BPS equations of monopole and dyon in \(S U(2)\) Yang-Mills-Higgs model, Nakamula-Shiraishi models, and their generalized versions from the BPS Lagrangian method (Q1719962)

Summary: We apply the BPS Lagrangian method to derive BPS equations of monopole and dyon in the \(S U \left(2\right)\) 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 \(j = \beta f\) with \(\beta\) 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 \(\beta\) is bounded to \(|\beta| < 1\) 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 \(4 b^2\), 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 \(w \left(G - \beta^2 w\right) = 1 - \beta^2\) which further gives lower bound to \(G\) as such \(G \geq | \beta \sqrt{1 - \beta^2}|\). We also write down the complete square-forms of all effective Lagrangians.