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 |α| ≃ |β| ≃ 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 π 0 final state the difference between the two methods is quite appreciable.

We test the convergence property of the chiral perturbation theory using a lattice QCD calculation of pion mass and decay constant with two dynamical quark flavors. The lattice calculation is performed using the overlap fermion formulation, which realizes exact chiral symmetry at finite lattice spacing. By comparing various expansion prescriptions, we find that the chiral expansion is well saturated at the next-to-leading order for pions lighter than approximately 450 MeV. Better convergence behavior is found, in particular, for a resummed expansion parameter xi, with which the lattice data in the pion mass region 290-750 MeV can be fitted well with the next-to-next-to-leading order formulas. We obtain the results in two-flavor QCD for the low energy constants l[over ]_{3} and l[over ]_{4} as well as the pion decay constant, the chiral condensate, and the average up and down quark mass.

We calculate the strange quark content of the nucleon N |ss|N in 2 + 1 -flavor lattice QCD.Chirally symmetric overlap fermion formulation is used to avoid the contamination from up and down quark contents due to an operator mixing between strange and light scalar operators,ss and uu +dd. At a lattice spacing a = 0.112(1) fm, we perform calculations at four values of degenerate up and down quark masses m ud , which cover a range of the pion mass M π ≃ 300 -540 MeV. We employ two different methods to calculate N |ss|N . One is a direct method where we calculate N |ss|N by directly inserting thess operator. The other is an indirect method where N |ss|N is extracted from a derivative of the nucleon mass in terms of the strange quark mass. With these two methods we obtain consistent results for N |ss|N with each other. Our best estimate f Ts = m s N |ss|N /M N = 0.009(15) stat (16) sys is in good agreement with our previous studies in two-flavor QCD.

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We study the axial U (1) symmetry at finite temperature in two-flavor lattice QCD. Employing the Möbius domain-wall fermions, we generate gauge configurations slightly above the critical temperature T c with different lattice sizes L = 2-4 fm. Our action allows frequent topology tunneling while keeping good chiral symmetry close enough to that of overlap fermions. This allows us to recover full chiral symmetry by an overlap/domain-wall reweighting. Above the phase transition, a strong suppression of the low-lying modes is observed in both of overlap and domainwall Dirac spectra. We, however, find a sizable violation of the Ginsparg-Wilson relation in the Möbius domain-wall Dirac eigenmodes, which dominates the signals of the axial U (1) symmetry breaking near the chiral limit. We also find that the use of overlap fermion only in the valence sector is dangerous since it suffers from the artifacts due to partial quenching. Reweighting the Möbius domain-wall fermion determinant to that of the overlap fermion, we observe the axial U (1) breaking to vanish in the chiral limit, which is stable against the changes of the lattice volume and lattice spacing.2