A renormalization group invariant scalar glueball operator in the (Refined) Gribov-Zwanziger framework

TL;DR

This paper provides an algebraic proof of the renormalizability of the scalar glueball operator in the Gribov-Zwanziger framework, establishing a renormalization group invariant crucial for studying the lightest scalar glueball.

Contribution

It introduces a complete algebraic analysis of operator mixing and constructs a renormalization group invariant operator within the GZ and RGZ formalisms.

Findings

Explicitly determines the mixing matrix to all orders.

Identifies a renormalization group invariant operator.

Lays groundwork for analyzing the scalar glueball in GZ formalism.

Abstract

This paper presents a complete algebraic analysis of the renormalizability of the $d=4$ operator $F^2_{\mu\nu}$ in the Gribov-Zwanziger (GZ) formalism as well as in the Refined Gribov-Zwanziger (RGZ) version. The GZ formalism offers a way to deal with gauge copies in the Landau gauge. We explicitly show that $F^2_{\mu\nu}$ mixes with other $d=4$ gauge variant operators, and we determine the mixing matrix $Z$ to all orders, thereby only using algebraic arguments. The mixing matrix allows us to uncover a renormalization group invariant including the operator $F^2_{\mu\nu}$. With this renormalization group invariant, we have paved the way for the study of the lightest scalar glueball in the GZ formalism. We discuss how the soft breaking of the BRST symmetry of the GZ action can influence the glueball correlation function. We expect non-trivial mass scales, inherent to the GZ approach, to enter the pole structure of this correlation function.