Improved lattice computation of proton decay matrix elements

We present an improved result for the lattice computation of the proton decay matrix elements in $N_f=2+1$ QCD. In this study, by adopting the error reduction technique of all-mode-averaging, a significant improvement of the statistical accuracy is achieved for the relevant form factor of proton (and also neutron) decay on the gauge ensemble of $N_f=2+1$ domain-wall fermions with $m_{\pi }=0.34–0.69\mathrm{GeV}$ on a $2.7{\mathrm{fm}}^3$ lattice, as used in our previous work [1]. We improve the total accuracy of matrix elements to 10–15% from 30–40% for $p\to \pi e^{+}$ or from 20–40% for $p\to K\overline{\nu }$. The accuracy of the low-energy constants $\alpha$ and $\beta$ in the leading-order baryon chiral perturbation theory (BChPT) of proton decay are also improved. The relevant form factors of $p\to \pi$ estimated through the “direct” lattice calculation from the three-point function appear to be 1.4 times smaller than those from the “indirect” method using BChPT with $\alpha$ and $\beta$. It turns out that the utilization of our result will provide a factor 2–3 larger proton partial lifetime than that obtained using BChPT. We also discuss the use of these parameters in a dark matter model.

Figure 1

$R_{\alpha }(t)$ (upper) and $R_{\beta }(t)$ (lower) as a function of lattice time slice. The cyan band denotes the line and statistical error of constant fitting within the fitting range.

Figure 2

Quark mass dependence of bare $\alpha$ and $\beta$. The straight lines are fits to the linear ansatz with three different fitting ranges: (i) containing all data [solid line (blue)], (ii) excluding the heaviest point [dashed-dotted line (red)], and (iii) excluding the lightest point [dashed line (green)] (see the text for details). Filled symbols represent the values in the chiral limit from the three fits.

Figure 3

Effective mass of the nucleon, pion, and kaon from the top to bottom panel. The different symbols denote data with different momentum values: zero (circle), ${\overrightarrow{n}}_p=(1,0,0)$ (square), ${\overrightarrow{n}}_p=(1,1,0)$ (diamond), and ${\overrightarrow{n}}_p=(1,1,1)$ (triangle). The quark mass is $m=0.005$.

Figure 4

Bare $W_0(t)$ for the $p\to {\pi }^0$ transition with $\mathrm{\Gamma }=R$ (top) and $\mathrm{\Gamma }=R$ (bottom) with $m=0.005$. The different symbols represent results with $t_s=22$ (red) and $t_s=18$ (blue). The left, middle, and right panels are the results at the momenta ${\overrightarrow{n}}_p=(1,0,0)$, (1,1,0), and (1,1,1), respectively. The source nucleon ($t=5$) and sink pion locations (27 or 23) have separations of 22 and 18, respectively.

Figure 5

Renormalized $W_0$ obtained from the plateau fit for data in Fig. 4 with three fitting ranges [13, 17], [11, 17], and [10, 18] with $m=0.005$. The left panel is the result for $\mathrm{\Gamma }=R$ and the right panel is the result for $\mathrm{\Gamma }=L$ in the $p\to \pi$ channel.

Figure 6

Renormalized $W_0$ in the $\overline{\mathrm{MS}}$ scheme with $\mu =2\mathrm{GeV}$ for $\mathrm{\Gamma }=R$ in each channel. The different symbols are for $m=0.005$ (circle), 0.01 (square), 0.02 (diamond), and 0.03 (triangle). We also show the chiral extrapolation line in the physical pseudoscalar mass as a cyan colored band. The orange colored band shows $W_0^{\alpha ,\beta }$ including the central value and error of the LECs obtained in our calculation. These error bands only include the statistical error.

Figure 7

Renormalized $W_0$ in the $\overline{\mathrm{MS}}$ scheme with $\mu =2\mathrm{GeV}$ for $\mathrm{\Gamma }=L$. Each symbol is the same as in Fig. 6.

Figure 8

Quark mass dependence of the renormalized $W_0$ after $q^2=0$ extrapolation for $\mathrm{\Gamma }=R$ in each channel. The cross symbol denotes $W_0$ at the physical quark mass after linear extrapolation. The band is the extrapolation line including statistical error.

Figure 9

Quark mass dependence of the renormalized $W_0$ after $q^2=0$ extrapolation for $\mathrm{\Gamma }=L$. Each symbol is the same as in Fig. 8.

Figure 10

Renormalized $W_1$ for $\mathrm{\Gamma }=R$. Each symbol is the same as in Fig. 6.

Figure 11

Renormalized $W_1$ for $\mathrm{\Gamma }=L$. Each symbol is the same as in Fig. 6.

Figure 12

Summary of the matrix elements obtained in our study. “$W_0$, $W_{\mu }$” are evaluated with the “direct” method, and “$W_0^{\alpha ,\beta }$, $W_{\mu }^{\alpha ,\beta }$” are evaluated with the “indirect” method, including the systematic error as discussed in the text.

Figure 13

Summary of the matrix elements “$W_{0,1}$, $W_{0,1}^{\alpha ,\beta }$” for a typical meson momentum $|\overrightarrow{p}|=1\mathrm{GeV}$ for the IND model.

Figure 14

The open circles denote the lattice results for $W_{\overrightarrow{p}=0}$ for $RL$ (left) and $LL$ (right) chirality for each quark mass, and the colored band shows the fitting function with the statistical error. In the chiral limit, the triangle and filled circle denote the value of BChPT and the extrapolated result in the soft-pion limit, respectively. The error bars denote the total error including the systematic one, which is obtained by the same procedure as in Sec. 5.