Patterns of chiral symmetry breaking and a candidate for a C-theorem in four dimensions

TL;DR

This paper investigates a proposed four-dimensional C-function across various gauge theories, finding it generally decreases along the renormalization group flow, with a notable exception suggesting potential new constraints on flavor numbers.

Contribution

It tests a candidate C-function in diverse gauge theories and identifies conditions under which it decreases or possibly violates the C-theorem, indicating new theoretical constraints.

Findings

The C-function decreases in most theories along RG flow.

A potential violation occurs only in SU(2) with fermions in a pseudo-real representation.

This may imply new restrictions on flavor numbers for chiral symmetry breaking.

Abstract

We test a candidate for a four-dimensional C-function. This is done by considering all asymptotically free, vectorlike gauge theories with N_f flavors and fermions in arbitrary representations of any simple Lie group. Assuming spontaneous breaking of chiral symmetry in the infrared limit and that the value of the C-function in this limit is determined by the number of Goldstone bosons, we find that only in the case of a theory with two colors and fermions in one single pseudo-real representation of SU(2) the C-theorem seems to be violated. Conversely, this might also be a sign of new constraints, restricting the number of flavors consistent with spontaneous chiral symmetry breaking. For all other groups and representations we find that this candidate C-function decreases along the renormalization group flow.