Constraints

6.3. Constraints 141

142

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

and three singly charged (W⁺H0, W⁺φ0, W⁺Z) combinations of two-particle initial and final states [362, 363, 456]. The perturbative unitarity limit can be derived by implement- ing the bound on the scattering amplitude M[362, 363, 456]

M<8π. (6.26)

The unitarity constraints are obtained as [362, 363, 456]

λ_H <4π, λ_φH <8π, and 1

4{12λ_H+λ_φ±q

16λ²_φH+ (λ_φ−12λ_H)²}<8π. (6.27)

• In addition, all relevant couplings in the framework should maintain the perturbativity limit. Perturbative conditions of relevant couplings in our set up appear as [354] ¹

λH < 2

3π, λφ< 2

3π, λφH <4π, λ <

√

4π, and c <

√

4π. (6.28)

We will ensure the perturbativity of the couplings present in the model till M_P energy scale by employing the renormalization group equations.

[B] Experimental Constraints:

• In the present singlet doublet dark matter model, dark matter candidate χ1 has coupling with the Standard Model HiggsH₁and neutral gauge bosonZ. Therefore, if kinematically allowed, the gauge boson and Higgs can decay into pair of dark matter particles. Hence we should take into account the bound on invisible decay width of Higgs and Z boson from LHC and LEP. The corresponding tree level decay widths of Higgs bosonH₁ and Z into DM is given as

Γ^inv_H1 = λ²_H

1χ1χ1

16π mH1 1−4m²_χ1 m²_H

1

!3/2

Γ^inv_Z = λ²_Zχ1χ1

24π m_Z 1−4m²_χ1 m²_Z

!3/2

, (6.29)

where the couplings, λH1χ1χ1 and λZχ1χ1 can be obtained from Eq.(6.22). The bound on Z invisible decay width from LEP is Γ^inv_Z ≤2 MeV at 95% C.L. [457] while LHC provides

1With a Lagrangian term likeλφiφjφkφl, the perturbative expansion parameter for a 2→2 process involving different scalarsφi,j,k,lturns out to beλ. Hence the limit isλ <4π[354]. Similarly with a termySfifjinvolving scalarSand fermionsf(i6=j), the corresponding expansion parameter is restricted byy²<4π[354]. Considering the associated symmetry factors (due to the presence of identical fields), we arrive at the limits mentioned at Eq.(6.28).

6.3. Constraints 143 bound on Higgs invisible decay and invisible decay branching fraction Γ^inv_H1/ΓH1 is 23%

[458].

• The mass of the SM gauge boson W gets correction from the scalar induced one loop diagram[364]. This poses stronger limit on the scalar mixing angle sinθ.(0.3−0.2) for 300 GeV< mH2 <800 GeV [371].

• Moreover, the Higgs production cross-section also gets modified in the present model due to mixing with the real scalar singlet. As a result, Higgs production cross-section at LHC is scaled by a factor cos²θ and the corresponding Higgs signal strength is given as R = _σ^σH1

SM

Br(H1→XX)

BrSM [370], where σ_SM is the SM Higgs production cross-section and Br_SM is the measure of the SM Higgs branching ratio to final state particles X. The simplified expression for the signal strength is given as [365–371, 459]

R = cos⁴θ Γ1

Γ^{T ot}_H

1

, (6.30)

where Γ1 is the decay width of H1 in the SM. In absence of any invisible decay (when m_χ1 > m_H1/2), the signal strength is simply given as R= cos²θ. SinceH₁ is the SM like Higgs with mass 125.09 GeV,R'1. Hence, this restricts the mixing between the scalars.

The ATLAS [457] and CMS [458] combined result provides

R= 1.09^+0.11_−0.10. (6.31)

This can be translated into an upper bound on sinθ.0.36 at 3σ.

Similarly, one can also obtain signal strength of the other scalar involved in the model expressed as R⁰ = sin⁴θ_Γ^ΓT ot²

H2

, where Γ₂ being the decay with of H₂ with massm_H2 in the SM and Γ^{T ot}_H2 is the total decay width of the scalar H2 given as Γ^{T ot}_H2 = sin²θ Γ2+ Γ^inv_H2 + Γ_H2→H1H1. The additional term Γ_H2→H1H1 appears whenm_H2 ≥2m_H1 and is expressed as ΓH2→H1H1 = ^λ

2 H1H1H2

32πmH2

r 1−^4m

2 H1

m²_H

2

, where λH1H1H2 can be obtained from Eq.( 6.25).

However due to small mixing with the SM Higgs H₁, R⁰ is very small to provide any significant signal to be detected at LHC [371].

• In addition, we include the LEP bound on the charged fermions involved in the singlet doublet model. The present limit from LEP excludes a singly charged fermion having mass below 100 GeV [460]. Therefore we consider m_ψ & 100 GeV. The LEP bound on the heavy Higgs state (having mass above 250 GeV) turns out to be weaker compared to the limit obtained from W boson mass correction [371].

144

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

• The presence of fermions in the dark sector and the additional scalar φ will affect the oblique parameters [461] S,T and U through changes in gauge boson propagators. How- ever only T parameter could have a relevant contributions from the newly introduced fields. Contributions to the T parameter by the additional scalar fieldφ can be found in [462]. However in the small mixing case, this turns out to be negligible [463] and can be safely ignored [432]. When we consider fermions, the corresponding T parameter in our model is obtained as [431, 464]

∆T =

3

X

i=1

h1

2(V_3i−V_2i)²A(m_ψ, m_i) +1

2(V_3i+V_2i)²A(m_ψ,−m_i)i

−

3

X

i,j=1

1

4(V_2iV_2j −V_3iV_3j)²A(m_i,−m_j), (6.32) where A(m_i, m_j) = _32α¹

emπv²

h

(m_i −m_j)²ln_m^Λ2⁴

im²_j −2m_im_j + ^2mimj(m

2

i+m²_j)−m⁴_i−m⁴_j m²_i−m²_j ln^m_m²ⁱ2

j

i and Λ is the cut off of the loop integral which vanishes during the numerical estimation.

• Furthermore, we also use the measured value of DM relic abundance by Planck experiment [39] and apply limits on DM direct detection cross-sections from LUX [185], XENON-1T [183], Panda 2018 [186] and XENON-nT [379] experiments to constrain the parameter space of the model. Detailed discussions on direct searches of dark matter have been presented later in Sec. 6.4.

In the above discussion, we infer that the scalar mixing angle sinθis restricted by sinθ.0.3, provided the mass of additional Higgs (m_H2) is around 300 GeV. For further heavierm_H2, sinθ is even more restricted, e.g. sinθ . 0.2 for m_H2 around 800 GeV. On the other hand, if we consider H1 to be lighter than the Higgs discovered at LHC, we need to identify H2 as the SM Higgs as per Eqs.(6.15,6.17-6.18) (where sinθ → 1 is the decoupling limit). In this case, the limit turns out to be sinθ&0.87 formH1 .100 GeV [371]. Note that this case is not interesting from vacuum stability point of view in this work for the following reason. From Eq. (6.19), we find the first term in right hand side serves as the threshold correction to the SM Higgs quartic coupling (contrary to the case with H₁ as the SM Higgs andH₂ as the heavier one, where the threshold correction is provided by the second term). However with m_H1 < m_H2 ≡ SM Higgs and sinθ&0.87, the contribution of the first term is much less compared to the second term.

Hence in this case, the SM Higgs quartic coupling λ_H cannot be enhanced significantly such

6.4. Dark matter phenomenology 145