Calculation of the Hadronic Vacuum Polarization Contribution to the Muon Anomalous Magnetic Moment

We present a first-principles lattice $\mathrm{QCD}+\mathrm{QED}$ calculation at physical pion mass of the leading-order hadronic vacuum polarization contribution to the muon anomalous magnetic moment. The total contribution of up, down, strange, and charm quarks including QED and strong isospin breaking effects is $a_{\mu }^{\text{HVP LO}}=715.4(18.7)\times {10}^{-10}$. By supplementing lattice data for very short and long distances with $R$-ratio data, we significantly improve the precision to $a_{\mu }^{\text{HVP LO}}=692.5(2.7)\times {10}^{-10}$. This is the currently most precise determination of $a_{\mu }^{\text{HVP LO}}$.

Figure 1

Quark-connected (left) and quark-disconnected (right) diagram for the calculation of $a_{\mu }^{\text{HVP LO}}$. We do not draw gluons but consider each diagram to represent all orders in QCD.

Figure 2

QED-correction diagrams with external pseudoscalar or vector operators. The figure parts (a)–(g) assign labels V, S, T, D1, D2, F, and D3 to the respective diagrams.

Figure 3

Strong isospin-breaking correction diagrams. The crosses denote the insertion of a scalar operator. The figure parts (a)–(c) assign labels M, R, and O to the respective diagrams.

Figure 4

Comparison of $w_{t}C(t)$ obtained using $R$-ratio data [10] and lattice data on our 64I ensemble.

Figure 5

We show results for the window method with $t_0=0.4\mathrm{fm}$ as a function of $t_1$. The top panel shows the combined $a_{\mu }$, the middle panel shows the partial contributions of the lattice and $R$-ratio data, and the bottom shows the respective uncertainties.

Figure 6

The individual lattice components of the window method with $t_0=0.4\mathrm{fm}$ as function of $t_1$.

Figure 7

Our results (RBC and UKQCD 2018) compared to previously published results. The green data points are pure lattice computations, the orange data point is our combined window analysis, and the purple data points are pure $R$-ratio results. The references are ETMC 2013 [46], HPQCD 2016 [47], Mainz 2017 [48], BMW 2017 [44], HLMNT 2011 [3], DHMZ 2012 [49], DHMZ 2017 [6], Jegerlehner 2017 [5], and No new physics [2]. The innermost error bar corresponds to the statistical uncertainty.