Gradient properties of $\varphi^3$ in $d=6-\varepsilon$

TL;DR

This paper demonstrates that the renormalization group flow of the $^3$ theory in $d=6-$ dimensions maintains a gradient structure, extending previous three-loop results to non-integer dimensions and revealing new constraints on beta function coefficients.

Contribution

It shows the gradient structure of the $^3$ theory's RG flow in $d=6-$, including new linear constraints on beta function coefficients.

Findings

Gradient structure persists in $d=6-$ dimensions.

New linear constraints on beta function coefficients.

Extension of three-loop results to non-integer dimensions.

Abstract

The renormalization group flow of the multiscalar interacting $\varphi^3$ theory in $d=6$ dimensions is known to have a gradient structure, in which suitable generalizations of the beta functions $B^{I}$ emerge as the gradient of a scalar function $A$, $\partial_I A = T_{IJ} B^J $, with a nontrivial tensor $T_{IJ}$ in the space of couplings. This has been shown directly to three loops in schemes such as $\overline{\rm MS}$ and can be argued in general by identifying $A$ with the coefficient of the topological term of the trace-anomaly in $d=6$ up to a normalization. In this paper we show that the same renormalization group has a gradient structure in $d=6-\varepsilon$. The requirement of a gradient structure is translated to linear constraints that the coefficients of the $\overline{\rm MS}$ beta functions must obey, one of which is new and pertinent only to the extension to $d \neq 6$.