Matching $A$ with $F$ in long-range QFTs

TL;DR

This paper demonstrates that the renormalization group flow in long-range multiscalar  theory satisfies a gradient structure up to third order, linking it to the sphere free-energy and Zamolodchikov's metric, supporting the -theorem.

Contribution

It shows the gradient flow structure in long-range  theory up to third loop order and connects it to conformal data, providing a perturbative proof of the -theorem.

Findings

RG flow satisfies gradient structure up to third order

Matching of A and G_{IJ} with  and C_{IJ} in specific models

Supports perturbative -theorem at leading nontrivial order

Abstract

Irreversibility theorems -- such as the $A$-theorem -- establish a hierarchy among fixed points of the renormalization group flow. The strongest thesis of this type of theorems would be that there exists a scalar function $A$ (generally suggested by the topological Weyl anomaly) and a positive definite metric $G_{IJ}$ in the space of couplings such that the renormalization group flow satisfies a gradient equation, $\partial_I A= G_{IJ}\beta^J$, in which case $A$ is locally monotonic along the flow. In this paper we consider the long-range multiscalar $\phi^4$ theory, a theory without a local energy-momentum tensor that is unitary in $d=2,3$ and that is believed to be conformally invariant at fixed points, and show that its renormalization group flow satisfies the gradient structure up to the third loop order in the coupling. We also show that $A$ and $G_{IJ}$ can be matched to the leading nontrivial order with the sphere free-energy $\tilde{F}$ and Zamolodchikov's metric $C_{IJ}$ of the corresponding conformal theory concentrating on the examples of the long-range vector $O(N)$ and hypercubic $H_N$ models. Our results imply a perturbative proof of the $\tilde{F}$-theorem at the leading nontrivial order. We conclude the paper discussing briefly whether this result should hold to the next orders in perturbation theory.