Isospin Breaking Effects in QCD

**Principal Investigator:**

Kálmán Szabó

**Affiliation:**

University of Wuppertal and Forschungszentrum Jülich, Germany

**Local Project ID:**

chfz04, pn56bu

**HPC Platform used:**

JUWELS of JSC, SuperMUC and SuperMUC-NG of LRZ

**Date published:**

**Introduction**

The standard model of particle physics describes the vast majority of experiments and observations involving basic constituents of visible matter. Any deviation from its predictions would be a sign of entirely new, fundamental physics.

A particularly important, long-standing discrepancy concerns the magnetic moment of an elementary particle called the muon. The current measurement and theoretical calculations of this property have similar precision, but disagree significantly.

To transform this disagreement into an actual discovery of new physics, an ongoing experiment at Fermilab (Batavia, USA), and one planned at J-PARC (Tokai, Japan), are aiming to reduce the measured uncertainty by a quite large factor of four.

On the theory side, the largest part of the error comes from the leading-order, hadronic vacuum polarization (LO-HVP) contribution. This term accounts for important quark, gluon and photon quantum fluctuations in the vacuum, which are described by quantum chromodynamics (QCD) and quantum electrodynamics (QED). Up until now, the most precise computations of this contribution have been performed using dispersion relation based phenomenological analyses, with input from electron-positron annihilation experiments.

Here we present a completely independent, ab-initio computation of the LO-HVP term, in which the equations of QCD and QED are solved using supercomputers and state-of-the-art numerical techniques. We reach a precision similar to that of the phenomenological approach for the first time.

Surprisingly, our result leads to a standard model prediction for the muon’s magnetic moment that is in agreement with the current experimental measurement, suggesting that new physics is not needed to explain this measurement at present levels of precision.

The methods developed here will be useful to continue improving the accuracy of the standard model prediction, as will be required to pursue the search for new physics in ongoing and future experiments designed to measure the magnetic moment of the muon.

**Isospin breaking effects**

The ab initio calculation means in the above context lattice quantum field theory. A space-time grid is introduced and and at every point of it the time evolution of various quantum operators are determined (to be more specific a path integral formalism is applied to that end). In some sense it reminds us to meteorology. Usually, people also introduce a three-dimensional grid, temperatures, pressures and wind velocities are measured and using the underlying equations the time evolution is determined. In both cases it is a heroic effort.

Let us come back to the magnetic moment of the muon. Obviously, the most important goal is to reach an accuracy which is compatible with the expected experimental errors. Only reaching this accuracy guarantees that the experimental findings of several hundred million dollars are fully utilized and only with this accuracy can we decide if and what sort of new physics is there. When we speak about precision a sub-percent error is needed. Electromagnetic and strong isospin breaking effects are on the percent level. Thus any reasonable result needs the inclusion of these effects. This is a very hard task.

The electromagnetic interaction is weak and long-ranged, whereas the strong interaction is strong and short-ranged. Keeping both of them in a system is more than just challenging.

**Proton-neutron mass difference**

In 2015, we published a paper in Science [1], in which we included both theories in order to determine the mass splittings between the proton and the neutron (and for many similar hadron pairs). The building blocks of these nucleons are up and down quarks. Most of the lattice QCD calculations are using the same very small mass for both of them, which is a very good approximation in most cases (no strong isospin breaking scenario).

In this isospin symmetric case without QED the protons and the neutrons have the same mass, they are mass degenerate. However, if one introduces electromagnetism (QED) the proton becomes slightly heavier than the neutron. The world, as we know it today, would not exist. No stars, no people, not even atoms. This is in full contradiction with observations.

One has to introduce strong isospin breaking, too. These two extensions of the isospin symmetric case are contributing similar amounts, but there is a huge cancellation between them. In our 2015 paper, we have shown how to include both effects. This also had a deep theoretical outcome. It was shown how to use renormalization theory simultaneously for QCD and QED. The calculation was fundamental and expensive. The main result, the isospin splittings for various baryons, is shown in Figure 1.

This figure is shown in many textbooks and year by year belongs to the 0.3% best cited lattice papers ever written. Nobel Prize Laureate Frank Wilczek writes about this work “This progress encourages us to predict a future in which nuclear physics reaches the level of precision and versatility that atomic physics has already achieved, with vast implications for astrophysics, and conceivably for technology. We can look forward to much more accurate modelling of supernovae and neutron stars than has so far been possible, and entertain dreams of refined nuclear chemistry, enabling, for example, dense energy storage and ultrahigh-energy lasers.”

**The muon’s magnetic moment**

Indeed, a similar level of precision is needed to decide if there is a new physics signal in the magnetic moment of the muon. After we have introduced full dynamical QED into our work, we were using it in a step-by-step procedure in our 2020 paper on the muon [2]. Electromagnetism and isospin breaking is introduced and controlled in a Taylor-series manner.

This approach made it possible to shuffle around with the CPU demands in a way that the CPU was invested in those parts of the calculation which minimized the overall error. Surprisingly, our results are in agreement with the experimental value for the magnetic moment of the muon and the long-standing discrepancy seems to have disappeared. Clearly, this work should be cross-checked and repeated by other groups.

After many years of hard work and with the inclusion of strong isospin breaking and electromagnetism, we reached a sub-percent accuracy, comparable to the experimental errors on the muon’s magnetic moment. We are waiting for the new experimental results from Fermilab (USA), which should come out in a few months, to see if there is new physics beyond the standard model.

**References and Links**

[1] Borsanyi et al, Science 347, 14521455 (2015).

[2] Borsanyi et al, arxiv:2002.12347 (2020).

**Research Team**

Sz. Borsanyi^{1}, Z. Fodor^{1,2,3,4,5}, J. N. Guenther^{6}, C. Hoelbling^{1}, S. D. Katz^{3}, L. Lellouch^{7}, T. Lippert^{1,2}, K. Miura^{7,8,9}, L. Parato^{7}, F. Stokes^{2}, Kalman K. Szabo(PI)^{1,2}, B. C. Toth^{1}, Cs. Torok^{2}, L. Varnhorst^{1}

^{1}Department of Physics, University of Wuppertal, Germany^{2}Jülich Supercomputing Centre, Forschungszentrum Jülich, Germany^{3}Institute for Theoretical Physics, Eötvös University, Budapest, Hungary,^{4}University of California, San Diego, USA^{5}Pennsylvania State University, Department of Physics, USA, 6Department of Physics, University of Regensburg,^{7}Aix Marseille Univ, Université de Toulon, CNRS, CPT, Marseille, France^{8}Helmholtz Institute Mainz, Germany^{9}Kobayashi-Maskawa Institute, Nagoya University, Japan

**Scientific Contact**

Prof. Dr. Kálmán Szabó

Forschungszentrum Jülich GmbH

Institute for Advanced Simulation (IAS), Jülich Supercomputing Centre (JSC)

Wilhelm-Johnen-Straße, D-52425 Jülich (Germany)

e-mail: szaboka [@] general.elte.hu

**NOTE: **This report was first published in the book "High Performance Computing in Science and Engineering – Garching/Munich 2020 (2021)" (ISBN 978-3-9816675-4-7)

*Local project ID: chfz04 (JSC) and pn56bu (LRZ)*

*September 2021*