Hadron Structure Observables on a Fine Lattice at the Physical Point

**Principal Investigator:**

Sara Collins

**Affiliation:**

Institute for Theoretical Physics, University of Regensburg

**Local Project ID:**

pn34xo

**HPC Platform used:**

SuperMUC-NG of LRZ

**Date published:**

**Introduction**

Protons and neutrons, known collectively as nucleons, are the building blocks of visible matter. Since their discovery in alpha particle scattering experiments in 1919 by Rutherford and in 1932 by Chadwick, for the case of the proton and neutron, respectively, their basic properties have been mapped out very accurately. For example, their masses and spins are known to a precision of at least one part in 10^{-8}. However, precise information on their internal structure has not yet been provided.

Thanks to deep inelastic scattering (DIS) experiments at SLAC in the USA in the late 1960s and others we know that nucleons are comprised of fundamental constituents, quarks and gluons. A very simple picture of the nucleons is one where they are made up of three light (u and d) quarks, uud (the proton) and udd (the nucleon). However, comparing the masses of the quarks (of the order of a few mega-electron volts (MeV)) with that of the nucleons (of the order of a thousand MeV) indicates that the internal structure of nucleons is far more complex and, for example, the gluon degrees of freedom play an important role. Similarly, early deep inelastic muon-nucleon scattering experiments showed that the spin of the u and d quarks may contribute as little as 20% to the spin of the nucleon, starting the so-called spin “crisis” or “puzzle”.

Extracting precise information on nucleon structure, how the quark and gluon constituents account for the properties of nucleons is difficult due to the nature of the strong interaction between them. At large distances (or equivalently low energies), the strong force increases such that the quarks and gluons are always confined to "hadronic" bound states. At present, the only first principles approach to calculating hadronic properties, i.e. using the theory of the strong interaction, Quantum Chromodynamics (QCD), without any additional assumptions, is via large scale numerical simulations (lattice QCD). Predictions of nucleon properties are a necessary input for interpreting the many experiments which use nucleons (or nuclei) as probes or targets. These experiments are searching for evidence of the physics that lies beyond our current understanding of particle physics (that is encapsulated in the Standard Model).

The Standard Model has a number of limitations, most notably that it only involves three of the four forces of nature and does not explain the origin of dark matter and energy. Astrophysical evidence suggests that dark matter (which does not interact electromagnetically and so is difficult to detect) comprises 85% of the total mass of the Universe. This project is concerned with computing nucleon properties relevant for experiments aiming to directly detect dark matter particles, for those experiments investigating the poorly understood neutrino sector of the Standard Model and for precision experiments searching for signals of new interactions in beta decay.

**Results and Methods**

In order to calculate the properties of nucleons, QCD is formulated on a space-time Euclidean grid, with a finite lattice spacing (a) and volume (V). Representative (gauge) configurations of the gluon and quark fields are generated by hybrid Monte Carlo simulation and the quantities of interest can be extracted (after a statistical average) from``correlation'' functions which are computed on top of the gauge configurations. The simulations must be repeated for several values of the lattice spacing to enable the continuum limit (a to zero) to be performed. Similarly, finite (spatial) volume effects must be explored. A large part of the computational cost is due to inverting large matrices of size *12Vx12V* (where *V* is of the order of 64^{3}x192) that are related to the propagation of quarks across the lattice. The condition number of these matrices increases inversely proportional to the mass of the light (u and d) quarks and for this reason unphysically large light quark masses are often employed in the simulations. This necessitates an additional extrapolation to the physical light quark mass value. Note that the light quark mass is proportional to the pion mass squared and this is used when displaying the light quark mass dependence of physical quantities.

This project is part of a larger analysis involving the computation of correlation functions relevant for determining nucleon structure observables. A unique feature of our calculation is that we use configurations generated with open boundary conditions which enable lattice spacings as low as 0.04 fm to be realized while maintaining ergodicity in the simulation [1]. We vary the light quark mass such that the physical point is approached along two trajectories with an additional trajectory where the light and strange quark masses are equal. This project completes the analysis by computing correlation functions on a physical quark mass ensemble with a fine lattice spacing of 0.06 fm. We are still in the process of analysing much of the data, however, first results are presented in Figs. 1 to 3. The masses of the Ξ, Λ and Σ baryons have been computed in addition to that of the nucleon. We are able to precisely fix the lattice spacing using the Ξ mass, with the other masses at the physical point being predictions. Note that the data points generated as part of this project (directly at the physical point) are consistent with the experimental results (filled circles in Fig. 1). The nucleon sigma term, given by the slope of the nucleon mass with respect to the light quark mass will be extracted from this analysis. The sigma term is required to compute dark matter-nucleon scattering cross-sections in direct dark matter detection experiments.

Figures 2 and 3 display the axial and induced pseudoscalar form factors of the nucleon after extrapolation to the continuum limit and to physical light quark mass. The axial form factor is a crucial input to Monte-Carlo generators which determine the energy distributions of neutrinos in long baseline neutrino oscillation experiments. The two form factors are linked via symmetry relations and our analysis is one of the first to show the lattice results are consistent with these relations and also agreement with the experimental results for the induced form factor at one particular point where this exists.

The data were generated on SuperMUC-NG using batch jobs involving 432 nodes with 48 tasks per node. While we compute many measurements per configuration the data is stored in HDF5 format and only around 1,000 files are generated, which were stored on the project WORK directory. As mentioned above, the main computational expense is the inversion of the large sparse Dirac matrices for which we used the locally deflated domain decomposition solver of openQCD [3].

**Ongoing Research / Outlook**

The data generated during the project are still being analysed. Final results, incorporating also earlier simulations, will include the contribution of the light and strange quark spin to the spin of the nucleon (needed to understand the spin puzzle) and the scalar and tensor charges which are used to provide bounds on the coupling strength of beyond the Standard Model interactions contributing to beta decay. © University of Regensburg

**References**

[1] M. Bruno et al., JHEP 02 (2015) 043.

[2] Particle Data Group collaboration, M. Tanabashi et al., Phys. Rev. D98 (2018) 030001.

[3] http://luscher.web.cern.ch/luscher/openQCD/

[4] RQCD collaboration, G. Bali et al., JHEP 05 (2020) 126.

**Research Team**

Gunnar Bali, Marius Löffler, Andreas Schäfer, Jakob Simeth, Wolfgang Söldner, Simon Weishäupl, Thomas Wurm (all: University of Regensburg)

**Scientific Contact**

Dr. Sara Collins

Institut für theoretische Physik

Universität Regensburg D-93040 Regensburg (Germany)

e-mail: sara.collins [@] ur.de

**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: pn34xo*

*August 2021*