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

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