1.3 Possible resolutions of drawbacks of the Big bang cosmology and the SM of par-
1.3.4 Higgs Vacuum Stability
30 Chapter 1. Introduction the effective neutrino mass matrix can be written as
Mν = 0 mTD mD MR
!
. (1.87)
H
¯lL
NR×NR
H
¯lL
ψ− ψ¯1
W+
W− H1,2
ψ− ψ¯1
W+
W−
Z
ψ− ψ¯1
ψ−
γ W−
ψ− ψ¯1
ψ1,2,3
W− H1,2
ψ− ψ¯1
ψ1,2,3
W− Z
ψ− ψ¯1
ψ+ Z W− Figure 1.8: Feynman diagram for Type-I seesaw.
For three generations of neutrinos, each entries in Eq.(1.87) are 3×3 matrices. Assuming mD to be much lighter than MR (mD << MR), after block diagonalisation, mass matrices for light and heavy neutrinos can be expressed as
mIν ' −mTDMR−1mD and Mheavy 'MR. (1.88) Hence from Eq.(1.88) it is easy to explain the smallness of the light neutrinos, provided MR is sufficiently large. This is commonly known as type-I seesaw mechanism of neutrino mass generation. A typical Feynman diagram for such mass generation is portrayed in Fig. 1.8. As an example, from Eq.(1.88), to yield neutrino mass of the order of 0.1 eV assuming YD∼ O(1), the required value of RH neutrino mass is MR ∼ 1014 GeV. Although there are several other mechanism of generating neutrino mass such as Type-II seesaw [22, 23], Type-III seesaw [24]
and radiative generation [198], we employ mainly Type-I seesaw in the thesis.
1.3. Possible resolutions of drawbacks of the Big bang cosmology and the SM of particle
physics 31
potential provided HInf >ΛSMI which is satised by most of the large eld ination models Below we discuss the fate of Higgs vacuum in the SM considering 3σ uncertainty of top mass.
1.3.4.1 Higgs Vacuum Stability in Standard Model
The tree level Higgs potential in the SM as narrated in Eq.(1.6) V(H) =µ2H|H2|+ λh
4 |H|4. (1.89)
At high energy (H0 >> v in Eq.(1.7)), one can safely ignore the quadratic term inV(H). Now for a correct description of Higgs potential we should consider the higher order loop contribu- tions [219–221]. Particles which have coupling with the SM Higgs will enter into the loop and contribute to the correction of the Higgs potential. Therefore, the effective Higgs potential with the approximation H) >> v, can be written as
Veff(H) = λeffh
4 H04, (1.90)
whereλeffh includes contribution of higher order loop correction in the SM Higgs potential which is read as [34, 219–221]
λeffh =e4Γ(h)h
λh(µ=H0) +λ1h(µ=h) +..i
, (1.91)
where Γ(h) = Rh
mtγ(µ)dlnµ, γ the anomalous dimension of Higgs field [34]. The one loop correction to the self coupling of Higgs field is given by [34, 219–221]
λ1h= 1 (4π)2
h3g24 8
lng24
4 −5 6+ 2Γ
+ 3
16(g22+g21)2
lng22+g12
4 −5
6 + 2Γ
−3yt4
lnyt2 2 −3
2 + 2Γ
+ 3λ2h(4lnλh−6 + 3ln3 + 8Γ) i
. (1.92)
In addition we also have to perform the renormalization group (RG) running of the SM couplings. Among all the fermionic couplings, top Yukawa coupling yt turns out to be the dominant one. Now for the purpose of RG running, evaluation of initial values of coupling constants is required. To find their values atmt, one has to consider various threshold corrections at different mass scales. This has been rigorously worked out in Ref [34]. Below we provide the initial values of all the SM couplings as function of mt(top mass),mh (Higgs mass) and strong
32 Chapter 1. Introduction coupling constant (αs) at µ=mt energy scale.
g1(µ=mt) = 0.35761 + 0.00011 mt
GeV −173.10
−0.00021MW −80.384 GeV
0.014 GeV , (1.93) g2(µ=mt) = 0.64822 + 0.00004 mt
GeV −173.10
−0.00011MW −80.384 GeV
0.014 GeV , (1.94) g3(µ=mt) = 1.16449 + 0.0005 mt
GeV −173.10
−0.00011MW −80.384 GeV 0.014 GeV + 0.0031α3−0.1184
0.0007
(1.95) λh(µ=mt) = 0.12711 + 0.00206
MH0
GeV −125.66
−0.00004 mt
GeV −173.10
, (1.96) y(µ=mt) = 0.93558−0.00550 mt
GeV−173.10
−0.00042α3(MZ)−0.1184 0.0007
−0.00042MW −80.384 GeV
0.014 GeV . (1.97)
Next in order to study the running, one should employ RG equations of all the SM couplings [34, 222–226] at three loop. Below we present one loop RG equations of relevant SM couplings.
βgSM1 = 1 16π2
n
−41 6 g13
o
, (1.98)
βgSM2 = 1 16π2
n
−19 6 g23
o
, (1.99)
βgSM3 = 1 16π2
n
−7g33o
, (1.100)
βλSM
h = 1
16π2 n
24λ2h+ 12y2tλh−9λhg12 3 +g22
−6yt4+9 8
g41
3 +g24+ 2
3g12g22o
, (1.101) βySMt = 1
16π2 n9
2yt3+
−17 12g12−9
4g22−8g32 yto
(1.102) where βCi = dCdti and t= lnµ. The stability condition of electroweak vacuum is λh(µ)> 0 for any energy scale. Using this criteria the stability region has been presented by light green color in left panel of Fig. 1.9. On the other hand, if there exists another deeper minimum other than the EW one, the estimate of the tunneling probability PT of the EW vacuum to the second minimum is essential. The Universe will be in metastable state only provided the decay time of EW vacuum is longer than the age of the Universe. The tunneling probability is given by [30, 34],
PT =TU4µ4Be−
8π2
3|λH(µB)|. (1.103)
1.3. Possible resolutions of drawbacks of the Big bang cosmology and the SM of particle
physics 33
where TU is the age of the Universe. µB is the scale at which probability is maximized, deter- mined from βλH = 0. Hence for metastable Universe requires [30, 34]
λH(µB) > −0.065 1−ln
v µB
. (1.104)
As noted in [34], for µB> MP, one can safely considerλH(µB) =λH(MP). This condition has been plotted in Fig. 1.9 with solid red line. Hence pink colored region below the solid red line is considered to be the instable region. Therefore, using Eqs.(1.93-1.97) and Eqs.(1.98-1.102) one can attain the RG evolutions of the relevant couplings.
Figure 1.9: Running ofλhas a function of energy scaleµfor [left panel:] for varyingmtwith fixedmH0 = 125.18 GeV and [righ panel] for varyingmH0 withmt= 173.2 GeV
In left panel of Fig. 1.9, we display the running ofλhwith energy scaleµ. As initial value of ytis ofO(1) (see Eq.(1.96)), fermionic effect dominates in Eq.(1.101) which brings downλhfrom the initial value in tis RG running. As a consequence,λhturns negative atµ∼1010= ΛSMI GeV for mt= 173.2 GeV withmH0 = 125.18 GeV in Fig. 1.9 (left panel). If we consider somewhat a lower value of top mass, initial value yt will be lesser than the earlier case. Therefore it is obvious that for mt = 171 GeV, the instability energy scale ΛI becomes larger than ΛSMI as viewed in Fig. 1.9 (left panel). On the other hand for a highermt, it turns out that ΛI <ΛSMI . In Fig. 1.9 (right panel) running of all the SM couplings have been shown.
Hence, from the discussion, it is quite prominent that any theoretical model having large fermionic couplings with the SM Higgs are dangerous in view of stability of EW vacuum, One ideal way to achieve absolute vacuum stability is to introduce additional scalars [58, 59] in the
34 Chapter 1. Introduction theory. In Chapter 3, Chapter 4 and Chapter 6 we will discuss in detail the role of scalars in EW vacuum stability.