A Simple Model of Dark Matter and CP Violation

Due to the mixing between the doublet and singlet scalars, the 125-GeV Higgs boson may have a new invisible decay channel, if $m_{h}\geq 2m_{\chi}$, and its coupling strength to the SM fields is universally reduced.

We first consider the constraint on the invisible decay rate given by CMS Sirunyan et al. (2019a):⁶⁶6The bound given by ATLAS Aaboud et al. (2019a), $BR(h_{1}\to{\rm inv})<0.26$ (95% C.L.), is weaker than the one given by CMS.

$$BR(h_{1}\to{\rm inv})<0.19~{}({\rm 95\%~{}confidence~{}level~{}(C.L.)})~{}.$$

(17)

For $m_{\chi}<m_{h_{1}}/2$, the $h_{1}\to\chi\overline{\chi}$ partial width is given by

$$\Gamma(h_{1}\to\chi\overline{\chi})=\frac{\epsilon^{2}}{8\pi}\frac{m_{\chi}^{2}}{v_{s}^{2}}m_{h_{1}}\Bigg{(}1-\frac{4m_{\chi}^{2}}{m_{h_{1}}^{2}}\Bigg{)}^{1/2}\Bigg{[}s_{23}^{2}\Bigg{(}1-\frac{4m_{\chi}^{2}}{m_{h_{1}}^{2}}\Bigg{)}+c_{23}^{2}\Bigg{]}~{}.$$

(18)

In FIG. 1, we show the constraint in the $v_{s}-m_{\chi}$ plane for $\epsilon=0.1$ and a few choices of $\theta_{23}$. The colored region is allowed by the invisible decay constraint. The small gray region at the bottom of the plot denotes the region where $\lambda_{\chi}=\sqrt{2}m_{\chi}/v_{s}>4\pi$ and violates the perturbativity bound. The mixing angle $\theta_{23}$ has a significant impact on the constraints when $m_{\chi}$ is close to $m_{h_{1}}/2$, in which region the phase space suppression becomes prominent. If $h_{1}$ contains more CP-even component, i.e., $s_{23}^{2}$ becomes larger, then the allowed phase space also becomes larger due to the extra factor of $1-4m_{\chi}^{2}/m_{h_{1}}^{2}$.

Next we consider the constraints coming from the measured Higgs signal strengths Zyla et al. (2020) listed in TABLE. 2.

In the model, the couplings of $h_{1}$ to the other SM fields are modified universally by a factor of $1-\epsilon^{2}/2$, leading to a reduction in the production rate by $1-\epsilon^{2}$. The branching ratios, however, remains the same unless new invisible decay channel opens up when $m_{h_{1}}>2m_{\chi}$. The strongest bound comes from the slightly enhanced signal strength in $\mu_{ZZ}$, which at 95% C.L. requires

$$|\epsilon|<0.125\ .$$

(19)

On the other hand, if $h_{1}\to\chi\overline{\chi}$ is kinematically allowed, then the signal strengths of the SM channels are modified to be

$$\mu=(1-\epsilon^{2})\big{[}1-BR(h_{1}\to\chi\overline{\chi})\big{]}=1-\epsilon^{2}-\frac{\Gamma(h_{1}\to\chi\overline{\chi})}{\Gamma_{h_{1}}^{\rm SM}}+\mathcal{O}(\epsilon^{4})~{},$$

(20)

where $\Gamma_{h_{1}}^{\rm SM}$ is the total decay width of the 125-GeV Higgs predicted by the SM. Seeing that such modifications would make the constraint on $\epsilon$ even stronger, we do not consider this case and only explore the case of $m_{\chi}>m_{h_{1}}/2$ in the benchmark studies, where we fix $\epsilon=0.1$.

We now consider the Peskin-Takeuchi $S$ and $T$ parameters defined in Ref. Peskin and Takeuchi (1992). The current fits given by PDG Zyla et al. (2020) are

	$\displaystyle\Delta S$	$\displaystyle=0.00\pm 0.07~{},$		(21)
	$\displaystyle\Delta T$	$\displaystyle=0.05\pm 0.06~{},$

with a correlation of $0.92$. In our model, because the vector bosons only couple to the physical scalars through their $\phi_{1}$-components, the mixing of which is determined entirely by $\epsilon$ and $\theta_{12}$ as shown in Eq. (10)⁷⁷7$\theta_{23}$ only parametrizes the mixing between $\phi_{2}$ and $a$, but not the gauge couplings, and hence does not take part in the oblique corrections., $S$ and $T$ parameters only depend on $m_{h_{2}},m_{h_{3}},\epsilon$ and, to a much less extent, $\theta_{12}$. We fix $\epsilon=0.1$ as in Sec. III.1, and choose two sets of heavy scalar masses:

$\displaystyle\begin{split}(m_{h_{2}},m_{h_{3}})=(280,420)~{}\mbox{GeV}~{},\\ (m_{h_{2}},m_{h_{3}})=(280,600)~{}\mbox{GeV}~{}.\end{split}$

(22)

The first set is chosen to allow the $h_{3}\to h_{1}h_{2}$, $h_{2}\to 2h_{1}$, and $h_{3}\to 2h_{1}$ decays, while the second further allows the $h_{3}\to 2h_{2}$ decay. For both mass sets, the above constraint can be satisfied within ${1-2}\sigma$ for all possible values of $\theta_{12}$. In fact the oblique corrections have very little dependence on $\theta_{12}$, whose contributions are suppressed by $\epsilon^{2}$. We show the 68% C.L. and 95% C.L. contours in the $\Delta T$-$\Delta S$ plane, as well as the values for both sets of masses, in FIG. 2. The detailed formulas for the electroweak oblique observables are given in Appendix B.

Here we consider constraints from direct searches of heavy neutral scalars at the LHC, focusing on the diboson final states: $WW$, $ZZ$ and $h_{1}h_{1}$. The $t\bar{t}$ channel is less stringent. The light decay channels such as $b\bar{b}$, $\tau^{+}\tau^{-}$, and $\gamma\gamma$ are also less stringent because of suppressed decay BRs. ⁸⁸8Unlike in the C2HDM, we do not have to consider $h_{i}\to h_{1}Z$, $i=2,3$, decays, which are absent in our model because the singlet scalar does not contain any “eaten” Goldstone bosons.

Because the singlet scalar $S$ does not couple to the SM gauge bosons and fermions directly, $h_{2}$ and $h_{3}$ couple to the SM gauge bosons and fermions only through their $\phi_{1}$ component. As such, their productions will go through the gluon-fusion (ggF) channel and are suppressed by the alignment parameter $\epsilon$:

$\displaystyle\begin{split}&\sigma(gg\to h_{i})=R_{i1}^{2}\ \sigma^{\rm SM}(gg\to h_{i})~{},\\ &\Gamma(h_{i}\to f_{\rm SM})=R_{i1}^{2}\ \Gamma^{\rm SM}(h_{i}\to f_{\rm SM})~{},\end{split}$

(23)

where $R_{11}=1-\epsilon^{2}/2$, $R_{21}=-\epsilon s_{12}$, and $R_{31}=-\epsilon c_{12}$. In addition, $\sigma^{\rm SM}(gg\to h_{i})$ and $\Gamma^{\rm SM}(h_{i}\to f_{\rm SM})$ denote the SM production rate and decay partial width at the mass $m_{h_{i}}$, which we obtain from Ref. de Florian et al. (2016). In addition to direct two-body decays from $h_{i}\to VV/h_{1}h_{1}$, we also include Higgs-to-Higgs decays such as $h_{3}\to(h_{2}\to 2h_{1})+h_{1}$.

We base our constraints on those in Refs. Aaboud et al. (2019b, 2018a); Sirunyan et al. (2018a); Aaboud et al. (2019c); Sirunyan et al. (2018b); Aaboud et al. (2018b, c); Sirunyan et al. (2019b); Aaboud et al. (2019d); Sirunyan et al. (2018c, 2019c); Aad et al. (2020a); Sirunyan et al. (2018d, 2020); Aad et al. (2020b, 2021). As seen in Eq. (23), the direct search constraints are all sensitive to $|\epsilon|$, which suppresses the production rates by a factor of $\epsilon^{2}$. Moreover, while the $WW$ and $ZZ$ constraints only depend on $\epsilon$ and $\theta_{12}$ in addition to $m_{h_{2}}$ and $m_{h_{3}}$, the $h_{1}h_{1}$ constraints further depend on $\theta_{23}$, $\delta_{2}$, and $\delta_{3}$ through their participation in the $g_{112}$ and $g_{113}$ couplings. Moreover, $\theta_{23}$ always shows up in $g_{112}$ and $g_{113}$ in the combination of $\theta_{\delta_{3}}-(\theta_{12}+2\theta_{23})$, as can be seen in Eq. (15). We choose to fix $|\delta_{3}|=3.5$ and focus on the effects of $\delta_{2}$ on the constraints. In order to maximize the $3h_{1}$ cross sections, we further maximize $|{g}_{123}|$ by choosing $\theta_{\delta_{3}}-2(\theta_{12}+\theta_{23})=-\pi/2$. Finally, we choose $\theta_{12}$ in a way that the ggF production rates of $h_{2}$ and $h_{3}$ are similar, which implies $\theta_{12}=0.73$ for the first mass set and $\theta_{12}=0.41$ for the second mass set in Eq. (22).

We examine the direct search constraints in the $\delta_{2}$-$|\epsilon|$ plane for the two mass sets, and in the $m_{h_{2}}$-$|\epsilon|$ plane with $(\delta_{2},m_{h_{3}})=(-0.5,420~{}{\rm GeV})$ and $(\delta_{2},m_{h_{3}})=(-0.5,600~{}{\rm GeV})$, respectively, focusing on the region where $\delta_{2}\in[-5,5]$, $m_{h_{2}}\in[260,600]$ GeV, and $|\epsilon|\in[0,0.5]$. For definiteness as well as to impose the constraints on the parameter space in the strictest manner, we also neglect the $h_{2,3}\to\chi\bar{\chi}$ decays. We present the results in FIG. 3, where we also include the $\mu_{ZZ}$ constraint at 95% C.L. in Eq. (19). Note that in FIGS. 3(c) and (d), there are two sudden jumps caused by the onset of the $h_{3}\to h_{2}h_{1}$ decay when $m_{h_{3}}\geq m_{h_{1}}+m_{h_{2}}$. It can be seen that the direct search constraints are all less stringent than the $\mu_{ZZ}$ constraint, and thus for our choice of $\epsilon=0.1$, both mass sets are completely safe from the LHC direct search constraints within the specified region.

We remark in this section that our CPVDM model does not generate new EDM contributions up to at least two-loop level, in sharp contrast with the C2HMDs. This is mainly due to the fact that new sources of CPV in our model are confined to the scalar and dark sectors: the CPV interactions take place among the scalars, or between $\chi\overline{\chi}$ and the messenger scalar $S$. There is no new source of CPV in the visible fermionic sector as the singlet scalar $S$ does not have Yukawa interactions with the SM fermions. Therefore, even though the 125-GeV SM-like Higgs could have a component in $S$ due to the mass mixing, such a component does not introduce any CP-odd coupling of the 125-GeV Higgs to the SM fermions. Similar consideration applies to the heavy scalars couplings to SM fermions, which are induced only through the doublet component and remain CP-even. This explains why no electron EDM appears at one loop.

Now we exhibit potential two-loop contributions to $e-e-\gamma$ coupling in our model in FIG. 4. To introduce CPV into the three-point electron-photon interaction, we must insert at least one trilinear/quartic CPV scalar vertex or one CPV scalar-$\chi\bar{\chi}$ vertex into the internal loops. In the first case, the scalars have to either all attach to the electron lines, such as that in FIG. 4(a), or form an internal loop, such as that in FIGS. 4(b) and (c). As for the second case, since $\chi$ only interacts with the scalars, it must form an internal loop, as shown in FIG. 4(d). For the cases of FIGS. 4(a), (b), and (c), no factor of $\gamma^{5}$ would appear, and hence no EDM would be induced. As to FIG. 4(d), after taking the Dirac trace of the fermion loop, no Lorentz-invariant terms of the form $\epsilon^{\alpha\beta\rho\sigma}p_{1\alpha}p_{2\beta}p_{3\rho}p_{4\sigma}$ would be induced, where $\epsilon^{\alpha\beta\mu\nu}$ is the rank-four Levi-Civita symbol and $p_{1,2,3,4}$ are generic four-momenta, and hence there is no EDM contribution either.

In this section, we briefly comment on the contributions to the muon anomalous magnetic dipole moment, $(g-2)_{\mu}$, in our model. The latest measurement was made in the E989 experiment at Fermilab, and the result was given by Abi et al. (2021)

$$a_{\mu}^{\rm FNAL}=116592040(54)\times 10^{-11}~{},$$

(24)

while the SM prediction is given by Aoyama et al. (2020)

$$a_{\mu}^{\rm SM}=116591810(43)\times 10^{-11}~{},$$

(25)

leading to a 4.2$\sigma$ discrepancy

$$\Delta a_{\mu}=(251\pm 59)\times 10^{-11}~{}.$$

(26)

The leading contributions are the one-loop diagrams and the two-loop Barr-Zee diagrams with top-, bottom-, $\tau$-, and $W$-loops running in the loop, as demonstrated in FIG. 5. Denoting their contributions by $\Delta a_{\mu}^{(1)}$ and $\Delta a_{\mu}^{(2)}$, respectively, we have Ilisie (2015); Chen et al. (2020); Chiang and Yagyu (2021); Chen et al. (2021)

$$\Delta a_{\mu}^{(1)}=\sum_{i}\epsilon^{2}\kappa_{i}\frac{m_{\mu}^{2}}{8\pi^{2}v^{2}}\int_{0}^{1}dx\frac{\tau^{\mu}_{i}x^{2}(2-x)}{1-x(1-\tau^{\mu}_{i}x)}~{},$$

(27)

	$\displaystyle\Delta a_{\mu}^{(2)}$	$\displaystyle=\sum_{i}\epsilon^{2}\kappa_{i}\frac{\alpha m_{\mu}^{2}}{8\pi^{3}v^{2}}\Bigg{[}\sum_{f=t,b,\tau}N_{c}^{f}Q_{f}^{2}\int_{0}^{1}dx\tau^{f}_{i}\frac{2x(1-x)-1}{\tau^{f}_{i}-x(1-x)}\ln\left(\frac{\tau^{f}_{i}}{x(1-x)}\right)$		(28)
		$\displaystyle\quad+\frac{1}{2}\int_{0}^{1}dx\frac{x\left[3x(4x-1)+10\right]\lambda_{i}-x(1-x)}{\lambda_{i}-x(1-x)}\ln\left(\frac{\lambda_{i}}{x(1-x)}\right)\Bigg{]}~{},$

where $\tau^{f}_{i}=m_{f}^{2}/m_{h_{i}}^{2}$, $\lambda_{i}=m_{W}^{2}/m_{h_{i}}^{2}$, and

$$\kappa_{1}=-1,~{}\kappa_{2}=s_{12}^{2},~{}\kappa_{3}=c_{12}^{2}~{}.$$

(29)

Note that they are both suppressed by $\epsilon^{2}$. With the two chosen scalar mass sets, we have $\Delta a_{\mu}\sim\mathcal{O}\left(10^{-13}\right)$, which is negligible compared to Eq. (26). Thus, $(g-2)_{\mu}$ cannot be addressed in our model.

The DM relic density is measured to be $\Omega_{\chi}h^{2}=0.1197\pm 0.0022$ Ade et al. (2016). The $\chi\overline{\chi}$ annihilation processes are shown in FIG. 6. We use micrOMEGAs Belanger et al. (2006, 2007, 2009, 2010, 2014) to calculate the relic density of $\chi$, which is mainly determined by $m_{\chi}=\lambda_{\chi}v_{s}/\sqrt{2}$ and $v_{s}$. Other input parameters such as $\delta_{2}$ do not have a significant impact on the constraints. In FIG. 7 we show the relic density constraints in the $v_{s}$-$m_{\chi}$ plane for the two mass sets mentioned in Eq. (22) with $\delta_{2}=-0.5$. The small blue regions denote the parameter space that has a relic density within the experimental 2$\sigma$ bounds, and the orange regions those below the lower 2$\sigma$ bound. We also plot the $m_{\chi}=m_{h_{1}}/2$ contour (red dotted), the $m_{\chi}=m_{h_{2}}/2$ contour (purple dotted), and the $m_{\chi}=m_{h_{3}}/2$ contour (magenta dotted) to show the resonance effect.

As can be seen in FIG. 7, the annihilation process is quite efficient for a DM mass that is sufficiently heavy, $m_{\chi}\gtrsim 100$ GeV, and/or small $v_{s}$. In particular, a small $v_{s}$ increases $\lambda_{\chi}$ for a fixed $m_{\chi}$, which makes the annihilation rate larger. When the DM mass is close to the resonance region, $m_{\chi}\approx m_{h_{i}}/2$, $i=1,2,3$, the annihilation rate becomes enhanced and the relic density reduced, as can be seen from the plot. For our benchmark study, we choose the following four DM masses: $m_{\chi}=156,187,280,420$ GeV, which will be further studied in Sec. IV.

As for the DM direct searches, we quote the results of XENON1T Aprile et al. (2018). Since $\delta_{2}$ is only relevant to scalar interactions, it does not have a significant impact on the DM-nucleon scattering at the leading order. Furthermore, both scalar mass sets in Eq. (22) give roughly the same results. In FIG. 8(a), we show the experimental constraint in the $m_{\chi}$-$v_{s}$ plane for mass set 1 and $\delta_{2}=-0.5$, taking the average of the spin-independent DM-proton and -neutron scattering cross sections.

Finally, for indirect DM searches, we quote the results of the Fermi-LAT+MAGIC combined analysis Ahnen et al. (2016) to constrain dark matter annihilation in the $b\bar{b}$ and $W^{+}W^{-}$ channels. In the $2h_{1}$ channel we use the recent results given by MAGIC Acciari et al. (2021). We perform a scan over $m_{\chi},v_{s},\delta_{2}$ for the $\chi$-pair annihilation rates into different final states. The dominant annihilation channel is largely determined by kinematics: for $m_{\chi}\lesssim 85$ GeV, $\chi$-pairs mainly annihilate into $b\bar{b}$; for $85~{}{\rm GeV}\lesssim$ $m_{\chi}\lesssim 200$ GeV, they mainly annihilate into $W^{+}W^{-}$ or $h_{1}h_{1}$; for $m_{\chi}\gtrsim 200$ GeV, they mainly annihilate into heavy scalars, and occasionally to $W^{+}W^{-}$ or $h_{1}h_{1}$.

Again $\delta_{2}$ does not have a considerable impact for annihilations into the SM particles, and we find the $b\bar{b}$ and $2h_{1}$ constraints are always satisfied. In FIG. 8(b) we show the $W^{+}W^{-}$ constraint in the $m_{\chi}$-$v_{s}$ plane for the two scalar mass sets in Eq. (22). As shown in the plot, the indirect detection constraints are mostly caused by heavy scalar resonances near $m_{\chi}=140,210,300$ GeV (black dashed lines).

Before concluding this section, we summarize the DM relic, direct, and indirect detection constraints in FIG. 9.