On BPS Equations of Generalized $SU(2)$ Yang-Mills-Higgs Model with Scalars-Dependent Coupling $\theta$-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 $\theta$-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 $\gamma$, while for BPS dyons there is an additional parameter which is the coupling of $\theta$-term. Interestingly even for a single BPS dyon we find the value of $\theta$-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.