Singlet-Doublet Fermionic Dark Matter, Neutrino Mass and Collider Signatures
SSinglet-Doublet Fermionic Dark Matter, Neutrino Mass and Collider Signatures
Subhaditya Bhattacharya ∗ Department of Physics, Indian Institute of Technology Guwahati, North Guwahati, Assam- 781039, India
Nirakar Sahoo † and Narendra Sahu ‡ Department of Physics, Indian Institute of Technology,Hyderabad, Kandi, Sangareddy, 502285, Telangana, India
We propose a minimal extension of the standard model (SM) by including a scalar triplet withhypercharge 2 and two vector-like leptons: one doublet and a singlet, to explain simultaneouslythe non-zero neutrino mass and dark matter (DM) content of the Universe. The DM emerges outas a mixture of the neutral component of vector-like lepton doublet and singlet, being odd undera discrete Z symmetry. After electroweak symmetry breaking the triplet scalar gets an inducedvev, which give Majorana masses not only to the light neutrinos but also to the DM. Due to theMajorana mass of DM, the Z mediated elastic scattering with nucleon is forbidden. However, theHiggs mediated direct detection cross-section of the DM gives an excellent opportunity to probe itat Xenon-1T. The DM can not be detected at collider. However, the charged partner of the DM(often next-to-lightest stable particle) can give large displaced vertex signature at the Large HadronCollider (LHC). PACS numbers: 98.80.Cq , 12.60.Fr ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] a r X i v : . [ h e p - ph ] J u l I. INTRODUCTION
Astrophysical evidences like galaxy rotation curves, gravitational lensing, large scale structure of the universe andanisotropies in Cosmic Microwave Background Radiation (CMBR) hint towards the existence of an unknown form ofnon-luminous matter, called dark matter (DM) in our Universe [1, 2]. However they imply only about the gravitationalproperty of DM, whose relic abundance is precisely measured by the satellite borne experiments such as WMAP [3]and PLANCK [4] to be Ω DM h = 0 . ± . LLHH/
Λ, where L and H arethe lepton and Higgs doublet of the SM and Λ is the scale of new physics. After electroweak (EW) symmetry breakingthe neutrino mass is given by M ν = (cid:104) H (cid:105) / Λ. Thus for tiny neutrino mass M ν ∼ . ∼ GeV when the involved couplings are order 1. However, Λ can be reduced down to TeVscales if the couplings are assumed to be smaller.In an attempt to bring dark matter and neutrino mass mechanisms under one umbrella (for some earlier attempts,see [11–19] ), we consider a minimal type-II seesaw extension of the SM by adding a TeV scale triplet scalar ∆ withhypercharge 2 and introduce two additional vector-like leptons: one doublet N ≡ ( N , N − ) T and a singlet χ . A Z symmetry is also imposed under which N and χ are odd while all other fields are even. As a result the DM emergesout to be a mixed state of singlet and neutral component of the doublet vector-like leptons. Such DM frameworks havebeen discussed earlier; see for example, refs. [20–31]. However, the presence of the triplet adds to some interestingDM phenomenology as we will discuss in this paper. Since the scalar triplet can be light, it contributes to therelic abundance of DM through s-channel resonance on top of Z and H mediation. Moreover, it relaxes the strongconstraints coming from direct detection.The triplet scalar not only couples to the SM lepton and Higgs doublets, but also to the additional vector-likelepton doublet N . The Majorana couplings of ∆ with N , L and H is then be given by f N ∆ N N + f L ∆ LL + µ ∆ † HH .Note that if the triplet is heavier than the DM and leptons, it can be integrated out and hence effectively generatingthe dimension five operators: (cid:18) LLHH
Λ +
N N HH Λ (cid:19) , where Λ ∼ M ∆ . After EW symmetry breaking ∆ acquires an induced vacuum expectation value (vev) of O (1) GeVwhich in turn give Majorana masses to light neutrinos as well as to N . Since N is a vector-like Dirac fermion,it can have a Dirac mass too. As a result N splits up into two pseudo-Dirac fermions, with a mass splitting ofsub-GeV order, whose elastic scattering with the nucleon mediated by Z -boson is forbidden. This feature of themodel leads to a survival of larger region of parameter space from direct search constraints given by the latest datafrom Xenon-100 [32] and LUX [33]. On the other hand, the Higgs mediated elastic scattering of the DM with thenucleon gives an excellent opportunity to detect it at future direct search experiments such as XENON1T [34]. It isharder to see the signature of only DM production at collider as they need to recoil against an ISR jet for missingenergy. However, the charged partner of the DM (which is next-to-lightest stable particle) can be produced copiouslywhich eventually decays to DM giving rise to leptons and missing energy. More interestingly, the charged companioncan also give large displaced vertex signature as we will elaborate.The paper is arranged as follows. In section II, we discuss about model and formalism, mixing in fermionic andin scalar sector. In section III, we explain non-zero neutrino mass in a type II see-saw scenario while section IVis devoted to illustrate the pseudo-Dirac nature of DM. The relic abundance of DM is obtained in section V. Theinelastic scattering of DM with the nucleus for direct search is presented in section VI. Section VII is devoted for directdetection of DM through elastic scattering and limits on model parameter space. The displaced vertex signatureof the charged partner of DM is discussed in sec VIII. With a summary of the analysis, we finally conclude in section IX. II. THE MODEL
As already been stated in the introduction, we extend the standard model (SM) by introducing two vector likefermions N T = ( N , N − ) (1,2,-1) and χ (1,1,0) and a scalar triplet ∆ (1,3,2), where the numbers inside the parenthesisare quantum numbers under the SM gauge group SU (3) c × SU (2) L × U (1) Y . A Z symmetry is imposed under which χ and N are odd, while other fields are even. The relevant Lagrangian involving the additional fields is given by: L new = N (cid:26)(cid:26) DN + χ (cid:1) ∂χ + ( D µ ∆) † ( D µ ∆) + M N N N + M χ χ χ + L yuk − V (∆ , H ) , (1)where D µ is the covariant derivative involving SU (2) ( W µ ) and U (1) Y ( B µ ) gauge bosons and is given by : D µ = ∂ µ − i g τ.W µ − ig (cid:48) Y B µ . The scalar potential involving SM doublet ( H ) and triplet (∆) in Eq. (1) is given by V (∆ , H ) = − µ H H † H + λ H ( H † H ) + µ (∆ † ∆) + λ ∆ (∆ † ∆) + λ H ∆ ( H † H )(∆ † ∆) + 12 (cid:2) µ ∆ † HH + h . c . (cid:3) , (2)where ∆ in matrix form is ∆ = (cid:32) ∆ + √ ∆ ++ ∆ − ∆ + √ (cid:33) . (3)We assume that µ is positive. So it doesn’t acquire a vacuum expectation value (vev). But it gets an induced vevafter EW phase transition. The vev of ∆ is given by (cid:104) ∆ (cid:105) ≡ u ∆ ≈ − µv √ µ + λ H ∆ v /
2) (4)where v is the vev of Higgs field and its value is 174 GeV.The Yukawa interaction in Eq. (1) is given by: L yuk = 1 √ (cid:2) ( f L ) αβ L cα iτ ∆ L β + f N N c iτ ∆ N + h . c (cid:3) + (cid:104) Y N (cid:101) Hχ + h . c . (cid:105) , (5)where L is the SM lepton doublet and α, β denote family indices. The Yukawa interactions importantly inherit thesource of neutrino masses (terms in first square bracket) and DM-SM interactions (terms in second square bracket). A. Singlet-doublet fermion mixing
After electroweak phase transition vev of Higgs field introduces a mixing between N and χ . The mass matrix isgiven by M = M χ m D m D M N (6)where m D = Y v . Diagonalizing the above mass matrix we get two mass eigenvalues: M ≈ M χ − m D M N − M χ M ≈ M N + m D M N − M χ (7)where we have assumed m D << M N , M χ . The corresponding mass eigenstates are given by: N = cos θχ + sin θN N = cos θN − sin θχ , (8)where the mixing angle (in small mixing limit) is given by:tan 2 θ = 2 m D M N − M χ . (9)Due to the imposed Z symmetry, the lightest odd particle remains stable. We choose N to be the lightest and hencebecomes a viable candidate for DM. The next-to-lightest Z odd particle is charged lepton N ± whose mass in termsof M , M and mixing angle θ is given by: M ± = M sin θ + M cos θ (cid:39) M N . (10)From Eq. 9, we see that Y and sin θ are not two independent parameters. They are related by: Y = ∆ M sin 2 θ v , (11)with ∆ M = M − M . We use sin θ as an independent parameter in our analysis. We will see that the mixing angleplays a vital role in the DM phenomenology. In particular, the relic abundance of DM gives an upper bound on thesinglet-doublet mixing angle to be sin θ (cid:46) .
4. For larger mixing angle the relic abundance is less than the observedvalue due to large annihilation cross-sections in almost all parameter space. We also found that a lower bound onsin θ coming from the decay of N and N − after they freeze out from the thermal bath. In principle these particlescan decay on, before or after the DM ( N ) freezes out depending on the mixing angle. In the worst case, N and N − have to decay before the onset of Big-Bang nucleosynthesis. In that case, the lower bound on the mixing angleis very much relaxed and the out-of-equilibrium decay of N and N − will produce an additional abundance of DM.Therefore, in what follows, we demand that N and N − decay on or before the freeze out of DM ( N ). As a resultwe get a stronger lower bound on sin θ , which of course depends on their masses.If the mass splitting between N − and N is larger than W ± -boson mass, then N − decay preferably through thetwo body process: N − → N + W − . However, if the mass splitting between N − and N is less than W ± -boson massthen N − decay through the three body process: N − → N (cid:96) − ν (cid:96) . For the latter case, we get a stronger lower boundon the mixing angle than the former. The three body decay width of N − is given by [20]:Γ = G F sin θ π M N I (12)where G F is the Fermi coupling constant and I is given as: I = 14 λ / (1 , a , b ) F ( a, b ) + 6 F ( a, b ) ln (cid:18) a a − b − λ / (1 , a , b ) (cid:19) . (13)In the above Equation F ( a, b ) and F ( a, b ) are two polynomials of a = M /M N and b = m (cid:96) /M N , where m (cid:96) is thecharged lepton mass. Up to O ( b ), these two polynomials are given by F ( a, b ) = (cid:0) a − a − a (1 + b ) + 10 a ( b −
2) + a (12 b −
7) + (3 b − (cid:1) F ( a, b ) = (cid:0) a + a + a (1 − b ) (cid:1) . (14)In Eq. (13), λ / = √ a + b − a − b − a b defines the phase space. In the limit b = m (cid:96) /M N → − a =∆ M/M N , λ / goes to zero and hence I →
0. The life time of N − is then given by τ ≡ Γ − . We take the freeze outtemperature of DM to be T f = M /
20. Since the DM freezes out during radiation dominated era, the correspondingtime of DM freeze-out is given by : t f = 0 . g − / (cid:63) m pl T f , (15)where g (cid:63) is the effective massless degrees of freedom at a temperature T f and m pl is the Planck mass. Demandingthat N − should decay before the DM freezes out (i.e. τ (cid:46) t f ) we getsin θ (cid:38) . × − (cid:18) . × − I (cid:19) / (cid:18) M N (cid:19) / (cid:16) g (cid:63) . (cid:17) / (cid:18) M (cid:19) . (16)Notice that the lower bound on the mixing angle depends on the mass of N − and N . For a typical value of M N = 200GeV, M = 180 GeV, we get sin θ (cid:38) . × − . Since τ is inversely proportional to M N , larger the mass, smallerwill be the lower bound on the mixing angle. We will come back to this issue while calculating the relic abundanceof DM in section V. B. Doublet-triplet scalar mixing
In the scalar sector, the model constitutes an usual Higgs doublet and an additional triplet. The quantum fluctua-tions around the vacuum is given as: H = 1 √ v + h + iξ ) , ∆ = 1 √ u ∆ + δ + iη ) (17)The mass matrix is given as : M sc = M H µv/ µv/ M (18)where M = µ + λ H ∆ v /
2. The two neutral Higgs fields (CP - even) mass eigenstates are given by H = cos θ h + sin θ δ , H = − sin θ h + cos θ δ (19)where H is the standard model like Higgs and H is the triplet like Higgs. The mixing angle is given bytan 2 θ = µv ( M − M H ) . (20)The corresponding mass eigenvalues are M H (SM Higgs like) and M H (triplet like) and are given as : M H ≈ M H − ( µv/ M − M H M H ≈ M + ( µv/ M − M H . (21)Since the addition of a scalar triplet can modify the ρ parameter, which is not differing from SM value: ρ =1 . ± . u ∆ as: u ∆ ≤ . . (22)For different values of M ∆ we have shown µ as a function of sin θ in Fig. (1). Here we see that smaller is the tripletscalar mass, the smaller is the dependence on mixing angle sin θ . θ μ ( G e V ) FIG. 1: Contours of different values of M ∆ (in GeV) in the plane of µ versus sin θ . From Eqs.(20), (4) and (22) we see that there exist an upper bound on the mixing anglesin θ < . (cid:18) v (cid:19) − . ( M H / ( M ∆ / . (23)We also get a constraint on sin θ from SM Higgs phenomenology, since the mixing can change the strength of theHiggs coupling to different SM particles. See for example [36, 37], in which the global fit yields a constraint on mixingangle sin θ (cid:46) .
5, which is much larger than the constraint obtained using ρ parameter. We may also get a constrainton sin θ from the decay of SM Higgs to different channels. For example, let us take the decay of H to τ leptons.The decay width is given by: Γ = M H π m τ v (cid:18) − m τ M H (cid:19) / (1 − sin θ ) (24)Comparing with the experimental branching fraction Br ( H → τ τ ) = 6 . × − , we found that (sin θ ) max = 0 . W + W −∗ , ZZ ∗ , which are much precisely measured at LHC. For example, if we choose Higgs decay to W + W −∗ state, the observed branching fraction is Br ( H → W + W −∗ ) : 2 . × − . In order to obtain a limit on thedoublet-triplet mixing angle sin θ , we need to calculate the decay width of this process process as given in [38] :Γ H → W W ∗ → W f ¯ f (cid:48) = 3 g M H π ( g sin θ u ∆ / (4 M w ) − cos θ ) F ( x ) , (25)where F ( x ) = −| − x | (cid:18) x −
132 + 1 x (cid:19) + 3(1 − x + 4 x ) | Ln ( x ) | + 3(1 − x + 20 x ) |√ x − | arccos (cid:20) x − x (cid:21) , with x = M W /M H . In the small mixing limit sin θ →
0, the decay reproduces same branching ratio as that of theSM prediction. However, as we increase the value of the mixing angle, the branching ratio to this particular final statereduces due to larger triplet contributions. For example, with sin θ = 0 . , . , . Br ( H → W W ∗ ) is changedby 0 . , . , .
04% respectively from the central value. Hence, in a conservative limit, if we take sin θ ∼ .
05 orsmaller, it is consistent with the experimental observation of Higgs decay to
W W ∗ final state.Thus we see that the bound obtained on the mixing angle from Higgs decay depends on the final state that wechoose, but is less constraining than that from the ρ parameter. Therefore, we will use the constraint on the mixingangle, obtained from ρ parameter, while calculating the DM-nucleon elastic scattering in section (VII). Since thedoublet-triplet scalar mixing is found to be small, we assume that the flavour eigenstates are the mass eigenstatesand treat M H = M H , M H = M ∆ throughout the calculation.We also assume for simplicity that there is no mixing between the neutral CP-odd states and in the charged scalarstates, so that ξ is absorbed by the gauge bosons after spontaneous symmetry breaking in unitary gauge and thecharged triplet scalar fields remain as mass eigen states. III. NON ZERO NEUTRINO MASS
The coupling of scalar triplet ∆ to SM lepton and Higgs doublets combinely break the lepton number by two unitsas given in Eq. (5). As a result the ∆ L α L β coupling yields Majorana masses to three flavors of active neutrinos as[8]: ( M ν ) αβ = √ f L ) αβ (cid:104) ∆ (cid:105) ≈ ( f L ) αβ − µv √ M . (26)Taking µ (cid:39) M ∆ (cid:39) O (10 ) GeV, we can explain neutrino masses of order 0 . f L (cid:39) M ∆ can be brought down to TeV scales by taking the smaller couplings.To get the neutrino mass eigen values, the above mass matrix can be diagonalised by the usual U P MNS matrix as : M ν = U PMNS M diagν U T PMNS , (27)where U P MNS is given by U P MNS = c c s c s e − iδ − s c − c s s e iδ c c − s s s e iδ s c s s − c c s e iδ − c s − s c s e iδ c c .U ph , (28)with c ij , s ij stand for cos θ ij and sin θ ij respectively and U ph is given by U ph = Diag (cid:0) e − iγ , e − iγ , (cid:1) . (29)Where γ , γ are two Majorana phases. The diagonal matrix M diagν = Diag ( m , m , m ) with diagonal entries are themass eigen values for the neutrinos. The current neutrino oscillation data at 3 σ confidence level give the constrainton mixing angles [35] :0 . < sin θ < . , . < sin θ < . , . < sin θ < . δ as well as the Majorana phases. Althoughthe absolute mass of neutrinos is not measured yet, the mass square differences have already been measured to a gooddegree of accuracy : ∆ m ≡ m − m = (6 . − . × − eV | ∆ m | ≡ | m − m | = (2 . − . × − eV (31)One of the main issues of neutrino physics lies in the sign of the atmospheric mass square difference | ∆ m | ≡| m − m | , which is still unknown. This yields two possibilities: normal hierarchy (NH) ( m < m < m ) or invertedhierarchy (IH) ( m < m < m ). Another possibility, yet allowed, is to have a degenerate (DG) neutrino massspectrum ( m ∼ m ∼ m ). Assuming that the neutrinos are Majorana, the mass matrix can be written as : M ν = a b cb d ec e f (32)Using equations 27, 28, 30 and 31, we can estimate the unknown parameters in neutrino mass matrix of Eq. (32).To estimate the parameters in NH, we use the best fit values of the oscillation parameters. For a typical value of m = 0 . a = 0 . , b = 0 . , c = 0 . d = 0 . , e = 0 . , f = 0 . m = 0 .
001 eV, we get the mass parameters (in eV) as : a = 0 . , b = − . , c = − . d = 0 . , e = − . , f = 0 . IV. PSEUDO-DIRAC NATURE OF DARK MATTERA. Pseudo-Dirac nature of Inert fermion doublet dark matter
Let us assume the case where the singlet fermion χ is absent in the spectrum. In this case, the imposed Z symmetry stabilizes the neutral component of the fermion doublet N ≡ ( N , N − ) T . From Eq. (5) we see that afterEW phase transition the induced vev of the triplet yields a Majorana mass to N and is given by: m = √ f N (cid:104) ∆ (cid:105) ≈ f N − µv √ M . (35)Thus the N has a large Dirac mass M N as given in Eq. (1) and a small Majorana mass m as shown in the aboveEq. (35). Therefore, we get a mass matrix in the basis { N L , ( N R ) c } as: M = (cid:18) m M N M N m (cid:19) (36)The presence of small Majorana mass of the doublet DM splits the Dirac state N into two pseudo-Dirac states: ψ , ,whose mass eigenvalues are given by M N ± m for mixing angle π/
4, which is the maximal mixing. Hence the masssplitting between the two states { N L , ( N R ) c } is: δM = 2 m = 2 √ f N u ∆ . (37)Notice that the above mass splitting δM << M N and hence does not play any role in the relic abundance calculation,where both the components act as degenerate DM components. However, the small mass splitting between the twopseudo-Dirac states prohibits N to interact to the detector through Z mediation in the non-relativistic inelasticscattering limit and is crucial to escape from the strong direct detection constraints mediated via Z -boson. Forexample, to explain the DAMA signal through the inelastic scattering of DM with the nuclei the required masssplitting should be O (100keV) [40–42].A crucial observation from Eq. (26) and (35) is that the ratio: R = ( M ν ) m = f L f N (38)is extremely small. In particular, if M ν ∼ O (eV) and m ∼ O (100KeV) then R ∼ − . In other words the tripletscalar coupling to SM sector is highly suppressed in comparison to the DM sector. Using this constraint, in sectionV A we will calculate the relic abundance of inert fermion doublet DM. B. Pseudo-Dirac nature of singlet-doublet fermion dark matter
Next we adhere to the actual scenario where DM is the lightest one among the mixed states of singlet and doubletfermions χ and N . As discussed in section (II A), the DM is assumed to be N = cos θχ + sin θN with a Diracmass M . However, from Eq. (5) we see that the vev of ∆ induces a Majorana mass to N due to singlet-doubletmixing and is given by: m = √ f N sin θ (cid:104) ∆ (cid:105) ≈ f N sin θ − µv √ M . (39)Thus the Majorana mass m splits the Dirac spinor N into two pseudo-Dirac states ψ a,b with masses M ± m . Themass splitting between the two pseudo-Dirac states ( ψ a,b ) is given by δM = 2 m = 2 √ f N sin θu ∆ (40)Note that again δM << M from the estimate of induced vev of the triplet and hence does not play any role in therelic abundance calculation. However, the sub-GeV order mass splitting plays a crucial role in direct detection byforbidding the Z-boson mediated DM-nucleon elastic scattering. We will come back to this issue while discussing theinelastic scattering of DM with nucleon in sec. VI. Now from Eq. (26) and (39) we see that the ratio: R = ( M ν ) m = f L f N sin θ . (41)Thus in comparison to Eq. (38), we see that the ratio between the two couplings R = f L /f N is improved by twoorders of magnitude ( i.e. R ∼ − ) if we assume sin θ = 0 .
1, which is the rough order of magnitude of singlet-doubletmixing being used in relic abundance calculation as we demonstrate in the next section.
V. RELIC ABUNDANCE OF DMA. Relics of Inert fermion doublet dark matter
In absence of the singlet fermion χ , the neutral component ( N ) of the fermion doublet is stable due to the imposed Z symmetry. However, this does not guarantee that the N alone is a viable dark matter candidate. Under thiscircumstance it is crucial to check if N can give rise correct relic abundance observed by WMAP and PLANCK.
10 100 10001e-061e-050.00010.0010.010.11 I nv i s i b l e Z d eca y li m it M N (GeV) W h FIG. 2: Relic abundance (green line) of N , the neutral component of the doublet as DM, plotted as a function of doubletmass ( M N ) in GeV. Black horizontal line shows the observed relic abundance by PLANCK data. The solid red vertical line isshown to mark M Z = 45 GeV; for M N > M Z the DM does not contribute to the invisible Z decay width. The relic abundance of a DM is characterised by the number changing processes in which the candidate is involved.In this case, on top of annihilations to SM particles, the DM ( N ) can also participate in co-annihilations withheavier particles N ± which are odd under the same Z symmetry. The relevant annihilation and co-annihilationchannels in order to keep the inert fermion doublet DM in the thermal equilibrium in the early universe are listed below. N N → HH, ZH, W + W − , ZZ, ∆ ++ ∆ −− , ∆ + ∆ − , ∆ ∆ , W ∓ ∆ ± , ∆ H, ∆ Z, f ¯ fN N ± → W ± γ, W ± H, W ± Z, ∆ ± Z, ∆ ± H, ∆ ± γ, W ± ∆ , ∆ ±± W ∓ , ∆ ∆ ± , f (cid:48) ¯ fN ± N ∓ → W ± W ∓ , ZH, γZ, γγ, ZZ, ∆ ++ ∆ −− , ∆ + ∆ − , W + ∆ − , Z ∆ , f ¯ f We use micrOMEGAs [43] to calculate the relic abundance of dark matter. In Fig. 2, we have shown the relicabundance of N dark matter as a function of its mass. In a conservative limit we take the mass splitting between N and its charged partner N − to be 1 GeV. The scalar triplet mass is fixed at 200 GeV and its coupling with thefermions is taken to be f L f N = 10 − . We see that the large annihilation and co-annihilation cross-sections always yieldmuch smaller relic density than required and hence the model is ruled out with the mass range of the order of TeV.The dominant channels are N N → HH, ZH, W + W − , ZZ and N ± N ∓ → W ± W ∓ . We can also clearly spot theresonance at M N = M Z , where the relic density drops due to enhancement in the cross-section due to s-channel Z mediation. The resonance drop at M N = 100 GeV specifies the presence of triplet. Thus we infer that the neutralcomponent of the doublet alone can not be a viable DM candidate as its relic abundance is much below the observedlimit. Therefore, in the next section we will consider a mixed singlet-doublet state as the candidate of DM. B. Relics of Singlet-Doublet mixed fermion dark matter
The singlet ( χ ) and neutral component of the doublet ( N ) fermion mix with each other after the electroweaksymmetry breaking. In this scenario, the lightest particle N = cos θχ + sin θN , which is stabilized by the imposed Z symmetry, serves as a viable candidate of DM. The relic abundance of N can be obtained through its annihilationsto SM particles as well as through co-annihilations with N − and N . The main processes which contribute to therelic abundance of DM without involving triplet scalar are [20] : N N → HH, ZH, W + W − , ZZ, f ¯ fN N → HH, ZH, W + W − , ZZ, f ¯ f N N → HH, ZH, W + W − , ZZ, f ¯ fN N ± → W ± γ, W ± H, W ± Z, f (cid:48) ¯ fN N ± → W ± γ, W ± H, W ± Z, f (cid:48) ¯ fN ± N ∓ → W ± W ± , ZH, γZ, γγ, ZZ, f ¯ f In presence of the light scalar triplet ∆, there will be additional s-channel processes through ∆ mediation as wellas processes involving ∆ particles in the final states. The relevant processes are : N N −−→ f ¯ f , HH, W + W − , ZZN N → ∆ ++ ∆ −− , ∆ + ∆ − ∆ ∆ , W ± ∆ ± , ∆ H, ∆ ZN N −−→ f ¯ f , HH, W + W − , ZZN N → ∆ ++ ∆ −− , ∆ ∆ , ∆ + ∆ − , W ± ∆ ± , ∆ H, ∆ ZN N + → ∆ − ∆ ++ , W − ∆ ++ , ∆ ∆ + , H ∆ + , Z ∆ + , A ∆ + , W + ∆ N N −−→ f ¯ f , HH, W + W − , ZZN N → ∆ ++ ∆ −− , ∆ + ∆ − ∆ ∆ , W ± ∆ ± , ∆ H, ∆ ZN N + → ∆ − ∆ ++ , W − ∆ ++ , ∆ ∆ + , H ∆ + , Z ∆ + , A ∆ + , W + ∆ N ± N ± → ∆ ++ ∆ −− , ∆ + ∆ − , W + ∆ − , Z ∆ Relic density for N is given by [44] Ω N h = 1 . × Gev − g / (cid:63) m pl J ( x f ) , (42)where J ( x f ) is given by J ( x f ) = (cid:90) ∞ x f (cid:104) σ | v |(cid:105) eff x dx, (43)where (cid:104) σ | v |(cid:105) eff is thermal average of annihilation and coannihilation cross-sections of the DM particle. The expressionfor effective cross-section can be written as : (cid:104) σ | v |(cid:105) eff = g g σ ( N N ) + 2 g g g σ ( N N )(1 + ω ) / exp ( − xω )+ 2 g g g σ ( N N − )(1 + ω ) / exp ( − xω )+ 2 g g g σ ( N N − )(1 + ω ) exp ( − xω ) + g g σ ( N N )(1 + ω ) exp ( − xω )+ g g σ ( N − N − )(1 + ω ) exp ( − xω ) . (44)In this equation g , g , g represent spin degrees of freedom for particles N , N , N − respectively and their valuesare 2 for all. ω stands for the mass splitting ratio, given by ω = M i − M M , where M i is the mass of N and N ± . Theeffective degrees of freedom denoted by g eff , and is given by g eff = g + g (1 + ω ) / exp ( − xω ) + g (1 + ω ) / exp ( − xω ) (45)To calculate the relic density of DM, we use the code micrOMEGAs [43]. We have shown in fig. 3 the relic densityas a function of DM mass keeping the mass difference fixed at M − M = 500 GeV, for three different values of themixing angle: sin θ = 0 . , . , .
3, shown in red (top), green (middle), purple (bottom) respectively in the plot. Inthe left panel of the fig. 3 we use M ∆ = 200 GeV, whereas in the right panel of fig. 3 we use M ∆ = 1000 GeV. Theblack horizontal line corresponds to the observed relic density: Ω DM h = 0 . ± . Z , h and ∆ . Fromthese figures it is clear that as sin θ increases relic density decreases. It is due to the fact that the Z and ∆ mediatedcross-section increases for increase in sin θ , and hence yield a low relic density. For both the plots in fig 3 we fixthe ratio of Majorana couplings to be: f L f N = 10 − . From the plots in the fig. 3, we conclude that the ∆ field iscontributing to the relic density only near the resonance points. Apart from the resonance region, the triplet does not1 Ω h M (GeV) 0.0001 0.001 0.01 0.1 1 10 100 1 10 100 1000 Ω h M (GeV) FIG. 3: Relic density of DM as a function of its mass M for different values of sin θ = 0 . , . , .
3, shown by red (top), green(middle) and purple (bottom) respectively. The value of the triplet mass: M ∆ = 200 , M − M = 500 GeV. Ratio of Majoranacouplings are fixed at : f L f N = 10 − for illustration. contribute significantly. This is because the total cross-section is dominated by N ¯ N → W + W − and the ∆-mediateds-channel contribution is suppressed due to the large triplet scalar mass present in the propagator. Therefore, wecan not expect any change in relic density allowed parameter space if we vary the ratio of Majorana couplings: f L f N .The cross-sections involving scalar triplet in the final states also do not affect the relic abundance since those aresuppressed by phase space due to heavy triplet masses and as in this region of parameter space ( M > M ∆ ) thecross-sections involving gauge bosons in the final state dominate. In summary, we don’t see almost any difference inrelic density of DM in left and right panel of Fig. 3 due to change in triplet masses. We can however see that theresonance drop due to s-channel triplet mediation is reduced for large triplet mass M ∆ = 1000 GeV (shown in rightpanel) in comparison to M ∆ = 200 GeV (shown in left panel) for obvious reasons. As the mass splitting between N and N is taken to be very large in the above cases, the dominant contribution to relic density comes from annihilationchannels while co-annihilation channels are Boltzmann suppressed.
50 100 500 1000 5000 10 M ( GeV ) Δ M ( G e V )
50 100 500 1000 5000 10 M ( GeV ) Δ M ( G e V ) FIG. 4: Scatter plot for correct relic density in the plane of M and ∆ M , shown by green, red, blue and purple coloured pointsfor sin θ = 0 . , . , . , . M ∆ = 200 , and 1000 GeV respectively for theleft and right panel plots. We fixed the value of Majorana coupling ratio: f L /f N = 10 − in both the figures for illustration. N and N on DM relic density. In fig. 4, we have shown ascatter plot for correct relic density in the plane of M and ∆ M = M − M . Green, red, blue and purple colouredpoints satisfy the constraint of relic density for sin θ = 0 . , . , . , . Z and H mediation at M N = M Z and at M N = M H respectively where the annihilation cross-section is independent of ∆ M . Annihilationcross-sections contribute significantly for large ∆ M to provide correct relic abundance. As the mass splitting decreasesco-annihilation channels contribute significantly to add to the annihilation channels. As seen from the figure 4, we candivide the relic density allowed parameter space into two regions with same sin θ value: i) The region in which ∆ M is increasing with DM mass to satisfy correct relic density constraint. In this region, the contribution to relic densitycomes from both annihilation and dominantly from co-annihilation channels as the mass splitting is small. Here,due to small, ∆ M , the Yukawa coupling Y (see Eq. 11) is small and so is the Higgs mediated cross-sections. Hence,co-annihilation channels provide with the rest of the requirement for correct relic density and allowed parameter spacerequires ∆ M ∼ M . ii) The second region corresponds to a large ∆ M while insensitive to DM mass satisfying thecorrect relic abundance. In this region, the dominant contribution to relic density comes from the annihilation channels(large ∆ M indicates large Yukawa Y and large Higgs mediation cross-sections), and the co-annihilation channels areBoltzmann suppressed. Z mediated annihilation cross-sections are fixed by the choice of a specific mixing angle (inthe DM mass region within ∼ TeV). Therefore, the larger is the mixing the larger is the Z mediated annihilation.This correctly balances the Higgs mediated annihilation cross-sections to yield correct relic density. That is why wenotice that a smaller mass splitting (∆ M ) is required for larger sin θ for a fixed value of DM mass. Hence green lineswith smaller mixing (sin θ = 0 .
1) requires larger ∆ M and appears on top. With larger mixing, red, blue and purplelines, the required ∆ M are smaller and appears below. It is easy to extend the analysis for even larger mixing angles,where the triangle becomes smaller and smaller in size and covers the innermost regions to yield the correct relicdensity. For sin θ (cid:38) . θ ) provide more than required annihilation andhence those are under abundant regions. Similarly just above those, the annihilation will not be enough to producecorrect density and hence are over abundant regions. Points below (above) the “correct co-annihilation regions”produce more (less) co-annihilations than required and hence depict under (over) abundant regions. There is notmuch difference in the parameter space if we vary the scalar triplet mass except few points in the resonance region.It can be clearly seen in left and right panel of the fig. 4 with scalar triplet mass 200 GeV and 1000 GeV respectively.The Yukawa coupling ratio f L /f N = 10 − is fixed for both the plots. Again, if we change this ratio to a differentvalue, no significant change in the allowed parameter space is expected. Ω h M (GeV) 0.0000010.0000100.0001000.0010000.0100000.1000001.00000010.000000100.000000 0 500 1000 1500 2000 Ω h M (GeV) FIG. 5: Left : Ω h versus DM mass M DM in GeV for sin θ = 0 . M = 10 , , , ,
100 GeV (Blue, Green, Orange,Purple, Red respectively from bottom to top). Right : Ω h versus DM mass M DM in GeV for sin θ = 0 . M =10 , , , ,
100 GeV (Blue, Green, Orange, Purple, Red respectively from bottom to top). Horizontal line shows the correctrelic density. We fixed the value f L /f N = 10 − and M ∆ = 200 GeV for all the plots. The ∆ M dependency on the relic density for a specific choice of mixing angle is shown in Fig. 5, particularly forsmall mixing regions where co-annihilations play a crucial role in yielding correct relic density. In the left panel we use3sin θ = 0 . θ = 0 . M = 10 , , , , M , the annihilation cross-section increases due to enhancement in Yukawa coupling Y ∝ ∆ M . However,co-annihilation decreases due to increase in ∆ M as σ ∝ e − ∆ M . Note that in the small sin θ limit the dominantcontribution to relic density comes from the channels involving only N and N ± in the initial state going to SM gaugebosons, as mentioned in the beginning of this section. The processes involving N N → SM are heavily suppressedwith small sin θ . As a result, we first get relics of N and N − which subsequently decay to N before N freezes out.In particular, if the mass splitting between N − and N is more than 80 GeV, then N − decays through two bodyprocess: N − → N + W − . However, if the mass splitting between N − and N is less than 80 GeV, than the formerdecays through the three body process, say N − → N + (cid:96) − + ν (cid:96) . Notice that the mixing angles sin θ = 0 . , . N − , namely sin θ > O (10 − ).For large ∆ M the co-annihilation cross-sections decrease, which are the dominant processes in the small sin θ limit.As a result relic abundance increases for a particular value of M with larger ∆ M . Hence we require a larger massdifference ∆ M for larger DM mass to account for correct co-annihilation so that the relic density will be in theobserved limit. VI. DIRECT DETECTION OF DM THROUGH INELASTIC SCATTERING WITH THE NUCLEI
As discussed in section (V A), the inert fermion doublet N alone does not produce correct relic abundance.Therefore, we refrain ourselves to consider the inelastic scattering of N only with the nuclei mediated via Z boson.Rather we will consider the inelastic scattering of DM N , which is an admixture of doublet N and singlet χ .From Eq. (1), the relevant interaction for scattering of N with nucleon mediated via the Z-boson is given by L Z − DM ⊃ N ( γ µ ∂ µ + ig z γ µ Z µ ) N , (46)where g z = g θ w sin θ . However the presence of scalar triplet, as discussed in section (IV B), splits the Dirac state N into two pseudo-Dirac states ψ a,b with a small mass splitting m . Therefore, the above interaction in terms of thenew eigenstates ψ a,b can be rewritten as: L Z − DM ⊃ ψ a iγ µ ∂ µ ψ a + ψ b iγ µ ∂ µ ψ b + ig z ψ a γ µ ψ b Z µ . (47)From the above expression the dominant gauge interaction is off-diagonal, and the diagonal interaction vanishes. Asa result there will be inelastic scattering possible for the DM with the nucleus. Note that the mass splitting betweenthe two mass eigen states ψ a,b is given by: δM = 2 √ f N sin θ u ∆ . In this case, the minimum velocity of the DMneeded to register a recoil inside the detector is given by [40–42, 45, 46] : v min = c (cid:114) m n E R (cid:18) m n E R µ r + δM (cid:19) , (48)where E R is the recoil energy of the nucleon and µ r is the reduced mass. If the mass splitting is above a fewhundred keV, then it will be difficult to excite ψ b with the largest possible kinetic energy of the DM ψ a . So theinelastic scattering mediated by Z -boson will be forbidden. As a result constraints coming from direct detection canbe relaxed significantly. This in an important consequence in presence of the scalar triplet ∆ in this model, whichmakes a sharp distinction with the existing analysis in this direction [20]. VII. DIRECT DETECTION OF DM THROUGH ELASTIC SCATTERING WITH THE NUCLEI
We shall now point out constraints on the model parameters from direct search of DM via Higgs mediation. Therelevant diagrams through which N interacts with the nuclei are shown in Fig. (6). In particular, our focus will be onXenon-100 [32] and LUX [33] which at present give strongest constraint on spin-independent DM-nucleon cross-sectionfrom the null detection of DM yet. In our model, this in turn puts a stringent constraint on the singlet-doublet mixingangle sin θ for spin independent DM-nucleon interaction mediated via the H and H -bosons (see in the Fig. (6)).The cross-section per nucleon is given by [47, 48] σ SI = 1 πA µ r |M| (49)4 FIG. 6: Feynman diagrams for direct detection of N DM via Higgs mediation. where A is the mass number of the target nucleus, µ r = M m n / ( M + m n ) ≈ m n is the reduced mass, m n is themass of nucleon (proton or neutron) and M is the amplitude for DM-nucleon cross-section. There are two t-channelprocesses through which DM can interact with the nucleus which is shown in the fig 6. The amplitude is given by: M = (cid:88) i =1 , (cid:2) Zf ip + ( A − Z ) f in (cid:3) (50)where the effective interaction strengths of DM with proton and neutron are given by: f ip,n = (cid:88) q = u,d,s f ( p.n ) T q α iq m ( p,n ) m q + 227 f ( p,n ) T G (cid:88) q = c,t,b α iq m p.n m q (51)with α q = Y sin 2 θ cos θ M H (cid:16) m q v (cid:17) (52) α q = − Y sin 2 θ sin θ M (cid:16) m q v (cid:17) . (53)In Eq. (51), the different coupling strengths between DM and light quarks are given by [1] f ( p ) T u = 0 . ± . f ( p ) T d = 0 . ± . f ( p ) T s = 0 . ± . f ( n ) T u = 0 . ± . f ( n ) T d = 0 . ± . f ( n ) T s = 0 . ± . f ( p,n ) T G = 1 − (cid:88) q = u,,d,s f ( p,n ) T q . (54)We have plotted the spin independent direct detection cross-section as a function of DM mass in the Fig.7 bytaking the value of M ∆ = 200 GeV for two different values of M − M = 100 ,
500 GeV in the left and rightpanel respectively. The plot is generated using different values of the singlet-doublet mixing angle: sin θ = { } (Purple), sin θ = { } (Pitch), sin θ = { } (Green), sin θ = { } (Gray), sin θ = { } (Orange),sin θ = { } (Red). The top Black dotted line shows the experimental limit on the SI nuclei-DM cross-sectionwith DM mass predicted from LUX 2016 and the one below shows the sensitivity of XENON1T. The constraint fromXENON 100 is loose and weaker than the LUX data and hence not shown in the figure. One of the main outcome ofthe figure in the left panel is that with larger sin θ , due to larger Yukawa coupling direct search cross-section throughHiggs mediation is larger. Hence, LUX data constrains the singlet-doublet mixing to sin θ ∼ . ∼ M = 100 GeV (on the left hand side of Fig. 7). The constraint on the mixing is even more weaker forlarger DM mass ∼
900 GeV and can be as large as sin θ ∼ .
4. This presents a strikingly different outcome thanwhat we obtained in absence of scalar triplet [20], the mixing angle was constrained there significantly to sin θ ≤ . Z mediated direct search5 LUX - - - - - - - - - - - - - - M ( GeV ) Log [ σ N + n → N + n ] LUX - - - - - - - - - - - - - - - - - - M ( GeV ) Log [ σ N + n → N + n ] FIG. 7: Spin Independent direct detection cross-section for DM as a function of DM mass for sin θ = { } (Purple),sin θ = { } (Pitch), sin θ = { } (Green), sin θ = { } (Gray), sin θ = { } (Orange), sin θ = { } (Red). Black dotted curves show the data from LUX and XENON 1T prediction. Value of ∆ M = 100 ,
500 GeV are fixed forleft and right panel figures respectively. The scalar triplet mass is fixed at M ∆ = 200 GeV and scalar mixing angle is fixed atsin θ = 0 .
05 for the calculation. processes due to the mass splitting generated by the triplet as discussed in the above section and hence allows theDM to live in a much larger region of relic density allowed parameter space. In the right panel of the Fig. 7 withlarger ∆ M = 500 GeV, the constraint on sin θ is more stringent than the left one. It is because the SI cross-sectionis enhanced due to the increase in Yukawa coupling Y ∝ ∆ M for larger ∆ M as expected. In the right panel, for DMmass of ∼
300 GeV: sin θ ∼ . ∼ θ ∼ .
15 can be accommodated.Since the mixing between ∆ − h is small: sin θ < × − , the contribution to the cross-section by the H mediateddiagram is suppressed. This is also further suppressed by the large mass of M ∆ present in the propagator. For thisreason no striking difference in direct search cross-section for higher values of M ∆ is found as the cross-section isdominated by H mediation only. VIII. DECAY OF N − AND THE DISPLACED VERTEX SIGNATURE
FIG. 8: Feynman diagrams for three body decay of N − to DM. The phenomenology of the charged partner of DM is quite interesting. If the mass splitting between N ± and N isless than mass of W − , then N − will decay via three body suppressed process: N − → N (cid:96)ν (cid:96) and N − → N +di − jets.The relevant Feynman diagrams for the decay is shown in Fig. 8. However the figure on the right side, mediated bytriplet (∆ − ), is suppressed due to the small coupling of triplet with the leptons and the large mass M ∆ present inthe propagator. So the dominant contribution for decay of N − is coming from the left diagram of Fig. 8 through W N − → N (cid:96)ν (cid:96) is given in Eq.12.In the left panel of Fig. 9, a scatter plot is shown taking relic abundance as a function of DM mass keeping the masssplitting less than 50 GeV. Here, we fix the singlet-doublet mixing angle to be sin θ = 3 × − , a moderately smallervalue. We have also shown the correct relic abundance as allowed by the PLANCK data with a horizontal solid blackline. We choose those set of points from the relic abundance data which are allowed by the PLANCK result and usethem to calculate the displaced vertex signature of N ± (Γ − ) and plotted as a function of M ± in the right-panelof Fig. (9). We observe that the displaced vertex becomes very small for larger values of M ± , as the inverse ofdecay width Γ − is inversely proportional to the mass of decaying charged particle. However, for smaller masses with M ± ∼
200 GeV, the displaced vertex can be as large as 2.5 mm to be detected in Large Hadron Collider (LHC). Theproduction cross sections for such excitations have already been discussed earlier [20] with possible leptonic signaturesand we refrain from discussing those here again. The important point to be noted here is that to get a large displacedvertex we need a small mixing angle between the singlet and doublet. In fact, the small mixing angle is favoured by allthe constraints we discussed in previous sections, such as correct relic abundance and null detection of DM at directsearch experiments. However, from Eq. (16) we also learnt that the singlet-doublet mixing can not be arbitrarilysmall and therefore, the displaced vertex can not be too large.
100 200 300 400 500 600 70010 - M ( GeV ) Ω h
100 150 200 250 3000.010.050.100.501510 M ± ( GeV ) Γ - ( c m ) FIG. 9: Left panel: Scatter plot showing relic abundance as a function of DM mass with mass splitting less than 50 GeV.Black solid line shows the correct relic abundance as allowed by PLANCK data. Right panel: Displaced vertex (Γ − ) in cm asa function of M ± (GeV) for relic density allowed points. Value of mixing angle sin θ = 3 × − is used in both the plots forillustration. IX. SUMMARY AND CONCLUSION
We explored the possibility of a singlet-doublet mixed vector-like fermion dark matter in presence of a scalar triplet.The mixing angle: sin θ between the singlet and doublet plays an important role in the calculation of relic abundanceas well as direct detection. We found that the constraint from null detection of DM at direct search experiments andrelic abundance can be satisfied in a large region of parameter space for mixing angle: sin θ ∼ . M ∆ (cid:46)
500 GeV, then it contributes to relic abundance only near the resonance i.e with M N ∼ M ∆ . On the other hand, if M ∆ (cid:38) Z mediated DM-nucleon cross-section relaxes the constraint on mixing angle sin θ as we can go as high as sin θ = 0 . M >
400 GeV for small mass splitting ∆
M <
100 GeV. This high value of sin θ is also well satisfied bythe correct relic abundance. So the spin independent direct detection cross-section does not put stronger constraint7on the mixing angle if the mass splitting is not so large and allows large region of parameter space unlike the modelin absence of a triplet.The ρ parameter in the SM restricts the vev of scalar triplet to u ∆ ≤ .
64 GeV. This in turn gives the mixingbetween the SM Higgs and ∆ to be sin θ O (10 − ) even if the M ∆ (cid:46)
500 GeV. Therefore, ∆ does not contributesignificantly to the spin independent direct detection cross-section. R e li c A b u n d a n c ea ll o w e d b y P L A N C K I n v i s i b l e Zd ecay li m i t LHC Search Bound D i r ec t S ea r c h L i m i tf r o m L U X M ( GeV ) M ( G e V ) R e li c A b u n d a n ce A ll o w e d b y P L A N C K I n v i s i b l e Zd ecay li m i t LHC Search Bound D i r e c t S e a r c h L i m i t f r o m L U X M ( GeV ) M ( G e V ) FIG. 10: Summary of all constraints in the plane of M − M using sin θ = 0 . θ = 0 . We summarize the constraints on the parameters in Fig. 10, where we have shown the allowed values in the planeof M − M using sin θ = 0 . θ = 0 . θ = 0 .
1, direct search constraint isless severe as has already been discussed and the whole relic density allowed points are consistent with direct searchbound. However, for larger sin θ = 0 .
3, on the right hand side of Fig. 10, a significant part of the relic density allowedspace is submerged into direct search bound excepting for the low DM mass region upto ∼
400 GeV. The directsearch bound gets more stringent with larger ∆ M and that is one of the primary reasons that relic density allowedparameter space with large sin θ = 0 . ≥ .
1, was completely forbidden by direct search data. There are other small regionswhich are disfavoured by various experimental searches. For example, the region in cyan colour is disfavoured by thecollider search of N ± and hence the allowed values are given by M ± ∼ M > N (DM), i.e., M >
45 GeV, is required in order to relax the severe constraints from the invisible Z boson decay [20]. The chargedpartner of the DM gives interesting signatures at colliders if M ± − M (cid:46)
80 GeV. As a result the two body decay of N ± is forbidden. The only way it can decay is the three body decay. For example, the notable one is N − → N (cid:96) − ν (cid:96) .In the small singlet-doublet mixing limit we get a displaced vertex of 10 cm for M ± ∼
100 GeV and a mass splittingof few tens of GeV while satisfying the constraint from observed relic abundance.
Acknowledgement
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