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

We compute the renormalised four-fermion operator $O^{\mathrm{\Delta }S=2}$ 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, $O^{\mathrm{\Delta }S=2}$ 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 $\text{<}{\bar{K}}^0\mathrm{\mid }{O}^{\mathrm{\Delta }S=2}\mathrm{\mid }{K}^0{}_{\text{latt}}$.