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Baryon chemical potential and in-medium properties of BPS skyrmions

We continue the investigation of thermodynamical properties of the BPS Skyrme model. In particular, we analytically compute the baryon chemical potential both in the full field theory and in a mean-field approximation. In the full field theory case, we find that the baryon chemical potential is always exactly proportional to the baryon density, for arbitrary solutions. We further find that, in the mean-field approximation, the BPS Skyrme model approaches the Walecka model in the limit of high density - their thermodynamical functions as well as the equation of state agree in this limit. This fact allows to read off some properties of the $ω$-meson from the BPS Skyrme action, even though the latter model is entirely based on the (pionic) $SU(2)$ Skyrme field. On the other hand, at low densities, at the order of the usual nuclear matter density, the equations of state of the two models are no longer universal, such that a comparison depends on some model details. Still, also the BPS Skyrme model gives rise to nuclear saturation in this regime, leading, in fact, to an exact balance between repulsive and attractive forces. The perfect fluid aspects of the BPS Skyrme model, which, together with its BPS properties, form the base of our results, are shown to be in close formal analogy with the Eulerian formulation of relativistic fluid dynamics. Within this analogy, the BPS Skyrme model, in general, corresponds to a non-barotropic perfect fluid.