Non-Perturbative Renormalisation of the Lattice $\Delta s=2$ Four-Fermion Operator

TL;DR

This paper presents a non-perturbative calculation of the renormalised $ riangle S=2$ four-fermion operator on the lattice, improving the chiral behavior of the kaon matrix element by accurately determining renormalisation constants.

Contribution

It introduces a non-perturbative method for renormalising lattice four-fermion operators, including operator mixing due to Wilson term effects, and applies it to the $ riangle S=2$ operator.

Findings

Non-perturbative renormalisation constants improve chiral behavior.

Mixing coefficients for operators with different chiralities are successfully determined.

Numerical results obtained on a $16^3 imes 32$ lattice at $eta=6.0$.

Abstract

We compute the renormalised four-fermion operator $O^{\Delta S=2}$ using a non-perturbative method recently introduced for determining the renormalisation constants of generic lattice composite operators. Because of the presence of the Wilson term, $O^{\Delta S=2}$ mixes with operators of different chiralities. A projection method to determine the mixing coefficients is implemented. The numerical results for the renormalisation constants have been obtained from a simulation performed using the SW-Clover quark action, on a $16^3 \times 32$ lattice, at $\beta=6.0$. We show that the use of the constants determined non-perturbatively improves the chiral behaviour of the lattice kaon matrix element $\<\bar K^0| O^{\Delta S=2} | K^0\>_{\latt}$.