ELEMENTARY PARTICLE PHYSICS

Confinement is the observation that quarks cannot be seen in isolation in nature. As a consequence the static potential V(r), which is defined as the energy of a system made of a static quark and a static anti-quark separated by a distance r, grows linearly with the separation r. When r is large enough, the potential V(r) flattens due to creation of a pair of light quarks, which combine into two static-light mesons. This so called “string breaking” phenomenon provides an intuitive example of a strong decay. It can be studied through the simulation of strong interactions between quarks and gluons on a supercomputer.

The strong interactions are described by quantum chromodynamics (QCD). The elementary particles that feel the strong force are quarks and gluons. Quarks possess a property called color charge. They cannot be observed isolated but they make up composite, color-neutral particles which are called hadrons (proton, neutron, pions, kaons, etc.). When quarks are produced in particle accelerators one sees jets of many color-neutral particles. This process is called hadronization, fragmentation, or string breaking; it is one of the least understood processes in particle physics. The mechanism responsible for the formation of hadrons is confinement. In order to study it, QCD is discretized on a Euclidean four-dimensional space-time, called the lattice. QCD on the lattice is a well-established non-perturbative approach to solving QCD, the theory of the strong force, and can be studied using Monte Carlo simulations on supercomputers.

Consider the potential between an infinitely heavy or static quark at spatial position x and a static anti-quark at position y, separated by a distance r=|y-x|. The static potential V(r) is defined as the energy of the ground state of this system. As a consequence of confinement, the energy between the quark-anti-quark pair is contained inside a color flux tube, the so-called string. As the quark-anti-quark pair is being pulled apart, the energy between them rises linearly. As soon as the energy is high enough, the gluonic string connecting the quarks breaks due to creation of a pair of light quarks, which combine with the static quarks into two static-light mesons. The potential exhibits screening after the string is broken and saturates towards twice the static-light meson mass 2E_B. String breaking is a mixing phenomenon. This means that the two states, the string-like and the two meson state, are both needed to describe the potential. After the string is broken, the two meson state is the new ground state of this system. In the neighborhood of the critical separation, the two states mix. Without mixing a plain level crossing of the two states would occur. If there is mixing, the ground state and first excited state are superpositions of the string state and the two meson state. The system undergoes an avoided level crossing, giving rise to an energy gap between the states.

String breaking has never been studied before in QCD simulated with up, down and strange quarks. The recent production of a suitable set of ensembles as part of the CLS (“Coordinated Lattice Simulations”) effort makes it a perfect time to study this phenomenon.

This project aims at computing the energy spectrum of a system containing a static quark-anti-quark pair using lattice QCD with N_f = 2 + 1 dynamical flavours. The first three energy levels of this system are determined up to and beyond the distance where it is energetically favourable for the vacuum to screen the static sources through light- or strange-quark pair creation, enabling both these screening phenomena to be observed, as shown in Figure 01 taken from [1]. This calculation provides invaluable first-principles input into models of coupled-channel scattering of heavy-light mesons and the decays of quarkonia near threshold. It can be used to shed light on the nature of recently-discovered exotic heavy-flavor hadrons.

The computationally expensive part of the calculation is the inversion of the Dirac operator, a very large sparse matrix, for example on a 192 x 64 x 64 x 64 lattice it is a 603,979,776 x 603,979,776 matrix. There are advanced highly parallelized algorithms to perform the inversion on a source vector. A smearing technique called distillation is used to create source vectors which have an optimal overlap with the physical states.

The results of the inversions (so-called sinks) are stored on disk and can be used for other projects. In [2] these quark sinks are used in a benchmark calculation of the scattering between pions, which are (in contrast to the static-light mesons) the lightest hadrons and contain light up- and down-quarks. Ref. [2] was the first such calculation to examine systematic errors due to the finite lattice volume and lattice spacing, an important step toward solid QCD calculations of these scattering amplitudes. Thanks to the re-usability of the quark sinks due to distillation, a large part of the computational effort spent for string breaking enabled the pion-pion scattering calculation in [2]. The scattering phase shift in the channel of the ρ-meson resonance is shown in the Figure 02. It was computed on the CLS J303 ensemble (192 x 64 x 64 x 64 lattice, 260 MeV pion mass, lattice spacing 0.05 fm). The Breit-Wigner fit is included and the legend shows all the irreducible representations of the reduced lattice symmetry group relevant for the ρ-resonance that were used. For a particular irreducible representation, the integer in the () is the total momentum squared in finite-volume units.