- Email: [email protected]
Leptogenesis, dark matter and Higgs phenomenology at TeV
Nuclear Physics B 789 (2008) 245–257
Leptogenesis, dark matter and Higgs phenomenology at TeV Pei-Hong Gu a,∗ , Utpal Sarkar b a The Abdus Salam International Centre for Theoretical Physics, Strada Costiera 11, 34014 Trieste, Italy b Physical Research Laboratory, Ahmedabad 380009, India
Received 21 May 2007; received in revised form 3 August 2007; accepted 15 August 2007 Available online 21 August 2007
Abstract We propose an interesting model of neutrino masses to realize leptogenesis and dark matter at the TeV scale. A real scalar is introduced to naturally realize the Majorana masses of the right-handed neutrinos. We also include a new Higgs doublet that contributes to the dark matter of the universe. The neutrino masses come from the vacuum expectation value of the triplet Higgs scalar. The right-handed neutrinos are not constrained by the neutrino masses and hence they could generate leptogenesis at the TeV scale without subscribing to resonant leptogenesis. In our model, all new particles could be observable at the forthcoming Large Hardon Collider or the proposed future International Linear Collider. © 2007 Elsevier B.V. All rights reserved.
1. Introduction Currently many neutrino oscillation experiments [1] have confirmed that neutrinos have tiny but nonzero masses. This phenomenon is elegantly explained by the seesaw mechanism [2], in which neutrinos naturally acquire small Majorana [2] or Dirac [3] masses. At the same time, the observed matter–antimatter asymmetry [1] in the universe can be generated via leptogenesis [3–12] in these models. In this scheme, the right-handed neutrinos or other virtual particles need have very large masses to realize a successful leptogenesis if we do not resort to the resonant effect [6,7] by highly fine tuning.
* Corresponding author.
E-mail address: [email protected] (P.-H. Gu). 0550-3213/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysb.2007.08.004
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Another big challenge to the standard model is the dark matter [1]. What is the nature of dark matter? Recently, it has been pointed out [13–16] that a new Higgs doublet can be a candidate for the dark matter if it does not decay into the standard model particles. Although the possibility of Higgs doublet to be a dark matter candidate was proposed many years back [17], following the recent proposal [13] a thorough analysis have been carried out [14,18] demonstrating its consistency with all the recent results. In this interesting scenario, the dark matter is expected to produce observable signals at the Large Hardon Collider (LHC) [14] and in the GLAST satellite experiment [18]. Combining this idea, the type-I seesaw and the concept [19] of generation of the cosmological matter–antimatter asymmetry along with the cold dark matter, the author of [15] successfully unified the leptogenesis and dark matter. However, this scenario need the righthanded neutrinos to be very heavy, around the order of 107 GeV. In this paper, we propose a new scheme to explain neutrino masses, baryon asymmetry and dark matter at TeV scale by introducing a Higgs triplet which is responsible for the origin of neutrino masses, a new Higgs doublet that can be a candidate for the dark matter, and a real scalar which can generate the Majorana masses of the right-handed neutrinos naturally. A discrete symmetry ensures that the new Higgs doublet cannot couple to ordinary particles. This same discrete symmetry will also prevent any connection between the right-handed neutrinos and lefthanded neutrino masses. This allows the right-handed neutrinos to decay at low scale generating the lepton asymmetry, which will be finally converted to the baryon asymmetry through the sphaleron processes [20]. This will then explain the observed matter–antimatter asymmetry in the universe, even if the Majorana masses of the right-handed neutrinos are not highly quasidegenerate. In our model, all new particles could be close to the TeV scale and hence should be observable at the forthcoming LHC or the proposed future International Linear Collider (ILC). 2. The model We extend the standard model with some new fields. The field content is shown in Table 1, in which 0 νL φ ψL = (1) , φ= lL φ− are the left-handed lepton doublet and Higgs doublet of the standard model, respectively, while η=
η0 η−
(2)
Table 1 The field content in the model. Here ψL , φ are the standard model lefthanded lepton doublets and Higgs doublet, η is the new Higgs doublet, νR is the right-handed neutrinos, χ is the real scalar and ΔL is the Higgs triplet. Here the other standard model fields and the family indices have been omitted for simplicity Fields SU(2)L U (1)Y
ψL
φ
η
2
2
2
− 12
− 12
− 12
νR
χ
ΔL
1 0
1 0
3 1
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is the new Higgs doublet that will be the dark matter candidate, νR is the right-handed neutrino, χ is the real scalar and 1 + √ δ δ ++ 2 ΔL = (3) δ0 − √1 δ + 2
is the Higgs triplet. We further introduce a discrete Z4 symmetry, under which the different fields transform as ψL → ψL ,
φ → φ,
νR → iνR ,
χ → −χ,
η → −iη, ΔL → ΔL .
(4)
Here the other standard model fields, which are all even under the Z4 , and the family indices have been omitted for simplicity. We write down the relevant Lagrangian for the Yukawa interactions, 1 1 c c yij ψLi ηνRj + gij χ νRi νRj + fij ψLi iτ2 ΔL ψLj + h.c. , −L ⊃ (5) 2 2 ij
where yij , gij , fij are all dimensionless. We also display the general scalar potential of φ, η, χ and ΔL , 2 1 1 V (χ, φ, η, ΔL ) = μ21 χ 2 + λ1 χ 4 + μ22 φ † φ + λ2 φ † φ 2 4 2 2 + μ23 η† η + λ3 η† η + MΔ Tr Δ†L ΔL 2
2 + λ4 Tr Δ†L ΔL + λ5 Tr Δ†L ΔL + α1 χ 2 φ † φ + α2 χ 2 η† η + α3 χ 2 Tr Δ†L ΔL + 2β1 φ † φ η† η + 2β2 φ † η η† φ + 2β3 φ † φ Tr Δ†L ΔL + 2β4 φ † Δ†L ΔL φ + 2β5 η† η Tr Δ†L ΔL + 2β6 η† Δ†L ΔL η + μφ T iτ2 ΔL φ + κχηT iτ2 ΔL η + h.c. ,
(6)
where μ1,2,3 and μ have the mass dimension-1, while λ1,...,5 , α1,2,3 , β1,...,6 and κ are all dimen2 is the positive mass-square of the Higgs triplet. Without loss of generality, μ and κ sionless, MΔ will be conveniently set as real after proper phase rotations. 3. The vacuum expectation values For λ1 > 0 and μ21 < 0, we can guarantee that before the electroweak phase transition, the real scalar χ acquires a nonzero vacuum expectation value (VEV), μ2 χ ≡ u = − 1 . (7) λ1 We can then write the field χ in terms of the real physical field σ as χ ≡ σ + u,
(8)
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so that the explicit form of the Yukawa couplings become 1 1 c ν c −L ⊃ yij ψLi ηνRj + Mij νRi Rj + fij ψLi iτ2 ΔL ψLj 2 2 1 c ν + μφ T iτ2 ΔL φ + μη ˜ T iτ2 ΔL η + gij σ νRi Rj 2 2 + κσ ηT iτ2 ΔL η + h.c. + MΔ Tr Δ†L ΔL ,
(9)
where we defined, Mij ≡ gij u
and μ˜ ≡ κu.
(10)
For convenience, we diagonalize gij → gi as well as Mij → Mi by redefining νRi and then simplify the Lagrangian (9) as 1 c iτ Δ ψ + μφ T iτ Δ φ −L ⊃ yij ψLi ηNj + fij ψLi 2 L Lj 2 L 2 + μη ˜ T iτ2 ΔL η + κσ ηT iτ2 ΔL η + h.c. 1 1 2 + gi σ Ni Ni + Mi Ni Ni + MΔ Tr Δ†L ΔL , 2 2
(11)
with c , Ni ≡ νRi + νRi
(12)
being the heavy Majorana neutrinos. After the electroweak symmetry breaking, we denote the different VEVs as φ ≡ √1 v , 2
ΔL ≡
√1 vL 2
and
χ ≡ u
√1 v, 2
η ≡
and then analyze the potential as a function of these VEVs,
1 1 1 1 V (u , v, v , vL ) = μ21 u 2 + λ1 u 4 + μ22 v 2 + λ2 v 4 2 4 2 4 1 2 2 1 1 1 2 2 + μ3 v + λ3 v 2 + MΔ vL + (λ4 + λ5 )vL4 2 4 2 4 1 1 1 + α1 u 2 v 2 + α2 u 2 v 2 + α3 u 2 vL2 2 2 2 1 1 + (β1 + β2 )v 2 v 2 + (β3 + β4 )v 2 vL2 2 2 1 1 1 + (β5 + β6 )v 2 vL2 + √ μv 2 vL + √ μ˜ v 2 vL , 2 2 2
(13)
with μ˜ ≡ κu . Using the extremum conditions, 0 = ∂V /∂u = ∂V /∂v = ∂V /∂v = ∂V /∂vL , we obtain, 1 0 = λ1 u 3 + μ21 u + α1 v 2 u + α2 v 2 u + α3 vL2 u + √ κv 2 vL , 2 √ 2 2 2 2 0 = μ2 + α1 u + (β1 + β2 )v + (β3 + β4 )vL + 2 2μvL + λ2 v 2 , √ 0 = μ23 + α2 u 2 + (β1 + β2 )v 2 + (β5 + β6 )vL2 + 2 2μ˜ vL + λ3 v 2 ,
(14) (15) (16)
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2 1 1 0 = √ μv 2 + √ μ˜ v 2 + MΔ + α3 u 2 + (β3 + β4 )v 2 2 2
+ (β5 + β6 )v 2 vL + (λ4 + λ5 )vL3 . For
λ3 > 0, √ μ23 + α2 u 2 + (β1 + β2 )v 2 + (β5 + β6 )vL2 + 2 2μ˜ vL > 0,
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(17)
(18)
2 , u 2 , the new Higgs doublet η gets a zero VEV, i.e., v = 0. We assume μ < MΔ and v 2 MΔ and then deduce
μv 2 μv 2 1 1 vL √ √ 2 + α u 2 + (β + β )v 2 2 + α u 2 2 MΔ 2 MΔ 3 3 4 3 1 μv 2 √ 2 2 MΔ
for
2 MΔ α3 u 2 .
Subsequently, u and v can be solved, 2 + α v2 + α v2 μ μ2 + α1 v 2 1 3 L u = − 1 − 1 , λ1 λ1 √ μ22 + α1 u 2 + (β3 + β4 )vL2 + 2 2μvL μ2 + α1 u 2 − 2 , v= − λ2 λ2 for
(19)
(20)
(21)
λ1 > 0, μ2 + α1 v 2 + α3 vL2 < 0, 1 λ2 > 0, √ μ22 + α1 u 2 + (β3 + β4 )vL2 + 2 2μvL < 0.
(22) (23)
We then obtain the masses of resulting physical scalar bosons after the electroweak symmetry breaking, 2 + α3 u 2 + (β3 + β4 )v 2 , Mδ2++ MΔ 1 2 + α3 u 2 + β3 + β4 v 2 , Mδ2+ MΔ 2 2 Mδ20 MΔ + α3 u 2 + β3 v,
m2η± m2η0 R
μ23 + α2 u 2 m ¯ 2η + δm2η ,
+ β1 v , 2
(24) (25) (26) (27) (28)
¯ 2η − δm2η , m2η0 m
(29)
¯ 2h − δm2h , m2h1 m
(30)
I
m2h2
m ¯ 2h
+ δm2h ,
(31)
with m ¯ 2η ≡ μ23 + α2 u 2 + (β1 + β2 )v 2 ,
(32)
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δm2η ≡
μ˜ μ μ˜ μ 2 2 v v , 2 + α u 2 2 MΔ MΔ 3
(33)
m ¯ 2h ≡ λ1 u 2 + λ2 v 2 , 2
1 δm2h ≡ λ1 u 2 − λ2 v 2 + 4α12 u 2 v 2 2 .
(34) (35)
0 are defined by Here η+ and ηR,I
η+ ≡ (η− )∗ , 1 0 + iηI0 . η 0 ≡ √ ηR 2
(36) (37)
In addition, the mass eigenstates h1,2 are the linear combinations of h and σ , i.e., h1 ≡ σ sin ϑ + h cos ϑ,
(38)
h2 ≡ σ cos ϑ − h sin ϑ, are defined by 1 v+h , φ≡√ 0 2
where h,
(39)
σ
χ ≡ u + σ ,
(40)
and the mixing angle is given by tan 2ϑ
2α1 u v . λ2 v 2 − λ1 u 2
(41)
4. Neutrino masses The first diagram of Fig. 1 shows the type-II seesaw approach to the generation of the neutrino masses. It is reasonable to take the scalar cubic coupling μ less than the triplet mass MΔ in (19). In consequence, the triplet VEV in (19) is seesaw-suppressed by the ratio of the electroweak scale v over the heavy mass MΔ . Substantially, the neutrinos naturally obtain the small Majorana masses,
mII ν
ij
μv 2 1 ≡ √ fij vL −fij . 2 2MΔ 2
(42)
For the zero VEV of new Higgs doublet η, we cannot realize the neutrino masses via the type-I seesaw. However, similar to [15], it is possible to generate the radiative neutrino masses at one-loop order due to the trilinear scalar interactions in (11). As shown in the second diagram of Fig. 1, the one-loop process will induce a contribution to the neutrino masses,
m ˜ Iν ij
m2 0 m2 0
m2 0 m2 0 1 ηR ηR ηI ηI = yik yj k Mk ln ln − 2 . 2 2 2 2 2 16π m 0 − Mk Mk m 0 − Mk Mk 2 k ηR
(43)
ηI
Here Mk ≡ uu Mk . For |μ21 | |α1 |v 2 , we have u u and then Mk Mk , so the above formula can be simplified as 2 2
I Mk Mk 1 1 2 2 y y ln ln m ˜ ν ij (44) m − m , ik j k 0 2 2 ηR ηI0 Mk 16π 2 m m 0 0 k ηR
ηI
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Fig. 1. The neutrino mass-generation. (a) Is the type-II seesaw. (b) Is the radiative contribution. 2 , we have δm2 m by taking m2 0 Mk2 . Moreover, from (32) and (33), if |μ˜ μ| MΔ ¯ 2η and η ηR,I
then obtain
m ˜ Iν
ij
2 δm2η Mk v2 1 y y yik yj k 1 − ln = −ξ ik j k 2 2 Mk 2Mk 8π m ¯η
(45)
2
2 Mk Mk 1 μ˜ μ 1 δm2η 1 − ln = O 1 − ln . 2 2 2 2 2 4π v m ¯η 4π MΔ m ¯ 2η
(46)
−
k
k
for ξ =O
Note that the above loop-contribution will be absent once the values of κ and then μ˜ are taken to be zero. 5. Baryon asymmetry We now demonstrate how the observed baryon asymmetry is generated in this model. In the Lagrangian (11), the lepton number of the left-handed lepton doublets and the Higgs triplet are 1 and −2, respectively, while those of the heavy Majorana neutrinos, the Higgs doublets and the real scalar are all zero. There are two sources of lepton number violation, one is the trilinear interaction between the Higgs triplet and the Higgs doublets, the other is the Yukawa couplings of the heavy Majorana neutrinos to the left-handed lepton doublet and the new Higgs doublet. Therefore, both the Higgs triplet and the heavy Majorana neutrinos could decay to produce the lepton asymmetry if their decays are CP-violation and out-of-equilibrium.1 We can obtain the CP asymmetry in the decay of Ni through the interference between the treelevel process and three one-loop diagrams of Fig. 2, in which the first two one-loop diagrams are the ordinary self-energy and vertex correction involving another heavy Majorana neutrinos, while the third one-loop diagram is mediated by the Higgs triplet [21]. So it is convenient to divide the 1 Note that there is an equivalent choice of lepton number: L = 1 for η and L = 0 for ν , which makes only the R μφ T iτ2 ΔL φ term to be lepton number violating. So, the CP asymmetry in the decays of Ni and ΔL can only create an asymmetry in the numbers of ψL and an equal and opposite amount of asymmetry in the numbers of η. Thus there is no net lepton number asymmetry at this stage. However, since only the left-handed fields take part in the sphaleron transitions, only the ψL asymmetry gets converted to a B–L asymmetry before the electroweak phase transition. After the electroweak phase transition, we are thus left with a baryon asymmetry equivalent to the B–L asymmetry generated from the ψL asymmetry and an equivalent amount of η asymmetry or lepton number asymmetry, which does not affect the baryon asymmetry of the universe. In the rest of the article we shall not discuss this possibility, since the final amount of baryon asymmetry comes out to be the same.
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Fig. 2. The Higgs triplets decay to the leptons at one-loop order.
total CP asymmetry into two independent parts, c ∗ j [Γ (Ni → ψLj η ) − Γ (Ni → ψLj η)] εi ≡ = εiN + εiΔ , Γi
(47)
where Γi ≡
1 † c Γ Ni → ψLj η∗ + Γ Ni → ψLj η = y y ii Mi 8π
(48)
j
is the total decay width of Ni , while
1 1 † 2 ak ak ai ai Im y y 1 − 1 + ln 1 + + , ik 8π (y † y)ii ai ai ak a i − ak k =i
3 ai 1 † † † μ˜ aΔ Im f y y ln 1 + 1 − εiΔ = j m ij im 2π (y † y)ii Mi ai aΔ
εiN =
(49) (50)
jm
are the contributions of the first two one-loop diagrams and the third one, respectively. Here the definitions ai ≡
Mi2
, 2
M1
aΔ ≡
2 MΔ
M12
(51)
have been adopted. Furthermore, as shown in Fig. 3, in the decay of ΔL , the tree-level diagram interferes with the one-loop correction to generate the CP asymmetry, c c ∗ ij [Γ (ΔL → ψLi ψLj ) − Γ (ΔL → ψLi ψLj )] εΔ ≡ 2 ΓΔ 2 /M 2 ) ˜ k ln(1 + MΔ 2 ij k (yki ykj fij )μM k = , (52) 2 † 2 2 π Tr(f f )MΔ + 4μ˜ + 4μ with ΓΔ ≡
c c Γ ΔL → ψLi ψLj ij
+ Γ (ΔL → ηη) + Γ (ΔL → φφ)
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≡
253
Γ Δ∗L → ψLi ψLj ij
+ Γ Δ∗L → η∗ η∗ + Γ Δ∗L → φ ∗ φ ∗
1 1 † μ˜ 2 + μ2 Tr f f + MΔ = 2 8π 4 MΔ
(53)
being the total decay width of ΔL or Δ∗L . Note that we have not considered the cases where σ directly decay to produce the leptons and anti-leptons through the imaginary Ni or ΔL if mσ > 2Mi , MΔ + 2mη with mσ and mη being the masses of σ and η, respectively. For simplicity, here we will not discuss these cases. It is straightforward to see that εΔ and εiΔ will both be zero for κ = 0 and then μ˜ = 0. In the following, to illustrate how to realize non-resonant TeV leptogenesis, we first focus on the simple case where εiN is the unique source of the CP asymmetry. Note that μ˜ = 0 for κ = 0, accordingly, the one-loop diagram of Fig. 1 is absent and Ni have no possibility for the neutrino masses, we thus obtain 3 Im[(y † y)21k ] M1 16π (y † y)11 Mk k=2,3 3 M 1 M1 − + sin δ 16π M2 M3
ε1N −
(54)
with δ being the CP phase. Here we have assumed N1 to be the lightest heavy Majorana neutrinos, 2 , M 2 . The final baryon asymmetry can be given by approximate relation [22] i.e., M12 M2,3 Δ ε1 (for K 1), g∗ nB 28 YB ≡ (55) − × 0.3ε1 s 79 (for K 1), g K(ln K)0.6 ∗
Fig. 3. The heavy Majorana neutrinos decay at one-loop order.
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where the factor 28/79 is the value of B/(B − L) and the parameter K is a measure of the departure from equilibrium and is defined by 1 2 M † Γ1 45 Pl K≡ = y y . 11 26 π 5 g H (T ) T =M1 M 1 ∗
(56)
1
Here H (T ) = (4π 3 g∗ /45) 2 T 2 /MPl is the Hubble constant with the Planck mass MPl ∼ 1019 GeV and the relativistic degrees of freedom g∗ ∼ 100. For example, inspecting MΔ = 10 TeV, |μ| = 1 GeV and f ∼ 10−6 to (43), we obtain mν ∼ O(0.1 eV) which is consistent with the neutrino oscillation experiments. Furthermore, let M1 = 0.1M2,3 = 1 TeV, y ∼ 10−6 and sin δ = 10−3 , we drive the sample predictions: K 48 and ε1 −1.2 × 10−5 . In consequence, we arrive at nB /s 10−10 as desired. For κ = 0 and then μ˜ , μ˜ = 0, ΔL and Ni will both contribute to the neutrino masses and the lepton asymmetry. In the limit of MΔ Mi , the final lepton asymmetry is expected to mostly produce by the decay of ΔL . However, because the electroweak gauge scattering should be out of thermal equilibrium, it is difficult for a successful leptogenesis to lower the mass of ΔL at TeV scale. Let us consider another possibility that Ni are much lighter than ΔL . In this case, 2 , M 2 and |μ 2 , εN leptogenesis will be dominated by the decay of Ni . For M12 M2,3 ˜ μ| MΔ Δ 1 and ε1Δ can be simplified as [10] ε1N −
† † ˜ I∗ 3 Im[(y † y)21k ] M1 3 M1 Im[(m ν )j k y1j y1k ] 1 − 16π 8π v 2 ξ (y † y)11 Mk (y † y)11 jk
k=2,3
− ε1Δ −
3 8π
˜ Imax M1 m v2
1 sin δ , ξ
† † II∗ ˜ 3 M1 μ˜ Im[(mν )j k y1j y1k ] 3 M1 mII max μ − μ sin δ , † 2 2 8π v μ 8π (y y)11 v
(57)
(58)
jk
˜ Imax are the maximal eigenstates of the neutrino where δ and δ are CP phases, mII max and m mass matrixes (42) and (45), respectively. Inputting y ∼ 10−7 , M1 = 1 TeV and M2,3 = 10 TeV, we obtain m ˜ Imax = O(10−3 eV). Similarly, mII max = O(0.1 eV) for MΔ = 10 TeV, |μ| = 1 GeV and f ∼ 10−6 . Under this setup, we deduce ξ 10−3 by substituting m ¯ η = 70 GeV, |μ˜ | = N 3 −12 10 TeV into (46) and then have ε1 −2 × 10 with the maximum CP phase. We also acquire ˜ |μ˜ | and sin δ = 0.15. We thus drive the sample predictions: K ε1Δ −3 × 10−8 for |μ| 0.5 and ε1 ε1Δ −3 × 10−8 . In consequence, we arrive at nB /s 10−10 consistent with the cosmological observations. 6. Dark matter and Higgs phenomenology 0 Since the new Higgs doublet cannot decay into the standard model particles, the neutral ηR 0 and ηI can provide the attractive candidates for dark matter [13–15]. In particular, to realize dark 0 and η0 should have the mass spectrum [14]: matter, ηR I
Δm (8–9) GeV for mL = (60–73) GeV,
(59)
Δm (9–12) GeV for mL = (73–75) GeV.
(60)
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Here Δm ≡ mN L − mL with mL and mN L being the lightest and the next lightest masses between 0 and η0 . Note ηR I 1 m ¯ η ≡ (mL + mN L ), 2 2 1 2 δm ≡ m − m2 , η NL L 2 we thus deduce, 1 |δm2η | ¯η 1− mL = m , 2 m ¯ 2η Δm =
|δm2η | m ¯η
(61) (62)
(63) (64)
.
In the previous discussions of TeV leptogenesis with κ = 0, we take MΔ = 10 TeV, |μ| = 1 GeV, |μ| ˜ = 103 TeV and m ¯ η = 70 GeV. It is straightforward to see |δmη | 25 GeV from (33). Therefore, we obtain mL 66 GeV and Δm 9 GeV, which is consistent with the mass spectrum (59). 0 and η0 are expected to be produced in pairs by the standard model gauge bosons W ± , Z ηR I 0 and a virtual or γ and hence can be verified at the LHC. Once produced, η± will decay into ηR,I 0 is lighter W ± , which becomes a quark–antiquark or lepton–antilepton pair. For example, if ηR 0 than ηI , the decay chain η+ → ηI0 l + ν,
0 + − then ηI0 → ηR l l ,
(65)
has 3 charged leptons and large missing energy, and can be compared to the direct decay 0 + l ν η + → ηR
(66)
to extract the masses of the respective particles. As for the phenomenology of the Higgs triplet at the LHC as well as the ILC, it has been discussed in [23]. The same-sign dileptons will be the most dominating modes of the δ ++ . Complementary measurements of |fij | at the ILC by the process e+ e+ (μ+ μ− ) → li− lj− would allow us to study the structure of the neutrino mass matrix in detail. For χ = O(TeV), which is natural to give the TeV Majorana masses of the right-handed neutrinos and then realize the TeV leptogenesis, the mixing angle ϑ and the splitting between h1,2 may be large. Furthermore, the couplings of h1,2 to W and Z bosons, quarks and charged leptons have essentially the same structure as the corresponding Higgs couplings in the standard model, however, their size is reduced by cos ϑ and sin ϑ , respectively. In the extreme case ϑ = π2 , the couplings of the lighter physical boson h1 to quarks and leptons would even vanish. In other words, this mixing could lead to significant impact on the Higgs searches at the LHC [24,25]. 7. Summary We propose a new model to realize leptogenesis and dark matter at the TeV scale. A real scalar is introduced to naturally realize the Majorana masses of the right-handed neutrinos. Furthermore, we also consider a new Higgs doublet to provide the attractive candidates for dark matter. Since the right-handed neutrinos have no responsibility to generate the neutrino masses,
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Leptogenesis, dark matter and Higgs phenomenology at TeV Pei-Hong Gu a,∗ , Utpal Sarkar b a The Abdus Salam International Centre for Theoretical Physics, Strada Costiera 11, 34014 Trieste, Italy b Physical Research Laboratory, Ahmedabad 380009, India
Received 21 May 2007; received in revised form 3 August 2007; accepted 15 August 2007 Available online 21 August 2007
Abstract We propose an interesting model of neutrino masses to realize leptogenesis and dark matter at the TeV scale. A real scalar is introduced to naturally realize the Majorana masses of the right-handed neutrinos. We also include a new Higgs doublet that contributes to the dark matter of the universe. The neutrino masses come from the vacuum expectation value of the triplet Higgs scalar. The right-handed neutrinos are not constrained by the neutrino masses and hence they could generate leptogenesis at the TeV scale without subscribing to resonant leptogenesis. In our model, all new particles could be observable at the forthcoming Large Hardon Collider or the proposed future International Linear Collider. © 2007 Elsevier B.V. All rights reserved.
1. Introduction Currently many neutrino oscillation experiments [1] have confirmed that neutrinos have tiny but nonzero masses. This phenomenon is elegantly explained by the seesaw mechanism [2], in which neutrinos naturally acquire small Majorana [2] or Dirac [3] masses. At the same time, the observed matter–antimatter asymmetry [1] in the universe can be generated via leptogenesis [3–12] in these models. In this scheme, the right-handed neutrinos or other virtual particles need have very large masses to realize a successful leptogenesis if we do not resort to the resonant effect [6,7] by highly fine tuning.
* Corresponding author.
E-mail address: [email protected] (P.-H. Gu). 0550-3213/$ – see front matter © 2007 Elsevier B.V. All rights reserved. doi:10.1016/j.nuclphysb.2007.08.004
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Another big challenge to the standard model is the dark matter [1]. What is the nature of dark matter? Recently, it has been pointed out [13–16] that a new Higgs doublet can be a candidate for the dark matter if it does not decay into the standard model particles. Although the possibility of Higgs doublet to be a dark matter candidate was proposed many years back [17], following the recent proposal [13] a thorough analysis have been carried out [14,18] demonstrating its consistency with all the recent results. In this interesting scenario, the dark matter is expected to produce observable signals at the Large Hardon Collider (LHC) [14] and in the GLAST satellite experiment [18]. Combining this idea, the type-I seesaw and the concept [19] of generation of the cosmological matter–antimatter asymmetry along with the cold dark matter, the author of [15] successfully unified the leptogenesis and dark matter. However, this scenario need the righthanded neutrinos to be very heavy, around the order of 107 GeV. In this paper, we propose a new scheme to explain neutrino masses, baryon asymmetry and dark matter at TeV scale by introducing a Higgs triplet which is responsible for the origin of neutrino masses, a new Higgs doublet that can be a candidate for the dark matter, and a real scalar which can generate the Majorana masses of the right-handed neutrinos naturally. A discrete symmetry ensures that the new Higgs doublet cannot couple to ordinary particles. This same discrete symmetry will also prevent any connection between the right-handed neutrinos and lefthanded neutrino masses. This allows the right-handed neutrinos to decay at low scale generating the lepton asymmetry, which will be finally converted to the baryon asymmetry through the sphaleron processes [20]. This will then explain the observed matter–antimatter asymmetry in the universe, even if the Majorana masses of the right-handed neutrinos are not highly quasidegenerate. In our model, all new particles could be close to the TeV scale and hence should be observable at the forthcoming LHC or the proposed future International Linear Collider (ILC). 2. The model We extend the standard model with some new fields. The field content is shown in Table 1, in which 0 νL φ ψL = (1) , φ= lL φ− are the left-handed lepton doublet and Higgs doublet of the standard model, respectively, while η=
η0 η−
(2)
Table 1 The field content in the model. Here ψL , φ are the standard model lefthanded lepton doublets and Higgs doublet, η is the new Higgs doublet, νR is the right-handed neutrinos, χ is the real scalar and ΔL is the Higgs triplet. Here the other standard model fields and the family indices have been omitted for simplicity Fields SU(2)L U (1)Y
ψL
φ
η
2
2
2
− 12
− 12
− 12
νR
χ
ΔL
1 0
1 0
3 1
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is the new Higgs doublet that will be the dark matter candidate, νR is the right-handed neutrino, χ is the real scalar and 1 + √ δ δ ++ 2 ΔL = (3) δ0 − √1 δ + 2
is the Higgs triplet. We further introduce a discrete Z4 symmetry, under which the different fields transform as ψL → ψL ,
φ → φ,
νR → iνR ,
χ → −χ,
η → −iη, ΔL → ΔL .
(4)
Here the other standard model fields, which are all even under the Z4 , and the family indices have been omitted for simplicity. We write down the relevant Lagrangian for the Yukawa interactions, 1 1 c c yij ψLi ηνRj + gij χ νRi νRj + fij ψLi iτ2 ΔL ψLj + h.c. , −L ⊃ (5) 2 2 ij
where yij , gij , fij are all dimensionless. We also display the general scalar potential of φ, η, χ and ΔL , 2 1 1 V (χ, φ, η, ΔL ) = μ21 χ 2 + λ1 χ 4 + μ22 φ † φ + λ2 φ † φ 2 4 2 2 + μ23 η† η + λ3 η† η + MΔ Tr Δ†L ΔL 2
2 + λ4 Tr Δ†L ΔL + λ5 Tr Δ†L ΔL + α1 χ 2 φ † φ + α2 χ 2 η† η + α3 χ 2 Tr Δ†L ΔL + 2β1 φ † φ η† η + 2β2 φ † η η† φ + 2β3 φ † φ Tr Δ†L ΔL + 2β4 φ † Δ†L ΔL φ + 2β5 η† η Tr Δ†L ΔL + 2β6 η† Δ†L ΔL η + μφ T iτ2 ΔL φ + κχηT iτ2 ΔL η + h.c. ,
(6)
where μ1,2,3 and μ have the mass dimension-1, while λ1,...,5 , α1,2,3 , β1,...,6 and κ are all dimen2 is the positive mass-square of the Higgs triplet. Without loss of generality, μ and κ sionless, MΔ will be conveniently set as real after proper phase rotations. 3. The vacuum expectation values For λ1 > 0 and μ21 < 0, we can guarantee that before the electroweak phase transition, the real scalar χ acquires a nonzero vacuum expectation value (VEV), μ2 χ ≡ u = − 1 . (7) λ1 We can then write the field χ in terms of the real physical field σ as χ ≡ σ + u,
(8)
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so that the explicit form of the Yukawa couplings become 1 1 c ν c −L ⊃ yij ψLi ηνRj + Mij νRi Rj + fij ψLi iτ2 ΔL ψLj 2 2 1 c ν + μφ T iτ2 ΔL φ + μη ˜ T iτ2 ΔL η + gij σ νRi Rj 2 2 + κσ ηT iτ2 ΔL η + h.c. + MΔ Tr Δ†L ΔL ,
(9)
where we defined, Mij ≡ gij u
and μ˜ ≡ κu.
(10)
For convenience, we diagonalize gij → gi as well as Mij → Mi by redefining νRi and then simplify the Lagrangian (9) as 1 c iτ Δ ψ + μφ T iτ Δ φ −L ⊃ yij ψLi ηNj + fij ψLi 2 L Lj 2 L 2 + μη ˜ T iτ2 ΔL η + κσ ηT iτ2 ΔL η + h.c. 1 1 2 + gi σ Ni Ni + Mi Ni Ni + MΔ Tr Δ†L ΔL , 2 2
(11)
with c , Ni ≡ νRi + νRi
(12)
being the heavy Majorana neutrinos. After the electroweak symmetry breaking, we denote the different VEVs as φ ≡ √1 v , 2
ΔL ≡
√1 vL 2
and
χ ≡ u
√1 v, 2
η ≡
and then analyze the potential as a function of these VEVs,
1 1 1 1 V (u , v, v , vL ) = μ21 u 2 + λ1 u 4 + μ22 v 2 + λ2 v 4 2 4 2 4 1 2 2 1 1 1 2 2 + μ3 v + λ3 v 2 + MΔ vL + (λ4 + λ5 )vL4 2 4 2 4 1 1 1 + α1 u 2 v 2 + α2 u 2 v 2 + α3 u 2 vL2 2 2 2 1 1 + (β1 + β2 )v 2 v 2 + (β3 + β4 )v 2 vL2 2 2 1 1 1 + (β5 + β6 )v 2 vL2 + √ μv 2 vL + √ μ˜ v 2 vL , 2 2 2
(13)
with μ˜ ≡ κu . Using the extremum conditions, 0 = ∂V /∂u = ∂V /∂v = ∂V /∂v = ∂V /∂vL , we obtain, 1 0 = λ1 u 3 + μ21 u + α1 v 2 u + α2 v 2 u + α3 vL2 u + √ κv 2 vL , 2 √ 2 2 2 2 0 = μ2 + α1 u + (β1 + β2 )v + (β3 + β4 )vL + 2 2μvL + λ2 v 2 , √ 0 = μ23 + α2 u 2 + (β1 + β2 )v 2 + (β5 + β6 )vL2 + 2 2μ˜ vL + λ3 v 2 ,
(14) (15) (16)
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2 1 1 0 = √ μv 2 + √ μ˜ v 2 + MΔ + α3 u 2 + (β3 + β4 )v 2 2 2
+ (β5 + β6 )v 2 vL + (λ4 + λ5 )vL3 . For
λ3 > 0, √ μ23 + α2 u 2 + (β1 + β2 )v 2 + (β5 + β6 )vL2 + 2 2μ˜ vL > 0,
249
(17)
(18)
2 , u 2 , the new Higgs doublet η gets a zero VEV, i.e., v = 0. We assume μ < MΔ and v 2 MΔ and then deduce
μv 2 μv 2 1 1 vL √ √ 2 + α u 2 + (β + β )v 2 2 + α u 2 2 MΔ 2 MΔ 3 3 4 3 1 μv 2 √ 2 2 MΔ
for
2 MΔ α3 u 2 .
Subsequently, u and v can be solved, 2 + α v2 + α v2 μ μ2 + α1 v 2 1 3 L u = − 1 − 1 , λ1 λ1 √ μ22 + α1 u 2 + (β3 + β4 )vL2 + 2 2μvL μ2 + α1 u 2 − 2 , v= − λ2 λ2 for
(19)
(20)
(21)
λ1 > 0, μ2 + α1 v 2 + α3 vL2 < 0, 1 λ2 > 0, √ μ22 + α1 u 2 + (β3 + β4 )vL2 + 2 2μvL < 0.
(22) (23)
We then obtain the masses of resulting physical scalar bosons after the electroweak symmetry breaking, 2 + α3 u 2 + (β3 + β4 )v 2 , Mδ2++ MΔ 1 2 + α3 u 2 + β3 + β4 v 2 , Mδ2+ MΔ 2 2 Mδ20 MΔ + α3 u 2 + β3 v,
m2η± m2η0 R
μ23 + α2 u 2 m ¯ 2η + δm2η ,
+ β1 v , 2
(24) (25) (26) (27) (28)
¯ 2η − δm2η , m2η0 m
(29)
¯ 2h − δm2h , m2h1 m
(30)
I
m2h2
m ¯ 2h
+ δm2h ,
(31)
with m ¯ 2η ≡ μ23 + α2 u 2 + (β1 + β2 )v 2 ,
(32)
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δm2η ≡
μ˜ μ μ˜ μ 2 2 v v , 2 + α u 2 2 MΔ MΔ 3
(33)
m ¯ 2h ≡ λ1 u 2 + λ2 v 2 , 2
1 δm2h ≡ λ1 u 2 − λ2 v 2 + 4α12 u 2 v 2 2 .
(34) (35)
0 are defined by Here η+ and ηR,I
η+ ≡ (η− )∗ , 1 0 + iηI0 . η 0 ≡ √ ηR 2
(36) (37)
In addition, the mass eigenstates h1,2 are the linear combinations of h and σ , i.e., h1 ≡ σ sin ϑ + h cos ϑ,
(38)
h2 ≡ σ cos ϑ − h sin ϑ, are defined by 1 v+h , φ≡√ 0 2
where h,
(39)
σ
χ ≡ u + σ ,
(40)
and the mixing angle is given by tan 2ϑ
2α1 u v . λ2 v 2 − λ1 u 2
(41)
4. Neutrino masses The first diagram of Fig. 1 shows the type-II seesaw approach to the generation of the neutrino masses. It is reasonable to take the scalar cubic coupling μ less than the triplet mass MΔ in (19). In consequence, the triplet VEV in (19) is seesaw-suppressed by the ratio of the electroweak scale v over the heavy mass MΔ . Substantially, the neutrinos naturally obtain the small Majorana masses,
mII ν
ij
μv 2 1 ≡ √ fij vL −fij . 2 2MΔ 2
(42)
For the zero VEV of new Higgs doublet η, we cannot realize the neutrino masses via the type-I seesaw. However, similar to [15], it is possible to generate the radiative neutrino masses at one-loop order due to the trilinear scalar interactions in (11). As shown in the second diagram of Fig. 1, the one-loop process will induce a contribution to the neutrino masses,
m ˜ Iν ij
m2 0 m2 0
m2 0 m2 0 1 ηR ηR ηI ηI = yik yj k Mk ln ln − 2 . 2 2 2 2 2 16π m 0 − Mk Mk m 0 − Mk Mk 2 k ηR
(43)
ηI
Here Mk ≡ uu Mk . For |μ21 | |α1 |v 2 , we have u u and then Mk Mk , so the above formula can be simplified as 2 2
I Mk Mk 1 1 2 2 y y ln ln m ˜ ν ij (44) m − m , ik j k 0 2 2 ηR ηI0 Mk 16π 2 m m 0 0 k ηR
ηI
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Fig. 1. The neutrino mass-generation. (a) Is the type-II seesaw. (b) Is the radiative contribution. 2 , we have δm2 m by taking m2 0 Mk2 . Moreover, from (32) and (33), if |μ˜ μ| MΔ ¯ 2η and η ηR,I
then obtain
m ˜ Iν
ij
2 δm2η Mk v2 1 y y yik yj k 1 − ln = −ξ ik j k 2 2 Mk 2Mk 8π m ¯η
(45)
2
2 Mk Mk 1 μ˜ μ 1 δm2η 1 − ln = O 1 − ln . 2 2 2 2 2 4π v m ¯η 4π MΔ m ¯ 2η
(46)
−
k
k
for ξ =O
Note that the above loop-contribution will be absent once the values of κ and then μ˜ are taken to be zero. 5. Baryon asymmetry We now demonstrate how the observed baryon asymmetry is generated in this model. In the Lagrangian (11), the lepton number of the left-handed lepton doublets and the Higgs triplet are 1 and −2, respectively, while those of the heavy Majorana neutrinos, the Higgs doublets and the real scalar are all zero. There are two sources of lepton number violation, one is the trilinear interaction between the Higgs triplet and the Higgs doublets, the other is the Yukawa couplings of the heavy Majorana neutrinos to the left-handed lepton doublet and the new Higgs doublet. Therefore, both the Higgs triplet and the heavy Majorana neutrinos could decay to produce the lepton asymmetry if their decays are CP-violation and out-of-equilibrium.1 We can obtain the CP asymmetry in the decay of Ni through the interference between the treelevel process and three one-loop diagrams of Fig. 2, in which the first two one-loop diagrams are the ordinary self-energy and vertex correction involving another heavy Majorana neutrinos, while the third one-loop diagram is mediated by the Higgs triplet [21]. So it is convenient to divide the 1 Note that there is an equivalent choice of lepton number: L = 1 for η and L = 0 for ν , which makes only the R μφ T iτ2 ΔL φ term to be lepton number violating. So, the CP asymmetry in the decays of Ni and ΔL can only create an asymmetry in the numbers of ψL and an equal and opposite amount of asymmetry in the numbers of η. Thus there is no net lepton number asymmetry at this stage. However, since only the left-handed fields take part in the sphaleron transitions, only the ψL asymmetry gets converted to a B–L asymmetry before the electroweak phase transition. After the electroweak phase transition, we are thus left with a baryon asymmetry equivalent to the B–L asymmetry generated from the ψL asymmetry and an equivalent amount of η asymmetry or lepton number asymmetry, which does not affect the baryon asymmetry of the universe. In the rest of the article we shall not discuss this possibility, since the final amount of baryon asymmetry comes out to be the same.
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Fig. 2. The Higgs triplets decay to the leptons at one-loop order.
total CP asymmetry into two independent parts, c ∗ j [Γ (Ni → ψLj η ) − Γ (Ni → ψLj η)] εi ≡ = εiN + εiΔ , Γi
(47)
where Γi ≡
1 † c Γ Ni → ψLj η∗ + Γ Ni → ψLj η = y y ii Mi 8π
(48)
j
is the total decay width of Ni , while
1 1 † 2 ak ak ai ai Im y y 1 − 1 + ln 1 + + , ik 8π (y † y)ii ai ai ak a i − ak k =i
3 ai 1 † † † μ˜ aΔ Im f y y ln 1 + 1 − εiΔ = j m ij im 2π (y † y)ii Mi ai aΔ
εiN =
(49) (50)
jm
are the contributions of the first two one-loop diagrams and the third one, respectively. Here the definitions ai ≡
Mi2
, 2
M1
aΔ ≡
2 MΔ
M12
(51)
have been adopted. Furthermore, as shown in Fig. 3, in the decay of ΔL , the tree-level diagram interferes with the one-loop correction to generate the CP asymmetry, c c ∗ ij [Γ (ΔL → ψLi ψLj ) − Γ (ΔL → ψLi ψLj )] εΔ ≡ 2 ΓΔ 2 /M 2 ) ˜ k ln(1 + MΔ 2 ij k (yki ykj fij )μM k = , (52) 2 † 2 2 π Tr(f f )MΔ + 4μ˜ + 4μ with ΓΔ ≡
c c Γ ΔL → ψLi ψLj ij
+ Γ (ΔL → ηη) + Γ (ΔL → φφ)
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≡
253
Γ Δ∗L → ψLi ψLj ij
+ Γ Δ∗L → η∗ η∗ + Γ Δ∗L → φ ∗ φ ∗
1 1 † μ˜ 2 + μ2 Tr f f + MΔ = 2 8π 4 MΔ
(53)
being the total decay width of ΔL or Δ∗L . Note that we have not considered the cases where σ directly decay to produce the leptons and anti-leptons through the imaginary Ni or ΔL if mσ > 2Mi , MΔ + 2mη with mσ and mη being the masses of σ and η, respectively. For simplicity, here we will not discuss these cases. It is straightforward to see that εΔ and εiΔ will both be zero for κ = 0 and then μ˜ = 0. In the following, to illustrate how to realize non-resonant TeV leptogenesis, we first focus on the simple case where εiN is the unique source of the CP asymmetry. Note that μ˜ = 0 for κ = 0, accordingly, the one-loop diagram of Fig. 1 is absent and Ni have no possibility for the neutrino masses, we thus obtain 3 Im[(y † y)21k ] M1 16π (y † y)11 Mk k=2,3 3 M 1 M1 − + sin δ 16π M2 M3
ε1N −
(54)
with δ being the CP phase. Here we have assumed N1 to be the lightest heavy Majorana neutrinos, 2 , M 2 . The final baryon asymmetry can be given by approximate relation [22] i.e., M12 M2,3 Δ ε1 (for K 1), g∗ nB 28 YB ≡ (55) − × 0.3ε1 s 79 (for K 1), g K(ln K)0.6 ∗
Fig. 3. The heavy Majorana neutrinos decay at one-loop order.
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where the factor 28/79 is the value of B/(B − L) and the parameter K is a measure of the departure from equilibrium and is defined by 1 2 M † Γ1 45 Pl K≡ = y y . 11 26 π 5 g H (T ) T =M1 M 1 ∗
(56)
1
Here H (T ) = (4π 3 g∗ /45) 2 T 2 /MPl is the Hubble constant with the Planck mass MPl ∼ 1019 GeV and the relativistic degrees of freedom g∗ ∼ 100. For example, inspecting MΔ = 10 TeV, |μ| = 1 GeV and f ∼ 10−6 to (43), we obtain mν ∼ O(0.1 eV) which is consistent with the neutrino oscillation experiments. Furthermore, let M1 = 0.1M2,3 = 1 TeV, y ∼ 10−6 and sin δ = 10−3 , we drive the sample predictions: K 48 and ε1 −1.2 × 10−5 . In consequence, we arrive at nB /s 10−10 as desired. For κ = 0 and then μ˜ , μ˜ = 0, ΔL and Ni will both contribute to the neutrino masses and the lepton asymmetry. In the limit of MΔ Mi , the final lepton asymmetry is expected to mostly produce by the decay of ΔL . However, because the electroweak gauge scattering should be out of thermal equilibrium, it is difficult for a successful leptogenesis to lower the mass of ΔL at TeV scale. Let us consider another possibility that Ni are much lighter than ΔL . In this case, 2 , M 2 and |μ 2 , εN leptogenesis will be dominated by the decay of Ni . For M12 M2,3 ˜ μ| MΔ Δ 1 and ε1Δ can be simplified as [10] ε1N −
† † ˜ I∗ 3 Im[(y † y)21k ] M1 3 M1 Im[(m ν )j k y1j y1k ] 1 − 16π 8π v 2 ξ (y † y)11 Mk (y † y)11 jk
k=2,3
− ε1Δ −
3 8π
˜ Imax M1 m v2
1 sin δ , ξ
† † II∗ ˜ 3 M1 μ˜ Im[(mν )j k y1j y1k ] 3 M1 mII max μ − μ sin δ , † 2 2 8π v μ 8π (y y)11 v
(57)
(58)
jk
˜ Imax are the maximal eigenstates of the neutrino where δ and δ are CP phases, mII max and m mass matrixes (42) and (45), respectively. Inputting y ∼ 10−7 , M1 = 1 TeV and M2,3 = 10 TeV, we obtain m ˜ Imax = O(10−3 eV). Similarly, mII max = O(0.1 eV) for MΔ = 10 TeV, |μ| = 1 GeV and f ∼ 10−6 . Under this setup, we deduce ξ 10−3 by substituting m ¯ η = 70 GeV, |μ˜ | = N 3 −12 10 TeV into (46) and then have ε1 −2 × 10 with the maximum CP phase. We also acquire ˜ |μ˜ | and sin δ = 0.15. We thus drive the sample predictions: K ε1Δ −3 × 10−8 for |μ| 0.5 and ε1 ε1Δ −3 × 10−8 . In consequence, we arrive at nB /s 10−10 consistent with the cosmological observations. 6. Dark matter and Higgs phenomenology 0 Since the new Higgs doublet cannot decay into the standard model particles, the neutral ηR 0 and ηI can provide the attractive candidates for dark matter [13–15]. In particular, to realize dark 0 and η0 should have the mass spectrum [14]: matter, ηR I
Δm (8–9) GeV for mL = (60–73) GeV,
(59)
Δm (9–12) GeV for mL = (73–75) GeV.
(60)
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255
Here Δm ≡ mN L − mL with mL and mN L being the lightest and the next lightest masses between 0 and η0 . Note ηR I 1 m ¯ η ≡ (mL + mN L ), 2 2 1 2 δm ≡ m − m2 , η NL L 2 we thus deduce, 1 |δm2η | ¯η 1− mL = m , 2 m ¯ 2η Δm =
|δm2η | m ¯η
(61) (62)
(63) (64)
.
In the previous discussions of TeV leptogenesis with κ = 0, we take MΔ = 10 TeV, |μ| = 1 GeV, |μ| ˜ = 103 TeV and m ¯ η = 70 GeV. It is straightforward to see |δmη | 25 GeV from (33). Therefore, we obtain mL 66 GeV and Δm 9 GeV, which is consistent with the mass spectrum (59). 0 and η0 are expected to be produced in pairs by the standard model gauge bosons W ± , Z ηR I 0 and a virtual or γ and hence can be verified at the LHC. Once produced, η± will decay into ηR,I 0 is lighter W ± , which becomes a quark–antiquark or lepton–antilepton pair. For example, if ηR 0 than ηI , the decay chain η+ → ηI0 l + ν,
0 + − then ηI0 → ηR l l ,
(65)
has 3 charged leptons and large missing energy, and can be compared to the direct decay 0 + l ν η + → ηR
(66)
to extract the masses of the respective particles. As for the phenomenology of the Higgs triplet at the LHC as well as the ILC, it has been discussed in [23]. The same-sign dileptons will be the most dominating modes of the δ ++ . Complementary measurements of |fij | at the ILC by the process e+ e+ (μ+ μ− ) → li− lj− would allow us to study the structure of the neutrino mass matrix in detail. For χ = O(TeV), which is natural to give the TeV Majorana masses of the right-handed neutrinos and then realize the TeV leptogenesis, the mixing angle ϑ and the splitting between h1,2 may be large. Furthermore, the couplings of h1,2 to W and Z bosons, quarks and charged leptons have essentially the same structure as the corresponding Higgs couplings in the standard model, however, their size is reduced by cos ϑ and sin ϑ , respectively. In the extreme case ϑ = π2 , the couplings of the lighter physical boson h1 to quarks and leptons would even vanish. In other words, this mixing could lead to significant impact on the Higgs searches at the LHC [24,25]. 7. Summary We propose a new model to realize leptogenesis and dark matter at the TeV scale. A real scalar is introduced to naturally realize the Majorana masses of the right-handed neutrinos. Furthermore, we also consider a new Higgs doublet to provide the attractive candidates for dark matter. Since the right-handed neutrinos have no responsibility to generate the neutrino masses,
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