Exploring the Landscape of Spontaneous CP Violation in Supersymmetric Theories

TL;DR

Addressing the strong CP problem, the paper investigates spontaneous CP violation in supersymmetric theories. It develops a spurion-based criterion for SCPV in the exact SUSY limit and constructs a concrete intermediate-scale SCPV model with pseudo-flat directions stabilized by soft SUSY breaking and non-perturbative dynamics, predicting light SCPV-sector scalars with masses set by the SUSY-breaking scale. It also embeds the CP-violating sector within a Nelson-Barr framework to transmit CP violation to the CKM matrix while suppressing the strong CP, and analyzes stability and parameter space. The work demonstrates how SUSY protects SCPV against radiative corrections and higher-dimensional operators and discusses cosmological implications like domain walls and dark-matter candidates.

Abstract

The strong CP problem remains one of the most important unresolved issues in the Standard Model. Spontaneous CP violation (SCPV) is a promising approach to the problem by assuming that CP is an exact symmetry of the Lagrangian but broken spontaneously at the vacuum, which enables the generation of the observed Cabibbo-Kobayashi-Maskawa (CKM) phase without reintroducing a nonzero strong CP phase. Supersymmetry (SUSY) provides a natural framework to accommodate such a mechanism, as SUSY can not only protect the scale of SCPV from radiative corrections but also suppress problematic higher-dimensional operators generating a strong CP phase. In the present study, we explore the realization of SCPV in two distinct SUSY scenarios. First, we investigate SCPV in the exact SUSY limit, extending the spurion formalism developed in non-supersymmetric theories and introducing a method to determine whether the given superpotential satisfies the necessary condition for SCPV. Second, we construct a model in which CP is spontaneously broken at an intermediate scale along pseudo-flat directions, stabilized by soft SUSY breaking and non-perturbative effects of a gauge theory. The latter setup predicts light scalars in the SCPV sector whose masses are determined by the SUSY breaking scale.

Figures (2)

• Figure 1: The parameter space in the plane of $\alpha_b$-$\alpha_m$ for $\gamma = 1, 2, 3,0.1$ and $10$ with $\beta = 0.75$, $N = 3$ and $\kappa_1 = \kappa_2 = 1$ fixed. In the pink shaded regions, the potential is unbounded from below, given by Eq. (\ref{['bounded_cond']}), while in the blue shaded regions, the potential is bounded from below but SCPV is not obtained. The white regions represent the viable parameter region that realizes correctly stabilized CP-violating vacua within a bounded-from-below potential. The inside of the red curves in panels (a)–(c) indicates that the maximal determinant of the Hessian matrix at the stationary points is positive.
• Figure 2: The parameter space of $\alpha_b$ and $\alpha_m$ is shown for $\gamma = 3$, with $\beta$ varying from $0.25$ to $1$ and $N = 2, 3$ and $4$. The couplings $\kappa_1$ and $\kappa_2$ are fixed to $1$. The color code is the same as that of Fig. \ref{['fig:β=0.75,Ν=3']}