Lattice QCD calculation of the proton decay matrix element in the continuum limit

We present a quenched lattice QCD calculation of the $\alpha$ and $\beta$ parameters of the proton decay matrix element. The simulation is carried out using the Wilson quark action at three values of the lattice spacing in the range $a\approx 0.1-0.064\mathrm{f}\mathrm{m}$ to study the scaling violation effect. We find only mild scaling violation when the lattice scale is determined by the nucleon mass. We obtain in the continuum limit, $|\alpha (\mathrm{N}\mathrm{D}\mathrm{R},2\mathrm{G}\mathrm{e}\mathrm{V})|=0.0090(09)(+5-19){\mathrm{G}\mathrm{e}\mathrm{V}}^3$ and $|\beta (\mathrm{N}\mathrm{D}\mathrm{R},2\mathrm{G}\mathrm{e}\mathrm{V})|=0.0096(09)(+6-20){\mathrm{G}\mathrm{e}\mathrm{V}}^3$ with $\alpha$ and $\beta$ in a relatively opposite sign, where the first error is statistical and the second is due to the uncertainty in the determination of the physical scale.