*J. High Energ. Phys.* 1909 (2019) 001 arXiv link

**Holographic OPE coefficients from AdS black holes with matters**

**Yue-Zhou Li**^{1},**Zhan-Feng Mai**^{1}^{,*}, **H. Lü**^{1}

* zhanfeng.mai@gmail.com

* 1. Center for Joint Quantum Studies and Department of Physics, School of ScienceTianjin UniversityTianjinChina*

**Abstract**

We study the OPE coefficients *c*_{Δ, 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 Δ_{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 Δ_{L}, although the OPE coefficients are not fully determined, we set up a framework to read off the coefficients *γ*_{Δ,J} of the log(*z*
z¯¯¯
$\stackrel{\xaf}{z}$) terms that are associated with the anomalous dimensions of the exchange operators and obtain a general formula for *γ*_{Δ,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 Δ_{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.