4-point function from conformally coupled scalar in AdS$_6$

TL;DR

This paper investigates the holographic 4-point function of a conformally coupled scalar in AdS$_6$, revealing its non-conformal nature except in specific momentum configurations, and introduces a method to compute higher-point functions consistent with conformal symmetry.

Contribution

It provides a detailed analysis of 4-point functions in AdS$_6$ for a conformally coupled scalar, including a new scheme to remove divergences and a rule for constructing $n$-point functions.

Findings

4-point function is not conformal generally but becomes conformal in the co-linear limit.

Holographic correlation functions are free from co-linear divergences.

A Feynman-like rule for constructing $n$-point functions reproduces conformal correlators in the co-linear limit.

Abstract

We explore conformally coupled scalar theory in AdS$_{6}$ extensively and their classical solutions by employing power expansion order by order in its self-interaction coupling $\lambda$. We study holographic correlation functions of scalar operator deformations to a certain 5-dimensional conformal field theory where the operators share the same scaling dimension $\Delta=3$, from the classical solutions. For our solutions, we choose a scheme where we remove co-linear divergences of momenta along the AdS boundary directions which frequently appear in the classical solutions. This shows clearly that the holographic correlation functions are free from the co-linear divergences. It turns out that this theory provides correct conformal 2- and 3- point functions of the $\Delta=3$ scalar operators as expected in previous literature. It makes sense since 2- and 3- point functions are determined by global conformal symmetry not being dependent on the details of the conformal theory. We also get 4-point function from this holographic model. In fact, it turns out that the 4-point correlation function is not conformal because it does not satisfy the special conformal Ward identity although it does dilation Ward identity and respect $SO(5)$ rotation symmetry. However, in the co-linear limit that all the external momenta are in a same direction, the 4-point function is conformal which means that it satisfy the special conformal Ward identity. We inspect holographic $n$-point functions of this theory which can be obtained by employing a certain Feynman-like rule. This rule is a construction of $n$-point function by connecting $l$-point functions each other where $l<n$. In the co-linear limit, these $n$-point functions reproduce the conformal $n$-point functions of $\Delta=3$ scalar operators in $d=5$ Euclidean space addressed in arXiv:2001.05379.