Two component dark matter with inert Higgs doublet: neutrino mass, high scale validity and collider searches
Subhaditya Bhattacharya, Purusottam Ghosh, Abhijit Kumar Saha, Arunansu Sil
TTwo component dark matter with inert Higgsdoublet: neutrino mass, high scale validity andcollider searches
Subhaditya Bhattacharya, a Purusottam Ghosh, a Abhijit Kumar Saha, b Arunansu Sil a a Department of Physics, Indian Institute of Technology Guwahati, North Guwahati, Assam- 781039,India b Theoretical Physics Division, Physical Research Laboratory, Ahmedabad 380009, India
E-mail: [email protected] , [email protected] , [email protected] , [email protected] Abstract:
The idea of this work is to investigate the constraints on the dark matter (DM)allowed parameter space from high scale validity (absolute stability of Higgs vacuum andperturbativity) in presence of multi particle dark sector and heavy right handed neutrinos toaddress correct neutrino mass. We illustrate a simple two component DM model, consistingof one inert SU (2) L scalar doublet and a scalar singlet, both stabilised by additional Z ×Z (cid:48) symmetry, which also aid to vacuum stability. We demonstrate DM-DM interaction helpsachieving a large allowed parameter space for both the DM components by evading directsearch bound. High scale validity puts further constraints on the model, for example, onthe mass splitting between the charged and neutral component of inert doublet, which hasimportant implication to its leptonic signature(s) at the Large Hadron Collider (LHC). Keywords:
Dark matter, perturbativity, vacuum stability, neutrinos. a r X i v : . [ h e p - ph ] F e b ontents φ or H
105 Two component DM set-up with φ and H β functions and RG running 288.3 Allowed parameter space of the model from high scale validity 35 Discovery of the ‘Higgs’ boson at Large Hadron Collider (LHC) in 2012 [1, 2] stronglyvalidates the Standard Model (SM) of particle physics as the fundamental governing theoryof Strong, Weak and Electromagnetic interactions. However many unresolved issues persist.For example, the electroweak (EW) vacuum turns out to be metastable [3–9] with the presentmeasured value of Higgs mass ( m h ∼ . GeV [10]) and top quark mass ( m t ∼ . GeV– 1 –10]). Large uncertainty in the measured value of m t can even make EW vacuum unstable,questioning the existence of our universe. It is well known that inclusion of additional scalarsin the theory can help stabilizing EW vacuum by compensating the negative contributionsof fermionic couplings in renormalization group (RG) running of Higgs quartic coupling λ H [11, 12]. This motivates us to look for extended scalar sector.On the other hand, the existence of dark matter (DM) in the Universe is convincinglysupported from the observations of various experiments around the globe, for example,WMAP [13] and Planck [14]. Extensions of SM to accommodate DM is therefore in-evitable. The simplest of its kind is Weakly Interacting Massive Particle (WIMP) [15] andmost economical is the presence of a singlet scalar dark matter [16–24] with Higgs portalinteraction with SM. Non observation of DM in direct search experiments like LUX [25],XENON 1T [25–27], Panda X [28] put a stringent bound on this model (with mass be-low 1 TeV getting disallowed) due to its prediction of large spin independent (SI) directsearch cross section. As an alternative, multi-component DM scenarios [29–48] are proposedwhere DM-DM interactions (see for example, [39, 40]) play an important role to evade directsearch bound.Such an attempt was made in [39], where two singlet scalars serve as two DMcomponents, that satisfy required DM constraints. However, the framework is unable toaccommodate the low mass region corresponding to both the DM’s masses, simultaneouslybelow 500 GeV, to satisfy DM direct search bounds.Singlet DM fields (even in multipartite framework consisting of two singlet DM candi-dates [39]) also have limited colliders search possibilities, due to small interaction (Higgsportal) with visible sector. The search strategy for such DM is therefore mainly limitedto mono- X signature with missing energy, where X can be jet, W , Z or Higgs [49–51].Such signal arises out of initial state radiation (ISR) and suffers from huge SM background.Therefore, searching such singlet DM candidates at collider turn out to be difficult, evenmore due to direct search constraint limiting the DM mass to a higher value. Higher mul-tiplet in dark sector, being equipped with charge components have better possibilities ofgetting unravelled in future collider search experiments, but tighter constraints arise fromdark sector. The simplest of its kind is to assume an inert Higgs SU (2) L doublet (IDM).However, due to gauge coupling, a single component IDM is severely constrained and isnot allowed between DM mass within − GeV, referred as the desert region for underabundance. However, it can produce hadronically quiet single and two lepton signatureswith missing energy at LHC [44, 52, 53]. Therefore, it is ideal to study a multipartiteDM framework, involving an inert doublet to satisfy DM constraints as well as to haveinteresting phenomenological consequences.Physics beyond the SM (BSM) is also strongly motivated by the presence of tiny neu-trino mass ( ∼ eV). In order to explain this tiny neutrino masses, type-I seesaw model isproposed where right handed SM singlet fermions are introduced with their Majorana massterms. One can then have SM gauge invariant dimension five effective operator of the form ∼ LLHH , where L represents SM lepton doublet, H represents SM Higgs doublet and Λ is the scale representative of RH neutrino mass scale. Different seesaw models other thantype-I are also invented. They all necessarily predict BSM physics with important andinteresting phenomenological consequences. Among these, type-I seesaw is the simplest– 2 –ossibility [54, 55], which indicates the presence of heavy right handed (RH) neutrinos toyield correct light neutrino mass through additional Yukawa coupling with SM Higgs. Thisin turn may alter the Higgs vacuum stability condition at high scale (larger than the RHneutrino masses) with large neutrino Yukawa coupling [56–68]. Inclusion of an inert Higgsdoublet along with right handed neutrinos can also generate the light neutrino mass ra-diatively [69]. In that case, the usual neutrino Yukawa coupling involving the SM Higgsdoublet can be forbidden with the choice of appropriate discrete symmetry, while a newYukawa interaction involving the inert Higgs doublet is present. However in such a scenario(with radiative light neutrino mass generation), the EW vacuum stability would not bemuch different from that of the SM as the running of the Higgs quartic coupling remainsunaffected by the presence of this new Yukawa interaction.Our model under scrutiny addresses some of the interesting features in view of abovediscussion. We address a multipartite dark sector consisting of a SU (2) L doublet scalar(the IDM) and a scalar singlet, both stabilized by additional Z × Z (cid:48) symmetry (for anearlier effort, see [41]) and provide a two component DM set up. The presence of DM-DMinteractions enlarge the available parameter space significantly (by reviving the below 500GeV mass region for both the DM candidates), while the inert DM can also produce leptoniccollider signature at LHC. We augment the model with heavy RH neutrinos to address thelight neutrino masses. Though the set-up with inert Higgs doublet and RH neutrinosopens up both the possibilities of generating light neutrinos masses, either through type-Iseesaw or via radiative generation, we consider here the neutrino mass generation throughtype-I seesaw and accordingly choose the appropriate discrete charges of the fields so as toforbid the Yukawa interaction involving SM lepton doublet, RH neutrinos and inert Higgsdoublet. The primary reason behind such a choice is to involve the study of the stabilityof the EW vacuum with large neutrino Yukawa couplings (involving SM Higgs doublet) asmentioned before. Then the parameter space obtained from DM phenomenology will befurther constrained from vacuum stability criteria. In brief, we want to study a multipartiteDM scenario which would be adversely affected by the non-zero light neutrino mass in termsof EW vacuum stability. Note that while the neutrino Yukawa coupling involving RHneutrinos and SM Higgs tends to destabilze the EW vacuum, the presence of the additionalscalars in the set-up tends to stabilize it [56–68]. So, we take up an interesting exercise ofvalidating the model from DM constraints, neutrino masses and high scale validity (absolutestability of the Higgs vacuum and perturbativity). This analysis provides some importantconclusions, which are phenomenologically viable at LHC.Let us finally discuss the plan of the paper. In section 2, we discuss the model constructin details. Section 3 presents possible theoretical and experimental constraints on the modelparameters. Then in sections 4,5 and 6 subsequently, we discuss DM phenomenology.Indirect search constraints are discussed in 7. In section 8, we investigate the high scalevalidity of the model. Section 9 summarises collider signature(s) in context of the proposedset up. Finally we conclude in section 10. Tree level unitarity condition is elaborated inAppendix A. The high scale stability condition on the single component DM frameworksin presence of RH Neutrino is chalked out in Appendix B for the sake of comparison. Allthe constraints together on the model parameter space along with different choices of RH– 3 –eutrino mass and Yukawa couplings are listed in two tables in Appendix C. The model is intended to capture the phenomenology of two already established DM frame-works involving that of a singlet scalar and that of an inert scalar doublet together withright handed neutrinos to address neutrino mass under the same umbrella. Therefore, weextend SM by an inert doublet scalar (Φ) and a real scalar singlet ( φ ) and include three RHMajorona neutrinos N i ( i = 1 , , in the set up. The lightest neutral scalar mode of theIDM and φ are the DM candidates provided an appropriate symmetry in addition to thatof SM stabilizes both of them. This is minimally possible by introducing an additional Z × Z (cid:48) discrete symmetry under which all SM fields along with the right handed neutrinostransform trivially and the other additional fields transform non-trivially as tabulated inthe Table 1. We also note the charges of SM Higgs ( H ) explicitly in Table 1, as it willbe required to form the scalar potential of the model. Note here, that charges of the twoDM candidates ( Φ and φ ) are complementary, i.e. odd under either Z or Z (cid:48) for theirstability. We also point out that the U (1) Y hypercharge assignment of Φ is identical to SMBSM and SM Higgs Fields SU (3) C × SU (2) L × U (1) Y × Z × Z (cid:48) ≡ G Φ ≡ H +1 √ ( H + iA ) φ N i ( i = 1 , , H ≡ w +1 √ ( h + v + iz ) Table 1 : Charge assignments of the BSM fields assumed in the model under G as well asthat of SM Higgs. The U (1) Y hypercharge is chosen as Q = T + Y / .doublet H . Therefore the only SU (2) L × U (1) Y invariant terms are H † H, Φ † Φ , H † Φ andits conjugate.The scalar Lagrangian reads as : L scalar = | D µ H | + | D µ Φ | + 12 ( ∂ µ φ ) − V ( H, Φ , φ ) , (2.1) where D µ = ∂ µ − ig σ a W aµ − ig Y B µ and g , g denote SU (2) L and U (1) Y coupling respectively. Introduction of a single Z could lead to a decay of one of the DM candidates, as seen from an allowedterm Φ † Hφ in that case. – 4 –he most relevant renormalizable scalar potential in this case is given by, V ( H, Φ , φ ) = − µ H ( H † H ) + λ H ( H † H ) + V ( H, Φ) + V ( H, φ ) + V (Φ , φ ) , (2.2)where, V ( H, Φ) = µ (Φ † Φ) + λ Φ (Φ † Φ) + λ ( H † H )(Φ † Φ)+ λ ( H † Φ)(Φ † H ) + λ H † Φ) + h.c. ] , (2.3) V ( H, φ ) = 12 µ φ φ + λ φ φ + 12 λ φh φ ( H † H ) , (2.4) V (Φ , φ ) = λ c φ ) (cid:16) Φ † Φ (cid:17) . (2.5)The Lagrangian involving right handed neutrinos can be written as, L ν = − ( Y ν ) ij ¯ l L i (cid:101) HN j − M N ij N Ci N j , (2.6)where ˜ H = iσ H ∗ . We have considered three generations of RH neutrinos with { i, j } =1 , , , which can acquire Majorana masses and can possess Yukawa interactions with SMlepton doublet l L . Note here, that the charge assignment of the N fields then aid us toobtain neutrino masses through standard Seesaw-I [54, 55] mechanism (as detailed later),while it also prohibits the operator like ¯ l L Φ N due to Z charge assignment, and hencediscards the possibility of generating the light neutrino mass radiatively. The ingredientsand interactions of the model set up is described in the cartoon as in Fig. 1. Figure 1 : A schematic diagram illustrating the different sectors of the model and theirconnection to SM. The dotted lines represent Higgs portal coupling, wavy line indicate gaugecoupling, while the thin solid line indicates direct DM-DM coupling through λ c φ (Φ † Φ) term.After spontaneous symmetry breaking, SM Higgs doublet acquires non-zero vacuumexpectation value (VEV) as H = (0 v + h √ ) T with v =246 GeV. Also note that neither of– 5 –he added scalars acquire VEV to preserve Z ×Z (cid:48) and act as DM components. Afterminimizing the potential V ( H, Φ , φ ) along different field directions, one can obtain thefollowing relations between the physical masses and the couplings involved: µ H = m h , µ = m H − λ L v , λ = v ( m H − m A ) ,λ = v ( m H + m A − m H ± ) and λ = 2 λ L − (m − m ± ) , (2.7)where λ L = ( λ + λ + λ ) and m h , m H , m A are the mass eigenvalues of SM-like neutralscalar found at LHC ( m h = 125 . GeV), heavy or light additional neutral scalar and theCP-odd neutral scalar respectively. m H ± denotes the mass of charged scalar eigenstate(s).The mass for φ DM will be rescaled as m φ = µ φ + λ φh v . The independent parameters ofthe model, those are used to evaluate the DM, neutrino mass constraints are as follows:Parameters : { m H , m A , m H ± , m φ , λ L , λ φh , λ Φ , λ φ , Y ν ij , M N ij } . (2.8) We would like to address possible theoretical and experimental constraints on model pa-rameters here. • Stability:
In oder to get the potential bounded from below, the quartic couplingsof the potential V ( H, Φ , φ ) must have to satisfy following co-positivity conditions (CPC)as given by [70, 71], CPC{1,2,3} : λ H ( µ ) ≥ , λ Φ ( µ ) ≥ , λ φ ( µ ) ≥ , CPC{4,5} : (cid:16) λ ( µ ) + λ ( µ ) ± λ ( µ ) (cid:17) + (cid:112) λ H ( µ ) λ Φ ( µ ) ≥ , CPC{6,7} : λ ( µ ) + 2 (cid:112) λ H ( µ ) λ Φ ( µ ) ≥ , λ φh ( µ ) + (cid:114) λ H ( µ ) λ Φ ( µ ) ≥ , CPC8 : λ c ( µ ) + (cid:114) λ Φ ( µ ) λ φ ( µ ) ≥ , (3.1)where CPC { i } denotes i th copositivity condition and µ is the running scale. The aboveconditions show that the model offers to choose even negative λ , , ,φh satisfying the aboveconditions. However, as demonstrated in Eqn. 2.8, we use λ L and physical masses to bethe parameters. Therefore, if we choose a specific mass hierarchy as: m H ± ≥ m A ≥ m H with positive λ L , we are actually using λ to be positive while λ , negative abiding by theabove conditions (see Eqn. 2.7). The conditions CPC { i } as in Eqn. 3.1, will be used laterin demonstrating stability of the potential in Sec. 8. • Perturbativity:
In oder to maintain perturbativity, the quartic couplings of the scalarpotential V ( H, Φ , φ ) , gauge couplings ( g i =1 , , ) and neutrino Yukawa coupling Y ν shouldobey: | λ H ( µ ) | < π, | λ Φ ( µ ) | < π, | λ φ ( µ ) | < π, – 6 – λ c ( µ ) | < π, | λ φh ( µ ) | < π, | λ ( µ ) | < π, | λ ( µ ) | < π, | λ ( µ ) | < π, | g i =1 , , | < √ π and (cid:12)(cid:12) Tr (cid:104) Y † ν ( µ ) Y ν ( µ ) (cid:105)(cid:12)(cid:12) < π . (3.2) • Tree Level Unitarity:
Next we turn to the constraints imposed by tree level unitarity ofthe theory, coming from all possible → scattering amplitudes as detailed in Appendix Afollows as [72, 73]: | λ H | < π, | λ Φ | < π, | λ c | < π, | λ φh | < π, | λ | < π, | λ + 2( λ + λ ) | < π | λ + λ + λ | < π, | λ − λ − λ | < π, | ( λ Φ + λ H ) ± (cid:112) ( λ + λ ) + ( λ H − λ Φ ) | < π, and | x , , | < π . (3.3)where x , , be the roots of the cubic equation as detailed in Appendix A. • Electroweak precision parameters:
There exists an additional SU (2) L doublet ( Φ )in our model in addition to a gauge singlet scalar ( φ ). Figure 2 : Constraints from ∆ S (left) and ∆ T (right) in m A − m H and m H ± − m H plane. For ∆ S , we have taken 1 σ limit for two different choices of m H = { , } GeV.For ∆ T scan, we show both 1 σ and 2 σ limits for a range of m H = { − } GeV.As the vev of Φ is zero, it does not alter the SM predictions of electroweak ρ parameter[74]. However IDM, being an SU (2) L doublet makes a decent contribution to S and T parameters [75, 76] which we will identify as ∆ S and ∆ T . The experimental bound fromthe global electroweak fit results on ∆ S and ∆ T using ∆ U = 0 are given by: ∆ S | ∆ U =0 = 0 . ± . , ∆ T | ∆ U =0 = 0 . ± . , (3.4)– 7 –t 1 σ level with correlation coefficient 0.91 [77].We show the constraint from ∆ S and ∆ T on the model parameter space in Fig. 2using the standard formula as presented in [75, 76]. In left plot, we scan 1 σ fluctuation on ∆ S in m A − m H versus m H ± − m H plane for two different values of m H = { , } GeV. We see that for smaller m H , the constraint is larger. In right panel, we show 1 σ and 2 σ limits from ∆ T in m A − m H versus m H ± − m H plane for a range of IDM mass m H = { − } GeV. We can clearly see, that ∆ T constrains the mass splitting muchmore than ∆ S . • Higgs invisible decay:
Whenever the DM particles are lighter than half of the SMHiggs mass, the Higgs can decay to DM and therefore it will contribute to Higgs invisibledecay. Therefore, in such circumstances, we have to employ the bound on the invisibledecay width of the 125.09 GeV Higgs as [10]: Br ( h → Inv ) < . h → Inv )Γ( h → SM ) + Γ( h → Inv ) < . . (3.5)where Γ( h → Inv ) = Γ( h → H H ) + Γ( h → φ φ ) , when m φ , m H < m h / ∼ . GeV ; and Γ( h → SM ) = 4 . MeV [10]. In this analysis, we have mostly focused in the regionwhere m φ , m H > m h / , actually larger than W mass, i.e. m φ , m H ≥ m W , so that theabove constraint is not applicable. • Collider search constraints:
Experimental searches for additional charged scalarsand pseudoscalar in LEP and LHC provide bound on IDM mass parameters and couplingcoefficients of IDM with SM particles. (i) Bounds from LEP:
The observed decay widths of Z and W bosons from LEP datarestrict the decay of gauge bosons to the additional scalars and therefore provide a boundon IDM mass parameters as m A + m H > m Z , m H ± > m Z and m H ± + m H ,A > m W .In addition, neutralino searches at LEP-II, provides a lower limit on the pseudoscalar Higgs( m A ) to 100 GeV when m H < m A [78]. The chargino search at LEP-II limits indicate abound on the charged Higgs to m H ± > GeV [79]. (ii) Bounds from LHC:
Due to the presence of SM Higgs and IDM interaction, chargedscalars H ± take part into the decay of SM Higgs to diphoton. Thus it contributes to Higgsto diphoton signal strength µ γγ which is defined as [80–83] µ γγ = σ ( gg → h → γγ ) σ ( gg → h → γγ ) SM (cid:39) Br( h → γγ ) IDM
Br( h → γγ ) SM . (3.6)Now when IDM particles are heavier than m h / , one can further write Br( h → γγ ) IDM
Br( h → γγ ) SM = Γ( h → γγ ) IDM Γ( h → γγ ) SM . (3.7)– 8 –he analytic expression of Γ( h → γγ ) IDM can be obtained as [80–83]: Γ( h → γγ ) IDM = (cid:12)(cid:12)(cid:12) A SM + α e m / h π / λ vm H ± F (cid:16) m h m H ± (cid:17)(cid:12)(cid:12)(cid:12) , (3.8)where A SM represents pure SM contribution (see [80–83]). And F ( x ) = − [ x − f ( x )] x − where f ( x ) = (sin − x ) , x ≤ − (cid:104) ln √ − x − −√ − x − − iπ (cid:105) x > . (3.9)Therefore it turns out that IDM contribution to µ γγ is function of both mass of the charged Figure 3 : µ γγ as function of m H ± for different values of λ as defined in the inset. σ and σ limits of µ γγ from ATLAS are also shown in blue and orange colours for comparisonpurpose.Higgs ( m H ± ) and the coefficient of the trilinear coupling hH + H − i.e. λ . The measuredvalue of µ γγ are given by µ γγ = 1 . ± . from ATLAS [84] and µ γγ = 1 . +0 . − . fromCMS [85]. In Fig. 3, we show the variation of µ γγ as function of m H ± for different values of λ . We also present the experimental limits on µ γγ from ATLAS in Fig. 3. Excepting forthe resonance at m h/ , we see that our choice of λ is consistent with experimental bound.The larger is m H ± GeV, µ γγ → i.e. approches to SM value. Also, we see that λ > diminishes µ γγ , while λ < tends to enhance it. In this analysis, we mostly consider m H ± > m h/ and positive λ within correct experimental limit. • Relic Density of DM:
The PLANCK experiment [14] provides the observed amount ofrelic abundance . ≤ Ω DM h ≤ . . (3.10)Furthermore, strong constraints exist from direct DM search experiments. In our analy-sis we will consider the most recent bound on direct detection cross section provided by– 9 –ENON 1T [26]. Relic density and direct search allowed parameter space of the model willbe evaluated in details. • Neutrino observables:
The parameters associated to the neutrino sector should satisfythe bounds provided by different ongoing neutrino experiments. Limit on sum of light neu-trino masses (cid:80) m ν i ≤ . eV as provided by PLANCK data [14, 86] is incorporated. Thepresent values of neutrino mass hierarchies and mixing angle can be found in [87, 88]. Dueto the presence of RH neutrino in the set up, the constraint from lepton flavor violatingdecay (LFV) (dominantly from µ → eγ ) will be applicable [89–91]. LFV constraint canbe successfully evaded for M N (cid:38) . GeV [92, 93] even for neutrino Yukawa coupling of O (1) . It is important to note that in our model, the IDM does not interact with SM leptonsand thus plays no role in LFV. φ or H φφ hh φφ φ hh φφ h SMSM
Figure 4 : Annihilation processes of real scalar singlet DM ( φ ) to SM particles. SM in thelast graph stands for W ± , Z, h and SM fermions.The model inherits two DM candidates: inert DM H and singlet scalar DM φ . Boththe DM components have been studied extensively in literature as individual candidatesto satisfy relic density and direct search bounds. Let us first revisit the single componentframeworks for these two cases here. The relic density of scalar singlet ( φ ) is obtainedvia thermal freeze out through annihilation to SM through the Feynman graphs shown inFig. 4. The direct search constraint for φ comes from the t - channel Higgs portal interaction(turning the last graph of Fig. 4 upside down). The relevant parameters of the model are[17, 20, 23]: φ as single component DM : { m φ , λ φh } . (4.1)The allowed parameter space of φ is depicted in left hand side (LHS) of Fig. 5 in m φ − λ φh plane by the red dots. Direct search allowed parameter space from XENON1Tdata [26, 27] using spin-independent DM-nucleon scattering cross section is shown by theblue dots. We therefore see that the model can only survive either in the Higgs resonanceregion ( ∼ m h / ) or at a very heavy mass (cid:38)
900 GeV. The under abundant (shown in yel-low) and over abundant regions are also indicated.– 10 – igure 5 : Relic density (red dots) and direct search/XENON1T (blue dots) allowed pa-rameter space of the single component DM; scalar singlet ( φ ) on left (in m φ − λ φh plane)and IDM ( H ) on right (in m H − λ L plane). For the right hand side plot, we have used: ≤ m A − m H ≤ GeV, ≤ m H ± − m H ≤ GeV, and λ Φ = 0 . . H H hh H H H hh H H h SMSM H H W + ( Z ) W − ( Z ) H H H ± ( A ) W ± ( Z ) W ∓ ( Z ) H H h W + ( Z ) W − ( Z ) Figure 6 : Annihilation processes of IDM ( H ) to SM particles. SM in the top right graphstands for W ± , Z, h and SM fermions.IDM ( H ) as a single component DM have annihilation and co-annihilation channelsfor freeze-out due to both gauge and Higgs portal interactions as shown by the Feynmangraphs in Figs. 6 and 7. The parameters, which govern the IDM phenomenology are [94]: H as single component DM: { m H , m A , m H ± , λ L } . (4.2)Relic density allowed parameter space for single component IDM is shown in right hand side(RHS) of Fig. 5 in m H − λ L plane by red dots. Direct search (XENON1T data) allowedpoints are shown by blue dots. The scan is obtained using ≤ m A − m H ≤ GeV, ≤ m H ± − m H ≤ GeV with self coupling λ Φ = 0 . kept constant. Here we see again thatallowed region from relic density and direct search constraint for IDM lies either in the small– 11 – A ( H ± ) Z ( W ± ) SMSM H H ± Z, AW ± H A H , A ( H ± ) h, Z ( W ± ) Z, h ( W ± ) H H ± H ± ( A ) W ± ( Z, h, A ) h, Z, A ( W ± ) Figure 7 : Co-annihilation processes of IDM ( H ) with A and H ± to SM particles. SMin the top left graph stands for W ± , Z, h and SM fermions in suitable combination.mass region m H (cid:46) m W ∼ GeV or in the heavy mass region m H (cid:38) GeV. It is a wellknown result coming essentially due to too much annihilation and co-annihilation of IDMto SM through gauge interactions [94]. The disallowed region GeV < m H < GeV isoften called desert region, which is obviously under abundant. Another important point ofsingle component IDM is that the direct search allowed parameter space beyond resonance( m H (cid:38) GeV) have significant co-annihilation dependence with m H ± − m H (cid:46) GeVand m A − m H (cid:46) GeV. φ and H In presence of two DM components ( φ and H ) DM-DM conversion plays a crucial role. Theheavier DM can annihilate to the lighter component and thus contribute to the freeze-outof heavier DM. The conversion processes are shown in Fig. 8, which shows that they aredictated by four point contact interactions as well as by Higgs portal coupling. It is clearthat H ± , A are not really DM, but belongs to dark sector, hence annihilation to them isbroadly classified within DM-DM conversion. More importantly none of them contributeto direct search. The two component DM set up therefore requires following parametersfor analysis: Two component DM : { m H , m A , m H ± , m φ , λ L , λ φh } . (5.1)There are two self interacting quartic couplings present in the model; namely λ Φ and λ φ , which do not play an important role in DM analysis, but appear in vacuum stabilityconstraint that we discuss later. – 12 – φ h H / A / H + H ( A ) / A / H − φφ H / A / H + H / A / H − Figure 8 : DM → DM conversion processes in a model with φ and H . We have assumed m φ > m H , m H ± , m A here. The reverse processes occur with reverse hierarchy.When m φ > m H , m H ± , m A , then φ can annihilate to all possible IDM components.The dominant s - wave DM-DM conversion cross-sections ( σv ) of φ in non-relativistic ap-proximation are given by: ( σv ) φ φ → H H = 164 πm φ (cid:104) λ c + 2 λ L λ φh v (4 m φ − m h ) (cid:105) (cid:115) − m H m φ Θ( m φ − m H )( σv ) φ φ → A A = 164 πm φ (cid:104) λ c + 2 λ S λ φh v (4 m φ − m h ) (cid:105) (cid:115) − m A m φ Θ( m φ − m A )( σv ) φ φ → H + H − = 132 πm φ (cid:104) λ c + λ λ φh v (4 m φ − m h ) (cid:105) (cid:115) − m H ± m φ Θ( m φ − m H ± ) , (5.2)where λ S = ( λ + λ − λ ) . On the other hand, when m φ < m H , the conversion processwill be as H H ( A ) → φφ or H + H − → φφ . The corresponding cross-sections can easilybe gauged from Eqn. 5.2.The evolution of DM number density for both components ( φ and H ) in early universeas a function of time is obtained by coupled Boltzmann equations (CBEQ) as described inEqn. 5.3: dn H dt + 3 Hn H = − (cid:88) X (cid:104) σv (cid:105) H X → SM SM (cid:16) n H n X − n eqH n eqX (cid:17) Θ( m H + m X − m SM ) , − (cid:88) X (cid:104) σv (cid:105) H X → φφ (cid:16) n H n X − n eqH n eqX n eqφ n φ (cid:17) Θ( m H + m X − m φ ) , + (cid:88) X,Y (cid:104) σv (cid:105) φφ → X Y (cid:16) n φ − n eqφ n eqX n eqY n X n Y (cid:17) Θ(2 m φ − m X + m Y ); dn φ dt + 3 Hn φ = −(cid:104) σv (cid:105) φφ → SM SM (cid:16) n φ − n eqφ (cid:17) Θ( m φ − m SM ) , − (cid:88) X,Y (cid:104) σv (cid:105) φφ → X Y (cid:16) n φ − n eqφ n eqX n eqY n X n Y (cid:17) Θ(2 m φ − m X + m Y ) , + (cid:88) X (cid:104) σv (cid:105) H X → φφ (cid:16) n H n X − n eqH n eqX n eqφ n φ (cid:17) Θ( m H + m X − m φ ) (5.3)– 13 –here { X, Y } = { H , A , H ± } . We can clearly spot DM-DM conversion contributions insecond and third lines of each equation, which actually make the two equations ‘coupled’.The freeze-out of two component DM is therefore obtained by numerically solving the aboveCBEQ and yields relic density (for a detailed discussion see for example [39]). The totalrelic density ( Ω DM ) will then have contributions from both DM components as: Ω DM h = Ω H h + Ω φ h . (5.4)Now let us turn to direct search of two component DM set up. Both the DM candidatescan be detected through the spin independent (SI) direct detection (DD) processes through t -channel Higgs mediation as depicted in Fig. 9. The SI DD cross section for H ( σ H ) and H H NN h φ φ N N h
Figure 9 : Spin independent direct detection processes for IDM (left) and scalar singletDM (right).for φ ( σ φ ) at tree level turn out to be [20, 39]: σ eff , tree H = (cid:16) Ω H h Ω DM h (cid:17) λ L f N π µ H ,N m N m h m H , σ eff , tree φ = (cid:16) Ω φ h Ω DM h (cid:17) λ φh f N π µ φ,N m N m h m φ , (5.5)where µ φ,N = m φ m N m φ + m N and µ H ,N = m H m N m H + m N are the reduced masses with nucleon, f N =0 . represents the form factor of nucleon [95, 96] and m N = 0 . GeV represents nucleonmass. Importantly, the effective direct search cross-section for each individual componentis modified by the fraction with which it is present in the universe , given by Ω H h Ω DM h for H and Ω φ h Ω DM h for φ . It is pertinent to note here that next to leading order (NLO) correctionsto SI direct detection cross section can turn dominant for some region of parameter space incase of IDM [98, 99]. The NLO corrections arise mainly from one loop triangular and boxdiagrams mediated by gauge bosons and also from Higgs vertex corrections. In [99], authorsextend the analysis of [98] to include gluon contribution at two loop and also contribution ofIDM self coupling. In our case, the self coupling of IDM can be considered to be sufficientlysmall (as it is not playing any significant role in DM phenomenology) and hence we canignore the effect of it. As shown in [99], in presence of loop effects, the coupling ( λ L ) thatenters into σ eff H gets replaced by λ eff L = λ L + δλ L where δλ L takes into account the loop A more comprehensive bound on multipartite DM from direct search can be obtained from the recoilrate of the nucleus [34, 97]. – 14 –ontributions effectively. Following this, we rewrite σ eff H as σ eff , H = σ eff , tree H (1 + κ ) , (5.6)where κ is the enhancement factor to SI DD cross section of IDM candidate due to loopcorrection. It is shown in [98], κ can take different values depending on λ L . For smallervalues of λ L , κ turns bigger; for example, with . (cid:46) λ L < . the correction turns outto be (cid:38) κ > ; for . (cid:46) λ L < . → (cid:38) κ > and κ (cid:46) for λ L (cid:38) . forsingle component IDM relic density allowed points. Here we follow similar scaling and take κ ∼ . λ L to accommodate NLO corrections for IDM direct search in our analysis.To obtain relic density and tree level DD cross sections of both the DM candidates nu-merically, we have used MicrOmegas [100]. It is noteworthy, that version 4.3 of
MicrOmegas is capable of handling two component DM and we have cross-checked the solution from thecode to match very closely to the numerical solution of CBEQ in Eqn. 5.3. For generatingthe model files compatible with
MicrOmegas , we have implemented the model in
LanHEP [101].
We first study the variation of relic density with respect to DM mass and other relevantparameters to extract the importance of DM-DM conversion in this two-component set upbefore elaborating on the relic density and direct search allowed parameter space of themodel.In Fig. 10 we plot relic densities of two DM candidates: Ω H h in left panel and Ω φ h in right panel figures as function of m H for different values of λ c . We also keep m φ fixedat 200 GeV here. Other parameters are chosen as mentioned in the inset of individual plots. • Top left of Fig. 10: In this figure we have shown the variation of Ω H h with m H for different values of λ c = 0 , − , . , . by yellow, blue, purple and black solid linesrespectively. Pure IDM case (in a single component framework) is depicted by red dottedline, where also evidently λ c = 0 . It is important to note the other parameters kept fixedfor this plot are: λ φh = 10 − , λ L = 0 . , m H ± − m A = m A − m H = m H < m φ , relic density of H changes significantly with the variation of λ c . We alsosee that λ c = 0 (in two component set-up) is way above the pure IDM case yielding a largerelic density. Now, with slight increase in λ c = 10 − , relic density goes further up andthen reduces significantly for larger λ c = 0 . , . . The interesting point is that the caseof λ c = 0 . lies very close to the pure IDM case. It is therefore evident that the presenceof the second DM component φ plays an important role in Ω H h through the coupling λ c .For m H < m φ , relic density of the heavier component φ can easily inherit the annihilationto other DM components { X, Y } = { H , A , H ± } in addition to SM as[39, 102]: Ω φ h (cid:39) . × − x f √ g ∗ ( (cid:104) σv (cid:105) φφ → SM SM + (cid:104) σv (cid:105) φφ → XY ) (cid:39) . (cid:104) σv (cid:105) φφ → SM SM + (cid:104) σv (cid:105) φφ → XY , (5.7)where in the last step, we have used g ∗ = 106 . and x f = 20 [102]. However it is difficult toenvisage the relic density for the lighter DM component due to such DM-DM conversion.– 15 – igure 10 : Relic Density of IDM, H (left panel) and scalar DM, φ (right panel) asa function of IDM mass m H , with different choices of λ c . We illustrate two differentcombinations of DM-SM couplings, in the top panel: { λ φh = 10 − , λ L = 0 . } and inbottom panel: { λ φh = 0 . , λ L = 10 − } . Other parameters kept fixed, are mentioned in theinset of each figure.This can be understood from the CBEQ for the two component DM as in Eqn. 5.3. Let usdefine the following notations first: (cid:104) σv (cid:105) H X → SM n H n X = F H ; (cid:104) σv (cid:105) φφ → XY n φ = F φ Φ ; (cid:104) σv (cid:105) φφ → SM n φ = F φ ; (cid:104) σv (cid:105) H X → φφ n H n X = F Φ φ . (5.8)Again { X, Y } = H , A , H ± ; as earlier. The CBEQ with above notation, turns out to be(assuming m φ > m H ): dn H dt + 3 Hn H (cid:39) −F H + F φ Φ ; dn φ dt + 3 Hn φ (cid:39) −F φ − F φ Φ . (5.9)In Eqn. 5.9, we neglected the equilibrium number densities as they are tiny near freeze-out, where the dynamics is under study. With λ c = 0 , and λ φh = 10 − , annihilationcross-sections for φ , ( (cid:104) σv (cid:105) φ φ → SM SM and (cid:104) σv (cid:105) φ φ → X Y ) are very small. Hence φ freezesout early and the number density of φ turns out to be large since n φ ∝ / (cid:104) σv (cid:105) eff φ where– 16 – σv (cid:105) eff φ (cid:39) (cid:16) (cid:104) σv (cid:105) φφ → SM SM + (cid:104) σv (cid:105) φφ → X Y (cid:17) following Eqn. 5.7. Now, it is easy to appreciatethat with λ c = 0 , and λ φh = 10 − , annihilation of φ to SM is larger than conversion toother DM ( H ), i.e. (cid:104) σv (cid:105) φφ → SM SM (cid:29) (cid:104) σv (cid:105) φφ → XY . However due to large φ abundance ( n φ ∼ . − ) , F φ Φ becomes comparable with F H . As these two terms ( F φ Φ and F H )appear in the evolution of n H (Eqn. 5.9) with opposite sign, it is quite evident that effectiveannihilation cross-section for H becomes small and hence n H after freeze out turns outto be much larger than the pure IDM case. Next let us consider non zero but small λ c ( = 10 − ). Then (cid:104) σv (cid:105) φφ → X Y increases compared to the earlier case of λ c = 0 . Howeverdue to smallness of the coupling λ c , this does not make any significant change in the numberdensity of φ and n φ ∼ . − remains the same (as φ φ → SM SM still dominantlycontributes to the total annihilation cross section of φ ). Therefore, F φ Φ increases andreduces the separation with F H . Hence the effective annihilation cross-section for H turns out to be even smaller than λ c = 0 case. Therefore, for λ c = 10 − relic densityincreases further than that of λ c = 0 . For larger value of λ c = 0 . , contribution fromDM-DM conversion, (cid:104) σv (cid:105) φφ → X Y significantly rises and therefore the number density of n φ drops to n φ ∼ − cm − , and therefore F φ Φ becomes much smaller than F H . Thisincreases the effective annihilation for H and reduces relic density. This trend continuesfor higher values of λ c and eventually leads to a vanishingly small F φ Φ to closely mimic thecase of single component IDM. The case of m H > m φ can also be understood from theCBEQ in this limit: dn H dt + 3 Hn H (cid:39) −F H − F Φ φ ; dn φ dt + 3 Hn φ (cid:39) −F φ + F Φ φ . (5.10)For m H > m φ , relic density of H can be written simply as Ω H h (cid:39) . (cid:104) σv (cid:105) H X → SM SM + (cid:104) σv (cid:105) H X → φφ . (5.11)The annihilation to SM ( (cid:104) σv (cid:105) H X → SM SM ) due to gauge coupling is much larger thanthe conversion cross-section (cid:104) σv (cid:105) H X → φφ for all the choices of λ c , so we do not find anydistinction between all those cases. • Top right of Fig. 10: In the top right panel, the same parameter space is used to showthe variation of Ω φ h with respect to m H . The dynamics is much simpler for m φ > m H (see BEQ. 5.9) where Ω φ h (cid:39) . (cid:104) σv (cid:105) φφ → SM SM + (cid:104) σv (cid:105) φφ → X Y . With larger λ c , the conversionto other DM ( F φ Φ ) becomes larger and relic density drops accordingly. For m φ < m H , wesee from Eqn. 5.10, that there is a competition between F φ and F Φ φ . With increasing m H , (cid:104) σv (cid:105) H X → φφ decreases, therefore F Φ φ decreases and eventually it becomes vanishingly smallfor m H (cid:38) GeV. The equation for n φ then becomes equivalent to the single componentcase of φ where m H is no more relevant for Ω φ h . • The bottom panel figures of Fig. 10 essentially indicate that with larger λ φh , annihi-lation of φ to SM becomes large, resulting a smaller n φ after freeze out. Therefore F φ Φ in n φ can be calculated by using the formula Ω φ = n φ m φ ρ c , where ρ c = 1 . × − h GeV cm − [102]. – 17 –qn. 5.9 turns insignificant. On the other hand, F Φ φ also becomes smaller than F φ in Eqn.5.10. Together, relic density of the lighter DM component is not affected by the presenceof a heavy DM component.Before we move on, let us summarise the outcome of Fig. 10. We see here that relicdensity of lighter DM component is affected by the heavier one, when the annihilationcross-section to SM is tiny. Figure 11 : Ω H h vs m H for λ c = 0 . (left panel) and λ c = 0 . (right panel) withdifferent choices of m A − m H which are depicted by different coloured lines in the figure.The other parameters kept fixed are mentioned in the inset of each figure.In Fig. 11. we plot the variation of IDM relic density as function of DM mass m H for different choices of ∆ m = m A − m H : { − } GeV shown by different colouredlines. The other parameters are kept fixed and mentioned explicitly in the figure insets.Let us first focus on the left panel plot. We notice that for m H < m W , with larger ∆ m = m A − m H , relic density becomes larger. Keeping in mind that in this case ( m H < m W ),co-annihilations (see Fig. 7) dominate the total (cid:104) σv (cid:105) , this can be explained from the usualconvention of co-annihilation cross-section being reduced by Boltzmann suppression ( ∼ e − ∆ m/T ) with larger splitting ( ∆ m ). However, for m H > m W , the phenomena seems to bereversed, i.e., with larger mass splitting, relic density decreases. Note that for m H > m W ,DM annihilations (in particular, annihilations into gauge bosons) become dominant. Itturns out that the contribution of (cid:104) σv (cid:105) H H → W + W − or ZZ increases with larger ∆ m (for therelevant cross-section, see [41]). This happens due to presence of the destructive interferencebetween the s and t channel contributions. With larger ∆ m , the destructive interferencereduces and hence the annihilation cross section increases which in turn lowers the relicdensity of H in this region. The same feature prevails in the right panel figure. Here,for a larger λ c , the relic density of H goes further down due to annihilation to φ beyond m H > m φ . One of the important motivations of this analysis is to study interacting multi-componentDM phenomenology for DM mass lying between ≤ m DM ≤ GeV for both the com-ponents in view of relic density ( . ≤ Ω DM h ( ≡ Ω H h + Ω φ h ) ≤ . [14]) and– 18 –irect detection (XENON 1T [26]) bounds. We would like to recall that for the individualscenarios, none of the DM components satisfy relic and direct detection bounds simultane-ously within this mass range (see section 4). Now in order to find a consistent parameterspace in the set up, we perform a numerical scan of the relevant parameters within thespecified ranges as mentioned below. ≤ m H ≤
500 GeV , ≤ m φ ≤
500 GeV , ≤ m A − m H ≤
200 GeV , ≤ m H ± − m A ≤
180 GeV , . ≤ λ L ≤ . , . ≤ λ φh ≤ . , . ≤ λ c ≤ . . (6.1)We also note here that as λ L and λ φh enters into direct search cross-sections for H and φ DMs respectively, we keep those couplings in a moderate range, while λ c governs DM-DMinteractions, but do not directly enter into direct search bounds, therefore we choose alarger range (with natural values) for scanning λ c . λ φ = 0 . and λ Φ = 0 . are keptfixed throughout the analysis since those are not relevant for DM phenomenology.There are two possible mass hierarchies for the two-component DM set up relevant forphenomenological analysis: (i) m φ > m H and (ii) m φ ≤ m H , which we address separatelybelow. Case I: m φ > m H Primarily, in such a scenario, the main physics arises due to annihilation of φ to H , on topof their individual annihilation to SM to govern the freeze-out.In Fig. 12, we show relic density allowed parameter space for the model in terms ofdifferent relevant parameters. In top left panel of Fig. 12, we have shown the relic densitysatisfied (considering contribution from both φ and H components) points in m φ − λ φh plane for different ranges of λ c , as depicted in the figure with different colour codes. Asseen from the plot, for a fixed m φ , there is a maximum λ φh . All possible values less thanthe maximum λ φh is also allowed subject to different choices of λ c . The larger is λ c , thesmaller is the required λ φh thanks to the conversion of φ → H to yield relic density. It isalso noted that for large λ c (cid:38) . , as the DM-DM conversion is very high, the DM mass m φ has to lie in the high mass region ( (cid:38) GeV) to tame the annihilation cross-sectionwithin acceptable range. To summarise, this figure shows that due to the presence of secondDM component, much larger parameter space (actually the over-abundant regions of thesingle component framework, compare Fig. 5) is allowed. Top right panel figure shows therelic density allowed parameter space in m H − λ L plane again for different ranges of λ c as in the left plot. It naturally depicts that λ L is insensitive to m H as the annihilationand co-annihilation cross-section of H is mainly dictated by gauge interaction. However,we see a mild dependence on λ c , such that when λ c (cid:38) . , the DM mass m H has to beheavy (cid:38) GeV. This is because with large λ c , DM-DM conversion is large; to achieverelic density of correct order, m φ requires to be large and the conversion can only be tameddown by phase space suppression, i.e. by choosing H mass as close as possible to φ mass( m H ∼ m φ ). Bottom left figure correlates the DM masses to obtain correct density within m φ > m H . We see that for small λ c , particularly with higher m φ , large m H values are– 19 – igure 12 : Relic Density allowed parameter space is shown in m φ − λ φh plane (topleft), m H − λ L plane (top right), m φ − m H plane (bottom left) and Ω φ h / Ω DM h (%) − Ω H h / Ω DM h (%) plane (bottom right) for the mass hierarchy m φ > m H .disfavoured in order to keep the DM-DM conversion in the right order. While for large λ c ,mass degeneracy is required ( m H ∼ m φ ) to tame the DM-DM conversion. Bottom rightfigure shows the relative contribution of relic density of the two DM components. First ofall, this shows that φ contributes with larger share of relic density, while the relic densityof H can at most be of the total. This is natural from the perspective of IDM tobe the lighter DM component, as we know it is always suppressed below 500 GeV (with amaximum contribution ) and the conversion channels do not alter IDM’s annihilationfor m φ > m H . Although for . < λ c (cid:46) . , individual relic abundance of IDM couldbe large for very small (and fine tuned) λ φh ∼ − , as shown in upper left of Fig. 10;with the increase of λ φh , this enhancement vanishes. Since we scan the parameter spacewhere λ φh (cid:38) . , such effect can not be found in Fig. 12. For small λ c , contributionfrom H is smaller, as relic density contribution from φ gets larger due to small DM-DMconversion. With larger λ c (cid:38) . , the DM-DM conversion for φ becomes larger and thereforethe relic density of φ can easily span between − . With very high λ c ∼ , DM-DMconversion becomes too large, therefore to keep relic density in the correct ballpark, theDM mass ( m φ ) has to be heavy and almost degenerate with the heavier DM ( m H ∼ m φ ).– 20 – φh in such cases, requires to be very small, which are validated by some dark blue pointswith Ω φ h / Ω DM h ∼ . Figure 13 : SI Direct detection cross section for IDM component is plotted against m H for (upper left) . (cid:46) λ L < . , (upper right) . (cid:46) λ L < . and (bottom) λ L (cid:38) . .All the figures show σ eff H both at tree level (light black) and at NLO level (orange). TheXENON 2018 bound (red dotted line) is shown for comparison purpose.Next in Fig. 13, we show the SI direct detection cross section for H at relic densitysatisfied points in σ eff H − m H plane. Direct search cross-section at tree level is shown bylight black points and NLO effects as prescribed in Eq. 5.6, is shown by orange points. Forcomparison, we also point out the most recent XENON 1T experimental bound in dottedred line. Upper left panel of Fig. 13 shows that for . (cid:46) λ L < . , the one loopcorrection to σ eff H is really significant considering κ ∼ . λ L and rules out some of the DDbound satisfied points at tree level. Similar kind of feature can be found for . (cid:46) λ L < . case (upper right panel figure) also. Once λ L (cid:38) . , as in the lower panel of Fig. 13, σ eff H at NLO shows minor difference to tree level computation.Relic density allowed parameters space consistent with direct search constraints atNLO where both DMs φ and H simultaneously satisfy XENON 1T 2018 [103] bound (fordifferent ranges of λ c ) are shown next in Fig. 14. This is illustrated in m φ − λ φh plane(top left panel), m H − λ L plane (top right panel) and in m φ − m H plane (bottom panel)– 21 –imilar to Fig. 12. We have already mentioned that spin independent (SI) DM-nucleoncross-section depends on square of Higgs portal couplings of the respective DM candidates, λ φh for φ and λ L for H scaled by a pre-factor of the relative number density Ω i h / Ω DM h ( i = φ, H ). Since in this two component scenario, the dominant contribution is comingfrom φ DM, the pre-factor Ω φ h / Ω DM h ∼ . On the other hand, for H the pre-factoris small, Ω H h / Ω DM h < , and will help H reducing the effective direct search cross-section. Therefore, portal coupling λ φh is tightly constrained from XENON 1T bound to λ φh (cid:46) . for DM mass m φ (cid:46) GeV, as shown in top left panel of Fig. 14 (compare itwith the top left panel of Fig. 12). Similarly in top right panel, the direct search allowed m H − λ L plane for H shows that a large region corresponding to higher λ L is excludedas a function of m H (again, compare it with top right panel of Fig. 12). A possible masscorrelation after direct search bound are plotted in bottom panel of Fig. 14 in m φ − m H plane. The main outcomes from this figure are: (i) For λ c ≤ . , small m H ∼ GeV isfavoured, (ii) for moderate values of λ c ∼ { . − . } , there is no correlation and (iii) forlarge λ c , only degenerate mass scenario ( m φ ∼ m H ) with large m φ ∼ GeV is allowed.
Figure 14 : Relic Density and direct detection (XENON 1T 2018) at NLO allowed param-eter space is shown in: m φ − λ φh plane (top left panel), m H − λ L plane (top right panel)and m φ − m H plane (bottom panel). The scans are performed for for the mass hierarchy m φ > m H . – 22 – ase II: m φ ≤ m H Naturally here the conversion of heavier H to the lighter component φ will mainly dictatethe relic density of DM components on top their annihilations to SM. Relic density allowed Figure 15 : Relic Density allowed parameter space is shown in m φ − λ φh plane (topleft), m H − λ L plane (top right), m φ − m H plane (bottom left) and Ω φ h / Ω DM h (%) − Ω H h / Ω DM h (%) plane (bottom right) for the mass hierarchy m φ ≤ m H .parameter space for m φ ≤ m H is shown in Fig. 15. Again, this is illustrated in m φ − λ φh plane (top left), m H − λ L plane (top right), m φ − m H plane (bottom left) and Ω φ h / Ω DM h (%) − Ω H h / Ω DM h (%) plane (bottom right). Different ranges of λ c areshown by the same colour code as in Fig. 12, 14. Let us first focus on the top left figure.It shows that for small values of λ c , relic density allowed parameter space points lie inthe vicinity of single component framework of φ (red points in figure). In absence of alighter mode, the relic density of φ is essentially governed by its annihilation to SM anddue to small conversion cross-section the production of φ is also not large enough to changethe conclusion. However, the situation changes significantly with larger λ c (cyan and bluepoints), where we see again that the overabundant region of the single component scenariois getting allowed by relic density. In order to understand this let us remind ourself of theCBEQ for m φ < m H as depicted in Eqn. 5.10. In particular, the number density of φ isdictated by ˙ n φ + 3 Hn φ (cid:39) −F φ + F Φ φ . With larger λ c and larger conversion, F Φ φ increases– 23 – igure 16 : Relic Density and direct detection (XENON 1T 2018) at NLO allowed param-eter space is shown in: m φ − λ φh plane (top left panel), m H − λ L plane (top right panel)and m φ − m H plane (bottom panel). The scans are performed for for the mass hierarchy m φ ≤ m H .to reduce the effective F φ that determines the relic of φ . Therefore, to keep the relicdensity of φ to correct order, F φ has to increase. Now recall, F φ = (cid:104) σv (cid:105) φφ → SM SM ( n φ ) ∼ / (cid:104) σv (cid:105) φφ → SM SM as n φ ∼ / (cid:104) σv (cid:105) φφ → SM SM . Therefore, to increase F φ , one has to reducethe annihilation cross-section (cid:104) σv (cid:105) φφ → SM SM . This is possible by reducing λ φh as we seehere in the plot. Next let us discuss the top right figure. This figure in m H − λ L planeessentially depicts that with larger λ c , larger m H is favoured to tame the DM conversionas well as annihilation cross-section to keep the relic within limit. The dependence howeveris not that much significant due to the presence of large number of co-annihilation channelswhich remain unaffected by λ c . In the bottom left panel, mass correlation has been plottedand carries no information. Lastly, bottom right figure shows the relative relic densitycontributions of these two components. It is well understood that an additional channel forannihilation of H only reduces the possibility of bringing Ω H h in the correct ballpark dueto already existing gauge mediated annihilation and co-annihilation channels. Therefore,for small λ c , it is still possible to get a contribution from Ω H h ∼ , but that becomesharder with large λ c , where the relic density contribution of H is further limited to Ω H h ∼ – 24 – .Direct search constraints from XENON 1T 2018 on the relic density allowed pointsare shown in Fig. 16. To emphasise again, the demand of these plots are simultaneoussatisfaction of XENON1T limit for both DM components. The main outcome of this plotis to see the absence of small λ c points (red dots) upto λ c ∼ . . This is simply due tothe fact that, with small λ c , the required λ φh is high enough for φ DM to be discarded byXENON1T data. The other important feature is that with larger λ c , larger DM massesare favoured. Lastly a very important conclusion comes from the bottom panel figure inthe mass correlation plot. This shows, as only high λ c region is allowed, the conversion of H to φ still needs to be restricted and therefore the mass difference between the two DMcomponents ( m H − m φ ) has to be very very small. These features are all distinct fromthat of m φ > m H region.So far our discussion has been focused on DM mass region m W ± ≤ m H , m φ ≤ GeV. But if m H < m W ± or m φ < m W ± , while other DM mass is heavier than m W ± ,the only region available for lighter DM with mass < m W ± are the Higgs and Z resonanceregions: m H ∼ m Z , m h and m φ ∼ m h . It is important to remind that the resonanceregions are already available in absence of second DM component and therefore brings nonew phenomenological outcome. It is possible to probe DM signals in the context of different indirect detection experimentsas well. The experiments look for astrophysical sources of SM particles produced throughDM annihilations or via DM decays. Amongst these final states, photon and neutrinos,being neutral and stable can reach indirect detection probes without getting affected muchby intermediate regions. For WIMP type DM, emitted photons lie in the gamma rayregime that can be measured at space-based telescopes like the Fermi-LAT or ground-basedtelescopes like MAGIC [104]. The photon flux in a specific energy range is written as Φ F = 14 π (cid:104) σv (cid:105) ann m DM (cid:90) E max E min dN γ dE γ dE γ (cid:90) LOS dxρ ( r ( b, l, x )) , (7.1)where r ( b, l, x ) is the distance of the DM halos from the galactic center. Galactic coordinatesare defined by b, l and ρ ( r ) represents the DM density profile. Eq.(7.1) is further rewrittenas Φ F = 14 π (cid:104) σv (cid:105) ann m DM (cid:90) E max E min dN γ dE γ dE γ × J, (7.2)where J = (cid:82) dxρ ( r ( b, l, x )) is conventionally known as J − factor and LOS means line ofsight. Now from the observed Gamma ray flux produced due to DM annihilations, onecan constrain the DM annihilation into different charged final states like µ + µ − , τ + τ − , W + , W − and b + b − . It is pertinent to mention here that along with the continuous gammaray spectrum, gamma ray lines originated from IDM annihilations are also of special interest[105]. It has been shown in [106] that such a feature, characteristic of inert doublet dark– 25 –atter (IDM) model, can indeed be important for DM mass above TeV where Sommerfeldenhancement becomes relevant. Since we mostly focus on the intermediate mass range ofIDM in this study ( ∼
80 - 500 GeV), aforementioned constraint is extremely subdued.In the multicomponent dark matter set up as we have here, it is expected that the γ ray flux will be the sum of the contributions from both φ and H . Then we can write thetotal γ ray flux can be written as: Φ TF = Φ φF + Φ H F , (7.3)where Φ φF and Φ H F are the individual contribution to Φ TF from φ and H . Following Eq.(7.2), it would be natural to write (cid:104) σv (cid:105) eff XX m ρ T = (cid:104) σv (cid:105) φφ → XX m φ ρ φ + (cid:104) σv (cid:105) H H → XX m H ρ H , (7.4)where X, X denote the annihilation products and m R indicates some effective mass scale.One can further write the effective annihilation cross-section in Eq. (7.4) considering m R to be the reduced mass of the two component system as (cid:104) σv (cid:105) eff XX = (cid:18) Ω H Ω T (cid:19) m φ ( m φ + m H ) (cid:104) σv (cid:105) H H → XX + (cid:18) Ω φ Ω T (cid:19) m H ( m φ + m H ) (cid:104) σv (cid:105) φφ → XX . (7.5) Figure 17 : DM annihilations φφ → W + W − (left) and H H → W + W − (right) nor-malised by branching fraction of individual relic densities compared against the latest in-direct detection bounds of Fermi-LAT [104].It turns out that for real singlet scalar dark matter indirect searches does not posestrong bound [18, 107]; while for single component IDM candidate, the indirect searchrules out the region of m H < GeV due to large thermal average cross section of H H → W + W − channel [106–109]. Hence in our proposed two component DM set upwe shall mostly focus on the (cid:104) σv (cid:105) DM , DM → W + W − and compare it with the indirect searchexperimental bound from Magcic+Fermi Lat bound [104]. In left panel of Fig. 17, we show– 26 –he contribution of φ DM in (cid:104) σv (cid:105) eff W + W − for all relic and SI DD cross section satisfied pointsand compare it with the limit coming from Fermi Lat experiment [104]. Similar comparisonis drawn for IDM ( H ) in right panel of Fig. 17. Finally in Fig. 18, we show the orderof magnitude of total (cid:104) σv (cid:105) eff W + W − for all relic density and SI DD (at NLO) satisfied pointsand find it to lie well below the upper limit imposed by Fermi Lat experiment. The relativeabundance of individual relic densities and ratio of reduced mass to the DM mass helps toevade the indirect search bound on (cid:104) σv (cid:105) DM , DM → W + W − successfully. We have also confirmedthat the bounds on (cid:104) σv (cid:105) eff for b ¯ b , µ + µ − , τ + τ − and other annihilation final products fromMagic+Fermi Lat experiment [104] are easily satisfied in our set up. Figure 18 : Effective DM annihilations to W + W − as defined in Eq.(7.4) in the two-component model for relic density and direct search allowed points compared against thelatest indirect detection bounds of Fermi-LAT [104]. One of the motivations of this work is to show that the presence of right handed neutrinos toyield correct neutrino masses in presence of multipartite DM. Although the neutrino sectorconsidered here seems decoupled from the dark sector, is not completely true. The effectof the RH neutrinos alter the allowed DM parameter space when the model is validated athigh scale. As already stated before, we employ type-I seesaw mechanism to generate thelight neutrino mass, for which three RH neutrinos are included in the set up.
We first describe the strategy in order to study their impact on RG evolution. For simplicity,the RH neutrino mass matrix M N is considered to be diagonal with degenerate entries, i.e. – 27 – N i =1 , , = M R . It is to be noted that in the RG equation, Tr [ Y † ν Y ν ] enters. In orderto extract the information on Y ν , we use the type-I seesaw formula for neutrino mass m ν = Y Tν Y ν v M R . Then, naively one would expect that large Yukawa couplings are possiblewith even heavier RH neutrino masses. For example with M R ∼ GeV, Y ν comes outto be . in order to obtain m ν (cid:39) . eV. However, contrary to this, it can be shownthat even with smaller M R one can achieve large values of Tr [ Y † ν Y ν ] , but effectively keeping Y Tν Y ν small, using a special flavor structure of Y ν through Casas-Ibarra parametrization[58]. Note that our aim is to study the maximum effect coming from the right handedneutrino Yukawa, i.e. with large Tr [ Y † ν Y ν ] , on EW vacuum stability. For this purpose, weuse the parametrisation by [110] and write Y ν as Y ν = √ √ M R v R (cid:113) m dν U † PMNS , (8.1)where m dν is the diagonal light neutrino mass matrix and U PMNS is the unitary matrixdiagonalizing the neutrino mass matrix m ν such that m ν = U ∗ PMNS m dν U † PMNS . Here R represents a complex orthogonal matrix which can be written as R = O exp ( i A ) with O asreal orthogonal matrix and A as real antisymmetric matrices. Using above parametrisation,then one gets, Tr [ Y † ν Y ν ] = 2 M R v Tr (cid:104)(cid:113) m dν e i A (cid:113) m dν (cid:105) . (8.2)Note that the real antisymmetric matrix A however does not appear in the seesaw expressionfor neutrino mass as m ν = Y Tν Y ν v M R . Therefore with any suitable choice of A , it wouldactually be possible to have relatively large Yukawa even with light M R . β functions and RG running To study the high scale validity of this multi-component DM model with neutrino mass(including perturbativity and vacuum stability), we need to consider the RG running of theassociated couplings. The framework contains few additional mass scales: one extra scalarsinglet, one inert doublet and three RH neutrinos with mass M N i =1 , , (= M R ) . Hence inthe study of RG running, different couplings will enter into different mass scales. In DMphenomenology, we have considered the physical masses of DM sector particles within ∼
500 GeV. Then for simplicity, we can safely ignore the small differences between the darksector particle masses and identify the single additional mass scale as m DM . On the otherhand, RH neutrinos are considered to be heavier than the scalars present in the model( M R > m DM ). Hence, for running energy scale µ > m DM , we need to consider the cou-plings associated to DM sector in addition to SM while while for µ > M N i =1 , , , the neutrinocouplings will additionally enter into the scenario. Below we provide the one loop β func-tions for the additional for the couplings involved in our model, when µ > M R . β functions of gauge couplings [111]: β g = 215 g , – 28 – g = − g ,β g = − g . (8.3) β functions of Yukawa couplings [111, 112]: β y t = 32 y t + y t (cid:16) y t − g − g − g + y t Tr [ Y † ν Y ν ] (cid:17) ,β Tr [ Y † ν Y ν ] = 3 Tr [( Y † ν Y ν ) ] + Tr [ Y † ν Y ν ] (cid:16) − g − g + 6 y t + 2 Tr [ Y † ν Y ν ] (cid:17) . (8.4) β functions of quartic scalar couplings [111]: β λ H = 27200 g + 920 g g + 98 g − g λ H − g λ H + 24 λ H + 12 λ H y t − y t + 2 λ + 2 λ λ + λ + λ + 12 λ φh + 4 λ H Tr[ Y † ν Y ν ] − Y † ν Y ν ) ] (8.5) β λ = − g λ − g λ + 8 λ λ + 12 λ λ + 4 λ λ H + 4 λ λ Φ + 6 λ y t + 2 λ Tr [ Y † ν Y ν ] ,β λ = + 95 g g − g λ − g λ + 8 λ λ + 4 λ + 8 λ + 4 λ λ H + 4 λ λ Φ + 6 λ y t + 2 λ Tr[ Y † ν Y ν ] ,β λ = 27100 g − g g + 94 g − g λ − g λ + 4 λ + 2 λ + 2 λ + 12 λ λ H + 4 λ λ H + 12 λ λ Φ + 4 λ λ Φ + 6 λ y t + λ c λ φh + 2 λ Tr[ Y † ν Y ν ] ,β λ Φ = 24 λ + 2 λ + 2 λ λ − g λ Φ + 27200 g + 920 g (cid:16) − λ Φ + g (cid:17) + 98 g + λ + λ + 12 λ c ,β λ φh = − g λ φh − g λ φh + 4 λ φh + 12 λ φh λ H + λ φh λ φ + 6 λ φh y t + 4 λ λ c + 2 λ λ c + 2 λ φh T r [ Y † ν Y ν ] ,β λ c = − g λ c − g λ c + 12 λ c λ Φ + 4 λ λ φh + 2 λ λ φh + λ c λ Φ + 4 λ c ,β λ φ = 3 (cid:16) λ c + 4 λ φh + λ φ (cid:17) . (8.6)The above β functions are generated using the model implementation in the code SARAH [113]. The running of λ H as in Eqn.(8.5) is influenced adversely mostly by top Yukawacoupling y t ∼ O (1) and right handed neutrino Yukawa coupling as Tr[ Y † ν Y ν ] . On the otherhand, multiparticle scalar DM couplings present in the model help in compensating thestrong negative effect from y t and Tr[ Y † ν Y ν ] . To evaluate Tr [ Y † ν Y ν ] , we employ Eq. 8.2. Asan example, let us consider magnitude of all the entries of A to be equal, say a with alldiagonal entries to be zero. Then using the best fit values of neutrino mixing angles andcosnidering the mass of lightest neutrino zero, one can find for M R = 1 TeV, Tr [ Y † ν Y ν ] canbe as large as 1 with a = 8 . [110, 114]. Equipped with all these β functions and strategy toestimate the relevant couplings present in them, let us now analyse the SM Higgs vacuumstability at high energy scale. Below we provide the stability and metastability criteria. • Stability criteria:
The stability of Higgs vacuum can be ensured by the condition λ H > at any scale. However, we have multiple scalars (SM Higgs doublet, one inert dou-blet and one gauge real singlet) in the model. Therefore we should ensure the boundedness– 29 –r stability of the entire scalar potential in any field direction. This can be guaranteed byusing the co-positivity criterion in Eqn. 3.1. Note that once λ H becomes negative the othercopositivity conditions no longer remain valid. • Metastability criteria:
On the other hand, when λ H becomes negative, theremay exist another deeper minimum other than the EW one. Then the estimate of thetunnelling probability P T of the EW vacuum to the second minimum is essential to confirmthe metastability of the Higgs vacuum. Obviously, the Universe can be in a metastable stateonly, provided the decay time of the EW vacuum is longer than the age of the Universe.The tunnelling probability is given by [4, 5], P T = T U µ B e − π | λH ( µB ) | , (8.7)where T U is the age of the Universe, µ B is the scale at which tunnelling probability ismaximized, determined from β λ H = 0 . Therefore, solving above equation, we see thatmetastable Universe requires [4, 5] : λ H ( µ B ) > − . − ln (cid:16) vµ B (cid:17) . (8.8)As noted in [4], for µ B > M P (Planck Scale, M P = 1 . × GeV), one can safely consider λ H ( µ B ) = λ H ( M P ) . One should also note, that even with meatstability of Higgs vacuum,the instability in other field direction may also occur. The conditions to avoid the possibleinstability along various field directions for λ H < are listed below [115]: ( i ) λ Φ > to avoid the unboundedness of the potential along H , A and H ± directions, ( ii ) λ > to ensure the stability along a direction between H ± and h , ( iii ) λ L > to ensure the stability along a direction between H and h , ( iv ) λ S > , to avoid unboundedness of potential along a direction between A and h , ( v ) λ φ > , otherwise the potential will be unbounded along φ direction, ( vi ) λ φh > , to ensure the stability along a direction between φ and h .Now to begin the vacuum stability analysis in the present multi-component DM sce-nario, we first choose two benchmark values of DM masses ( φ and H ) which satisfy boththe relic density and direct detection bounds. These along with corresponding values ofother relevant parameters are denoted by the set of two benchmark points, BP1 and BP2,as tabulated in Table 2. These parameters are mentioned in the caption of Table 2 for bothbenchmark points. We also show the value of electroweak parameters and µ γγ for thesetwo benchmark points in Table 3. We run the two loops RG equations for all the SMcouplings and one loop RG equations for the other relevant couplings in the model from µ = m t to M P energy scale. We use the inital boundary values of all the SM couplingsas provided in [4]. The boundary values have been evaluated at µ = m t in [4] by takingvarious threshold corrections and mismatch between top pole mass and M S renormalisedcouplings into account. – 30 – Ps m H m φ m A m H ± λ L λ φh λ c Ω H h Ω φ h BP1 330 343 333 339 0.043 0.065 0.2 0.033 0.086BP2 427 449 438 440 0.086 0.017 0.3 0.027 0.088
Table 2 : Benchmark points used to analyse EW vacuum stability in our model. Allmasses are in GeVs. The other couplings used in these benchmark points play an importantrole; they are: BP1: { λ = 0 . , λ = − . , λ = − . } , BP2: { λ = 0 . , λ = − . , λ = − . } . BPs ∆ S (10 − ) ∆ T (10 − ) µ γγ (10 − ) BP1 -12 5.1 3BP2 -9 2.5 3
Table 3 : Estimate of electroweak precision parameters and µ γγ for the two benchmarkpoints as chosen in Table 2. Figure 19 : RG running of λ H with energy scale for BP1. In left panel, we have shownthe effect of different right handed neutrinos masses, M R (indicated by different coloursand mentioned in figure inset) for a fixed top quark mass m t = 173 . GeV. The blackdotted line corresponds to the case when right handed neutrinos are absent in the scenario.In right panel, we choose a specific M R = 10 GeV and consider top mass in σ range: m t = 173 . ± . GeV. Tr [ Y † ν Y ν ] = 0 . is kept constant in both plots.In Fig. 19, we show the running of λ H for BP1 as a function of energy scale µ . Theleft panel shows running of λ H for different values of RH neutrino mass M R , consideringtop quark mass m t = 173 . GeV, Higgs mass m h = 125 . GeV and Tr [ Y † ν Y ν ] = 0 . . As itis visible that for low value of M R ∼ GeV, λ H enters into unstable region at very earlystage of its evolution (blue line in the figure). This has happened as the scalar couplings in– 31 – igure 20 : RG running of λ H with energy scale µ for different values of Tr [ Y † ν Y ν ] (shownin different colours and values are mentioned in the figure inset) for the benchmark pointsBP1 (left) and BP2 (right). Here we have chosen M R = 10 GeV for illustration. Theblack dotted line here corresponds to the case when right handed neutrinos are absent inthis scenario. β λ H (Eqn. 8.5) are not sufficiently large to counter the strong negative impact of Tr [ Y † ν Y ν ] which brings down the λ H shraply towards unstable region. On contrary, for large valueof M R ∼ GeV, the effect of Y ν in β λ H starts at a comparatively larger energy scalethan the earlier case. As a consequence, although λ H becomes negative at some highenergy scale, it stays in metastable region till M P energy scale (violate line). Green regionhere describes stable, white region describes metastable (see Eqn. 8.8) and the red regionindicates unstable solution for the potential. For comparison, we also display the evolutionof λ H (black dotted curve) in absence of RH neutrinos in the theory with the inclusion ofscalar couplings (related to DM) only. This clearly shows that in absence of RH neutrinos,EW vacuum could be absolutely stable till M P energy scale. In right panel of Fig. 19, westudy the evolution of λ H considering σ uncertainty of measured top mass m t , keeping m h = 125 . GeV, M R = 10 GeV and Tr [ Y † ν Y ν ] = 0 . fixed. It is trivial to find that withthe increase of top mass, λ H crosses zero at earlier stage in its evolution.Next we show the effect of Tr [ Y † ν Y ν ] in the RG evolution of λ H in Fig. 20 for BP1 (left) andBP2 (right). For the purpose we fix the RH neutrino mass scale M R = 10 GeV. It can beseen from left panel that large value of Tr [ Y † ν Y ν ] ∼ . brings down λ H towards the negativevalues at earlier energy scale. This is obvious from the β function of λ H as the amount ofscalar couplings for BP1 are not that effective in presence of such large value of Tr [ Y † ν Y ν ] .With comparatively lesser value of Tr [ Y † ν Y ν ] ∼ . , the EW vacuum might be in metastablestate provided other conditions ( ( i ) − ( vi ) ) are satisfied as shown in left panel of Fig. 21. Ifwe further reduce the value of Tr [ Y † ν Y ν ] ∼ . the EW vacuum might be absolutely stable.For the stability case we also show the satisfaction of all the copositivity criterias (Eqn. 3.1)in left panel of Fig. 22. This ensures the boundness of the scalar potential in any arbitaryfield direction. The analysis for BP2 turns out to be similar as observed from right panelsof Fig. 20-22. Note that due to comparatively larger values of the scalar couplings in BP2,– 32 – igure 21 : RG running of all the quartic couplings in metastability criteria for BP1 (left)and BP2 (right) to ensure the boundedness of the scalar potential in various field directionswith energy scale µ for Tr [ Y † ν Y ν ] = 0 . (left) and 0.84 (right). The choices of Tr [ Y † ν Y ν ] aredemonstrated in cyan (0.3) and in orange (0.84) in left and right panels of Fig. 20 to yieldmetastability. Figure 22 : RG running of all copositivity criteria in Eqn. 3.1 for BP1 (left) and BP2(right) to ensure the boundedness of the scalar potential in any field direction with energyscale µ for Tr [ Y † ν Y ν ] = 0 . (left) and 0.7 (right) The choices of Tr [ Y † ν Y ν ] are shown in darkblue and purple colours in left and right panels of Fig. 20 respectively to yield stability.we achieve a better results in stability point of view than BP1.Based on the inputs from above analysis, now we constrain Tr [ Y † ν Y ν ] − M R parameterspace using the stability, metastability and instability criteria (green, white and red regionsrespectively) for BP1 (left panel) and BP2 (right panel) in Fig. 23. The criteria has beenset at Planck scale ( M P ). We use α S = 0 . and m h = 125 . GeV for both the plots.The solid lines indicate the contour for m t = 173 . GeV while the dotted lines denote 2 σ uncertainty of the measured value of m t . It can be concluded from Fig. 23, that to have astable/metastable EW vacuum, smaller values of M R requires low Tr [ Y † ν Y ν ] and vice versa.One important distinction between the left and right panel figures, corresponding to two– 33 – igure 23 : Stability, metastability and instability regions plot on M R − Tr [ Y † ν Y ν ] planefor the benchmark point BP1 (left panel) and BP2 (right panel). For illustration we haveconsidered top mass variation in σ range : . ± . GeV.different benchmark points, is that BP2 has significantly larger parameter space availablefrom high scale stability. This is because of the larger values of λ , , parameters used inBP2 compared to BP1 (see Table 4 for details). Therefore, it can be concluded, that largeris the mass splitting in IDM sector, the more favourable it is from the stability point ofview. However there is an important catch to this statement, which we will illustrate next. Figure 24 : RG running of relevant coupling parameters for BP1 (left panel) and BP2(right panel). M R = 10 GeV, Tr [ Y † ν Y ν ] = 0 . and m t = 173 . GeV have been kept fixed inboth plots.In Fig. 24 we plot the running of all the individual couplings present in the set up. Wesee that (fortunately) for the two benchmark points used in the analysis, we are still okaywith the perturbative limit at the high scale, i.e. all the couplings obey the perturbativelimit, | λ i | < π, | Tr [ Y † ν Y ν ] | < π at Planck scale. However, for BP1, with M R = 10 GeV, and Tr [ Y † ν Y ν ] = 0 . as shown in the left panel yields unstable solution to EW vacuum– 34 –ith λ H running negative. On the other hand, BP2 with same choices of right handedneutrino mass and Yukawa yields a stable vacuum. Therefore, once again we find thatlarger splitting in IDM sector is more favourable for EW vacuum stability, as we have alsoinferred before from Fig. 23. But, it turns out that as larger splitting in the IDM sector alsouses larger values of λ , , , it is possible, that many of those points become non-perturbativei.e. | λ i | > π when run upto Planck scale. We will show in the next section, that this veryphenomena disallows many of the relic density and direct search allowed parameter space ofthe model. Another point to end this section is to note that our available parameter spacefrom DM constraints heavily depend on large DM-DM conversion, which naturally comesfrom large choices of the conversion coupling λ c . It is natural, that some of those cases willalso be discarded by the perturbative limit | λ c | < π , when we evaluate the validity of themodel at high scale. Finally, we come to the point where we can assimilate all the constraints together, fromDM constraints to high scale validity. In order to do that, we choose relic density and directsearch at NLO allowed parameter space of the model as discussed in Section 6 and imposethe high scale validity of the model till some energy scale µ by demanding: • Stability of the scalar potential (Eqn.(3.1)) determined by satisfying copositivity condi-tions for any energy scale µ . • Non-violation of perturbativity and unitarity of all the relevant couplings present in themodel as defined in Eqns. 3.2 and 3.3.Note that the high scale validity of the models does not depend on the structure of mass
Figure 25 : Relic density, direct search and indirect search allowed points in λ + λ + λ +2 λ λ versus λ plane (left figure) and m H ± − m H versus λ plane (right figure) at EWscale (orange points). We also find high scale validity of the model following Eqns. 3.1, 3.2and 3.3 by considering the high scale to be GeV (green), GeV (blue) and GeV (red). Right handed neutrino mass and Yukawa couplings are kept fixed at M R = 10 GeV, Tr [ Y † ν Y ν ] = 0 . with m t = 173 . GeV.hierarchy of the DM candidates ( i.e. on the sign of mass difference m H − m φ ). To study– 35 –he EW vacuum stability we demand the positivity of λ H at each energy scale from EWto high scale µ . As evident from β λ H in Eqn. 8.5, a particular combination of the scalarcouplings in the form of λ + λ + λ + 2 λ λ + λ φh determines the positivity of λ H during its running. However the factor λ φh (cid:46) . is strongly constrained from the SIDD cross section bound for m φ < GeV. Hence, we can assume safely that the factor λ + λ + λ + 2 λ λ without λ φh effectively determines the stability of Higgs vacuum inrelic density direct search and indirect search allowed parameter space. It turns out thatwhen λ H > . all other copositivity conditions for all relic and DD cross section satisfiedpoints in our model stays positive from µ = m t to µ = M P .In Fig. 25, we constrain relic density direct search and indirect search allowed points of Figure 26 : Relic density direct search and indirect search allowed points in λ − λ planefor different values of λ (left). Allowed points in the same parameter space from DM con-straints (orange), stability and perturbativity conditions following Eqns. 3.1-3.3 consideringthe high scale µ = µ EW , GeV (green), GeV (blue) and GeV (purple) (right).the model in λ + λ + λ + 2 λ λ − λ plane to additionally satisfy perturbativity andvacuum stability conditions following Eqns. 3.1-3.3. Orange points satisfy relic density, DDbounds and indirect search bound, while the green, blue and red points on top of that sat-isfy perturbativity and vacuum stability conditions upto high energy scales µ = 10 GeV, GeV and GeV respectively. This very figure essentially shows that all those pointswith either small values of λ (i.e. small m H ± − m H )are discarded due to stability of EWvacuum, while those with large λ (i.e. large m H ± − m H )are discarded by perturbativelimits of the coupling at high scale.In Fig. 26 we study the correlation between the individual scalar couplings to satisfythe DM constraints, perturbativity limits and vacuum stability criteria. In left panel ofFig. 26, we show the DM relic density and DD cross section satisfied points in λ − λ plane for different values of λ . In right panel we first identify the relevant parameter spacein the same plane which satisfy the DM constraints, the perturbativity bound and vacuumstability criteria till EW energy scale. Then we further impose perturbativity bound and– 36 – igure 27 : Relic density direct search and indirect search allowed points (orange) in m A − m H (top left), m H ± − m H (top left) and ∆ m (= m A − m H ) − ∆ M (= m H ± − m H )(bottom). We further apply stability and perturbativity conditions following Eqns. 3.1-3.3at different energy scales µ = 10 GeV (light blue), GeV (dark blue) and GeV(red).vacuum stability conditions considering the high scale as µ = 10 GeV, GeV and GeV in addition to the DM constraints. It is seen that the lower portion of the availableparameter space gets discarded by vacuum stability or high value of λ c while perturbativitybounds constrain the higher values of the couplings. In this plot also we kept M R = 10 GeV, Tr [ Y † ν Y ν ] = 0 . with m t = 173 . GeV. We must also note that with larger M R andsmaller Yukawa Tr [ Y † ν Y ν ] , we could obtain a larger available parameter space from highscale validity.In Fig. 27, we show the high scale validity of relic density and direct search allowedparameter space of the model in m H − m A (top left), m H − m H ± (top right) and ∆ m (= m A − m H ) − ∆ M (= m H ± − m H ) planes with M R = 10 GeV and Tr [ Y † ν Y ν ] = 0 . at different high energy scales µ = { , , } GeVs denoted by light blue, dark blueand red points. The orange points are corresponding to relic and direct search allowedparameter space at EW scale. We see that larger mass difference between inert higgs com-ponents, ∆ m − ∆ M which are related with quartic couplings, λ , , (see Eqn. 2.7), are– 37 –iscarded from perturbativity conditions mentioned in Eqn. 3.2. While the small mass dif-ferences between inert componets are also excluded from stability criteria of Higgs potential.Till now, while discussing the effect of stability and high scale validity of the proposedset up, we have considered fixed right handed neutrino mass, M R = 10 GeV and corre-sponding Yukawa coupling Tr [ Y † ν Y ν ] = 0 . . For sake of completeness we extend our studyfor few different values of RH neutrino mass and Tr [ Y † ν Y ν ] and find out the allowed rangesof the relevant parameters considering both the hierarchies m φ > m H and m φ ≤ m H separately. We note the corresponding results in Table 9 and Table 10 of Appendix C. Inert doublet has been an attractive DM framework, due to the possibility of collider de-tection [52, 53]. Here, we relook into the possible collider search strategies of IDM at LHCin presence of a second scalar singlet DM component. It is also worth mentioning here thatthe real scalar singlet, which interacts with SM only through Higgs portal coupling, doesnot have any promising collider signature excepting mono- X signature arising out of initialstate radiation (ISR), where X stands for W, Z, jet. Such signals are heavily submergedin SM background due to weak production cross-section of DM in relic density and directsearch allowed parameter space [51]. The charge components H + , H − of inert DM can beproduced at LHC via Drell-Yan Z and γ mediation as well as through Higgs mediation.Further decay of H ± to DM ( H ) and leptonic final states through on/off shell W ± yieldshadronically quiet opposite sign di-lepton plus missing energy (OSDL+ /E T ) , as shown inleft panel of Fig. 28. In this study, we focus on this particular signal of inert dark matteras detailed below:Signal :: OSDL + /E T ≡ (cid:96) + (cid:96) − + ( /E T ) : p p → H + H − , ( H − → (cid:96) − ν (cid:96) H ) , ( H + → (cid:96) + ν (cid:96) H ); where (cid:96) = { e, µ } . (9.1)In the right panel of Fig. 28, we show variation of charged pair ( H + H − ) production cross-section at LHC for center-of-mass energy √ s = 14 TeV as a function of m H ± = m H + ∆ M ,where ∆ M indicates the mass difference with the inert DM and serves as a very importantvariable for the signal characteristics. The plot on RHS show that production cross-sectionis decreasing with larger charged scalar mass m H ± , where we have demonstrated threefixed values of m H ± − m H = 5 , and GeV. Around m H ± ∼ m h / , there is asharpe fall of production cross-section. This is because, for m H ± ≤ m h / , there is asignificant contribution arising from Higgs production and its subsequent decay to thecharged scalar components, which otherwise turns into an off-shell propagator to yield asubdued contribution to Drell-Yan production. Following [116], a conservative bound onthe charge scalars is applied here as m H ± ≥ GeV, as indicated in the RHS plot of Fig. 28. There are other possible signatures (for example, three lepton final state) of inert DM arising from thethe combination of H ± , A production and their subsequent decays, for a detailed list see [44, 53]. – 38 – p γ/ Z/ h H − H + W − W + H lH lν l ν l Figure 28 : [Left] Feynman graph for OSDL+ /E T signature of IDM at LHC. [Right] Varia-tion of production cross-section σ pp → H + H − (in fb) with m H ± ( = m H + ∆ M ) in GeV fordifferent choices of m H ± − m H for center-of-mass energy √ s = 14 TeV at LHC. LEP limiton charged scalar is shown by the shaded region.
BPs { m H , m φ , m A , m H ± , λ L , λ φh , λ c } Ω H h Ω φ h σ effH (cm ) σ effφ (cm )BPC1 { , , , , . , . , . } . . . × − . × − BPC2 { , , , , . , . , . } . . . × − . × − BPC3 { , , , , . , . , . } . . . × − . × − BPs m H ± − m H { λ , λ , λ } { M R , Tr [ Y † ν Y ν ] } Validity Scale ( µ )BPC1
30 ( < m W ) { . , − . , − . } { , . } . × ( M pl )BPC2
51 ( < m W ) { . , − . , − . } { , . } . × ( M pl )BPC3
78 ( ∼ m W ) { . , − . , − . } { , . } ∼ Table 4 : DM masses, quartic couplings, relic densities and spin independent effective DM-neucleon cross-section of selected benchmark points for collider study. All benchmark pointschosen here have m H ± − m H < m W ± for off-shell production of W ± . The maximum scale( µ ) of Higgs vacuum satability and peturbativity in presence of right handed neutrinos arealso noted. All masses and scales are in GeV.We next choose a set of benchmark points (BPs) allowed from DM relic, direct searchconstraints as well as from Higgs invisible decay constraints for performing collider simula-tion, shown in Table 4 and Table 5. The BPs are also allowed from absolute Higgs vacuumstability and perturbativity limits in presence of right handed neutrinos, valid upto scale µ as mentioned in the tables. The benchmark points are divided into two categories: (BPC1-BPC3) in Table 4 correspond to ∆ M = m H ± − m H (cid:46) m W ± where the charged scalar, H ± decay through off-shell W ± . On the other hand, benchmark points (BPD1-BPD2) inTable 5 correspond to ∆ M = m H ± − m H > m W ± , where the charged scalar H ± decaythrough on-shell W ± . Each table (Table 4 and Table 5) consists of two parts: the first– 39 – Ps { m H , m φ , m A , m H ± , λ L , λ φh , λ c } Ω H h Ω φ h σ effH (cm ) σ effφ (cm )BPD1 { , , , , . , . , . } . . . × − . × − BPD2 { , , , , . , . , . } . . . × − . × − BPs m H ± − m H { λ , λ , λ } { M R , Tr [ Y † ν Y ν ] } Validity Scale ( µ )BPD1
106 ( > m W ) { . , − . , − . } { , . } BPD2
143 ( > m W ) { . , − . , − . } { , . } Table 5 : DM masses, quartic couplings, relic densities and spin independent effective DM-neucleon cross-section of selected benchmark points for collider study. All benchmark pointschosen here have m H ± − m H > m W ± for on-shell production of W ± . The maximum scale( µ ) of Higgs vacuum satability and peturbativity in presence of right handed neutrinos arealso noted. All masses and scales are in GeV.part contains all the relevant dark sector masses, couplings, relic density and direct searchcross-sections of both DM components. The second part demonstrates the mass difference( ∆ M = m H ± − m H ), choice of right handed neutrino mass, neutrino Yukawa and themaximum scale of validity ( µ ) of the Higgs vacuum.The simulation technique adopted here is as follows. We first implemented the modelin FeynRule [117] to generate UFO file which is required to feed into event generator
Madgraph [118]. Then these events are passed to
Pythia [119] for hadronization. All partonlevel leading order (LO) signal events and SM background events are generated in Madgraph at √ s = 14 TeV using cteq6l1 [120] parton distribution. Leptons ( (cid:96) = e, µ ) isolation, jetand unclustered event formation to mimic to the actual collider environment are performedas follows:(i) Lepton isolation: The minimum transverse momentum required to identify a lepton ( (cid:96) = e, µ ) has been kept as p T > GeV and we also require the lepton to be producedin the central region of detector followed by pseudorapidity selection as | η | < . . Twoleptons are separated from each other with minimum distance ∆ R ≥ . in η − φ plane. Toseparate leptons from jets we further imposed ∆ R ≥ . .(ii) Jet formation: For jet formation, we used cone algorithm PYCELL in built in
Pythia .All partons within a cone of ∆ R ≤ . around a jet initiator with p T > GeV is identifiedto form a jet. It is important to identify jets in our case because we require the final statesignal to be hadronically quiet i.e. to have zero jets.(iii) Unclustered Objects: All final state objects with . < p T < GeV and . < | η | < are considered as unclustered objects. Those objects neither form jets nor identified asisolated leptons and they only contribute to missing energy.The main idea is to see if the signal events rise over SM background. For that there arethree key kinematic variables where the signal and background show different sensitivity. There are several SM process which contribute to the chosen (cid:96) + (cid:96) − + ( /E T ) signal, dominant processesare: tt , W + W − , ZZ and W + W − Z . – 40 –hey are: • Missing Energy ( /E T ): The most important signature of DM being produced at col-lider. This is defined by a vector sum of transverse momentum of all the missingparticles (those are not registered in the detector); this in turn can be estimatedform the momentum imbalance in the transverse direction associated to the visibleparticles. Thus missing energy (MET) is defined as: /E T = − (cid:115) ( (cid:88) (cid:96),j p x ) + ( (cid:88) (cid:96),j p y ) , (9.2)where the sum runs over all visible objects that include the leptons, jets and theunclustered components. • Transverse Mass ( H T ): Transverse mass of an event is identified with the scalarsum of the transverse momentum of objects reconstructed in a collider event, namelylepton and jets as defined above. H T = (cid:88) (cid:96),j (cid:113) ( p x ) + ( p y ) . (9.3) • Invariant mass ( m (cid:96)(cid:96) ): Invariant mass of opposite sign dilepton hints to the parentparticle, from which the leptons have been produced and thus helps segregating signalfrom background. This is defined as: m (cid:96) + (cid:96) − = (cid:115) ( (cid:88) (cid:96) + (cid:96) − p x ) + ( (cid:88) (cid:96) + (cid:96) − p y ) + ( (cid:88) (cid:96) + (cid:96) − p z ) . (9.4)The distribution of missing energy ( /E T ), invariant mass of opposite sign dilepton ( m (cid:96) + (cid:96) − )and transverse mass ( H T ) for the BPs along with dominant SM background events areshown in Fig. 29 top left, top right and bottom panel respectively. All BPs depicted inFig. 29 correspond to m H ± − m H ≡ ∆ M < m W ± where opposite sign di-lepton areproduced from off-shell W ± mediator. All the distributions are normalised to one event.Missing energy (as well as H T ) distributions of BPs (BPC1-BPC3) show that the peak ofdistribution for the signal is on the left of SM background. This is because the benchmarkpoints are characterised by small ∆ M , where the charged scalars and inert DM have smallmass splitting. Therefore, such situations are visibly segregated from SM background byMET and H T distribution. Clearly, when ∆ M becomes m W ± (for example, BPC3) thedistribution closely mimic SM background. Therefore the signal events for this class ofbenchmark points can survive for a suitable upper /E T and H T cut while reducing SM back-grounds. It is important to take a note that OSDL events coming from ZZ backgroundnaturally peaks at m Z in m (cid:96)(cid:96) distribution. Therefore, we use invariant mass cut in the Z mass window to get rid of this background.The situation is reversed for larger splitting between the charged scalar component withDM, i.e ∆ M ≡ m H ± − m H > m W ± corresponding to BPs (BPD1-BPD2) as in Table 5.– 41 – igure 29 : Distribution of missing energy ( /E T ), invariant mass of OSDL ( m (cid:96) + (cid:96) − ) andtransverse mass ( H T ) for signal events (cid:96) + (cid:96) − + ( /E T ) and dominant SM background eventsat LHC with √ s = 14 TeV .The distributions of /E T , m (cid:96)(cid:96) and H T therefore become flatter and peak of the distributionshifts to higher value as shown in Fig. 30. In such cases the signal events for large ∆ M canbe separated from SM background at a suitable lower end cut of /E T and H T .Therefore the selection cuts used in this analysis are summarised as follows: • Invariant mass ( m (cid:96)(cid:96) ) cuts: m (cid:96)(cid:96) < ( m Z − GeV and m (cid:96)(cid:96) > ( m Z + 15) GeV. • H T cuts: – H T < when m H ± − m H < m W ± . – H T > , when m H ± − m H > m W ± . • /E T cuts: – /E T < , when m H ± − m H < m W ± . – /E T > , when m H ± − m H > m W ± .We next turn to signal and background events that survive after the selection cuts areemployed. The signal events are listed in Table 6 for BPC1-BPC3.– 42 – igure 30 : Distribution of missing energy ( /E T ), invariant mass of OSDL ( m (cid:96) + (cid:96) − ) andtransverse mass ( H T ) for signal events (cid:96) + (cid:96) − + ( /E T ) and dominant SM background eventsat LHC with √ s = 14 TeV . BPs σ OSD (fb) /E T (GeV) H T (GeV) σ OSDeff (fb) N OSDeff @ L = 10 fb − BPC1 9.16 < <
70 0 . <
40 0 . < <
70 0 . <
40 0 . < <
70 0 . <
40 0 . Table 6 : Signal cross-section for BPC1-BPC3 after the selection cuts are employed.Similarly, signal events for BPD1-BPD2 are listed in Table 7 using a lower cut on /E T and H T . The signal cross-section and event numbers for this class of points are much smallerdue to the fact that the charged scalar masses are on higher side as for all of the cases ∆ m > m W independent of DM masses. However, the SM background events get even– 43 – Ps σ OSD (fb) /E T (GeV) H T (GeV) σ OSDeff (fb) N OSDeff @ L = 10 fb − BPD1 . > > .
75 75 > .
48 48 > > .
23 23 > .
21 21
BPD2 . > > . > . > > .
34 34 > .
29 29
Table 7 : Signal cross-section for BPD1-BPD2 after the selection cuts are employed.more suppressed with the set of /E T , H T cuts. The SM background cross-section and eventnumbers after cut flow is mentioned in Table 8. Therefore, in spite of smaller signal eventsin this region of parameter space with BPD benchmark points, the discovery potential ofthe signal requires similar luminosity to that of BPC cases.Finally we present the discovery reach of the signal events in terms of significance σ = S √ S + B ,where S denotes signal events and B denotes SM background events in terms of luminos-ity. This is shown in Fig. 31. This shows that the benchmark points that characterisethe two component DM framework, can yield a visible signature at high luminosity with L ∼
500 fb − depending on the charged scalar mass and its splitting with DM for the case m H ± − m H > m W ± . Figure 31 : Signal significance of some select benchmark points at LHC for √ s = 14 TeV,in terms of Luminosity (fb − ). 3 σ and 5 σ lines are shown. Left: Points with ∆ M < m W ;Right: Points with ∆ M > m W . – 44 – M Bkg. σ OSD (fb) /E T (GeV) H T (GeV) σ OSDeff (fb) N OSDeff @ L = 10 fb − < < < t ¯ t . × > >
150 10.64 1064 >
200 4.40 440 > >
150 1.47 147 >
200 1.10 110 < < < W + W − . × > >
150 7.72 772 >
200 4.97 4.97 > >
150 1.23 123 >
200 1.18 118 < < < Z Z . × > >
150 0.18 18 >
200 0.05 5 > >
150 0.09 9 >
200 0.04 4 < < < W + W − Z . > >
150 0.06 6 >
200 0.04 4 > >
150 0.03 3 >
200 0.02 2
Table 8 : Dominant SM background contribution to (cid:96) + (cid:96) − + ( /E T ) signal events for √ s =14 TeV at LHC. The effective number of final state background events with different /E T , H T and m (cid:96)(cid:96) cuts are tabulated for luminosity L = 100 f b − . To incorporate the Next-to-Leading order (NLO) cross section of SM background we have used appropriate K-factors[121].
10 Summary and Conclusions
We have studied a two component scalar DM model in presence of right handed neutrinosthat address neutrino mass generation through type I seesaw. The DM components are (i) a– 45 –inglet scalar and (ii) an inert scalar doublet, both studied extensively as single componentDM in literature. We show that the presence of second component enlarges the availableparameter space significantly considering relic density, direct search and indirect searchconstraints. In particular, the inert scalar DM will now be allowed in the so called ‘desertregion’: { m W − } GeV. Also for singlet scalar, we can now revive it below TeV, whichis otherwise discarded (except Higgs resonance) from direct search in single componentframework. The results obtained for DM analysis crucially depends on DM-DM conversion,which have been demonstrated in details.We also study the high scale perturbativity and vacuum stability of the Higgs potentialby analysing two loop RGE β functions. This in turn puts further constraints on theavailable DM parameter space of the model. One of the important conclusions obtainedare that the mass splitting of the charged scalar component to the corresponding DMcomponent of inert doublet is crucially tamed depending on the absolute stability scale ofthe scalar potential, coming from the perturbativity constraint on the quartic and Yukawacoupling. For example, we find : • Validity scale ( µ ) ∼ intermediate scale ( GeV): ∆ M = m H ± − m H ∼ { − } GeV, • Validity scale ( µ ) ∼ Planck scale ( GeV): ∆ M = m H ± − m H ∼ { − } GeV,with RH neutrino mass M R = 10 GeV and Yukawa Tr [ Y † ν Y ν ] = 0 . . The presence ofRHNs in the model not only helps us addressing the neutrino masses but also controls thehigh scale validity of the model parameters, for example, low ∆ M regions. This is how theneutrino and dark sector constraints affect each other.Inert Higgs having charged scalars have collider detectability. We point out that thecollider search prospect of the charged components are not only limited to low DM masses( < m W ), but is open to a larger mass range in presence of the second DM component, evenafter taking the high scale validity constraints. We exemplified this at LHC for hadronicallyquiet dilepton channel with missing energy, where ∆ M , turns out to be a crucial kinematicparameter, constrained from DM, high scale validity and neutrino sector. At LHC, due toits t ¯ t background, high ∆ M regions can be segregated from SM background more efficientlyat the cost of small production cross section. Low ∆ M regions are more affected by SMbackground, although the signal cross-section is high. Therefore the model can be probedwithin High Luminosity reach of LHC. On the other hand, e + e − annihilation have betterpossibility to explore low ∆ M regions absent t ¯ t background. Here, we would like to commentthat there are several studies that have been done in this direction, but the high scale validityconstraint may alter the conclusion significantly as we demonstrate.Finally, we would like to mention that the analysis performed here, although focus ona specific model set up, but there are some generic conclusions that can be borrowed. Forexample, if the two DM components have sufficient interaction in between, the availableparameter space will be enlarged significantly from both relic density and direct search.The conversion of one DM into the other may also affect the collider outcome of the DMsignificantly. It is obvious that richer signal is obtained when we have larger multiplets in– 46 –ark sector (as scalar doublet produces two lepton final state in the analysis). It is alsopossible that the dark sector and neutrino sector although may not inherit a common origin,the high scale validity of the model can bring them together. Acknowledgments :
PG would like to thank Basabendu Barman and Rishav Roshanfor useful discussions and also acknowledges MHRD, Government of India for researchfellowship. SB would like to acknowledge the DST-INSPIRE research grant IFA13-PH-57at IIT Guwahati.
A Tree Level Unitarity Constraints
In this section, we perform the analysis to find the tree level unitarity limits on quarticcouplings present in our model at high energy. The scattering amplitude for any 2 → M → = 16 π ∞ (cid:88) l =0 a l (2 l + 1) P l (cos θ ) , (A.1)where θ is the scattering angle and P l (cos θ ) is the Legendre polynomial of order l . In thehigh-energy limit, only the s-wave ( l = 0 ) partial amplitude a will determine the leadingenergy dependence of the scattering processes. The unitarity constraint turns out to be[72, 73, 122] Re | a | < . (A.2)The constraint in Eqn.(A.2) can be further converted to a bound on the scattering amplitude M [72, 73, 122]: |M| < π. (A.3)In the present set up, we have multiple possible 2 → M → i ; j = M i → j ) by considering all possible two particle states.Finally, we calculate the eigenvalues of M and employ the bound as in Eqn. (A.3). In thehigh-energy limit, we express the SM Higgs doublet as H T = (cid:16) w + h + iz √ (cid:17) . Then, the scalarpotential in Eqn.(2.2) gives rise to 19 neutral combinations of two particle states: w + w − , H + H − , hh √ , zz √ , H H √ , A A √ , φφ √ , h z, H A , w + H − , H + w − ,h H , h A , z H , z A , h φ, z φ, H φ, A φ . (A.4)and 10 singly charged two-particle states: h w + , z w + , H H + , A H + , h H + , z H + , H w + , A w + , φ w + , φ H + . (A.5)– 47 –herefore, we can write the scattering amplitude matrix (M) in block-diagonal form bydecomposing it into a neutral ( N C ) and singly charged( SC ) sector as M = (cid:32) M NC × M SC × (cid:33) . (A.6)where the sub-matrices are given by M NC × = ( M NC ) × M NC ) × M NC ) ×
00 0 0 ( M NC ) × (A.7)with M NC = λ H λ + λ + λ √ λ H √ λ H λ √ λ √ λφh √ λ + λ + λ λ Φ λ √ λ √ √ λ Φ √ λ Φ λc √ √ λ H λ √ λ H λ H (cid:16) λ + λ + λ (cid:17) (cid:16) λ + λ + λ (cid:17) λφh √ λ H λ √ λ H λ H (cid:16) λ + λ + λ (cid:17) (cid:16) λ + λ + λ (cid:17) λφh λ √ √ λ Φ (cid:16) λ + λ + λ (cid:17) (cid:16) λ + λ + λ (cid:17) λ Φ λ Φ λc λ √ √ λ Φ (cid:16) λ + λ + λ (cid:17) (cid:16) λ + λ + λ (cid:17) λ Φ λ Φ λc λφh √ λc √ λφh λφh λc λc λφ , (A.8) M NC = λ + λ + λ λ + λ iλ + iλ − iλ − iλ λ + λ λ + λ + λ λ + λ − iλ − iλ iλ + iλ λ + λ λ + λ λ + λ (cid:16) λ + λ + λ (cid:17) − iλ − iλ iλ + iλ (cid:16) λ + λ + λ (cid:17) iλ + iλ − iλ − iλ (cid:16) λ + λ + λ (cid:17) λ + λ λ + λ (cid:16) λ + λ + λ (cid:17) , (A.9) M NC = (cid:16) λ H
00 2 λ Φ (cid:17) , M NC = λ φh λ φh λ c
00 0 0 λ c . (A.10)and M SC = λ H λ + λ iλ + iλ λ H − iλ − iλ λ + λ λ + λ iλ + iλ λ Φ − iλ − iλ λ + λ λ Φ λ λ + λ − iλ − iλ λ iλ + iλ λ + λ λ + λ − iλ − iλ λ iλ + iλ λ + λ λ λ φh
00 0 0 0 0 0 0 0 0 λ c (A.11)– 48 –fter determining the eigenvalues of Eqn.(A.6) we conclude that tree level unitarity con-straints in this set up are following: | λ H | < π, | λ Φ | < π, | λ c | < π, | λ φh | < π, | λ | < π, | λ + 2( λ + λ ) | < π | λ + λ + λ | < π, | λ − λ − λ | < π, | ( λ Φ + λ H ) ± (cid:112) ( λ + λ ) + ( λ H − λ Φ ) | < π, and | x , , | < π (A.12)where x , , be the roots of following cubic equation x + x ( − λ H − λ Φ − λ φ ) + x ( − λ − λ λ − λ λ − λ − λ λ − λ − λ c + 144 λ H λ Φ + 12 λ H λ φ + 12 λ Φ λ φ − λ φh ) + 16 λ λ φ + 16 λ λ λ φ + 16 λ λ λ φ − λ λ c λ φh + 4 λ λ φ + 8 λ λ λ φ − λ λ c λ φh + 4 λ λ φ − λ λ c λ φh + 48 λ c λ H − λ H λ Φ λ φ + 48 λ Φ λ φh = 0 (A.13) B High Scale Validity of Single component DM models
Here we show the allowed parameter space for single component DM when the vacuumstability conditions are taken into account in presence of RH neutrinos. In left panel ofFig. 32 the parameter space for scalar singlet DM ( φ ) is shown considering M R = 10 GeV and Tr [ Y † ν Y ν ] = 0 . , while in right panel we present the same for IDM ( H ) with M R = 10 GeV and Tr [ Y † ν Y ν ] = 0 . . We see that for φ , the DM mass is allowed beyond 900GeV considering the absolute stability of the EW vacuum upto GeV [93]. However forIDM, we notice that the absolute EW vacuum stability can be extended even upto Planckscale (with similar Yukawa) due to the presence of several scalar degrees of freedom.
Figure 32 : Parameter space scan for (left) scalar singlet DM and (right) IDM cosnideringsatisfaction of relic density bound, direct detection cross section limit, indirect cross-sectionlimit and high scale validity in presence of RH neutrinos.– 49 –
Available parameters of the model from high scale validity for differentchoices of RH Neutrino mass and Yukawa coupling
Here, we would like to show the available parameter space of the two component DM modelwith RH neutrinos, viable from high scale validity constraint after choosing different possibleRH neutrino mass and Neutrino Yukawa coupling together with relic density, direct searchand indirect search bound. In the main text, we elaborated only the case of Tr [ Y † ν Y ν ] = 0 . with RH neutrino mass of GeV. We show different possibilities in two tables, in Table9, we depict the case of m φ > m H and in Table 10, the reverse hierarchy m φ ≤ m H ispresented. m φ > m H (in GeV) RH Neutrinos Relic + DD (XENON 1T)+ ID (Fermi LAT+MAGIC) + Stability + PerturbativityRelic + DD (XENON 1T)+ID (Fermi LAT+MAGIC) M R Tr [ Y † ν Y ν ] µ = 10 GeV µ = 10 GeV (GUT) µ = 10 GeV( M pl ) m H ∼ { − } m φ ∼ { − } ∆ m ≡ m A − m H ∼ { − } ∆ M ≡ m H ± − m H ∼ { − } λ c ∼ { . − . } λ L ∼ { . − . } λ φh ∼ { . − . } GeV 0.1 m H ∼ { − } m φ ∼ { − } ∆ m ∼ { − } ∆ M ∼ { − } λ c ∼ { . − . } λ L ∼ { . − . } λ φh ∼ { . − . } m H ∼ { − } m φ ∼ { − } ∆ m ∼ { − } ∆ M ∼ { − } λ c ∼ { . − . } λ L ∼ { . − . } λ φh ∼ { . − . } m H ∼ { − } m φ ∼ { − } ∆ m ∼ { − } ∆ M ∼ { − } λ c ∼ { . − . } λ L ∼ { . − . } λ φh ∼ { . − . } m H ∼ { − } m φ ∼ { − } ∆ m ∼ { − } ∆ M ∼ { − } λ c ∼ { . − . } λ L ∼ { . − . } λ φh ∼ { . − . } No parameter spaceavialable No parameter spaceavialable GeV 0.1 m H ∼ { − } m φ ∼ { − } ∆ m ∼ { − } ∆ M ∼ { − } λ c ∼ { . − . } λ L ∼ { . − . } λ φh ∼ { . − . } m H ∼ { − } m φ ∼ { − } ∆ m ∼ { − } ∆ M ∼ { − } λ c ∼ { . − . } λ L ∼ { . − . } λ φh ∼ { . − . } m H ∼ { − } m φ ∼ { − } ∆ m ∼ { − } ∆ M ∼ { − } λ c ∼ { . − . } λ L ∼ { . − . } λ φh ∼ { . − . } m H ∼ { − } m φ ∼ { − } ∆ m ∼ { − } ∆ M ∼ { − } λ c ∼ { . − . } λ L ∼ { . − . } λ φh ∼ { . − . } m H ∼ { − } m φ ∼ { − } ∆ m ∼ { − } ∆ M ∼ { − } λ c ∼ { . − . } λ L ∼ { . − . } λ φh ∼ { . − . } No parameter spaceavialable GeV 0.1 No effective contributionfromRH Neutrinos m H ∼ { − } m φ ∼ { − } ∆ m ∼ { − } ∆ M ∼ { − } λ c ∼ { . − . } λ L ∼ { . − . } λ φh ∼ { . − . } m H ∼ { − } m φ ∼ { − } ∆ m ∼ { − } ∆ M ∼ { − } λ c ∼ { . − . } λ L ∼ { . − . } λ φh ∼ { . − . } m H ∼ { − } m φ ∼ { − } ∆ m ∼ { − } ∆ M ∼ { − } λ c ∼ { . − . } λ L ∼ { . − . } λ φh ∼ { . − . } m H ∼ { − } m φ ∼ { − } ∆ m ∼ { − } ∆ M ∼ { − } λ c ∼ { . − . } λ L ∼ { . − . } λ φh ∼ { . − . } Table 9 : Allowed ranges of relevant parameters considering m φ > m H for different valuesof RH neutrino mass and Tr[ Y † ν Y ν ] . – 50 – φ ≤ m H (in GeV) RH Neutrinos Relic + DD (XENON 1T) +ID (Fermi LAT+MAGIC)+ Stability + PerturbativityRelic + DD (XENON 1T)+ID (Fermi LAT+MAGIC) M R Tr [ Y † ν Y ν ] µ = 10 GeV µ = 10 GeV (GUT) µ = 10 GeV( M pl ) m H ∼ { − } m φ ∼ { − } ∆ m ≡ m A − m H ∼ { − } ∆ M ≡ m H ± − m H ∼ { − } λ c ∼ { . − . } λ L ∼ { . − . } λ φh ∼ { . − . } GeV 0.1 m H ∼ { − } m φ ∼ { − } ∆ m ∼ { − } ∆ M ∼ { − } λ c ∼ { . − . } λ L ∼ { . − . } λ φh ∼ { . − . } m H ∼ { − } m φ ∼ { − } ∆ m ∼ { − } ∆ M ∼ { − } λ c ∼ { . − . } λ L ∼ { . − . } λ φh ∼ { . − . } m H ∼ { − } m φ ∼ { − } ∆ m ∼ { − } ∆ M ∼ { − } λ c ∼ { . − . } λ L ∼ { . − . } λ φh ∼ { . − . } m H ∼ { − } m φ ∼ { − } ∆ m ∼ { − } ∆ M ∼ { − } λ c ∼ { . − . } λ L ∼ { . − . } λ φh ∼ { . − . } No parameter spaceavailable No parameter spaceavialable GeV 0.1 m H ∼ { − } m φ ∼ { − } ∆ m ∼ { − } ∆ M ∼ { − } λ c ∼ { . − . } λ L ∼ { . − . } λ φh ∼ { . − . } m H ∼ { − } m φ ∼ { − } ∆ m ∼ { − } ∆ M ∼ { − } λ c ∼ { . − . } λ L ∼ { . − . } λ φh ∼ { . − . } m H ∼ { − } m φ ∼ { − } ∆ m ∼ { − } ∆ M ∼ { − } λ c ∼ { . − . } λ L ∼ { . − . } λ φh ∼ { . − . } m H ∼ { − } m φ ∼ { − } ∆ m ∼ { − } ∆ M ∼ { − } λ c ∼ { . − . } λ L ∼ { . − . } λ φh ∼ { . − . } No parameter spaceavailable No parameter spaceavailable GeV 0.1 No effective contributionfromRH Neutrinos m H ∼ { − } m φ ∼ { − } ∆ m ∼ { − } ∆ M ∼ { − } λ c ∼ { . − . } λ L ∼ { . − . } λ φh ∼ { . − . } m H ∼ { − } m φ ∼ { − } ∆ m ∼ { − } ∆ M ∼ { − } λ c ∼ { . − . } λ L ∼ { . − . } λ φh ∼ { . − . } Table 10 : Allowed ranges of relevant parameters considering m φ ≤ m H for different valuesof RH neutrino mass and Tr[ Y † ν Y ν ] . References [1]
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