Toward the Anomalous Magnetic Moment of the Muon from 2+1‑Flavour Lattice QCD

**Principal Investigator:**

Georg von Hippel

**Affiliation:**

Institut für Kernphysik, Johannes Gutenberg-Universität Mainz (Germany)

**Local Project ID:**

hmz23

**HPC Platform used:**

JUQUEEN of JSC

**Date published:**

*Lattice Quantum Chromodynamics (Lattice QCD) is a first-principles, non-perturbative formulation of the theory of the strong interaction that allows for numerical simulations with systematic control of theoretical uncertainties, and which has a long and successful history of providing the information required for a quantitative understanding of strong interaction physics at low energies. Nevertheless, a number of quantities could not be studied so far with the desired level of control of statistical and systematic uncertainties; this includes the hadronic contribution to the anomalous magnetic moment of the muon, a precise determination of which is currently the most promising avenue in the search for physics beyond the Standard Model (SM) of particle physics. Here, reserachers investigate this quantity, among others, using lattice QCD simulations on fine and large lattices in order to control systematic uncertainties and enable a precise theoretical prediction.*

**Project Report**

Lattice Quantum Chromodynamics (Lattice QCD) is a first-principles, non-perturbative formulation of the theory of the strong interaction that allows for numerical simulations with systematic control of theoretical uncertainties. It has a long and successful history of providing the information required for a quantitative understanding of strong interaction physics at low energies, where the strength of the gauge coupling renders perturbation theory inapplicable and non-perturbative phenomena, such as confinement and dynamical chiral symmetry breaking, emerge. One area of particular success has been precision flavour physics, where Lattice QCD input has been crucial for the interpretation of experimental results. Furthermore Lattice QCD correctly postdicted and in part predicted the low-lying hadron spectrum of QCD. State-of-the-art QCD+QED simulations nowadays also allow for a precision determination of quantities related to isospin splitting, such as the proton-neutron mass difference, which is vital for our very existence.

Nevertheless, a number of quantities could not be studied so far with the desired level of control of statistical and systematic uncertainties. Two examples of such quantities are structural properties, such as form factors and parton distribution amplitudes, of mesons and baryons, as well as calculations of the hadronic contributions to the anomalous magnetic moment (*g-2*)_{μ} of the muon, which is currently the most promising avenue in the search for physics beyond the Standard Model (SM) of particle physics: there is currently a significant discrepancy between highly precise measurements of (*g-2*)_{μ} and equally precise theoretical calculations within the SM. To resolve this discrepancy, or to confirm it to the level of an indirect discovery of new physics beyond the SM, new experiments are currently under way. To match the expected precision of these experiments, all sources of uncertainty on the theory side must be brought under control. Currently, the main uncertainties arise from the hadronic contributions, *i.e.* the contributions described by QCD, lattice predictions for which must therefore be significantly improved.

The Coordinated Lattice Simulations (CLS) effort combines the human and computational resources of several European teams in order to generate a set of lattice QCD ensembles that allow for a controlled extrapolation to the physical point (*i.e.* zero lattice spacing, physical pion mass and infinite lattice volume). For this end, it is essential to have ensembles both at fine lattice spacing and at (near-)physical pion mass, all while maintaining a large enough overall lattice volume. Simultaneously fulfilling these competing demands is very expensive computationally, so large HPC systems, such as JUQUEEN at JSC, are required.

In Figure 1, we show the landscape of CLS ensembles in terms of lattice spacing and pion mass. All ensembles satisfy the condition *m*_{π}*L>4* required for good control of finite-size effects. Highlighted as pink circles are the ensembles that have been generated (N302) or thermalized (E250) as part of this project. The pink diamond shows the ensemble (E300) which is currently in production. As can be seen from the plot, the E250 and E300 ensembles are the closest to the physical point (shown as a red dot) and will have a large impact on the extrapolation in lattice spacing and pion mass, respectively.

In Figures 2 and 3, we show how the contribution to the anomalous magnetic moment coming from the charm and strange quarks, respectively, depends on the lattice spacing and pion mass used in the simulations. In each case, the horizontal axis corresponds to the square of the pion mass, and the vertical axis corresponds to the respective contribution to (*g-2*)_{μ}, with the lattice spacing dependence shown through the colour-coding of the data points. The coloured curves indicate the pion-mass dependence at the corresponding lattice spacing, and the black curve indicates the pion-mass dependence in the continuum limit (*a=0*), with the black circle indicating the predicted result at the physical point (*a=0, m*_{π}=*m*_{π}^{phys}), where the black dashed vertical line indicates the physical pion mass *m*_{π}=*m*_{π}^{phys}. The curves have been obtained from a fit to the data points left of the red vertical dashed line only, where the pion masses are light enough for the functional form of the fit to be considered appropriate. The ensembles E250 (red, second finest lattice spacing, physical pion mass) and N302 (green, finest lattice spacing, intermediate pion mass), which were generated as part of the present project, are shown by enlarged symbols, highlighting their importance in constraining the extrapolation in the pion mass and lattice spacing, respectively. The point where the E300 ensemble currently in production would be expected to fall is highlighted by the pink diamond; it can be seen that this ensemble will further constrain the fit, making it a necessary ingredient on the path towards achieving the required sub-percent accuracy in a lattice prediction for (*g-2*)_{μ}.

Beyond their importance for predicting the hadronic contribution to the anomalous magnetic moment of the muon, the lattice QCD ensembles generated as part of this project will have a significant impact on the accuracy of lattice studies of nucleon structure, and of charmonia and the nature of the so-called XYZ resonances.

**References**

A. Gérardin, T. Harris, G. von Hippel, B. Hörz, H. Meyer, D. Mohler, K. Ottnad and H. Wittig, *The hadronic vacuum polarization contribution to (g-2) _{μ} from 2+1 flavours of O(a)-improved Wilson quarks*, to appear in:

*D. Mohler, S. Schaefer and J. Simeth, CLS 2+1 flavor simulations at physical light- and strange-quark masses*, EPJ Web Conf. 175 (2018) 02010, doi:10.1051/epjconf/201817502010; arXiv:1712.04884 [hep-lat].

**Project team **

Dalibor Djukanovic, Antoine Gérardin, Tim Harris, Ben Hörz, Georg von Hippel, Harvey B. Meyer, Daniel Mohler, Konstantin Ottnad, Hartmut Wittig

**Scientific Contact:**

PD Dr. Georg von Hippel

Institut für Kernphysik

Johannes Gutenberg-Universität Mainz

Johann-Joachim-Becher-Weg 45, D-55099 Mainz (Germany)

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

*JSC Project ID: hmz23*

*March 2019*