Nuclei as near BPS Skyrmions

Abstract

We study a generalization of the Skyrme model with the inclusion of a sixth-order term and a generalized mass term. We first analyze the model in a regime where the nonlinear σ and Skyrme terms are switched to zero which leads to well-behaved analytical BPS-type solutions. Adding contributions from the rotational energy, we reproduce the mass of the most abundant isotopes to rather good accuracy. These BPS-type solutions are then used to compute the contributions from the nonlinear sigma and Skyrme terms when these are switched on. We then adjust the four parameters of the model using two different procedures and find that the additional terms only represent small perturbations to the system. We finally calculate the binding energy per nucleon and compare our results with the experimental values.

Near BPS skyrmions and restricted harmonic maps

“…In fact, every hedgehog skyrme field is restricted harmonic (though stability is an open question). By contrast, the axially symmetric BPS skyrmions ϕ B used in [3,4,2,11,10] are never restricted harmonic, for |B| > 1, so there certainly exist fields in their SDiff orbits with lower E 2 , which better approximate near-BPS skyrmions in the model E BP S + c 2 E 2 . Another natural family of skyrme fields, those within the rational map ansatz, can also be shown to have no restricted harmonic members with |B| > 1.…”

“…These problems were addressed in a pair of papers by Marleau and collaborators [11,10]. In [11] the model with a certain massless potential is studied.…”

“…In particular, ϕ should minimize E 2 within all fields in its SDiff(R 3 ) orbit. It is not hard to show that, for |B| > 1, the axially symmetric BPS skyrmions ϕ B used in [3,4,2,10,11] do not have this property, and that their failure to minimize E 2 gets worse as B grows. Recall that a function ϕ : (M, g) → (N, h) between Riemannian manifolds which locally extremizes the Dirichlet energy E 2 with respect to all smooth variations is called a harmonic map.…”

“…Second, in the case M = R k , if one is (much) less ambitious, and seeks to minimize E 2 only over the orbit of the finite dimensional subgroup of SDiff(R k ) consisting of linear volume preserving diffeomorphisms, SL(k, R), the problem has an easy and neat explicit solution. This allows us to construct a better sequence of maps ϕ ′ B = ϕ B • A B by linearly deforming those used in previous analytic studies [3,4,2,11,10]. These are still not restricted harmonic, but they have much lower E 2 than ϕ B , particularly at large…”

“…In [11] the model with a certain massless potential is studied. A sequence of axially symmetric charge B BPS skyrmions is constructed which have finite E 2 and E 4 .…”