Spontaneous CP-violation in the Simplest Little Higgs Model and its Future Collider Tests: the Scalar Sector

TL;DR

This paper explores spontaneous CP-violation in the Simplest Little Higgs model, highlighting how scalar mixing occurs and proposing future collider experiments to detect CP-violation effects in the scalar sector.

Contribution

It introduces a novel mechanism for spontaneous CP-violation within the Simplest Little Higgs model and discusses potential collider signatures for future experiments.

Findings

Spontaneous CP-violation occurs via pseudoscalar vacuum expectation value.

The model remains viable under current experimental constraints.

Future collider measurements can probe CP-violation through specific scalar vertices.

Abstract

We proposed the spontaneous CP-violation in the Simplest Little Higgs model. In this model, the pseudoscalar field can acquire a nonzero vacuum expected value. It leads to a mixing between the two scalars with different CP-charge, which means spontaneous CP-violation happens. It is also a connection between composite Higgs mechanism and CP-violation. Facing the experimental constraints, the model is still alive for both scenarios in which the extra scalar appears below or around the electro-weak scale. We also discussed the future collider tests on CP-violation in the scalar sector through measuring $h_2ZZ$ and $h_1h_2Z'$ vertices (see the definitions of the particles in the text) which provides new motivations on future $e^+e^-$ and $pp$ colliders. It also shows the importance of the vector-vector-scalar- and vector-scalar-scalar-type vertices to discover CP-violation effects in the scalar sector.

Figures

Equations

$$\Phi_{1}=\textrm{e}^{\textrm{i}\Theta^{\prime}}\textrm{e}^{\textrm{i}t_{\beta}\Theta}\left(\begin{array}[]{c}\mathbf{0}_{1\times 2}\\ fc_{\beta}\end{array}\right),\quad\quad\Phi_{2}=\textrm{e}^{\textrm{i}\Theta^{\prime}}\textrm{e}^{-\textrm{i}\Theta/t_{\beta}}\left(\begin{array}[]{c}\mathbf{0}_{1\times 2}\\ fs_{\beta}\end{array}\right);$$

$$\Theta\equiv\frac{1}{f}\left(\frac{\eta\mathbb{I}_{3\times 3}}{\sqrt{2}}+\left(\begin{array}[]{cc}\mathbf{0}_{2\times 2}&\phi\\ \phi^{{\dagger}}&0\end{array}\right)\right),\quad\textrm{and}\quad\Theta^{\prime}\equiv\frac{1}{f}\left(\frac{G^{\prime}\mathbb{I}_{3\times 3}}{\sqrt{2}}+\left(\begin{array}[]{cc}\mathbf{0}_{2\times 2}&\varphi\\ \varphi^{{\dagger}}&0\end{array}\right)\right),$$

$$\mathcal{L}={\sum }_{i=1,2}{\left(D^{\mu }{\Phi }_i\right)}^{†}\left(D_{\mu }{\Phi }_i\right)$$

$$D_{\mu }\equiv {\partial }_{\mu }-\text{i}g{\mathbb{G}}_{\mu }.$$

$$\mathbb{G}_{\mu}=\frac{A^{3}_{\mu}}{2}\left(\begin{array}[]{ccc}1&&\\ &-1&\\ &&\end{array}\right)+\frac{A^{8}_{\mu}}{2\sqrt{3}}\left(\begin{array}[]{ccc}1&&\\ &1&\\ &&-2\end{array}\right)+\frac{1}{\sqrt{2}}\left(\begin{array}[]{ccc}&W^{+}&Y^{0}\\ W^{-}&&X^{-}\\ \bar{Y}^{0}&X^{+}&\end{array}\right)_{\mu}+\frac{t_{W}B_{\mu}}{3\sqrt{1-t^{2}_{W}/3}}\mathbb{I}$$

$$m_W=\frac{gv}{2}\text{and}{m}_X=m_Y=\frac{gf}{\sqrt{2}}.$$

$$\left(\begin{array}[]{c}A\\ Z\\ Z^{\prime}\end{array}\right)=\left(\begin{array}[]{ccc}-s_{W}&s_{X}c_{W}&c_{X}c_{W}\\ c_{W}&s_{X}s_{W}&c_{X}s_{W}\\ 0&c_{X}&-s_{X}\end{array}\right)\left(\begin{array}[]{c}A^{3}\\ A^{8}\\ B\end{array}\right)_{\mu}$$

$$m_A=0,m_Z=\frac{gv}{2c_W},\text{and}{m}_{Z^{\prime }}=\sqrt{\frac{2}{3-t_W^2}}gf.$$

$$\delta m_Z^2=-\delta m_{Z^{\prime }}^2=\frac{g^2{v}^2{c}_{2W}^2}{32c_W^6}{\left(\frac{v}{f}\right)}^2,\text{and}\delta m_{Y^2}^2=0.$$

$${\mathbb{R}}_{Z^{\prime }Z}=\frac{\sqrt{3}{c}_{2W}{c}_X}{8c_W^3}{\left(\frac{v}{f}\right)}^2,{\mathbb{R}}_{Y^{2}Z}=\frac{\sqrt{2}}{3t_{2\beta }{c}_W}{\left(\frac{v}{f}\right)}^3,\text{and}{\mathbb{R}}_{Y^2{Z}^{\prime }}=\frac{2c_X}{\sqrt{6}{t}_{2\beta }}{\left(\frac{v}{f}\right)}^3;$$

$${\mathcal{L}}_{\text{kin}}=\frac{1}{2}\left({\partial }^{\mu }h{\partial }_{\mu }h+{\partial }^{\mu }{y}^1{\partial }_{\mu }{y}^1+{\mathbb{K}}_{ij}{\partial }^{\mu }{G}_i{\partial }_{\mu }{G}_j\right)$$

$$\left(\tilde{\eta },{\tilde{G}}_p\right)=\left(\frac{\eta }{\sqrt{{\left({\mathbb{K}}^{-1}\right)}_{11}}},\frac{{\left(\mathbb{RF}\right)}_{pi}}{m_p}{G}_i\right)$$

$${\mathcal{L}}_y$$

$$L_i={\left({\nu }_L,{\ell }_L,\text{i}{N}_L\right)}_i^T,Q_1={\left(d_L,-u_L,\text{i}{D}_L\right)}^T,$$

$$Q_2={\left(s_L,-c_L,\text{i}{S}_L\right)}^T,Q_3={\left(t_L,b_L,\text{i}{T}_L\right)}^T.$$

$$m_N^j={\lambda }_N^{j}f{s}_{\beta },m_Q=\sqrt{{\left|{\lambda }_q^a{c}_{\beta }\right|}^2+{\left|{\lambda }_q^b{s}_{\beta }\right|}^2}f,$$

$$m_{\nu }^j=0,m_q=\frac{v}{\sqrt{2}}\frac{\left|{\lambda }_q^a{\lambda }_q^b\right|}{\sqrt{{\left|{\lambda }_q^a{c}_{\beta }\right|}^2+{\left|{\lambda }_q^b{s}_{\beta }\right|}^2}}=\frac{{\lambda }_{q}v}{\sqrt{2}}.$$

$$m_{\ell }^j=\frac{v}{4\sqrt{2}\pi }{y}_{\ell }^j,m_b=\frac{v}{4\sqrt{2}\pi }{\lambda }_{b,3},m_{u,c}=\frac{v}{4\sqrt{2}\pi }{y}_{u,c},$$

$$\delta V_h=-\delta m^2\left(h^{†}h\right)+\delta \lambda {\left(h^{†}h\right)}^2.$$

$${\left(\delta m^2\right)}_{\text{1-loop}}$$

$${\left(\delta \lambda \right)}_{\text{1-loop}}$$

$$\delta V=-{\mu }^2{\Phi }_1^{†}{\Phi }_2+\text{H.c.}$$

$$m_{\eta }^2=\frac{{\mu }^2}{s_{\beta }{c}_{\beta }}\cos \left(\frac{v}{\sqrt{2}f{s}_{\beta }{c}_{\beta }}\right)\approx \frac{{\mu }^2}{s_{\beta }{c}_{\beta }}$$

$${\left(\delta V_h\right)}_{\mu }$$

$$\delta V=-{\mu }^2{\Phi }_1^{†}{\Phi }_2+ϵ{\left({\Phi }_1^{†}{\Phi }_2\right)}^2+\text{H.c.}$$

$$V_{\eta }=-{\mu }^2{f}^2{s}_{\beta }{c}_{\beta }{c}_{\alpha }\cos \left(\frac{\eta }{\sqrt{2}f{s}_{\beta }{c}_{\beta }}\right)+ϵf^4{s}_{\beta }^2{c}_{\beta }^2{c}_{\alpha }^2\cos \left(\frac{\sqrt{2}\eta }{f{s}_{\beta }{c}_{\beta }}\right).$$

$${\mu }^2<4ϵf^2\left|s_{\beta }{c}_{\beta }{c}_{\alpha }\right|,$$

$$v_{\eta }\equiv ⟨\eta ⟩=\pm \sqrt{2}f{s}_{\beta }{c}_{\beta }\arccos \left(\frac{{\mu }^2}{4ϵf^2{s}_{\beta }{c}_{\beta }{c}_{\alpha }}\right),$$

$$\mathcal{L}_{\textrm{m}}=-\frac{1}{2}\left(h,\eta\right)\left(\begin{array}[]{cc}M^{2}&\epsilon f^{2}s_{2\alpha}s_{2\xi}\\ \epsilon f^{2}s_{2\alpha}s_{2\xi}&4\epsilon f^{2}c^{2}_{\alpha}s_{\xi}^{2}\end{array}\right)\left(\begin{array}[]{c}h\\ \eta\end{array}\right).$$

$$\left(\begin{array}[]{c}h_{1}\\ h_{2}\end{array}\right)\equiv\left(\begin{array}[]{cc}c_{\theta}&-s_{\theta}\\ s_{\theta}&c_{\theta}\end{array}\right)\left(\begin{array}[]{c}h\\ \eta\end{array}\right),$$

$$\theta =\frac{1}{2}\arctan \left(\frac{2ϵf^2{s}_{2\alpha }{s}_{2\xi }}{M^2-4ϵf^2{c}_{\alpha }^2{s}_{\xi }^2}\right)$$

Full text

I Introduction

The discovery of a 125 GeV Higgs boson mass ; PDG by the ATLAS and CMS collaborations higdisc in 2012 implies the success of the standard model (SM) because the measured signal strengths are consistent with those predicted by the SM stra ; strc . However, the electro-weak symmetry breaking (EWSB) mechanism is an important topic and researches on physics beyond the SM (BSM) are still necessary and attractive.

For example, to solve the little hierarchy problem, Arkani-Hamed et al. proposed the Little Higgs (LH) framework LH in which the collective symmetry breaking (CSB) mechanism LH was used to forbid the quadratic divergences in the Higgs potential at one-loop level. The LH framework contains a lot of models. All of them are special kinds of composite Higgs models comp thus each of them must contain a global symmetry which is spontaneously broken at a high scale $f\gg v$ where $v=246~{}\textrm{GeV}$ is the vacuum expected value (VEV) of the Higgs field. The SM-like Higgs boson is treated as a pseudo-Nambu-Goldstone boson corresponding to one of the broken generators and EWSB is generated dominantly through quantum correction thus the Higgs boson can be naturally light LH ; comp . Usually the gauge group is also enlarged thus there are extra gauge bosons with their masses at $\mathcal{O}(f)$ scale. LH models are effective field theories (EFT) below a cut off scale $\Lambda\sim 4\pi f$. Below the scale $\Lambda$, a LH model is weakly coupled, but we do not know what would happen above $\Lambda$. Among those models, the simplest Little Higgs (SLH) model SLH ; SLHi ; SLH2 has the minimal extended scalar sector in which their are only two scalars. In the SLH model, a global symmetry $\left[\textrm{SU}(3)\times\textrm{U}(1)\right]^{2}$ is spontaneously broken to $\left[\textrm{SU}(2)\times\textrm{U}(1)\right]^{2}$ at scale $f$. The gauge symmetry is enlarged to $\textrm{SU}(3)\times\textrm{U}(1)$ and spontaneously broken to the electro-weak (EW) gauge symmetry $\textrm{SU}(2)_{L}\times\textrm{U}(1)$ at scale $f$ as well. And at the EW scale $v$, the gauge symmetry is further broken to $\textrm{U}(1)_{\textrm{em}}$ like what happens in the SM. If CP-violation is absent in the scalar sector, one of the scalars is the SM-like Higgs boson (denoted as $h$), and the other is a pseudoscalar.

CP-violation is another important topic in both SM and BSM physics. In 1964, CP-violation was first discovered through the $K_{L}\rightarrow\pi\pi$ rare decay process CPVdisc . More CP-violation effects have been discovered in K- and B-meson sectors PDG . All these measured CP-violation effects can be successfully explained by the Kobayashi-Maskawa (K-M) mechanism KM which was proposed by Kobayashi and Maskawa in 1973. They showed that a nontrivial CP-phase can appear in the quark mixing matrix (named CKM matrix KM ; Cab ) if there exist three generations of fermions. However, the succeed of K-M mechanism is not the end of CP-violation studies. For example, the observed matter-antimatter asymmetry in the universe PDG ; Plank requires new sources of CP-violation because SM itself cannot generate such a large asymmetry EWBG ; EWBG2 . Thus it is attractive to study new CP-violation sources. Till now, the scalar sector is still an unfamiliar world for us and there may be lots of hidden new physics, including new sources of CP-violation. Thus in this paper, we focus on extra CP-violation in the scalar sector.

Theoretically, there are already many extensions of the SM which contains new CP-violation sources. For example, if we add more complex scalar singlets or doublets, there may be CP-violation in scalar sector 2HDM ; example1 ; Lee ; example2 ; example3 which can leads to a CP-mixing Higgs boson 222For the 125 GeV Higgs boson, LHC measurements preferred a CP-even one and excluded a CP-odd one at over $3\sigma$ level through the final distribution of $h\rightarrow ZZ^{*}\rightarrow 4\ell$ decay assuming no CP-violation in the Higgs interactions higCP . However, a CP-mixing Higgs boson is still allowed since the contribution from pseudoscalar component should be loop suppressed.. Some of these models may be CP-conserving at the Lagrangian level and CP-violation can arise only from a complex vacuum, which was called the spontaneous CP-violation mechanism Lee . This mechanism was proposed by Lee in 1973 Lee as the first kind of two-Higgs-doublet model (2HDM) 2HDM . Moreover, spontaneous CP-violation mechanism is also a possible solution of the strong-CP problem strong , and it may have further connection with lightness of the Higgs boson as well mao . Besides these models, spontaneous CP-violation in the scalar sector can also arise from the composite framework. There are already two examples, one is the next-to-minimal composite Higgs model ($\textrm{SO}(6)/\textrm{SO}(5)$, or equivalently $\textrm{SU}(4)/\textrm{Sp}(4)$) NMC , and the other is the Littlest Higgs model ($\textrm{SU}(5)/\textrm{SO}(5)$) CPVLH . In each model, CP-violation occurs when the pseudoscalar field acquires a nonzero VEV. In this paper, we will propose the possibility of spontaneous CP-violation in the SLH model through the realization of the same mechanism. This model can also appear as one of the candidates to solve strong-CP problem as mentioned above. More details on this topic will appear in a forthcoming paper strongmao .

Phenomenologically, we can test new CP-violation effects directly or indirectly. The indirect effects may appear in the electric dipole moments (EDM) of electron and neutron EDM , modifications in meson mixing parameters modify , or anomalous $ZZZ$ couplings anoZ ; while the direct effects may be discovered in $h\tau^{+}\tau^{-}$ or $ht\bar{t}$ vertices through measuring the final state distributions dist . If another scalar is discovered and we denote the scalars as $h_{1,2}$ ($h_{1}$ is the SM-like Higgs boson and $h_{2}$ is the extra scalar), we can also discover CP-violation in the scalar sector through directly measuring tree-level vector-vector-scalar- ($VVS$-) and vector-scalar-scalar- ($VSS$-) type vertices, such as $h_{2}VV$ and $Vh_{1}h_{2}$ vertices 333Here $V$ denotes a massive gauge boson. For the SM gauge group, $V=W$ or $Z$; while for LH gauge groups, $V$ can also denotes extra heavy gauge bosons., according to the CP-properties analysis mao . Based on this idea, the author and his collaborators recently proposed a model-independent method to measure the CP-violation effects in the scalar sector through $e^{+}e^{+}\rightarrow Z^{*}\rightarrow Zh_{1},Zh_{2},h_{1}h_{2}$ associated production processes at future $e^{+}e^{-}$ colliders testCPV . In that research, the product of the three vertices was used as a quantity to measure the magnitude of CP-violation testCPV ; K . However, in the SLH model, the author and his collaborators recently showed the $Zh_{1}h_{2}$ vertex is suppressed by a factor $(v/f)^{3}$ Zh1h2 which means it is difficult to test. Thus to test CP-violation in the SLH model, we can turn to extra heavy gauge bosons for help.

As a summary, the model studied in this paper is attractive both theoretically and phenomenologically. This paper is organized as following: in section II we briefly review the CP-conserving SLH model, build the SLH model with spontaneous CP-violation, and obtain the domain interactions; in section III we consider the constraints on this model, especially in the scalar sector; in section IV we discuss the tests on CP-violation effects in this model at future $e^{+}e^{-}$ or $pp$ colliders; and in section V we present our conclusions and further discussions. In the appendix Appendix A, we also presented the improved SLH formalism Zh1h2 which is very helpful for the model building.

II Model Construction

In this section, we first briefly review the CP-conserving SLH model and then construct the spontaneous CP-violation SLH model. We will also derive the useful vertices in the spontaneous CP-violation SLH model. In both models, we have the same nonlinear realization for Goldstone bosons. We also have the same particle spectra in both models, while in the CP-violation model, the scalars are both CP-mixing states. The CSB mechanism and loop corrections in the Higgs potential are also similar in both models. The difference comes from an extra explicit global $[\textrm{SU}(3)\times\textrm{U}(1)]^{2}$ breaking term which is absent in the CP-conserving model.

II.1 A Brief Review of the CP-conserving SLH Model

The SLH model contains two scalar triplets $\Phi_{1,2}$ which transform as $(\mathbf{3},\mathbf{1})$ and $(\mathbf{1},\mathbf{3})$ respectively under the global $[\textrm{SU}(3)\times\textrm{U}(1)]^{2}$ transformation SLH ; SLHi ; SLH2 ; SLH3 . At a scale $f\gg v$, $[\textrm{SU}(3)\times\textrm{U}(1)]^{2}$ breaks to $[\textrm{SU}(2)\times\textrm{U}(1)]^{2}$ and ten Nambu-Goldstone bosons are generated, eight of which should be eaten by massive gauge bosons during spontaneous gauge symmetry breaking $\textrm{SU}(3)\times\textrm{U}(1)\rightarrow\textrm{SU}(2)_{L}\times\textrm{U}(1)\rightarrow\textrm{U}(1)_{\textrm{em}}$. Two physical scalars are finally left. The nonlinear realized scalar triples can be written as SLH3

[TABLE]

where $\beta$ is a mixing-angle between the two scalar triplets. The matrix fields $\Theta$ and $\Theta^{\prime}$ are separately

[TABLE]

in which $\phi\equiv\left((v_{h}+h-\textrm{i}G)/\sqrt{2},G^{-}\right)^{T}$ is the usual Higgs doublet and $\varphi\equiv(y^{0},x^{-})^{T}$ is another complex doublet for Goldstones corresponding to heavy gauge bosons following the conventions in SLH3 .

The covariant derivative term is

[TABLE]

where

[TABLE]

$g$ is the weak coupling constant and the gauge fields matrix is SLH ; SLHi ; SLH3

[TABLE]

where $\theta_{W}$ is the EW mixing angle 444In this paper, we denote $s_{\alpha}\equiv\sin\alpha$, $c_{\alpha}\equiv\cos\alpha$, and $t_{\alpha}\equiv\tan\alpha$, for any angle $\alpha$. and complex fields $Y^{0}(\bar{Y}^{0})\equiv\left(Y^{1}\pm\textrm{i}Y^{2}\right)/\sqrt{2}$. The terms including $\mathbb{G}_{\mu}\mathbb{G}^{\mu}$ in (3) give the masses of gauge bosons. Before EWSB, $v_{h}=0$; while after EWSB, $v_{h}$ is generated through quantum correction. It must be close to $v$ and their difference arises at $\mathcal{O}\left((v/f)^{2}\right)$ level. To the leading order of $(v/f)$, we have SLH3

[TABLE]

The other three neutral degrees of freedom will mix with each other at leading order of $(v/f)$ through the matrix SLH3

[TABLE]

where $\theta_{X}\equiv\arcsin\left(t_{W}/\sqrt{3}\right)$. The corresponding masses at leading order of $(v/f)$ are then

[TABLE]

The massless gauge boson is photon. If we go beyond leading order of $(v/f)$, the gauge bosons will have further mixing with each other. For example, in charged sector, $W^{\pm}$ and $X^{\pm}$ will mix with each other at $\mathcal{O}\left((v/f)^{3}\right)$ level, and $W(X)^{\pm}$ will acquire their relative mass corrections at $\mathcal{O}\left((v/f)^{2}\right)$ level. While in neutral sector, the off-diagonal elements of the mass matrix $\mathbb{M}^{2}_{V}$ in the basis $(Z,Z^{\prime},Y^{2})$ are nonzero. Using an orthogonal matrix $\mathbb{R}$, it can be diagonalized as $\left(\mathbb{R}\mathbb{M}^{2}_{V}\mathbb{R}^{T}\right)_{pq}=m_{p}\delta_{pq}$ where $m_{p}$ are the gauge bosons’ masses. The neutral gauge bosons acquire their mass corrections as

[TABLE]

We denote the corresponding mass eigenstates as $\tilde{Z}$, $\tilde{Z}^{\prime}$, and $\tilde{Y}^{2}$. Their mixing angles (which are also approximately the rotation matrix elements)

[TABLE]

to the leading order of $(v/f)$. $A$ and $Y^{1}$ do not participate further mixing.

The six neutral scalar degrees of freedom can be divided into CP-even ($h$ and $y^{1}$) and CP-odd ($\eta$, $G$, $G^{\prime}$, and $y^{2}$) parts where $y^{0}(\bar{y}^{0})\equiv\left(y^{1}\pm\textrm{i}y^{2}\right)/\sqrt{2}$. A straightforward calculation showed that after EWSB, the kinetic terms can be written as

[TABLE]

with $G_{i}$ runs over the four CP-odd scalar degrees of freedom and $\mathbb{K}_{ij}\neq\delta_{ij}$ means the CP-odd part is not canonically-normalized 555Details on the improved formalism to treat this case can be found in the appendix Appendix A and Zh1h2 .. To find out the canonically-normalized basis, we should consider the gauge fixing term together. The two-point transitions between gauge bosons and scalars arise from the cross terms of $\partial_{\mu}\Phi_{i}$ and $\mathbb{G}_{\mu}\Phi_{i}$. These transitions can be parameterized as $V_{p}^{\mu}\mathbb{F}_{pi}\partial_{\mu}G_{i}$ and their contributions should be canceled by $\left(\partial_{\mu}V_{p}^{\mu}\right)\mathbb{F}_{pi}G_{i}$ from the gauge fixing term. It can be checked straightforwardly that (see appendix Appendix A or Zh1h2 for more details) a new basis

[TABLE]

is canonically-normalized. $\tilde{G}_{p}$ is just the corresponding Goldstone of $\tilde{V}_{p}$.

In the fermion sector, each left-handed doublet must be extended to a triplet thus there must be additional heavy fermions. In lepton sector, a heavy neutrino $N_{i}$ should be added for each generation. While in the quark sector, choosing the “anomaly-free embedding” af , $T$ with $Q=2/3$ is added as the parter of $t$, $D$ and $S$ with $Q=-1/3$ are added as the parters of $d$ and $s$ separately. The Yukawa interactions are then SLH ; SLHi ; SLH2 ; SLH3

[TABLE]

where the left-handed triplets are SLH3

[TABLE]

The first line is for leptons where $\ell_{R,j}$ runs over $(e,\mu,\tau)_{R}$; the second line is for the third generation of quarks where $d_{R,j}$ runs over $(d,s,b,D,S)_{R}$, and the last line is for the first two generations of quarks where $u_{R,j}$ runs over $(u,c,t,T)_{R}$. $\Lambda\sim 4\pi f$ is a cut-off scale. A right-handed quark with index $a$ or $b$ must be a mixing state between an additional quark and its SM partner, for example, $u_{R,3}^{a,b}$ are mixing states between $t_{R}$ and $T_{R}$. To the leading order of $(v/f)$, The heavy fermions’ masses are SLHi ; SLH3

[TABLE]

for $Q=T,D,S$ and $q=t,d(d_{1}),s(d_{2})$. To the leading order, the corresponding partners in SM sector have the masses

[TABLE]

CSB mechanism keeps all neutrinos massless 666In the first term of (13), we can also use $\Phi_{1}$ instead of $\Phi_{2}$, but we cannot have both terms together if we assume massless neutrinos. If we perform this replacement, $m_{N}^{j}$ in (15) should also be changed to $\lambda_{N}^{j}fc_{\beta}$.. Other fermions require their masses (similarly, to the leading order)

[TABLE]

in which $y^{j}_{\ell}$ are eigenvalues of matrix $\lambda_{\ell}^{jk}$ and $y_{u,c}$ are eigenvalues of matrix $\lambda_{u}^{jk}$. To this step, we ignored small mixing between $q$ and $Q$. Consider this kind of mixing $\Delta_{qQ}$, a mass correction $\delta m_{q}/m_{q}\sim\mathcal{O}\left(\Delta_{qQ}^{2}/m^{2}_{Q}\right)$ is generated.

Last, let’s turn to the scalar potential. In the discussions above, we assume the Higgs doublet acquire a correct VEV to derive the particle spectra everywhere. However, at tree-level, $\left|\Phi_{1}^{{\dagger}}\Phi_{2}\right|^{2}$ term is forbidden due to the CSB mechanism. The Higgs potential can be generated through Coleman-Weinberg mechanism CW at loop-level as

[TABLE]

The CSB mechanism forbids quadratic divergence in (18) thus SLH ; SLHi ; SLH2

[TABLE]

Here $\Lambda\sim 4\pi f$ is a cut-off scale and $\lambda_{t}\equiv\sqrt{2}m_{t}/v$, which means the contributions from the first and second generations of fermions are ignorable. When $m_{T}$ is heavy enough, EWSB can be generated through these loop corrections.

Now the pseudoscalar $\eta$ is still massless due to an accidental global $\textrm{U}(1)$ symmetry. Adding a term

[TABLE]

in the potential 777This term breaks the CSB mechanism explicitly which means a quadratic divergence in the Higgs potential can be generated at one-loop level. Thus numerically $\mu$ should be very small comparing with $f$. In the convention of this paper (which is the same as that in SLH3 ), the degrees of freedom in $\Theta^{\prime}$ cancels with each other thus $\eta$ dose not acquire additional mixing with $y^{2}$., $\eta$ acquires its mass SLH2

[TABLE]

and the Higgs potential acquires another correction SLH2

[TABLE]

Two-loop contributions to $\delta m^{2}$ can be absorbed into the possible contributions from unknown physics at the cut-off scale $\Lambda$ LH ; SLHa which can be parameterized as $\left(\delta m^{2}\right)_{\textrm{2-loop}}=-cf^{2}$. We can roughly estimate $|c|\sim\mathcal{O}(10^{-2})$.

II.2 Spontaneous CP-violation in the SLH Model

In (21), $\mu$-term provides the $\eta$ mass. In general, $\mu^{2}$ can be complex, but its argument can always be absorbed into the shift of $\eta$ (which is equivalent to a rotation of $\Phi_{i}$). Besides this, $\eta$ cannot acquire a nonzero VEV, thus there is no CP-violation in the scalar potential. Comparing with the CP-conserving case in subsection II.1, we can add another term and (21) becomes

[TABLE]

Here $\epsilon$ is also required to be small (for example, $\epsilon\lesssim\mathcal{O}\left((v/f)^{2}\right)$ thus the CSB mechanism is not significantly broken). In general, $\mu^{2}$ and $\epsilon$ can be complex, but we can shift $\eta$ to make at least one of them real. If we choose $\mu^{2}$ real, when $\epsilon$ is still complex, CP-symmetry would be explicitly broken in the scalar sector. However, if both $\mu^{2}$ and $\epsilon$ are real, $\eta$ is also possible to acquire a nonzero VEV which means spontaneous CP-violation happens. In this paper, we focus on the spontaneous CP-violation case.

According to (24), denote $\alpha\equiv v_{h}/(\sqrt{2}fs_{\beta}c_{\beta})$, we have

[TABLE]

Minimize this potential, we found that when

[TABLE]

$\langle\eta\rangle=0$ becomes unstable thus $\eta$ would acquire a nonzero VEV

[TABLE]

which means spontaneous CP-violation is possible. For simplify, we choose “+” in the equation above from now on. We denote $\xi\equiv v_{\eta}/(\sqrt{2}fs_{\beta}c_{\beta})$, and the scalar mass term is

[TABLE]

Here $M$ should be close to $125~{}\textrm{GeV}$ and it includes all the quantum correction effects from (19) and (20) 888These quantum corrections are not affected by the CP properties of the scalar sector which means (19) and (20) derived in the CP-conserving model can be simply transported into the CP-violation case. Nonzero off-diagonal elements means the mass eigenstates cannot be CP eigenstates. Define the mass eigenstates (in which $h_{1}$ is SM-like)

[TABLE]

we have the mixing angle

[TABLE]

and scalar masses

[TABLE]

We can see that only when both $\mu^{2}$ and $\epsilon$ are nonzero, CP-violation can occur, which means in this model, CP-symmetry is also collectively broken 999The case $\epsilon$ absents was already discussed above. The case $\mu^{2}$ absents allows a nonzero $v_{\eta}$, but $\xi=\pi/2$ that the off-diagonal elements in (31) are still zero. A shift of $\eta$ (rotation of $\Phi$) can remove this $\xi$ hence it is trivial. A nontrivial $\xi$ requires nontrivial $\mu^{2}$ and $\epsilon$..

For the Yukawa couplings, we can also choose all the couplings real thus there is no explicit CP-violation. Complex CKM matrix can arise from the mixing between a SM quark and an extra quark, which is the same mechanism as that in example1 .

II.3 Some Useful Interactions in this Model

In the CP-violation SLH model, mixing between $h$ and $\eta$ can modify some of the vertices in the CP-conserving model. The $hVV$ couplings can be parameterized as

[TABLE]

where $\tilde{V}$ denote the mass eigenstates. For real vector fields, $\tilde{V}^{*}=\tilde{V}$. To the leading order of $(v/f)$, we have

[TABLE]

while $c_{i,Y}$ remains zero to all order of $(v/f)$.

For the antisymmetric type $Vh\eta$ couplings 101010We don’t consider the symmetric type couplings $(h_{1}\partial^{\mu}h_{2}+h_{2}\partial^{\mu}h_{1})$ here because they cannot contribute anything in the processes with on-shell gauge boson(s)., we parameterize it as

[TABLE]

The results to the leading order of $(v/f)$ are

[TABLE]

which are the same as the CP-conserving case, since $h_{1}\partial^{\mu}h_{2}-h_{2}\partial^{\mu}h_{1}=h\partial^{\mu}\eta-\eta\partial^{\mu}h$.

The scalar trilinear interactions should be

[TABLE]

where to the leading order of $(v/f)$, the dimensionless coefficients

[TABLE]

$\lambda$ in the equations is the Higgs self-coupling constant.

The Yukawa couplings for SM leptons and quarks $f=\ell,q$ can be parameterized as

[TABLE]

For $f=u,c,b,\nu,\ell$, the pseudoscalar degree of freedom dose not couple to these fermions, thus we have

[TABLE]

while for $q=d,s,t$, the coupling coefficients

[TABLE]

Here $\delta_{q}=-1$ for the third generation ($q=t$) and $\delta_{q}=+1$ for the first two generations ($q=d,s$). The imaginary parts are generated by the left-handed mixing between light and heavy quarks. $\theta_{R}=\arctan\left(t^{-1}_{\beta}\lambda_{1}/\lambda_{2}\right)$ at the leading order of $v/f$ is the right-handed mixing angle. Here we don’t consider the possible flavor changing couplings. The Yukawa couplings including a heavy quark should be

[TABLE]

where $P_{L/R}=(1\mp\gamma^{5})/2$. The coefficients

[TABLE]

Here $\delta_{Q}=+1$ for the third generation ($Q=T$) and $\delta_{Q}=-1$ for the first two generations ($Q=D,S$), which different with those for SM fermions. $s_{2\theta_{R}}\propto\delta_{Q}m_{q}/m_{Q}$ thus for the first two generations, we have $s_{2\theta_{R}}\ll 1$. The other four coefficients including both light and heavy quarks are

[TABLE]

In the calculation of $c_{iR,q}$, the improved formalism affects on their imaginary parts since the $\eta$ component in $G$ cannot be ignored due to the improved SLH formalism Zh1h2 . For the third generation, $m_{t}/m_{T}\sim\mathcal{O}(v/f)$, thus $c_{iL,q}$ can reach $\mathcal{O}(1)$. But for the first two generations, $m_{q}/m_{Q}\ll v/f$ means $c_{iL,q}\ll 1$.

III Recent Constraints on the Model

As a BSM model, SLH always face many direct and indirect constraints, such as collider searches for new particles predicted by the model and EW precision tests. The scalar sector contains an extra scalar $h_{2}$, whose properties are quite different from the SM-like scalar. If it is light enough ($m_{2}<m_{1}/2$), it should also face the $h_{1}$ cascade decay constraint. As a model with new CP-violation source, we should also discuss the EDM constraints EDM . In this paper, we don’t discuss more details about quark flavor physics.

III.1 Direct and Indirect Constraints on $f$

In the SLH model, the modifications on $S$ and $T$ parameters are sensitive to the new scale $f$. Thus before LHC Run II, the $S$ and $T$ parameter constraint obl1 ; obl2 ; obl3 on $f$ used to be the strictest one. $f\gtrsim(4-7)~{}\textrm{TeV}$ at $95\%$ C.L. when $t_{\beta}\sim(1-10)$ SLH4 ; SLH5 . In the SLH with spontaneous CP-violation, this constraint is similar, because the $S$ and $T$ parameters are note sensitive to $m_{2}$ and $c_{2,W/Z}$ when $c_{2,W/Z}\ll 1$.

However, since LHC Run II began, the lower limits on exotic particles increase quickly hence the corresponding new physics scales are pushed higher. In the SLH model, $\tilde{X}^{\pm}$ and $\tilde{Y}^{0}(\tilde{\bar{Y}}^{0})$ gauge bosons couple to SM fermions with a suppression factor $v/f$, thus they are difficult to be produced at LHC. However, couplings between $\tilde{Z}^{\prime}$ and SM fermions have the same order with those in SM 111111These couplings are the same in CP-conserving and CP-violation models., thus $\tilde{Z}^{\prime}$ searches at LHC can provide a direct constraint on $f$. Recently, using $36.1~{}\textrm{fb}^{-1}$ luminosity at $\sqrt{s}=13~{}\textrm{TeV}$, ATLAS collaboration set a new constraint $m_{Z^{\prime}}\gtrsim 4.5~{}\textrm{TeV}$ at $95\%$ C.L. Zprime for the sequential standard model (SSM) SSM in which $Z^{\prime}$ couples to SM fermions with the strengths in the SM.

In the SLH model with “anomaly free embedding”, the gauge couplings for fermions are fixed, which can be found in SLHi and SLH3 . The signal strength are then PDG ; SLH2 ; SLH3 ; SSM

[TABLE]

in which

[TABLE]

for $m_{Z^{\prime}}=(4-4.5)~{}\textrm{TeV}$, using the MSTW2008 PDF MSTW . Comparing with the results shown in Zprime and assuming $m_{T,D,S,N_{i}}>m_{Z^{\prime}}/2$, it can be roughly estimated that $f\gtrsim 7.5~{}\textrm{TeV}$ at $95\%$ C.L 121212Recently, Dercks et al. reported new lower limit $f\gtrsim 1.3~{}\textrm{TeV}$ for littlest Higgs model with T-parity TParity , which is quite lower than the limit in SLH model. That is because in the T-parity model, extra $Z^{\prime}$ boson is T-odd thus it cannot have sizable coupling with SM fermion pairs. Thus in that model, direct searches on $Z^{\prime}$ cannot lead to strict constraint on the scale $f$. Comparing with the indirect constraints discussed above, we can see that the $Z^{\prime}$ direct searching experiments can provide the strictest constraint on $f$ in the SLH model for most $\beta$ region.

III.2 Constraints on the Properties of Extra Scalar $h_{2}$

$h_{2}$ couples to SM particles dominantly through its $h$ component, since the couplings between $\eta$ component and SM sector are highly suppressed by the high scale $f$. Experimentally, for a light $h_{2}$, it mainly face the direct searches through $e^{+}e^{-}\rightarrow Zh_{2}$ at LEP; while for a heavy $h_{2}$, it mainly faces the direct searches through $gg\rightarrow h_{2}\rightarrow W^{+}W^{-}/ZZ$ at LHC. Both production cross sections a suppressed by a factor $s^{2}_{\theta}$. When $m_{2}<m_{1}/2$, it should also face the $h_{1}\rightarrow 2h_{2}$ rare decay constraint. Theoretically, the allowed parameter region also depend on the details of EWSB.

For $m_{2}\sim(15-80)~{}\textrm{GeV}$, experimentally, LEP direct searches through $e^{+}e^{-}\rightarrow Z^{*}\rightarrow Zh_{2}$ associated production process gave LEP

[TABLE]

at $95\%$ C.L. assuming $\textrm{Br}_{h_{2}\rightarrow b\bar{b}}=1$. $\tilde{c}_{Zh_{1}h_{2}}$ cannot be constrained at LEP since it is suppressed by a factor $(v/f)^{3}$. When $m_{2}<m_{1}/2$, it must face the $h_{1}$ rare decay constraint as well. In SLH model, the dominant exotic decay channel is $h_{1}\rightarrow 2h_{2}$ with a branching ratio $\textrm{Br}_{h_{1}\rightarrow 2h_{2}}\equiv\Gamma_{h_{1}\rightarrow 2h_{2}}/\Gamma_{1}$. The partial decay width

[TABLE]

while the $h_{1}$ total decay width

[TABLE]

Based on the Higgs signal strengths measurements using full 2016 data stra ; strc , we perform a global-fit and obtain an estimation

[TABLE]

at $95\%$ C.L., which is a bit stricter than the previous constraint from LHC Run I exo . We show the branching ratio distribution in Figure 1. According to the figures, when $m_{2}\sim(20-60)~{}\textrm{GeV}$, we have $s_{\theta}\lesssim(0.04-0.16)$ which is a stricter constraint than that from LEP direct searches. The numerical results are not sensitive to $f$ and $\beta$.

Theoretically, the allowed parameter region also depends on the details of EWSB, especially the contributions from cut-off scale, $\delta m^{2}=-cf^{2}$. In the CP-violation case, (23) becomes

[TABLE]

leaving the other contributions to $\delta V_{h}$ unchanged. For $f=8~{}\textrm{TeV}$, in the light $h_{2}$ scenario, $c$ is favored in the region $(0.01-0.02)$ since larger $c_{2}$ is excluded by the Higgs data. However, if $c\lesssim 0.01$, EWSB requires larger $s_{\theta}$ which was excluded by the Higgs rare decay constraints, thus smaller $c_{2}$ would lead to the exclusion of light $h_{2}$ scenario. Larger $f$ requires smaller $c$, for example, if $f=12~{}\textrm{TeV}$, the lower limit of $c$ reaches about $4\times 10^{-3}$.

For a heavy $h_{2}$ (with $m_{2}\gtrsim 200~{}\textrm{GeV}$), experimentally it is constrained by LHC direct searches. At LHC, the gluon fusion process acquires dominant contribution through top quark loop, and the amplitudes through heavy quark loops are suppressed by $(v/f)^{2}$, so thus $\sigma_{h_{2}}/\sigma_{h_{2},\textrm{SM}}\approx s^{2}_{\theta}$. If $m_{2}<2m_{1}$, the branching ratios of $h_{2}$ are the same as those of a SM-like Higgs boson with the mass $m_{2}$. For $m_{2}>2m_{1}$, another decay channel $h_{2}\rightarrow 2h_{1}$ opens with a partial width

[TABLE]

Its branching ratio can reach $(20-30)\%$ when $m_{2}\gtrsim 300~{}\textrm{GeV}$. If $m_{2}\gtrsim 350~{}\textrm{GeV}$, $h_{2}\rightarrow t\bar{t}$ decay channel can also open. Recently, ATLAS collaboration performed the direct searches through the channels $pp\rightarrow h_{2}\rightarrow W^{+}W-,ZZ$ for $m_{2}>200~{}\textrm{GeV}$ with $36.1~{}\textrm{fb}^{-1}$ luminosity at $\sqrt{s}=13~{}\textrm{TeV}$ h2ZZ ; h2WW . If $m_{2}\lesssim 1~{}\textrm{TeV}$, the strictest constraints come from the $h_{2}\rightarrow ZZ$ decay channel. Comparing with the SM theoretical predictions hig12 ; LHCW , we have a rough estimation

[TABLE]

at $95\%$ C.L. These constraints are a bit weaker than those in the light $m_{2}$ region.

Theoretical constraints here are similar to those in the case with light $h_{2}$. $c\sim(0.005-0.03)$ is favored in the heavy $h_{2}$ scenario. In this scenario, the results are not sensitive to $f$ or $\beta$. Bound on $m_{2}$ is sensitive to $c$, but not sensitive to $s_{\theta}$, which is different from the properties in light $h_{2}$ scenario.

III.3 EDM Constraints

The EDM effective interaction can be written as

[TABLE]

which violated P- and CP-symmetries. In the SM, CP-violation comes only from complex CKM matrix so that the leading contributions to the EDMs of electron and neutron arise at four- and three-loop level respectively. It is estimated that EDM

[TABLE]

both of which are far below the recent experimental constraints EDMe ; EDMn

[TABLE]

at $90\%$ C.L. However, in some BSM models, electron or neutron EDM can be generated at one- or two- loop level, which means it may face strict experimental constraints.

In the SLH model with spontaneous CP-violation, the leading contribution to electron EDM comes from the two-loop “Barr-Zee” type diagrams BZ with $F=t,T,D,S$ running in the loop, see the left diagram in Figure 2.

Following the calculations in BZ ; BZ2 , we have the analytical expression for the EDM of an electron as

[TABLE]

in which the function

[TABLE]

Numerical results showed that $d_{e}$ is not sensitive to the masses of extra heavy quarks. For $0.2\lesssim t_{\beta}\lesssim 8$, in the whole mass region $m_{2}\sim(20-600)~{}\textrm{GeV}$, we have

[TABLE]

The constraints from electron EDM are not strict due to the suppressions by $\theta$ and $f$.

For a neutron, its EDM comes from not only quarks’ EDM, but also their color EDM (CEDM) operator EDM ; BZ ; BZ2

[TABLE]

where $\tilde{d}_{q}$ is the CEDM of the quark, $t^{a}$ denotes the color $\textrm{SU}(3)$ generator, and $i,j$ are color indices. The $u$ quark EDM comes only from the left diagram in Figure 2, just like that for electron; while the $d$ quark EDM acquire contributions from both the left and middle diagrams in Figure 2, because of the left-handed mixing between $d$ and $D$ quarks. The CEDM of quarks come from the right diagram in Figure 2. Calculate at the EW scale, the quarks’ EDM and CEDM in the SLH model with spontaneous CP-violation are BZ2

[TABLE]

in which the function

[TABLE]

After the running to hadron scale, the neutron EDM BZ2

[TABLE]

Numerically, for $0.2\lesssim t_{\beta}\lesssim 8$, in the whole mass region $m_{2}\sim(20-600)~{}\textrm{GeV}$, we have

[TABLE]

which is still below the experimental limit. The constraint from neutron EDM is weaker than that from electron EDM.

Besides the “Barr-Zee” type diagram, there are also one-loop diagrams and Weinberg operator wein contributing to neutron EDM, see the Feynman diagrams in Figure 3.

Following (45)-(48), we can estimate the one-loop contribution to neutron EDM (the left and middle diagrams in Figure 3) as

[TABLE]

where the scale $\mu\sim\mathcal{O}(v)$. This result is sensitive to $m_{D}$ and $m_{2}$. For $m_{D}\sim\mathcal{O}(f)$ and $m_{2}\lesssim\mathcal{O}(v)$, $\left|\delta d_{n,\textrm{1-loop}}\right|\lesssim\mathcal{O}(10^{-29}-10^{-27})~{}e\cdot\textrm{cm}$. The Weinberg operator (the right diagram in Figure 3) wein ,

[TABLE]

in which $f^{abc}$ is the structure constant of $\textrm{SU}(3)$ group, contribute to neutron EDM as BZ2

[TABLE]

In the SLH model with spontaneous CP-violation, we have

[TABLE]

where the function BZ2

[TABLE]

Typical $|\delta d_{n,W}|\lesssim\mathcal{O}(10^{-28})~{}e\cdot\textrm{cm}$. Thus we can conclude that for neutron EDM, the contributions from Figure 3 are sub-dominant.

There are also upper limits on heavy atoms’ EDM. The recent measurement on 199Hg atom’s EDM set new limit $|d_{\textrm{Hg}}|<7.4\times 10^{-30}~{}e\cdot\textrm{cm}$ at $95\%$ C.L. dHg which provides an indirect constraint $d_{n}\lesssim 1.6\times 10^{-26}~{}e\cdot\textrm{cm}$ dnind . The SLH model with spontaneous CP-violation is still allowed by this new indirect constraints. The theoretical estimation on the EDM of Hg contains rather large uncertainties atomEDM , thus it cannot directly provide further constraint on this model.

IV Future Collider Tests of the CP-violation Effects

Recent Higgs data have already confirmed the $0^{+}$ component of $h_{1}$ higCP . Following the idea in mao ; testCPV , we should try to measure tree level $h_{2}VV$ and $h_{1}h_{2}V$ vertices to confirm CP-violation in the scalar sector. For different $h_{2}$ mass, we need different future colliders.

IV.1 Measuring $h_{2}VV$ Vertex

If $h_{2}$ is light (for example, $m_{2}\ll v$), it is difficult to be discovered at LHC because of the large QCD backgrounds at low mass region. To test this scenario, we need future $e^{+}e^{-}$ colliders. For example, at CEPC CEPC or TLEP TLEP with $\sqrt{s}\sim(240-250)~{}\textrm{GeV}$, the $h_{2}VV$ vertex can be measured through the $e^{+}e^{-}\rightarrow Z^{*}\rightarrow Zh_{2}$ associated production process. Its cross section is LEP ; sig

[TABLE]

in which the function

[TABLE]

With $5~{}\textrm{ab}^{-1}$ luminosity at CEPC, the inclusive discovery potential on $s_{\theta}$ can reach $5\sigma$ if $s_{\theta}\sim 0.15$ at low mass region ($m_{2}\lesssim 70~{}\textrm{GeV}$) testCPV through the “recoil mass” technique CEPC ; rec ; rec2 . This result does not depend on the decay channel of $h_{2}$, and it is not sensitive to $m_{2}$ in this region. With the the help of “$p_{T}$ balance cut” method ptb to reduce large backgrounds with photons, the $5\sigma$ discovery bound on $s_{\theta}$ can reach about $0.1$ with a tiny breaking of inclusiveness. If we completely give up the inclusiveness in this measurement and consider only the $h_{2}\rightarrow b\bar{b}$ decay channel, the $5\sigma$ discovery bound on $s_{\theta}$ can be suppressed to about $(4-5)\times 10^{-2}$ according to testCPV 131313Simulation details about the cross sections of the background channels were not shown in the text of testCPV .. This result means the allowed regions obtained in Figure 1 are still possible to be discovered at $5\sigma$ level at CEPC with $5~{}\textrm{ab}^{-1}$ luminosity. For larger $m_{2}$ when it is close to $Z$-peak, large $ZZ$ background will decrease the sensitivity on $s_{\theta}$ measured though this channel.

For light $h_{2}$, we can also measure $s_{\theta}$ through $Z\rightarrow Z^{*}(f\bar{f})h_{2}$ rare decay, if an $e^{+}e^{-}$ collider runs at $Z$-pole ($\sqrt{s}=m_{Z}$). The branching ratio PDG

[TABLE]

where $q$ is the invariant mass of $Z^{*}$. The momentum of $h_{2}$ in initial $Z$ frame and the relative velocity between $h_{2}$ and $Z^{*}$ are respectively

[TABLE]

With $10^{12}$ $Z$-boson events as the goal of a “Tera-$Z$” factory, the typical sensitivity to this rare decay branching ratio is about $(10^{-8}-10^{-7})$ Zrare , which means it has a better sensitivity to discover nonzero $s_{\theta}$ comparing with the $Zh_{2}$ associated production channel in the whole mass region $m_{2}\lesssim 70~{}\textrm{GeV}$.

For a heavy $h_{2}$ (for example, $m_{2}\sim\mathcal{O}(v)$), LHC future direct searches will discover it or set a stricter limit on $s_{\theta}$, through its $ZZ$ decay channel CMSFUT . Through merely visible leptonic decay channel, with $3~{}\textrm{ab}^{-1}$, the $5\sigma$ discovery bounds would be around $s_{\theta}\sim(0.1-0.2)$, which is similar to the current upper limits using the combination of $h_{2}\rightarrow 4\ell$ and $h_{2}\rightarrow 2\ell 2\nu$ channels LHCW ; h2ZZ ; CMSFUT . We also expect the $2\ell 2\nu$ channel can help to increase the sensitivity on $s_{\theta}$ at future LHC. When $m_{2}\gtrsim 0.6~{}\textrm{TeV}$, the $2\ell 2\nu$ channel would become more sensitive than the $4\ell$ channel h2ZZ .

IV.2 Measuring $h_{1}h_{2}V$ Vertex

Based on the improved formalism of SLH model Zh1h2 , we obtain the $Zh_{1}h_{2}$ vertex in (36). $\tilde{c}^{as}_{Zh_{1}h_{2}}$ is suppressed by a factor $(v/f)^{3}\lesssim\mathcal{O}(10^{-5})$ and thus the associated production channels cannot be used to measure this vertex. Similarly, precision measurements on $h_{1}\rightarrow Z^{(*)}h_{2}$ are also useless to test this vertex, since the typical $5\sigma$ discovery bounds for such rare decay channels are of $\mathcal{O}(10^{-3})$ CEPC ; rare . That means we must turn to the heavy neutral gauge boson sector for help.

According to (36), $\tilde{c}^{as}_{Z^{\prime}h_{1}h_{2}}$ is suppressed by a factor $(v/f)$, and there is no suppression in $\tilde{c}^{as}_{Yh_{1}h_{2}}$. These vertices will become helpful to confirm the $0^{-}$ component in at least one of the scalars. Since $m_{Z^{\prime}}\gg m_{1,2}$, the decay branching ratio

[TABLE]

Assuming the heavy quark masses $m_{F}>m_{Z^{\prime}}/2$ thus $\tilde{Z}^{\prime}\rightarrow F\bar{F}$ decay channels cannot be opened. The total width $\Gamma_{Z^{\prime}}\approx 6.5\times 10^{-3}f$ if we choose the “anomaly free” embedding SLHi . Numerically, we have

[TABLE]

When $\beta\sim\pi/4$, this decay channel vanishes, while if $\beta$ is close to [math] or $\pi/2$, there is an enhancement by $t^{-2}_{2\beta}$. It decreases quickly when $f$ increases.

For this process, we need future $pp$ colliders with larger $\sqrt{s}$, for example, $(50-100)~{}\textrm{TeV}$ CEPC ; 100T . Since at LHC, when $m_{Z^{\prime}}\gtrsim 5~{}\textrm{TeV}$, the event number of $pp\rightarrow\tilde{Z}^{\prime}\rightarrow h_{1}h_{2}$ cannot reach $\mathcal{O}(1)$ with $3~{}\textrm{ab}^{-1}$ luminosity 100T . However, with the same luminosity at $\sqrt{s}=100~{}\textrm{TeV}$ $pp$ collider, the events number can reach $N_{pp\rightarrow Z^{\prime}\rightarrow h_{1}h_{2}}\sim\mathcal{O}(10^{2}-10^{3})$ for $m_{Z^{\prime}}\sim 5~{}\textrm{TeV}$, and $N_{pp\rightarrow Z^{\prime}\rightarrow h_{1}h_{2}}\sim\mathcal{O}(10-10^{2})$ for $m_{Z^{\prime}}\sim 10~{}\textrm{TeV}$ 100T . This implies the $Z^{\prime}h_{1}h_{2}$ vertex in the SLH model is testable at $\sqrt{s}=100~{}\textrm{TeV}$ $pp$ collider.

If we can discover nonzero values for both $h_{2}ZZ$ and $Z^{\prime}h_{1}h_{2}$ couplings, we can confirm the CP-violation effects in the scalar sector.

V Conclusions and Discussions

We proposed the possibility of spontaneous CP-violation in the scalar sector of the SLH model in this paper. Through adding a new interaction term, $\epsilon\left(\Phi_{1}^{{\dagger}}\Phi_{2}\right)^{2}+\textrm{H.c.}$, in the scalar potential, the pseudoscalar field $\eta$ can acquire a nonzero VEV which means CP-violation happens spontaneously. Both scalars then become CP-mixing states. In this paper, we denote $h_{1}$ as the SM-like Higgs boson with its mass $m_{1}=125~{}\textrm{GeV}$, and $h_{2}$ is the extra scalar. Based on the improved SLH formalism (see Appendix A in the appendix), we derived the interactions in this model.

Facing strict experimental constraints, the spontaneous CP-violation SLH model is still not excluded. LHC Run II data have already push the lower limit of the scale $f$ to about $7.5~{}\textrm{TeV}$, which means the EW precision tests only provide sub-dominant constraints on $f$. For the extra scalar $h_{2}$, we have two scenarios based on its mass, $m_{2}\sim\mathcal{O}(v)$ or $m_{2}\ll v$. For a light $h_{2}$, the most strict constraint comes from $h_{1}\rightarrow 2h_{2}$ rare decay channel. The $95\%$ C.L. upper limit on $s_{\theta}$ is $(0.04-0.16)$ for $m_{2}\sim(20-60)~{}\textrm{GeV}$. While for large $m_{2}\sim\mathcal{O}(v)$, the $95\%$ C.L. upper limit on $s_{\theta}$ varies in the region $(0.1-0.4)$, especially when $m_{2}\sim(300-700)~{}\textrm{GeV}$, the $95\%$ C.L. upper limit on $s_{\theta}$ is about $0.2$. In both scenarios, tiny but nonzero contributions from cut-off scale are necessary. As a CP-violation model, it must also face the EDM constraints. Since the effects are suppressed by $s_{\theta}v/f$, the constraints are weak. The most strict EDM constraint comes from electron, which favors $0.2\lesssim t_{\beta}\lesssim 8$ in the whole $m_{2}\sim(20-600)~{}\textrm{GeV}$ mass region.

We also discussed the future collider tests of this model. The basic idea is to discover nonzero $h_{2}VV$ and $Vh_{1}h_{2}$ vertices. For a light $h_{2}$ , we can test $h_{2}ZZ$ vertex at future $e^{+}e^{-}$ colliders, as Higgs factories or $Z$-factory. With $5~{}\textrm{ab}^{-1}$ at CEPC, for $m_{2}\lesssim 70~{}\textrm{GeV}$, $s_{\theta}\sim(4-5)\times 10^{-2}$ can be discovered at $5\sigma$ level; while with $10^{12}$ $Z$-boson events at $Z$-pole, we can have a better sensitivity in the same mass region. For a heavy $h_{2}$ with $m_{2}\sim\mathcal{O}(v)$, the vertex can be tested through $gg\rightarrow h_{2}\rightarrow ZZ$ channel at LHC. With $3~{}\textrm{ab}^{-1}$ luminosity, the $5\sigma$ discovery bound is around $(0.1-0.2)$ through merely the $4\ell$ decay channel. The $2\ell 2\nu$ decay channel is also expected to help increase the sensitivity on $s_{\theta}$, especially in large $m_{2}$ region. Based on the improved formalism, we know the $Zh_{1}h_{2}$ vertex is suppressed by $(v/f)^{3}$, thus we must ask a heavy gauge boson, such as $Z^{\prime}$, for help. Since $\textrm{Br}_{Z^{\prime}\rightarrow h_{1}h_{2}}\lesssim\mathcal{O}(10^{-4}-10^{-3})$, it is difficult to be tested at LHC. We need $pp$ colliders with larger $\sqrt{s}$. For example, if $\sqrt{s}=100~{}\textrm{TeV}$, with $3~{}\textrm{ab}^{-1}$ luminosity, we can obtain $\mathcal{O}(10^{2}-10^{4})$ events for $pp\rightarrow\tilde{Z}^{\prime}\rightarrow h_{1}h_{2}$ process in the mass region $m_{Z^{\prime}}\sim(5-10)~{}\textrm{TeV}$ which means it may become testable. CP-violation in the scalar sector will be confirmed if both nonzero $h_{2}ZZ$ and $Z^{\prime}h_{1}h_{2}$ vertices are discovered.

This model is attractive both theoretically and phenomenologically. Theoretically, in this model, we proposes a new possible CP-violation source, which may provide new understanding of the matter-antimatter asymmetry problem in the Universe. Besides this, the spontaneous CP-violation mechanism is also a possible solution to the strong-CP problem, which is worthy to study further. This model is also a candidate to connect between the composite Higgs mechanism and CP-violation in the scalar sector. Based on this, new CP-violation effects are naturally suppressed by the global symmetry breaking scale $f$, as shown in the calculation of electron and neutron EDM.

Phenomenologically, it is an application of the basic idea to measure $h_{2}VV$ and $Vh_{1}h_{2}$ vertices. It provides an example to show how extra scalars and gauge bosons can help to confirm new CP-violation sources, which also implies the importance to search for $VVS$- and $VSS$-type vertices. It also shows another motivation for future $e^{+}e^{-}$ and $pp$ colliders.

Acknowledgement

We thank Jordy de Vries, Shi-ping He, Fa-peng Huang, Gang Li, Jia Liu, Lian-tao Wang, Ke-pan Xie, Ling-xiao Xu, Wen Yin, Felix Yu, Chen Zhang, and Shou-hua Zhu for helpful discussions. This work was partly supported by the China Postdoctoral Science Foundation (Grant No. 2017M610992).

Appendix A Improved Formalism of the SLH Model

In this section show the improved formalism for the SLH model based on Zh1h2 . The neutral scalar sector (including six degrees of freedom) can be divided into CP-even and CP-odd parts. The CP-odd part, denoting as $G_{i}$ running over $\eta$, $G$, $G^{\prime}$, and $y^{2}$, is not canonically-normalized. We can write the kinetic term as

[TABLE]

The matrix elements of $\mathbb{K}$ are calculated to $\mathcal{O}\left((v/f)^{3}\right)$ in Zh1h2 . If we rewrite this term in another basis $S_{i}=U_{ij}G_{j}$ which is canonically-normalized

[TABLE]

thus we can define a inner product $\langle S_{i}|S_{j}\rangle=\delta_{ij}$ in the linear space spanned by the scalars $S_{i}$. A straightforward calculation shows that

[TABLE]

The VEVs in $\Phi_{1,2}$ will lead to two-point transitions between gauge bosons and pseudo-scalars as

[TABLE]

where $V_{p}$ denotes a gauge boson running over $Z$, $Z^{\prime}$, and $Y^{2}$, and $\mathbb{F}$ is a $4\times 3$ matrix. The matrix elements of $\mathbb{F}$ are also calculated to $\mathcal{O}\left((v/f)^{3}\right)$ in Zh1h2 . The gauge fixing term must provide the two-point transition like

[TABLE]

to cancel all contributions from (A.4). Define

[TABLE]

in the convention of SLH3 (which is also the convention of this paper), we can derive that

[TABLE]

through a straightforward calculation where $\mathbb{M}^{2}_{V}$ is the mass matrix for gauge bosons in the basis $(Z,Z^{\prime},Y^{2})$. Calculate to the leading order of $(v/f)$ for every matrix element, we have

[TABLE]

Using an orthogonal matrix $\mathbb{R}$, we can diagonalize $\mathbb{M}^{2}_{V}$ as

[TABLE]

where $\tilde{V}_{p}$ denotes the mass eigenstate of a gauge boson and $m_{p}$ is its mass. The matrix elements of $\mathbb{R}$ are calculated to $\mathcal{O}\left((v/f)^{3}\right)$ in Zh1h2 as well. For simplify, to this order, the off-diagonal elements can also be expressed as

[TABLE]

It is natural for us to define

[TABLE]

According to $\langle\eta|\eta\rangle=\left(\mathbb{K}^{-1}\right)_{11}\approx 1+(2/t^{2}_{2\beta})(v/f)^{2}$, we should also define

[TABLE]

It is easy to check that in the basis $\left(\tilde{\eta},\tilde{G}_{p}\right)$, the kinetic part is canonically-normalized. (A.5) also becomes $m_{p}\left(\partial_{\mu}\tilde{V}_{p}^{\mu}\right)\tilde{G}_{p}$, thus it is natural to choose the gauge fixing term as

[TABLE]

It is now clear that $\tilde{G}_{p}$ is the corresponding Goldstone eaten by $\tilde{V}_{p}$, and its mass should be $\sqrt{\xi_{p}}m_{p}$ where $\xi_{p}$ is the corresponding gauge parameter. To this step, we have already built the formalism to treat a model with non-canonically-normalized scalar sector and the SLH model is one of the examples. The main point is that all the two-point transitions must be carefully canceled if we don’t want these kind of Feynman diagrams appear during calculation.

Because of the $\eta$ components in the Goldstone fields, the interactions including $\eta$ must be changed comparing with the naively calculated case. We divide $\mathbb{F}$ into

[TABLE]

where $\tilde{f}_{p}=\mathbb{F}_{p1}$ is a $1\times 3$ vector and $\tilde{\mathbb{F}}$ is a $3\times 3$ matrix. Thus for any kind of couplings including the pseudo-scalar degrees of freedom, if we write the coefficients as $\left(c_{\eta},c_{j}\right)$ in $G_{i}$ basis where $c_{j}$ runs for the couplings including $G$, $G^{\prime}$ and $y^{2}$, the physical coupling should be

[TABLE]

For example, the anti-symmetric type $Vh\eta$ couplings in mass eigenstates can be parameterized as

[TABLE]

With the improved formalism, we can calculate to the leading order of $(v/f)$ as

[TABLE]

The first two results are quite different from those appearing in previous papers SLH2 ; SLH3 . Similarly, the Yukawa couplings between $\eta$ and SM fermions can be parameterized as

[TABLE]

According to (A.15), $c_{\eta,f}=0$ to all order of $(v/f)$ for $f=\nu,\ell,u,c,b$. This result is also quite different from that in previous papers SLH2 ; SLH3 . For $f=t,d,s$, to the leading order of $(v/f)$, we have

[TABLE]

which is generated by the left-handed mixing between SM fermion and additional heavy fermion. Formally all these results can be calculated to all order of $(v/f)$, though some of the results are extremely lengthy.