2.2 The SM Gauge Interactions and Symmetry Breaking
2.2.3 The Higgs Mechanism
The minimal Higgs potential that (minimal in the sense that only one Higgs doublet is involved) invariant under the SM gauge transformation is given by
V(H, H†) =−m2H†H+λSM
4 (H†H)2, (2.31)
with λSM > 0 to ensure vacuum stability and such that the extremum of the potential located at
This leads to two solutions of the extremum point as
⟨H⟩ = ⟨H†⟩ = 0 , (2.34)
⟨H†H⟩ = 2m2 λSM
. (2.35)
The first solution corresponds to the local maximum while the second one is the true minimum of the potential. The vacua are degenerate and form a circle of radius
2m2 λSM
1/2
. The stability of the theory requires us to take the second solution. We choose the minima of the form
⟨H⟩ = 0
and furthermore expand the theory around the ground state (minima of the potential) by shifting the Higgs field as
By doing this we fix the vacuum and therefore single out a particular direction in the ground state. Thus, there is a special direction in the vacuum of our theory and this particular point clearly breaks the corresponding symmetry that we have in the beginning. This mechanism refers to the spontaneously breakdown of the symmetry (SSB) or the Higgs mechanism. This occurs when the symmetry of the theory is broken by the ground state.
The mass spectrum of the Standard Model can be obtained by substituting the Higgs field as in Eq. (2.37) and retain only the quadratic terms of the fields (taking into account the mixing terms as well). This will give three massive gauge bosons as
Wµ± = (Wµ1∓ iWµ2)
√2 with mass mW± = gv
2, (2.38)
Zµ = −sinθWBµ+ cos θWWµ3 with mass mZ =p
g2+ g′2v
2 , (2.39) where the weak mixing angle θW is related to the weak couplings as
sin θW = g′
pg2+ g′2 . (2.40)
One can immediately see that cos θW = mmW ±
Z . Note that after the SSB, our gauge fields have been redefined as
Wµ±∓ i
mW∂µφ±→ Wµ±, (2.41)
Zµ− 1
mZ∂µχ→ Zµ, (2.42)
where χ is defined as χ ≡ √i2 ¯φ0− φ0
. Thus, after the SSB, the massless gauge field receives one additional degree of freedom and become massive. The gauge bosons are said to have eaten the Goldstone bosons during the SSB and hence become massive. In addition to the massive gauge fields, there is one massless degree of freedom written as
Aµ = cos θWBµ+ sin θWWµ3, (2.43) which is what we called the photon.
Furthermore, when the Higgs field generates non-zero vacuum expectation value (VEV), the Yukawa interactions induce the mass matrices for quarks and charged leptons with the generic form Mf = Yfv/√
2, where f denotes the charged leptons or quarks fields. On the other hand, due to the fact that only the left-chirality of the neutrinos are present, the neutrinos are massless in this SM framework. As an additional remark, the general form of the charged fermion mass matrices Mf are given by 3× 3 complex matrices which can be brought into
diagonal forms by using two unitary matrices via the following transformations The primed fields denote the physical fields with definite masses.
Next, the interaction between the gauge fields and the matter fields are categorized into three parts: the electromagnetic current, the neutral current, and the charged current. The expression of the first two currents are described by
Lem=− e
The electric charges of up-type quarks, down-type quarks and charged leptons are denoted as Qu, Qd and Qerespectively. These interaction currents are unchanged under the transforma-tions given in Eq. (2.47).
Finally, under the transformation in Eq. (2.47), the charged current is modified into the following form
LCC=− e
√2 sin θW
u¯′LγµWµ+VCKMdL′ + ¯νLγµWµ+(VLe)†e′L + h.c.
(2.50)
where VCKM= VLu(VLd)†is the usual Cabibbo–Kobayashi–Maskawa (CKM) mixing matrix in the quark sector [15, 16]. Due to the massless neutrinos, there is no corresponding mixing matrix in the lepton sector. We note that the neutral current interaction of the neutrinos with
the Z boson is invariance under this field redefinition. Thus we can drop all the primes in the physical fermion fields in the SM Lagrangian.
Chapter 3
The G2HDM Model
The Gauged Two Higgs Doublet Model (G2HDM) proposed by Huang et al. [9] will be discussed in this chapter. The model was constructed to explain the observed dark matter relic abundance by embedding the two Higgs doublets into a non Abelian gauge group SU (2)H in such a way that the DM stability is protected by the SU (2)H gauge group instead of the usual ad hoc Z2discrete group. First, we will discuss the important motivation behind the creation of the G2HDM model. We further discuss the particle contents in this model and their role. The potential consistent with the underlying symmetry of the gauge group as well as its SSB breaking pattern will be briefly reviewed. As a bonus, the existence of the accidental Z2in the model will be elaborated. Lastly, we will summarize the corresponding theoretical constraints coming from the vacuum stability, perturbative unitarity as well as Higgs phenomenology on the scalar sector.
3.1 Motivation
The discovery of the 125 GeV Higgs boson at the LHC was certainly a milestone in the history of particle physics. It completes, at least, the Standard Model particle contents discussed in the chapter 2. However, the SM alone can not answer the remaining real open problems in physics such as the origin of the neutrino mass, the observed dark energy that dominates our universe, and the existence of the unknown matter, which constitutes almost one third of the energy density budget in our universe, dubbed as the dark matter problem.
In order to explain these problems, many authors have tried to construct various models by enlarging the particle contents of the SM either by adding new fermions into the SM, extending the SM scalar sector as well as embedding the SM gauge group into higher groups.
Among the many beyond the Standard Model (BSM) models available in the market, the general two Higgs doublet model (2HDM) is a simple extension [17] by just adding one
more Higgs doublet to the SM. One class of 2HDM model, the inert Higgs doublet model (IHDM) [18–21], advertised the lightest neutral component of the second Higgs doublet to be the DM candidate. The stability of the DM candidate in this model is achieved by assigning a discrete Z2symmetry on the scalar potential of the model. However, according to the study of reference [22, 23], the discrete and continuous global symmetry is strongly disfavored by the gravitational effects. In order to avoid these unpleasant features as well as providing the DM candidate, a recent study in [9] has constructed a model called Gauged Two Higgs Doublet model (G2HDM) in which the two Higgs doublets H1and H2are put together into a doublet H = (H1, H2)T of a new non-abelian SU (2)H gauge group. The neutral component of H2is stable thanks to the SU (2)H gauge symmetry and hence it is a viable DM candidate. Other particles are incorporated in the G2HDM model including a SU(2)H doublet, a SU (2)Htriplet, and heavy SU (2)Lsinglet Dirac fermions. Their roles will be explained in the following session. In addition, The SM right-handed fermions are paired with new heavy right-handed fermions to form SU (2)H doublets. After the SSB, the VEV of SU(2)H doublet induces the heavy fermions masses. In order to simplify the scalar potential, a U (1)X group is employed. Let us list some of the important properties of G2HDM:
• It is free of gauge and gravitational anomalies;
• It is renormalizable;
• The stability of inert DM candidate (H2) is protected by SU (2)Hgauge symmetry;
• After SSB, the accidental Z2symmetry survives such that all the SM particles belong to the Z2even particles while some of the new scalars, W′ and new heavy fermions are odd. The lightest odd particle if neutral can be a DM candidate whose stability is protected by the accidental Z2symmetry;
• No flavour changing neutral currents at tree level for the SM sector;
• the VEV of the triplet induces SU (2)L symmetry breaking while that of ΦH generate masses to the new fermions via SU (2)H-invariant Yukawa couplings;
• etc.
Some phenomenology of G2HDM at the LHC had been studied previously in [9, 24]
for Higgs physics and in [25] for the search of the new gauge bosons. Recently, systematic studies on theoretical and phenomenological constraints for both the Higgs and gauge sectors in G2HDM have been presented in [26] and [27] respectively.