CFTs with $U(m)\times U(n)$ Global Symmetry in 3D and the Chiral Phase Transition of 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 $d=4-\epsilon$ and $d=3$. This includes perturbative computations in the $\epsilon$ 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 $\epsilon$ expansion. Similar evidence is found for the case of three-flavor massless QCD, where we observe a pronounced kink.

Comments: 35 pages, 15 figures. v2: Comments added. Misidentification of $U_{\pm }$ fixed points fixed. v3: Discussion on applicability of bootstrap results amended