A Mini-review on Vector-like Leptonic Dark Matter, Neutrino Mass and Collider Signatures
Subhaditya Bhattacharya, Purusottam Ghosh, Nirakar Sahoo, Narendra Sahu
MMini Review on Vector-like Leptonic Dark Matter, Neutrino Mass and ColliderSignatures
Subhaditya Bhattacharya ∗ and Purusottam Ghosh, † Department of Physics, Indian Institute of Technology Guwahati, North Guwahati, Assam- 781039, India
Nirakar Sahoo ‡ Institute of Physics, Sachivalaya Marg, Bhubaneswar, Odisha 751005, Indiaand Homi Bhabha National Institute, Training School Complex, Anushakti Nagar, Mumbai 400085, India
Narendra Sahu § Department of Physics, Indian Institute of Technology,Hyderabad, Kandi, Sangareddy, 502285, Telangana, India
We review a class of models in which the Standard Model (SM) is augmented by vector-like leptons:one doublet and a singlet, which are odd under an unbroken discrete Z symmetry. As a result, theneutral component of these additional vector-like leptons are stable and behave as dark matter. Westudy the phenomenological constraints on the model parameters and elucidate the parameter spacefor relic density, direct detection and collider signatures of dark matter. In such models, we furtheradd a scalar triplet of hypercharge two and study the consequences. In particular, after electroweak symmetry breaking (EWSB), the triplet scalar gets an induced vacuum expectation value(vev), which yield Majorana masses not only to the light neutrinos but also to vector-like leptonicdoublet DM. Due to the Majorana mass of DM, the Z mediated elastic scattering with nucleon isforbidden and hence allowing the model to survive from stringent direct search bound. The DMwithout scalar triplet lives in a small singlet-doublet leptonic mixing region (sin θ ≤ .
1) due to largecontribution from singlet component and have small mass difference (∆ m ∼
10 GeV) with chargedcompanion, the NLSP (next to lightest stable particle), to aid co-annihilation for yielding correctrelic density. Both these observations change to certain extent in presence of scalar triplet to aidobservability of hadronically quiet leptonic final states at LHC, while one may also confirm/rule-outthe model through displaced vertex signal of NLSP, a characteristic signature of the model in relicdensity and direct search allowed parameter space.
PACS numbers: ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] § Electronic address: [email protected] a r X i v : . [ h e p - ph ] M a y I. INTRODUCTION
The existence of dark matter (DM) in a large scale ( > a few kpc) has been proven irrefutably by various astrophysicalobservations. The prime among them are galaxy rotation curves [1], gravitational lensing [2] and large scale structureof the Universe. See for a review [3, 4]. In the recent years the satellite borne experiments like Wilkinson MicrowaveAnisotropy Probe (WMAP) [5] and PLANCK [6] precisely determined the relic abundance of DM by measuring thetemperature fluctuation in the cosmic microwave background radiation (CMBR). All the above said evidences of DMemerge via gravitational interaction in astrophysical environments, which make a challenge to probe the existence ofDM in a terrestrial laboratory where density of DM is feeble.Alternatively one can explore other elementary properties of DM which can be probed at an earth based laboratory.In fact, one can assign a weak interaction property to DM through which it can be thermalised in the early Universeat a temperature above its mass scale. As the temperature falls due to adiabatic expansion of the Universe, the DMgets decoupled from the thermal bath below its mass scale. As a result the ratio: n DM /s , where s is the entropydensity, remains constant and is precisely measured by PLANCK in terms of Ω DM h = 0 . ± . etc . This leads to huge uncertainty in search of DM. Despite this, many experimentsare currently operational, that uses direct, indirect and collider search methods. Xenon-100 [7], LUX [8], Xenon-1T [9], PANDA [10] are some of the direct DM search experiments which are looking for signature of DM via nuclearscattering, while PAMELA [11], AMS-2 [12], Fermi gamma ray space telescope [13], IceCube [14] etc are some of theindirect DM search experiments which are looking for signature of DM in the sky. The search of DM is also goingon at collider experiments like large Hadron Collider (LHC) [15, 16]. Except some excess in the antiparticle flux inthe indirect search data, direct and collider searches for DM has produced null observation so far. This in turn put astrong bound on the DM mass and coupling with which it can interact to the visible sector of the universe.After the Higgs discovery in 2012 at CERN LHC, the standard model (SM) seems to be complete. However, it isfound that none of the particles in SM can be a candidate of DM, which is required to be stable on cosmological timescale. While neutrinos in SM are stable, but their relic density is far less than the required DM abundance and isalso disfavored from the structure formation. Moreover, neutrinos are massless within the SM. A tiny but non-zeroneutrino mass generation requires the SM to be extended. This opens up the possibility of exploring new models ofDM, while explaining non-zero masses for neutrinos in the same framework and thus predict a measurable alternationto DM phenomenology, which can be examined in some of the above said experiments.In this review we explore the possibility of leptonic DM and non-zero neutrino masses [17] in a framework beyondthe SM. The simplest leptonic DM can arise by augmenting the SM with an additional singlet fermion [18] χ ,stabilized by a Z symmetry. However, unless we assume the presence of an additional scalar singlet, which acquiresvacuum expectation value (vev) and thus mixes with SM Higgs, the lepton singlet DM can not possess renormalisableinteraction with visible sector. The next possibility is to introduce a vector-like leptonic doublet: N = ( N N − ) T ,which is also odd under the Z symmetry. The annihilation cross-section of such fermions are large due to Z mediationand correct relic density can only be achieved at a very high DM mass. However, the combination of singlet χ withthe doublet vector-like lepton N provides a good candidate of DM [19], which has been discussed in details here. Wediscuss the phenomenological constraints on model parameters and then elucidate the allowed parameter space ofsuch models from relic density and direct detection constraints. We also indicate collider search strategies for suchDM. It turns out that the displaced vertex of the charged fermion: N ± (also called next to lightest stable particle(NLSP)) is a natural signature of such DM model.In an attempt to address neutrino mass generation in the same framework, we further add a scalar triplet ∆ ofhypercharge 2 and study the consequences [20]. In particular, after electro weak symmetry breaking (EWSB), thetriplet scalar acquires an induced vacuum expectation value (vev) which give rise sub-eV Majorana masses to lightneutrinos through the Type II Seesaw mechanism [22]. The scalar triplet also generates a Majorana mass for theneutral component of the vector-like lepton doublet: N , which constitues a minor component of the DM. Due toMajorana mass of DM, the Z mediated elastic scattering with nucleon is forbidden [23]. As a result, the model survivesfrom the stringent direct search bound. In absence of scalar triplet, the singlet-doublet DM is allowed to have only atiny fraction of doublet component (sin θ ≤ .
1) to evade direct search bound. In this limit, due to large contributionfrom the singlet component, the annihilation cross section of the DM becomes smaller than what it requires to achievecorrect relic density. To make it up for correct relic density, the DM additionally requires to co-annihilate with itscharged and heavy neutral companions and therefore requires small mass difference (∆ m ∼
10 GeV) with chargedcompanion or NLSP. However, in presence of the scalar triplet, we show that both the singlet-doublet mixing (sin θ )and the mass difference with NLSP (∆ m ) can be relaxed and larger parameter space is available for correct relicdensity and being compatible with the latest direct detection bounds. Moreover, the scalar triplet aid observabilityof hadronically quiet leptonic final states at LHC while one may also confirm/rule-out the model through displacedvertex of NLSP, a characteristic signature of the model in relic density and direct search allowed parameter space.The paper is arranged as follows. In section-II, we briefly discuss about a vector-like singlet leptonic DM. In section-III, we discuss the viable parameter space of a vector-like inert doublet lepton DM. It is shown that an inert leptondoublet DM alone is ruled out due to large Z-mediated elastic scattering with the nucleus. However, in presence of ascalar triplet of hyper charge-2, the inert lepton doublet DM can be reinstated in a limited parameter space, which wediscuss in section-IV. Moreover in section-IV, we discuss how the scalar triplet can give rise non-zero masses to activeneutrinos via type-II seesaw [22]. In section-V we discuss how an appropriate combination of singlet and doubletvector like leptons can give rise a nice possibility of DM in a wide range of parameter space. A triplet extension ofsinglet-doublet leptonic DM is further discussed in section-VI. We discuss collider signature of singlet-doublet leptonicDM in presence of a scalar triplet in section-VII and conclude in section-VIII. We provide some vertices of inert leptondoublet (ILD) DM in presence of scalar triplet in Appendix-A. II. VECTOR-LIKE LEPTONIC SINGLET DARK MATTER
A simplest possibility to explain DM content of the present Universe is to augment the SM by adding a vector-likesinglet lepton χ . The stability of χ can be ensured by imposing an additional discrete Z symmetry, under which χ isodd while all other particles are even. In fact, a singlet DM has been discussed extensively in the literature [18]. Herewe briefly recapitulate the main features to show the allowed parameter space by observed relic density and latestconstraint from direct detection experiments.The Lagrangian describing the singlet leptonic DM χ can be given as: L = χ ( iγ µ ∂ µ − m χ ) χ − (cid:16) H † H − v (cid:17) χ χ. (1) FIG. 1: [Left] Relic density as a function of singlet DM mass, m χ for different values of Λ mentioned in the figure inset. [Right]SI DD cross-section vs DM mass, m χ for different values of Λ. Notice that the Lagrangian introduces two new parameters: DM mass, m χ and the new physics scale Λ connectingDM to the SM through effective dimension five operator, on which the DM phenomenology depends. In the earlyUniverse, χ freezes out via the interaction χχ → SM particles to give rise a net relic density that we observe today.We use micrOmegas [24] to calculate the relic density as well as spin independent elastic cross-section with nucleonof χ . In Fig. 1, we show relic abundance (left-panel) and spin independent direct detection (SIDD) cross section (rightpanel) as a function of DM mass ( m χ ) for different values Λ ∼ { − } GeV . We observe that the constrainton SIDD cross section favors large values of Λ, while large Λ values yield over abundance of DM. We also note in theright panel figure 1, that the SIDD cross section is very less sensitive to DM mass. This is because the direct searchcross-section is proportional to the effective DM-nucleon reduced mass square ( µ r = m N m χ m N + m χ ) (see Eq. 58 or Eq. 59),where m N < m χ yields a mild dependence on DM mass. This feature is observed in rest of the analysis as well. The scale Λ is a priori unknown and should be validated from experimental constraints. In effective theory consideration, we ensurethat m χ < Λ. Therefore, a singlet leptonic DM alone is almost ruled out. However, the dark sector of the Universe may notbe simple as in the case of singlet leptonic DM. In the following we discuss a few more models with larger numberof parameters, yet predictive. We end this section by noting that one can think of a pseudoscalar propagator toyield an effective DM-SM interaction of the form ( ¯ χγ χ )( H † H ) / Λ. In this case, the relic density and direct searchcross-sections become velocity dependent (see for example in [25]). Please also see section V below Eq. 35 for moredetails.
III. INERT LEPTON DOUBLET DARK MATTER
Let us assume that the dark sector is composed of a vector-like lepton doublet: N = ( N N − ) T , which is oddunder an extended Z symmetry (hence called inert lepton doublet (ILD)), while all the Standard Model (SM) fieldsare even. As a result the neutral component of the ILD N is stable. The quantum numbers of dark sector fields andthat of SM Higgs under the SM gauge group, augmented by a Z symmetry, are given in Table I. We will check if N can be a viable candidate of DM with correct relic abundance while satisfying the direct detection constraints fromthe null observation at various terrestrial laboratories. Fields SU (3) C × SU (2) L × U (1) Y × Z N = (cid:18) N N − (cid:19) H = (cid:18) H + H (cid:19) G ≡ SU (3) C × SU (2) L × U (1) Y × Z . The Lagrangian of the model is given as: L IL = N [ iγ µ ( ∂ µ − ig σ a W aµ − ig (cid:48) Y B µ ) − m N ] N . (2)Thus the only new parameter introduced in the above Lagrangian is the mass of N , i.e. m N . Expanding the covariantderivative of the above Lagrangian L IL , we get the interaction terms of N and N ± with the SM gauge bosons as: L ILint = N iγ µ ( − ig σ a W aµ + i g (cid:48) B µ ) N = (cid:16) e θ W cos θ W (cid:17) N γ µ Z µ N + e √ θ W N γ µ W + µ N − + e √ θ W N + γ µ W − µ N − e N + γ µ A µ N − − (cid:16) e θ W cos θ W (cid:17) cos 2 θ W N + γ µ Z µ N − . (3)where g = e / sin θ W and g (cid:48) = e / cos θ W with e being the electromagnetic coupling constant and θ W being theWeinberg angle.Since N is a doublet under SU (2) L , it can contribute to invisible Z -decay width if its mass is less than 45 GeVwhich is strongly constrained. Therefore, in our analysis we will assume m N >
45 GeV.
A. Relic abundance of ILD Dark Matter
The number changing annihilation and co-annihilation processes which control freeze-out and hence relic densityof DM N are shown in Figs. 2, 3 and 4.To estimate the relic density of DM in this framework one needs to solve the relevant Boltzmann equation: dn N dt + 3 Hn N = −(cid:104) σv (cid:105) N N → SMSM (cid:16) n N − n eqN (cid:17) −(cid:104) σv (cid:105) N N ± → SMSM (cid:16) n N n N ± − n eqN n eqN ± (cid:17) . (4)To find the relic density of ILD DM, here we adopted micrOmegas [24] and implemented the model in it. Noticethat the relic density of DM is mainly controlled by DM mass, m N and SM gauged couplings. Since SM gauge N N N /N ± Z/W + Z/W − N N Z f/W + /hf/W − /Z FIG. 2: Annihilation of ILD DM to SM particles . N − N + N W + W − N − N + A f/W + f/W − N − N + N ± Z/A/AZ/Z/A N − N + Z f/W + /h/N f/W − /Z/N FIG. 3: Annihilation of charged partner of ILD DM to SM particles which contributes as co-annihilation with ILD DM. N N − N − W − Z/A N N − W f/h/W/Wf (cid:48) /W/A/ZN N − N ZW − FIG. 4: Co-annihilation processes of DM N with its charged partner N ± to SM particles. couplings are fixed, the only relevant parameter which controls the relic density is the DM mass, m N . The behaviorof relic density with DM mass shown in Fig. 5 along with correct relic density bound which is shown in grey patch.Note here that a sharp drop around DM mass m N ∼ m h / m N ∼ N we get overabundance of DM (due to small cross section), while for lighter mass of N we get under abundance of DM (due tolarge cross-section). B. Direct search constraint on ILD Dark Matter
In a direct detection experiment, the DM N scatters with the nucleon through t-channel Z mediated diagram,as shown schematically in the left panel of Fig. 6. Like relic density we obtain the DM-nucleon cross-section using micrOmegas [24]. Since N N Z interaction is coming from SM gauge coupling and which is large, so the outcome ofspin independent direct detection (SIDD) cross section becomes large. The SIDD cross-section in this case is plotted FIG. 5: Variation of relic density of DM N with mass m N . The gray patch corresponds to relic density allowed limit fromPLANCK: 0 . ≤ Ω h ≤ . with DM mass, m N , which is shown in orange colored patch in right panel of Fig. 6. The green patch in this Fig. 6indicates relic density allowed mass range in the same plane. LUX 2017 [8] and XENON 1T [9] direct detection limitsare also plotted in the same figure (right panel of Fig. 6). Thus we see that an ILD DM is completely ruled out bydirect detection bound. However, as we discuss in section IV the ILD DM can be resurrected in presence of a scalartriplet of hyper charge 2. Moreover, the scalar triplet will generate sub-eV masses of active neutrinos through type-IIseesaw. N N nn Z FIG. 6: Left: Feynman diagram for direct detection (DD) of DM, N . Right: Spin independent(SI) DM-nucleon cross-sectionis plotted in m N − σ SIN plane. Relic density allowed DM mass region is indicated here by green patch.
IV. TRIPLET EXTENSION OF THE ILD DARK MATTER
We now extend the ILD dark matter model with a scalar triplet, ∆ ( Y ∆ = 2) which is even under the discrete Z symmetry. The Lagrangian of this extended sector is given as: L II = Tr[( D µ ∆) † ( D µ ∆)] − V ( H, ∆) + L II Yuk , (5)where the covariant derivative is defined as D µ ∆ = ∂ µ ∆ − ig (cid:2) σ a W aµ , ∆ (cid:3) − ig (cid:48) Y ∆ B µ ∆and in the adjoint representation the triplet ∆ can be expressed in a 2 × (cid:32) H + √ H ++ δ − H + √ (cid:33) .Similarly the scalar doublet H can be written in component form as: H = (cid:18) φ + φ (cid:19) . (6)The modified scalar potential including ∆ and H can be given as: V ( H, ∆) = − µ H ( H † H ) + λ H (cid:0) H † H (cid:1) + M Tr[∆ † ∆] + λ ( H † H )Tr[∆ † ∆]+ λ (cid:0) Tr[∆ † ∆] (cid:1) + λ Tr[(∆ † ∆) ] + λ H † ∆∆ † H + (cid:2) µ (cid:0) H T iσ ∆ † H (cid:1) + h . c . (cid:3) , (7)where we assume that M is positive. So the scalar triplet ∆ does not acquire any vev. However, it acquires aninduced vev after EW phase transition. The vevs of the scalar fields are given by: (cid:104) ∆ (cid:105) = (cid:18) v t / √ (cid:19) and (cid:104) H (cid:105) = (cid:18) v/ √ (cid:19) . (8)Since the addition of a scalar triplet can modify the ρ parameter, whose observed value: ρ = 1 . ± . ρ = 1, so we have a constraint on the vev v t as: v t ≤ . . (9)On the other hand electroweak symmetry breaking gives (cid:112) v + 2 v t = 246GeV. This implies that v t << v .Now minimizing the scalar potential, V ( H, ∆) (in Eq. 7) we get: M = 2 µv − √ λ + λ ) v v t − √ λ + λ ) v t √ v t ,µ H = λ H v λ + λ ) v t − √ µv t . (10)In the limit v t << v , from Eq. 10 we get the vevs, v t = µv M + ( λ + λ ) v / v ≈ µ H √ λ H . (11)In presence of the scalar triplet ∆, the Yukawa interactions in Eq. 5 are given by: L IIY uk = 1 √ (cid:2) ( f L ) αβ L cα iσ ∆ L β + f N N c iσ ∆ N + h.c. ] , (12)where L is the SM lepton doublet and α, β denote family indices. The Yukawa interactions importantly inherit thesource of neutrino masses ( first term in square bracket) and DM-SM interactions (second term in square bracket). A. Scalar doublet-triplet mixing
The quantum fluctuations around the minimum of scalar potential, V ( H, ∆) can be given as:∆ = (cid:32) H + √ H ++ v t + h t + iA √ − H + √ (cid:33) and H = (cid:18) v + h √ (cid:19) . (13)Thus the scalar sector constitute two CP-even Higgses: h and h t , one CP-odd Higgs: A , one singly charged scalar: H ± and one doubly charged scalar: H ±± . In the limit v t << v , the mass matrix of the CP-even Higgses: h and h t ,is given by: M = m h −√ µv −√ µv m T , (14)where m h ≈ λ H v / m T = µv / √ v t . Diagonalizing the above mass matrix we get two neutral physical Higgses: H and H : H = cos α h + sin α h t , H = − sin α h + cos α h t , (15)where H is the standard model like Higgs and H is the triplet like scalar. The corresponding mass eigenvalues are m H (SM like Higgs) and m H (triplet like scalar) are given by: m H ≈ m h − ( µv/ √ m T − m h ,m H ≈ m T + ( µv/ √ m T − m h . (16)The mixing angle is given by tan 2 α = −√ µv ( m T − m h ) . (17)From Eqs.(17), (11) and (9) we see that there exist an upper bound on the mixing anglesin α < . (cid:18)
174 GeV v (cid:19) − . ( m h /
125 GeV) ( m T /
200 GeV) . (18)We also get a constraint on sin α from SM Higgs phenomenology, since the mixing can change the strength of theHiggs coupling to different SM particles. See for example [27, 28], in which the global fit yields a constraint on mixingangle sin α (cid:46) .
5, which is much larger than the above constraint obtained using ρ parameter.From Eq. 7, all the couplings λ H , λ , λ , λ , λ and µ can be expressed in terms of physical scalar masses: m H , m H , m H ± , m H ±± , m A and the vevs v and v t as [29] : λ H = 2 v (cid:16) m H cos α + m H sin α (cid:17) ,λ = 4 m H ± v + 4 v t − m A v + 4 v t + sin 2 α v t v (cid:0) m H − m H (cid:1) ,λ = 1 v t (cid:104) (cid:16) m H sin α + m H cos α (cid:17) + v v + 4 v t ) m A − v v + 4 v t m H ± + m H ±± (cid:105) ,λ = 1 v t (cid:104) v v + 2 v t m H ± − v v + 4 v t m A − m H ±± (cid:105) ,λ = 4 m A v + 4 v t − m H ± v + 2 v t ,µ = √ v t v + 4 v t m A . (19)where m A is the mass of pseudo scalar. It is important to note that the quartic couplings λ and λ are inverselyproportional to the triplet vev v t which has important consequences for dark matter relic abundance that we discussin section IV D. B. Non-zero neutrino masses
The coupling of scalar triplet ∆ to SM lepton and Higgs doublet combinely break the lepton number by two unitsas given in Eq. 12. As a result the ∆ L α L β coupling yields Majorana masses to three flavors of active neutrinos as [22]:( M ν ) αβ = √ f L ) αβ (cid:104) ∆ (cid:105) ≈ ( f L ) αβ − µv √ M . (20)Assuming µ (cid:39) M ∆ (cid:39) O (10 ) GeV, we can explain neutrino masses of order 1eV with a coupling strength f L (cid:39) M ∆ can be brought down to ∼ TeV by taking the coupling to be much smaller f L (cid:39) − , andindeed represents a bit of fine tuning in the neutrino sector.To get the neutrino mass eigen values, the above mass matrix can be diagonalized by the usual U P MNS matrix as : M ν = U PMNS M diagν U T PMNS , (21)where U P MNS is given by U P MNS = c c s c s e − iδ − s c − c s s e iδ c c − s s s e iδ s c s s − c c s e iδ − c s − s c s e iδ c c .U ph , (22)with c ij , s ij stand for cos θ ij and sin θ ij respectively and U ph is given by U ph = Diag (cid:0) e − iγ , e − iγ , (cid:1) . (23)Where γ , γ are two Majorana phases. The diagonal matrix M diagν = Diag ( m , m , m ) with diagonal entries are themass eigen values for the neutrinos. The current neutrino oscillation data at 3 σ confidence level give the constrainton mixing angles [26] :0 . < sin θ < . , . < sin θ < . , . < sin θ < . . (24)However little information is available about the CP violating Dirac phase δ as well as the Majorana phases. Althoughthe absolute mass of neutrinos is not measured yet, the mass square differences have already been measured to a gooddegree of accuracy : ∆ m ≡ m − m = (6 . − . × − eV | ∆ m | ≡ | m − m | = (2 . − . × − eV . (25)One of the main issues of neutrino physics lies in the sign of the atmospheric mass square difference | ∆ m | ≡| m − m | , which is still unknown. This yields two possibilities: normal hierarchy (NH) ( m < m < m ) or invertedhierarchy (IH) ( m < m < m ). Another possibility, yet allowed, is to have a degenerate (DG) neutrino massspectrum ( m ∼ m ∼ m ). Assuming that the neutrinos are Majorana, the mass matrix can be written as : M ν = a b cb d ec e f . (26)Using equations (21), (22), (24) and (25), we can estimate the unknown parameters in neutrino mass matrix of Eq. 26.To estimate the parameters in NH, we use the best fit values of the oscillation parameters. For a typical value of thelightest neutrino mass of m = 0 . a = 0 . , b = 0 . , c = 0 . ,d = 0 . , e = 0 . , f = 0 . . (27)Similarly for IH case, choosing the lightest neutrino mass m = 0 .
001 eV, we get the mass parameters (in eV) as : a = 0 . , b = − . , c = − . ,d = 0 . , e = − . , f = 0 . . (28)In both the cases, we put the Dirac and Majorana phases to be zero for simplicity.The analysis of neutrino mass is more indicative here than being exhaustive. This is essentially to build theconnection between the dark sector and neutrino sector advocated in the model set up. One can easily perform ascan over the mass matrix parameters to obtain correct ranges of the neutrino observables, and that of course liesin the vicinity of the aforementioned values. But, we do not aim to elaborate that in this draft. We have not alsoadhered to a specific lepton mixing matrix pattern (say tri-bi-maximal mixing) coming from a defined underlyingflavour symmetry (say A ), which will be able to correlate different parameters of the mass matrix.The mass of the scalar triplet can also be brought down to TeV scale by choosing appropriate Yukawa coupling asexplained before. If the triplet mass is order of a few hundreds of GeV, then it can give interesting dilepton signalsin the collider. See for example, [30] for a detailed discussion regarding the dilepton signatures of the scalar triplet atcollider.0 C. Pseudo-Dirac nature of ILD Dark Matter
From Eq. (12) we see that the vev of ∆ induces a Majorana mass to N which is given by: m = √ f N (cid:104) ∆ (cid:105) ≈ f N − µv √ M . (29)Thus the N has a large Dirac mass M N (as in Eq. 2) and a small Majorana mass m as shown in the above Eq. 29.Therefore, we get a mass matrix in the basis { N L , ( N R ) c } as: M = (cid:18) m M N M N m (cid:19) . (30)Thus the Majorana mass m splits the Dirac spinor N into two pseudo-Dirac states N , with mass eigenvalues M N ± m . The mass splitting between the two pseudo-Dirac states N , is given by δm = 2 m = 2 √ f N (cid:104) ∆ (cid:105) . (31)Note that δm << M N from the estimate of induced vev of the triplet and hence does not play any role in therelic abundance calculation. However, the sub-GeV order mass splitting plays a crucial role in direct detection byforbidding the Z-boson mediated DM-nucleon elastic scattering. Now from Eq. (20) and (29) we see that the ratio: R = ( M ν ) m = f L f N (cid:46) − , (32)where we assume M ν ∼ m ∼
100 keV. Here the mass splitting between the two states N and N is chosento be O (100) keV in order to forbid the Z -mediated inelastic scattering with the nucleons in direct detection. Thuswe see that the ratio R (cid:46) − is heavily fine tuned. In other words, the scalar triplet strongly decay to ILD darkmatter, while its decay to SM leptons is suppressed. D. Effect of scalar triplet on relic abundance of ILD dark matter
In presence of a scalar triplet, when the DM mass is larger than the triplet mass, a few additional annihilation andco-annihilation channels open up as shown in Figs. 7, 8, 9 and 10 in addition to the previously mentioned Feynmandiagrams given in Figs. 2,3 and 4. These additional channels also play a key role in number changing processes ofDM, N to yield a modified freeze-out abundance. We numerically calculate relic density of N DM once again byimplementing the model in the code micrOmegas [24]. The parameter space, in comparison to the ILD dark matteralone, is enhanced due to the additional coupling of N with ∆. In particular, the new parameters are: triplet scalarmasses m H , m A , m H ± , m H ±± , vev of scalar triplet v t , coupling of scalar triplet with ILD dark matter N , i.e. f N ,scalar doublet-triplet mixing sin α . N N N ± H + H − N N Z H + /H ± /A /H { , } H − /W/H { , } /ZN N N A /H { , } /H { , } A /A /H { , } FIG. 7: Additional annihilation N N , in presence of scalar triplet. N ( N ) c H H ++ H −− N − ( N ) c H + H −− H + N − ( N − ) c H ++ H −− / H − H /H − FIG. 8: Dominant annihilation ( N ( N ) c )and co-annihilation ( N − ( N ) c , N − ( N − ) c ) processes of ILD DM ( N ) to scalartriplet in final states. N − N N ± H −− H + N − N W H −− , H − , H { , } H + , W, A , H { , } , Z, AN − N N H − /H − A /H { , } FIG. 9: Co-annihilation channels of ILD DM ( N ), with charged fermions N − in presence of scalar triplet. N − N + N − /N H −− /H − H ++ /H + N − N + A/Z H ++ , H ± , A , H { , } H −− , H − , W, H { , } , Z FIG. 10: Co-annihilation processes involving only charged partner of ILD DM, N ± in presence of scalar triplet. To understand the effect of the triplet scalar on relic density of ILD DM, we show in Fig. 11 the variation of relicdensity as a function of DM mass ( m N ) with different choices of triplet vev ( v t ) while keeping a fixed f N and sin α .In the left panel of Fig. 11 we choose the scalar doublet-triplet mixing to be sin α = 0 .
001 while in the right-panelof Fig. 11 we choose sin α = 0 .
1. In both cases we set the physical CP even, CP odd and charged triplet scalarmasses respectively at m H = 280 , m A = 280 , m H ± = 300 and m H ±± = 310 GeV and f N = 0 .
1. We note thatthere is a resonance drop near m N ∼ m Z / Z mediated s-channel diagrams. Additionally we find thatnear m N ∼
280 GeV (which is the mass of H ), relic density drops suddenly because of new annihilation processes N N → ∆∆ start to contribute (see diagrams in Figs. 7, 8, 9 and 10). With larger triplet vev v t ∼ GeV, theeffect of annihilation to scalar triplet becomes subdued. This can be understood as follows. First of all, we seethat the quartic couplings involving the triplet, as given in Eq. 19, are inversely proportional to the triplet vev ( v t ).In a typical annihilation process: N − N − c → H − H − , mediated by H −− , the vertex H −− H − H − is proportional to2 FIG. 11: Relic density vs m N plot for different choices triplet vev, v t , keeping sin α = 0 .
001 (left) and sin α = 0 . f N = 0 . √ v t λ ∼ /v t , which diminishes with larger v t . On the other hand, let us consider the process: N N c → H ++ H −− ,which has a significant contribution to the total relic density. This process is mediated by H and H . In small sin α limit, the H mediated diagram is vanishingly small as the N ¯ N H ∼ sin α . So, H mediation dominates here.However, the vertex involving H H ++ H −− is proportional to (2 cos αλ v t − sin αλ v d ). One can see that for small v t , the first term is negligible, while for a larger v t , the first term becomes comparable to that of the second one andhas a cancellation. This cancellation therefore decreases the annihilation cross-section to the chosen final state. Sucha phenomena is also present for co-annihilation processes like N − N − c → H −− H etc., where the vertices involve acombination of λ , and λ . On the other hand, for smaller values of triplet vev ( v t ∼ .
01 GeV), there is a largerdrop in relic density due to the additional annihilation channels (to the triplet scalars as mentioned). Therefore, theDM N achieves correct relic density for larger DM mass m N (as compared to that of the case in absence of triplet).Moreover, we set f N = 0 . f N . Therefore with larger f N , the drop in relic density in the vicinity of triplet scalar mass decreaseseven further. In summary, the presence of scalar triplet shifts the relic density of ILD DM to a higher DM mass regionwhich crucially depends on the choice of the triplet vev as well as ∆ N N coupling f N .An important conclusion about ILD dark matter is that the mass of DM ( m N ) is around 1 TeV which satisfies theobserved relic abundance. This implies the mass of N − , the charged partner of N , is about 1 TeV as well. However,the electroweak correction induces a small mass splitting between N and N − to be around 162 MeV. Therefore, N − can give rise a displaced vertex signature through the 3-body decay N − → N (cid:96) − ¯ ν (cid:96) [31]. But the main drawbackis that the production cross-section of N ± of mass ∼ TeV is highly suppressed at LHC as this can only be possiblethrough Drell-Yan. Therefore, in section-V we discuss a more predictive model by enlarging the dark sector with anadditional singlet fermion χ . E. Effect of scalar triplet on direct detection of ILD dark matter
As discussed in section III B, the ILD dark matter alone is ruled out due to large Z-mediated elastic scattering withnucleus. However, it can be reinstated in presence of the scalar triplet, which not only forbids the Z-mediated elasticscattering [23] but also provides a new portal for the detection of ILD dark matter via the doublet-triplet mixing aswe discuss below.The interaction of DM with the Z boson with the kinetic term is given as L Z − DM ⊃ i ¯ N ( γ µ ∂ µ − ig Z γ µ Z µ ) N , (33)where ig Z = g θ W . After the symmetry breaking the scalar triplet ∆ gets an induced vev and hence gives Majoranamass to the ILD dark matter N as shown in Eq. 29. The presence of such Majorana mass term splits the Dirac DMstate into two real Majorana states N and N with a mass splitting of δm as discussed in sec IV C. Now we rewritethe Lagrangian involving DM- Z interaction in terms of the new Majorana states as: L Z − DM ⊃ N iγ µ ∂ µ N + N iγ µ ∂ µ N − ig z N γ µ N Z µ . (34)3We can see that the dominant gauge interaction becomes off-diagonal. The absence of diagonal interaction term N N nn H , H FIG. 12: Elastic scattering of the ILD DM( N ) with the nucleon through scalar mediation due to doublet triplet mixing. for the DM- Z vertex leads to the vanishing contribution to elastic scattering of the DM with the nucleus. Howeverthere could be an inelastic scattering through Z mediation, which is suppressed if the mass splitting between twostates is of the order O (100) keV or less. But the Yukawa term involving DM and ∆ is still diagonal in the new basisand hence can lead to elastic scattering through a mixing between the doublet-triplet Higgs. Assuming N to be thelightest among the two Majorana states, hence being the DM, the relevant diagram for the elastic scattering is shownin Fig 12.The direct detection cross-section mainly depends on m H , f N and sin α . We have plotted the spin independentdirect detection cross-section as a function of the DM mass m N in Fig 13. Keeping m H = 280 GeV (Solid) and m H = 600 GeV (Dashed) fixed, we have shown the cross-section for three different values { f N = 0 . , . , } in Red,Blue and Green color respectively. Since there is a relative negative sign between the two amplitudes, the destructiveinterference is more for m H comparable to SM Higgs. Hence cross section for m H = 280 GeV turns out to be smallerthan m H = 600 GeV. But if we increase the mass of m H to TeV scale then H mediated process will be suppresseddue to the large mass in the propagator and only H mediated process will contribute. The direct search cross-sectionincreases with larger Yukawa coupling f N from 0 .
01 to 1 and can be is easily seen from the Fig 13. Since the DMcouples dominantly to the triplet scalar, the more the mixing angle (sin α ), the more is the cross-section which can beclearly seen from the left and right panel of Fig 13. But all these cross-sections are well below the present experimentalbound of LUX and Xenon-1T. Note that the relic density allowed parameter space of ILD DM in presence of a scalartriplet live in a very high DM mass region ∼ T eV with moderate f N and Higgs data unambiguously indicates that FIG. 13: SI direct detection cross-section σ SI as a function of DM mass m N for sin α = 0 .
001 (Left) and sin α = 0 .
01 (Right).Different choices of the coupling { f N = 0 . , . , } are shown in Red, Blue and Green color respectively. Dashed and solidlines corresponds to different values of the heavy Higgs m H = 600 ,
280 GeVs respectively. The bound from LUX 2017 andXENON1T 2018 are shown. α ) should be kept small, as we have shown in Fig 13. Therefore, the ILD becomes a viable DMcandidate in the presence of a triplet scalar allowed by both relic density and direct search constraints. V. SINGLET-DOUBLET LEPTONIC DARK MATTER
Now let us assume that the dark sector is composed of two vector like leptons : a doublet, N = ( N N − ) T anda singlet χ , which are odd under an extended Z symmetry while all the Standard Model (SM) fields are even. As aresult the lightest odd particle in the dark sector is stable and behave as a candidate of DM. The quantum numbersof dark sector fields and that of SM Higgs under the SM gauge group, augmented by a Z symmetry, are given inTable I.The Lagrangian of the model can be given as follows: L V F = N [ iγ µ ( ∂ µ − ig σ a W aµ − ig (cid:48) Y B µ ) − m N ] N + χ ( iγ µ ∂ µ − m χ ) χ − ( Y N (cid:101) Hχ + h.c ) . (35)Note that here we have assumed to have a CP conserving interaction between the additional vector like fermion tothe SM Higgs. One may also think of a coupling − ( Y N γ (cid:101) Hχ + h.c ) that will violate CP. Now, it is a bit intriguing tothink of such interactions before the debate on the SM Higgs to be a scalar or a pseudoscalar is settled. The outcomeof such an interaction will alter the subsequent phenomenology significantly. For example, it is known that advocatinga pseudoscalar ( S ) interaction to a vector like DM ( ψ ), for example, with a term like − yS ¯ ψγ ψ indicates that DM-nucleon scattering becomes velocity dependent and therefore reduces the direct search constraint significantly to allowthe model live in a larger allowed parameter space. However, the Yukawa interaction term itself ( − yS ¯ ψγ ψ ) do notviolate parity as S is assumed to be pseudoscalar by itself (see for example in [25]).After Electroweak symmetry breaking (EWSB) the SM Higgs acquires a vacuum expectation value v . The quantumfield around the vacuum can be given as: H = (cid:16) √ ( v + h ) (cid:17) T where v = 246 GeV. The presence of the Yukawaterm: Y N (cid:101) Hχ term in the Lagrangian (Eq. 35), arises an admixture between N and χ . The bare mass terms of thevector like fermions in L V F then take the following form: −L V Fmass = m N N N + m N N + N − + m χ χχ + Y v √ N χ + Y v √ χN = (cid:0) χ N (cid:1)(cid:32) m χ Y v √ Y v √ m N (cid:33)(cid:18) χN (cid:19) + m N N + N − . (36)The unphysical basis, (cid:0) χ N (cid:1) T is related to physical basis, (cid:0) N N (cid:1) T through the following unitary transfor-mation: (cid:18) χN (cid:19) = U (cid:18) N N (cid:19) = (cid:18) cos θ − sin θ sin θ cos θ (cid:19) (cid:18) N N (cid:19) , (37)where the mixing angle tan 2 θ = − √ Y vm N − m χ . (38)The mass eigenvalues of the physical states N and N are respectively given by: m N = m χ cos θ + m N sin θ + Y v √ θm N = m χ sin θ + m N cos θ − Y v √ θ (39)For small sin θ (sin θ →
0) limit, m N and m N can be further expressed as: m N (cid:39) m χ + Y v √ θ ≡ m χ − ( Y v ) ( m N − m χ ) ,m N (cid:39) m N − Y v √ θ ≡ m N + ( Y v ) ( m N − m χ ) . (40)5Here we have considered Y v/ √ (cid:28) m χ < m N . Hence m N < m N . Therefore N becomes the stable DMcandidate. From Eqs.38 and 39, one can write : Y = − ∆ m sin 2 θ √ v ,m N = m N sin θ + m N cos θ. (41)where ∆ m = m N − m N is the mass difference between the two mass eigenstates and m N is the mass of electricallycharged component of vector like fermion doublet N − .Therefore, one can express interaction terms of L V F in mass basis of N and N as L V Fint = (cid:16) e θ W cos θ W (cid:17)(cid:104) sin θN γ µ Z µ N + cos θN γ µ Z µ N + sin θ cos θ ( N γ µ Z µ N + N γ µ Z µ N ) (cid:105) + e √ θ W sin θN γ µ W + µ N − + e √ θ W cos θN γ µ W + µ N − + e √ θ W sin θN + γ µ W − µ N + e √ θ W cos θN + γ µ W − µ N − (cid:16) e θ W cos θ W (cid:17) cos 2 θ W N + γ µ Z µ N − − e N + γ µ A µ N − − Y √ h (cid:104) sin 2 θ ( N N − N N ) + cos 2 θ ( N N + N N ) (cid:105) (42)The relevant DM phenomenology of the model then mainly depend on following three independent parameters : { m N , ∆ m, sin θ } (43) A. Constraints on the model parameters
The model parameters are not totally free from theoretical and experimental bounds. Here we would like to discussbriefly the constraints coming from Perturbativity, invisible decay widths of Z and H , and corrections to electroweakparameters. • Perturbativity : The upper limit of perturbativity bound on quartic and Yukawa couplings of the model aregiven by, | Y | < √ π . (44) • Invisible decay width of Higgs : If the mass of DM is below m h /
2, then Higgs can decay to two invisibleparticles in final state and will yield invisible decay width. Recent Large Hadron Collider (LHC) data put strongconstraint on the invisible branching fraction of Higgs to be Br ( h → inv ) ≤ .
24 [32], which can be expressedas: Γ( h → inv. )Γ( h → SM ) + Γ( h → inv. ) ≤ . , (45)where Γ( h → SM ) = 4 . m h = 125 .
09 GeV, obtained from recent measured LHC data [26].Therefore, the invisible Higgs decay width is given byΓ( h → inv. ) ≤ .
32 MeV , (46) We would like to remind the readers that N and N are not same as N and N . N , are the physical eigenstates arising out of thesinglet ( χ ) doublet ( N ) admixture. Here, N is the DM while N is the NLSP. Where as, N and N are the two pseudo Dirac statesthat emerge from the neutral component ( N ) of the vector-like lepton doublet ( N ) due to the majorana mass term acquired by N inpresence of the scalar triplet ∆. h → inv. ) = Γ( h → N N ) . = 116 π (cid:16) Y sin 2 θ (cid:17) m h (cid:16) − m N m h (cid:17) Θ( m h − m N ) , (47)where the step function Θ( m h − m N ) = 1 if m N ≤ m h / m N > m h /
2. The decay width of Higgsto DM is proportional to Y = − ∆ m sin 2 θ √ v . Therefore, it depends on the mass splitting with NLSP (∆ m ) as wellas on the doublet singlet mixing (sin θ ). The invisible Higgs decay constraint on the model for m N < m h / M N − sin θ plane for different choices of ∆ m ranging from10 GeV to 500 GeV. The inner side of the contour is excluded. This essentially shows that with large ∆ m ∼ m N ≤ m h / m ∼
10 GeV, the constraint is milder dueto less Higgs decay width and excludes only regions for DM masses M N ≤
30 GeV within sin θ ∼ { . − . } .Therefore, even in m N ≤ m h / m and sin θ are small, which turns out to be the case for satisfyingrelic density and direct search as we demonstrate later, then are allowed by the Higgs invisible decay constraint. FIG. 14: Constraints from Higgs invisible decay width is shown in m N − sin θ plane. Here each contour line correspond todifferent value of ∆ m depicted in the figure. Inner region of each contour line excluded from Higgs invisible decay width,Γ( h → inv. ) ≤ .
32 MeV for a particular value of ∆ m . • Invisible decay width of Z : If the masses of dark sector particles are below m Z /
2, then Z can decay to darksector particles leading to an increase of Z -decay width. However, from current observation, invisible decaywidth of Z boson is strongly bounded. The upper limit on invisible Z -decay width is given by [26]:Γ( Z → inv. ) = 499 . ± . , (48)where the decay of Z to N DM is given as:Γ( Z → N N ) = 148 π (cid:16) g sin θ cos θ W (cid:17) m Z (cid:16) m N m Z (cid:17)(cid:115) − m N m Z Θ( m Z − m N ) . (49)The Z − invisible decay does not play a crucial role in small sin θ regions, which are required for the DM toachieve correct relic density, thus allowing almost all of the M N ≤ m Z / B. Corrections to the electroweak precision parameters
Addition of a vector like fermion doublet to the SM gives correction to the electroweak precision test parameters
S, T and U [33, 34]. The values of these parameters are tightly constrained by experiments. The new observed parametersare infect four in number ˆ S , ˆ T , W and Y [35], where the ˆ S , ˆ T are related to Peskin-Takeuchi parameters S , T asˆ S = αS/ θ w , ˆ T = αT , while W and Y are two new set of parameters. The measured values of these parametersat LEP-I and LEP-II put a lower bound on the mass scale of vector like fermions. The result of a global fit of theparameters is presented in the table II for a light Higgs [35] . ˆ S ˆ T W Y Light Higgs 0 . ± . . ± . . ± . − . ± . In the present scenario, we have a vector like doublet and a singlet fermion field are added to the SM. But thephysical states are a charged fermion N − , and two singlet doublet mixed neutral fermions N (dominant singletcomponent) and N (dominant doublet component). Therefore, the contribution to the precision parameters alsodepends on the mixing angle sin θ . The expression for ˆ S in terms m N , m N , m N and sin θ of is given as [21]:ˆ S = g π (cid:20) (cid:26) ln (cid:18) µ ew m N (cid:19) − cos θ ln (cid:18) µ ew m N (cid:19) − sin θ ln (cid:18) µ ew m N (cid:19)(cid:27) − θ cos θ (cid:26) ln (cid:18) µ ew m N m N (cid:19) + m N − m N m N + m N m N − m N ) + ( m N + m N )( m N − m N m N + m N )6( m N − m N ) ln (cid:18) m N m N (cid:19) + m N m N ( m N + m N )2( m N − m N ) + m N m N ( m N − m N ) ln (cid:18) m N m N (cid:19)(cid:27)(cid:21) (50)where µ ew is at the EW scale. = 0.05 Sin θ = 0.075
Sin θ = 0.1
Sin θ = 0.15
Sin θ = 0.2 m N (GeV) S ^ θ = 0.05 Sin θ = 0.1
Sin θ = 0.2 m N (GeV) m N - m N ( G e V ) FIG. 15: In the left panel, ˆ S is shown as a function of m N for m N = 200 GeV and sin θ = 0 .
05 (Green colour), sin θ = 0 . θ = 0 . θ = 0 .
15 (Black color) and sin θ = 0 . S is plotted in m N − m N versus m N plane for sin θ = 0 .
05 (Green Color), sin θ = 0 . θ = 0 . In the left panel of Fig. 15, we have plotted ˆ S as a function of m N keeping m N = 200 GeV for different values ofthe mixing angle. In the right panel, we have shown the allowed values of ˆ S in the plane of m N − m N versus m N for sin θ = 0 .
05 (Green Color), sin θ = 0 . θ = 0 . S does not The value ˆ S , ˆ T , W and Y are obtained using a Higgs mass m h = 115 GeV. However, we now know that the SM Higgs mass is 125 GeV.Therefore, the value of ˆ S , ˆ T , W and Y are expected to change. But this effect is nullified by the small values of sin θ . m N and m N . Moreover, small values of sin θ allows a small mass splitting between N and N − which relaxes the constraint on ˆ T parameter as we discuss below. The expression for ˆ T is given as [21]:ˆ T = g π M W (cid:2) θ cos θ Π( m N , m N , − θ Π( m N , m N , − θ Π( m N , m N , (cid:3) , (51)where Π( a, b,
0) is given by:Π( a, b,
0) = −
12 ( M a + M b ) (cid:18) Div + ln (cid:18) µ ew M a M b (cid:19)(cid:19) −
14 ( M a + M b ) − ( M a + M b )4( M a − M b ) ln M b M a + M a M b (cid:26) Div + ln (cid:18) µ ew M a M b (cid:19) + 1 + ( M a + M b )2( M a − M b ) ln M b M a (cid:27) , (52)with Div= (cid:15) + ln4 π − γ (cid:15) contains the divergent term in dimensional regularisation method. From the left panel of θ = 0.05 Sin θ = 0.075
Sin θ = 0.1
Sin θ = 0.15
Sin θ = 0.2 m N (GeV) T ^ θ=0.05 Sin θ=0.1
Sin θ = 0.2 m N (GeV) m N - m N ( G e V ) FIG. 16: In the left panel, ˆ T is shown as a function of m N for m N = 200 GeV and sin θ = 0 .
05 (Green colour), sin θ = 0 . θ = 0 . θ = 0 .
15 (Black color) and sin θ = 0 . T is plotted in m N − m N versus m N plane for sin θ = 0 .
05 (Green color), sin θ = 0 . θ = 0 . Fig. (16) we see that for sin θ < .
05 we don’t get strong constraints on m N and m N . Moreover, small values ofsin θ restricts the value of m N − m N to be less than a GeV. As a result large m N values are also allowed. Near m N ≈ m N , ˆ T vanishes as expected. The value of Y and W are usually suppressed by the masses of new fermions.Since the allowed masses of N , N and N ± are above 100 GeV by the relic density constraint (to be discussed later),so Y and W are naturally suppressed. C. Relic density of singlet-doublet leptonic dark matter
As stated earlier, the lightest stable physical state N is the DM, which is an admixture of a singlet vector-likefermion ( χ ) and the neutral component of a vector-like fermionic doublet ( N ). Due to presence of mass hierarchybetween dark sector particles N , N and N − , the lightest component N not only annihilate with itself but alsoco-annihilate with N and N − to yield a net a relic density. The relevant diagrams are shown in Figs. 17, 18, 19.We assume all the heavier particles: N and N − in the dark sector ultimately decay to lightest stable particle N .So in this scenario one can write the Boltzmann equation in terms of total number density n = n N + n N + n N ± as dndt + 3 Hn = −(cid:104) σv (cid:105) eff (cid:16) n − n eq (cid:17) , (53)9 N i N j N k Z/Z/hh/Z/h N i N j h f/W + /Z/hf/W − /Z/hN i N j N − W + W − N i N j Z f/W + /hf/W − /Z FIG. 17: Annihilation ( i = j ) and co-annihilation ( i (cid:54) = j ) of vector-like fermion DM. Here ( i, j = 1 , N i N − N j Z/hW − /W − N i N − W f/h/W/Wf (cid:48) /W/A/ZN i N − N − W − A/Z
FIG. 18: Co-annihilation process of N i ( i = 1 ,
2) with the charge component N − to SM particles. N + N − N i W + W − N + N − A/Z f/W + f/W − N + N − N − A/ZA/Z N + N − Z hZ
FIG. 19: Co-annihilation process of charged fermions N ± to SM particles in final states . (cid:104) σv (cid:105) eff = g g eff (cid:104) σv (cid:105) N N + 2 g g g eff (cid:104) σv (cid:105) N N (cid:16) mm N (cid:17) e − x ∆ mmN + 2 g g g eff (cid:104) σv (cid:105) N N − (cid:16) mm N (cid:17) e − x ∆ mmN + 2 g g g eff (cid:104) σv (cid:105) N N − (cid:16) mm N (cid:17) e − x ∆ mmN + g g eff (cid:104) σv (cid:105) N N (cid:16) mm N (cid:17) e − x ∆ mmN + g g eff (cid:104) σv (cid:105) N + N − (cid:16) mm N (cid:17) e − x ∆ mmN . (54)In above equation, g eff , defined as effective degrees of freedom, which is given by g eff = g + g (cid:16) mm N (cid:17) e − x ∆ mmN + g (cid:16) mm N (cid:17) e − x ∆ mmN , (55)where g , g and g are the degrees of freedom of N , N and N − respectively and x = x f = m N T f , where T f is thefreeze out temperature. Then the relic density of the N DM can be given by [36, 37]Ω N h = 1 . × GeV − g / (cid:63) M P L J ( x f ) , (56)where J ( x f ) is given by J ( x f ) = (cid:90) ∞ x f (cid:104) σ | v |(cid:105) eff x dx . (57)We note here that the freeze-out abundance of N DM is controlled by the annihilation and co-annihilation channelsas shown in Fig. 17, 18 and 19. Therefore, the important parameters which decide the relic abundance of N aremass of DM ( m N ), the mass splitting (∆ m ) between the DM and the next-to-lightest stable particle (NLSP) andthe singlet-doublet mixing angle sin θ . Here we use MicrOmega [24] to calculate the relic density of N DM.Variation of relic density of N DM is shown in Fig. 20 as a function of its mass, for a fixed ∆ m = 10 −
100 GeV (inleft and right panels of Fig. 20 respectively) and different choices of mixing angle sin θ . We note that the annihilationcross-section increases with sin θ , due to larger SU (2) component, resulting in smaller relic density. The resonancedrop at m Z / m h / s -channel Z and H mediated contributions to relic abundance. Anotherimportant feature of Fig. 20 is that when ∆ m is small, relic density is smaller due to large co-annihilation contribution(less Boltzmann suppression followed from Eq. 54). This feature can also be corroborated from Fig. 20, where wehave shown relic density as a function of DM mass by keeping a fixed range of sin θ and chosen different possible∆ m . Alternatively in Fig. 21, we have shown relic density as a function of DM mass by keeping a fixed range of ∆ m ,while varying sin θ . We see from the left panel of Fig. 21 that for small ∆ m co-annihilation dominates and hence theeffect of sin θ on relic abundance is quite negligible. On the other hand, from the right panel of Fig. 21, we see thatfor large ∆ m , where co-annihilation is suppressed, the effect of sin θ on relic abundance is clearly visible. For smallsin θ , the effective annihilation cross-section is small which leads to large relic abundance, while for large sin θ therelic abundance is small provided that the ∆ m is big enough to avoid co-annihilation contributions.From Fig. 22, we see that for a wide range of singlet-doublet mixing (sin θ ), we can get correct relic abundance inthe plane of m N versus ∆ m . Different ranges of sin θ are indicted by different color codes. To understand our result,we divide the plane of m N versus ∆ m into two regions: (i) the bottom portion with small ∆ m , where ∆ m decreaseswith larger mass of N , (ii) the top portion with larger mass splitting ∆ m , where ∆ m increases slowly with larger DMmass m N . In the former case, for a given range of sin θ , the annihilation cross-section decreases for large mass of N .Therefore, we need more co-annihilation contribution to compensate, which requires ∆ m to decrease. This also implythat the region below to each colored zone is under abundant (small ∆ m implying large co-annihilation for a givenmass of N ), while the region above is over abundant (large ∆ m implying small co-annihilation for a given mass of N ). To understand the allowed parameter space in region (ii), we first note that co-annihilation contribution is muchsmaller here due to large ∆ m , so the annihilation processes effectively contribute to relic density. Now, let us recall1 FIG. 20: Variation of relic density with DM mass m N keeping fixed range of sin θ : 0 . ≤ sin θ ≤ .
05 (left pannel) and0 . ≤ sin θ ≤ .
12 (right panel). The different color patches corresponds to different ∆ m region : 1 ≤ ∆ m (inGeV) ≤ < ∆ m (inGeV) ≤
30 (Blue), 30 < ∆ m (inGeV) ≤
50 (Red) and 50 < ∆ m (inGeV) ≤
100 (Gray). Correct relicdensity, 0 . ≤ Ω h ≤ . m N keeping fixed region of ∆ m : 1 ≤ ∆ m ≤
10 GeV (left panel) and50 ≤ ∆ m ≤
100 GeV (right panel). The different color patches are corresponding to different sin θ region : 0 . ≤ sin θ ≤ . . < sin θ ≤ .
08 (Blue), 0 . < sin θ ≤ .
12 (Red). Correct relic density, 0 . ≤ Ω h ≤ . that the Yukawa coupling Y ∝ ∆ m sin θ . Therefore, for a given sin θ , larger ∆ m can lead to larger Y and thereforelarger annihilation cross-section to yield under abundance, which can only be tamed down to correct relic density byhaving a larger DM mass. Hence in case-(ii), the region above to each colored zone (allowed region of correct relicdensity) is under abundant, while the region below to each colored zone is over abundant. Thus the over and underabundant regions of both cases (i) and (ii) are consistent with each other. D. Constraints on parameters from direct search of singlet-doublet leptonic dark matter
Let us now turn to constraints on parameters from direct search of N DM in terrestrial laboratories. Due tosinglet-doublet mixing, the N DM in direct search experiments can scatter off the target nucleus via Z and Higgsmediated processes as shown by the Feynman graphs in Fig. 23. The cross-section per nucleon for Z -boson mediationis given by [38, 39] σ Z SI = 1 πA µ r |M| (58)2 FIG. 22: Relic density allowed region are shown in m N − ∆ m plane for different range of range of sin θ : { . − . } (Red), { . − . } (Green) and { . − . } (Blue). where A is the mass number of the target nucleus, µ r = M m n / ( M + m n ) ≈ m n is the reduced mass, m n is the massof nucleon (proton or neutron) and M is the amplitude for Z -boson mediated DM-nucleon cross-section given by M = √ G F [ ˜ Z ( f p /f n ) + ( A − ˜ Z )] f n sin θ , (59)where f p and f n are the interaction strengths (including hadronic uncertainties) of DM with proton and neutronrespectively and ˜ Z is the atomic number of the target nucleus. On the other hand, the spin-independent DM-nucleoncross-section per nucleon mediated via the exchange of SM Higgs is given by: σ h SI = 1 πA µ r [ Zf p + ( A − Z ) f n ] (60)where the effective interaction strengths of DM with proton and neutron are given by: f p,n = (cid:88) q = u,d,s f ( p.n ) T q α q m ( p,n ) m q + 227 f ( p,n ) T G (cid:88) q = c,t,b α q m p.n m q (61)with α q = Y sin 2 θM h (cid:16) m q v (cid:17) . (62)In Eq. 61, the different coupling strengths between DM and light quarks are given by [3] f ( p ) T u = 0 . ± . f ( p ) T d = 0 . ± . f ( p ) T s = 0 . ± . f ( n ) T u = 0 . ± . f ( n ) T d = 0 . ± . f ( n ) T s = 0 . ± . f ( p,n ) T G = 1 − (cid:88) q = u,,d,s f ( p,n ) T q . (63)Thus from Eqs. (60,61,62,63) the spin-independent DM-nucleon cross-section is given to be: σ h SI = 4 πA µ r Y sin θM h (cid:20) m p v (cid:18) f pT u + f pT d + f pT s + 29 f pT G (cid:19) + m n v (cid:18) f nT u + f nT d + f nT s + 29 f nT G (cid:19)(cid:21) . (64)In the above equation the only unknown quantity is Y or sin 2 θ which can be constrained by requiring that σ h SI is lessthan the current DM-nucleon cross-sections. Now we make a combined analysis by taking both Z and H mediateddiagrams taken into account together. In Fig. 24, we show the spin-independent cross-section for N DM within itsmass range m N : 1 − θ within { . − . } with sin θ = { . − . } (Cyan), sin θ = { . − . } (Blue), sin θ = { . − . } (Red), sin θ = { . − . } (Green). It clearly showsthat the larger is sin θ , the stronger is the interaction strength (through larger contribution from Z mediation) and3 N N nn Z N N nn h FIG. 23: Feynman diagrams of SI direct detection of singlet-doublet N DM.FIG. 24: Spin independent direct detection cross-section σ SI as a function of DM mass m N for different range of sin θ : { . − . } (Cyan), { . − . } (Blue), { . − . } (red) and { . − . } (Green). The direct search limits from LUX,PANDA, XENON1T are shown while that of the future sensitivity of XENONnT is also indicated. hence the larger is the DM-nucleon cross-section. Hence, it turns out that direct search experiments constraints sin θ to a large extent. For example, we see that sin θ ≤ .
04, for the DM mass m N >
300 GeV. The effect of ∆ m onDM-nucleon cross-section is less. However, we note that ∆ m plays a dominant role in the relic abundance of DM.Approximately, sin θ ≤ .
05 (Blue points) are allowed for most of the parameter space except for smaller DM masses.Cyan points indicate sin θ < .
01. The discrete bands correspond to specific values of sin θ chosen for the scan andessentially one can consider the whole region together to fall into this category which evidently have sensitivity closeto neutrino floor. Note here, the scanned points in Fig. 24, do not satisfy relic abundance.Now let us turn to the parameter space, simultaneously allowed by observed relic density and latest constraintsfrom direct DM search experiments such as Xenon-1T. In Fig. 25, we have shown the allowed parameter space againin m N − ∆ m plane. We see that null observation from direct search crucially tames down the relic density allowedparameter space to sin θ < .
05 (Purple). Fig. 25 also shows that large singlet-doublet mixing, i.e. sin θ (cid:38) . θ ) in accordance with DM mass ( m N ) and the splitting with charged fermioncontent (∆ m ) to yield a large available parameter space for correct relic density. However, due to Z -mediated processcontributing to direct detection of singlet-doublet leptonic dark matter, a stringent constraint on sin θ ≤ .
05 arises.This leads the DM to be allowed only in small ∆ m region (as in Fig. 25) to achieve correct relic density throughco-annihilation processes. However, this constraint can be relaxed in presence of a scalar triplet as we discuss below.Moreover, the triplet can also give rise Majorana masses to light neutrinos (see section IV B) through type-II seesawto address DM and neutrinos in the same framework.4 FIG. 25: Relic density and direct detection ( XENON 1T 2018 ) allowed parameter space plotted in m N − ∆ m plane. VI. TRIPLET EXTENSION OF SINGLET-DOUBLET LEPTONIC DARK MATTERA. Pseudo-Dirac nature of singlet-doublet leptonic dark matter
As discussed in section -V, the DM is assumed to be N = cos θχ + sin θN with a Dirac mass m N . However, fromEq. 12 we see that the vev of ∆ induces a Majorana mass to N due to singlet-doublet mixing and is given by: m = √ f N sin θ (cid:104) ∆ (cid:105) ≈ f N sin θ − µv √ M . (65)Thus the N has a large Dirac mass m N and a small Majorana mass m as shown in the above Eq. 65. Therefore, weget a mass matrix in the basis { N L , ( N R ) c } as: M = (cid:18) m m N m N m (cid:19) . (66)Thus the Majorana mass m splits the Dirac spinor N into two pseudo-Dirac states ψ , with mass eigenvalues m N ± m . The mass splitting between the two pseudo-Dirac states ψ , is given by δm = 2 m = 2 √ f N sin θ (cid:104) ∆ (cid:105) . (67)Note that δm << m N from the estimate of induced vev of the triplet and hence does not play any role in therelic abundance calculation. However, the sub-GeV order mass splitting plays a crucial role in direct detection byforbidding the Z-boson mediated DM-nucleon elastic scattering. Now from Eq. 20 and (65) we see that the ratio: R = M ν m = f L f N sin θ . (68)Thus we see that for R ∼ − the ratio, f L /f N ∼ − if we assume sin θ = 0 .
1, which is much larger than thesinglet-doublet mixing being used in section V D.
B. Effect of scalar triplet on relic abundance and direct search of singlet-doublet dark matter
We already have noted the diagrams that are present due to the addition of a scalar triplet for the ILD DM tofreeze-out (see Section IV D). The main features of having a additional scalar triplet in the singlet-doublet DM modelis very similar to what we have discussed before in case of ILD DM. The additional freedom that we have in case ofsinglet-doublet leptonic DM is to play with the mixing parameter sin θ and ∆ m .Let us first study relic density as a function of DM mass in presence of scalar triplet. This is shown in Fig. 26,where we choose two fixed values of ∆ m = 10 ,
200 GeV in left and right panel respectively for a scalar triplet mass5
FIG. 26: Variation of relic density with DM mass m N keeping fixed region of ∆ m : ∆ m = 10 GeV (left panel) and ∆ m = 200GeV (right panel) in presence of scalar triplet. Different color patches correspond to different sin θ region : 0 . ≤ sin θ ≤ . . < sin θ ≤ . . < sin θ ≤ . . ≤ Ω h ≤ . m N keeping fixed region of ∆ m : ∆ m = 10 GeV (left panel) and ∆ m = 200GeV (right panel) for heavy scalar triplet mass, m H = 1000 GeV. Different color patches correspond to different sin θ region :0 . ≤ sin θ ≤ . . < sin θ ≤ . . < sin θ ≤ . . ≤ Ω h ≤ . around 280 GeV. Different possible ranges of sin θ are shown by different color codes. The main feature is again tosee a drop in relic density near the value of the triplet mass, where the additional annihilation channel to the scalartriplet reduces relic density significantly. For small ∆ m , co-annihilation channels play an important part and thereforedifferent mixing angles do not affect relic density significantly (compare left and right panel figures). Also due to largeco-annihilation for small ∆ m as in the left panel, the relic density turns out to be much smaller than the right panelfigure where ∆ m is large and do not offer the co-annihilation channels to be operative. An additional resonance dropat half of the triplet scalar mass is observed here ( ∼
140 GeV) due to s-channel triplet mediated processes.A similar plot is shown in Fig. 27 with larger value of the scalar triplet mass ∼ ∼ m N − ∆ m plane is shown in left panel of Fig. 28. Thebottom part of the allowed parameter space is again due to co-annihilation. For small sin θ ∼ . θ , the resonancedrops of doublet and triplet scalars also yield correct relic density. There is an under-abundant region when the tripletchannel opens up, which is then reduced with larger DM mass. Therefore, it ends up with two different patches (bothfor blue and purple points) to be allowed below and above the scalar triplet mass. The direct search constraint inpresence of scalar triplet thankfully omits the Z mediated diagram due to the pseudo-Dirac splitting and allows alarger sin θ ∼ .
3. However, in addition to the Yukawa coupling (Y) initiated SM Higgs mediation, there is an added6
FIG. 28: Relic density (left panel) and both relic density and direct search (XENON 1T 2018) (right panel) allowed parameterspace plotted in m N − ∆ m plane with different range of sin θ : 0 . ≤ sin θ ≤ . . < sin θ ≤ . . < sin θ ≤ . contribution from the heavy Higgs due to the doublet triplet mixing. We have already seen before that the effect ofthe additional contribution to direct search cross-section is small in the small sin α limit with a moderate choice of f N . Therefore, we have omitted such contributions in generating the direct search allowed parameter space of themodel as shown in the right panel of Fig. 28. This again depicts that the model in presence of scalar triplet earnsmore freedom in relaxing ∆ m and sin θ to some extent. VII. COLLIDER SIGNATURES
Finally, we discuss the collider signature of the model, which can be subdivided into two categories: (i) Displacedvertex signature and (ii) Excess in leptonic final states.
A. Displaced Vertex signature
In the small sin θ limit, the charged inert fermion can show a displaced vertex signature. If the mass differencebetween the N − and N is greater than W − mass then N − can decay via a two body process. But if the massdifference is smaller than M W , then N − can decay via three body process say N − → N l − ¯ ν l . The three body decaywidth is given as [20]: Γ = G F sin θ π m N I (69)where G F is the Fermi coupling constant and I is given as: I = 14 λ / (1 , a , b ) F ( a, b ) + 6 F ( a, b ) ln (cid:18) a a − b − λ / (1 , a , b ) (cid:19) . (70)In the above Equation F ( a, b ) and F ( a, b ) are two polynomials of a = m N /m N and b = m (cid:96) /m N , where m (cid:96) is thecharged lepton mass. Up to O ( b ), these two polynomials are given by F ( a, b ) = (cid:0) a − a − a (1 + b ) + 10 a ( b −
2) + a (12 b −
7) + (3 b − (cid:1) F ( a, b ) = (cid:0) a + a + a (1 − b ) (cid:1) . (71)In Eq. 70, λ / = √ a + b − a − b − a b defines the phase space. In the limit b = m (cid:96) /m N → − a = δM/m N , λ / goes to zero and hence I →
0. The life time of N − is then given by τ ≡ Γ − . We take the freeze outtemperature of DM to be T f = m N /
20. Since the DM freezes out during radiation dominated era, the correspondingtime of DM freeze-out is given by : t f = 0 . g − / (cid:63) m pl T f , (72)7where g (cid:63) is the effective massless degrees of freedom at a temperature T f and m pl is the Planck mass. Demandingthat N − should decay before the DM freezes out (i.e. τ (cid:46) t f ) we getsin θ (cid:38) . × − (cid:18) . × − I (cid:19) / (cid:18) m N (cid:19) / (cid:16) g (cid:63) . (cid:17) / (cid:16) m N (cid:17) . (73)The lower bound on the mixing angle depends on the mass of N − and N . For a typical value of m N = 200 GeV, m N = 180 GeV, we get sin θ (cid:38) . × − . Since τ is inversely proportional to m N , larger the mass, smaller will bethe lower bound on the mixing angle. - m N [ GeV ] Ω N h Δ m ≤
50 GeV m H =
300 GeV m A =
300 GeV m H ± =
300 GeV m H ±± =
300 GeVSin α = v t = = θ =
50 100 500 10000.050.50550 m N [ GeV ] Γ - [ c m ] FIG. 29: Variation of relic density with DM mass m N keeping fixed ∆ m ≤
50 GeV (left panel). Black dashed lines correspondto measured value of relic density by PLANCK. Displaced vertex (Γ − ) is plotted as a function of m N (right panel). For thedisplaced vertex we choose the set of parameters satisfying relic density from the left panel figure. To explore more whether we can get the relic abundance and displaced vertex simultaneously, we have shown inFig. 29 relic abundance as a function of DM mass keeping the mass splitting ∆ M ≤
50 GeV and sin θ = 10 − . In thissmall mixing angle limit there are coannihilation channels (see Fig. 19) which are independent of sin θ contributes torelic density. We choose the set of points which are giving us the correct relic density and tried to find the displacedvertex value. We have plotted in the right panel of Fig 29 displaced vertex ( Γ − ) as a function of m N . We can seethat in the large mass of m N , the displaced vertex is very small as expected as Γ − decreases with increase in mass.For larger mixing angles displaced vertex is suppressed. Again sin θ can not be arbitrarily small as shown in Eq. 73,so Γ − will not be very large. B. Hadronically quiet dilepton signature
Since our proposed scenario have one vector like leptonic doublet, there is a possibility of producing charge partnerpair of the doublet ( N + N − ) at proton proton collider (LHC). The decay of N ± further produce leptonic final statesthrough on-shell/off-shell W ± mediator to yield opposite sign dilepton plus missing energy as is shown in Fig. 30.Obviously, W can decay to jets as well, to yield single lepton plus two jets and missing energy signature or that offour jets plus missing energy signature. But, LHC being a QCD machine, the jet rich final states are prone to veryheavy SM background and can not be segregated from that of the signal. We therefore refrain from calculating theother two possibilities here. A detailed analysis of collider signature of this model will be addressed in [44]. BPs { m N , sin θ } ∆ m Ω N h σ SIN (in cm ) DM modelsBP1 { , . }
10 0.1201 7 . × − Doublet Singlet DMBP2 { , . }
147 0.1165 1 . × − Doublet Singlet DM + Triplet ScalarTABLE III: DM mass, sin θ , ∆ m = m N ± − m N , relic density and SI direct search cross-sections of two benchmark points arementioned for collider study. BP1 correspond to singlet doublet fermion DM scenario. BP2 depicts the case of singlet doubletDM model with an additional triplet in the picture. Note here that others parameters for BP2 remains same mentioned insetof the Fig. 28. pp A/Z N − N + W − W + N lN lν l ν l FIG. 30: Feynman diagram for producing hadronically quiet opposite sign dilepton plus missing energy ( (cid:96) + (cid:96) − + ( /E T )) signalevents at LHC. Doublet-singlet fermion DM in absence or in presence of scalar triplet do not distinguish to yield a different finalstate from that of (cid:96) + (cid:96) − +( /E T ) shown in Fig. 30. However, there is an important distinction that we discuss briefly here. N + N − production cross-section depends on the charge lepton masses and nothing else, leptonic decay branchingfraction is also fixed. However, the splitting between DM ( N ) and its charged partner ( N ± ) (∆ m = m N ± − m N )is seen in the missing energy distribution. The signal can only be segregated from that of the SM background whenthe splitting is large and it falls within the heaps of SM background when ∆ m is small. This feature can distinguishbetween the two cases of singlet doublet DM in presence and in absence of scalar triplet. To illustrate, we choosetwo benchmark points from two scenarios: i ) doublet singlet leptonic DM (BP1) in absence of scalar triplet and ii )doublet singlet leptonic DM in presence of triplet (BP2), shown in Table III. For BP1, we see that ∆ m = 10 GeV,has to be very small because relic density and direct search (XENON 1T 2018) put strong constraint on ∆ m ( ≤ m ∼
150 GeV for low DM mass ( ∼
50 GeV) and obey both relic density and direct search constraint,as indicated in BP2. Again, we note here, that such a low DM mass is still allowed by the Invisible Higgs data dueto small sin θ that we have taken here.To study the collider signature of the model, we first implemented the model in FeynRule [40]. To generate eventsfiles, we used
Madgraph [41] and further passed to
Pythia [42] for analysis. We have imposed further selectioncuts on leptons ( (cid:96) = e, µ ) and jets as follows to mimic the actual collider environment: • Lepton isolation: Leptons are the main constituent of the signal. We impose transverse momentum cut of p T >
20 GeV, pseudorapidity of | η | < . R ≥ . R ≥ . R = (cid:112) (∆ η ) + (∆ φ ) . • Jet formation and identification is performed in Pythia. We use cone-algorithm and impose that the jet initiatorparton must have p T ≥
20 GeV and forms a jet within a cone of ∆ R ≤ .
4. Jets are required to be defined forour events as to have zero jets.Using above basic cuts, we have studied hadronically quite opposite signed dilepton final states :Signal :: (cid:96) + (cid:96) − + ( /E T ) : p p → N + N − , (N − → (cid:96) − ν (cid:96) N ) , (N + → (cid:96) + ν (cid:96) N ) , where (cid:96) = e , µ . The distribution of signal events with respect to missing Energy ( /E T ), invariant mass of OSD ( m (cid:96)(cid:96) ) and effectivemomentum ( H T ) is shown in Fig. 31 respectively top left, top right and bottom panel. We see that each of thedistribution becomes flatter and the peak is shifted to higher energy value with larger ∆ m . As we have alreadymentioned that SM background yields a very similar distribution to that of BP1 and therefore can not be segregatedfrom small ∆ m cases. For details of background estimate and distribution, see for example, [43]. Without furtherselection cuts, the signals constitute a very tiny fraction of hadronically quiet dilepton channel at LHC. To reduceSM background, further selection cuts must be employed: • m (cid:96)(cid:96) < | m z − | and m (cid:96)(cid:96) > | m z + 15 | ,9 FIG. 31: Missing energy ( /E T ), invariant mass of dilepton ( m (cid:96)(cid:96) ) and transverse mass ( H T ) distributions of the hadronicallyquite dilepton signal events ( (cid:96) + (cid:96) − + /E T ) for C.O.M. energy, √ s = 14 TeV at LHC . • H T > ,
200 GeV, • /E T > ,
200 GeV.We see that the signal events for BP1, after such a cut is reduced significantly, while for BP2, we are still left withmoderately large number of events as shown in Table IV.
BPs ∆ m (GeV) σ pp → N + N − (fb) /E T (GeV) H T (GeV) σ OSD (fb) N OSDeff = σ OSD × L
BP1 10 12.01 > > < < > >
100 0.711 71 >
200 0.250 25BP2 147 33.11 > >
100 0.040 4 >
200 0.039 4TABLE IV: Signal events for above mentioned benchmark points with √ s = 14 TeV at the LHC for the luminosity L = 100 fb − after /E T , H T and m (cid:96)(cid:96) cuts. To summarize, we point out that singlet-doublet fermion DM can possibly yield a displaced vertex signature outof the charged fermion decay, thanks to small mass splitting and small sin θ , while due to the same reason, seeingan excess in leptonic final state will be difficult. On the other hand, the model where singlet-doublet fermion DM isextended with a scalar triplet satisfy relic density and direct search with a larger mass splitting between the DM andcharged companion which allows such a case to yield a lepton excess to be probed at LHC, but the displaced vertexsignature may get subdued due to this. The complementarity of the two cases will be elaborated in [44]. We also notethat scalar triplet extension do not allow the fermion DM to have any mass to also accommodate large ∆ m . This0is only possible in the vicinity of Higgs resonance. We can however, earn a freedom on choosing a large ∆ m at anyfermion mass value in the presence of a second DM component and see a lepton signal excess as has been pointed outin [44]. VIII. CONCLUSIONS AND FUTURE DIRECTIONS
Vector like leptons stabilised by a symmetry, provide a simple solution to DM problem of the universe. The relicdensity allowed parameter space provide a wide class of phenomenological implications to be explored in DM directsearch experiments and in collider searches through signal excess or displaced vertex. In this article, we have provideda thorough analysis of different possible models in such a category. The results have been illustrated with parameterspace scans, taking into account the constraints coming from non-observation of DM in present direct search data,constraints from electroweak precision tests, vacuum stability, invisible decay widths of Higgs and Z etc. to see theallowed region where the model(s) can be probed in upcoming experiments.We first reviewed the possibility of vector-like leptonic singlet χ , doublet N and their combination χ − N as viablecandidates of DM. First we discussed about a vector-like singlet leptonic DM χ . In this case, the DM can only coupleto visible sector through non-renormalisable dimension-5 operator χχH † H/ Λ, where Λ denotes a new physics (NP)scale. We find relic density allowed parameter space of the model requires Λ to be 500 GeV or less for DM massranging between 100 GeV to 500 GeV. However, the direct search cross-section for such Λ is much larger than theconstraints obtained from XENON1T data. Therefore, a singlet lepton is almost ruled out being a viable candidateof DM.We then discussed the possibility of neutral component of a vector-like inert lepton doublet (ILD) N to be aviable DM acndidate. Since the doublet has only gauge interaction, the correct relic abundance can be obtained onlyat heavy DM mass around ∼ TeV. Again, the doublet DM suffers a stringent constraint from Z -mediated elasticscattering at direct search experiments. The relic density allowed parameter space therefore lies way above thanthe XENON1T bound of not observing a DM in direct search experiment. Therefore, an ILD DM alone is alreadyruled out. However, we showed that in presence of a scalar triplet, an ILD DM can be reinstated by forbidding theZ-mediated elastic scattering with the nucleons thanks to pseudo Dirac splitting. Due to additional interaction ofILD in presence of a scalar triplet, the mass of ILD DM is pushed to a higher side to achieve correct relic density.Moreover, the scalar triplet mixes with the SM doublet Higgs and paves a path for detecting the ILD DM at terrestriallaboratories. The presence of scalar triplet also yield a non-zero neutrino mass to three flavors of active neutrinoswhich are required by oscillation data. However, we noticed that the parameter space of an ILD DM is very limitedto a very high mass due to its gauge coupling.We then searched for a combination of singlet χ and neutral component of doublet N = ( N − , N ) being a viablecandidate of DM. This is possible if both of the fermion fields possess same Z symmetry. They mix after electroweaksymmetry breaking. In fact, we found that the appropriate combination of a singlet-doublet can be a viable DMcandidate in a large parameter space spanning DM mass between Z resonance to ∼
700 GeV. The singlet-doubletmixing plays a key role in deciding the relic abundance of DM as well as detecting it in terrestrial laboratories. In fact,we found that a large singlet component admixture with a small doublet component is an appropriate combinationto be a viable candidate for DM, particularly to meet direct search bounds (sin θ ≤ . N and its partners N ± , N . In particular, if the mixing angle is very small(around sin θ ∼ − ), the decay of NLSP ( N − → N + l − + ¯ ν l ) gives a measurable displaced vertex signature atLHC, aided by a small mass difference of N − with the DM ( N ). However, this typical feature makes it difficult toidentify any signal excess from production of the NLSP at LHC.The situation becomes more interesting in presence of a scalar triplet. The latter, not only enhances the allowedparameter space of singlet-doublet mixed DM (by allowing a larger mixing sin θ < ∼ . A , then non-zero value of θ can be obtained from the flavour charge of DM,which has been elaborated in [45, 46].1 Acknowledgement
We thank Basabendu Barman for helpful discussions. SB would like to acknowledge the support from DST-INSPIRE faculty grant IFA-13-PH-57 at IIT Guwahati. PG also like to thank MHRD,Government of India forresearch fellowship.
Appendix-A: Couplings of ILD dark matter with scalar triplets and SM particles
Trilinear vertices involving ILD and triplet Scalar:( N − ) c N − H ++ : √ f N ( N − ) c N H + : f N ( N ) c N H : − sin α f N ( N ) c N H : − cos α f N ( N ) c N A : − i f N (74)Trilinear vertices involving triplet scalars: H ++ H − H − : √ v t λ ∝ /v t H ++ H −− H : − (cid:16) α v t λ − sin α v λ (cid:17) sin α → −−−−−→ − v t λ ∝ /v t H ++ H −− H : − (cid:16) cos α vλ + 2 sin α v t λ (cid:17) (75)Trilinear vertices involving ILD and SM particles: N N − W + : e √ θ W γ µ N − N + Z : − e sin 2 θ W cos 2 θ W γ µ N − N + A : − e γ µ N N Z : e sin 2 θ W γ µ (76) [1] F. Zwicky, Helv. Phys. Acta 6, 110 (1933), [Gen. Rel. Grav.41,207(2009)]; V. C. Rubin and W. K. Ford, Jr., Astrophys. J.159, 379 (1970).[2] D. Clowe, M. Bradac, A. H. Gonzalez, M. Markevitch, S. W. Randall, C. Jones and D. Zaritsky, Astrophys. J. 648, L109(2006), astro-ph/0608407.[3] G. Bertone, D. Hooper and J. Silk, Particle Dark Matter: Evidence, Candidates and Constraints , Phys. Rept. 405, 279(2005), arXiv:hep-ph/0404175.[4] G. Jungman, M. Kamionkowski and K. Griest,
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