Invisible dark matter decays of a non-Standard Model like CP-even scalar boson

2.1 Overview of the N2HDM

The N2HDM comprises two-Higgs-doublets ( ${\Phi }_1$ and ${\Phi }_2$ ) and a real singlet ( ${\Phi }_S$ ), and its potential reads (following the conventions in Mühlleitner et al. (2017a)),

For the case where the ${\mathbb{Z}}_2^{\prime }$ symmetry is intact, only the Higgs doublets acquire VEVs. This case is named “the Dark Singlet Phase (DPS)” by the authors of Mühlleitner et al. (2017a), and is the one we are considering (we will denote the model in the subsequent sections by DSP-N2HDM). Moreover, the vacuum structure is,

The couplings of $H_1$ and $H_2$ to SM particles are the same as in Type-I 2HDM. For $H_2$ , which is our focus in this paper, $c\left(H_2\bar{f}f\right)=sin\alpha /sin\beta$ , while $c\left(H_{2}VV\right)=cos\left(\alpha -\beta \right)$ . In addition to these couplings, we have the triple-Higgs couplings that are responsible for the possible decays of the CP-even Higgs scalars into DM. Particularly,

Finally, the DSP-N2HDM is represented by 11 input parameters, $$v,tan\beta ,m_{H_1},m_{H_2},m_{H_D},m_A,m_{H^{\pm }},\alpha ,{\lambda }_6,{\lambda }_7,{\lambda }_8,$$ where $m_{H_1}$ is taken to be the mass of the SM Higgs boson, while $m_{H_D}$ is the mass of the DM candidate, $m_A$ is the mass of the CP-odd Higgs boson, $m_{H^{\pm }}$ is the mass of the charged Higgs bosons. The other parameters were defined earlier.

2.2 Overview of the NMSSM

The NMSSM is a well-known SUSY model that solves the $\mu -$ problem in the MSSM due to adding a singlet superfield $S$ . This singlet couples to the Higgs doublets, extending the well-known MSSM superpotential. Following the convention in Ellwanger et al. (2010), the superpotential is,

Furthermore, the soft SUSY breaking lagrangian of the NMSSM is,

The non-SM CP-even Higgs couplings to two neutralinos (with a focus on the NLSP and the LSP) is,

Finally, the input parameters of the NMSSM, which are specified at the GUT scale, are $$m_0,m_{1/2},A_0,A_{\lambda },A_{\kappa },\lambda ,\kappa ,tan\beta ,{\mu }_{\text{eff}}.$$ comprising scalar and gaugino mass parameters, trilinear couplings, the singlet-Higgs coupling $\lambda$ , the cubic singlet self-coupling $\kappa$ , and ${\mu }_{\text{eff}}\equiv \lambda s$ . Both $m_{H_u}$ and $m_{H_d}$ can have the same value as $m_0$ at the GUT scale, but the case where they differ from $m_0$ is called the Non-Universal Higgs NMSSM (NUH-NMSSM), which we consider here. However, due to the nature of the tool we use (described in Section 3) these two parameters are computed.

3.1 The DSP-N2HDM

To generate the mass-spectrum generator of the N2HDM we use $N2HDECAY$ (Engeln et al., 2019), and is embedded in the state-of-the-art scanning tool $ScannerS$ (Mühlleitner et al., 2022). Specifically, we consider the dark-singlet phase of the N2HDM and scan the parameter space over the following ranges: $$m_{H_2}=[10-1500]\mathrm{GeV},m_{H_D}=[1-1500]\mathrm{GeV},m_A=[1-1500]\mathrm{GeV},m_{H^{\pm }}=[150-1500]\mathrm{GeV},\alpha =[-1.5-1.6],tan\beta =[0.8-25],m_{12}^2=[1\times 10^{-3}-5\times 10^5]{\mathrm{GeV}}^2,{\lambda }_6=[0-20],{\lambda }_{7,8}=[-30-30].$$ The package applies stringent theoretical and experimental constraints on the model. The former includes perturbative unitarity, boundedness, vacuum stability, constraints from electroweak precision, and flavor conservation. The latter include Higgs searches and measurements via interfacing with $HiggsBounds$ $(v.5.9)$ (Bechtle et al., 2020) and $HiggsSignals$ $(v.2.6)$ (Bechtle et al., 2021), and Dark matter constraints using $MicrOMEGAs$ $(v.5)$ (Belanger et al., 2021). It is worth mentioning that HiggsBounds provides the production cross-section of $H_2$ via gluon fusion at 13 TeV via tabulated results from $SusHi1.6.1$ (Harlander et al., 2017).

3.2 The NUH-NMSSM

The parameter space of the NMSSM was scanned using $NMSSMTools$ $(v.5.6)$ (Ellwanger et al., 0000, 2005; Ellwanger and Hugonie, 2006; Das et al., 2012). A combination of random and Markov Chain Monte Carlo (MCMC) sampling methods was deployed. The scanned ranges of the input parameters are: $$m_0=[1-4000]\mathrm{GeV},m_{1/2}=[1-4000]\mathrm{GeV},A_0,A_{\lambda },A_{\kappa }=[-3000-3000]\mathrm{GeV},tan\beta =[1-30]\lambda ,\kappa =[0.01-0.7],{\mu }_{eff}=[100-1500]\mathrm{GeV}.$$ As mentioned in Section 2.2, the parameters $m_{H_u}$ and $m_{H_d}$ are computed, and hence what we are considering here is the NUH-NMSSM. The constraints implemented in $NMSSMTools$ are similar to those in $ScannerS$ if not more comprehensive (see the tool’s History Section for all constraints included and the corresponding references). These include non-tachyonic masses, successful Electroweak Symmetry Breaking, and the existence of a global minimum, which represents the theoretical constraints. In contrast, the phenomenological ones include: Flavor physics, LHC constraints on sparticles, satisfying the upper limit on dark matter relic density using $MicrOMEGAs$ $(v.5)$ . As for the SM-Higgs couplings, we consider recent results from the LHC, and select points to be within $3\sigma$ away from the central values of the combination of ATLAS and CMS results (Tumasyan et al., 2022; ATLAS Collaboration, 2022), which makes our results subject to the latest LHC limits. Finally, the production cross-section of $H_2$ via gluon fusion is calculated through the method described in Ellwanger and Hugonie (2022), where the data for the BSM production cross-section at 13 TeV is obtained from CERN Yellow Report (0000) and subsequently multiplied by the relevant reduced coupling squared. Thus, the calculation accounts for a significant portion of the radiative Quantum Chromodynamics corrections, leaving theoretical uncertainties of the order of $O\left(10\text{\%}\right)$ .

4.1 The DSP-N2HDM

The results of the N2HDM with a singlet dark matter are shown in Figures 1.a. to 1.e. We have restricted our analysis to points where $m_{H_2}\le 1000$ GeV, for which a total of 20k successful points are collected. Fig. 1.a. shows a scatter plot of the branching ratio of $H_2$ to a pair of dark matter particles ( $Br(H_2\to H_D{H}_D)$ ) versus $m_{H_2}$ . The color indicates the dark matter mass, $m_{H_D}$ . As can be seen, $m_{H_2}$ ranges from 222 GeV to 1000 GeV, while $m_{H_D}$ ranges from $100$ GeV to $497$ GeV. The branching ratio $Br(H_2\to H_D{H}_D)$ can reach values close to one, especially for regions where the mass of the dark singlet is below 200 GeV. As the mass increases, the branching ratio reduces significantly, as seen in the plot’s red corner. A representative point where $Br(H_2\to H_D{H}_D)\sim 1$ corresponds to the parameter space point: $m_{H_2}=337$ GeV, $m_{H_D}=127$ GeV, $tan\beta$ $=$ 3, $\sigma (gg\to H_2)=1.1$ pb, and $\Omega h^2=O\left(10^{-9}\right)$ .

Close

Fig. 1

Two dimensional scatter plots of (a) $Br(H_2\to H_D{H}_D)$ versus $m_{H_2}$ with $m_{H_D}$ shown in color (b) $\sigma (gg\to H_2)$ versus $m_{H_2}$ (c) $Br(H_2\to H_D{H}_D)$ versus $tan\beta$ (d) $\sigma \times Br$ versus $m_{H_2}$ with $m_{H_D}$ shown in color (e) The relic density.

Export to PPT

Next, Fig. 1.b. displays the production of $H_2$ via gluon fusion ( $\sigma (gg\to H_2)$ ) versus $m_{H_2}$ . It ranges from $3\times 10^{-6}$ pb to 5.8 pb. The maximum value occurs at $m_{H_2}=378$ GeV, $m_{H_D}=170$ GeV, $tan\beta$ $=$ 1.5, $Br(H_2\to H_D{H}_D)=0.95$ , and $\Omega h^2=O\left(10^{-8}\right)$ . The dependence on $tan\beta$ is shown in Fig. 1.c. First, in the allowed regions, $tan\beta$ takes values between 0.9 and 28. We note that most successful points lie in regions where $tan\beta <5$ . The distribution of the branching ratio to dark matter appears to be nearly uniform. However, smaller values of $tan\beta$ tend to be associated with branching ratio below 0.1. As $tan\beta$ slightly increases from 0.9 to 1.5, the branching ratio exceeds 0.6.

Fig. 1.d. shows the branching ratio times the cross-section. The values range between $O\left(10^{-12}\right)$ pb and 5.6 pb. Moreover, for a given mass of $m_{H_2}$ , the maximum value of $Br\times \sigma$ slightly decreases with the increase of $m_{H_2}$ . The maximum value of $Br\times \sigma$ found in the allowed parameter space is achieved at $m_{H_2}=378$ GeV, $m_{H_D}=170$ GeV, $tan\beta$ $=$ 1.5, $Br(H_2\to H_D{H}_D)=0.95$ , and $\Omega h^2=O\left(10^{-8}\right)$ . At this point, $H_2$ mainly decays into $t\bar{t}$ with a percentage of 3%, followed by $h_1{h}_1$ at 0.5% and $W^{+}{W}^{-}$ at 0.4%.

Finally, we note that the relic density in the scanned parameter space is almost always below the lower Planck bound, as shown in Fig. 1.e. Hence, while this parameter space provides a candidate for dark matter, it cannot explain all dark matter phenomena. A representative point where the relic density is fully explained by this model corresponds to $m_{H_2}=914$ GeV, $m_{H_D}=180$ GeV, $tan\beta$ $=$ 1.4, $Br(H_2\to H_D{H}_D)\approx 0.11$ , $\sigma (gg\to H_2)\approx 0.11$ pb and $\Omega h^2\approx 0.11$ . In this case, the dominant decay of $H_2$ is to $t\bar{t}$ pair at 88%.

4.2 The NUH-NMSSM

In this subsection, we present the results for the NMSSM with boundary conditions set at the GUT scale. A total of 120k valid points are collected, and the data corresponds to $m_{H_2}<1\mathrm{TeV}$ . Fig. 2.a shows the results of our scans in the usual $m_0$ - $m_{1/2}$ plane, where the first parameter ranges from $\sim 0$ to $5961$ GeV, while the second ranges between $690$ GeV and $10460$ GeV. As can be seen in the Figure, values of $m_{H_2}\le 350$ GeV correspond to portions of the parameter space where $m_{1/2}<2500$ GeV, while $m_0$ can take the full range of its allowed values. Regions where $m_{1/2}>2500$ GeV are associated with $m_{H_2}>350$ GeV. The shape of the parameter space reflects the fact that we have used both random scanning method (especially the box where $m_{1/2},m_0<2500$ GeV) as well as the MCMC method described in Section 3.

As mentioned in Section 1, the main production channel of the neutral scalars is via the fusion of two gluons. The results are displayed in Fig. 2.b where the minimum and the maximum values of the production as a function of $m_{H_2}$ are $\mathcal{O}\left(10^{-10}\right)$ pb and 2.2 pb, respectively. In the subsequent analysis, we require that $Br(H_2\to {\tilde{\chi }}_1{\tilde{\chi }}_1)>0$ . By doing so, some points in the parameter space are removed, and what remains is a parameter space where ${\sigma }_{gg\to H_2}^{\text{max}}\sim 0.51$ pb. This corresponds to $m_{H_2}=210$ GeV, $m_{{\tilde{\chi }}_1}=101$ GeV, and $Br(H_2\to {\tilde{\chi }}_1{\tilde{\chi }}_1)=0.012$ .

Moving on to the branching ratio, Fig. 2.c depicts the allowed ranges of $Br(H_2\to {\tilde{\chi }}_1{\tilde{\chi }}_1)$ as a function of $m_{H_2}$ and $m_{{\tilde{\chi }}_1}$ (indicated by color). The branching ratio takes values ranging from $\sim 0$ to 0.87. The highest value occurs at $m_{H_2}=212$ GeV, and $m_{{\tilde{\chi }}_1}=98$ GeV. These values correspond to $tan\beta =17$ , and are associated with $\Omega h^2\sim 0.01$ and ${\sigma }_{gg{H}_2}\sim 0.1$ pb. In this case, $H_2$ decays to $b\bar{b}$ with Br $\sim 0.1$ , and to $W^{+}{W}^{-}$ and $\tau \bar{\tau }$ with Br $\sim 0.01$ .

Fig. 2.d shows the dependence of the invisible decay branching ratio on $tan\beta$ , ranging from 1.8 to 30. The maximum value of $Br(H_2\to {\tilde{\chi }}_1{\tilde{\chi }}_1)$ is obtained only for $tan\beta >5$ .

Fig. 2.e displays $\sigma (gg\to H_2)\times Br(H_2\to {\tilde{\chi }}_1{\tilde{\chi }}_1)$ . The maximum value is 0.1 pb, which takes place at $m_{H_2}=216$ GeV, $m_{{\tilde{\chi }}_1}=102$ GeV, $tan\beta =12$ , $\Omega h^2=0.001$ , $Br\approx 0.3$ and $\sigma \approx 0.3$ pb. In this case, $H_2$ is more likely to decay into $W^{+}{W}^{-}$ , which accounts for 50% of the decays, while $ZZ$ accounts for 20%.

For completeness, Fig. 2.f shows the relic density as a function of the mass of the DM particle. All points in the parameter space are associated with values below the lower Planck limit, in which case the DM particle is insufficient to account for all of the observed DM relic density.

We turn to the case where ${\tilde{\chi }}_2$ and ${\tilde{\chi }}_1$ are mass degenerate. This interesting case can lead to ${\tilde{\chi }}_2$ being long-lived and hence escaping detection. There are two cases here. The first is when $H_2$ decays to ${\tilde{\chi }}_2$ and ${\tilde{\chi }}_1$ , and the second is when it decays to a pair of ${\tilde{\chi }}_2$ . The results for both cases are shown in Fig. 3. Starting with $H_2\to {\tilde{\chi }}_2{\tilde{\chi }}_1$ , we can see in Fig. 3.a that the branching ratio can reach a maximum value of 0.01. Here the most likely decays are to ${\tilde{\chi }}_1{\tilde{\chi }}_1$ at 45%, to ${\chi }^{+}{\chi }^{-}$ at 40%, to $W^{+}{W}^{-}$ at 6.5%, to $b\bar{b}$ at 3%, to $ZZ$ at 3%, and to ${\tilde{\chi }}_2{\tilde{\chi }}_2$ at 2%. Furthermore, the maximum value of $\sigma \times Br$ is 0.001 pb, which is displayed in Fig. 3.c. This occurs at $m_{H_2}=456$ GeV, $m_{{\tilde{\chi }}_1}\approx m_{{\tilde{\chi }}_2}=107$ GeV, $\sigma (gg\to H_2)\sim 0.23$ pb, and $Br(H_2\to {\tilde{\chi }}_2{\tilde{\chi }}_1)\sim 0.003$ . The other relevant decays of $H_2$ account for branching ratios of 0.48 to $b\bar{b}$ , 0.35 to $t\bar{t}$ , 0.065 to $\tau \bar{\tau }$ , 0.041 to $H_1{H}_1$ , and 0.02 to $W^{+}{W}^{-}$ and ${\chi }^{+}{\chi }^{-}$ .

Finally, we consider the second case, displayed in Fig. 3.b. The maximum value of $Br(H_2\to {\tilde{\chi }}_2{\tilde{\chi }}_2)$ is 0.25. while it is 0.5 to ${\chi }^{+}{\chi }^{-}$ , and 0.25 to ${\tilde{\chi }}_1{\tilde{\chi }}_1$ . And the maximum value of $\sigma \times Br$ is 0.01 pb, which is achieved at $m_{H_2}=291$ GeV, $m_{{\tilde{\chi }}_1}\approx m_{{\tilde{\chi }}_2}=109$ GeV, $\sigma (gg\to H_2)\sim 0.13$ pb, and $Br(H_2\to {\tilde{\chi }}_2{\tilde{\chi }}_1)\sim 0.1$ . Other relevant branching ratios of $H_2$ are 0.29 to $W^{+}{W}^{-}$ , 0.2 to ${\chi }^{+}{\chi }^{-}$ , 0.18 to $H_1{H}_1$ , 0.14 to both ${\tilde{\chi }}_1{\tilde{\chi }}_1$ , and 0.12 to $ZZ$ .

Close

Fig. 2

Two dimensional scatter plots of (a) $m_{1/2}$ versus $m_0$ (b) $\sigma (gg\to H_2)$ as a function of $m_{H_2}$ (c) $Br(H_2\to {\tilde{\chi }}_1{\tilde{\chi }}_1)$ as a function of $m_{H_2}$ (d) The dependence of $Br$ on $tan\beta$ (e) $\sigma \times Br$ as a function of $m_{H_2}$ (f) The relic density.

Export to PPT

Close

Fig. 3

Two dimensional scatter plots of (a) $Br(H_2\to {\tilde{\chi }}_2{\tilde{\chi }}_1)$ versus $m_{H_2}$ (b) $Br(H_2\to {\tilde{\chi }}_2{\tilde{\chi }}_2)$ versus $m_{H_2}$ (c) $\sigma \times Br(H_2\to {\tilde{\chi }}_2{\tilde{\chi }}_1)$ versus $m_{H_2}$ (d) $\sigma \times Br(H_2\to {\tilde{\chi }}_2{\tilde{\chi }}_2)$ versus $m_{H_2}$ .

Export to PPT