The role of the Callan–Witten anomaly density as a Chern–Simons term in Skyrme model^*

Abstract

We consider axially symmetric solutions of the U(1) gauged Skyrme model supplemented with a Callan-Witten (CW) anomaly density term. The main properties of the solutions are studied, several specific features introduced by the presence of the CW term being identified. We find that the solitons possess a nonzero angular momentum proportional to the electric charge, which in addition to the usual Coulomb part, acquires an extra (topological) contribution from the CW term. Specifically, it is shown that the slope of mass/energy M vs. electric charge Q and angular momentum J can be both positive and negative.
Furthermore, it is shown that the gauged Skyrmion persists even when the quartic (Skyrme) kinetic term disappears.

The effect of Skyrme–Chern–Simons dynamics on gauged Skyrmions in 2 + 1 dimensions

“…(Indeed, the definition of that SCS term is inspired by the CW anomaly.) Such a study was carried out recently in [12], where it was found that the effect of the CW term on the U(1) gauged Skyrmion is similar to the effect of the SCS term employed in the 2 + 1 dimensional model studied here, namely it results in the negative slopes in (E, Q e ) and (E, J) plots. But the CW term in 3 + 1 dimensions is parametrised by the Maxwell field and the O(4) Skyrme scalar ϕ a , a = 1, 2, 3, 4 (or the group element U) as the ones that parametrise the Lagrangian of the Abelian gauged Skyrmion.…”

“…However, the SCS term employed in section 2 is that resulting from the contraction 12 of the SO(4) connection A ab µ ; a = (α, A) ; α = 1, 2 ; A = 3, 4 ; to the SO(2) × SO(2) connections…”