$1\mathbf{+}1\mathbf{+}1$ flavor $\mathrm{QCD}\mathbf{+}\mathrm{QED}$ simulation at the physical point

We present the results of $1+1+1$ flavor $\mathrm{QCD}+\mathrm{QED}$ simulation at the physical point, in which the dynamical quark effects in QED and the up-down quark mass difference are incorporated by the reweighting technique. The physical quark masses together with the lattice spacing are determined with $m_{{\pi }^{+}}$, $m_{K^{+}}$, $m_{K^0}$ and $m_{{\Omega }^{-}}$ as physical inputs. Calculations are carried out using a set of $2+1$ flavor QCD configurations near the physical point generated by the nonperturbatively $O(a)$-improved Wilson quark action and the Iwasaki gauge action at $\beta =1.9$ on a ${32}^3\times 64$ lattice. We evaluate the values of the up, down and strange quark masses individually with nonperturbative QCD renormalization.

Figure 1

Noise dependence of the odd part in $e$ for $|W_q^{-1}{\eta }_i{|}^2-|{\eta }_i{|}^2$ ($q=\mathrm{u}$, d, s) with the first breakup interval from $e=0$ to $e_{\mathrm{ph}}/400$. Total contributions including both the odd and even parts in $e$ are also presented (black circles).

Figure 2

Configuration dependence of the reweighting factor from $2+1$ flavor QCD to $1+1+1$ flavor $\mathrm{QCD}+\mathrm{QED}$ at the physical point.

Figure 3

Effective mass plots of $m_{{\pi }^{+}}$, $m_{K^{+}}$, $m_{K^0}$ and $m_{{\Omega }^{-}}$. Blue horizontal bars represent the fit results with 1 standard deviation error band.

Figure 4

Hadron mass ratios $m_{{\pi }^{+}}/m_{{\Omega }^{-}}$, $m_{K^{+}}/m_{{\Omega }^{-}}$ and $m_{K^0}/m_{{\Omega }^{-}}$ (black filled circles) as a function of the number of noises for each determinant breakup. Red broken lines indicate the experimental values.

Figure 5

Ratio of $K^0$ to $K^{+}$ propagators to detect the mass difference $m_{K^0}-m_{K^{+}}$. Our results (black symbol) are consistent with the expected slope from the experimental value of $m_{K^0}-m_{K^{+}}$ (straight red line).