Mathematical Description of the Standard Model

In the following sections the mathematical formulation of SM interactions is described.

Quantum Chromodynamics

Quantum Chromodynamics (QCD) [22, 30] is the non-Abelian gauge theory behind the strong interaction, which is based on the SU(3) symmetry group of colour. The eight generators of this group correspond to the eight massless gluons, which mediate the in-teraction of coloured quarks. The quarks are described by colour triplets

q^T_f ≡(q¹_f, q²_f, q_f³), (2.4) with 1,2,3 representing the three colour states: red, green and blue.

The Lagrangian density of QCD is given by

LQCD =

nf

X

j=1

¯

qj(iDµγ^µ−mj)qj

| {z } quarks

−1 4

8

X

A=1

F^AµνF_µν^A

| {z } gluons

(2.5)

with the quark-field spinors, qj, and the quark masses mj. The γ^µ represent the Dirac matrices andDµ=∂µ−igsTAA^Aµ is the covariant derivative, whereA^Aµ correspond to the

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2.1. Introduction into the Standard Model of Particle Physics

gluon fields and TA to the eight generators of the SU(3) symmetry group. Finally, F_µν^A represents the field strength tensor based on the gluon fieldA^Aµ

F_µν^A =∂µA^Aν −∂νA^Aµ −gsfABCA^BµA^Cν, (2.6) with the structure constants of the SU(3) symmetry group, fABC, and the QCD coupling constant, gs =√

4παs.

Due to the fact that the SU(3) is a non-Abelian group, the 3rd term of Eq. 2.6 does not vanish and thus gluon fields are able to self-interact. Due to this self-interaction, the effective coupling constant of the strong interaction decreases with increasing energy, leading to asymptotic freedom. Here, for short distances the strong coupling constant converges asymptotically against zero, so that quarks and gluons can be treated as free and their interactions can be calculated within perturbation theory. On the other hand with increasing distance between two quarks, the quarks become bounded in hadrons through a process called confinement.

Electroweak Theory

The electroweak (EW) theory [17–19] is the gauge theory behind the electroweak in-teraction. It describes the unification of the weak interaction with the electromagnetic interaction under theSU(2)_L⊗U(1)_Y symmetry group. The SU(2) group involves three gauge fields and theU(1) group one gauge field. The corresponding gauge bosons areW_µⁱ, i = 1,2,3 for SU(2) and Bµ for U(1). The EW theory, also known as GSW theory, was introduced by S. Glashow, A. Salam and S. Weinberg. The Lagrangian of the EW theory is

L^EW =

3

X

j=1

iψ¯j(x)γ^µDµψj(x)

| {z } part for the fermions

−1

4BµνB^µν− 1

4W_µν^j W_j^µν

| {z }

part for the gauge field

. (2.7)

Dµ describes the covariant derivative Dµ=∂µ−igσj

2 W_µ^j(x)−ig⁰Y

2Bµ(x), (2.8)

with the coupling constants g corresponding to SU(2)L and g⁰ corresponding to U(1)Y. The fermionic part of the Lagrangian describes the kinetic energy of the fermions and their interactions, while the covariant derivative describes the interaction with the gauge field. It is worth mentioning that no explicit mass term for the fermions is allowed. If there is an explicit mass term, there will be a mixture of left-handed multiplets with right-handed singlets. Therefore, the local gauge invariance would be violated, since the weak interaction only couples to left-handed fermions.

The second part of the Lagrangian describes the gauge fields. Again, there is a term for the kinetic energy and a term that describes the self interaction between the gauge fields. In this part of the Lagrangian no explicit mass term is included, in order to avoid a violation of the invariance of local gauge transformations.

2. Z Boson and Higgs Boson Production in the Context of the Standard Model

The four gauge bosons of theSU(2)L⊗U(1)Y symmetry group do not translate directly inW^±, Z and γ. W^± are linear combinations of W_µ¹ and W_µ²

W_µ^±= (1/√

2)(W_µ¹∓iW_µ²), (2.9)

representing the charged part of the interaction. The neutral part of the interaction, represented by Z and γ, evolves from the mixing of the two neutral fields W_µ³ and Bµ

A_µ Zµ

=

cosθ_W sinθ_W

−sinθW cosθW

B_µ W_µ³

, (2.10)

with the weak mixing angle θW, which has been experimentally determined to sin²θW = 0.23116±0.00012 at the Z scale [22].

Higgs Mechanism

The existence of massive gauge bosons within the EW theory requires an additional mech-anism which is able to accommodate those masses in a gauge invariant and renormalisable way. The most popular and minimal solution of this problem is through the Higgs mech-anism [31–36].

The Higgs mechanism leaves the fundamental symmetry of the EW theory unchanged and generates the masses by spontaneous symmetry breaking of the quantum vacuum ground state. Within the theory a complex scalar SU(2) doublet φ with a hypercharge Y = 1 is introduced

φ(x) =

φ⁽⁺⁾(x) φ⁽⁰⁾(x)

= r1

2

φ₁(x) +iφ₂(x) φ3(x) +iφ4(x)

. (2.11)

A gauge invariant Lagrangian is obtained by coupling φ to the gauge bosons

LHiggs= (Dµφ)^†D^µφ−V(φ), (2.12)

using the covariant derivativeD_µdefined in Eq. 2.8. Here,V(φ) describes the most general renormalisable potential, which is invariant under anSU(2)L⊗U(1)Y gauge transforma-tion

V(φ) =µ²φ^†φ+λ(φ^†φ)². (2.13) The potential depends on the choice of µ and λ. For µ² < 0 and λ > 0 the potential is bounded from below, with a rotationally symmetric degenerate ground state

−µ² 2λ = v²

2 , (2.14)

v describes the vacuum expectation value, which is related to the Fermi constant GF [22]:

v =

s 1

√2GF

≈246.22 GeV. (2.15)

φ(x) is expanded using Eq. 2.14 by means of perturbation theory. Regarding a rotation in phase space, the choice of the ground state is arbitrary. Therefore it can be fixed to φ1 =φ2 =φ4 = 0 andφ3 =v at

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2.1. Introduction into the Standard Model of Particle Physics

φ0(x) = 1

√2 0

v

. (2.16)

The ground state is invariant with respect to a U(1)_em symmetry, which is a subgroup of SU(2)L⊗U(1)Y. Then one expandsφ(x) around the ground state φ0(x), resulting in

φ(x) = 1

√2

0 v+H(x)

(2.17) for the complex and scalar Higgs SU(2) doublet.

Once the vacuum state of Eq. 2.16 is chosen, the underlyingSU(2)L⊗U(1)Y symmetry is spontaneously broken. Only theU(1)_em symmetry remains, leaving the photon massless.

The electroweak theory has four degrees of freedom, three of them are absorbed by the longitudinal polarization of the gauge bosons to form massive particles (W^±, Z⁰). The remaining degree of freedom implies the existence of one additional neutral scalar particle, the so-called Higgs boson. Non-minimal models are based on a more complex Higgs sector and therefore predict additional neutral and charged Higgs bosons [37].

Summarising the considerations above, the Lagrangian of the Higgs field after sponta-neous symmetry breaking is

LHiggs = 1

2∂µH∂^µH

| {z } kinetic part

+const

+1

4g²v²W_µ⁺W^−µ+ 1

8(g²+g⁰²)v²ZµZ^µ−λv²H²

| {z }

mass terms +1

2g²vHW_µ⁺W^−µ+ 1

4(g²+g⁰²)vHZµZ^µ

| {z }

trilinear HW⁺W⁻ and HZZ coupling +1

4g²H²W_µ⁺W^−µ+1

8(g²+g⁰²)H²ZµZ^µ

| {z }

quartic HHW⁺W⁻ and HHZZ coupling

−λvH³−1 4λH⁴

| {z }

Self-coupling of the Higgs field

. (2.18)

The mass terms of the gauge bosons at tree level can be determined directly from the Lagrangian

MW = 1

2vg= ev 2 sinθW

, (2.19)

MZ = 1 2

pg²+g⁰²v = ev

2 sinθ_W cosθ_W = MW

cosθ_W, (2.20)

Mγ = 0, (2.21)

M_H =v√

2λ. (2.22)

2. Z Boson and Higgs Boson Production in the Context of the Standard Model

The masses of theW andZ boson depend directly on the vacuum expectation value, thus it could be determined by measuring both masses. But the Higgs boson mass cannot be calculated from the vacuum expectation value, since it also depends on λ, which is a free parameter in the SM.

Mass terms for the fermions have to be added via trilinear Yukawa couplings of the fermions to the Higgs fields, which results in extra terms for the Lagrangian. Fermion masses are given by

m_f = 1

√2g_fv, (2.23)

with the coupling constants gf being free parameters of the SM.

Im Dokument Physics with Jets in Association with a Z Boson in pp-collisions with the ATLAS Detector at the Large Hadron Collider (Seite 16-20)