# The Search for the *H* Dibaryon in Lattice QCD

**Principal Investigator:** Hartmut Wittig, Institute for Nuclear Physics and PRISMA Cluster of Excellence, Johannes Gutenberg University of Mainz (Germany)

**HPC Platform:** JUQUEEN of JSC

**Date published: **October 2018

**Project-ID @ JSC:** hmz21

The familiar protons and neutrons consist of different combinations of “up” (*u*) and “down” (*d*) quarks. There is, however, a very rich and complex spectrum of similar particles (collectively referred to as hadrons) involving also heavier variants of quarks, such as the “strange” (*s*), “charm” (*c*) or “bottom” (*b*) quarks. One particular example is the Λ-baryon with quark content *uds*. In the same way that protons and neutrons form the known nuclei that are listed in the chart of nuclides, it is conceivable that nuclei exist in which one (or more) nucleons are replaced by a Λ-baryon. The first such hypernucleus has indeed been discovered in 1952, and a great effort has since been invested to investigate and understand these systems in detail. In 1977, the theorist Bob Jaffe argued that a bound state consisting of two Λ-baryons – the so-called *H* dibaryon – may exist. Since his findings were obtained using a simple theoretical model, it is important to establish whether the underlying theory of QCD predicts the existence of such a state. This is the main focus of our project.

The question of the existence of the *H* dibaryon is all the more challenging, since experimental searches have remained largely inconclusive. Studying the *H* dibaryon also offers the opportunity to understand, in terms of the underlying fundamental theory of QCD, the interaction between hadrons, which is ultimately responsible for the formation of nuclear matter.

To study the *H* dibaryon we employ Lattice QCD, i.e. the formulation of the theory on a four-dimensional, Euclidean space-time grid. In this way the strongly coupled theory of QCD is amenable to a numerical treatment based on Monte Carlo integration. Lattice QCD has been used successfully to determine the masses of low-lying hadrons from the exponential fall-off rate of correlation functions. However, when applied to the *H* dibaryon, one is faced with a major challenge: The correlation function describing a multi-hadron state such as the *H* dibaryon shows a very unfavourable signal-to-noise ratio. This implies that huge statistics are required to resolve the energy levels reliably. This is all the more important since the binding energy (i.e. the energy difference between the *H* dibaryon and a pair of Λ-baryons) is of the order of a few MeV and thus quite small.

Our initial study was performed for QCD with a mass-degenerate doublet of dynamical up and down quarks. In a first step we have studied the energy spectrum in a situation where the strange quark is mass-degenerate with the up and down quarks. In order to achieve high statistics at moderate numerical cost, we have employed the technique of “all-mode-averaging”. In addition, we have used a method known as “distillation” to construct correlator matrices for the *H* dibaryon, which allow for a reliable and efficient determination of the energy levels of the ground state and the first few excitations. Another numerical challenge is the effort associated with performing the large number of necessary contractions to form the various entries in the correlator matrix.

The requirement to perform a high-statistics calculation, comprising up to 100000 individual samples per ensemble, together with the numerically demanding tasks associated with the construction of the correlator matrices implies that the use of a powerful supercomputer is indispensable. The JUQUEEN HPC system at JSC is ideally suited for running the highly parallelised and optimised code that we have developed.

*Fig. 01: Effective masses for various six-quark states grouped according to the quark flavour symmetry. The *H *dibaryon corresponds to the lowest-lying state. Its energy is below that of two individual Λ-baryons, represented by the horizontal band, indicating the existence of a bound state, in accordance with the original prediction. (©) University of Mainz*

In Figure 1 we plot the effective mass against the Euclidean time separation. The domination of the ground state manifests itself as a plateau. The energy level of the *H* dibaryon (labelled by ‘singlet’ in the legend) lies below the one that corresponds to a pair of Λ-baryons, which provides first evidence that the *H* dibaryon is energetically favoured, thus confirming Jaffe's conjecture.

In Figure 2 we show the estimator of the energy difference in a finite volume between the *H* dibaryon and two non-interacting Λ-baryons, computed using the “distillation” technique and compare it to the results obtained using the conventional "smearing" technique. It is evident that distillation is far superior, as the results are statistically much more precise and stable.

*Fig 02: Estimates for the binding energy in a finite volume of the *H *dibaryon computed using our preferred method of distillation (shown in green), compared to the conventional method (open blue circles).(©) University of Mainz*

In order to account for the effects of performing these calculations in a finite volume and thus to provide a more realistic determination of the binding energy we have – for the first time in the case of a two-baryon system – employed Lüscher's finite-volume quantisation condition. This elegant and sophisticated formalism allows for the calculation of the scattering phase shifts of two interacting hadrons and, ultimately, for the determination of the binding energy from the pole of the scattering amplitude. This is shown in Figure 3 where the phase shift is plotted versus the scattering momentum. The pole of the scattering amplitude is then determined as the intersection of the phase shift with the dashed brown curve. From this analysis we extract a binding energy of 19 ± 10 MeV. While this provides strong evidence for the existence of a *H* dibaryon, our estimate of the binding energy is much smaller than Jaffe's original analysis.

*Fig 03: Determination of the binding energy via Lüscher's finite-volume quantisation condition: The intersection of the computed scattering phase shift, shown as the grey band, with the dashed curve defines the pole of the scattering amplitude, which corresponds to the binding energy. (©) University of Mainz*

It should be realised that our estimate extracted in this first study is valid for the theory with degenerate *u*,* d* and *s* quarks, whose masses deviate significantly from the physical situation. In our ongoing study we have started to trace out the mass dependence of the binding energy as the quark masses are tuned towards their physical values. First preliminary results have been presented at the international conference on lattice field theory, “Lattice 2018”.

**References:**

J. Green, A. Francis, P. Junnarkar, C. Miao, T. Rae and H. Wittig, Search for a bound H-dibaryon using local six-quark interpolating operators, PoS LATTICE 2014 (2014) 107 doi:10.22323/1.214.0107, arXiv:1411.1643 [hep-lat]. https://pos.sissa.it/214/107

P. Junnarkar, A. Francis, J. Green, C. Miao, T. Rae and H. Wittig, Search for the H-Dibaryon in two flavor Lattice QCD, PoS LATTICE 2015 (2016) 082 doi:10.22323/1.253.0079, arXiv:1511.01849 [hep-lat]. https://pos.sissa.it/253/079

A. Francis, J.R. Green, P.M. Junnarkar, C. Miao, T.D. Rae and H. Wittig, Lattice QCD study of the H dibaryon using hexaquark and two-baryon interpolators, arXiv:1805.03966 [hep-lat]. https://arxiv.org/abs/1805.03966

R.L. Jaffe, Perhaps a stable dihyperon, Phys. Rev. Lett. 38 (1977) 195 http://dx.doi.org/10.1103/PhysRevLett.38.195

T. Blum, T. Izubuchi and E. Shintani, New class of variance-reduction techniques using lattice symmetries, Phys. Rev. D88 (2013) 094503, http://dx.doi.org/10.1103/PhysRevD.88.094503

Hadron Spectrum Collaboration (M. Peardon et al.), A Novel quark-field creation operator construction for hadronic physics in lattice QCD, Phys. Rev. D80 (2009) 054506, http://dx.doi.org/10.1103/PhysRevD.80.054506

M. Lüscher, Two-particle states on a torus and their relation to the scattering matrix, Nucl. Phys. B354 (1991) 531, http://dx.doi.org/10.1016/0550-3213(91)90366-6

**Scientific Contact:**

Professor Dr. Hartmut Wittig

Institute of Nuclear Physics

Johannes Gutenberg-Universität Mainz

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

wwwth.kph.uni-mainz.de/wittig-hartmut

e-mail: hartmut.wittig (AT) uni-mainz.de

*October 2018 *

*Project ID: hmz21*