QFT and Spontaneous Symmetry Breaking

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U PPSALA U NIVERSITY

B

ACHELOR

D

EGREE

P

ROJECT

15 HP

DEPARTMENT OFPHYSICS AND ASTRONOMY

DIVISION OF THEORETICAL PHYSICS

QFT and Spontaneous Symmetry Breaking

Abstract

The aim of this project is to understand the structure of the Standard Model of the particle physics. Therefore quantum field theories (QFT) are studied in the both cases of abelian and non-abelian gauge theories i.e. quantum electrody- -namics (QED), quantum chromodynamics (QCD) and electroweak interaction are reviewed. The solution to the mass problem arising in these theories i.e.

spontaneous symmetry breaking is also studied.

Sammanfattning

Syftet med detta projekt är att förstå strukturen för partikelfysikens standard- -modell. Därför studeras kvantfältsteorier (QFT) i båda fallen av abelska och icke-abelska gaugeteorier, dvs kvantelektrodynamik (QED), kvantkromodynamik (QCD) och elektrosvag växelverkan granskas. Lösningen på massproblemet som uppstår i dessa teorier, dvs. spontant symmetribrott studeras också.

Author:

Seyede Shila Seyedi

Supervisor:

Konstantina Polydorou Subject reader:

Joseph Minhan

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1 Introduction

In the second half of the 20th century physicists accomplished to describe many different occurring phenomena in nature by using a model called the Standard Model of particle physics. Developing this model helped to classify all elementary particles (or fundamental particles).

In the Standard Model, particles are divided into matter which consists of twelve elementary particles and radiation including four particles known as bosons. These are responsible for there out of four fundamental forces in nature. These four forces are the strong force, the weak force, the electromagnetic force and gravity.

The Standard Model does not include gravity and therefore is not complete unified theory of nature. Although general relativity is used to describe gravity in a classical level its unification with Standard Model in quantum level faces problems.

The mathematical framework of the Standard Model is provided by Quantum Field Theory (QFT) where each elementary particle is described by a field. In the 1950s, physicists used QFT to make excellent descriptions of fermions (particles such as electron) and bosons. However it predicted that all degrees of freedom should be massless which contradicts reality. We know for example that fermions are massive.

The photon (boson) is massless but on the other hand measurements show that the bosons of the weak interaction (Z, W^±) are massive.

The question is then why does for example the electron has mass and the photon has not? To answer this question among others, physicists suggested the Higgs mechanism [1]. According to it, there should be another field called the Higgs field which induces spontaneous symmetry breaking to give particles masses. The Higgs field is associated with an elementary particle called the Higgs boson.

In 2012 it was announced that experiments performed in the Large Hadron Collider at CERN, had succeeded in finding a particle consistent with the Higgs particle, [2].

In this work, we will explain the above statements and we will also explore the idea of spontaneous symmetry breaking so that we can deepen our understanding of the Standard Model. This work is a literature review.

2 Standard Model of Particle Physics

This work is mainly be based on [3],[4] and [5]. This introductory section is mainly based on [6].

Elementary particles i.e. particles with no internal structure, are classified into two classes of fermions and bosons by their spin. Particles with half-integer spins are fermions and obey Fermi-Dirac statistics which shows that each state can only be

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occupied by one particle (Pauli exclusion principle). Bosons have integer spins and obey Bose-Einstein statistics where a state can be occupied by any number of particles i.e. bosons are radiation and fermions consist matter. [7].

Fermions are divided further into two fundamental families of quarks and leptons each with six members. Quarks consist of u (up), d (down), c (charmed), s (strange), t (top) and b (bottom). These are also called flavors and are paired into so called generations as doublets, as follows

u d

, c

s

, t

b

. (1)

the u, c, and t quarks have charge +²₃ while d, s and b have charge −¹₃.

Leptons consist of the six different flavors of e⁻ (electron), νe (electron neutrino), µ⁻ (muon), νµ(muon neutrino), τ⁻(tau) and ντ (tau neutrino). These accur also as generations as

νe

e⁻

, νµ

µ⁻

, ντ

τ⁻

. (2)

e⁻, µ⁻, τ⁻ have electric charge q = −e but the other three νe, νµ, ντ are all neutral.

All of these particles (quarks and leptons) have a corresponding anti-particle i.e.

anti-quarks and anti-leptons.

Bosons are grouped into two families of spin-1 and spin-0 particles. Spin-1 bosons are called gauge bosons and are defined as force carriers and mediate interactions among matter particles or fermions. With interactions we mean how elementary particles influence each other or interact with each other. These interactions happen when fermions exchange other particles (i.e. force carriers).

Fundamental interactions can be described by a fundamental symmetry principle called gauge invariance which states that the laws of physics described by Lagrangians are invariant under some symmetries and the symmetry transformations can be performed locally and not just globally. By local symmetry transformation we mean that it is possible to make different transformations in each point of space-time.

There are four kinds of different gauge bosons for three out of four fundamental interactions which are electromagnetic interaction, weak interaction and strong interaction. The fourth fundamental interaction is gravity, however it is highly unexplored in the quantum level and it will not be included in this report.

The gauge bosons are: photon (γ) (mediating electromagnetic interaction) and can be described by quantum electrodynamics, QED; 8 different types of gluons, (g) (mediating strong interaction) which can be described by quantum chromodynamics, QCD; and W and Z bosons (W^±, Z) (mediating weak interaction). W^± and Z bosons together with photon mediate electroweak interaction. Photon and gluons are massless while W and Z bosons are massive.

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Charged leptons interact both electromagnetically and via weak interactions while the other three neutral leptons only interact via the weak interaction. Different from leptons, quarks have one more degree of freedom called color. Since only quarks carry color charge, they are the only particles that interact via strong interaction.

They can interact by the weak and electromagnetic interactions as well.

Lastly we have Higgs boson which is a scalar boson with spin-0. Higgs boson explains why only photon and gluons are massless particles of the elementary particles.

In the following section we will talk about QED, QCD and electroweak gauge theories and finally introduce the Higgs mechanism which solves the problems arising from these theories.

2.1 Abelian Gauge Theory (QED)

We will work in the so called natural units where

c = ~ = 1. (3)

In this system the unit of energy is also fixed, so for some physical quantities we have

[energy] = [mass] = [length]⁻¹ = [time]⁻¹

If needed, factors of c and ~ can be restored by dimensional analysis. The metric throughout this report is Lorentzian in which the Minkowski metric signature is mostly minuses, (+, −, −, −).

We want to first look at a simple case of spin-¹₂ fields and see how these fields interact.

2.1.1 Constructing the QED Lagrangian for Spin-¹₂ Fields

The Free Lagrangian for spin-¹₂ fields are given as [3],

L_Dirac= iΨ /∂Ψ − mΨΨ, (4)

where Ψ = Ψ^†γ⁰ and /∂ = γ^µ∂_µ (Feynman slash notation). This Lagrangian is referred as Dirac Lagrangian (read more on free Lagrangians in appendix C).

The Dirac Lagrangian does not include any interactions. We mentioned before that using a specific symmetry principle called gauge invariance we can introduce interactions. This principle states that under some transformations, in this case under U (1) transformations, the Lagrangian is invariant. Let us look at the global transformation of the Lagrangian. First we note that elements of U (1) group can be presented by e^iqα (see appendix A) where q is a real constant and α is

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space-time independent (since we have global transformation). Then under U (1) transformation the spinor fields become

Ψ −→ Ψ⁰ = U Ψ = e^iqαΨ,

Ψ −→ Ψ⁰ = ΨU^†= Ψe^−iqα. (5)

Substituting (5) in (4) we see that the Lagrangian is invariant i.e. L −→ L⁰ = L.

Now, the more interesting case is local transformations rather than global transformations, i.e making α space-time dependent, as α(x^µ). The transformations are then

Ψ −→ Ψ⁰ = U (x^µ)Ψ = e^iqα(x^µ⁾Ψ,

Ψ −→ Ψ⁰ = ΨU^†(x^µ) = Ψe^−iqα(x^µ⁾. (6) Substituting (6) into (4) the Lagrangian becomes

L −→ L⁰= iΨe^−iqα(x^µ⁾γ^µ∂µe^iqα(x^µ⁾Ψ

| {z }

?

−mΨ e^−iqα(x^µ⁾e^iqα(x^µ⁾

| {z }

=1

Ψ, (7)

where ? = iq(∂µα(x^µ))e^iqα(x^µ⁾Ψ + e^iqα(x^µ⁾∂_µΨ. This shows that the Lagrangian is not invariant under local transformations. In order to make it invariant one replaces the ordinary derivative with the gauge covariant derivative, [4]

∂µ−→ D_µ:= ∂_µ+ iqAµ, (8)

where Aµis a four-vector potential or gauge field Aµ:= (φ, − ~A).

By introducing the gauge covariant derivative we can now rewrite the Lagrangian as

L = iΨ /DΨ − mΨΨ = iΨ /∂Ψ − qΨ /AΨ − mΨΨ, (9) where /A = γ^µAµ.

We can show that under U (1) transformation of the spinor fields, the gauge field Aµ

must transform as

Aµ−→ A⁰_µ= Aµ− ∂_µα(x^µ), (10) in order to have a gauge invariant Lagrangian. Substituting all of these into the Lagrangian we get

L → L⁰= iΨe^−iqα(x^µ⁾∂e/ ^iqα(x^µ⁾Ψ − qΨe^−iqα(x^µ⁾A/⁰e^iqα(x^µ⁾Ψ − mΨ · 1 · Ψ

= iΨ^−iqα(x

µ)

γ^µh

iq(∂_µα(x^µ))e^iqα(x^µ⁾Ψ + e^iqα(x^µ⁾∂_µΨi

− qΨ /A⁰Ψ − mΨΨ

= iΨ /∂Ψ − qΨγ^µ(∂µα(x^µ))Ψ − qΨ /A⁰Ψ − mΨΨ

= iΨ /∂Ψ − qΨγ^µ(∂_µα(x^µ))Ψ − qΨγ^µ(A_µ− ∂_µα(x^µ))Ψ − mΨΨ {using (10)} = iΨ/∂Ψ − qΨ /AΨ − mΨΨ = iΨ /DΨ − mΨΨ = L.

(11)

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We have claimed that by introducing covariant derivative we can make an invariant term under the gauge transformations different from the usual derivative. It means that the covariant derivative of Ψ must transform such that the Ψ /DΨ is gauge invariant. Then the transformation is as follows

DµΨ −→ e^iqα(x^µ⁾DµΨ. (12)

We can show that this is true:

D_µΨ −→ (∂_µ+ iq [A_µ− ∂_µα(x^µ)])(e^iqα(x^µ⁾Ψ)

= e^iqα(x^µ⁾

iq(∂_µα(x^µ))Ψ + ∂_µΨ + iqA_µΨ − iq∂_µα(x^µ)Ψ

= e^iqα(x^µ⁾(∂_µ+ iqA_µ)Ψ

= e^iqα(x^µ⁾D_µΨ.

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This shows that (12) holds. The transformation rule for Dµcan also be written more explicitly using the 1 × 1 unitary matrix,

D_µ−→ D⁰_µ= e^+iqα(x^µ⁾D_µe^−iqα(x^µ⁾= U (x^µ)D_µU^†(x^µ). (14) Thus

DµΨ −→

U (x^µ)DµU^†(x^µ)

U (x^µ)

Ψ = U (x^µ)DµΨ = e^iqα(x^µ⁾DµΨ, (15) which is the same as (12). Note that U^†(x^µ)U (x^µ) = 1.

Going back to (11), we see that by introducing the covarinat derivative which itself introudces a new field Aµ, we can make the Lagrangian invariant under U (1) transformation. But this is actually not the only thing that we have gained here.

Looking at the term −qΨ /AΨwe see that we have an interaction between the spinor field and the gauge field (it is the vertex of QED) which was not there in the case of global transformations. This can be written as

L_int= qΨ /AΨ = J^µAµ, (16)

where J^µis called the current density [3]. We have then introduced an interaction as mentioned before using the gauge principle.

Note that in this Lagrangian there are no terms proportional to the derivative of Aµ

hence we have no information about its propagation yet. We want to make Aµfield dynamical. We know that Aµis a vector field with spin-1 and the free Lagrangian called the Proca Lagrangian for spin-1 fields is [3]

L_proca= − 1

16πF^µνF_µν+ 1

8πm²A^µA_µ, (17)

where F^µν := ∂^µA^ν − ∂^νA^µ is called the field-strength tensor and it is totally anti-symmetric (F^µν = −F^νµ) with components F⁰ⁱ = ∂⁰Aⁱ − ∂ⁱA⁰ = −Eⁱ and F^ij = ∂ⁱA^j− ∂ⁱA^j = −^ijB^k.

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We need to make sure that the Proca Lagrangian is invariant under the local transformation with A^µ−→ A^0µ= A^µ− ∂^µα(x^µ):

L_{P roca} → L⁰_{P roca} = − 1

16π(∂^µA^0ν− ∂^νA^0µ)η_µλη_νβ(∂^λA^0β− ∂^βA^0λ) + 1

8πm²A^0νη_µνA^0µ

= − 1

16π(∂^µA^ν −∂^µ∂^να − ∂^νA^µ+∂^ν∂^µα) η_µλη_νβ(∂^λA^β−∂^λ∂^βα − ∂^βA^λ+∂^β∂^λα)

+ 1

8πm²(A^ν− ∂^να)ηµν(A^µ− ∂^µα)

= − 1

16πF^µνFµν+ 1

8πm²(A^ν− ∂^να)ηµν(A^µ− ∂^µα)

(18) The first term is invariant but the second term is not, therefore one can use the Proca Lagrangian only if mAµ = 0. Hence we can rewrite the full Lagrangian by adding the massless Proca Lagrangian to it as

L = iΨ /DΨ − mΨΨ − 1

16πF^µνFµν, (19)

which is invariant under local U (1) transformations.

The U(1) symmetry group is identified as the gauge group of electromagnetism and (19) is the QED Lagrangian. The gauge field Aµmediates the electromagnetic interaction. q in Dµterm is the coupling strength of Ψ to the electromagnetic field i.e. charge. [3] [4] [5]

The above example can be summarized as follows:

One first begins with a free matter Lagrangian with some global symmetry transformation. Then one goes from the global symmetry to a local gauge symmetry which will be possible only if one includes a compensating gauge field. This promotion of global symmetry to local symmetry, will lead to interactions. One needs to promote the ordinary derivatives to the covariant derivatives as well. Lastly by adding a field strength (kinetic term of the guage field) i.e. −_16π¹ F^µνFµν to the Lagrangian, the gauge field can propagate as well. We also found out that in order to maintain the local symmetry the Aµfield must be massless.

In the two following examples we will derive the equations of motion for the fermion field Ψ and the gauge field Aµfrom the QED Lagrangian that we have constructed.

2.1.2 Equation of Motion for fermions

Let us first find the equation of motion for the fermion field Ψ using the Euler-Lagrange equation (see appendix C)

∂L

∂Ψa

− ∂_µ ∂L

∂(∂µΨa) = 0. (20)

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We obtained the Lagrangian in (19). Expanding the terms we get

L = iΨγ^µ∂^µΨ − qΨγ^µA_µΨ − mΨΨ − F^µνF_µν (21) Inserting the Lagrangian in (20) with Ψa = Ψand using ∂µ((∂L)/(∂(∂_µΨ)) = 0, we obtain

∂L

∂Ψ = −(−iγ^µ∂µΨ + qγ^µAµΨ + mΨ) = 0, (22) which results to

(iγ^µ∂µ− m)Ψ = qγ^µAµΨ. (23) This is Dirac’s equation and the interaction with electric field. Note that if there is also the external magnetic field Bµ, one would obtain (iγ^µ∂_µ−m)Ψ = qγ^µ(A_µ+B_µ)Ψ which indicates an interaction with electromagnetic field. Using the Gauge covariant derivative we can rewrite (23) as

iγ^µ(∂_µ+ iqA_µ)Ψ − mΨ = 0 (24)

(iγ^µD_µ− m)Ψ = 0. (25)

2.1.3 Equation of Motion for Gauge Fields

Now we will look at the equations of motion for the gauge fields which will lead to Maxwell’s equations. Using (204), the Euler-Lagrange equation for Aµis

∂L

∂Aν

= −q∂A_µ

∂Aν

Ψγ^µΨ = −gδ^ν_µΨγ^µΨ = −qΨγ^νΨ (26) Now the second term

∂µ

∂L

∂(∂µAν) = ∂µ

− 1 16π

∂F_αβF^αβ

∂(∂µAν)

= − 1 16π∂µ

∂F_αβ

∂(∂µAν)F^αβ+ Fαβ

∂F^αβ

∂(∂µA^ν)

= − 1 16π∂µ

∂Fαβ

∂(∂_µA_ν)F^αβ+ F_αβ∂(g^αρg^βσFρσ)

∂(∂_µA^ν)

= − 1 16π∂µ

∂F_αβ

∂(∂µAν)F^αβ+ g^αρg^βσFαβ

∂Fρσ

∂(∂µA^ν)

= − 1

8π∂_µ ∂F_αβ

∂(∂µAν)F^αβ

= − 1

8π∂_µ∂(∂αA_β− ∂_βAα)

∂(∂µAν) F^αβ

= − 1

8π∂(δ_α^µδ_β^ν− δ_β^µδ_α^ν)F^αβ = − 1

8π∂µ(F^µν− F^νµ) = − 1

4π∂µ(F^µν) (27) Now by setting the results in (20), we obtain

−qΨγ^νΨ + 1

4π∂µ(F^µν) = 0. (28)

The Lorentz gauge condition states that ∂µA^µ= 0, which simplifies (28) to

∂_µ∂^µA^ν = 4πqΨγ^νΨ = 4πJ^µ, (29)

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which is the relativistic Maxwell’s equations in cgs units (J^µ= qΨγ^νΨis the electric current density).

Next we will move on to the nonabelian gauge theory also called Yang-Mills Theory which basically will follow the same procedure as here.

2.2 Nonabelian Gauge Theory (Yang-Mills Theory)

Let us consider a scalar or spinor Ψ where Ψ is a N-tuple of fields given by

Ψ =





 Ψ1

Ψ₂ ... Ψ_N







(30)

We want to construct a Lagrangian with Ψ such that it is invariant under an continuous SU (N ) symmetry, i.e we want it to have a global symmetry.

SU (N )is an example of a compact Lie group and it therefore has an underlying Lie algebra g with generators T^a. The commutator of any two generator matrices is

[Ta, T_b] = if_ab^cTc (31) where f_ab^c ∈ R are called the structure coefficients of the group. (For proof of f_ab^c ∈ R see appendix A.2). If f_ab^c 6= 0, the group is nonabelian.

We want this Lagrangian to also be invariant under a local SU (N ) transformation

Ψ −→ Ψ⁰= U (x)Ψ, (32)

where

U (x) = e^−igα^a^(x)T^a. (33)

Note that a = 1, · · · , N² − 1, where N²− 1 is the number of generators of SU (N ).

Moreover the T^aare the generators of SU (N ) and these are Hermitian and traceless matrices.

We can find how the gauge field transforms using (8) and (14), (∂_µ+ igA⁰_µ)U (x)Ψ = U (x)(∂_µ+ igA_µ)Ψ (∂µU (x))Ψ +

U (x)∂µΨ + igA⁰_µΨ =

U (x)∂µΨ + igU (x)AµΨ

⇒ ∂_µU (x) + igA⁰_µU (x) = igU (x)A_µ.

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Multiplying by U^†(x)from right

(∂_µU (x))U^†(x) + igA⁰_µ= igU (x)A_µU^†(x)

⇒A⁰_µ= U (x)A_µU^†(x) + i

g(∂_µU (x))U^†(x). (35)

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Note that Aµ is a N × N Hermitian matrix. If we choose U (x) = e^igα(x), then (35) reduces to the abelian gauge transformation A⁰_µ = A_µ− ∂_µα(x)we studied in the previous section.

The covariant derivative is [8]

D_µ= ∂_µ· 1N ×N + igA_µ. (36)

Since there are N fields then each field will be transformed as

Ψi−→ Ψ⁰_i = U (x)ijΨj, (37) where repeating indices are summed.

There are generators T^a for each representation of g and each gauge field Aµ can be expanded on a basis {T^a} of generators of the Lie algebra. Then the covariant derivative acting on Ψ (30) can explicitly be written as

D_µΨ_j = ∂_µΨ_j+ ig(T_aA^a_µ)_jlΨ_l. (38)

Now we can construct a Lagrangian that is invariant under a local SU (N ) symmetry by replacing the ordinary derivatives with covariant derivatives as we did in the local U (1) symmetry example.

However, we also need to add a field strength to the Lagrangian. In order to do this we will firstly show that the gauge invariant field strength can be expressed in terms of the covariant derivative: [4]

[D^µ, D^ν] Ψ = (∂^µ+ iqA^µ)(∂^ν+ iqA^ν)Ψ − (∂^ν + iqA^ν)(∂^µ+ iqA^µ)Ψ

= (∂^µ+ iqA^µ)(∂^νΨ + iqA^νΨ) − (∂^ν + iqA^ν)(∂^µΨ + iqA^µΨ)

= ∂^µ∂^νΨ + iqA^µ∂^νΨ − q²A^µA^νΨ + iq(∂^µA^ν)Ψ + iqA^ν∂^µΨ

− ∂^ν∂^µΨ − iqA^ν∂^µΨ + q²A^νA^µΨ − iq(∂^νA^µ)Ψ − iqA^µ∂^νΨ

= iq(∂^µA^ν − ∂^νA^µ)Ψ

→ [D^µ, D^ν] Ψ = iqF^µνΨ

(39)

More explicitly this can be written as F^µν = −i

q[D^µ, D^ν] . (40)

Now we will expand both Aµand Fµν in terms of the generator matrices as [8]

A_µ= A^a_µT_a, (41)

and

F_µν = F_µν^a T_a. (42)

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Inserting these into (40) yields F_µνΨ = −i

q [D_µ, D_ν] Ψ

= −i

g(∂_µ+ igA^a_µT_a)(∂_νΨ + igA^b_νT_b) − i

g(∂_ν + igA^b_νT_b)(∂_µΨ + igA^a_µT_a)

− i

g∂µ∂νΨ + ∂µ(A_ν^bTb)Ψ + A^b_νTb∂µΨ + A^a_µTa∂νΨ + ig(A_µ^aTaA^b_νTb)Ψ + i

g∂ν∂µΨ − ∂ν(Aâ_µTa)Ψ − Aâ_µTa∂νΨ − A^b_νT_b∂µΨ − ig(A_ν^bT_bAâ_µTa)Ψ

= ∂_µ(A^a_νT_a)Ψ − ∂_ν(A^a_µT_a)Ψ + igh

A^a_µT_a, A^b_νT_bi Ψ

(43)

Then

F_µν^c Tc= Tc(∂µA^c_ν − ∂_νA^c_µ) + ig [Ta, Tb] A^a_µA^b_ν. (44) So using (31) yields

F_µν^c = ∂µA^c_ν − ∂_νA_µ^c − gf_ab^cA^a_µA^b_ν. (45) In previous section we considered abelian gauge theory which means that the structure constant is equal to zero. We can therefore clearly see that the field strength in that case becomes

F_µν = ∂_µA_ν − ∂_νA_µ. (46)

The kinetic term is [8]

L_kin = −1

4Tr(F^cµνF_cµν). (47)

One should note that since we consider SU (N ) to be nonabelian, then there will be interactions among the gauge fields. In the next section we will go through the example of Yang-Mills theory known as QCD.

2.3 Yang-Mills Theory with SU (3) symmetry (QCD)

We will first construct a free Lagrangian where each field corresponds to a flavor of a quark. There are six flavors where each comes in three colors: red, blue and green.

These colors are notations for three distinct states of charge that each particular quark has. The free Lagrangian can then be written as

L = iΨ_il∂/_µΨ_il− m_lΨ_ilΨ_il, (48) where i = r, b, g is the color index and l = 1, · · · , 6 is the flavor index.

To simplify this Lagrangian, we can consider a 3-component quark field as [3]

Ψ =



 Ψr

Ψ_b Ψg



, Ψ = Ψ_r, Ψ_b, Ψ_g , (49)

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and rewrite the Lagrangian as follows

L = iΨ_l∂/_µΨl− m_lΨlΨl. (50) This is invariant under global SU (3) transformations¹,

Ψ_l−→ Ψ⁰_l= e^−iqλâ^αâΨ_l, (51) where λâdenotes the eight generators of SU (3) and αaare arbitrary parameters.

We now require the Lagrangian to be invariant under a local SU (3) transformation.

For local transformations as usual we make the parameters to be space-time dependent. The transformation will be

Ψl−→ Ψ⁰_l = e^−igλ^a^α^a^(x)Ψl. (52) In order to fulfill this requirement, we will change the ordinary derivative with the covariant derivative

∂µ−→ D_µ= ∂µ+ igλaG^a_µ. (53) Here we need eight gauge fields, one for each generator. These gauge fields are called gluons hence the notation G.

The transformation rules for covariant derivatives Dµand gauge fields Gµas shown in previous sections are

D_µ−→ D_µ⁰ = U (x)D_µU^†(x), (54) and

λ_aG^a_µ−→ λ_aG_µ^0a = U (x)λ_aG^a_µU^†(x) + i

g(∂_µU (x))U^†(x). (55) The modified Lagrangian is

L = iΨ_lD/_µΨ_l− mΨ_lΨ_l= iΨ_l∂/_µΨ_l− mΨ_lΨ_l− gΨ_lγ^µλaG^a_µΨ_l. (56) As before we notice that interactions between Ψ and Gµhas been introduced, Lint= gΨ_lγ^µλaG^a_µΨ_l, which describe quark fields Ψl interacting with gluons Gµ. There is only one unique coupling constant despite the fact that there are three types of charge²[4][9].

1Since the Lagrangian is similar to Dirac Lagrangian but with a 3-component field, it will have a U (3)symmetry. Unitary matrix can be written as U = e^Hwhere H^†= Hand also H = θ · 1 + λiaⁱ, i = 1, ..8where λi are Gell-Mann matrices (generators of SU (3)). Thus U = eîθeîλⁱâⁱ. eîλⁱâⁱ has determinant 1, hence by definition is a SU (3) symmetry.[3]

2The group is SU (3) and therefore the gauge fields transform irreducibly under SU (3) transformations hence they have the same coupling constant.

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Now, we will give the gauge fields Gµtheir gauge invariant kinetic term using the field strength F_µν^a as in (47)

L_G_µ = −1

4F_µν^a F_a^µν = −1

4(∂_µGâ_ν − ∂_νGâ_µ− gf_bcâG^b_µG^c_ν)(∂^µG^ν_a− ∂^νG^µ_a− gf_adeG^µdG^νe)

= −1

4(∂µG^a_ν − ∂_νG^a_µ)(∂^µG^ν_a− ∂^νG^µ_a) +1

4f_de^aG^d_µG^e_ν(∂^µG^ν_a− ∂^νG^µ_a) +1

4f_bc^aG^µbG^νc(∂µGν a− ∂_νGµa)

−g²

4 f_bc^afadeG_µ^bG_ν^cG^µdG^νe

(57) The first term is the usual kinetic term but the three other terms indicate gluon-gluon interactions. We can clearly see that if we had an abelian theory such as example 1 in section (2.1), we would have no interactions between gauge fields which is the case for the photons in that example (classically). However here with non-zero structure constants f_bc^a 6= 0 and we get interactions among gauge fields as well. [4]

In this example we have shown a way to introduce interactions between the quark fields and the gluon fields i.e. we have shown how to develop the theory of the strong interactions and derived the Lagrangian of the QCD.

2.4 Weak Interactions

In order to formulate the weak interaction in terms of a gauge symmetry, one has to unify the electromagnetism and weak interaction. This unified description of these two fundamental interactions is called electroweak interaction. But the weak interactions have several differences with QCD and electromagnetism. [3]

In [3], [4] and [5] these differences are pointed out. It is stated that one of the differences is that the weak interaction is mediated by the force mediators that are extremely heavy unlike the massless photon and gluons. Also, different from both strong and electromagnetic interactions the weak interaction violates the parity, charge conjugation and charge-parity (CP) symmetry. Weak interactions are also the only interactions that can change one type of a particle into another. It is also stated that every matter particle can interact weakly unlike other two interactions (electromagnetic and strong interactions) that only act on a subgroup of the matter particles.

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2.4.1 Yang-Mills Theory with SU (2)L ⊗ U (1)_Y symmetry, (Electroweak Interaction)

The underlying gauge symmetry group of the electroweak interaction is³

SU (2)_L⊗ U (1)_Y, (58)

where Y denotes the hypercharge [3]. SU (2)L only acts on "left-handed" spinor fields, hence the reason for the notation L. These notations will be explained later.

Let us first introduce the Dirac Lagrangian in terms of 4-component objects ΨLand Ψ_R. The Dirac Lagrangian as stated before in (4) is

L_D = iΨ /∂_µΨ − mΨΨ, (59)

where Ψ is a 4-component spinor. But this spinor is actually composed of two 2-component spinors, ψLand ψR, called Weyl or Chiral spinors,

Ψ =ψL

ψ_R

, (60)

where L stands for left and R for right. ψLand ψRexhibit a certain symmetry called Chiral Symmetry:

ψL−→ ψ_Le^iα ψ_R−→ ψ_R,

or

ψ_L−→ ψ_L ψR−→ ψ_Re^iα⁰.

This symmetry states that one gets the same theory by an independent rotation of the left-handed ψLand the right-handed ψRcomponents.

Left- and right-handed components (i.e Chiral spinors) can be projected out using P_Land PRprojection operators [8]

P_L= 1

2(1 − γ₅) =12×2 0

0 0

, (61)

and

P_R= 1

2(1 + γ₅) =0 0 0 12×2

(62) with

γ5 =−12×2 0 0 12×2

. (63)

3SU (2)is the simplest group with doublet representations, to include electromagnetic interactions one also need one additional U (1) group. (This statement is taken from [9].)

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Then

P_LΨ =ψ_L 0

:= Ψ_L (64)

and

P_RΨ = 0 ψR

:= Ψ_R (65)

Using equations (64) and (65), we have

Ψ = ΨL+ ΨR (66)

and

Ψ = ΨL+ ΨR (67)

(67) can be shown as follows:

Ψ = Ψ^†PLγ⁰+ Ψ^†PRγ⁰ (68) P_Lγ⁰ = γ⁰P_R, P_Rγ⁰ = γ⁰P_Lwith γ⁰ = 0 1

1 0

(69)

⇒Ψ = Ψ^†γ⁰PR+ Ψ^†γ⁰PL= ΨPR+ ΨPL (70) wih

ΨP_R= ψ^∗_R ψ^∗_L0 0 0 1

= 0 ψ^∗_L := Ψ_L, (71) ΨPL= ψ^∗_R ψ^∗_L1 0

0 0

= ψ^∗_R 0 := Ψ_R (72)

Then

ΨPR+ ΨPL= ΨL+ ΨR. (73)

We can now rewrite the iΨ /∂_µΨterm:

iΨ /∂_µΨ = i(Ψ_L+ Ψ_R) /∂_µ(Ψ_L+ Ψ_R)

= iΨL∂/_µΨL+ iΨL∂/_µΨR+ iΨR∂/_µΨL+ iΨR∂/_µΨR

= iΨ_L∂/_µΨ_L+ iΨ_R∂/_µΨ_R

(74)

since γ^µ = 0 σ^µ σ^µ 0

, where σ^µ := (1, σ) and σ^µ := (1, −σⁱ)in which 1 is a 2 × 2 identity matrix and σⁱfor (i= 1, 2, 3) denote Pauli matrices:

σ₁ =0 1 1 0

, σ₂ =0 −i i 0

, σ₃=1 0 0 −1

, (75)

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and:

ΨL∂/_µΨR= 0 ψ_L^∗ 0 σ^µ σ^µ 0

0

∂µψ_R^∗

= 0 Ψ_R∂/_µΨ_L= ψ_R^∗ 0 0 σ^µ

σ^µ 0

∂_µψ_L^∗ 0

= 0

(76)

Let us now look at the mass term of (59)

ΨΨ = (Ψ_L+ Ψ_R)(Ψ_L+ Ψ_R) = Ψ_LΨ_L

| {z }

=0

+Ψ_LΨ_R+ Ψ_RΨ_L+ Ψ_RΨ_R

| {z }

=0

⇒ ΨΨ = Ψ_LΨR+ ΨRΨL

(77)

If we only transform the left handed component, the term iΨ /∂_µΨremains invariant but the mass term as seen in (77) will not be invariant. One can therefore conclude that massive fermions do not exhibit Chiral symmetry as the mass term breaks the Chiral symmetry. Hence a Chiral theory of fermions only allows for massless fermions!

So in terms of (66) i.e by decomposing Dirac field Ψ into left- and right-handed fields, the fermionic Lagrangian becomes

L_Dirac= iΨL∂/_µΨL+ iΨR∂/_µΨR (78) So there are left-handed and right-handed particles. Left-handed particles are paired into left-handed doublets where the SU (2)Lwhich has a doublet representation can act on them [4]. These are [3, 4, 5]

χL→ν_e e

L

,ν_µ µ

L

,ν_τ τ

L

,u d

L

,c s

L

,t b

L

. (79)

The right-handed singlets needed for U (1)Y to act on (the symmetry group is SU (2)_L× U (1)_Y) are [3, 4]

eR, µR, τR, uR, dR, cR, sR, tR, bR. (80) To construct the Lagrangian we should include all of the right- and left-handed fields. For simplicity we will only look at the first generation of leptons, i.e.

Ψ₁ = νe

e

L

,and Ψ2 = e_R. The reason for this is that the only difference between these 6 doublets and 6 singlets are their masses when we only look at electroweak interactions. [3, 4].

Then the Lagrangian becomes

L_Dirac= iΨ₁∂/_µΨ₁+ iΨ₂∂/_µΨ₂=

2

XiΨ_j∂/_µΨ_j (81)

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Since symmetry group of Electroweak interactions is (58), the Lagrangian given by (81) will be invariant under global transformations

e^igσⁱ^αⁱ, i = {1, 2, 3} for SU (2)L (82) and

e^ig⁰^y^j^β, j = {1, 2} for U (1)Y. (83) Earlier we mentioned that the Y denotes the hypercharge and here we can see that the U (1)Ltransformations is similar to the transformation in QED, hence the name hypercharge. [9]

The fields will then transform as

Ψ₁ −→ Ψ⁰₁ = e^ig⁰^y¹^βe^igσⁱ^αⁱΨ₁

Ψ₂ −→ Ψ⁰₂ = e^ig⁰^y²^βΨ₂. (84)

We will now promote the global symmetry to a local symmetry with αⁱ = αⁱ(x) and β = β(x) using the same methods as before, which means that we will need covariant derivatives and the transformations of the gauge fields introduced through the covariant derivatives.

It is however important to note that we have αⁱ(x), where i = {1, 2, 3} and β(x) hence four different gauge fields are needed. Three for SU (2) and one for U (1).

(Recall also from section 2.2 that the number of generators of SU (N ) is (N²− 1). In this case N = 2 hence 3 generators and therefore there are 3 associated gauge fields).

This means that the covariant derivatives are

∂µΨ1−→ D_µΨ1 = ∂µΨ1+ igσiW_µⁱΨ1+ ig⁰y1BµΨ1

∂_µΨ₂−→ D_µΨ₂ = ∂_µΨ₂+ ig⁰y₂B_µΨ₂, (85) where we have denoted the 3 SU (2)L gauge fields W_µⁱ, i = {1, 2, 3}, and denoted the gauge field of U (1) as Bµ.

Transformation rules for the gauge fields are identical to the transformation rules we found in (35) and (10), i.e.

σiW_µ⁰ⁱ= e^igσⁱ^αⁱσiW_µⁱe^−igσⁱ^αⁱ + i

g∂µ(e^igσⁱ^αⁱ)e^−igσⁱ^αⁱ B_µ⁰ = B_µ− ∂_µβ(x).

(86)

As we did before, we make the gauge fields dynamical as well using the invariant term of the Proca Lagrangian. Hence we need to find out the gauge invariant field strength for each gauge field using (45). For the U (1)Y we get

B_µν= ∂_µB_ν − ∂_νB_µ, (87)

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and for SU (2)L

W_µνâ = ∂_µW_νâ− ∂_νW_µâ− gf_bcâW_µ^bW_ν^c. (88) We can now write the electroweak Lagrangian with the kinetic term Lkinas

L_Dirac+ L_kin= iΨ_L∂/_µΨ_L+ iΨ_R∂/_µΨ_R−1

4B_µνB^µν−1

4W_µνⁱ W_i^µν (89) We can see in (88) that just like QCD there will be self-interactions between the SU (2)_Lgauge bosons. [4].

We want to know how the SU (2)Lmatrix look in order to find how the gauge bosons {W_µⁱ} act on the doublets (79). If we expand the term σiW_µⁱ = σ₁W_µ¹+ σ₂W_µ²+ σ₃W_µ³ we can find the matrix as

0 W_µ¹ W_µ¹ 0

+

0 −iW_µ² iW_µ² 0

+W_µ³ 0 0 −W_µ³

=

W_µ¹ W_µ¹− iW_µ² W_µ¹+ iW_µ² W_µ¹

(90)

By defining

W_µ⁺ = W_µ¹+ iW_µ² and W_µ⁻= W_µ¹− iW_µ², (91) we get the following matrix:

W_µ³ W_µ⁻ W_µ⁺ −W_µ³

. (92)

By looking at this matrix we can see that when W_µ^±acts on a doublet, there will be a change in flavor. For example with the doubletνe

e

L

, W⁺e = ν_eand W⁻ν_e = e.

On the other hand we see that W_µ³will not change the flavor. It means that W^±are positively/negatively charged gauge bosons and W_µ³ is a neutral gauge boson. [4]

[9]

In order for gauge symmetry to be the principle underlying the weak interactions, the gauge fields must be massless. This will not cause a problem in QED and QCD since the gauge bosons are the massless photons and gluons. But it has been shown experimentally that that W^±and Z⁰bosons are massive [6]. In this section we also encountered another problem where the fermions must be massless otherwise the Chiral symmetry breaks. There must be a solution to these issues to be able to use this theory at all.

Physisicts S. Glashow, A. Salam and S. Wienberg were able to solve these problems by introducing a process called spontaneous symmetry breaking. According to [3]

(p. 345-346), Glashow used his theory based on spontaneous symmetry breaking and was able to produce the massless photon and the massive Z⁰ boson by mixing the W_µ³and Bµas

Aµ= Bµcos θW + W_µ³sin θW

Z = −B sin θ + W³cos θ (93)

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where θW is called weak mixing angle.

Going back to (82) and (83), We have two different coupling constants g and g⁰ for SU (2)_L and for U (1)Y respectively. In the same section of [3], the relation between these two coupling constants, through the weak coupling constant, is shown as

g sin θW = g⁰cos θW. (94)

In the following section we will go through the basic ideas of spontaneous symmetry breaking and the of Higgs mechanism.

2.5 Spontaneous Symmetry Breaking

Here we want to go through a simple case to demonstrate the spontaneous symmetry breaking following the example given in [3] and [4].

Let us consider the following Lagrangian L = 1

2(∂µφ)(∂^µφ) + 1

2µ²φ²− 1

4λ²φ⁴, (95)

where λ²and µ²are constants, λ² is a positive number and µ²can be either positive or negative. The sign of µ² being positive will lead to the wrong sign of the mass term in the Lagrangian, as in (95). We need therefore go through a procedure to find the correct mass term. (Compare to the mass term in the Klein-Gordan Lagrangian, see appendix C).

The potential term in this Lagrangian is U (φ) = −1

2µ²φ²+1

4λ²φ⁴, (96)

and is symmetric under

φ −→ −φ. (97)

Figure 1: Illustration of the potential U (φ).

We can rewrite the potential as

U = λ

4(φ²− a)², (98)

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with a = ^µ_λ²2 and plus a constant which is not relevant.

The Euler-Lagrange equation is

∂L

∂φ − ∂_µ ∂L

∂(∂_µφ)

= −∂U

∂φ − ∂_µ(∂µφ) = 0. (99) The static solutions require ∂µφ = 0, then

∂U

∂φ = −µ²φ + λ²φ³= 0. (100)

This gives the three solutions φ1,2 = ±^µ_λ and φ3 = 0. But the minimum of the potential occurs at φ1and φ2 (clear form both the formula (98) and figure (1)), these are called the vacuum expectation value of φ. These are the only two ground states (states of lowest energy), i.e. φ3 is not a ground state. One should also note that the φ1,2 solutions do not share the symmetry of the potential. This can be shown more explicitly by looking at the very small fluctuations around the solutions φ1,2,3. We consider φ(x) = φ0+ δφ(x)where φ0is one of the static solutions. To determine the equation of motion for the fluctuations δφ(x), we plug in this into (95)

L(δφ) = 1

2∂µ(φ0+ δφ)∂^µ(φ0+ δφ) +1

2µ²(φ0+ δφ)²−1

4λ²(φ0+ δφ)⁴

= 1

2(∂_µδφ)(∂^µδφ) +1

2µ²(φ₀+ δφ)²−1

4λ²(φ₀+ δφ)⁴.

(101)

If we choose φ0 = φ₃ = 0, then the Lagrangian will become

Figure 2: Choosing φ0 = φ₃ = 0 as the static solution and perturbing it in small fluctuations about that solution.

L_φ₃ = 1

2(∂_µδφ)(∂^µδφ) + 1

2µ²δφ²−1

4λ²δφ⁴ (102)

This still admits the same symmetry as the original Lagrangian (95) i.e. Lφ3(δφ) = L_φ₃(−δφ), (see figure (2)).

If we choose φ0 = φ₁(the choice between φ1 or φ2is irrelevant since they will result the same physics because of the symmetry (97)) the Lagrangian will be

QFT and Spontaneous Symmetry Breaking

U PPSALA U NIVERSITY

B

D

P

15 HP

QFT and Spontaneous Symmetry Breaking

Contents

1 Introduction

2 Standard Model of Particle Physics