Low-energy theorem and OPE in the conformal window of massless QCD

TL;DR

This paper introduces a new nonperturbative technique based on a low-energy theorem to analyze the conformal window in massless QCD-like theories, revealing constraints on correlators and explaining the origin of divergent contact terms in the operator product expansion.

Contribution

It develops a novel LET-based method to study IR/UV fixed points in massless QCD and clarifies the nature of divergent contact terms in the OPE of $Tr F^2$.

Findings

Constraints on 3-point correlators involving $Tr F^2$ and anomalous dimensions.

Connection between divergent contact terms and anomalous dimensions.

Verification that the divergence matches the nontrivial anomalous dimension.

Abstract

We develop a new technique, based on a low-energy theorem (LET) of NSVZ type derived in arXiv:1701.07833, for the nonperturbative investigation of SU(N) QCD with N${}_f$ massless quarks - or, more generally, of massless QCD-like theories - in phases where the beta function, $\beta(g)$, with $g=g(\mu)$ the renormalized gauge coupling, admits an isolated zero, $g_*$, in the infrared (IR) or ultraviolet (UV). We point out that the LET sets constraints on 3-point correlators involving the insertion of $Tr\, F^2$, its anomalous dimension $\gamma_{F^2}$, and the anomalous dimensions of multiplicatively renormalizable operators at $g_*$. These constraints intertwine with the exact conformal scaling for $g(\mu)\rightarrow g_*$ with $\mu\neq 0,+\infty$ fixed and the IR/UV asymptotics - which may or may not coincide with the IR/UV limit of the aforementioned conformal scaling - for $\Lambda_{\scriptscriptstyle{IR/UV}}$ fixed. In the conformal case we also discuss how the LET for bare correlators is the rationale for the existence in massless QCD of the mysterious divergent contact term in the OPE of $Tr\,F^2$ with itself discovered in perturbation theory in arXiv:1209.1516, arXiv:1407.6921 and computed to all orders in arXiv:1601.08094. Specifically, if $\gamma_{F^2}$ does not vanish, the divergent contact term in the rhs of the LET for the 2-point correlator of $Tr\,F^2$ has to match - and we verify by direct computation that it actually does - the divergence in the lhs due to the nontrivial anomalous dimension of $Tr\,F^2$. Hence, remarkably, the additive renormalization due to the divergent contact term in the rhs is related by the LET to the multiplicative renormalization in the lhs, in such a way that a suitably renormalized version of the LET has no ambiguity for additive renormalization.