A challenge to the $a$-theorem in six dimensions

TL;DR

This paper investigates the behavior of the  ext{-theorem} in six-dimensional  ext{-theory} and finds that the expected monotonic decrease of  ext{-like} quantities does not hold in perturbation theory, challenging previous assumptions.

Contribution

It provides the first perturbative analysis of the  ext{-theorem} in six dimensions, showing that  ext{-like} quantities can increase along RG flows, which is contrary to lower-dimensional cases.

Findings

 ext{-like} quantity  ext{tilde} increases monotonically in perturbation theory.

Calculated anomalous dimensions and beta functions to two loops.

Results suggest the need for new intuition about the  ext{-theorem} in higher dimensions.

Abstract

The possibility of a strong $a$-theorem in six dimensions is examined in multi-flavor $\phi^3$ theory. Contrary to the case in two and four dimensions, we find that in perturbation theory the relevant quantity $\tilde{a}$ increases monotonically along flows away from the trivial fixed point. $\tilde{a}$ is a natural extension of the coefficient $a$ of the Euler term in the trace anomaly, and it arises in any even spacetime dimension from an analysis based on Weyl consistency conditions. We also obtain the anomalous dimensions and beta functions of multi-flavor $\phi^3$ theory to two loops. Our results suggest that some new intuition about the $a$-theorem is in order.