Theoretical framework

Quantum field theory provides the mathematical framework for the Standard Model (SM). The system Lagrangian dictates the dynamics and kinematics of particle fields that permeate space-time. The gauge theory is built upon a set of postulated symmetries by developing the most general renormalizable Lagrangian from its particle content that observes these symmetries [62].

Noether’s theorem states that every differentiable symmetry of the La-grangian of a physical system has a corresponding conservation law [63].

Lorentz invariance, consisting of time, translation and rotation invariance, is the global Poincaré symmetry group central to special relativity. It is neces-sary to guarantee energy, momentum and angular momentum conservations.

Each term of the SM Lagrangian must obey Lorentz invariance.

The fundamental objects of the SM are quantum fields, defined every-where in spacetime. Contrary to classical fields, they are operator valued, i.e. they do not represent a physical quantity but rather act upon quantum

states of the system. There is an operator for every point in spacetime that is merely used to exhibit some aspect of the state, at the point to which they belong. A single quantum field is associated with each particle type included in the theory. The excited states of the field are interpreted as particles in the classical framework. The amount of energy required to produce an excited state of the field is related to the massof the associated particle.

Figure2.3 compiles the Standard Model particles and their properties.

Quarks Leptons

Fermions Bosons

125.1 GeV 0

0

H

⁰

Higgs boson

u c t γ

d s b g

ν

_e

ν

_µ

ν

_τ

Z

⁰

e µ τ W

^±

up charm top photon

down strange bottom gluon

eneutrino µneutrino τneutrino Zboson

electron muon tau Wboson

4.7 MeV 1.274 GeV 173.1 GeV 0

2.2 MeV 94.6 MeV 4.176 GeV 0

0 0 0 91.2 GeV

0.511 MeV 105.7 MeV 1.777 GeV 80.4 GeV

+²₃ +²₃ +²₃ 0

−¹3 −¹3 −¹3 0

0 0 0 0

−1 −1 −1 ±1

1

2 1

2 1

2 1

1

2 1

2 1

2 1

1

2 1

2 1

2 1

1

2 1

2 1

2 1

Figure 2.3:Particles of the Standard Model. The figures in each tile represent the rest mass, the charge and the spin of the particle in natural units. The fermions that exhibit both helicities are represented with a slant across them.

The internal symmetry group of the SM is defined as a local

SU(3)C⊗SU(2)L⊗U(1)Y (2.1) gauge symmetry [64]. The SU(3)C group represents the colour symmetry observed by the system. Particles that participate in the strong interaction, the quarks, carry a colour charge that is conserved by the system. The gauge bosons of the strong interaction are the gluon field tensors,G^a, witharunning over the eight possible representation of the group. The SU(2)L×U(1)Y

electroweak symmetry group corresponds to the conservation of weak isospin, T, and weak hypercharge,YW. The electroweak gauge bosons areW1,W2

andW3 representingSU(2)L andB representingU(1)Y. The fermions that make up all matter are each represented by fermion fieldsψ.

The free particle Lagrangian defines the evolution of the fermion and

gauge fields in the absence of interaction through Lkin=iψ /¯∂ψ−1

4BµνB^µν−1

2trWµνW^µν−1

2trGµνG^µν, (2.2) with∂/=γ^µ∂µ and γ^µ the Dirac matrices, ¯ψ=ψ^†γ⁰,Fµν =∂µAν−∂νAµ, F = B, W, G the field strength tensors and the trace accounting for the summation over the different representations of the boson fields. In the absence of interaction, the principle of stationary action requires that

∂L_kin

∂ψ − ∂

∂x^µ ∂L

∂(∂µψ)

= 0, (2.3)

which yields the Dirac equation of a free massless field, i /∂ψ = 0. The evolution of theU(1)Y gauge field reads∂µB^µν = 0 which is analogous to the evolution of the electromagnetic field tensor in Maxwell’s equations.

In the interaction picture fermions may scatter or be created or destroyed through the exchange of gauge bosons. This changes the free kinematic derivative,∂µ, to its dynamic form

Dµ=∂µ−igsG^a_µT^a+ig⁰1

2YWBµ+ig1

2τWµ, (2.4) withg_s,g⁰ andgthe coupling constants of the strong and electroweak forces, T^a the tensor generators of theSU(3)C symmetry andτ the Pauli matrices generators of theSU(2)L symmetry. The second term, related to the strong force, exists only for quarks. The electroweak couplings are present for all fermions. The last term maximally violates parity and only couples to left-handed fermions. At this point the Lagrangian describes a theory of massless fermions interacting through the exchange of massless gauge bosons.

One could give mass to all fermions in the theory by adding a self-coupling term to each of them by hand. There is no way to give mass to gauge bosons that way because it would violate Lorentz invariance. The Higgs mechanism postulates the existence of a scalar boson,φ, with Lagrangian

Lh= (D^µφ)^†(D_µφ)−V(φ), (2.5) where D_µφ does not include the strong coupling term. Consider a com-plex scalar field in the spinor representation ofSU(2)L, φ= φ⁺ φ^0T. Renormalizability ofSU(2)L⊗SU(1)Y requires the potential to be of the form

V(φ) =−µ²φ^†φ+λ φ^†φ²

, (2.6)

withλ > 0. Ifµ² >0, the field admits an infinity of minimums atφ^†φ6= 0. Although the field is globally symmetric about zero, the symmetry is spontaneously broken to a non-zero vacuum expectation value (VEV) by

the potential. Since the potential depends only on the combinationφ^†φ, the VEV is arbitrarily chosen as

hφi= 1

√2 0

v

, (2.7)

withv=µ/√

λ. Due to the observed conservation of electric charge, only a neutral scalar field can acquire a VEV. TheU(1)Y symmetry is unbroken by the scalar VEV, i.e. it yields a breaking scheme

SU(2)L⊗U(1)Y →U(1)Q, (2.8) withQ=T3+YW/2 the electric charge. It is the symmetry of quantum electrodynamics and is by construction the true realized vacuum symmetry.

One can parametrize the second component of the scalar field asv+hwith hthe Higgs boson, a perturbation around the VEV. Using this expression in the first term of equation2.5yields

(D^µφ^†)(D_µφ) = v² 8

g² (W_µ¹)²+ (W_µ²)²

+ (gW_µ³−g⁰B_µ)²

. (2.9) Considering that the physical bosons realized in vacuum are expressed as











W_µ^±=^√1₂ W_µ¹±iW_µ² Z_µ=√ ¹

g²+g⁰² gW_µ³−g⁰B_µ Aµ=√ ¹

g²+g⁰² g⁰W_µ³+gBµ

, (2.10)

the coupling term in equation2.9can be rewritten (D^µφ^†)(D_µφ) = 1

2 gv

2 ²

W_µ^†W^µ+1 2

vp g²+g⁰²

2

!²

Z_µ^†Z^µ, (2.11) giving mass to theW^± andZ bosons but leaving the photon massless.

The fermions acquire mass in the Standard Model through the so-called Yukawa Lagrangian which describes the coupling between the scalar fieldφ and the fermion fields of the form

LYuk=−ψyφψ,¯ (2.12)

withy the Yukawa couplings. As the scalar potential has a minimum in vacuum athφi 6= 0, it introduces a term

− yv

√2ψψ¯ ≡ −mψψ,¯ (2.13)

which is a mass term for each fermion and function of their individual Yukawa couplings, free parameters of the Standard Model.

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