diff --git "a/data_all_eng_slimpj/shuffled/split2/finalzzixph" "b/data_all_eng_slimpj/shuffled/split2/finalzzixph" new file mode 100644--- /dev/null +++ "b/data_all_eng_slimpj/shuffled/split2/finalzzixph" @@ -0,0 +1,5 @@ +{"text":"\\section{Introduction}\n\nThe quantum field theory (QFT) on the expanding portion of the de Sitter space-time can be studied extensively since the equations of the principal free fields can be solved analytically in various local charts. In the co-moving charts with Cartesian space coordinates there are plane waves solutions which are eigenfunctions of the momentum operator, as in special relativity, but with more complicated time modulation functions. Moreover, in this geometry one cannot use the Hamiltonian operator for separating the frequencies since this does not commute with the momentum operator. Thus the main task here is just the criterion of separating the frequencies, defining the particle and antiparticle modes and, implicitly, the vacuum \\cite{BD}. \n\nThe principal method used so far focused mainly on the asymptotic states which are somewhat similar to the usual Minkowskian particle or antiparticle states, as in the case of the adiabatic vacuum of the Bunch-Davies type \\cite{BuD} largely used in applications. The problem arising in this vacuum is that one cannot reach the rest limit, for vanishing momentum, since this limit is undefined for the corresponding scalar mode functions or Dirac spinors. This problem was considered sporadically by few authors which found similar results \\cite{nach,born,CPasc,CD}. \n\nUnder such circumstances, we proposed recently a method of separating the frequencies in rest frames defining thus the rest frame vacuum (r.f.v.) of the massive Klein-Gordon \\cite{CrfvKG}, Dirac \\cite{CrfvD} and Proca \\cite{CrfvP} fields. What is new here is that the bosons can have either a tardyionic behavior or even a tachyonic one if the mass is less than a given limit which depends on the type of coupling (minimal or conformal). Fortunately, the tachyonic modes are eliminated in a natural manner since all their mode functions have null norms \\cite{CrfvKG}. In contrast, the Dirac field of any non-vanishing mass survives in this vacuum \\cite{CrfvD}. In other respects, we must specify the r.f.v. can be defined only for massive particles since the massless ones do not have rest frames. This is not an impediment since the massless fields of physical interest, namely the Maxwell and neutrino ones, have conformally covariant field equations whose solutions in the co-moving de Sitter chart with conformal time can be taken over from special relativity \\cite{Max,CD1}. \n\nTechnically speaking, for defining the r.f.v. we introduced suitable phases depending on momentum for assuring the correct limits of the mode functions in rest frames \\cite{CrfvKG,CrfvD,CrfvP}. Unfortunately, these phases are not enough for defining the other important limit, namely the flat one, when the de Sitter Hubble constant tends to zero. In general, this limit is undefined because of some singularities arising in the phases of the mode functions defined in the adiabatic vacuum. For removing them a regularization procedure was applied by adding the convenient phase factors which, in general, depend on momentum \\cite{nach,born,CPasc,CD}. \n\nSince this problem was not yet considered for the recently defined r.f.v., the aim of this paper is to study the flat limits of the Klein-Gordon and Dirac fundamental solution in this vacuum, deriving the regularized phases which guarantee that these limits are well-defined. In this manner, the rest and flat limits determine completely the form of the mode functions or spinors of the de Sitter QFT . This is important since there are quantities whose expressions are strongly dependent on the momentum-dependent phase factors as, for example, the one-particle Hamiltonian operator \\cite{nach,CD1,Cpop}. We prove that the phase factors derived here determine the correct Minkowskian flat limit of this operator. \n\nWe must specify that in what concerns the regularized phases we recover previous results \\cite{nach,born,CPasc,CD} but the complete expressions of the scalar mode functions or Dirac spinors are presented here for the first time. Moreover, the approximations we obtain here are also new results that can be used in concrete calculations.\n\nWe start presenting in the next section the plane wave fundamental solutions of the Klein-Gordon and Dirac fields in the adiabatic and rest frame vacua pointing out how the last one solves the rest limits. The third section is devoted to the flat limits which are derived by using a new uniform asymptotic expansion we propose here based on numerical arguments. This enables us to derive the regularized phases which assure the flat limits of the fundamental solutions. Thus we obtain for the first time the complete expressions of the scalar mode functions and Dirac spinors in the r.f.v. as well as useful approximations of them. Moreover, we show that in this approach the flat limit of the one-particle Hamiltonian operator is just the corresponding operator of the Minkowskian QFT. Finally we present our concluding remarks. \n\n\\section{Free fields in adiabatic and rest frame vacua}\n\nThe $(1+3)$-dimensional de Sitter expanding universe, $M$, is the expanding portion of the de Sitter space-time where we may choose the co-moving local charts $\\{t,{\\bf x}\\}$ whose coordinates $x^{\\kappa}$ (labelled by the natural indices $\\kappa,\\nu,...=0,1,2,3 $) are the proper (or cosmic) time, $t$, and the Cartesian space coordinates $x^i$ ($i,j,k...=1,2,3$) for which we may use the vector notation ${\\bf x}=(x^1,x^2,x^3)$. The geometry is given by the scale factor $a(t)=\\exp(\\omega t)$ depending on the Hubble de Sitter parameter denoted here by $\\omega$. Another useful chart is that of the conformal time,\n\\begin{equation}\\label{tct}\nt_c=\\int \\frac{dt}{a(t)}=-\\frac{1}{\\omega}e^{-\\omega t}~\\to~ a(t_c)=a[t(t_c)]=-\\frac{1}{\\omega t_c}\\,,\n\\end{equation}\nand the same Cartesian space coordinates, denoted by $\\{t_c,\\vec{x}\\}$. The line elements of these charts are, \n\\begin{eqnarray}\nds^2=g_{\\kappa\\nu}(x)dx^{\\kappa}dx^{\\nu}&=&dt^2-e^{2\\omega t} d{\\vec x}\\cdot d{\\vec x}\\nonumber\\\\\n&=&\\frac{1}{(\\omega t)^2}(dt_c^2-d{\\vec x}\\cdot d{\\vec x})\\,,\n\\end{eqnarray}\nNote that on the expanding portion we have $t_c\\in (-\\infty,0]$ and $t\\in(-\\infty,\\infty)$.\n\nFor writing down the Dirac equation we chose the diagonal tetrad gauge in which the vector fields $e_{\\hat\\alpha}=e_{\\hat\\alpha}^{\\kappa}\\partial_{\\kappa}$ defining the local orthogonal frames, and the 1-forms $\\omega^{\\hat\\alpha}=\\hat e_{\\kappa}^{\\hat\\alpha}dx^{\\kappa}$ of the dual coframes (labeled by the local indices, $\\hat\\kappa,\\hat\\nu,...=0,1,2,3$) are defined as \n\\begin{eqnarray}\n&e_0=\\partial_t=\\frac{1}{a(t_c)}\\,\\partial_{t_c}\\,,\\qquad & \\omega^0=dt=a(t_c)dt_c\\,,\\label{tetrad}\\label{T1} \\\\\n&~~~e_i=\\frac{1}{a(t)}\\,\\partial_i=\\frac{1}{a(t_c)}\\,\\partial_i\\,, \\qquad & \\omega^i=a(t)dx^i=a(t_c)dx^i\\,,\\label{T2}\n\\end{eqnarray}\nin order to preserve the global $SO(3)$ symmetry allowing us to use systematically the $SO(3)$ vectors. We remind the reader that the metric tensor of $M$ can be expressed now as $g_{\\kappa\\nu}=\\eta_{\\hat\\alpha\\hat\\beta}\\hat e^{\\hat\\alpha}_{\\kappa}\\hat e^{\\hat\\beta}_{\\nu}$ where $\\eta={\\rm diag}(1,-1,-1,-1)$ is the Minkowski metric. \n\n\n\\subsection{Klein-Gordon field}\n\nIn the chart $\\{t,\\vec{x}\\}$ the scalar field $\\Phi : M\\to {\\Bbb C}$ of mass $m$, minimally coupled to the de Sitter gravity, satisfies the Klein-Gordon equation,\n\\begin{equation}\\label{KG1}\n\\left( \\partial_t^2-e^{-2\\omega t}\\Delta +3\\omega\n\\partial_t+m^2\\right)\\Phi(x)=0\\,,\n\\end{equation}\nwhose general solutions can be expanded as\n\\begin{equation}\\label{field1}\n\\Phi(x)=\\Phi^{(+)}(x)+\\Phi^{(-)}(x)=\\int d^3p \\left[f_{\\vec{p}}(x)a(\\vec{p})+f_{\\vec{p}}^*(x)b^{\\dagger}(\\vec{p})\\right] \\,,\n\\end{equation}\nin terms of field operators, $a(\\vec{p})$ and $b(\\vec{p})$, and fundamental solutions, $f_{\\vec{p}}$ and $f_{\\vec{p}}^*(x)$, of positive and respectively negative frequencies. These solutions must satisfy the orthonormalization relations\n\\begin{eqnarray}\n\\langle f_{\\vec{p}},f_{\\vec{p}'}\\rangle_{KG}=-\\langle f_{\\vec{p}}^*,f_{\\vec{\np}'}^*\\rangle_{KG}&=&\\delta^3(\\vec{p}-\\vec{p}')\\,,\\label{ff}\\\\\n\\langle f_{\\vec{p}},f_{\\vec{p}'}^*\\rangle_{KG}&=&0\\,,\n\\end{eqnarray}\nand a completeness condition with respect to the relativistic scalar product \\cite{BD}\n\\begin{equation}\\label{SP}\n\\langle f,f'\\rangle_{KG}=i\\int d^3x\\, a(t)^3\\, f^*(x)\n\\stackrel{\\leftrightarrow}{\\partial_{t}} f'(x)\\,.\n\\end{equation}\nThe fundamental mode functions can be expressed as \n\\begin{equation}\\label{fp}\nf_{\\vec{p}}(t,\\vec{x})=\\frac{e^{i \\vec{x}\\cdot \\vec{p}}}{[2\\pi a(t)]^{\\frac{3}{2}}}{\\cal F}_p(t)\\,,\n\\end{equation}\nin terms of the time modulation functions ${\\cal F}_p: D_t\\to {\\Bbb C}$ which depend on $p=|\\vec{p}|$ satisfying the equation\n\\begin{equation}\\label{KGred}\n\\left[\\frac{d^2}{dt^2}+\\frac{p^2}{a(t)^2}+m^2-\\frac{9}{4}\\,\\omega^2\\right] {\\cal F}_p(t)=0\\,.\n\\end{equation}\nand the normalization condition \n\\begin{equation}\\label{normF}\n\\left({\\cal F}_p, {\\cal F}_p\\right)\\equiv i\\,{\\cal F}_p^*(t)\\stackrel{\\leftrightarrow}{\\partial}_{t}{\\cal F}_p(t)=1\\,.\n\\end{equation}\n which guarantees the condition (\\ref{ff}).\n\nThe most general solution of Eq. (\\ref{KGred}) can be derived easily in the chart $\\{t_c,\\vec{x}\\}$ obtaining \\cite{Cpop,CrfvKG}\n\\begin{equation}\\label{fdS}\n{\\cal F}_p(t_c)=\\ c_1\\phi_p(t_c) + c_2\\phi_p^*(t_c)\\,,\\quad \\phi_p(t_c)=\\frac{1}{\\sqrt{\\pi\\omega}}\\,K_{\\nu}(ipt_c)\\,,\n\\end{equation}\nwhere $K$ is the modified Bessel function of the index\n\\begin{equation}\\label{ndS}\n\\nu=\\left\\{\\begin{array}{lll}\n\\sqrt{\\frac{9}{4}-\\mu^2}&{\\rm for} & \\mu<\\frac{3}{2}\\\\\ni\\kappa\\,,\\quad \\kappa= \\sqrt{\\mu^2-\\frac{9}{4}}&{\\rm for} & \\mu>\\frac{3}{2}\n\\end{array} \\right. \\,, \\quad \\mu=\\frac{m}{\\omega}\\,.\n\\end{equation}\nThe particular solution $\\phi_p(t_c)$ is normalized, satisfying $(\\phi_p,\\phi_p)=1$, such that the condition (\\ref{normF}) is fulfilled only if we take\n\\begin{equation}\\label{norC}\n\\left|c_1\\right|^2-\\left|c_2\\right|^2=1\\,.\n\\end{equation}\nThus we remain with an undetermined integration constant that may be fixed by giving a criterion of frequencies separation setting thus the vacuum. \n\nThe most popular vacuum is the adiabatic Bunch-Davies one \\cite{BuD}, with $c_1=1$ and $c_2=0$, that holds for any mass, regardless the real or imaginary value of the index (\\ref{ndS}). Despite of this advantage here we face with the problem of the rest limit which cannot be defined as long as the functions $K_{i\\kappa} (ipt_c)$ have an ambiguous behavior,\n\\begin{equation}\n K_{i\\kappa} (ipt_c)\\propto \\frac{1}{\\Gamma(\\frac{1}{2}-i\\kappa)}\\left(\\frac{ipt_c}{2}\\right)^{-i\\kappa}-\\frac{1}{\\Gamma(\\frac{1}{2}+i\\kappa)}\\left(\\frac{ipt_c}{2}\\right)^{i\\kappa}\\,,\n\\end{equation}\nfor $p\\to 0$, as it results from Eq. (\\ref{I0}). A possible solution is to redefine these functions replacing $i\\kappa\\to i\\kappa \\pm \\epsilon$ for eliminating one of the above terms and introducing a convenient phase factor for the remaining one \\cite{nach,born,CPasc}. However, this procedure is palliative since this affects the physical meaning of the mass which gets an imaginary part.\n\nFor avoiding these difficulties we defined recently the r.f.v., separating the frequencies in the rest frames just as in special relativity \\cite{CrfvKG}. Thus we found that the rest energy,\n\\begin{equation}\\label{Mko}\nM=\\kappa\\omega=\\sqrt{m^2-\\frac{9}{4}\\omega^2}\\,, \\quad m>\\frac{3}{2}\\omega\\,,\n\\end{equation}\nwhich plays the role of a dynamical mass, does make sense only for $\\mu>\\frac{3}{2}$ since for $\\mu<\\frac{3}{2}$ the mode functions do not have a physical meaning being of tachyonic type but with null norms. We have shown that in the tardyonic domain this vacuum is stable corresponding to the integration constants,\n\\begin{equation}\nc_1=-i\\left(\\frac{p}{2\\omega}\\right)^{-i\\kappa}\\frac{e^{\\pi\\kappa}}{\\sqrt{e^{2\\pi\\kappa}-1}}\\,, \\quad\nc_2= i\\left(\\frac{p}{2\\omega}\\right)^{-i\\kappa}\\frac{1}{\\sqrt{e^{2\\pi\\kappa}-1}}\\,,\n\\end{equation} \ndetermining the time modulation functions of positive energy as, \\cite{CrfvKG}\n\\begin{equation}\\label{Ftc}\n{\\cal F}_p(t_c)=\\sqrt{\\frac{\\pi}{\\omega}}\\left(\\frac{p}{2\\omega}\\right)^{-i\\kappa}\\frac{I_{i\\kappa}(ipt_c)}{\\sqrt{e^{2\\pi\\kappa}-1}}\\,.\n\\end{equation}\nWe must specify that the above phase factor is introduced for assuring the correct rest limit \n\\begin{equation}\n\\lim_{p\\to 0} {\\cal F}_p(t_c) =\\frac{1}{\\sqrt{2M}}\\,e^{-i M t}\\,,\n\\end{equation}\ncalculated according to Eqs. (\\ref{tct}) and (\\ref{I0}). \n\n\\subsection{Dirac field}\n\nIn the tetrad-gauge defined by Eqs. (\\ref{T1}) and (\\ref{T2}) the massive Dirac field $\\psi$ of mass $m$ satisfies the field equations $(D_x-m) \\psi (x)=0$ given by the Dirac operator \n\\begin{equation}\\label{ED}\nD_x=i\\gamma^0\\partial_{t}+i e^{-\\omega t}\\gamma^i\\partial_i\n+\\frac{3i \\omega}{2} \\gamma^{0}\\,.\n\\end{equation}\nwhere $\\gamma^{\\hat\\alpha}$ are the Dirac matrices labeled by local indices. It is known that the last term of this operator can be removed at any time by substituting $\\psi \\to [a(t)]^{-\\frac{3}{2}}\\psi$. Similar results can be written in the conformal chart.\n\nThe general solution of the Dirac equation may be written as a mode integral, \n\\begin{eqnarray}\n\\psi(t,{\\bf x}\\,)& =& \n\\psi^{(+)}(t,{\\bf x}\\,)+\\psi^{(-)}(t,{\\bf x}\\,)\\nonumber\\\\\n& =& \\int d^{3}p\n\\sum_{\\sigma}[U_{\\vec{p},\\sigma}(x){\\frak a}(\\vec{p},\\sigma)\n+V_{\\vec{p},\\sigma}(x){\\frak b}^{\\dagger}(\\vec{p},\\sigma)]\\,,\\label{p3}\n\\end{eqnarray}\nin terms of the fundamental spinors $U_{\\vec{p},\\sigma}$ and $V_{\\vec{p},\\sigma}$ of positive and respectively negative frequencies which are plane waves solutions of the Dirac equation depending on the conserved momentum $\\vec{p}$ and an arbitrary polarization $\\sigma$. These spinors form an orthonormal basis being related through the charge conjugation, \n\\begin{equation}\\label{chc}\nV_{\\vec{p},\\sigma}(t,{\\bf x})=U^c_{\\vec{p},\\sigma}(t,{\\bf x}) =C\\left[{U}_{\\vec{p},\\sigma}(t,{\\bf x})\\right]^* \\,, \\quad C=i\\gamma^2\\,,\n\\end{equation}\n(see the Appendix A), and satisfying the orthogonality relations\n\\begin{eqnarray}\n\\langle U_{\\vec{p},\\sigma}, U_{{\\vec{p}\\,}',\\sigma'}\\rangle_D &=&\n\\langle V_{\\vec{p},\\sigma}, V_{{\\vec{p}\\,}',\\sigma'}\\rangle_D=\n\\delta_{\\sigma\\sigma^{\\prime}}\\delta^{3}(\\vec{p}-\\vec{p}\\,^{\\prime})\\label{ortU}\\\\\n\\langle U_{\\vec{p},\\sigma}, V_{{\\vec{p}\\,}',\\sigma'}\\rangle _D&=&\n\\langle V_{\\vec{p},\\sigma}, U_{{\\vec{p}\\,}',\\sigma'}\\rangle_D =0\\,, \\label{ortV}\n\\end{eqnarray}\nwith respect to the relativistic scalar product \\cite{CD1}\n\\begin{equation}\n\\langle \\psi, \\psi'\\rangle_D=\\int d^{3}x\n\\sqrt{|g|}\\,e^0_0\\,\\bar{\\psi}(x)\\gamma^{0}\\psi(x) \n=\\int d^{3}x\\,\na(t)^{3}\\bar{\\psi}(x)\\gamma^{0}\\psi(x)\\,, \n\\end{equation}\nwhere $g={\\rm det}(g_{\\kappa\\nu})$ and $\\bar{\\psi}=\\psi^+\\gamma^0$ is the Dirac adjoint of $\\psi$. Moreover, this basis is supposed to be complete complying with a completeness condition \\cite{CD1}. This is the basis of the momentum representation in which the particle $({\\frak a},{\\frak a}^{\\dagger})$ and antiparticle (${\\frak b},{\\frak b}^{\\dagger})$ operators satisfy the canonical anti-commutation relations \\cite{CD1}. \n\nIn the standard representation of the Dirac matrices (with diagonal $\\gamma^0$) the general form of the fundamental spinors in momentum representation, \n\\begin{eqnarray}\nU_{\\vec{p},\\sigma}(t,\\vec{x}\\,)&=&\\frac{e^{i\\vec{p}\\cdot\\vec{x}}}{[2\\pi a(t)]^{\\frac{3}{2}}}\\left(\n\\begin{array}{c}\nu^+_p(t) \\,\n\\xi_{\\sigma}\\\\\nu^-_p(t) \\,\n \\frac{{p}^i{\\sigma}_i}{p}\\,\\xi_{\\sigma}\n\\end{array}\\right)\n\\label{Ups}\\\\\nV_{\\vec{p},\\sigma}(t,\\vec{x}\\,)&=&\\frac{e^{-i\\vec{p}\\cdot\\vec{x}}}{[2\\pi a(t)]^{\\frac{3}{2}}} \\left(\n\\begin{array}{c}\nv^+_p(t)\\,\n\\frac{{p}^i{\\sigma}_i}{p}\\,\\eta_{\\sigma}\\\\\nv^-_p(t) \\,\\eta_{\\sigma}\n\\end{array}\\right)\n\\,,\\label{Vps}\n\\end{eqnarray}\nis determined by the modulation functions $u^{\\pm}_p(t)$ and $v^{\\pm}_p(t)$ that depend only on $t$ and $p=|\\vec{p}|$. The Pauli spinors $\\xi_{\\sigma}$ and $\\eta_{\\sigma}= i\\sigma_2 (\\xi_{\\sigma})^{*}$ have to be correctly normalized, $\\xi^+_{\\sigma}\\xi_{\\sigma'}=\\eta^+_{\\sigma}\\eta_{\\sigma'}=\\delta_{\\sigma\\sigma'}$, satisfying a natural completeness equation.\n \nThe time modulation functions $u_p^{\\pm}(t)$ and $v_p^{\\pm}(t)$ must be related as \\cite{CrfvD}\n\\begin{equation}\\label{VU}\nv_p^{\\pm}=\\left[u_p^{\\mp}\\right]^*\\,,\n\\end{equation}\nfor assuring the charge conjugation symmetry (\\ref{chc}). Moreover, the normalization conditions\n\\begin{equation}\n|u_p^+|^2+|u_p^-|^2=|v_p^+|^2+|v_p^-|^2 =1 \\label{uuvv}\\\\\n\\end{equation}\nguarantee that Eqs. (\\ref{ortU}) and (\\ref{ortV}) are accomplished. \n\nThe time modulation functions can be derived easily in the conformal chart where we have to solve the system\n\\begin{eqnarray}\n\\left[i\\partial_{t_c}\\mp m\\, a(t_c)\\right]u_p^{\\pm}(t_c)&=&{p}\\,u_p^{\\mp}(t_c)\\,,\\label{sy1c}\\\\\n\\left[i\\partial_{t_c} \\mp m\\, a(t_c)\\right]v_p^{\\pm}(t_c)&=&-{p}\\,v_p^{\\mp}(t_c)\\,,\\label{sy2c}\n\\end{eqnarray}\nobtaining the general form \\cite{CrfvD}\n\\begin{eqnarray}\nu^{+}_p(t_c)&=&\\sqrt{-\\frac{p t_c}{\\pi}}\\left[c_1 K_{\\nu_{-}}\\left(i p t_c\\right)+c_2 K_{\\nu_{-}}\\left(-i p t_c\\right)\\right]\\,,\\label{coco1}\\\\\nu^{-}_p(t_c)&=&\\sqrt{-\\frac{p t_c}{\\pi}}\\left[c_1 K_{\\nu_{+}}\\left(i p t_c\\right)-c_2 K_{\\nu_{+}}\\left(-i p t_c\\right)\\right]\\,,\\label{coco2}\n\\end{eqnarray}\nwhere the orders of the modified Bessel functions $K$ are $\\nu_{\\pm}=\\frac{1}{2}\\pm i \\kappa$ after we re-define $ \\kappa= \\frac{m}{\\omega}$. \n\n The particular solutions of Eqs. (\\ref{coco1}) and Eqs. (\\ref{coco2}) are normalized and orthogonal to each other such that the normalization condition (\\ref{uuvv}) is satisfied if \n\\begin{equation}\n|c_1|^2+|c_2|^2=1\\,.\n\\end{equation}\nThe functions $v_p^{\\pm}$ result from Eq. (\\ref{VU}). \n\nThe adiabatic vacuum is defined simply by choosing $c_1=1$ and $c_2=0$ as in Refs. \\cite{nach,CD1}. The major difficulty of this vacuum is that in the momentum-spin representation we cannot reach the rest frame limit even though the functions $K$ have now defined limits for $p\\to 0$. This is because of the term $\\frac{\\vec{p}\\cdot\\vec{\\sigma}}{p}$ whose limit is undefined \\cite{CrfvD}. Moreover, if we force the limit vanishing this term by hand then we affect the normalization \\cite{CD,CD1}. \n\nThe solution is to adopt the r.f.v. imposing the conditions \\cite{CrfvD}\n\\begin{equation}\\label{rfv}\n\\lim_{p\\to 0} u^{-}_p(t)=\\lim_{p\\to 0} v^{+}_p(t)=0\\,,\n\\end{equation}\nwhich drop out the contribution of the mentioned terms in rest frames. These conditions are accomplished only if we take\n\\begin{equation}\\label{con}\nc_1=\\frac{e^{\\pi\\kappa}p^{-i\\kappa}}{\\sqrt{1+e^{2\\pi \\kappa}}}\\,, \\quad c_2=\\frac{i\\,p^{-i\\kappa}}{\\sqrt{1+e^{2\\pi \\kappa}}}\\,,\n\\end{equation}\ndetermining the definitive form of the modulation functions of positive frequencies as \\cite{CrfvD}\n\\begin{equation}\\label{uuI}\nu_p^{\\pm}(t_c)=\\pm \\frac{\\sqrt{-\\pi t_c}\\, p^{\\nu_-}}{\\sqrt{1+e^{2\\pi\\kappa}}}\\, I_{\\mp\\nu_{\\mp}}(ipt_c)\\,.\n\\end{equation}\nThe modulation functions of the negative frequencies have to be calculated according to Eq.(\\ref{VU}). Thus we obtain fundamental spinors whose rest limits\n\\begin{eqnarray}\n\\lim_{\\vec{p}\\to 0} U_{\\vec{p},\\sigma}(t,{\\bf x})&=&\\frac{e^{-i mt}}{[2\\pi a(t)]^{\\frac{3}{2}}}\\left(\n\\begin{array}{c}\n\\xi_{\\sigma}\\\\\n0\n\\end{array}\\right)\\,,\\label{Ur}\\\\\n\\lim_{\\vec{p}\\to 0}V_{\\vec{p},\\sigma}(t,{\\bf x})&=&\\frac{e^{i mt}}{[2\\pi a(t)]^{\\frac{3}{2}}}\\left(\n\\begin{array}{c}\n0\\\\\n\\eta_{\\sigma}\n\\end{array}\\right)\\,,\\label{Vr}\n\\end{eqnarray} \nindicate that the rest energy of the Dirac field is $m$ just as in special relativity.\n\n\\section{Flat limits}\n\nThe time modulation functions studied above depend on the variable \n\\begin{equation}\\label{var}\nx=-\\omega t_c=\\frac{p}{\\omega}e^{-\\omega t} \n\\end{equation}\nwhich in the rest limit tends to $0$ but in the flat limit, when $\\omega \\to 0$ and $-\\omega t_c \\to 1$, this tends to infinity. Therefore, for analyzing the behavior of the time modulation functions in the flat limit we need to use an uniform expansion of the Bessel function $J_{i\\kappa+\\lambda}(\\kappa x)$ for large values of $\\kappa>0$, any $x>0$ and $\\lambda=0,\\pm\\frac{1}{2}$. Unfortunately, we have a rigorous proof only for $\\lambda=0$ such that we are forced to generalize this case based on some analytical and numerical arguments. \n\n\n\\subsection{Approximating method}\n\n{ \\begin{figure}\n\\centering\n \\includegraphics[scale=0.37]{.\/F1}\n \\caption{The functions $\\Re\\,J_{i\\kappa+\\lambda}(\\kappa x)$ (red) and $\\Re\\,{\\cal J}(\\kappa,\\lambda, x)$} (blue) for $\\kappa=1$ and $\\lambda=-\\frac{1}{2},0, \\frac{1}{2}$.\n \\end{figure}}\n\nWe propose a generalization of the standard uniform asymptotic expansion (\\ref{unexA}) to $ J_{i\\kappa+\\lambda}(\\kappa x)$ observing that this is analytic in $\\kappa$ such that we can replace $i\\kappa \\to i\\kappa+\\lambda$ but without affecting the variable $x$ or expressions containing it, as $\\kappa x$ or $\\kappa\\sqrt{1+x^2}$. Therefore, we assume that the following approximation,\n\\begin{equation}\\label{unex0}\nJ_{i\\kappa+\\lambda}(\\kappa x) \\simeq {\\cal J}(\\kappa,\\lambda, x)=\\frac{e^{\\frac{\\pi\\kappa}{2}-\\frac{i\\pi\\lambda}{2}}}{\\sqrt{2\\kappa\\pi}}\\,\\frac{e^{i \\kappa\\sqrt{1+x^2}-\\frac{i\\pi}{4}}}{ (1+x^2)^{\\frac{1}{4}}}\n\\left(\\frac{1}{x}+\\sqrt{1+\\frac{1}{x^2}}\\right)^{-i\\kappa-\\lambda}\\,,\n\\end{equation}\nin which we neglected the terms of the order ${\\cal O}(\\kappa^{-1})$, holds even for non-vanishing values of $\\lambda\\in {\\Bbb R}$. However, the crucial point is to verify if this approximation is numerically satisfactory, comparing the functions $J$ and ${\\cal J}$. \n\n{ \\begin{figure}\n\\centering\n \\includegraphics[scale=0.37]{.\/F3}\n \\caption{The function ${\\cal E}(\\kappa,\\lambda, x)$ for $\\kappa=5$ (upper panels) and $\\kappa=10$ (lower panels).}\n \\end{figure}}\n \nWe start with the observation that in our case the variable (\\ref{var}) is positively defined without reaching the value $x=0$ if $p\\not=0$ such that we can restrict our graphical study to the interval $0.1\\leq x \\leq 10$. Then we may ask what it means `large values of $\\kappa$' plotting the functions $J$ and ${\\cal J}$ on this interval. Thus we see that their graphics tend to approach to each other even for modest values of $\\kappa$ as in Fig. (1) where $\\kappa=1$ and $\\lambda=0,\\pm \\frac{1}{2}$. For larger values of $\\kappa$ (e.g. $\\kappa>4$) the graphics of these functions are overlapping such that we need to resort to the function, \n\\begin{equation}\n{\\cal E}(\\kappa,\\lambda,x)=1-\\frac{{|\\cal J|}(\\kappa,\\lambda,x)|}{|J_{i\\kappa+\\lambda}(\\kappa x)|}\\,,\n\\end{equation}\nfor pointing out the errors of our approximation. In Fig. (2) we see how the errors are diminishing when $\\kappa$ is increasing from $5$ to $10$. \n\nThe conclusion is that our approximation is numerically satisfactory even for $\\lambda \\not= 0$. In practice it is convenient to substitute $\\kappa x \\to x$ for getting the more homogeneous approximation, \n\\begin{equation}\\label{unex}\nJ_{i\\kappa+\\lambda}(x)\\simeq\\frac{e^{\\frac{\\pi\\kappa}{2}-\\frac{i\\pi\\lambda}{2}}}{\\sqrt{2\\pi}}\\,\\frac{e^{i \\sqrt{\\kappa^2+x^2}-\\frac{i\\pi}{4}}}{ (\\kappa^2+x^2)^{\\frac{1}{4}}}\n\\left(\\frac{\\kappa}{x}+\\sqrt{1+\\frac{\\kappa^2}{x^2}}\\right)^{-i\\kappa-\\lambda}\\,,\n\\end{equation}\nwhich is useful in investigating the flat limits of the scalar and spinor fields.\n\n\n \n\\subsection{Flat limit of the Klein-Gordon field in r.f.v.}\n\nLet us briefly analyze the flat limits, for $\\omega\\to 0$, of the mode functions in r.f.v. for $\\kappa>\\frac{3}{2}$ starting with the time modulation functions (\\ref{Ftc}) rewritten, according to Eq. (\\ref{IJ}), as \n\\begin{equation}\\label{Ft}\n{\\cal F}_p(t)=e^{i\\delta_{KG}(p)}\\left(\\frac{p}{2\\omega}\\right)^{-i\\kappa}\\sqrt{\\frac{\\pi}{\\omega}}\\frac{e^{\\frac{1}{2}\\pi\\kappa}}{\\sqrt{e^{2\\pi\\kappa}-1}}\\,J_{i\\kappa}\\left(\\frac{p}{\\omega}e^{-\\omega t}\\right)\\,.\n\\end{equation}\nHere we introduced the auxiliary phase $\\delta_{KG}(p)$ we need for removing the pole in $\\omega=0$ of the general phase. Note that the second phase factor assures the correct rest frame limit for $p\\to 0$ \\cite{CrfvKG}. The flat limit can be evaluated by using the uniform expansion of this Bessel functions (\\ref{unex}) where we substitute $x$ as in Eq. (\\ref{var}). Then, according to Eq. (\\ref{unex}), we may approximate\n\\begin{equation}\n{\\cal F}_p(t_c)\\simeq\\rho(p,t)e^{i\\theta(p,t)}\\,,\n\\end{equation}\nwhere\n\\begin{eqnarray}\n\\rho(p,t)&=&\\frac{1}{\\sqrt{2}(M^2+p^2e^{-2\\omega t})^{\\frac{1}{4}}}\\frac{e^{\\pi\\kappa}}{\\sqrt{e^{2\\pi\\kappa}-1}}\\,,\\label{rhoD}\\\\\n\\theta(p,t)&=&\\delta_{KG}(p)-\\frac{\\pi}{4}-\\frac{M}{\\omega} \\ln\\left(\\frac{1}{2\\omega}\\right)-Mt\\nonumber\\\\\n&& -\\frac{M}{\\omega}\\ln\\left(M+\\sqrt{M^2+p^2e^{-2\\omega t}}\\right)+\\frac{1}{\\omega}\\sqrt{M^2+p^2e^{-2\\omega t}}\\,.\\label{theta}\n\\end{eqnarray}\nFurthermore, we compute the series of $\\theta(p,t)$ around $\\omega=0$ where this function has a pole that can be removed by setting \n\\begin{equation}\\label{deltaKG}\n\\delta_{KG}(p)=\\frac{\\pi}{4}+\\frac{M}{\\omega}\\ln \\frac{M+\\sqrt{M^2+p^2}}{2\\omega}-\\frac{\\sqrt{M^2+p^2}}{\\omega}\\,.\n\\end{equation} \nIn this manner we generate the phase factor,\n\\begin{equation}\ne^{i\\delta_{KG}(p)}=e^{\\frac{i \\pi}{4}}\\left(\\frac{M+\\sqrt{M^2+p^2}}{2\\omega} \\right)^{\\frac{iM}{\\omega}}e^{-i\\frac{\\sqrt{M^2+p^2}}{\\omega}}\\,,\n\\end{equation}\nwithout physical significance, but necessary for deriving the convenient approximation that can be used in applications,\n\\begin{equation}\\label{Ffin}\n{\\cal F}_p(t)\\simeq\\frac{e^{\\pi\\kappa}}{\\sqrt{e^{2\\pi\\kappa}-1}}\\frac{e^{i\\theta_{KG}(p,t)}}{\\sqrt{2}(M^2+p^2e^{-2\\omega t})^{\\frac{1}{4}}}\\,. \n\\end{equation}\nThe regularized phase, \n\\begin{eqnarray}\n\\theta_{KG}(p,t)&=&-Mt\n -\\frac{M}{\\omega}\\ln\\left(\\frac{M+\\sqrt{M^2+p^2e^{-2\\omega t}}}{M+\\sqrt{M^2+p^2}}\\right)\\nonumber\\\\\n&+&\\frac{1}{\\omega}\\left(\\sqrt{M^2+p^2e^{-2\\omega t}}-\\sqrt{M^2+p^2}\\right),.\\label{thetaKG}\n\\end{eqnarray}\nis obtained by substituting in Eq. (\\ref{theta}) the phase $\\delta_{KG}$ defined by Eq. (\\ref{deltaKG}). For small values of $\\omega$ we may use the Taylor series \n\\begin{equation}\\label{thetaKGex}\n\\theta_{KG}(p,t)=-\\sqrt{M^2+p^2}\\,t+\\frac{\\omega p^2 t^2}{2\\sqrt{M^2+p^2}}-\\frac{\\omega^2 p^2(2M^2+p^2)t^3}{6(M^2+p^2)^\\frac{3}{2}}+{\\cal O}(\\omega^3)\\,.\n\\end{equation}\nfinding that in the flat limit, when $\\lim_{\\omega\\to 0}M=m$, and\n\\begin{equation}\n\\lim_{\\omega\\to 0}{\\cal F}_p(t)=\\frac{e^{-iE(p)t}}{\\sqrt{2E(p)}}\\,, \\quad E(p)=\\sqrt{m^2+p^2}\\,,\n\\end{equation}\nwe recover just the Minkowkian time modulation functions. \n\n\\subsection{Flat limit of the Dirac field in r.f.v.}\n\nFor analyzing how the flat limit can be reached in the case of the Dirac field it is convenient to rewrite the time modulation functions (\\ref{uuI}) in the chart $\\{t,\\vec{x}\\}$ as\n\\begin{equation}\nu_p^{\\pm}(t)=\\pm e^{i\\delta_D(p)\\pm \\frac{i\\pi}{4}} p^{-i\\kappa}\\frac{e^{\\frac{\\pi\\kappa}{2}}}{\\sqrt{e^{2\\pi \\kappa}+1}}\\sqrt{\\frac{\\pi}{\\omega}\\, p e^{-\\omega t}}\\, J_{i\\kappa\\mp\\frac{1}{2}}\\left(\\frac{p}{\\omega}e^{-\\omega t}\\right)\\,,\n\\end{equation}\nafter introducing the phase $\\delta_D(p)$ which should take over the singularities of the general phase as in the previous case. The uniform expansion (\\ref{unex}) with $\\lambda=\\pm\\frac{1}{2}$ helps us to approximate\n\\begin{equation}\nu_p^{\\pm}(t)\\simeq\\rho^{\\pm}(p,t) e^{i\\theta(p,t)}\\,,\n\\end{equation} \nwhere\n\\begin{eqnarray}\n\\rho^+(p,t)&=&\\frac{\\sqrt{\\sqrt{m^2+p^2e^{-2\\omega t}}+m}}{\\sqrt{2}(m^2+p^2 e^{-2\\omega t})^{\\frac{1}{4}}}\\frac{e^{\\pi\\kappa}}{\\sqrt{e^{2\\pi \\kappa}+1}}\\,,\\label{RP}\\\\\n\\rho^-(p,t)&=&\\frac{p e^{-\\omega t}}{\\sqrt{2}(m^2+p^2 e^{-2\\omega t})^{\\frac{1}{4}}\\sqrt{\\sqrt{m^2+p^2e^{-2\\omega t}}+m}}\\frac{e^{\\pi\\kappa}}{\\sqrt{e^{2\\pi \\kappa}+1}}\\,,\\label{RM}\\\\\n\\theta(p,t)&=&\\delta_D(p)+\\frac{\\pi}{4}-mt\\nonumber\\\\\n&& -\\frac{m}{\\omega}\\ln\\left(m+\\sqrt{m^2+p^2e^{-2\\omega t}}\\right)+\\frac{1}{\\omega}\\sqrt{m^2+p^2e^{-2\\omega t}}\\,.\\label{thetaD}\n\\end{eqnarray}\nWe observe that the obvious identity \n\\begin{equation}\npe^{-\\omega t}=\\sqrt{\\sqrt{m^2+p^2e^{-2\\omega t}}+m}\\,\\sqrt{\\sqrt{m^2+p^2e^{-2\\omega t}}-m}\n\\end{equation}\ncan be substituted in Eq. (\\ref{RM}) for getting a more symmetric and compact form.\nFurthermore, we expand the function $\\theta(p,t)$ around $\\omega=0$ and we chose\n\\begin{equation}\n\\delta_D(p)=-\\frac{\\pi}{4}+\\frac{m}{\\omega}\\ln(\\sqrt{m^2+p^2}+m)-\\frac{1}{\\omega}\\sqrt{m^2+p^2}\\,,\n\\end{equation}\nfor eliminating the effects of the pole in $\\omega=0$. Thus we arrive at the final expansions for large values of $\\kappa$ (when $\\omega \\to 0$) that reads\n\\begin{equation}\\label{uaprox}\nu_p^{\\pm}(t)\\simeq \\frac{e^{\\frac{\\pi m}{\\omega}}}{\\sqrt{e^{2 \\frac{\\pi m}{\\omega}}+1}}\\frac{\\sqrt{\\sqrt{m^2+p^2e^{-2\\omega t}}\\pm m}}{\\sqrt{2}(m^2+p^2 e^{-2\\omega t})^{\\frac{1}{4}}}\\, e^{i\\theta_D(p,t)}\\,.\n\\end{equation}\nThe regularized phase, \n\\begin{eqnarray}\n\\theta_D(p,t)&=&-m t -\\frac{m}{\\omega}\\ln\\left(\\frac{m+\\sqrt{m^2+p^2e^{-2\\omega t}}}{m+\\sqrt{m^2+p^2}}\\right)\\nonumber\\\\\n&+&\\frac{1}{\\omega}\\left(\\sqrt{m^2+p^2e^{-2\\omega t}} -\\sqrt{m^2+p^2}\\right)\\,,\n\\end{eqnarray}\nobtained after substituting $\\delta_D$ in Eq. (\\ref{thetaD}), can be expanded as \n\\begin{equation}\\label{thetaD1}\n\\theta_{D}(p,t)=-\\sqrt{m^2+p^2}\\,t+\\frac{\\omega p^2 t^2}{2\\sqrt{m^2+p^2}}-\\frac{\\omega^2 p^2(2m^2+p^2)t^3}{6(m^2+p^2)^\\frac{3}{2}}+{\\cal O}(\\omega^3)\\,,\n\\end{equation}\nlaying out a similar form as the phase (\\ref{thetaKG}) of the scalar field but with the usual mass $m$ instead of the dynamical one.\n\nFinally we verify that in the flat limit we obtain the usual Minkowskian time modulation functions\n\\begin{equation}\n\\lim_{\\omega\\to 0}u_p^{\\pm}(t) =\\sqrt{\\frac{E(p)\\pm m}{2 E(p)}}\\, e^{-iE(p)t}\\,. \n\\end{equation}\n\n\\subsection{Physical consequences}\n\nSolving the problem of the flat limit we derived the suitable phases that complete the phases we introduced previously for defining the r.f.v.. We obtain thus the final form the scalar time modulation functions can be written now as \n\\begin{equation}\\label{Fdef}\n{\\cal F}_p(t)=e^{i\\alpha_{KG}(p)}\\sqrt{\\frac{\\pi}{\\omega}}\\frac{e^{\\frac{1}{2}\\pi\\kappa}}{\\sqrt{e^{2\\pi\\kappa}-1}}\\,J_{i\\kappa}\\left(\\frac{p}{\\omega}e^{-\\omega t}\\right)\\,,\n\\end{equation}\nwhere $\\kappa=\\frac{M}{\\omega}$ while the global phase,\n\\begin{eqnarray}\n\\alpha_{KG}(p)&=&\\delta_{KG}(p)-\\kappa \\ln\\left(\\frac{p}{2\\omega}\\right) \\nonumber\\\\\n&=&\\frac{\\pi}{4}+\\frac{M}{\\omega}\\ln \\frac{M+\\sqrt{M^2+p^2}}{p}-\\frac{\\sqrt{M^2+p^2}}{\\omega}\\,,\n\\end{eqnarray}\ndepends on the dynamical mass ({\\ref{Mko}).\n\nFor the Dirac time modulation functions we may write a similar result,\n \\begin{equation}\\label{ufin}\nu_p^{\\pm}(t)=\\pm e^{i\\alpha_D(p)\\pm \\frac{i\\pi}{4}} \\frac{e^{\\frac{\\pi\\kappa}{2}}}{\\sqrt{e^{2\\pi \\kappa}+1}}\\sqrt{\\frac{\\pi}{\\omega}\\, p e^{-\\omega t}}\\, J_{i\\kappa\\mp\\frac{1}{2}}\\left(\\frac{p}{\\omega}e^{-\\omega t}\\right)\\,,\n\\end{equation}\nwhere now $\\kappa=\\frac{m}{\\omega}$ and\n\\begin{eqnarray}\n\\alpha_D(p)&=&\\delta_{D}(p)-\\kappa \\ln p\\nonumber\\\\\n&=&-\\frac{\\pi}{4}+\\frac{m}{\\omega}\\ln\\left(\\frac{\\sqrt{m^2+p^2}+m}{p}\\right)-\\frac{\\sqrt{m^2+p^2}}{\\omega}\\,,\n\\end{eqnarray}\nis very similar with the scalar phase but with the genuine mass $m$ instead of the dynamical one, $M$.\n\nWhy the phases are so important in the de Sitter spacetime as long as these do not affect the scalar products and do not contribute to the expressions of the transition probabilities. A specific feature of the de Sitter QFT is that the forms of some one-particle operators including the Hamiltonian (or energy) one are strongly dependent on the phases which are functions of $p$. We remind the reader that, after canonical quantization, the one-particle Hamiltonian operator of any quantum field, $\\Psi$, can be calculated by using the corresponding relativistic scalar product, ${\\cal H}=:\\langle \\Psi, H\\Psi\\rangle:$, in which we respect the normal ordering of the field operators \\cite{BDR}. The energy operator,\n\\begin{equation}\nH=i\\partial_t+\\omega \\vec{x}\\cdot\\vec{P}=i\\partial_t-i\\omega \\vec{x}\\cdot \\nabla\n\\end{equation}\nis the same for any free field since it does not have spin parts \\cite{nach,CGRG}. \n\nIn the case of the Klein-Gordon field we consider the normalized mode functions (\\ref{fp}) with the time modulation functions (\\ref{Fdef}). Then it is not difficult to verify the identity \n\\begin{equation}\\label{H}\n(Hf_{\\vec{p}})=\\left[-i\\omega \\left(p^i\\partial_{p_i}+{\\frac{3}{2}}\\right)-\\omega p^i\\partial_{p^i}\\alpha_{KG}(p)\\right]f_{\\vec{p}}\n\\end{equation}\nwhich allows us to derive the form of the one-particle Hamiltonian operator as,\n\\begin{eqnarray}\n{\\cal H}_{KG}&=&:\\langle \\Phi, H\\Phi\\rangle_{KG}:\\,= \\int d^3 p\\, \\sqrt{M^2+p^2}\n\\left[a^{\\dagger}({\\bf p}) a({\\bf p})\n+{b}^{\\dagger}({\\bf p}){b}({\\bf p})\\right]\\nonumber\\\\\n&&~~~+\\frac{i\\omega}{2}\\int d^3p\\, p^i \\left\\{ \\left[\\, a^{\\dagger}({\\bf\np})\\stackrel{\\leftrightarrow}{\\partial}_{p_i} a({\\bf p})\\right]+ \\left[\\,\nb^{\\dagger}({\\bf p}) \\stackrel{\\leftrightarrow}{\\partial}_{p_i} b({\\bf\np})\\right]\\right\\}\\,,\\label{Ham}\n\\end{eqnarray}\nsince\n\\begin{equation}\\label{Eap}\n\\omega p^i\\partial_{p^i}\\alpha_{KG}(p)=-\\sqrt{M^2+p^2}\\,.\n\\end{equation}\nWe have thus the nice surprise to see that by fixing the correct phases requested by the rest and flat limits we obtain an operator whose flat limit,\n\\begin{equation}\n\\lim_{\\omega \\to 0} {\\cal H}_{KG}= \\int d^3 p\\, \\sqrt{m^2+p^2}\n\\left[a^{\\dagger}({\\bf p}) a({\\bf p})\n+{b}^{\\dagger}({\\bf p}){b}({\\bf p})\\right]\\,,\n\\end{equation}\nis just the well-known Hamiltonian operator of the Minkowskian QFT. A similar result can be obtained for the Dirac field. Therefore, the flat limit of the entire de Sitter QFT is the just the QFT of special relativity. \n\n\\section{Concluding remarks}\n\nWe derived the definitive forms of the fundamental solutions of the Klein-Gordon and Dirac fields whose frequencies are separated in the rest frames as in special relativity having, in addition, Minkowskian flat limits. \n\nSimilar results concerning the phase factors or mode expansions as the second term of Eq. (\\ref{Ham}) were obtained in premiere for the scalar and spinor fields long time ago in Ref. \\cite{nach} where the adiabatic vacuum was considered. Subsequent studies refined these results \\cite{born,CPasc,CD} such that now we can conclude that the regularizes phases derived so far are very similar to those obtained here in r.f.v.. This is because in the adiabatic vacuum, where the rest limits are undefined, one forced some {\\em ad hoc} changes of the time modulation functions, introducing phase factors proportional to $p^{-i\\kappa}$ similar to those arising naturally in r.f.v.. \n\nHowever, apart from the known regularized phases recovered here, we report new results as the final form of the time modulation functions (\\ref{Fdef}) and (\\ref{ufin}) in r.f.v. and the approximations (\\ref{Ffin}) and (\\ref{uaprox}) that can be used in applications for deriving transition amplitudes between states defined in this vacuum.\n \nThe final conclusion is that the rest and flat limits appear as the cornerstones of the QFT on the de Sitter expanding universe determining the form of the time modulation functions in a natural manner and setting thus the structure of the one-particle operators. In our opinion, the results presented here are an argument that the r.f.v. could be universal.\n\n\\subsection*{Acknowledgments}\n\nThis work is partially supported by a grant of the Romanian Ministry of Research and Innovation, CCCDI-UEFISCDI, project number PN-III-P1-1.2-PCCDI-2017-0371. \n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\nRecently several string actions which\nnaturally describe curved space-times with singularities\nhave been obtained from WZWN models.\nFollowing the coset construction \\cite{witten,HH,bars},\nin Ref.~\\cite{ch} we analyzed a particular WZWN action\nin the Poincar\\'e $ISO(2,1)$ group.\n\\par\nIn a recent paper \\cite{stern} a very general family of Lagrangians\nin the Poincar\\'e group $ISO(d-1,1)$ was studied and shown to describe\ndiverse closed, bosonized, spinning strings in $(d-1)+1$-dimensional\nMinkowski space-time depending on the values of the constants which\nparameterize the family.\nIn this paper, we start from the family of actions cited above and add\nfurther contributions amounting to terms which are symmetric in the\nworld-sheet indices.\nThen we apply a gauging procedure which generalizes the one used in\nRef.~\\cite{ch}:\nwe raise a subgroup $H$ to a local symmetry of the action, where\n$H$ turns out to be necessarily isomorphic to $\\mbox{{\\rm I\\hspace{-2truemm} R}}^n$, $n\\le d$,\nbecause of the prescription that the gauge field belongs to the algebra\nof $H$ itself, and show that the gauged action generates a family\nof effective actions for $\\sigma$-models with $N=d\\,(d+1)\/2-\\mbox{{\\rm dim}}(H)$\ndegrees of freedom.\nThe latter can be viewed as effective theories describing the\ndynamics of a bosonic string moving in a (generally) curved\nbackground which can also contain an axion field.\n\\par\nFurther, since both the metric and axion fields are independent\nof at least $d-\\mbox{{\\rm dim}}(H)$ out of $N$ degrees of freedom,\nit is easy to prove that the usual T-duality\nconsiderations apply \\cite{giveon}.\nOne can include a dilaton field at a higher order in the loop\nexpansion and build dual spaces.\n\\par\nThe main idea behind the procedure we use is actually quite simple,\nand it is worth displaying the way it works on a {\\em toy model\\\/}\nto show its main features.\nConsider the following 2-dimensional action,\n\\begin{equation}\nS(x^1,x^2)=\n{1\\over 2}\\,\\int dt\\,\\left[(\\partial_t x^1)^2+(\\partial_t x^2)^2\\right]\n\\ ,\n\\end{equation}\nand gauge one of the coordinates, {\\em e.g.\\\/} $x^2$,\nby the minimal coupling prescription,\n$\\partial_t x^2\\to \\partial_t x^2+A^2$, where $A^2$ is a gauge field,\n\\begin{equation}\nS_g(x^1,x^2,A^2)=\nS+{1\\over 2}\\,\\int dt\\,A^2\\left(A^2+2\\,\\partial_t x^2\\right)\n\\ .\n\\end{equation}\nSince $S_g$ is quadratic in $A^2$, one can define an effective\naction by integrating out $A^2$ in the path integral,\n\\begin{equation}\n\\int [dx^1]\\,[dx^2]\\,[d A^2]\\,e^{-S_g(x^1,x^2,A^2)}\\equiv\n\\int [dx^1]\\,e^{-S_{eff}(x^1)}\n\\ .\n\\end{equation}\nThis is equivalent to solving the equation of motion for $A^2$,\n$\\delta_{A^2} S_g=0$, and substituting back the result into\n$S_g$.\nThus one obtains\n\\begin{equation}\nS_{eff}(x^1)={1\\over 2}\\,\\int dt\\,(\\partial_t x^1)^2\n\\ ,\n\\end{equation}\nwhich is a trivial result and equals the one which we get\nby assuming $A^2$ is a pure gauge, $A^2=-\\partial_t x^2$.\nBut suppose we now perform the following (canonical)\ntransformation,\n\\begin{equation}\n\\left[\\begin{array}{c}\nx^1 \\\\\nx^2\n\\end{array}\\right]\n\\equiv\n\\left[\\begin{array}{cc}\n\\theta_{11} & \\theta_{12} \\\\\n\\theta_{21} & \\theta_{22}\n\\end{array}\\right]\n\\,\n\\left[\\begin{array}{c}\nx \\\\\n\\tilde x\n\\end{array}\\right]\n\\ ,\n\\label{can}\n\\end{equation}\nwith $\\theta_{11}\\,\\theta_{22}-\\theta_{12}\\,\\theta_{21}\\not=0$,\nthen we gauge {\\em e.g.\\\/} $\\tilde x$ introducing a gauge field\n$\\tilde A$ and repeat the process above.\nThis time we obtain\n\\begin{equation}\nS_{eff}(x)={1\\over 2}\\,\\int dt\\,\n{(\\theta_{11}\\,\\theta_{22}-\\theta_{12}\\,\\theta_{21})^2\\over\n\\theta_{12}^2+\\theta_{22}^2}\\,(\\partial_t x)^2\n\\ ,\n\\end{equation}\nwhich is different from the result one would get by setting\n$\\tilde A=-\\partial_t \\tilde x$, namely\n\\begin{equation}\nS(x,\\tilde x=const)={1\\over 2}\\,\\int dt\\,(\\theta_{11}^2+\\theta_{21}^2)\\,\n(\\partial_t x)^2\n\\ .\n\\end{equation}\nWe can rephrase this conclusion by saying that the canonical\ntransformation in Eq.~(\\ref{can}) introduces {\\em cross terms\\\/}\nof the form $\\partial_t x\\,\\partial_t \\tilde x$ in the action\nand these in turn generate the following mapping:\n\\begin{equation}\n(\\theta_{11}^2+\\theta_{21}^2)\\to\n{(\\theta_{11}\\,\\theta_{22}-\\theta_{12}\\,\\theta_{21})^2\\over\n\\theta_{12}^2+\\theta_{22}^2}\n\\ ,\n\\label{toy_map}\n\\end{equation}\nwhich becomes trivial for\n$\\theta_{11}\\,\\theta_{12}+\\theta_{21}\\,\\theta_{22}=0$\n(this also rules out rotations).\n\\par\nOf course, the previous model applies to quadratic actions only.\nWhenever we encounter Lagrangians which are linear in the gauge\nfield we will revert to the pure gauge sector, as we did in\nRef.~\\cite{ch}, and obtain degenerate metrics (apart from two\nexceptional cases).\n\\par\nThis in brief describes both our coset construction\nand compactification to get T-dual solutions.\nHowever,\ndue to the high degree of generality of the model we start with,\nwe were not able to draw any explicit conclusions other than\nformal mappings like the one shown in Eq.~(\\ref{toy_map})\nfor the toy model above.\nIn particular, we cannot say much about the reduced space-time\nin general, although we show that,\nwith particular choices of the parameters\ninvolved and for $d=2,3$, it is actually possible to complete\nthe analysis.\n\\par\nIn the next Section we describe the ungauged model,\nwith particular attention to the derivation of the equations\nof motion and their comparison with the models introduced in\nRef.~\\cite{stern}.\nIn Section~\\ref{gauged} we give the general formal treatment of\nthe action when one gauges a subgroup $H\\sim\\mbox{{\\rm I\\hspace{-2truemm} R}}^n$,\nidentify the whole set of its symmetries and introduce an effective\naction in the form of a $\\sigma$-model.\nWe obtain expressions for actions which can be\neither quadratic or\nlinear in the gauge field, but in the latter case we show there\nare only two cases with non degenerate metrics.\nIn Section~\\ref{T-d} we describe the T-dual procedure as applied\nto our model and show the properties of our model which are related\nto its multiple isometries.\nWe then study the properties of the effective action under T-duality\ntransformations.\nIn Section~\\ref{d=2} we specialize to the simplest, 2-dimensional,\nnon degenerate case and perform an explicit analysis to find one\neffective background and one of its T-duals.\nIn Section~\\ref{iso21} we prove that the (non degenerate) metrics\nfor all the models with $d=3$ which are linear in the gauge field\nreduce to the case already treated in Ref.~\\cite{ch},\nwhich we now revise.\n\\setcounter{equation}{0}\n\\section{The ungauged action}\n\\label{ungauged}\nWe recall here that the Poincar\\'e group in $(d-1)+1$ space-time\ndimensions, $ISO(d-1,1)$, is the semidirect product of the Lorentz\ngroup $SO(d-1,1)$ with the space-time translation group\n$T(d-1,1)\\sim\\mbox{{\\rm I\\hspace{-2truemm} R}}^{(d-1,1)}\\sim\\mbox{{\\rm I\\hspace{-2truemm} R}}^d$.\nTherefore we write its elements $g$ using the notation\n$g=\\left(\\Lambda,x\\right)$, where $\\Lambda\\in SO(d-1,1)$ and\n$x\\in \\mbox{{\\rm I\\hspace{-2truemm} R}}^d$.\n\\par\nGiven the map $g:\\ {\\cal M}\\mapsto ISO(d-1,1)$ from the\n2-dimensional manifold ${\\cal M}$, parametrized by the\ncoordinates $\\sigma^\\alpha$ ($\\sigma^0\\equiv\\tau$,\n$\\sigma^1\\equiv\\sigma$), to $ISO(d-1,1)$,\nwe consider the very general action given by\n\\begin{equation}\nS(\\Lambda,x;K)=S_1+S_2+S_3\n\\ ,\n\\label{S}\n\\end{equation}\nwhere\n\\begin{eqnarray}\nS_1&=&{1\\over2}\\,\\int_{\\cal M}\\,d^2\\sigma\\,\n(g^{\\alpha\\beta}+\\epsilon^{\\alpha\\beta})\\,\nK_{ij}^{(1)}\\,V^i_\\alpha\\,V^j_\\beta\n\\nonumber \\\\\nS_2&=&\\int_{\\cal M}\\,d^2\\sigma\\,(g^{\\alpha\\beta}+\\epsilon^{\\alpha\\beta})\\,\nK_{ijk}^{(2)}\\,V^i_\\alpha\\,W^{jk}_\\beta\n\\label{S123} \\\\\nS_3&=&{1\\over8}\\,\\int_{\\cal M}\\,d^2\\sigma\\,\n(g^{\\alpha\\beta}+\\epsilon^{\\alpha\\beta})\\,\nK_{ijkl}^{(3)}\\,W^{ij}_\\alpha\\,W^{kl}_\\beta\n\\ . \\nonumber\n\\end{eqnarray}\nSummation is assumed among upper and lower repeated latin indices\n$i,j,\\dots=0,\\dots,d-1$ according to the usual Lorentzian scalar product\nrule $A_i\\,B^i\\equiv A^i\\,\\eta_{ij}\\,B^j$, where\n$\\eta_{ij}=(-,+,\\dots,+)$\nis the Minkowski tensor in $(d-1)+1$-dimensions.\n\\par\nThe Lagrangians in Eqs.~(\\ref{S123}) contain two kinds of contribution:\nthe first one is proportional to the area element on ${\\cal M}$,\n$d^2\\sigma\\,\\epsilon^{\\alpha\\beta}=d\\sigma^\\alpha\\wedge d\\sigma^\\beta$,\nwith $\\epsilon^{\\alpha\\beta}=-\\epsilon^{\\beta\\alpha}$\n($\\epsilon^{\\tau\\sigma}=+1$) the Levi Civita symbol in two dimensions;\nthe second one is proportional to the constant symmetric 2-dimensional\nmatrix $g^{\\alpha\\beta}$ with $|\\det g^{\\alpha\\beta}|=1$.\nSince the constants $K$ are assumed to satisfy\n\\begin{eqnarray}\nK^{(1)}_{ij}&=&-K^{(1)}_{ji}\n\\nonumber \\\\\nK^{(2)}_{ijk}&=&-K^{(2)}_{ikj}\n\\nonumber \\\\\nK^{(3)}_{ijkl}&=&-K^{(3)}_{jikl}=-K^{(3)}_{ijlk}=-K^{(3)}_{klij}\n\\ ,\n\\label{K_cond}\n\\end{eqnarray}\nit turns out that the contributions proportional to $g^{\\alpha\\beta}$\ndrop out both of $S_1$ (because of the skewsymmetry of $K^{(1)}_{ij}$\nin the indices $i,j$)\nand $S_3$ (because of the skewsymmetry of $K^{(3)}_{ijkl}$\nunder the exchange of the pairs of indices $(i,j),(k,l)$)\nand thus only $S_2$ contains it.\nWhen $g^{\\alpha\\beta}\\equiv 0$, the action $S(K)$ coincides with the\nmodel introduced in Ref.~\\cite{stern} and describes a different\nkind of closed bosonized spinning string moving in $(d-1)+1$ Minkowski\nspace-time with coordinates $x^k$, $k=0,\\dots,d-1$ depending\non the values of the constants $K$.\n\\par\nThe 1-forms $V^i$, $W^{ij}$, with components\n\\begin{equation}\n\\begin{array}{l}\nV^i_\\alpha\\equiv(\\Lambda^{-1})^i_{\\ r}\\,\\partial_\\alpha x^r \\\\\n\\\\\nW^{ij}_\\alpha\\equiv\n(\\Lambda^{-1})^i_{\\ r}\\,\\partial_\\alpha\\Lambda^{rj}\n\\ ,\n\\end{array}\n\\label{gdg}\n\\end{equation}\nare obtained by projecting the (left invariant)\nMaurer-Cartan form $g^{-1}\\,dg$ on the basis of the Poincar\\'e\nalgebra $iso(d-1,1)$.\nThus it immediately follows that the action $S$ in Eq.~(\\ref{S})\nis invariant under the left rigid action of the Poincar\\'e group,\n$g\\to g'\\,g$, $g'=(\\theta,y)\\in ISO(d-1,1)$,\n\\begin{equation}\nS(\\theta\\,\\Lambda,\\theta\\,x+y;K)=S(\\Lambda,x;K)\n\\ .\n\\end{equation}\nIt is also invariant under the right rigid action,\n$g\\to g\\,g'$, $g'=(\\theta,y)\\in ISO(d-1,1)$,\n\\begin{equation}\nS(\\Lambda\\,\\theta,\\Lambda\\,y+x;K)=S(\\Lambda,x;K')\n\\ ,\n\\end{equation}\nprovided the constants\n$K\\equiv(K^{(1)}_{ij},K^{(2)}_{ijk},K^{(3)}_{ijkl})$\nmap to new values $K'$ according to an expression given in\nRef.~\\cite{stern}.\nThe action $S_2$ alone, depending on the choice of $g^{\\alpha\\beta}$,\nmay actually be invariant under a {\\em semi\\\/}-local transformation,\nas we report in Section~\\ref{iso21}.\n\\par\nIf we use the convention $\\Lambda_i^{\\ j}\\equiv(\\Lambda^{-1})_{\\ i}^j$,\nthe action $S$ can be written more explicitely in terms of the\nelements $\\Lambda$ and $x$ as\n\\begin{eqnarray}\nS_1&=&{1\\over2}\\,\\int_{\\cal M}\\,d^2\\sigma\\,\\epsilon^{\\alpha\\beta}\\,\nK_{ij}^{(1)}\\,\n\\Lambda^{\\ i}_r\\,\\Lambda^{\\ j}_s\\,\n\\partial_\\alpha x^r\\,\\partial_\\beta x^s\n\\nonumber \\\\\nS_2&=&\\int_{\\cal M}\\,d^2\\sigma\\,(g^{\\alpha\\beta}+\\epsilon^{\\alpha\\beta})\\,\nK_{ijk}^{(2)}\\,\n\\Lambda^{\\ i}_r\\,\\Lambda^{\\ j}_s\\,\n\\partial_\\alpha x^r\\,\\partial_\\beta \\Lambda^{sk} \\\\\nS_3&=&{1\\over 8}\\,\\int_{\\cal M}\\,d^2\\sigma\\,\\epsilon^{\\alpha\\beta}\\,\nK_{ijkl}^{(3)}\\,\n\\Lambda^{\\ i}_r\\,\\Lambda^{\\ k}_s\\,\n\\partial_\\alpha \\Lambda^{rj}\\,\\partial_\\beta \\Lambda^{sl}\n\\nonumber\n\\ .\n\\end{eqnarray}\n\\par\nThe equations of motion $\\delta_x S=0$, which follow from the\nvariation $x\\to x+\\delta x$, with $\\delta x$ an infinitesimal\n$(d-1)+1$ vector, amount to linear momentum conservation,\n\\begin{equation}\n\\partial_\\alpha{\\cal P}^\\alpha_i\\equiv\n\\partial_\\alpha\\left(\\epsilon^{\\alpha\\beta}\\,{\\cal P}_{\\beta\\,i}^{(1)}\n+(g^{\\alpha\\beta}+\\epsilon^{\\alpha\\beta})\\,\n{\\cal P}_{\\beta\\,i}^{(2)}\\right)\n=0\n\\ ,\n\\label{dp=0}\n\\end{equation}\nwhere the only two linear momentum currents that are not identically\nzero follow from $S_1$ and $S_2$ and are respectively given by\n\\begin{equation}\n\\begin{array}{l}\n{\\cal P}_{i}^{(1)\\alpha}=\\epsilon^{\\alpha\\beta}\\,\n\\Lambda^{\\ r}_i\\,K_{rs}^{(1)}\\,V^s_\\beta\n\\equiv {\\cal V}^{\\,\\alpha\\beta}_{is}\\,\\partial_\\beta x^s \\\\\n \\\\\n{\\cal P}_{i}^{(2)\\alpha\\,}=\n\\left(g^{\\alpha\\beta}+\\epsilon^{\\alpha\\beta}\\right)\\,\n\\Lambda^{\\ r}_i\\,K_{rst}^{(2)}\\,\nW^{st}_\\beta\\equiv\n{\\cal W}^{\\,\\alpha\\beta}_{ist}\\,\\partial_\\beta\\Lambda^{st}\n\\ .\n\\end{array}\n\\label{T}\n\\end{equation}\nUpon integrating on a fixed $\\tau$ slice of the world-sheet,\none finds that the conserved charges are given by\n\\begin{equation}\nP^{(1)}_i+P^{(2)}_i=\\int d\\sigma\\ \\left[{\\cal P}^{(1)}_{\\sigma\\,i}\n+(1+g^{\\tau\\sigma})\\,{\\cal P}^{(2)}_{\\sigma\\,i}\n+g^{\\tau\\tau}\\,{\\cal P}^{(2)}_{\\tau\\,i}\\right]\n\\ ,\n\\end{equation}\nwhere $\\tau$ and $\\sigma$ are the world sheet coordinates and use has\nbeen made of the periodicity in $\\sigma$ to discard boundary terms.\nFor the particular choice we will make in Section~\\ref{iso21},\n$g^{\\alpha\\beta}=\\pm\\eta^{\\alpha\\beta}=\n\\pm\\mbox{{\\rm diag}}(-1,1)$ (the Minkowski metric tensor on the world-sheet),\none finds that the conserved linear momentum following from $S_2$\ncoincides with the spatial integral of ${\\cal P}^{(2)}_\\pm$,\nwhere\n\\begin{equation}\n\\sigma^\\pm\\equiv\\tau\\pm\\sigma\n\\label{light}\n\\end{equation}\nare light-cone coordinates on the world-sheet.\n\\par\nSimilarly, from the variation $\\Lambda\\to\\Lambda+\\delta\\Lambda$,\n$\\delta\\Lambda=\\Lambda\\,\\rho$ and $\\delta x=\\rho\\,x$,\nwith $\\rho_{ij}=-\\rho_{ji}$ an infinitesimal $so(d-1,1)$\nmatrix, the equations $\\delta_\\Lambda S=0$ lead to\nangular momentum conservation\n\\begin{equation}\n\\partial_\\alpha{\\cal J}^\\alpha_{ij}\\equiv\n\\partial_\\alpha\\left(\\epsilon^{\\alpha\\beta}\\,{\\cal J}^{(1)}_{\\beta\\,ij}+\n(g^{\\alpha\\beta}+\\epsilon^{\\alpha\\beta})\\,{\\cal J}_{\\beta\\,ij}^{(2)}+\n\\epsilon^{\\alpha\\beta}\\,{\\cal J}_{\\beta\\,ij}^{(2)}\\right)\n=0\n\\ ,\n\\label{dj=0}\n\\end{equation}\nwhere the three angular momentum currents\nfollowing from $S_1$, $S_2$ and $S_3$ read\n\\begin{eqnarray}\n{\\cal J}_{ij}^{(1)\\alpha}&=&{\\cal L}_{ij}^{(1)\\alpha}=\nx_i\\wedge{\\cal P}_{j}^{(1)\\alpha}\n\\nonumber \\\\\n{\\cal J}_{ij}^{(2)\\alpha}&=&{\\cal L}_{ij}^{(2)\\alpha}\n+{\\cal S}_{ij}^{(2)\\alpha}\n\\ ,\\ \\ \\ \\\n\\left\\{\\begin{array}{l}\n{\\cal L}_{ij}^{(2)\\alpha}=x_i\\wedge{\\cal P}_{j}^{(2)\\alpha} \\\\\n \\\\\n{\\cal S}_{ij}^{(2)\\alpha}=2\\,{\\cal W}_{rij}^{\\alpha\\beta}\\,\n\\partial_{\\beta}x^r\n\\end{array}\\right.\n\\nonumber \\\\\n{\\cal J}_{ij}^{(3)\\alpha}&=&{\\cal S}_{ij}^{(3)\\alpha}=\n-{1\\over2}\\,\\epsilon^{\\alpha\\beta}\\,\\Lambda_i^{\\ r}\\,\\Lambda_j^{\\ s}\\,\nK_{rstp}^{(3)}\\,W_\\beta^{tp}\n\\ .\n\\label{J}\n\\end{eqnarray}\nIt is thus clear that one obtains terms which can be interpreted as\nnon zero intrinsic angular momentum\n(spin) ${\\cal S}_{ij}$ without the use of Grassmann variables.\nIf one expands Eq.~(\\ref{dj=0}) one finds that the conserved charges\nare given by\n\\begin{equation}\nJ^{(1)}_{ij}+J^{(2)}_{ij}+J^{(3)}_{ij}=\n\\int d\\sigma\\left[{\\cal J}^{(1)}_{\\sigma\\,ij}\n+(1+g^{\\tau\\sigma})\\,{\\cal J}^{(2)}_{\\sigma\\,ij}\n+g^{\\tau\\tau}\\, {\\cal J}^{(2)}_{\\tau\\,ij}\n+{\\cal J}^{(3)}_{\\sigma\\,ij}\\right]\n\\ ,\n\\end{equation}\nand, for $g^{\\alpha\\beta}=\\pm\\eta^{\\alpha\\beta}$, one obtains\n$J^{(2)}=\\int d\\sigma{\\cal J}^{(2)}_\\mp$.\nAgain, this will be the case for the model studied in\nSection~\\ref{iso21}.\n\\par\nTo summarize, the difference between the models in Ref.~\\cite{stern}\nand ours is given by the contribution to linear and angular momentum\nproportional to $g^{\\alpha\\beta}$ (see Eqs.~(\\ref{T}), (\\ref{J})).\n\\setcounter{equation}{0}\n\\section{The gauged action}\n\\label{gauged}\nWe can modify the action $S$ in Eq.~(\\ref{S}) in such a way as to make\nit invariant under the {\\em local\\\/} left action of a subgroup $H$\nof the whole Poincar\\'e group,\n\\begin{equation}\ng\\to h\\,g=(\\theta\\,\\Lambda, \\theta\\,x+y)\n\\ ,\n\\end{equation}\nwith $h(\\tau,\\sigma)=(\\theta,y)\\in H$.\nFor this purpose we introduce a gauge connection\n$A_\\alpha(\\tau,\\sigma)=(\\omega_\\alpha,\\xi_\\alpha)$\nand the corresponding covariant derivative\n$D_\\alpha g\\equiv\\partial_\\alpha g+A_\\alpha$.\n\\par\nWe require that $A_\\alpha$ belongs to the algebra ${\\cal H}$\nof the group $H$ (so that it has as many components as the\nelements of $H$ have). Since for every element $g\\in ISO(d-1,1)$\none has\n\\begin{equation}\nD_\\alpha(h\\,g)\\simeq\\partial_\\alpha g+\\partial_\\alpha(\\delta h\\,g)\n+A_\\alpha\n\\ ,\n\\end{equation}\nwhere $\\delta h=(\\delta\\theta,\\delta y)\\in{\\cal H}$, it follows that\n$H$ must act invariantly from the left on the elements of $ISO(d-1,1)$,\nthat is\n\\begin{equation}\n\\delta_{_L} g=\\delta h\\,g\n=(\\delta\\theta\\,\\Lambda,\\delta\\theta\\,x+\\delta y)\n\\in {\\cal H}\\ ,\n\\ \\ \\ \\forall\\, g=(\\Lambda,x)\\in ISO(d-1,1)\n\\ .\n\\end{equation}\nThe only possible non trivial choices for $H$ are then subgroups\nof the translation group $\\mbox{{\\rm I\\hspace{-2truemm} R}}^d$, that is\n\\begin{equation}\n\\left\\{\\begin{array}{l}\n\\theta=0 \\\\\n \\\\\ny\\in \\mbox{{\\rm I\\hspace{-2truemm} R}}^n\\oplus\\mbox{{\\rm 1\\hskip-1truemm I}}_{d-n}\\sim\\mbox{{\\rm I\\hspace{-2truemm} R}}^n\n\\ ,\n\\end{array}\\right.\n\\end{equation}\nwith $n\\le d$, $\\mbox{{\\rm 1\\hskip-1truemm I}}_{d-n}$ being the identity in $d-n$ dimensions,\nfor which $\\delta_{_L} g=\\delta h$, $\\forall\\, g\\in ISO(d-1,1)$.\nThus one also has the following form for the gauge field\n\\begin{equation}\n\\left\\{\\begin{array}{ll}\n\\omega_\\alpha^k\\equiv 0\\ \\ &\\ \\ k=0,\\dots,d-1 \\\\\n& \\\\\n\\xi_\\alpha^k\\equiv 0\\ \\ &\\ \\ k\\not\\in H\n\\ ,\n\\end{array}\\right.\n\\label{xi}\n\\end{equation}\nwhere $k\\not\\in H$ is a shorthand notation for $x^k\\not\\in H$.\n\\par\nFurther, the gauge field must change under an infinitesimal\n$ISO(d-1,1)$ transformation according to\n\\begin{equation}\n\\left\\{\\begin{array}{ll}\n\\xi_\\alpha^i\\to \\xi_\\alpha^i-\\partial_\\alpha(\\delta x^i)\n\\ \\ &\\ \\ i\\in H \\\\\n& \\\\\n\\xi_\\alpha^i\\to \\xi_\\alpha^i=0\n\\ \\ &\\ \\ i\\not\\in H\n\\ ,\n\\end{array}\\right.\n\\label{cond}\n\\end{equation}\nwhere $\\delta x$ is any allowed infinitesimal variation of $x\\in\\mbox{{\\rm I\\hspace{-2truemm} R}}^d$,\nincluding $(d-1)+1$ Lorentz transformations.\n\\par\nThe gauged action then reads\n\\begin{equation}\nS_g(\\Lambda,x,\\xi;K)=S_{1g}+S_{2g}+S_{3g}\n\\ ,\n\\label{S_g}\n\\end{equation}\nwith\n\\begin{eqnarray}\nS_{1g}&=&{1\\over 2}\\,\\int d^2\\sigma\\,{\\cal V}^{\\alpha\\beta}_{rs}\\,\n\\left(\\partial_\\alpha x+\\xi_\\alpha\\right)^r\\,\n\\left(\\partial_\\beta x+\\xi_\\beta\\right)^s\n\\equiv S_1+\\Delta S_1(\\xi)\n\\nonumber \\\\\nS_{2g}&=&\\int d^2\\sigma\\,{\\cal W}^{\\alpha\\beta}_{rsk}\\,\n\\left(\\partial_\\alpha x+\\xi_\\alpha\\right)^r\n\\partial_\\beta\\Lambda^{sk}\n\\equiv S_2+\\Delta S_2(\\xi) \\label{S_123g}\\\\\nS_{3g}&=&S_3\n\\ , \\nonumber\n\\end{eqnarray}\nwith ${\\cal V}$ and ${\\cal W}$ defined in Eq.~(\\ref{T}).\nThe new term\n\\begin{equation}\n\\Delta S_1=\\int d^2\\sigma\\,\\sum\\limits_{s\\in H}\\,\n\\left[{\\cal P}^{(1)\\alpha}_s\\,\\xi_\\alpha^s+\n{1\\over 2}\\,\\sum\\limits_{r\\in H}\\,\n{\\cal V}^{\\,\\alpha\\beta}_{rs}\\,\\xi^r_\\alpha\\,\\xi_\\beta^s\n\\right]\n\\ ,\n\\label{ds1}\n\\end{equation}\nis bilinear in the gauge field $\\xi$ while\n\\begin{equation}\n\\Delta S_2=\\int d^2\\sigma\\,\\sum\\limits_{s\\in H}\\,\n{\\cal P}^{(2)\\alpha}_s\\,\\xi_\\alpha^s\n\\ ,\n\\label{ds2}\n\\end{equation}\nis linear in $\\xi$ ($\\sum_{r,s,\\dots\\in H}$ means the indices $r,s,\\dots$\nare summed only over $r,s,\\dots\\in H$, while all other latin indices are\nnot restricted).\n\\subsection{Symmetries}\nWe now describe the symmetries of the new action.\nTo simplify the notation, we momentarily turn to the Euclidean case\n$ISO(d)$ which is the semidirect product of the rotation group\n$SO(d)$ with $\\mbox{{\\rm I\\hspace{-2truemm} R}}^d$ (this can be achieved by complexifying\nthe time-like variable $x^0\\mapsto i\\,x^d$).\nWe then consider the following four subgroups:\n\\begin{itemize}\n\\item\nthe two groups of $n$- and $(d-n)$-dimensional translations,\nwith $n\\le d$,\n\\begin{equation}\n\\begin{array}{lcr}\nH_n\\equiv\\mbox{{\\rm I\\hspace{-2truemm} R}}^n\\oplus\\mbox{{\\rm 1\\hskip-1truemm I}}_{d-n}\\ &\\ {\\rm and}\\ &\nH_{d-n}\\equiv\\mbox{{\\rm 1\\hskip-1truemm I}}_n\\oplus\\mbox{{\\rm I\\hspace{-2truemm} R}}^{n-d}\n\\ ,\n\\end{array}\n\\end{equation}\nsuch that $\\mbox{{\\rm I\\hspace{-2truemm} R}}^d\\sim H_n\\oplus H_{d-n}$,\n\\end{itemize}\nand\n\\begin{itemize}\n\\item\nthe following two rotation subgroups of the whole rotation group:\n\\begin{equation}\n\\begin{array}{lcr}\nR_n\\equiv SO(n)\\oplus\\mbox{{\\rm 1\\hskip-1truemm I}}_{d-n}\\ &\\ {\\rm and}\\ &\nR_{d-n}\\equiv \\mbox{{\\rm 1\\hskip-1truemm I}}_n\\oplus SO(d-n)\n\\ .\n\\end{array}\n\\end{equation}\n\\end{itemize}\nWe also write the Euclidean connection $\\bar \\xi$ as a $d$-dimensional\nvector $\\bar\\xi\\equiv(\\xi^k,\\phi^\\mu)$, $k=1,\\dots,n$,\n$\\mu=n+1,\\dots,d$, so that we separate its components into $\\xi\\in H_n$\nand $\\phi\\in H_{d-n}$.\n\\par\nThe ungauged action obtained by Euclideanizing $S$ in Eq.~(\\ref{S})\nis invariant under the left rigid action of $ISO(d)$.\nThe Euclideanized action $S_g$ obtained by gauging the group $H=H_n$\nbecomes invariant under the left local action of $H_n$, and it\nis still invariant under the left rigid action of $H_{d-n}$.\nHowever it is no longer invariant under the left rigid action of the\nwhole $SO(d)$ group because $\\Delta S_1(\\xi)$ in Eq.~(\\ref{ds1})\nand $\\Delta S_2(\\xi)$ in Eq.~(\\ref{ds2}) are not.\nIn fact, in order to preserve the invariance in $\\Delta S_1$\nand $\\Delta S_2$, the gauge field $\\bar\\xi$ must transform\naccording to the (Euclidean version) of the first constraint\nin Eq.~(\\ref{cond}),\n\\begin{equation}\n\\bar\\xi_\\alpha^i\\to\\bar\\xi_\\alpha^i-\\theta^i_{\\ l}\\,\\partial_\\alpha x^l\n\\ ,\\ \\ \\ \\ \\ \\forall\\, i=1,\\dots,d\n\\ ,\n\\end{equation}\nbut this would mix $\\xi$ components with $\\phi$ components of the\nconnection for a general rotation $\\theta\\in SO(d)$.\nAt the same time, according to Eq.~(\\ref{xi}) or, equivalently, the\nsecond constraint in Eq.~(\\ref{cond}), it must be possible to\nset $\\phi$ to zero (or $\\bar\\xi^i=0$, $\\forall\\, i=n+1,\\dots,d$),\nsince these components correspond to the group $H_{d-n}$ that we are not\ngauging.\nThis implies that the only rigid rotations that leave $S_g$ left invariant\nare the ones which do not mix $\\xi$ with $\\phi$ and thus belong\nto $R_n$ or $R_{d-n}$.\nTo summarize:\n\\begin{equation}\nS_g(\\theta\\,\\Lambda,\\theta\\,x+y,\\xi;K)=S_g(\\Lambda,x,\\xi;K)\n\\Leftrightarrow\\left\\{\n\\begin{array}{l}\ny(\\tau,\\sigma)\\in H_n\\ \\vee\\ y\\in H_{d-n} \\\\\n\\\\\n\\theta\\in R_n\\ \\vee \\ \\theta\\in R_{d-n}\n\\ .\n\\end{array}\\right.\n\\end{equation}\n\\par\nThe same argument above applied to right rigid transformations\nwould require\n\\begin{equation}\n\\bar\\xi_\\alpha\\to\\bar\\xi_\\alpha-(\\partial_\\alpha\\Lambda)\\,y\n\\ ,\\ \\ \\ \\ \\ \\forall\\,\\Lambda\\in SO(d)\n\\ ,\n\\end{equation}\nwhich necessarily mixes $\\xi$ and\n$\\phi$ components if $y\\not\\equiv 0$.\nThis singles out the whole $SO(d)$ subgroup, so that\n\\begin{equation}\nS_g(\\Lambda\\,\\theta,\\Lambda\\,y+x,\\xi;K)=S_g(\\Lambda,x,\\xi;K')\n\\Leftrightarrow\\left\\{\n\\begin{array}{l}\ny\\equiv 0 \\\\\n\\\\\n\\theta\\in SO(d)\n\\ .\n\\end{array}\\right.\n\\end{equation}\n\\par\nIt is now straightforward to translate these conclusions back to the\nLorentzian framework.\n\\par\nAs a simple corollary, one obtains that the action\n$S_g$ in Eq.~(\\ref{S_g}) is invariant under the left rigid\naction of $ISO(d-1,1)$ iff $n=0$ or $n=d$.\n\\subsection{Equations of motion}\nIn order to study the equations of motion following from the\naction $S_g$ it is more convenient to rewrite\n\\begin{equation}\nS_g=S^{(g)}+S^{(H)}+S^{(I)}\n\\ ,\n\\end{equation}\nwhere\n\\begin{equation}\nS^{(g)}=S_3+\\int d^2\\sigma\\,\\sum\\limits_{r\\not\\in H}\\,\n\\left[{\\cal W}^{\\,\\alpha\\beta}_{rpq}\\,\\partial_\\alpha x^r\\,\n\\partial_\\beta\\Lambda^{pq}+{1\\over2}\\,\\sum\\limits_{s\\not\\in H}\\,\n{\\cal V}^{\\,\\alpha\\beta}_{rs}\\,\\partial_\\alpha x^r\\,\\partial_\\beta x^s\n\\right]\n\\ ,\n\\end{equation}\ndoes not contain elements $x\\in H$,\n\\begin{equation}\nS^{(H)}=\\int d^2\\sigma\\,\\sum\\limits_{r\\in H}\\,\n\\left[{\\cal W}^{\\,\\alpha\\beta}_{rpq}\\,\\partial_\\beta\\Lambda^{pq}\n+{1\\over2}\\,\\sum\\limits_{s\\in H}\\,{\\cal V}^{\\,\\alpha\\beta}_{rs}\\,\n(\\partial_\\beta x^s+\\xi_\\beta^s)\\right]\n\\,(\\partial_\\alpha x^r+\\xi_\\alpha^r)\n\\ ,\n\\end{equation}\ncontains only terms proportional to elements of $H$, and\n\\begin{equation}\nS^{(I)}=\\int d^2\\sigma\\,\\sum\\limits_{r\\in H,s\\not\\in H}\\,\n{\\cal V}^{\\,\\alpha\\beta}_{rs}\\,(\\partial_\\alpha x^r+\\xi_\\alpha^r)\n\\,\\partial_\\beta x^s\n\\ ,\n\\end{equation}\ncontains mixed terms.\n\\par\nThe equations of motion $\\delta_x S_g=0$ together with\nthe transformation law for the gauge field in Eq.~(\\ref{cond})\nsplit the theory into two sectors:\n\\begin{enumerate}\n\\item\nwhen $\\delta x^i\\not\\in H$ one requires $\\delta_x S^{(g)}=0$ and\nobtains\n\\begin{equation}\n\\partial_\\alpha{\\cal P}^{(g)\\alpha}_{i}\\equiv\n\\partial_\\alpha\\left({\\cal P}^{(1g)\\alpha}_i\n+{\\cal P}^{(2g)\\alpha}_i\\right)=0\n\\ ,\\ \\ \\ \\ i\\not\\in H\n\\ ,\n\\label{dpg=0}\n\\end{equation}\nso that the $d-\\mbox{{\\rm dim}}(H)=d-n$ linear momentum currents\n${\\cal P}^{(g)\\alpha}_i$ with $i\\not\\in H$ and\n\\begin{equation}\n\\begin{array}{l}\n{\\cal P}^{(1g)\\alpha}_{i}=\\sum\\limits_{j\\not\\in H}\\,\n{\\cal V}^{\\,\\alpha\\beta}_{ij}\\,\\partial_\\beta x^j \\\\\n{\\cal P}_{i}^{(2g)\\alpha}={\\cal P}_{i}^{(2)\\alpha}\n\\ ,\n\\end{array}\n\\end{equation}\nare still conserved;\n\\item\nwhen $\\delta x^i\\in H$ one gets $\\delta_x S^{(H)}=\\delta_x S^{(I)}=0$\nidentically and thus the corresponding currents\n\\begin{equation}\n{\\cal P}^{(H)\\alpha}_i\\equiv{\\cal P}^{\\alpha}_i-\n{\\cal P}^{(g)\\alpha}_r=\\sum\\limits_{s\\in H}\\,\n{\\cal V}^{\\,\\alpha\\beta}_{is}\\,\\partial_\\beta x^s\n\\ ,\\ \\ \\ \\ i\\in H\n\\label{Ph}\n\\end{equation}\nare not conserved.\n\\end{enumerate}\n\\par\nFrom the variation $\\delta_\\Lambda S_g=0$ and Eq.~(\\ref{cond})\none obtains an analogous splitting into three sectors:\n\\begin{enumerate}\n\\item\nin the sector for which\n$\\delta \\Lambda_{kj}=\\Lambda_k^{\\ i}\\,\\rho_{ij}$, $i,j\\not\\in H$\none has\n\\begin{equation}\n\\partial_\\alpha\\left[{\\cal J}^{\\alpha}_{ij}\n+2\\,\\sum\\limits_{r\\in H}\\,\n{\\cal W}_{rij}^{\\,\\alpha\\beta}\\,\\xi_\\beta^r\n+x_i\\wedge \\sum\\limits_{r\\in H}\\,{\\cal V}_{jr}^{\\,\\alpha\\beta}\\,\n\\xi_\\beta^r\\right]=0\n\\ \\ \\ \\ \\ i,j\\not\\in H\n\\ .\n\\label{djg=0}\n\\end{equation}\nThus the angular momentum currents ${\\cal J}^{\\alpha}_{ij}$\nwith both indices $i,j\\not\\in H$ are no longer conserved\nbut couple to the gauge field;\n\\item\nwhen $\\delta \\Lambda_{kj}=\\Lambda_k^{\\ i}\\,\\rho_{ij}$, $i,j\\in H$\none obtains an analogous relation\n\\begin{equation}\n\\partial_\\alpha\\left[{\\cal S}_{ij}^\\alpha\n+2\\,\\sum\\limits_{r\\in H}\\,{\\cal W}^{\\,\\alpha\\beta}_{rij}\\,\n\\xi_\\beta^r\\right]=\n\\left({\\cal P}^\\alpha_i+\\sum\\limits_{r\\in H}\\,\n{\\cal V}_{ir}^{\\,\\alpha\\beta}\\,\\xi_\\beta^r\\right)\\wedge\n\\left(\\partial_\\alpha x_j+\\xi_{\\alpha j}\\right)\n\\ \\ \\ \\ i,j\\in H\\ ;\n\\label{H_mix}\n\\end{equation}\nin which only the spin ${\\cal S}_{ij}$ appears on the L.H.S. because\nnow $\\partial_\\alpha{\\cal P}^\\alpha_i\\not=0$, $i\\in H$;\n\\item\nfinally, in the sector in which\n$\\delta \\Lambda_{kj}=\\Lambda_k^{\\ i}\\,\\rho_{ij}$, $i\\not\\in H$, $j\\in H$\none obtains the following non trivial equation\n\\begin{eqnarray}\n&\\partial_\\alpha\\left[{\\cal J}^{\\alpha}_{ij}\n+2\\,\\sum\\limits_{r\\in H}\\,\n{\\cal W}_{rij}^{\\,\\alpha\\beta}\\,\\xi_\\beta^r\n+x_i\\wedge \\sum\\limits_{r\\in H}\\,{\\cal V}_{jr}^{\\,\\alpha\\beta}\\,\n\\xi_\\beta^r\\right]=&\n\\nonumber \\\\\n&\\left({\\cal P}^\\alpha_i+\\sum\\limits_{r\\in H}\\,\n{\\cal V}_{ir}^{\\,\\alpha\\beta}\\,\\xi_\\beta^r\\right)\\,\n\\left(\\partial_\\alpha x_j+\\xi_{\\alpha j}\\right)\n-\\partial_\\alpha x_i\\,\n\\left({\\cal P}^\\alpha_j+\\sum\\limits_{r\\in H}\\,\n{\\cal V}_{jr}^{\\,\\alpha\\beta}\\,\\xi_\\beta^r\\right)\\, ,\n\\ \\ i\\not\\in H,\\, j\\in H&\n\\ .\n\\label{H_sec}\n\\end{eqnarray}\nwhich mixes terms from the two previous sectors.\n\\end{enumerate}\n\\par\nTo summarize, whenever one considers only quantities which\ndo not contain elements of $H$, the equations of motion for the linear\nmomenta look the same as the ungauged ones and amount again to linear\nmomentum conservation.\nThe angular momentum, instead, is no longer conserved, even\nin that sector of the theory,\nbecause of the presence of $S^{(H)}$ and $S^{(I)}$.\nFurther, due to these latter contributions to the action,\nthe constraints displayed in Eqs.~(\\ref{H_mix}), (\\ref{H_sec})\nmust also be satisfied, together with the equations of motion\nfor the gauge field, $\\delta_\\xi S_g=0$,\nwhich we will study in the following Subsection.\n\\subsection{Eliminating $\\xi$: the quadratic case}\nNow that we have introduced the gauge field $\\xi$,\nwe want to eliminate it from the action.\nOne way to achieve this goal is to integrate out $\\xi$ first\nin the path integral\n\\begin{equation}\n\\int[d\\Lambda]\\,[dx\\not\\in{\\cal H}]\\,[dx\\in{\\cal H}]\\,[d\\xi]\\,\ne^{-S(\\Lambda,x)-\\Delta S(\\Lambda,x,\\xi)}\n\\equiv\n\\int[d\\Lambda]\\,[dx\\not\\in{\\cal H}]\\,\\,e^{-S_{eff}(\\Lambda,x)}\n\\ ,\n\\label{path}\n\\end{equation}\nwhere, from Eqs.~(\\ref{ds1}), (\\ref{ds2}) one has\n\\begin{equation}\n\\Delta S\\equiv\\Delta S_1+\\Delta S_2=\\int d^2\\sigma\\,\n\\sum\\limits_{s\\in H}\\,\\left[{\\cal P}^\\alpha_s\\,\\xi^s_\\alpha+\n{1\\over 2}\\,\\sum\\limits_{r\\in H}\\,\n{\\cal V}^{\\,\\alpha\\beta}_{rs}\\,\\xi^r_\\alpha\\,\\xi^s_\\beta\n\\right]\n\\ ,\n\\end{equation}\nwith ${\\cal P}^{\\alpha}_s$ defined in Eq.~(\\ref{dp=0}).\n\\par\nWhen ${\\cal V}^{\\,\\alpha\\beta}_{rs}\\not=0$, for some $r,s\\in H$,\n$\\Delta S$ is quadratic in $\\xi$ and the above mentioned integration\ncorresponds to solving the equations of motion for $\\xi$,\nnamely $\\delta_\\xi\\Delta S=0$,\nand substituting back the result into the action.\nWe also notice that, since\n${\\cal V}^{\\,\\alpha\\beta}=-{\\cal V}^{\\,\\beta\\alpha}$,\nthe condition ${\\cal V}^{\\,\\alpha\\beta}_{rs}\\not=0$\nimplies that $\\mbox{{\\rm dim}}(H)=n \\ge 2$ and thus excludes\nthe ungauged case.\n\\par\nIt is easy to find that $\\delta_\\xi\\Delta S=0$ implies\n\\begin{equation}\n{\\cal P}^{\\alpha}_s+\\sum\\limits_{r\\in H}\\,\n{\\cal V}^{\\,\\alpha\\beta}_{sr}\\,\\xi^r_\\beta=0\n\\ ,\\ \\ \\ \\ s\\in H\n\\ .\n\\label{xi=P}\n\\end{equation}\nOne can try to go further if we assume that\n${\\cal V}$ is invertible inside the $H$ sector\n(this will put constraints on the constants $K^{(1)}$\nand could also exclude part of the subgroup $SO(d-1,1)$)\nand define ${\\cal V}_{\\alpha\\beta}^{\\,rs}$ such that\n\\begin{equation}\n\\sum\\limits_{a\\in H}\\,{\\cal V}_{\\alpha\\gamma}^{\\,ra}\\,\n{\\cal V}^{\\,\\gamma\\beta}_{as}=\\delta_\\alpha^\\beta\\,\\delta^r_s\n\\ ,\\ \\ \\ \\ \\forall\\,r,s\\in H\n\\ .\n\\end{equation}\nThen Eq.~(\\ref{xi=P}) above can be inverted to give\n\\begin{equation}\n\\xi^r_\\alpha=-\\sum\\limits_{s\\in H}\\,\n{\\cal V}_{\\alpha\\beta}^{\\,rs}\\,{\\cal P}^{\\beta}_s\n\\ ,\\ \\ \\ \\ r\\in H\n\\ ,\n\\label{xi=}\n\\end{equation}\nand the correction to the (ungauged) action becomes\n\\begin{equation}\n\\Delta S=-{1\\over 2}\\,\\int d^2\\sigma\\,\n\\sum\\limits_{r,s\\in H}\\,{\\cal P}^{\\alpha}_r\\,\n{\\cal V}_{\\alpha\\beta}^{\\,rs}\\,{\\cal P}^{\\beta}_s\n\\ .\n\\label{corr}\n\\end{equation}\nBy simply expanding the linear momentum current ${\\cal P}$\naccording to Eq.~(\\ref{Ph}) one then easily proves\nthat $\\Delta S$ cancels out every dependence on $x\\in H$ from\nthe (ungauged) action, since\n\\begin{eqnarray}\n&S_1(x\\not\\in H,x\\in H)-{1\\over 2}\\,\n\\int d^2\\sigma\\,\\sum\\limits_{r,s\\in H}\\,{\\cal P}^{(1H)\\alpha}_r\\,\n{\\cal V}^{rs}_{\\alpha\\beta}\\,\\left({\\cal P}^{(1H)\\beta}_s\n+2\\,{\\cal P}^{(1g)\\beta}_s\\right)=\nS_1(x\\not\\in H)&\n\\nonumber \\\\\n&S_2(x\\not\\in H, x\\in H)-\n\\int d^2\\sigma\\,\\sum\\limits_{r,s\\in H}\\,{\\cal P}^{(1H)\\alpha}_r\\,\n{\\cal V}^{rs}_{\\alpha\\beta}\\,{\\cal P}^{(2g)\\beta}_s=\nS_2(x\\not\\in H)&\n\\ .\n\\end{eqnarray}\nThe remaining terms in Eq.~(\\ref{corr}) lead to corrections\nto the three (ungauged) contributions of the action\naccording to the following scheme\n\\begin{eqnarray}\nS^q_{1eff}(x\\not\\in H)&=&S_1(x\\not\\in H)-{1\\over 2}\\,\\int\nd^2\\sigma\\,\\sum\\limits_{r,s\\in H}\\,{\\cal P}^{(1g)\\alpha}_r\\,\n{\\cal V}^{rs}_{\\alpha\\beta}\\,{\\cal P}^{(1g)\\beta}_s\n\\nonumber \\\\\nS^q_{2eff}(x\\not\\in H)&=&S_2(x\\not\\in H)-\n\\int d^2\\sigma\\,\\sum\\limits_{r,s\\in H}\\,{\\cal P}^{(1g)\\alpha}_r\\,\n{\\cal V}^{rs}_{\\alpha\\beta}\\,{\\cal P}^{(2g)\\beta}_s\n\\nonumber \\\\\nS^q_{3eff}&=&S_3-\n{1\\over 2}\\,\\int d^2\\sigma\\,\\sum\\limits_{r,s\\in H}\\,\n{\\cal P}^{(2g)\\alpha}_r\\,{\\cal V}^{rs}_{\\alpha\\beta}\\,\n{\\cal P}^{(2g)\\beta}_s\n\\ .\n\\label{cor_q}\n\\end{eqnarray}\nThis defines a total effective action\n\\begin{equation}\nS^q_{eff}(\\Lambda,x\\not\\in H,K)=S^q_{1eff}+S^q_{2eff}+S^q_{3eff}\n\\ ,\n\\end{equation}\nwhich differs from the one obtained by simply setting the terms\ncontaining $x\\in H$ to zero.\nThis is due to the presence of {\\em cross terms} of the kind\ndiscussed in the Introduction.\n\\par\nWe observe that the number of degrees of freedom in the\neffective action $S^q_{eff}$ is at most\n$N\\equiv\\mbox{{\\rm dim}}(ISO(d-1,1))-\\mbox{{\\rm dim}}(H)=d\\,(d+1)\/2-n$, $n1}\\,\\left[\\tilde D^{\\alpha\\beta}_{1b}\\,\n\\partial_\\beta\\tilde x^b-2\\,\\Sigma_{1b}^{\\alpha\\beta}\\,\n\\partial_\\beta x^b\\right]\n\\ ,\n\\end{equation}\nfor the corresponding $U(1)$ symmetry, in which we singled out\nthe first term in the sum.\n\\par\nIf the terms with $b>1$ vanish in the expression for the current\nabove, the ungauged action (before one doubles $x^1$) reads\n\\begin{equation}\nS_{eff}=S_t+\\int d^2\\sigma\\,N^{\\alpha\\beta}_{1i}\\,\\partial_\\alpha x^1\\,\n\\partial_\\beta t^i\n\\ ,\n\\label{1-dim}\n\\end{equation}\nwhile the gauged action becomes\n\\begin{equation}\nS_{N+d-n}^g(x,\\tilde x,\\tilde A^1)=S_{N+d-n}-\n\\int d^2\\sigma\\,\\tilde A_\\alpha^1\\,g^{\\alpha\\beta}\\,\\Sigma^S_{11}\\,\n\\left[\\partial_\\beta \\tilde x^1+\\partial_\\beta x^1\n+\\tilde A_\\beta^1\\right]\n\\ ,\n\\end{equation}\nand one obtains\n\\begin{equation}\nS_N(x,t)=S_t+\\int d^2\\sigma\\,\\left[g^{\\alpha\\beta}\\,\\Sigma^S_{11}\\,\n\\partial_\\alpha x^1\\,\\partial_\\beta x^1+\nN^{\\alpha\\beta}_{1i}\\,\\partial_\\alpha x^1\\,\\partial_\\beta t^i\\right]\n\\ ,\n\\end{equation}\nwhere the metric tensor $\\bar G$ has now acquired the new component\n$G_{xx,11}=2\\,\\Sigma^S_{11}$.\n\\setcounter{equation}{0}\n\\section{The lowest dimensional case: $ISO(1,1)$}\n\\label{d=2}\nWe now consider the $(d=2)$-dimensional case for which the\nalgebra is simple enough to allow one to carry the computation\nto the end.\n\\par\nEvery element $\\Lambda\\in SO(1,1)$ can be written as function\nof the only boost parameter $t\\in\\mbox{{\\rm I\\hspace{-2truemm} R}}$,\n\\begin{equation}\n\\begin{array}{lcr}\n\\Lambda^i_{\\ j}=\\left[\\begin{array}{cc}\n\\cosh t&\\sinh t \\\\\n\\sinh t&\\cosh t\n\\end{array}\\right]\n&\\ \\ \\ \\ \\ \\ \\ &\n\\Lambda^{\\ i}_j=\\left[\\begin{array}{cc}\n\\cosh t&-\\sinh t \\\\\n-\\sinh t&\\cosh t\n\\end{array}\\right]\n\\ .\n\\end{array}\n\\end{equation}\nThe relevant 1-forms in Eq.~(\\ref{gdg}) become\n\\begin{equation}\n\\begin{array}{lr}\nV^i=\\left[\\begin{array}{l}\n\\cosh t\\,dx^1-\\sinh t\\,dx^2 \\\\\n\\cosh t\\,dx^2-\\sinh t\\,dx^1\n\\end{array}\\right]\n\\ \\ &\\ \\\nW^{ij}=\\left[\\begin{array}{cc}\n0&+1\\\\\n-1&0\n\\end{array}\\right]\\,dt\n\\ .\n\\end{array}\n\\end{equation}\nIt is then easy to find that\n\\begin{eqnarray}\nS_1&=&S_3=0\n\\nonumber \\\\\nS_2&=&\\int d^2\\sigma\\,g^{\\alpha\\beta}\\,\n\\left[\\left(K_2\\,\\sinh t-K_1\\,\\cosh t\\right)\\,\n\\partial_\\alpha x^1\\,\\partial_\\beta t \\right.\n\\nonumber \\\\\n&&\\left.\\phantom{2\\,\\int d^2\\sigma\\,\\epsilon^{\\alpha\\beta}\\,[}\n+\\left(K_1\\,\\sinh t-K_2\\,\\cosh t\\right)\\,\n\\partial_\\alpha x^2\\,\\partial_\\beta t\\right]\n\\ ,\n\\label{S(1,1)}\n\\end{eqnarray}\nwhere total derivatives are discarded as usual and\n$(K_1,K_2)$ are the only relevant independent constants satisfying\nEq.~(\\ref{K_cond}) in $d=2$.\n\\par\nThe (conserved) linear momentum currents related to $S_2$ are given by\n\\begin{eqnarray}\n{\\cal P}^\\alpha_1&=&g^{\\alpha\\beta}\\,\n\\left(K_1\\,\\cosh t-K_2\\,\\sinh t\\right)\\,\\partial_\\beta t\n\\nonumber \\\\\n{\\cal P}^\\alpha_2&=&g^{\\alpha\\beta}\\,\n\\left(K_2\\,\\cosh t-K_1\\,\\sinh t\\right)\\,\\partial_\\beta t\n\\ .\n\\end{eqnarray}\nUpon varying the parameter $t$ one finds that the following quantity is\nalso conserved\n\\begin{eqnarray}\n{\\cal J}^\\alpha&=&{\\cal P}^\\alpha_1\\,x^2- {\\cal P}^\\alpha_2\\,x^1\n+g^{\\alpha\\beta}\\,\n\\left(K_1\\,\\cosh t-K_2\\,\\sinh t\\right)\\,\\partial_\\beta x^1\n\\nonumber \\\\\n&&+\ng^{\\alpha\\beta}\\,\n\\left(K_2\\,\\cosh t-K_1\\,\\sinh t\\right)\\,\\partial_\\beta x^2\n\\ .\n\\end{eqnarray}\nIt is quite obvious that, since only the term proportional to\n$g^{\\alpha\\beta}$ survives, no axion field will appear\nin the resulting $\\sigma$-models.\nFurther, one can gauge only 1-dimensional subgroups,\nsince eliminating both $x^1$ and $x^2$ leads to $S^{pg}_{eff}=0$.\n\\subsection{Gauging a 1-dimensional subgroup}\nSince $S_2$ is linear in both $x^1$ and $x^2$,\nif we gauge a 1-dimensional subgroup, {\\em e.g.\\\/} the one corresponding\nto $x^2$, we obtain that the gauge field $A^2$ is a pure gauge,\n$A^2=-\\partial x^2$.\nThis is actually the first of the two exceptional cases listed\nin Section~\\ref{linear}.\n\\par\nThe conserved currents which survive are given by\n${\\cal P}_1$ and ${\\cal J}(x^2=0)$ above.\nWe then define\n\\begin{equation}\n\\left\\{\\begin{array}{l}\nX\\equiv x^1+t \\\\\n\\\\\nT\\equiv x^1-t\n\\ ,\n\\end{array}\\right.\n\\end{equation}\nand we obtain\n\\begin{eqnarray}\nS^{pg}_{2eff}&=&{1\\over 2}\\,\\int d^2\\sigma\\,\ng^{\\alpha\\beta}\\,f(X-T)\\,\\left(\n\\partial_\\alpha X\\,\\partial_\\beta X-\n\\partial_\\alpha T\\,\\partial_\\beta T \\right)\n\\ ,\n\\end{eqnarray}\nwhere $f(t)\\equiv K_2\\,\\sinh t-K_1\\,\\cosh t$.\nThe diagonal form of the metric tensor is thus given by\n\\begin{equation}\n\\bar G=\\left[\\begin{array}{cc}\nf & 0 \\\\\n0 & -f\n\\end{array}\\right]\n\\ ,\n\\label{G2}\n\\end{equation}\nwhich becomes singular (both components vanish)\nfor $f(t_s)=0\\ \\Leftrightarrow \\ \\tanh t_s=K_1\/K_2$, $K_2\\not=0$.\n\\par\nThe curvature of space-time and the Ricci tensor are zero\neverywhere, so $\\bar G$ above represents a vacuum solution.\nThe singularity $t=t_s$ is a light-like volume singularity whose location\ndepends on the ratio of the constants $K_1\/K_2$.\nFor example, when $K_1=0$ and $K_2=1$, one finds that\n$\\bar G$ becomes singular along the light-cone $X=T$.\n\\subsection{T-dual form}\nThe action $S^{pg}_{eff}$ is of the form given in Eq.~(\\ref{1-dim})\nwith $S_t=\\int d^2\\sigma\\,\\Phi\\,R^{(2)}$ and\n$N^{\\alpha\\beta}_{11}=2\\,g^{\\alpha\\beta}\\,f(t)$.\nIf we introduce a coordinate $x_2^1$ and define $x$ and\n$\\tilde x$ according to Eq.~(\\ref{x=x}),\nwe can then dualize with respect to the coordinate $\\tilde x$ and\nobtain a new metric tensor whose diagonal form is given by\n\\begin{equation}\n\\bar G=\\left[\\begin{array}{cc}\n\\Sigma^S_{11}-\\sqrt{(\\Sigma^S_{11})^2+f^2(t)}\n& 0 \\\\\n0 &\n\\Sigma^S_{11}+\\sqrt{(\\Sigma^S_{11})^2+f^2(t)}\n\\end{array}\\right]\n\\ .\n\\label{G2T}\n\\end{equation}\nRegardless of the explicit form of $\\Sigma^S_{11}=\\Sigma^S_{11}(t)$,\n$\\det(\\bar G)=f^2$ and one obtains the same volume singularity for\n$f(t_s)=0$.\nThe scalar curvature is again zero everywhere.\n\\par\nThe causal structure determined by $\\bar G$ in Eq.~(\\ref{G2T})\nis different from the one given by the metric tensor in Eq.~(\\ref{G2}),\nsince in the latter case one has an overall change of sign when\ngoing through $f=0$, however in the former this cannot happen.\n\\par\nFurther, according to Eq.~(\\ref{Phi}), since the Ricci tensor is\nstill zero, the presence of a non-vanishing $\\Sigma^S_{11}(t)$\ndoes not affect the dilaton.\n\\setcounter{equation}{0}\n\\section{$S=S_2$ in $d=3$ dimensions}\n\\label{iso21}\nThe second exceptional case listed in Section~\\ref{linear}\nhas $d=3$ and $n=0$.\nSince we are mainly interested in the metric structure of the\neffective theory, we only consider $S=S_2(K)$ according to the\ngeneral form given in Eq.~(\\ref{Gpg}).\n\\par\nFirst we prove that, when $g^{\\alpha\\beta}=\\eta^{\\alpha\\beta}$\nthere is actually only one such action,\nnamely the one with $K^{(2)}_{ijk}=\\epsilon_{ijk}$,\nand one recovers the model previously studied in Ref.~\\cite{ch}.\nIn fact, every matrix $K^{(2)}_{ijk}$ with the symmetry properties\ndisplayed in Eq.~(\\ref{K_cond}) can be written\n\\begin{equation}\nK^{(2)}_{ijk}=\\sum\\limits_{l=0}^2\\,\nA^{(l)}_i\\,\\Sigma^{(l)}_{jk}\n\\ ,\n\\end{equation}\nwhere\n\\begin{equation}\n\\begin{array}{ccc}\n\\Sigma^{(0)}=\n\\left[\\begin{array}{ccc}\n0 & 1 & 0 \\\\\n-1 & 0 & 0 \\\\\n0 & 0 & 0\n\\end{array}\\right]\\ ,\\ \\ &\n\\Sigma^{(1)}=\n\\left[\\begin{array}{ccc}\n0 & 0 & -1 \\\\\n0 & 0 & 0 \\\\\n1 & 0 & 0\n\\end{array}\\right]\\ ,\\ \\ &\n\\Sigma^{(2)}=\n\\left[\\begin{array}{ccc}\n0 & 0 & 0 \\\\\n0 & 0 & 1 \\\\\n0 & -1 & 0\n\\end{array}\\right]\\ ,\\\n\\end{array}\n\\end{equation}\nand $A_i^{(l)}$ are nine arbitrary constants such that\n$\\det(A_i^{(l)})\\not=0$.\nThe action $S_2$ then can be written\n\\begin{eqnarray}\nS_{(2,1)}(\\Lambda,y;K)&=&\n{1\\over 2}\\,\\int d^2\\sigma\\,\nA^{(l)}_i\\,\\Sigma^{(l)}_{jk}\\,\\partial_-y^i\n(\\partial_+\\Lambda\\,\\Lambda^{-1})^{jk}\n\\nonumber \\\\\n&=&{1\\over 2}\\,\\int d^2\\sigma\\,A^{(l)}_i\\partial_-y^i\n\\,\\mbox{{\\rm Tr}}\\left[\\Sigma^{(l)}\\,\n(\\partial_+\\Lambda\\,\\Lambda^{-1})\\right]\n\\ ,\n\\end{eqnarray}\nwhere $y\\in\\mbox{{\\rm I\\hspace{-2truemm} R}}^3$ and $\\sigma^\\pm$ have been defined in\nEq.~(\\ref{light}).\nThe trace in the integrand above can now be evaluated assuming\na specific parameterization of the Lorentz group $SO(2,1)$.\nAs in Ref.~\\cite{ch}, we write any matrix $\\Lambda^i_{\\ j}$\nas a product of two rotations (of angles $\\alpha$ and $\\gamma$)\nand a boost ($\\beta$),\n\\begin{equation}\n\\Lambda=\n\\left[\\begin{array}{ccc}\n1 & 0 & 0 \\\\\n0 & \\cos\\alpha & -\\sin\\alpha \\\\\n0 & \\sin\\alpha & \\cos\\alpha\n\\end{array}\\right]\\,\n\\left[\\begin{array}{ccc}\n\\cosh\\beta & 0 & \\sinh\\beta \\\\\n0 & 1 & 0 \\\\\n\\sinh\\beta & 0 & \\cosh\\beta\n\\end{array}\\right]\\,\n\\left[\\begin{array}{ccc}\n1 & 0 & 0 \\\\\n0 & \\cos\\gamma & -\\sin\\gamma \\\\\n0 & \\sin\\gamma & \\cos\\gamma\n\\end{array}\\right]\n\\ ,\n\\end{equation}\nand we obtain\n\\begin{eqnarray}\n\\mbox{{\\rm Tr}}\\left[\\Sigma^{(0)}\\,\n(\\partial_+\\Lambda\\,\\Lambda^{-1})\\right]\n&\\equiv&\n{\\cal P}^0_+=\\partial_+t^0+\\cosh t^1\\,\\partial_+ t^2 \\\\\n\\nonumber \\\\\n\\mbox{{\\rm Tr}}\\left[\\Sigma^{(1)}\\,\n(\\partial_+\\Lambda\\,\\Lambda^{-1})\\right]\n&\\equiv&\n{\\cal P}^1_+=\\cos t^0\\,\\partial_+ t^1\n+\\sin t^0\\,\\sinh t^1\\,\\partial_+t^2 \\\\\n\\nonumber \\\\\n\\mbox{{\\rm Tr}}\\left[\\Sigma^{(2)}\\,\n(\\partial_+\\Lambda\\,\\Lambda^{-1})\\right]\n&\\equiv&\n{\\cal P}^2_+=\\sin t^0\\,\\partial_+ t^1\n-\\cos t^0\\,\\sinh t^1\\,\\partial_+t^2\n\\ .\n\\label{P21}\n\\end{eqnarray}\nFinally, on defining\n\\begin{equation}\nx^i\\equiv\\sum\\limits_{j=0}^2\\,A^{(i)}_j\\,y^j\n\\ ,\n\\label{x=y}\n\\end{equation}\none gets\n\\begin{equation}\nS_{(2,1)}(\\Lambda,y;K)=\n\\int d^2\\sigma\\,{\\cal P}_+^k\\,\\partial_-x_k\n=S_{(2,1)}(\\Lambda,x;\\epsilon_{ijk})\n\\ ,\n\\label{2+1}\n\\end{equation}\nas claimed.\n\\par\nThis result is very peculiar and follows from the fact that\nthere are $d^2\\,(d-1)\/2$ independent elements in $K^{(2)}_{ijk}$.\nThis number is equal to the square of the space-time dimension\nfor $d=3$ and one can thus use all these constants to build\nthe linear combination given in Eq.~(\\ref{x=y}).\nIn general (for $d>3$) one has\n\\begin{equation}\nd^2\\,(d-1)\/2>d^2\n\\ ,\n\\end{equation}\nand one can not eliminate in this way $d^2\\,(d-3)\/2$ elements of\n$K^{(2)}_{ijk}$.\n\\par\nThe three linear momentum currents in Eq.~(\\ref{P21})\ndefine a metric tensor $\\bar G$ in 6 dimensions of the\nform given in Eq.~(\\ref{Gpg}) with\n\\begin{equation}\nU_{xt}=\\left[\\begin{array}{ccc}\n1 & 0 & \\cosh t^1 \\\\\n0 & \\cos t^0 & \\sin t^0\\,\\sinh t^1 \\\\\n0 & \\sin t^0 & -\\cos t^0\\,\\sinh t^1\n\\end{array}\\right]\n\\ .\n\\end{equation}\n\\par\nThe Ricci tensor computed from the metric $\\bar G$ has the following\nnon-zero components,\n\\begin{eqnarray}\n\\bar R_{t^0 t^0}&=&-1\n\\nonumber\\\\\n\\bar R_{t^0 t^2}&=&-\\cosh(t^1)\n\\nonumber\\\\\n\\bar R_{t^1 t^1}&=&1\n\\nonumber\\\\\n\\bar R_{t^2 t^2}&=&-1\n\\ ,\n\\end{eqnarray}\nand its trace $\\bar R=0$.\nThe subspace $t^1=0$ is a volume singularity, since\n\\begin{equation}\n\\det(\\bar G)=f^2(t^1)\n\\ ,\n\\label{g}\n\\end{equation}\nwith $f\\equiv\\sinh t^1$.\n\\par\nWe now analyze two degenerate cases following from $S_{(2,1)}$.\n\\subsection{Gauging a 1-dimensional translation}\nWe notice that $S_{(2,1)}$ is already invariant under the following\n{\\em semi\\\/}-local action of the Poincar\\'e group:\n\\begin{equation}\ng\\to h_{_L}(\\sigma^+)\\,g\\,h_{_R}^{-1}(\\sigma^-)\n\\ ,\n\\end{equation}\nwhere $h_{_{L\/R}}=(\\theta_{_{L\/R}},y_{_{L\/R}})\\ \\in\\ ISO(2,1)$.\nHowever, it is not invariant under the {\\em fully\\\/}\nlocal action of any subgroup $H$ of $ISO(2,1)$ given by\n$g\\to h_{_L}\\,g\\,h_{_R}^{-1}=\n\\left(\\theta_{_L}\\,\\Lambda\\,\\theta^{-1}_{_R},\n-\\theta_{_L}\\,\\Lambda\\,\\theta^{-1}_{_R}\\,y_{_R}+\n\\theta_{_L}\\,x+y_{_L}\\right)$,\nwhere $h_{_{L\/R}}=h_{_{L\/R}}(\\sigma^-,\\sigma^+)=\n(\\theta_{_{L\/R}},y_{_{L\/R}})\\in H$,\ndue to the dependence of $h_{_L}$ on $\\sigma^-$\nand of $h_{_R}$ on $\\sigma^+$.\nTo promote $H$ to a gauge symmetry of the action we introduce again\nthe gauge field $A_\\pm=\\left(\\omega_\\pm,\\xi_\\pm\\right)\\in iso(2,1)$,\nand the covariant derivatives $D_\\pm g=\\partial_\\pm g+A_\\pm$.\nThe requirement that $H$ acts invariantly,\n$\\delta g=h_{_L}\\,g=(0,y_{_L})$,\nleads to $h_{_{L\/R}}=(0,y_{_{L\/R}}\\in\\mbox{{\\rm I\\hspace{-2truemm} R}}^n)$, $n\\le 3$,\nso that\n\\begin{equation}\n\\left\\{\\begin{array}{lr}\n\\omega_\\pm=\\xi_+ \\equiv 0 &\\\\\n \\\\\n\\xi_-^k \\equiv 0 & k\\not\\in H\n\\ .\n\\end{array}\\right.\n\\end{equation}\n\\par\nIf we gauge the 1-dimensional subgroup corresponding to $x^0$,\nthe gauged action $S_g(x,t,\\xi^0_-)$ is linear in $\\xi_-^0$\nand after we eliminate $t^0$ as an irrelevant parameter, we obtain the\neffective action \\cite{ch}\n\\begin{eqnarray}\nS^{pg}_{eff}(x^1,x^2,t^1,t^2)&=&\\int d^2\\sigma\\,\\left[\n-\\partial_+ t^1\\,\\partial_- x^1+\n\\sinh t^1\\,\\partial_+ t^2\\,\\partial_- x^2\\right]\n\\nonumber \\\\\n&=&\\int d^2\\sigma\\,\\left(\\eta^{\\alpha\\beta}\n+\\epsilon^{\\alpha\\beta}\\right)\\,\\left[\n\\partial_\\alpha t^1\\,\\partial_\\beta x^1\n-f\\,\\partial_\\alpha t^2\\,\\partial_\\beta x^2\\right]\n\\ ,\n\\label{S_4}\n\\end{eqnarray}\nwhere again $f=\\sinh t^1$.\nThe metric tensor is $N-N_d=5-1=4$-dimensional and\nis given by\n\\begin{equation}\n\\bar G=\\left[\\begin{array}{cccc}\n0 & 0 & 1 & 0 \\\\\n0 & 0 & 0 & -f \\\\\n1 & 0 & 0 & 0 \\\\\n0 & -f & 0 & 0\n\\end{array}\\right]\n\\ .\n\\label{G21}\n\\end{equation}\nThe axion field potential in this non-degenerate 4-dimensional\nsubspace is given by\n\\begin{equation}\n\\bar B=2\\,\\left[\\begin{array}{cccc}\n0 & 0 & 1 & 0 \\\\\n0 & 0 & 0 & -f \\\\\n-1 & 0 & 0 & 0 \\\\\n0 & f & 0 & 0\n\\end{array}\\right]\n\\ ,\n\\label{B21}\n\\end{equation}\nits field strength $\\bar H$ having $\\bar H_{124}=-2\\,\\cosh t^1$ as\nthe only non-zero component.\n\\par\nThe Ricci tensor in this frame of reference has one\nnon-vanishing component,\n\\begin{equation}\n\\bar R_{t^1 t^1}={1\\over 2}\\,{\\sinh^2 t^1-1\\over\\sinh^2 t^1}\n\\ ,\n\\end{equation}\nand the scalar curvature $\\bar R$ is zero.\n\\par\nThe signature of the metric is $2+2$ and never changes,\nas can be inferred by noting that the determinant of\n$\\bar G$ is again given by the expression in Eq.~(\\ref{g}),\nand the eigenvalues of $\\bar G$ are given by $(\\pm 1,\\pm f)$.\nThis corrects an erroneous statement in the last part of\nRef.~\\cite{ch}.\n\\subsection{T-dual form}\nBoth the metric tensor $\\bar G$ and the axion field $\\bar B$ given\nabove depend only on $t^1$, so that in this case we have a Lorentz\nparameter ($t^2$) which is irrelevant, as are the two translational\nparameters $x^1$ and $x^2$.\n\\par\nThe dual effective theory obtained upon introducing\n$x$ and $\\tilde x$ as given in Eq.~(\\ref{x=x}) with $x^1_1\\equiv x^1$\ncontains the same axion field potential $\\bar B$ given in\nEq.~(\\ref{B21}) above, but the metric tensor acquires a new component\n\\begin{equation}\n\\bar G_{xx}=2\\,\\Sigma(t^1)\n\\ .\n\\end{equation}\nThis extra term generates new non-zero components for the Ricci\ntensor,\n\\begin{eqnarray}\n\\bar R_{x x}&=&2\\,\\Sigma\\,\\dot\\Sigma\\,{\\sqrt{f^2+1}\\over f}\n+\\ddot\\Sigma\n\\nonumber\\\\\n\\bar R_{x t^1}&=&-\\ddot\\Sigma\n-\\dot\\Sigma\\,{f\\over\\sqrt{f^2+1}}\n\\nonumber\\\\\n\\bar R_{x^2 t^2}&=&\\Sigma\\,f\n+\\dot\\Sigma\\,\\sqrt{f^2+1}\n\\ ,\n\\end{eqnarray}\nwhere $\\dot\\Sigma\\equiv\\partial_{t^1}\\Sigma$,\nand a non-vanishing scalar curvature,\n\\begin{equation}\n\\bar R=-3\\,\\Sigma-2\\,\\ddot\\Sigma+{\\Sigma\\over f^2}\n-4\\,\\sqrt{f^2+1}\\,{\\dot\\Sigma\\over f}\n\\ .\n\\end{equation}\nHowever the determinant of the metric is still given by Eq.~(\\ref{g}),\nwhile the eigenvalues of $\\bar G$ become\n$(\\Sigma\\pm\\sqrt{\\Sigma^2+1},\\pm f)$.\n\\subsection{Gauging a 2-dimensional translation}\nUpon gauging a 2-dimensional subgroup one obtains an effective action\nin $N-N_d=4-2=2$ dimensions of the same type as the one in\nEq.~(\\ref{S_eff}), but with $\\bar G$ a constant symmetric matrix with\nsignature 1+1 and $\\bar B$ a constant antisymmetric matrix.\n\\par\nOn dualizing with respect to the unique translational coordinate\nwhich is left, one obtains a metric tensor whose diagonal form\nis\n\\begin{equation}\n\\bar G=\\left[\\begin{array}{cc}\n\\Sigma-\\sqrt{\\Sigma^2+1} & 0 \\\\\n0 & \\Sigma+\\sqrt{\\Sigma^2+1}\n\\end{array}\\right]\n\\ ,\n\\end{equation}\nin which there are no singularities regardless of the specific\nform of $\\Sigma=\\Sigma(t)$.\n\\section{Conclusions}\nIn this paper we have examined a new class of $\\sigma$-models, which are\ngenerated by gauging a subgroup of the Poincar\\'e group $ISO(d-1,1)$\nwhich acts invariantly from the left.\nThe fact that this group is a noncompact, semi-direct product group\ndifferentiates our coset model from all such models studied heretofore.\nThere are several intriguing results in this investigation.\nOur starting point is a model with values in $ISO(d-1,1)$,\nwhich describes spinning strings in flat $(d-1)+1$ dimensions.\nAfter promoting a translation subgroup to a gauge symmetry,\nhowever, the resulting action describes spinless strings moving\nin curved space-times and interacting with an axion field.\nIf the effective action is obtained from a pure gauge field,\nthe resulting metric tensor is in general degenerate.\nThe degeneracy is equal to the difference between the number\nof relevant (Lorentz) coordinates and the number of isometries.\nFinally, the effective actions inherently possess T-duals.\n\\bigskip\n\\par\n\\begin{center}\n{\\bf Acknowledgements}:\n\\end{center}\n\\par\nWe would like to thank A. Stern for many useful discussions.\nThis work was supported in part by the U.S. Department of\nEnergy under Grant No. DE-FG02-96ER40967.\n\\bigskip\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nIt is a remarkable success of the theory of General Relativity (GR) that it has so far withstood the onslaught of ever more precise astrophysical and cosmological data, and at present remains our best low-energy description of gravity. That is not to say, however, that GR is without its problems, particularly in providing a robust description of the dark sector of the universe. These outstanding issues have motivated research into modifications of GR, the prototypical example being scalar-tensor theory, in which one introduces an additional scalar degree of freedom into the gravitational sector. These are typically described by Horndeski theory \\cite{Horndeski:1974wa,Deffayet:2009wt,Deffayet:2009mn,Deffayet:2011gz}, or its more recent generalisation, Degenerate Higher Order Scalar Tensor (DHOST) theory \\cite{Langlois:2015cwa,Crisostomi:2016czh,BenAchour:2016fzp}. Such an approach has proven particularly popular in recent years for constructing novel descriptions of dark energy. The recent gravitational wave data \\cite{Monitor:2017mdv,Creminelli:2017sry,Sakstein:2017xjx,Ezquiaga:2017ekz,Baker:2017hug,Bettoni:2016mij} has dealt a strong blow to such models, proving that in the late Universe light and gravity propagate at identical speeds, and thus ruling out large regions of the parameter space of these theories. However, a simple subclass of Horndeski Lagrangians passes this test. Some models even provide an interesting self-tuning mechanism. \n\nSelf-tuning is the ability of a model to screen an arbitrary cosmological constant at the level of the action, and possibly to yield an independent, effective cosmological constant at the level of the solutions. Examples of self-tuning towards Minkowski spacetime were first provided in \\cite{Charmousis:2011bf} and dubbed `Fab Four' models; the mechanism was later generalized to non-zero effective cosmological constants (e.g. \\cite{Appleby:2012rx,Martin-Moruno:2015bda,Charmousis:2014mia}). This is particularly interesting in the context of the large cosmological constant problem; as is the case for any parameter in a quantum field theory, the cosmological constant that we observe should be the sum of a bare contribution and of quantum corrections. However, the bare value of the cosmological constant and the associated quantum corrections must compensate at a level of at least one part in $10^{55}$ \\cite{Martin:2012bt} in order to yield the observed value. Self-tuning is thus extremely useful, as a way to trade this huge contribution for an effective cosmological constant that matches the observations. \n\nOn the other hand, with gravitational wave detectors and electromagnetic observations, we have entered an era that makes it possible to explore the strong field regime of gravity. It is thus more important than ever to understand the behaviour of scalar-tensor theories in this sector. Of particular interest is the study of astrophysical black holes, to which current observational and detector experiments are sensitive to. Indeed, the incoming data provides a new testing ground in which to probe the validity of such modifications to GR. Determining black hole solutions within scalar-tensor theories is, however, a complicated endeavour, due to the presence of non-minimal couplings between curvature terms and the additional scalar degree of freedom in the gravity sector. Nevertheless, the problem of embedding black hole solutions in self-tuned universes has already been tackled. Such solutions have been obtained through approximate \\cite{Babichev:2012re}, numerical \\cite{Babichev:2016fbg} or even exact methods \\cite{Babichev:2013cya,Kobayashi:2014eva,Babichev:2016rlq,Babichev:2016kdt}. Star solutions were also obtained numerically in similar contexts \\cite{Cisterna:2015yla,Cisterna:2016vdx,Maselli:2016gxk,Ogawa:2019gjc}. Exact black hole solutions were only obtained in the framework of shift-symmetric theories, when the action is invariant under an overall shift of the scalar field across spacetime. In this paper, we also focus on the subclass of shift-symmetric theories; this allows us to maintain tractable calculations, while capturing the higher-derivative effects that are essential to self-tuning.\n\nThe structure of the paper is as follows. In sec.~\\ref{sec:model}, we introduce our model and the associated field equations. Then, in sec.~\\ref{sec:cosmo}, we discuss its self-tuning properties. We present the only self-tuning model left in the class of Horndeski theories that pass the gravitational wave tests, together with an exact cosmological solution. Section \\ref{sec:bhs} describes how black holes can be embedded in such a universe, both using analytic expansions and numerical techniques. We conclude in sec.~\\ref{sec:discussion}.\n\n\n\n\\section{A shift-symmetric self-tuning model}\n\\label{sec:model}\n\nThe subset of Horndeski theory that passes the gravitational wave test can be parametrized as \\cite{Monitor:2017mdv,Creminelli:2017sry,Sakstein:2017xjx,Ezquiaga:2017ekz,Baker:2017hug,Bettoni:2016mij}\n\\begin{equation}\\label{eq:action}\n\\int\\mathrm{d}^4 x \\sqrt{-g}\\Big[K(\\phi,X)- G_3(\\phi,X)\\Box\\phi+ G_4(\\phi)R - M_{\\text{Pl}}^2\\Lambda\\Big], \n\\end{equation}\nwhere $M_{\\text{Pl}}$ is the reduced Planck mass, $\\Lambda$ is the bare cosmological constant, and $K$, $G_3$ and $G_4$ are a priori arbitrary functions of the scalar field $\\phi$ and its kinetic density $X = -\\frac{1}{2}\\nabla^{\\mu}\\phi\\nabla_{\\mu}\\phi$. Note that $G_4$ is taken to be a function of $\\phi$ alone, in order for the theory to remain consistent with the gravitational wave constraints. \nThe element that is essential to self-tuning is the $X$ dependence in $G_3$. Thus, for tractability of the calculations, we will consider the simpler shift-symmetric subset of \\eqref{eq:action}:\n\t\\begin{equation}\n\tK(\\phi,X) = X,\\quad G_{3}(\\phi,X) = G_3(X),\\quad G_4(\\phi) = \\frac{M_{\\text{Pl}}^2}{2}\\,,\n\t\\end{equation}\nwhich results in the following action:\n\\begin{equation}\\label{eq:shift symmetric action}\nS = \\int\\mathrm{d}^4 x \\sqrt{-g}\\Big[X - G_3(X)\\Box\\phi + \\frac{M_{\\text{Pl}}^2}{2}R - M_{\\text{Pl}}^2\\Lambda\\Big].\n\\end{equation}\nThe above action was actually first studied in \\cite{Deffayet:2010qz}, where it was dubbed Kinetic Gravity Braiding (KGB). It was later incorporated in the generic framework of covariant Galileons \\cite{Deffayet:2009wt,Deffayet:2009mn,Deffayet:2011gz}, which were in turn found to be an equivalent formulation of Horndeski's theory \\cite{Horndeski:1974wa}. The associated metric and scalar field equations of motion are then given by:\n\\begin{align}\n\t\\begin{split}\n\t\\mathcal{E}_{(g)}^{\\mu\\nu} =\\dfrac{1}{\\sqrt{-g}}\\,\\dfrac{\\delta S}{\\delta g_{\\mu\\nu}} &=G_{3X}\\left(\\nabla^{(\\mu}X\\nabla^{\\nu)}\\phi + \\frac{1}{2}\\Box\\phi\\nabla^{\\mu}\\phi\\nabla^{\\nu}\\phi - \\frac{1}{2}g^{\\mu\\nu}\\nabla_{\\lambda}X\\nabla^{\\lambda}\\phi\\right)\n\t\\\\ \n\t&\\quad +\\frac{M_{\\text{Pl}}^2}{2}(G^{\\mu\\nu} + g^{\\mu\\nu}\\Lambda) - \\frac{1}{2}\\nabla^{\\mu}\\phi\\nabla^{\\nu}\\phi - \\frac{1}{2}g^{\\mu\\nu}X = 0,\n\t\t\\label{eq:metric eom}\n\t\\end{split}\n\t\\end{align}\n\t\\begin{equation}\n\t\\mathcal{E}_{(\\phi)} =\\dfrac{1}{\\sqrt{-g}}\\,\\dfrac{\\delta S}{\\delta \\phi} = \\nabla_{\\mu}J^\\mu = 0 ,\n\t\\label{eq:conservation eq}\n\t\\end{equation}\nwhere \n\\begin{equation}\n\\label{eq:conserved current}\nJ^\\mu = \\nabla^\\mu\\phi - G_{3X}(\\nabla^\\mu X + \\Box\\phi\\nabla^\\mu\\phi)\n\\end{equation}\nis the conserved Noether current associated to the shift-symmetry, and the subscript $X$ denotes a partial derivative with respect to $X$ (e.g. $G_{3X}=\\partial G_3\/\\partial X$).\n\n\n\n\\section{Cosmological solutions to the field equations}\n\\label{sec:cosmo}\n\nLet us first study the cosmological solutions of the model, within which we aim to embed black holes. Any black hole solution should match this behaviour asymptotically. We start with the Friedmann Lema\\^{i}tre Robertson Walker (FLRW) geometry:\n\\begin{equation}\\label{eq:flrw}\n\t\\mathrm{d} s^2 = - \\mathrm{d} \\tau^2 + a^2(\\tau)\\mathrm{d}\\mathbf{x}^2,\n\\end{equation}\nwhere $\\tau$ is the cosmic time, $a(\\tau)$ is the cosmological scale factor, and we adopt the metric signature $(-+++)$ throughout. The scalar field is also assumed to be spatially homogeneous at this scale, i.e. $\\phi=\\phi(\\tau)$. Given this, the metric and scalar equations of motion, eqs.~\\eqref{eq:metric eom}-\\eqref{eq:conservation eq}, are \n\\begin{subequations}\\label{eq:cosmological eoms}\n\t\\begin{flalign}\n\t\t0 &= 6M_{\\text{Pl}}^2H^2 - 6H\\dot{\\phi}^3G_{3X} - \\dot{\\phi}^2 - 2M_{\\text{Pl}}^2\\Lambda,\\label{eq:Friedmann constraint}\n\t\t\\\\\n\t\t0 &= 2M_{\\text{Pl}}^2\\Lambda -4M_{\\text{Pl}}^2\\dot{H} - 6M_{\\text{Pl}}^2H^2 +2\\dot{\\phi}^2\\ddot{\\phi}G_{3X} - \\dot{\\phi}^2, \\label{eq:Hubble eom}\n\t\t\\\\\n\t\t0 &= 3H\\dot\\phi(1+3G_{3X}H\\dot\\phi)+3G_{3X}\\dot\\phi^2\\dot{H}+ \\big[1 + 3H\\dot{\\phi}(2G_{3X} + G_{3XX}\\dot\\phi^2)\\big]\\ddot{\\phi}, \\label{eq:cosmo scalar eom}\n \t\\end{flalign}\n\\end{subequations}\nwhere a dot denotes a derivative with respect to cosmic time $\\tau$, $H=\\dot{a}\/a$ is the Hubble parameter, and $G_3$ depends implicitly on $\\dot\\phi$ through $X=\\dot{\\phi}^2\/2$. We now fix the geometry to be de Sitter, i.e. $\\dot{H}=0$, $H = H_0$. It immediately follows from eq.~\\eqref{eq:Friedmann constraint} that $\\dot{\\phi}$ is also constant. The $G_3$ dependence then drops from eq.~\\eqref{eq:Hubble eom}, from which we can determine that $\\dot{\\phi}$ is given by\n\\begin{equation}\\label{eq:phidot sol}\n\t\\dot{\\phi}_0 = \\pm\\sqrt{2}M_{\\text{Pl}}\\sqrt{\\Lambda- 3H_0^2}.\n\\end{equation}\nOnce substituted in eq.~\\eqref{eq:Friedmann constraint}, one realises that $H_0$ will always depend on $\\Lambda$, unless $\\dot\\phi G_{3X}$ is itself insensitive to the value of $\\dot\\phi$. Thus, to ensure self-tuning, i.e. that the cosmological properties of the metric do not depend on the bare cosmological constant, we set\n\\begin{equation}\\label{eq:G3}\nG_3(X) = - \\frac{1}{3}\\,\\frac{\\sqrt{2X}}{\\mu} = -\\frac{1}{3}\\,\\frac{\\dot{\\phi}}{\\mu}\\,,\n\\end{equation}\nwhere $\\mathcal{\\mu}$ is a priori an arbitrary parameter in the action~\\eqref{eq:action}, with mass dimension $[\\mu]=1$. Interestingly, the particular choice of $G_3$ in eq.~\\eqref{eq:G3} corresponds to a constant diffusion coefficient in the imperfect fluid interpretation of the KGB model \\cite{Pujolas:2011he}. The on-shell field equations then impose that $H_0=\\mu$, such that the value of the Hubble parameter is independent of the cosmological constant, i.e. the model admits self-tuning solutions. Having to introduce the current Hubble scale of $\\mu\\sim10^{-33}$~eV directly at the level of the action can appear unnatural. However, it is counterbalanced by the advantage that $H_0$ is completely independent on $\\Lambda$; thus, the tuning will survive phase transitions, that are otherwise extremely problematic in the context of the standard model \\cite{Martin:2012bt}.\n\nWe have established the cosmology to which we will match black hole solutions. We now need to consider how we can achieve such a matching. We will look for static and spherically symmetric black hole solutions. Thus, we will use the following system of coordinates:\n\\begin{equation}\\label{eq:static metric}\n\t\\mathrm{d} s^2 = - f(r)\\,\\mathrm{d} t^2 + \\frac{1}{g(r)}\\,\\mathrm{d} r^2 + r^2\\mathrm{d}\\Omega^2 ,\n\\end{equation}\nwhere $\\mathrm{d}\\Omega^2 = \\mathrm{d}\\theta^2 + \\sin^2\\theta\\,\\mathrm{d}\\varphi^2$, and $f(r)$ and $g(r)$ are a priori arbitrary functions to be determined by the field equations\\footnote{Note that unlike in GR, the functions $f(r)$ and $g(r)$ do not have to be equal on astrophysical scales, and indeed, they often are not for scalar-tensor theories of gravity.}. The de Sitter metric, eq.~\\eqref{eq:flrw} with $a(\\tau)=e^{H_0\\tau}$, can actually be matched to its well-known static patch through the following invertible change of coordinates \\cite{Babichev:2012re}:\n\\begin{subequations}\\label{eq:FLRW coord transform}\n\t\\begin{align}\n\t\t\\tau &= t + \\dfrac{1}{2H_0} \\ln\\left[1 - (H_0 r)^2\\right],\n\t\t\\label{eq:taut}\n\t\t\\\\\n\t\t\\rho &= \\dfrac{e^{-Ht}}{\\sqrt{1-(H_0r)^2}} \\,r.\n\t\\end{align}\n\\end{subequations}\nUnder this change of coordinate, the metric \\eqref{eq:flrw} with $a(\\tau)=e^{H_0\\tau}$ is mapped to \n\\begin{equation}\n\tf(r) = g(r) = 1 - (H_0r)^2.\n\\end{equation} \n\n\n\n\\section{Static black hole solutions}\n\\label{sec:bhs}\n\n\\subsection{Ansatz and field equations}\n\nHaving determined how a black hole solution within this shift-symmetric model should behave asymptotically, i.e. on cosmological scales, we shall now move on to determine the details of such solutions. For simplicity, we shall consider static and spherically symmetric metric configurations, as given by the ansatz \\eqref{eq:static metric}. The scalar field, on the other hand, is taken to depend not only on the radial coordinate $r$, but also linearly on time. This emerges naturally from the combination of eqs.~\\eqref{eq:phidot sol}-\\eqref{eq:taut}, and was shown to be fully consistent in \\cite{Babichev:2015rva}. The stress-energy tensor associated with $\\phi$ only involves derivatives of the scalar field; thus it is itself static and one can consistently require a static geometry\\footnote{This is certainly restrictive, and dynamical branches of solutions should exist. However, no such solution has been obtained in the literature so far, due to technical difficulty mostly.}. Furthermore, although a naive counting of the field equations and free functions tells us that the system seems overconstrained, ref.~\\cite{Babichev:2015rva} proved this is not the case; the $(tr)$ component of the metric equation is always proportional to the radial component of the current, $J^r$. This allows us to look for solutions with the following ansatz for $\\phi$:\n\\begin{equation}\\label{eq:spherical symmetry ansatz}\n\t\\phi(t,r) = qt + \\int^r\\mathrm{d} r'\\,\\frac{\\chi(r')}{f(r')}\\,,\n\\end{equation}\nwhere $q=\\dot\\phi$ is a free, non-zero constant for now, and $\\chi(r)$ is an arbitrary function to be determined by the field equations. The precise form of the second term on the right hand side is chosen for convenience. Notably, the $f(r)$ denominator allows us to absorb the diverging contribution of the $r$-dependent part of $\\phi$ when approaching the black hole horizon. \nLet us now write down the independent field equations. We will work with the $(tr)$ component of the metric field equations. Additionally, we use specific linear combinations of the $(tt)$ and $(tr)$ field equations on one hand, and $(rr)$ and $(tr)$ field equations on the other. Due to spherical symmetry, $\\smash{\\mathcal{E}_{(g)}^{\\theta\\theta}=\\sin^2\\theta\\,\\mathcal{E}_{(g)}^{\\varphi\\varphi}}$, and $\\smash{\\mathcal{E}_{(g)}^{\\theta\\theta}}$ itself can be deduced from the previous three equations, due to the diffeomorphism invariance identity \\smash{$\\mathcal{E}_{(\\phi)}\\nabla^\\nu\\phi+2\\nabla_{\\mu}\\mathcal{E}_{(g)}^{\\mu\\nu}=0$}. The $(tr)$ metric equation reads (up to an overall non-zero factor):\n\\begin{equation}\n\\label{eq: psi constraint from scalar eom}\n\\frac{\\chi}{q} + \\frac{q}{2f}G_{3X}\\Big[f' - \\frac{g}{r^4f}\\big(r^4f)'\\Big(\\frac{\\chi}{q}\\Big)^2\\Big] = 0,\n\\end{equation}\nwith\n\\begin{equation}\nG_{3X} = -\\frac{f}{3q\\mu}\\Big[f - g\\Big(\\frac{\\chi}{q}\\Big)^2\\Big]^{-1\/2},\n\\end{equation}\nThen, the $(rr)$-$(tr)$ and $(tt)$-$(tr)$ combinations read respectively:\n\t\\begin{align}\n\t0 &= M_{\\text{Pl}}^2\\,\\frac{g\\big(rf\\big)' + r^2f\\Lambda -f}{2r^2f} - \\frac{q^2}{4f}\\Big[1 - \\frac{g}{f}\\Big(\\frac{\\chi}{q}\\Big)^2\\Big], \\label{eq:Err}\n\t\\\\\n\t\\begin{split}\n\t0 &= M_{\\text{Pl}}^2\\,\\frac{\\big(rg\\big)' + r^2\\Lambda - 1}{2r^2} - \\frac{q^2}{2r^2f}\\sqrt{\\frac{g}{f}}G_{3X}\\Big[1 - \\frac{g}{f}\\Big(\\frac{\\chi}{q}\\Big)^2\\Big]\\Big(r^2\\sqrt{\\frac{g}{f}}\\chi\\Big)' \n\t\\\\\n\t&\\quad+ \\frac{q^2}{4f}\\Big[1 - \\frac{g}{f}\\Big(\\frac{\\chi}{q}\\Big)^2\\Big]. \n\t\\label{eq:Ett}\n\t\\end{split}\n\t\\end{align}\nNote that the $G_3$ dependence drops out from the $(rr)$-$(tr)$ combination, eq.~\\eqref{eq:Err}. It is not possible to solve this system fully analytically, and therefore, we will ultimately have to appeal to numerical techniques. The difficulty of finding exact black hole solutions for cubic Horndeski models was already noted in \\cite{Babichev:2016fbg}. In general, it appears that no exact black hole solution is known for Horndeski models that do not possess the reflection symmetry $\\phi\\to-\\phi$, particularly the simplest cubic and quintic models. Indeed, typically the scalar field equation can be integrated once, but it then becomes a high-order algebraic equation for $\\phi'$. Exact solutions are known only for models that possess both shift and reflection symmetry \\cite{Babichev:2013cya,Kobayashi:2014eva,Babichev:2017guv} (apart from standard scalar-tensor theories). In order to integrate the system numerically, it proves convenient to introduce a set of dimensionless quantities. To this end, we note that the Lagrangian contains three dimensionful parameters: $M_{\\text{Pl}}$, $\\mu$ and $\\Lambda$, along with the scalar field velocity $q$, introduced in the ansatz for $\\phi$. Let us further introduce a length scale $r_0$, which physically corresponds to the event horizon radius of the black hole, such that we can define a dimensionless radius $x=r\/r_0$. Then, given the parameters of the Lagrangian, the scalar field velocity $q$ and the length scale $r_0$, we can express the field equations using only the following three dimensionless constants:\n\\begin{equation}\n\\beta_1 = \\frac{1}{r^2_0\\mu^2}\\,, \\quad \\beta_2 = \\frac{r_0^2q^2}{M_{\\text{Pl}}^2} \\,, \\quad \\beta_3 = r_0^2\\Lambda.\n\\label{eq:betas}\n\\end{equation}\nEquations \\eqref{eq: psi constraint from scalar eom}, \\eqref{eq:Err} and \\eqref{eq:Ett} can then be rewritten:\n\t\\begin{align}\n\t0 &= 6x^4\\frac{\\chi}{q} - \\sqrt{\\beta_1}\\Big[x^4\\frac{\\mathrm{d} f}{\\mathrm{d} x} - \\frac{g}{f}\\frac{\\mathrm{d}}{\\mathrm{d} x}\\big(x^4f)\\Big(\\frac{\\chi}{q}\\Big)^2\\Big]\\Big[f - g\\Big(\\frac{\\chi}{q}\\Big)^2\\Big]^{-1\/2}, \n\t\\label{eq:alpha 1 scalar eom} \n\t\\\\\n\t0 &= 2\\Big[g\\frac{\\mathrm{d}}{\\mathrm{d} x}\\big(xf\\big) + \\beta_3x^2f- f\\Big] - \\beta_2x^2\\Big[1 - \\frac{g}{f}\\Big(\\frac{\\chi}{q}\\Big)^2\\Big] ,\n\t\\label{eq:alpha 1 Err}\n\t\\\\\n\t\\begin{split}\n\t0 &= 6\\Big[\\frac{\\mathrm{d}}{\\mathrm{d} x}\\big(xg\\big) + \\beta_3x^2 - 1 \\Big] + \\frac{3\\beta_2x^2}{f}\\Big[1 + \\frac{g}{f}\\Big(\\frac{\\chi}{q}\\Big)^2\\Big] \n\t\\\\\n\t&\\quad + \\frac{2\\sqrt{\\beta_1g}\\beta_2}{f}\\Big[1 -\\frac{g}{f}\\Big(\\frac{\\chi}{q}\\Big)^2\\Big]^{1\/2}\\frac{\\mathrm{d}}{\\mathrm{d} x}\\Big(x^2\\sqrt{\\frac{g}{f}}\\frac{\\chi}{q}\\Big).\n\t\\label{eq:alpha 1 Ett}\n\t\\end{split}\n\t\\end{align}\nThese will be the equations that we integrate numerically.\n\n\\subsection{Analytical approximations in the small \\& large $r$ limits}\n\\label{sec:rexpansion}\n\nBefore solving eqs.~\\eqref{eq:alpha 1 scalar eom}, \\eqref{eq:alpha 1 Err} and \\eqref{eq:alpha 1 Ett} numerically, we shall study the asymptotic behaviour of $f$, $g$ and $\\chi$ in the small and large $r$ limits. Solving the system of equations near the origin $r= 0$, we find that \n\\begin{subequations}\n\\label{eq:smallrexp}\n\t\\begin{flalign}\n\t\tf(r) &\\underset{r\\to0}{=} - \\frac{a}{r^4} + \\mathcal{O}(r^{-3}),\n\t\t\\\\\n\t\tg(r) &\\underset{r\\to0}{=} - \\frac{1}{3} - br + \\mathcal{O}(r^2),\n\t\t\\\\\n\t\t\\chi(r) &\\underset{r\\to0}{=} - \\frac{c}{r^{9\/2}} + \\mathcal{O}(r^{-7\/2}),\n\t\\end{flalign}\n\\end{subequations}\nwhere $a$, $b$ and $c$ are fixed by the field equations \\eqref{eq: psi constraint from scalar eom}, \\eqref{eq:Err} and \\eqref{eq:Ett} (we omit their exact expressions here in the interest of succinctness). We note that, as was found in~\\cite{Babichev:2016fbg}, the $g(r)$ component of the metric is finite at the origin, unlike in GR. It is possible that this is a generic feature of any cubic Horndeski model for which such black hole solutions exist. \n\nLet us now consider the large $r$ asymptotic behaviour of the solutions. We assume that, as $r\\to \\infty$, the solution has an analytic expansion in powers of $1\/r$.\nWe find that the asymptotic behaviour of the solution is given by:\n\\begin{subequations}\\label{eq:large r sol}\n\t\\begin{flalign}\n\t\tf(r) &\\underset{r\\to\\infty}{=} \\Big(\\frac{q}{q_0}\\Big)^2\\left[1 - (H_0r)^2\\right] + \\mathcal{O}(r^{-1}),\n\t\t\\\\\n\t\tg(r) &\\underset{r\\to\\infty}{=} 1 - (H_0r)^2 + \\mathcal{O}(r^{-1}),\n\t\t\\\\\n\t\t\\chi(r) &\\underset{r\\to\\infty}{=} - \\dfrac{\\mu q^2}{q_\\mathrm{0}} r + \\mathcal{O}(r^{-1}),\n\t\\end{flalign}\n\\end{subequations}\nwhere $q_0=\\dot\\phi_0$ is given by eq.~\\eqref{eq:phidot sol}. Note that, at this point, the velocity parameter $q$ remains arbitrary, and may not necessarily coincide with $q_0$, i.e., the velocity parameter corresponding to the cosmological solution. However, it is interesting to check whether the scalar field in the asymptotic solution~\\eqref{eq:large r sol} is homogeneous in the set of cosmological coordinates $(\\tau,\\rho)$. From eqs.~\\eqref{eq:spherical symmetry ansatz} and~\\eqref{eq:large r sol}, it follows that \n\\begin{equation}\n\t\\phi(t,r)\\underset{r\\to\\infty}{=} qt +\\frac{\\mu q}{H_0^2}\\,\\text{ln}(r) +\\mathcal{O}(r^{-2}).\n\\end{equation}\nSimilarly, using the coordinate transformation~\\eqref{eq:FLRW coord transform}, $\\phi$ can be expanded as\n\\begin{equation}\n\t\\phi(\\tau,\\rho)\\underset{\\rho\\to\\infty}{=}q\\tau +\\dfrac{q}{\\mu}\\Big(\\dfrac{q_0}{q}-1\\Big)\\text{ln}(\\rho) +\\mathcal{O}(\\rho^{-2})\n\\end{equation}\nin cosmological coordinates. Thus, we see that $\\phi$ is asymptotically homogeneous only when the local velocity $q$ matches the asymptotic cosmological value, as was already the case in \\cite{Babichev:2016fbg}. Whether the solutions with $q\\neq q_0$ are physically acceptable is arguable, since only the gradient of $\\phi$ is physically relevant, and the latter always decays as $1\/\\rho$. In our numerical analysis, still, we chose to impose that $q=q_0$ to ensure a safe cosmological behaviour.\n\nFinally, let us comment briefly on the fate of the scalar field at the (black hole and cosmological) horizons. It is well known \\cite{Babichev:2013cya} that in the system of coordinates \\eqref{eq:static metric}, the $r$-dependent part of $\\phi$ diverges at both horizons. However, this is a coordinate effect, and this divergence can be taken care of by working in adapted Eddington-Finkelstein coordinates. Note that it is not in general possible to absorb the divergence for both horizons. However, recent works have shown that generalising to the rotating case can cure the divergence of $\\phi$ on both horizons for this type of solution \\cite{Charmousis:2019vnf}.\n\n\\subsection{Numerical integration of the field equations}\n\nHaving studied the asymptotics of the solutions, we shall now proceed to carry out a numerical integration of this system. Note that eqs.~\\eqref{eq:alpha 1 scalar eom} and~\\eqref{eq:alpha 1 Err} are algebraic equations in $g$ and $\\chi$, and can thus, in principle, be solved to express them in terms of $f$ and $\\mathrm{d} f\/\\mathrm{d} x$. Upon substitution into eq.~\\eqref{eq:alpha 1 Ett}, we are then left with a second-order ordinary differential equation (ODE) for $f$. In practice, it turns out the system is more readily solved numerically by first solving eq.~\\eqref{eq:alpha 1 Err} to determine $\\chi$ in terms of $f$, $\\mathrm{d} f\/\\mathrm{d} x$ and $g$, and then solving the remaining two equations as a system of ODEs for $f$ and $g$. This requires us to specify several boundary conditions in order to obtain the unique solution. By imposing that both $f$ and $g$ vanish at $x=1$, we define the black hole horizon to be located at this point (corresponding to $r=r_0$ as mentioned earlier). Moreover, having imposed $f\\vert_{x=1}=0$, we can then expand eq.~\\eqref{eq:alpha 1 scalar eom} around $x=1$ to determine \\smash{$\\frac{\\mathrm{d} f}{\\mathrm{d} x}\\vert_{x=1}$} in terms of \\smash{$\\frac{\\mathrm{d} g}{\\mathrm{d} x}\\vert_{x=1}$} (or vice versa). This leaves us with one remaining boundary condition that we are free to choose arbitrarily, although ultimately the choice is dictated by requiring that at large $x$, the solution has the desired asymptotic cosmological behaviour. Thus, for given parameters, we use the shooting method to pick the value of \\smash{$\\frac{\\mathrm{d} g}{\\mathrm{d} x}\\vert_{x=1}$} such that \\smash{$f\\underset{x\\to\\infty}{\\sim}g$}. We start the numerical integration slightly outside the black hole horizon. Then, the code breaks down when approaching both the black hole and cosmological horizons. It is however very easy to perform a series expansion around these points, and to restart the numerical integration on the other side of the horizon(s). Thus, we are able to compute the solution over the full range $0 2$,\n$N_{\\rm cluster} > 17$,\n$E_{\\rm vis} \/ \\sqrt{s} > 0.6$,\n$E_{\\rm neutral} \/ \\sqrt{s} < 0.5$,\n$E_{\\gamma} < 30$~GeV,\nthrust $ < 0.92$.\nThe number of simulated events for the signal and for each background \nchannel, as well as the remaining events after the preselection are given in \nTable~\\ref{tab:pre}.\n\n\\begin{table}[hp]\n\\caption{\\label{tab:pre} Number of simulated signal and background events before and \n after the preselection.}\n\\begin{center}\n\\begin{tabular}{|c|c|c|c|c|c|c|c|c|}\\hline\nChannel & bbA & qq & WW &eW$\\nu$& tt & ZZ & eeZ & hA \\\\ \\hline\n(in 1000) & 50 &6250 & 3500 & 2500 & 350 & 300 & 3000 & 50 \\\\ \\hline\nAfter presel.& 73\\% &20991 &7481&0 &89983&10278 & 145 & 12665 \\\\ \\hline\n\n\\hline\n\\end{tabular}\n\\end{center}\n\\end{table}\n\n\\section{Iterative Discriminant Analysis}\nIn order to separate the signal from the background, the following selection variables\nare defined: highest and third significant jet b-tagging, thrust value,\nsecond Fox-Wolfram moment, \ny-cut value to form four jets, minimum and maximum jet energy,\nisolation angle of the most energetic jet, minimum angle and invariant mass between\nany jet pair, minimum jet and event charge multiplicity, charged and neutral event\nenergy.\nFigure~\\ref{fig:btag} shows the simulated b-tagging variable for\nthe third significant jet and the second Fox-Wolfram moment after the preselection for\n141544 remaining background events.\nThe thrust value and the IDA output variable are shown in Fig.~\\ref{fig:ida1}.\nHalf of these events and half of the signal events are used to train the \nIDA~\\cite{ida}.\nIn a first step, a cut on the IDA output variable is applied such that the \nefficiency is reduced by 30\\%. The remaining signal and 6745 background events \nare again passed through the IDA. Figure~\\ref{fig:ida2} shows the\nIDA output variable and the resulting number of background events as a function \nof the signal efficiency.\n\n\\begin{figure}[hp]\n\\caption{\\label{fig:btag} b-tagging \nvariable and Fox-Wolfram moment after the preselection.}\n\\begin{center}\n\\vspace*{-0.7cm}\n\\mbox{\\epsfig{file=btag.eps,width=0.49\\textwidth}}\n\\mbox{\\epsfig{file=fox.eps,width=0.49\\textwidth}}\n\\end{center}\n\\vspace*{-0.2cm}\n\\end{figure}\n\n\\begin{figure}[hp]\n\\vspace*{-5mm}\n\\caption{\\label{fig:ida1} Thrust value and first step IDA output.}\n\\begin{center}\n\\vspace*{-0.7cm}\n\\mbox{\\epsfig{file=thrust.eps,width=0.49\\textwidth}}\n\\mbox{\\epsfig{file=ida1.eps,width=0.49\\textwidth}}\n\\end{center}\n\\vspace*{-0.9cm}\n\\end{figure}\n\n\\begin{figure}[tp]\n\\caption{\\label{fig:ida2} Final IDA output and background vs. signal \nefficiency.}\n\\begin{center}\n\\vspace*{-0.7cm}\n\\mbox{\\epsfig{file=ida2.eps,width=0.49\\textwidth}}\n\\mbox{\\epsfig{file=perf_curv_bbh.eps,width=0.49\\textwidth}}\n\\end{center}\n\\end{figure}\n\n\\vspace*{0.5cm}\n\\section{Results}\nWe have determined the expected background rate for a given signal\nefficiency and evaluated that the sensitivity for a 100 GeV \npseudoscalar Higgs boson in the process $\\mbox{$\\rm e^+e^-$}\\rightarrow\\mbox{$\\rm b\\bar{b}$}\\rightarrow\\mbox{$\\rm \\bb A$}$ suffices to\ndetermine the value of $\\tan\\beta$.\nThe sensitivity $N_{\\rm signal} \/ \\sqrt{N_{\\rm background}}$\nis almost independent of the working point signal efficiency \nin the range 5\\% to 50\\%.\nFor a working point of 10\\% efficiency, the total simulated background of about \n16 million events is reduced to 100 background events.\nThe resulting error on $\\tan\\beta= 50$ is 7\\%:\n$$\n\\Delta\\tan^2\\beta \/ \\tan^2\\beta = \\Delta N_{\\rm signal} \/ N_{\\rm signal}\n=\\sqrt{N_{\\rm signal} + N_{\\rm background}} \/ N_{\\rm signal} = 0.14.\n$$\nFor smaller values of $\\tan\\beta$ the sensitivity reduces quickly.\nA $5\\sigma$ signal detection is possible for $\\tan\\beta= 35$.\n\nIn conclusion, an IDA anlalysis based on experience at LEP2\nwas applied and gave sufficient sensitivity to detect the signal process. \nA high-luminosity linear collider is essential and unique \nfor this channel and allows the value of $\\tan\\beta$ to be measured with \nprecision.\n\n\\section*{References}\n\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}} +{"text":"\\section{Introduction}\n\nCourts of common law systems rely on statutes and precedents for deciding the law applicable to a case. A court opinion on a case, being a decision taken by a court when litigating a specific case, is a text assembled by the court judges using the parties' arguments and rebuttals of arguments. Previously published opinions (also known as case law) are cited to support the argumentation of the parties and the opinion of the court on the litigated case.\n\nThereby a legal discourse is made of a mix of factual details from the case as well as legal discussions arising from the qualification of the facts, driven by the past discussions with a similarity, either in facts or in reasoning. We provide an example, taken from the VerbCL dataset that illustrates how case law citation is used in practice, in Table~\\ref{table:sample}.\n\nFor any legal-domain practitioner, search for an appropriate citation is a key task in preparing the argumentation for a case, which needs to be facilitated by an information retrieval system. This task was considered in case law retrieval~\\cite{coliee2018, coliee2019} as an information retrieval task where the query is a draft of an opinion and the result is a ranked list of past opinions. Relevance in this case may be guided by the similarity between the query and the retrieved opinion, or by the fact that the retrieved opinions address legal facets or questions in the query, as well as by the consideration whether the retrieved opinions might be worth being cited or not in the context of the query. \n\nIn this work, we introduce the new task of \\textit{highlight extraction}: given the text of an opinion, predict the subset of text spans that are likely to be cited in the future. \nIn this task, the question is whether we can successfully predict which parts of an opinion will have an impact on future litigation, i.e., will be cited.\n\nThe motivation behind this task is to facilitate the work done by a legal commentator, who is tasked to identify interesting points in an opinion at the time of its publication. The legal commentator inspects all recently published opinions looking for potentially valuable excerpts that are worth being cited. The evaluation whether a part of an opinion is of a particular interest, or might be of interest in future cases is informed by the knowledge of the jurisprudential landscape, so the commentator can contrast in a new opinion what makes part of this landscape, and what is novel.\n\nWe aim to inform the task of highlight extraction through the construction of a large-scale dataset that will allow the development and evaluation of data-driven models able to solve the task. To this end, we use a public dataset of US court opinions, CourtListener\\footnote{\\url{https:\/\/www.courtlistener.com}}, to extract a citation network of court opinions.\n\nThe citation network naturally emerges from the repository of opinion documents since each opinion usually quotes multiple previously published opinions for the purpose of the legal argumentation of the case.\nWe consider this network as a directed acyclic graph, where each node stands for an opinion document.\nEdges represent the citations made in opinions, directed from the citing opinion's node towards the cited opinion's node. \nIn this paper, we consider the citing-cited relation as follows: a citing opinion is making an argument by referring to a cited opinion.\n\nIn the text of the citing opinion, a citation is introduced by an \\textit{anchor}, that we consider to be the supported legal argument.\nMore specifically, we differentiate between two types of anchors: abstractive anchors, where the argument is cited as a paraphrase of the original opinion document, and extractive anchors (or \\textit{verbatim quotes}) where the argument is directly extracted from the cited opinion's text by copying the relevant text span.\nWe focus on the verbatim quotes in this work since they are much easier to detect and reproduce than paraphrases.\n\nWe further consider that verbatim quotes emphasize which part of the cited opinion have a legal weight, bringing novelty and importance into the legal landscape at the time of the publication of an opinion document.\nWe define a \\textit{highlight} of an opinion as the set of all text spans that are later cited as verbatim quotes.\nTherefore, we can produce the highlights for the previously published opinions by considering the citation graph.\nThe highlight of an opinion can be constructed by considering all the verbatim quotes of this opinion used in the citing opinion documents (see an example of the highlight in Table~\\ref{table:sample}).\n\nTo aggregate information about all citing opinions, the citation graph is used.\nWe illustrate the citation graph around the cited opinion from Table~\\ref{table:sample} in Figure~\\ref{fig:citation-graph}, which shows all the citing opinions as well as the identifiers of their verbatim quotes as the edge labels.\n\n\\begin{figure}[h]\n \\centering\n \\includegraphics[width=\\linewidth]{citation_graph.pdf} \n \\caption{Citation graph, where each node is an opinion document with edges pointing from a citing opinion to a cited opinion. Every edge is labeled with the identifier of a sentence of the cited opinion that was quoted verbatim. For example, sentence 96 of opinion 1239944 is cited by the opinion 1058281, which contains sentence 44 cited by opinion 1058321.}\n \\label{fig:citation-graph} \n \n\\end{figure}\n\nThe goal of highlight extraction is to predict the highlights for new opinions, i.e., predicting the future citations based on the opinion text.\nWe proceed by establishing the first baseline for the highlight extraction task using the state-of-the-art single-document summarization approaches: TextRank~\\cite{barrios2016variations} and PreSumm~\\cite{presumm}, as well as a sentence-level classifier based on DistilBERT~\\cite{sanh2020distilbert} . The goal of our experiments is to verify to which extent the highlight extraction task in the citation network can be addressed using single-document summarization approaches without any information about the citation graph structure.\n\n\\begin{samepage}\n Our main contributions are two-fold:\n \\begin{enumerate}\n \\item the dataset with a citation graph, verbatim quotes and opinion highlights;\n \\item the task of highlight extraction with baselines.\n \\end{enumerate}\n\\end{samepage}\n\nWe start by describing how VerbCL was constructed in Section~\\ref{section:dataset-construction} and discuss its main characteristics in Section~\\ref{section:dataset-statistics}. \nWe formalize the task of highlight extraction and report on the experiments we conducted with the baseline models for this task in Section~\\ref{section:highlight-extraction}. The overview of existing datasets for case law retrieval, and related work on highlight extraction and citation-based summarization in other domains is given in Section~\\ref{section:related-work}. We conclude and propose directions for future work in Section~\\ref{section:conclusion}.\n\n\\begin{table*}[h]\n \\begin{tabularx}{\\linewidth}{|l|l|X|}\n \\hline\n Field Name & Type & Comment \\\\\n \\hline\n \\texttt{citing\\_opinion\\_id} & \\texttt{int} & The \\texttt{opinion\\_id} of the \\textbf{CITING} opinion. \\\\\n \\texttt{cited\\_opinion\\_id} & \\texttt{int} & The \\texttt{opinion\\_id} of the \\textbf{CITED} opinion. \\\\\n \\texttt{sentence\\_id} & \\texttt{int} & The \\texttt{sentence\\_id} of the source sentence in the CITED opinion. \\\\\n \\texttt{verbatim} & \\texttt{str} & The span of text in the CITING opinion that we consider to be a potential verbatim quote from the CITED opinion \\\\\n \\texttt{snippet} & \\texttt{str} & The span of text from the CITING opinion around the in-text citation of the CITED opinion. It contains 100 words before the in-text citation, and 100 words after the in-text citation. \\\\\n \\texttt{score} & \\texttt{float} & The score of \\texttt{verbatim} for being an actual verbatim quote from the CITED opinion, placed in the CITING opinion. A score of $-1$ indicates our model does not consider it as being a quote from the CITED opinion. \\\\\n \\hline\n \\end{tabularx}\n \\caption{Fields of the documents in VerbCL Citation Graph.}\n \\label{table:citation-graph-fields}\n\\end{table*}\n\n\\begin{table*}[h]\n \\begin{tabularx}{\\linewidth}{|l|l|X|}\n \\hline\n Field Name & Type & Comment \\\\\n \\hline\n \\texttt{opinion\\_id} & \\texttt{int} & The unique identifier of the opinion. \\\\\n \\texttt{sentence\\_id} & \\texttt{int} & The unique identifier of the sentence: its index within the list of sentences of the opinion. \\\\\n \\texttt{raw\\_text} & \\texttt{str} & The complete text of the sentence. \\\\\n \\texttt{highlight} & \\texttt{bool} & The binary label of the sentence. $True$ means the sentence has been quoted verbatim in a citing opinion. \\\\\n \\texttt{count\\_citations} & \\texttt{int} & The number of times the sentence has been quoted verbatim. \\\\\n \\hline\n \\end{tabularx}\n \\caption{Fields of the documents in VerbCL Highlights.}\n \\label{table:highlights-fields}\n\\end{table*}\n\n\n\\section{Dataset Construction}\n\\label{section:dataset-construction}\n\nVerbCL is based on the Court Listener dataset, a collection, which is publicly available for free.\nIn this section, we describe the process of constructing the dataset. \n\n\\subsection{Court Listener}\nCourtListener is a project of the Free Law Project\\footnote{\\url{https:\/\/free.law\/}}, a US non-profit organization started in 2010, with the aim of \\say{making the legal world more fair and efficient}.\n\nThe original Court Listener dataset is a collection of every court opinion published by every court in the United States. It covers 406 jurisdictions (out of 423), with opinions from the year 1754 up to now. It is constantly updated with newly filed opinions, and digitized archives. \n\nWe obtained the Court Listener dataset by downloading the bulk data on September 1st 2019.\\footnote{\\url{https:\/\/www.courtlistener.com\/api\/bulk-info\/}}. More recent versions will have more opinions available, the whole processing will apply as long as the underlying relational database scheme remains unchanged. \n\nThe unit of data in Court Listener is the court opinion. Each court opinion is a single unique decision made by a specific court at a specific date on a specific case. A single case can be litigated by many courts, in case of appeals for example, so opinions are clustered per case. The Court Listener dataset follows the scheme of a relational database, where each opinion belongs to one cluster. Opinions are also clustered in dockets, a single docket gathering the different opinions made by a unique court on a unique case, as it happens that the same court may emit multiple opinions on the case at different dates.\n\nAt a more granular level, Court Listener represents each opinion as an HTML document, where specific tags mark where an in-text citation of case law appears in the opinion. Free Law implements a citation parsing and matching program\\footnote{https:\/\/github.com\/freelawproject\/courtlistener} that allows for an automated annotation of the unstructured text of the original court opinion. \n\n\\subsection{Data Processing}\n\nFor the construction of VerbCL, we focus on the opinions of CourtListener whose full text is available as HTML code including tagged in-text citations (looking for the field \\texttt{html\\_with\\_citations}). The HTML code for each opinion has specific tags to identify citations made in the text of the opinion, each citation being identified with the unique identifier of the cited opinion. \n\nWe consider that each citation is introduced by an anchor, which is located in the vicinity of the citation tag. The anchor is the ``raison d'\u00eatre'' of the citation, it is an argument to the current case, it is participating to the litigation of the case, and its validity is affirmed by invoking a similar argument that has already been made in an existing court opinion. \n\nWe focus specifically on the anchors that are extracts of the text of the cited opinion (\\textit{verbatim quotes}), similar to the example provided in Table~\\ref{table:sample}. \n\n\\begin{algorithm}[htbp]\n\\SetAlgoLined\n\\KwResult{Verbatim Quotes}\n initialization\\;\n \\ForAll{opinions}{\n \\ForAll{citations}{\n snippet = N words before \/ after in-text citation\\;\n candidates = potential verbatim quotes\\;\n \\ForAll{candidates}{\n \\If{candidate is verbatim}{\n identify original sentences in cited opinion\\; \n store data point\\;\n }\n }\n }\n }\n \\caption{Data Processing Pipeline}\n\\end{algorithm}\n\nOur data processing procedure includes extracting the text around the in-text citation, identifying potential verbatim quotes (often marked by quotation marks) and eventually classifying those who actually are verbatim quotes from the text of the cited opinion. Our code includes a framework for parallel distributed processing of massive dataset, using \\texttt{pyarrow}, Elasticsearch\\footnote{\\url{https:\/\/www.elastic.co\/}} through its Python API, and MongoDB\\footnote{\\url{www.mongodb.com}} together with \\texttt{pymongo}.\n\n\n\\subsubsection{Snippets extraction}\n\nAlthough the anchor is in the vicinity of the in-text citation, we observe a wide range of variations in the presentation and wording of these anchors, in relation to the in-text citation. Our strategy is to reduce this problem to identify a verbatim quote in the direct surroundings of the in-text citation, defined as the N words before and after the citation. We opted for a value $N=100$, which is a compromise between high recall (the more text around the citation we consider, the more likely it is we will identify the anchor), and computational footprint. Experiments included: considering entire paragraphs, but it yielded too much text; identifying nearby sentences, but the high numbers of acronyms and abbreviations around in-text citations renders sentence splitting useless.\n\n\n\\subsubsection{Verbatim candidates}\n\nThe identification of potential verbatim quotes is entirely rule-based, around the usage of different quote characters. Each snippet will generate multiple spans that were enclosed between those quote characters.\n\n\n\\subsubsection{Qualifying verbatim quotes}\n\nAt this stage, we want to confirm for each verbatim candidate whether it is a text that originates from the cited opinion. In the context of studying a specific in-text citation in a citing opinion, the cited opinion is known, so we have to evaluate whether a text span is lifted from the full text of the cited opinion.\n\nThe problem of identifying segments of text within a long text has proven to be difficult. As most of the opinions originate from printed books, OCR artifacts are expected, such as misplaced spaces or misspellings. This will affect the quality of the text in both cited and citing opinion. The common practice of using ellipsis in the citing opinion's text (as can be seen in the sample shown in Table~\\ref{table:sample}) will result in the suppression of fragments of the original text. \n\nTraditional approaches to this fuzzy matching problem rely on heuristics based on the computation of editing distances, such as the Levenshtein Distance~\\cite{1966SPhD...10..707L}. The design of the heuristics is a first challenge, given the sheer variety of differences that could be observed between the original text in the cited opinion and its counterpart in the citing opinion. More importantly, the computational footprint of calculating an editing distance renders this method inapplicable to our dataset.\n\nWe built a classifier based on ``Interval Queries'' offered by Elasticsearch. We refer the reader to the provided source code for the exact parameters of this query. The classifier predicts that a span of text belongs to the positive class when the query returns a non-empty result, and predicts the negative class when this query does not produce any search result. We manually annotated a test dataset for this classifier which revealed it had high precision and recall for both classes. The reader can refer to Chapter~\\ref{subsection:dataset-quality}.\n\nAs a result of this stage, we have built the VerbCL citation graph, a directed graph where nodes are opinions and edges are verbatim quotes.\n\n\\subsubsection{Identifying highlights}\n\nHighlights in the cited opinion are the sentences that were used for verbatim quotes. For each span of text identified as a verbatim quote of the cited opinion, we identify which sentence or sentences, are actually quoted from the text of the cited opinion. This is achieved through another set of Elasticsearch queries that are run over an indexed collection of opinion sentences. For sentence tokenization, we used the Punkt sentence tokenizer from \\texttt{NLTK}~\\cite{bird2009natural}.\n\nAs a result, we have built the VerbCL Highlights dataset, a dataset of opinions annotated at sentence level with a binary label where the positive class is made of sentences that were later on cited as verbatim quotes.\n\n\n\\subsubsection{Dataset structure}\n\nFollowing this stage, we have produced the following data collections:\n\\begin{itemize}[]\n \\item \\textbf{VerbCL Highlights}: An annotated collection of all opinion sentences, with a binary label for which True indicates that the sentence was cited verbatim;\n \\item \\textbf{VerbCL Citation Graph}: An annotated citation graph for all case law citations with a verbatim quote of the cited opinion. It is a directed acyclic graph $(V, E)$ where $V$ is the set of all opinion documents and $E$ is the set of verbatim quotes.\n\\end{itemize}\n\nVerbCL Highlights is used to inform the highlight extraction task since it contains the original opinion text and the subset we consider as the correct highlight.\nVerbCL Citation Graph was used to produce VerbCL Highlights and provides auxiliary information about the opinion text reuse that can be used for the highlight extraction task.\n\nSee Tables~\\ref{table:citation-graph-fields}-\\ref{table:highlights-fields} for the structure of these subsets.\nA tutorial on loading the data is available as a \\texttt{Jupyter Notebook} in the code repository\\footnote{\\url{https:\/\/github.com\/j-rossi-nl\/verbcl}}.\n\n\\subsection{Quality Assurance}\n\\label{subsection:dataset-quality}\n\nWe manually annotated 180 random anchors, coming from random opinions, and evaluated whether or not the candidate snippet was an actual verbatim quote out of the cited opinion. The code for this manual annotation task is included in our codebase, it is based on the \\texttt{Django} framework.\\footnote{\\url{https:\/\/www.djangoproject.com\/}}\n\nOn this random sample of 180 texts, with 60 of them from the positive class (actual verbatim quotes from the cited opinion), our binary classifier had a Precision $P>0.96$ and Recall $R>0.92$ for both classes. We consider our solution for this problem to offer a sufficiently good compromise between computation and accuracy.\n\n\\section{Dataset Statistics}\n\\label{section:dataset-statistics}\n\nIn the following we describe the main characteristics of the VerbCL Citation Graph and the VerbCL Highlights.\n\n\\subsection{VerbCL Citation Graph}\n\nFrom Court Listener, we consider the subset of opinions that either cite other opinions with a verbatim quote, or are cited verbatim by other opinions. This subset contains circa 1.5M opinions.\n\nThe main characteristics of our dataset are summarized in Table~\\ref{table:stats}. Considering that we are studying the nodes of a citation network, we also disclose our analysis of the corresponding graph. \n\n\\begin{table}[h]\n \\begin{tabularx}{\\linewidth}{|X|r|}\n \\multicolumn{2}{c}{\\textbf{Court Listener (CL)}} \\\\\n \\hline\n Opinions & 4,265,231 \\\\\n \\hline \n Citing opinions & 3,062,334 \\\\\n Cited opinions & 2,020,779 \\\\\n Citations & 30,318,321 \\\\\n \\hline\n \\multicolumn{2}{c}{} \\\\\n \\multicolumn{2}{c}{\\textbf{VerbCL Citation Graph}} \\\\\n \\hline\n Opinions & 1,493,561 \\\\\n & 35\\% of opinions in CL \\\\\n \\hline \n Verbatim quotes & 6,210,703 \\\\\n & 20\\% of citations in CL \\\\\n Verbatim citing opinions & 1,086,238 \\\\\n & 35\\% of citing opinions in CL \\\\\n Verbatim cited opinions & 946,962 \\\\\n & 47\\% of cited opinions in CL \\\\\n \\hline\n Edges in the citation graph & 4,002,137 \\\\\n Graph density & $1.8 \\times 10^{-6}$ \\\\\n \\hline\n Number of words in a verbatim & $[5-100]$ \\\\\n Average & 15 \\\\\n Quartiles 25-50-75 & 12 - 20 - 30 \\\\\n \\hline\n \\end{tabularx}\n \\caption{Dataset statistics of Court Listener (CL) and VerbCL.}\n \\label{table:stats}\n\\end{table}\n\nIn the citation network, nodes are opinions and directed edges materialize the citations, from the citing opinion towards the cited opinion.\nThereby, the in-degree of a node $v$ is the number of times this opinion is cited by other opinions, while the out-degree indicates the number of opinions that the opinion corresponding to the node $v$ cites.\n\nWe plot the distribution of in-degree of the nodes in the citation network in Figure~\\ref{fig:distrib_indegree}, as well as the Zipf's law, a power law with an exponent $k=-1$, which is a common distribution in the field of text mining. This shows us that these verbatim quotes have a behavior closer to the appearance of a term in a collection of documents, than to the expected distribution of links in a web graph (a power law with exponent $k \\approx 2.1$).\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=\\linewidth]{outdegrees.pdf} \n \\caption{Distribution of number of citations for an opinion.}\n \\label{fig:distrib_indegree} \n\\end{figure}\n\nWe imported the citation network, using \\texttt{networkx}~\\cite{networkx}, in order to compute some descriptive centrality statistics of the graph. For computing time reasons, we used approximations instead of the actual values, following the experiments in~\\cite{brandes2007centrality}.\n\n\\begin{figure}\n \\centering\n \\includegraphics[width=\\linewidth]{btw_ctr.pdf}\n \\caption{Distribution of betweenness centrality.}\n \\label{fig:distrib_btwctr}\n\\end{figure}\n\nThe distribution of betweenness centrality of the nodes of the citation graph is plotted in Figure~\\ref{fig:distrib_btwctr}, where we observe that the network contains a few very central nodes, corresponding to the group with the highest values.\nA node with a high betweenness centrality is a node which is on the shortest path between many nodes of the graph, reflecting its importance in the network.\nBetweenness centrality~\\cite{brandes2001faster} of a node $v$ is defined as the sum of the fraction of all-pairs shortest paths that pass through $v$, see Equation~\\ref{equation:betweenness-centrality}.\nSince the exact calculation of betweenness centrality for every node of the network is too computationally expansive, we made use of an approximation algorithm~\\cite{brandes2007centrality}.\n\n\\begin{equation}\n c_B(v) =\\sum_{s,t \\in V} \\frac{\\sigma(s, t|v)}{\\sigma(s, t)}\n \\label{equation:betweenness-centrality}\n\\end{equation}\n\n\nWhen applied to academic literature and the corresponding citation network, betweenness centrality is considered to measure importance of a paper in the field, i.e., seminal academic papers were reported to have a relatively high betweenness centrality~\\cite{leydesdorff2018betweenness,ORTEGA2014728}. \nIn contrast, our citation network has only a few nodes with high relative betweenness centrality but the absolute values are still very low (maximum: $3 \\times 10^{-4}$).\n\nWe hypothesize that this difference can be explained by much higher content redundancy in the court opinions in comparison with academic literature.\nNovelty and originality of contributions are the main characteristics for an academic paper, which makes the paper take a unique and irreplaceable place in the network.\nIn contrast, courts of the justice system serve any valid case presented regardless of its originality.\nTherefore, it is only natural that multiple cases can often make for equally good candidates in support of the same argumentation line.\nThis has an effect of the more even citation distribution across the whole network of court opinions in comparison with an academic network, which also reduces the centrality measures for court opinions as our data confirms.\n\nComparing the ranking of nodes by centrality with the ranking by outdegree (i.e. the number of times this opinion is cited by others), we compute the Kendall's Tau and Spearman's Rho coefficients of rank correlation: $\\tau = 0.21, \\, \\rho = 0.27$, both with p-value $p = 0.0$, showing weak but significant correlation. We attribute this weak correlation to the historicity factor: contrary to a webpages network, each node in our citation network is constrained by the capacity to cite only past opinions; we also consider the novelty factory, where a legal practitioner is biased towards most recent opinions when citing, therefore pushing the ``central'' opinions out of the shortest paths.\n\n\\subsection{VerbCL Highlights}\n\nThe main characteristics of our dataset are summarized in Table~\\ref{table:stats-highlights}. These statistics come from a random sample of circa 10k opinions.\n\nSentences from one opinion that are cited verbatim by other opinions are considered as highlights of the cited opinion. The task of identifying the highlights from the full text of an opinion is introduced in this paper under the name ``Highlight Extraction'', in Section~\\ref{section:highlight-extraction}.\n\n\\begin{table}[h]\n \\begin{tabularx}{\\linewidth}{|X|r|}\n \\multicolumn{2}{c}{\\textbf{VerbCL Highlights}} \\\\\n \\hline\n Opinions & 1,493,561 \\\\\n \\hline \n Number of sentences per opinion & $99.5\\%$ under $5000$ sentences \\\\\n Average & 437 \\\\\n \\hline\n Number of tokens per sentence & $99.9\\%$ under $171$ tokens \\\\\n Average & 24 \\\\\n \\hline\n \\% highlight sentences (per opinion, the number of highlights divided by the number of sentences) & $99.5\\%$ under $21.8\\%$ \\\\\n Average & $3.5\\%$ \\\\\n \\hline\n Number of citations per highlight & $99\\%$ under $434$ citations \\\\\n Average & 34 \\\\\n \\hline\n \\end{tabularx}\n \\caption{Dataset statistics for VerbCL Highlights.}\n \\label{table:stats-highlights}\n\\end{table}\n\nCourt opinions are typically large documents, we can illustrate this by showing the distribution of the number of words per opinion in Figure~\\ref{fig:token-per-opinion}.\n\n\\begin{figure}[h]\n \\centering\n \\includegraphics[width=\\linewidth]{tokens_per_opinion.pdf} \n \\caption{Number of words per opinion.}\n \\label{fig:token-per-opinion} \n\\end{figure}\n\n\\section{Highlight Extraction}\n\\label{section:highlight-extraction}\n\nIn this section, we will formally define the task of highlight extraction, then describe the empirical experiments we conducted and their results.\n\n\\subsection{Task Description}\n\nWe formally define the task of highlight extraction as follows.\nGiven a set of opinions $V$, where each opinion $v \\in V$ is associated with an opinion text $T_v$ split into a set $S_v$ of $n_v$ sentences.\nThe task of highlight extraction is to predict a subset of sentences $\\widehat{S_v} \\subseteq S_v$ which will be cited in the future.\nThe input of this system is the text of a single opinion. It is implied that the system's parameters are learned from the complete collection of opinions, and the system's training should not be restricted to document level tasks.\nThereby, the task of identifying highlights, when formulated at sentence level can be considered as a sentence binary classification (ranking) problem.\n\nIn this work, we define this task as a sentence highlighting task. The text spans we will consider are sentences from the opinion, we want to build model that predicts which sentence(s) in a given opinion might be cited, considering the historical observations as a ground truth that can be used for training.\n\nThis task involves the detection of network and timeline effects, as novelty arises from being the first to make a certain argument, or the first to legally qualify some facts as being compliant or not with generic statutes. As such we do expect it to be solvable at document level, at text level, if and only if we can assume that the opinion drafters are aware of the novelty of elements in the opinion, and draft them textually in a way that the task can be reduced to a Language Modeling task.\n\n\\subsection{Evaluation Metrics}\nIn line with the formulation of highlight extraction as a sentence ranking task, we will evaluate the different models with ranking metrics, such as Precision at 1 (P@1), Precision at R (P@R), Mean Average Precision (MAP) and Mean Reciprocal Rank (MRR). \n\nIn the context of supporting the work of a legal practitioner, we value the presence of relevant sentences early in the ranked list of sentences. P@1 measures the capacity of placing a highlight sentence at the top of the ranked list, P@R is a precision metric suitable for queries with a diverse number of relevant answers, MAP balances equally important metrics precision and recall and MRR is an indicator of the effort of the system's user, in relation to the rank of the first relevant sentence.\n\nWe also consider ROUGE~\\cite{Lin2004}, a family of metrics for summary evaluation. For each opinion, from the list of ranked sentences, we consider the hypothesis summary to be made of the top 5 sentences. The reference summary of an opinion is made of the highlight sentences in the opinion. ROUGE will score favorably a ranker which identified sentences lexically similar to the reference sentences.\n\n\\subsection{Data}\nWe use a sample out of VerbCL Highlights as training material. We make use of a random sample of 88k opinions, from the pool of opinions that have at least one sentence marked as highlight. The data is split between training material and test material. The train collection contains circa 70k opinions.\n\nAs we have to respect the sequence length limitations of the deep neural models, we restricted the test collection to the opinions for which there is at least one highlighted sentence in the first $N=512$ words of the opinion, and we consider the original text as made of these N first words. This test collection contains circa 1,000 opinions.\nSince the data is split based on opinions, the data in the test set is unseen for a model trained on the train set. \n\nAs we focus on identifying highlights, we make use of documents for which we observe that they contain highlights. We leave apart in this work the task of deciding whether or not an opinion contains highlight. From the perspective of the highlight extraction task, we consider having an oracle that has already stated the existence of highlights in the dataset we use.\n\n\\begin{table*}[h]\n \\begin{tabularx}{\\linewidth}{|X|r|r|r|r|r|r|r|}\n \\hline\n \\textbf{Model} & \\textbf{P@1} & \\textbf{MAP} & \\textbf{P@R} & \\textbf{MRR} & \\textbf{Rouge1-F} & \\textbf{Rouge2-F} & \\textbf{RougeL-F} \\\\\n \\hline\n DistilBERT & 0.05 & 0.11 & 0.05 & 0.12 & 0.31 & 0.20 & 0.26 \\\\\n TextRank & 0.14 & 0.23 & 0.14 & 0.24 & 0.41 & 0.36 & 0.38 \\\\\n PreSumm & 0.31 & 0.37 & 0.30 & 0.38 & 0.52 & 0.48 & 0.48 \\\\\n \\hline\n \\end{tabularx}\n \\caption{Results for the Highlight Extraction Baselines}\n \\label{table:highlight-results}\n\\end{table*}\n\n\\subsection{Baselines}\n\nConsidering the highlight task as a sentence ranking task, we propose two different types of baseline using\n\\begin{itemize}[]\n \\item binary classification at a sentence level;\n \\item extractive summarization at a document level.\n\\end{itemize}\n\nWhile it makes sense to consider the binary classification task, we argue that this approach will fail in this case, as most of the signals that a sentence in a legal opinion might be of interest for citation lie outside of the sentence itself. We produce this experiments to verify our assumptions that the sentence itself does not contain the signal of its importance.\n\nAn extractive summarization model will have to take the whole document into account in order to rank the sentences. On the one hand, deterministic methods will focus on one aspect for the importance of a sentence within a document. On the other hand, trainable models can be fine-tuned towards golden summaries, based on sentence-level annotations. We will use both types of models with our annotated data from VerbCL Highlights, and observe whether our annotations for importance can be learned this way. The assumption we want to confirm is that the document contains more information about the important sentence than the sentence itself, but still not enough information to lead to an accurate extraction.\n\nWe select the following models as our baselines:\n\\begin{itemize}[]\n \\item \\textbf{DistilBERT}~\\cite{sanh2020distilbert}: a generalized language model that we fine-tune for the binary classification task at sentence level. We use the \\texttt{transformers} library from HuggingFace~\\footnote{\\url{https:\/\/huggingface.co\/transformers\/}}. We derive a sentence ranker from this model by considering the estimated probability of belonging to the positive class as the sentence score.\n \n \\item \\textbf{TextRank}~\\cite{barrios2016variations}: an extractive summarization deterministic algorithm. Although it can not be considered as state of the art, TextRank has proven to be a reliable baseline for summarization with a reasonable computation footprint. Considering the task of highlight detection as an extractive summarization task, we evaluate how well the standard summarization approach that ranks sentences by their importance is able to capture what is considered salient from the perspective of citing an opinion. Using the Python implementation of \\texttt{PyTextRank}\\cite{PyTextRank}, we retrieve the score of each sentence.\n \n \\item \\textbf{PreSumm}~\\cite{presumm}: an extractive summarization model, based on Bert~\\cite{devlin-etal-2019-bert}. PreSumm has the capacity to be fine-tuned on any dataset annotated at sentence level. PreSumm implementation~\\footnote{\\url{https:\/\/github.com\/nlpyang\/PreSumm}} has been adapted to our dataset, and we retrieve the score of each sentence after inference. The model was trained for 50k steps, which took approximatively 14 hours with 4 nVidia GPUs.\n\\end{itemize}\n\n\n\\subsection{Results}\n\nWe present our results in Table~\\ref{table:highlight-results}. They confirm the initial assumptions with regard to the difficulty of the task. We formulate the highlight extraction as a task where the relevant context is the context of the current jurisprudence, therefore we make the assumption that this task can only be solved at corpus level, and not at document level. We observe from our experiences that using summarizers improves on the performance, but only to a certain degree. \n\nThe baselines we selected are introduced in their order of complexity. The sentence ranking we derive from a binary relevance classifier based on DistilBERT performs similarly to a random ranker. This validates our assumption that highlights can not be discovered at sentence level only. \n\nOn the other hand, the improvements shown with the summarizing models emphasizes that some of the highlights can be captured at the document level. We hypothesize that this is linked to the opinion drafter's awareness of the importance of the sentence, and its potential to be cited later. An opinion drafter is a person skilled in the legal domain, they are in position to have this intuition. They would reflect the importance of the sentence at the text level, and therefore be detected by models from the BERT lineage, who excel at isolating signals from the text~\\cite{tenney2019bert,clark2019does,lin-etal-2019-open}.\n\nThe $\\textrm{P}@1 = 0.31$ observed on PreSumm is a promising result, but we also have to consider that it is restricted to short documents, while a majority of documents are longer than the maximum input length. Despite the observed improvements, we conclude that none of these systems has solved the task sufficiently. \n\n\\section{Related Work}\n\\label{section:related-work}\n\nIn this section, we present an overview of existing work in relation with the dataset and the tasks we want to support, from within the legal domain as well as outside of this legal domain. We provide a brief overview of the previously proposed datasets, and describe the recent advancements on the tasks of highlight extraction, citation-based summarization and case law retrieval.\n\nIt is important to note that similar tasks often arise also in patent retrieval~\\cite{sarica2019engineering, hofstatter2019enriching, shalaby2019patent, shalaby2018toward, rossi2018query} and academic document retrieval domains~\\cite{schaer2020overview,li2017topic,xiong2017explicit,li2017investigating}. \n\nWhen drawing parallels to the tasks outside of the legal domain, the following dimensions are of a special importance: \n\\begin{itemize}[]\n \\item emergence of a directed citation network;\n \\item time-based dependencies (it is possible to cite only previously published sources); \n \n \\item highlight extraction for document summarization;\n \\item predicting citations and text reuse.\n\\end{itemize}\n\nWe observe recent work in the field of citation analysis for scientific literature, enabled by S2ORC~\\cite{lo-wang-2020-s2orc}, which we consider a similar dataset to VerbCL Citation Graph and VerbCL Highlights in content and intent. Applied to academic literature, rating novelty~\\cite{hua2021extraction}, identifying contributions~\\cite{hayashi2020whats} or summarizing~\\cite{cachola2020tldr} share similarities with the highlight extraction task. We depart from the work done on academic literature, as we argue that the novelty is the core motivation of academic publication, while court opinions are motivated by the obligations of a public service which has to serve justice whenever it is requested. In that matter, novelty is expected to be sparser, while abundant redundancy should be the norm.\n\n\\subsection{Case Law Datasets}\n\nCourtListener was also the original source of CaseLaw~\\cite{Locke2018}, for the purpose of testing case law retrieval models and systems, while the main contribution was a test collection of 2,500 relevance assessments and an annotation tool. We observe similarly sized datasets in other languages, such as CAIL2019-SCM \\cite{DBLP:journals\/corr\/abs-1911-08962} in Chinese. Recent meta-reviews of the field \\cite{zhong-etal-2020-nlp} provide pointers to existing datasets\\footnote{\\url{https:\/\/github.com\/thunlp\/LegalPapers}}, we observe that none of them has the size required for training a complex model on a specific task. \n\nCOLIEE~\\cite{coliee2018, coliee2019} is also a relevant competition, task and dataset for legal purposes, of reduced size. It is made of multiple tasks, tackling case law retrieval with tasks 1 and 2, within the legal domain of Immigration and Citizenship in canadian courts.\n\nTask 1 describes case law retrieval as the task of identifying which past cases should be mentioned in a query case from which citations have been removed. The rationale behind the relevance judgment, the reason why cases have been cited in the past is left unknown. Task 2 will focus on identifying relevancy at paragraph level, which is a step forward to clarifying relevance for the human reader, although the dataset does not include annotations with regard to relevance arising from either situation and facts or reasoning.\n\n\\subsection{Highlight Extraction}\n\nGiven a long document, the task of highlight extraction aims at finding the snippets of text that contain the information that a user will find useful in this document, making it a task ripe for user query biased extraction. For example, \\cite{49929}, \\cite{7950943} and \\cite{doi:10.1080\/10494820.2017.1282878} show the importance of highlighting for learning. Highlighting is described as a manual annotation task, for the benefit of meeting future information needs. The concept is put in practice also by \\cite{hardy-etal-2019-highres} and \\cite{cheng-lapata-2016-neural}, where highlights support the generation or the evaluation of text summaries. We observe only few attempts at creating an automated highlighting system for text, in contrast to the flourishing field of video highlight generation. \n\nThe key concept we articulate our task around is meeting future information needs. Although we have plenty of historical data available, we can only speculate now on what is important in any opinion that is published, and what will be picked up in cases that courts will litigate in the near or far future. \n\nThe highlight extraction task we present in this work focus on identifying the novel legal content in a new case that would potentially be cited in future cases, and evaluate how this could stem from text level analysis of the opinion, rather than identifying how an opinion would suit an argument made in another one. We refer to the Section~\\ref{section:highlight-extraction} for more details.\n\n\\subsection{Case Law Retrieval}\nOur work is also related to the previous work on the task of case law retrieval. In this task, a legal practitioner retrieves a list of past cases that are legally relevant to a case at hand.\n\nCOLIEE~\\cite{coliee2018, coliee2019} has hosted a yearly competition including such a retrieval task, restricted to cases from the Immigration and Citizenship Courts of Canada. The target is to identify cases that could be cited in a query case. \n\nThe task is reduced to identifying the similarities between court opinions, similarities in details of the case, in the underlying storylines and reasoning. Participants reformulated the retrieval task as a ranking task based on a pairwise relevancy classifier, with various text representation techniques applied, from lexical representations~\\cite{ubirled, tran2018jnlp} to deep representations~\\cite{rossi2019legal, iitpcoliee2019}. It is observed that the scarcity of data is hindering the use of advanced data-driven models. This domain relevant task has shown the difficulty to work at text level to evaluate a relevancy that professionals had annotated.\n\nWe see a potential application of our dataset in a case law retrieval task guided by arguments, where the query is an argument, and the search results a list of past opinions that could support the argument, based on the highlights that were extracted.\n\n\\subsection{Citation-based Summarization}\n\nThe task of highlight extraction finds its roots in the task of summarizing long documents. We observed many efforts in the field of summarizing by using citations, in the domain of academic literature summarization, for example. Citation-based summarization was first introduced in the context of scholarly data processing for summarizing scientific articles~\\cite{IbrahimAltmami2020}.\nIn citation-based summarization, citation anchors (called \\say{catchphrases}) for one target document over multiple citing document form a summary for this target document. The CL-SciSumm shared task~\\cite{Chandrasekaran2019} aims at producing summaries of academic papers guided by citation anchors (called \\say{citances}), by identifying spans of text relevant to the anchor. The approaches are evaluated against a set of golden summaries, using standard metrics of the ROUGE family~\\cite{Lin2004}.\n\nThe court opinions will show a predetermined structure, as academic papers do, although the articulation in different parts or chapters will not be signaled explicitly in the text. Our task definition differs as we focus on actual verbatim quotes from cited documents which are less biased towards the writer's view of the cited document than paraphrases. Our work considers these verbatim quotes to be the target summary, as opposed to considering them as a way to build a summary similar to the golden summary.\n\nWe consider that the spans of an opinion which are later used in citations form a summary of this opinion, which a specific type of summary guided by the search for novel and important legal aspects addressed in the opinion document.\nThis summary characterizes what the change an opinion makes in terms of the legal argumentation, i.e., its network effect (global context), not just a summary of all the elements present in the opinion (local context).\n\n\n\\section{Conclusion}\n\\label{section:conclusion}\n\nIn this paper we introduced the task of highlight extraction from court opinions and the large dataset VerbCL of annotated court opinions aimed at supporting training and evaluation for the highlight extraction task.\nOur dataset focuses on citations made in a court opinions by quoting verbatim preceding court opinions in support of a legal argument.\nVerbCL is sufficiently large and can be used for training machine learning models on the highlight extraction task.\nWe also see the potential of VerbCL for other legal information retrieval tasks that can be informed by the citation network. \n\nWe also demonstrated the difficulty of the highlight extraction task, which escapes the reduction to a sentence-level or document-level task.\nNote that the citation network is the result of a massively distributed expert work from the date of the first citation until now.\nEvery citation is the result of an expert's retrieval of a relevant opinion to support the argument in the context of a specific case at hand. \nHowever, we can verify the importance of every specific text span from an opinion only post-hoc, i.e., not at the time of the publication but later on as we observe how the same arguments made in the opinion are subsequently re-used within other opinion documents in our dataset.\nOnly this reuse over time indicates the document's importance in terms of the impact it made to help shape other documents.\nFuture work should aim to address such network and historical effects, distilling knowledge of the previously published documents into the processing of a new one.\nWe believe that similar approaches will also prove useful for patent retrieval and academic search scenarios.\n\nFuture work should also consider extending the analysis to abstractive anchors, which requires a text generation task setup instead of purely retrieval-based one considered here.\nFinally, considering the full document length remains a challenge for the current neural ranking approaches that should be addressed to be applicable in the legal domain. \n\n\n\\begin{acks}\nThis research was supported by\nthe NWO Innovational Research Incentives Scheme Vidi (016.Vidi.189.039),\nthe NWO Smart Culture - Big Data \/ Digital Humanities (314-99-301),\nthe H2020-EU.3.4. - SOCIETAL CHALLENGES - Smart, Green And Integrated Transport (814961),\nthe Amsterdam Business School PhD program.\nAll content represents the opinion of the authors, which is not necessarily shared or endorsed by their respective employers and\/or sponsors.\n\\end{acks}\n\n\\balance\n\\clearpage\n\\bibliographystyle{ACM-Reference-Format}\n","meta":{"redpajama_set_name":"RedPajamaArXiv"}}