Light-like retarded correlators and the horizon
We show that retarded correlators in conformal field theories of scalar primary operators evaluated at finite temperature using $AdS/CFT$ scale anomalously at large light like momenta. The anomalous scaling depends on the curvature of the horizon. Setting the momenta equal to the frequency $ω$, the retarded correlator for black holes with planar horizons scales as $ω^{\frac{2}{d+2} (2Δ- d) }$ as opposed to the scaling behaviour of $ω^{2Δ- d}$ expected by dimensional analysis at generic fixed momenta and large frequencies. $Δ$ is the dimension of the primary and $d$, the number of space-time dimensions. For black holes with spherical and hyperbolic horizons when the frequency squared equals the Casimir along the horizon, the retarded correlator scales as $ω^{\frac{2Δ-d}{2} }$. We establish this using exact results in $d=2$, and a numerical analysis of the Heun equation for the $AdS_5$ planar black hole and finally using the WKB approximation in general $d$. The exact result for black holes with hyperbolic horizon as well as the analysis at large $d$ provides additional checks for the anomalous scaling behaviour. Finally we evaluate the anomalous scaling exponent for the stress tensor correlator in all its 3 channels.