Hadronic Light-by-Light Scattering Contribution to the Muon Anomalous Magnetic Moment from Lattice QCD

We report the first result for the hadronic light-by-light scattering contribution to the muon anomalous magnetic moment with all errors systematically controlled. Several ensembles using $2+1$ flavors of physical mass Möbius domain-wall fermions, generated by the RBC and UKQCD collaborations, are employed to take the continuum and infinite volume limits of finite volume lattice $\mathrm{QED}+\mathrm{QCD}$. We find $a_{\mu }^{\text{HLbL}}=7.87(3.06{)}_{\text{stat}}(1.77{)}_{\text{sys}}\times {10}^{-10}$. Our value is consistent with previous model results and leaves little room for this notoriously difficult hadronic contribution to explain the difference between the standard model and the BNL experiment.

Figure 1

Leading contributions from hadronic light-by-light scattering to the muon anomaly. The shaded circles represent quark loops containing QCD interactions to all orders. Horizontal lines represent muons. Quark-connected (left) and disconnected (right) diagrams are shown. Ellipsis denote diagrams obtained by permuting the photon contractions with the muons and diagrams with three and four quark loops with photon couplings (See Fig. 3).

Figure 2

Connected diagrams. Sums over $x$ and $y$ are computed stochastically. The third internal vertex $z$ and the external vertex $x_{\mathrm{op}}$ are summed over exactly. The sums on the muon line are done exactly using fast Fourier transforms. Strong interactions to all orders are not shown.

Figure 3

Disconnected diagrams contributing to the muon anomaly. The top leftmost is the leading one, and does not vanish in the SU(3) flavor limit. Strong interactions to all orders, including gluons connecting the quark loops, are not shown.

Figure 4

QED light-by-light scattering contribution from the muon loop to the muon anomaly. The lattice spacing decreases from bottom to top. Solid lines are from a fit using Eq. (4).

Figure 5

Infinite volume extrapolation. Connected (top), disconnected (middle), and total (bottom). We have use the hybrid method to calculate the continuum limit for the connected contribution.

Figure 6

Cumulative contributions to the muon anomaly, connected (upper) and disconnected (lower). $r$ is the distance between the two sampled currents in the hadronic loop (the other two currents are summed exactly) and the horizontal axis is the cumulative contributions from $r$ and below: ${24}^3$ IDSDR (squares), ${24}^3$ IDSDR (squares), ${32}^3$ IDSDR (crosses), ${48}^3$ Iwasaki (diamonds), and ${64}^3$ Iwasaki (plusses).