Strong-Isospin-Breaking Correction to the Muon Anomalous Magnetic Moment from Lattice QCD at the Physical Point

All lattice-QCD calculations of the hadronic-vacuum-polarization contribution to the muon’s anomalous magnetic moment to date have been performed with degenerate up- and down-quark masses. Here we calculate directly the strong-isospin-breaking correction to $a_{\mu }^{\mathrm{HVP}}$ for the first time with physical values of $m_u$ and $m_d$ and dynamical $u$, $d$, $s$, and $c$ quarks, thereby removing this important source of systematic uncertainty. We obtain a relative shift to be applied to lattice-QCD results obtained with degenerate light-quark masses of $\delta a_{\mu }^{\mathrm{HVP},m_u\ne m_d}=+1.5(7)\%$, in agreement with estimates from phenomenology.

Figure 1

Leading hadronic contribution to $a_{\mu }$. 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

Valence-quark-mass dependence of the light-quark-connected contribution to $a_{\mu }^{\mathrm{HVP}}$ on the $N_f=1+1+1+1$ ensemble without rescaling (open symbols) and with rescaling each ${\mathrm{\Pi }}_j^{(qq)}$ by $(E_0/M_{{\rho }^0}{)}^{2j}$ (closed symbols). From left to right, the pairs of data points correspond to $m_u$, $m_l=(m_u+m_d)/2$, and $m_d$; each pair of data points is horizontally offset for clarity. The values of $a_{\mu }^{qq}$ include the charge factor $q_u^2+q_d^2=5/9$ appropriate for the isospin-symmetric case.