Singlettriplet fermionic dark matter and LHC phenomenology
Abstract
It is well known that for the pure standard model triplet fermionic WIMPtype dark matter (DM), the relic density is satisfied around 2 TeV. For such a heavy mass particle, the production crosssection at 13 TeV run of LHC will be very small. Extending the model further with a singlet fermion and a triplet scalar, DM relic density can be satisfied for even much lower masses. The lower mass DM can be copiously produced at LHC and hence the model can be tested at collider. For the present model we have studied the multi jet (\(\ge 2\,j\)) + missing energy ( Open image in new window ) signal and show that this can be detected in the near future of the LHC 13 TeV run. We also predict that the present model is testable by the earth based DM direct detection experiments like Xenon1T and in future by Darwin.
1 Introduction
The Standard Model (SM) of elementary particles is a very well established and successful theory. With the discovery of the Higgs boson at the LHC, the last missing piece of the SM has been found. So far, all observations at the collider experiments are reasonably consistent with the SM cementing its position even further. However, despite this success story it is well accepted that the SM is not the full theory of nature. Rather, SM is widely looked as a low energy effective limit of a more complete underlying theory. The reasons to believe that SM needs to be extended are both theoretical as well as observational. Amongst the most compelling observational evidences of physics Beyond the SM (BSM) is the issue of the Dark Matter (DM). The DM, if it is a particle, should be massive, chargeless and weakly interacting such that its relic abundance should be consistent with the observational data on DM. The SM fails to provide any such candidate. The only weakly interacting chargeless particle in the SM is the neutrino and it is postulated to be massless. Of course experimental data have now given conclusive evidence that neutrinos are massive  which is another compelling reason to extend the SM to accommodate neutrino mass. However neutrinos, even if massive in a BSM theory, can only be hot dark matter candidate and it is well known that hot dark matter is inconsistent with structure formation of the universe. Therefore, we need a BSM theory that can provide either cold or warm dark matter candidate.
Particle content and their corresponding charges under various symmetry groups
Gauge group  Baryon fields  Lepton fields  Scalar fields  

\(Q_{L}^{i}=(u_{L}^{i},d_{L}^{i})^{T}\)  \(u_{R}^{i}\)  \(d_{R}^{i}\)  \(L_{L}^{i}=(\nu _{L}^{i},e_{L}^{i})^{T}\)  \(e_{R}^{i}\)  \(\rho \)  \(\phi _{h}\)  
\(SU(3)_{c}\)  3  3  3  1  1  1  1 
\(SU(2)_{L}\)  2  1  1  2  1  3  2 
\(U(1)_{Y}\)  1 / 6  2 / 3  \(\,1/3\)  \(\,1/2\)  \(\,1\)  0  1 / 2 
\(\mathbb {Z}_{2}\)  \(+\)  \(+\)  \(+\)  \(+\)  \(+\)  −  \(+\) 
Many different classes of DM candidates like the Weakly Interacting Massive Particle (WIMP), Strongly Interacting Massive Particle (SIMP) and Feebly Interacting Massive Particle (FIMP), have been proposed in the literature. Each type can solve the DM puzzle in its unique way. In this work we will consider a model with a WIMPtype fermionic DM candidate by extending the SM particle spectrum and study in detail the DM phenomenology. We will also study the collider signal of the DM at the 13 TeV run of the LHC and its effect will basically manifest as the missing energy associated with hard jets. In [7, 8, 9, 10, 11, 12, 13], a model has been proposed where the fermionic DM belongs to the triplet representation of the SM. An extra discrete \(\mathbb {Z}_{2}\) symmetry stabilizes the DM making the neutral part of the triplet fermion as the viable candidate for the DM. If the SM is extended by only the triplet fermion [10, 12], then the main coannihilation processes take part in the DM relic density calculation are mediated by the charged gauge boson \(W^{\pm }\), and for this case the correct value of DM relic density is obtained for DM mass around 2.3 TeV [10, 12]. This high mass makes it difficult to produce the purely triplet fermionic DM at the 13 TeV collider, and hence to test this model at the LHC. Of course with a higher energy collider one might be able to produce these heavy fermions. Another major drawback of such high mass DM is that when the DM annihilate to gamma rays via \(W^{\pm }\)mediated one loop diagrams, then for such high DM mass, the annihilation is increased by the Sommerfeld enhancement (SE) factor [8, 9, 10]. This is ruled out from the indirect search of the HESS data [32]. In this paper, we propose an extension of the SM that accommodates both high as well as low mass fermionic DM such that it can be produced and tested at the 13 TeV run of the LHC. The low mass DM regime do not have any significant SE enhancement (because the DM mass becomes comparable to the mediator mass inside the loops) and hence are safe from the gamma ray indirect detection bounds put by the FermiLAT collaboration [35]. Our proposed extension of the particle content includes one SM singlet fermion and SM triplet fermion [14, 15, 16, 17, 18]. The scalar sector is also extended to include a SM triplet scalar. The \(\mathbb {Z}_{2}\) charge of these BSM particles is arranged in such a way that there is a mixing between the neutral component of the triplet fermion and the singlet fermion, that generates two mass eigenstates for the neutral fermions. The lower mass eigenstate becomes the viable DM candidate. The neutral and charged components of the SM doublet and triplet scalars also mix, that gives rise to two physical neutral Higgs scalars and one charged Higgs scalar. The presence of these extra scalars opens up additional annihilation and coannihilation processes between the two DM candidates which effectively reduces the mass of the DM for which the current DM relic density bound can be easily satisfied. For low mass DM we give the prediction for the annihilation of the DM to two gamma rays by one loop process. In addition, these lower mass DM fermions (\(\sim \) 100 GeV) can be observed with large production crosssection at the 13 TeV LHC. We perform a detailed collider phenomenology of the DM model. In this work we will consider multi jets + missing energy signal in the final state for searching the DM. We study in detail the dominant backgrounds for such type of signal. The SM backgrounds are reduced by applying suitable cuts that increases the statistical significance of detection for the fermionic DM with the low luminosity run of the LHC. A final comment is in order. It is possible to embed our model in a SO(10) GUT where the SU(2) triplet would belong to the 45 representation of SO(10) and would help in the gauge coupling unification, as was shown in [19, 20].
Particle content and their corresponding charges under various symmetry groups
Gauge group  Baryon fields  Lepton fields  Scalar fields  

\(Q_{L}^{i}=(u_{L}^{i},d_{L}^{i})^{T}\)  \(u_{R}^{i}\)  \(d_{R}^{i}\)  \(L_{L}^{i}=(\nu _{L}^{i},e_{L}^{i})^{T}\)  \(e_{R}^{i}\)  \(N^{\prime }\)  \(\rho \)  \(\phi _{h}\)  \(\Delta \)  
\(SU(3)_{c}\)  3  3  3  1  1  1  1  1  1 
\(SU(2)_{L}\)  2  1  1  2  1  1  3  2  3 
\(U(1)_{Y}\)  1 / 6  2 / 3  \(\,1/3\)  \(\,1/2\)  \(\,1\)  0  0  1 / 2  0 
\(\mathbb {Z}_{2}\)  \(+\)  \(+\)  \(+\)  \(+\)  \(+\)  −  −  \(+\)  \(+\) 
2 Triplet fermionic dark matter
Note that, while the model can be tested in direct and indirect detection experiments, due to its heavy mass it is difficult to produce this DM candidate at the 13 TeV or 14 TeV LHC search. One will need a very high energy collider to test this DM model. Minimal extension of the model by adding a gauge singlet fermion and a triplet scalar opens up the possibility to test the model at collider. Below, we discuss in detail the required extensions and the model predictions.
3 Singlet triplet mixing
A further discussion is in order. For the present model we can generate the neutrino mass by Type I seesaw mechanism just by introducing SM singlet right handed neutrinos. In other variants of the triplet fermionic DM model, neutrino masses were generated by using the Type III seesaw mechanism and radiatively by the authors of [24] and [12, 14, 15, 16], respectively.
4 Constraints used in dark matter study
4.1 SI direct detection cross section
4.2 Invisible decay width of Higgs
4.3 Planck limit
5 Dark matter relic abundance
In analysing the DM phenomenology we implement the model in Feynrules [39]. We generate Calchep files using Feynrules and feed the output files into micrOmegas [40]. The relevant Feynman diagrams that determine the DM relic abundance are shown in Fig. 4. In presence of triplet as well as singlet states, additional channels mediated by the neutral and charged Higgs state opens up.

In the left panel (LP) of Fig. 5, we show the variation of the DM relic density with DM mass for three different values of the singlettriplet mixing angle \(\sin \beta \). The thin magenta band shows the \(2\sigma \) experimentally allowed range of the DM relic density reported by the Planck collaboration. From the figure, this is evident, that there are four dip regions with respect to the DM mass. The first resonance occurs at \(M_{\rho _{1}^{0}} \simeq M_{h_1}/2 \sim 62.5 \) GeV. The SMlike Higgs mediated diagrams shown in Fig. 4 give the predominant contribution in this mass range. The second resonance occurs at \(M_{\rho _{1}^{0}} \sim 150\) GeV, when the DM mass is approximately half the BSM Higgs mass (\(M_{\rho _{1}^{0}} \simeq M_{h_2}/2\)) assumed in this figure. The third dip is due to the tchannel diagram \(\rho _{1}^{0}\,\rho _{1}^{0} \rightarrow W^{\pm } H^{\mp }\) mediated by the \(\rho ^{\pm }\). This dip occurs when the DM mass satisfies the relation \(M_{\rho _{1}^{0}} = \frac{M_{W^{\pm }} + M_{H^{\mp }}}{2}\) and happens due to the destructive interference term of the \(W^{\pm } H^{\mp }\) final state. The fourth dip happens because of the threshold effect of the \(W^{\pm } H^{\mp }\) final state and clear from the fact that with the variation of the charged scalar mass (\(M_{H^{\mp }}\)), this dip also changes its position with respect to the DM mass. For DM masses greater than this, the DM relic abundance is mainly dominated by the schannel annihilation diagram where the final state contains \(H^{+}H^{}\), \(h_2 h_2\).

This is to emphasize, in the present scenario even relatively lighter DM is in agreement with the observed relic density. The low mass DM can be copiously produced at LHC and hence can further be tested in the ongoing run of LHC. The lowering of DM mass is possible due to the addition of the extra SM gauge singlet fermion \(N^{\prime }\) and the extra SM triplet Higgs \(\Delta \). This opens up additional annihilation and coannihilation diagrams shown in Fig. 4. As described before, this allows the three resonance regions and make the model compatible with the experimental constraint from Planck for DM masses accessible at LHC. This should be contrasted with the pure triplet model discussed in Sect. 2, where the DM mass compatible with the Planck data is 2.37 TeV, well outside the range testable at LHC due to small production crosssection. In the next section, we will discuss in detail the prospects of testing the DM at LHC (see Fig. 11).

The singlet triplet mixing angle \(\beta \) has significant effect on the relic density. With the increase of the mixing angle \(\beta \), the DM relic density decreases. This happens because the \(g_{\rho ^0_1 \rho ^0_1 h_i}\) (\(i=1,2\)) coupling increases with \(\beta \) (cf. Eq. (22)), thereby increasing the crosssection of the annihilation processes. Since the relic density is inversely proportional to the velocity times crosssection \(\langle \sigma v \rangle \), where \(\sigma \) is the annihilation crosssection of the DM particles and v is the relative velocity, increase of \(\sin \beta \) causes the relic density of DM to decrease.
 Additionally, we also explore the effect of the Higgs mixing angle \(\alpha \). In the right panel (RP) of Fig. 5, we show the variation of the DM relic density for three different values of the doublettriplet Higgs mixing angle \(\alpha \). The first resonance peak is seen to be nearly unaffected by any change in \(\sin \alpha \). As the DM mass increases, the impact of \(\sin \alpha \) increases and we see an increase in the DM relic density with increase of \(\sin \alpha \). These features can be explained as follows. Inserting \(Y_{\rho \Delta }\) from Eq. (19) into Eq. (22), and replacing \(v_\Delta \) in terms of \(\tan \delta \) using Eq. (14), we get$$\begin{aligned} g_{\rho ^{0}_{1} \rho ^{0}_{1} h_{1}}= & {} \frac{\Delta M_{21} \sin 2 \beta }{2 v} \frac{\sin \alpha }{\tan \delta }, \nonumber \\ g_{\rho ^{0}_{1} \rho ^{0}_{1} h_{2}}= & {} \frac{\Delta M_{21} \sin 2 \beta }{2 v} \frac{\cos \alpha }{\tan \delta }. \end{aligned}$$(27)
Additionally, we also show the variation of relic density for different mass difference \(\Delta M_{21}\) in the LP of Fig. 6. The first and second resonance regions show very little dependence on the mass difference \((M_{\rho _{2}^{0}}  M_{\rho _{1}^{0}})\), with the relic abundance being marginally less for higher \((M_{\rho _{2}^{0}}  M_{\rho _{1}^{0}})\). However, for the high DM mass we see that the decrease in DM relic abundance with increasing values of \((M_{\rho _{2}^{0}}  M_{\rho _{1}^{0}})\) is visible. The reason for this can be understood as follows. From Eq. (19), one can see that the singlettriplet Yukawa coupling \(Y_{\rho \Delta }\) is directly proportional to the mass difference \((M_{\rho _{2}^{0}}  M_{\rho _{1}^{0}})\). Both DM couplings \(g_{\rho ^{0}_{1} \rho ^{0}_{1} h_{2}}\) and \(g_{\rho ^{0}_{1} \rho ^{0}_{1} h_{1}}\) (see Eq. (22)) depend on the Yukawa coupling \(Y_{\rho \Delta }\) and hence in first and second resonance regions where the schannel processes dominate, viz., at the resonance regions mainly, controlled by resonance, hence less effect. On the other hand for higher \(M_{\rho ^0_1}\) regions and tchannel dominated regions no such resonance exists, so vary linearly with the mass differences. Close to the third resonance region, the tchannel process dominates and here, the crosssection is suppressed due to the propagator mass \(M_{\rho ^{\pm }}\). Therefore, for regions of the parameter space where the tchannel process dominates, the relic abundance is seen to increase as \((M_{\rho _{2}^{0}}  M_{\rho _{1}^{0}})\) (here we considered \(M_{\rho ^{\pm }}  M_{\rho _{2}^{0}} = 160\) MeV) increases for a given \(M_{\rho ^0_1}\). One can see that there is clear cross over between the tchannel and schannel diagrams for \(M_{\rho _{1}^{0}} > M_{H^{\pm }},\,M_{h_2}\), because after this value of DM mass \(\rho _{1}^{0} \rho _{1}^{0}\) mainly annihilates to \(h_{2}h_{2}\) and \(H^{+}H^{}\) by the schannel process mediated by the Higgses.
Finally, we also explore the dependency on the mass of the neutral Higgs \(h_2\). In the RP of Fig. 6, we show the variation of the relic density with DM mass for three different values of the BSM Higgs mass: \(M_{h_2} = 200\) GeV, 300 GeV and 400 GeV, respectively. From the figure we see that the first resonance remains unchanged at \(M_{\rho _{1}^{0}} \sim 62.5\) because the SMlike Higgs mass is fixed at \(M_{h_1} = 125.5\) GeV. However, the second resonance occurs at three different values of the DM mass depending on the values of \(M_{h_2}\), as the resonance occurs at \(M_{\rho _{1}^{0}} \sim \frac{M_{h_2}}{2}\). Since here we vary only the BSM Higgs mass \(M_{h_2}\), the couplings which are related to the Higgses remain unaffected, and all three curve merge for greater values of DM mass.
Parameter varied in the above mentioned range at the time of generating the scatter plots
Model parameters  Range 

\(M_{\rho _{1}^{0}}\)  110–300 [GeV] 
\(M_{h_2}\)  (2 \(M_{\rho _{1}^{0}}\))\(^{100}_{50}\) [GeV] 
\(\sin \beta \)  \(10^{3}\)–1 
6 Correlation between parameters
In the LP of Fig. 7 we show the allowed regions in \(M_{h_2}\) and \(M_{\rho _{1}^{0}}\), where all the dots satisfy the relic density bound as given in Eq. 1. The three colors correspond to three different benchmark choices for the Higgs mixing angle \(\alpha \). From Figs. 5 and 6, one can see that the DM relic density can always be satisfied near the resonance regions. Hence, for a given BSM Higgs mass, there is only a range of DM masses that are allowed by the Planck bound. In generating the scatter plots we have varied the model parameters as shown in Table 3. We have kept the values of \(M_{h_2}\) near the resonance region. As expected, we get a sharp correlation between the mass of DM and the BSM Higgs mass as stressed above. On the other hand, in the RP of Fig. 7 we have shown the allowed region in the sine of singlettriplet mixing angle (\(\sin \beta \)) and the DM mass (\(M_{\rho _{1}^{0}}\)) plane. Here we keep \(\Delta M_{21} = 50\) GeV (\(\Delta M_{21}\) as defined before), and the allowed region shows that for the given ranges as in Table 3, the DM relic density can be satisfied for \(0.025< \sin \beta < 0.27\). One interesting point to note here is that in the LP of Fig. 7 for \(\sin \alpha \), \(\sin \delta \) = 0.03, correlation in the \(M_{\rho _1^{0}}M_{h_2}\) is wider compared to the other two lower values of \(\sin \alpha \), \(\sin \delta \). We can understand this as follows. From the RP of Fig. 7 for \(\sin \alpha \), \(\sin \delta \) = 0.03, the DM relic density is satisfied for higher values of \(\sin \beta \) (\(\sim 0.3\)). From the LP of Fig. 5 we see that near the second resonance region (\(M_{\rho _1^{0}} \sim M_{h_2}/2\)) the DM relic density is satisfied for a wider range of \(M_{\rho ^0_1}\) for higher values of \(\sin \beta \). Since, for \(\sin \alpha \), \(\sin \delta \) = 0.03, we get higher values of \(\sin \beta \) (as seen from the RP of Fig. 7), so the correlation in \(M_{\rho _1^{0}}M_{h_2}\) planes becomes wider.
7 Indirect detection of dark matter by \(\gamma \gamma \) observation
In Fig. 10, we show the variation of \(\langle \sigma v \rangle \) with the DM mass, \(M_{\rho _1^0}\) by considering the relevant one loop diagrams. As the DM relic density for the pure triplet fermion is satisfied for DM mass of about 2.4 TeV and this is already ruled out by the FermiLAT data when we the Sommerfeld enhancement is taken into consideration. In the current work, we have taken the triplet fermion mixing with the singlet fermion with the help of the triplet scalar and DM relic density can be satisfied around the 100 GeV order DM mass. For such low mass range of the DM where \(M_{\rho _1^0} \sim M_{W} \sim M_{H^{\pm }}\), the Sommerfeld enhancement factor will have no significant role in the increment of \(\langle \sigma v \rangle _{\gamma \gamma }\). We have shown the FermiLAT2013 [34] and FermiLAT2015 [35] data in the \(\langle \sigma v \rangle _{\gamma \gamma }  M_{\rho _1^0}\) plane by the red and green dash line, respectively. By blue solid line we have shown the \(\langle \sigma v \rangle _{\gamma \gamma }\) variation with the DM mass which is suppressed by the one loop factor for the present model.
8 LHC phenomenology
Although there has been no dedicated search for such a model at the LHC, one can in principle, derive limits on the masses of the exotic fermions (\(\rho ^0_{1,2}\), \(\rho ^{\pm }\)) and the additional scalar states(\(h_2\), \(H^{\pm }\)) from existing LHC analyses looking for similar particles. LHC has extensively searched for heavy neutral Higgs boson similar to \(h_2\) and the nonobservation of any such states puts stringent constraints on masses and branching ratios of such particles provided their decay modes are similar to that of the SMlike Higgs [42, 43, 44]. However, in our case, these bounds are significantly weakened because the decays of \(h_2\) here are quite different compared to the conventional modes. \(h_2\) mostly decays into \(h_1h_1\), \(\rho ^0_2\rho ^0_1\) or \(\rho ^0_1\rho ^0_1\) pair depending on the availability of the phase space. In absence of \(\rho ^0_2\rho ^0_1\) mode, \(\rho ^0_1\rho ^0_1\) always has a large (30%  40%) branching ratio, which is a completely invisible mode and thus leads to weaker event rates in the visible final states. In the presence of \(\rho ^0_2\rho ^0_1\) and(or) \(h_1h_1\) modes, a \(b\bar{b}\) final state study can constrain the \(h_2\) mass since \(\rho ^0_2\) always decays dominantly via \(b\bar{b}\). However, the net branching ratio suppression results in weaker limits from the existing studies. Charged Higgs search at the LHC concentrates on the \(\tau \bar{\nu }\), \(c\bar{s}\), \(c\bar{b}\) and \(t\bar{b}\) decay modes depending on the mass of \(H^{\pm }\) [45, 46, 47, 48]. None of these decay modes are significant in our present scenario. Here \(\rho ^{\pm }\) decays via \(\rho ^0_1\rho ^{\pm }\) and(or) \(W^{\pm }Z\) depending on the particle masses. Thus the existing charged Higgs mass limits do not apply here. Instead, a dilepton or trilepton search would be more suitable for such particles although the charged leptons originating solely from the gauge boson decays will be hard to distinguish from those coming from the SM. Constraints on the masses of \(\rho ^0_2\) and \(\rho ^{\pm }\) can be drawn from searches of winolike neutralino and chargino in the context of supersymmetry [49, 50]. However, production crosssection of this pair at the LHC is smaller compared to the gauginos leading to weaker mass limits. Moreover, the decay pattern of \(\rho ^0_2\) is quite different from that of a winolike neutralino. The most stringent gaugino mass bounds are derived from the trilepton final state analysis. Such a final state cannot be expected in our present scenario since \(\rho ^0_2\) dominantly decays into a \(b\bar{b}\) pair along with \(\rho ^0_1\). However, \(\rho ^{\pm }\) always decays into \(\rho ^0_1\) associated with an onshell or offshell Wboson, similar to a winolike chargino. Thus the bounds derived on the chargino masses in such cases [49, 50] can be applied to \(m_{\rho ^{\pm }}\) as well if appropriately scaled to its production crosssection and subjected to \(m_{\rho ^0_1}\). We have taken this constraint into account while constructing our benchmark points.
8.1 Production crosssection and choice of benchmark points
For this we consider production of \(\rho ^{\pm } \rho _{2}^{0}\) which further decay into \(\rho _{1}^{0}\) associated with quarks resulting in a multijet + Open image in new window signal. Similar collider signal can also arise from other production modes, namely, \(\rho _{2}^0 \rho _2^0\) and \(\rho ^+\rho ^\). While the \(\rho _2^0\) pair production crosssection is smaller by orders of magnitude, the other two production channels have comparable crosssections as shown in Fig. 11. For Fig. 11, we have kept the mass gap between \(\rho _2^{0}\) and \(\rho _{\pm }\) fixed at the pion mass and the crosssection is computed at 13 TeV centreofmass energy. Clearly, \(\sigma (pp\rightarrow \rho ^{\pm } \rho _{2}^{0})\) is almost twice to that of \(\sigma (pp\rightarrow \rho ^+\rho ^)\) making the former one the most favored production channel to probe for the present scenario. However, the latter one can also contribute significantly to boost the multijet + Open image in new window signal event rate given the fact that \(\rho _2^0\) and \(\rho ^{\pm }\) are mass degenerate from the collider perspective. The degeneracy of \(\rho _2^0\) and \(\rho ^{\pm }\) results in their decay products to have very similar kinematics. Therefore, in our study of the multijet final state we have included both the production channels \(pp\rightarrow \rho _2^0\rho ^{\pm }\) and \(pp\rightarrow \rho ^+\rho ^\). \(\rho _2^0\) further decays into \(\rho _1^0\) mostly via \(h_2\) whereas \(\rho ^{\pm }\) also decays into \(\rho _1^0\) via Wboson. Regardless of whether the intermediate scalar or the gauge bosons are onshell or offshell, we always consider their decays into pair of bquarks or light quarks. In the former case, the decay of \(h_2\) is likely to give rise to bjets in the final state whereas the latter one results in light jets arising from W decay. Hence in order to combine the event rates arising from these two production channels, we do not demand any btagged jets in the final states. Besides, demanding btagged jets in the final state can also hinder the signal event rates specially for cases where the mass difference between \(\rho ^0_2\) and \(\rho ^0_1\), i.e., \(\Delta M_{21}\) is small.
Benchmark points to study LHC phenomenology. We fixed other BSM parameters as \(\sin \alpha = 0.03\), \(\sin \beta = 0.1\)
Parameters  \(M_{\rho ^{0}_{1}}\) [GeV]  \(M_{\rho ^{0}_{2}}\) [GeV]  \(M_{\rho ^{+}}\) [GeV]  \(M_{h_2}\) [GeV]  \(M_{H^{\pm }}\) [GeV]  \(\sigma _{SI}\) [pb]  \(\Omega h^{2}\) 

BP1  87.6  128.0  128.2  195.5  195.5  2.1 \(\times 10^{12}\)  0.1207 
BP2  132.0  172.0  172.2  300.0  300.0  4.1 \(\times 10^{12}\)  0.1208 
BP3  171.1  211.0  211.2  400.0  400.0  4.8 \(\times 10^{12}\)  0.1197 
BP4  86.7  200.0  200.2  194.1  194.1  1.8 \(\times 10^{11}\)  0.1186 
BP5  119.0  230.0  230.2  280.0  280.0  2.9 \(\times 10^{11}\)  0.1195 
8.2 Simulation details
Since the number of hard jets obtained in the cascade are expected to vary for the different benchmark points depending on the choice of \(\Delta M_{21}\), we have chosen our final state with an optimal number of jet requirement along with missing energy: \(\ge 2\)jets + Open image in new window . The dominant SM background contributions for such a signal can arise from QCD, V+ jets, \(t\bar{t}\)+ jets and VV + jets channels, where \(V = W^{\pm }\) and Z. For collider analysis of this final state we have followed strategy similar to that adopted in, for example [62, 63].
Selection cuts

Leptons are selected with \(p_{T}^{l} > 10\) GeV and the pseudorapidity \(\eta ^{\ell } < 2.5\), where \(\ell = e, \mu \).

We used \(p_{T}^{\gamma } > 10\) GeV and pseudorapidity \(\eta ^{\gamma } < 2.5\) as the basic cuts for photon.

We have chosen the jets which satisfy \(p_{T}^{j} > 40\) GeV and \(\eta ^{j} < 2.5\).

We have considered the azimuthal separation between all reconstructed jets and missing energy must be greater than 0.2 i.e. Open image in new window .
Cutflow table for the obtained signal cross section at 13 TeV LHC corresponding to \(\rho ^0_2 \rho ^{\pm }\) channel. The five benchmark points are referred as BP1BP5. See the text for the details of the cuts A0–A5
Signal at 13 TeV  Effective cross section after applying cuts (fb)  

BP  Crosssection (pb)  A0 + A1  A2  A3  A4  A5 
BP1  6.757  1005.05  175.08  138.45  22.02  19.15 
BP2  2.279  385.22  69.16  56.51  11.87  10.85 
BP3  1.052  189.71  34.63  29.19  7.36  6.82 
BP4  1.296  1047.86  145.67  116.94  14.19  9.82 
BP5  0.760  616.00  89.60  72.63  9.80  7.40 
Cutflow table for the obtained signal cross section at 13 TeV LHC corresponding to \(\rho ^{+} \rho ^{}\) channel. The five benchmark points are referred as BP1BP5. See the text for the details of the cuts A0–A5
Signal at 13 TeV  Effective cross section after applying cuts (fb)  

BP  Crosssection (pb)  A0 + A1  A2  A3  A4  A5 
BP1  3.419  2639.30  74.36  59.18  8.54  7.31 
BP2  1.156  880.60  28.77  23.87  4.95  4.43 
BP3  0.532  402.24  14.80  12.62  3.18  2.95 
BP4  0.652  446.80  63.99  45.54  5.72  3.76 
BP5  0.380  258.55  34.40  28.07  3.99  3.08 
 A1:
Since we are studying a hadronic final state, we have imposed a lepton and photon veto in the final state. This cut coupled with a large Open image in new window cut helps to reduce background events particularly arising from W + jets when W decays leptonically.
 A2:
\(p_T\) requirements on the hardest and second hardest jets: \(p_{T}^{j_1} > 130\) GeV and \(p_{T}^{j_2} > 80\) GeV. This cut significantly reduce the V + jets (where V = \(W^{\pm }\), Z) and QCD backgrounds.
 A3:
The QCD multijet events have no direct source of missing energy. Therefore, any contribution to Open image in new window in these events must arise from the mismeasurement of the jet \(p_T\)s. In order to minimise this effect, we have ensured that the Open image in new window and the jets are well separated, i.e., Open image in new window where i = 1, 2. For all the other jets, Open image in new window .
 A4:
We demand a hard cut on the effective mass variable, \(M_{Eff} > 800\) GeV.
 A5:
We put the bound on the missing energy Open image in new window GeV. \(M_{Eff}\) and Open image in new window are the two most effective cuts to reduce SM background events for multijet analyses. As shown in Fig. 12, these variables clearly separates the signal kinematical region from most of the dominant backgrounds quite effectively and can reduce the backgrounds in a significant amount. Most importantly, these cuts along with A1 and A2, reduces the large QCD background to a negligible amount.
9 Results
Cutflow table for the obtained crosssections corresponding to the relevant SM background channels for the cuts A0–A5 as mentioned in the text at the LHC with 13 TeV centerofmass energy
SM Backgrounds at 13 TeV  Effective cross section after applying cuts (pb)  

Channels  Crosssection (pb)  A0 + A1  A2  A3  A4  A5 
Z \(+\le \) 4 jets  5.7\(\times 10^4\)  5.5 \(\times 10^{3}\)  361.90  241.60  11.40  2.20 
W\(^{\pm }\) + \(\le \) 4 jets  1.9\(\times 10^5\)  9.1 \(\times 10^{3}\)  783.20  504.00  18.90  1.50 
QCD (\(\le \) 4 jets)  2.0\(\times 10^{8}\)  1.5 \(\times 10^{7}\)  3.5 \(\times 10^{5}\)  2.4 \(\times 10^{5}\)  2.5 \(\times 10^{3}\)  – 
t \(\bar{\mathrm{t}} + \) \(\le \) 2 jets  722.94  493.73  171.46  120.63  13.89  1.94 
W\(^{\pm }\)Z + \(\le \) 2 jets  51.10  19.66  5.37  3.59  0.50  0.12 
Z Z + \(\le \) 2 jets  13.71  4.99  0.80  0.53  0.06  0.02 
Total backgrounds  5.78 
Statistical significance of the multijet signal corresponding to different benchmark points for \({\mathcal {L}}=100~\mathrm{fb}^{1}\) integrated luminosity along with the required luminosity to achieve \(3\sigma \) statistical significance at 13 TeV run of the LHC
Signal at 13 TeV  Statistical significance (\(\mathcal {S}\))  Required luminosity \(\mathcal {L}\) (\(\mathrm{fb^{1}}\))  

BP  DM mass [GeV]  \(\mathcal {L} = 100 \, \mathrm{fb^{1}}\)  \(\mathcal {S} = 3\sigma \) 
BP1  87.6  3.5  74.4 
BP2  132.0  2.0  223.0 
BP3  171.1  1.3  545.3 
BP4  86.7  1.8  282.3 
BP5  119.0  1.4  473.9 
As evident from Tables 5, 6, 7 and 8, the used kinematical cuts are efficient enough to reduce the SM background contributions to the multijet channel. At the same time sufficient number of signal events survive leading to discovery potential of such a scenario at the 13 TeV run of the LHC with realistic integrated luminosities. The cuts A2, A4 and A5 are particularly useful in reducing the dominant background contributions arising from W + jets, Z + jets and \(t\bar{t}\) + jets. A combination of cuts A2–A5 has reduced the QCD contribution to a negligible amount. As the numbers indicate in Table 8, BP1 can be probed at the 13 TeV run of the LHC with 3\(\sigma \) statistical significance with relatively low luminosity owing to the large production crosssection. As expected, the signal significance declines as the mass of \(\rho _2^0\) (\(\rho ^{\pm }\)) is increased while its mass gap with \(\rho _1^0\) is kept same as represented by the numbers corresponding to the two subsequent benchmark points (BP2 and BP3). The last two benchmark points, BP4 and BP5 represent the scenario when the parent particles have masses significantly higher than the DM candidate. As a result, one would expect the cut efficiencies to improve for these benchmark points. This is reflected for example in the case of BP5 which has a signal significance very similar to BP3 in spite of having the smallest production crosssection. It can be inferred from our analysis that \(\rho _2^0\) (\(\rho ^{\pm }\)) masses \(\sim \) 250 GeV can easily be probed at the 13 TeV LHC with a reasonable luminosity.
10 Conclusion and summary
For the WIMPtype DM, its relic density, detection at direct detection experiments, and detection at collider experiments are intimately interrelated. In this work we have proposed a fermion DM model that can successfully explain the DM relic density, can be tested in future direct detection experiments, and can be produced and tested at the 13 TeV run of the LHC.
The model we propose extends the SM particle content by a SM triplet fermion and a SM singlet fermion, as well as by a SM triplet scalar. Both new fermions are given Majorana masses. An overall discrete \(\mathbb {Z}_{2}\) symmetry is imposed and the corresponding charges of the particles under this symmetry is arranged in such as way that the only Yukawa coupling involving the new fermions is the one which includes the triplet fermion, the singlet fermion and the triplet scalar. This gives rise to the mixing between the neutral component of the SM triplet fermion mixing with the SM singlet fermion. The lighter of the two mass eigenstates becomes the DM candidate in the model, stabilised by the \(\mathbb {Z}_{2}\) symmetry. There is also mixing between the neutral as well as charged scalar degrees of freedom belonging to the SM doublet and triplet representations. Finally, we get two physical neutral scalars  one SMlike Higgs \(h_1\) of mass 125.5 GeV and another heavier BSM Higgs \(h_2\) whose mass we keep as a free parameter in the model. From the charged scalar sector, we get physical charged scalars \(H^\pm \) while the other degree of freedom becomes the charged Goldstone boson, which is ‘eaten up’ to give mass to the \(W^{\pm }\) bosons. The presence of the triplet scalar as well as the mixing between the triplet and singlet fermions lead to additional schannel diagrams mediated by \(h_1\) and \(h_2\) as well as tchannel diagrams mediated by the new fermions \(\rho ^0_{1,2}\) and \(\rho ^{\pm }\) with \(H^\pm \) or \(h_2\) in their final states. These additional diagrams allow for resonant production of DM at (1) \(h_1\) mediated schannel processes, (2) \(h_2\) mediated schannel processes, and (3) tchannel diagrams with \(H^\pm \) or \(h_2\) in their final states. This allows us to satisfy the observed DM relic density by Planck with DM masses in the 100 GeV range. We study the impact of the model parameters on the DM relic density. We also study the possibility of testing this model at the current and future direct detection experiments, Xenon 1T and Darwin.
Finally we study the LHC phenomenology for few benchmark points (BP) and show that this model is testable in the very near future run of LHC. The model proposed in [12] had only the triplet fermion (and an inert doublet scalar) where the neutral component of the triplet becomes the DM stabilised by the \(\mathbb {Z}_{2}\) symmetry. The relic density of the fermionic DM in that model was governed by the tchannel processes involving only the \(W^\pm \), SMlike Higgs and SM fermions. As a result, the DM mass was seen to be 2.37 TeV for the Planck bound to be satisfied. Hence, this model cannot be tested at the LHC. Since the DM mass in our model is \(\mathcal{O} (100\) GeV), therefore, we can produce them in the collider at the 13 TeV run of LHC with a reasonably large crosssection. In this work we analysed the multi jets + missing energy signal. We also considered the low mass difference between the DM (\(\rho _{1}^{0}\)) and the nexttolightest neutral particle (\(\rho _{2}^{0}\)), that might lead to softer jets. However, high \(p_T\) jets may come from the ISR. Corresponding to this signal we figured out the dominant SM backgrounds for multi jets + missing energy signal. By suitably choosing the cuts we have reduced the SM backgrounds and simultaneously increased the statistical significance of the signal. We showed that for 100 fb\(^{1}\), we could observe the DM with \(3.5\sigma \) statistical significance for one of the BP. The statistical significance was seen reduce as the mass of \(\rho ^0_2\) and \(\rho ^\pm \) was increased. We also studied how much luminosity would be required to probe this model with a \(3\sigma \) statistical significance for our BPs.
A final comment on our model is in order. While we have focussed only on explaining the DM relic density within the context of the present model, we can easily extend it to generate the neutrino masses by Type I seesaw mechanism. This can be done by introducing righthanded neutrinos.
In conclusion, the present model allows for low mass fermionic DM that satisfactorily produces the observed the relic density of the universe. It can be tested at the current and nextgeneration DM direct detection experiments. More importantly the 100 GeV mass range of the DM candidate in this model allows its production and detection at the LHC. The 13 TeV LHC can discover this fermionic DM candidate for with more than \(3\sigma \) statistical significance with reasonable luminosity.
Notes
Acknowledgements
We acknowledge the HRI cluster computing facility (http://cluster.hri.res.in). The authors would like to thank the Department of Atomic Energy (DAE) Neutrino Project of HarishChandra Research Institute. This project has received funding from the European Union’s Horizon 2020 research and innovation programme InvisiblesPlus RISE under the Marie SklodowskaCurie Grant Agreement No. 690575. This project has received funding from the European Union’s Horizon 2020 research and innovation programme Elusives ITN under the Marie SklodowskaCurie Grant Agreement No. 674896. SK would also like to thank IOP, Bhubaneswar for hospitality during the initial stage of this work. MM acknowledges the support of DSTINSPIRE FACULTY research grant. SM would also like to thank HRI and RECAPP, HRI for hospitality and financial support during the initial phase of this work.
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