# Holographic OPE Coefficients from AdS Black Holes with Matters

## TL;DR

This paper computes holographic OPE coefficients for heavy-light scalar four-point functions in AdS black hole backgrounds, revealing how matter fields and scalar hair influence these coefficients and their relation to black hole properties.

## Contribution

It introduces a framework for calculating OPE coefficients in charged AdS black holes with matter, including recursion formulas and explicit examples, extending previous pure gravity results.

## Key findings

- Verified $c_{d,0}=0$ for pure gravity black holes.
- Derived explicit OPE coefficients for charged black holes with matter.
- Established a method to extract anomalous dimensions from $ ext{log}(zar{z})$ terms.

## Abstract

We study the OPE coefficients $c_{\Delta, J}$ for heavy-light scalar four-point functions, which can be obtained holographically from the two-point function of a light scalar of some non-integer conformal dimension $\Delta_L$ in an AdS black hole. We verify that the OPE coefficient $c_{d,0}=0$ for pure gravity black holes, consistent with the tracelessness of the holographic energy-momentum tensor. We then study the OPE coefficients from black holes involving matter fields. We first consider general charged AdS black holes and we give some explicit low-lying examples of the OPE coefficients. We also obtain the recursion formula for the lowest-twist OPE coefficients with at most two current operators. For integer $\Delta_L$, although the OPE coefficients are not fully determined, we set up a framework to read off the coefficients $\gamma_{\Delta,J}$ of the $\log(z\bar{z})$ terms that are associated with the anomalous dimensions of the exchange operators and obtain a general formula for $\gamma_{\Delta,J}$. We then consider charged AdS black holes in gauged supergravity STU models in $D=5$ and $D=7$, and their higher-dimensional generalizations. The scalar fields in the STU models are conformally massless, dual to light operators with $\Delta_L=d-2$. We derive the linear perturbation of such a scalar in the STU charged AdS black holes and obtain the explicit OPE coefficient $c_{d-2,0}$. Finally, we analyse the asymptotic properties of scalar hairy AdS black holes and show how $c_{d,0}$ can be nonzero with exchanging scalar operators in these backgrounds.