# Towards non-perturbative matching of three/four-flavor Wilson coefficients with a position-space procedure

## TL;DR

This paper introduces a non-perturbative method using position-space techniques and spherical averaging to match Wilson coefficients across different flavor theories, demonstrated through an exploratory lattice calculation.

## Contribution

It presents a novel position-space approach combined with spherical averaging for non-perturbative matching of Wilson coefficients in multi-flavor theories.

## Key findings

- Successful implementation of spherical averaging to convert lattice two-point functions into continuous functions.
- Exploratory calculation of two-point functions for $ ext{Δ}S=1$ operators $Q_7$ and $Q_8$.
- Evidence supporting the feasibility of the proposed non-perturbative matching method.

## Abstract

We propose a strategy to non-perturbatively match the Wilson coefficients in the three- and four-flavor theories, which uses two-point Green's functions of the corresponding four-quark operators at long distances. The idea is refined by combining with the spherical averaging technique, which enables us to convert two-point functions calculated on the lattice into continuous functions of the distance $|x-y|$ between two operators. We also show the result for an exploratory calculation of two-point functions of the $\Delta S=1$ operators $Q_7$ and $Q_8$ that are in the $(8_L,8_R)$ representation of ${\rm SU(3)}_L\times{\rm SU(3)}_R$ and mix with each other.