Two-loop renormalization of multiflavor $\phi^3$ theory in six dimensions and the trace anomaly

TL;DR

This paper computes two-loop counterterms and the trace anomaly for multiflavor $\,\phi^3$ theory in six dimensions, revealing a violation of the strong $a$-theorem and providing a three-loop expression for the Euler term coefficient.

Contribution

It presents the first two-loop renormalization of multiflavor $\,\phi^3$ theory in curved spacetime with spacetime-dependent couplings and derives implications for the trace anomaly and $a$-theorem.

Findings

Two-loop counterterms for the theory are obtained.

A general expression for the trace anomaly is derived.

The strong $a$-theorem is shown to be violated in this context.

Abstract

We use the background-field method and the heat kernel to obtain all counterterms to two-loop order of conformally-coupled multiflavor $\phi^3$ theory in six spacetime dimensions, defined in curved spacetime and with spacetime-dependent couplings. We also include spacetime-dependent mass terms for completeness. We use these results to write a general expression for the trace anomaly. With the use of Weyl consistency conditions we are able to show that the strong $a$-theorem for a certain natural candidate quantity $\tilde{a}$ is violated in this theory, and obtain a three-loop expression for the coefficient $a$ of the Euler term in the anomaly.