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Phenomenology of Spontaneous CP-violation in 2HDMs

Tanmoy Mondal, Kei Yagyu

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Abstract

Spontaneous CP violation (SCPV) provides an attractive possibility for explaining the origin of CP violation and the observed CP-violating phenomena in the quark sector. We investigate SCPV in the two-Higgs-doublet model as one of its simplest realizations, in which the additional Higgs bosons are necessarily non-decoupling and flavor misalignment in the Yukawa sector is required. The non-decoupling Higgs sector predicts sizable deviations from the Standard Model predictions even in the Higgs alignment limit: the decay rates of the discovered Higgs boson into γγγγ and ZγZγ can deviate by at least about −10%-10\% and −4%-4\%, respectively, while the trilinear Higgs self-coupling can receive O(100)%\mathcal{O}(100)\% corrections. We consider a scenario in which the down-type quark and charged-lepton Yukawa matrices are flavor aligned, while flavor-violating interactions are allowed in the up-type quark sector to reproduce the Kobayashi-Maskawa phase. We explore the viable parameter space by imposing theoretical constraints together with experimental constraints from electroweak precision observables, flavor measurements, electric dipole moments, and LHC data. We find that a sizable flavor-violating top-charm Yukawa coupling remains allowed, giving rise to distinctive LHC signatures, including same-sign top-quark and three-top production as well as scalar cascade decays. We also find that the deviation in the trilinear Higgs self-coupling is correlated with the decay patterns of the additional neutral Higgs bosons, providing a connection between precision Higgs measurements and direct searches for the additional Higgs bosons. Our results demonstrate the importance of combining searches for additional Higgs bosons with precision Higgs measurements, complemented by flavor and electric-dipole-moment experiments, to probe SCPV.

Introduction

CP violation (CPV) was first established through the discovery of the KL→ππK_L \to\pi\pi decay, in addition to the dominant πππ\pi\pi\pi mode, by Cronin and Fitch in 1964 [1]. Since then, various CP-violating phenomena, including direct CPV, have been observed in the BB-meson system at the BaBar [2] and Belle [3, 4] experiments. More recently, the angle γ\gamma of the Cabibbo–Kobayashi–Maskawa (CKM) unitarity triangle has been precisely measured by the LHCb experiment [5, 6]. All these observations of CPV are successfully accommodated by the Kobayashi–Maskawa (KM) phase in the CKM matrix within the three-generation Standard Model (SM) [7].

Despite this remarkable success, the origin of CPV remains unknown. In particular, CPV can arise either explicitly or spontaneously. In the SM, only the former possibility is realized, since the Higgs potential is CP invariant and, apart from the strong CP phase, the only source of CPV originates from the complex phase in the quark Yukawa sector. On the other hand, both explicit CP violation (ECPV) and spontaneous CP violation (SCPV) can generally be realized in models with extended Higgs sectors because of their richer vacuum structure compared with that of the SM. Extended Higgs sectors are also frequently introduced in new-physics scenarios addressing phenomena unexplained within the SM, such as tiny neutrino masses, dark matter, and the baryon asymmetry of the Universe.

Scenarios with SCPV are particularly well motivated for several reasons. First, SCPV provides a dynamical origin of CPV through the vacuum structure, in close analogy with electroweak symmetry breaking [8]. Second, SCPV can provide an alternative solution to the strong CP problem [9, 10] to the Peccei–Quinn mechanism [11]. Third, SCPV naturally leads to the non-decoupling nature of additional Higgs bosons [12], which can play an important role in realizing a strongly first-order electroweak phase transition, as required for successful electroweak baryogenesis (EWBG) [13] [14]. Moreover, SCPV can provide additional sources of CPV beyond the KM phase, which may also play an important role in EWBG. Clarifying the origin of CPV is therefore an important avenue for probing physics beyond the SM.

SCPV was first proposed by T. D. Lee in the framework of the two-Higgs-doublet model (2HDM) [8], in which the vacuum expectation values (VEVs) of the Higgs doublet fields can develop a nontrivial relative complex phase. The vacuum structure of the 2HDM has since been extensively studied; see, e.g., Refs. [15, 16] for reviews. In particular, Refs. [17, 18, 19] have shown that a CP-violating vacuum, if it exists, is necessarily the global minimum of the potential. In Ref. [20], conditions for SCPV were formulated in a basis-independent manner. The phenomenology of the 2HDM with SCPV has also been studied in Refs. [21, 22, 23], with particular emphasis on scenarios with a Z2Z_2 symmetry. It was pointed out in Ref. [24], however, that reproducing the KM phase in the 2HDM with SCPV is incompatible with flavor alignment of the Yukawa matrices, owing to the reality of the Yukawa couplings before electroweak symmetry breaking. Flavor misalignment is therefore an unavoidable feature of such a scenario. In our previous work [12], we showed that, in a viable SCPV 2HDM, the properties of the discovered Higgs boson (hh) can exhibit sizable deviations from the SM predictions as a consequence of the non-decoupling nature of the additional Higgs bosons. In particular, the decay rate of h→γγh \to\gamma\gamma and the trilinear Higgs self-coupling hhhhhh can deviate from their SM predictions by about −10%-10\% and O(100)%O(100)\%, respectively.

In this paper, we investigate the collider phenomenology of the SCPV 2HDM, focusing not only on the properties of the discovered Higgs boson but also on direct signatures of the additional Higgs bosons. In particular, we study their flavor-violating decays, which arise from the constrained structure of the Yukawa sector inherent to SCPV. By imposing theoretical constraints from vacuum stability and perturbative unitarity, together with experimental constraints from electroweak precision observables, flavor measurements, electric dipole moments (EDMs), Higgs measurements, and direct searches at the LHC, we clarify that an off-diagonal coupling in the up-type quark Yukawa sector, particularly the top–charm flavor-violating interaction, can be sizable and give rise to distinctive signatures at the LHC. We systematically investigate the possible production and decay channels of the additional Higgs bosons and evaluate the corresponding cross sections for representative benchmark points satisfying all relevant constraints. We demonstrate that searches for the additional Higgs bosons at the High-Luminosity LHC (HL-LHC) can probe a significant portion of the viable parameter space of the SCPV 2HDM. Furthermore, we identify a characteristic correlation between deviations in the trilinear Higgs self-coupling from the SM prediction and the decay patterns of the additional Higgs bosons.

This paper is organized as follows. In Sec. II, we briefly review SCPV in the 2HDM. We first discuss the vacuum structure and then present the Yukawa interactions, which inevitably contain flavor-violating interactions in the SCPV scenario. In § 1.2, we discuss theoretical and experimental constraints on the model parameter space. In Sec. IV, we investigate the collider phenomenology of the additional Higgs bosons at the LHC. Finally, conclusions are given in Sec. V.

Model

Higgs potential and spontaneous CP-violation

We briefly review the 2HDM with SCPV. The Higgs sector consists of two SU(2)LSU(2)_{L} scalar doublets, Φ1\Phi_{1} and Φ2\Phi_{2}, both with hypercharge Y=1/2Y = 1/2. The most general renormalizable Higgs potential is given by

V=m12∣Φ1∣2+m22∣Φ2∣2−(m32Φ1†Φ2+h.c.)+λ12∣Φ1∣4+λ22∣Φ2∣4+λ3∣Φ1∣2∣Φ2∣2+λ4∣Φ1†Φ2∣2+[λ52(Φ1†Φ2)+λ6∣Φ1∣2+λ7∣Φ2∣2](Φ1†Φ2)+h.c.,(1)\begin{aligned} V ={}& m_{1}^{2}|\Phi_{1}|^{2} + m_{2}^{2}|\Phi_{2}|^{2} - \left(m_{3}^{2}\Phi_{1}^{\dagger}\Phi_{2} + \text{h.c.}\right) + \frac{\lambda_{1}}{2}|\Phi_{1}|^{4} + \frac{\lambda_{2}}{2}|\Phi_{2}|^{4} + \lambda_{3}|\Phi_{1}|^{2}|\Phi_{2}|^{2} + \lambda_{4}|\Phi_{1}^{\dagger}\Phi_{2}|^{2} \\ &+ \left[\frac{\lambda_{5}}{2}(\Phi_{1}^{\dagger}\Phi_{2}) + \lambda_{6}|\Phi_{1}|^{2} + \lambda_{7}|\Phi_{2}|^{2}\right](\Phi_{1}^{\dagger}\Phi_{2}) + \text{h.c.}, \tag*{(1)} \end{aligned}

where m32m_{3}^{2} and λ5,6,7\lambda_{5,6,7} are generally complex parameters. Since we focus on SCPV, we work in a basis where all the parameters in the Higgs potential are real.1 Assuming an electrically neutral vacuum that preserves U(1)emU(1)_{\rm em}, the VEVs of the two doublets can be parameterized as [15]

⟨Φ1⟩=(0v12),⟨Φ2⟩=(0v22eiξ),v1,v2,ξ∈R.(2)\langle\Phi_{1}\rangle= \begin{pmatrix} 0 \\ \frac{v_{1}}{\sqrt{2}} \end{pmatrix}, \qquad\langle\Phi_{2}\rangle= \begin{pmatrix} 0 \\ \frac{v_{2}}{\sqrt{2}}e^{i\xi} \end{pmatrix}, \qquad v_{1},v_{2},\xi\in\mathbb{R}. \tag*{(2)}

A nontrivial relative phase, ξ≠nπ\xi\ne n\pi with n∈Zn \in\mathbb{Z}, allows for SCPV [20]. The VEVs satisfy v2≡v12+v22=(2GF)−1v^{2} \equiv v_{1}^{2}+v_{2}^{2}=(\sqrt{2}G_{F})^{-1}, where GFG_{F} is the Fermi constant. The stationary conditions evaluated at this vacuum are given by

∂V0∂v1=v1T1−v2cξT3=0,∂V0∂v2=v2T2−v1cξT3=0,∂V0∂ξ=v1v2sξT3=0,(3)\begin{aligned} \frac{\partial V_{0}}{\partial v_{1}} &= v_{1}T_{1}-v_{2}c_{\xi}T_{3}=0, \\ \frac{\partial V_{0}}{\partial v_{2}} &= v_{2}T_{2}-v_{1}c_{\xi}T_{3}=0, \\ \frac{\partial V_{0}}{\partial\xi} &= v_{1}v_{2}s_{\xi}T_{3}=0, \tag*{(3)} \end{aligned}

where we use the shorthand notation cX≡cos⁡Xc_{X}\equiv\cos X and sX≡sin⁡Xs_{X}\equiv\sin X. Here, V0V_{0} denotes the Higgs potential evaluated at the vacuum defined in Eq. (2), and T1,2,3T_{1,2,3} are defined as

T1=m12+v122λ1+v222(λ3+λ4−λ5)+v1v2cξλ6,(4)T_{1}=m_{1}^{2}+\frac{v_{1}^{2}}{2}\lambda_{1}+\frac{v_{2}^{2}}{2}(\lambda_{3}+\lambda_{4}-\lambda_{5})+v_{1}v_{2}c_{\xi}\lambda_{6}, \tag*{(4)}
T2=m22+v222λ2+v122(λ3+λ4−λ5)+v1v2cξλ7,(5)T_2 = m_2^2 + \frac{v_2^2}{2}\lambda_2 + \frac{v_1^2}{2}(\lambda_3 + \lambda_4 - \lambda_5) + v_1v_2c_{\xi}\lambda_7, \tag*{(5)}
T3=m32−v1v2cξλ5−v122λ6−v222λ7.(6)T_3 = m_3^2 - v_1v_2c_{\xi}\lambda_5 - \frac{v_1^2}{2}\lambda_6 - \frac{v_2^2}{2}\lambda_7. \tag*{(6)}

For v1v2sξ≠0v_1v_2s_{\xi} \ne0, the stationary conditions in Eq. (3) reduce to T1=T2=T3=0T_1 = T_2 = T_3 = 0. These conditions can be solved for the three quadratic mass parameters mi2m_i^2 (i=1,2,3i = 1, 2, 3), yielding

\begin{pmatrix} m_1^2 \\ m_2^2 \\ m_3^2 \end{pmatrix} = \begin{pmatrix} -\frac{\lambda_1}{2} & -\frac{1}{2}(\lambda_3 + \lambda_4 - \lambda_5) - \lambda_6 \\ -\frac{1}{2}(\lambda_3 + \lambda_4 - \lambda_5) & -\frac{\lambda_2}{2} & -\lambda_7 \\ \frac{\lambda_6}{2} & \frac{\lambda_7}{2} & \lambda_5 \end{pmatrix} \begin{pmatrix} v_1^2 \\ v_2^2 \\ v_1v_2c_{\xi} \end{pmatrix}. \tag*{(7)}\end{pmatrix}

Eq. (7) shows that the quantities v12v_1^2, v22v_2^2, and v1v2cξv_1v_2c_{\xi} are uniquely determined by the parameters of the Higgs potential, provided that the 3×33 \times3 coefficient matrix is invertible. Consequently, the CP-violating stationary point is unique up to symmetry transformations, as discussed in Ref. [17]. Furthermore, it has been shown that a CP-violating minimum cannot coexist with a deeper CP-conserving or charge-breaking minimum [17, 18, 19, 25]. Thus, if the CP-violating stationary point is a local minimum, it is also the global minimum, up to possible symmetry-related degeneracies. This feature contrasts with the CP-conserving case, where sξ=0s_{\xi} = 0. In this case, the third stationary condition in Eq. (3) is automatically satisfied, and multiple CP-conserving stationary points may exist. Another important consequence of Eq. (7) is the absence of a decoupling limit in the SCPV scenario. All three quadratic mass parameters mi2m_i^2 are determined by the VEVs and the quartic couplings λj\lambda_j (j=1,…,7j = 1, \ldots, 7). Since the quartic couplings are bounded by perturbative unitarity and perturbativity [22, 24], the quadratic mass parameters cannot be made arbitrarily large compared with the electroweak scale. As a result, the additional Higgs bosons cannot be arbitrarily heavy, and their masses are typically constrained to lie below the TeV scale. This non-decoupling property can be explicitly understood from the Higgs boson mass formulas presented below.

We expand the scalar doublets around the CP-violating vacuum, and they can be expressed in the Higgs basis [26] as

\begin{pmatrix} \Phi_1 \\ \Phi_2 \end{pmatrix} = \begin{pmatrix} 1 & 0 \\ 0 & e^{i\xi} \end{pmatrix} \begin{pmatrix} \cos\beta& -\sin\beta\\ \sin\beta& \cos\beta\end{pmatrix} \begin{pmatrix} \Phi\\ \Phi' \end{pmatrix}, \tag*{(8)}\end{pmatrix}

where tan⁡β=v2/v1\tan\beta= v_2/v_1 and

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