Leading hadronic contribution to $(g-2{)}_{\mu }$ from lattice QCD with $N_{\mathrm{f}}=2+1$ flavors of $\mathrm{O}(a)$ improved Wilson quarks

The comparison of the theoretical and experimental determinations of the anomalous magnetic moment of the muon $(g-2{)}_{\mu }$ constitutes one of the strongest tests of the Standard Model at low energies. We compute the leading hadronic contribution to $(g-2{)}_{\mu }$ using lattice QCD simulations employing Wilson quarks. Gauge field ensembles at four different lattice spacings and several values of the pion mass down to its physical value are used. We apply the $\mathrm{O}(a)$ improvement program with two discretizations of the vector current to better constrain the approach to the continuum limit. The electromagnetic current correlators are computed in the time-momentum representation. In addition, we perform auxiliary calculations of the pion form factor at timelike momenta in order to better constrain the tail of the isovector correlator and to correct its dominant finite-size effect. For the numerically dominant light-quark contribution, we rescale the lepton mass by the pion decay constant computed on each lattice ensemble. We perform a combined chiral and continuum extrapolation to the physical point, and our final result is $a_{\mu }^{\mathrm{hvp}}=(720.0\pm 12.4_{\mathrm{stat}}\pm 9.9_{\mathrm{syst}})\times {10}^{-10}$. It contains the contributions of quark-disconnected diagrams, and the systematic error has been enlarged to account for the missing isospin-breaking effects.

Figure 1

Integrand of Eq. (1) in the time-momentum representation for the connected light, strange and charm contributions. (Left panel) Ensemble D200 with a pion mass of 200 MeV. (Right panel) Ensemble E250 at the physical pion mass. For better visibility, the strange and charm contributions have been scaled by a factor of 6. The displayed discretization is the local-local one for the light and strange contributions, and the local-conserved one for the charm. The muon mass is the $f_{\pi }$ rescaled one for the light integrand and the physical one for the strange and charm integrands.

Figure 2

Reconstruction of the TMR correlator at long distances using a dedicated spectroscopy analysis on ensemble D200. The same gauge configurations are used for the spectroscopy and for the TMR correlator calculation.

Figure 3

(Left panel) Bounding method with the contribution of $N=0$ [method (1)] and (right panel) $N=2$ [method (2)] states subtracted on ensemble D200 for the local-local correlator and the $f_{\pi }$-rescaled muon mass. Results are based on 1100 gauge configurations.

Figure 4

Determination of $a_{\mu }^{\mathrm{hvp},l}$ with the $f_{\pi }$-rescaled muon mass using the extension of the connected light (local-local) correlator using $N=2$ energy levels on ensemble D200. On the left [method (3)], the amplitudes corresponding to energy levels were predetermined in a spectroscopy calculation, while on the right [method (4)], they are fitted to the TMR correlator. Results are based on 1100 gauge configurations.

Figure 5

Testing the finite-size correcting procedure described in the main text on the ensembles N101 and H105 at a pion mass of 280 MeV. The scale-setting uncertainty is not displayed, since both ensembles have the same lattice spacing.

Figure 6

Integrand of Eq. (1) in the time-momentum representation for the disconnected contribution on ensemble N200, using the local-local discretization and the physical muon mass.

Figure 7

Extrapolation of the connected strange and charm contributions to $a_{\mu }^{\mathrm{hvp}}$ with a muon mass fixed to its physical value. The black curve represents the chiral dependence in the continuum, and the black point the final result at the physical pion mass.

Figure 8

Extrapolation of the connected light contribution to $a_{\mu }^{\mathrm{hvp}}$, (left panel) using the physical value of the muon mass in the kernel $\tilde{K}(t)$ on all ensembles, and (right panel) using the rescaled mass $m_{\mu }^{\mathrm{phys}}·\frac{f_{\pi }^{\mathrm{latt}}}{f_{\pi }^{\mathrm{phys}}}$. The result of the fit based on Eq. (29c) is shown. The black curve represents the chiral dependence in the continuum, and the black point the final result at the physical pion mass.

Figure 9

Extrapolation of the disconnected contribution to $a_{\mu }^{\mathrm{hvp}}$ in the SU(3)-breaking variable ${\mathrm{\Delta }}_2\equiv m_K^2-m_{\pi }^2$. The data points for the local-local and the local-conserved discretizations are shown. A linear fit (straight black line), as well as a fit based on Ansatz (31) are shown.

Figure 10

Compilation of lattice results for the connected contributions to $a_{\mu }^{\mathrm{hvp}}$ from individual charm, strange and light quarks (left to right). In the rightmost panel, the full results, including (where available) the contributions from quark-disconnected diagrams and corrections due to isospin breaking, are compared to the phenomenological determination of Ref. [55], represented by the red vertical band. Our result is compared to the calculations labeled FNAL-HPQCD-MILC 19 [56, 57, 58], PACS 19 [59], ETMC 19 [60, 61, 62], RBC/UKQCD 18 [39], BMW 17 [38], as well as our previous calculation in two-flavor QCD [20] (Mainz/CLS 17).