\sigma,\\nonumber\\\\\n & & \\left.\\sigma^*=\\left\\{\\begin{array}[c]{ll}\n                \\widehat{Y}_{\\gamma_1k_1,\\gamma_2k_2}\\sigma\\widehat{Y}_{\\gamma_1k_1,\\gamma_2k_2}^{T} & ,\\ \\beta\\in\\{1,4\\} \\\\\n\t\t\\widehat{Y}_{k_1k_2}\\sigma^*\\widehat{Y}_{k_1k_2}^{T} & ,\\ \\beta=2\n          \\end{array}\\right\\}\\right\\}\\label{2.17}\n\\end{eqnarray}\nwhere\n\\begin{equation}\\label{2.18}\n \\fl\\left.\\widehat{Y}_{pq}\\right|_{\\beta=1}=\\left[\\begin{array}{cc}   \\leavevmode\\hbox{\\small1\\kern-3.8pt\\normalsize1}_{p} & 0 \\\\ 0 & Y_s\\otimes \\leavevmode\\hbox{\\small1\\kern-3.8pt\\normalsize1}_{q} \\end{array}\\right]\\ ,\\ \\ \\left.\\widehat{Y}_{pq}\\right|_{\\beta=2}=  \\leavevmode\\hbox{\\small1\\kern-3.8pt\\normalsize1}_{p+q}\\ \\ \\ {\\rm and}\\ \\ \\ \\left.\\widehat{Y}_{pq}\\right|_{\\beta=4}=\\left[\\begin{array}{cc} Y_s\\otimes \\leavevmode\\hbox{\\small1\\kern-3.8pt\\normalsize1}_{p} & 0 \\\\ 0 &  \\leavevmode\\hbox{\\small1\\kern-3.8pt\\normalsize1}_{q} \\end{array}\\right].\n\\end{equation}\nThe set ${\\rm Mat}(p\/q)$ is the set of $(p+q)\\times(p+q)$ supermatrices on the complex Grassmann algebra $\\bigoplus\\limits_{j=0}^{2pq}\\Lambda_j$. The entries of the diagonal blocks of an element in ${\\rm Mat}(p\/q)$ lie in $\\Lambda^0$ whereas the entries of the off-diagonal block are elements in $\\Lambda^1$.\n\nThe rectangular supermatrix $\\widehat{V}^\\dagger=V^\\dagger U^\\dagger$ is composed of real, complex or quaternionic supervectors whose adjoints form the rows. They are given by\n\\begin{equation}\\label{2.19}\n \\fl\\Psi_j^\\dagger=\\left\\{\\begin{array}{ll}\n                        \\left(\\left\\{\\sqrt{2}{\\rm Re\\,} z_{jn},\\sqrt{2}{\\rm Im\\,} z_{jn}\\right\\}_{1\\leq n\\leq k_1},\\left\\{\\zeta_{jn},\\zeta^*_{jn}\\right\\}_{1\\leq n\\leq k_2}\\right) & ,\\ \\beta=1,\\\\\n                        \\left(\\left\\{z_{jn}\\right\\}_{1\\leq n\\leq k_1},\\left\\{\\zeta_{jn}\\right\\}_{1\\leq n\\leq k_2}\\right) & ,\\ \\beta=2,\\\\\n                        \\left(\\left\\{\\begin{array}{cc} z_{jn} & -z_{j+N,n}^* \\\\ z_{j+N,n} & z_{jn}^* \\end{array}\\right\\}_{1\\leq n\\leq k_1},\\displaystyle\\left\\{\\begin{array}{cc} \\zeta_{jn}^{(-)} & \\zeta_{jn}^{(+)} \\\\ \\zeta_{jn}^{(-)*} & \\zeta_{jn}^{(+)*} \\end{array}\\right\\}_{1\\leq n\\leq k_2}\\right) & ,\\ \\beta=4,\n                       \\end{array}\\right.\n\\end{equation}\nrespectively, where $\\zeta_{jn}^{(\\pm)}=\\imath^{(1\\pm1)\/2}(\\zeta_{jn}\\pm\\zeta_{j+N,n}^*)\/\\sqrt{2}$. Then, the supermatrix $\\widehat{B}$ acquires the form\n\\begin{equation}\\label{2.20}\n \\widehat{B}=\\frac{1}{\\tilde{\\gamma}}\\sum\\limits_{j=1}^N\\Psi_j\\Psi_j^\\dagger .\n\\end{equation}\nThe integrand in Eq.~\\eref{2.12}\n\\begin{equation}\\label{2.21}\n F\\left(\\widehat{B}\\right)=\\Phi\\left(\\widehat{B}\\right)\\exp\\left(-\\imath{\\rm Str\\,} E\\widehat{B}\\right)\n\\end{equation}\ncomprises a symmetry breaking term,\n\\begin{equation}\\label{2.22}\n \\exists\\ U\\in{\\rm U\\,}^{(\\beta)}(\\gamma_1k_1\/\\gamma_2k_2)\\ {\\rm\\ that\\ \\ }F\\left(\\widehat{B}\\right)\\neq F\\left(U\\widehat{B}U^\\dagger\\right) ,\n\\end{equation}\naccording to the supergroup\n\\begin{equation}\\label{2.23}\n {\\rm U\\,}^{(\\beta)}(\\gamma_1k_1\/\\gamma_2k_2)=\\left\\{\\begin{array}{ll}\n                 {\\rm UOSp\\,}^{(+)}(2k_1\/2k_2) & ,\\ \\beta=1\\\\\n                 {\\rm U\\,}(k_1\/k_2) & ,\\ \\beta=2\\\\\n                 {\\rm UOSp\\,}^{(-)}(2k_1\/2k_2) & ,\\ \\beta=4\n                \\end{array}\\right. .\n\\end{equation}\nWe use the notation of Refs.~\\cite{KohGuh05,KKG08} for the representations ${\\rm UOSp\\,}^{(\\pm)}$ of the supergroup ${\\rm UOSp\\,}$. These representations are related to the classification of Riemannian symmetric superspaces by Zirnbauer \\cite{Zir96}. The index ``$+$'' in Eq.~\\eref{2.23} refers to real entries in the Boson--Boson block and to quaternionic entries in the Fermion--Fermion block and ``$-$'' indicates the other way around.\n\n\\section{Supersymmetric Wishart--matrices and some of their properties}\\label{sec3}\n\nWe generalize the integrand \\eref{2.21} to arbitrary sufficiently integrable superfunctions on rectangular $(\\gamma_2c+\\gamma_1d)\\times(\\gamma_2a+\\gamma_1b)$ supermatrices $\\widehat{V}$ on the complex Grassmann--algebra $\\Lambda=\\bigoplus\\limits_{j=0}^{2(ad+bc)}\\Lambda_j$. Such a supermatrix\n\\begin{equation}\\label{3.1}\n \\fl\\widehat{V}=\\left(\\Psi_{11}^{({\\rm C})},\\ldots,\\Psi_{a1}^{({\\rm C})},\\Psi_{12}^{({\\rm C})},\\ldots\\Psi_{b2}^{({\\rm C})}\\right)=\\left(\\Psi_{11}^{({\\rm R})*}\\ldots,\\Psi_{c1}^{({\\rm R})*},\\Psi_{12}^{({\\rm R})*},\\ldots\\Psi_{d2}^{({\\rm R})*}\\right)^{T_{\\rm S}}\n\\end{equation}\nis defined by its columns\n\\begin{eqnarray}\n \\Psi_{j1}^{({\\rm C})\\dagger} & = & \\left\\{\\begin{array}{ll}\n                        \\left(\\left\\{x_{jn}\\right\\}_{1\\leq n\\leq c},\\left\\{\\chi_{jn},\\chi_{jn}^*\\right\\}_{1\\leq n\\leq d}\\right) & ,\\ \\beta=1,\\\\\n                        \\left(\\left\\{z_{jn}\\right\\}_{1\\leq n\\leq c},\\left\\{\\chi_{jn}\\right\\}_{1\\leq n\\leq d}\\right) & ,\\ \\beta=2,\\\\\n                        \\left(\\left\\{\\begin{array}{cc} z_{jn1} & -z_{jn2}^* \\\\ z_{jn2} & z_{jn1}^* \\end{array}\\right\\}_{1\\leq n\\leq c},\\left\\{\\begin{array}{c} \\chi_{jn} \\\\ \\chi_{jn}^* \\end{array}\\right\\}_{1\\leq n\\leq d}\\right) & ,\\ \\beta=4,\n                       \\end{array}\\right.\\label{3.2a}\\\\\n \\Psi_{j2}^{({\\rm C})\\dagger} & = & \\left\\{\\begin{array}{ll}\n                        \\left(\\left\\{\\begin{array}{c} \\zeta_{jn} \\\\ \\zeta_{jn}^* \\end{array}\\right\\}_{1\\leq n\\leq c},\\left\\{\\begin{array}{cc} \\tilde{z}_{jn1} & -\\tilde{z}_{jn2}^* \\\\ \\tilde{z}_{jn2} & \\tilde{z}_{jn1}^* \\end{array}\\right\\}_{1\\leq n\\leq d}\\right) & ,\\ \\beta=1,\\\\\n                        \\left(\\left\\{\\zeta_{jn}\\right\\}_{1\\leq n\\leq c},\\left\\{\\tilde{z}_{jn}\\right\\}_{1\\leq n\\leq d}\\right) & ,\\ \\beta=2,\\\\\n                        \\left(\\left\\{\\zeta_{jn},\\zeta^*_{jn}\\right\\}_{1\\leq n\\leq c},\\left\\{y_{jn}\\right\\}_{1\\leq n\\leq d}\\right) & ,\\ \\beta=4,\n                       \\end{array}\\right.\\label{3.2b}\n\\end{eqnarray}\nor by its rows\n\\begin{eqnarray}\n \\Psi_{j1}^{({\\rm R})\\dagger} & = & \\left\\{\\begin{array}{ll}\n                        \\left(\\left\\{x_{nj}\\right\\}_{1\\leq n\\leq a},\\left\\{ \\zeta_{nj}^*, -\\zeta_{nj} \\right\\}_{1\\leq n\\leq b}\\right) & ,\\ \\beta=1,\\\\\n                        \\left(\\left\\{z_{nj}^*\\right\\}_{1\\leq n\\leq a},\\left\\{\\zeta_{nj}^*\\right\\}_{1\\leq n\\leq b}\\right) & ,\\ \\beta=2,\\\\\n                        \\left(\\left\\{\\begin{array}{cc} z_{nj1}^* & z_{nj2}^* \\\\ -z_{nj2} & z_{nj1} \\end{array}\\right\\}_{1\\leq n\\leq a},\\left\\{\\begin{array}{c} \\zeta_{nj}^* \\\\ -\\zeta_{nj} \\end{array}\\right\\}_{1\\leq n\\leq b}\\right) & ,\\ \\beta=4,\n                       \\end{array}\\right.\\label{3.3a}\\\\\n \\Psi_{j2}^{({\\rm R})\\dagger} & = & \\left\\{\\begin{array}{ll}\n                        \\left(\\left\\{\\begin{array}{c} -\\chi_{nj}^* \\\\ \\chi_{nj} \\end{array}\\right\\}_{1\\leq n\\leq a},\\left\\{\\begin{array}{cc} \\tilde{z}_{nj1}^* & \\tilde{z}_{nj2}^* \\\\ -\\tilde{z}_{nj2} & \\tilde{z}_{nj1} \\end{array}\\right\\}_{1\\leq n\\leq b}\\right) & ,\\ \\beta=1,\\\\\n                        \\left(\\left\\{-\\chi_{nj}^*\\right\\}_{1\\leq n\\leq a},\\left\\{\\tilde{z}_{nj}^*\\right\\}_{1\\leq n\\leq b}\\right) & ,\\ \\beta=2,\\\\\n                        \\left(\\left\\{-\\chi_{nj}^*,\\chi_{nj}\\right\\}_{1\\leq n\\leq a},\\left\\{y_{nj}\\right\\}_{1\\leq n\\leq b}\\right) & ,\\ \\beta=4\n                       \\end{array}\\right.\\label{3.3b}\n\\end{eqnarray}\nwhich are real, complex and quaternionic supervectors. We use the complex Grassmann variables $\\chi_{mn}$ and $\\zeta_{mn}$ and the real numbers $x_{mn}$ and $y_{mn}$. Also, we introduce the complex numbers $z_{mn}$, $\\tilde{z}_{mn}$, $z_{mnl}$ and $\\tilde{z}_{mnl}$. The $(\\gamma_2c+\\gamma_1d)\\times(\\gamma_2c+\\gamma_1d)$ supermatrix $\\widehat{B}=\\tilde{\\gamma}^{-1}\\widehat{V}\\widehat{V}^\\dagger$ can be written in the columns of $\\widehat{V}$ as in Eq.~\\eref{2.20}. As this supermatrix has a form similar to the ordinary Wishart--matrices, we refer to it as supersymmetric Wishart--matrix. The rectangular supermatrix above fulfills the property\n\\begin{equation}\\label{3.4}\n \\widehat{V}^*=\\widehat{Y}_{cd}\\widehat{V}\\widehat{Y}_{ab}^T .\n\\end{equation}\nThe corresponding generating function \\eref{2.1} is an integral over a rotation invariant superfunction $P$ on a superspace, which is sufficiently convergent and analytic in its real independent variables,\n\\begin{equation}\\label{3.5}\n  Z_{cd}^{ab}(E^-) =\\int\\limits_{\\widetilde{\\Sigma}_{\\beta,ab}^{(-\\psi)}} P(\\sigma) {\\rm Sdet\\,}^{-1\/\\tilde{\\gamma}}\\left(\\sigma\\otimes\\widehat{\\Pi}_{2\\psi}^{({\\rm C})}- \\leavevmode\\hbox{\\small1\\kern-3.8pt\\normalsize1}_{\\gamma_2a+\\gamma_1b}\\otimes E^-\\right)d[\\sigma],\n\\end{equation}\nwhere\n\\begin{equation}\\label{3.5b}\n \\fl E^-={\\rm diag\\,}\\left(E_{11}\\otimes \\leavevmode\\hbox{\\small1\\kern-3.8pt\\normalsize1}_{\\gamma_2},\\ldots,E_{c1}\\otimes \\leavevmode\\hbox{\\small1\\kern-3.8pt\\normalsize1}_{\\gamma_2},E_{12}\\otimes \\leavevmode\\hbox{\\small1\\kern-3.8pt\\normalsize1}_{\\gamma_1},\\ldots,E_{d2}\\otimes \\leavevmode\\hbox{\\small1\\kern-3.8pt\\normalsize1}_{\\gamma_1}\\right)-\\imath\\varepsilon \\leavevmode\\hbox{\\small1\\kern-3.8pt\\normalsize1}_{\\gamma_2c+\\gamma_1d} .\n\\end{equation}\nLet $\\widetilde{\\Sigma}_{\\beta,ab}^{0(\\dagger)}$ be a subset of $\\widetilde{\\Sigma}_{\\beta,ab}^{(\\dagger)}$. The entries of elements in $\\widetilde{\\Sigma}_{\\beta,ab}^{0(\\dagger)}$ lie in $\\Lambda_0$ and $\\Lambda_1$. The set $\\widetilde{\\Sigma}_{\\beta,ab}^{(-\\psi)}=\\widehat{\\Pi}_{-\\psi}^{({\\rm R})}\\widetilde{\\Sigma}_{\\beta,ab}^{0(\\dagger)}\\widehat{\\Pi}_{-\\psi}^{({\\rm R})}$ is the Wick--rotated set of $\\widetilde{\\Sigma}_{\\beta,ab}^{0(\\dagger)}$ by the generalized Wick--rotation $\\widehat{\\Pi}_{-\\psi}^{({\\rm R})}={\\rm diag\\,}( \\leavevmode\\hbox{\\small1\\kern-3.8pt\\normalsize1}_{\\gamma_2a},e^{-\\imath\\psi\/2} \\leavevmode\\hbox{\\small1\\kern-3.8pt\\normalsize1}_{\\gamma_1b})$. As in Ref.~\\cite{KGG08}, we introduce such a rotation for the convergence of the integral \\eref{3.5}. The matrix $\\widehat{\\Pi}_{2\\psi}^{({\\rm C})}={\\rm diag\\,}( \\leavevmode\\hbox{\\small1\\kern-3.8pt\\normalsize1}_{\\gamma_2c},e^{\\imath\\psi} \\leavevmode\\hbox{\\small1\\kern-3.8pt\\normalsize1}_{\\gamma_1d})$ is also a Wick--rotation.\n\nIn the rest of our work, we restrict the calculations to a class of superfunctions. These superfunctions has a Wick--rotation such that the integrals are convergent. We have not explicitly analysed the class of such functions. However, this class is very large and sufficient for physical interests. We consider the probability distribution\n\\begin{equation}\\label{3.5c}\n P(\\sigma)=f(\\sigma)\\exp(-{\\rm Str\\,}\\sigma^{2m}),\n\\end{equation}\nwhere $m\\in\\mathbb{N}$ and $f$ is a superfunction which does not increase so fast as $\\exp({\\rm Str\\,}\\sigma^{2m})$ in the infinty, in particular\n\\begin{equation}\\label{3.5d}\n \\underset{\\epsilon\\to\\infty}{\\lim}P\\left(\\epsilon e^{\\imath\\alpha}\\sigma\\right)=0 \\Leftrightarrow \\underset{\\epsilon\\to\\infty}{\\lim}\\exp\\left(-\\epsilon e^{\\imath\\alpha}{\\rm Str\\,}\\sigma^{2m}\\right)=0\n\\end{equation}\nfor every angle $\\alpha\\in[0,2\\pi]$. Then, a Wick--rotation exists for $P$.\n\nTo guarantee the convergence of the integrals below, let $\\widehat{V}_{\\psi}=\\widehat{\\Pi}_{\\psi}^{({\\rm C})}\\widehat{V}$, $\\widehat{V}_{-\\psi}^\\dagger=\\widehat{V}^\\dagger\\widehat{\\Pi}_{\\psi}^{({\\rm C})}$ and $\\widehat{B}_{\\psi}=\\widehat{\\Pi}_{\\psi}^{({\\rm C})}\\widehat{B}\\widehat{\\Pi}_{\\psi}^{({\\rm C})}$. Considering a function $f$ on the set of supersymmetric Wishart--matrices, we give a lemma and a corollary which are of equal importance for the superbosonization formula and the generalized Hubbard--Startonovich transformation. For $b=0$, the lemma presents the duality relation between the ordinary and superspace \\eref{2.11} which is crucial for the calculation of \\eref{2.1}. This lemma was proven in Ref.~\\cite{LSZ07} by representation theory. Here, we only state it.\n\\begin{lemma}\\label{c1}\\ \\\\\n Let $f$ be a superfunction on rectangular supermatrices of the form \\eref{3.1} and invariant under\n \\begin{equation}\\label{c1.1}\n  f(\\widehat{V}_{\\psi},\\widehat{V}_{-\\psi}^\\dagger)=f\\left(\\widehat{V}_{\\psi}U^\\dagger,U\\widehat{V}_{-\\psi}^\\dagger\\right)\\ ,\n \\end{equation}\n for all $\\widehat{V}$ and $U\\in{\\rm U\\,}^{(\\beta)}(a\/b)$. Then, there is a superfunction $F$ on the ${\\rm U\\,}^{(\\beta)}(c\/d)$--symmetric supermatrices with\n \\begin{equation}\\label{c1.2}\n  F(\\widehat{B}_{\\psi})=f(\\widehat{V}_{\\psi},\\widehat{V}_{-\\psi}^\\dagger)\\ .\n \\end{equation}\n\\end{lemma}\n\nThe ${\\rm U\\,}^{(\\beta)}(c\/d)$--symmetric supermatrices are elements of $\\widetilde{\\Sigma}_{\\beta,ab}^{(\\dagger)}$. The invariance condition \\eref{c1.1} implies that $f$ only depends on the rows of $\\widehat{V}_{\\psi}$ by $\\Psi_{nr}^{({\\rm R})\\dagger}\\Psi_{ms}^{({\\rm R})}$ for arbitrary $n,m,r$ and $s$. These scalar products are the entries of the supermatrix $\\widehat{V}_{\\psi}\\widehat{V}_{-\\psi}^\\dagger$ which leads to the statement.\n\nThe corollary below is an application of integral theorems by Wegner \\cite{Weg83} worked out in Refs.~\\cite{Con88,ConGro89} and of the Theorems III.1, III.2 and III.3 in Ref.~\\cite{KKG08}. It states that an integration over supersymmetric Wishart--matrices can be reduced to integrations over supersymmetric Wishart--matrices comprising a lower dimensional rectangular supermatrix. In particular for the generating function, it reflects the equivalence of the integral \\eref{3.5} with an integration over smaller supermatrices \\cite{KKG08}. We assume that $\\tilde{a}=a-2(b-\\tilde{b})\/\\beta\\geq0$ with\n\\begin{equation}\\label{3.6}\n \\tilde{b}=\\left\\{\\begin{array}{ll}\n\t\t1 & ,\\ \\beta=4\\ {\\rm and}\\ b\\in2\\mathbb{N}_0+1\\\\\n\t\t0 & ,\\ {\\rm else}\n\t   \\end{array}\\right. .\n\\end{equation}\n\\begin{corollary}\\label{c2}\\ \\\\\n Let $F$ be the superfunction of Lemma \\ref{c1}, real analytic in its real independent entries and a Schwartz--function. Then, we find\n \\begin{equation}\\label{c2.1}\n  \\displaystyle\\int\\limits_{\\mathfrak{R}}F(\\widehat{B}_{\\psi})d[\\widehat{V}]=C\\int\\limits_{\\widetilde{\\mathfrak{R}}}F(\\widetilde{B}_{\\psi})d[\\widetilde{V}]\n \\end{equation}\n where $\\widetilde{B}=\\tilde{\\gamma}^{-1}\\widetilde{V}\\widetilde{V}$. The sets are $\\mathfrak{R}=\\mathbb{R}^{\\beta ac+4bd\/\\beta}\\times\\Lambda_{2(ad+bc)}$ and $\\widetilde{\\mathfrak{R}}=\\mathbb{R}^{\\beta\\tilde{a}c+4\\tilde{b}d\/\\beta}\\times\\Lambda_{2(\\tilde{a}d+\\tilde{b}c)}$, the constant is\n \\begin{equation}\\label{c2.2}\n  C=\\left[-\\frac{\\gamma_1}{2}\\right]^{(b-\\tilde{b})c}\\left[\\frac{\\gamma_2}{2}\\right]^{(a-\\tilde{a})d}\n \\end{equation}\n and the measure\n\\begin{equation}\\label{c2.3}\n d[\\widehat{V}]=\\underset{1\\leq l\\leq \\beta}{\\underset{1\\leq n\\leq c}{\\underset{1\\leq m\\leq a}{\\prod}}}dx_{mnl}\\underset{1\\leq l\\leq 4\/\\beta}{\\underset{1\\leq n\\leq d}{\\underset{1\\leq m\\leq b}{\\prod}}}dy_{mnl}\\underset{1\\leq n\\leq c}{\\underset{1\\leq m\\leq b}{\\prod }}d\\zeta_{mn}d\\zeta_{mn}^*\\underset{1\\leq n\\leq d}{\\underset{1\\leq m\\leq a}{\\prod }}d\\chi_{mn}d\\chi_{mn}^*\\ .\n\\end{equation}\nThe $(\\gamma_2c+\\gamma_1d)\\times(\\gamma_2\\tilde{a}+\\gamma_1\\tilde{b})$ supermatrix $\\widetilde{V}$ and its measure $d[\\widetilde{V}]$ is defined analogous to $\\widehat{V}$ and $d[\\widehat{V}]$, respectively. Here, $x_{mna}$ and $y_{mna}$ are the independent real components of the real, complex and quaternionic numbers of the supervectors $\\Psi_{j1}^{({\\rm R})}$ and $\\Psi_{j2}^{({\\rm R})}$, respectively.\n\\end{corollary}\n\\textbf{Proof:}\\\\\nWe integrate $F$ over all supervectors $\\Psi_{j1}^{({\\rm R})}$ and $\\Psi_{j2}^{({\\rm R})}$ except $\\Psi_{11}^{({\\rm R})}$. Then, \n \\begin{equation}\\label{c2.4}\n  \\displaystyle\\int\\limits_{\\mathfrak{R}^\\prime}F(V_{\\psi}V_{-\\psi}^\\dagger)d[\\widehat{V}_{\\neq11}]\n \\end{equation}\nonly depends on $\\Psi_{11}^{({\\rm R})\\dagger}\\Psi_{11}^{({\\rm R})}$. The integration set is $\\mathfrak{R}^\\prime=\\mathbb{R}^{\\beta a(c-1)+4bd\/\\beta}\\times\\Lambda_{2(ad+b(c-1))}$ and the measure $d[\\widehat{V}_{\\neq11}]$ is $d[\\widehat{V}]$ without the measure for the supervector $\\Psi_{11}^{({\\rm R})}$. With help of the Theorems in Ref.~\\cite{Weg83,Con88,ConGro89,KKG08}, the integration over $\\Psi_{11}^{({\\rm R})}$ is up to a constant equivalent to an integration over a supervector $\\widetilde{\\Psi}_{11}^{({\\rm R})}$. This supervector is equal to $\\Psi_{11}^{({\\rm R})}$ in the first $\\tilde{a}$--th entries and else zero. We repeat this procedure for all other supervectors reminding that we only need the invariance under the supergroup action ${\\rm U\\,}^{(\\beta)}\\left(b-\\tilde{b}\/b-\\tilde{b}\\right)$ on $f$ as in Eq.~\\eref{c1.1} embedded in ${\\rm U\\,}^{(\\beta)}(a\/b)$. This invariance is preserved in each step due to the zero entries in the new supervectors. \\hfill$\\square$\n\nThis corollary allows us to restrict our calculation on supermatrices with $b=1$ only to $\\beta=4$ and $b=0$ for all $\\beta$. Only the latter case is of physical interest. Thus, we give the computation for $b=0$ in the following sections and consider the case $b=1$ in Sec. \\ref{sec7}. For $b=0$ we omit the Wick--rotation for $\\widehat{B}$ as it is done in Refs.~\\cite{Guh06,KGG08} due to the convergence of the integral \\eref{3.5}.\n\n\\section{The superbosonization formula}\\label{sec4}\n\nWe need for the following theorem the definition of the sets\n\\begin{eqnarray}\n \\fl\\Sigma_{1,pq} = \\left\\{\\left.\\sigma=\\left[\\begin{array}{ccc} \\sigma_1 & \\eta & \\eta^* \\\\ -\\eta^\\dagger & \\sigma_{21} & \\sigma_{22}^{(1)} \\\\ \\eta^T  & \\sigma_{22}^{(2)} & \\sigma_{21}^T \\end{array}\\right]\\in{\\rm Mat}(p\/2q)\\right|\\sigma_1^\\dagger=\\sigma_1^*=\\sigma_1\\ {\\rm with\\ positive}\\right.\\nonumber\\\\\n \\left.{\\rm definite\\ body},\\ \\sigma_{22}^{(1)T}=-\\sigma_{22}^{(1)},\\ \\sigma_{22}^{(2)T}=-\\sigma_{22}^{(2)}\\right\\},\\label{4.1}\\\\\n \\fl\\Sigma_{2,pq} = \\left\\{\\left.\\sigma=\\left[\\begin{array}{cc} \\sigma_1 & \\eta \\\\ -\\eta^\\dagger & \\sigma_2 \\end{array}\\right]\\in{\\rm Mat}(p\/q)\\right|\\sigma_1^\\dagger=\\sigma_1\\ {\\rm with\\ positive\\ definite\\ body}\\right\\},\\label{4.2}\\\\\n \\fl\\Sigma_{4,pq} = \\left\\{\\sigma=\\left[\\begin{array}{ccc} \\sigma_{11} & \\sigma_{12} & \\eta \\\\ -\\sigma_{12}^* & \\sigma_{11}^* & \\eta^* \\\\ -\\eta^\\dagger  & \\eta^T & \\sigma_2 \\end{array}\\right]\\in{\\rm Mat}(2p\/q)\\right|\\sigma_1^\\dagger=\\sigma_1=\\left[\\begin{array}{cc} \\sigma_{11} & \\sigma_{12} \\\\ -\\sigma_{12}^* & \\sigma_{11}^* \\end{array}\\right]\\nonumber\\\\\n {\\rm with\\ positive\\ definite\\ body},\\ \\sigma_2=\\sigma_2^T\\Biggl\\}.\\label{4.3}\n\\end{eqnarray}\nAlso, we will use the sets\n\\begin{equation}\\label{4.4}\n \\Sigma_{\\beta,pq}^{(\\dagger)} = \\left\\{\\left.\\sigma\\in\\Sigma_{\\beta,pq}\\right|\\sigma_2^\\dagger=\\sigma_2\\right\\}=\\widetilde{\\Sigma}_{\\beta,pq}^{(\\dagger)}\\cap\\Sigma_{\\beta,pq}\n\\end{equation}\nand\n\\begin{equation}\\label{4.5}\n \\Sigma_{\\beta,pq}^{({\\rm c})} = \\left\\{\\left.\\sigma\\in\\Sigma_{\\beta,pq}\\right|\\sigma_2\\in{\\rm CU\\,}^{(4\/\\beta)}\\left(q\\right)\\right\\}\n\\end{equation}\nwhere ${\\rm CU\\,}^{(\\beta)}\\left(q\\right)$ is the set of the circular orthogonal (COE, $\\beta=1$), unitary (CUE, $\\beta=2$) or unitary-symplectic (CSE, $\\beta=4$) ensembles,\n\\begin{equation}\\label{4.6}\n \\fl{\\rm CU\\,}^{(\\beta)}\\left(q\\right)=\\left\\{A\\in{\\rm Gl}(\\gamma_2q,\\mathbb{C})\\left| \\begin{array}{ll} A=A^T\\in{\\rm U\\,}^{(2)}(q) & ,\\ \\beta=1 \\\\ A\\in{\\rm U\\,}^{(2)}(q) & ,\\ \\beta=2 \\\\ A=(Y_s\\otimes\\leavevmode\\hbox{\\small1\\kern-3.8pt\\normalsize1}_q)A^T(Y_s^T\\otimes\\leavevmode\\hbox{\\small1\\kern-3.8pt\\normalsize1}_q)\\in{\\rm U\\,}^{(2)}(2q) & ,\\ \\beta=4 \\end{array}\\right.\\right\\}\n\\end{equation}\nThe index ``$\\dagger$'' in Eq.~\\eref{4.4} refers to the self-adjointness of the supermatrices and the index ``${\\rm c}$'' indicates the relation to the circular ensembles. We notice that the set classes presented above differ in the Fermion--Fermion block. In Sec. \\ref{sec6}, we show that this is the crucial difference between both methods. Due to the nilpotence of $B$'s Fermion--Fermion block, we can change the set in this block for the Fourier--transformation. The sets of matrices in the sets above with entries in $\\Lambda_0$ and $\\Lambda_1$ are denoted by $\\Sigma_{\\beta,pq}^{0}$, $\\Sigma_{\\beta,pq}^{0(\\dagger)}$ and $\\Sigma_{\\beta,pq}^{0({\\rm c})}$, respectively.\n\nThe proof of the superbosonization formula \\cite{Som07,LSZ07} given below is based on the proofs of the superbosonization formula for arbitrary superfunctions on real supersymmetric Wishart--matrices in Ref.~\\cite{Som07} and for Gaussian functions on real, complex and quaternionic Wishart--matrices in Ref.~\\cite{Som08}. This theorem extends the superbosonization formula of Ref.~\\cite{LSZ07} to averages of square roots of determinants over unitary-symplectically invariant ensembles, i.e. $\\beta=4$, $b=c=0$ and $d$ odd in Eq.~\\eref{3.5}. The proof of this theorem is given in  \\ref{app1}.\n\\begin{theorem}[Superbosonization formula]\\label{t1}\\ \\\\\n Let $F$ be a conveniently integrable and analytic superfunction on the set of $\\left(\\gamma_2c+\\gamma_1d\\right)\\times\\left(\\gamma_2c+\\gamma_1d\\right)$ supermatrices and\n\\begin{equation}\\label{t1.3}\n \\kappa=\\frac{a-c+1}{\\gamma_1}+\\frac{d-1}{\\gamma_2} .\n\\end{equation}\nWith\n\\begin{equation}\\label{t1.0}\n a\\geq c\\ ,\n\\end{equation}\nwe find\n \\begin{equation}\\label{t1.1}\n  \\fl\\int\\limits_{\\mathfrak{R}}F(\\widehat{B})\\exp\\left(-\\varepsilon{\\rm Str\\,} \\widehat{B}\\right)d[\\widehat{V}]=C_{acd}^{(\\beta)}\\int\\limits_{\\Sigma_{\\beta,cd}^{0({\\rm c})}}F(\\rho)\\exp\\left(-\\varepsilon{\\rm Str\\,}\\rho\\right){\\rm Sdet\\,}\\rho^{\\kappa}d[\\rho] ,\n \\end{equation}\n where the constant is\n\\begin{eqnarray}\n \\fl C_{acd}^{(\\beta)}\n =  \\left(-2\\pi\\gamma_1\\right)^{-ad}\\left(-\\frac{2\\pi}{\\gamma_2}\\right)^{cd}2^{-c}\\tilde{\\gamma}^{\\beta ac\/2}\\frac{{\\rm Vol}\\left({\\rm U\\,}^{(\\beta)}(a)\\right)}{{\\rm Vol}\\left({\\rm U\\,}^{(\\beta)}(a-c)\\right)}\\times\\nonumber\\\\\n \\times\\prod\\limits_{n=1}^d\\frac{\\Gamma\\left(\\gamma_1\\kappa+2(n-d)\/\\beta\\right)}{\\imath^{4(n-1)\/\\beta}\\pi^{2(n-1)\/\\beta}} .\\label{t1.2}\n\\end{eqnarray}\nWe define the measure $d[\\widehat{V}]$ as in Corollary \\ref{c2} and the measure on the right hand side is $d[\\rho]=d[\\rho_1]d[\\rho_2]d[\\eta]$ where\n\\begin{eqnarray}\n \\fl d[\\rho_1] & = & \\prod\\limits_{n=1}^{c}d\\rho_{nn1}\\times\\left\\{\\begin{array}{ll}\n                  \\prod\\limits_{1\\leq n< m\\leq c}d\\rho_{nm1} & ,\\ \\beta=1,\\\\\n                  \\prod\\limits_{1\\leq n< m\\leq c}d{\\rm Re\\,}\\rho_{nm1}d{\\rm Im\\,}\\rho_{nm1} & ,\\ \\beta=2,\\\\\n                  \\prod\\limits_{1\\leq n< m\\leq c}d{\\rm Re\\,}\\rho_{nm11}d{\\rm Im\\,}\\rho_{nm11}d{\\rm Re\\,}\\rho_{nm12}d{\\rm Im\\,}\\rho_{nm12} & ,\\ \\beta=4,\n                 \\end{array}\\right.\\label{t1.4}\\\\\n \\fl d[\\rho_2] & = & {\\rm FU}_{d}^{(4\/\\beta)}|\\Delta_{d}(e^{\\imath\\varphi_j})|^{4\/\\beta}\\prod\\limits_{n=1}^{d}\\frac{de^{\\imath\\varphi_n}}{2\\pi\\imath}d\\mu(U)\\label{t1.5} ,\\\\\n \\fl d[\\eta]    & = & \\prod\\limits_{n=1}^{c}\\prod\\limits_{m=1}^{d}(d\\eta_{nm}d\\eta_{nm}^*)\\label{t1.6}.\n\\end{eqnarray}\nHere, $\\rho_2=U{\\rm diag\\,}\\left(e^{\\imath\\varphi_1},\\ldots,e^{\\imath\\varphi_{d}}\\right)U^\\dagger$, $U\\in{\\rm U\\,}^{(4\/\\beta)}\\left(d\\right)$ and $d\\mu(U)$ is the normalized Haar-measure of ${\\rm U\\,}^{(4\/\\beta)}\\left(d\\right)$. We introduce the volumes of the rotation groups\n\\begin{equation}\\label{t1.7}\n {\\rm Vol}\\left({\\rm U\\,}^{(\\beta)}(n)\\right)=\\prod\\limits_{j=1}^n\\frac{2\\pi^{\\beta j\/2}}{\\Gamma\\left(\\beta j\/2\\right)}\n\\end{equation}\nand the ratio of volumes of the group flag manifold and the permutation group\n\\begin{equation}\\label{t1.8}\n {\\rm FU}_{d}^{(4\/\\beta)}=\\frac{1}{d!}\\prod\\limits_{j=1}^d\\frac{\\pi^{2(j-1)\/\\beta}\\Gamma(2\/\\beta)}{\\Gamma(2j\/\\beta)}\\ .\n\\end{equation}\nThe absolute value of the Vandermonde determinant $\\Delta_{d}(e^{\\imath\\varphi_j})=\\prod\\limits_{1\\leq n<m\\leq d}\\left(e^{\\imath\\varphi_n}-e^{\\imath\\varphi_m}\\right)$ refers to a change of sign in every single difference $\\left(e^{\\imath\\varphi_n}-e^{\\imath\\varphi_m}\\right)$ with ``$+$'' if $\\varphi_m<\\varphi_n$ and with ``$-$'' otherwise. Thus, it is not an absolute value in the complex plane.\n\\end{theorem}\n\nThe exponential term can also be shifted in the superfunction $F$. We need this additional term to regularize an intermediate step in the proof.\n\nThe inequality \\eref{t1.0} is crucial. For example, let $\\beta=2$ and $F(\\rho)=1$. Then, the left hand side of Eq.~\\eref{t1.1} is not equal to zero. On the right hand side of Eq.~\\eref{t1.1}, the dependence on the Grassmann variables  only stems from the superdeterminant and we find\n\\begin{equation}\\label{4.7}\n  \\int\\limits_{\\Lambda_{2cd}}{\\rm Sdet\\,}\\rho^\\kappa d[\\eta]=\\int\\limits_{\\Lambda_{2cd}}\\frac{\\det\\left(\\rho_1+\\eta\\rho_2^{-1}\\eta^\\dagger\\right)^\\kappa}{\\det\\rho_2^\\kappa} d[\\eta]=0\n\\end{equation}\nfor $\\kappa<d$. The superdeterminant ${\\rm Sdet\\,}\\rho$ is a polynomial of order $2c$ in the Grassmann variables $\\{\\eta_{nm},\\eta_{nm}^*\\}$ and the integral over the remaining variables is finite for $\\kappa\\geq0$. Hence, it is easy to see that the right hand side of Eq.~\\eref{t1.1} is zero for $\\kappa<d$. This inequality is equivalent to $a<c$.\n\nThis problem was also discussed in Ref.~\\cite{BEKYZ07}. These authors gave a solution for the case that \\eref{t1.0} is violated. This solution differs from our approach in Sec. \\ref{sec7}.\n\n\\section{The generalized Hubbard--Stratonovich transformation}\\label{sec5}\n\nThe following theorem is proven in a way similar to Refs.~\\cite{Guh06,KGG08}. The proof is given in \\ref{app2}. We need the Wick--rotated set $\\Sigma_{\\beta,cd}^{(\\psi)}=\\widehat{\\Pi}_\\psi^{({\\rm C})}\\Sigma_{\\beta,cd}^{0(\\dagger)}\\widehat{\\Pi}_\\psi^{({\\rm C})}$, particularly $\\Sigma_{\\beta,cd}^{(0)}=\\Sigma_{\\beta,cd}^{0(\\dagger)}$. The original extension of the Hubbard--Stratonovich transformation \\cite{Guh06,KGG08} was only given for $\\gamma_2c=\\gamma_1d=\\tilde{\\gamma}k$. Here, we generalize it to arbitrary $c$ and $d$.\n\\begin{theorem}[Generalized Hubbard--Stratonovich transformation]\\ \\label{t2}\\\\\n Let $F$ and $\\kappa$ be the same as in Theorem \\ref{t1}. If the inequality \\eref{t1.0} holds, we have\n \\begin{eqnarray}\n   \\fl\\int\\limits_{\\mathfrak{R}}F(\\widehat{B})\\exp\\left(-\\varepsilon{\\rm Str\\,} \\widehat{B}\\right)d[\\Psi]=\\nonumber\\\\\n   \\fl= \\widetilde{C}_{acd}^{(\\beta)}\\int\\limits_{\\Sigma_{\\beta,cd}^{(\\psi)}}F\\left(\\hat{\\rho}\\right)\\exp\\left(-\\varepsilon{\\rm Str\\,}\\hat{\\rho}\\right)\\det\\rho_1^\\kappa \\left(e^{-\\imath\\psi d}D_{dr_2}^{(4\/\\beta)}\\right)^{a-c}\\frac{\\delta(r_2)}{|\\Delta_d(e^{\\imath\\psi}r_2)|^{4\/\\beta}}e^{-\\imath\\psi cd}d[\\rho]=\\nonumber\\\\\n    \\fl= \\widetilde{C}_{acd}^{(\\beta)}\\int\\limits_{\\Sigma_{\\beta,cd}^{(0)}}\\det\\rho_1^\\kappa\\frac{\\delta(r_2)}{|\\Delta_d(r_2)|^{4\/\\beta}} \\left((-1)^dD_{dr_2}^{(4\/\\beta)}\\right)^{a-c}\\left.F\\left(\\hat{\\rho}\\right)\\exp\\left(-\\varepsilon{\\rm Str\\,}\\hat{\\rho}\\right)\\right|_{\\psi=0}d[\\rho]\\label{t2.1}\n \\end{eqnarray}\n with\n \\begin{equation}\\label{t2.0}\n  \\hat{\\rho}=\\left[\\begin{array}{c|c} \\rho_1 & e^{\\imath\\psi\/2}\\rho_\\eta \\\\ \\hline -e^{\\imath\\psi\/2}\\rho_\\eta^\\dagger & e^{\\imath\\psi}\\left(\\rho_2-\\rho_\\eta^\\dagger\\rho_1^{-1}\\rho_\\eta\\right) \\end{array}\\right].\n \\end{equation}\n The variables $r_2$ are the eigenvalues of the supermatrix $\\rho_2$. The measure $d[\\rho]=d[\\rho_1]d[\\rho_2]d[\\eta]$ is defined by Eqs. \\eref{t1.4} and \\eref{t1.6}. For the measure $d[\\rho_2]$ we take the definition \\eref{t1.4} for $4\/\\beta$. The differential operator in Eq.~\\eref{t2.1} is an analog of the Sekiguchi--differential operator \\cite{OkoOls97} and has the form \\cite{KGG08}\n \\begin{equation}\\label{t2.2}\n  D_{dr_2}^{(4\/\\beta)}=\\frac{1}{\\Delta_d(r_2)}\\det\\left[r_{n2}^{d-m}\\left(\\frac{\\partial}{\\partial r_{n2}}+(d-m)\\frac{2}{\\beta}\\frac{1}{r_{n2}}\\right)\\right]_{1\\leq n,m\\leq d} .\n \\end{equation}\n The constant is\n \\begin{equation}\\label{t2.3}\n  \\widetilde{C}_{acd}^{(\\beta)}=2^{-c}\\left(2\\pi\\gamma_1\\right)^{-ad}\\left(\\frac{2\\pi}{\\gamma_2}\\right)^{cd}\\tilde{\\gamma}^{\\beta ac\/2}\\frac{{\\rm Vol}\\left({\\rm U\\,}^{(\\beta)}(a)\\right)}{{\\rm Vol}\\left({\\rm U\\,}^{(\\beta)}(a-c)\\right){\\rm FU}_d^{(4\/\\beta)}} .\n \\end{equation}\n\\end{theorem}\n\nSince the diagonalization of $\\rho_2$ yields an $|\\Delta_d(r_2)|^{4\/\\beta}$ in the measure, the ratio of the Dirac--distribution with the Vandermonde--determinant is for Schwartz--functions on ${\\rm Herm\\,}(4\/\\beta,d)$ well--defined. Also, the action of $D_{dr_2}^{(4\/\\beta)}$ on such a Schwartz--function integrated over the corresponding rotation group is finite at zero.\n\nThe distribution in the Fermion--Fermion block in Eq.~\\eref{t2.1} takes for $\\beta\\in\\{1,2\\}$ the simpler form \\cite{Guh06,KGG08}\n\\begin{eqnarray}\n \\fl\\left(D_{dr_2}^{(4\/\\beta)}\\right)^{a-c}\\frac{\\delta(r_2)}{|\\Delta_d(r_2)|^{4\/\\beta}} =\\nonumber\\\\\n ={\\rm FU}_d^{(4\/\\beta)}\\prod\\limits_{n=1}^{d}\\frac{\\Gamma\\left(a-c+1+2(n-1)\/\\beta \\right)}{(-\\pi)^{2(n-1)\/\\beta}\\Gamma\\left(\\gamma_1\\kappa\\right)}\\prod\\limits_{n=1}^{d}\\frac{\\partial^{\\gamma_1\\kappa-1}}{\\partial r_{n2}^{\\gamma_1\\kappa-1}}\\delta(r_{2n})\\label{5.1}\\ .\n\\end{eqnarray}\nThis expression written as a contour integral is the superbosonization formula \\cite{BasAke07}. For $\\beta=4$, we do not find such a simplification due to the term $|\\Delta(r_2)|$ as the Jacobian in the eigenvalue--angle coordinates.\n\n\\section{Equivalence of and connections between the two approaches}\\label{sec6}\n\nAbove, we have argued that both expressions in Theorems \\ref{t1} and \\ref{t2} are equivalent for $\\beta\\in\\{1,2\\}$. Now we address all $\\beta\\in\\{1,2,4\\}$. The Theorem below is proven in  \\ref{app3}. The proof treats all three cases in a unifying way. Properties of the ordinary matrix Bessel--functions are used.\n\\begin{theorem}[Equivalence of Theorems \\ref{t1} and \\ref{t2}]\\ \\label{t3}\\\\\n The superbosonization formula, \\ref{t1}, and the generalized Hubbard--Stratonovich transformation, \\ref{t2}, are equivalent for superfunctions which are Schwartz--functions and analytic in the fermionic eigenvalues.\n\\end{theorem}\n\nThe compact integral in the Fermion--Fermion block of the superbosonization formula can be considered as a contour integral. In the proof of Theorem \\ref{t3}, we find the integral identity\n\\begin{eqnarray}\n  & & \\int\\limits_{[0,2\\pi]^{d}}\\widetilde{F}\\left(e^{\\imath\\varphi_j}\\right)\\left|\\Delta_d\\left(e^{\\imath\\varphi_j}\\right)\\right|^{4\/\\beta}\\prod\\limits_{n=1}^d\\frac{e^{\\imath(1-\\gamma_1\\kappa)\\varphi_n}d\\varphi_n}{2\\pi}=\\nonumber\\\\\n  & = & \\prod\\limits_{n=1}^d\\frac{\\imath^{4(n-1)\/\\beta}\\Gamma(1+2n\/\\beta)}{\\Gamma(2\/\\beta+1)\\Gamma(\\gamma_1\\kappa-2(n-1)\/\\beta)}\\left.\\left(D_{dr_2}^{(4\/\\beta)}\\right)^{a-c}\\widetilde{F}(r_2)\\right|_{r_2=0}\\label{6.1}\n\\end{eqnarray}\nfor an analytic function $\\widetilde{F}$ on $\\mathbb{C}^d$ with permutation invariance. Hence, we can relate both constants \\eref{t1.2} and \\eref{t2.3},\n\\begin{equation}\\label{6.2}\n \\frac{\\widetilde{C}_{acd}^{(\\beta)}}{C_{acd}^{(\\beta)}}=(-1)^{d(a-c)}\\prod\\limits_{n=1}^d\\frac{\\imath^{4(n-1)\/\\beta}\\Gamma(1+2n\/\\beta)}{\\Gamma(2\/\\beta+1)\\Gamma(\\gamma_1\\kappa-2(n-1)\/\\beta)}.\n\\end{equation}\nThe integral identity \\eref{6.1} is a reminiscent of the residue theorem. It is the analog of the connection between the contour integral and the differential operator in the cases $\\beta\\in\\{1,2\\}$, see Fig. \\ref{f1}. Thus, the differential Operator with the Dirac--distribution in the generalized Hubbard--Stratonovich transformation restricts the non-compact integral in the Fermion--Fermion block to the point zero and its neighborhood. Therefore it is equivalent to a compact Fermion--Fermion block integral as appearing in the superbosonization formula.\n\\begin{figure}[ht]\n \\centering\n \\includegraphics[width=0.7\\textwidth]{Superbosonization.eps}\n \\caption{In the superbosonization formula, the integration of the fermionic eigenvalues is along the unit circle in the complex plane (dotted circle). The eigenvalue integrals in the generalized Hubbard--Stratonovich transformation are integrations over the real axis (bold line) or on the Wick--rotated real axis (thin line at angle $\\psi$) if the differential operator acts on the superfunction or on the Dirac--distribution at zero (bold dot, $0$), respectively.}\\label{f1}\n\\end{figure}\n\n\\section{The general case for arbitrary positive integers $a$, $b$, $c$, $d$ and arbitrary Dyson--index $\\beta\\in\\{1,2,4\\}$}\\label{sec7}\n\nWe consider an application of our results. The inequality \\eref{t1.0} reads\n\\begin{equation}\\label{7.1}\n  N\\geq\\gamma_1k\n\\end{equation}\nfor the calculation of the $k$--point correlation function \\eref{2.3} with help of the matrix Green function. For $\\beta=1$ , a $N\\times N$ real symmetric matrix has in the absence of degeneracies $N$ different eigenvalues. However, we can only calculate $k$--point correlation functions with $k<N\/2$. For $N\\to\\infty$, this restriction does not matter. But for exact finite $N$ calculations, we have to modify the line of reasoning.\n\nWe construct the symmetry operator\n\\begin{equation}\\label{7.2}\n \\mathfrak{S}\\left(\\sigma\\right)=\\mathfrak{S}\\left(\\left[\\begin{array}{cc} \\sigma_{11} & \\sigma_{12} \\\\ \\sigma_{21} & \\sigma_{22} \\end{array}\\right]\\right)=\\left[\\begin{array}{cc} -\\sigma_{22} & -\\sigma_{21} \\\\ \\sigma_{12} & \\sigma_{11} \\end{array}\\right]\n\\end{equation}\nfrom $(m_1+m_2)\\times(n_1+n_2)$ supermatrix to $(m_2+m_1)\\times(n_2+n_1)$ supermatrix. This operator has the properties\n\\begin{eqnarray}\n \\mathfrak{S}(\\sigma^\\dagger) & = & \\mathfrak{S}(\\sigma)^\\dagger\\quad, \\label{7.3}\\\\\n \\mathfrak{S}(\\sigma^*) & = & \\mathfrak{S}(\\sigma)^*, \\label{7.4}\\\\\n \\mathfrak{S}^2(\\sigma) & = & -\\sigma, \\label{7.5}\\\\\n \\fl\\mathfrak{S}\\left(\\left[\\begin{array}{cc} \\sigma_{11} & \\sigma_{12} \\\\ \\sigma_{21} & \\sigma_{22} \\end{array}\\right]\\left[\\begin{array}{cc} \\rho_{11} & \\rho_{12} \\\\ 0 & 0 \\end{array}\\right]\\right) & = & \\mathfrak{S}\\left(\\left[\\begin{array}{cc} \\sigma_{11} & \\sigma_{12} \\\\ \\sigma_{21} & \\sigma_{22} \\end{array}\\right]\\right)\\mathfrak{S}\\left(\\left[\\begin{array}{cc} \\rho_{11} & \\rho_{12} \\\\ 0 & 0 \\end{array}\\right]\\right). \\label{7.6}\n\\end{eqnarray}\n\nLet $a,\\ b,\\ c$, $d$ be arbitrary positive integers and $\\beta\\in\\{1,2,4\\}$. Then, the equation \\eref{7.6} reads for a matrix product of a $(\\gamma_2c+\\gamma_1d)\\times(0+\\gamma_1b)$ supermatrix with a $(0+\\gamma_1b)\\times(\\gamma_2c+\\gamma_1d)$ supermatrix\n\\begin{equation}\\label{7.7}\n \\fl\\left[\\begin{array}{c} \\zeta^\\dagger \\\\ \\tilde{z}^\\dagger \\end{array}\\right]\\left[\\begin{array}{cc} \\zeta & \\tilde{z} \\end{array}\\right]=\\mathfrak{S}\\left(\\left[\\begin{array}{c} \\tilde{z}^\\dagger \\\\ -\\zeta^\\dagger \\end{array}\\right]\\right)\\mathfrak{S}\\left(\\left[\\begin{array}{cc} \\tilde{z} & \\zeta \\end{array}\\right]\\right)=\\mathfrak{S}\\left(\\left[\\begin{array}{c} \\tilde{z}^\\dagger \\\\ -\\zeta^\\dagger \\end{array}\\right]\\left[\\begin{array}{cc} \\tilde{z} & \\zeta \\end{array}\\right]\\right).\n\\end{equation}\nWith help of the operator $\\mathfrak{S}$, we split the supersymmetric Wishart--matrix $\\widehat{B}$ into two parts,\n\\begin{equation}\\label{7.8}\n \\widehat{B}=\\widehat{B}_1+\\mathfrak{S}(\\widehat{B}_2)\n\\end{equation}\nsuch that\n\\begin{equation}\\label{7.9}\n \\fl\\widehat{B}_1=\\tilde{\\gamma}^{-1}\\sum\\limits_{j=1}^{a}\\Psi_{j1}^{({\\rm C})}\\Psi_{j1}^{({\\rm C})\\dagger}\\qquad{\\rm and}\\qquad\\widehat{B}_2=\\tilde{\\gamma}^{-1}\\sum\\limits_{j=1}^{b}\\mathfrak{S}\\left(\\Psi_{j2}^{({\\rm C})}\\right)\\mathfrak{S}\\left(\\Psi_{j2}^{({\\rm C})}\\right)^\\dagger .\n\\end{equation}\nThe supervectors $\\mathfrak{S}\\left(\\Psi_{j2}^{({\\rm C})}\\right)$ are of the same form as $\\Psi_{j1}^{({\\rm C})}$. Let $\\sigma$ be a quadratic supermatrix, i.e. $m_1=n_1$ and $m_2=n_2$. Then, we find the additional property\n\\begin{equation}\\label{7.10}\n {\\rm Sdet\\,}\\mathfrak{S}(\\sigma)=(-1)^{m_2}{\\rm Sdet\\,}^{-1}\\sigma .\n\\end{equation}\n\nLet $\\widehat{\\Sigma}_{\\beta,pq}^{(0)}=\\mathfrak{S}\\left(\\Sigma_{\\beta,pq}^{(0)}\\right)$, $\\widehat{\\Sigma}_{\\beta,pq}^{0({\\rm c})}=\\mathfrak{S}\\left(\\Sigma_{\\beta,pq}^{0({\\rm c})}\\right)$ and the Wick--rotated set $\\widehat{\\Sigma}_{\\beta,pq}^{(\\psi)}=\\widehat{\\Pi}_{\\psi}^{({\\rm C})}\\widehat{\\Sigma}_{\\beta,pq}^{(0)}\\widehat{\\Pi}_{\\psi}^{({\\rm C})}$. Then, we construct the analog of the superbosonization formula and the generalized Hubbard--Stratonovich transformation.\n\\begin{theorem}\\label{t4}\\ \\\\\n Let $F$ be the superfunction as in Theorem \\ref{t1} and \n \\begin{equation}\\label{t4.1}\n  \\kappa=\\frac{a-c+1}{\\gamma_1}-\\frac{b-d+1}{\\gamma_2} .\n \\end{equation}\n Also, let $e\\in\\mathbb{N}_0$ and\n \\begin{equation}\\label{t4.2}\n \\tilde{a}=a+\\gamma_1e\\qquad{\\rm and}\\qquad\\tilde{b}=b+\\gamma_2e\n \\end{equation}\n with\n \\begin{equation}\\label{t4.3}\n \\tilde{a}\\geq c\\qquad\\tilde{b}\\geq d .\n \\end{equation}\n We choose the Wick--rotation  $e^{\\imath\\psi}$ such that all integrals are convergent. Then, we have\n \\begin{eqnarray}\n   \\fl\\int\\limits_{\\mathfrak{R}}F(\\widehat{B}_{\\psi})\\exp\\left(-\\varepsilon{\\rm Str\\,} \\widehat{B}_{\\psi}\\right)d[\\widehat{V}]=\\nonumber\\\\\n   \\fl= \\left(-\\frac{2}{\\gamma_1}\\right)^{\\gamma_2ec}\\left(\\frac{2}{\\gamma_2}\\right)^{\\gamma_1ed} \\int\\limits_{\\widetilde{\\mathfrak{R}}}F(\\widetilde{B}_{\\psi})\\exp\\left(-\\varepsilon{\\rm Str\\,} \\widetilde{B}_{\\psi}\\right)d[\\widetilde{V}]=\\nonumber\\\\\n  \\fl= C_{{\\rm SF}}\\int\\limits_{\\Sigma_{\\beta,cd}^{0({\\rm c})}}\\int\\limits_{\\widehat{\\Sigma}_{4\/\\beta,dc}^{0({\\rm c})}}d[\\rho^{(2)}]d[\\rho^{(1)}]F(\\rho^{(1)}+e^{\\imath\\psi}\\rho^{(2)})\\exp\\left[-\\varepsilon{\\rm Str\\,}(\\rho^{(1)}+e^{\\imath\\psi}\\rho^{(2)})\\right]\\times\\nonumber\\\\\n  \\fl\\times {\\rm Sdet\\,}^{\\kappa+\\tilde{b}\/\\gamma_2}\\rho^{(1)}{\\rm Sdet\\,}^{\\kappa-\\tilde{a}\/\\gamma_1}\\rho^{(2)}=\\label{t4.5a}\\\\\n  \\fl =C_{{\\rm HS}}\\int\\limits_{\\Sigma_{\\beta,cd}^{(0)}}\\int\\limits_{\\widehat{\\Sigma}_{4\/\\beta,cd}^{(0)}}d[\\rho^{(2)}]d[\\rho^{(1)}]\\frac{\\delta\\left(r_2^{(1)}\\right)}{\\left|\\Delta_d\\left(r_2^{(1)}\\right)\\right|^{4\/\\beta}}\\frac{\\delta\\left(r_1^{(2)}\\right)}{\\left|\\Delta_c\\left(r_1^{(2)}\\right)\\right|^{\\beta}}{\\det}^{\\kappa+b\/\\gamma_2}\\rho_1^{(1)}{\\det}^{a\/\\gamma_1-\\kappa}\\rho_2^{(2)}\\times\\nonumber\\\\\n  \\fl\\times \\left(D_{dr_2^{(1)}}^{(4\/\\beta)}\\right)^{\\tilde{a}-c}\\left(D_{cr_1^{(2)}}^{(\\beta)}\\right)^{\\tilde{b}-d}F(\\hat{\\rho}^{(1)}+e^{\\imath\\psi}\\hat{\\rho}^{(2)})\\exp\\left[-\\varepsilon{\\rm Str\\,}(\\hat{\\rho}^{(1)}+e^{\\imath\\psi}\\hat{\\rho}^{(2)})\\right], \\label{t4.5b}\n \\end{eqnarray}\n where the constants are\n \\begin{eqnarray}\n  C_{{\\rm SF}} & = & (-1)^{c(b-d)}e^{\\imath\\psi(\\tilde{a}d-\\tilde{b}c)}\\left(\\frac{2}{\\gamma_1}\\right)^{\\gamma_2ec}\\left(\\frac{2}{\\gamma_2}\\right)^{\\gamma_1ed}C_{\\tilde{a}cd}^{(\\beta)}C_{\\tilde{b}dc}^{(4\/\\beta)} ,\\\\\n  C_{{\\rm HS}} & = & (-1)^{d(a-c)}e^{\\imath\\psi(\\tilde{a}d-\\tilde{b}c)}\\left(-\\frac{2}{\\gamma_1}\\right)^{\\gamma_2ec}\\left(-\\frac{2}{\\gamma_2}\\right)^{\\gamma_1ed}\\widetilde{C}_{\\tilde{a}cd}^{(\\beta)}\\widetilde{C}_{\\tilde{b}dc}^{(4\/\\beta)} .\n \\end{eqnarray}\n Here, we define the supermatrix\n \\begin{equation}\\label{t4.6}\n  \\fl\\hat{\\rho}^{(1)}+e^{\\imath\\psi}\\hat{\\rho}^{(2)}=\\left[\\begin{array}{c|c} \\rho_1^{(1)}+e^{\\imath\\psi}\\left(\\rho_1^{(2)}-\\rho_{\\tilde{\\eta}}^{(2)}\\rho_2^{(2)-1}\\rho_{\\tilde{\\eta}}^{(2)\\dagger}\\right) & \\rho_\\eta^{(1)}+e^{\\imath\\psi}\\rho_{\\tilde{\\eta}}^{(2)} \\\\ \\hline -\\rho_\\eta^{(1)\\dagger}-e^{\\imath\\psi}\\rho_{\\tilde{\\eta}}^{(2)\\dagger} & \\rho_2^{(1)}-\\rho_\\eta^{(1)\\dagger}\\rho_1^{(1)-1}\\rho_\\eta^{(1)}+e^{\\imath\\psi}\\rho_2^{(2)} \\end{array}\\right]\n \\end{equation}\n The set $\\widetilde{\\mathfrak{R}}$ is given as in corollary \\ref{c2}. The measures $d[\\rho^{(1)}]=d[\\rho_1^{(1)}]d[\\rho_2^{(1)}]d[\\eta]$ and $d[\\rho^{(2)}]=d[\\rho_1^{(2)}]d[\\rho_2^{(2)}]d[\\tilde{\\eta}]$ are given by Theorem \\ref{t1}. The measures \\eref{t1.4} for $\\beta$ and $4\/\\beta$ assign $d[\\rho_1^{(1)}]$ and $d[\\rho_2^{(2)}]$ in Eqs. \\eref{t4.5a} and \\eref{t4.5b}, respectively. In Eq.~\\eref{t4.5a}, $d[\\rho_2^{(1)}]$ and $d[\\rho_1^{(2)}]$ are defined by the measure \\eref{t1.5} for the cases $\\beta$ and $4\/\\beta$, respectively, and, in Eq.~\\eref{t4.5b}, they are defined by the measure \\eref{t1.4} for the cases $4\/\\beta$ and $\\beta$, respectively. The measures $d[\\eta]$ and $d[\\tilde{\\eta}]$ are the product of all complex Grassmann pairs as in Eq.~\\eref{t1.6}.\n\\end{theorem}\nSince this Theorem is a consequence of corollary \\ref{c2} and Theorems \\ref{t1} and \\ref{t2}, the proof is quite simple.\\\\\n\\textbf{Proof:}\\\\\nLet $e\\in\\mathbb{N}_0$ as in Eq.~\\eref{t4.2}. Then, we use corollary \\ref{c2} to extend the integral over $\\widehat{V}$ to an integral over $\\widetilde{V}$. We split the supersymmetric Wishart--matrix $\\widehat{B}$ as in Eq.~\\eref{7.8}. Both Wishart--matrices $\\widehat{B}_{1}$ and $\\widehat{B}_{2}$ fulfill the requirement \\eref{t1.0} according to their dimension. Thus, we singly apply both Theorems \\ref{t1} and \\ref{t2} to $\\widehat{B}_{1}$ and $\\widehat{B}_{2}$. \\hfill$\\square$\n\nOur approach of a violation of inequality \\eref{t1.0} is quite different from the solution given in Ref.~\\cite{BEKYZ07}. These authors introduce a matrix which projects the Boson--Boson block and the bosonic side of the off-diagonal blocks onto a space of the smaller dimension $a$. Then, they integrate over all of such orthogonal projectors. This integral becomes more difficult due to an additional measure on a curved, compact space. We use a second symmetric supermatrix. Hence, we have up to the dimensions of the supermatrices a symmetry between both supermatrices produced by $\\mathfrak{S}$. There is no additional complication for the integration, since the measures of both supermatrices are of the same kind. Moreover, our approach extends the results to the case of $\\beta=4$ and odd $b$ which is not considered in Ref.~\\cite{BEKYZ07}.\n\n\\section{Remarks and conclusions}\\label{sec8}\n\nWe proved the equivalence of the generalized Hubbard--Stratonovich transformation \\cite{Guh06,KGG08} and the superbosonization formula \\cite{Som07,LSZ07}. Thereby, we generalized both approaches. The superbosonization formula was proven in a new way and is now extended to odd dimensional supersymmetric Wishart--matrices in the Fermion--Fermion block for the quaternionic case. The generalized Hubbard--Stratonovich transformation was here extended to arbitrary dimensional supersymmetric Wishart--matrices which not only stem of averages over the matrix Green functions. \\cite{GMW98,Zir06,Guh06,KKG08} Furthermore, we got an integral identity beyond the restriction of the matrix dimension, see Eq.~\\eref{t1.0}. This approach distinguishes from the method presented in Ref.~\\cite{BEKYZ07} by the integration of an additional matrix. It is, also, applicable on the artificial example $\\beta=4$ and odd $b$ which has not been considered in Ref.~\\cite{BEKYZ07}.\n\nThe generalized Hubbard--Stratonovich transformation and the superbosonization formula reduce in the absence of Grassmann variables to the ordinary integral identity for ordinary Wishart--matrices. \\cite{Fyo02,LSZ07} In the general case with the restriction \\eref{t1.0}, both approaches differ in the Fermion--Fermion block integration. Due to the Dirac--distribution and the differential operator, the integration over the non-compact domain in the generalized Hubbard--Stratonovich transformation is equal with help of the residue theorem to a contour integral. This contour integral is equivalent to the integration over the compact domain in the superbosonization formula. Hence, we found an integral identity between a compact integral and a differentiated Dirac--distribution.\n\n\\section*{Acknowledgements}\n\nWe thank Heiner Kohler for fruitful discussions. This work was supported by Deutsche Forschungsgemeinschaft within Sonderforschungsbereich Transregio 12 ``Symmetries and Universality in Mesoscopic Systems''.\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nThe most successful method for detecting faint companions around nearby stars is undoubtedly the radial velocity (RV) technique \\citep{Perryman00}, with more than 500 extrasolar planets detected up to date. Nevertheless, RV has three major drawbacks. First, it cannot be applied to all kind of stars: pulsating stars, as well as young, rapidly rotating and\/or active stars have an intrinsic radial velocity jitter that generally precludes planet-search programmes \\citep{Udry07}. The discovery of extrasolar planets orbiting high-mass main-sequence stars (O, B, and A spectral types) and young solar-type stars (T Tauri stars) therefore remains rare in the literature. Second, RV can only access companions with periods shorter than a few tens of years, owing to the limited time span of the observations obtained with high-precision spectrographs. Third, RV is inherently insensitive to the orbital inclination $i$. This implies that additional astrometric observations are mandatory to unveil the real companion mass from the measured quantity $M_p\\sin{i}$. For these reasons, the development of direct imaging techniques for low-mass companions is currently one of the highest priorities in instrumental astronomy \\citep{Oppenheimer09,Absil10b}.\n\nDirect imaging techniques should be tailored to the typical angular separations for which low-mass companions are to be found. For instance, the search for young planets in nearby star-forming regions ($\\geq 100$\\,pc) should be optimised for typical linear separations from 0.1 to a few 10\\,AU for Sun-like stars, where planets are supposed to be forming and migrating in their early history \\citep{Bodenheimer02}. This translates into typical angular separations of 1--100\\,milli-arcsecs (mas). In a more general context, more than 90\\% of the sub-stellar companions detected to date by all techniques have expected apparent separations smaller than 100\\,mas \\citep{Schneider11}. This domain is hardly accessible to single-pupil imaging techniques, even with adaptive-optics (AO) assisted facilities \\citep{Burrows05}. Nowadays, with 10-m class telescopes, only interferometric aperture-masking observations routinely break the 100\\,mas limit. However, their search space is still restricted to angular separations larger than about 40\\,mas for dynamic ranges of the order of 500:1 \\citep{Kraus08,Lacour:2011}.\n\nThanks to its higher angular resolution, optical long baseline interferometry is the ideal tool for exploring separations in the range 1--50\\,mas. The simplest strategy for faint companion detection with optical\/infrared interferometry is to obtain closure phase measurements that are directly sensitive to the asymmetry in the brightness distribution of the target source and hence to possible off-axis companions. However, owing to the sparse structure of the point spread function associated with the diluted aperture of an interferometer, the depth to which a companion can be detected strongly depends on the relative orientation of the companion and the interferometric baselines. Consequently, the sensitivity limit should be defined as a two-dimensional map and\/or for various completeness levels.\n\nThe goal of this paper is to provide quantitative estimations of the detection efficiency versus the companion contrast and separation considering realistic observations. In Sect.~\\ref{sec:theory}, we derive a linear expression for the closure phase signature of a faint companion. We also define a simple, general expression for the sensitivity limit in terms of companion contrast, which we use in Sect.~\\ref{sec:array} to compute the efficiency of various interferometric configurations. We compare the efficiency of the four configurations currently offered at the Very Large Telescope Interferometer. To add generality, we also discuss the efficiency of three ``standard'' interferometric arrays (two linear and one Y-shaped). In Sect.~\\ref{sec:accuracy}, we discuss the typical achievable dynamic range for the closure phase accuracy provided by existing instruments.\n\n\n\n\\section{Detecting faint companions with closure phase}\n\\label{sec:theory}\n\nOptical long baseline interferometers provide several observables, which are all sensitive to the presence of a faint companion: visibilities, differential phases, and closure phases. The latter has the main advantage of being, to first order, uncorrupted by telescope-specific phase errors, including pointing errors, atmospheric piston, and longitudinal dispersion due to air and water vapour. The noise floor is only limited by the accuracy to which the instrumental systematics can be mastered. As a consequence, the high-precision closure phase is the favoured observational strategy to directly detect faint companions with modern interferometers \\citep{Vannier06,Zhao08,Renard08}.\n\n\n\\subsection{Closure-phase signal of a faint companion} \\label{sec:signal}\n\nMeasuring a closure phase requires the use of an interferometric array composed of (at least) three telescopes. Each pair of telescopes defines a geometrical vector referred to as the interferometric baseline. The closure phase is then defined as the sum of the phases measured on the three baselines, or equivalently, as the argument of the bispectrum, formed through the triple product of the measured complex visibilities around the triangle \\citep{Monnier03}. Assuming that the stellar diameters are not resolved at the considered interferometric baseline lengths, the closure phase signature of a binary object can be expressed as a function of the three baseline vectors projected onto the sky \\{\\ensuremath{\\vec{B}_{12}},\\ensuremath{\\vec{B}_{23}},\\ensuremath{\\vec{B}_{31}}\\}, the wavelength of observation \\ensuremath{\\lambda{}}{}, the binary flux ratio \\ensuremath{\\rho}{} (referred to as the contrast in the following), and the apparent binary separation vector $\\ensuremath{\\vec\\Delta}$\n\\begin{equation}\n\\ensuremath{\\phi} = \\arg \\left( \\frac{(1+\\ensuremath{\\rho}\\,e^{i\\ensuremath{\\alpha_{12}}}) (1+\\ensuremath{\\rho}\\,e^{i\\ensuremath{\\alpha_{23}}}) (1+\\ensuremath{\\rho}\\,e^{i\\ensuremath{\\alpha_{31}}})} {(1+\\ensuremath{\\rho})^3} \\right) \\, , \\label{eq:1}\n\\end{equation}\nwhere\n\\begin{equation}\n\\ensuremath{\\alpha_{12}} = 2\\pi \\frac{\\ensuremath{\\vec{B}_{12}} \\cdot \\ensuremath{\\vec\\Delta}}{\\ensuremath{\\lambda{}}} \\; ; \\;\\;\n\\ensuremath{\\alpha_{23}} = 2\\pi \\frac{\\ensuremath{\\vec{B}_{23}} \\cdot \\ensuremath{\\vec\\Delta}}{\\ensuremath{\\lambda{}}} \\; ; \\;\\;\n\\ensuremath{\\alpha_{31}} = 2\\pi \\frac{\\ensuremath{\\vec{B}_{31}} \\cdot \\ensuremath{\\vec\\Delta}}{\\ensuremath{\\lambda{}}} \\; . \\label{eq:1b}\n\\end{equation}\nThe geometrical terms \\ensuremath{\\alpha_{12}}{}, \\ensuremath{\\alpha_{23}}{}, and \\ensuremath{\\alpha_{31}}{} describe the relative orientation of the companion compared to the spatial frequencies explored by the interferometric array. We note that $\\ensuremath{\\alpha_{31}}=-(\\ensuremath{\\alpha_{12}}+\\ensuremath{\\alpha_{23}})$ since the vectorial sum of the three interferometric baselines is zero by definition.\n\nTo model the signal of a high contrast binary, we can either use Eq.~\\ref{eq:1} with small ($\\ll$$1$) or large ($\\gg$$1$) values of $\\ensuremath{\\rho}$. In the first case, the interferometer points toward the primary, while in the second case it points toward the companion. Since the closure-phase is independent of the pointing position, these two approaches are formally identical. Assuming $\\ensuremath{\\rho}\\ll1$, we can develop and simplify Eq.~\\ref{eq:1} to be\n\\begin{equation}\n\\ensuremath{\\phi} \\approx \\ensuremath{\\rho} \\, (\\sin\\ensuremath{\\alpha_{12}} + \\sin\\ensuremath{\\alpha_{23}} - \\sin(\\ensuremath{\\alpha_{12}}+\\ensuremath{\\alpha_{23}})) \\, . \\label{eq:2}\n\\end{equation}\nThe quantity $\\ensuremath{m}=\\sin\\ensuremath{\\alpha_{12}} + \\sin\\ensuremath{\\alpha_{23}} - \\sin(\\ensuremath{\\alpha_{12}}+\\ensuremath{\\alpha_{23}})$ is referred to as the magnification factor because a high value of $\\ensuremath{m}$ corresponds to a stronger companion signature in the closure phase signal.\n\nThe validity range of our approximate formula can be explored numerically by computing the difference between Eq.~\\ref{eq:1} and~\\ref{eq:2} for various values of \\{\\ensuremath{\\rho},\\ensuremath{\\alpha_{12}},\\ensuremath{\\alpha_{23}}\\}. We found that Eq.~\\ref{eq:2} stays within a factor $\\lesssim 1.5$ of the value given by Eq.~\\ref{eq:1} as long as $\\ensuremath{\\rho} \\leq 10^{-1}$, which we consider to be the validity range of our work. Therefore, the following results apply only to relatively faint companions, and cannot be immediately transposed to binaries with flux ratios close to unity.\n\nApart from the computational gain, the main advantage of using Eq.~\\ref{eq:2} is to define a proportional relation between the closure phase and the companion contrast. This reduces the number of parameters to be explored and allows the results to be presented in a synthetic way. Additionally, we note that in Eq.~\\ref{eq:2}, the magnification factor $m$ takes a maximum value of about 2.6\\,rad ($\\approx149\\deg$) for purely geometrical reasons. This magnification is only achieved in the optimal geometrical case, where all interferometric baselines add together to amplify the companion signal. Therefore, as a quantitative example, we can already conclude that reaching a dynamic range of $10^{-3}$ with one single closure phase measurement requires a closure phase accuracy \\emph{better} than $0.15\\deg$. In practice, this accuracy is generally reached after appropriate uv-plane or temporal averaging of a series of individual closure phase measurements.\n\n\\subsection{The close-companion limit}\nIn the specific case where the binary is not resolved by any individual interferometric baselines (all $\\alpha<1$), it is possible to simplify Eq.~\\ref{eq:2} by expanding the sines for small values of $\\alpha$\n\\begin{equation}\n  \\frac{\\phi}{\\ensuremath{\\rho}} \\approx \\frac{\\ensuremath{\\alpha_{12}}^2\\,\\ensuremath{\\alpha_{23}} + \\ensuremath{\\alpha_{12}}\\,\\ensuremath{\\alpha_{23}}^2}{2} \\;+\\; o(\\alpha^5) \\; .\n\\label{eq:close1}\n\\end{equation}\nWe can arbitrarily choose $\\ensuremath{\\alpha_{12}}$ to be associated with the most resolving baseline and $\\ensuremath{\\alpha_{23}}{}$ with the least resolving baseline in the direction of the considered binary. We introduce the $r$ ratio of the two geometrical terms: $\\ensuremath{\\alpha_{23}}=-r\\,\\ensuremath{\\alpha_{12}}$ (the negative sign is introduced for the triangle to close with positive values of $r$). Rewriting Eq.~\\ref{eq:close1}, we obtain\n\\begin{equation}\n  \\frac{\\phi}{\\ensuremath{\\rho}} \\approx \\frac{1}{2}\\,\\ensuremath{\\alpha_{12}}^3\\,r\\,(1-r)\n  \\label{eq:close2}\n\\end{equation}\nThis equation already shows two interesting aspects: (i) the closure phase signature of an unresolved companion is proportional to the baseline length at the third power, and (ii) the arrays that are redundant when projected onto the binary direction $(r=0.5)$ seem optimal for detecting a close companion using a given long baseline.\n\n\n\\subsection{Deriving sensitivity limits}\nA typical interferometric observation is not composed of a single closure phase measurement. We now derive sensitivity limits in terms of companion contrast for a set of $n$ linearly independent\\footnote{Using $m$ apertures, one can form $(m-1)(m-2)\/2$ linearly independent closure phases. This is equivalent to holding one aperture fixed and forming all possible triangles with that aperture \\citep{Monnier03}.} closure phase measurements $\\ensuremath{\\phi}_i$ of accuracy $\\sigma_i$. We base our analysis on the $\\chi^2$ of the data with respect to a model of an unresolved source with no companion, for which all closure phases are zero\n\\begin{equation}\n\\chi^2 = \\sum_{i=1}^n \\frac{\\ensuremath{\\phi}^2_i}{\\sigma_i^2} \\; .\n\\end{equation}\nWe introduce the simplification presented in Eq.~\\ref{eq:2} and assume that all the measurements have similar accuracies (a valid assumption when observing an unresolved target) to obtain\n\\begin{equation}\n\\chi^2 = \\frac{\\rho^2}{\\sigma^2} \\; \\sum_{i=1}^n m_i^2 \\; .\n\\end{equation}\nThe probability that the data set is compatible with the single-star model is given by the complement of the cumulative probability distribution function (CDF) with $n$ degrees of freedom\n\\begin{equation}\nP = 1-\\mathrm{CDF}_n(\\chi^2) \\; .\n\\end{equation}\nIf $P$ is below a predefined threshold, the data set allows the model with no companion to be rejected. The threshold can generally be fixed at a 3-$\\sigma{}$ level, i.e., at a probability of 0.27\\%. The choice of the threshold actually depends on the context of the observations. If a large region of parameter space (or indeed number of targets) is searched for a companion, then a 5-$\\sigma{}$ threshold may be more appropriate. A higher threshold may even be needed in the case of sparse data sets dominated by non-Gaussian systematic errors. Taking the 3-$\\sigma{}$ level as an example, this threshold can be converted into a sensitivity limit in terms of companion contrast\n\\begin{equation}\n  \\rho = \\sigma \\sqrt{\\frac{\\mathrm{CDF}^{-1}_n(1-0.27\\%)}{\\sum_{i=1}^n m_i^2}} \\; ,\n\\end{equation}\nwhere $\\mathrm{CDF}^{-1}_n(1-0.27\\%)$ is simply the $\\chi^2$ value for a 3-$\\sigma{}$ detection with $n$ degrees of freedom. As a consequence of the approximation presented in Eq.~\\ref{eq:2}, the sensitivity limit is directly proportional to the accuracy of the closure phase measurements.\n\nBecause of the sparse structure of the point spread function associated with the diluted aperture of an interferometer, the depth to which a companion can be detected strongly depends on the relative orientation between the companion and the interferometric baselines (information embedded in the magnification factors $m_i$). For some lucky separations, the greatest dynamic range is achieved, while in the worst cases even obvious binaries with equal brightnesses can be missed. Consequently, the sensitivity limit should be defined as a two-dimensional map or for various completeness levels on a given search region.\n\n\\subsection{Validity limit of this study}\nIn the case of wide companions, care should be taken regarding the chromaticity limit of our study. The results presented here are indeed formally valid only for monochromatic light. The main effect of wavelength smearing inside Eq.~\\ref{eq:1} is to degrade the dynamic range. To avoid significant smearing, the spectral resolution should be higher than the $\\alpha$ quantities defined in Eq.~\\ref{eq:1b}. This translates into a spectral resolution $R>10$ for a $40\\,$mas binary observed with a $100\\,$m baseline at $1.7\\,\\mu$m (H band). Such a low spectral resolution is available in most modern interferometric beam combiners. In the case of spatially filtered beam combiners, a similar effect may occur because of baseline smearing, in the case where the telescope size cannot be neglected in Eq.~\\ref{eq:1}. For the $1.8$-m Auxiliary Telescopes, this corresponds to angular separations of about $175\\,$mas for H-band observations. For $10$-m class telescopes, this corresponds to angular separations of about $45\\,$mas. In practice, these limitations are not very relevant to our study because AO-assisted spare aperture-masking imaging on 10-m class telescopes becomes more efficient than long-baseline interferometry for separations larger than about $40\\,$mas \\citep[see e.g.][]{Kraus08,Lacour:2011}.\n\nFinally, Eq.~\\ref{eq:1} assumes that the angular diameters of the binary components are unresolved by the interferometric baselines. Resolving the diameter of the faint component indeed appears unrealistic, although maybe not for the central star, especially in the case of bright late-type giants or very nearby stars. In this latter situation, Eq.~\\ref{eq:1} underestimates the closure phase signal, which peaks for a fully resolved primary star. This effect, referred to as \\emph{closure phase nulling}, can lead to larger magnification factors than the limit $m<149\\deg$ presented in Sect.~\\ref{sec:signal}. This specific observing technique is discussed in detail in \\citet{Chelli:2009}, while on-sky applications can be found in \\citet{Monnier:2006}, \\citet{Lacour08}, \\citet{Zhao08}, and \\citet{Duvert:2010}.\n\n\n\\section{Optical interferometric array}\n\\label{sec:array}\n\nWe use the formalism introduced previously to compute and compare the capabilities of various four-telescope interferometric configurations.\n\n\\subsection{Performances of VLTI configurations}\n\n\\begin{figure}\n    \\centering \n    \\includegraphics[scale=0.58]{visa}\n    \\caption{Interferometric configuration offered at VLTI. The configurations using the four relocatable Auxiliary Telescopes are represented by colours. The configuration using the four fixed Unit Telescopes is represented in black.}\n    \\label{fig:visa}\n\\end{figure}\n\n\\begin{figure*}\n  \\centering \n  \\includegraphics[scale=0.52]{A1G1K0I1_rhoangle1}\\hspace{0.7cm}\n  \\includegraphics[scale=0.49]{A1G1K0I1_distance1}\n  \\includegraphics[scale=0.52]{A1G1K0I1_rhoangle3}\\hspace{0.7cm}\n  \\includegraphics[scale=0.49]{A1G1K0I1_distance3}\n  \\includegraphics[scale=0.52]{A1G1K0I1_rhoangle5}\\hspace{0.7cm}\n  \\includegraphics[scale=0.49]{A1G1K0I1_distance5}\n  \\caption{{\\it Left:} Map of the 3-$\\sigma{}$ sensitivity with the A1-K0-G1-I1 configuration from VLTI.  Radial zones are the bins in separation used for the plots in the right panel. {\\it Right:}  Sensitivity as a function of angular distance, for various completeness levels. {\\it From top to botom:}  Simulations for a single snapshot pointing (top), and for three pointing (middle), and for five pointing (bottom). The contrast axes can be scaled for any accuracy on the closure phase (here $\\sigma=0.25\\deg$).}\n  \\label{fig:det_map}\n\\end{figure*}\n\n\\begin{figure*}\n    \\centering \n    \\includegraphics[scale=0.55]{all_contdist}\\hspace{2cm}\n    \\includegraphics[scale=0.55]{all_compcont}\n    \\caption{{\\it Left:} 3-$\\sigma{}$ sensitivity versus separation for a completeness level of 80\\%. {\\it Right:} Completeness in the separation range $6-40\\,$mas versus contrast. Simulations are for a single snapshot pointing (dashed lines) and for five pointings separated by one hour (solid lines). Colours are for the four configurations of VLTI displayed in Fig.~\\ref{fig:visa}. The contrast axes can be scaled for any given accuracy of the closure phase measurements (here $\\sigma=0.25\\deg$).}\n    \\label{fig:cont_sep}\n    \\label{fig:comp_cont}\n\\end{figure*}\n\nWe compute the map of the 3-$\\sigma{}$ sensitivity limit using all the VLTI configurations displayed in Fig.~\\ref{fig:visa}, assuming a target at declination $-35\\,$deg. The limits are computed considering data sets consisting of respectively one pointing at an hour angle $\\mathrm{HA}=-2\\,$h, three pointings at hour angles $\\mathrm{HA}=-2\\,$h, $-1\\,$h and $0\\,$h, and five pointings at hour angles $\\mathrm{HA}=-2\\,$h, $-1\\,$h, $0\\,$h, $+1\\,$h, and $+2\\,$h. A closure phase accuracy of $0.25\\deg$ is assumed (see Sect.~\\ref{sec:accuracy}). We consider a maximum binary separation of $40\\,$mas, which is the separation where AO-assisted spare aperture-masking imaging on 10-m class telescopes becomes more efficient than long-baseline interferometry.\n\nDetailed results of the widest AT configuration (A1-K0-G1-I1) are displayed in Fig.~\\ref{fig:det_map} for illustration. The figure shows that the detection completeness for a given sensitivity level increases drastically with the number of pointings. This is clearly illustrated by the decreasing number of ``blind spots'' (white zones) in the left-hand side plots. The sensitivity level is mostly flat for angular separations larger than 2\\,mas, while the detection performance drops considerably within this inner working angle (IWA). The median sensitivity levels in the region 2--40 mas are respectively about $6\\times 10^{-3}$, $4.5\\times 10^{-3}$ and $4\\times 10^{-3}$ for the three considered data sets. For more than five pointings, the median sensitivity level would continue to improve slightly, but the shape of the sensitivity curve would no longer significantly change.\n\nThe relative performances of the different configurations are illustrated in Fig.~\\ref{fig:cont_sep} (left: sensitivity versus separation, right: completeness level versus contrast). All configurations provide flat performances for large separations, down to their respective IWA where the performances dramatically drop. The IWA are respectively $2\\,$mas for the A1-K0-G1-I1 and U1-U2-U3-U4 configurations, $3\\,$mas for the D0-H0-I1-G1 configuration, and $6\\,$mas for the A1-B2-C1-D0 configuration. They correspond to the spatial resolution of the smallest baseline of the array. Given the similarity of the results for all configurations, we discuss them together in two different regimes: (i) the close-companion and (ii) the wide-companion regimes.\n\n\\subsection{Close-companion regime}\n\nClose companions are defined here as companions with angular separations that are not fully resolved by \\emph{at least} one baseline (that is $\\vec{B}\\cdot\\vec{\\Delta}\/\\lambda<1$). In this regime, the achievable contrast for a given completeness follows a power law of the angular separation $\\ensuremath{\\rho} \\propto \\Delta^{-3}$, as predicted by Eq.~\\ref{eq:close2}. The exact factor entering into this law depends on the array geometry but, as expected, the longest arrays provide the highest spatial resolution and are thus able to detect both the deepest and the closest binaries.\n\nThis well-known result was discussed by \\citet{Lachaume03} in the context of partially resolved interferometric observations. We emphasize that our study additionally provides a quantitative estimation. As a typical example, we now detail the case of a faint companion with a contrast of $5\\times10^{-3}$. We first consider the companion to be located at 2\\,mas from the central star.  A closure-phase accuracy of $0.25\\deg$ results in a detection efficiency of about 50\\% using three pointings with the configuration A0-K0-G1-I1, according to Fig.~\\ref{fig:det_map} (middle-right plot). However, if we now consider the companion to be located at $1\\,$mas from the central star, the closure-phase accuracy should be $0.025\\deg$ to reach the same efficiency. Since the angular separation is only marginally resolved by the interferometer, the lack of spatial resolution has to be compensated for by an increase in the accuracy on the signal (super-resolution effect).\n\n\\subsection{Wide-companion regime}\n\nWe note that those companions are \\emph{wide} only in the interferometric sense, corresponding to separations larger than about 4\\,mas for the typical $\\sim100\\,$m baselines available in modern interferometric facilities. \n\nIn this regime, the detection efficiency becomes independent of the companion separation. Interestingly, all arrays have the same efficiency. In other words, as long as the companion is expected to be resolved by the interferometric baselines, the choice of array configuration does not matter. We conclude that there is no reason to favour a given VLTI configuration when looking for faint unknown companion with separations in the range $6-40\\,$mas. More quantitatively, Fig.~\\ref{fig:cont_sep} (right) displays the detection efficiency in this annular region for the four VLTI configurations versus the companion contrast, and for two observing scenarios (snapshot and long integration). The combination of five observations separated by one hour provides a detection efficiency higher than 95\\% for companion contrasts of $10^{-2}$, assuming a realistic closure phase accuracy of $0.25\\deg$. \n\nWe note that the curve of completeness versus contrast become significantly sharper when accumulating observations. As shown by the solid lines in the right panel of Fig.~\\ref{fig:cont_sep}, when accumulating five pointings, the efficiency drops from 80\\% for a contrast of $5\\times10^{-3}$ to less than 10\\% for a contrast of $3\\times10^{-3}$. The constraints provided by this dataset can thus be presented as a sensitivity limit and an inner working-angle, as for a classical imaging observation.\n\nQuantitatively, when accumulating several pointings, these detection limits computed from the derivation of Sect.~\\ref{sec:theory} are compatible with the blind-test analyses presented by \\citet[Fig.~5]{Absil:2011} and the Monte-Carlo simulations of \\citet[Fig.~4 and Eq.~2]{Lacour:2011}.\n\n\\subsection{Performances of standard configurations}\n\\label{sec:fake}\n\nTo add generality to the results presented in the previous section, we now study configurations that are not specifically linked to any existing interferometric array. We select the configurations presented in Fig.~\\ref{fig:fake}, which is a non-redundant linear configuration, a fully redundant linear configuration and a Y-shaped configuration. All configurations have their longest baseline of the same size. The results are the following:\n\n\\begin{enumerate}\n\\item The linear non-redundant and Y-shaped configurations have similar detection limits as the currently offered (irregular) VLTI configurations for snapshot observations. Surprisingly, they also have similar performances when accumulating several pointings, while we may have expected that the Y-shape would unveil faster the remaining blind-spots.\n\\item For snapshot observations, the linear redundant configuration favours the \\emph{dynamic range} with respect to the \\emph{completeness}: it has a fainter detection limit for completeness levels below 50\\%, but becomes significantly worse for higher completeness levels. When accumulating several pointings, both the highest completeness and largest dynamic range are reached, although the gain is never higher than 20\\%.\n\\item Y-shaped arrays have smaller inner working angles than linear configurations of identical maximum baseline length, even when considering the accumulation of several pointings. The gain is almost a factor of two in terms of angular resolution.\n\\end{enumerate}\n\n\\begin{figure}\n    \\centering \n    \\includegraphics[scale=0.58]{fake}\n    \\caption{Fake VLTI interferometric configurations used in this paper: non-redundant linear D0-E0-H0-K0 (red), redundant linear D9-G2-H9-K9 (green, fake stations), and Y-shaped E0-J3-J2-H0 (blue).}\n    \\label{fig:fake}\n\\end{figure}\n\n\\begin{figure*}\n    \\centering \n    \\includegraphics[scale=0.55]{all_contdist_fake}\\hspace{2cm}\n    \\includegraphics[scale=0.55]{all_compcont_fake}\n    \\caption{{\\it Left:} 3-$\\sigma{}$ sensitivity versus separation for a completeness level of 80\\%. {\\it Right:} Completeness in the separation range $6-40\\,$mas versus contrast. Simulations are for a single snapshot pointing (dashed lines) and for five pointing separated by one hour (solid lines). Colours are for the four configurations displayed in Fig.~\\ref{fig:fake}. The black curves are for the U1-U2-U3-U4 configuration displayed in Fig.~\\ref{fig:visa}. The contrast axes can be scaled for any accuracy on the closure phase (here $\\sigma=0.25\\deg$).   }\n    \\label{fig:cont_sep_fake}\n    \\label{fig:comp_cont_fake}\n\\end{figure*}\n\n\n\n\n\\section{Closure phase accuracy and achievable dynamic range}\n\\label{sec:accuracy}\n\n\\subsection{Photon and piston noises}\nWe consider the theoretical photon noise limit for an observation of 1\\,h on a star of sixth magnitude using one-metre class telescopes (e.g., the $1.8\\,$m auxiliary telescopes of the VLTI). The choice of a sixth-magnitude star is driven by the current sensitivity limit of most interferometric instruments world-wide. A crude estimation of the photon noise is given by\n\\begin{equation}\n\\sigma_{phot} \\approx \\frac{360\\deg}{\\sqrt{N}} \\; ,\n\\end{equation}\nwhere $N$ is the total number of detected photons. If we consider that the 1\\,h observing time should include the overheads and the observation of a calibration star, the effective integration time on target will be of the order of 20\\,min. Assuming a realistic total transmission of 2\\%, including both the instrumental and atmospheric contributions, the number of detected photons is $N\\approx 10^8$ and the resulting photon noise is $\\sigma_{phot} \\approx 0.1\\deg$. For a $1^\\mathrm{mag}$ star, the resulting photon noise would be $\\sigma_{phot} \\approx 0.01\\deg$.\n\nIn theory, closure phase is a robust observable against the telescope phase errors \\citep{Monnier03}. However, in the context of non-zero exposure times and the presence of atmospheric turbulence, closure phase measurements are also affected by piston noise. A proper estimation of its amplitude is beyond the scope of this paper but it is still possible to provide a rough upper limit. Since piston noise is independent of the number of incident photons, it is expected to dominate the final statistical uncertainty for very bright stars. As an example, in the case of the PIONIER instrument at VLTI, the statistical uncertainty for bright stars is typically of the order of $0.25\\deg$ to $2.5\\deg$ for an integration time of 1\\,min. This uncertainty depends on the atmospheric conditions as expected for piston noise. In decent atmospheric conditions, integrating over 20\\,min allows the piston noise contribution to be reduced below $0.2\\deg$, as it decreases with the square root of the integration time.\n\n\n\t\\subsection{Calibration accuracy}\n\nIt is interesting to compare these fundamental limits to published accuracies that include the calibration of the instrumental closure phase (also called the transfer function):\n\\begin{description}\n\\item[VLTI\/AMBER:] \\citet{Absil:2010} reported calibration errors of between $0.20\\deg$ and $0.37\\deg$ depending on the night, using this three-telescope combiner in its medium spectral resolution mode ($R=1\\,500$). With the low spectral resolution mode ($R=35$), typical calibration errors range from one to a few degrees \\citep[see for instance][]{kraus:2009apr,le-bouquin:2009mar}.\n\\item[VLTI\/PIONIER:] Typical calibration errors range from $0.25\\deg$ to $1\\deg$ \\citep{LeBouquin:2011,Absil:2011} for this four-telescope combiner. Sequences with closure phases stable down to $0.1\\deg$ have been recorded. Systematic discrepancies have been noted when calibration stars were separated by more than $10\\deg$ on the sky.\n\\item[CHARA\/MIRC:] The typical accuracy obtained with this four-telescope combiner is between $0.1$ and $0.2\\deg$, which makes this instrument the most accurate of the currently available suite. Calibration uncertainties dominate the final accuracy at this level \\citep{Zhao08,Zhao:2010,Zhao11}.\n\\end{description}\n\nAltogether, a typical noise floor of $\\sim0.25\\deg$ seems to appear for the calibration of the closure phases in long baseline interferometry. Two results indicate that the major cause is probably longitudinal dispersion: (i) that the accuracy depends on the spectral resolution and (ii) the dependence on position of the calibration star on the sky. This is also the finding of \\citet{Zhao11}, who proposed an elaborate calibration scheme for MIRC. Although this is clearly a very promising way of characterizing already known substellar companions, this method is probably not suited to surveying a large number of stars with a standard calibration procedure. Interestingly, $0.25\\deg$ is also the noise floor reported by \\citet{Lacour:2011} for the calibration of the closure phase of the spare aperture masking mode of NACO at VLT. This calibration noise floor of $0.25\\deg$ theoretically does not prevent us from reaching very high dynamic ranges, by accumulating a large number of observations and\/or baselines, as for instance in sparse aperture masking or spectrally dispersed observations. Care should however be taken to ensure that individual closure phase measurements are statistically independent. In particular, one should avoid repeating the same systematic errors in individual data sets, e.g., by choosing different calibrator stars, instrumental setups, etc., in order not to reach a true noise floor in the observations.\n\nConcerning future instruments, the announced accuracy on the closure phases is $1\\deg$ for the K-band four-telescope combiner GRAVITY \\citep[document \\mbox{VLT-SPE-ESO-15880-4853} and][]{Gillessen:2010} and from $1\\deg$ to $5\\deg$ for the L-band four-telescope combiner MATISSE \\citep[Florentin Millour, private communication and][]{Lopez:2008}. Although these performances may be conservative, we conclude that the next generation of VLTI instruments is unlikely to break the $0.25\\deg$ limit.\n\n\n\t\\subsection{Discussion}\n\nIt appears realistic to reach a closure phase accuracy of $0.25\\deg$ within less than one hour on one-metre class telescopes for stars of magnitude six and brighter. According to the results of Sect.~\\ref{sec:array}, such performances allow a dynamic range of $5\\times 10^{-3}$ ($\\Delta \\mathrm{mag}=5.75$) to be reached with 80\\% completeness when five pointings are obtained with a four-telescope interferometer. The same performance would probably be reached within a snapshot using an interferometric instrument combining six telescopes or more at a time.\n\nThis result can be compared with the survey for stellar and sub-stellar companions on the ten-metre Keck and five-metre Palomar telescopes using aperture masking techniques in the K band \\citep{Kraus08,Kraus:2011}. The achieved dynamic range is $\\Delta K\\approx5.5$ for separation as small as $25\\,$mas (slightly worse for Palomar). A similar dynamic range and inner working angle have  been achieved within an ongoing survey of massive stars using aperture masking at VLT\/NACO in the H band, e.g. $\\Delta H\\approx5$ down to $25\\,$mas with this eight-metre diameter telescope (Hugues Sana, private communication). All close companions presented in these near-infrared surveys would have been detected by interferometry with an efficiency higher than 90\\%, provided that they reside within the interferometric field-of-view. In addition, this efficiency would have been achieved down to about 2\\,mas. At longer wavelengths, the aperture masking technique has a dynamic range of about $\\Delta L\\approx 7.5$ \\citep{Hinkley:2011}, although the inner working angle in that case is only $70\\,$mas. There is currently no L-band interferometric beam-combiner with closure phase capabilities to which these performances could be compared.\n\nSeveral observing programs related to faint companion detection would benefit significantly from the capabilities of closure phase measurements on long-baseline interferometric instruments. An example is the search for low-mass (sub)-stellar companions around main-sequence stars residing in nearby young associations. Considering associations with ages between 10\\,Myr and 200\\,Myr, and a limiting magnitude $K=6$ for the instrument, one could survey stars up to about 15-20\\,pc for stellar type M0V, 40\\,pc for type G0V, and 120\\,pc for type A0V. With an estimated median dynamic range of $\\Delta K \\simeq 6$, we computed the masses of the faintest companions that could be detected within a survey of nearby moving groups, using the (sub-)stellar cooling models of \\citet{Baraffe98,Baraffe03}. The results are given in Table~\\ref{tab:associations}, showing that the $13 M_{\\rm Jup}$ ($=0.012 M_{\\odot}$) limit between the brown dwarf and planetary regimes can be reached for young late-type dwarfs. In the case of A-type stars, one would be sensitive to companions in the range M3V-M7V depending on the age. For even younger stars, located in nearby star forming regions, closure phase measurements have the potential to reveal the formation of planetary-mass objects, as suggested by \\citet{Kraus12}.\n\n\\begin{table}[t]\n\\caption{Sensitivity limits in terms of companion masses around young main-sequence stars.}\n\\centering\n\\begin{tabular}{c c c c}\n\\hline\\hline\nAge & A0V & G0V & M0V \\\\\n\\hline\n10 Myr & $0.09 M_{\\odot}$ & $0.017 M_{\\odot}$ & $0.012 M_{\\odot}$ \\\\\n50 Myr & $0.22 M_{\\odot}$ & $0.043 M_{\\odot}$ & $0.013 M_{\\odot}$ \\\\\n200 Myr & $0.35 M_{\\odot}$ & $0.08 M_{\\odot}$ & $0.030 M_{\\odot}$ \\\\\n\\hline\n\\end{tabular}\n\\label{tab:associations}\n\\end{table}\n\nAnother program is the determination of the binary fraction for massive stars. The interest is that despite the preponderance of multiple stars, the mechanism that produces multiple stars rather than single stars is still uncertain. The measurement of the mass distribution, and how it evolves with the mass of the primary, is an appropriate tool for disentangling between \\emph{capture} and \\emph{fragmentation} models. Stellar companions to B-type stars have been investigated using AO \\citep[e.g. ][]{Roberts:2007}, although the stars observed typically have a large range of distances, limiting the statistical significance of the results. Radial velocity measurement of massive stars is challenging owing to the lack of suitable spectral lines and their intrinsic broadening. With the limiting magnitude $K=6$ presented in this paper, it is possible to observe interferometrically all the B stars within a distance of 75\\,pc ($\\sim$$100$ objects for the southern hemisphere), providing the first comprehensive study of massive binaries in the $0.25-5\\,$AU separation range. \n\nLast but not least, one of the main selling argument for high-precision closure phases in optical interferometry has been the direct detection of hot extrasolar giant planets (EGP). Several hot EGP host stars are indeed bright enough to be observed with state-of-the-art interferometric instruments. For mature planetary systems ($>10$\\,Myr), the expected contrast between the planet and the star is however generally too low ($<10^{-3}$) to be currently accessible with closure phase measurements \\citep{Zhao08,Zhao11}. To routinely reach the hot EGP regime ($\\Delta K \\simeq 8 - 10$), a gain of two to four magnitudes is required in the dynamic range, which would translate into a noise floor of between $0.04\\deg$ and $0.006\\deg$ on the closure phase. Achieving such an accuracy would probably require a significant breakthrough in the instrumental domain.\n\n\n\\section{Conclusions}\n\nIn summary, optical interferometric surveys designed to detect faint companions have the following properties:\n\\begin{enumerate}\n\\item  The observable (closure phase) is robust against unstable atmospheric seeing conditions \\citep{Monnier03}. Integrating over $20\\,$min is sufficient to consistently reduce the photon and atmospheric noises below $0.25\\deg$, which appears as a hard limit for the calibration of current instruments.\n\\item A single snapshot with four telescopes provides a 80\\% detection efficiency at \\mbox{$\\Delta\\mathrm{mag}=4.5$} as soon as the binary separation is fully resolved. The only requirement of the interferometric array is to use baselines as long as possible to improve the inner working angle, which is typically of the order of a few milli-arcseconds.\n\\item Accumulating more observations (several pointing and\/or recombining more telescopes) allows a dynamic range \\mbox{$\\Delta\\mathrm{mag}=6$} to be reached, which appears to be a realistic limit in respect to published performances. Going deeper would require us to break the current limit of $0.25\\deg$ on the closure phase accuracy, or to massively increase the number of observations.\n\\item The achievable dynamic range scales linearly with the closure phase accuracy.\n\\end{enumerate}\n\nIn conclusion, interferometric closure phase surveys would be well-suited as filler programs for service-mode interferometric facilities, such as the VLTI. They can be considered as a useful complement to the AO-assisted imaging surveys currently carried out on ten-metre class telescopes. In particular, the search space of long-baseline interferometry bridges the gap between the wide companions found in direct imaging and the close companions detected by RV measurements. Moreover, interferometry could nicely complement RV studies in the particular cases where RV measurements are quite inappropriate. Young stars, for instance, are especially promising targets since their (sub)stellar companions are supposed to be relatively bright compared to their host stars.\n\n\\begin{acknowledgements} \nThe authors thank the referee for his helpful comments. This work has made use of the Smithsonian\/NASA Astrophysics Data System (ADS) and of the Centre de Donnees astronomiques de Strasbourg (CDS). All calculations and graphics were performed with the freeware \\texttt{Yorick}\\footnote{\\texttt{http:\/\/yorick.sourceforge.net}}.\n\\end{acknowledgements}\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nDiscovering new particles would entail the Standard Model (SM) being falsified in Popper's sense~\\cite{Popper:1934} and force us to extend it. Absent such discovery, \nthe SM is still falsifiable upon finding new forces among the known particles. Because the SM has a characteristic energy scale of 100 GeV (the mass of the Higgs boson at $m_h=125$ GeV, as well as the vacuum constant $v=246$ GeV exemplify it), but no new particles below 1000 GeV have been found, there is a scale separation that begs the use of Effective Field Theory.\n\nThe popular SMEFT extension of the SM Electroweak Symmetry Breaking Sector (EWSBS) adds to it operators classified by their mass dimension,\n\\begin{equation}\n    \\mathcal{L}_{\\rm SMEFT} =\n    \\mathcal{L}_{\\rm SM} + \n    \\sum_{n=5}^{\\infty}\n    \\sum_i\n    \\frac{c_i^{(n)}}{\\Lambda^{n-4}} \\mathcal{O}_i^{(n)}(H) \\ .\n \\label{SMEFTL}\n \\end{equation}\nThese operators $\\mathcal{O}_i^{(n)}(H)$,  whose intensity is controlled by the  Wilson coefficients $c_i^{(n)}$ for a given reference scale $\\Lambda$, are the potentials of those new forces being sought. \nA nonzero $c_i^{(n)}$ would signal departure from the SM, that then would need to be extended,\n{perhaps by new resonances~\\cite{Dobado:2017lwg}}.\n\nBut it is easy to ask oneself how the whole framework of SMEFT can be tested. \nEffective theories include all the possible interactions that are compatible with the known particle content and symmetries believed to hold. Would it not be that any separation from the SM could be recast in SMEFT form? In that case, absent some new light particle, any phenomena could be described by adding an operator with a parameter to the SM. This is not so, as we will detail. \n\nThe particle content of the electroweak sector is packaged in a Higgs doublet field in the SM as well as in SMEFT\n\\begin{equation}\nH = \n\\frac{1}{\\sqrt{2}} \\begin{pmatrix} \\varphi_1+i\\varphi_2 \\\\   \\varphi_0 + i\\varphi_3\n\\end{pmatrix} \n= \nU(\\boldsymbol{\\omega}) \\begin{pmatrix} 0 \\\\   (v+h_{\\rm SMEFT})\/\\sqrt{2}\n\\end{pmatrix} \\, ,\n\\end{equation}     \nwhere the Cartesian coordinates $\\varphi_a$ can be rearranged to the polar decomposition in terms the $\\omega_i$ Goldstone bosons (which set the orientation of $H$ through the unitary matrix $U(\\boldsymbol{\\omega})$) and the radial coordinate $h_{\\rm SMEFT}$ (with $|H|=(v+h_{\\rm SMEFT})\/\\sqrt{2}$). \n\n\n\n\\begin{figure}[!b] \n\\centering\n     \\begin{tikzpicture}[scale=1]\n     \\draw[decoration={aspect=0, segment length=1.8mm, amplitude=0.7mm,coil},decorate] (-1,1) -- (0,0)-- (-1,-1);\n     \\draw[] (0,0)-- (1,1);\n     \\draw[] (1.25,1) node {$h_1$};\n          \\draw[] (1.25,-1) node {$h_n$};\n     \\draw[] (0,0)-- (.98,0.6);\n          \\draw[] (1.25,0.6) node {$h_2$};\n     \\draw[] (.75,-.15) node {.};\n     \\draw[] (.75,0.) node {.};\n     \\draw[] (.75,-0.3) node {.};\n     \\draw[] (0,0)-- (1,-1);\n       \\draw[] (3.25,0) node {$\\displaystyle{  \\, =\\,  -\\frac{n!a_n}{2v^n} \\,  s }$};  \n\\end{tikzpicture}\n\\caption{\\label{fig:Feynman}\\small\nThe $\\omega\\omega\\to nh$ processes can be the key to disentangling the nature of the EWSBS. They give direct access to the $a_i$ coefficients of the flare function $\\mathcal{F}$, and hence to their correlations, as listed in Table~\\ref{tab:correlations}.\n}\n\\end{figure}\n\n\n\nAn additional non-linear redefinition of $h_{\\rm SMEFT}$ allows us to rearrange the SMEFT Lagrangian in the form of a  more general theory, HEFT: \n\\begin{align}  \\label{HEFTL}\n{\\cal L}_{\\rm HEFT} = \\frac{1}{2}\\partial_\\mu h_{\\rm HEFT}\\partial^\\mu h_{\\rm HEFT}-V(h_{\\rm HEFT}) + \\nonumber \\\\ \n\\frac{1}{2}\\!\\mathcal{F}(h_{\\rm HEFT})\n\\partial_\\mu \\omega^i \\partial^\\mu \\omega^j\\!\\left(\\!\\delta_{ij}\\!+\\!\\frac{\\omega^i\\!\\omega^j}{v^2\\!-\\!\\boldsymbol{\\omega}^2}\\!\\right)\\ .\n\\end{align}\nOf current focus therein is the flare function~\\cite{Alonso:2016oah, Grinstein:2007iv} \n\\begin{equation} \\label{Fexpansion}\n    {\\mathcal F}(h_{\\rm HEFT})=1+\\sum_{n=1}^{\\infty}{a_n}\\Big(\\frac{h_{\\rm HEFT}}{v}\\Big)^n \\,,\n\\end{equation}\nwhich amounts to a radial ``scale'' (think of $a(t)$ in a Friedmann-Robertson-Walker cosmology) in the field space of the $(h,\\omega_i)$ electroweak bosons (with $\\omega_i$ analogous to the spatial coordinates).  \nWhat we call attention to in this letter is that the Taylor-series coefficients of $\\mathcal{F}$ as defined in Eq.~(\\ref{Fexpansion}) must satisfy experimental correlations or constraints as given in Table~\\ref{tab:correlations} and in the companion extended manuscript~\\cite{Gomez-Ambrosio:2022giw} \n{\\it if SMEFT is a valid description}. It is clear that an experimental program aimed at these correlations\nvia the key process to access $\\mathcal{F}$, $\\omega\\omega\\to nh$ as sketched in Figure~\\ref{fig:Feynman}, or $mh\\to nh$ to access $V(h_{\\rm HEFT})$, can test the validity of SMEFT itself, and not only its parameters.\n\n\n\n\n\\begin{figure}[!t]\n    \\centering\n    \\includegraphics[width=0.8\\columnwidth]{Bandasa1a2.png}\n    \\caption{\\small The correlation $a_2=2a_1-3$ that SMEFT predicts at order $1\/\\Lambda^2$ is plotted against the current 95\\% confidence intervals for these two HEFT parameters~\\cite{ATLAS:2020qdt,CMS:2022cpr}.}\n    \\label{fig:a1a2}\n\\end{figure}\n\n\n\n\\begin{table}[!t] \n\\setlength{\\arrayrulewidth}{0.3mm}\n\\setlength{\\tabcolsep}{0.2cm} \n\\renewcommand{\\arraystretch}{1.4}\n    \\caption{\\small Correlations between the $a_i$ HEFT coefficients necessary for SMEFT to exist, at order $\\Lambda^{-2}$ (first and second columns, with the numbers in the second consistent with 95\\% confidence-level experimental bounds $a_1\/2 \\in [0.97,1.09]$~\\cite{ATLAS:2020qdt}).\n    The right column provides the corresponding numerical values at the next order~\\cite{Gomez-Ambrosio:2022giw}. \n   \n    They are quoted in terms  of $\\Delta a_1:=a_1-2$ and $\\Delta a_2:=a_2-1$, so that all  objects in the table vanish in the Standard Model, with all the equalities becoming $0=0$.    \n    \\label{tab:correlations}}\n    \\centering\n    \\begin{tabular}{|c|c|c|}\\hline\n        $\\mathcal{O}(1\/\\Lambda^2)$ & $\\mathcal{O}(1\/\\Lambda^2)$ & $\\mathcal{O}(1\/\\Lambda^4)$ \\\\ \\hline \n        $\\Delta a_2=2\\Delta a_1$ & $\\Delta a_2\\in [-0.12,0.36] $  &  \\\\\n        $a_3=\\frac{4}{3} \\Delta a_1$ &  $a_3\\in[-0.08,0.24]$  & $a_3\\in [-3.1,1.7]$  \\\\  \n        $a_4=\\frac{1}{3} \\Delta a_1$ &  $a_4\\in [-0.02,0.06]    $ &  $a_4\\in [-3.3,1.5]$ \\\\\n        $a_5=0$ &  & $a_5 \\in[-1.5,0.6]$ \\\\\n        $a_6=0$ &  & $ a_6=a_5$ \\\\ \n        \\hline\n    \\end{tabular}\n\\end{table}\n\n\n\n\nAs an example, the correlation predicted by SMEFT among $a_1$ and $a_2$ is shown in Figure~\\ref{fig:a1a2}.\n\n\n\n\n\n\nThere are several reasons why the experimental tests of those coefficients need large energies and statistics, at the limit of what is possible today at the LHC and beyond.\nFirst, the $\\mathcal{F}$ function multiplies terms with derivatives of the Goldstone bosons $\\partial_\\mu \\omega_i\\to q_\\mu \\omega_i$ that yield couplings proportional to their four-momenta, and become more relevant at higher energies. \nSecond, \n the equivalence theorem~\\cite{Veltman:1989ud,Dobado:1993dg} tells us that the scattering of longitudinal gauge bosons is related to scattering of Goldstone boson ($\\omega_i \\sim W^\\pm_L,\\ Z_L$): the EW gauging of the HEFT Lagrangian~(\\ref{HEFTL}) leads to the $WW\\to n h$ interaction $\\Delta\\mathcal{L}_{\\rm HEFT}^{WW\\to n h} = \\left(\\frac{1}{2}m_Z^2 Z_\\mu Z^\\mu + m_W^2 W_\\mu^+ W_\\mu^-\\right) \\mathcal{F}(h_{\\rm HEFT})$, which for longitudinal gauge bosons clearly dominates over the non-derivative interactions from $V$ only  at high energies~\\cite{Dicus:1987ez,Kallianpur:1988cs}.  \nAnd third, an increasing number of Higgs bosons  (necessary to access each $h^n$ order of $\\mathcal{F}$, the Higgs-flare function) requires an ample phase space and, thus, high energy.  \n\nHowever, as we shortly show after\nEq.~(\\ref{changeofvariable}) below, the correlations arise from the need for\nconsistency of the SMEFT formulation when a change of variable $h_{\\rm HEFT}\\to h_{\\rm SMEFT}$ is performed. This change affects any other piece of the Lagrangian that involves the Higgs bosons, such as the Yukawa couplings to fermions (that we leave for future works) but also the interactions among Higgs bosons themselves. Such interactions do not require derivative couplings and first appear even in the renormalizable SM, \n\\begin{equation}\n{\\mathcal{L}}_{\\rm SM} = |\\partial H|^2 -   \\underbrace{\\left( \\mu^2 |H|^2 +\\lambda|H|^4\\right)}_{V(H)} \\, ,\n\\end{equation}\nin the much discussed $V(H)$  \nHiggs-potential, accessible already at low $\\sqrt{s}$. In HEFT, this potential has additional non-renormalizable couplings and can be organized in a power-series expansion\n\\begin{align}\\label{expandV}\n    V_{\\rm HEFT}= \\frac{m_h^2 v^2}{2}  \\Bigg[&   \\left(\\frac{h_{\\rm HEFT}}{v}\\right)^2 +  v_3   \\left(\\frac{h_{\\rm HEFT}}{v}\\right)^3 \\nonumber+ \\\\  &+ v_4    \\left(\\frac{h_{\\rm HEFT}}{v}\\right)^4 + \\dots \\Bigg]\\,,\n\\end{align}\nwith $v_3=1$, $v_4=1\/4$ and $v_{n\\geq 5}=0$ in the SM. \nIts coefficients also need to satisfy constraints that are exposed in Table~\\ref{tab:corV} and Figure~\\ref{fig:a1a22} if and when SMEFT applies.\n\n\nLet us then see, very briefly, how these correlations come about. Instead of relying on the powerful geometric methods of~\\cite{Alonso:2016oah,Alonso:2015fsp,Alonso:2016btr,Alonso:2021rac,Alonso:2022ffe} we use the more pedestrian coordinate-dependent approach, more familiar to phenomenologists working on LHC physics.\nThe goal is to see when is it possible to cast  \nEq.~(\\ref{HEFTL}) into the specific SMEFT one, Eq.~(\\ref{SMEFTL}).\nThis we write as \n\\begin{align}\n \\mathcal{L}_{\\rm SMEFT}= |\\partial H|^2-V(|H|^2)\n+\\frac{1}{2} B(|H|^2)(\\partial (|H|^2))^2 +\\dots  \n\\label{eq:SMEFTL-kin+V+B}\n\\end{align}\nwhere the  non-derivative and derivative terms, respectively given by $V$ and $B$, \ncollect typical SMEFT operators of Eq.~(\\ref{SMEFTL}) such as, at lowest $1\/\\Lambda^2$ order,\n\\begin{equation}\\label{operatorsSMEFT}\n  \\mathcal{O}_H := (H^\\dagger H)^3  \\, , \\ \\ \\ \n  \\mathcal{O}_{H \\Box} := (H^\\dagger H) \\Box (H^\\dagger H)\\ .\n\\end{equation} \n There are also other operators, such as, e.g., \n$  \\mathcal{O}_{HD} = (H^\\dagger D_{\\mu} H)^*  (H^\\dagger D^{\\mu} H) $, but they break custodial symmetry, and LEP studies suggest that the $SU(2)\\times SU(2)\\to SU(2)$ electroweak symmetry breaking mechanism is the appropriate pattern, leaving the residual custodial $SU(2)$ as a good approximate global symmetry of the scalar sector.  \nThe additional $A(H)$ structure pointed out in~\\cite{Cohen:2020xca} for Lagrangian~(\\ref{eq:SMEFTL-kin+V+B}) can be eliminated through partial integration and the use of the equations of motion~\\cite{Gomez-Ambrosio:2022giw}. \n\n \n \n\n\n\\begin{figure}[!t]\n    \\centering\n    \\includegraphics[width=0.8\\columnwidth]{Bandasv3v4.png}\n    \\caption{\\small The correlation $v_4=\\frac{3}{2} v_3 -\\frac{5}{4} -\\frac{1}{6}\\Delta a_1$ that SMEFT predicts at $\\mathcal{O}(1\/\\Lambda^2)$ is plotted making use of current 95\\% confidence interval for $v_3\\in[-2.5,5.7]$~\\cite{ATLAS:2021jki}. The experimental $a_1$ uncertainty~\\cite{ATLAS:2020qdt,CMS:2022cpr},  $a_1\/2\\in[0.97,1.09]$, is numerically negligible and allows to  predict a SMEFT band given by the solid black line. An experimental measurement for $v_4$ is still missing. }\n    \\label{fig:a1a22}\n\\end{figure}\n\\begin{table}[!t]\n\\setlength{\\arrayrulewidth}{0.3mm} \n\\setlength{\\tabcolsep}{0.2cm}  \n\\renewcommand{\\arraystretch}{1.4} \n\\caption{\\small Correlations among the coefficients $\\Delta v_3:=v_3-1$, $\\Delta v_4:=v_4-1\/4$, $v_5$ and $v_6$ of the HEFT Higgs potential expansion in Eq.~(\\ref{expandV}) that need to hold, at $\\mathcal{O}(1\/\\Lambda^2)$, if SMEFT is a valid description of the electroweak sector.\nBased  on the current bound $\\Delta v_3\\in[-2.5,5.7]$ in Ref.~\\cite{ATLAS:2021jki}, $\\mathcal{O}(1\/\\Lambda^2)$ SMEFT predicts the coefficient intervals in the last column, testable in few-Higgs final states. A coupling \n$c_{H\\Box}\\neq 0$ induces the correction $\\Delta a_1\\propto c_{H\\Box}$, nevertheless numerically negligible since $v_3$ experimental uncertainties much exceed those of $a_1$.}\n\\label{tab:corV}\n\\begin{center}\n \\begin{tabular}{|c|c|} \\hline \n        $\\Delta v_4=\\frac{3}{2}\\Delta v_3  -\\frac{1}{6}\\Delta a_1$ & $\\Delta v_4\\in[-3.8,8.6]$\\\\[2ex]\n        $ v_5=6v_6=\\frac{3}{4}\\Delta v_3 -\\frac{1}{8}\\Delta a_1 $ \n        & $v_5=6 v_6 \\in[-1.9,4.3]$\n           \\\\\n         \\hline    \n    \\end{tabular}\n\\end{center}  \n\\end{table}\n \nTo proceed, we need to perform the following conversion to pass from SMEFT to HEFT and viceversa:  \n \\begin{align}\n& |\\partial H|^2 +  \n\\frac{1}{2} B(|H|^2)(\\partial (|H|^2))^2 \\longleftrightarrow \\nonumber \\\\\n&\\frac{v^2}{4} \\mathcal{F}(h_{\\rm HEFT}) \\,   {\\rm Tr}\\{ \\partial_\\mu U^\\dagger \\partial^\\mu U\\} \n+\n\\frac{1}{2} (\\partial h_{\\rm HEFT})^2 \\,,\n\\end{align}  \nThe change from SMEFT to HEFT is straightforward and always possible, with the canonical, nonlinear change of variables given in differential form as\n\\begin{equation}  \\label{changeofvariable}  \ndh_{\\rm HEFT}\\, =\\, \\sqrt{1+(v+h_{\\rm SMEFT})^2 B(h_{\\rm SMEFT})}\\,\\, dh_{\\rm SMEFT} \\,,\n\\end{equation}\nwhere the flare-function is provided by the relation \n\\begin{equation}\n\\mathcal{F}(h_{\\rm HEFT}) \\,=\\, \\left(1+h_{\\rm SMEFT}\/v\\right)^2\\, .\n\\end{equation}  \nHowever,  \nthe reverse conversion from HEFT to SMEFT, \n\\begin{equation}\nh_{\\rm HEFT}   \\,=\\, \\mathcal{F}^{-1}\\left((1+h_{\\rm SMEFT}\/v)^2\\right) \\,,\n\\end{equation}\nruns into difficulty. \nThis is because of the need to reconstruct squared operators of the Higgs doublet field $H$ that is the basis of SMEFT, such as\n\\begin{eqnarray}\n|H|^2 &=& \\frac{(v+h_{\\rm SMEFT})^2}{2}\\, ,\n\\nonumber \\\\\n(\\partial|H|^2)^2 &=& (v+h_{\\rm SMEFT})^2 \\,   (\\partial h_{\\rm SMEFT})^2 \\nonumber \\\\ &=&\\, 2 |H|^2 \\,   \n(\\partial h_{\\rm SMEFT})^2 \\, .\n\\end{eqnarray} \nThe extra $|H|^2$ on the right hand side of the second equation \nends in a denominator\n\\begin{align}\n&\\mathcal{L}_{\\rm SMEFT} =  \\underbrace{ |\\partial H|^2}_{= \\mathcal{L}_{\\rm SM}} \\quad +\\quad\n\\label{eq:HEFT2SMEFT}\n\\\\ \\nonumber \n&\\underbrace{ \\frac{1}{2} \\bigg[  \\frac{8|H|^2}{v^2}\\bigg(   (\\mathcal{F}^{-1})'\\left(2| H |^2\/v^2\\right)  \\bigg)^2 \\,\\,\\,-\\,\\,\\, 1\\bigg] \\, \\frac{(\\partial| H |^2)^2}{2| H |^2}  }_{=\\Delta \\mathcal{L}_{\\rm BSM}}\\, .\n\\end{align}\nAs SMEFT is assumed to have the analytical power expansion\nin Eq.~(\\ref{SMEFTL}), such singularity precludes its existence and needs to be cancelled by the preceding bracket in the second line of Eq.~(\\ref{eq:HEFT2SMEFT}). \n\nThe result is the same as that obtained by geometric methods~\\cite{Cohen:2020xca}, there must be a double zero of $\\mathcal{F}$, a symmetric point with respect to the global $SU(2)\\times SU(2)$ group so that the SMEFT expansion can be performed.  Furthermore,\nanalyticity requires that all its odd derivatives vanish at the symmetric point.\n\nThe particular case of the SM is given by $\\mathcal{F}=(1+h_{\\rm SMEFT}\/v)^2 $. As already pointed out,\nat higher orders in $h\/v$, the existence of SMEFT requires that the odd derivatives of $\\mathcal{F}$ at the symmetric point $h_\\ast$ vanish. \n\nThe correlations from Table~\\ref{tab:correlations} can then be obtained by matching the Taylor expansion of $\\mathcal{F}$ around such symmetric point \n$ h_{\\rm HEFT}=  h_\\ast$ with the expansion around our physical vacuum $h_{\\rm HEFT}=0$.\nInstead of that matching, one can also obtain the correlations by eliminating the SMEFT Wilson coefficients order by order. For example, at $\\mathcal{O}(1\/\\Lambda^2)$ there is only one operator, $\\mathcal{O}_{H\\Box}$,  in Eq.~(\\ref{operatorsSMEFT}), that controls all the HEFT coefficients of $\\mathcal{F}$: \n\\begin{align}\n&&\na_1 = 2a=2\\left(1 + v^2\\frac{c_{H\\Box}}{\\Lambda^2}\\right)\\,, \\quad \na_2 = b=1+{4v^2}\\frac{c_{H\\Box}}{\\Lambda^2} \\,, \n\\nonumber \\\\ \n&&\na_3 = \\frac{8v^2}{3}\\frac{c_{H\\Box}}{\\Lambda^2}\\,, \\qquad \na_4 = \\frac{2v^2}{3}\\frac{c_{H\\Box}}{\\Lambda^2}\\,,\\qquad \na_{n\\geq 5} = 0 \\, .\n\\end{align}\nThe potential $V(h_{\\rm HEFT})$ is in turn also affected by $\\mathcal{O}_H$,\n\\begin{eqnarray}\n&&\nv_3 =1 +  \\frac{3v^2 c_{H\\Box}}{\\Lambda^2} +\\epsilon_{c_H}\n\\,, \\,\\,\\, \nv_4 = \\frac{1}{4} +  \\frac{25v^2c_{H\\Box} }{6\\Lambda^2} +\\frac{3}{2}\\epsilon_{c_H}\n\\,,\n\\nonumber\\\\\n&& \nv_5 =    \\frac{2v^2c_{H\\Box} }{\\Lambda^2} +\\frac{3}{4}\\epsilon_{c_H}\n\\,,\\,\\,\\, \nv_6 =  \\frac{v^2c_{H\\Box} }{3\\Lambda^2} +\\frac{1}{8}\\epsilon_{c_H}\n\\,,\n\\nonumber\\\\\n&& \nv_{n\\geq 7} = 0 \\,, \\qquad\\qquad\\qquad \n\\end{eqnarray}\nwith  $m_h^2=\\, -2\\mu^2 \\left(1+\\frac{2 c_{H\\Box}v^2}{\\Lambda^2}+ \\frac{3}{4}\\epsilon_{c_H}\\right)$, $2\\langle |H|^2\\rangle =v^2= -\\frac{\\mu^2}{\\lambda}\\left(1-\\frac{3}{4}\\epsilon_{c_H}\\right) $ and $\\epsilon_{c_H}= - \\frac{2 v^4 c_H}{m_h^2\\Lambda^2}= \\frac{\\mu^2 c_H}{\\lambda^2  \\Lambda^2}$. \n\nVarious authors, see {\\it e.g.}~\\cite{Brivio:2016fzo} have pointed out to differences between the  SMEFT and HEFT formulations~\\cite{Dobado:2019fxe}.\nFor example, in SMEFT the Goldstone\n $\\omega_i$  and Higgs  $h_{\\rm SMEFT}$ bosons are arranged in a left-$SU(2)$ doublet while in HEFT\n$h_{\\rm HEFT}$ is an $SU(2)\\times SU(2)$ singlet, independent of the Goldstone triplet $\\omega_i$.  \nAlso, in SMEFT the Higgs field always appears  in the combination $ (h_{\\rm SMEFT} + v)$ \nand thus, HEFT deploys more independent higher-dimension effective operators  (in exchange, it is less model dependent).\nThis means that SMEFT is natural when $h_{\\rm SMEFT}$ is a fundamental field while HEFT is typical for composite models of the EWSBS (such as those with $h_{\\rm HEFT}$ as a Goldstone boson).\nAnd finally, the counting of SMEFT is based in a cutoff $\\Lambda$ expansion taking the canonical operator dimensions, $\\mathcal{O}(d)\/\\Lambda^{d-4}$ (independently of $N_{\\rm loops}$) whereas HEFT is a derivative expansion (independently of $N_{\\rm particles})$ like the older Electroweak Chiral Lagrangian, with $\\mathcal{F}(h)$ inserted in the derivative Goldstone term.\n\n\n\nAmong the two types of correlations that we have presented in tables~\\ref{tab:correlations} and~\\ref{tab:corV}, the first ones for the coefficients of $\\mathcal{F}$ are more interesting for large values of the energy $\\sqrt{s}\\gg m_h\\sim m_W \\sim m_Z$ whereas the second, that do not involve Goldstone bosons, are therefore of greater interest at low energies, when  and additionally \nthe potential competes with the derivative operators on equal ground, as $\\sqrt{s} \\sim m_i$.  \n\\\\\nNevertheless, a lot of this is cosmetic and can be reorganized by changing variables $h_{\\rm SMEFT}\\leftrightarrow h_{\\rm HEFT}$. What is key is the San Diego criterion~\\cite{Alonso:2015fsp,Alonso:2016oah}:  \n$\\mathcal{F}(h_{\\rm HEFT})$ must have a point $h_\\ast$ symmetric under the global $SU(2)\\times SU(2)$  group  and due to its existence and convergence in the $h$ field space, SMEFT is deployable if and only if  (which is a statement about the HEFT Lagrangian)\n\\begin{itemize}\n\\item   $\\exists h_\\ast \\in\\mathbb{R}$ where $\\mathcal{F} (h_\\ast)=0$, and \n\\item  Because of the need for $\\mathcal{L}_{\\rm SMEFT}$ analyticity, $\\mathcal{F}$ is analytic between our vacuum $h=0$ and $h_\\ast$, particularly around $h_\\ast$. Moreover its odd derivatives vanish.\n\\end{itemize}\nWe have presented new relations that implement this criterion \nup to $\\mathcal{O}(1\/\\Lambda^2)$ and $\\mathcal{O}(1\/\\Lambda^4)$ in the $1\/\\Lambda$ counting;\nmore precision is unnecessary until (if) separations from the Standard Model are found. Then only, with the scale $\\Lambda$ at hand, out of separations of EFT coefficients from the SM, can we decide how relevant the corrections due to the higher orders are expected to be, and whether further work is warranted.\n\nIn conclusion, we have newly translated these conditions into correlations among HEFT coefficients \nwhose violation falsifies SMEFT. \nMoreover, since many extensions of the Standard Model incorporating supersymmetry, supergravity, or other possibilities, can be cast as a SMEFT, they can be likewise simultaneously falsified.\n\n\nFor the time being, \n{no separations from the SM have been found~\\cite{Eboli:2021unw} and} one can only infer direct experimental bounds on  the first  terms, $a_1$ and, perhaps, $a_2$, so we have to wait for data with a larger number of Higgs boson before assessing them. But when this will be done, the correlations will allow to falsify SMEFT in experiment\neven without new particles. We believe that this possibility improves the standing of SMEFT as a scientific theory.\n\n\n\\vspace{.2cm}\n\\begin{acknowledgments}\nSupported by spanish MICINN PID2019-108655GB-I00 grant, and Universidad Complutense de Madrid under research group 910309 and the IPARCOS institute; \nERC Starting Grant REINVENT-714788; UCM CT42\/18-CT43\/18;\nthe Fondazione Cariplo and Regione Lombardia, grant 2017-2070.\n\\end{acknowledgments}\n\\vspace*{-0.5cm}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}