One-loop matching of the LEFT to the QCD gradient flow

TL;DR

This paper derives the one-loop matching between the low-energy effective field theory and QCD gradient flow, facilitating precise lattice-QCD calculations for low-energy phenomenology.

Contribution

It provides the complete one-loop matching coefficients and short-flow-time expansion for LEFT operators using gradient flow, including systematic treatment of evanescent operators and chiral symmetry restoration.

Findings

Verified cancellation of chiral-symmetry-violating terms.

Provided matching coefficients before and after operator redefinitions.

Established a perturbative link enabling high-precision low-energy calculations.

Abstract

We present the complete one-loop matching of the baryon- and lepton-number-conserving low-energy effective field theory (LEFT) to the QCD gradient flow. Using Euclidean conventions and the background-field formulation of the gradient flow, we derive the short-flow-time expansion for the full LEFT operator basis up to mass dimension six. The matching is performed in dimensional regularization in the algebraically consistent 't Hooft-Veltman scheme, including a systematic treatment of evanescent operators and the finite counterterms required to restore chiral symmetry in the spurion sense. Keeping fully generic flavor structures, we verify the cancellation of spurious chiral-symmetry-violating terms with the known finite symmetry-restoring counterterms. This demonstrates that the gradient flow as a gauge-invariant ultraviolet regulator enables an efficient extraction of both divergent and finite counterterms in addition to the matching contributions. We provide the matching coefficients both before and after field redefinitions that remove redundant operators, as well as power-divergent mixings into lower-dimensional operators. Our results establish a consistent perturbative link between continuum LEFT calculations and gradient-flow-based lattice-QCD matrix elements, enabling precision low-energy phenomenology beyond leading-logarithmic accuracy.