Vacuum Stability

Now that we have developed the scalar sector by finding the spectrum of scalars for a particular set of values of the VEVs and obtained some constraints over the parameters of the potential due to Higgs invisible decay and lepton flavor violation, it is the moment to investigate the stability of the vacuum by finding the bound from below conditions and calculating the running of the self coupling of the Higgs.

4.7.1 Bounded from below conditions

In order to assure that the scalar potential of the 1-2-3 model is bounded from below at large field strength, where the potential is generically dominated by the Quartic terms, we need to find the set of conditions that guarantee that the parameters of the Quartic Couplings of the potential are positive when the fields go to infinity. We find the whole set of conditions and paved the way for similar models. In this subsection, we

techniques employed in [92, 93, 94].

Firstly, we separate the quartic couplings of the potential,

V⁴ =λ1(Φ^†Φ)²+λ2[tr(∆^†∆)]²+λ3Φ^†Φtr(∆^†∆) +λ4tr(∆^†∆∆^†∆) +λ5(Φ^†∆^†∆Φ) +β₁(σ^∗σ)²+β₂Φ^†Φσ^∗σ+β₃tr(∆^†∆)σ^∗σ−κ(Φ^T∆Φσ^†+H.c.),(4.45) and then build the following parametrization:

r² = Φ^†Φ +tr(∆^†∆) +σ^∗σ, Φ^†Φ =r²cos²γsin²θ,

tr(∆^†∆) =r²sin²γsin²θ,

σ^∗σ =r²cos²θ, (4.46)

where 0≤r ≤ ∞, 0≤γ ≤ π

2 and 0≤θ≤ π 2. We also need to develop the following parameters,

ζ = tr(∆^†∆∆^†∆) [tr(∆^†∆)]² , ξ = Φ^†∆∆^†Φ

Φ^†Φtr(∆^†∆),

α = Re(Φ^T∆Φσ)

tr(∆^†∆)σ^∗σ+ Φ^†Φσ^∗σ+tr(∆^†∆)Φ^†Φ, (4.47) where 1

2 ≤ ζ ≤ 1, 0 ≤ ξ ≤ 1 and −1 ≤ α ≤ 1. Two of them are already knew in the literature. The third one is a new parameter. We can see in detail in Appendix A how we can limit this parameter.

Let also define new variables x and y that must vary between 0 and 1 in the following way:

y =sin²θ,

x =sin²γ. (4.48)

Replacing Eq. (4.48) in Eq. ( 4.45) we get,

V⁴

r⁴ =y²[λ₁(1−x)²+λ₂x²+λ₃(1−x)x+ζλ₄x²+ξλ₅(1−x)x−2καx(1−x)]

+(1−y)²β₁+ (1−y)y[β₂(1−x) +β₃x−2κα]. (4.49) We manage things such that we can express these quartic terms in the following

66 way,

V⁴

r⁴ =A_xy²+B_x(1−y)²+C_x(1−y)y where,

A_x =λ₁(1−x)²+ (λ₂ +ζλ₄)x²+ (λ₃+ξλ₅−2κα)(1−x)x, B_x =β₁,

C_x =β₂(1−x) +β₃x−2κα. (4.50)

We can fix y = 0 or y = 1 to obtain the cases when the quartic couplings of the potential is positive. When we do this, we obtain the following conditions

A_x >0, (4.51)

B_x >0, (4.52)

C_x+ 2^qA_xB_x >0. (4.53)

ForA_x >0 we need to use the same argument as before. Fixing x= 0 and x= 1 we have similar conditions for the inequalities

λ₁ >0, λ₂+ζλ₄ >0,

λ₃+ξλ₅ −2κα+ 2^qλ₁(λ₂+ζλ₄)>0. (4.54) These new conditions depends of the parameters in Eq. (4.50). They vary in different ranges, but we only need to study the boundary values of these intervals. In this case the new conditions are:

λ1 >0, λ₂+λ₄ >0, λ₂+1

2λ₄ >0, λ₃+ 2κ+ 2

s

λ₁(λ₂+ 1

2λ₄)>0, λ₃+ 2κ+ 2^qλ₁(λ₂+λ₄)>0, λ₃+λ₅+ 2κ+ 2

s

λ₁(λ₂+ 1

2λ₄)>0, λ₃+λ₅+ 2κ+ 2^qλ₁(λ₂+λ₄)>0, λ₃−2κ+ 2

s

λ₁(λ₂+ 1

2λ₄)>0, λ₃−2κ+ 2^qλ₁(λ₂+λ₄)>0, λ₃+λ₅−2κ+ 2

s

λ₁(λ₂+ 1

2λ₄)>0,

λ₃+λ₅−2κ+ 2^qλ₁(λ₂+λ₄)>0. (4.55) Using the same argument for the condition in Eq. ( 4.52), it turns easy to see that

β₁ >0. (4.56)

Using the condition in Eq. (4.53) and the same fact that C_x can have x = 0 or x= 1, we obtain

β₂−2κα+ 2^qβ₁λ₁ >0,

β₃−2κα+ 2^qβ₁(λ₂+ζλ₄))>0. (4.57) The first inequality has two different solutions while the second has four ones. At the end of the day, we have

β₂+ 2κ+ 2^qβ₁λ₁ >0, β₂−2κ+ 2^qβ₁λ₁ >0, β₃+ 2κ+ 2

s

β₁(λ₂+ 1

2λ₄)>0, β₃+ 2κ+ 2^qβ₁(λ₂+λ₄)>0, β₃−2κ+ 2

s

β₁(λ₂+1

2λ₄)>0,

β₃−2κ+ 2^qβ₁(λ₂+λ₄)>0. (4.58)

68

Figure 4.1: Running ofλΦ≈λ1 at one-loop level as a function of the energy scaleµ forλ5 =λ3 =β2

=0.001 with yt = 0.9965, gY = 0.4627 and g = 0.6535. The doted line represents the expectation of Standard Model and the red line represents the expectation for our model for two values ofκ.

So, those are the set of condition that guarantee the potential in Eq. (4.3) is bounded from below. In what follow we obtain the running of the self coupling related of the standard-like Higgs.

4.7.2 RGE-evolution of the self coupling of the standard-like Higgs

The standard model predicts that the self coupling of the Higgs becomes negative at an energy scale around Λ = 10¹¹GeV. This means that the standard model can not assure the stability of the vacuum up to the Planck scale. This must be remedied by means of new physics in the form of new particles with appropriate interactions. This issue has been extensively investigated in the literature[50, 95, 96, 97, 98, 99]. As we already mentioned in chapter 2, the minimum type 2 seesaw can solve this problem, but a natural problem appears in such a way that the two cannot be solved simultaneously.

Within 123 model we show that the right behavior of the self coupling of the Higgs that guarantees absolute stability for the Electroweak Vacuum depends strongly on the coupling κ, not only of the quartic couplings between Φ and ∆ (in this case, λ₃ and λ₅).

We do our analysis by implementing the model in SARAH 4.13.0 [63] and evaluating the β function for λ₁ ≈λ_SM at one-loop level.

The main contributions for the beta function of λ₁ involve the following terms β_λ1 = 27

100g_Y⁴ + 9

4g⁴ + 9

20g²_Y(g²−2λ₁)− 9

5g²λ₁+ 12λ²₁+ 12λ₁y_t^†y_t−6y_t^†y_ty^†_ty_t+ +β₂²+ 3λ²₃ + 3λ₃λ₅+5

4λ²₅+ 2κ², (4.59)

where g and g_Y are the gauge couplings of the standard gauge group SU(2) and U(1)_Y whiley_t is the Yukawa coupling of the quark top.

Observe that the couplingsβ₂,λ_3,5andκgive positive contributions to the running of λ₁. However, as showed above, the invisible Higgs decay requires β₂ , λ_3,5 minor than 10⁻² which turns insignificant their contributions to the RGE-evolution. Rest us the contribution of the parameter κ. In Fig. 4.1 we show the plot of the running of λ₁ with

Figure 4.2: Running ofλ_∆≈λ₂+λ₄at one-loop level as a function of the energy scaleµforλ₅=λ₃=β₂

=0.001 with y_t = 0.9965, g_Y = 0.4627 and g = 0.6535. The doted lines represents the expectation for our model for three values ofκ. The couplingβ₃is important in the same level asκfor this running.

Figure 4.3: Running ofλ_σ ≈β₁ at one-loop level as a function of the energy scale µfor λ₅ =λ₃ =β₂

=0.001 with y_t = 0.9965, g_Y = 0.4627 and g = 0.6535. The doted lines represents the expectation for our model for three values ofβ₃.

energy scale for two possible values of κ. We see that the running ofλ₁ may get positive up to Planck scale for κ > 0.3. Thus, the model may have the vacuum stable thanks to the contribution of the parameterκ.

For sake of completeness, in Figs. 4.2 and 4.3 we present the running at one-loop level of the other self-quartic couplingsλ_2,4 and β₁. As we can see in those plots, they do not develop negative values and are weakly influenced byκ.