The scalar glueball operator, the a-theorem, and the onset of conformality

TL;DR

This paper investigates how the anomalous dimension of the scalar glueball operator signals the mechanism behind the onset of conformality in non-Abelian gauge theories, revealing new insights and analogies with supersymmetric theories.

Contribution

It establishes a relation between the glueball anomalous dimension and the conformal transition mechanism, and compares non-Abelian gauge theories with supersymmetric models using the $a$-theorem.

Findings

The anomalous dimension distinguishes between different conformal transition mechanisms.

An exact relation between $\gamma_G$ and $\gamma_m$ is derived in SQCD.

The $a$-theorem constrains the existence of fixed points and the merging mechanism.

Abstract

We show that the anomalous dimension $\gamma_G$ of the scalar glueball operator contains information on the mechanism that leads to the onset of conformality at the lower edge of the conformal window in a non-Abelian gauge theory. In particular, it distinguishes whether the merging of an UV and an IR fixed point -- the simplest mechanism associated to a conformal phase transition and preconformal scaling -- does or does not occur. At the same time, we shed light on new analogies between QCD and its supersymmetric version. In SQCD, we derive an exact relation between $\gamma_G$ and the mass anomalous dimension $\gamma_m$, and we prove that the SQCD exact beta function is incompatible with merging as a consequence of the $a$-theorem; we also derive the general conditions that the latter imposes on the existence of fixed points, and prove the absence of an UV fixed point at nonzero coupling above the conformal window of SQCD. Perhaps not surprisingly, we then show that an exact relation between $\gamma_G$ and $\gamma_m$, fully analogous to SQCD, holds for the massless Veneziano limit of large-N QCD. We argue, based on the latter relation, the $a$-theorem, perturbation theory and physical arguments, that the incompatibility with merging may extend to QCD.