Leptogenesis, radiative neutrino masses and inert Higgs triplet dark ...

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Sep 6, 2016 - Björkeroth, F.J. de Anda, I. de Medeiros Varzielas, and. S.F. King, JHEP ... Zhang, JCAP 1601, 012 (2016); J.M. Cline, A. Diaz-. Furlong, and J.
Leptogenesis, radiative neutrino masses and inert Higgs triplet dark matter Wen-Bin Lu∗ and Pei-Hong Gu†

arXiv:1603.05074v1 [hep-ph] 16 Mar 2016

Department of Physics and Astronomy, Shanghai Jiao Tong University, 800 Dongchuan Road, Shanghai 200240, China We extend the standard model by three types of inert fields including Majorana fermion singlets/triplets, real Higgs singlets/triplets and leptonic Higgs doublets. In the presence of a softly broken lepton number and an exactly conserved Z2 discrete symmetry, these inert fields together can mediate a one-loop diagram for a Majorana neutrino mass generation. The heavier inert fields can decay to realize a successful leptogenesis while the lightest inert field can provide a stable dark matter candidate. As an example, we demonstrate the leptogenesis by the inert Higgs doublet decays. We also perform a systematic study on the inert Higgs triplet dark matter scenario where the interference between the gauge and Higgs portal interactions can significantly affect the dark matter properties. PACS numbers: 98.80.Cq, 14.60.Pq, 95.35.+d

I.

INTRODUCTION

The atmospheric, solar, accelerator and reactor neutrino experiments have established the phenomena of neutrino oscillations, which reveal that three flavors of neutrinos should be massive and mixed [1]. Hence we need new physics beyond the SU (3)c × SU (2)L × U (1)Y standard model (SM) where the neutrinos are massless. Meanwhile, the cosmological observations indicate that the neutrino masses should be below the eV scale [1]. Along the lines to naturally generate the tiny neutrino masses, the famous seesaw [2] mechanism stands out as one of the most compelling paradigms. In the usual seesaw models [2–6], the neutrino masses are induced by some lepton-number-violating interactions, through which a lepton asymmetry could be produced and then converted to a baryon asymmetry by virtue of the sphaleron [7] processes. Thanks to this framework, one gets to understand the cosmic matter-antimatter asymmetry which is tantamount to the baryon asymmetry. This baryogensis scenario within the seesaw context is the well known leptogenesis [8] mechanism and has been extensively studied [9–21]. On the other hand, the existence of non-baryonic dark matter (DM) poses another challenge to particle physics and cosmology [1]. There have been a number of DM candidates in the literature. The particles for the DM may also play an essential role in the generation of the neutrino masses [22–27] and even the origin of the baryon asymmetry [23]. For example [23], one extends the SM by two or more gauge-singlet fermions and a second isodoublet Higgs scalar, which are odd under an exactly conserved Z2 discrete symmetry, to simultaneously explain the puzzles of the neutrino masses, the baryon asymmetry and the DM. Specifically, the new Higgs doublet can provide a real scalar to be a stable DM particle, while the new fermions with heavy Majorana masses can highly

∗ Electronic † Electronic

address: [email protected] address: [email protected]

suppress the radiative neutrino masses and their decays can realize a successful leptogenesis. In this paper, we will consider a class of models with Majorana fermion singlets/triplets, real Higgs singlets/triplets and leptonic Higgs doublets. Our models respect a softly broken lepton number and an exactly conserved Z2 discrete symmetry, so that they can only give the Majorana neutrino masses at one-loop level. The interactions for the neutrino mass generation can also allow the decays of the heavier non-SM fields to produce a lepton asymmetry stored in the SM leptons and then realize the leptogenesis. Furthermore, the lightest non-SM field can provide a stable DM candidate. As an example, we will demonstrate the leptogenesis scenario by the inert Higgs doublet decays. We will also perform a systematic study on the inert Higgs triplet DM scenario where the Higgs portal interaction can significantly affect the dark matter annihilation and scattering.

II.

THE MODELS

The non-SM gauge-singlet/iso-triplet fermions, gaugesinglet/iso-triplet Higgs scalars and iso-doublet Higgs scalars are denoted by √ TL0 / 2

# TL+ NR (1, 1, 0)/TL (1, 3, 0) = ; √ TL− −TL0 / 2 # " 0 √ Σ / 2 Σ+ = Σ† ; χ(1, 1, 0)/Σ(1, 3, 0) = √ − 0 Σ −Σ / 2 # " 0 # " 1 0 √ (η + iη 0 ) η I 2 R 1 η(1, 2, − 2 ) = . (1) = η− η− "

Here and thereafter the brackets following the fields describe the transformations under the SU (3)c × SU (2)L × U (1)Y gauge groups. Our models also contain a Z2 discrete symmetry under which these non-SM fields are odd

2 while the SM fields are even, i.e. Z

Z

2 2 (NR /TL ; χ/Σ; η) −→ −(NR /TL ; χ/Σ; η) , SM −→ SM . (2)

Among the non-SM fields, the Higgs doublets η carry a lepton number the same as that of the SM leptons, while the others do not. We require the Z2 symmetry to be exactly conserved while allowing the lepton number to be softly broken. With all the considerations stated, four options to extend the SM emerge: • the fermion singlets + Higgs singlets + Higgs doublets (SSD) model, 1 ¯ N c − (M 2 ) η † η LSSD ⊃ − (MN )ij N Ri Rj η ij i j 2 † −ρij ηi χj φ − yαij ¯ lLα NRi ηj + H.c. 1 − (Mχ2 )ij χi χj . 2

(3)

• the fermion singlets + Higgs triplets + Higgs doublets (STD) model, 1 ¯ N c − (M 2 ) η † η LSTD ⊃ − (MN )ij N Ri Rj η ij i j 2 √ † lLα NRi ηj + H.c. − 2ρij ηi Σj φ − yαij ¯ 1 (4) − (MΣ2 )ij Tr(Σi Σj ) . 2 • the fermion triplets + Higgs singlets + Higgs doublets (TSD) model, LTSD

1 c ⊃ − (MT )ij Tr(T¯Li iτ2 TLj iτ2 ) − (Mη2 )ij ηi† ηj 2 √ c iτ2 ηj + H.c. −ρij ηi† φχj − 2yαij ¯lLα iτ2 TLi 1 − (Mχ2 )ij χi χj . (5) 2

• the fermion triplets + Higgs triplets + Higgs doublets (TTD) model, LTTD

1 c ⊃ − (MT )ij Tr(T¯Li iτ2 TLj iτ2 ) − (Mη2 )ij ηi† ηj 2 √ √ c lLα iτ2 TLi iτ2 ηj + H.c. − 2ρij ηi† Σj φ − 2yαij ¯ 1 − (MΣ2 )ij Tr(Σi Σj ) . (6) 2

Here φ and lL are the SM Higgs and lepton doublets, " 0 # φ 1 , φ(1, 2, − ) = 2 φ− " # νLα 1 (α = e , µ , τ ) . (7) lLα (1, 2, − ) = 2 eLα

For simplicity, we shall not write down the full SM Lagrangian where the Higgs doublet φ has the potential as below, LSM ⊃ −µ2φ φ† φ − λ(φ† φ)2 .

(8)

We would like to emphasize that the lepton number is only allowed to be softly broken by the cubic coupling among the SM Higgs scalar φ and the non-SM Higgs scalars η and χ/Σ, i.e. the ρ-term in Eqs. (3-6). Meanwhile, the exactly conserved Z2 discrete symmetry will not be broken at any scales and hence the non-SM Higgs scalars will not develop any nonzero vacuum expectation values (VEVs). This Z2 symmetry has also forbidden the other gauge invariant terms involving the non-SM fields. In this sense, we will refer to the non-SM fields (1) as the inert fermion singlets/triplets, the inert Higgs singlets/triplets and the inert Higgs doublets, respectively. Without loss of generality and for the sake of convenience, we can choose the basis in which the Majorana mass matrix of the inert fermion singlets NR is real and diagonal, i.e. MN = diag{MN1 , ...} with MN1 < MN2 < ... .

(9)

Accordingly we can define the Majorana fermions as below, Ni = NRi + NRc i = Nic .

(10)

Similar procedures for the inert fermion triplets TL lead to MT = diag{MT1 , ...} with MT1 < MT2 < ... ,

(11)

and hence the physical states, Ti0 = TL0i + (TL0i )c = (Ti0 )c , Ti± = TL±i + (TL∓i )c = (Ti∓ )c .

(12)

For the same reason, we can rotate the inert Higgs scalars χ/Σ and η to diagonalize their mass terms, Mχ2 = diag{Mχ21 , ...} with Mχ21 < Mχ22 < ... , MΣ2 = diag{MΣ2 1 , ...} with MΣ2 1 < MΣ2 2 < ... ; Mη2 = diag{Mη21 , ...} with Mη21 < Mη22 < ... . (13) III.

INERT HIGGS TRIPLET DARK MATTER

The present models will invariantly select a stable particle from the inert fields. Since this stable particle leaves a relic density in the universe, it should be neutral and could be a viable DM candidate. Firstly, we consider the fermionic DM. In the SSD and STD models, the lightest inert fermion singlet NR can be the DM particle [22–25]. In this case, the DM fermion can annihilate into the SM leptons through the t-channel

3 exchange of the inert Higgs doublets. Therefore, the inert Higgs doublets cannot be too heavy while the Yukawa couplings cannot be too small. The detailed studies can be found in [28]. In the TSD and TTD models, the neutral component TL0 of the lightest inert fermion triplet TL can be the DM particle [26, 29]. Since the DM fermion now have the SU (2)L gauge couplings, its annihilation can be free of the Yukawa interactions. This means the inert Higgs doublets can be very heavy and the Yukawa couplings can be quite small, which indeed falls in the context of the minimal DM scenario [29]. Alternatively, the DM particle can be a scalar from the inert Higgs singlets/triplets and doublets. In fact, after the SM Higgs scalar φ develops its VEV to spontaneously break the electroweak symmetry, it can be written as " 1 # √ (h + v) 2 φ= , (14) 0 with h being the Higgs boson and v ≃ 246 GeV being the VEV. This gives rise to the mixing between the inert Higgs scalars χ/Σ and η due to the ρ-term in Eqs. (3-6), from which the DM scalar should be the lightest mass eigenstate obtained. To be simple and instructive, we hereby will only consider some limiting cases where the mixings could be essentially ignored. If the inert Higgs singlets χ or triplets Σ are much heavier than the lightest inert Higgs doublet η, we actually arrive at the inert Higgs doublet DM scenario which has been studied in a lot of literature [23, 29–31]. In the usual inert Higgs doublet DM scenario, the inert Higgs doublet η has a quartic coupling with the SM Higgs doublet φ, i.e. L ⊃ −ǫ(η † φ)2 + H.c. .

(15)

This term will induce the required mass split between the real and imaginary parts of η’s neutral component after the electroweak symmetry breaking. By integrating out the heavy inert Higgs singlets χ in the SSD and TSD models, or the heavy inert Higgs triplets Σ in the STD and TTD models, we can obtain the coupling, ǫ = −ρ

1 T 2 ρ . Mχ,Σ

(16)

In the SSD and TSD models, the lightest inert Higgs singlet χ can dominate the DM scalar if it is much lighter than the inert Higgs doublets η. Consequently, the gauge interactions of the DM scalar are negligible compared to the more significant Higgs portal interaction between the DM scalar and the SM Higgs scalar. This simple DM scenario has attracted many people [32, 33]. In the STD and TTD models, the lightest inert Higgs triplet Σ can be set much lighter than the inert Higgs doublets η. In this case we can work with the inert Higgs triplet Σ as an approximately physical state, i.e. ˆ 0 ≃ Σ0 , Σ ˆ ± ≃ Σ± . Σ

(17)

The SU (2)L radiative corrections will induce a mass split, making the charged components slightly heavier than the neutral one [29], mΣ ≫ ∆m = mΣ± − mΣ0      g 2 mΣ mW mZ 2 = f − cos θ f , W 16π 2 mΣ mΣ (18) where i  h √ 3 − 21 r4 ln r − r(r2 − 4) 2 ln r+ 2r2 −4 for r ≥ 2 , h i f (r) = √ + 1 r4 ln r + r(4 − r2 ) 23 arctan 4−r2 for r ≤ 2 . 2 r

(19)

For mΣ ≫ mZ,W , the radiative mass split ∆m can arrive at a determined value,   g2 θW 2 ∆m = mW = 167 MeV . (20) sin 4π 2 The Σ± − Σ0 mass difference can be also induced at tree level due to the mixing between the inert Higgs triplet Σ and the inert Higgs doublets η. However, we have checked that this tree-level contribution should be far below the radiative corrections provided that the inert Higgs doublets η are much heavier than the inert Higgs triplet Σ. With the radiative mass split (20), the charged Σ± can decay into the neutral Σ0 with a virtual W ± before the Big Bang Nucleosynthesis (BBN) epoch. Therefore, the stable Σ0 can serve as the DM particle. In the following we shall perform a systematic study on this inert Higgs triplet DM scenario [29, 34, 35]. Particular emphasis will be placed on investigating some interesting implications arising from the Higgs portal interaction between the inert Higgs triplet Σ and the SM Higgs doublet φ, i.e. 1 1 LSTD/TTD ⊃ − κ1 φ† φTrΣ2 − κ3 φ† Σ2 φ . 2 2

(21)

Subjected to the stability and perturbativity requirements, the Higgs portal coupling κ should be in the range as below, p 1 −2.6 . −2 λξ ≤ κ ≡ κ1 + κ3 < 4π for 2 1 (22) ξ ≡ ξ1 + ξ3 < 4π , λ ≃ 0.13 . 2 Here the new parameters ξ1,3 are the self quartic couplings of the Higgs triplet Σ, i.e. 1 1 LSTD/TTD ⊃ − ξ1 [Tr(Σ2 )]2 − ξ3 Tr(Σ4 ) . 4 4 As for the choice λ ≃ 0.13, it is given by λ≃

(23)

m2h ≃ 0.13 for mh = 125 GeV , v = 246 GeV .(24) 2v 2

We will explain this point later.

4 30

Dark matter mass

Since the Σ± and Σ0 scalars now are highly quasidegenerate, all the following annihilation and coannihilation channels, Σ0 Σ0 Σ0 Σ± Σ± Σ∓ Σ± Σ±

→ → → →

W + W − , φ0∗ φ0 , φ+ φ− ; W 3 W ± , f ′ f¯ , φ0 φ+ , φ0∗ φ− ; W ± W ∓ , f f¯ , φ0∗ φ0 , φ+ φ− ; W ±W ± ,

and then obtain an effective cross section [38], 4 1 hσ 0 0 v i + hσΣ0 Σ± vrel i 9 Σ Σ rel 9 2 2 + hσΣ+ Σ− vrel i + hσΣ± Σ± vrel i 9 9     g4 47 κ2 3 = 1 − + 1 − , 8πm2Σ 12x 48πm2Σ x (27)

hσeff vrel i =

where we have defined (28)

The DM relic density then can be well approximated to [36, 39] ΩDM h2 ≃

1.07 × 109 GeV−1 1/2

J(xF )g∗ MPl

,

(29)

with MPl ≃ 1.22 × 1019 GeV being the Planck mass, g∗ ≃ 106.75 being the number of the relativistic degrees of freedom at the freeze-out point, while J(xF ) being an integral, Z ∞ hσeff vrel i J(xF ) = dx , (30) x2 xF determined by the freeze-out point, xF = ln

3 × 0.038 MPlmΣ hσeff vrel i 1/2 1/2

g∗ xF

,

(31)

20 15 10 5

(25)

should be taken into account in determining the relic density of the DM particle Σ0 [36, 37]. Here f and f ′ denote the SM fermions. Up to the p-wave contributions, we calculate the thermally averaged cross sections of the above annihilations and co-annihilations,     g4 5 κ2 3 hσΣ0 Σ0 vrel i = 1− + 1− , 4πm2Σ x 16πm2Σ x   g4 5 hσΣ0 Σ± vrel i = 1− , 16πm2Σ 4x     51 3 κ2 3g 4 1− 1− + , hσΣ+ Σ− vrel i = 16πm2Σ 4x 16πm2Σ x   5 g4 1 − , (26) hσΣ± Σ± vrel i = 8πm2Σ x

m x≡ Σ. T

25

mΣ (TeV)

A.

0 -5

0

5

10

15

κ FIG. 1: The correlation between the dark matter mass mΣ and the Higgs portal coupling κ. The dot, dash and solid vertical lines correspond to κ = −2.6, 0 and 4π, respectively. We have mΣ = 5.2 TeV, mΣ = 2 TeV and mΣ = 23.6 TeV for κ = −2.6, κ = 0 and κ = 4π, respectively.

at which the annihilations and co-annihilations become slower than the expansion rate of the universe. From Eqs. (27-31), we can easily understand that the present DM relic density ΩDM h2 = 0.1188 ± 0.0010 [40] only depends on two parameters: the mass of the DM scalar Σ0 and the Higgs portal coupling between the inert Higgs triplet Σ and the SM Higgs doublet φ. In Fig. 1, we show the correlation between the DM mass mΣ and the Higgs portal coupling κ. Specifically, mΣ will decrease from 5.2 TeV to 2 TeV when κ increases from −2.6 to 0, subsequently, mΣ will increase to 23.6 TeV when κ increases to 4π.

B.

Dark matter direct detection

As shown in Fig. 2, the DM scalar Σ0 can scatter off a nuclei at tree and loop level. Note the tree-level effect is only induced by the Higgs portal interaction between the inert Higgs triplet Σ and the SM Higgs doublet φ. We have performed an improved computation on the spin-independent DM-nucleon scattering cross section incorporating all interfering channels, σSI

  2 2 g 8 fN 16πκ mW 1 m4N 1 = + 2 1− 4 256π 3 m2W m2W mh g mΣ for mΣ ≫ mW ≫ mN . (32)

Here mN ≃ 1 GeV is the nucleon mass, fN ≃ 0.3 [41] is the effective coupling of the Higgs boson to the nucleon. The cross section σSI is a function of the two correlated parameters: the DM mass mΣ and the Higgs portal coupling κ. Remarkably, Eq. (32) gives a zero point, σSI = 0 ⇒ κ =

g 4 mΣ 16πmW

  m2 1 + 2h . mW

(33)

5 Σ±

Σ0 W

Σ0 W

q q

q



W

q

Σ0

Σ0

W

W q



Σ0 W

Σ0 W

q q

Σ0

W

Σ±

Σ0

W q

Σ±

Σ0

Σ±

Σ0

q



Σ0

Σ0

W h

h

h

q

q

q

q

q

q

10-43

10-43

10-44

10-44

10-45

10-45

σSI (cm2)

σSI (cm2)

FIG. 2: The effective couplings of the dark matter scalar to the SM quarks. The tree-level effect is only induced by the Higgs portal interaction.

10-46 10-47

10-46 10-47

10-48

10-48

10-49

10-49

10-50 -5

0

5

10

15

κ

10-50

0

5

10

15

20

25

mΣ (TeV)

FIG. 3: The dependence of the DM-nucleon scattering cross section σSI on the Higgs portal coupling κ and the DM mass mΣ . In the left panel, the dot, dash and solid vertical lines correspond to κ = −2.6, κ = 0 and κ = 4π, respectively. In the right panel, the dot curve is for κ < 0 and the solid curve is for κ > 0, while the dot, dash and solid vertical lines correspond to mΣ = 2 TeV, mΣ = 5.2 TeV and mΣ = 23.6 TeV, respectively. We have σSI = 1.8 × 10−44 cm2 for κ = −2.6 and mΣ = 5.2 TeV, σSI = 0.9 × 10−45 cm2 for κ = 0 and mΣ = 2 TeV, while σSI = 5 × 10−45 cm2 for κ = 4π and mΣ = 23.6 TeV.

Although the above extreme condition would not be exactly accessible constrained by the correlation between the DM mass mΣ and the coupling κ, our result indeed exhibits the intriguing property that the spinindependent cross section σSI might be highly suppressed for some choice of κ and mΣ . In Fig. 3, we show the dependence of the DM-nucleon scattering cross section σSI on the Higgs portal coupling κ and the DM mass mΣ . We find the coupling κ can significantly affect the cross section σSI . For example, we read σSI ≃ 1.8 × 10−44 cm2 , 0.9 × 10−45 cm2 and 5 × 10−45 cm2 for κ = −2.6, 0 and 4π, respectively. The cross section σSI could even drastically decrease to an extremely small value for κ → 0.3 or mΣ → 2.05 TeV. C.

Higgs phenomenology

With the presence of the Higgs portal interaction, we can realize a one-loop diagram mediated by the inert Higgs triplet Σ to give a dimension-6 operator of the SM

Higgs doublet φ [43]. By integrating out the inert Higgs triplet Σ, we obtain V = µ2φ φ† φ + λ(φ† φ)2 +

3κ3 (φ† φ)3 . 16π 2 m2Σ

(34)

By minimizing this potential, we have µ2φ + 3λv 2 +

9κ3 v 4 = 0. 64π 2 m2Σ

(35)

Then the quadratic and trilinear terms of the Higgs boson h could be extracted, 1 L ⊃ − m2h h2 − λeff vh3 with 2 9κ3 v 4 2 mh = 2λv 2 + , 16π 2 m2Σ λeff = λ +

3κ3 v 2 m2 15κ3 v 2 = h2 + . (36) 2 2 32π mΣ 2v 16π 2 m2Σ

6 0.04

0.03

0.03

0.02

0.02





0.04

0.01

0.01

0.00

0.00

-0.01 -5

0

5

10

15

-0.01 0

5

10

15

20

25

mΣ (TeV)

κ

FIG. 4: The deviation Rλ of the trilinear coupling of the SM Higgs boson from its SM value versus the Higgs portal coupling κ and the DM mass mΣ . In the left panel, the dot, dash and solid vertical lines correspond to κ = −2.6, κ = 0 and κ = 4π, respectively. In the right panel, the solid curve is for κ > 0 and the dot curve is for κ < 0, while the dot, dash and solid vertical lines correspond to mΣ = 2 TeV, mΣ = 5.2 TeV and mΣ = 23.6 TeV, respectively. We have Rλ = −0.014 for κ = −2.6 and mΣ = 5.2 TeV, Rλ = 0 for κ = 0 and mΣ = 2 TeV, while Rλ = 0.032 for κ = 4π and mΣ = 23.6 TeV.

The trilinear coupling of the Higgs boson yields a deviation from the SM value, 3 4

Rλ =

3κ v λeff − λSM = 2 2 2 with λSM = λSM 8π mΣ mh

m2h 2v 2

. (37)

We also check the Higgs to diphoton decay [42],

Γ (h → γγ) ΓSM (h → γγ) 2 A0 (τΣ ) κ v2 = 1 + with 4 2 2 mΣ A1 (τW ) + 3 A 1 (τt )

Rγγ ≡

2

A0 (x) = A1 (x) = A 1 (x) = 2

RADIATIVE NEUTRINO MASSES

As shown in Fig. 6, the left-handed neutrinos can obtain a Majorana mass term,

In Fig. 4, we show the dependence of this deviation Rλ on the Higgs portal coupling κ and the DM mass mΣ . We find these deviations are consistent with the experimental limits [44]. Specifically, we note Rλ = −1.4% for κ = −2.6 and mΣ = 5.2 TeV, Rλ = 0 for κ = 0 and mΣ = 2 TeV, while Rλ = 3.2% for κ = 4π and mΣ = 23.6 TeV. It is clear now that these numerical results should be in the convincing magnitude to explain why Eq. (24) is a good approximation to determine the low limit κ = −2.6.

τX =

IV.

m2 4 X , m2h   1 2 2 1 , −x − arcsin √ x x     2 3 2 2 1 2 −x , + +3 − 1 arcsin √ x2 x x x     1 1 1 . (38) 2x2 + − 1 arcsin2 √ x x x

As shown in Fig. 5, the deviation from the SM prediction is always negligible.

1 L ⊃ − ν¯L mν νLc + H.c. , 2

(39)

after the electroweak symmetry breaking. We take a unitary rotation as below,  0   0   ηˆR ηR Uη0 ηˆ0 Uη0 ηˆ0 Uη0 Sˆ R R R I R  0   0    η  =  Uη0 ηˆ0 Uη0 ηˆ0 U 0 ˆ   ηˆI  , (40) ηI S     I   I R I I US ηˆ0 US ηˆ0 US Sˆ S Sˆ R I to diagonalize the mass matrix of the neutral scalars χ/Σ 0 and ηR,I . The neutrino masses then can be exactly computed by ! (" Uη0 ηˆ0 m2ηˆ0 UηT0 ηˆ0 m2ηˆ0 1 R R R R R R mν = yMF ln 16π 2 m2ηˆ0 − MF2 MF2 R ! m2ηˆ0 Uη0 ηˆ0 m2ηˆ0 UηT0 ηˆ0 I R I I R I ln + m2ηˆ0 − MF2 MF2 I ! 2 Uη0 Sˆ m2Sˆ UηT0 Sˆ m ˆ R S  + R2 ln mSˆ − MF2 MF2 ! " m2ηˆ0 Uη0 ηˆ0 m2ηˆ0 UηT0 ηˆ0 R I R R I R ln − m2ηˆ0 − MF2 MF2 R ! Uη0 ηˆ0 m2ηˆ0 UηT0 ηˆ0 m2ηˆ0 I + I 2I I 2I I ln mηˆ0 − MF MF2 I ! 2  Uη0 Sˆ m2Sˆ UηT0 Sˆ m ˆ S I  yT , (41) ln + I2  m ˆ − MF2 MF2 S

1.0004

1.0004

1.0002

1.0002

Rγγ

Rγγ

7

1.0000 0.9998

1.0000 0.9998

0.9996

0.9996

-5

0

5

10

0

15

5

10

15

20

25

mΣ (TeV)

κ

FIG. 5: The deviation Rγγ of the Higgs decay to diphoton from its SM value versus the Higgs portal coupling κ and the DM mass mΣ . In the left panel, the dot, dash and solid vertical lines correspond to κ = −2.6, κ = 0 and κ = 4π, respectively. In the right panel, the dot curve is for κ < 0 and the solid curve is for κ > 0, while the dot, dash and solid vertical lines correspond to mΣ = 2 TeV, mΣ = 5.2 TeV and mΣ = 23.6 TeV, respectively.

hφ0 i

hφ0 i

hφ0 i

hφ0 i

χ/Σ0 η0

χ/Σ0 η0

νL

η0 νL

NR

νL

NR

η0 TL0

TL0

νL

FIG. 6: The one-loop diagrams for generating the Majorana neutrino masses.

and

where S denotes χ/Σ while F standards for N/T .

v 2 X yαij yαik MFi ρjl ρkl 32π 2 Mη2j − Mη2k ijkl ! " MS2 1 l ln × Mη2j − MS2 Mη2j l !# MS2 1 l − 2 ln Mηk − MS2 Mη2k

(mν )αβ ≃ − The above formula can be simplified under some limiting conditions. For example, we can obtain

l

for Mη2 , MS2 ≫ MF2 . (mν )αβ

By further assuming

v 2 X yαij yαik MFi ρjl ρkl ≃ − 32π 2 Mη2j − Mη2k ijkl ! " MF2 1 i ln × Mη2j − MF2 Mη2j i !# MF2 1 i ln − 2 Mηk − MF2 Mη2k

Mη2 ≫ MF2 , MS2 ,

(44)

the neutrino masses can have a more simplified form, v 2 X yαij yαik MFi ρjl ρkl for 16π 2 Mη2j Mη2k ijkl ! Mη = O(1) . (45) ln MF,S

(mν )αβ ≃ −

i

for Mη2 , MF2 ≫ MS2 ,

(43)

(42)

8 We then can parametrize the Yukawa couplings y by y = i

1 4π p 1 ˆ ν O Mη2 p U m with v ρ MF

mν = U m ˆ ν U T = U diag{m1 , m2 , m3 }U T , OOT = OT O = 1 .

(46)

As the inert Higgs doublets η mediate a quartic coupling between the inert Higgs singlets/triplets χ/Σ and the SM Higgs doublet φ, i.e. L ⊃ −ρ†

1 ρφ† S 2 φ , Mη2

• In the SSD model, Γηi ≡ Γηi∗

(47)

the cubic coupling ρ should favor a perturbative requirement, √ (48) ρ < 4πMη . Applying this constraint to Eq. (45), we find   v2 MF /1012 GeV 2 13 2 Mη . M = (2.5 × 10 GeV) mν F mν /0.1 eV √ (49) for y < 4π . Note the simple formula (45) means the models should contain at least (i) one inert fermion singlet/triplet, one inert Higgs singlet/triplet and two inert Higgs doublets; (ii) one inert fermion singlet/triplet, two inert Higgs singlets/triplets and one inert Higgs doublet; (iii) two inert fermion singlets/triplets, one inert Higgs singlet/triplet and one inert Higgs doublet, in order to give two or more non-zero neutrino mass eigenvalues. V.

doublets can generate a lepton asymmetry stored in the SM leptons. We calculate the decay widths at tree level and the CP asymmetries at one-loop order in the SSD, STD, TSD and TTD models, respectively.

LEPTOGENESIS

It is well known that we can realize a leptogenesis through the CP-violating decays of the inert fermion singlets/triplets into the inert Higgs doublets and the SM lepton doublets if these inert fermions are heavy enough [23]. Another possibility also holds for the decays of the inert Higgs singlets/triplets into the inert Higgs doublets and the SM Higgs doublet. The produced asymmetry stored in the inert Higgs doublet pairs will eventually turn into a lepton asymmetry stored in the SM leptons after the inert Higgs doublets decay into the SM lepton doublets and the inert fermion singlets/triplets. Alternatively, we can implement a leptogenesis making use of the decays of the inert Higgs doublets. We will focus on this leptogenesis scenario and illustrate its main aspects in the following. As shown in Fig. 7, the inert Higgs doublets can have two decay channels: one is into the inert Higgs singlets/triplets and the SM Higgs doublet, the other is into the SM lepton doublets and the inert fermion singlets/triplets. Therefore, the decays of the inert Higgs

Γηi ≡ Γηi∗ ≡

εηi ≡ =

X αj

X αj

P

# " (ρρ† )ii 1 † (y y)ii + M = 16π ηi Mη2i q 1 (y † y)ii (ρρ† )ii with ≥ 8π c [Γ(ηi → lLα + NRj ) + Γ(ηi → φ + χj )] , c [Γ(ηi∗ → lLα + NRj ) + Γ(ηi∗ → φ∗ + χj )] ,

(50)

αj [Γ(ηi

c c → lLα + NRj ) − Γ(ηi → lLα + NRj )]

Γηi

1 X Im[(y † y)ij (ρρ† )ji ] 1 (ρρ† )ii M 2 − M 2 † 4π ηi ηj j6=i (y y)ii + M 2 η



i



Mηi 1 X Im[(y y)ij (ρρ )ji ] p ≤ 2 † † 8π (y y)ii (ρρ )ii Mηi − Mη2j j6=i

(51)

• In the STD model, Γηi ≡ Γηi∗

Γηi ≡ Γηi∗ ≡

εηi ≡

X αj

X αj

P

# " † (ρρ ) 1 ii (y † y)ii + 3 M = 16π ηi Mη2i √ q 3 ≥ (y † y)ii (ρρ† )ii with 8π c [Γ(ηi → lLα + NRj ) + Γ(ηi → φ + Σj )] , c [Γ(ηi∗ → lLα + NRj ) + Γ(ηi∗ → φ∗ + Σj )] ,

αj [Γ(ηi

(52)

c c → lLα + NRj ) − Γ(ηi → lLα + NRj )]

Γηi

3 X Im[(y † y)ij (ρρ† )ji ] 1 = †) 2 2 (ρρ ii 4π (y † y) + 3 2 Mηi − Mηj j6=i



ii †



i

Mηi 3 X Im[(y y)ij (ρρ† )ji ] p ≤ . 2 8π (y † y)ii (ρρ† )ii Mηi − Mη2j j6=i

(53)

9 • In the TSD model,

where the parameters C, m ˜ i and mmax are defined by "

(ρρ† )ii 1 Mηi 3(y † y)ii + 16π Mη2i √ q 3 ≥ (y † y)ii (ρρ† )ii with 8π

Γηi ≡ Γηi∗ =

Γηi ≡ Γηi∗ ≡

εηi ≡

X αj

X αj

#

c c [Γ(ηi∗ → lLα + TLj ) + Γ(ηi∗ → φ∗ + χj )] ,

(54)

P

αj [Γ(ηi

c c → lLα + TLj ) − Γ(ηi → lLα + TLj )]

Γηi





1 3 X Im[(y y)ij (ρρ )ji ] †) 2 2 (ρρ ii M 4π † ηi − Mηj j6=i 3(y y)ii + Mη2 i √ Mηi 3 X Im[(y † y)ij (ρρ† )ji ] p ≤ . 2 − M2 † † 8π M (y y)ii (ρρ )ii ηi ηj j6=i

• In the TTD model,

Γηi∗ ≡

εηi ≡ =

αj

X

(55)

#

[Γ(ηi∗

αj [Γ(ηi



c lLα

+

c TLj )

+

Γ(ηi∗



→ φ + Σj )] ,

c c → lLα + TLj ) − Γ(ηi → lLα + TLj )]

Γηi



1 3 X Im[(y y)ij (ρρ )ji ] (ρρ† )ii M 2 − M 2 † 4π ηi ηj j6=i (y y)ii + M 2 η

i

Mηi 3 X Im[(y † y)ij (ρρ† )ji ] p ≤ . 2 − M2 † † 8π M (y y)ii (ρρ )ii ηi ηj j6=i

(60)

Instead of fully integrating the Boltzmann equations to determine the final baryon asymmetry, we adopt an instructive and reliable estimation for demonstration, assuming a hierarchical spectrum for the inert Higgs doublets. Consequently, the final baryon asymmetry should be mostly produced by the decays of the lightest inert Higgs doublet η1 . We define Γη1 K = , (61) 2H(T ) T =Mη1 where H(T ) is the Hubble constant,  3  12 8π g∗ T2 H= , 90 MPl

with zf =

(56)



m ˜ i ≡ (O† m ˆ ν O)ii ∈ (mmin , mmax ) ,  1 in the SSD models ,   √ C= 3 in the STD/TSD models ,    3 in the TTD models .

(62)

with g∗ being the relativistic degrees of freedom during the leptogenesis epoch. For 1 ≪ K . 106 , the final baryon asymmetry can well approximate to [39] εη1 28 n ×2 ηB = B ≃ − × s 79 g∗ Kzf

[Γ(ηi → lLα + TLj ) + Γ(ηi → φ + Σj )] ,

αj

P

"

(ρρ† )ii 3 Mηi (y † y)ii + 16π Mη2i q 3 (y † y)ii (ρρ† )ii with ≥ 8π

Γηi ≡ Γηi∗ =

X

mmin = min{m1 , m2 , m3 } ,

[Γ(ηi → lLα + TLj ) + Γ(ηi → φ + χj )] ,

=

Γηi ≡

mmax = max{m1 , m2 , m3 } ,

(57)

By making use of the parametrization (46), the above decay widths and CP asymmetries may be simplified. For example, in the models with one inert fermion singlet/triplet, one inert Higgs singlet/triplet and two or more inert Higgs doublets, we can derive s s 2 2 ˆ ν O)ii m ˜i C Mηi C Mηi (O† m = , (58) Γηi ≥ 2 v MF 2 v MF r C Mηi mmax for Mη2i ≪ Mη2j ,(59) |εηi | < εmax = ηi 2 v MF

Mη1 Tf

≃ 4.2(ln K)0.6 .

(63)

Here nB and s, respectively, are the baryon number density and the entropy density, Tf corresponds to the temperature when the processes damping the lepton asym28 is the sphaleron leptonmetry freeze out, the factor − 79 to-baryon coefficient, while the factor 2 appears because the decaying particle η1 is a doublet. To provide a numerical illustration, we consider the STD model with one inert fermion singlet, one inert Higgs triplet, two or more inert Higgs doublets. After taking g∗ = 109.75 (the SM fields plus one inert Higgs triplet) and setting the inputs, Mη1 = 2 × 1013 GeV , MF = 2 × 1012 GeV , mmax = 0.05 eV , m ˜ 11 = 0.01 eV ,

(64)

in Eqs. (58), (59) and (61), we read εmax = 0.352 , K = 2761 , ηi zf = 14.6 , Tf = 1.4 × 1012 GeV ,

(65)

and then obtain an expected baryon asymmetry [1], ! εη1 −10 ηB = 10 . (66) −0.002 εmax η1

10

φa

a lLα

ηia

+

ηia

Nk /Tk0

ηia

ηia

ηia

+

ηja

ηia

ηja

φb

b lLα

ηia

φa

a lLα

Σ± l

Nk /Tk0

φa +

+

ηja χl /Σ0l

b lLα

φb

a lLα

Nk /Tk0

χl /Σ0l

b lLα

φb ηia

ηja

Tk±

χ/Σ0l

Tk±

Σ± l

Tk±

Σ± l

ac lLα

φa

ac lLα

φb

ac lLα

φa∗

ηia∗

+

ηia∗ χl /Σ0l

Nk /Tk0

ηia∗

+ Tk∓

ηia∗

φb +

Tk∓

ηia∗

ηja Σ± l

bc lLα

ηja χ/Σ0l

ηia∗

Nk /Tk0

φa

ac lLα

+

ηja

ηia∗

Nk /Tk0

χl /Σ0l

bc lLα

φb∗ ηia∗

ηja Σ± l

Tk∓

Σ∓ l

FIG. 7: The decays of the inert Higgs doublets η into the inert Higgs singlets/triplets χ/Σ, the inert fermion singlets/triplets N/T as well as the SM Higgs doublet φ and the SM lepton doublets lL .

VI.

SUMMARY

In this paper, we have built a class of models by introducing the inert fermion singlets/triplets, the inert Higgs singlets/triplets and the inert Higgs doublets. In our models, the Majorana neutrino masses could only be induced through a one-loop diagram mediated by these inert fields, as a consequence of the softly broken lepton number and the exactly conserved Z2 discrete symmetry. The interactions for generating the neutrino masses can also accommodate the decays of the heavier inert fields into the lighter ones and the SM fields. Such decays can realize a successful leptogenesis to explain the baryon asymmetry in the universe. As an example, we have considered the inert Higgs doublet decays. While

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on the other side, the lightest inert field could provide a stable DM candidate. We have performed a systematic study on the inert Higgs triplet DM scenario with emphasis on investigating some phenomenological effects from the Higgs portal interaction. Our computation shows the interference between the Higgs portal and gauge interactions can result in a drastic decrease of the DM-nucleon scattering cross section. Acknowledgement: This work was supported by the Recruitment Program for Young Professionals under Grant No. 15Z127060004, the Shanghai Jiao Tong University under Grant No. WF220407201 and the Shanghai Laboratory for Particle Physics and Cosmology under Grant No. 11DZ2260700.

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