Finite size effect on vector meson and baryon sectors in $2+1$ flavor QCD at the physical point

We investigate the finite size effect on the vector meson and the baryon sectors using a subset of the “PACS10” configurations which are generated, keeping the space-time volumes over ${(10\mathrm{fm})}^4$ in $2+1$ flavor QCD at the physical point. Comparing the results on ${(5.5\mathrm{fm})}^4$ and ${(10.9\mathrm{fm})}^4$ lattices, the ground states of octet baryons, which are stable on the lattice, show no finite size effect within less than 0.5% level of statistical errors. For those of vector mesons, which are unstable on the lattice, we observe that the effective masses are well below the experimental resonance levels both on ${(5.5\mathrm{fm})}^4$ and ${(10.9\mathrm{fm})}^4$ lattices. For the decuplet baryon sector, we have found that the time dependence of the effective mass looks quite similar to that for the vector meson sector, including the $\mathrm{\Omega }$ baryon channel. We discuss its origin due to a possible mixing with the nearby multihadron states. Since the $\mathrm{\Xi }$ baryon mass can be determined with the smallest ambiguity among the vector meson and the baryon masses, we use it together with the pion and kaon masses as the physical inputs to determine the physical point.

Figure 1

Cross section of the unit cell with SC, BCC, and FCC crystal structures. Green areas indicate the balls.

Figure 2

Comparison of effective masses for $\rho$, $K^{*}$, and $\varphi$ (from top to bottom) channels on $64^4$ and $128^4$ lattices. The horizontal line denotes the experimental resonance level in the lattice unit, whose scale is determined by the $\mathrm{\Xi }$ baryon mass.

Figure 3

Comparison of effective masses for $\rho$, $K^{*}$, and $\varphi$ (from top to bottom) channels on the $64^4$ lattice at the original hopping parameters and the reweighted ones $({\kappa }_{\mathrm{ud}}^{*},{\kappa }_{\mathrm{s}}^{*})=(0.126119,0.124902)$. The horizontal line is the same as in Fig. 2.

Figure 4

Comparison of effective masses for $N$, $\mathrm{\Lambda }$, $\mathrm{\Sigma }$, and $\mathrm{\Xi }$ (from top to bottom) baryon channels on $64^4$ and $128^4$ lattices. The horizontal line denotes the experimental mass in the lattice unit.

Figure 5

Comparison of effective masses for $\mathrm{\Delta }$, ${\mathrm{\Sigma }}^{*}$, ${\mathrm{\Xi }}^{*}$, and $\mathrm{\Omega }$ (from top to bottom) baryon channels on $64^4$ and $128^4$ lattices. The horizontal line denotes the experimental resonance level for $\mathrm{\Delta }$, ${\mathrm{\Sigma }}^{*}$, and ${\mathrm{\Xi }}^{*}$ and the experimental mass for $\mathrm{\Omega }$.

Figure 6

Comparison of effective mass for $\mathrm{\Omega }$ baryon channel on $64^4$ and $128^4$ lattices. Open symbols are the same as in the bottom panel of Fig. 5. The horizontal line denotes the experimental $\mathrm{\Omega }$ baryon mass.

Figure 7

AWI quark mass dependence of $m_{\pi }^2-m_{\pi }^2{|}_{\mathrm{sim}}$, $m_K^2-m_K^2{|}_{\mathrm{sim}}$ and $m_{\mathrm{\Xi }}-m_{\mathrm{\Xi }}{|}_{\mathrm{sim}}$ (from top to bottom) on $64^4$ lattice together with the fit results.

Figure 8

Physical point in $m_{\mathrm{ud}}\text{−}{m}_{\mathrm{s}}$ plane. The light and strange quark masses at the reweighted hopping parameters $({\kappa }_{\mathrm{ud}}^{*},{\kappa }_{\mathrm{s}}^{*})=(0.126119,0.124902)$ on $64^4$ lattice are consistent with those at the original hopping parameters on $128^4$ lattice.