The Standard Model

SM, combining the electroweak theory proposed by Glashow, Weinberg and Salam [1,2]

to describe the electromagnetic and weak interactions, and quantum chromodynamics (QCD) describing the strong interaction, provides a beautiful description of the properties and interactions of the elementary particles. The SM is based on the following gauge group

SU(3)_C×SU(2)_L×U(1)_Y.

Here the subscript C represents color, the strong interaction charge, L represents left- handed chirality andY represents weak-hypercharge. There are total twelve gauge bosons with spin 1 related to the local symmetry of the above gauge group. They can be described in the following way:

• SU(3)_C is the special unitary gauge group representing color charges with eight gluons (G^A_µ), as gauge bosons (G^A_µ) describe strong interactions among quarks.

• SU(2)_L is the gauge group of weak isospin, with three gauge bosons W_µ^a, describing weak interaction quarks and leptons.

• U(1)_Y is the unitary gauge group of hypercharge (Y) with one gauge bosonB_µ. The building blocks of matter are fermions (spin-¹₂ particles) called leptons and quarks.

Both leptons and quarks are divided into three families with different masses. Leptons have only electro-weak interactions while quarks have both electroweak and strong inter- actions. Left-handed leptons are doublets underSU(2)_Lcontaining one charged leptonl_L and one neutrino νL, whereas the right-handed charged leptons (eR, µR, τR) are singlets

1.1. The Standard Model 3 Field content SU(3)_C SU(2)_L U(1)_Y

G^A_µ 8 1 0

Gauge W_µ^a 1 3 0

B_µ 1 1 0

Lepton L_i = νL

e_L

i

1 2 -1

eRi 1 1 -2

Qi = u_L

dL

i

3 2 ¹₃

Quark u_Ri 3 1 ⁴₃

dRi 3 1 −²₃

Scalar Φ = φ⁺

φ⁰

1 2 1

Table 1.1: Standard Model fields are represented along with quantum numbers asso- ciated with their symmetry groups. Here, i= 1, 2, 3 denotes the generation index;

e_i = e, µ, τ;u_i = u, c, t and d_i = d, s, b respectively. In the subscript of G_A, the value of A goes from 1 to 8, representing 8 gluon fields, and a = 1, 2, 3 for three weak gauge

bosons.

under SU(2)L. In the SM, there are no right-handed neutrinos. The quarks come in six flavours in the SM. Three of them are up-type quarks, up (u), charm (c), top (t) and rest three are down type quarks, down (d), strange (s) and bottom (b). Here again, the left-handed up type (u) and down type (d) quarks form a doublet underSU(2)_L, whereas the right-handed partner of these quarks are singlet underSU(2)Lsymmetry. Mass terms of gauge bosons and the fermions in the above set up are not permitted, as they break gauge invariance explicitly. On the other hand, experimentally we know some of these have non-zero mass. A way out of this is the Higgs mechanism, which breaks the symme- try spontaneously, and thereby generate mass terms for gauge boson. Fermions get their mass through their Yukawa interactions with the Higgs field. Higgs mechanism requires introduction of a scalar (spin-zero) field.

The quantum numbers of the SM fields corresponding to various gauge group can be summarised as in Table 1.1. The particle content of the SM with respective masses, electric charges and spins are given in Figure1.1.

Figure 1.1: Particle content with mass, charge and spin of the Standard Model.

The SM Lagrangian that is consistent with the gauge group and the field content may be written as

L_SM =L_quark+L_lepton+L_Gauge+L_{Y uk}+L_φ. (1.1) The Lagrangian L_quark includes the quark sector of the SM, is represented as follows

L_quark=Q_jiγ^µD_µ^QQj+q_jiγ^µD_µ^qqj, (1.2) where

D^Q_µ = (∂_µ−ig_sT^AG^A_µ −igτ^a

2 W_µ^a−ig⁰Y B_µ), (1.3) D^q_µ = (∂_µ−ig_sT^AG^A_µ −ig⁰Y B_µ), (1.4)

1.1. The Standard Model 5 with Q_j the left-handed quark doublets andq_j the right-handed quark singlets, andj = 1,2,3 representing the three families. Similarly, L_lepton for lepton sector can be written as

L_lepton=Ljiγ^µD^L_µLj+`jiγ<sup>µ</sup>D<sup>`</sup>_µ`j, (1.5) where

D^L_µ = (∂_µ−igτ^a

2 W_µ^a−ig⁰Y B_µ), (1.6)

D_µ^` = (∂_µ−ig⁰Y B_µ), (1.7)

with L_j the left-handed lepton doublets and `<sub>j</sub> the right-handed lepton singlets (`_j = eR, µR, τR). The kinetic term for the gauge fieldL_Gauge under the gauge groupSU(3)C× SU(2)_L×U(1)_Y can be written as

L_Gauge=−1

4G^AµνG^A_µν−1

4W^aµνW_µν^a − 1

4B^µνB_µν, (1.8) where the field strength tensorG^A_µν,W_µν^a andB_µν represents the field content ofSU(3)_C, SU(2)LandU(1)Y gauge groups, respectively. G^A_µν,W_µν^a andBµν can be defined explicitly in terms of the corresponding gauge field as

G^A_µν =∂µG^A_ν −∂νG^A_µ −gsfABCG^B_µG^C_ν , (1.9)

W_µν^a =∂µW_ν^a−∂νW_µ^a−gabcW_µ^bW_ν^c , (1.10)

Bµν =∂µBν−∂νBµ , (1.11)

wherefABC andabcare real, called structure constants, antisymmetric under interchange of any two indices.

The L_Φ and L_{Y uk} terms in Eq. 1.1 introduce the interaction of a scalar field, Φ with the gauge bosons and the fermions, thereby generating the masses to the corresponding particles. We describe this in the next subsection.

1.1.1 The standard Higgs mechanism

The symmetry of the electroweak gauge group followed by SM forbids mass terms in the Lagrangian for fermions and bosons. These masses can be generated through the well known Higgs Mechanism proposed by R. Brout, F. Englert, P. Higgs, G. S. Gurlanik, C.

R. Hagen and T.W.B. Kibble [3–5] in 1964 preserving the renormalizability of the theory.

This is achieved by introducing aSU(2)_Ldoublet of complex scalar field with hypercharge Y = 1,

Φ =



 φ⁺

φ⁰



, (1.12)

where φ⁺ and Φ⁰ both are complex scalar fields. The Lagrangian involving the scalar field Φ can be written as

L_Φ= (D_µΦ)^†(D^µΦ)−V(Φ), (1.13) where the covariant derivativeD_µ=∂_µ+ig^τ₂^aW_µ^a+ig^0Y₂B_µ, withgis theSU(2)_Lcoupling and g⁰ is the U(1)Y coupling. The scalar potential of the Lagrangian is given by

V(Φ) =µ²Φ^†Φ +λ(Φ^†Φ)². (1.14) If λ > 0 and µ² <0, the minimization condition on the potential V, i.e. _∂(Φ^∂V†Φ) = 0, for the Φ field, gives a non zero vacuum expectation value (VEV) which can be expressed as

D Φ^†Φ

E

=v² =−µ²

λ. (1.15)

Now we can expand the Φ field around the minimum VEV and takes the following form

Φ = 1

√ 2





φ1(x) +iφ2(x) v+h(x) +iφ₃(x)



. (1.16)

Here φ1(x), φ2(x), φ3(x) and h(x) are all real fields withh(x) corresponding to the field of physical Higgs particle. Using a gauge transformation, the fields φ₁, φ₂ and φ₃ can be removed from the scalar field Φ, and transferred to the gauge bosons, W_µ^a and B^µ. Notice that we are now fixing the gauge by our choice of this transformation. This gauge in

1.1. The Standard Model 7 which the physical spectrum is explicitly present is known as the unitary gauge or physical gauge. This addition of degrees of freedom is now manifested in the gauge bosons, leaving two massive charged gauge bosons,W^±= ^√1

2(W_µ¹∓iW_µ²) and one massive neutral gauge boson, Zµ. The neutral gauge boson is a suitable combination ofW_µ³ andBµ given by

Zµ=W_µ³cosθ_W −Bµsinθ_W. (1.17) The orthogonal combination,

A_µ=W_µ³sinθ_W +B_µcosθ_W (1.18) corresponds to the massless photon, the mediator of EM interactions. The angle, θ_W = tan⁻¹(^g_g⁰) is the weak mixing angle, also known as the Weinberg angle.

In this way, theSU(2)_L×U(1)_Y symmetry is broken down to theU(1)_EM symmetry, with the corresponding gauge boson, the photon, indeed massless. The masses of the gauge bosons in terms of the VEV of the Φ and the gauge couplings are given by M_W^± = ¹₂vg and M_Z0 = ¹₂vp

(g²+g⁰²). Electromagnetic coupling constant ecan be written in terms of parameters g,g⁰ and θ_W :

e=g⁰sinθ_W =gcosθ_W. The Higgs field itself, in this (unitary) gauge, can be written as

Φ(x) = 1

√2



 0 v+h(x)



. (1.19)

As we mentioned earlier, the three degrees of freedom φ₁(x), φ₂(x), φ₃(x) of the scalar doublet reappear as the longitudinal components of the massive weak-isospin triplet gauge bosons. After the spontaneous symmetry breaking, the mass of the remaining degrees of freedom in Φ appears as SM Higgs boson with mass,M_h =p

−2µ². Given that v= 246 GeV from the measurement of Fermi coupling constant, G_F = ^√v2

2, all other couplings in the scalar sector are fixed in terms of v and M_h. Thus the trilinear and quartic self interaction couplings of Higgs boson can be written as

λ_hhh=−^µ_v² = ^M_2v^h2, λ_hhhh = ^−µ_v2² = ^M_8v^h2².

The interaction between the Higgs field and fermion field, the so called Yukawa cou- pling, gives masses to the fermions in the SM. The Yukawa Lagrangian

L_{Y uk} =Y_ij^dQ¯i φ dRj+Y_ij^uQ¯i φ u˜ Rj+Y_ij<sup>`</sup>L¯i φ `Rj+h.c., (1.20) where ˜φ=iτ2φ and i, j = 1,2,3 represents three generations of fermions, and Y_ij^d,u,` are the complex Yukawa couplings. In addition to providing masses to the fermions, these terms also dictate the interaction of the Higgs boson, h, with the fermions. The predicted physical Higgs boson h is discovered by ATLAS and CMS collaborations [6, 7] in 2012, leading to the nobel prize for F. Englert and P. W. Higgs in 2013. The mass of the Higgs boson is measured to be M_h ∼125 GeV. This most awaited discovery of Higgs boson is a grand success for Particle Physics community to complete the SM predictions and has began a new era of Particle Physics.

Inspite of being a very successful model including the very stringent experimental tests, many hints from theory and experiments clearly indicate that there are many un- explainable phenomena which the SM is not capable of addressing. In the next section, we are going to expose some of the open questions as well as limitations of the SM.