Right-handed Neutrino Dark Matter with Radiative Neutrino Mass in Gauged B-L Model

Right-handed Neutrino Dark Matter with Radiative Neutrino Mass in Gauged Model

Debasish Borah dborah@iitg.ac.in Department of Physics, Indian Institute of Technology Guwahati, Assam 781039, India    Dibyendu Nanda dibyendu.nanda@iitg.ac.in Department of Physics, Indian Institute of Technology Guwahati, Assam 781039, India    Nimmala Narendra ph14resch01002@iith.ac.in Indian Institute of Technology Hyderabad, Kandi, Sangareddy, 502285, Telangana, India    Narendra Sahu nsahu@iith.ac.in Indian Institute of Technology Hyderabad, Kandi, Sangareddy, 502285, Telangana, India
Abstract

We study the possibility of right-handed neutrino dark matter (DM) in gauged extension of the standard model augmented by an additional scalar doublet, being odd under the symmetry, to give rise to the scotogenic scenario of radiative neutrino masses. Due to lepton portal interactions, the right-handed neutrino DM can have additional co-annihilation channels apart from the usual annihilations through which give rise to much more allowed mass of DM from relic abundance criteria, even away from the resonance region like . This enlarged parameter space is found to be consistent with neutrino mass constraints while remaining sensitive to direct detection experiments of DM. Due to the possibility of the odd scalar doublet being the next to lightest stable particle that can be sufficiently produced in colliders by virtue of its gauge interactions, one can have interesting signatures like displaced vertex or disappearing charged tracks.

I Introduction

It is quite well known, thanks to several evidences gathered in the last few decades, starting from the galaxy cluster observations by Fritz Zwicky Zwicky (1933) back in 1933, observations of galaxy rotation curves in 1970’s Rubin and Ford (1970) and the more recent observation of the bullet cluster Clowe et al. (2006) to the latest cosmology data provided by the Planck satellite Aghanim et al. (2018), that the present Universe is composed of a mysterious, non-luminous and non-baryonic form of matter, known as dark matter (DM). The latest data from the Planck mission suggest that the DM constitutes around of the total energy density of the present Universe. In terms of density parameter and , the present DM abundance is conventionally reported as Aghanim et al. (2018):

(1)

at 68% CL. However, in spite of such overwhelming evidence from astrophysics and cosmology based experiments, very little is known about the particle nature of DM. The typical list of criteria, that a particle DM candidate has to satisfy Taoso et al. (2008), already rules out all the standard model (SM) particles from being a DM candidate. This implies that we need physics beyond standard model (BSM) to incorporate the cosmic DM abundance. The most widely studied DM scenario so far has been is the weakly interacting massive particle (WIMP) paradigm. Here, the DM particle, having mass and interactions typically in the electroweak scale, can give rise to the correct relic abundance after thermal freeze-out, a remarkable coincidence often referred to as the WIMP Miracle Kolb and Turner (1990). For a recent review, one may refer to Arcadi et al. (2017). Such electroweak scale mass and interactions make this WIMP paradigm very appealing from direct detection point of view as well Liu et al. (2017).

Apart from DM, another equally appealing motivation for BSM physics is the observed neutrino mass and mixing which have been confirmed by several experiments for more than a decade till now Fukuda et al. (2001); Ahmad et al. (2002a, b); Abe et al. (2008, 2011, 2012); An et al. (2012); Ahn et al. (2012); Adamson et al. (2013); Patrignani et al. (2016). Among them, the more recent experimental results from the T2K Abe et al. (2011), Double Chooz Abe et al. (2012), Daya Bay An et al. (2012), RENO Ahn et al. (2012) and MINOS Adamson et al. (2013) experiments have not only confirmed the results from earlier experiments but also discovered the non-zero reactor mixing angle . For a recent global fit of neutrino oscillation data, we refer to Esteban et al. (2017). Apart from neutrino oscillation experiments, the neutrino sector is constrained by the data from cosmology as well. For example, the latest data from the Planck mission constrains the sum of absolute neutrino masses eV Aghanim et al. (2018). Similar to the observations related to DM, these experimental observations also can not be addressed by the SM as neutrinos remain massless at the renormalisable level. The Higgs field, which lies at the origin of all massive particles in the SM, can not have any Dirac Yukawa coupling with the neutrinos due to the absence of the right-handed neutrino. Even if the right handed neutrinos are included, one needs the Yukawa couplings to be heavily fine tuned to around in order to generate sub-eV neutrino masses from the same Higgs field of the SM. At non-renormalisable level, one can generate a tiny Majorana mass for the neutrinos from the same Higgs field of the SM through the dimension five Weinberg operator Weinberg (1979). However, the unknown cut-off scale in such operators points towards the existence of new physics at some high energy scale. Several BSM proposals, known as seesaw mechanism Minkowski (1977); Gell-Mann et al. (1979); Mohapatra and Senjanovic (1980); Schechter and Valle (1980), attempts to provide a dynamical origin of such operators by incorporating additional fields. Apart from the conventional type I seesaw, there exist other variants of seesaw mechanisms also, namely, type II seesaw Mohapatra and Senjanovic (1981); Lazarides et al. (1981); Wetterich (1981); Schechter and Valle (1982); Brahmachari and Mohapatra (1998), type III seesaw Foot et al. (1989) and so on.

Although the origin of neutrino mass and DM may look unrelated to each other, it is highly appealing and economical to find a common origin of both. Motivated by this here we study a very well motivated BSM framework based on the gauged symmetry Mohapatra and Marshak (1980); Marshak and Mohapatra (1980); Masiero et al. (1982); Mohapatra and Senjanovic (1983); Buchmuller et al. (1991), where and correspond to baryon and lepton numbers respectively. The most interesting feature of this model is that the inclusion of three right-handed neutrinos, as it is done in type I seesaw mechanism of generating light neutrino masses, is no longer a choice but arises as a minimal possible way to make the new gauge symmetry anomaly free. 111For other exotic and non-minimal solutions to such anomaly cancellation conditions, please refer to Montero and Pleitez (2009); Wang and Han (2015); Patra et al. (2016); Nanda and Borah (2017); Bernal et al. (2018) and references therein. The model has also been studied in the context of dark matter by several groups Rodejohann and Yaguna (2015); Okada and Seto (2010); Dasgupta and Borah (2014); Okada and Okada (2017); Klasen et al. (2017); Sahu and Yajnik (2006); Kohri and Sahu (2013); Kohri et al. (2009). DM in scale invariant versions of this model was also studied by several authors Okada and Orikasa (2012); Guo et al. (2015). Although the scalar DM in such models can be naturally stable by virtue of its charge, the fermion DM can not be realized in the minimal model except for the possibility of a keV right-handed neutrino DM which is cosmologically long lived Biswas et al. (2018). One can introduce additional discrete symmetries, such as that can stabilize one of the right-handed neutrinos Basak and Mondal (2014); Okada and Okada (2016) while the other two neutrinos take part in the usual type I seesaw mechanism, giving rise to solar and atmospheric neutrino mixing. Since the right-handed neutrino DM in this case annihilates into the SM particles only through the gauge bosons, the relic density is typically satisfied only near the resonance . Since the experimental limits from LEP II constrain such new gauge sector by giving a lower bound on the ratio of new gauge boson mass to the corresponding gauge coupling TeV Carena et al. (2004); Cacciapaglia et al. (2006), typically one gets a lower on mass to be around 3 TeV for generic gauge coupling similar to electroweak gauge couplings. This constrains the allowed DM mass to be more than a TeV. The DM sector also gets decoupled from the neutrino mass generation mechanism in such a case.

In this work, we consider the SM augmented by symmetry. In addition to three right-handed neutrinos: , we introduce one scalar doublet which are all odd under the discrete symmetry. The gauged symmetry is broken by introducing a singlet scalar . As a result the low energy phenomenology of this model is similar popular BSM framework that provide a common origin of neutrino mass and DM, known as the scotogenic scenario as proposed by Ma Ma (2006), where the odd particles take part in radiative generation of light neutrino masses. We consider the lightest right-handed neutrino to be lightest odd particle and hence the DM candidate. Due to the existence of new Yukawa interactions, we find that the parameter space giving rise to correct relic abundance is much larger than the resonance region for usual right-handed neutrino DM in model. This is possible due to additional annihilation and co-annihilation channels that arise due to Yukawa interactions. We also check the consistency of this enlarged DM parameter space with constraints from direct detection as well as neutrino mass. Since the odd scalar doublet can be the next to lightest stable particle (NLSP) in this case, it’s charged component can be sufficiently produced at the Large Hadron Collider (LHC) by virtue of its electroweak gauge interactions, provided it is in the sub-TeV regime. Due to the possibility of small mass splitting between NLSP and DM as well as within the components of the odd scalar doublet, we can have interesting signatures like displaced vertex or disappearing charged track (DCT) which the LHC is searching for.

This article is organized as follows. In section II, we discuss the model followed by neutrino mass in section III. The details of dark matter is discussed in section IV. We briefly discuss some collider signatures of the model in section V and finally conclude in section VI.

Ii The Model

Gauged extension of the SM is one of the most popular BSM frameworks in the literature. Since the charges of all the SM fields are already known, it is very much straightforward to write the details of such a model. However uplifting the global of the SM to a gauged one brings in anomalies. This is because the triangle anomalies for both and the mixed diagrams are non-zero. These triangle anomalies for the SM fermion content turns out to be

(2)

These anomalies can be cancelled minimally by introducing three right-handed neutrinos: with unit lepton number each, which is exactly what we need in the SM for realizing neutrino masses. These right-handed neutrinos contribute leading to vanishing total of triangle anomalies. As pointed out before, there exists alternative and non-minimal ways to cancel these anomalies as well Montero and Pleitez (2009); Wang and Han (2015); Patra et al. (2016); Nanda and Borah (2017); Bernal et al. (2018).

We then extend the minimal gauged model by introducing an additional symmetry and a scalar doublet so that the right-handed neutrinos: and are odd under the unbroken symmetry. The BSM particle content of the model is shown in table 1. The singlet scalar field is introduced in order to break the gauge symmetry spontaneously after acquiring a non-zero vacuum expectation value (VEV). Due to the imposed symmetry the neutrinos can not acquire masses at tree level, making way for radiative neutrino masses as we discuss in the next section.

Fields SU(3) SU(2) U(1) U(1) Z
1 1 0 -1 -
1 1 0 2 +
1 2 0 -
Table 1: New particles and their quantum numbers under the imposed symmetry.

The corresponding Lagrangian can be written as,

(3)

Where

(4)

We consider the mass squared term so that the neutral component of only acquire non-zero VEV’s v and u respectively. Expanding around the VEV, we can write

(5)

The minimization conditions of the above scalar potential will give

(6)

As a result the neutral scalar mass matrix becomes:

(7)

The mass eigenstates h and h are linear combinations of h and s and can be written as

(8)
(9)

where

(10)

Such a mixing can be tightly constrained by LEP as well as LHC Higgs exclusion searches as shown recently by Dupuis (2016). These constraints are more strong for low mass scalar and the upper bound on the mixing angle can be as low as Dupuis (2016). We consider a conservative upper limit on the mixing parameter for our analysis. This can be easily satisfied by suitable tuning of the parameters involved in the expression for mixing given in (10).

Physical masses at tree level for all the scalars can be written as:

(11)
(12)
(13)
(14)
(15)

Thus, the scalar sector consists of one SM Higgs like scalar , one singlet scalar , one charged scalar , another neutral scalar and one pseudoscalar .

Iii Neutrino Mass

As mentioned earlier, neutrinos do not acquire mass through Yukawa couplings of the type as they are forbidden by the unbroken symmetry. Therefore, type I seesaw is forbidden here. However, the term: allows us to get radiative neutrino mass at one loop level, as shown by the Feynman diagram in figure 1,

Figure 1: Radiative neutrino mass in scotogenic fashion.

By the exchange of Re() and Im() we can analytically calculate the one-loop diagram similar to Ma (2006) which gets a non-zero contribution after the electroweak symmetry breaking . The one-loop expression for neutrino mass is

(16)

. With different set of choices for Yukawa couplings we try to fit the observed light neutrino masses and mixing Esteban et al. (2017) as given below. A typical choice of Yukawa couplings is

(17)

The choices for mass parameters are

(18)

Thus the light neutrino mass matrix compatible with two mass squared differences and three mixing angles is given (in eV unit) as:

Note that in the above calculation, the CP phases are taken to be zero.

Iv Dark Matter

The relic abundance of a dark matter particle , which was in thermal equilibrium in the early Universe, can be calculated by solving the required Boltzmann equation:

(19)

where is the number density of the dark matter particle , is the equilibrium number density of , is the Hubble expansion rate of the Universe and is the thermally averaged annihilation cross section of . In terms of partial wave expansion one can write, . Numerical solution of the above Boltzmann equation gives Kolb and Turner (1990); Scherrer and Turner (1986)

(20)

where , is the freeze-out temperature, is the mass of dark matter, is the number of relativistic degrees of freedom at the time of freeze-out and and GeV is the Planck mass. Dark matter particles with electroweak scale mass and couplings freeze out at temperatures approximately in the range . More generally, can be calculated from the relation

(21)

which can be derived from the equality condition of DM interaction rate with the rate of expansion of the Universe . There also exists a simpler analytical formula for the approximate DM relic abundance Jungman et al. (1996)

(22)

The thermal averaged annihilation cross section is given by Gondolo and Gelmini (1991)

(23)

where ’s are modified Bessel functions of order and is the temperature.

If there exists some additional particles having mass difference close to that of DM, then they can be thermally accessible during the epoch of DM freeze out. This can give rise to additional channels through which DM can co-annihilate with such additional particles and produce SM particles in the final states. This type of co-annihilation effects on dark matter relic abundance were studied by several authors in Griest and Seckel (1991); Edsjo and Gondolo (1997); Bell et al. (2014); Bhattacharya et al. (2016); Chatterjee and Sahu (2014). Here we summarize the analysis of Griest and Seckel (1991) for the calculation of the effective annihilation cross section in such a case. The effective cross section can given as

where and and

(25)

The masses of the heavier components of the inert Higgs doublet are denoted by . The thermally averaged cross section can be written as

We first implement our model in micrOMEGAs package Belanger et al. (2014) to calculate the relic abundance of DM, the results of which we discuss in the following subsections.

iv.1 Relic Density of in Minimal Model

First, we show the the relic abundance of lightest right-handed neutrino DM in the minimal model so that we can later compare it with the modifications obtained in the scotogenic extension. In the minimal model, DM annihilates into SM particles either through the gauge boson or through singlet scalar mixing with the SM Higgs. In figure 2 we show the relic density (left) and corresponding annihilation cross-section (right) as a function of DM mass. The singlet scalar and masses are taken as 400 GeV and 2 TeV respectively. The singlet-SM Higgs mixing is taken to be and the gauge coupling is . The three resonance corresponding to the SM Higgs, singlet scalar and the boson are clearly seen in this figure. It is also clear that the correct relic abundance (corresponding to the Planck 2018 bound shown as the horizontal band in the left panel of figure 2) is satisfied only near these resonance regions. This is a typical feature of fermion singlet DM in minimal model.

(a)
(b)
Figure 2: (a). Relic density of as a function of DM mass, the horizontal band corresponds to the central value of Planck 2018 limit (1). (b) DM annihilation cross-section of as function of DM mass.

iv.2 Relic Density of in Scotogenic model

Apart from the usual annihilation channels of DM in minimal model discussed above, there arises a few more annihilation and co-annihilation channels after extending the model in scotogenic fashion. The corresponding annihilation and co-annihilation channels are shown in figure 3 and 4 respectively.

(a)
(b)
(c)
(d)
(e)
Figure 3: DM annihilation channels.
(a)
(b)
(c)
(d)
(e)
(f)
Figure 4: DM co-annihilation channels.

We first show the effects of co-annihilations on DM relic abundance by considering five different mass splittings where NLSP is the scalar doublet and its components. In figure 5 we show relic abundance as a function of DM mass for mass splittings GeV and with the singlet scalar-SM Higgs mixing . As can be seen from this figure, co-annihilation effects become more as we decrease the mass splitting from 500 GeV to 1 GeV. This results in smaller final relic abundance for GeV (sky blue line) and maximum relic abundance for mass splitting GeV (orange line). Since thermal relic abundance typically goes as inverse of annihilation cross section, the sky blue and orange lines in figure 5 correspond to maximum and minimum co-annihilation cross sections respectively. To generate this plot, the scalar mass and the mass have been fixed as GeV, GeV. The gauge coupling is fixed at and the Yukawa couplings with all three generation of leptons are taken to be universal . The value of Yukawa coupling has been taken from the maximum possible value in (17). The effects of co-annihilations will decrease slightly if we consider non-universal Yukawa couplings with first two generations have smaller couplings, in order to be in agreement with neutrino mass, as discussed before.

Figure 5: Relic abundance of dark matter with different mass splittings GeV respectively from sky blue line to orange line. The horizontal band corresponds to the central value of Planck 2018 limit (1). The singlet scalar- SM Higgs mixing is = 0.1.

In figure 6, we show the parameter space allowed by the relic abundance criteria for three different values of singlet scalar- SM Higgs mixing (blue points), (green points), (orange points). Here we have fixed the masses GeV, , =400 GeV, =2000 GeV, and varied the with 1 GeV to 1000 GeV mass splitting. The resonance corresponding to the singlet scalar is clearly visible at , which disappears for tiny mixing (orange coloured line).

Figure 6: Allowed parameter space in terms of the mass splitting and the dark matter mass. The points satisfying the relic density constraint correspond to different values of sin (blue), (green), (orange). The other parameters are fixed at =400 GeV, =2000 GeV, , GeV and GeV. is being varied corresponding to mass splitting 1 to 1000 GeV.

We then show the effects of Yukawa coupling of DM with leptons and scalar doublet on the relic abundance of DM. This is shown in figure 7 where the allowed parameter space from DM relic density point of view are shown in terms of for ten different values of Yukawa couplings. The mass of corresponds to which is being varied from 1 GeV to 1000 GeV. The lowermost line corresponds to smallest value of Yukawa coupling while the uppermost line corresponds to . This behaviour can be understood from the fact that for smaller Yukawa coupling, the mass splitting has to be smaller to enhance the co-annihilation cross section while larger Yukawa couplings can give similar co-annihilations even for larger mass splitting. The other parameters are fixed at same benchmark values as before.

Figure 7: Allowed parameter space in terms of the mass splitting and the dark matter mass. The points satisfying the relic density constraint correspond to different values of Yukawa coupling varying from 0.1 (bottom) to 1 (top). The singlet scalar - SM Higgs mixing taken as sin. Other parameters are fixed at =400 GeV, =2000 GeV, , GeV and GeV. is being varied as mass splittings is varied from 1 GeV to 1000 GeV.
Figure 8: Allowed parameter space in terms of the mass splitting and the dark matter mass. The parameters are fixed at sin. GeV and GeV, =400 GeV, =2000 GeV, . is being varied with respect to DM mass by considering mass splittings from 0.5 MeV to 1.777 GeV.

We then show the allowed parameter space in parameter space in figure 8, for and the DM mass splittings in the range 0.5 MeV to 1.777 GeV, for a mixing of and for a fixed values of flavour Yukawa couplings given in (17). Such mass splittings are chosen keeping in view of the collider analysis later where we consider -DM mass splitting to be below the tau lepton mass threshold so that can decay to first two generation leptons with displaced vertex signatures (and hence having tiny Yukawa couplings) but can co-annihilate due to large tau lepton Yukawa couplings.

In the above results, the effects of co-annihilation between DM and were clearly visible, specially when the mass splitting between them was relatively small. The co-annihilations between DM and heavier right-handed neutrinos on the other hand, were subdominant as the corresponding mass splittings were kept high. To see the effects of co-annihilation between DM and we fix the masses at some high values and vary the mass splittings between and . To be more specific, we choose GeV, GeV and GeV and vary which is defined as . We choose five different benchmark values of and vary from 1 GeV to 1000 GeV to calculate the corresponding relic, while keeping all other parameters fixed at previously chosen values. The corresponding result for relic abundance as a function of dark matter mass is summarized in the plot shown in figure 9. As we can see, the effects of changing the mass splitting on relic abundance is very minimal, with all values of gives rise to the same relic. On the other hand, the effect of changing the corresponding mass splitting between and was very significant, as shown in the plot of figure 5. This is expected, as the co-annihilations between and are through s-channel diagrams while that between and are through t-channel diagrams as shown in figure 4, except the one that is suppressed by singlet scalar-SM Higgs mixing. As a final check, we set all the mass splittings to be same namely, and find the relic abundance as a function of DM mass. This results in the same results that we had shown earlier in figure 5. This again validates our conclusion that the co-annihilations are dominated by DM- initiated diagrams rather than the ones between DM and .

Figure 9: Relic abundance of dark matter with different mass splittings GeV respectively from sky blue line to orange line. The horizontal band corresponds to the central value of Planck 2018 limit (1). The singlet scalar- SM Higgs mixing is = 0.1.

iv.3 Direct Detection of Dark Matter

Apart from the relic abundance constraints from Planck experiment, there exists strict bounds on the dark matter nucleon cross section from direct detection experiments like LUX Akerib et al. (2017), PandaX-II Tan et al. (2016); Cui et al. (2017) and Xenon1T Aprile et al. (2017, 2018). For right-handed neutrino DM in our model, there are two ways DM can scatter off nuclei: one is mediated by gauge boson and the other is mediated by scalars. The scalar mediated interactions occur due to mixing of singlet scalars of the model with the SM Higgs boson. Due to the Majorana nature of DM, the mediated diagram contribution to the spin-independent direct detection cross section turns out to be velocity suppressed and hence remains within experimental bounds. The scalar mediated diagram shown in figure 10 can however, saturate the latest experimental bounds. For the scalar mediated case, the spin-independent elastic scattering cross-section of DM per nucleon can be written as,

(27)

where A and Z are the mass and atomic number of the target nucleus respectively. is the reduced mass. The interaction strengths of proton and neutron with DM can be written as,

(28)

and

(29)

Where in the above Eq.28, the are given by  Ellis et al. (2000).

Using these, the spin-independent cross section is

Figure 10: DM-nucleon scattering mediated by scalars.
Figure 11: Spin independent DM-nucleon scattering cross section mediated by scalars in comparison to the latest Xenon1T bounds.

We show the DM-nucleon cross section mediated by scalars in figure 11 in comparison to the latest Xenon1T bound Aprile et al. (2018). The only unknown parameter in (LABEL:SI_cross) is . In figure 11 the blue points show the spin-independent DM-nucleon cross-section for the values of in between from bottom to top at a step of . As can be seen from this plot, the model remains sensitive to present direct detection experiments, specially when .

(GeV) , M, M (GeV) (pb)
100 105, 120, 120 0.189
200 205, 220, 220 1.65
300 305, 320, 320 3.46
400 405, 420, 420 1.04
500 505, 520, 520 3.817
600 605, 620, 620 1.593
700 705, 720, 720 7.286
800 805, 820, 820 3.568
900 905, 920, 920 1.828
1000 1005, 1020, 1020 9.794
Table 2: Production cross sections of from collisions at TeV LHC. Here we have kept fixed the mass splittings as M=5 GeV and M=M=15 GeV
(GeV) , M, M (GeV) (pb)
100 101, 120, 120 0.2176
200 201, 220, 220 1.782
300 301, 320, 320 3.65
400 401, 420, 420 1.087
500 501, 520, 520 3.957
600 601, 620, 620 1.647
700 701, 720, 720 7.523
800 801, 820, 820 3.656
900 901, 920, 920 1.879
1000 1001, 1020, 1020 1.004
Table 3: Production cross sections of from collisions at TeV LHC. Here we have kept fixed the mass splittings as M=1 GeV and M=M=19 GeV
(GeV) , M, M (GeV) (pb)
100 101.2, 101, 101.2 0.2473
200 201.2, 201, 201.2 2.057
300 301.2, 301, 301.2 4.359
400 401.2, 401, 401.2 1.341
500 501.2, 501, 501.2 5.001
600 601.2, 601, 601.2 2.141
700 701.2, 701, 701.2 9.938
800 801.2, 801, 801.2 4.91
900 901.2, 901, 901.2 2.546
1000 1001.2, 1001, 1001.2 1.367
Table 4: Production cross sections of from collisions at TeV LHC. Here we have kept fixed the mass splittings as M=1 GeV and M=M=200 MeV

V Collider Signatures

Collider signatures of models have been discussed extensively in the literature. Since all the SM fermions are charged under this gauge symmetry, the production of gauge boson in proton proton collisions can be significant Okada and Okada (2016); Basso et al. (2009), if the corresponding gauge coupling is of the same strength as electroweak gauge couplings. Such heavy gauge boson, if produced at colliders, can manifest itself as a narrow resonance through its decay into dileptons, say. The latest measurement by the ATLAS experiment at 13 TeV LHC constrains such gauge boson mass to be heavier than TeV depending on whether the final state leptons are of muon or electron type Aaboud et al. (2017). The corresponding bound for tau lepton final states measured by the CMS experiment at 13 TeV LHC is slightly weaker, with the lower bound on mass being 2.1 TeV Khachatryan et al. (2017). In deriving the bounds for final states, the corresponding gauge coupling was chosen to be . Therefore, such bounds can get weaker if we consider slightly smaller values of gauge couplings. For a recent discussion on such signatures, please refer to Nanda and Borah (2017). For other possible signatures say, right-handed neutrinos in model among others, please see references Basso et al. (2009, 2011a, 2011b, 2012a, 2012b); Accomando et al. (2013); Okada and Okada (2016); Das et al. (2018a, b)

Instead of such conventional searches, here we consider two interesting signatures our present version of model can have. This is related to the production and subsequent decay of the charged component of odd scalar doublet which can be the NLSP or next to NLSP in such a case while LSP, the lightest right-handed neutrino is the DM. The production cross section of charged pairs as well as at 14 TeV proton proton collisions are shown in table 2, 3, 4 for different benchmark values of parameters. For this calculation, we implemented the model in FeynRule Alloul et al. (2014) and used MADGRAPH Alwall et al. (2014) for the cross section calculations. If the NLSP is long lived, it can give rise to a displaced vertex signature at colliders. Since such signatures are very much clean, one can search for such particles at colliders with relatively fewer events. Here we make some crude estimates at the cross section level and decay length without going into the details of event level analysis. For recent searches of displaced vertex type signatures at the LHC, one may refer to Aaboud et al. (2018a, b). For a recent discussion on such signatures in type I seesaw model and active-sterile neutrino mixing case, please see Jana et al. (2018) and Cottin et al. (2018).

Figure 12: Decay length of as a function of mass.

The decay width of can be written as

(31)

where is the Yukawa coupling of the interaction . The corresponding decay length as a function of mass for different benchmark values of are shown in figure 12. At high luminosity LHC, decay length of a few cm can be searched for, if the decaying particle has production cross section of the order a few fb or more Jana et al. (2018), which is clearly satisfied for several benchmark masses as shown in table 2, 3, 4. Although such tiny Yukawa couplings required for displaced vertex signatures will not induce any co-annihilations between and the components of , we can still have strong co-annihilations due to tau lepton couplings while decay into DM and tau lepton can be kinematically forbidden. In such a case, DM can be sufficiently light due to strong co-annihilations via tau lepton sector couplings but at the same time we can have displaced vertex signatures of into first two generation charged leptons. Future proposed experiments like the Large Hadron electron Collider (LHeC), Future Circular electron-hadron Collider (FCC-eh) will be able to search for even shorter decay lengths and cross sections, than the ones discussed here.

Another interesting possibility arises when the mass splitting between and is very small, of the order of 100 MeV. For such mass splitting, the dominant decay mode of can be , if the corresponding Yukawa coupling of vertex is kept sufficiently small for the leptonic decay mode to be subdominant. The corresponding decay width is given by

(32)

where , g, m are the form factor , gauge coupling, and W boson mass respectively. Such tiny decay width keeps the lifetime of considerably long enough that it can reach the detector before decaying. In fact, the ATLAS experiment at the LHC has already searched for such long-lived charged particles with lifetime ranging from 10 ps to 10 ns, with maximum sensitivity around 1 ns Aaboud et al. (2018b). In the decay , the final state pion typically has very low momentum and it is not reconstructed in the detector. On the other hand the neutral scalar in the final state eventually decays into DM and a light neutrino and hence remain invisible throughout. Therefore, it gives rise to a signature where a charged particle leaves a track in the inner parts of the detector and then disappears leaving no tracks in the portions of the detector at higher radii. The corresponding decay length as a function of mass is shown in the left panel plot of figure 13. The right panel plot of figure 13 shows a comparison of the decay length in our model with the ATLAS bound Aaboud et al. (2018b). In figure 14, we show the comparison between the leptonic decay mode and pionic decay mode for different benchmark values of Yukawa couplings.

Figure 13: Decay length corresponding to the pionic decay for fixed mass splitting of 200 MeV (left panel) and its comparison with the ATLAS bound for different benchmark values of mass splitting (right panel).
Figure 14: Decay length corresponding to the pionic decay leading to DCT and its comparison with the decay responsible for displaced vertex signature.

Vi Conclusions

We have studied a simple extension of the minimal gauged with three right-handed neutrinos in order to realise fermion singlet dark matter. The minimal model is extended by a scalar doublet and an additional symmetry so that the right-handed neutrinos and are odd under this symmetry while all other fields are even. Neutrinos remain massless at tree level but acquires a radiative contribution with the odd fields going in the look, in a way similar to scotogenic scenarios. The lightest odd particle, considered to be the lightest right-handed neutrino, is the dark matter candidate in the model. Due to lepton portal interactions and hence several co-annihilation channels, there exists enlarged parameter space in terms of dark matter mass so that the correct relic abundance is obtained. This is in sharp contrast with minimal fermion singlet dark matter scenarios in such models where relic is usually satisfied only in the vicinity of resonance regions. We also find that the co-annihilation between right-handed neutrino DM and the odd scalar doublet remains dominant over that between DM and heavier right-handed neutrinos.

After showing the parameter space allowed from relic abundance criteria, we incorporate the constraints from neutrino mass and dark matter direct detection. While the direct detection scattering mediated by the gauge bosons remain velocity suppressed, the scalar mediated contribution can saturate the current limits on spin independent direct detection cross section. We then point out two interesting collider signatures this model can have apart from the usual collider prospects of models. These are namely, displaced vertex and disappearing charged track signatures which are considered to be very clean at colliders like the LHC. This possibility arises due to the fact that the charged component of the odd scalar doublet can have masses slightly above DM mass, can be sufficiently produced at colliders by virtue of electroweak gauge interactions and have a long lifetime due to tiny Yukawa couplings or compressed mass spectrum. We show that for the parameter space allowed from dark matter considerations, LHC can remain sensitive to probing the model for some region of parameter space while future experiments can be playing a decisive role in verifying or ruling out most of the parameter space. We leave a detailed collider study to future works.

Acknowledgements.
We thank the organizers of WHEPP XV at IISER Bhopal (14-23 December, 2017), where this work was initiated. DB acknowledges the support from IIT Guwahati start-up grant (reference number: xPHYSUGI-ITG01152xxDB001) and Associateship Programme of IUCAA, Pune. DN would like to thank Shibananda Sahoo for useful discussions.

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