Hadronic vacuum polarization contribution to the muon $g-2$ with ($2+1$)-flavor lattice QCD on a larger than ${(10\mathrm{fm})}^4$ lattice at the physical point

We study systematic uncertainties in the lattice QCD computation of the hadronic vacuum polarization (HVP) contribution to the muon $g-2$. We investigate three systematic effects: the finite volume (FV) effect, the cutoff effect, and integration scheme dependence. We evaluate the FV effect at the physical pion mass on two different volumes of ${(5.4\mathrm{fm})}^4$ and ${(10.8\mathrm{fm})}^4$ using the PACS10 configurations at the same cutoff scale. For the cutoff effect, we compare two types of lattice vector operators, which are local and conserved (point-splitting) currents, by varying the cutoff scale on a larger than ${(10\mathrm{fm})}^4$ lattice at the physical point. For the integration scheme dependence, we compare the results between the coordinate- and momentum-space integration schemes at the physical point on a ${(10.8\mathrm{fm})}^4$ lattice. Our result for the HVP contribution to the muon $g-2$ is given by $a_{\mu }^{\mathrm{hvp}}=737(9)({}_{-18}^{+13})\times {10}^{-10}$ in the continuum limit, where the first error is statistical and the second one is systematic.

Figure 1

(Left panel) Ratio of relative error for the vector-vector current correlator $C^{\mathrm{CL}}(t)$ on ${128}^4$ and ${64}^4$ lattices using the same number of measurements. The straight line shows the expected volume scaling. (Right panel) Relative error of the vector-vector current correlator on ${128}^4$ and ${64}^4$ lattices in $a^{-1}=2.33\mathrm{GeV}$ and ${160}^4$ lattice at $a^{-1}=3.06\mathrm{GeV}$.

Figure 2

Effective mass for the vector-vector current correlator at the physical pion mass on the (top panel) ${64}^4$, (middle panel) ${128}^4$, and (bottom panel) ${160}^4$ lattices. Solid lines denote the physical $\rho$ meson mass, and dashed ones are for the free two-pion energies $E_{\pi \pi }^{\mathrm{free}}=2\sqrt{m_{\pi }^2+(2\pi /L{)}^2}$ on each lattice volume.

Figure 3

Comparison of $C^{\mathrm{LL}}(r)W_r(r)$ in Eq. (18) between different spatial volumes with $L/a=128$ and 64 in the light quark sector.

Figure 4

(Left panel) Difference of $a_{\mu }^{\mathrm{hvp}}(r_{\mathrm{cut}})$ on the ${128}^4$, $64^4$, and ${64}^3\times 128$ lattices in the light quark sector. The hopping parameters are the same on both lattices. The solid (dashed) curve denotes the leading order of the ChPT prediction for the FV effect between ${(10.8\mathrm{fm})}^3$ and ${(5.4\mathrm{fm})}^3$ spatial volumes with $m_{\pi }=135\mathrm{MeV}$ on the [$L/a=128$, $T/a=128$] lattice, and 139 MeV on the [$L/a=64$, $T/a=128$] ([$L/a=64$, $T/a=64$]) lattice. (Right panel) Ratio of the FV effect between the LQCD and ChPT estimates with the same symbol as the left panel.

Figure 5

Difference of $a_{\mu }^{\mathrm{hvp}}(r_{\mathrm{cut}})$ on the ${128}^4$ and ${64}^4$ lattices in the strange quark sector.

Figure 6

(Top panels) $C^{\mathrm{LL}}(r)W_r(r)$ and (bottom panels) $C^{\mathrm{CL}}(r)W_r(r)$ in Eq. (18) in the light quark sector as a function of distance $r$ on the ${128}^4$ lattice at $a^{-1}=2.33\mathrm{GeV}$ (circles) and on the ${160}^4$ lattice at $a^{-1}=3.06\mathrm{GeV}$ (triangles).

Figure 7

(Left panel) $r$ dependence of ${\mathrm{\Delta }}_r(r)\equiv 1-C^{\mathrm{CL}}(r)/C^{\mathrm{LL}}(r)$ on the ${128}^4$ lattice at $a^{-1}=2.33\mathrm{GeV}$ (circles) and on the ${160}^4$ lattice at 3.06 GeV (crosses). (Right panel) The cutoff dependence of ${\mathrm{\Delta }}_r(r)$ at $r\simeq 1.5\mathrm{fm}$. The straight line and the band denote the central value and statistical error of the linear fitting function of ${\mathrm{\Delta }}_r(r)\propto a$.

Figure 8

LQCD results for $[a_{\mu }^{\mathrm{hvp}}{]}_{\mathrm{lat}}(r_{\mathrm{cut}})$ of Eq. (18) in the (L, L) and (C, L) channels on the ${128}^4$ lattice at $a^{-1}=2.33\mathrm{GeV}$, and on the ${160}^4$ lattice at $a^{-1}=3.09\mathrm{GeV}$. The left (right) panel pertains to the light (strange) quark sector.

Figure 9

Cutoff dependence of (top panel) $[a_{\mu }^{\mathrm{hvp}}{]}_{\mathrm{lat}}^l(r_{\mathrm{cut}})$, (middle panel) $[a_{\mu }^{\mathrm{hvp}}{]}_{\mathrm{lat}}^s(r_{\mathrm{cut}})$, and (bottom panel) $[a_{\mu }^{\mathrm{hvp}}{]}_{\mathrm{lat}}^c(r_{\mathrm{cut}})$ in the (L, L) and (C, L) channels with $r_{\mathrm{cut}}\approx 3.5\mathrm{fm}$. The extrapolated result in the continuum limit (diamond) has two kinds of errors: the inner one is statistical and the outer one denotes the total error, including the systematic error explained in the text.

Figure 10

$Q^2$ dependence of the VPF in (left panel) light, (middle panel) strange, and (right panel) charm quark sectors. Solid lines denote the fit results, including the statistical error with (left and middle panels) the Padé [1,1] approximation and (right panel) the linear function.

Figure 11

$Q^2$ dependence of the integrand in Eq. (1) up to $Q^2=0.5{\mathrm{GeV}}^2$ at the light and strange quark sectors. The horizontal axis is rescaled to a dimensionless quantity $1/(1+\ln (Q_c^2/Q^2))$ [52] with $Q_c^2=0.5{\mathrm{GeV}}^2$ in the right panel. Curved bands show the fit results including the statistical error. The shaded vertical band denotes the fitting range for $\mathrm{\Pi }(Q)$.

Figure 12

$Q_{\mathrm{fit}}^2$ dependence in zero-momentum extrapolation with linear function and Padé approximation for the VPF in (top panel) light, (middle panel) strange, and (bottom panel) charm quark sectors. The horizontal lines are the central value (solid) and statistical error (dashed) of the result in the coordinate-space integration scheme with $r_{\mathrm{cut}}\approx 3.5\mathrm{fm}$.

Figure 13

The same as Fig. 12 but on a ${64}^4$ lattice.

Figure 14

(Left panel) Summary plot of the connected $a_{\mu }^{\mathrm{hvp}}$ in the light quark sector $[a_{\mu }^{\mathrm{hvp}}{]}^l$ and (right panel) the full result of $a_{\mu }^{\mathrm{hvp}}$ in comparison with recent LQCD results ($N_f\ge 3$) of the BMW [16], ETMC [18], HPQCD [14], and RBC-UKQCD [17] collaborations, and phenomenological estimate obtained with the experimental R-ratio by DHMZ [10] and KNT [11]. The shaded vertical band shows the experimental $a_{\mu }^{\mathrm{hvp}}$ estimated as the difference between the BNL experimental value of $a_{\mu }$ and the theoretical value with QED and EW, including the light-by-light scattering contribution. The error bar for $[a_{\mu }^{\mathrm{hvp}}{]}^l$ in this work represents the combined error with the statistical one and the systematic one due to the cutoff effect. Additional uncertainties of the missing disconnected diagram and the IB effect are included in the error bar of $a_{\mu }^{\mathrm{hvp}}$ in this work.