One-loop Matching and Running with Covariant Derivative Expansion

TL;DR

This paper introduces a gauge-covariant functional approach and a new covariant derivative expansion technique for efficient one-loop effective field theory calculations, including matching and running analyses involving heavy and light fields.

Contribution

It develops a novel covariant derivative expansion method and clarifies the treatment of mixed heavy-light one-loop diagrams in EFT matching and running calculations.

Findings

New covariant derivative expansion technique for functional traces.

Method to isolate heavy-only from mixed heavy-light one-loop contributions.

Explicit examples demonstrating the application of the methods.

Abstract

We develop tools for performing effective field theory (EFT) calculations in a manifestly gauge-covariant fashion. We clarify how functional methods account for one-loop diagrams resulting from the exchange of both heavy and light fields, as some confusion has recently arisen in the literature. To efficiently evaluate functional traces containing these "mixed" one-loop terms, we develop a new covariant derivative expansion (CDE) technique that is capable of evaluating a much wider class of traces than previous methods. The technique is detailed in an appendix, so that it can be read independently from the rest of this work. We review the well-known matching procedure to one-loop order with functional methods. What we add to this story is showing how to isolate one-loop terms coming from diagrams involving only heavy propagators from diagrams with mixed heavy and light propagators. This is done using a non-local effective action, which physically connects to the notion of "integrating out" heavy fields. Lastly, we show how to use a CDE to do running analyses in EFTs, \textit{i.e.} to obtain the anomalous dimension matrix. We demonstrate the methodologies by several explicit example calculations.