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

TL;DR

This work delivers the complete one-loop matching of the LEFT to the QCD gradient flow in Euclidean space using the background-field method and the HV scheme. By deriving a full short-flow-time expansion for the LEFT operator basis up to mass dimension six and carefully handling evanescent structures with finite counterterms, the authors establish a robust, gauge-invariant pipeline for connecting continuum LEFT calculations to gradient-flow lattice matrix elements. The method of regions enables efficient extraction of both divergent and finite counterterms, while field redefinitions reveal the physical operator content and power-divergent mixings. The results provide a consistent perturbative bridge to lattice QCD inputs for precision low-energy phenomenology beyond leading-log accuracy, and offer a foundation for future extensions to QED, baryon/lepton-number-violating sectors, and higher-order matching.

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.