We construct a modified non-Bogomol’nyi-Prasad-Sommerfield sine-Gordon theory which supports stable static kinks of arbitrary topological degree <math><mi>N</mi></math>. We use this toy model to study problems that are interesting for higher-dimensional soliton theories supporting multisolitons. We construct a 2-kink collective coordinate model and use it to generate scattering trajectories, which are compared to full-field dynamics. We find that the approximation works well, but starts to fail as radiation becomes more important, due to our model becoming less Bogomol’nyi-Prasad-Sommerfield or when the initial kink velocities are large. We also construct the quantum 2-kink using various approximations and consider how these quantum corrections affect the binding energy of the 2-kink.

Sweden

Chris Halcrow

India

Renjan Rajan John

Anusree N.

Modified sine-Gordon theory with static multikinks

We provide a step-by-step method to construct skyrmions from instanton ADHM data, including when the exact ADHM data is unknown. The configurations look like clusters of smaller skyrmions, and can be used to build manifolds of skyrmions with or without symmetries. Nuclei are described by quantum states on these manifolds. We describe the construction and quantization procedure generally, then apply the methods in detail to the 8-skyrmion which describes the Beryllium-8 nucleus.

United Kingdom

Josh Cork

Quantization of skyrmions using instantons

A major problem in the Skyrme model is that the binding energy of skyrmions, which model nuclei, is too high by an order of magnitude. We show that the most popular solution to this problem, to construct models with zero classical binding energy, still produces large binding energies when spin energy is included. We argue that it is thus necessary to include quantum effects. We calculate the binding energy of skyrmions including the most simple quantum correction, that of vibrational modes in a harmonic approximation. We show that this can give physically reasonable results for nucleon numbers <math><mi>N</mi><mo>=</mo><mn>1</mn></math>-8 thanks to a remarkable cancellation between the strongly binding classical energy and the strongly unbinding zero-point energy from vibrational modes.

China

Sven Bjarke Gudnason

Quantum binding energies in the Skyrme model

We derive the nucleon-nucleon interaction from the Skyrme model using the instanton and product approximations to skyrmion dynamics. In doing so, we also calculate the classical potential and metric for skyrmion dynamics in each of the approximations. This is the first time they have been compared in detail and the results show major disagreements between the approximations. We derive the eight low-energy nucleon-nucleon interaction potentials and compare them with the Paris model. For the instanton approximation we find strong negative isoscalar and isovector spin-orbit potentials, matching phenomenological models and our geometric intuition. Results for the other potentials are mixed, in part due to the zero pion mass limit used in this approximation.

Derek Harland

Nucleon-nucleon potential from instanton holonomies

Ireland

Corrigendum to “Nucleon-nucleon potential from skyrmion dipole interactions” [Nucl. Phys. B 967 (2021) 115430]

We study static solutions of the standard Skyrme model with a pion mass. Using approximately 105 pseudo-random initial configurations made of single Skyrmions in the non-symmetrized product Ansatz and an automatic detection of repeated solutions, we find 409 local energy minimizers (Skyrmions) of the model with baryon numbers 1 through 16, of which 383 are new. In particular, we find new solutions for baryon numbers 5, 8, 9, 10, 11, 12, 13, 14, 15, and 16. Our results for the number of solutions per baryon number suggest that this number could grow either polynomially or exponentially. We identify new families of solutions: sheets of Skyrmions in synchronized and antisynchronized hexagonal layers (which we call graphene); chains of 2- and 3-tori; chain-like solutions containing a hinge and many clustered Skyrmions. Contrary to common lore, only the B = 12 global energy minimizer is made of alpha particles or some chunk of a cubic crystal, whereas the B = 9, 11, 14, 15 minimizers contain the B = 7 icosahedrally symmetric Skyrmion as a component. The B = 10, 13, 16 are symmetric graphene-like solutions. We find B = 5 and B = 8 minimizers with numerically indistinguishable energies. The B = 8 candidates are the chain of two cubes, which is a chunk of the cubic Skyrme crystal and the fullerene-type ball found originally by the rational map approximation. The B = 5 global minimizer is either the well-known D 2d  symmetric fullerene or a new C 2v  symmetric solution. Finally, our findings show a large number of solutions have no discrete symmetries or just one symmetry, contrary to the common lore that Skyrmions are highly symmetric configurations.

Sven Gudnason

A Smörgåsbord of Skyrmions

A spectral wall is a surface in a moduli space of classically BPS (Bogomol’nyi–Prasad–Sommerfield) solitons where an internal excitation crosses the continuum mass threshold. It has recently been shown that spectral walls in classical field theory repel solitons whose corresponding internal excitations are excited. We investigate, for the first time, the fate of spectral walls in a quantum theory, calculating the instantaneous acceleration of a wave packet of classically BPS antikinks in the presence of an impurity. We find that, in the quantum theory, the antikinks are repelled even when the bound mode is not excited. Perhaps due to this repulsive force, the dramatic classical effects of the spectral wall are not seen in our quantum calculation.

Jarah Evslin

Poland

Tomasz Romańczukiewicz

Andrzej Wereszczyński

Spectral walls at one loop

To study a nuclear system in the Skyrme model one must first construct a space of low energy Skyrme configurations. However, there is no mathematical definition of this configuration space and there is not even consensus on its fundamental properties, such as its dimension. Here, we propose that the full instanton moduli space can be used to construct a consistent skyrmion configuration space, provided that the Skyrme model is coupled to a vector meson which we identify with the ρ-meson. Each instanton generates a unique skyrmion and we reinterpret the 8N instanton moduli as physical degrees of freedom in the Skyrme model. In this picture a single skyrmion has six zero modes and two non-zero modes: one controls the overall scale of the solution and one the energy of the ρ-meson field. We study the N = 1 and N = 2 systems in detail. Two interacting skyrmions can excite the ρ through scattering, suggesting that the ρ and Skyrme fields are intrinsically linked. Our proposal is the first consistent manifold description of the two-skyrmion configuration space. The method can also be generalised to higher N and thus provides a general framework to study any skyrmion configuration space.

Thomas Winyard

A consistent two-skyrmion configuration space from instantons

We consider the quantum multisoliton scattering problem. For BPS theories one truncates the full field theory to the moduli space, a finite dimensional manifold of energy minimising field configurations, and studies the quantum mechanical problem on this. Non-BPS theories — the generic case — have no such obvious truncation. We define a quantum soliton scattering manifold as a configuration space which satisfies asymptotic completeness and respects the underlying classical dynamics of slow moving solitons. Having done this, we present a new method to construct such manifolds. In the BPS case the dimension of the n-soliton moduli space ℳ n  is n multiplied by the dimension of ℳ1. We show that this scaling is not necessarily valid for scattering manifolds in non-BPS theories, and argue that it is false for the Skyrme and baby-Skyrme models. In these models, we show that a relative phase difference can generate a relative size difference during a soliton collision. Asymptotically, these are zero and non-zero modes respectively and this new mechanism softens the dichotomy between such modes. Using this discovery, we then show that all previous truncations of the 2-Skyrmion configuration space are unsuitable for the quantum scattering problem as they have the wrong dimension. This gives credence to recent numerical work which suggests that the low-energy configuration space is 14- dimensional (rather than 12-dimensional, as previously thought). We suggest some ways to construct a suitable manifold for the 2-Skyrmion problem, and discuss applications of our new definition and construction for general soliton theories.

Quantum soliton scattering manifolds

We derive the nucleon-nucleon isoscalar spin-orbit potential from the Skyrme model and find good agreement with the Paris potential. This solves a problem that has been open for more than 30 years and gives a new geometric understanding of the spin-orbit force. Our calculation is based on the dipole approximation to skyrmion dynamics and higher order perturbation theory.