Hadronic Contributions to the Anomalous Magnetic Moment of the Muon from Lattice QCD

**Principal Investigator:**

Hartmut Wittig

**Affiliation:**

Institute for Nuclear Physics and PRISMA Cluster of Excellence, Johannes Gutenberg University of Mainz

**Local Project ID:**

GCS-HQCD

**HPC Platform used:**

Hazel Hen and HAWK of HLRS

**Date published:**

The Standard Model of Particle Physics provides a quantitative and precise description of the properties of the known constituents of matter in terms of a uniform theoretical formalism. However, despite its enormous success, the Standard Model (SM) offers no explanations for some of the most pressing problems in particle physics, such as the nature of dark matter or the asymmetry between matter and antimatter. The world-wide quest for discovering physics beyond the SM involves several complementary strategies: (1) the search for new particles and interactions that are not described by the SM, (2) the search for the enhancement of rare processes by new interactions, and (3) the comparison of precision measurements with theoretical, SM-based predictions of the same quantity. These complementary activities form an integral part of the future European strategy for particle physics. Precision observables, such as the anomalous magnetic moment of the muon, *a*_{μ}, have attracted a lot of attention recently, chiefly because of the persistent tension of 3.7 standard deviations between the experimental measurement and its theoretical prediction. As the community prepares for the announcement of a new and even more precise measurement by the E989 experiment at Fermilab, the precision of the theoretical prediction must be improved to a similar level. Since the main uncertainties in the SM prediction are associated with the effects from the strong interaction, current efforts are focussed on quantifying the contributions from hadronic vacuum polarisation and hadronic light-by-light scattering. This has also been emphasised in a recent white paper in which the status of the theoretical prediction is reviewed.

Our project is focussed on calculations of the hadronic contributions to the muon anomalous magnetic moment from first principles, using the methodology of Lattice QCD. In this formalism, the hadronic vacuum polarisation contribution is accessible via the 2-point correlation function of the electromagnetic current. Our goal is the determination of this crucial observable with a total error below 1%. At this level of precision, lattice calculations face enormous technical challenges. These are associated with the rapidly increasing statistical noise in the infrared region, sizeable corrections due to finite-volume effects, and the contributions from so-called quark-disconnected diagrams that exhibit a high intrinsic level of statistical noise. Furthermore, the effects of unequal “up” and “down” quark masses, as well as electromagnetic corrections must be taken into account. Moreover, in order to minimise the systematic effects arising from extrapolations in the quark mass, calculations should preferably be carried out at the physical value of the pion mass.

In our calculation, we employed *O(a)* improved Wilson fermions with dynamical up, down and strange quarks. In order to minimise the systematic effects arising from extrapolations in the quark mass, we generated a gauge ensemble at the physical value of the pion mass. Numerically, this is extremely demanding: not only is the system size extremely large (in our case the lattice contained *96*^{3}*×192* sites), but also special care is required to stabilise the simulation algorithm. Addressing the problem of the unfavourable signal-to-noise ratio, we have performed ancillary calculations of the hadron spectrum in the isovector channel which dominates the correlation function at large distances. Finally, in order to determine quark-disconnected contributions with good statistical accuracy, we have adapted the so-called “one-end trick” which allows for the calculation of this important contribution at much reduced numerical cost. The enormous computational resources required for our project imply that it is ideally suited for massively parallel computing platforms such as Hazel Hen at HLRS.

Figure 1 shows the correlator of the electromagnetic current multiplied by the kernel function which absorbs the effects from the muon and the external photon. The abscissa represents the Euclidean time variable, and the hadronic vacuum polarisation contribution is obtained from the area underneath the data points. In the case of the light quark contribution, one clearly sees that the statistical signal becomes worse in the long-distance (infrared) regime. A dedicated calculation of the spectrum in the isovector channel will allow for a much more precise determination of the contributions at large Euclidean times.

Figure 2 shows the extrapolation of our results of the light quark contribution obtained at several different pion masses and lattice spacings to the physical point. The availability of a data point directly at the physical pion mass is of crucial importance to control the strong dependence on the quark mass among the results.

Finally, in Figure 3 we show our results for the quark-disconnected contribution to the correlation function, which was obtained on just 50 gauge configurations of the ensemble at the physical pion mass. The plot demonstrates that the new stochastic technique allows us to compute this intrinsically noisy contribution with good statistical accuracy up to large Euclidean times. In the future, we will further refine our calculation, by accumulating more statistics, including more gauge ensembles at or near the physical pion mass and by including the effects of strong and electromagnetic isospin breaking.

In addition, we also address the hadronic light-by-light scattering contribution to the muon’s anomalous magnetic moment. So far, this important contribution has been estimated mostly using hadronic models, and a determination from first principles is highly desirable. Technically, lattice calculations of light-by-light scattering are even more involved, and there are several complementary approaches. The leading contribution comes from the pion pole and can be quantified via a lattice calculation of the form factor for the process *π*^{0}*→**γ*^{*}*γ*^{*}. A direct calculation of the full HLbL scattering contribution can also be performed by combining lattice results for a four-point correlation function with an analytic kernel function. A first calculation based on this approach in the limit of degenerate light and strange quark masses has recently been performed by our group.

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**Scientific Contact**

Univ.-Prof. Dr. Hartmut Wittig

Institute of Nuclear Physics

Johannes Gutenberg-Universität Mainz

Johann-Joachim-Becher Weg 45, D-D-55099 Mainz

e-mail: wittigh [@] uni-mainz.de

*HLRS project ID: GCS-HQCD*

*August 2020*