To understand how the smallest known particles in our universe form structures, scientists need to use sophisticated mathematical models and techniques. These help scientists to estimate the energies of these particles, to work out how they combine and interact. In a recent paper, Dr Eef Van Beveren from the Centre for Physics of the University of Coimbra and Dr George Rupp from the Centre of Physics and Engineering of Advanced Materials of the University of Lisbon review the techniques that have led to scientific discoveries about mesons – subatomic particles that exist for tiny fractions of a second. They also discuss how such techniques may evolve into the future. More

Everything around us is made up of atoms. However, if we begin to look at matter at a sub-atomic level, we see that it is constructed of quarks. Quarks are one of the smallest particles we can break matter down into. However, quarks can only exist inside protons, neutrons, mesons, and more exotic states – a property that is appropriately called confinement. These particles come in a range of ‘flavours’, each of which has different properties, such as mass and charge. Quarks also have complementary particles, known as antiquarks, which have the same mass as their namesake quark, but opposite charge.

We can structure these quarks in different ways to form a range of particles. Mesons are a class of particles that are composed of one quark and one antiquark. However, mesons are highly unstable particles – they only exist for tiny fractions of a second.

In their article, Dr Van Beveren and Dr Rupp review the models used to probe these mesons and discover more about their structure. They do this through their consideration of meson spectroscopy.

Using spectroscopy, scientists investigate how mesons absorb and emit energy, which can tell us more about their structure. A key feature in meson spectroscopy is the scattering matrix – or the ‘S-matrix’ for short.

The S-matrix describes the initial and final states of particles undergoing a scattering process. Scattering processes occur when particles move away from the free path they would have otherwise taken, if they had not interacted with each other. Here, the particles can be scattered by electromagnetic radiation, such as light, or by collision with other particles.

All unstable particles like mesons should ideally be described using an S-matrix. An S-matrix encompasses all the information on a scattering process that leads to the temporary formation of an unstable particle, before it decays again. A meson then shows up as a resonance peak in what is called the ‘cross section’ – a measurable quantity that is directly related to the S-matrix. A fundamental mathematical property of the S-matrix is unitarity, which expresses the conservation of probability for the different modes in a scattering process.

Scientists often use the Godfrey-Isgur model, or GI model for short, to describe the structure of mesons and the forces – or ‘potentials’ – that confine the quarks. This model represents the forces between the quarks using a funnel-shaped potential. The GI model has been successful in identifying mesons composed of heavier quarks. However, the model has shortcomings in identifying mesons with lighter quarks.

In the light-quark sector, the GI model strongly overpredicts mass splittings between excited states, especially in the energy range of 1 to 2 gigaelectronvolts, or GeVs for short. Moreover, below 1 GeV, certain mesons called scalars are not even reproduced at all. In contrast, models developed by Drs Van Beveren and Rupp already predicted these mesons 40 years ago, owing to effects from S-matrix unitarity, which can give rise to extra states in the spectrum.

So, there is reason to question the funnel-shaped potential in the GI and similar models, at least for light quarks. This is because the mass dependence of the potential due to quantum effects cannot account for the observed discrepancies. Even unitarity, neglected in the GI and other models, will not be capable of compensating for the too-large predicted mass splittings.

More generally speaking, all modern models of mesons should somehow account for their instability, because the dynamical effects of strong decay can lead to large mass shifts and even to new states not present in the spectrum of the potential itself, as in the case of the scalar mesons. Ideally, this should be done in an S-matrix framework, which guarantees unitarity. Nevertheless, approximative approaches may be useful as well, in order to get a rough idea of the size of such effects. But taking such corrections due to the mesons’ instability into account is crucial to draw more decisive conclusions on the shape of the confining potential.

Several models for meson spectroscopy that one way or another account for the dynamical effects of strong decay have been presented by many authors over the years. The most recent one developed by Drs Van Beveren and Rupp is called the Resonance-Spectrum Expansion, or RSE for short. The RSE is formulated in terms of particle momenta, so is dependent on masses and velocities, in contrast to the configuration approach in their earlier work. Both formulations produce very similar results, though the latter is more intuitive and consumes considerably less computer time.

Overall, Drs Van Beveren and Rupp outline a number of developments that allow scientists to study mesons. They identify the important historical role of the GI model in trying to describe all possible mesons as combinations of the various quark flavours. However, they also outline the limitations of the funnel-shaped potential, and how this and similar models fail to account for mass shifts, decay widths, and dynamical states resulting from a unitary approach.

They demonstrate the predictive power of a simple model such as the RSE formalism, by incorporating S-matrix unitarity from the start. As all mesons are unstable and many of them are even broad resonances, unitarity must be taken into account for reliable predictions, no matter what confining potential is used. This has recently been confirmed in several cases of enigmatic mesons studied via massive computations in numerical simulations on extremely powerful computers.

Developing meson models further offers scientists a way to deepen their understanding of these particles, and may even allow us to predict more exotic mesons that could then be searched for in large-scale particle physics experiments. These models can ultimately help us learn more about the subatomic particles that make up everything around us.