Evidence for the Strongest Version of the 4d a-Theorem, via a-Maximization Along RG Flows

TL;DR

This paper provides evidence supporting the strongest form of the 4d a-theorem by extending a-maximization to RG flows, proposing that RG flows are gradient flows of an a-function with a positive metric, and confirming this in perturbation theory.

Contribution

The authors extend a-maximization to RG flows away from fixed points, proposing a monotonic a-function and the gradient flow nature of RG flows, strengthening the evidence for the strongest 4d a-theorem.

Findings

a-maximization yields a monotonic a-function along RG flows

RG flows are gradient flows of the a-function with a positive metric

Perturbative RG flow metric matches previous computations by Osborn et al.

Abstract

In earlier work, we (KI and BW) gave a two line "almost proof" (for supersymmetric RG flows) of the weakest form of the conjectured 4d a-theorem, that a_{IR}<a_{UV}, using our result that the exact superconformal R-symmetry of 4d SCFTs maximizes a=3Tr R^3-Tr R. The proof was incomplete because of two identified loopholes: theories with accidental symmetries, and the fact that it's only a local maximum of \it{a}. Here we discuss and extend a proposal of Kutasov (which helps close the latter loophole) in which a-maximization is generalized away from the endpoints of the RG flow, with Lagrange multipliers that are conjectured to be identified with the running coupling constants. a-maximization then yields a monotonically decreasing "a-function" along the RG flow to the IR. As we discuss, this proposal in fact suggests the strongest version of the a-theorem: that 4d RG flows are gradient flows of an a-function, with positive definite metric. In the perturbative limit, the RG flow metric thus obtained is shown to agree precisely with that found by very different computations by Osborn and collaborators. As examples, we discuss a new class of 4d SCFTs, along with their dual descriptions and IR phases, obtained from SQCD by coupling some of the flavors to added singlets.