Proton lifetime bounds from chirally symmetric lattice QCD

We present results for the matrix elements relevant for proton decay in grand unified theories (GUTs). The calculation is performed at a fixed lattice spacing $a^{-1}=1.73(3)\mathrm{GeV}$ using $2+1$ flavors of domain-wall fermions on lattices of size ${16}^3\times 32$ and ${24}^3\times 64$ with a fifth dimension of length 16. We use the indirect method which relies on an effective field theory description of proton decay, where we need to estimate the low-energy constants, $\alpha =-0.0112(25){\mathrm{GeV}}^3$ and $\beta =0.0120(26){\mathrm{GeV}}^3$. We relate these low-energy constants to the proton decay matrix elements using leading order chiral perturbation theory. These can then be combined with experimental bounds on the proton lifetime to bound parameters of individual GUTs.

Figure 1

(a) Effective mass plot; (b) effective amplitude [Eq. (26)] plot for the nucleon. Both are calculated on the ${24}^3\times 64$ data set with $a{m}_u=0.01$. The different colors in the effective mass plot correspond to different smearings. Data sets are labeled with the smearing (i.e. LL). Those data sets labeled with a 2 use the operator $f_{A_{4}S,A_{4}S}(t)$, the rest use $f_{PS,PS}(t)$. (c) Linear extrapolation of the ground state mass to the chiral limit.

Figure 2

Effective mass plots for the nucleon on the $a{m}_u=0.005$, $V={24}^3\times 64$ ensemble. The different colors in the effective mass plot correspond to different smearings. Data sets are labeled with the smearing (i.e. LL). Those data sets labeled with a 2 use the operator $f_{A_{4}S,A_{4}S}(t)$, the rest use $f_{PS,PS}(t)$. (a) Fit before scaling the errors; (b) fit after rescaling the errors.

Figure 3

The ratio $R_{\alpha }$ in Eq. (30) for the ${24}^3\times 64$ data set with $a{m}_u=0.005$, 0.01, 0.02, and 0.03, respectively. The different colors correspond to different source smearing. Horizontal lines show the fit to the plateau.

Figure 4

Linear chiral extrapolation for the ratios $R_{\alpha }$ (a) $R_{\beta }$ (b) for the ${24}^3\times 64$ data set.

Figure 5

The mixing matrix $M$ in Eq. (39) in the chirality basis, $\Gamma {\Gamma }^{\prime }=\{LL,RL,A(LV)\}$, as a function of the lattice momentum squared for the ${16}^3\times 32$ lattices with $a{m}_u=0.01$, 0.02, and 0.03 (from left to right, respectively). The off-diagonal mixing between operators is highly suppressed. It is worthwhile to note that the mass dependence of the mixing matrix is very mild.

Figure 6

The figure on the left shows the average and the difference of the amputated local axial vector and vector bilinear currents as a function of $(ap{)}^2$. In the figure on the right, the black points show the MOM scheme renormalization factor in the chiral limit for the ${\mathcal{O}}^{LL}$ operator normalizsed by the axial current renormalization factor $Z_A$ as a function of the renormalization scale $(ap{)}^2$. The red points show the renormalization factor in the $\overline{\mathrm{MS}}$ scheme at a scale $\mu =1/a$ as a function of the matching scale. The red line shows the linear extrapolation in $(ap{)}^2$, where the blue points are those included in the extrapolation.

Figure 7

The LEC $\alpha$ measured on the two different volumes. There are no noticeable finite size effects. The chiral extrapolations from the two volumes are shown as white filled circles and also agree within errors.

Figure 8

An extrapolation for $\alpha$ and $\beta$ both with and without the value from the lightest valence quark mass point. This gives results differing by 18% for $\alpha$ and 17% for $\beta$.

Figure 9

Summary of computations of the hadronic matrix element $\alpha$, as given in Table 5. Square points correspond to QCD model calculations, blue circles correspond to $N_f=0$ lattice QCD calculations, the green circle is from $N_f=2$, and the result from our $N_f=2+1$ calculation is shown in red.