Higher-order hadronic-vacuum-polarization contribution to the muon $g-2$ from lattice QCD

We introduce a new method for calculating the $\mathrm{O}({\alpha }^3)$ hadronic-vacuum-polarization contribution to the muon anomalous magnetic moment from ab initio lattice QCD. We first derive expressions suitable for computing the higher-order contributions either from the renormalized vacuum polarization function $\widehat{\mathrm{\Pi }}(q^2)$ or directly from the lattice vector-current correlator in Euclidean space. We then demonstrate the approach using previously published results for the Taylor coefficients of $\widehat{\mathrm{\Pi }}(q^2)$ that were obtained on four-flavor QCD gauge-field configurations with physical light-quark masses. We obtain ${10}^{10}{a}_{\mu }^{\mathrm{HVP},\mathrm{HO}}=-9.3(1.3)$ in agreement with, but with a larger uncertainty than, determinations from $e^{+}{e}^{-}\to \text{hadrons}$ data plus dispersion relations.

Figure 1

Leading hadronic contribution to the muon $g_{\mu }-2$. The shaded circle denotes all corrections to the internal photon propagator from the vacuum polarization of $u$, $d$, $s$, $c$, and $b$ quarks in the leading one-loop muon vertex diagram.

Figure 2

Higher-order hadronic-vacuum-polarization contributions to $g_{\mu }-2$. For contribution (a), diagrams that are reflections across the vertical axis through the center and diagrams in which the tree and corrected photon propagators are interchanged are not shown.

Figure 3

Left: integrands of Eqs. (a1) (blue dots), (a6) (green dashes), and (a13) (purple dot-dashes) obtained from the $N=2+1+1$ Mellin-Barnes approximant for $\widehat{\mathrm{\Pi }}(Q^2)$ given in Ref. [25], which employs preliminary moments of $R_{\gamma }(s)$ provided by Keshavarzi et al. [16]. The leading-order integrand is also shown as a solid magenta line for comparison. Right: integrands of Eqs. (a3) (blue squares) and (a11) (green diamonds) obtained from the parameterization of $R_{\gamma }(s)$ provided by Jegerlehner in his public alphaQED FORTRAN package [30]. The leading-order integrand is also shown as magenta circles for comparison.

Figure 4

First four Padé approximants (“[1,0]–[2,2] Padé”) and Mellin-Barnes approximants (“$\mathrm{N}=1-\mathrm{N}=2+1+1\mathrm{M}\text{−}\mathrm{B}$”) of the renormalized vacuum polarization function calculated from the moments of $R_{\gamma }(s)$ analysis of Keshavarzi et al. [16]. The exact result is shown as a solid black line for comparison [34].

Figure 5

Continuum extrapolation of $\mathrm{O}({\alpha }^3)$ quark-connected contribution to $a_{\mu }^{\mathrm{HVP}}$. The filled cyan band shows the result of our preferred constant fit, while the solid blue lines show the result of a linear fit to Eq. (3.3) with the slope $c_{a^2}$ constrained with a Gaussian prior $0\pm 1$.