Using Gradient Flow to Renormalise Matrix Elements for Meson Mixing and Lifetimes

TL;DR

This paper proposes a novel non-perturbative renormalisation method using fermionic gradient flow for matrix elements related to meson mixing and lifetimes, aiming to improve accuracy and consistency in lattice QCD calculations.

Contribution

It introduces a gradient flow-based renormalisation technique combined with perturbative matching to the MS scheme for four-quark operators in meson physics.

Findings

Demonstrates the feasibility of using gradient flow for non-perturbative renormalisation.

Provides a framework to combine non-perturbative and perturbative methods.

Addresses complications in renormalising four-quark operators with mixing and disconnected diagrams.

Abstract

Neutral meson mixing and meson lifetimes are theory-side parametrised in terms four-quark operators which can be determined by calculating weak decay matrix elements using lattice Quantum Chromodynamics. While calculations of meson mixing matrix elements are standard, determinations of lifetimes typically suffer from complications in renormalisation procedures because dimension-6 four-quark operators can mix with operators of lower mass dimension and, moreover, quark-line disconnected diagrams contribute. We present work detailing the idea to use fermionic gradient flow to non-perturbatively renormalise matrix elements describing meson mixing or lifetimes, and combining it with a perturbative calculation to match to the $\overline{\rm MS}$ scheme using the shoft-flow-time expansion.