Proton decay matrix elements on the lattice

Abstract

Hadronic matrix elements of proton decays are essential ingredients for bridging the grand unification theory to low-energy observables like the proton lifetime. In this paper we nonperturbatively calculate the matrix elements, relevant for the process of a nucleon decaying into a pseudoscalar meson and an antilepton through generic baryon-number-violating four-fermi operators. Lattice QCD with $2+1$ flavor dynamical domain-wall fermions with the direct method---which is the direct measurement of the matrix elements from the three-point function without using chiral perturbation theory---is used for this study in order to have good control over the errors due to lattice discretization effects, operator renormalization, and chiral extrapolation. The relevant form factors for possible transition processes from an initial proton or neutron to a final pion or kaon induced by all types of three-quark operators are obtained through three-point functions of the (nucleon)-(three-quark operator)-(meson) with physical kinematics. In this study all the relevant systematic uncertainties of the form factors are taken into account for the first time, and the total error is found to be in the range 30%--40% for the $\ensuremath{\pi}$ and 20%--40% for the $K$ final states.