^{1,2}

^{3}

^{4,5}

^{6,2}

^{1,2}

^{,*}

^{6}

^{7,6}

^{3}.

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

The anomalous magnetic moment of the muon is providing an important test of the standard model. The current discrepancy between experiment and theory stands between three and four standard deviations. An ongoing experiment at Fermilab (E989) and one planned at J-PARC (E34) aim to reduce the uncertainty of the BNL E821 value

The magnetic moment is an intrinsic property of a spin-

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.

Here the muon, photons, quarks, and gluons are treated on a finite, discrete lattice. The method is described in detail in Ref.

Connected diagrams. Sums over

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.

Two additional, but related, parts of the method bear mentioning. First, the form dictated by the right-hand side of Eq.

The quark-disconnected diagrams that occur at

As for the connected case, two point sources (at

The simulation parameters are given in Table

The muons and photons take discrete free-field forms. The muons are DWFs with infinite size in the extra fifth dimension, and the photons are noncompact in the Feynman gauge. In the latter all modes with

Before moving to the hadronic case, the method was tested in pure QED

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.

Our physical point calculation

The results are displayed in Fig.

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.

The systematic errors mostly result from the higher order discretization and finite volume effects which are not included in the fitting formula Eq.

Central value and various systematic errors. Numbers in parentheses are statistical error for the corresponding values.

While the large relative error on the total is a bit unsatisfactory, we emphasize that our result represents an important estimate on the hadronic light-by-light scattering contribution to the muon anomaly, with all systematic errors controlled. It appears that this contribution cannot bring the standard model and the E821 experiment in agreement.

In fact we can do even a bit better with the data on hand. As seen in Fig.

Cumulative contributions to the muon anomaly, connected (upper) and disconnected (lower).

Central value and various systematic errors, use the hybrid continuum limit for the connected diagrams. Numbers in parentheses are statistical error for the corresponding values.

We have presented results for the hadronic light-by-light scattering contribution to the muon

tWe would like to thank our RBC and UKQCD collaborators for helpful discussions and critical software and hardware support. This work is partially supported by the U.S. Department of Energy. T. B. and L. C. J. are supported by U.S. DOE Grant No. DE-SC0010339. N. C. is supported by U.S. DOE Grant No. DE-SC0011941. M. H. is supported in by Japan Grants-in-Aid for Scientific Research, No. 16K05317. T. I. and C. L. are supported in part by U.S. DOE Contract No. DESC0012704(BNL). T. I. is also supported by JSPS KAKENHI Grants No. JP26400261 and No. JP17H02906. C. L. is also supported by a DOE Office of Science Early Career Award. We developed the computational code based on the Columbia Physics System (CPS) and Grid

We continue to use the distribution as described in our previous Letter