The SM Gauge Interactions and Symmetry Breaking

• The gravitational interaction is characterized by graviton with spin 2 and zero mass.

This interaction occurs between all forms of matter and radiation. It is also long range. However, its strength among elementary particles is too weak as compared to other interactions. For example, if one takes the strong interaction strength as 1, then approximately the relative strength for electromagnetic interaction will be 10 ², for weak interaction it will be 10 ⁷but for gravitational interaction it will be 10 ³⁹. This is the reason why we can omit gravity interaction among elementary particles in the SM.

It is important to note that the electromagnetic interaction and weak interaction are indeed different aspects of a single unified interaction, the so-called electroweak (EW) interaction.

The merging of two interactions occurs when the universe is hot enough or the energy is above order of 246 GeV.

There is another boson in SM called scalar Higgs boson with zero spin which was discovered in 2012 [4, 5]. Its role is to implement the Higgs mechanism, giving mass to all particles as well as itself. A combined result of the Higgs boson at Run1 LHC showed that it has a mass around 125 GeV [14] and very likely it is the SM Higgs boson but we still do not know for sure. Future experiments carried out at the colliders will hopefully give us the answer soon.

2.2 The SM Gauge Interactions and Symmetry Breaking

Gauge theory is used to describe all the fundamental interactions in Nature. Gauge symme-tries are the invariance of the Lagrangian density under gauge transformations which are realized on the elementary fields as local phase rotations. The SM gauge group is identified as

G_SM=SU(3)C⇥ SU(2)L⇥U(1)Y, (2.1) where the color group SU(3)C describes strong interaction with the gauge coupling gs, while electroweak interaction obeys SU(2)L⇥U(1)Y with gauge couplings g and g⁰, respectively.

Because parity is violated in weak interaction as opposed to electromagnetic and strong interactions, one expects to use Weyl spinors in SM. Indeed, within the SM, neutrinos have only left-handed components, in contrast to quarks and charged leptons. The left-handed fermions transform as SU(2)L doublets, while the right-handed components are all singlets.

The leptons are not involved with strong interaction, therefore they are singlets under SU(3)C, while the quarks are triplets under SU(3)C. Note that the generator of the U(1)Y hypercharge is given by Y /2 = Q T³, where Q is the electric charge in a unit of e and T³is the third

8 An Overview of the Standard Model

Fields SU(3)C SU(2)L U(1)Y

LLi=✓ nL

e_L

◆

i 1 2 ¹₂

e_Ri 1 1 1

Q_Li=

✓ u_L d_L

◆

i 3 2 ¹₆

u_Ri 3 1 ²₃

d_Ri 3 1 ¹₃

Table 2.1 Fermion fields and their assignments of Standard Model quantum numbers. The index i stands for the generation of fermions.

component of weak isospin SU(2)L. In Table 2.1, we summarize the fermion fields and their quantum number assignments under the SM gauge group.

The Standard Model Lagrangian is defined as

LSM= LGauge+ LFermion+ LHiggs+ LYukawa, (2.2) where LGauge, LFermion, LHiggsand LYukawaare Lagrangian for the gauge, fermion, Higgs and Yukawa interactions, respectively.

The Lagrangian for the gauge interaction could be written down as follows

L_Gauge= 1

4G^a_µnG_a^µn 1

4W_µnⁱ W_i^µn 1

4B_µnB^µn (2.3)

where G^a_µn, W_µnⁱ and B_µn are gauge field strengths defined as

G^a_µn ⌘ ∂µG^a_n ∂_nG^a_µ+g_sf^abcG^b_µG^c_n, (2.4) W_µnⁱ ⌘ ∂µW_nⁱ ∂_nW_µⁱ+ge^{i jk}W_µ^jW_n^k, (2.5)

B_µn ⌘ ∂µB_n ∂_nB_µ, (2.6)

with f^abc ande^{i jk} the structure constants of SU(3)Cand SU(2)L, respectively.

The fermion Lagrangian L_Fermionis given by

L_Fermion= ¯L_Lii /DLLi+¯eRii /DeRi+ ¯Q_Lii /DQLi+¯uRii /DuRi+ ¯d_Rii /DdiR, (2.7)

2.2 The SM Gauge Interactions and Symmetry Breaking 9

where /D ⌘ D^µg_µ and D^µ is the gauge covariant derivative. For a fermion f under the representation of SM gauge group given in Table 2.1, we have

D^µf =⇣

∂_µ ig_sl^a

2 G^a_µ igTⁱW_µⁱ ig⁰Y_f 2B_µ⌘

f , (2.8)

withl^a(a = 1,··· ,8) are the eight Gell-Mann matrices, Tⁱ(i = 1,2,3) are the SU(2) gener-ators (for doublet Tⁱ=sⁱ/2 wheresⁱare the Pauli matrices, while for singlet Tⁱ=0) and Y_f/2 is given by the last column in Table 2.1.

The Higgs Lagrangian LHiggsis defined as

L_Higgs=⇣

D_µH⌘†

D^µH V (H) (2.9)

with V (H) = µ²H^†H +l_SM(H^†H)² being the Higgs potential for the Higgs doublet H.

The shape of the potential V (H) depends on the choice of the parametersµ and l_SM. While l_SM>0 will ensure stability of the vacuum, the potential will have a local maximum at the origin and degenerate minima on a circle around it when µ²<0. Because the Higgs field H transforms as SU(3)Csinglet, SU(2)L doublet and has U(1)Y hypercharge Y = 1, one can write down its covariant derivative as

D_µH =⇣

∂_µ igsⁱ

2W_µⁱ ig⁰ 2B_µ⌘

H . (2.10)

The last piece is Yukawa Lagrangian, LYukawa, which is given by

LYukawa= Q¯_LiY_d^{i j}d_{R j}H Q¯_LiY_u^{i j}u_{R j}H˜ ¯LLiY_e^{i j}e_RiH + h.c. (2.11) where ˜H = is₂H^⇤.

Recall that gauge invariance leads to massless gauge bosons. However it is not the case for the electroweak gauge bosons in reality. The experimentally observed W^± and Z bosons are massive. Thus the EW gauge symmetry SU(2)L⇥U(1)Y has to be broken to give masses to these gauge bosons. This can be achieved by giving a non-vanishing vacuum expectation value (VEV) of the Higgs field, hHi = (0,v/p

2)^T. As a result, the gauge symmetry SU(2)L⇥ U(1)Y breaks down to U(1)EM, the electromagnetism gauge group. This is called spontaneous symmetry breaking (SSB) of the EW gauge group. After

10 An Overview of the Standard Model

the SSB, there are three massive vector bosons which are given as follows

W_µ^± = (W_µ¹⌥ iW_µ²)

p2 with mass m_W^±=gv

2, (2.12)

Z_µ = sinqWB_µ+cosqWW_µ³ with mass mZ=p

g²+g⁰²v

2, (2.13) where the weak mixing angleq_W is related to the coupling constants in the following way:

sinq_W = g⁰

pg²+g⁰², (2.14)

It is easy to see that cosqW =^m_m^{W ±}_Z .

Furthermore, the photon field defined as

A_µ=cosqWB_µ+sinqWW_µ³, (2.15) remains massless.

On the other hand, when the Higgs field develops the VEV, the Yukawa interactions generate mass matrices for the charged leptons and quarks of the form Mf =Y_fv/2, where f = u,d,e. Because neutrinos have only left-handed components, they are massless in the SM. Note that the fermions mass matrices Mf are in general 3 ⇥ 3 complex matrices and can be diagonalized by bi-unitary transformations as

V_L^uM_u(V_R^u)^†=diag(mu,m_c,m_t) , (2.16) V_L^dM_d(V_R^d)^†=diag(md,m_s,m_b) , (2.17) V_L^eMe(V_R^e)^†=diag(me,m_µ,m_t) , (2.18) with V_L^u, V_R^u, V_L^d, V_R^d, V_L^eand V_R^e are unitary matrices in the flavor space, i.e.

u_L=V_L^uu⁰_L, u_R=V_R^uu⁰_R, d_L =V_L^dd_L⁰ , d_R=V_R^dd⁰_R,e_L =V_L^ee⁰_L, e_R=V_R^ee⁰_R, (2.19) where the primed fields are physical fields of definite masses.

The electromagnetic and the neutral current interactions are given by Lem= e⇣

Q_u(¯uLg^µu_L+¯uRg^µu_R) +Q_d d¯_Lg^µd_L+ ¯d_Rg^µd_R +Q_e(¯eLg^µe_L+¯eRg^µe_R)⌘

A_µ ,

(2.20)

2.2 The SM Gauge Interactions and Symmetry Breaking 11

and

LNC= e

sinq_Wcosq_W h1

2¯uLg^µu_L 1

2d¯_Lg^µd_L+1

2n¯Lg^µnL 1

2¯eLg^µe_L sin²qW

⇣Q_u(¯uLg^µu_L+¯uRg^µu_R) +Q_d d¯_Lg^µd_L+ ¯d_Rg^µd_R

+ Qe(¯eLg^µeL+¯eRg^µeR)⌘i

Z_µ , (2.21)

respectively. Here, Qu, Qd and Qe are the electric charges of the up-type quarks, down-type quarks and charged leptons. Note that these interactions are invariant under the field redefinitions in Eq. 2.19.

Lastly, under these field redefinitions the charged gauge boson W^±couples to the physical fermions become

L_CC= e

p2sinqW

⇣¯u_0Lg^µW_µ⁺V_CKMd_L⁰ + ¯nLg^µW_µ⁺(V_L^e)^†e⁰_L +h.c.⌘

(2.22)

where VCKM=V_L^u(V_L^d)^†is the famous Cabibbo–Kobayashi–Maskawa (CKM) quark mixing matrix [15, 16]. In SM where the neutrinos are left-handed and strictly massless, the mixing matrix V_L^ehas no physical consequences since one can define a new neutrino fieldn_L⁰ =V_L^en_L to rotate this mixing matrix away. 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 Electroweak-Scale n R Model

The electroweak-scale right-handed neutrino (EW-n_R) model proposed by Hung [9] will be briefly reviewed in this chapter. The crucial feature of this model is non-sterile right-handed neutrinos having masses at the electroweak scale. In the first section we will review the main motivation of the EW-n_Rmodel. Next we will present the gauge structure and particle content of the model. In the third section we will review the SSB of this model which provide masses and mixings of various particles, especially that of Dirac and Majorana neutrinos and their mixings. Then we will mainly focus on low energy phenomenological constraints of the EW-n_Rmodel.

3.1 Motivation

The motivation of introducing mirror fermions in [9] was manifold. First of all, it is aestheti-cally satisfactory to have parity restoration at a higher energy scale while the maximal parity violating interaction (V A interaction) in the SM emerges from spontaneous symmetry breaking. This is one of the main reasons for various left-right symmetric models in the literature [17–20]. Secondly, it is important to study non-perturbative effects in the SM by discretizing it on the lattice. However it is well known that putting chiral fermion on the lattice is plagued by fermion doubling - an unavoidable consequence of the no-go theorem proved by Nielsen and Ninomiya [21]. Sophisticated techniques like using Wilson fermion, Wilson-Ginsparg fermion, staggered fermion, or domain wall fermion etc., which by violating at least one of the assumptions in the no-go theorem gets rid of the unwanted species, are often employed to handle this problem in practise. For new physics model builders, it is attractive to add mirror fermions to the SM which makes the theory becomes vector-like at a higher scale and hence one can avoid the fermion doubling problem if formulating on the lattice. Chiral gauge anomalies will then be cancelled automatically in this class of

14 The Electroweak-Scalen_R Model

models. The third motivation is the electroweak scale non-sterile right-handed neutrinos introduced in [9]. For each generation, the right-handed neutrino is introduced together with a right-handed heavy charged fermion partner to form a SM SU(2) doublet. Similarly a left-handed heavy mirror charged lepton will be introduced for each right-handed SM charged lepton. Majorana masses can then be given to these right-handed neutrinos via the vacuum expectation value (VEV) of a hypercharge Y /2 = 1 Higgs triplet with mass at the electroweak scale, rather than the grand unification scale in the usual scheme. Tiny Dirac masses can also be given via small VEVs of Higgs singlets with Y = 0. This is the electroweak scale see-saw mechanism in mirror fermion model which is testable at the LHC [22, 23].

Recently, many phenomenological implications of the mirror model [9] have been ex-plored further. We summarize what we have been done in a series of works involving various collaborations: In [24], the model was challenged by the electroweak precision measurements.

It was shown that the dangerously large contributions to the oblique parameters from the mirror fermions (especially the S parameter) can be tamed by the opposite contributions from the Higgs triplets. In [25], the original mirror model was extended by adding a mirror Higgs doublet so as to accommodate the LHC data for the SM Higgs signal strengths of various channels. Searches for mirror fermions at the LHC were studied in [22] for mirror quarks and [23] for mirror leptons. In [28], the neutrino and charged lepton masses and mixings were discussed in the mirror model with a horizontal A4(the discrete tetrahedron symmetry group) symmetry imposed on the lepton sector. Subsequently, in [37], the charged lepton flavor violating (CLFV) radiative decayµ ! eg was studied in details in this mirror model with the horizontal A4symmetry extension, updating an earlier calculation [43] done for the original model. Moreover, theµ e conversion in nuclei was also studied [44]. In [45], the CLFV Higgs decay h(125GeV) ! µt was studied for the extended mirror model with a mirror Higgs doublet [25], while the electron electric dipole moment has been performed in [46] for the model. In [47], a study on top quark rare decays in the EW-n_Rmodel has also been investigated. Some of these works will be presented in the following chapter.