A New Constraint on Strongly Coupled Field Theories

TL;DR

The paper proposes a universal constraint for asymptotically free four-dimensional field theories: the infrared degrees of freedom, encoded in $f_{IR}$, cannot exceed the ultraviolet count $f_{UV}$. This is formalized via the finite-temperature free energy, with $f_{IR} = - \lim_{T\to 0} \mathcal F/T^4 \cdot 90/\\pi^2$ and $f_{UV} = - \lim_{T\to \infty} \mathcal F/T^4 \cdot 90/\\pi^2$. The authors test the inequality in a broad set of examples, including SUSY and non-SUSY QCD-like theories, dual descriptions, and QED in 2+1 dimensions, finding consistency and deriving new constraints on low-energy spectra, such as bounds on the onset of chiral symmetry breaking. They explore potential connections to RG c-theorems, discuss temperature dependence and possible proofs, and note that violations can occur in theories with nontrivial UV fixed points. Overall, the work provides a novel, quantitative constraint on RG flows with practical implications for the structure of strongly coupled theories and their low-energy spectra.

Abstract

We propose a new constraint on the structure of strongly coupled field theories. The constraint takes the form of an inequality limiting the number of degrees of freedom in the infrared description of a theory relative to the number of underlying, ultraviolet degrees of freedom. We apply the inequality to a variety of theories (both supersymmetric and nonsupersymmetric), where it agrees with all known results and leads to interesting new constraints on low energy spectra. We discuss the relation of this constraint to Renormalization Group c-theorems.

Figures (1)

• Figure 1: Plot of $f_{UV}$ and $f_{IR}$ in units of $N^2$ as functions of $F/N$. We have taken the large-$N$ and $F$ limit and show only the neighborhood of the interesting point $F/N=3/2$. For $F/N<3/2$ one sees that $f_{IR}<f_{UV}$, at $F/N=3/2$ the two $f$'s touch, and for $F/N>3/2$ we again find $f_{IR}<f_{UV}$, but only after taking into account the interactions. For comparison we also show $f^0_{IR}$, the expression for $f_{IR}$ with no interactions included.