We study the one-loop corrections to the four-point function in the Anti de Sitter space-time
for a

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0$ and $y_3 > 0$. Hence, inside the loops in Euclidean signature $\bar{y}$ plays no role. On the other hand, in Lorentzian signature, $s$ and $\bar{s}$ vanish on the light-cones whose tips are $y$ and, respectively, $\bar{y}$. Therefore, even though $\bar{y}$ does not belong to the space-time manifold its light-cone penetrates into it (see figure~\ref{fig:mirror}). That leads to a different UV renormalization at the leading order, as we will see below. \begin{figure}[t] \centering \includegraphics[width=6.4cm]{4.pdf} ]]>

0 $ so that $\zeta \geq 1$). Here $d\left(Z, Z'\right)$ is the length of the arc of the hyperbola obtained by intersecting $EAdS_4$ with the plane containing $Z$, $Z'$ and the origin of the ambient space. In the Lorentzian cases, however, not always the two points in eq.~(\ref{invsc}) can be joined by an arc of geodesic. For instance in the dS case, where $Z= (i Y_0, \vec X)$, the geodesic distance between two points is implicitly defined as in eq.~(\ref{geohyp}) for $\zeta>1$ (i.e for $Z$ and $Z'$ timelike separated), as in eq.~(\ref{geospher}) {for} $|\zeta|\leq 1$ and not defined at all for $\zeta< -1$. As before $d\left(Z, Z'\right)$ is the arc length of either the hyperbola or the ellipse joining the two points $Z$ and $Z'$. Similar remarks apply to the AdS case. Both in dS and AdS $\zeta$ takes all the values between plus and minus infinity. Consider now a two-point function $W(Z,Z') = w(\zeta)$ depending only on the invariant variable $\zeta$ and defined in some domain of the complex $d$-dimensional sphere. We may extend the function $w$ to the ambient space by homogeneity of degree zero by considering the function $w_{\rm ext}\left(\frac{ Z\cdot Z'} {\sqrt {Z^2}{\sqrt{Z'}^2}}\right)$, where $Z$ and $Z'$ are no more constrained to the complex sphere. The complex Laplace operator applied to $w_{\rm ext}$ (in general dimension $d$) coincides with the the Laplace-Beltrami operator on the sphere applied to $w$: \begin{equation} \left. \nabla^2 w_{\rm ext} \right|_{CS_d} = \left. \frac \partial {\partial Z_\mu}\frac {\partial w_{\rm ext}} {\partial Z^\mu} \right |_{CS_d} = \left(1-\zeta^2 \right)\, \partial_\zeta^2 w(\zeta)- d \, \zeta \, \partial_\zeta w(\zeta). \end{equation} The following equation in the invariant variable: \beq\label{wightprop} \left[\left(1-\zeta^2 \right)\, \partial_\zeta^2 - d \, \zeta \, \partial_\zeta +\sigma(d-1-\sigma) \right] \, w(\zeta) = 0 \eeq represents therefore either the Laplace or the Klein-Gordon equation (possibly also with a delta source at the r.h.s.). The parameter $\sigma$ is related to the mass as follows: \beq \pm \left(m \, R\right)^2 =\sigma(d-1-\sigma) \eeq i.e.\ $$\sigma =\frac{d-1} 2 + \ \sqrt{\left(\frac{d-1}{2}\right)^2 \pm \left(m \, R\right)^2}= \frac{d-1} 2 + \nu. $$ The plus sign is chosen for the AdS and Lobachevsky cases, while the minus sign is chosen for the dS case and the sphere. However in both cases $\left(m\, R\right)^2$ can also take negative values, as we recall in appendix~\ref{apb}. Changing to either $ \frac{1+\zeta}{2}$ or $ \frac{1-\zeta}{2}$ gives to eq.~(\ref{wightprop}) the standard hypergeometric form. Correspondingly the general solution may be written in general dimension $d$ as follows: \begin{equation} w(\zeta) = \frac{ {2}^{\frac{d-2}2} A_\pm \, {}_2F_1 \left(\frac{1} 2 + \nu, \, \frac{1} 2 - \nu; \frac{d}{2}; \frac{1+\zeta}{2}\right) }{\left({1-\zeta}{}\right)^{\frac{d-2}2} }+ \frac { 2^{\frac{d-2}2} B_\pm {}_2F_1 \left(\frac{1} 2 + \nu, \, \frac{1} 2 - \nu ; \frac{d}{2}; \frac{1-\zeta}{2}\right)}{\left({1+\zeta}\right)^{\frac{d-2}2} } \label{propgen} \end{equation} where ${}_2F_1$ is the hypergeometric function and we used Kummer's relation. $A_\pm$ and $B_\pm$ are some complex coefficients that are chosen according with the sign of $\Im \zeta$. There are indeed cuts on the real axis of the complex $\zeta$-plane which come from the quantum commutation relations; one has to specify suitable analyticity properties for $w$ which depend on the chosen geometry (i.e.\ dS or AdS) in such a way that the upper and lower boundary values of the Wightman functions in~(\ref{propgen}) coincide at spacelike separated pairs. In particular, in the dS case two (real) points $Z$ and $Z'$ are time-like separated when $(Z-Z')^2<0$ and therefore in the complex plane of the invariant variable $\zeta$ there is cut (at least) on the positive reals starting from $\zeta=1$. On the other hand, in the AdS case two (real) points $Z$ and $Z'$ are time-like separated when $(Z-Z')^2>0$ and therefore the cut is opposite to the previous i.e.\ it is the half-line $(-\infty ,1]$. In some special periodic cases, when $\nu = \sqrt{\left(\frac{d-1}{2}\right)^2 + \left(m \, R\right)^2}= \frac 12 , \frac 32 , \frac 52 \ldots $, the two contributions in~(\ref{propgen}) compensate and cut reduces to the interval $[ -1,1]$. The inverse image --- in the complex sphere at fixed $Z'$ --- of the first singularity $\zeta=1$ is the intersection of the complex sphere $Z^2=1$ with the complex cone $(Z-Z')^2= 0$. This surface includes coinciding points on either the sphere or the Lobachevsky space and lightlike separated pairs on either the dS or the AdS manifold. On the other hand the singularities of the second term in~(\ref{propgen}) are on the intersection of the complex sphere $Z^2=1$ with the complex cone $(Z+Z')^2= 0$; the latter is the cone having its tip at $-{Z}'$, the antipodal point of $Z'$. Every point $Z$ on the complex sphere has an antipode $-Z$. This reflection gives the antipodal points also on the real submanifolds we are considering in this paper (on the Lobachevsky space the antipodal point belongs to the other sheet of the two-sheeted hyperboloid). The crucial difference is the following: while in the dS case antipodal pairs are spacelike separated in the AdS case they are not. Actually, in the AdS case all the timelike geodesics issued from an event $Z$ focus at the antipodal point $-Z$; this fact remains true also on a general covering of the (real) AdS space. This is the reason why in the AdS case the second singularity in eq.~(\ref{propgen}) is always present. In the dS case there is a special choice of vacuum, namely the Bunch-Davies or Euclidean vacuum~\cite{Bunch-ml-1978yq} which is maximally analytic~\cite{bros,bros2,bros3} and corresponds to the choice $A_+=A_-$, $B_\pm=0$ in~(\ref{propgen}). The maximal analyticity properties implies that the Schwinger function on the sphere is the analytic continuation of the two-point function on the real dS manifold and that there are no singularities in between. However there are other choices, namely the so called alpha-vacua {\cite{Spindel,Mottola-ml-1984ar,Allen-ml-1985ux}} (see~\cite{Akhmedov-ml-2013vka} for a review) which are dS invariant (at least at tree-level) but they are not analytic precisely because of the presence of the additional singularity. In the AdS case the situation is a little more involved due to the presence of a non trivial topology of the real manifold. As we show in the appendix, the Feynman propagator can be represented both in global AdS and in its covering $\widetilde{AdS}$ manifold as follows: \begin{eqnarray} \label{FAdS} && F_\nu\left(X,X'\right) = \non\\ && -\frac{i \, \Gamma\left(\frac {d-1}2-\nu \right) \Gamma\left(\frac {d-1} 2 +\nu\right)}{ 2\,{(2\pi)^{\frac{d}2}} \Gamma\(\frac {d}2\)} \left[ \left(\frac{1}{\xi -1 +i\epsilon}\right)^{\frac {d-2}2} \phantom{|}_1F_2\left(\frac 1 2-\nu ,\frac 1 2+\nu,\frac {d}2;\frac {1+ \xi+ i \epsilon} 2 \right)\right. \non\\ && - \left. e^{- i\(\nu-\frac 1 2\)\pi} \left(\frac{1}{\xi+1+i\epsilon}\right)^{\frac {d-2}2} \phantom{|}_1F_2\left(\frac 1 2-\nu,\frac 1 2+\nu,\frac {d}2;\frac {1-\xi - i\epsilon}2 \right)\right] \label{prop0} \end{eqnarray} Here $X,X'$ denote two events of the real AdS manifold~(\ref{ads}) and $\xi$ denotes their invariant scalar product~(\ref{invsc}) (see also~(\ref{ambientmetricads})). Both contributions on the r.h.s.\ of this expression separately grow when $\xi \to \infty$, i.e.\ as the distance between $X$ and $X'$ is increasing. But in the linear combination appearing here the growing contributions compensate each other and the resulting expression is decaying with the distance. That agrees with~\cite{Avis-ml-1977yn,kent2013quantum} (see also~\cite{Polyakov-ml-2007mm} and~\cite{Akhmedov-ml-2009ta}). The corresponding Wightman function is maximally analytic~\cite{bbgm,bem}. Note also that \begin{equation} (X-X')^2-i \epsilon = 2R^2(1 - (\xi+i\epsilon)) \end{equation} and therefore the prescription $\xi \to \xi + i \epsilon$ corresponds to the standard prescription in Minkowski space. Details on the analiticity properties of AdS propagators are given in appendix. The relation of the above $i \eps$ prescription with the local time ordering of the AdS manifold and the global time ordering of its covering is discussed below and in appendix. ]]>

0,\ \ \epsilon(Z) = \mbox{sign} (Y^0 X^{d}- X^0 Y^{d}) = \pm\}. \label{Tubes} \end{equation} The following \emph{normal analyticity property} is equivalent to the positivity of the spectrum of the energy operator: \begin{center}\emph{ $W\left( X, X'\right)$ is the boundary value of a function $ W\left(Z, Z'\right) $ holomorphic in $\widetilde{T}^-\times \widetilde{T}^+$} \end{center} where $\widetilde{T}^\pm $ are the coverings of the above chiral tubular domains. By using the complex coordinates (see above) the domains~(\ref{Tubes}) are seen to be semi-tubes invariant under translations in the variable $\Re t$. They can be described by the following inequalities: \begin{equation}\pm \sinh \Im t> \left[\frac{(\sin\Im \psi)^2 + \left( (\cosh\Re \psi)^2 - (\cos \Im \psi)^2 \right)(\Im \omega)^2}{ (\cosh \Re \psi)^2 - (\sin\Im \psi)^2}\right]^\frac{1}{2}. \label{retubes}\end{equation} To grasp more intuitively the meaning of the above statement, in the case of real $\psi$ and $\omega$ i.e.\ when only the time coordinate $t$ is complexified it simply amounts to require that $W\left[Z(t, \psi,{\omega}), \, Z'(t', \psi',{\omega'})\right]$ has an analytic continuation to complex pairs such that $\Im t<0$ and $\Im t'>0$ (flat tubes); this analyticity property is equivalent to the positivity of the energy spectrum. AdS invariance further implies that $W(Z,Z')$ is actually a function $w(\zeta)$ of a single complex variable $\zeta$ which can be identified with $Z\cdot Z'$ when $Z$ and $Z'$ are both in the fundamental sheet of $\widetilde{CAdS}_{\dd}$; therefore AdS invariance and the spectrum condition together imply the following \emph{maximal analyticity property}: \begin{center} \emph{$w(\zeta)$ is analytic on the covering $\widetilde\Theta$ of the cut-plane $\Theta = \{{\mathbb C} \setminus[-1,1]\}$.} \end{center} For fields periodic in the time coordinate (semi-integer $\nu$ case) $w(\zeta)$ is in fact analytic in $\Theta$ itself. One can now introduce all the standard Green functions. The permuted two-point function ${W}(X',X)$ is the boundary value of ${ W}(Z,Z')$ from the opposite domain $\{(Z,Z'): Z\in { \widetilde T}^{+}, \; Z'\in {\widetilde T}^{-}\}$. The retarded propagator ${R}(X,X')$ is introduced by splitting the support of the commutator ${C}(X,X')$ as usual ($X,X' \in AdS_\dd$ real) \begin{eqnarray} C(X,X')&=&{W}(X,X')-{W}(X',X), \\ {R}(X,X')&=& i\theta (t - t') { C}(X,X'). \end{eqnarray} Finally, the chronological (Feynman) propagator is given by \begin{equation} {\tilde{F}} (X,X')= -i \theta (t - t') {W}(X,X')- i \theta (t' - t) {W}(X',X). \label{chronological} \end{equation} This definition refers to the global time-ordering of the covering manifold. The uncovered AdS manifold is not globally time ordered. We take into account this property by modifying the previous definition as follows \begin{equation} {F} (X,X')= -i \theta\left[\sin(t - t')\right]\, {W}(X,X')- i \theta\left[\sin(t' - t)\right]\, {W}(X',X). \label{chronologicalads} \end{equation} The two prescriptions coincide on the fundamental sheet $|t-t'|<\pi$. ]]>

0$ or $\Im \zeta< 0$}~\mbox{\cite{Bateman}}. Here the parameter $\nu$ is related to the mass as follows \begin{equation} \nu^2 = \frac {(d-1)^2} {4} + \left(m \, R\right)^2, \label{nu} \end{equation} and the normalization of $W_\nu$ is chosen by imposing the short-distance Hadamard behavior. {For $d=4$ the coefficient at the r.h.s.\ of eq.~(\ref{kgtp3}) is divergent when $\nu = \frac 32 , \frac 52,\ldots $}. At the same time for $\nu = \frac 32 , \frac 52,\ldots $, the phases multiplying the second term are equal in the upper and lower half-planes. The hypergeometric series become sums and the constant terms exactly cancel. Therefore for $\nu = \frac 12, \frac 32 , \frac 52 , \ldots $, etc.\ we may extract two-point functions which are periodic in the time variable (and therefore live on the true AdS manifold) and analytic in the cut plane $\Theta$. This may be done by analytic continuation in the dimension $d$. For instance for $\nu = \frac 32$ (the massless $m=0$) case we have \begin{eqnarray} w^d_{\frac 32}(\zeta) &=& \frac{\Gamma\left(\frac {d-1}2-\frac 32 \right) \Gamma\left(\frac {d-1} 2 +\frac 32\right)}{ 2\,{(2\pi)^{\frac{d}2}} \Gamma\(\frac {d}2\)} \left[ \left(\frac{1}{\zeta -1 }\right)^{\frac {d-2}2} \left(1- \frac{4}{d}\frac {1+\zeta}2 \right)\right. \non\\ && + \left. \left(\frac{1}{\zeta+1}\right)^{\frac {d-2}2} \left(1 - \frac 4 {d} \frac {1-\zeta }2 \right)\right] . \end{eqnarray} Taking the limit at $d=4$ we get \begin{eqnarray} w_{\frac 3 2}(\zeta) =-\frac{1}{ 2{(2\pi)^{2}} } \log \left(\frac {\zeta+1}{\zeta -1}\right)+ \frac{ 1}{ 2\,(2\pi)^{2}} \left[ \left(\frac{1}{\zeta -1 }\right) + \left(\frac{1}{\zeta+1}\right) \right]. \end{eqnarray} Note that for $\nu = \frac 12$ the coefficient at the r.h.s.\ of eq.~(\ref{kgtp3}) is finite; both hypergeometric series are equal to 1 and the r.h.s.\ is just the difference of the two poles: \begin{eqnarray} w_{\frac 1 2}(\zeta) =\frac{ 1}{ 2\,(2\pi)^{2}} \left[ \left(\frac{1}{\zeta -1 }\right) - \left(\frac{1}{\zeta+1}\right) \right] . \label{12} \end{eqnarray} ]]>

0.$$
On the other hand when $-\pi < t<0$ the real event $X(t, r,{\omega})$ has to be considered at the boundary of $T^+$;
$$\zeta = \cosh\psi\cos (t+i \epsilon ) =\xi - i \epsilon \sin t, \ \ \ \ \epsilon>0.$$
Therefore as long as we have $-\pi