On BPS equations of generalized SU(2) Yang-Mills-Higgs model with scalars-dependent coupling θ-term

We consider a most general SU(2) Yang-Mills-Higgs model consist of terms up to quadratic in first-derivative of the fields, that is the generalized SU(2) Yang-Mills-Higgs with additional scalars-dependent coupling θ-term. Using the BPS Lagrangian method we try to find Bogomolnyi’s equations for BPS monopoles and dyons by taking most general BPS Lagrangian density. We obtain more general Bogomolnyi’s equations and a relation between all scalars dependent couplings. From these equations we can see there is a family of BPS monopole solutions parameterized by a real constant γ, while for BPS dyons there is an additional parameter which is the coupling of θ-term. Interestingly even for a single BPS dyon we find the value of θ-term’s coupling only gives additional contribution to electric charge of BPS Dyons, which is in accordance with Witten’s result in Phys.Lett. B 86 (1979), and thus can determine whether we get BPS monopoles or BPS dyons. In the presence of the coupling of θ-term, it is now possible to obtain Julia-Zee BPS dyons where all scalar-dependent couplings are non-constants, which is the extension to the BPS Julia-Zee dyons proposed in Eur.Phys.J. C 82, 602 (2022).