BPS Equations of Monopole and Dyon in SU(2) Yang-Mills-Higgs Model, Nakamula-Shiraishi Models, and Their Generalized Versions from the BPS Lagrangian Method

Abstract

We apply the BPS Lagrangian method to derive BPS equations of monopole and dyon in the SU2 Yang-Mills-Higgs model, Nakamula-Shiraishi models, and their generalized versions. We argue that, by identifying the effective fields of scalar field, f, and of time-component gauge field, j, explicitly by j=βf with β being a real constant, the usual BPS equations for dyon can be obtained naturally. We validate this identification by showing that both Euler-Lagrange equations for f and j are identical in the BPS limit. The value of β is bounded to β<1 due to reality condition on the resulting BPS equations. In the Born-Infeld type of actions, namely, Nakamula-Shiraishi models and their generalized versions, we find a new feature that, by adding infinitesimally the energy density up to a constant 4b2, with b being the Born-Infeld parameter, it might turn monopole (dyon) to antimonopole (antidyon) and vice versa. In all generalized versions there are additional constraint equations that relate the scalar-dependent couplings of scalar and of gauge kinetic terms or G and w, respectively. For monopole the constraint equation is G=w-1, while for dyon it is wG-β2w=1-β2 which further gives lower bound to G as such G≥2β1-β2. We also write down the complete square-forms of all effective Lagrangians.