A Generalized False Vacuum Skyrme model

We propose a generalization of the False Vacuum Skyrme model for any simple compact Lie groups $G$ that leads to Hermitian symmetric spaces. The Skyrme field that in the original theory takes its values in $SU(2)$ Lie group, now takes its values in $G$, while the remaining fields correspond to the entries of a symmetric, positive, and invertible $\dim G \times \dim G$-dimensional matrix $h$. This model is also an extension of the generalized BPS Skyrme model. We prove that the global minima correspond to the $h$ fields being self-dual solutions of the generalized BPS Skyrme model, together with a particular field configuration for the Skyrme field that leads to a spherically symmetric topological charge density. As in the case of the original model, the minimization of the energy leads to the so-called reduced problem, defined in the context of false vacuum decay. This imposes a condition on the Skyrme field, which, if satisfied, makes the total energy of the global minima, as well as the main properties of the model, equivalent to those obtained for the $G=SU(2)$ case. We study this condition and its consequences within the generalized rational map ansatz and show how it can be satisfied for $G=SU(p+q)$, where $p$ and $q$ are positive integers, with the Hermitian symmetric spaces being $SU(p+q)/SU(p) \otimes SU(q) \otimes U(1)$. In such a case, the model is completely equivalent to the $G=SU(2)$ False Vacuum Skyrme model, independent of the values of $p$ and $q$. We also provide a numerical study of the baryon density, RMS radius, and binding energy per nucleon, which deepens the previous analysis conducted for the $SU(2)$ False Vacuum Skyrme model. Additionaly, in the case of $G = SU(3)$, we have studied the application of our model to the description of the binding energies and masses of the $Λ$-hypernuclei.

💡 Research Summary

This paper presents a significant generalization of the False Vacuum Skyrme model, extending it from its original formulation based on the SU(2) group to any simple compact Lie group G that leads to a Hermitian symmetric space. The model describes baryons as topological solitons, with the topological charge Q interpreted as the baryon number. In the generalized framework, the Skyrme field U takes values in the group G, while the model incorporates additional scalar fields encoded in a symmetric, positive-definite, and invertible matrix h of dimension dim G × dim G. This construction also serves as an extension of the previously established generalized BPS Skyrme model.

The authors’ central achievement is proving that the global minima of the total energy functional, defined as the sum of the BPS-like term E1 and an extension term E2, correspond to specific field configurations. These configurations require the h fields to be self-dual solutions of the generalized BPS Skyrme model, simultaneously with the Skyrme field being the minimizer of the E2 term alone. The term E2 includes kinetic and potential terms that are functions of the topological charge density Q, plus a topological term approximating the Coulomb interaction. This minimization process leads to a “reduced problem” analogous to Coleman’s false vacuum decay in quantum field theory. A critical consequence of this reduction is that it imposes a condition on the Skyrme field: the topological charge density Q must be spherically symmetric.

The paper investigates this spherical symmetry condition within the context of the generalized rational map ansatz for the Skyrme field. A key finding is that for groups of the form G = SU(p+q), corresponding to Hermitian symmetric spaces SU(p+q)/SU(p)⊗SU(q)⊗U(1), a special Skyrme field configuration can be explicitly constructed that satisfies the condition. Remarkably, in this case, the model becomes completely equivalent to the original SU(2) False Vacuum Skyrme model, independent of the positive integer values of p and q. This equivalence implies that all primary features and numerical results derived for the SU(2) case can be directly carried over to the SU(N) generalization.

The work includes a detailed numerical analysis for the SU(2) case, deepening prior studies by examining baryon density profiles, root-mean-square (RMS) radii, and binding energy per nucleon across a range of baryon numbers. Furthermore, the paper explores a concrete application of the generalized model for G = SU(3). Leveraging the three-flavor symmetry of SU(3), the authors investigate the model’s potential to describe the binding energies and masses of Λ-hypernuclei, which are nuclei containing one or more strange hyperons. They note that at the classical level, this approach is most reliable for describing very heavy hypernuclei, where the mass per baryon varies slowly with the baryon number A.

In summary, this research provides a robust mathematical generalization of a key Skyrme-type model, proves the equivalence of a large class (SU(p+q)) to the foundational SU(2) version, and opens new avenues for applying the model to systems involving hyperons, thereby broadening the scope of Skyrme model-based nuclear physics.