Finite size effect on pseudoscalar meson sector in $2+1$ flavor QCD at the physical point

We investigate the finite size effect on pseudoscalar meson masses and decay constants 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. We have tried two kinds of analyses, fixing $\kappa$ values or measured axial Ward identity quark masses. Comparing the results on $(5.4\mathrm{fm}{)}^4$ and $(10.8\mathrm{fm}{)}^4$ lattices, we have found a sizable finite size effect on the pseudoscalar meson sector in the former analysis: a 2.1(8)%, 4.8(1.6)%, and 0.36(31)% finite size effect on $m_{\pi }$, $m_{\mathrm{ud}}$, and $f_{\pi }$, respectively, on the $(5.4\mathrm{fm}{)}^4$ lattice. For the latter analysis, the finite size effect on the pseudoscalar meson decay constants is 0.66(33)% for $f_{\pi }$, 0.26(13)% for $f_K$, and 0.40(32)% for $f_K/f_{\pi }$. These values with two-sigma error bars are consistent with the predictions from the full one-loop SU(3) chiral perturbation theory, which are 0.20% for $f_{\pi }$, 0.08% for $f_K$, and 0.13% for $f_K/f_{\pi }$. The finite size effect on the pseudoscalar meson masses is hardly detected under the current statistical precision.

Figure 1

Binsize dependence in jackknife analysis for the plaquette values on $64^4$ and $128^4$ lattices. Data for $128^4$ lattice are horizontally shifted for clarification.

Figure 2

Comparison of the effective masses for $\pi$ (top) and $K$ (bottom) mesons on $64^4$ and $128^4$ lattices.

Figure 3

Comparison of the decay constants for $\pi$ (top), $K$ (middle) mesons, and $f_K/f_{\pi }$ (bottom) on $64^4$ and $128^4$ lattices. A shaded band denotes the error band of the $128^4$ result. A red triangle symbol denotes the result reweighted to the point where the AWI quark masses on $64^4$ lattice are equal to those on $128^4$ lattice, while a red cross symbol represents the interpolated result onto the point where the reweighted pion mass reproduces the ChPT prediction (see Sec. 3c).

Figure 4

Configuration dependence of the reweighting factor from $({\kappa }_{\mathrm{ud}},{\kappa }_{\mathrm{s}})=(0.126117,0.124902)$ to $({\kappa }_{\mathrm{ud}}^{*},{\kappa }_{\mathrm{s}}^{*})=(0.126119,0.124902)$.

Figure 5

Reweighting factor from $({\kappa }_{\mathrm{ud}},{\kappa }_{\mathrm{s}})=(0.126117,0.124902)$ to $({\kappa }_{\mathrm{ud}}^{*},{\kappa }_{\mathrm{s}}^{*})=(0.126119,0.124902)$ as a function of a stout-smeared plaquette value.

Figure 6

Reweighted values for $m_{\mathrm{ud}}$ (top), $m_{\pi }$ (middle) and $f_{\pi }$ (bottom) as a function of $N_{\eta }$.

Figure 7

$L$ dependence of $|R_X|$ ($R_{m_{\mathrm{PS}}}>0$ and $R_{f_{\mathrm{PS}}}<0$) for $X=m_{\pi }$, $m_K$, $f_{\pi }$, $f_K$, $f_K/f_{\pi }$ at the physical point. Dotted vertical lines indicate $L=5.4\mathrm{fm}$ and 10.8 fm.