On BPS Equations of Generalized $SU(2)$ Yang-Mills-Higgs Model with Scalars-Dependent Coupling $θ$-term

TL;DR

The paper addresses Bogomolnyi equations for BPS monopoles and dyons in a generalized $SU(2)$ Yang-Mills-Higgs model with scalar-dependent couplings and a CP-violating theta-term. It employs a highly general BPS Lagrangian framework with auxiliary fields to derive first-order Bogomolny equations, imposes energy-momentum and constraint conditions, and expresses the couplings in terms of a single parameter $\gamma$. The main results show a family of BPS monopoles parameterized by $\gamma$ and, for nonzero theta-term coupling, BPS dyons with modified electric charges, with the theta-term playing a crucial role in distinguishing monopole from dyon sectors. The findings extend prior work by clarifying how CP-violating terms enter the BPS structure and suggesting CP-violating effects can be tuned via $H$ and $\gamma$, including scenarios with spatially varying $H$ that could yield space-dependent electric fields.

Abstract

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.

Figures (4)

• Figure 1: Parameter space of the CP-violation coupling, $H_0$. We have Dyon solutions for the cyan region and anti-Dyon solutions for the red region. The solid curves represent the special values of $H_0$ for monopole (magenta) and anti-monopole (green) solutions. The dashed orange line is the limit where $B_i=0$.
• Figure 2: The blue line denotes BPS dyons for $H_0=0$ while the red lines represents BPS dyons with the same magnetic charge, but opposite electric charge. The magenta dashed curve corresponds to BPS monopoles. The figure also illustrates some fixed $\gamma$'s, highlighting the location of BPS dyons, represented by blue dots, and their corresponding BPS dyons, represented by red dots. The relative distance between these dots, represented by black dashed line, describes how much we must shift $H_0$ to change from BPS dyons to their corresponding BPS dyons.
• Figure 3: The blue line denotes BPS dyons for $H_0=\pi$ while the red lines represents BPS dyons with the same magnetic charge, but opposite electric charge. The magenta dashed curve corresponds to BPS monopoles. The figure also illustrates some fixed $\gamma$'s, highlighting the location of BPS dyons, represented by blue dots, and their corresponding BPS dyons, represented by red dots. The relative distance between these dots, represented by black dashed line, describes how much we must shift $H_0$ to change from BPS dyons to their corresponding BPS dyons.
• Figure 4: The blue line denotes BPS dyons for $H_0=-\pi$ while the red lines represents BPS dyons with the same magnetic charge, but opposite electric charge. The magenta dashed curve corresponds to BPS monopoles. The figure also illustrates some fixed $\gamma$'s, highlighting the location of BPS dyons, represented by blue dots, and their corresponding BPS dyons, represented by red dots. The relative distance between these dots, represented by black dashed line, describes how much we must shift $H_0$ to change from BPS dyons to their corresponding BPS dyons.