Useful trick to compute correlation functions of composite operators

TL;DR

The paper introduces a practical method for calculating correlation functions of gauge-invariant composite operators by adding a non-dynamical auxiliary field, simplifying the computational process.

Contribution

It proposes a novel trick that enables the use of standard techniques for elementary fields to compute correlation functions of complex composite operators.

Findings

Simplifies the calculation of correlation functions of composite operators.

Maintains gauge invariance and positivity properties in the computed correlators.

Facilitates perturbative analysis of gauge-invariant observables.

Abstract

In general, in gauge field theories, physical observables are represented by gauge-invariant composite operators, such as the electromagnetic current. As we recently demonstrated in the context of the $U\left(1\right)$ and $SU\left(2\right)$ Higgs models \cite{Dudal:2019pyg,Dudal:2020uwb,Maas:2020kda}, correlation functions of gauge-invariant operators exhibit very nice properties. Besides the well-known gauge independence, they do not present unphysical cuts, and their K\"{a}ll\'{e}n-Lehmann representations are positive, at least perturbatively. Despite all these interesting features, they are not employed as much as elementary fields, mainly due to the additional complexities involved in their computation and renormalization. In this article, we present a useful trick to compute loop corrections to correlation functions of composite operators. This trick consists of introducing an additional field with no dynamics, coupled to the composite operator of interest. By using this approach, we can employ the traditional algorithms used to compute correlation functions of elementary fields.