Window observable for the hadronic vacuum polarization contribution to the muon $g-2$ from lattice QCD

Euclidean time windows in the integral representation of the hadronic vacuum polarization contribution to the muon $g-2$ serve to test the consistency of lattice calculations and may help in tracing the origins of a potential tension between lattice and data-driven evaluations. In this paper, we present results for the intermediate time window observable computed using $\mathrm{O}(a)$ improved Wilson fermions at six values of the lattice spacings below 0.1 fm and pion masses down to the physical value. Using two different sets of improvement coefficients in the definitions of the local and conserved vector currents, we perform a detailed scaling study which results in a fully controlled extrapolation to the continuum limit without any additional treatment of the data, except for the inclusion of finite-volume corrections. To determine the latter, we use a combination of the method of Hansen and Patella and the Meyer-Lellouch-Lüscher procedure employing the Gounaris-Sakurai parametrization for the pion form factor. We correct our results for isospin-breaking effects via the perturbative expansion of $\mathrm{QCD}+\mathrm{QED}$ around the isosymmetric theory. Our result at the physical point is $a_{\mu }^{\mathrm{win}}=(237.30\pm 0.79_{\mathrm{stat}}\pm 1.22_{\mathrm{syst}})\times {10}^{-10}$, where the systematic error includes an estimate of the uncertainty due to the quenched charm quark in our calculation. Our result displays a tension of $3.9\sigma$ with a recent evaluation of $a_{\mu }^{\mathrm{win}}$ based on the data-driven method.

Figure 1

Integrands used to compute the intermediate window $a_{\mu }^{\mathrm{win}}$ for the isovector, isoscalar and charm-quark contributions. The isoscalar contribution does not include the charm-quark contribution. The data have been obtained on ensemble E250, which has close-to-physical quark masses, using two local vector currents and set 1 of renormalization and improvement coefficients.

Figure 2

Continuum extrapolation for the isovector quark contribution at the ${\mathrm{SU}(3)}_{\mathrm{f}}$-symmetric point. Left: using $f_{\pi }$ rescaling. Right: with $t_0$ to set the scale. The blue and green points correspond to the two different sets of improvement coefficients (see Sec. 3). For clarity, the extrapolated results have been shifted to the left.

Figure 3

Left: one typical extrapolation of the isovector contribution using $f_{\mathrm{ch}}(\tilde{y})=1/\tilde{y}$. The data correspond to the local-conserved discretization of the correlator using set 1 of improvement coefficients. Error bands are the results from the fit for each of the six lattice spacings. The black line is the chiral extrapolation in the continuum limit. The black point is the result at the physical point. Right: the same for the isoscalar contribution but using $f_{\mathrm{ch}}(\tilde{y})=0$.

Figure 4

Comparison of the isovector and isoscalar contributions (without the charm) using different variations (either using $f_{\pi }$ or $t_0$ to set the scale, and with both sets of improvement coefficients). The blue point is our final estimate obtained from the rescaling method with set 1 of improvement coefficients.

Figure 5

Extrapolation to the physical point for the quark-disconnected contribution using Eq. (38). The vertical dashed line represents the physical point in our isosymmetric QCD setup. The black point is the result of the extrapolation, and the gray band represents the extrapolation to the continuum limit with $X_K=X_K^{\star }$. Points with dashed error bars are not included in the fit.

Figure 6

Left panel: study of the continuum extrapolation of the charm-quark contribution to $a_{\mu }^{\mathrm{win}}$ at the $\mathrm{SU}(3{)}_{\mathrm{f}}$-symmetric point using the local-conserved discretization of the correlation function. The black and green points are obtained using two independent sets of improvement coefficients, as explained in Sec. 3b. Right panel: example of a typical extrapolation to the physical point of the charm-quark contribution. The error from the scale setting, which is highly correlated between ensembles, is not shown. The plain lines are obtained from the fit function (40) without any cut in the pion mass.

Figure 7

Comparison of our results (in units of ${10}^{-10}$) with other lattice calculations [13, 18, 20, 21, 22, 23, 24] in isosymmetric QCD. The four panels on the left show compilations of the individual quark-disconnected, charm, strange and light-quark contributions. The total result for $a_{\mu }^{\mathrm{win}}$ in the isosymmetric case is shown in the rightmost panel. Our results are represented by green circles and vertical bands.

Figure 8

Comparison of our result for $a_{\mu }^{\mathrm{win}}$ including isospin-breaking corrections with the estimates by ETMC [22], BMW [20], and RBC/UKQCD [13]. The estimate based on the data-driven method of Ref. [48] is shown in red.

Figure 9

Derivatives of the isovector and the strange-connected contributions to the window observable with respect to $X_{\pi }$. The gray areas illustrate the priors that are used in the global extrapolation.