Hadronic-vacuum-polarization contribution to the muon’s anomalous magnetic moment from four-flavor lattice QCD

We calculate the contribution to the muon anomalous magnetic moment hadronic vacuum polarization from the connected diagrams of up and down quarks, omitting electromagnetism. We employ QCD gauge-field configurations with dynamical $u$, $d$, $s$, and $c$ quarks and the physical pion mass, and analyze five ensembles with lattice spacings ranging from $a\approx 0.06$ to 0.15 fm. The up- and down-quark masses in our simulations have equal masses $m_l$. We obtain, in this world where all pions have the mass of the ${\pi }^0$, $10^{10}{a}_{\mu }^{ll}(\mathrm{conn}.)=637.8(8.8)$, in agreement with independent lattice-QCD calculations. We then combine this value with published lattice-QCD results for the connected contributions from strange, charm, and bottom quarks, and an estimate of the uncertainty due to the fact that our calculation does not include strong-isospin breaking, electromagnetism, or contributions from quark-disconnected diagrams. Our final result for the total $\mathcal{O}({\alpha }^2)$ hadronic-vacuum polarization to the muon’s anomalous magnetic moment is $10^{10}{a}_{\mu }^{\mathrm{HVP},\mathrm{LO}}=699(15{)}_{u,d}(1{)}_{s,c,b}$, where the errors are from the light-quark and heavy-quark contributions, respectively. Our result agrees with both ab-initio lattice-QCD calculations and phenomenological determinations from experimental $e^{+}{e}^{-}$-scattering data. It is $1.3\sigma$ below the “no new physics” value of the hadronic-vacuum-polarization contribution inferred from combining the BNL E821 measurement of $a_{\mu }$ with theoretical calculations of the other contributions.

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. Diagrams in which the photon creates a quark-antiquark pair, which propagate while interacting via the strong and electromagnetic forces, and subsequently annihilate back into a photon, are called “quark-connected” diagrams. Those in which the quark-antiquark pair annihilates into gluons are referred to as “quark-disconnected” diagrams.

Figure 2

Local-local vector-current correlator on the two $a\approx 0.15\mathrm{fm}$ ensembles with similar parameters but differing statistics. Based on this plot, we choose $t_{\max }/a=15$ and $t_{\max }/a=24$ for the correlator fits on the low- and high-statistics ensembles, respectively. Plots for other ensembles look similar to the low-statistics $a\approx 0.15\mathrm{fm}$ data.

Figure 3

Stability of $a_{\mu }^{ll}(\mathrm{conn}.)$ calculated from the mixed correlator $G_{\mathrm{data}}(t\le 2.0\mathrm{fm})$ and $G_{\mathrm{fit}}(t>2.0\mathrm{fm})$ on the $a\approx 0.09\mathrm{fm}$ ensemble. For each value of $t_{\min }$, the results for $a_{\mu }^{ll}(\mathrm{conn}.)$ from fits with 2–5 pairs of oscillating and nonoscillating states are shown with a slight horizontal displacement for clarity; $t_{\max }/a=30$ for all fits. For this ensemble, we select $t_{\min }/a=8$ and three pairs of states.

Figure 4

$a_{\mu }^{ll}(\mathrm{conn}.)$ versus the transition time $t^{*}$ in the mixed $\mathrm{data}+\mathrm{fit}$ correlator on the two ensembles with $a\approx 0.15\mathrm{fm}$. The dashed horizontal lines show the $1\sigma$ error band for the value of $a_{\mu }^{ll}(\mathrm{conn}.)$ calculated entirely from data on the high-statistics ensemble.

Figure 5

Time-dependence of fake-data correlator $G_{\mathrm{fake}}(t)$ (blue) created from the chiral model used to calculate finite-volume corrections for $a=0.06\mathrm{fm}$ simulations compared with the result from a least-squares fit with two exponentials (red). The agreement between the data and fit is so close that the blue curve obscures the red curve over most of the figure.

Figure 6

Top two panels: contributions to $a_{\mu }^{ll}(\mathrm{conn}.)$ from all $\pi \pi$ states with energies $E<2\mathrm{GeV}$ in the chiral model used to calculate finite-volume corrections for the $a=0.06\mathrm{fm}$ simulations. The top panel uses a linear $y$-scale; the second panel uses a log scale so that a more complete set of energy levels can be displayed. Bottom panel: contributions to $a_{\mu }^{ll}(\mathrm{conn}.)$ from the states in a two-exponential fit to the fake data created from the chiral model. Summing all contributions in each case gives results that agree to within $0.9\times {10}^{-10}$, which is roughly $1/10$ the fit error.

Figure 7

Comparison of $a_{\mu }^{ll}(\mathrm{conn}.)$ from our noise-reduction strategy with BMW’s bounding method [41] on the low-statistics $a\approx 0.15\mathrm{fm}$ ensemble. The $x$-axis shows either the value of $t^{*}$ employed in the $\mathrm{data}+\mathrm{fit}$ method or of $t_c$ used in the bounding method. For the bounding method, we plot the average of $a_{\mu }^{ll}(\mathrm{conn}.)$ obtained from the upper and lower bounds; the two bounds meet at $t_c\sim 2.5\mathrm{fm}$.

Figure 8

Quark-connected Taylor coefficients of the renormalized vacuum-polarization function before (empty) and after (filled symbols) lattice corrections compared with the R-ratio determination from Keshavarzi et al. [7]. We take the $s$-, $c$-, and $b$-quark connected ${\mathrm{\Pi }}_i\mathrm{s}$ from HPQCD’s companion calculations on the MILC HISQ ensembles [19, 21].

Figure 9

Lattice-spacing dependence of $a_{\mu }^{ll}(\mathrm{conn}.)$ before (open blue squares) and after (filled red circles) finite-volume, taste-breaking, and $M_{\pi }$ corrections are applied. The horizontal light-red band and red dotted line show the continuum limit result $a_{\mu }^{ll}(\mathrm{conn}.)=637.8(8.8)$ obtained by fitting the corrected data points with the function in Eq. (3.1).

Figure 10

Stability of continuum-limit result for $a_{\mu }^{ll}(\text{conn.})$ against various fit variations. From top to bottom, the alternate fits (open circles) correspond to modifying the central fit (closed circle) by (i) adding to the fit function terms proportional to ${\chi }_f^2$, ${\chi }_{a^2}^2$, and ${\chi }_f{\chi }_{a^2}$, where ${\chi }_f\equiv \sum _{f=l,l,s,c}\delta m_f/\mathrm{\Lambda }$ and ${\chi }_{a^2}\equiv (a\mathrm{\Lambda }{)}^2/{\pi }^2$; (ii) removing from the fit function the terms proportional to $c_s$ and $c_{a^2}$; (iii) doubling the prior widths on all fit parameters; (iv) removing constraints on the fit parameters altogether; (v) removing data from the coarsest $a$ ($\approx 0.15$ fm); (vi) removing data from the finest $a$ ($\approx 0.06$ fm); and (vii) removing correlations between the data points.

Figure 11

Comparison of our result in Eq. (4.1) for the light-quark connected contribution to $a_{\mu }^{\mathrm{HVP},\mathrm{LO}}$ with recent unquenched lattice-QCD results [14, 15, 16, 17, 18, 23, 48]. All values correspond to isospin-symmetric QCD without electromagnetism. Results for $a_{\mu }^{ll}(\mathrm{conn}.)$ from four-, three, and two-flavor QCD simulations are denoted by squares, circles, and triangles, respectively. Note that the RBC/UKQCD Collaboration employed three-flavor QCD gauge-field configurations and then added the charm sea-quark contribution estimated from perturbation theory a posteriori.

Figure 12

Comparison of our results in Eqs. (4.2) and (4.3) for the light-quark connected contribution to the slope and curvature of $\widehat{\mathrm{\Pi }}(Q^2)$ with published unquenched lattice-QCD results from Refs. [15, 41, 48]. All values correspond to isospin-symmetric QCD without electromagnetism. Note that we multiplied the Taylor coefficients quoted in Refs. [41, 48] by the charge factor $q_u^2+q_d^2=5/9$ so that they correspond to our normalization convention.

Figure 13

Comparison of our result in Eq. (5.1) for the leading-order hadronic-vacuum-polarization contribution to the muon anomalous magnetic moment (magenta square) with results from $N_f\ge 2$ lattice QCD [14, 15, 16, 17, 23, 48, 54] (blue and purple squares) and from experimental $e^{+}{e}^{-}$ cross-section data [5, 6, 7, 55] (red and orange triangles). The filled black circle shows the value of $a_{\mu }^{\mathrm{HVP},\mathrm{LO}}$ that is implied by the measurement of $a_{\mu }$ by BNL experiment E821 [1] assuming no contributions beyond the Standard Model; vertical dashed lines denote the $\pm 1\sigma$ range [25].