A Generalized False Vacuum Skyrme model

TL;DR

This work generalizes the False Vacuum Skyrme framework to any simple compact Lie group $G$ that yields a Hermitian symmetric space, introducing an $h$-matrix and a two-term energy $E=E_1+E_2$ in which $E_1$ comes from a generalized BPS Skyrme sector and $E_2$ governs a non-derivative $\,\psi$ potential plus a topological term. It proves that global minima are achieved by $h$ solving the self-duality equations and by a Skyrme field that minimizes $E_2$, reducing the problem to a Coleman false vacuum bounce, with a spherically symmetric topological density enforced via a generalized rational map ansatz. For $SU(p+q)$ with Hermitian symmetric cosets $SU(p+q)/SU(p)\, imes SU(q)\, imes U(1)$ one can construct maps yielding $F=1$, making the full theory reduce to the $SU(2)$ case up to coupling rescalings, so SU($N$) results for binding energies, radii, and densities extend to higher $N=p+q$. Numerical analyses for $G=SU(2)$ and $G=SU(3)$ demonstrate accurate predictions of baryon density, RMS radii, and binding energies, including $\Lambda$-hypernuclei, within a two-energy-scale regime where $E_1$ dominates and $E_2$ provides fine-structure corrections. Overall, the framework broadens the Skyrme paradigm to a wider class of symmetry groups, preserving key phenomenology of the SU(2) False Vacuum model while enabling applications to hypernuclei and larger groups.

Abstract

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.