CFTs with $U(m)\times U(n)$ Global Symmetry in 3D and the Chiral Phase Transition of QCD

TL;DR

This paper investigates 3D conformal field theories with $U(m) imes U(n)$ symmetry using perturbative and non-perturbative methods, providing new bounds and insights relevant to the chiral phase transition in QCD.

Contribution

It presents new perturbative results, extensive bootstrap bounds, and the analysis of fixed point transitions, advancing understanding of symmetry and phase transitions in QCD-related CFTs.

Findings

Bootstrap bounds are saturated by large-$n$ results.

Evidence suggests the chiral transition in 2-flavor QCD may be second order.

A pronounced kink indicates a potential universality class in 3-flavor QCD.

Abstract

Conformal field theories (CFTs) with $U(m)\times U(n)$ global symmetry in $d=3$ dimensions have been studied for years due to their potential relevance to the chiral phase transition of quantum chromodynamics (QCD). In this work such CFTs are analyzed in $d=4-\varepsilon$ and $d=3$. This includes perturbative computations in the $\varepsilon$ and large-$n$ expansions as well as non-perturbative ones with the numerical conformal bootstrap. New perturbative results are presented and a variety of non-perturbative bootstrap bounds are obtained in $d=3$. Various features of the bounds obtained for large values of $n$ disappear for low values of $n$ (keeping $m<n$ fixed), a phenomenon which is attributed to a transition of the corresponding fixed points to the non-unitary regime. Numerous bootstrap bounds are found that are saturated by large-$n$ results, even in the absence of any features in the bounds. A double scaling limit is also observed, for $m$ and $n$ large with $m/n$ fixed, both in perturbation theory as well as in the numerical bootstrap. For the case of two-flavor massless QCD existing bootstrap evidence is reproduced that the chiral phase transition may be second order, albeit associated to a universality class unrelated to the one usually discussed in the $\varepsilon$ expansion. Similar evidence is found for the case of three-flavor massless QCD, where we observe a pronounced kink.