The Model

• The presence of the singlet scalar with non-zero vev helps in achieving the absolute stability of the EW vacuum. Here the mixing between singlet doublet scalars (we call it scalar mixing) plays an important role. Hence the combined analysis of DM phenomenology (where this scalar mixing also participates) and vacuum stability results in constraining this scalar mixing at a level which is even stronger than the existing limits on it from experiments.

The Chapter is organized as follows. In Sec. 6.2, we describe the singlet scalar extended SDDM model. Various theoretical and observational limits on the specified model are presented in Sec. 6.3. In the next section, we present our strategy, the related expressions including Feynman diagrams for studying dark matter phenomenology of this model. The discussion on the allowed parameter space of the model in terms of satisfying the DM relic density and direct detection limits are also mentioned in this Sec. 6.4. In Sec. 6.5, the strategy to achieve vacuum stability of the scalar enhanced singlet doublet model is presented. In Sec. 6.6, we elaborate on how to constrain parameters of the model while having a successful DM candidate with absolute vacuum stability within the framework. Finally the work is concluded with conclusive remarks in Sec. 6.7.

6.2 The Model

Like the usual singlet doublet dark matter model[430–433], here also we extend the SM frame- work by introducing two doublet Weyl fermions, ψ_D1, ψ_D2 and a singlet Weyl fermion fieldψ_S. The doublets are carrying equal and opposite hypercharges (Y = ¹₂(−¹₂) forψ_D1(D2)) as required from gauge anomaly cancellation. Additionally the scalar sector is extended by including a SM real singlet scalar field,φ. There exists aZ4 symmetry, under which only these additional fields are charged which are tabulated in Table 6.1. The purpose of introducing this Z₄ is two fold:

firstly it avoids a bare mass term for the ψS field. Secondly, although theZ4 is broken by the vev of theφfield, there prevails a residualZ₂under which all the extra fermions are odd. Hence the lightest combination of them is essentially stable. Note that with this construction, not only the DM mass involves vev ofφbut also the dark matter phenomenology becomes rich due to the involvement of two physical Higgs (as a result of mixing between φ and the SM Higgs doublet H). Apart from these,φis also playing a crucial role in achieving electroweak vacuum stability.

The purpose of φwill be unfolded as we proceed. For the moment, we split our discussion into two parts first as extended fermion and next as scalar sectors of the model.

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Chapter 6. Study of Higgs vacuum stability in a scalar extended singlet doublet dark matter model

Symmetry ψ_D1 ψ_D2 ψ_S φ

Z₄ -i i i -1

Table 6.1: Particle multiplets and their transformation properties underZ4 symmetry.

6.2.1 Extended fermion sector

The dark sector fermions ψ_D1, ψ_D2 and ψ_S are represented as,

ψD1 = ψ₁⁰ ψ₁⁻

!

: [2,1

2], ψD2 = ψ₂⁺ ψ₂⁰

!

: [2,−1

2], ψS : [1,0]. (6.1) Here field transformation properties under the SM (SU(2)_L×U(1)_Y) are represented within square brackets. The additional fermionic Lagrangian in the present framework is therefore given as

L_Dark =iψ_D^†

1σ¯^µDµψD1 +iψ_D^†

2σ¯^µDµψD2+ iψ^†_Sσ¯^µ∂_µψ_S−(m_ψ^abψ_D1aψ_D2b+1

2cφψ_Sψ_S+h.c.) , (6.2) whereD_µis the gauge covariant derivative in the Standard Model,D_µ=∂_µ−igW_µ^aσ₂^a−ig⁰Y B_µ. From Eq.(6.2) it can be easily observed that after φ gets a vev, the singlet fermion ψS in the present model receives a Majorana mass, m_ψS =chφi.

Apart from the interaction with the singlet scalarφ, the dark sector doubletψ_D1 and singlet ψ_S can also have Yukawa interactions with the Standard Model Higgs doublet,H. This Yukawa interaction term is given as

− L_Y=λψ_Sψ_D1H+h.c. . (6.3) Note that due toZ₄charge assignment,ψ_D2 does not have such Yukawa coupling in this present scenario. Once φ gets a vev vφ and the electroweak symmetry is broken (with H acquires a vev v/√

2, with v= 246 GeV), Eq.(6.3) generates a Dirac mass term for the additional neutral fermions. Hence including Eq.(6.2) and Eq.(6.3), the following mass matrix (involving the neutral fermions only) results

M=









m_ψS ^√1

2λv 0

√1

2λv 0 m_ψ

0 m_ψ 0









. (6.4)

6.2. The Model 137 The matrix is constructed with the basis X^T = (ψS, ψ⁰₁, ψ₂⁰). On the other hand, the charged components have a Dirac mass term, m_ψψ⁻₁ψ₂⁺+h.c..

In general the mass matrix M could be complex. However for simplicity, we consider the parameters m_ψS, λ and m_ψ to be real. By diagonalizing this neutral fermion mass matrix, we obtain V^TMV =diag(mχ1, mχ2, mχ3), where the three physical states P^T = (χ1, χ2, χ3) are related to X by,

X_i =V_ijP_j, (6.5)

where V is diagonalizing matrix ofM. Then the corresponding real mass eigenvalues obtained at the tree level are given as [432, 454]

mχ1 =−B 3A− 2

3A R

2 1/3

cosθm, (6.6)

m_χ2 =−B 3A+ 1

3A R

2 1/3

(cosθ_m−√

3 sinθ_m), (6.7)

m_χ3 =−B 3A+ 1

3A R

2 1/3

(cosθ_m+√

3 sinθ_m), (6.8)

where A= 1, B=−m_ψS, C =−(m²_ψ+^λ2₂^v2), D=m²_ψm_ψS (provided the discriminant (∆) of M is positive). NowR and the angleθ_m can be expressed as

R=p

P²+Q², tan 3θ_m = Q

P , (6.9)

whereP = 2B³−9ABC+ 27A²Dand Q= 3√

3∆A, ∆ = 18ABCD−4B³D+B²C²−4AC³− 27A²D² is the discriminant of the matrix M. The lightest neutral fermion, protected by the unbroken Z2, can serve as a potential candidate for dark matter.

At this stage, one can form usual four component spinors out of these physical fields [430–

433]. Below we define the Dirac fermion (F⁺) and three neutral Majorana fermions (Fi=1,2,3) as,

F⁺ = ψ⁺_α (ψ⁻)^†α˙

!

, Fi = χ_iα (χ_i)^†α˙

!

, (6.10)

whereψ⁻₁ andψ⁺₂ are identified withψ⁻andψ⁺respectively. In the above expressions ofF⁺and F_i, α( ˙α) = 1,2 refers to upper (lower) two components of the Dirac spinor that distinguishes the left handed Weyl spinor from the right handed Weyl spinor [248]. Hence m_F⁺ = −m_ψ corresponds to the tree level Dirac mass for the charged fermion. Masses of the neutral fermions are then denoted as mFi = mχi. As we have discussed earlier, although the Z4 symmetry is broken by hφi, a remnant Z₂ symmetry prevails in the dark sector which prevents dark sector

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Chapter 6. Study of Higgs vacuum stability in a scalar extended singlet doublet dark matter model

fermions to have direct interaction with the SM fermions. This can be understood later from the Lagrangian of Eq.(6.22) which remains invariant if the dark sector fermions are odd under the remnant Z2 symmetry.

Now we need to proceed for finding out various interaction terms involving these fields which will be crucial in evaluating DM relic density and finding direct detection cross-sections.

However as in our model, there exists an extra singlet scalar, φ with non-zero vev, its mixing with the SM Higgs doublet also requires to be included. For that purpose, we now discuss the scalar sector of our framework.

6.2.2 Scalar sector

As mentioned earlier, we introduce an additional real singlet scalar φthat carries Z₄ charge as given in Table 6.1. The most general potential involving the SM Higgs doublet and the newly introduced scalar is given as

L_scalar(H, φ) =−µ²_H|H|²+λ_H|H|⁴−µ²_φ

2 φ²+λ_φ

4 φ⁴+λ_φH

2 |H|²φ². (6.11) After electroweak symmetry is broken andφ gets vev, these scalar fields can be expressed as

H= 0

√1

2(v+H₀)

!

, φ=v_φ+φ₀ . (6.12)

Minimization of the scalar potential leads to the following vevs of φand H given by v²_φ= 4µ²_φλ_H −2µ²_Hλ_φH

4λHλφ−λ²_φH , (6.13)

v²= 4µ²_Hλ_φ−2µ²_φλ_φH

4λ_Hλ_φ−λ²_φH . (6.14)

Therefore, after φ gets the vev and electroweak symmetry is broken, the mixing between the neutral component ofH and φwill take place (the mixing is parametrized by angleθ) and new mass or physical eigenstates will be formed. The two physical eigenstates (H1 and H2) can be obtained in terms of H₀ and φ₀ as

H1 =H0cosθ−φ0sinθ,

H2 =H0sinθ+φ0cosθ, (6.15)

6.2. The Model 139 where θis the scalar mixing angle defined by

tan 2θ= λ_φHvv_φ

−λ_Hv²+λ_φv_φ². (6.16)

Similarly the mass eigenvalues of these physical scalars at tree level are found to be

m²_H1 =λφv_φ²(1−sec 2θ) +λHv²(1 + sec 2θ), (6.17) m²_H2 =λφv_φ²(1 + sec 2θ) +λHv²(1−sec 2θ). (6.18) Using Eqs.(6.16-6.18), the couplings λ_H, λ_φ and λ_φH can be expressed in terms of the masses of the physical eigenstates H1 and H2, the vevs (v,v_φ) and the mixing angle θas

λ_H =m²_H

1

4v² (1 + cos 2θ) +m²_H

2

4v² (1−cos 2θ), (6.19)

λ_φ=m²_H

1

4v²_φ(1−cos 2θ) +m²_H

2

4v²_φ (1 + cos 2θ), (6.20) λφH = sin 2θ

m²_H2 −m²_H1 2vv_φ

. (6.21)

Note that withH1as the SM Higgs, second term in Eq. (6.19) serves as the threshold correction to the SM Higgs quartic coupling. This would help λ_H to maintain its positivity at high scale.

Before proceeding for discussion of how this model works in order to provide a successful DM scenario and the status of electroweak vacuum stability, we first summarize relevant part of the interaction Lagrangian and the various vertices relevant for DM phenomenology and study of our model.

6.2.3 Interactions in the model

Substituting the singlet and doublet fermion fields of Eqs.(6.2-6.3) in terms of their mass eigen- states following Eq.(6.5) and using the redefinition of fields given in Eq.(6.10), gauge and Yukawa

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Chapter 6. Study of Higgs vacuum stability in a scalar extended singlet doublet dark matter model

interaction terms can be obtained as Lint = eA_µF¯⁺γ^µF⁺+ g

2cW

(c²_W−s²_W)Z_µF¯⁺γ^µF⁺+ g

√2 X

i

W_µ⁻(V_3i^∗F¯_iγ^µP_LF⁺−V_2iF¯_iγ^µP_RF⁺)

+ g

√2 X

i

W_µ⁺(V_3iF¯⁺γ^µP_LF_i−V_2i^∗F¯⁺γ^µP_RF_i)−1 2

X

ij

ReX_ijZ_µF¯_iγ^µγ⁵F_j+1 2

X

ij

ImX_ijZ_µF¯_iγ^µF_j

−1 2

X

i

F¯_iF_i[ReY_iicosθ−Re(cV_1i²) sinθ]H₁−1 2

X

i6=j

F¯_iF_j[Re(Y_ij+Y_ji) cosθ−Re(cV_1iV_1j) sinθ]H₁

−1 2

X

i

F¯_iF_i[ReY_iisinθ+ Re(cV_1i²) cosθ]H₂−1 2

X

i6=j

F¯_iF_j[Re(Y_ij+Y_ji) sinθ+ Re(cV_1iV_1j) cosθ]H₂

+1 2

X

i

F¯iiγ⁵Fi[ImYiicosθ−Im(cV_1i²) sinθ]H1+1 2

X

i

F¯iiγ⁵Fi[ImYiisinθ+ Im(cV_1i²) cosθ]H2

+1 2

X

i6=j

F¯_iiγ⁵F_j[Im(Y_ij+Y_ji) cosθ−Im(cV_1iV_1j) sinθ]H₁

+1 2

X

i6=j

F¯iiγ⁵Fj[Im(Yij+Yji) sinθ+ Im(cV1iV1j) cosθ]H2 . (6.22)

Here the expressions of different couplings are given as X_ij = g

2cW

(V_2i^∗V_2j−V_3i^∗V_3j), (6.23)

Yii =

√

2λV1iV2i, Yij+Yji =

√

2λ(V1iV2j+V1jV2i). (6.24) With the consideration that all the couplings involved in M are real, the elements of diago- nalizing matrix V in Eq.(6.5) become real[454] and hence the interactions proportional to the imaginary parts in Eq.(6.22) will disappear. Only real parts of X_ij, Y_ij will survive. From Eq.(6.22) we observe that the Largrangian remains invariant if a Z2 symmetry is imposed on the fermions. Therefore this residual Z₂ stabilises the lightest fermion that serves as our dark matter candidate.

The various vertex factors involved in DM phenomenology, generated from the scalar La- grangian, are

H₁ff , H¯ ₂ff¯ : m_f

v cosθ,m_f v sinθ H₁ZZ, H₂ZZ : 2m²_Z

v cosθg^µν,2m²_Z

v sinθg^µν H₁W⁺W⁻, H₂W⁺W⁻ : 2m²_Z

v cosθg^µν,2m²_Z

v sinθg^µν

H₁H₁H₁ : [6vλ_Hcos³θ−3v_φλ_φHcos²θsinθ+ 3vλ_φHcosθsin²θ−6v_φλ_φsin³θ]

H₂H₂H₂ : [6vλ_Hsin³θ+ 3v_φλ_φHcosθsin²θ+ 3vλ_φHcos²θsinθ+ 6v_φλ_φcos³θ]

6.3. Constraints 141