Hadron Scattering and Resonance Properties from Lattice QCD

**Principal Investigator:**

Prof. Carsten Urbach

**Affiliation:**

Helmholtz-Institut für Strahlen- und Kernphysik (Theorie) and Bethe Center for Theoretical Physics, Universität Bonn, Germany

**Local Project ID:**

chbn28

**HPC Platform used:**

JUWELS/JURECA/JUQUEEN at JSC

**Date published:**

It is a long lasting dream in nuclear physics to study nuclei like, for instance, carbon directly from Quantum Chromodynamics (QCD), the underlying fundamental theory of strong interactions. Such a theoretical investigation from first principles is difficult for several reasons: first, QCD describes a strong interaction for which approximate methods based on perturbation theory break down. By contrast, in lattice QCD the problem can be treated from first principles non-perturbatively. To do so, space-time is discretised in a regular lattice and the interactions of the underlying degrees of freedom, quarks and gluons, are determined by numerical simulations of the so-called path integral.

Second, while nuclei can be described reasonably well as bound states of protons and neutrons, these themselves consist of three quarks each. The computational complexity in lattice QCD is proportional to the factorial of the number of involved quarks. Thus, a nucleus with more than five protons and neutrons, i.e. more than 15 quarks represents a major challenge. This challenge requires the usage of most modern supercomputer resources available for instance at the Jülich Supercomputing Centre (JSC). Third, bound states like nuclei can be studied in lattice QCD only indirectly. This indirect approach is named Lüscher

method and can be understood as follows: imagine two, for simplicity fully equal particles in a box with finite edge length L. If this length L is much larger than the typical interaction range of the two particles one expects little interaction between them. Any measurement of the two particle system will, hence, yield twice what one measures for a single particle. As soon as L becomes close to the interaction range, i.e. the particles feel each other, one expects small modifications compared to the non-interacting case. It turns out that these deviations are directly related to the interaction properties of the two particles. For three particles the principle is similar, but the investigation of the corresponding two particle system is required as a prerequisite. Here the additional challenge is to disentangle those small modifications from the background.

Scientists of the Rheinische Friedrich-Wilhelms-Universität Bonn together with collaborators from the Extended Twisted Mass Collaboration (ETMC) have been able to investigate various two-meson, meson-baryon and three-meson systems. In particular, pion-pion, pion-kaon and kaon-kaon systems could be studied with focus also a careful evaluation of the relevant uncertainties, leading also to a number of important publications. As one of the highlights, the resources available at Jülich Supercomputer Centre made it possible to study two- and three-pion systems at physical pion mass value (dubbed the physical point).

Since this was a longer running compute project, the JSC supercomputing resources used for reaching this result are JUQUEEN, JURECA and JUWELS, including also JUWELS Booster. The scientists had to develop highly optimised software to make optimal use of these precious resources. On the one hand, this involved adapting and making use of the

GPU-accelerated QUDA lattice QCD library and its highly efficient solvers [3] leading to reductions in computing time requirements by orders of magnitude at the physical point. On the other hand, the factorial growth of complexity described above was tackled through the factorisation and task-based parallelisation of the resulting expressions as well as the usage

symmetries and the caching of common subexpressions wherever possible. The generated data can be used for other projects, too, and is still being anlysed.

In the figure results for the isospin-2 pion scattering length a<sub>0</sub> multiplied with the mass of the pion M<sub>π</sub> as a function of the mass of the pion divided by its decay constant f<sub>π</sub> obtained in 2015 with the latest results obtained in 2021 are compared. The latest results [1] (orange triangles) make the long extrapolation to the physical point using chiral perturbation theory (ChPT, dashed line) superfluous, which was still mandatory in 2015 [2]. The 2021 result is also fully compatible with experimental results.

This result represents on the one hand a significant technical advancement as it proves the feasibility of the investigation of three hadron systems at the physical point, therefore allowing for a direct comparison with experimental results. On the other hand the result further fosters our confidence in QCD as a powerful theory to describe the strong interactions in nature.

[1] M. Fischer et al. (ETMC), Eur.Phys.J.C 81 (2021) 5, 436

[2] C. Helmes et al. (ETMC), JHEP 09 (2015) 109

[3] B. Kostrzewa et al. (ETMC), PoS LATTICE2022 (2023) 340