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+{"text":"\\section{Introduction}\nThe work of Seiberg and Witten \\cite{sewi,sewi2} on the exact solution \nto the low-energy effective action of $N=2$ supersymmetric Yang-Mills theory \nhas afforded not only a renewed insight into the charge confinement \n\\cite{sewi2} and the chiral symmetry breaking \\cite{sewi2}, \nbut also a marvelous insight into topological \ninvariants \\cite{wi} and conformal field theories in four \ndimensions \\cite{scft}. The key ingredients in obtaining these exact results\nare duality and the appearance of massless monopoles\/dyons in the strong\ncoupling regions of the theory. \nIn the weak coupling region, the exact solution enables us to\ndetermine full non-peturbative instanton corrections to the \neffective coupling constant, whose evaluation is otherwise quite cumbersome\nin the standard framework of quantum field theory.\n\nThe low-energy effective action is described in terms of a prepotential\nwhich is a single holomorphic function of superfields of  $N=2$ \n$U(1)$ vector multiplets. \nThe exact solution for the prepotential may be \ncharacterized by the period integrals of \nthe special type of one-form on a hyper-elliptic curve.\nThe curves associated with a variety of $N=2$ \nsupersymmetric Yang-Mills theories and QCD \nhave been studied extensively \\cite{KlLeThYa}$-$\\cite{itsa}.\nThe moduli space of the curve contains singularities, at which \nsome solitons become massless. \nIn order to investigate the strong coupling physics, one needs to \nevaluate the period integrals near the singular locus.\nAn efficient approach to study this problem is to use the Picard-Fuchs \nequation, the differential \nequation which the periods obey \\cite{Cer,KlLeTh,Ma,ItYa}.\n\nIn the present article, we shall investigate the quantum moduli space\nof $N=2$ supersymmetric Yang-Mills theories with massless hypermultiplets\nand gauge groups $SU(2)$ and $SU(3)$.\nWe shall derive the Picard-Fuchs equation for the scalar part of $N=2$ \n$U(1)$ vector multiplets and their duals.\nBy solving the non-linear differential equation obeyed by the prepotential\nfor $G=SU(2)$, we will explicitly evaluate\nthe non-perturvative contributions in the prepotential \nin both weak and strong coupling regions.\n\n\n\\section{$N=2$ Supersymmetric Yang-Mills Theory with Massless Hypermultiplets}\nWe begin with reviewing some basic properties of the low-energy \neffective action of the $N=2$ supersymmetric $G=SU(N_{c})$ QCD\n\\cite{sewi2,arplsh}. \nIn $N=1$ superfield formulation, \nthe theory contains chiral \nmultiplets $\\Phi^{a}$ and  chiral field strength $W^{a}$ \n($a=1,\\ldots, {\\rm dim}G$) both in the adjoint representation of $G$, \nand chiral superfields $Q^{i}$ in the $N_{c}$ and \n$\\tilde{Q}^{i}$ ($i=1,\\cdots, N_{f}$)\nin the $\\bar{N}_{c}$ representation of the gauge group.\nThe superpotential is given by \n${\\cal W}=\\tilde{Q}T^{a}\\Phi^{a}Q+M^{i}_{j}\\tilde{Q}_{i} Q^{j}$\nwhere $T^{a}$ is the generator of $G$ and $M^{i}_{j}$ is the mass matrix.\nAlong the flat direction the scalar fields $\\phi$ of $\\Phi$ get\nvacuum expectation values, which break the gauge group to the Cartan subgroup \n$U(1)^{r}$ where $r=N_{c}-1$ is the rank of $G=SU(N_{c})$. \nWhen the squark fields do not have vacuum expectation values, the low-energy\neffective theory is in \nthe Coulomb branch and contains $r$ $U(1)$ vector multiplets \n$(A^{i}, W_{\\alpha}^{i})$ $(i=1,\\cdots,r)$, where $A^{i}$ are $N=1$\nchiral superfields and $W_{\\alpha}^{i}$ are $N=1$ vector superfields. \nThe quantum moduli space may be characterized by \nthe low-energy effective Lagrangian ${\\cal L}$ \nwith the prepotential ${\\cal F}(A)$\n\\begin{equation}\n{\\cal L}={1\\over 4\\pi} {\\rm Im }\n\\left(\\int d^2\\theta d^2\\bar{\\theta}A_{D i}\\bar{A}^{i}+{1\\over2}\n\\int d^2\\theta \\tau^{i j}W_{\\alpha}^{i} W^{j \\alpha}\\right), \n\\end{equation}\nwhere $A_{D i}={\\partial {\\cal F}\\over \\partial A^{i}}$ is a field dual to $A^{i}$ \nand \n$\\tau^{i j}={\\partial^{2}{\\cal F}\\over \\partial A^{i} \\partial A^{j}}$ the effective\ncoupling constants.\nWe denote the scalar component of $A^{i}$, $A_{D i}$ by $a^{i}$, $a_{D i}$.\nThe pairs $(a_{D i},a_{i})$ are the $Sp(2r,{\\bf Z})$ section over the \nspace of gauge invariant parameters $s_{i}$ $(i=2,\\cdots, N_{c})$\ndefined by ${\\rm det}(x-\\phi)=x^{N_{c}}-\\sum_{i=2}^{N_{c}} s_{i} x^{N_{c}-i}$.\nThe quantum moduli space of the Coulomb branch is parametrized by the \ngauge invariants $s_{i}$ and the eigenvalues $m_{1},\\ldots, m_{N_{f}}$ \nof the mass matrix.\n\nThe sections $(a_{D i},a_{i})$ are obtained as\nthe period integrals of a meromorphic differential\n$\\lambda$ over the hyper-elliptic curve ${\\cal C}$ \\cite{haoz}:\n\\begin{eqnarray}\ny^{2}&=&C(x)^2-G(x), \\nonumber \\\\\nC(x)&=& x^{N_{c}}-\\sum_{i=2}^{N_{c}} s_{i} x^{N_{c}-i}\n       +{\\Lambda^{2N_{c}-N_{f}}\\over 4} \\sum_{i=0}^{N_{f}-N_{c}}\n t_{i}(m) x^{N_{f}-N_{c}-i},\n\\nonumber \\\\\nG(x)&=& \\Lambda^{2N_{c}-N_{f}}\\prod_{i=1}^{N_{f}}(x+m_{i}),\n\\end{eqnarray}\nwhere $\\Lambda$ stands for the QCD scale parameter and $t_{i}(m)$ is defined by\n$\\prod_{i=1}^{N_{f}}(x+m_{i})=\\sum_{i=0}^{N_{f}} t_{i}(m) x^{N_{f}-i}$ with\n$t_{0}(m)=1$.\nThe terms proportional to $\\Lambda^{2N_{c}-N_{f}}$ in $C(x)$ are absent \nin the case of $N_{f}<N_{c}$. Notice that for $N_c=2$, $N_f=2$ theory\n$C(x)$ should read\n\\begin{equation}\nC(x)=x^2-s_2+{\\Lambda^2\\over 8}.\n\\end{equation}\n\n\nThe meromorphic differential $\\lambda$ is given by\n\\begin{equation}\n\\lambda={x\\over 2\\pi i} d\\log {C-y\\over C+y}.\n\\end{equation}\nThis differential satisfies the relations\n\\begin{equation}\n{\\partial \\lambda \\over \\partial s_{i}}=-\\omega_{N_{c}+1-i}+d(*),\n\\end{equation}\nwhere $\\omega_{i}={x^{i-1}dx \\over y}$ is the basis of the holomorphic \none-forms of the hyper-elliptic curve ${\\cal C}$.\nChoosing the basis of homology cycles $(A_{i},B_{i})$ on ${\\cal C}$ with\nthe canonical intersection form $A_{i}\\cdot B_{j}=\\delta_{i j}$, \n$A_{i}\\cdot A_{j}=B_{i}\\cdot B_{j}=0$, one can express \nthe pairs $(a_{D i},a_{i})$ as\n\\begin{eqnarray}\n\\alpha_{i}\\cdot a_{D} &= & \\int_{B_{i}}\\lambda, \\nonumber \\\\\n(\\lambda_{i}-\\lambda_{i-1})\\cdot a &=& \\int_{A_{i}}\\lambda,  \\hskip7mm\ni=1,\\cdots,r,\n\\end{eqnarray}\nwhere $\\alpha_{i}$ are the simple roots of $G$ and $\\lambda_{i}$ \nthe fundamental weights of $G$ ($\\lambda_{0}\\equiv0$).\n\nThe BPS mass formula $M\\geq |q_{i}a_{i}+h_{i}a_{D i}|$\nshows that the massless soliton may be characterized by the vanishing cycle\n$q_{i}A_{i}+h_{i}B_{i}\\sim 0$.\nTherefore the singularities of the quantum moduli space are determined by the \ndiscriminant $\\Delta$ of the curve.\nLet us enumerate explicit forms of the discriminants $\\Delta^{N_{c}}_{N_{f}}$ \nfor the $SU(N_c)$ theories ($N_c =2,\\, 3$) with $N_{f}(<2 N_{c})$ matter\nhypermultiplets. In the following \n$\\Lambda_{N_{f}}$ denotes the scale parameter corresponding to \nthe $N_{f}$-flavor theory.\n\\begin{itemize}\n\\item $N_{c}=2$ ($u\\equiv s_{2}$)\n\\begin{eqnarray}\n\\Delta^{2}_{0}&=& 256 \\Lambda_{0}^{8} (u-\\Lambda_{0}^2)(u+\\Lambda_{0}^2)\n\\nonumber \\\\\n\\Delta^{2}_{1}&=& -\\Lambda_{1}^{6} (27 \\Lambda_{1}^{6}+256 u^3) \\nonumber \\\\\n\\Delta^{2}_{2}&=& \\Lambda_{2}^4 (8 u-\\Lambda_{2}^2)^2 (8 u+\\Lambda_{2}^2)^2\/16\n\\nonumber \\\\\n\\Delta^{2}_{3}&=& \\Lambda_{3}^{2} (\\Lambda_{3}^{2}-256 u) u^4 \\nonumber \\\\\n\\end{eqnarray}\n\\item $N_{c}=3$ ($u\\equiv s_{2}$, $v\\equiv s_{3}$)\n\\begin{eqnarray}\n\\Delta^{3}_{0}&=& 64 \\Lambda_{0}^{18}\n       (-27 (v-\\Lambda_{0}^3)^2+4 u^3)\n                    (-4 u^3+27 (v+\\Lambda_{0}^3)^2)\n\\nonumber \\\\\n\\Delta^{3}_{1}&=&\\Lambda_{1}^{15}\n (-3125 \\Lambda_{1}^{15}+ 256 \\Lambda_{1}^{5}  u^{5}\n  + 22500 \\Lambda_{1}^{10}   u v - 1024 u^{6}  v \\nonumber \\\\\n& & -\n 43200 \\Lambda_{1}^{5}  u^2  v^2  + 13824 u^3  v^3  - 46656 v^5 ) \\nonumber \\\\\n\\Delta^{3}_{2}&=&  64 \\Lambda_{2}^{12} v^2 (-4 (\\Lambda_{2}^2-u)^3 - 27 v^2) \n(-4 (\\Lambda_{2}^2+u)^3+ 27 v^2)\n\\nonumber \\\\\n\\Delta^{3}_{3}&=& \\Lambda_{3}^9  (-\\Lambda_{3}^3  + 4 v)^3\n   (729 \\Lambda_{3}^6  u^3  - 256 u^6  - 3888 \\Lambda_{3}^3  u^3  v \\nonumber \\\\\n  & & + 3456 u^3  v^2  +\n      2916 \\Lambda_{3}^3  v^3  - 11664 v^4 )) \/ 16 \\nonumber \\\\\n\\Delta^{3}_{4}&=& 4 \\Lambda_{4}^6 v^4 (\\Lambda_{4}^4 u - 8 \\Lambda_{4}^2 u^2 +\n 16 u^3 + 2 \\Lambda_{4}^3 v - 72 \\Lambda_{4} u v - 108 v^2)\\nonumber  \\\\\n& &    (-\\Lambda_{4}^4 u + 8 \\Lambda_{4}^2 u^2 - 16 u^3 + 2 \\Lambda_{4}^3 v\n- 72 \\Lambda_{4} u v + 108 v^2) \\nonumber \\\\\n\\Delta^{3}_{5}&=& 4 \\Lambda_{5}^3  v^5  (\\Lambda_{5}^2  u^5  - 256 u^6  +\n2 \\Lambda_{5}^3  u^3  v - 528 \\Lambda_{5} u^4  v +\n \\Lambda_{5}^4  u v^2 \\nonumber \\\\\n& &  -\n300 \\Lambda_{5}^2  u^2  v^2  + 3456 u^3  v^2  + 4 \\Lambda_{5}^3  v^3  -\n4212 \\Lambda_{5} u v^3  - 11664 v^4 ). \\nonumber \\\\\n\\end{eqnarray}\n\\end{itemize}\nNote that for the gauge group $SU(2)$ the zeros of the discriminant realize \nthe discrete\n${\\bf Z}_{4-N_{f}}$ symmetry which arises from the $U(1)_{\\cal R}$ \nanomaly and the parity transformation in the flavor symmetry $O(2 N_{f})$.\nOrder of the zeros corresponds to the number\nof massless hypermultiplets appearing at the singularities. \nMassless monopoles obey the spinor representation of the flavor \nsymmetry $SO(2 N_{f})$. \nFor $SU(3)$ symmetries are less obvious  particularly for odd flavors.\n\n\n\\section{The Picard-Fuchs equations}\nLet us study the Picard-Fuchs equations for \n$(a_{D i},a_{i})$ and examine their solutions.\nWe first discuss the $SU(2)$ case in detail for simplicity. \nOne can change the curve of fourth order to the third one by \nusing the M\\\"obius transformation. \nThe curves are given by\n\\begin{equation}\ny^2=x^2(x-u)+{1 \\over 4} \\Lambda_0^4 \\, x\n\\end{equation}\nfor $N_f=0$ and\n\\begin{equation}\ny^2=x^2(x-u)-{1 \\over 64} \\Lambda_{N_f}^{2(4-N_f)} (x-u)^{N_f-1}\n\\end{equation}\nfor $N_f=1,\\, 2,\\, 3$.\nThe meromorphic differential $\\lambda$ for this curve\nis given by \n\\begin{equation}\n\\lambda ={\\sqrt{2} \\over 8\\pi}\\, {2u-(4-N_f)x \\over y}\\, dx\n\\end{equation}\nwhich satisfies\n\\begin{equation}\n{d\\lambda \\over du}={\\sqrt{2} \\over 8\\pi} {d x \\over y}+d(*).\n\\end{equation}\nIt is not difficult to see that the period $\\Pi=\\oint \\lambda$ satisfies the \ndifferential equation \n\\begin{equation}\np(u){d^2 \\Pi \\over du^2}+ \\Pi =0,\n\\label{pfsu2}\n\\end{equation}\nwhere \\cite{Cer,KlLeTh,ItYa}\n\\begin{equation}\np(u)=\\left\\{ \\begin{array}{ll}\n\\displaystyle{4 (u^2-\\Lambda_{0}^4)} & \\qquad N_{f}=0 \\\\\n\\displaystyle{4u^2+{27 \\Lambda_1^6\\over 64u}} & \\qquad N_{f}=1 \\\\\n\\displaystyle{\\ 4\\Big(u^2-{\\Lambda_2^4\\over 64}\\Big)} & \\qquad N_{f}=2 \\\\\n\\displaystyle{u\\Big(4u-{\\Lambda_3^2\\over 64}\\Big)} & \\qquad N_{f}=3.\n\\end{array}\n\\right.\n\\end{equation}\nNote that the regular singularities of (\\ref{pfsu2}) correspond to the\nzeros of the discriminant. Introducing a  new variable\n\\begin{equation}\nz=\\left\\{\\begin{array}{ll}\n\\displaystyle{\\left({u\\over \\Lambda_{0}^2}\\right)^2} & \\qquad N_{f}=0 \\\\\n\\ \\displaystyle{-{256\\over 27} \\left({u\\over \\Lambda_{1}^2}\\right)^3}\n & \\qquad N_{f}=1 \\\\\n\\displaystyle{64 \\left({u\\over \\Lambda_{2}^2}\\right)^2 }& \\qquad N_{f}=2 \\\\\n\\displaystyle{256 \\left({u\\over \\Lambda_{3}^2}\\right)} & \\qquad N_{f}=3, \\\\\n\\end{array}\n \\right.\n\\end{equation}\nit is shown that the Picard-Fuchs equation (\\ref{pfsu2}) turns out to be the\nGauss's hypergeometric system\n\\begin{equation}\nz(1-z) {d^{2} \\Pi \\over d z^{2}}\n+(\\gamma-(\\alpha+\\beta+1)z){d \\Pi \\over d z}-\\alpha\\beta \\Pi=0,\n\\label{hyperg}\n\\end{equation}\nwhere\n\\begin{equation}\n\\alpha=\\beta={-1\\over 2(4-N_{f})}, \\hskip10mm  \n\\gamma={3-N_{f} \\over 4-N_{f}}.\n\\end{equation}\nUsing the fundamental solutions of (\\ref{hyperg}), we may evaluate \nthe power series solutions for $(a_{D},a)$ near the singularities, \nwhich has been  explicitly done in our previous work \\cite{ItYa}.\n\nNext we write down the Picard-Fuchs equations for $SU(3)$ theory \nwith flavors $N_{f}\\leq 5$.\nThe periods $\\Pi=\\oint \\lambda$ now depend on two parameters $u$ and $v$.\nThe Picard-Fuchs equations take the form\n\\begin{equation}\n{\\cal L}_{1}\\Pi=0, \\quad {\\cal L}_{2}\\Pi=0,\n\\end{equation}\nwhere ${\\cal L}_{1}$ and ${\\cal L}_{2}$ are given by\n\\begin{itemize}\n\\item $N_{f}=0$\n\\begin{eqnarray}\n{\\cal L}_{1}&=& {1\\over 3} (-27\\Lambda_{0}^{6}+4 u^3+27 v^2)\\partial_{v}^{2}\n          +12 u v \\partial_{u}\\partial_{v}\n          +3 v \\partial_{v}+1 \\nonumber \\\\\n{\\cal L}_{2}&=& {1\\over u } (-27\\Lambda_{0}^{6}+4 u^3+27 v^2)\\partial_{u}^{2}\n          +12 u v \\partial_{u}\\partial_{v}\n          +3 v \\partial_{v}+1\n\\end{eqnarray}\n\\item $N_{f}=1$\n\\begin{eqnarray}\n{\\cal L}_{1}&=& {-25\\Lambda_{1}^{5}u^2+84 u^3 v +405 v^3 \\over 45 v}\\partial_{v}^{2}\n          +{64 u^4-1125 \\Lambda_{1}^{5} v+2160 u v^{2}\\over 180 v}\n \\partial_{u}\\partial_{v} \\nonumber \\\\\n       & &+ {16 u^3+135 v^2 \\over 45 v} \\partial_{v}+1  \\nonumber \\\\\n{\\cal L}_{2}&=& {4 (-25\\Lambda_{1}^{5}u^2+84 u^3 v +405 v^3)\\over\n            -25 \\Lambda_{1}^5 +84 u v} \\partial_{u}^{2} \\nonumber \\\\\n& &\n+{(-25 \\Lambda_{1}^5 +36 u v) (-25 \\Lambda_{1}^5+96 u v)\\over -25 \\Lambda_{1}^{5} +84 u v}\n \\partial_{u}\\partial_{v} \\nonumber \\\\\n & &    +  {3 v (-25 \\Lambda_{1}^5 +36 u v)\\over -25 \\Lambda_{1}^5 +84 u v} \\partial_{v}\n          +1\n\\end{eqnarray}\n\\item  $N_{f}=2$\n\\begin{eqnarray}\n{\\cal L}_{1}&=& {-8\\Lambda_{2}^{4}u+8 u^3  +27 v^2\\over 3}\\partial_{v}^{2}\n+{4 (2 \\Lambda_{2}^8 -4 \\Lambda_{2}^4 u^2 +2 u^4+27 u v^2)\\over 9 v}\n \\partial_{u}\\partial_{v} \\nonumber \\\\\n  & &     + { -8 \\Lambda_{2}^4 u +8 u^3 +27 v^2 \\over 9 v} \\partial_{v}+1  \\nonumber \\\\\n{\\cal L}_{2}&=& {-8\\Lambda_{2}^{4}u+8 u^3  +27 v^2\\over 2 u} \\partial_{u}^{2}\n          + {3v (\\Lambda_{2}^4+3 u^2)\\over u} \\partial_{u}\\partial_{v}     +  1\n\\end{eqnarray}\n\\item  $N_{f}=3$\n\\begin{eqnarray}\n{\\cal L}_{1}&=& {( 4 v-\\Lambda_{3}^3) (4 u^3+9 v^2)\\over 4 v} \\partial_{v}^{2}\n+{u (64 u^3-81 \\Lambda_{3}^3 v +432  v^2) \\over 36 v}  \\partial_{u}\\partial_{v} \\nonumber \\\\\n       & &     + { 16 u^3 +27 v^2 \\over 9 v} \\partial_{v}+1  \\nonumber \\\\\n{\\cal L}_{2}&=& {4 u^3+9 v^2\\over u} \\partial_{u}^{2}\n          +{u (-9 \\Lambda_{3}^3 +32  v)\\over 4} \\partial_{u}\\partial_{v}\n          - v  \\partial_{v}\n                +1\n\\end{eqnarray}\n\n\\item  $N_{f}=4$\n\\begin{eqnarray}\n{\\cal L}_{1}&=& { -\\Lambda_{4}^4 u -56 \\Lambda_{4}^2 u^2+240 u^3+324 v^2\\over 36} \\partial_{v}^{2}\n+ {-8 \\Lambda_{4}^2 u^2 +32 u^3+27 v^2 \\over 9 v} \\partial_{v} \\nonumber \\\\\n& &\n +{2 \\Lambda_{4}^4 u^2 -16 \\Lambda_{4}^2 u^3 +32 u^4\n  -9 \\Lambda_{4}^2 v^2  +108 u v^2\\over 9 v} \\partial_{u}\\partial_{v}\n         +1  \\nonumber \\\\\n{\\cal L}_{2}&=&\n{ -\\Lambda_{4}^4 u -56 \\Lambda_{4}^2 u^2+240 u^3+324 v^2 \\over \\Lambda_{4}^2+60 u}\n \\partial_{u}^{2} \\nonumber \\\\\n& & +{v(\\Lambda_{4}^2+12 u) (-\\Lambda_{4}^2+36 u) \\over \\Lambda_{4}^2+60 u}\\partial_{u}\\partial_{v}\n     +{3 v (\\Lambda_{4}^{2}-36 u)\\over \\Lambda_{4}^2+60 u} \\partial_{v}+1 \\nonumber \\\\\n\\end{eqnarray}\n\n\\item\n $N_{f}=5$\n\\begin{eqnarray}\n{\\cal L}_{1}&=& {v (-2\\Lambda_{5}^2 u^2 +528 u^3+57\\Lambda_{5} u v +324 v^2)\\over\n 5 \\Lambda_{5} u + 36 v}\\partial_{v}^{2} \\nonumber \\\\\n& &     +{ -5 \\Lambda_{5}^2 u^3 +1280 u^4 -\\Lambda_{5}^3 u v +332 \\Lambda_{5} u^2 v\n-4 \\Lambda_{5}^2 v^2+1728 u v^2 \\over 4 (5 \\Lambda_{5} u\n + 36 v)} \\partial_{u}\\partial_{v} \\nonumber \\\\\n & & +\n{-5 \\Lambda_{5}^2 u^2 +1280 u^3+88 \\Lambda_{5} u v \n+432 v^2 \\over 4(5 \\Lambda_{5} u + 36 v)} \\partial_{v}+\n1  \\nonumber \\\\\n{\\cal L}_{2}&=& \n{-2\\Lambda_{5}^2 u^2 +528 u^3+57\\Lambda_{5} u v +324 v^2\\over 132 u}\n \\partial_{u}^2 \\nonumber \\\\\n& & +{-\\Lambda_{5} u^3-5 \\Lambda_{5}^2 u v+864 u^2 v-36 \\Lambda_{5} v^{2}\n\\over 132 u}\\partial_{u}\\partial_{v}\n -{\\Lambda_{5} u +324 v\\over 132} \\partial_{v}+1. \\nonumber \\\\\n\\end{eqnarray}\n\\end{itemize}\nThe Picard-Fuchs equations  for $N_{c}=3$, $N_{f}=0$ theory\nhave been investigated in detail by Klemm, Lerche and Theisen \\cite{KlLeTh}.\nThey show that the system reduces to the Appell's hypergeometric system of \ntype $F_{4}$. Although other cases are not classified as the Appell's system, \nthe solutions of\nthese equations may be obtained in the form of the power series expansion \nin $u$ and $v$. \n\n\n\\section{Calculation of the Prepotentials}\nSolving the system of the Picard-Fuchs equations, we evaluate the \nnon-perturbative  corrections to the prepotentials, which can be \nregarded as the multi-instanton contributions in the weak coupling \nregion and the threshold corrections in the strong coupling regions.\nIn the previous work \\cite{ItYa}, we have calculated some \nnon-trivial  corrections by using the solutions of the hypergeometric system. \nIn this article, we shall instead take more direct approach by solving the \nnon-linear differential equation for the prepotential in $SU(2)$ theory. \nThe present approach was proposed originally by Matone for $SU(2)$ pure Yang-\nMills theory \\cite{Ma}.\n\nWe start with the Picard-Fuchs equation (\\ref{pfsu2}). \nNotice that the equation carries no first derivative term. \nThis implies that the Wronskian of the system\n\\begin{equation}\nW=\\left|\n\\begin{array}{cc}\na & a_{D} \\\\\n\\partial_{u}a & \\partial_{u}a_{D}\n\\end{array} \\right|\n\\end{equation}\nis independent of $u$; $\\partial_{u}W=0$. When we integrate this with respect to\n$u$ there appears a constant, the explicit value of which \nis evaluated at any singularity in the moduli space. One finds that\n\\begin{equation}\nW={i b \\over 4\\pi},\n\\label{wrons}\n\\end{equation}\nwhere $b=4-N_{f}$ is the coefficient of the beta-function of $SU(2)$ $N=2$ \nQCD.  \nIn the present case, we can further simplify the equation (\\ref{wrons}) since\nthe Wronskian is written in the form of the total derivative\n\\begin{equation}\nW=\\partial_{u} \\left( a {\\partial {\\cal F} \\over \\partial a} -2 {\\cal F}\\right).\n\\label{totald}\n\\end{equation}\nThus we get the following relation between the prepotential and the \ngauge invariant parameter $u$\n\\begin{equation}\na {\\partial {\\cal F} \\over \\partial a} -2 {\\cal F}={i b \\over 4\\pi}u + {\\rm const.}\n\\label{scaling}\n\\end{equation}\nThe constant term may be absorbed in the definition of ${\\cal F}$ by shifts,\nwhich does not affect the form of the low-energy effective action.\nThis identity has been generalized for any $N=2$ Yang-Mills theories with \nor without matter hypermultiplets \\cite{sothya,egya} \nand is important in view of \nits relation to the soliton theory \\cite{whto}.\n\nThe relation (\\ref{scaling}) allows us to express $u$ in terms of $a$\n\\begin{equation}\nu={\\cal G}(a)\\equiv {4\\pi \\over i b}\n\\left(a {\\partial {\\cal F} \\over \\partial a} -2 {\\cal F}\\right).\n\\end{equation}\nThe Picard-Fuchs equation becomes\n\\begin{equation}\n-p({\\cal G}(a)){d^2 {\\cal G} \\over d a^2}+\na \\left({d {\\cal G}\\over d a}\\right)^3=0.\n\\label{nonl}\n\\end{equation}\nThis non-linear equation is utilized to  determine the instanton coefficients \nrecursively.\nNear the singularity at $u=\\infty$, the prepotential takes the form\n\\begin{equation}\n{\\cal F}(a)=\n\\left\\{\\begin{array}{ll}\n{i a^{2}\\over 2\\pi}\n\\left\\{ 2 \\log {a^{2}\\over \\Lambda_{0}^{2}}\n+\\sum_{k=0}^{\\infty} {\\cal F}_{k} (0)\n\\left( {\\Lambda_{0}\\over a} \\right)^{4k}\\right\\}\n & N_{f}=0 \\\\\n{i a^{2}\\over 4\\pi}\n\\left\\{ (4-N_{f}) \\ln {a^{2}\\over \\Lambda_{N_{f}}^{2}}\n+\\sum_{k=0}^{\\infty} {\\cal F}_{k} (N_{f})\n\\left( {\\Lambda_{N_{f}}^{2} \\over a^{2}} \\right)^{k(4-N_{f})}\\right\\}\n& N_{f}=1,2,3.\n\\end{array}\\right.\n\\end{equation}\nThe first four coefficients are listed in the table\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|}\n\\hline\n$N_{f}$ & ${\\cal F}_{1}$ & ${\\cal F}_{2}$ & ${\\cal F}_{3}$ & ${\\cal F}_{4}$ \\\\\n\\hline\n        &                              & & & \\\\\n0       &  $-{1\\over 2^5}$              & $-{5\\over 2^{14}}$    &\n        $-{3\\over 2^{18}}$ & $-{1469\\over 2^{18}}$  \\\\\n        &                              & & & \\\\\n1       &  ${3\\over 2^{12}}$             & -${3\\cdot 17 \\over 2^{288}}$  &\n           ${5\\cdot 7 \\cdot 11\\over 3 \\cdot 2^{38}}$ &\n           ${3^2 \\cdot  5 \\cdot 7\\cdot 9679 \\over 2^{59}}$  \\\\\n        &                              & & & \\\\\n2       &  $-{1\\over 2^{11}}$\n        &  $ -{5\\over 2^{26}}$\n        &  $-{3\\over 2^{36}}$\n        &  $ -{13\\cdot 113 \\over 2^{55}}$ \\\\\n        &                              & & & \\\\\n3       &  $-{1\\over 2^{5}}$\n        &  $-{1\\over 2^{24}}$\n        &  $0$\n       &   $-{5\\over 2^{51}}$\\\\\n\\hline\n\\end{tabular}\n\\end{center}\nand summarized by the formulas ($N_{f}=1,2,3$)\n\\footnote{We have corrected an error of  ${\\cal F}_{2}(N_{f})$ in the \nprevious work \\cite{ItYa}.}:\n\\begin{eqnarray}\n{\\cal F}_{1} (N_{f})&=&-{1  \\over 2^{b} c(N_{f})}\n                         {1\\over 2\\, b^2}, \\nonumber \\\\\n{\\cal F}_{2} (N_{f})&=&-{1\\over 2^{2b} c(N_{f})^{2}}\n                       {5-8b+4b^{2} \\over\n                         8^{2}\\, b^{4}}, \\nonumber \\\\\n{\\cal F}_{3} (N_{f})&=&{1\\over 2^{3b}c(N_{f})^{3}}\n                      {   (1-b)(1-2b)\n                         (-23+30b-16b^{2}) \\over\n                       1728\\, b^{6}}, \\nonumber \\\\\n{\\cal F}_{4} (N_{f})&=&{1\\over 2^{4b} c(N_{f})^{4}}\n{(1 - 2b )(-677 + 3910b - 8124b^2 +\n      8456b^3 - 4672b^4 + 1152b^5) \\over\n294912\\, b^8}, \\nonumber \\\\\n\\label{weakcorre}\n\\end{eqnarray}\nwhere \n$c(N_{f})=\\eta_{N_{f}} 2^{8} (4-N_{f})^{N_{f}-4}$ ($N_{f}=1,2,3$) and\n$\\eta_{1}=-1,\\, \\eta_{2}=\\eta_{3}=1$.\nThe $N_{f}=0$ result \\cite{KlLeTh,Ma} may be \nobtained by replacing $c(N_{f})$ with 1 in the $N_{f}=2$ case.\nThe results (\\ref{weakcorre}) indeed agree with those obtained previously\n\\cite{ItYa}.\n\nSimilarly we may solve the non-liner differential equation \nin the strong coupling region where monopoles become massless.\nIn this case the dual field $a_{D}$ is a good coordinate around the \nsingularity $u=u_{0}$ where\n\\begin{equation}\nu_{0}=\\left\\{\\begin{array}{ll}\n\\Lambda_{0}^2 & \\qquad N_{f}=0 \\\\\n\\displaystyle{-3\\cdot 2^{-{8\\over 3}}\\Lambda_{1}^2 }\n & \\qquad N_{f}=1 \\\\\n\\displaystyle{\\Lambda_{2}^2\/8} & \\qquad N_{f}=2 \\\\\n0 & \\qquad N_{f}=3. \\\\\n\\end{array}\n \\right.\n\\end{equation}\nIntroduce the dual prepotential ${\\cal F}_{D}(a_{D})$ satisfying\n$a={d {\\cal F}_{D}\\over d a_{D}}$. Then, instead of (\\ref{totald}), the\nWronskian is expressed in terms of ${\\cal F}_{D}$. Hence\naround $u=u_{0}$ we get\n\\begin{equation}\nu=u_{0}+{4\\pi\\over b} \\left( \n2  {\\cal F}_{D} -a_{D}{d  {\\cal F}_{D} \\over d a_{D}} \\right).\n\\end{equation}\nSimilarly to the weak-coupling computation we now obtain the non-linear\ndifferential equation which makes it possible to calculate ${\\cal F}_{D}$\nrecursively. Using the expansion\n\\begin{equation}\n {\\cal F}_{D}={i \\over 8\\pi} a_{D}^2\\left\\{ k \\log \n\\left({a_{D}^2\\over \\Lambda_{N_{f}}^2}\\right)\n+\\sum_{n\\geq-1} {\\cal F}_{D n}(N_{f}) \n\\left({a_{D}\\over \\Lambda_{N_{f}}}\\right)^n \\right\\}\n\\end{equation}\nwith $k=1$ ($N_{f}=0$) and $k=2^{N_{f}-1}$ ($N_{f}=1,2,3$), \nwe find the coefficients as follows\n\\begin{eqnarray}\n{\\cal F}_{D\\ -1}(N_{f})&=& 16 k b^2 \\tilde{c}(N_{f}), \\nonumber \\\\\n{\\cal F}_{D \\ 1}(N_{f})&=&{2\\cdot 2 k\n \\over 3 \\tilde{c}(N_{f})}\n                         {(1-2b)(-5+2b)\\over 16 b^{2}}, \\nonumber \\\\\n{\\cal F}_{D \\ 2}(N_{f})&=&{-2 k\\over 2 \\tilde{c}(N_{f})^{2}}\n{(1-2b)(-61+190b-92 b^{2}+8b^{3})\n\\over 1152 b^{4}}, \\nonumber \\\\\n{\\cal F}_{D \\ 3}(N_{f})&=&{2\\cdot 2k\\over 5 \\tilde{c}(N_{f})^{3}}\n{(1-2b)(1379-8162b+14596 b^{2}-7288b^{3}+960b^{4})\n\\over 110592 b^{6}}, \\nonumber \\\\\n\\end{eqnarray}\nwhere\n$\\tilde{c}(1)=-3^{-1\/2}\\cdot 2^{-17\/6}$,$\\tilde{c}(2)=-i 2^{-9\/2}$ and\n$\\tilde{c}(3)=2^{-13\/2}$.\nThe $N_{f}=0$ result is obtained by putting $b=2$ and \n$\\tilde{c}(0)=-i\/4$.\nHere we have corrected some misprints in the previous work \\cite{ItYa}.\n\nFor $SU(3)$ theory the prepotential ${\\cal F}$ obeys the basic identity\nanalogous to ({\\ref{scaling}) \\cite{sothya,egya}. \nIn contrast to the $SU(2)$ case, combining\nthis with the $SU(3)$ Picard-Fuchs equations does not yield a simple\nrecursion relation. It is not clear at present if the existence of such\nrecursion relation is peculiar to $SU(2)$ theory. It will be interesting\nif one could find an additional constraint equation for ${\\cal F}$ which\neventually leads to the direct evaluation of instanton corrections.\n\n\\vskip5mm\n\\section*{Acknowledgments}\nThe work of K.I. is supported in part by University of Tsukuba\nResearch Projects and the Grant-in-Aid for Scientific Research from\nthe Ministry of Education (No. 07210210).\nThe work of S.-K.Y. is supported in part by Grant-in-Aid for Scientific\nResearch on Priority Area 231 ``Infinite Analysis'',\nthe Ministry of Education, Science and Culture, Japan.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\\label{intro}\n\nPhase transitions occurring in the bulk, but driven by specific conditions at\nits boundaries, are called {\\it boundary-induced phase\n  transitions}~\\cite{HenkelSchuetz94}.  Examples include diffusive\ntransport~\\cite{Krug91,Schuetz00} and traffic flow~\\cite{PopkovEtAl01}\nmodels. A simple example for this is provided by the one-dimensional totally\nasymmetric simple exclusion process \\cite{Derrida93}, where particles enter\nthe system at the left boundary, jump to the right in the bulk, and exit at\nthe right boundary. Depending on the entering and exiting rate values, the\nsystem exhibits qualitatively different phenomenologies (maximal current,\nlarge current and low density, or small current and high density), with\nstraightforward applications to traffic flow problems.\n\nIn the present work, we are interested in boundary-induced phase transitions in\nsystems with absorbing states.  An absorbing state is a dynamical trap which\ncan be accessed but cannot be left~\\cite{Hinrichsen00,Lubeck04,Odor04}.\nSystems with absorbing phase transitions are controlled by a parameter,\ndepending on which the system either enters the absorbing state with certainty\nor survives in a stationary fluctuating\/active state. The most prominent\nfamily of phase transitions into an absorbing state is the very robust direct\npercolation (DP) universality class.  A recent breakthrough has been the\nexperimental observation of DP critical behavior for the first\ntime~\\cite{TakeuchiEtAl07}.\n\nA paradigmatic model in the DP universality class is the contact process (CP)\n\\cite{MarroDickman99}. It can be viewed as a simple model for the propagation\nof a disease where sick individuals can infect healthy neighbors or become\nhealthy spontaneously. More precisely, in the CP in $d$ spatial dimensions, a\nparticle (infected individual) can be created at an ``empty'' site with a rate\n$\\lambda n\/2d$, where $n$ is the number of nearest neighbors occupied by a\nparticle, and an occupied site can become empty at rate $1$.  The empty\nconfiguration is an absorbing state.  For $\\lambda$ larger than a certain\ncritical threshold, $\\lambda_c$, the process is able to sustain (in an\ninfinite lattice) a non-vanishing density of particles, while for\n$\\lambda<\\lambda_c$ the dynamics ends up, ineluctably, in the absorbing state.\n\nAs continuous phase transitions involve long-range correlations, boundary\neffects may play an important role. In the context of absorbing phase\ntransitions, previous studies focused primarily on DP confined to\nparabolas~\\cite{KaiserTurban94,KaiserTurban95}, active\nwalls~\\cite{HinrichsenKoduvely98}, as well as absorbing walls and\nedges~\\cite{Froedh98a,Froedh01a}. Although such boundaries influence the\ndynamics deep into the bulk, the universality class of the bulk transition is\nnot inherently changed, rather it is extended by an additional independent\nexponent describing the order parameter near the boundary.  Therefore, the\nquestion arises whether it is possible to find boundary-induced absorbing\nphase transitions, absent in the corresponding systems without boundaries,\nconstituting independent universality classes.\n\nIn this paper we present a detailed discussion of two slightly different\nmodels introduced in Ref.~\\cite{BaratoHinrichsen08} and\n~\\cite{DeloubriereWijland02}, respectively.  Both of them exhibit a\nboundary-induced nonequilibrium phase transition into an absorbing\nstate. These are one-dimensional particle systems consisting of a single site\n(and, at most, its nearest neighbor), where creation and annihilation of\nparticles occur, and a bulk, where particles move diffusively. While in the\nfirst reference \\cite{BaratoHinrichsen08}, the dynamics at the boundary is a\ncontact process, in the second one~\\cite{DeloubriereWijland02} particles at\nthe origin annihilate only pairwise ($A+A \\rightarrow 0$).  We study different\nversions of these models to compare them and scrutinize the relevance of\nrelaxing the fermionic constraint (i.e. occupation number not restricted to\n$0$ or $1$) both at the bulk and at the boundary.  Our study includes mean\nfield approximations, numerical analysis, some field theoretical arguments, as\nwell as the relation with a (0+1)-dimensional non-Markovian\nmodel~\\cite{BaratoHinrichsen09}.\n\nThe paper is organized as follows: In the next section we define the first\nmodel and present numerical results. In Sec.~\\ref{mean_field} we discuss\nvarious types of mean field approximations and show that this model has indeed\na non-trivial behavior. Sec.~\\ref{related_models} is concerned with two\nbosonic versions of the first model and the study of models with pair\nannihilation at the boundary. One of the bosonic versions is solved exactly\nand it is shown to have trivial critical behavior. Instead, the other bosonic\nversion and models with pair annihilation are shown to share the same critical\nbehavior as the first model. In Sec.~\\ref{non-Markovian} we discuss the\nrelation with a (0+1)-dimensional non-Markovian\nmodel~\\cite{BaratoHinrichsen09} and, finally, we present our main conclusions.\n\n\\section{Basic model definition and simulations}\n\\label{simulations}\n\\subsection{Definition of the model}\n\\label{definition}\n\\begin{figure}\n\\includegraphics[width=70mm]{.\/fig1.eps}\n\\vspace{-2mm}\n\\caption{(Color online) Model on a semi-infinite lattice with symmetric\n  diffusion in the bulk and special dynamical rules at the left boundary (see\n  main text).\n \\label{model}}\n\\end{figure}\nThe model presented in \\cite{BaratoHinrichsen08} is defined on a\none-dimensional semi-infinite discrete lattice where each site is either\noccupied by a particle ($s_i=1$) or empty ($s_i=0$). All lattice sites have\ntwo neighbors, except for the boundary ($i=0$) with a single one. The dynamics\nis a combination of an unbiased random walk in the bulk and a contact\nprocess-like dynamics at the left boundary. It is implemented as follows:\n\\begin{enumerate}\n\\item[(a)] A particle is randomly selected.\n\\item[(b)] If it is located at the leftmost site, it generates another particle\n  at site $1$ with probability $p$, provided that it is empty ($s_1=0$), or it\n  dies ($s_0=0$) with probability $1-p$.\n\\item[(c)] Particles in the bulk perform a symmetric exclusion process,\n  moving to any of their two neighbors with equal probability, provided that\n  the destination site is empty (otherwise nothing happens).\n\\end{enumerate}\nStarting with a single particle at the leftmost site in an otherwise absorbing\n(i.e. empty) configuration, the process evolves as follows: the initial\nparticle at site $0$ either dies or generates another particle at the\nneighboring site $1$. This last performs a random walk in the bulk until,\neventually, it returns to the origin to create another offspring or\ndisappear. A critical point, located at $p_c= 0.74435(15)$ has been reported\nto separate the absorbing phase, in which the total number of particles\nvanishes, from another with indefinitely sustained activity\n\\cite{BaratoHinrichsen08}.\n\\subsection{Order parameters}\n\\label{order_parameters}\n\\begin{figure}\n\\includegraphics[width=87mm]{.\/fig2.eps}\n\\vspace{-2mm}\n\\caption{(Color online) Density of particles at the leftmost site\n  $\\rho_0$ (left) and the average total number of particles $N$\n  (right) as functions of time for different values of $\\Delta$,\n  below, above, and at criticality. At the critical point,\n  $\\rho_0(t)$ decays as $t^{-1\/2}$ while and $\\langle N(t)\\rangle$ is\n  essentially constant.\n  \\label{fig:decay}}\n\\end{figure}\nA possible order parameter for this model is the average density of\nparticles at the leftmost site:\n\\begin{equation}\n  \\rho_0= \\langle s_0 \\rangle\\,,\n\\end{equation}\nwhere $\\langle \\rangle$ stands for ensemble averages. This quantity is plotted\nin Fig.~\\ref{fig:decay} as a function of $\\Delta:=p-p_c$. At the critical\npoint, $\\rho_0(t)$ decays algebraically in time, as:\n\\begin{equation} \n\\label{rhodecay}\n\\rho_0(t)\\;\\sim\\; t^{-\\alpha}\n\\end{equation}\nwith an exponent $\\alpha= 0.50(1)$, compatible with a rational value $\\alpha=\n1\/2$.\n\nAnother possibility is to choose as an order parameter the average total\nnumber of particles, $\\langle N(t) \\rangle$, which, as shown in\nFig.~\\ref{fig:decay}, goes to zero for $p<p_c$ and increases steadily for\n$p>p_c$ (actually, it is limited only by the system size).  At criticality,\n$\\langle N (t) \\rangle$ is found to be constant in the large time limit.\n\nIn the usual scaling picture of absorbing phase transitions, the critical\nexponent $\\beta$ is related to the probability that a given site belongs to an\ninfinite cluster generated from a fully occupied lattice at $t=-\\infty$. This\nquantity tends to zero as the control parameter approaches the critical value\nfrom above. Similarly, the exponent $\\beta'$ is related to the probability\nthat a localized seed generates an infinite cluster extending to\n$t=+\\infty$. Therefore, in the supercritical phase $(\\Delta >0)$, the averaged\nactivity of the site at the origin for $t\\to\\infty$ \\textit{measured in seed\n  simulations} averaging over all runs, scales as $\\rho^s \\sim\n\\Delta^{\\beta+\\beta'}$, where the superscript `s' stands for `stationary'.  At\ncriticality, this function is expected to decay as $\\rho(t) \\sim\nt^{-(\\beta+\\beta')\/\\nu_\\parallel}$, where $\\nu_\\parallel$ is the correlation\ntime exponent.  Moreover, in the DP class a special {\\it time reversal\n  symmetry} implies that $\\beta=\\beta'$ \\cite{Hinrichsen00}.\n\nAs shown in~\\cite{DeloubriereWijland02}, time reversal symmetry also holds in\nthe present type of models. This implies that, in supercritical seed\nsimulations, the density of active sites at the boundary is expected to\nsaturate as:\n\\begin{equation}\n  \\rho_0^s\\sim \\Delta^{2\\beta}\\,,\n\\end{equation}\nwhile, at criticality:\n\\begin{equation}\n  \\rho_0(t) \\sim t^{-2\\beta\/\\nu_\\parallel},\n\\end{equation}\nimplying that $\\alpha$ in Eq.~(\\ref{rhodecay}) is\n\\begin{equation}\n\\alpha=2\\beta\/\\nu_\\parallel\\,.\n\\label{alpha}\n\\end{equation}\nAssuming that $\\alpha=1\/2$, then $\\beta\/\\nu_\\parallel = 1\/4$.\n\\begin{figure}\n\\includegraphics[width=87mm]{.\/fig3.eps}\n\\vspace{2mm}\n\\caption{(Color online) Left panel: Density of particles at the leftmost site,\n  at the time when it reaches its minimum value, as a function of the distance\n  from criticality $\\Delta$. This gives the exponent $\\beta= 0.68(5)$.  Right\n  panel: The same quantity, at criticality and in the stationary state, as a\n  function of the external field $h$, giving $\\delta_h^{-1}= 0.29(5)$, compatible\n  with the conjectured value $1\/3$.\n\\label{fig:off}}\n\\end{figure}\n\n\\subsection{Stationary properties}\n\\label{stationary}\nIn numerical simulations in the active phase, it takes a very long time,\nspecially for small values of $\\Delta$, to reach the steady state.  Moreover,\nwe observed the unusual fact that, for $\\Delta>0$, the density $\\rho_0$ goes\nthrough a minimum before reaching the stationary state (see\nFig.~\\ref{fig:decay} and also \\cite{FES}, where similar non-monotonous curves\nwere reported). However, it turns out that the value $\\rho_0^m$ at the minimum\nand the saturation value $\\rho_0^s$ differ by a constant factor, entailing\nthat both quantities scale in the same way, i.e.:\n\\begin{equation}\n\\rho_0^m\\sim \\Delta^{2\\beta} \\,.\n\\end{equation}\nNote that this can be true only if the density $\\rho_0(t)$ in seed simulations\nobeys the scaling relation:\n\\begin{equation}\n\\rho_0(t) = \\Delta^{2\\beta}\\, R(t\\Delta^{\\nu_\\parallel})\n\\end{equation}\ni.e. if it is possible to collapse the data by plotting\n$\\rho_0\\Delta^{-2\\beta}$ versus $t\\Delta^{4\\beta}$. Indeed, this will be shown\nto be the case in Sec.~\\ref{non-Markovian} for a $0$-dimensional non-Markovian\nprocess argued to be in the same universality class.\n\nRelying on this observation, one can determine the value of the exponent\n$\\beta$ by measuring the density $\\rho_0^m$ at the minimum, which is reached\nmuch earlier than the stationary state. In Fig.~\\ref{fig:off} we plot $\\rho_m$\nas a function of $\\Delta$, inferring $\\beta= 0.68(5)$.\n\n\\subsection{External field}\n\\label{external}\nIn ordinary directed percolation, an external field, conjugate to the order\nparameter, can be implemented by creating active sites at some constant rate\n$h$, thereby destroying the absorbing nature of the empty configuration. At\ncriticality, the external field is known to drive a $d$+1-dimensional DP\nprocess towards a stationary state with $\\rho^s \\sim h^{1\/\\delta_h}$ where\n$\\delta_h^{-1}=\\beta\/(\\nu_\\parallel+d\\nu_\\perp-\\beta')$, and $\\nu_\\perp$ is the\ncorrelation length critical exponent.\n\nIn the present model, the external field, conjugate to the order parameter\n$\\rho_0$, corresponds to spontaneous creation of activity at the leftmost site\nat rate $h$. The above hyperscaling relation for $\\delta_{h}$ is thus expected\nto be fulfilled by taking $d=0$:\n\\begin{equation}\n  \\rho_0^s\\sim h^{1\/\\delta_h}.\n\\end{equation}\nwith\n\\begin{equation}\n  \\delta_h^{-1}= \\beta\/(\\nu_\\parallel-\\beta').\n\\end{equation}\nFrom this expression, exploiting the fact that $\\beta=\\beta'$ and using\nEq.(\\ref{alpha}) as well as the conjectured rational value $\\alpha= 1\/2$, a\nprediction $\\delta_h^{-1}=1\/3$ is obtained. Our numerical estimate, $\\delta_h^{-1}=\n0.29(5)$ (see Fig.~\\ref{fig:off}) is compatible with this result.\n\n\\subsection{Survival probability}\n\\label{survival}\n\n\n\\begin{figure}\n\\includegraphics[width=87mm]{.\/fig4.eps}\n\\vspace{-2mm}\n\\caption{(Color online) Survival probability $P_s(t)$ as function of time\n  below, above, and at criticality.  At the critical point, it decays with the\n  exponent $\\delta=0.15(2)$, different from $\\beta\/\\nu_\\parallel$, and in\n  agreement with the conjectured value $1\/6$.\n\\label{fig:survival}}\n\\end{figure}\nThe survival probability $P_s(t)$ is defined as the fraction of runs that,\nstarting with a single seed at the boundary, survive \\textit{at least} until\ntime $t$. At criticality, this quantity is expected to decay algebraically:\n\\begin{equation}\nP_s(t)\\sim t^{-\\delta},\n\\end{equation}\nwith the so-called survival exponent $\\delta$, while in the super-critical\nregime it saturates in the long time limit. Since $P_s(\\infty)$ coincides with\nthe probability for a seed to generate an infinite cluster, the saturation\nvalue of the survival probability as a function of the distance from\ncriticality gives the exponent $\\beta'$. As in DP, one expects $P_s(t)$ to\ndecay in time with an exponent $\\delta=\\beta'\/\\nu_\\parallel =1\/4$. However, as\nshown in Fig.~\\ref{fig:survival}, one finds a much smaller exponent $\\delta=\n0.15(2)$. Therefore, the usual relation $\\delta= \\beta'\/\\nu_\\parallel$ does\nnot hold. We also observed that it is not possible to collapse different\ncurves of $P_s(t)$ for different values of $\\Delta$, i.e. the survival\nprobability seems to exhibit an anomalous type of scaling behavior. We expect\nthat off-critical simulations of the survival probability give the exponent\n$\\beta'$ but the simulation times needed to reach steady state are\nprohibitively long.\n\nAn explanation for the value $\\delta= 0.15(2)$, differing from\n$\\beta\/\\nu_\\parallel$, is given in the following subsection.\n\n\n\n\\subsection{Time reversal symmetry}\n\n\\label{time_reversal}\n\nIn ordinary bond DP, the statistical weight of a configuration of percolating\npaths does not depend on the direction of time. More specifically, the\nprobability to find an open path from at least one site at time $t=0$ to a\nparticular site at time $t$ coincides with the probability to find an open\npath from a particular site at time $t=0$ to at least one site at time\n$t$. This implies that, in bond DP, {\\it i)} the density $\\rho(t)$ in simulations with\nfully occupied initial state and {\\it ii)} the survival probability $P_s(t)$ in seed\nsimulations coincide; hence $\\beta=\\beta'$. In other realizations of DP\n(e.g. site DP), this time reversal symmetry is not exact but only\nasymptotically realized.\n\nApplying the same arguments to the present model, the survival probability\n$P_s(t)$ in seed simulations should scale in the same way as the density of\nactive sites at the boundary $\\rho_0(t)$ in a process starting with a\n\\textit{fully occupied lattice} in the bulk. A numerical test, which\napproximates such a situation, confirms this conjecture, i.e. one has\n$\\rho_0(t)\\sim t^{-\\delta}$ with $\\delta \\approx 0.15$ for a fully occupied\ninitial state.\n\nFollowing the arguments of~\\cite{BaratoHinrichsen09} in a related model, this\nobservation can be used to provide an heuristic explanation for the fact that\n$\\delta\\neq\\beta\/\\nu_\\parallel$.\n\nIt is known that, if the boundary acts as a sink or perfect trap (e.g.  if\n$p=0$), then, in a process starting with a fully occupied lattice, one\nobserves a growing depletion zone around the boundary whose linear size\n$l'(t)$ increases as $l'\\sim t^{\\alpha_{l}}$, with $\\alpha_{l}=1\/2$ (see\n\\cite{rabbits} and the next subsection). Thus, the density of active sites\ndecays as $t^{-1\/2}$. Hence, the influx of particles from the bulk to the\nleftmost site may be considered as an effective time-dependent external field\n$h(t)\\sim t^{-1\/2}$. Making the assumption that this field varies so slowly\nthat the response of the process (i.e. the actual average activity at the\nboundary) behaves adiabatically, as if the field was constant, then in a\ncritical process starting from an initially fully occupied state:\n\\begin{equation}\n  \\rho_0(t) \\sim t^{-\\frac{1}{2\\delta_h}} \\sim t^{-1\/6}.\n\\end{equation}\nOwing to the time reversal property, this quantity should decay as the\nsurvival probability. This chain of heuristic arguments leads to the\nconjecture that the survival exponent is given by $ \\delta = 1\/6$, in\nagreement with the numerical estimate $\\delta=0.15(2)$.\n\nThis unusual value of the exponent $\\delta$ is clearly related to the fact\nthat the present problem is inhomogeneous. The argumentation presented above\ndoes not work for the CP, for example, since there is no special site and,\ntherefore, a fully occupied lattice cannot be interpreted as a time dependent\nfield acting on a special site.\n\n\n\\subsection{Density profile}\n\\label{density}\n\\begin{figure}\n\\includegraphics[width=87mm]{.\/fig5.eps}\n\\vspace{-2mm}\n\\caption{(Color online) Left: Data collapse of the rescaled profiles of the\n  particle density at criticality for $t_0=64,128,\\ldots, 8192$ (blue) compared\n  to a Gaussian distribution (red). Inset: The same data collapse in a\n  double-logarithmic representation. Right: Density of particles (blue) and\n  pairs (green) at $t_0=10^6$, showing the presence of correlations which\n  decay in space as $x^{-1\/2}$, indicating that $\\beta\/\\nu_\\perp=1\/2$.\n  \\label{fig:profile}}\n\\end{figure}\nNow, we consider the density profile $\\rho(x,t)$ in the bulk, where $x \\in\n\\mathbb{N}$ is the spatial coordinate (distance to the boundary), computed at\nthe critical point. In the left panel of Fig.~\\ref{fig:profile}, we compare\nthe data collapse of the curves $\\rho(x,t)t^{1\/2}$ as a function of\n$x\/t^{1\/2}$ with a Gaussian and observe an excellent agreement, indicating\nrandom-walk like behavior with a dynamical exponent $z=2$.  However, in\ncontrast to a simple random walk, particles are mutually correlated. This is\nillustrated in the right panel of Fig.~\\ref{fig:profile}, where the connected\ncorrelation function between two nearest neighbors:\n\\begin{equation}\n  \\rho^{\\rm pair}(x,t)= \\langle\\rho(x+1,t)\\rho(x,t)\\rangle- \n  \\langle\\rho(x+1,t)\\rangle\\langle\\rho(x,t)\\rangle\n\\end{equation}\nin a system at the critical point is plotted against time. One observes an\nalgebraic decay, $x^{-1\/2}$, with distance. According to the standard scaling\ntheory this implies that $\\beta\/\\nu_\\perp=1\/2$ , confirming that\n$z=\\nu_\\parallel\/\\nu_\\perp=2$.  Moreover, these results are in full agreement\nwith field theoretical calculations presented in\nRef.~\\cite{DeloubriereWijland02} (see section \\ref{vanWijland}), which predict\n$z=2$ and $\\alpha=1\/2$.\n\n\\section{Mean field approximation}\n\\label{mean_field}\nHere, we  study mean field approximations at different levels. Let us denote by\n$\\eta_i$ the probability to find a particle at site~$i$; the temporal\nevolution within a simple (one-site) mean field approximation is given by:\n\\begin{eqnarray}\n\\frac{d\\eta_0}{dt}&=& -(1-p)\\eta_0+ \\frac{1}{2}\\eta_1(1-\\eta_0),\n\\label{eqmf1}\\\\\n\\frac{d\\eta_1}{dt}&=& p\\eta_0(1-\\eta_1)+\n\\frac{1}{2}\\left(\\eta_{2}+\\eta_{0}\\eta_1-2\\eta_{1}\\right),\n\\label{eqmf2}\\\\\n\\frac{d\\eta_i}{dt}&=& \\frac{1}{2}\\left(\\eta_{i+1}+\n\\eta_{i-1}-2\\eta_{i}\\right),~~ \\mbox{ for } i=2,3,\\hdots\\,\n\\label{eqmf3}\n\\end{eqnarray}\nNote that the equations for the boundary site and its neighbor,\nEq.~(\\ref{eqmf1}) and Eq.~(\\ref{eqmf2}), include quadratic terms due to the\nexclusion constraint, while the equation for sites at the bulk,\nEq.~(\\ref{eqmf3}), describes in this approximation a symmetric random walk,\n{\\it i.e.} it is a diffusion equation. The critical point within simple mean\nfield theory (where the equation for $\\eta_1$ also becomes a diffusion\nequation) is $p_c= 1\/2$.\n\nConsidering a localized initial condition at the boundary,\n$\\eta_i=\\delta_{i,0}$, after a transient time the densities at sites $0$ and\n$1$ should, approximately, coincide.  Therefore, from Eq.~(\\ref{eqmf1}) with\n$\\eta_0 \\approx \\eta_1$, it follows that, at criticality, $\\eta_0\\sim\nt^{-1\/2}$.\n\nIn the stationary regime, Eq.~(\\ref{eqmf1}) leads to $\\eta_0\\sim (p-1\/2)$ for\n$p\\ge 1\/2$. From these results we have:\n\\begin{equation}\n\\alpha^{MF}=1\/2\\,,\\qquad \\beta^{MF}=1\\,.\n\\end{equation}\nTo obtain the survival exponent, $\\delta$, we follow the arguments of the\npreceding section and study the decay of activity from a fully occupied\nlattice, $\\eta_i=1$ for all $i$.  Integrating Eqs.~(\\ref{eqmf1}),\nEq.~(\\ref{eqmf2}) and Eq.~(\\ref{eqmf3}) numerically with this initial\ncondition, we obtain an exponent in agreement with\n\\begin{equation}\n\\delta^{MF}= 1\/4\\,.\n\\end{equation}\nA more accurate approximation can be obtained by keeping the correlation\nbetween the first two sites, which is expected to be more relevant than the\ncorrelation between other neighboring sites.  Such a pair-approximation was\nused recently in a model where a boundary site also plays a special\nrole~\\cite{Sugden07}. In this approximation, the master equation reads:\n\\begin{eqnarray}\n\\frac{d\\sigma_{00}}{dt}&=& (1-p)\\sigma_{10}+\n\\frac{1}{2}[\\sigma_{01}(1-\\eta_2)- \\sigma_{00}\\eta_2],\n\\\\ \\frac{d\\sigma_{01}}{dt}&=& (1-p)\\sigma_{11}+\n\\frac{1}{2}[\\sigma_{00}\\eta_2-\\sigma_{01}(2-\\eta_2)],\\nonumber\n\\\\ \\frac{d\\sigma_{10}}{dt}&=& -\\sigma_{10}+\n\\frac{1}{2}[\\sigma_{01}-\\sigma_{10}\\eta_2+\\sigma_{11}(1-\\eta_2)],\\nonumber\n\\\\ \\frac{d\\sigma_{11}}{dt}&=& p\\sigma_{10}- (1-p)\\sigma_{11}+\n\\frac{1}{2}[\\sigma_{10}\\eta_2-\\sigma_{11}(1-\\eta_2)],\\nonumber\n\\\\ \\frac{d\\eta_2}{dt}&=& \\frac{1}{2}\\left(\\eta_{3}+ \\sigma_{11}+ \\sigma_{01}\n-2\\eta_{2}\\right),\\nonumber \\\\ \\frac{d\\eta_i}{dt}&=&\n\\frac{1}{2}\\left(\\eta_{i+1}+ \\eta_{i-1}-2\\eta_{i}\\right) \\mbox{ for }\ni=3,4,\\hdots\\,\\nonumber\n\\end{eqnarray}\nwhere $\\sigma_{s_0s_1}$ is the probability that the occupation numbers of the\nfirst two sites are $s_0$ and $s_1$. Numerical integration of these equations\nleads to an improved critical point estimation, $p_c\\approx 0.634$, but to the\nsame mean-field exponents as above.\n\\section{Related models and field theoretical approaches}\n\\label{related_models}\n\n\\subsection{Bosonic variant}\n\\label{bosonic}\nThe model defined above is fermionic in the sense that each site can be\noccupied by, at most, one particle. We now consider a bosonic variant without\nsuch a constraint.  This means that diffusion is independent of the\nconfiguration of particles and that particles can be created at the boundary\nsite without restriction.  More specifically, the update rules are:\n\\begin{enumerate}\n\\item[(a)] A particle is chosen randomly.\n\\item[(b)] If the particle is located at the leftmost site it can:\ncreate another particle at the leftmost site ($s_0=s_0+1$) at rate $\\lambda$,\ndie ($s_0=s_0-1$) at rate $\\sigma$, or\ndiffuse to the next neighbor at rate $D$.\n      \\item[(c)] If the particle is located in the bulk, it diffuses to the\n        right or to the left at equal rates $D$.\n\\end{enumerate}\nThe corresponding master equation is:\n\\begin{eqnarray}\n\\frac{dP(\\{n\\},t)}{dt} &=& \\lambda\\bigl[(n_0-1)P(n_0-1,...,t)-\n  n_0P(\\{n\\},t)\\bigr]\\nonumber \\\\\n& +&\\sigma\\bigl[(n_0+1)P(n_0+1,...,t)- n_0P(\\{n\\},t)\\bigr]\\nonumber \\\\\n&+& D \\Bigl[ \\sum_{\\langle ij\\rangle}P(...,n_i-1,n_j+1,...,t) \\\\\n&+&P(...,n_i+1,n_j-1,...,t)-2P(\\{n\\},t) \\Bigr] \\nonumber\n\\label{eqboson1}\n\\end{eqnarray} \nwhere $P(\\{n\\},t)$ is the probability to find a given configuration $\\{n\\}=\nn_0, n_1, n_2\\ldots$ and the sum runs over all nearest neighbors, $j$, of site\n$i$ (recall that site $0$ has only one neighbor). Defining the state vector:\n\\begin{equation}\n|\\psi(t)\\rangle= \\sum_{\\{n\\}}P(\\{n\\},t)|\\{n\\}\\rangle,\n\\end{equation}\nwhere $|\\{n\\}\\rangle=\\otimes_i|n_i\\rangle$ denotes the usual configuration\nbasis, the master equation can be expressed in the form\n\\begin{equation}\n\\frac{d}{dt} |\\psi(t)\\rangle= -\\hat{H}|\\psi(t)\\rangle\\,,\n\\end{equation}\nwhere $\\hat{H}$ is the time evolution operator. Using bosonic creation and\nannihilation operators, defined by $\\hat{a}_i|n_i\\rangle= n_i|n_i-1\\rangle$\nand $\\hat{a}_i^\\dagger|n_i\\rangle= |n_i+1\\rangle$, the master equation\nEq.~(\\ref{eqboson1}) can be shown to correspond to the time evolution\noperator:\n\\begin{eqnarray}\n\\hat{H}&=& D\\sum_{\\langle\n  ij\\rangle}(\\hat{a}^\\dagger_i-\\hat{a}^\\dagger_j)(\\hat{a}_i-\\hat{a}_j)\n\\label{eqbosonH}\n\\\\ &&+\\sigma(\\hat{a}_0^\\dagger-1)\\hat{a}_0 +\\lambda\n\\hat{a}_0^\\dagger(1-\\hat{a}_0^\\dagger)\\hat{a}_0.\\nonumber\n\\end{eqnarray}\nIn this formalism, the expectation value of an operator $\\hat{B}$ is given by\n$\\langle\\hat{B}\\rangle= \\langle 1|\\hat{B} |\\psi(t)\\rangle$ where $\\langle 1|=\n\\sum_{\\{n\\}}\\langle \\{n\\}|$. As is the case for the bosonic contact process\n\\cite{Baumann05}, the equations for the time evolution of the density of\nparticles close. From the Heisenberg equation of motion, $\\frac{d\\hat{B}}{dt}=\n[\\hat{H},\\hat{B}]$ and Eq.~(\\ref{eqbosonH}), one obtains:\n\\begin{eqnarray} \n\\frac{d\\rho_0}{dt} &=& D(\\rho_1-\\rho_0)+ \\Delta\\rho_0 \\label{eqboson1b}\n\\\\ \\frac{d\\rho_i}{dt} &=& D(\\rho_{i+1}+\\rho_{i-1}-2\\rho_i)\\qquad\ni=1,2,3\\ldots\\nonumber\n\\end{eqnarray}\nwhere $\\rho_i(t)= \\langle a^\\dagger_i(t)a_i(t)\\rangle= \\langle a_i(t)\\rangle$\nand $\\Delta= \\lambda-\\sigma$.  Alternatively, one could have written a\nLangevin equation equivalent to Eq.(\\ref{eqbosonH}), and from it, averaging\nover the resulting noise, one readily arrives at the same set of equations\nEq.(\\ref{eqboson1b}).\n\nFrom these equations, we can see that the critical point is $\\Delta= 0$, where\nEq.(\\ref{eqboson1b}) is a diffusion equation.  In the continuum limit,\nEq.~(\\ref{eqboson1b}) reads:\n\\begin{equation}\n\\frac{\\partial\\rho(x,t)}{\\partial t}= \\frac{\\partial^2\\rho(x,t)} {\\partial\n  x^2}+ \\Delta\\delta(x)\\rho(x,t)\\,\n\\label{eqboson2}\n\\end{equation}\nwhere $x$ is the spatial coordinate and, without loss of generality, we have\nset $D=1$. We note that in order to take the continuum limit in equation (\\ref{eqboson1b}), a site $-1$, with $\\rho_{-1}= \\rho_{0}$, has to be introduced, so that appropriate boundary conditions are satisfied. The solution of this inhomogeneous diffusion equation is:\n\\begin{equation}\n\\rho(x,t)= \\int_0^\\infty \\delta(\\zeta)G(x,\\zeta,t)d\\zeta+\\nonumber\n\\end{equation}\n\\begin{equation}\n\\int_0^t\\int_0^\\infty\\Delta\\delta(\\zeta)\\rho(\\zeta,\\tau)G(x,\\zeta,t-\\tau),\nd\\zeta d\\tau \\,\n\\label{eqboson3}\n\\end{equation}\nwhere $G(x,\\zeta,t)= (e^{-(x+\\zeta)^2\/(4t)}+e^{-(x-\\zeta)^2\/(4t)})\/(\\sqrt{\\pi t})$ is the Green\nfunction and the first term in the right hand side comes from the initial\ncondition $\\rho(x,0)= \\delta(x)$. From Eq.~(\\ref{eqboson3}) we have\n\\begin{equation}\n\\rho_0(t)= \\frac{2}{\\sqrt{\\pi t}}+2\\Delta\\frac{d^{-1\/2}}{dt^{-1\/2}}\\rho_0(t) \\,\n\\label{eqboson4}\n\\end{equation}\nwhere $\\rho_0(t)= \\rho(0,t)$, and the operator $\\frac{d^{-1\/2}}{dt^{-1\/2}}$,\ndefined by\n\\begin{equation}\n\\frac{d^{-1\/2}}{dt^{-1\/2}}f(t)= \\int_0^t\\frac{f(\\tau)}{\\sqrt{\\pi (t-\\tau)}}d\\tau,\n\\end{equation}\nis a half integral operator \\cite{Oldham}.  Equation~(\\ref{eqboson4}) involves\n(owing to the delta function in the interaction term in Eq.~(\\ref{eqboson2}))\nonly the density at the leftmost site. This justifies the mapping of this\nmodel onto an effective one-site non-Markovian process (see next\nsection). Using some rules for half integration \\cite{Oldham} to solve\nEq.~(\\ref{eqboson4}), we find:\n\\begin{equation}   \n\\rho_0(t)= \\frac{2}{\\sqrt{\\pi t}}+ 4\\Delta\\exp(4\\Delta^2t)\\mbox{erf}(-2\\Delta\\sqrt{t}),\n\\label{eqboson5}\n\\end{equation}\nwhere $\\mbox{erf(x)}$ is the error function. This implies that, above the\ncritical point, $\\rho_0$ grows exponentially in the long time limit, and does\nnot reach a stationary value, {\\it i.e.} there is a first order transition\nand, hence, $\\beta= 0$ in this bosonic model.  From equation\nEq.~(\\ref{eqboson5}), we deduce $\\beta'=1$ and $\\nu_\\parallel=2$. We have not\nbeen able to calculate the survival-probability exponent exactly, but\nnumerical simulations suggest $\\delta= 1\/4$, in agreement with the mean field\nexponent.\n\n\\subsection{Partially bosonic variant}\n\\label{partially_bosonic}\n\nLet us now introduce a {\\it partially bosonic} variant of the previous model\nby retaining the exclusion constraint only at the boundary, but not in the\nbulk.  The rules, in this case, are:\n\\begin{enumerate}\n\\item[(a)] A particle is randomly chosen.\n\\item[(b)] If it is at the leftmost site, it can generate a particle at site\n  $1$ (provided that $s_1=0$) with probability $p$ or die ($s_0:=0$) with\n  probability $1-p$.\n\\item[(c)] Particles in the bulk diffuse to the right or to the left with the\n  same probability, $1\/2$.\n\\end{enumerate}\nNumerical simulations show that this variant exhibits the same critical\nbehavior as the original model, even if the critical point is shifted to $p_c=\n0.6973(1)$. This shows that the fermionic constraint is relevant only at the\nboundary, where it induces a saturation of the particle density and leads the\ntransition to become continuous.\n\n\\subsection{Models with pair annihilation at the boundary}\n\\label{vanWijland}\nIn the models discussed so far, particles at the boundary either create an\noffspring or die spontaneously at some rate. Instead, a very similar model was\nintroduced in Ref.~\\cite{DeloubriereWijland02}, for which particles at the\nboundary annihilate only in \\textit{pairs}.  In its fermionic variant,\nparticles at sites $0$ and $1$ annihilate with each other (provided that both\nsites are occupied) at some rate, while isolated particles at the boundary\ncannot disappear:\n\\begin{eqnarray*}\n\\mbox{present models:} & A\\to2A \\,,\\quad A\\to\\emptyset\\,, \\\\ \\mbox{models of\n  Ref.~\\cite{DeloubriereWijland02}:} & A\\to2A \\,,\\quad 2A\\to\\emptyset\\,.\n\\end{eqnarray*}\nAnalogously, one can define a bosonic version, in which two particles at the\nboundary can annihilate. In the following discussion we consider these two\nvariants in $d$ spatial dimensions where, as is the case $d=1$, only a single\nsite has ``special\" dynamics.\n \nA detailed field theoretical analysis of these pair-annihilating models was\npresented in \\cite{DeloubriereWijland02}. In the bosonic case, proceeding as\nabove (see Eq.(\\ref{eqbosonH})) one obtains the following time evolution\noperator:\n\\begin{eqnarray}\n\\hat{H}&=& D\\sum_{\\langle ij\\rangle}(\\hat{a}^\\dagger_i-\\hat{a}^\\dagger_j)\n(\\hat{a}_i-\\hat{a}_j) \\nonumber\n\\\\ &&+\\sigma[(\\hat{a}_0^\\dagger)^2-1]\\hat{a}_0^2 +\\lambda\n\\hat{a}_0^\\dagger(1-\\hat{a}_0^\\dagger)\\hat{a}_0.\n\\label{H2}\n\\end{eqnarray}  \nwhich, after eliminating higher order terms and taking the continuum limit, is\nequivalent to a Langevin equation identical to the one for DP except for the\nfact that\nall terms, except for the Laplacian, are multiplied by a $\\delta$ function at\nthe boundary; {\\it i.e.} the non-diffusive part of the dynamics operates only\nat the boundary.  An $\\epsilon$-expansion analysis of Eq.(\\ref{H2}) (see\n\\cite{DeloubriereWijland02}) leads to $\\alpha=1\/2$ and $z=2$ as exact results\nin all orders of perturbation theory, and to $\\beta=1-3(4-3d)\/8$, up to first\norder in $\\epsilon =4\/3-d$ around the critical dimension $d_c=4\/3$.  Also, it\nwas shown that the time reversal symmetry is preserved.\n\nWe have verified all these predictions in computer simulations of the bosonic\nannihilation model. For instance, from the time decay of $\\rho_0(t)$, as shown\nin Fig.~\\ref{FSS}, we determine $\\delta = 0.21(3)$, while from a finite size\nscaling analysis of the saturation values of the order parameter at\ncriticality we measure $\\beta\/\\nu_{\\perp}=0.51(2)$ (see Fig.~\\ref{FSS}), in\nreasonable agreement with the expected results, $\\delta= 1\/6$ and\n$\\beta\/\\nu_{\\perp}=1\/2$, respectively. Moreover, from spreading simulations\n(not shown) we estimate $\\alpha \\approx 1\/2 $ and $z \\approx 2$. All the\nexponents are in agreement with the ones presented in the previous section for\nsingle particle annihilation models.\n\nActually, a simple argument explains why the model of section\n\\ref{simulations} and the pair-annihilation model share the same\ncritical behavior. This is plausible because the chain reaction $A \\to\n2A \\to \\emptyset$ in the model with pair annihilation generates\neffectively the reaction $A\\to\\emptyset$ of the model considered with\nCP-like dynamics.\n\nHence, the field theoretical predictions discussed above\n\\cite{DeloubriereWijland02,BaratoHinrichsen08} apply also to the CP-like\nmodel. In $d=1$, the one-loop prediction $\\beta= 5\/8= 0.625$\n\\cite{DeloubriereWijland02}, is not far from the exponent measured in section\n\\ref{simulations}, $\\beta=0.68(5)$.\n\nOn the other hand, the fermionic version of the pair-annihilating model has\nbeen conjectured to yield in a different universality class, and a prediction\nfor its critical exponents is made in \\cite{DeloubriereWijland02} (for\ninstance, $\\beta=1$).  Our numerical simulations disprove such a claim; all\nthe measured critical exponents for the fermionic variant of the\npair-annihilation model are numerically indistinguishable from their bosonic\ncounterparts (see Fig.~\\ref{FSS}).\n\\begin{figure}[t]\n\\begin{center}\n  \\includegraphics[height=5cm]{fig6.eps}\n\\end{center}\n\\caption{\\footnotesize{(Color online) Temporal behavior of $\\rho_{0}$ for the bosonic\n    pair-annihilating model, starting from a homogeneous initial condition for\n    different system sizes (from $L=64$ to $L=2048$).  The exponent\n    $\\beta\/\\nu_{\\perp}$ can be measured from the scaling of the different\n    saturation values as a function of system size (see inset; yellow\n    line). Also, in the inset (dashed green line), we show the scaling of saturation\n    values for the fermionic version of the same model, showing the same type\n    of scaling. }}\n\\label{FSS}\n\\end{figure}\n\nIn summary, all the defined models, either with single particle annihilation\nor with pair-annihilation, fermionic or bosonic, exhibit a boundary induced\nphase transitions and, except for one of them, they all are continuous and\nshare the same critical behavior. The exception to this rule is the CP-like\nmodel without a fermionic constraint at the boundary, which lacks of a\nsaturation mechanism in the active phase, leading to unbounded growth of \nparticle density at the leftmost site above the critical point and to a discontinuous transition.\n\\section{Relation to a $(0+1)$-dimensional non-Markovian process}\n\\label{non-Markovian}\n\nIn Ref.~\\cite{DeloubriereWijland02}, by integrating out the fields related to\ndiffusion in the bulk from the corresponding action, it was shown that the\nclass of boundary-induced phase transitions into an absorbing state considered\nhere can be related to a non-Markovian single site process. The properties of\nsuch a spreading process on a time line has been studied in further detail in\nRef.~\\cite{BaratoHinrichsen09}.\n\nOn an heuristic basis, the relation can be explained as follows: consider the\nCP-like model only from the perspective of the leftmost site. A particle at\nthe origin may die or create a new particle that will go for a random walk\ncoming back to the origin after a time $\\tau$. What happens during this random\nwalk is irrelevant from the perspective of the leftmost site; the only\nrelevant aspect is the time needed for a created particle to come back to the\nboundary.  Once it returns it may die or create new offsprings which, on their\nturn, will undergo random walks in the bulk.\n\nOur simulations above show that the fermionic constraint is irrelevant in the\nbulk. Therefore, we can consider without lost of generality the bulk-bosonic\nversion in which there is no effective interaction among diffusing\nparticles. In this case, the probability distribution of the returning time to\nthe origin has the well-known asymptotic form~\\cite{Redner01}:\n\\begin{equation}\n\\label{WaitingTimeDistribution}\nP(\\tau)\\sim \\tau^{-3\/2}\\,.\n\\end{equation}\nTaking all these elements into account we define the following\nnon-Markovian model on a single site~\\cite{BaratoHinrichsen08}:\n\\begin{enumerate}\n\\item[(a)] Set initially $s(t):= \\delta_{t,0}$ for all times, $t$.\n\\item[(b)] Select the lowest $t$ for which $s(t)=1$.\n\\item[(c)] With probability $\\mu$, generate a waiting time $\\tau$ according to\n  the distribution~Eq.~(\\ref{WaitingTimeDistribution}), truncate it to an\n  integer, and set $s(t+\\tau):=1$; otherwise (with probability $1-\\mu$) set\n  $s(t):=0$.\n\\item[(d)] Go back to (b).\n\\end{enumerate}\nThe process runs until the system enters the absorbing state ($s(t')=0$ for\nall $t'>t$) or a predetermined maximum time is exceeded.\n\\begin{figure}\n\\includegraphics[width=87mm]{.\/fig7.eps}\n\\vspace{-2mm}\n\\caption{(Color online) Off-critical data collapse with the one-site model:\n  $\\langle s(t)\\rangle\\Delta^{-2\\beta}$ as a function of $t\\Delta^{4\\beta}$\n  for different values of $\\Delta$, with $\\beta= 0.71(2)$.\n  \\label{fig:datacollapse}}\n\\end{figure}\nThe density of particles at the leftmost site of the original model is related\nto $\\langle s(t)\\rangle$ in the single-site model, the survival probability at\ntime $t$ is given by the fraction of runs surviving at least up to $t$, and\nthe initial condition $s(t):= \\delta_{t,0}$ corresponds to start with a single\nparticle at the boundary in the full model.  Critical exponents can be defined\nas in the original model. However, the simulation results for the single-site\nnon-Markovian model are more reliable because it is possible to perform much\nlonger runs and, in the case of off-critical simulations, one can work with\nsmaller values of $\\Delta$. With time-dependent simulations at the critical\npoint $\\mu_c=0.574262(2)$, we obtained $\\alpha=0.500(5)$ and\n$\\delta=0.165(3)$, in good agreement with the conjectured values $\\alpha= 1\/2$\nand $\\delta= 1\/6$.  As an example, we show the results of supercritical\nsimulations in Fig.~\\ref{fig:datacollapse}, where we obtained a convincing\ndata collapse by plotting $\\langle s(t)\\rangle\\Delta^{-2\\beta}$ as a function\nof $t\\Delta^{4\\beta}$ for different values of $\\Delta$ with $\\beta=\n0.71(2)$. The latter estimate is in agreement with $\\beta= 0.68(5)$, coming\nfrom the original model.\n\nAs shown in previous studies (see e.g.~\\cite{Hinrichsen07} and references\ntherein), a non-Markovian time evolution with algebraically distributed\nwaiting times $P(\\tau) \\sim \\tau^{-1-\\kappa}$ is generated by so-called\nfractional derivatives $\\partial_t^\\kappa$ which are defined by:\n\\begin{equation}\n\\label{eq:IntegralTime}\n\\partial_t^\\kappa \\, \\rho(t)  \\;=\\;  \\frac{1}{\\mathcal{N}_\\parallel(\\kappa)}\n\\int_0^{\\infty} {\\rm d}t' \\, {t'}^{-1-\\kappa} [\\rho(t)-\\rho(t-t')]\\,,\n\\end{equation}\nwhere $\\kappa\\in[0,1]$ and $\\mathcal{N}_\\parallel(\\kappa)=-\\Gamma(-\\kappa)$ is\na normalization constant. Hence, we expect this model to be described by a\nDP-like $0$-dimensional Langevin equation with a half-time derivative, instead\nof the usual one, to account for the non-Markovian character of the model:\n\\begin{equation}\n\\label{Langevin}\n\\partial_t^{1\/2} \\rho(t) = a \\rho(t) - \\rho(t)^2 + \\xi(t)\\,\n\\end{equation}\nwhere $a$ is proportional to the distance from criticality and $\\xi$ is a\nmultiplicative noise with correlations $\\langle \\xi(t)\n\\xi(t')\\rangle=\\rho(t)\\delta(t-t')$. This equation can be obtained from the\neffective action that arises when the fields related to diffusion in the bulk\nare integrated out, and the relation of the order of the fractional derivative\nin a generalized one-site model with the dimension in the full model is\n$\\kappa= (2-d)\/2$ \\cite{DeloubriereWijland02}. An analysis of this one-site\nmodel with general $\\kappa$ and a comparison with the results coming from\nfield theory is presented in \\cite{BaratoHinrichsen09}.\n\\section{Conclusion}\n\\label{conclusions}\n\nWe have studied boundary-induced phase transitions into an absorbing\nstate in one-dimensional systems with creation\/annihilation dynamics\nat the boundary and simple diffusive dynamics in the bulk.  The\nnon-trivial dynamics at the boundary induces a phase transition in the\nbulk. We have analyzed such a transition for different though similar\nmodels, including different ingredients: either single-particle\nannihilation or pairwise annihilation, fermionic constraint or lack of\nit, etc.\n\nA particular bosonic version can be exactly solved; owing to the lack of any\nsaturation mechanism, the density of particles grows unboundedly in the active\nphase, leading to a discontinuous transition with trivial critical exponents.\n\nThe rest of the analyzed models exhibit a continuous transition and define a\nunique universality class. At the bulk, the dynamics is governed by\nrandom-walks, entailing the exponent values $z=2$ and $\\alpha=1\/2$. On the\nother hand, some critical exponents take non-trivial values:\n{\\it i)} the survival probability from a localized seed at\nthe boundary exponent, which from an heuristic argument supported by\nsimulations results, turns out to be $\\delta=1\/6$, as well as {\\it ii)} the\norder parameter exponent, $\\beta= 0.71(2)$.  The remaining exponents can be\nobtained from these ones using scaling relations.\n\nFinally, it has been shown that the class of boundary induced phase\ntransitions studied here can be related to a single-site non-Markovian\nprocess.  This process is particularly suitable for numerical\nsimulations and it is also of conceptual interest in the sense that it\nshows that nonequilibrium phase transitions can occur even in $0+1$\ndimensions by choosing an adequate non-Markovian dynamics. It is also\nconvenient for the comparison of the results obtained form the\n$\\epsilon$-expansion and simulations\n\\cite{BaratoHinrichsen09,DeloubriereWijland02}.\n\nThe models studied here possibly constitute the simplest universality class of\nnonequilibrium phase transition into an absorbing state, in the sense that the transition occurs \nbecause of the special dynamics of just one site and, in contrast to DP, some critical exponents\ncan be obtained exactly from the field theory.\n\n\\begin{acknowledgments}\n  We thank X. Durang and M. Henkel for helpful discussions. \n  Financial support by the Deutsche Forschungsgemeinschaft (HI\n  744\/3-1), by the Spanish MEyC-FEDER, project FIS2005-00791, and from\n  Junta de Andaluc{\\'\\i}a as group FQM-165 is gratefully acknowledged.\n\\end{acknowledgments}\n\n\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\nAt the eve of data taking at the LHC, the electroweak standard\nmodel~(SM) with a fundamental scalar Higgs doublet remains an\nextremely successful effective description of all data collected in\nparticle physics experiments at colliders.  Nevertheless, the\nmicroscopic dynamics of the electroweak symmetry breaking~(EWSB)\nsector has not yet been tested directly.  Therefore, detailed studies\nof the SM and realistic alternative scenarios for EWSB are an\nessential part of the LHC experimental program.\n\nIn the past decade, additional dimensions of space-time at the\nTeV-scale have become an important paradigm for electroweak~(EW) model\nbuilding.  Planck-scale extra dimensions have long been a solid\nprediction of superstring theory, but they are outside of the\nexperimental range of collider experiments.  In contrast, TeV-scale\nextra dimensions will be tested at the LHC.\n\nModels with just one additional space dimension that have the geometry of\na five dimensional Anti-de Sitter space~($\\text{AdS}_5$)~\\cite{Randall:1999ee},\nplay a special role, because the conjectured AdS\/CFT\ncorrespondence~\\cite{Maldacena:1997re} reveals such models as dual to conformal\nfield theories~(CFT) on the four dimensional boundary branes.  In\nparticular, a weakly interacting~$\\text{AdS}_5$ model turns out to be\ndual to a strongly interacting Technicolor~(TC) like model for EWSB.\nIf the conjectured AdS\/CFT correspondence is exact, the extra\ndimension can be viewed as a technically convenient description\nof a strongly interacting dynamics.\n\nModel building for EWSB with extra dimensions does not require them to\nbe continuous, instead they can be\ndeconstructed~\\cite{ArkaniHamed:2001ca} as a discrete lattice with a\nfinite number of sites.  In this approach, the extra dimensions play a\nmetaphorical role as a organization principle for gauge theories with\nlarge non-simple gauge groups and complicated matter representations,\nsimilar to moose models.  It turns out that a minimal version of\nwarped higgsless models can be (de)constructed on just three lattice\nsites in the extra dimension and is known as the Three Site Higgsless\nModel~(3SHLM)~\\cite{SekharChivukula:2006cg}.\n\nIn order to be compatible with the EW precision tests~(EWPT), any\nadditional heavy gauge bosons should couple weakly to the SM fermions.\nThe 3SHLM ensures this by ``ideal fermion\ndelocalization''~\\cite{SekharChivukula:2006cg,SekharChivukula:2005xm}\nand the predominant production mechanism at the LHC will be in vector\nboson fusion~\\cite{He:2007ge}.  However, it has been pointed out\nrecently~\\cite{Abe:2008hb}, that the EWPT actually require a small,\nbut nonvanishing coupling of the heavy gauge bosons to SM fermions.\nThis allows their production in the $s$-channel at the LHC.  In fact,\na measurement of the relative strengths of the production mechanisms\nfor heavy vector bosons will be required to constrain higgsless models\nof EWSB.\n\nIn this paper, we complement the existing phenomenological\nstudies~\\cite{He:2007ge} of the 3SHLM by allowing for non ideal\ndelocalization and the production of~$W'$ and $Z'$~bosons in the\n$s$-channel at LHC.  We perform parton level Monte Carlo studies to\nidentify the regions of parameter space where the coupling of the\n$W'$~boson to SM fermions can be measured at the LHC.  We show how the\ncontributions from the nearly degenerate ~$W'$ and $Z'$~bosons can be\nseparated for this purpose.\n\nThis paper is organized as follows: in section~\\ref{sec:3shl} we\nreview the features of the 3SHLM that are relevant for our\ninvestigation.  In section~\\ref{sec-cpl} we discuss the relevant\ncouplings, masses and widths that are used in our Monte Carlo\neventgenerator described in section~\\ref{sec:whizard}.\nIn sections~\\ref{sec-hzprod} and~\\ref{sec-hwprod} we discuss our\nresults for the production of heavy~$Z'$ and $W'$~bosons, respectively.\nLimitations arising from finite jet mass resolutions are described\nin~\\ref{sec-jetres}.  We conclude in section~\\ref{sec:concl}.\n\n\\section{The Three Site Higgsless Model}\n\\label{sec:3shl}\n\nThe Three Site Higgsless Model~\\cite{SekharChivukula:2006cg} can be\nviewed as a warped 5D model of EWSB,\ndimensionally deconstructed~\\cite{ArkaniHamed:2001ca}\nto three lattice sites. The structure and field content of the model\nis shown in moose notation in figure~\\ref{fig-moose}.\n\\begin{figure}\n\\centerline{\\includegraphics{moose}}\n\\caption{The field content and structure of the 3SHLM in moose notation. The dashed lines\nconnecting fermions represent Yukawa couplings, the dotted blob illustrates the nontrivial\n$\\mathbf{U}(1)_2$ charge carried by all fermions.}\n\\label{fig-moose}\n\\end{figure}\nThe gauge group consists of two $\\mathbf{SU}(2)$ group factors located at the\nlattice sites~$0$ and~$1$ with gauge fields $A^\\mu_{0\/1}$ and gauge couplings $g_{0\/1}$ and a\n$\\mathbf{U}(1)$ gauge group located at the third lattice site with the gauge field\n$A^\\mu_2$ and gauge coupling $g_2$.  Note that the continuous 5D\nanalogue of this is a bulk $\\mathbf{SU}(2)$\nbroken to $\\mathbf{U}(1)$ on one brane by boundary conditions. The lattice sites are\nlinked by $\\mathbf{SU}(2)$ valued Wilson line fields $\\Sigma_{0\/1}$\nthat transform bi-unitarily under\ngauge transformations as\n\\begin{equation*}\n \\Sigma_0 \\longrightarrow U_0\\Sigma_0 U_1^\\dagger \\quad,\\quad\n \\Sigma_1 \\longrightarrow U_1\\Sigma_0 e^{-i\\theta\\frac{\\sigma_3}{2}}\\,.\n\\end{equation*}\nIf the potential for the Wilson line fields is arranged such that these acquire a vacuum\nexpectation value\n\\begin{equation*}\n  \\left<\\Sigma_{0\/1}\\right> = \\sqrt{2}v\\,,\n\\end{equation*}\nthe symmetry group is broken spontaneously\nto the electromagnetic~$\\mathbf{U}(1)_\\text{em}$. The kinetic terms for~$\\Sigma_{0\/1}$\ncontain covariant derivatives which produce mass terms for the gauge bosons;\nafter diagonalization we find a massless photon, two massive charged\ngauge bosons~$W$ and~$W^\\prime$ and two neutral massive gauge bosons~$Z$ and~$Z^\\prime$. \n\nChoosing~$g_1\\gg g_0,g_2$, the mass gap between the massive\ngauge bosons becomes large and the lighter ones can be identified\nwith the SM~$W$  and $Z$~bosons. These are mostly localized at the\nbrane sites, while the heavy modes\nare strongly localized at the bulk site. This symmetry breaking setup is similar to the\nBESS model~\\cite{Casalbuoni:1985kq}. After fixing the electric charge and\nthe~$W$ and $Z$~masses from the observed values,\nthe only remaining free parameter in the gauge sector is the $W^\\prime$ mass~$m_{W'}$.\n\nFermions are incorporated into the model by putting left-handed $\\mathbf{SU}(2)$\ndoublets $\\Psi_{0\/1,L}$ on the sites~$0$ and~$1$, a right-handed doublet $\\Psi_{1,R}$ on\nsite~$1$ and singlets $\\Psi^{u\/d}_{2,R}$ on site~$2$ for every SM\nfermion (cf.~figure~\\ref{fig-moose}).  The $\\mathbf{U}(1)_2$~charges\nof the $\\Psi^{u\/d}_{2,R}$ fermions are taken from the SM hypercharge\nassignments for the corresponding righthanded singlets, whereas the\n$\\mathbf{U}(1)_2$~charges of all other left- and righthanded fermions\nare taken from the SM hypercharge assignments for the corresponding\n\\emph{left}handed doublets.\n\nIn addition to the kinetic terms,\nYukawa couplings are added to the fermion Lagrangian\n\\begin{multline}\n\\label{equ-lgr-fermion}\n\\mathcal{L}_\\text{Yukawa} = \\\\\n\\lambda\\sum_i\n\\left[\\epsilon_L\\overline{\\Psi}_{0,L}^i\\Sigma_0\\Psi_{1,R}^i +\n\\sqrt{2}v\\overline{\\Psi}_{1,L}^i\\Psi_{1,R}^i +\n\\overline{\\Psi}_{1,L}^i\\Sigma_1\n  \\begin{pmatrix}\n     \\epsilon_{u,R}^i & 0 \\\\\n     0                & \\epsilon_{d,R}^i\n  \\end{pmatrix}\n\\Psi_{2,R}^i\\right]\n\\end{multline}\nwith the index~$i$ running over all SM fermions.\nThe parameter $\\epsilon_L$ is chosen universally for all fermions and\nsuch that the tree-level corrections to the EWPT\nvanish.  This will be referred to as ``ideal fermion\ndelocalization''~\\cite{SekharChivukula:2006cg,SekharChivukula:2005xm}.\nThe parameter $\\lambda$\nis also chosen universally for all fermions; only the~$\\epsilon_{u\/d,R}$ have a nontrivial\nflavor structure and are used to implement the mass splitting of\nquarks and leptons, as well as CKM flavor mixing. The\nvacuum expectation value $v$ breaks the symmetry and the mass\neigenstates are the SM\nfermions (localized mostly at the branes) and heavy partner fermions (localized\nmostly in the bulk).\n\nThe only remaining free parameter in the fermion sector after fixing the SM fermion\nmasses and $\\epsilon_L$ is the heavy fermion mass scale $m_\\text{bulk}=\\sqrt{2}\\lambda v$.\nTherefore the model is fixed uniquely by setting the SM parameters,\n$m_{W'}$ and $m_\\text{bulk}$ and \nby the requirement of ideal delocalization.\nLoop corrections to the EWPT\nand other phenomenological bounds limit the minimal values for these two parameters,\nrequiring $m_{W'}>\\unit[380]{GeV}$ and\n$m_\\text{bulk}>\\unit[2]{TeV}$~\\cite{SekharChivukula:2006cg}.\n\nThe spectrum of the model consists of the SM gauge bosons and\nfermions, the $W^\\prime$ and $Z^\\prime$ and a heavy partner fermion\nfor each SM fermion. The masses of the two new heavy gauge bosons are\nquasi-degenerate ($|m_{W'}-m_{Z'}|=\\mathcal{O}(\\unit[1]{GeV})$), and the\nmasses of the partner fermions are of the order $m_\\text{bulk}$ with\nthe~$t^\\prime$ being slightly heavier than the rest.\n\n\\section{Couplings, widths and branching ratios}\n\\label{sec-cpl}\n\nIdeal fermion delocalization implies that the couplings of the light SM fermions\nto the $W^\\prime$ vanish and that those to the $Z^\\prime$ are small\n($\\mathcal{O}(10^{-2}))$. The both heavy gauge bosons also\ncouple to the SM $Z$ and $W$~bosons with couplings of order $\\mathcal{O}(10^{-2})$.\n\n\\begin{figure}\n\\centerline{\\includegraphics[angle=270,width=8cm]{widths}}\n\\caption{\\label{plot-widths}%\nThe relative width~$\\Gamma_V\/m_V$ of the\n$W^\\prime$ and $Z^\\prime$ bosons as a function of $m_{W'}$ with ideal\ndelocalization and $m_\\text{bulk}=\\unit[5]{TeV}$.}\n\\end{figure}\nTherefore, the only decay channel for the $W^\\prime$ in the ideally delocalized scenario is\nthe decay into a $W$ and a $Z$. The $Z^\\prime$ can in principle also decay into SM\nfermions; however, the decay of the longitudinal mode enhances the $WZ$ decay\nchannel by a factor of\n\\begin{equation*}\n  \\frac{m_{Z^\\prime}^4}{16m_W^2m_Z^2}\n\\end{equation*}\nover the decay into a fermion pair causing the latter decay to be highly\nsuppressed by a factor of the order of $\\mathcal{O}(10^{-2})$ (cf.~\\cite{He:2007ge}).\nLooking at figure~\\ref{plot-widths} we find that the resonances are rather narrow\n($\\Gamma_V\/m_V\\approx1-3\\%$) improving the prospects for observing these particles at the LHC.\n\nThe new heavy fermions decay into their light partner and a gauge boson, the resulting\nwidths being of the order $\\Gamma_f\/m_f\\approx0.1$, which, combined with their large mass\n($>\\unit[2]{TeV}$), will make the direct detection as a resonance at a collider rather\nchallenging.\n\nFor a massless SM fermion, the Yukawa coupling between the sites~$1$ and~$2$ vanishes.\nFrom~$\\mathcal{L}_{\\text{Yukawa}}$ we find that the wave function is\ncompletely fixed by the delocalization parameter $\\epsilon_L$. Therefore, the influence of\n$m_\\text{bulk}$ on the wave functions of the light SM fermions and their couplings is very\nsmall and the dependence of the cross section on $m_\\text{bulk}$ is\nalmost negligible at LHC energies.\n\nAlthough ideal delocalization guarantees compatibility with the\nconstraints from EWPT at tree level~\\cite{SekharChivukula:2005xm},\na recent 1-loop analysis~\\cite{Abe:2008hb} has shown that a deviation from\nideal delocalization is necessary to comply with the EWPT constraints at loop\nlevel. According to the authors of~\\cite{Abe:2008hb}, this deviation corresponds to an\non-shell coupling between the $W^\\prime$ and the light fermions as large as $1-2\\%$ of\nthe isospin gauge coupling $g_W\\approx g_0$.\n\n\\begin{figure}\n\\centerline{\\includegraphics{operator1}}\n\\caption{The tree-level diagram generating the operator~$O_1$~(cf.~(\\ref{equ-op1})) after\nintegrating out the bulk fermions.}\n\\label{fig-op1}\n\\end{figure}\nThe coupling $g_{W'ff}$ to which the bounds derived in~\\cite{Abe:2008hb} apply is defined\nin the effective theory obtained by integrating out the bulk fermions and is renormalized\nat the $W^\\prime$ mass shell. There are two operators contributing to this coupling in the\none loop analysis in addition to the coupling of the left-handed\nfermions to the component of\nthe $W^\\prime$ sitting at site $0$. The first one\n\\begin{equation} \n\\label{equ-op1}\n  O_1 = \\overline{\\Psi}_{0,L}\\Sigma_0\\fmslash{A_1}\\Sigma_0^\\dagger\\Psi_{0,L}\n\\end{equation}\nencodes a coupling between the component of the $W^\\prime$ sitting at site $1$ and the\nleft-handed SM fermion and is generated by integrating out the bulk fermion\nfrom the diagram in figure~\\ref{fig-op1} (see\nalso~\\cite{SekharChivukula:2006cg}). The second operator\n\\begin{equation}\n\\label{equ-op2}\n   O_2 = \n    \\overline{\\Psi}_{0,L}\\left(D_\\nu\\left(\\Sigma_0 F_1^{\\mu\\nu}\\Sigma_0^\\dagger\\right)\n      \\right)\\gamma_\\mu\\Psi_{0,L}\n\\end{equation}\narises from loop corrections. Although this operator also contains a contribution to the\ncoupling between the left-handed fermion and the gauge bosons at site $1$, it has a\nnontrivial momentum structure. However, using a non-linear field redefinition in the\nspirit of on-shell effective field theory~\\cite{Georgi:1991ch}, the\ncorresponding part of~$O_2$ \ncan be converted to the same form as~$O_1$ at the price of introducing additional\nhigher dimensional operators coupling at least two gauge bosons to two fermions whose\ncontributions are suppressed by another power of the gauge couplings. This allows the\noperator~$O_2$ to be included into $g_{W^\\prime ff}$ where it contributes to\nthe bounds derived by the authors of~\\cite{Abe:2008hb}.\n\nTherefore, the contributions of both these operators can be accounted for by adjusting the\ndelocalization parameter $\\epsilon_L$ in the tree level\nLagrangian $\\mathcal{L}_{\\text{Yukawa}}$\nto generate the coupling $g_{W^\\prime ff}$. The model parameters then should be understood\nto be renormalized at the $W^\\prime$~mass.\n\n\\begin{figure}\n\\centerline{\\includegraphics[angle=270,width=8cm]{gWff_500}}\n\\caption{$g_{W'ff}$, $g_{Z'uu,L}$ and $g_{Z'dd,L}$ normalized to the site $0$ gauge\ncoupling as a function of the delocalization parameter $\\epsilon_L$.\nThe gray rectangle marks the range for $g_{W^\\prime ff}$ allowed by the EWPT as\nderived by the authors of~\\cite{Abe:2008hb}.}\n\\label{fig-gwff-500}\n\\end{figure}\nIn the case of light SM fermions and their partners, only the wave functions of the\nleft-handed fermions depend on the delocalization parameter $\\epsilon_L$. Therefore, the\nright-handed couplings between the new gauge bosons and the light SM fermions are not\naffected by the departure from ideal delocalization. Denoting the wave functions\nby~$\\phi_{f,L,i}$ and~$\\phi_{Z',i}$ and using the normalization of the fermion\nwave functions, the left-handed coupling of a fermion\nto the~$Z^\\prime$ can be written as\n\\begin{equation}\n  \\sum_{i=0}^{1}\\phi_{f,L,i}^2\\left(\\pm\\frac{1}{2} g_i\\phi_{Z',i}\n      + Y g_2\\phi_{Z',2}\\right)\n    = \\pm\\frac{1}{2}\\sum_{i=0}^1 g_i\\phi_{Z',i}\\phi_{f,L,i}^2 + Yg_2\\phi_{Z',2}\\,,\n\\label{equ-cpl-zff}\\end{equation}\nwith the sign depending on the isospin of the fermion and $Y$ denoting the hypercharge.\nAs tuning away from ideal delocalization shifts the light mode of the\nfermion towards the heavy $Z^\\prime$ sitting at site $1$, the isospin dependent part\nin~(\\ref{equ-cpl-zff}) grows, while the correction to~$g_{Z'uu}$ differs only in sign\nfrom the correction to~$g_{Z'dd}$.\n\nFigure~\\ref{fig-gwff-500} shows the dependence of\n$g_{W'ff}$, $g_{Z'dd}$ and $g_{Z'uu}$ on the delocalization parameter and clearly\ndemonstrates this behavior. Considering that we have both $u\\bar u$\nand $d\\bar d$ initial states at\nthe LHC, that both couplings start with positive values of the same order of magnitude at\nthe point of ideal delocalization and that we also have right-handed couplings of the same\norder of magnitude which don't depend on $\\epsilon_L$ at all, we don't\nexpect a large impact from changing $\\epsilon_L$ on $Z^\\prime$~production in the $s$-channel.\nOn the other hand, the effect on the $s$-channel $W^\\prime$~production\nshould be sizable, because $\\epsilon_L$ interpolates between this channel\nbeing forbidden and being about the same order of magnitude as\n$Z^\\prime$ production.\n\\begin{figure}\n\\centerline{\\begin{tabular}{cc}\n\\includegraphics[angle=270,width=6.5cm]{gWff_380} &\n\\includegraphics[angle=270,width=6.5cm]{gWff_600}\n\\end{tabular}}\n\\caption{\\label{fig-gwff-380-600}%\nThe same plots as figure~\\ref{fig-gwff-500}, but for the other parts of parameter\nspace probed by our Monte Carlo simulations.}\n\\end{figure}\n\nFigure~\\ref{fig-gwff-380-600} shows the same plot as\nfigure~\\ref{fig-gwff-500} for the other\nregions of parameter space probed in our Monte Carlo simulations. In all three plots,\nchanging $m_\\text{bulk}$ doesn't generate a visible change of the actual couplings, but\ndoes move the gray band of acceptable $g_{W'ff}$ values.\n\n\\section{Implementation}\n\\label{sec:whizard}\n\nWe have coded a FORTRAN 90 module which diagonalizes the lagrangian of the model\nand calculates all masses and couplings. Furthermore, the module calculates the tree\nlevel widths of all new particles. Non ideal delocalization is implemented by tuning the\nparameter $\\epsilon_L$ away from the value required for vanishing $g_{W'ff}$.\nFor the automatized generation of tree level matrix\nelements, we encoded the model in unitarity gauge into the optimizing\nmatrix element generator O'Mega~\\cite{Moretti:2001zz,Kilian:2007gr}\nwhich is part of the Monte Carlo eventgenerator generator WHIZARD~\\cite{Kilian:2007gr}.\nThe results presented below are based on Monte\nCarlo simulations using WHIZARD~\\cite{Kilian:2007gr}.\n\nWe checked the couplings calculated by our FORTRAN code against all the couplings for\nwhich analytic expressions are given in~\\cite{SekharChivukula:2006cg}. To check the\nvalidity of our implementation of the model, we compared the cross sections for a number of\n$2\\rightarrow 2$ processes to the SM, taking $m_{W^\\prime}$,\n$m_\\text{bulk}$ and $m_\\text{Higgs}$ to be huge. The widths calculated by the FORTRAN\nmodule using analytic formulae were checked against numeric results obtained from\namplitudes generated by O'Mega.\n\nWe also checked gauge invariance by numerically checking the Ward Identities in the model \nobtained by taking the limit\n\\begin{equation*}\n  \\sqrt{2}v=\\left<\\Sigma_{0\/1}\\right>\\rightarrow 0\\,,\n\\end{equation*}\nwhere the exact $\\mathbf{SU}(2)_0\\times\\mathbf{SU}(2)_1\\times\\mathbf{U}(1)_2$\ngauge symmetry is restored.\n\nIn addition, we compared several $2\\rightarrow 2$ cross sections\nto the CalcHep~\\cite{Pukhov:2004ca} implementation of the model used by the authors\nof~\\cite{He:2007ge}. After plugging in the correct $W^\\prime$ and $Z^\\prime$ widths,\nthe results turn out to be in perfect agreement.\n\n\\section{$Z^\\prime$ production in the $s$-channel}\n\\label{sec-hzprod}\n\nIn the ideally delocalized scenario, only the $Z^\\prime$ has nonvanishing tree level\ncouplings to the SM fermions, while the $W^\\prime$ is perfectly\nfermiophobic. As explained above, the $Z^\\prime$ decays with a branching ratio of over\n$95\\%$ into a $W^+W^-$ pair, rendering the resulting four fermion final state highly\nfavored over the two lepton one. This is in sharp contrast to many new heavy neutral\ngauge bosons predicted by other extensions of the SM (Little Higgs, GUTs\netc.) which usually have larger fermion couplings but small or vanishing couplings to the\nSM gauge bosons, because they typically originate from different gauge group factors and have little\nor no mixing with the SM gauge bosons~\\cite{Rizzo:2006nw,Langacker:2008yv}.\n\nThe most interesting final states for $Z^\\prime$ production are thus $jjjj$, $l\\nu jj$ and\n$l\\nu l\\nu$. The four jet final state however is highly contaminated from\nbackgrounds containing gluon jets, and the two neutrino final state suffers from\nthe momentum information missing for the two neutrinos, leaving $l\\nu jj$ as\nthe most promising candidate assuming one can cope with the missing neutrino momentum.\n\\begin{figure}\n\\centerline{\\includegraphics{hzprod}}\n\\caption{Representative of the class of diagrams contributing to the $Z^\\prime$\nproduction signal in $pp\\rightarrow l\\nu jj$.}\n\\label{fig-diag-hzprod}\n\\end{figure}\nFigure~\\ref{fig-diag-hzprod} shows a representative of the class of diagrams contributing to\nthe signal in this process.\nIn addition to the signal, there are also reducible backgrounds from events with\nneutral jet pairs and an irreducible background from diagrams not of the type\nfigure~\\ref{fig-diag-hzprod} contributing to the same final state.  In\nthis and the next section, we assume that a veto on forward tagging\njets is effective in suppressing the background from vector boson fusion.\n\nFor the construction of an observable that can deal with the missing longitudinal neutrino\nmomentum, consider the decay of an on-shell $W$ into a lepton with momentum $p_l=q$ and a\nneutrino with momentum $p_\\nu=p$. The mass shell conditions of neutrino and $W$ boson then give\ntwo equations involving the neutrino energy $p_0$ and longitudinal\nmomentum~$p_L$\n\\begin{subequations}\n\\begin{align}\n\\label{equ-nurec1}\n  p_0^2 - p_L^2 - \\left|\\vec p_\\perp\\right|^2 &= 0 \\\\\n\\label{equ-nurec2}\n  p_0 q_0 - p_L q_L - \\vec p_\\perp \\vec q_\\perp &= \\frac{m_W^2}{2}\n\\end{align}\n\\end{subequations}\n(assuming the lepton to be massless), with~$\\vec p_\\perp$ and~$\\vec\nq_\\perp$ the projections of the momenta onto the transverse plane.\n(\\ref{equ-nurec1}) describes a hyperbola in the $p_L-p_0$\nplane and~(\\ref{equ-nurec2}) describes a straight line with the modulus of the slope\nsmaller than $1$. These curves are parametrized by $\\vec p_\\perp$, $q$ and $m_W$ and one\nof their (two in general) intersections gives the neutrino energy and longitudinal\nmomentum as a function of these quantities. This geometrical situation is depicted in\nfigure~\\ref{fig-nurec}.\n\\begin{figure}\n\\centerline{\\includegraphics[angle=270,width=8cm]{mnurec}}\n\\caption{The two curves generated by the mass shell conditions for $W$ and neutrino in the\ncase of a $W$ decaying to $l\\nu_l$.}\n\\label{fig-nurec}\n\\end{figure}\n\nThis construction allows us to reconstruct the full neutrino momentum from the lepton\nmomentum and the missing $p_T$ for the events coming from the decay of a quasi-on-shell\n$W$. However, owing to the modulus of the slope of the straight line being smaller than\none, we always have two solutions, none of which is preferred on kinematical\ngrounds. We have elected to deal with this by counting \\emph{both} solutions in the histograms,\neffectively doubling the amount of background events while preserving the size of the signal.\nThe two points of intersection can be obtained analytically by\n\\begin{equation}\np_0 = \\frac{q_0^2\\left(m_W^2 + 2\\vec{p}_\\perp\\vec{q}_\\perp\\right) \\pm q_L A}\n\t{2q_0\\left(q_0^2  - q_L^2\\right)} \\quad,\\quad\np_L = \\frac{q_L\\left(m_W^2 + 2\\vec{p}_\\perp\\vec{q}_\\perp\\right) \\pm A}\n\t{2\\left(q_0^2 - q_L^2\\right)}\\,,\n\\label{equ-nurec3}\n\\end{equation}\nwith the abbreviation\n\\[\nA = q_0\\sqrt{\\left(m_W^2 + 2\\vec{p}_\\perp\\vec{q}_\\perp\\right)^2 +\n\t4\\vec{p}_\\perp^2\\left(q_L^2 - q_0^2\\right)} \\,.\n\\]\n\nTo investigate the possibility of discovering the $Z^\\prime$ in\n$pp\\rightarrow jjl+p_{T,\\text{miss}}$ at the LHC we have performed full parton-level Monte\nCarlo simulations for an integrated luminosity of $\\int\\mathcal{L}=\\unit[100]{fb^{-1}}$,\nthe lepton being either an electron or a\nmuon and each jet being either a quark (excluding the top) or a gluon. To suppress the\nbackgrounds, we have applied $p_T$-cuts to all visible particles and to\n$p_{T,\\text{miss}}$\n\\[ p_T \\ge \\unit[50]{GeV}\\,. \\]\nIn addition, we have required the polar and intermediary angles of all visible particles\nto lie within\n\\[ -0.95 \\le \\cos\\theta \\le 0.95 \\]\nand also applied a small-$x$ cut to the ingoing partons\n\\[ x \\ge 1.4\\cdot 10^{-3} \\]\nto avoid infrared singularities in the amplitude. For identifying the intermediary $W$ we\napplied a cut to the invariant mass of the two jets\\footnote{See\n  section \\ref{sec-jetres} for a discussion of the effects of finite jet \n  resolution on the identification of the $W$.}\n\\[ \\unit[75]{GeV} \\le m_{jj} \\le \\unit[85]{GeV}\\,. \\]\nWe used~(\\ref{equ-nurec3}) to reconstruct the neutrino momentum, counting both solutions\ninto the histograms and discarding those with negative neutrino energy.\n\n\\begin{figure}\n\\centerline{\\begin{tabular}{cc}\n\\includegraphics[angle=270,width=6.5cm]{hist_hz_nurec} &\n\\includegraphics[angle=270,width=6.5cm]{hist_hz_deloc}\n\\end{tabular}}\n\\caption{\\emph{Left:} Invariant mass distribution in $pp\\rightarrow l\\nu_ljj$ obtained\nfrom the reconstructed neutrino momenta vs. the distribution obtained from $p_\\nu$\ntaken from Monte Carlo data. \\emph{Right:} The effect of tuning $\\epsilon_L$ away from\nideal delocalization (cf.~figure~\\ref{fig-gwff-500}).}\n\\label{hist-hzprod-nurec}\n\\end{figure}\nThe plot on the left of figure~\\ref{hist-hzprod-nurec} compares  the\ninvariant mass distribution obtained\nfrom the reconstructed neutrino momenta to that obtained from\nthe unobservable neutrino momenta taken from Monte Carlo\ndata for $m_{W^\\prime}=\\unit[500]{GeV}$ and $m_\\text{bulk}=\\unit[3.5]{TeV}$.\nIn both cases the peak from the $Z^\\prime$ is clearly visible. As expected, counting\nboth solutions obtained from the reconstruction doubles the amount of background events\nwhile the number of events contained in the peak stays roughly the same. However, the peak is\nbroadened by the reconstruction, which can been seen when comparing to a\nSM simulation (dotted line). The broadening at the center of the peak is mainly\ncaused by the mismatch between reconstructed and true neutrino momentum of the signal events\ndue to the $W$ not being exactly on-shell; the sidebands of the peak are caused by the\nsecond solutions for $p_\\nu$ of events at the center of the peak.\n\nThe plot on the right of figure~\\ref{hist-hzprod-nurec} shows the\neffect of changing the delocalization parameter\n$\\epsilon_L$ in the range allowed by the EWPT at one loop\n(cf.~section \\ref{sec-cpl}),\nagain for $m_{W^\\prime}=\\unit[500]{GeV}$ and $m_\\text{bulk}=\\unit[3.5]{TeV}$. As argued\nbefore, the impact on the invariant mass distribution is not strong, the peak\nstaying clearly visible over the whole range of allowed values of $\\epsilon_L$.\n\n\\begin{figure}\n\\centerline{\\includegraphics[angle=270,width=8cm]{hist_hz_fullrange}}\n\\caption{Invariant mass distribution in $pp\\rightarrow l\\nu_ljj$ for different values of\n$m_{W^\\prime}$ and $m_\\text{bulk}$.}\n\\label{hist-hzprod-fullrange}\n\\end{figure}\nFigure~\\ref{hist-hzprod-fullrange} shows the invariant mass distributions obtained for\n\\[ m_{W^\\prime}\\in\\left\\{\\unit[380]{GeV},\\unit[500]{GeV},\\unit[600]{GeV}\\right\\}\\,,\\]\nwhich covers the whole range of values allowed by the EWPT\nat one loop\\footnote\n{We also changed $m_\\text{bulk}$ as shown in figure~\\ref{hist-hzprod-fullrange} to\ncomply with the EWPT; however, as explained in section \\ref{sec-cpl}, this has no\nnoticeable effect on the cross section.}~\\cite{SekharChivukula:2006cg,Abe:2008hb}. As\nthe masses of the $Z^\\prime$ and $W^\\prime$ are quasi-degenerate,\nthe $Z^\\prime$ peak moves with\nchanging $m_{W^\\prime}$. The histogram shows that the peak stays clearly observable,\nalthough it decreases in size as $m_{W^\\prime}$ becomes larger owing\nto the smaller parton\ndistribution functions for the sea quarks at larger values of~$x$.\n\nTo get a quantitative handle on the significance of the signal and to estimate the minimal\nluminosity necessary for discovering the $Z^\\prime$, we define the raw signal $N$ to be the\nnumber of events in the $\\pm\\unit[20]{GeV}$ region around the peak. To estimate the\nbackground we have generated SM events for an integrated luminosity of\n$\\int\\mathcal{L}=\\unit[400]{fb^{-1}}$, analyzed this data the in the same way\nas the Monte Carlo data for the three site model and then downscaled the resulting\ndistributions by a factor of $4$ to reduce the error coming\nfrom fluctuations in the background. We denote the number of background events in the\n$\\pm\\unit[20]{GeV}$ region around the peak obtained this way by $N_b$.\n\nWe define the signal $N_s$ as\n\\begin{equation}\n   N_s = N - N_b\\,.\n\\end{equation}\nThe number of background events in the original Monte Carlo data $N^\\prime_b$\nis roughly doubled by our momentum reconstruction\n\\begin{equation*}\n  N_b = 2N^\\prime_b\n\\end{equation*}\nand the standard deviation of $\\sigma_{N_b}$ of $N_b$ must scale\naccordingly, resulting in\n\\begin{equation*}\n \\sigma_{N_b} = 2\\sigma_{N^\\prime_b} = 2\\sqrt{N^\\prime_b} = \\sqrt{2N_b}\\,.\n\\end{equation*}\nWe then define the significance in the usual way:\n\\begin{equation}\ns = \\frac{N_s}{\\sigma_{N_b}} = \\frac{N - N_b}{\\sqrt{2N_b}}\\,.\n\\label{equ-sgn-rec}\\end{equation}\n\n\\begin{figure}\n\\centerline{\\includegraphics[angle=270,width=8cm]{significance_hz}}\n\\caption{The significance as defined in the text as a function of the integrated\nluminosity. The dotted lines mark the $3\\sigma$ resp. $5\\sigma$ discovery thresholds.}\n\\label{fig-sig-hz}\n\\end{figure}\nThe significance of the signal in the ideally delocalized scenario thus calculated is\nshown in figure~\\ref{fig-sig-hz} together with the $5\\sigma$ and $3\\sigma$ discovery\nthresholds. The $5\\sigma$ thresholds are approx.~$\\unit[1]{fb^{-1}}$,\n$\\unit[2]{fb^{-1}}$, $\\unit[5]{fb^{-1}}$ for\n$m_{W^\\prime}=\\unit[380]{GeV}$, $\\unit[500]{GeV}$, $\\unit[600]{GeV}$, respectively.\nConsidering the fact that tuning\n$\\epsilon_L$ into the region allowed by the EWPT does not\nsignificantly change the signal,\nthe three-site $Z^\\prime$ may be discovered as early as in the first\n$\\unit[1-2]{fb^{-1}}$ and even in the worst case can be expected to\nmanifest itself in the first\n$\\unit[10-20]{fb^{-1}}$ of data.\n\n\\section{$W^\\prime$ production in the $s$-channel without ideal delocalization}\n\\label{sec-hwprod}\n\n\\begin{figure}\n\\centerline{\\begin{tabular}{cc}\n\\includegraphics{hwjj} &\n\\includegraphics{hwll}\n\\end{tabular}}\n\\caption{\\emph{Left: }Representative of the class of diagrams contributing to the\n$W^\\prime$ production signal in $pp\\rightarrow l\\nu_ljj$. \\emph{Right: }One of the signal\ndiagrams in the $lljj$ decay channel of the $W^\\prime$.}\n\\label{fig-diag-hwprod}\n\\end{figure}\nAs discussed in section \\ref{sec-cpl}, the deviation from ideal delocalization required by\nthe EWPT at one loop leads to non-vanishing couplings of the\n$W^\\prime$ to the SM fermions of the same order of magnitude as the $Z^\\prime ff$\ncouplings. This allows for the possibility of producing the $W^\\prime$ in the $s$-channel\nat the LHC.\n\nThere are two possible decay channels for the $W^\\prime$ that are promising candidates for\ndiscovering this resonance. The first possibility is the decay $W^\\prime\\rightarrow\nWZ\\rightarrow l\\nu_ljj$ (cf.~the left plot in figure~\\ref{fig-diag-hwprod}),\nwhich is the final state already discussed in the last section\nand which can be treated the same way (replacing the cut on the $W$ mass with a cut on the\n$Z$ mass). The second possibility is the decay of the $ZW$ pair into two leptons and two\njets (cf.~the right plot in figure~\\ref{fig-diag-hwprod}). The absence\nof missing $p_T$ is a clear advantage\nof this decay mode allowing for background suppression by cutting on the invariant mass of\nthe lepton pair; unfortunately, the branching ratio is smaller than that for the $l\\nu_ljj$\nmode.\n\nTo probe the $l\\nu_ljj$ final state we have used the same Monte Carlo data and cuts as in\nsection \\ref{sec-hzprod} replacing the cut on the invariant mass\\footnote{See\n  section~\\ref{sec-jetres} for a discussion of the effects of finite jet \n  resolution on the separation of $W$ and $Z$.}\nof the jet pair with\n\\[ \\unit[86]{GeV} \\le m_{jj} \\le \\unit[96]{GeV}\\,. \\]\nFor probing the $lljj$ final state we again performed Monte Carlo simulations\nfor an integrated luminosity of $\\int\\mathcal{L}=\\unit[100]{fb^{-1}}$. We applied the\nsame $p_T$, $x$ and angular cuts as in the last section together with the\nidentification cuts\n\\begin{equation}\n\\unit[75]{GeV} \\le m_{jj} \\le \\unit[85]{GeV} \\quad,\\quad\n\\unit[86]{GeV} \\le m_{ll} \\le \\unit[96]{GeV}\n\\label{equ-cut-wzident}\\end{equation}\non the invariant mass of the jet pair and on that of the dilepton system.\n\n\\begin{figure}\n\\centerline{\\begin{tabular}{cc}\n\\includegraphics[angle=270,width=6.5cm]{hist_hwjj_fullrange} &\n\\includegraphics[angle=270,width=6.5cm]{hist_hwll_fullrange}\n\\end{tabular}}\n\\caption{\\emph{Left: }Invariant mass distribution for $W^\\prime$ production in\n$pp\\rightarrow l\\nu_ljj$ for different $W^\\prime$ masses and large $g_{W'ff}$.\n\\emph{Right: }The same distribution for the $lljj$ final state.}\n\\label{hist-hw-fullrange}\n\\end{figure}\nFigure~\\ref{hist-hw-fullrange} shows the invariant mass distributions obtained for both final\nstates for $m_{W^\\prime}=\\unit[380]{GeV}$, $\\unit[500]{GeV}$, $\\unit[600]{GeV}$ and\n$\\epsilon_L$ chosen from the allowed range such as to give large\nvalues\\footnote{Even larger values of $g_{W'ff}$ are allowed by\n  increasing $m_\\text{bulk}$, but we are more interested in the lowest\n  possible value for which the $W^\\prime$ might still be detected in\n  this channel at the LHC.}\nof~$g_{W'ff}$ (cf.~figures~\\ref{fig-gwff-500} and~\\ref{fig-gwff-380-600}). For both final\nstates, the resonance peaks can be clearly seen for all three values of $m_{W^\\prime}$.\nThe total number of events for $lljj$ is much smaller compared to $l\\nu_l jj$\nowing to the smaller branching\nratio, but the cuts on both $m_Z$ and $m_W$ and the absence of the double counting\nintroduced by the neutrino reconstruction significantly improve the\nsignal to background ratio.\n\n\\begin{figure}\n\\centerline{\\begin{tabular}{cc}\n\\includegraphics[angle=270,width=6.5cm]{hist_hwjj_38_deloc} &\n\\includegraphics[angle=270,width=6.5cm]{hist_hwll_38_deloc} \\\\\n\\includegraphics[angle=270,width=6.5cm]{hist_hwjj_50_deloc} &\n\\includegraphics[angle=270,width=6.5cm]{hist_hwll_50_deloc} \\\\\n\\includegraphics[angle=270,width=6.5cm]{hist_hwjj_60_deloc} &\n\\includegraphics[angle=270,width=6.5cm]{hist_hwll_60_deloc}\n\\end{tabular}}\n\\caption{\\emph{Left column: }The $W^\\prime$ resonance peak in the invariant mass\ndistribution for $pp\\rightarrow l\\nu_ljj$ for different values of the delocalization\nparameter. \\emph{Right column: }The same distributions in the case of the $lljj$ final\nstate.}\n\\label{hist-hw-deloc}\n\\end{figure}\nThe dependence of the resonance peak on the delocalization parameter $\\epsilon_L$ is shown\nin figure~\\ref{hist-hw-deloc}. The left column shows the $\\pm\\unit[50]{GeV}$ region around\nthe peak for the $l\\nu_ljj$ final state for different values of $\\epsilon_L$ \nand for the case of ideal delocalization. For\n$m_{W^\\prime}=\\unit[500]{GeV}$ and $m_{W^\\prime}=\\unit[600]{GeV}$ the peak vanishes in the\ncase of ideal delocalization which demonstrates that the cut~(\\ref{equ-cut-wzident}) is\nsufficient to discriminate between jets coming from the decay of $W$ and those coming from\na $Z$. In the case of\n$m_{W^\\prime}=\\unit[380]{GeV}$, a small peak remains even in the case of ideal\ndelocalization which stems from jets coming from\n$pp\\rightarrow Z^\\prime\\rightarrow l\\nu_ljj$ misidentified as a $Z$ (we will discuss the\npossibility of unfolding these two contribution in the next section).\n\nThe histograms show\nthat tuning $\\epsilon_L$ towards the point of ideal delocalization quickly decreases the\nsize of the peak making it invisible for the lowest chosen values of~$\\epsilon_L$.\nThe right column shows the same region around the peak for the final state\n$lljj$ and the same values of $\\epsilon_L$. As should be expected, the same decrease of\nthe peak size is visible.\n\nTo obtain a numerical estimate for the integrated luminosity required\nfor a $s=5\\sigma$ or~$3\\sigma$\ndiscovery of the $W^\\prime$ at some given value of the delocalization\nparameter $\\epsilon_L$ we exploit the fact that the significance of\nthe signal scales as~$g_{W'ff}^2$\nwith the coupling of $W^\\prime$ to left-handed SM fermions. This allows\nus to estimate the integrated luminosity required for\nobtaining a signal with significance $s_0$ in terms of the significance of the signal\nfor other values of coupling and integrated luminosity.\n\nFor the actual determination of $s$ from Monte Carlo data we define the signal as in section\n\\ref{sec-hzprod}. In case of the $l\\nu_ljj$ final state we calculate~$s$\nvia~(\\ref{equ-sgn-rec}), while for the case of $lljj$ it can be calculated simply as\n\\begin{equation}\n  s = \\frac{N_s}{\\sqrt{N_b}}\\,,\n\\end{equation}\nbecause we don't have the additional doubling of the\nbackground events by the neutrino momentum reconstruction in this case\\footnote\n{Because of the lower number of events in the final state $lljj$, the background for this\ncase was calculated for an integrated luminosity of $\\int\\mathcal{L}=\\unit[1000]{fb^{-1}}$\nand scaled down.}.\n\n\\begin{table}\n\\centerline{\n\\begin{tabular}{|c|c|c||c|}\n\\hline\\multicolumn{4}{|c|}{$W^\\prime\\rightarrow l\\nu_ljj$} \\\\\\hline\\hline\n$m_{W^\\prime}\\:\\left[\\unit{GeV}\\right]$ & $m_\\text{bulk}\\:\\left[\\unit{TeV}\\right]$\n& $\\epsilon_L$ & $s$ \\\\\n\\hline 380 & 3.5 & 0.338 & 5.6 \\\\\n\\hline 500 & 3.5 & 0.254 & 8.6 \\\\\n\\hline 600 & 4.3 & 0.211 & 6.4 \\\\ \\hline\n\\end{tabular}\\hspace{1cm}\n\\begin{tabular}{|c|c|c||c|}\n\\hline\\multicolumn{4}{|c|}{$W^\\prime\\rightarrow lljj$} \\\\\\hline\\hline\n$m_{W^\\prime}\\:\\left[\\unit{GeV}\\right]$ & $m_\\text{bulk}\\:\\left[\\unit{TeV}\\right]$\n& $\\epsilon_L$ & $s$ \\\\\n\\hline 380 & 3.5 & 0.338 & 5.1 \\\\\n\\hline 500 & 3.5 & 0.254 & 8.5 \\\\\n\\hline 600 & 4.3 & 0.211 & 8.3 \\\\ \\hline\n\\end{tabular}}\n\\caption{The significance of the signal calculated at different points in parameter space\nfor both final states.}\n\\label{tab-sgn-hw}\n\\end{table}\nThe significances calculated this way at different points in parameter\nspace are shown in\ntable~\\ref{tab-sgn-hw}. For $m_{W^\\prime}=\\unit[380]{GeV}$ and\n$m_{W^\\prime}=\\unit[500]{GeV}$, both final states seem to do equally well at revealing the\nfermionic couplings of the $W^\\prime$; however, for $m_{W^\\prime}=\\unit[600]{GeV}$ the\ndilepton final state appears to give a slightly better signal owing to the\nbetter ratio of signal to background.\n\\begin{figure}\n\\centerline{\\begin{tabular}{cc}\n\\multicolumn{2}{c}{\\includegraphics[angle=270,width=6.5cm]{sign_hw_380}} \\\\\n\\includegraphics[angle=270,width=6.5cm]{sign_hw_500} &\n\\includegraphics[angle=270,width=6.5cm]{sign_hw_600}\n\\end{tabular}}\n\\caption{The integrated luminosity required for a $5\\sigma$ resp. $3\\sigma$ discovery of the\n$W^\\prime$ in the $s$-channel.}\n\\label{fig-lumi-hw}\n\\end{figure}\nThe integrated luminosity necessary for a $5\\sigma$ resp. $3\\sigma$ discovery of the\n$W^\\prime$ in the $s$-channel is shown in figure~\\ref{fig-lumi-hw} together with the range of the\ndelocalization parameter $\\epsilon_L$ allowed for the different choices of $m_{W^\\prime}$\nand $m_\\text{bulk}$. Taking the integrated luminosity collected over the full LHC running\ntime to be around $\\unit[400]{fb^{-1}}$ and considering the fact that the band of allowed\n$\\epsilon_L$ (and $g_{W^\\prime ff}$) can be moved further towards smaller values by\nlowering $m_\\text{bulk}$, it is evident from figure~\\ref{fig-lumi-hw} that there is a part of\nthe allowed parameter space in which the $W^\\prime$ would appear perfectly fermiophobic\nat the LHC. However, there also is a big region of parameter space in which the coupling of\nthe $W^\\prime$ to the SM fermions eventually should be discovered, although\nthis still would take several years of running time as the lowest integrated luminosity\nrequired for $3\\sigma$ is around $\\unit[10]{fb^{-1}}$ even at the point in parameter space\nmost easily accessible.\n\n\\section{Finite jet resolution and $W$\/$Z$ identification}\n\\label{sec-jetres}\n\nSince flavor tagging is impossible for light quark flavors, we have to\nrely on invariant mass cuts for the jet pairs to be able to separate\nthe case of the two jets in $l\\nu_ljj$ coming from the decay of a~$W$\nin $Z^\\prime$~production from that of the jets being produced by a\ndecaying $Z$ in $W^\\prime$~production.\nHowever, it may very well be impossible to obtain a resolution of order\n$\\pm\\unit[5]{GeV}$ in the jet invariant mass from experimental data.\nIn this section, we discuss the effect of a gaussian smearing of\nthe invariant mass of the jets on our analysis.\n\nIn the ideal case of exact $m_{jj}$ measurement, events coming from the decay of a\nintermediary $W$\/$Z$ are distributed according to a Breit-Wigner distribution\n\\[ p_b(x,m,\\Gamma)\\:dx =\n\\frac{n_b(m,\\Gamma)^{-1}}{\\left(x^2-m^2\\right)^2+\\Gamma^2 m^2}\\:dx\\,, \\]\nwith the normalization factor\n\\[ n_b(m,\\Gamma) = \\frac{\\pi}{4m^3}\\left(1+\\frac{\\Gamma^2}{m^2}\\right)^{-\\frac{3}{4}}\n\\sin^{-1}\\left(\\frac{1}{2}\\atan\\frac{\\Gamma}{m}\\right)\\,. \\]\nEmulating the measurement error in the jet mass by convoluting $p_\\text{bw}$ with a\ngaussian of standard deviation $\\sigma$\n\\[ \np_\\text{g}(x,\\sigma)\\:dx = \\frac{1}{\\sqrt{2\\pi}\\sigma}e^{-\\frac{x^2}{2\\sigma^2}}\\:dx\n\\]\nwe obtain the smeared distribution\n\\[ p_\\text{sm}(x,m,\\Gamma,\\sigma)\\: dx = \\int_0^\\infty dy\\:p_b(y,m,\\Gamma)\np_\\text{g}(x-y,\\sigma)\\,.\n\\]\n\\begin{figure}\n\\centerline{\\includegraphics[angle=270,width=8cm]{smearedpeaks}}\n\\caption{The effect of a gaussian smearing on the Breit-Wigner shape of the $W$ and $Z$\nresonances for various widths $\\sigma$ of the gaussian.}\n\\label{fig-smearedpeaks}\n\\end{figure}\n\nFigure~\\ref{fig-smearedpeaks} shows the effect of this smearing on the Breit-Wigner peaks of\nthe $Z$ and the $W$. Turning on the smearing and increasing $\\sigma$ causes the sharp\nBreit-Wigner peaks to decay rapidly, and for $\\sigma=\\unit[10]{GeV}$, only two very broad\nbumps are left. The result is that, if a cross section has one contribution which stems\nfrom the decays of a virtual $Z$ and one coming from a virtual $W$, any attempt to isolate\nthe $Z$ contribution by cutting on the resonance will inevitably also select events coming\nfrom the $W$ decay contaminating the sample (and vice versa). Therefore, our analysis of\nthe $l\\nu_ljj$ final state will show a $W^\\prime$ peak even in the case of ideal\ndelocalization which is caused by jet pairs from a decaying $W$ misidentified as a $Z$.\n\nIf we try to isolate the $W$ peak with a cut on the invariant mass $m_{jj}$\n\\[ L_W \\le m_{jj} \\le U_W \\]\nand the $Z$ peak with a cut\n\\[ L_Z \\le m_{jj} \\le U_Z\\,, \\]\nthen the resulting event counts $\\widetilde{N}_W, \\widetilde{N}_Z$ can be\ncalculated from the true event counts $N_W, N_Z$  coming from a decaying $W$ or $Z$\nvia a matrix $T$ as\n\\[ \\begin{pmatrix} \\widetilde{N}_W \\\\ \\widetilde{N}_Z \\end{pmatrix} =\n\\begin{pmatrix} T_{WW} & T_{WZ} \\\\ T_{ZW} & T_{ZZ} \\end{pmatrix}\n\\begin{pmatrix} N_W \\\\ N_Z \\end{pmatrix} \\]\nwith entries\n\\[ T_{ij} = \\int_{L_i}^{U_i}dm\\:p_\\text{sm}(m,m_j,\\Gamma_j,\\sigma)\\,. \\]\nInverting $T$ we can calculate the event counts $N_W$ and $N_Z$\n\\begin{equation} \\begin{pmatrix} N_W \\\\ N_Z \\end{pmatrix} = T^{-1}\n\\begin{pmatrix} \\widetilde{N}_W \\\\ \\widetilde{N}_Z \\end{pmatrix}\\,.\n\\label{equ-trans-mat}\\end{equation}\nThe entries of $T$ give the probability of misidentifying an event and can be readily\ncalculated numerically; for example, choosing cuts\n\\[ L_W=\\unit[60]{GeV} \\quad,\\quad U_W=\\unit[85]{GeV} \\quad,\\quad\nL_Z=\\unit[86]{GeV} \\quad,\\quad U_Z=\\unit[111]{GeV} \\]\nyields\n\\[ T \\approx \\begin{pmatrix} 0.64 & 0.27 \\\\ 0.29 & 0.62 \\end{pmatrix}\n\\quad,\\quad\nT^{-1} \\approx \\begin{pmatrix} 1.9 & -0.85 \\\\ -0.89 & 2.0 \\end{pmatrix}\\,. \\]\nThis way, we can in principle use $T$ to disentangle the\ncontributions from $W$ and $Z$ resonances\nto the signal in the presence of a measurement error which causes the Breit-Wigner\npeaks to lose their shape. However, to apply this to actual data, it is vital\nto separate the signal from both the reducible and the irreducible\nbackgrounds, because they don't follow a Breit-Wigner distribution.\n\nIn order to estimate the significance of a signal obtained this way, we calculate the\nstandard deviation $\\sigma_{N_i}$ of $N_i$ according to\n\\[ \\sigma_{N_i} = \\sqrt{\\sum_{j\\in W,Z}\\left(T^{-1}_{ij}\\right)^2\\sigma_{\\widetilde{N}_j}^2}\\,. \\]\nIn our analysis, we obtain the signal events inside the smeared Breit-Wigner peaks\n$\\widetilde{N}_i$ by subtracting the background $N_{b,i}$ from the total number of events\n$N_{t,i}$. The error on $N_{t,i}$ is\n\\[ \\sigma_{N_{t,i}} = \\sqrt{N_i + 2N_{b,i}} = \\sqrt{N_{t,i} + N_{b,i}}\\,, \\]\nbecause of the neutrino momentum reconstruction doubling the amount of background events\n(cf.~section~\\ref{sec-hzprod}), and we finally arrive at\n\\begin{equation}\n\\sigma_{N_i} = \\sqrt{\\sum_{j\\in W,Z}\\left(T^{-1}_{ij}\\right)^2\\left(N_{t,j} + N_{b,j}\\right)}\n\\label{equ-sigma-after-transfer}\\end{equation}\nFor a simulation of the effect of the measurement error our analysis we have\nrandomly distributed the invariant mass of the jet pairs within a gaussian with width\n$\\sigma=\\unit[10]{GeV}$ centered around the correct value calculated from Monte Carlo data. We\nthen did the same analysis as in sections \\ref{sec-hzprod} and \\ref{sec-hwprod}\nwith $m_{W^\\prime}=\\unit[500]{GeV}$ and $m_\\text{bulk}=\\unit[3.5]{TeV}$ both for\n$\\epsilon_L=0.254$ and for the ideally delocalized scenario. The only difference to the\nprevious analysis are the cuts on $m_{jj}$ which we enlarged to\n\\[ \\unit[60]{GeV}\\le m_{jj}\\le\\unit[85]{GeV} \\quad\\text{resp.}\\quad\n\\unit[86]{GeV}\\le m_{jj} \\le\\unit[111]{GeV}\\,. \\]\n\\begin{figure}\n\\centerline{\\begin{tabular}{cc}\n\\includegraphics[angle=270,width=6.5cm]{hist_w_smear_ideloc} &\n\\includegraphics[angle=270,width=6.5cm]{hist_w_smear_254}\n\\end{tabular}}\n\\caption{\\emph{Left: }Signal in the $W^\\prime$ detection channel for the case of ideal\ndelocalization smeared with a gaussian error. \\emph{Right: }The same for the case\nof nonzero $g_{W'ff}$}\n\\label{hist-wsmear}\n\\end{figure}\nFigure~\\ref{hist-wsmear} shows the resulting effect on the $W^\\prime$ peak for the cases\nof ideal delocalization (left) and for $\\epsilon_L=0.254$ (right). In both cases a peak is\nclearly visible, which in the ideally delocalized scenario is only composed of events\nwith jets coming from a decaying $W$ misidentified as a $Z$.\n\nThe number of signal events\n$\\widetilde{N}_{W\/Z}$ after smearing, the significance $s_{W\/Z}$ of these\ncalculated via~(\\ref{equ-sgn-rec}), $N_{W\/Z}$ obtained from applying the transfer\nmatrix $T^{-1}$ (\\ref{equ-trans-mat}) and the resulting significance\n$N_i\/\\sigma_{N_i}$ obtained from~(\\ref{equ-sigma-after-transfer}) are shown in\ntable \\ref{tab-sgn-wzsep}. All peaks are significant with $s>5\\sigma$; however, after\napplying the transfer matrix, the $W^\\prime$ peak vanishes within one standard deviation\nfor ideal delocalization,\nwhile in the case of $\\epsilon_L=0.254$ a residue as big as $2\\sigma$ remains. The\n$Z^\\prime$ peak remains significant after applying the transfer matrix, however, the\nsignificance is reduced because the transfer matrix enlarges the error.\n\\begin{table}\n\\centerline{\n\\begin{tabular}{|c||c|c|c|c|}\n\\hline\\multicolumn{5}{|c|}{ideal delocalization}\\\\\\hline\\hline\n & $\\widetilde{N}_i$ & $s_i$ & $N_i$ & $\\frac{N_i}{\\sigma_{N_i}}$ \\\\\\hline\\hline\n$i=W$ & $3193$ & $17$ & $5126$ & $13$ \\\\\\hline\n$i=Z$ & $1371$ & $7.5$ & $-96.10$ & $0.24$ \\\\\\hline\n\\end{tabular}\n\\hspace{1cm}\n\\begin{tabular}{|c||c|c|c|c|}\n\\hline\\multicolumn{5}{|c|}{$\\epsilon_L=0.254$}\\\\\\hline\\hline\n & $\\widetilde{N}_i$ & $s_i$ & $N_i$ & $\\frac{N_i}{\\sigma_{N_i}}$ \\\\\\hline\\hline\n$i=W$ & $3767$ & $21$ & $5628$ & $14$ \\\\\\hline\n$i=Z$ & $2083$ & $11$ & $811.6$ & $2.0$ \\\\\\hline\n\\end{tabular}\n}\n\\caption{Comparison of the signals $\\widetilde{N}_{W\/Z}$ obtained with an gaussian\nsmearing of the invariant mass of the jets with $\\sigma=\\unit[10]{GeV}$ to the ``true''\nsignals $N_{W\/Z}$ calculated from the measured ones via the transfer matrix $T^{-1}$.}\n\\label{tab-sgn-wzsep}\n\\end{table}\n\nWhat are the consequences for the detection of $Z^\\prime$ and $W^\\prime$ in the $l\\nu_ljj$\nfinal state? The detection of the $Z^\\prime$ is not affected by inaccuracies in the\njet mass resolution as the peak is always present with little variations of its size over\nthe whole parameter space, and we can always compensate for the smearing of the jet\nmass by enlarging the cut window on $m_{jj}$. However, the separation of a possible $W^\\prime$\ncontribution to the peak (which depends heavily on the point in parameter space) by\ncutting on $m_{jj}$ alone is spoiled by the error in $m_{jj}$; we have to apply additional\ntricks like the transfer matrix~(\\ref{equ-trans-mat}) to disentangle the two contributions.\nWhile this seems to work in principle, the significance of the $W^\\prime$ signal is\nreduced by this analysis, rendering this final state much less suitable for detecting a\ncoupling between $W^\\prime$ and SM fermions than the decay into $lljj$ which\nis not contaminated by a contribution of the $Z^\\prime$.\n\n\\section{Conclusions}\n\\label{sec:concl}\n\nWe have studied the production of the heavy~$W'$ and $Z'$ bosons of\nthe three site higgsless model in the $s$-channel at the LHC.  Unlike\nvector boson fusion, this production mode allows to directly measure\nthe couplings of the new bosons to standard model fermions.  These\ncouplings are constrained by electroweak precision tests and their\nmeasurement is therefore crucial for consistency checks of models of\nelectroweak symmetry breaking with extended gauge sectors.\n\nWe have found a method that will allow the separation of~$W'$ from\n$Z'$~processes at the parton level.  Our results show that the\nobservation of $s$-channel production of $Z'$ bosons will not require\na lot of integrated luminosity for all of the allowed parameter space.\nIn contrast, $W'$ production in the $s$-channel is much more sensitive\nto the model parameters and there are regions of parameter space where\nan observation will be very challenging, if not impossible.  A more\ndetailed experimental analysis should investigate the effects of\nhadronization and detector response on our results.\n\n\n\n\\section*{Acknowledgments}\nThis research is supported by Deutsche Forschungsgemeinschaft through\nthe Research Training Group 1147 \\textit{Theoretical Astrophysics and\nParticle Physics}, by Bundesministerium f\\\"ur Bildung und Forschung\nGermany, grant 05HT6WWA and by the Helmholtz Alliance \\textit{Physics\nat the Terascale}.\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\n\n\\indent Throughout this paper, the term ``polyhedron''  refers only to convex polyhedron in Euclidean 3-space $\\mathbb{E}^3$ (convex 3-polytope). For any polyhedron $P$, we denote the set of all vertices, edges and faces of $P$ by $F_0(P)$, $F_1(P)$ and $F_2(P)$ respectively. These sets together with the empty set $\\emptyset$ and $P$ itself, form a lattice $F(P)$ under inclusion, with $P$ as maximum and $\\emptyset$ as minimum. The lattice $F(P)$ of $P$ is determined by the {\\it incidence matrix} $M_P$ of $P$, where we label the vertices by $v_1,\\ldots, v_r$ and the polygonal faces by $A_1,\\ldots,A_s$, say, and define $M_P$ to be the $r \\times s$ matrix whose $(i,j)th$ element $m_{i,j}$ is 1 if $v_i \\in A_j$ and 0 otherwise. The number $\\mu=\\mu(M_P)=\\mu(P)$, of nonzero elements of $M_P$ is called the {\\it multiplicity} of $P$. Since each edge determines four incidences of the adjacent faces and vertices and each incidence corresponds to two edges, we have $\\mu=\\mu(P)=2e$, where $e=e(P)$ is the number of edges of $P$.\n\n\\indent Two polyhedra $P$ and $Q$ are face equivalent (also called combinatorially equivalent), written $P \\approx Q$, if there is a lattice isomorphism from $F(P)$ to $F(Q)$. The set $[P]$ of all $Q \\approx P$ is the \\emph{face type} or, as often called, the \\emph{isomorphism type} of $P$. Since $P$ is the convex hull of the set $F_0(P)$ of its vertices and since $P$ determines $\\F$, we can identify $P$ with the point  $(v_1,\\ldots,v_r)$ in $\\mathbb{E}^{3r}=(\\mathbb{E}^3)^r$, where $\\F = \\{ v_1,\\ldots, v_r \\} $. Thus we may identify a neighborhood of $P$ in $[P]$ with a neighborhood of $(v_1,\\ldots, v_r)$ in $\\mathbb{E}^{3r}$, since any $P'\\approx P$ sufficiently close to $P$ may have its vertices labeled $v'_1,\\ldots v'_r$ in a unique way so that $v'_i$ is close to $v_i$, $i=1,\\ldots,r$.\n\nA graph  $\\mathscr{G}=(V,E)$   with vertex set  $V$  and edge set $E$  is said to be $3$-connected if any two vertices are connected by three internally disjoint paths (each pair of paths have only the two vertices in common); $\\mathscr{G}$  is planar if it can be embedded in the plane such that no two edges intersect internally.\n\nNow, for  any polyhedron $P$ each  edge has exactly two vertices. Therefore we can define the  $1$-skeleton, or edge graph, $\\mathscr{G}_P$ of any  polyhedron  $P$ as  follows. The vertices of $\\mathscr{G}_P$, are vertices of   $P$, and two vertices are connected by an edge in graph $\\mathscr{G}_P$ if they are vertices of the same edge of $P$. This planar graph which is called ``\\emph{Schlegel diagram}'' of $P$, can be obtained by projecting $P$ on one of its faces (outside face).\n\nDue to Steinitz's famous Theorem called ``Fundamental theorem of convex types'' a graph $\\mathscr{G}_P=(V,E)$  is the edge graph of  a polyhedron $P$ if and only if it is planar and $3$-connected. Beyond this remarkable result, it is interesting that, Steinitz discovered some important facts about the topological structure of $[P]$ which is called the realization  space  of $P$ (see Section 2 for the relevant definitions).\n\nIndeed, a careful inspection of the proof of Steinitz's Theorem shows that $[P]$  is smooth manifold with dimension $\\dim[P]=e-1$, up to similarities, where $e=e(P)$ is the number of edges of $P$ .\n\nPerhaps the naturality of the above result of Steinitz was the main reason for other mathematicians to believe that the realization  space of polytopes, in dimensions higher than three, might also be a smooth manifolds. Theorem, on page 18 of \\cite{Rob}, asserts that $[P]$ is manifold in general case, where $P$ is an $n$-dimensional convex polytope.  But a striking result of Mnev \\cite{RG,Mnev} shows that, from the topological point of view, the realization  space of a convex polytope may be arbitrarily complicated. The argument given in \\cite{Rob} does not take full account of coplanarity conditions on the intersections of the faces, but works for $n=3$ (see Section 3).\n\nThe structure of this note is as follows. In Section 2 it is introduced the explicit definition of the {\\it realization space} of a polyhedron in terms of a relevant metric. Then, we give a proof that the realization space is in fact a smooth manifold with a specific dimension. The proof, which is independent of implicit proof of Steinitz himself, is illustrated with  an example adopted from (\\cite{Rob}, p. 21).\n\n\nIn Section 3, for any polyhedron $P$ we define $\\langle P \\rangle$ the {\\it symmetry type} of $P$, together with giving some basic facts about the actions of transformation groups on manifolds. Then, we recall the {\\it stratification} of $\\mathbb{E}^3$  and the decomposition of the manifold  of the space of all polyhedra into {\\it strata} of symmetry types under the action of the point groups.\n\nIn Section 4, we first consider finite point groups generated by reflections. Then, with the help of the familiar notation of {\\it fundamental region} of a reflection group, we prove our main theorem  which relates the dimension of the symmetry type of polyhedron $P$ to the number of its edge orbits, under the action of $G(P)$, the symmetry group of $P$ on its face lattice $F(P)$. This is the proof of Deicke's conjecture in  \\cite{Rob} as well, for polyhedra having reflection symmetry groups. For polyhedra with rotation groups we refer to \\cite{Ros1}, and for the basic properties of polyhedra with symmetry we refer to Robertson's book \\cite{Rob}.\n\n\n\n\\section{The realization space of a polyhedron}\n\n\\indent Our purpose in this Section is to define realization space and determine its topological structure. But first we present some basic definitions.\n\nLet $\\left( \\mathbb{E}^n , \\, d \\right)$ be the $n$-dimensional Euclidean space with induced norm\n$$\nd(x,y) \\, = \\, \\|x-y \\|, \\, x,y\\in \\mathbb{E}^n \\, .\n$$\n\nDenote by $\\mathbb{K}^n$ hyperspace of all nonempty compact convex subsets (convex bodies) of $\\mathbb{E}^n$. Then, $\\mathbb{K}^n$ is endowed with the following familiar metric, called \\emph{Hausdorff metric}.\n\n\\begin{defi}\nFor $P\\in \\mathbb{K}^n$ and $\\varepsilon >0$, let\n$$\n\\mathcal{U}_{\\varepsilon}(P) \\, = \\, \\left\\{ x\\in \\mathbb{E}^n \\, \\mid \\, d(x,P) < \\varepsilon \\right\\} \\, ,\n$$\n\n\\noindent where $d(x,P) \\, = \\, \\stackrel[p\\in P]{}{\\inf} \\left\\{ d(x,p)=\\|x-p \\| \\right\\}$ .\\\\\n\nNow, for $P, \\, P' \\in \\mathbb{K}^n$, let $\\rho(P,P') \\, = \\, \\inf  \\left\\{ \\varepsilon \\, \\mid \\, P' \\subseteq \\, \\, \\mathcal{U}_{\\varepsilon}(P) \\right\\}$. Then,\n$$\\,\n\\,\n\\begin{array}{rcl}\n  d_{\\mathcal{H}}(P,P') & = & \\max  \\left\\{ \\rho(P,P') \\, , \\, \\rho(P',P) \\right\\} \\\\[4pt]\n                        & = & \\inf \\left\\{ \\varepsilon >0 \\, \\mid \\, P \\subseteq \\, \\, \\mathcal{U}_{\\varepsilon}(P') \\text{ and } P' \\subseteq \\, \\, \\mathcal{U}_{\\varepsilon}(P) \\right\\} \\, ,\n\\end{array}\n$$\n\n\\noindent is a metric on $\\mathbb{K}^n$ which is called Hausdorff metric (distance).\n\\end{defi}\n\n\nWe denote by $\\mathscr{P}$ the set of all polyhedra in $\\mathbb{E}^3$ as a topological subspace of the metric space $\\left( \\mathbb{K}^3, \\, d_{\\mathcal{H}} \\right)$.\n\n\n\\begin{defi}\nLet $P$ be a polyhedron (convex 3-polytope). Denote by $[P]$ the face type of $P$, the set of all polyhedra $Q$ face equivalent to $P$.\n$$\n[P] \\, = \\, \\left\\{ Q \\, \\mid \\, Q {\\text{ is polyhedron and }} Q\\approx P \\right\\} \\, .\n$$\n\n\\noindent Then, $[P]$ together with its natural subspace topology induced by Hausdorff metric $d_{\\mathcal{H}}$ is called the realization space of $P$.\n\\end{defi}\n\nAlternatively, any polyhedron $P$ determines, and it is determined by, the set $F_0(P) = \\{ v_1,\\ldots,v_r \\}$ of its vertices. Since we can identify $P$ with the point $(v_1,\\ldots,v_r)\\in \\left( \\mathbb{E}^3 \\right)^r$, $[P]$ can be interpreted as a topological subspace of $\\mathbb{E}^{3r}$ with its topology induced by the vertices.\n\nGiven $\\varepsilon >0$, sufficiently small, there is an open neighborhood $V_{\\varepsilon}(P)$ of $P$ in $[P]$ such that, for all $P' \\in V_{\\varepsilon}(P)$ we have $F_0(P') = \\{ v_1^{'},\\ldots,v_r^{'} \\}$, where $\\| v_i - v_i^{'} \\| < \\varepsilon$, $i=1, \\ldots , r$.\n\nTherefore, $[P]$ can be topologized locally by $V_{\\varepsilon}(P)$ neighborhood of $P$. Thus $V_{\\varepsilon}(P)$, in fact, is the open space of the small perturbations of the vertices of $P$.\n\n\\vspace*{0,2cm}\nNow, let $\\Pi_j$ be the plane containing $A_j$. Then, we may suppose without loss of generality that for all $j=1,\\ldots,s$, the origin $O$ does not lie in $\\Pi_j$. Otherwise translate  $P$ to ensure this condition. Then, for all $j$ there is a unique $a_j \\in \\mathbb{E}^3$ such that $\\Pi_j$ is given by the equation $\\langle x,a_j\\rangle=1, x\\in \\mathbb{E}^3$. Thus $\\Pi_j$ is given by $a_j$ and $P$ itself by the point\n$$(v_1,\\ldots,v_r,a_1,\\ldots,a_s)\\in \\mathbb{E}^{3r}\\times \\mathbb{E}^{3s}=\\mathbb{E}^{3(r+s)} \\, ,$$\n\n\\noindent where $\\langle v_i,a_j\\rangle=1$ for all $i,j$ such that $m_{i,j}=1$. Now let $P'$ be the polyhedron with $P\\approx P'$, having vertices $v'_1,\\ldots,v'_r$ and faces given by $a'_1,\\ldots,a'_s$ where the plane $\\Pi'_j$ of $A'_j$ has equation $\\langle x,a'_j\\rangle= 1$ and again $\\langle v'_i,a'_j \\rangle=1$ if $m_{ij}^{'}=1$ (with obvious notation).\nSuppose that $P'$ is close to $P$, so that\n$$v'_i=v_i+\\xi_i,\\,\\,\\,\\,a'_j=a_j+\\eta_j \\, ,$$\n\n\\noindent for some $\\xi_j,\\eta_j \\in \\mathbb{E}^3$ with $||\\xi_i||$ and $||\\eta_j||$ small ($i=1,\\ldots,r;j=1,\\ldots,s$). Then,\n\\begin{equation}\\label{eq1}\n\\bf{\\Phi_{[i,j]}(\\xi,\\eta):=\\langle v_i,\\eta_j\\rangle+\\langle \\xi_i,a_j\\rangle+\\langle\\xi_i,\\eta_j\\rangle=0},\n\\end{equation}\n\n\\noindent if $m_{ij}=1$ (and hence $m_{ij}^{'}=1$).\n\n\\vspace*{0,2cm}\nConversely, for any sufficiently small $\\xi \\in \\mathbb{E}^{3r}$ and $\\eta \\in \\mathbb{E}^{3s}$, the point\n$$(\\nu_1+\\xi_1,\\ldots,\\nu_r+\\xi_r,a_1+\\eta_1,\\ldots,a_s+\\eta_s)$$\n\n\\noindent represents a unique polyhedron $P'\\approx P$, provided equations (\\ref{eq1}) hold. Let us now consider the polynomial map $\\Phi: \\mathbb{E}^{3r}\\times \\mathbb{E}^{3s}\\rightarrow \\mathbb{E}^\\mu$ given by  $\\Phi(\\xi,\\eta)=(y_1,\\ldots,y_\\mu)$ where the pairs $(i,j)$ with $m_{ij}=1$ are arranged in lexicographical order, and if $(i,j)$ is the $[i,j]th$ such pair then, $y_{[i,j]}=\\Phi_{[i,j]}(\\xi,\\eta)$ is defined as in (\\ref{eq1}). We may for convenience suppress component suffices of $\\xi_i$ and $\\eta_j$, and write\n$$\\frac{\\partial y_{[i,j]}}{\\partial \\xi_i}=a_j+\\eta_j,\\,\\,\\,\\,\\frac{\\partial y_{[i,j]}}{\\partial \\eta_j}=\\nu_i+\\xi_i \\, .$$\n\nThus the Jacobian matrix $J_{\\Phi}(0,0)$, has order  $\\mu\\times(3(r+s))$ with $a_j$ in the $[i,j]$th row and the $i$th triplet of columns, and $\\nu_i$ in the $[i,j]$th row and the $(r+j)$th triplet of columns. All the other elements of $J_{\\Phi}(0,0)$ are $0$.\n\nA simple example may helps to clarify these remarks. Let $P$ be the polyhedron shown, with its accompanying Schlegel diagram in the following figure.\n{ \\begin{figure}[!htb] \\centering\n\\hspace*{.75cm} \\includegraphics[scale=.2]{FIG_1.eps}\n\\vspace*{-0,4cm}\\caption{{\\small{Schlegel diagram of $P$ with ``outside face'' $A_1$}}}\n{\\label{im1}}\n\\end{figure}}\n\n\\newpage\nIn this example, $r=7,s=8$ and $\\mu=26$. Notice that $3(r+s)=45>26=\\mu$. In fact, for any polyhedron, $3(r+s)=3(e+2)=3e+6=\\mu+e+2>\\mu$, by Euler's Theorem. Thus $\\mu$ is the largest value of the rank of $J_{\\Phi}(0,0)$. In the example, $J_{\\Phi}(0,0)$ has 26 rows and 45 columns, shown below in the truncated $26\\times 15$ form.\n\n\\begin{center} {\\scriptsize{\\begin{tabular}{|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|c|}\n  \\hline\n\n    & 1 & 2 & 3 & 4 & 5 & 6 & 7  & 1 & 2 & 3 & 4 & 5 & 6 & 7   & 8  \\\\\n    \\hline\n 15 & $a_5$ &   &   &   &   &   &     &   &   &   & $\\nu_1$   &  &  &  & \\\\\n  \\hline\n  16 & $a_6$ &   &   &   &   &   &     &   &   &   &   & $\\nu_1$ &   & & \\\\\n  \\hline\n  17 & $a_7$ &   &   &   &   &   &     &   &   &   &   &   &  & $\\nu_1$ & \\\\\n  \\hline\n  18 & $a_8$ &   &   &   &   &   &     &   &   &   &   &   &   &   &$\\nu_1$ \\\\\n  \\hline\n  22 &   & $a_2$ &   &   &   &   &     &  & $\\nu_2$  &   &   &   &   &   & \\\\\n  \\hline\n  23 &  & $a_3$ &   &   &   &   &      &         &  &  $\\nu_2$ &   &   &   &   & \\\\\n  \\hline\n  25 &   & $a_5$ &   &   &   &   &     &   &   &  &  & $\\nu_2$  &   &   & \\\\\n  \\hline\n  26 &   & $a_6$ &   &   &   &   &     &   &   &   &   &  & $\\nu_2$  &   & \\\\\n  \\hline\n  32 &   &   & $a_2$ &   &   &   &     &  & $\\nu_3$  &   &   &   &   &   & \\\\\n  \\hline\n  34 &   &   & $a_4$ &   &   &   &     &   &  &   & $\\nu_3$ &   &   &   & \\\\\n  \\hline\n  35 &   &   & $a_5$ &   &   &   &     &   &   &   &  & $\\nu_3$  &   &   & \\\\\n  \\hline\n  38 &   &   & $a_8$ &   &   &   &    &   &   &   &   &   &   &   & $\\nu_3$\\\\\n  \\hline\n  41 &   &   &   & $a_1$ &   &   &   & $\\nu_4$   &   &   &   &   &   &  & \\\\\n  \\hline\n  44 &   &   &   & $a_4$ &   &   &   &     &   &  & $\\nu_4$  &   &   &   & \\\\\n  \\hline\n  47 &   &   &   & $a_7$ &   &   &   &     &   &   &   &   &  & $\\nu_4$ & \\\\\n  \\hline\n  48 &   &   &   & $a_8$ &   &   &   &     &   &   &   &   &   &   &$\\nu_4$\\\\\n  \\hline\n  51 &   &   &   &   & $a_1$ &   &   & $\\nu_5$   &   &   &   &   &   &   & \\\\\n  \\hline\n  53 &   &   &   &  &  $a_3$ &   &   &     &  & $\\nu_5$  &   &   &   &   & \\\\\n  \\hline\n  56 &   &   &   &   & $a_6$ &   &   &      &   &   &   &  &  $\\nu_5$ &   & \\\\\n  \\hline\n  57 &   &   &   &   & $a_7$ &   &   &     &   &   &   &   &  & $\\nu_5$  & \\\\\n  \\hline\n  61 &   &   &   &   &   & $a_1$ &   &  $\\nu_6$  &   &   &   &   &   &   &\\\\\n  \\hline\n  62 &   &   &   &   &   & $a_2$ &     &  &  $\\nu_6$ &   &   &   &   &   &\\\\\n  \\hline\n  63 &   &   &   &   &   &$ a_3$ &     &   &  & $\\nu_6$  &   &   &   &   &\\\\\n  \\hline\n  71 &   &   &   &   &   &   & $a_1$ &  $\\nu_7$ &   &   &   &   &   &   & \\\\\n  \\hline\n  72 &   &   &   &   &   &   & $a_2$ &   & $\\nu_7 $  &   &   &   &   &   & \\\\\n  \\hline\n  74 &   &   &   &   &   &   & $a_4$ &    &   &  &   $\\nu_7$ &   &   &   &\\\\\n  \\hline\n\\end{tabular}}} \\end{center}\n\n\\begin{center}\n[truncated $J_{\\Phi}$]\n\\end{center}\n\n\nRecall that a similarity is a map\n\\vspace*{-.3cm}\n$$ s:\\mathbb{E}^3 \\rightarrow \\mathbb{E}^3$$\n\n\\vspace*{-.3cm}\n\\noindent such that for some real $r>0$, and all $x,y\\in \\mathbb{E}^3$\n\\vspace*{-.3cm}\n$$\nrd(x,y)=d(s(x), s(y)) \\, .\n$$\n\n\\vspace*{-.3cm}\nThe set of all similarities is a group which is denoted by $Sim(3)$, and called similarity group of $\\mathbb{E}^3$.\n\nThe map $\\varphi: Sim(3)\\rightarrow \\mathbb{R}_\\ast$ defined by $\\varphi(f)=r$, where $r$ is as above and $\\mathbb{R}_\\ast$ is multiplicative group of positive real numbers, is a group homomorphism.\nIt is well known that the kernel of this homomorphism is the isometry group or the Euclidian group $\\mathbb{E}(3)$. We may identify $Sim(3)$ as $\\mathbb{E}(3) \\times \\mathbb{R}_\\ast$. Since the manifold $\\mathbb{E}(3)$ has dimension $6$, the dimension of $Sim(3)$ is equal to $7$.\n\n\\vspace*{.2cm}\nNow we are ready to prove our main theorem in this Section:\n\\begin{teo} Let $P$ be a convex 3-polyhedron. Then, $[P]\/Sim(3)$, the realization space of $P$ modulo similarities, is a manifold of dimension $\\dim \\left( [P]\/Sim(3) \\right) \\, = \\, e-1$, where $e=e(P)$ the number of edges of $P$.\n\\end{teo}\n\\vspace*{-.1cm} {\\bf Proof:} To show, equivalently, that for any $P$, the face type $[P]$ is a manifold of dimension $e+6$, it is enough to show that $J_{\\Phi}(0,0)$ has rank $\\mu$. The result follows by the implicit function Theorem, since $e-1=3(r+s)-\\mu-7$.\n\nWe want to show that the rows of $J_{\\Phi}(0,0)$ are linearly independent. Suppose then, that for some real number $\\alpha_{[i,j]}$, we have\n\\vspace*{-.1cm}\n\\begin{equation}\\label{eq2}\n\\sum_j \\alpha_{[i,j]}a_j=\\sum_i \\alpha_{[i,j]}\\nu_i=0 \\, ,\n\\end{equation}\n\n\\vspace*{-.3cm}\n\\noindent for each $i=1,\\ldots,r$ and each $j=1,\\ldots,s$. We know that for any $i$ and any three values of $j_1,j_2,j_3$ of $j$ with $m_{ij}=1$, $a_{j_1},a_{j_2},a_{j_3}$ are linearly independent. Hence for each $i$, the equations $\\sum  \\alpha_{[i,j]}a_j=0$ have solution space of dimension $s_i-3$ where $s_i$  is the number of the values of $j$ with $m_{i,j}=1$, and we can express any three of the  numbers  $\\alpha_{[i,j]}$ as linear functions  of the remaining $s_i-3$. Hence the $\\mu$ numbers $\\alpha_{[i,j]}$ are expressible as linear functions of $\\sum_{i=1}^r(s_i-3)=\\mu-3r$ independent variables. In particular, for each $i$ for which $s_i=3$, all the numbers $\\alpha_{[i,j]}$ are 0.\n\nNow consider the equations $\\sum \\alpha_{[i,j]}\\nu_i=0$. Again, for each $i$, any three of the vertices $v_i$ are linearly independent. So again we can express any three of the numbers $\\alpha_{[i,j]}$ as linear combinations of the remaining $r_j-3$. So the numbers $\\alpha_{[i,j]}$ are expressible as linear functions of $\\sum_{i=1}^s(v_j-3)=\\mu-3s$ independent variables.\n\nBut the labels $\\alpha_{[i,j]}$ for each $j$ are already expressed as linear functions of $\\mu-3r$ independent variables, as described above. Hence the numbers $\\alpha_{[i,j]}$ are over determined by the two sets of equations in (\\ref{eq2}), since the solution space of the equations in (\\ref{eq2}) has dimension $\\mu-3(r+s)$. But  $\\mu-3(r+s)<0$ by Euler formula. It follows that $\\alpha_{[i,j]}=0$ for all $i,j$ where $m_{i,j}=1$.\\hfill$\\square$\n\n\n\\vspace*{.5cm}\nReferring to the above example, we may express each of, say  $\\alpha_{[1,6]}$, $\\alpha_{[1,7]}$ and $\\alpha_{[1,8]}$ as linear function of $\\alpha_{[15]}$ that is as a multiple of $\\alpha_{[15]}$. Thus $\\alpha_{[16]}=\\lambda_{16}\\alpha_{[15]}$, $\\alpha_{[17]}=\\lambda_{17}\\alpha_{[15]}$, $\\alpha_{[18]}=\\lambda_{18}\\alpha_{[15]}$. Likewise, taking the first number $\\alpha_{[i,j]}$ in each block parameter where possible we have  $\\alpha_{[23]}=\\lambda_{23}\\alpha_{[22]}$, $\\alpha_{[25]}=\\lambda_{25}\\alpha_{[22]}$ and $\\alpha_{[26]}=\\lambda_{26}\\alpha_{[22]}$, and so for each of the values $i=1,2,3,4,5$. For $i=6$ and for $i=7$, $s_i=3$ and immediately we find that $\\alpha_{[6i]}=\\alpha_{[7j]}=0$ for each $j$ with $m_{6j}=1$ or $m_{7j}=1$.\n\nNow repeat this procedure with numbers $\\alpha_{[i,j]}$ with $j$ fixed. Thus  $\\alpha_{[51]}=\\mu_{51}\\alpha_{[41]}$, $\\alpha_{[61]}=\\mu_{61}\\alpha_{[41]}$, and $\\alpha_{[71]}=\\mu_{71}\\alpha_{[41]}$, and likewise, for $j=2$. For $j=3,4,5,6,7$ and $8$, only three values of $i$  corresponds to each values of $j$, because the associated faces $A_j$ are triangles. So all $\\alpha_{[ij]}=0$ for all $i,j$ with $m_{i,j}=1$, as in general case.\n\n\n\\begin{obs}\n{\\label{obs1}}\n{\\rm It is worth mentioning that Richter-Gebert in (\\cite{RG}, Section 13.3) by fixing a suitable affine basis, proves that the realization space of a polyhedron $P$ (denoted there by $\\mathcal{R}(P)$), that is the space of coordinatization for the combinatorial type of $P$ with $e(P)=e$ edges is a smooth open manifold of dimension $e-6$, module the natural action of $12$-dimensional affine transformation group.\n\nThe realization space $\\mathcal{R}(P)$ is understood as a subspace of $\\mathbb{E}^{3n}$ by identifying the $3n$ coordinates of the $n$ vertices of $P$ with points in $\\mathbb{E}^{3n}$. It is described by the set of all solutions of a collection of polynomial equations and inequalities with integer coefficients. Such sets are called a simple semi-algebraic variety.\n\nFurthermore, by fixing an affine basis in the definition of the realization space, one makes sure that the ``reflection'' (mirror images in Steinitz's proof) do not create a second component of the realization space. Therefore, $\\mathcal{R}(P)$ is indeed (path connected) \\emph{contractible} and has \\emph{Steinitz's isotopy property} (\\cite{Stei}, Section 69), i. e., any two realizations $P_1$ and $P_2$ of $P$ can be continuously deformed into each other while maintaining the same structure throughout.\n\nHowever, our approach to study the face type manifold $[P]$ of a polyhedron $P$, is different. Although the realization spaces such as $[P]$ are usually (for example as above) defined modulo affine or Euclidian groups, in this work we consider $[P]\/Sim(3)$, $[P]$ modulo similarity group. The topology of the realization space $[P]$ of polyhedron $P$ is induced by Hausdorff metric, and since the action $Sim(3)$ on $[P]$ is not (fixed-point) free (see Section 3 for definition), the manifold $[P]\/Sim(3)$ is not contractible, and clearly does not satisfy Steinitz's isotopy property.}\n\\end{obs}\n\n\\begin{obs}\n{\\rm Let $P$ be a polyhedron with $f_0(P)=r$ vertices, $f_1(P)=e$ edges and $f_2(P)=s$ faces. Consider the face type $[P]$ of $P$. In (\\cite{Rob}, p. 75) the dimension of this manifold is given by\n$$\\dim [P]=3(r+s)-\\mu(P) \\, .$$\n\nAn intuitive derivation of the above formula may be given as follows. If the vertices and faces of $P$ were allowed to move independently in $\\mathbb{E}^3$ then, they would have $3(r+s)$ \\emph{degrees of freedom} (see page 11 for definition). But they are not independent. In fact for each incidence relation $(v,F)\\in F_0(P)\\times F_2(P)$ with $v \\in F$, the whole space loses one degree of freedom. Hence we have\n$$\\dim[P] \\, = \\, 3(r+s)-\\mu(P)\\,.$$\n\n\nNote that, since the similarity Lie group $Sim(3)$ has dimension 7, by Euler's formula\n\\vspace*{-.1cm}\n$$n-e+f=2 \\: , $$\n\n\\vspace*{-.1cm}\n\\noindent we have\n\\vspace*{-.1cm}\n$$\n\\begin{array}{rcl}\n  \\dim \\left( [P]\/Sim(3) \\right) & = & 3(r+s)-\\mu(P)-7 \\\\\n  \\, & = &  3(r+s)-2e-7 \\, = \\, e-1\\, (\\text{\\small{\\emph{module similarities}}}) \\, .\n\\end{array}\n$$}\n\\end{obs}\n\n\n\\section{Symmetry type of a polyhedron and stratifications}\n\n{\\bf Transformation groups}\n\n\\vspace*{0,2cm}\n\\indent We start this Section by introducing some basic definitions and standard results from the theory of Lie groups acting on smooth manifolds and refer the reader to \\cite{Dei} and \\cite{Kaw} for more details.\n\n\\vspace*{0,2cm}\nLet $G$ be a Lie group, with identity element $e$, and $M$ a smooth manifold. A smooth action of $G$ on $M$ is a $C^{\\infty}$ mapping:\n\\vspace*{-0,2cm}\n$$\\phi: G\\times M \\rightarrow M \\: , \\:\\: \\phi(g,x)=\\phi_g(x)=g(x) \\, ,$$\nsuch that\n$$\n\\begin{array}{rcll}\n  e(x) & = & x, & x\\in M \\, ,\\\\\n  (g_1g_2)(x) & = & g_1(g_2(x)), & g_1, g_2 \\in G \\, , \\: x\\in M \\, .\n\\end{array}\n$$\n\n\nIn this case we say that $M$ is a $G$-manifold. For any $x\\in M$ the subgroup\n\\vspace*{-.2cm}\n$$G_x=\\{g \\in G \\mid g(x)=x\\} \\, ,$$\nof $G$ is called the \\emph{stabilizer} or \\emph{isotropy} subgroup at $x$.\n\nThe set $ \\, G(x)=\\{ g(x) \\mid g\\in G \\} \\,$ is called the \\emph{$G$-orbit} of $x$. The \\emph{orbit space} of the action of $G$ on $M$ is the space $M\/G$, the space of all $G$-orbits endowed with the quotient topology given by canonical projection\n\\vspace*{-.2cm}\n$$\n\\begin{array}{ccccl}\n  \\pi & : & M & \\rightarrow & M\/G\\\\\n   &  & x & \\mapsto & G(x)\n\\end{array} \\, ,\n$$\n\\noindent and the differentiable structure of $M\/G$ is induced by the same structure of $M$.\n\n\nThe action is called \\emph{free} if, for each $x\\in M$, $ \\: G_x=\\{ e \\} \\, $ .\n\nIf a Lie group $G$ acts on a smooth manifold $M$ via $\\phi$, we call $(M,G):=(M,G,\\phi)$ a transformation group, and $M$ is said to be a $G$-manifold.\n\nNow, for each polyhedron $P$, a symmetry of  $P$ is a rigid  transformation  (or self isometry) $f:\\mathbb{E}^3\\rightarrow \\mathbb{E}^3$  such that $f(P)=P$. Any such symmetry maps vertices to vertices, edges to edges and faces to faces and preserves inclusions (incidences). Hence any symmetry induces an  automorphism on $F(P)$. The set $G(P)$  of all symmetries of $P$ is a finite subgroup of the Euclidean group  $\\mathbb{E}(3)$ acting on $F(P)$ as a group of automorphisms. We may assume that the centroid is $O$, so $G(P)$ is a finite subgroup of the orthogonal group $\\mathcal{O}(3)$. If a finite subgroup $G$ of $\\mathcal{O}(3)$ is the symmetry group of a convex polyhedron $P$, we also call $P$ a $G$-polyhedron.\n\n\n\n\\begin{defi}\nTwo polyhedra $P$ and $Q$ are \\emph{symmetry equivalent}, and write $P\\cong Q$, if there is an isomorphism\n$$\\lambda: F(P)\\rightarrow F(Q)$$\n\n\\noindent of the face lattices and some isometry $f:\\mathbb{E}^3 \\rightarrow \\mathbb{E}^3$ such that for all\n$g \\in G(P)$ and all $x \\in F(P)$,\n$$\\lambda(gx)=(f g f^{-1})(\\lambda(x)) \\, .$$\n\\end{defi}\n\nIf we further assume that $P$ and $Q$ are both $G$-polyhedra, that is having the same (rather than conjugate) subgroups then, with the above condition, we say that $P$ and $Q$ are $G$-equivalent. Hence, in this case\n$$\\lambda(gx)=g\\lambda(x) \\, .$$\n\n\\begin{defi}\nLet $P$ be a $G$-polyhedron. The symmetry type $\\langle P \\rangle$ of $P$ is defined by\n$$\\langle P \\rangle = \\{ Q \\, | \\, Q \\text{ is $G$-polyhedron and $Q$ is $G$-equivalent to $P$}, \\,\\, Q\\cong P \\} \\, . $$\n\\end{defi}\n\nNow consider $\\mathscr{P}$ the space of all convex polyhedra in $\\mathbb{E}^3$. Since the subdivision of $\\mathscr{P}$ into face types and symmetry types both respect the Euclidian similarities, it is convenient to look at the action of the similarity group $Sim(3)$ on the space $\\mathscr{P}$ of all polyhedra in $\\mathbb{E}^3$. This action partitions the quotient space\n$$ \\, \\mathscr{S}:=\\mathscr{P}\/Sim(3) \\, ,$$\n\n\\vspace*{-.2cm}\n\\noindent of similarity classes or ``shapes'' of polyhedra, into orbit types, where each orbit type consists of all those orbits on which the isotropy subgroups at any polyhedron in the orbit are conjugate.\n\nThus the symmetry types partitions each $[P]$ into mutually disjoint subsets refining the partitions of $\\mathscr{S}$ into face types.\n\nBut the isotropy subgroup at $P$ is just the symmetry group $G(P)$ itself. It follows that the symmetry types are composed of components of the orbit types. The principal orbit type corresponds to the trivial isotropy subgroup, that is to say to the symmetry type of any polyhedron $Q$ in $[P]$ with trivial symmetry group $G(Q)=\\{ e \\}$. This type is open in $[P]\/Sim(3)$ of dimension $e(P)-1$. All other symmetry types have lower dimensions.\n\nAs an example, let $P$ be a polyhedron combinatorially equivalent to cube. Then, $[P]\/Sim(3)$ has dimension 11. The principal orbit type corresponds to the realization space of polyhedron $Q\\approx P$ with trivial group, is an open and dense submanifold of $[P]\/Sim(3)$ of dimension $e(Q)-1=11$.\n\nNow, since each orbit type is a submanifold of $[P]$, by the Slice Theorem of transformation groups (see \\cite{Kaw}, Th. 4.11, and \\cite{Rob}, p.42), we have the following well known theorem.\n\n\n\n\\begin{teo} {\\rm (in \\cite{Rob}, p. 42 and \\cite{Ros2})}  Let $P$ be a polyhedron in $\\mathbb{E}^3$. Then, $\\langle P \\rangle$, the  symmetry type of $P$, is a smooth manifold.\\end{teo}\n\nIn order to study the dimension of the symmetry type of $P$, $\\dim \\langle P \\rangle$, we simply study those polyhedra $Q$ that lie in some neighborhood of $P$ in $\\mathscr{S}$ and are symmetry equivalent to $P$. We can simplify our discussion by restricting our attention to those $Q$ whose symmetry group is not merely conjugate to $G(P)$ but $G(P)$ itself. Along this restriction we also factor out components that came from similarities. In this way, we may determine the value of  $\\dim \\langle P \\rangle$. We first look at a simple example again.\n\nLet $P$ be the right pyramid over a square (Figure \\ref{image1}) with symmetry group $G(P)$ which is dihedral reflection group.\n\n\nSuppose we fix the group $G = G(P)$. Then, vertex $v_1$ can be chosen only on the axis of $G$. Therefore it has one degree of freedom (see page 11). Likewise $v_2$  must lie on reflection plane, hence has only two degrees of freedom. Having chosen  $v_2$, the vertices $v_3$,$v_4$ and $v_5$  which are on the same orbit of $v_2$ , have no degree of freedom at all, since they are determined by our choice of G and $v_2$ . Hence the vertices have a total of $1 + 2 = 3$ degrees of freedom. Similarly for the faces, each triangular face or equivalently the plane that contains it has only two degrees of freedom in the space of affine plane in $\\mathbb{E}^3$, since each plane is invariant under a reflection element of $G$. But the square face has only one degree freedom, because it is orthogonal to the axis of rotation of $G$. Therefore the faces have just $2 + 1 = 3$ degrees of freedom. Of course the faces and vertices can not be chosen independently of one another. The incidence of $v_1$ with respect to any of the four triangular faces adjacent to it, determines the incidence of that vertex to the other three faces under the action of $G$. Hence $v_1$  has only one ``independent'' incidence. The vertices $v_2$, $v_3$, $v_4$ and $v_5$ are in the same $G$-orbit and each one is incident with two triangles and one square faces. Take one of them say $v_2$. There is a reflection which fixes $v_2$ and sends adjacent triangular faces each one to the other. Thus the number of independent multiplicity (to be defined later) of $P$ is $1 + 2 = 3$. Each such incidence relation in the form of the condition that a vertex lies in a particular face, reduces the dimension of the symmetry type by one. Now, if we take into account the fact that the center of $P$ can be chosen only on the fixed point set of $G$, which is one dimensional and considering also the dilation of $P$ which in each case reduces $\\dim\\langle P\\rangle $ by one, we get\n$$\\dim\\langle P\\rangle \\, = \\, (1 + 2) + (1 + 2) - (1 + 2) - 2 \\, = \\, 1 \\, .$$\n\n\\vspace*{.5cm}\n{ \\begin{figure}[!htb] \\centering\n\\vspace*{-.5cm} \\hspace*{.75cm} \\includegraphics[scale=.25]{figura1.eps}\n\\vspace*{-0,4cm}\\caption{$\\,$}\n\\label{image1}\n\\end{figure}}\n\n\\vspace*{.5cm}\nIn fact, in Figure \\ref{image1} or in any right pyramid with a regular polygon as base, if we denote the height and radius of the base of P by $h$ and $r$, respectively and consider the ratio $\\zeta=\\frac{h}{r}$  the two such pyramids are similar if they have the same ratio $\\zeta$. Therefore we can parameterize the symmetry type of $P$ by $\\zeta$ with $0<\\zeta$. Hence $\\langle P \\rangle$ has the structure of the open interval and $\\dim \\langle P\\rangle  = 1$. Indeed the action of $G(P)$ on $F(P)$ has $\\epsilon=2$ edge orbits. Hence\n$\\dim \\langle P\\rangle =\\epsilon -1 = 1$.  Note that in the right pyramid with regular base the ratio $\\zeta$ is similarity invariant and the symmetry type is a connected 1-manifold with boundary 0-dimensional symmetry types one for regular base and the other a segment (1-polytope). The following figure illustrates this idea.\n\n\\vspace*{.5cm}\n\\begin{center}\n\\begin{tikzpicture}[scale=0.5]\n\\filldraw [black]  (0,0) circle (2pt)\n            (15,0) circle (2pt);\n\\draw (15,0) -- (0,0) node[above] {\\small{$h\\rightarrow 0$}};\n\\draw (15,0) -- (0,0) node[below] {\\footnotesize{regular base}};\n\\draw (0,0) -- (15,0) node[above] {\\small{$r\\rightarrow 0$}};\n\\draw (0,0) -- (15,0) node[below] {\\footnotesize{1-polytope}};\n\\end{tikzpicture}\n\\end{center}\n\n\\begin{center}\n\\vspace*{-1.1cm} {\\footnotesize{right pyramid}}\n\\end{center}\n\n\n\\begin{center}\n{ \\begin{figure}[!htb] \\centering\n\\vspace*{-.55cm} \\includegraphics[scale=.075]{extra.eps}\n\\caption{$\\,$}\n\\label{extra}\n\\end{figure}}\n\\end{center}\n\n\\vspace*{-1cm}\nThe idea of this example can be applied in general to find the dimension of the symmetry type of any polyhedron $P$.\n\nLet us denote by $F_0(P), F_1(P)$ and $F_2(P)$ the set of all vertices, edges and faces of $P$, respectively, with symmetry group $G=G(P)$.\n\n\\begin{defi}\\label{TwoP}\nTwo ordered pairs $(v, F)$   and   $(v', F')$   in   $F_0(P)\\times F_2(P)$   are  called  $G$-independent incidences or simply independent incidences, if and only if, there exists no $g \\in G$   such   that  $g(v)=v'$    and $g(F)=F'$.  By  $\\mu_*(P)$   we  mean  the  number  of  $G$-independent incidences $(v, F)$ where $v \\in F_0(P)$,  $F \\in F_2(P)$  and $v \\in F$. Therefore $G(P)$ acts on the set of all such incident pairs $(v, F)$ with  $\\mu_*(P)$  orbits of independent incidences or ``incident orbits''.\n\\end{defi}\n\nFor example let $P$ be a rhombic dodecahedron (Figure \\ref{f2}) then, $\\mu(P) = 2e = 48$ but  $\\mu_*(P)=2$.\n\\vspace*{-.25cm}\n{ \\begin{figure}[!htb] \\centering\n\\hspace*{.5cm} \\includegraphics[scale=0.15]{figura2.eps}\n\\vspace*{-0,4cm}\\caption{$\\,$}\n\\label{f2}\n\\end{figure}}\n\n\\vspace*{.5cm}\n{\\bf Stratifications}\n\n\n\nFor a transformation group $(M,G)$ the structure of the orbit space $M\/G$ usually is complicated, for example it is not necessarily a manifold.\n\nHowever, when the Lie group $G$ is compact and the manifold $M$ is without boundary it can be shown that they are stratified into smooth manifolds.\n\nWe now describe briefly a stratification of  $\\mathbb{E}^3$ associated with a finite subgroup $G$ of the orthogonal group $O(3)$ that will help us to understand the relationship between the action of $G$ and the number $\\dim \\langle P\\rangle $.\n\n\\begin{defi} ({\\rm Stratification}) Let $X$ be a topological subspace of some Euclidian space  $\\mathbb{E}^3$.\n\nA partition  $\\sum=\\{M_i \\, | \\, i=1,\\ldots,k\\}$ of (pairwise disjoint) subsets of $X$ is called a stratification of $X$ if $\\sum$ satisfies the followings:\n\n\\begin{enumerate}\n\\item Each $M_i,\\,i=1,\\ldots,k$ is a connected smooth submanifold of $\\mathbb{E}^3$, called a \\newline $\\sum$-stratum.\n\n\\item For each $i$, the closure $\\overline{M_i}$ is the union of $M_i$ and the $M_j$'s with lower dimensions than the dimension of $M_i$, that is, the relative closure $X \\cap \\overline{M_i}$ is the union of elements of $\\sum$, one being $M_i$ itself and the others being of dimension less than the $\\dim M_i$.\n\n\\end{enumerate}\n\nThis condition is called {\\it frontier condition}. The dimension $\\dim(X)$ is\n\\vspace*{-0.4cm}\n$$\\max \\{\\dim(M_i) \\mid i=1,\\ldots,k\\} \\ .$$\n\\end{defi}\n\nThe stratification mainly is done by the help of the theorem so called Slice Theorem which is fundamental in studding the structure of the transformation groups (see \\cite{Kaw}, Th. 4.11).\n\n\\vspace*{.25cm}\nLet $X=\\mathbb{E}^3$ and $G$ a compact subgroup of $O(3)$, acting via $\\phi$ on $\\mathbb{E}^3$ as above.\nFor each  $x \\in \\mathbb{E}^3$ let $\\: \\: G_x = \\{g \\in G \\mid g(x)=x\\} \\: \\:$ be the isotropy subgroup of $G$ at $x$ and $F_x=Fix \\left( G_x \\right)$ be the set of all fixed points of $G_x$. Thus\n\\vspace*{-0,2cm}\n$$F_x=\\{y \\in \\mathbb{E}^3 \\mid \\text{ for all } g \\in G_x, \\, g(y)=y\\} \\, .$$\n\n\nSince $x \\in F_x$ and for all $g \\in G$, $g(0)=0$  and for $y,z$ in $F_x$  and  $\\lambda,\\mu\\in \\mathbb{R}$ we have $g(\\lambda y+\\mu z)=\\lambda y+\\mu z$, $F_x$  is a linear subspace of $\\mathbb{E}^3$. Define an equivalence relation $\\sim_{G}$  on $\\mathbb{E}^3$  as follows. Put $x\\sim_{G} y$  if $F_x=F_y$.\n\nNow let $x \\in \\mathbb{E}^3$ and $y \\in F_x$. If $F_x=F_y$ then, $y \\in [x]$, the equivalence class of $x$ in  $\\sim_{G}$.  However $y \\in F_x$  implies that $F_y \\subseteq F_x$, since  $G_x \\subseteq G_y$. Thus $ F_y$  is a linear subspace of  $F_x$. For $z\\in \\mathbb{E}^3-F_x$, we cannot have $z \\sim_{G} x$. Therefore $[x]=\\{y\\in F_x: F_y=F_x\\}$. So $[x]$ is complement in  $F_x$ of finitely many subspaces of $F_x$. Hence the equivalence classes $[x]$, $x \\in \\mathbb{E}^3$ stratify $\\mathbb{E}^3$ with finitely many such strata (orbit types) (see \\cite{Dei}, Th. 5.11). The dimension of $F_x$ denoted by $\\delta(x)$ is called the \\emph{degree of freedom} of $x$.\n\nFor example, let $G$ be a group generated by rotation matrix\n$$A=\\left(\n    \\begin{array}{ccc}\n      \\cos(\\theta) & -\\sin(\\theta) & 0 \\\\\n      \\sin(\\theta)& \\cos(\\theta) & 0 \\\\\n      0 & 0 & 1 \\\\\n    \\end{array}\n  \\right),\\,\\,\\,\\,\\,\\,\\,\\theta=\\frac{2\\pi}{n},\n$$\n\n\\noindent about $z$-axis through $\\theta$.\n\nThen, there are just four strata under this group action on $\\mathbb{E}^3$, namely the $0$-stratum $\\{0\\}$, the open rays $x=y=0, z>0$ and $x=y=0, z<0$, and the complement of the $z$-axis.\n\nNow let $P$ be a polyhedron with $G=G(P)$ some finite subgroup of $\\mathcal{O}(3)$ and $Q\\in \\langle P \\rangle$ as above.\nUnder the restriction imposed on $Q$ within the symmetry type of $P$, each vertex of $Q$ may be moved along a line, or within a plane, or in any direction in  $\\mathbb{E}^3$ near (without changing the symmetry type) its initial position in $P$ itself, having one, two or three degrees of freedom. Likewise, each face $F$ of $Q$ may have one, two or three degrees of freedom close to the corresponding face of $P$, according as F intersects a 1-stratum in an interior point of $F$ (necessarily at right angles), or intersects a 2-stratum in the interior of $F$ (again at right angles), or neither of these.\n\nConsidering the action of $G=G(P)$ on $F(P)$, let\n$$\n\\overline{v} \\, = \\, G(v) \\, = \\, \\left\\{ g(v) \\, \\mid \\, g\\in G \\right\\} \\: , \\: \\: \\overline{F} \\, = \\, G(F) \\, = \\, \\left\\{ g(F) \\, \\mid \\, g\\in G \\right\\} \\, , \\, \\,\\, \\text{ and }\n$$\n\\vspace*{-.6cm}\n$$\nF_0(P)\/ G \\, = \\, \\left\\{ \\overline{v} \\, \\mid \\, v\\in F_0(P) \\right\\} \\: , \\: \\: F_2(P)\/ G \\, = \\, \\left\\{ \\overline{F} \\, \\mid \\, F\\in F_2(P) \\right\\}\n$$\n\n\\noindent be the collection of orbits under the action of $G$ on face lattice $F(P)$.\n\nDefine $\\mathcal{M}_P$ to be the set of all pairs $(v,F)\\in F_0(P) \\times F_2(P)$ for which $v\\in F$ and\n\\vspace*{-.2cm}\n$$\n\\mathcal{M}_P\/ G \\, = \\, \\left\\{ \\left( g(v), g(F) \\right) \\, \\mid \\, g\\in G \\text{ and } (v,F)\\in \\mathcal{M}_P \\right\\} \\, .\n$$\n\nClearly $\\mu (P)$ and $\\mu _{*}(P)$ are the cardinalities of $\\mathcal{M}_P$ and $\\mathcal{M}_P\/ G$, respectively.\n\n\\vspace*{.2cm}\nWe observe that if $v,u\\in F_0(P)$ and $\\overline{v}=\\overline{u}$ then,\n\\vspace*{-.2cm}\n$$\n\\dim(v) \\, = \\, \\dim(u) \\, .\n$$\n\n\\vspace*{-.2cm}\nTherefore we can define the fixed dimensions $\\delta (\\xi)$, $\\xi \\in F_0(P)\/ G$, $\\delta (\\xi)=\\dim(v)$, the degree of freedom of $v$ for an arbitrary $\\overline{v}\\in \\xi$.\nThe same holds for $\\delta (\\zeta)$, $\\zeta \\in F_2(P)\/ G$.\n\nOur aim is to count the number of the vertices and the faces with dimensions $k=1,2,3$ and then, by subtracting the independent incidences $\\mu _{*}(P)$, express the $\\dim \\langle P \\rangle$ in terms of edge orbits alone.\n\n\n\n\n\n\n\\section{Fundamental regions and main theorem}\n\nIn this Section we consider finite subgroups of isometries which are generated by reflections namely $[q], [2, q], [3, 3], [3, 4]$ and $[3,5]$ (Table 2).\n\n\\vspace*{.2cm}\nThe finite subgroups of $\\mathbb{E}(3)$ which are generated by reflections in the plane, are given in the following table. They are called reflection groups, for obvious reason.\n\\begin{center}\n\\vspace*{-.2cm} {\\small{\\begin{tabular}{|c|l|c|}\n  \\hline\n \n  Symbol & description & order \\\\\n  \\hline\n                  & $q=1$: One plane of reflection; $q\\geq 2$: $q$ equally inclined planes  &   \\\\\n    $[q],q\\geq 1$ & of reflection passing through a $q$-fold axis of rotation, dihedral \\hspace{11cm}& $2q$ \\\\\n                  & reflection group. &   \\\\\n \\hline\n                  & $q$ equally inclined planes of reflection passing through a $q$-fold &   \\\\\n    $[2,q]$       & axis of rotation and reflection in a equatorial plane.   & 4q \\\\\n                  & $q$ 2-fold axes of rotation. The group of $q$-prism.&   \\\\\n\\hline\n     $[3,3]$      & Four 3-fold and three 2-fold axes. Six planes of reflection. &  \\\\\n                  & Symmetry group of the regular tetrahedron. & 24  \\\\\n\\hline\n                  & Three 4-fold and four 3-fold and six 2-fold axes of rotation. &   \\\\\n     $[3,4]$      & Nine planes of reflection. Symmetry group of the cube. & 48 \\\\\n\\hline\n                  & Six 5-fold, ten 3-fold and fifteen 2-fold axes of rotation.  &   \\\\\n     $[3,5]$      & Fifteen planes of reflection. Symmetry group of the icosahedron. & 120 \\\\\n  \\hline\n\\end{tabular}}}\n\\end{center}\n\n\\begin{center} Table 2: Reflection groups \\end{center}\n\nIt is well known that the \\emph{fundamental region} $\\Delta$ for the action of $[3, 3], [3, 4], [3, 5]$ and $[2, q]$ on the sphere $S^2$  are spherical triangles [1]. For $[q]$ the dihedral reflection group generated by two reflections, the fundamental region is a ``lune'' of angle $\\frac{\\pi}{q}$.\nWe may use the fundamental region of a reflection group to construct a stratification of $\\mathbb{E}^3$. For instance consider the tetrahedron $OABC$ (or its spherical projection) as a fundamental region of [3, 4] in Figure \\ref{f3} (\\cite{Rob}, p. 81).\nWe take the origin $O$ as a $0{\\text{-stratum}}$. By removing the origin from the rays $OA$, $OB$ and $OC$ we get three $1{\\text{-strata}}$, the interiors of the region $AOB$, $AOC$ and $BOC$ are $0{\\text{-strata}}$.\n\nFinally, the interior points of $\\mathbb{E}^3$  bounded by sectors $AOB$, $AOC$ and $BOC$ is $3{\\text{-strata}}$. By transferring these strata under the action of [3, 4] we obtain the required stratification of $\\mathbb{E}^3$.\n\nFor other reflection groups a stratification of $\\mathbb{E}^3$ is constructed in analogues fashion.\nReturning to our main problem, we now consider the following notion. Suppose that a finite reflection group $G$ in $\\mathcal{O}(3)$ has its fundamental region a spherical triangle $\\Delta$ and, let $P$ be a polyhedron with $G(P)=G$.\n{ \\begin{figure}[!htb] \\centering\n\\hspace*{.5cm} \\includegraphics[scale=.2]{figura3.eps}\n\\vspace*{-0,4cm}\\caption{$\\,$}\n\\label{f3}\n\\end{figure}}\n\nWe denote by  $\\Delta_p$ that portion of the surface of $P$ (namely those vertices, edges and subpolygonal faces) which lie within $\\Delta$, and call $\\Delta_p$ a \\emph{basic region} of $P$ (Figure \\ref{f4}).\n{ \\begin{figure}[!htb] \\centering\n\\hspace*{.75cm} \\includegraphics[scale=.225]{figura4.eps}\n\\vspace*{-0,4cm}\\caption{$\\,$}\n\\label{f4}\n\\end{figure}}\n\n\\vspace*{-0.1cm}\nHence $\\Delta_p$ is a simple closed planar polygonal region. If $n_{\\Delta_p}, e_{\\Delta_p}$ and $f_{\\Delta_p}$ denote the total number of distinct vertices, edges and subpolygonal faces of  $\\Delta_p$ respectively then, from Euler's formula by stereographic projection we get\n\\vspace*{-0,2cm}\n$$n_{\\Delta_p} - e_{\\Delta_p} + f_{\\Delta_p}=1 \\, .$$\n\nAs illustrated example, let $P$ be truncated cuboctahedron (Figure \\ref{f4}) with symmetry group $G(P)=[3,4]$ of cube, with order 48. Thus the basic region $\\Delta_p$ has $n_{\\Delta_p}=7$, $e_{\\Delta_p}=9$, $f_{\\Delta_p}=3$, and $n_{\\Delta_p} - e_{\\Delta_p} + f_{\\Delta_p}=7-9+3=1$.\n\n\\vspace*{0.1cm}\nNow having our necessary tools, we are in the position to state and prove our main theorem in this Section.\n\n\\begin{teo}Let $G$ be a finite reflection group in $\\mathbb{E}(3)$ and $P$ a polyhedron with $G(P)=G$. Then, $\\dim \\langle P\\rangle= \\epsilon-1$ , where $\\epsilon$ is the number of edge orbits of the action of $G$ on the set of edges of $P$.\n\\end{teo}\n\nFirst we prove the following lemma.\n\n\\begin{lema} Assuming the hypothesis of the theorem, let  $\\Delta_p$  be a basic region for $P$ such that the corners of fundamental region of $\\Delta$ of $G$ are vertices of $P$ (see Figure \\ref{f5}). Then, the number of incident pairs of vertices and faces of  $\\Delta_p$, the multiplicity $\\mu(\\Delta_p)$  of  $\\Delta_p$  is given by  $\\mu(\\Delta_p)=2e-\\beta$ where $e$ is the total number of edges of   $\\Delta_p$ and $\\beta$ the number of vertices on the boundary of the fundamental region.\n\\end{lema}\n{\\bf Proof:} By adjoining an extra face, say $K$, to  $\\Delta_p$, namely the complement of   $\\Delta_p$ itself with respect to the sphere we get a map $M_P$ on sphere. But the number of edges (and vertices) of $M_P$ is equal to the number of edges (and vertices) of  $\\Delta_p$. Hence  $\\mu(\\Delta_p)=2e$. Since there are $\\beta$ vertices on boundary of  $\\Delta$, with respect to that extra face $K$, we have  $\\mu(\\Delta_p)=\\mu(P)-\\beta=2e - \\beta$.\n\\vspace*{-0.3cm}\n{ \\begin{figure}[!htb] \\centering\n\\hspace*{.75cm} \\includegraphics[scale=.15]{figura5.eps}\n\\vspace*{-0,1cm}\\caption{$\\,$}\n\\label{f5}\n\\end{figure}}\n\n\\vspace*{-0,5cm}\n$\\:$ \\hfill$\\square$\n\n\\vspace*{-0,1cm}\n \\begin{obs}\n  {\\rm Let $P$ be a polyhedron and $v$ a vertex of $P$. According to stratification of  $\\mathbb{E}^3$ with reflection group $G(P)$ of $P$, $\\delta(v)$, the degree of freedom of $v$,  is one, two or three if $v$ is on a corner or side or within the interior of   $\\Delta_p$ respectively. Similarly if a face $F$ of $P$ has as its interior point a corner of $\\Delta$ then, $\\delta(F)=1$. If $F$ is orthogonal to a side of  $\\Delta_p$  or lies inside  $\\Delta_p$  then, $\\delta(F)=2 $ or 3, respectively.}\n \\end{obs}\n\n \\noindent {\\bf Proof of theorem:} Let  $\\eta(1),\\eta(2)$ and $\\eta(3)$ be the number of vertices of $\\Delta_p$   and   $\\phi(1),\\phi(2)$ and $\\phi(3)$ the number of faces of $\\Delta_p$ with one, two and three degrees of freedom respectively.\n\n First we assume that $P$ has no face $F$ with $\\delta(F)$ equal to one or two. Then, $\\mu_*(P)=\\mu(\\Delta_p)=2e-\\beta$. After factoring out the effect of dilation we get\n $$\n \\begin{array}{rcl}\n   \\dim\\langle P \\rangle & = & 1\\eta(1)+2\\eta(2)+3\\eta(3)+3\\phi(3)-\\mu(\\Delta_p)-1 \\\\[8pt]\n     & = & 3(n_{\\Delta_p}+\\phi_{\\Delta_p})-2e_{\\Delta_p}-4 \\, ,\n \\end{array}$$\n\\noindent since $\\eta(1)=3$.\n\nBut $n_{\\Delta_p}+\\phi_{\\Delta_p}=e_{\\Delta_p}+1$. Hence,\n$$\\dim\\langle P \\rangle=3(e_{\\Delta_p}+1)-2e_{\\Delta_p}-4= e_{\\Delta_p}-1 \\, .$$\n\\noindent Now clearly the number of edges of  $\\Delta_p$  is exactly the number of edge orbits of $P$. Therefore the theorem follows in this case.\n\nNext suppose $P$ has one face $F$ with $\\delta(F)=1$. This means that there is a face $F$ such that one corner say $v$ of $\\Delta_p$   is an interior point of $F$.\n\nLet $P'$ be a polyhedron which we get, by changing ``fake'' edges in $\\Delta_p$ into real ones (see Figure \\ref{f6}). This is done as follows.\n\nWe remove the constraint that the plane of  $F \\cap \\Delta_p$  is perpendicular to the ray $ov$. The edges of  $F \\cap \\Delta_p$  that lie in the boundary of  $\\Delta_p$  are also edges of $P'$ where $P'$ has a basic region  $\\Delta_{p'}$ , say, with the same combinatorial structure as $\\Delta_p$, and with vertices arbitrarily close to those of $\\Delta_p$.\nSince we have substituted a face with one degree of freedom by a face with three degrees of freedom and since the new vertex and its incidence cancel each other and hence do not effect our calculation for  $ \\dim\\langle P \\rangle $, we have $ \\dim\\langle P' \\rangle = \\dim\\langle P \\rangle  + 2$. But $\\dim\\langle P' \\rangle = \\epsilon'-1$ and $\\epsilon'=\\epsilon+2$ with obvious notations. Hence\n$$\\dim\\langle P \\rangle = \\dim\\langle P' \\rangle -2= \\epsilon'-1-2=\\epsilon-1 \\, .$$\n\nThe process will continue if $P$ has two or three (on other corners of $\\Delta$) faces of degree one.\n{ \\begin{figure}[!htb] \\centering\n\\vspace*{-.5cm} \\hspace*{.25cm} \\includegraphics[scale=.175]{figura6.eps}\n\\vspace*{-0,4cm}\\caption{\\small{\\emph{Broken lines represent ``fake'' edges and ``o'' a fake vertex.}}}\n\\label{f6}\n\\end{figure}}\n\nHere we remark that, in the process of changing ``fake'' edges in $\\Delta_p$  into real ones, since the transforms of $\\Delta_{p'}$ under corresponding group action is a polyhedral graph (planar and 3-connected), by the Theorem of Steinitz \\cite{Stei}, there exists a polyhedron P' which geometrically realizes $\\Delta_{P'}$.\n\n\\vspace*{.25cm}\nFinally, suppose $P$ has a face $F$ with $\\delta(F)=2$ (Figure \\ref{f7}). We construct $P'$ by adjoining the fake edge to $\\Delta_p$. Then, $\\dim\\langle P' \\rangle $ differs by one from\n$\\dim\\langle P \\rangle$ for replacement of a face with two degrees of freedom, by a face of three degrees of freedom. Hence\n$\\dim\\langle P' \\rangle = \\dim\\langle P \\rangle +1$. Because of $\\dim\\langle P' \\rangle =\\epsilon'-1$ and $\\epsilon=\\epsilon'-1$, we have\n\\vspace*{-0,3cm}\n$$\\dim\\langle P \\rangle +1=\\epsilon'-1=\\epsilon\\,\\,\\,\\,\\text{ and }\\,\\,\\,\\, \\dim\\langle P \\rangle =\\epsilon-1 \\, .$$\n\n{ \\begin{figure}[!htb] \\centering\n\\hspace*{.25cm} \\includegraphics[scale=.175]{figura7.eps}\n\\vspace*{-0,4cm}\\caption{$\\,$}\n\\label{f7}\n\\end{figure}}\n\n\\newpage\nNow this inductive process can be continued if $P$ has any number of faces with two degrees of freedom. In each step of construction, $\\dim\\langle P \\rangle$ and $\\epsilon$ each increase by one, while the operation leaves every other quantity in our calculation fixed. For the case of the group $[q]$ where $\\Delta$ is a ``lune'', the proof proceeds in similar way and is omitted here to avoid repetition. The proof of the theorem now is complete. \\hfill$\\square$\n\n\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}
+{"text":"\\section{Introduction}\nIn Sun-as-a-star helioseismology, it is common practice to fix the amplitude ratios between the $m$-components of the $l = 2$ and 3 multiplets (the so-called $m$-amplitude ratio) during the peak-fitting procedure when estimating the p-mode characteristics (e.g., Salabert et al. [1]), while the amplitudes of the $l=1$, 2, and 3 modes relative to the $l=0$ modes are left free (the so-called mode visibility). However, in asteroseismology, the mode visibilities are fixed to theoretical values due to lower signal-to-noise ratio (SNR) and shorter time series, and the $m$-amplitude ratios are expressed as a function of the inclination of the rotation axis only (Appourchaux et al. [2]; Garc\\'\\i a et al. [3]). In both cases, they are supposed to not depend on the star magnetic activity. However, in the near future, this situation could change when stellar activity cycles will be measured in asteroseismic targets (e.g., Garc\\'\\i a et al. [4]). After several years of observations collected by the Kepler mission, the SNR will be high enough to measure these parameters in a wide range of solar-like stars in the HR diagram at different evolution stages (Bedding et al. [5]; Chaplin et al. [6]). Moreover, simultaneous observations from SONG (Doppler velocity) and Kepler (intensity) will be extremely useful to better understand stellar atmospheres. \nAlthough, variations with the height in the solar atmosphere at which the measurements are obtained have been observed in the intensity VIRGO\/SPM data at the beginning of the SoHO mission (Fr{\\\"o}hlich et al. [7]), it has never been verified if these values change in the radial velocity GOLF measurements between the blue- and red-wing observing periods. \n\n\\section{Observations and analysis}\nWe used observations collected by the space-based instruments Global Oscillations at Low Frequency (GOLF) and Variability of Solar Irradiance and Gravity Oscillations (VIRGO) onboard the {\\it Solar and Heliospheric Observatory} (SoHO) spacecraft. GOLF (Gabriel et al. [8]) measures the Doppler velocity at different heights in the solar atmosphere depending on the wing -- Blue or Red -- of the sodium doublet  -- D1 and D2 -- in which the observations were performed (Garc\\'\\i a et al. [9]). VIRGO (Fr{\\\"o}hlich et al. [10]) is composed of three Sun photometers (SPM) at 402~nm (blue), 500~nm (green) and 862~nm (red). A total of 5021 days of GOLF and VIRGO observations starting on 1996 April 11 and ending on 2010 January 8 were analyzed, with respective duty cycles of  95.4\\% and 94.7\\%. The power spectra of the time series were fitted to extract the mode parameters (Salabert et al. [11]) using a standard likelihood maximization function (power spectrum with a $\\chi^2$ with 2 d.o.f. statistics). Each mode component was parameterized using an asymmetric Lorentzian profile. Since SoHO observes the Sun equatorwards, only the $l+|m|$ even components are visible in Sun-as-a-star observations of GOLF and VIRGO. In order to obtain observational estimates of the $m$-amplitude ratios, the $m = \\pm2$ and $m = 0$ components of the $l = 2$ multiplet, and the $m = \\pm3$ and $m = \\pm1$ components of the $l = 3$ multiplet were fitted using independent amplitudes, assuming that components with opposite azimuthal order $m$ have the same amplitudes ($H_{l,n,-m}=H_{l,n,+m}$). Note that the blue and red periods of GOLF were also analyzed separately, as well as the mean power spectrum of the three VIRGO  SPMs.\n\n\n\n\\section{Mode visibilities and $m$-amplitude ratios}\nThe amplitude of a given multiplet $(l,n)$ is defined as the sum of the amplitudes of its $m$-components, as $H_{l,n} = \\sum_{m=-l}^{m=+l} H_{l,n,m}$. \nThen, the visibilities of the $l=1$, 2, and 3 modes relative to the $l=0$ mode are respectively defined as the ratios $H_{l=1,n} \/ H_{l=0,n}$, $H_{l=2,n-1} \/ H_{l=0,n}$, and $H_{l=3,n-1}$. The left panel of Fig.~\\ref{fig:visi} shows these visibilities for both radial velocity (GOLF) and intensity (VIRGO) measurements as a function of frequency. These raw mode visibilities present a variation with frequency -- especially for the $l=1$ mode -- that is due to the large variation of the mode amplitudes with frequency, even over half a large frequency separation.\nThus, the visibilities are biased and in order to correct them we interpolated (using a spline interpolation) the amplitudes of the $l=1$, 2, and 3 modes to the frequencies of the $l=0$ mode (right panel of Fig.~\\ref{fig:visi}). \nFigure~\\ref{fig:meanvisi} shows the mode visibilities averaged over frequency for both GOLF and VIRGO observations as a function of $l$  (see Table~\\ref{tab:visigolf}). The amplitude ratios between the $m$-components of the $l = 2$ and $l = 3$ multiplets, defined as $H_{l=2,m=0}\/H_{l=2,m=\\pm2}$ and $H_{l=3,m=\\pm1}\/H_{l=3,m=\\pm3}$ respectively, are represented on Fig.~\\ref{fig:mratio} in the case of the radial velocity GOLF measurements and are also reported in Table \\ref{tab:mratiogolf}.\n\n\n\n\\begin{figure*} \n\\begin{center} \n\\includegraphics[scale=0.31]{GOLF_visi_nocorrection_forpaper.eps}  \\includegraphics[scale=0.31]{GOLF_visi_withcorrection_forpaper.eps}\n\\end{center} \n\\caption{\\label{fig:visi} Raw (left) and corrected (right) mode visibilities of $l = 1$ ($\\opencircle$), $l=2$ ($\\opensquare$), and $l=3$ ($\\opendiamond$) relative to $l = 0$ as a function of frequency in GOLF ($\\full$) and VIRGO ($\\dashed$) observations.} \n\\end{figure*} \n\n\n\\begin{figure}\n\\includegraphics[width=2.5in]{visib_vs_degree_forpaper.eps}\\hspace{0.5pc}%\n\\begin{minipage}[b]{22pc}\\caption{\\label{fig:meanvisi} Mode visibilities as a function of angular degree $l$ in GOLF ($\\full$, $\\fullcircle$) and VIRGO ($\\dashed$, $\\opensquare$) measurements. For comparison, the mode visibilities of the CoRoT target HD49385 measured by Deheuvels et al. [12] are also represented ($\\dotted$, $\\opentriangledown$).}\n\\end{minipage}\n\\end{figure}\n\n\n\\begin{figure}\n\\includegraphics[scale=0.25]{GOLF_mratio_forpaper.eps}\\hspace{0.5pc}%\n\\begin{minipage}[b]{20.5pc}\\caption{\\label{fig:mratio} $m$-amplitude ratios of the $l = 2$ (top) and $l = 3$ (bottom) modes as a function of frequency in the GOLF measurements.}\n\\end{minipage}\n\\end{figure}\n\n\n\\begin{table}\n\\caption{\\label{tab:visigolf} Mode visibilities in radial velocity GOLF and intensity VIRGO measurements.}\n\\begin{center}\n\\begin{tabular}{lllll}\n\\br\nMode visibility & GOLF & GOLF & GOLF& \\\\\nRadial velocity &         & Blue wing & Red wing&\\\\\n\\mr\n$H_{l=1}\/H_{l=0}$ & 1.69$\\pm$0.04 & 1.60$\\pm$0.05 & 1.85$\\pm$0.06&\\\\\n$H_{l=2}\/H_{l=0}$ & 0.81$\\pm$0.03 & 0.74$\\pm$0.04 & 0.98$\\pm$0.05&\\\\\n$H_{l=3}\/H_{l=0}$ & 0.17$\\pm$0.01 & 0.14$\\pm$0.02 & 0.28$\\pm$0.03&\\\\\n\\br\nMode visibility & VIRGO & VIRGO & VIRGO & VIRGO \\\\\nIntensity        &               & Blue      & Green   & Red \\\\\n\\mr\n$H_{l=1}\/H_{l=0}$ & 1.53$\\pm$0.05 & 1.55$\\pm$0.05 & 1.52$\\pm$0.05 & 1.39$\\pm$0.05\\\\\n$H_{l=2}\/H_{l=0}$ & 0.59$\\pm$0.03 & 0.63$\\pm$0.03 & 0.57$\\pm$0.03 & 0.42$\\pm$0.03\\\\\n$H_{l=3}\/H_{l=0}$ & 0.09$\\pm$0.02 & 0.10$\\pm$0.02 & 0.09$\\pm$0.02 & 0.05$\\pm$0.02\\\\\n\\br\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\n\\begin{table}\n\\caption{\\label{tab:mratiogolf} $m$-amplitude ratios in radial velocity GOLF and intensity VIRGO measurements.}\n\\begin{center}\n\\begin{tabular}{lll}\n\\br\n$m$-amplitude ratio & GOLF & VIRGO\\\\\n\\mr\n$H_{l=2,m=0}\/H_{l=2,m=\\pm2}$ & 0.63$\\pm$0.03 & 0.75$\\pm$0.06\\\\\n$H_{l=3,m=\\pm1}\/H_{l=3,m=\\pm3}$ & 0.40$\\pm$0.02 & 0.63$\\pm$0.06\\\\\n\\br\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\n\n\n\\section{Models}\nWhen the contribution of a solar-disk element to the total flux depends only on its distance to the limb, the mode visibility and the $m$-amplitude ratio are decoupled (e.g. Gizon \\& Solanki [13]; Ballot et al. [14]). For VIRGO observations, this is verified since the contribution depends mainly on the limb-darkening. However, for GOLF, this is no more the case and we have performed complete computation taking into account the instrumental response, which differ for the blue and red wings. Results of these computations are listed in Table~\\ref{tab:visimodel} . The limb-darkening law of Neckel \\& Labs [15] has been used. Indeed, visibility values for GOLF vary with frequency by a few percents due to the horizontal motions of modes that increase at low frequency.\nIn general, these predictions agree with the observations. There is nevertheless some shortcomings: (i) even if the trend is correct, the difference between blue and red wings for GOLF is larger than expected; (ii) the visibility of the $l = 3$ modes in VIRGO are sensitively higher than expected. That could be explained by stronger effects of limb-darkening.\n\n\n\\begin{table}\n\\caption{\\label{tab:visimodel} Modeled visibilities and $m$-amplitude ratios in intensity VIRGO and radial velocity GOLF measurements.}\n\\begin{center}\n\\begin{tabular}{llll}\n\\br\nMode visibility \\& & VIRGO & GOLF            & GOLF\\\\\n $m$-amplitude ratio       &               & Blue  wing   &   Red wing\\\\\n\\mr\n$H_{l=1}\/H_{l=0}$ & 1.51 & 1.84 & 1.86\\\\\n$H_{l=2}\/H_{l=0}$ & 0.53 & 1.09 & 1.14\\\\\n$H_{l=3}\/H_{l=0}$ & 0.025 & 0.27 & 0.31\\\\\n\\mr\n$H_{l=2,m=0}\/H_{l=2,m=\\pm2}$ & 0.67 & 0.59 & 0.58\\\\\n$H_{l=3,m=\\pm1}\/H_{l=3,m=\\pm3}$ & 0.60 & 0.43 & 0.40\\\\\n\\br\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\n\\ack\nThe authors want to thank Catherine Renaud and Antonio Jim\\'enez for the calibration and preparation of the GOLF and VIRGO datasets. The GOLF and VIRGO instruments onboard SoHO are a cooperative effort of many individuals, to whom we are indebted. SoHO is a project of international collaboration between ESA and NASA. DS acknowledges the support of the grant PNAyA2007-62650 from the Spanish National Research Plan. This work has been partially supported by the CNES\/GOLF grant at the SAp\/CEA-Saclay.\n\n\\section*{References}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}