A purely algebraic construction of a gauge and renormalization group invariant scalar glueball operator

TL;DR

This paper provides an algebraic proof of the all-order renormalizability of the gauge-invariant $F_{}^2$ operator in pure Yang-Mills theory, identifying operator mixing and constructing a renormalization group invariant.

Contribution

It introduces a purely algebraic method to prove the renormalizability and operator mixing of the scalar glueball operator in Yang-Mills theory, enabling future mass calculations.

Findings

Established all-order renormalization of $F_{}^2$ operator.

Explicitly identified operator mixing matrix.

Constructed a renormalization group invariant scalar glueball operator.

Abstract

This paper presents a complete algebraic proof of the renormalizability of the gauge invariant $d=4$ operator $F_{\mu\nu}^2(x)$ to all orders of perturbation theory in pure Yang-Mills gauge theory, whereby working in the Landau gauge. This renormalization is far from being trivial as mixing occurs with other $d=4$ gauge variant operators, which we identify explicitly. We determine the mixing matrix $Z$ to all orders in perturbation theory by using only algebraic arguments and consequently we can uncover a renormalization group invariant by using the anomalous dimension matrix $\Gamma$ derived from $Z$. We also present a future plan for calculating the mass of the lightest scalar glueball with the help of the framework we have set up.