  Hadronic matrix elements of operators relevant to nucleon decay in grand
unified theories are calculated numerically using lattice QCD. In this context,
the domain-wall fermion formulation, combined with non-perturbative
renormalization, is used for the first time. These techniques bring reduction
of a large fraction of the systematic error from the finite lattice spacing.
Our main effort is devoted to a calculation performed in the quenched
approximation, where the direct calculation of the nucleon to pseudoscalar
matrix elements, as well as the indirect estimate of them from the nucleon to
vacuum matrix elements, are performed. First results, using two flavors of
dynamical domain-wall quarks for the nucleon to vacuum matrix elements are also
presented to address the systematic error of quenching, which appears to be
small compared to the other errors. Our results suggest that the representative
value for the low energy constants from the nucleon to vacuum matrix elements
are given as |alpha| simeq |beta| simeq 0.01 GeV^3. For a more reliable
estimate of the physical low energy matrix elements, it is better to use the
relevant form factors calculated in the direct method. The direct method tends
to give smaller value of the form factors, compared to the indirect one, thus
enhancing the proton life-time; indeed for the pi^0 final state the difference
between the two methods is quite appreciable.
