Quantum Electrodynamics and the Higgs Mechanismcp3-origins.dk/content/thesis/2009-01-joergensen.pdf · QED and the Higgs Mechanism 1. Introduction 1 Introduction This report comprises

Quantum Electrodynamics and the Higgs Mechanism

Jakob Jark Jørgensen

4. januar 2009

QED and the Higgs Mechanism INDHOLD

Indhold

1 Introduction 2

2 Quantum Electrodynamics 32.1 Obtaining a Gauge Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 32.2 The S-Matrix Expansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.3 The Interaction Term . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.4 Identifying Processes in the S-matrix expansion . . . . . . . . . . . . . . . 92.5 Transition Amplitudes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

3 The Higgs Mechanism 173.1 Spontaneous Symmetry Breaking . . . . . . . . . . . . . . . . . . . . . . . 173.2 The Local U(1) Invariant Model . . . . . . . . . . . . . . . . . . . . . . . 20

3.2.1 Applying the Higgs Mechanism . . . . . . . . . . . . . . . . . . . . 223.3 The Physical Local SU(2)× U(1) Invariant Model . . . . . . . . . . . . . 23

3.3.1 The SU(2) Group and SU(2)× U(1) Invariance . . . . . . . . . . 243.3.2 Applying the Higgs Mechanism . . . . . . . . . . . . . . . . . . . . 27

4 Conclusion 33

5 Appendices 345.1 Appendix A1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345.2 Appendix B1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355.3 Appendix B2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365.4 Appendix B3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375.5 Appendix B4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395.6 Appendix B5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 405.7 Appendix B6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

6 Bibliography 43

Jakob Jark Jørgensen 1

QED and the Higgs Mechanism 1. Introduction

1 Introduction

This report comprises two main parts. The first part is about quantum electrodynamics.The second part is about spontaneous symmetry breaking and the Higgs mechanism.We start out by considering quantum electrodynamics which deals with the interactionsbetween electrons, positrons and photons. We shall obtain the electromagnetic intera-ction by requiring invariance of the Lagrangian density for the electron-positron fieldunder local U(1) phase transformations. We shall then develop a perturbative theoryknown as the S-matrix expansion, and use this to derive transition amplitudes for ele-mentary electromagnetic processes. We will see that interactions can be interpreted asbeing brought about by exchange of virtual intermediate particles.In the second part of the report we focus on the Higgs mechanism. We start out byexplaining the concept of spontaneous symmetry breaking using a simple model. Thissimple model will be ’gauged’ and we shall apply the Higgs mechanism thereby makingthe gauge bosons massive, while preserving the gauge symmetry of the theory. In thefinal section we shall see that the W± and Z0 bosons of the electroweak theory beco-me massice by application of the Higgs mechanism. We shall obtain expressions for theinvariant masses of the W±, Z0 and Higgs bosons.


QED and the Higgs Mechanism 2. Quantum Electrodynamics

2 Quantum Electrodynamics

In this first part of the report we will consider quantum electrodynamics which deals withthe interactions between electrons, positrons and photons. We will develop a perturbativetheory known as the S-matrix expansion, which to a given order can be used to derivetransition amplitudes for elementary processes of QED. We will see that this perturbativetheory leads one to interpret interactions as being brought about by exchange of virtualintermediate particles.

2.1 Obtaining a Gauge Theory

We start out by considering the classical Lagrangian density for the electron-positronfield:

L (x) = ψ(x)(iγµ∂µ −m

)ψ(x). (2.1)

The notation is as follows: The four-spinor fields ψ(x) and ψ(x) = ψ(x)†γ0 satisfy theDirac equation and the adjoint Dirac equation respectively. These equations of motioncan be derived from the Lagrangian density by the Euler Lagrange equations. In thequantized theory, the quanta of the fields will be electrons and positrons, both of massm. γµ =

(γ0, γ1, γ2, γ3

)is a vector consisting of the Dirac 4×4 gamma matrices. ∂µ = ∂

∂xµ

is the covariant four gradient operator. xµ = (t, ~x)1 is the contravariant space-time four-vector and will be denoted as x. Repeated indices implies summation. 2

We will consider the symmetry of the field theory specified by 2.1. We first note thatthe Lagrangian density is invariant under global U(1) transformations3:

ψ(x)→ ψ′(x) = eiαψ(x), ψ†(x)→ ψ′†(x) = e−iαψ†(x). (2.2)

Invariance under 2.2 allows one to change the phase of the fields by the same amount ateach point in space. This seems very restrictive for a field theory and we shall demandinvariance under the more general local U(1) phase transformations. Doing so we obtainwhat is called a gauge theory 4

Under local U(1) gauge transformations the fields transform according to 2.2, withα = α(x), ie. the transformation is dependent on the space-time position. 2.1 trans-form according to:

L → L ′ = ψe−iα(iγµ∂µ −m

)eiαψ

= ψe−iα(iγµ[eiα∂µψ + i(∂µα)eiαψ]−meiαψ

)= L − ψγµ(∂µα)ψ. (2.3)

1We use natural units, ie. ~ = c = 1.2The metric used to define the covariant fourvector xµ from the contravariant xµ ie. xµ = gµνg

ν isspecified by g00 = −g11 = −g22 = −g33 = 1, gµν = 0 if µ 6= ν.

3This symmetry implies current conservation so that a conserved charge can be associated with thefield

4Gauge theories have the advantage of being for example renomalizable, these points will not bediscussed here.


QED and the Higgs Mechanism 2.1 Obtaining a Gauge Theory

The Lagrangian density is not invariant under U(1) gauge transformation.Gauge invariance can be obtained by coupling a gauge field Aµ to the matter field. Thiscoupling is made by replacing the ordinary differential ∂µ in the Lagrangian density bythe covariant differential Dµ:

Dµ =(∂µ − ieAµ(x)

). (2.4)

Aµ(x) is a real vector field, and will be called the gauge field. e is a constant specifying thecoupling. Under U(1) transformation the covariant derivative of ψ transform accordingto:

Dµψ →(Dµψ

)′ =(∂µ − ieA′µ)ψ′. (2.5)

We see that the theory will be gauge invariant if the covariant derivative transform asthe fields themselves. This will be demanded:

Dµψ →(Dµψ

)′ = eiα(Dµψ

).

This requirement determines how the gauge field must transform under local U(1) trans-formations. From 2.5 we obtain:(

Dµψ)′ =(∂µ − ieA′µ)eiα = eiα

(∂µ − ie

(A′µ −

1e

(∂µα))ψ.

From this and 2.4 we see that the gauge field must transform according to:

A′µ −1e

(∂µα) = Aµ ⇔ A′µ = Aµ +1e

(∂µα).

Demanding gauge invariance we have been let to couple a gauge field to the matter field.The coupling being made by introducing the covariant differential in the Lagrangiandensity. Replacing ∂µ in 2.1 by Dµ we obtain:

L = ψ(iγµDµ −m

)ψ = ψ

(iγµ[∂µ − ieAµ

]−m

)ψ

ψ(iγµ∂µ −m

)ψ︸︷︷︸

Free e+/e− field.

+ eψ /Aψ︸︷︷︸Interaction.

. (2.6)

5 2.6 contains the free electron-positron field Lagrangian density and a term which isinterpreted as describing the interaction between the matter field and the gauge field.To obtain the full Lagrangian density for the system we must add a term for the freegauge field. This term must of course be U(1) gauge invariant as well.We note that Aµ has 4 components and is a real field. In the quantized theory thisimplies the quanta of the gauge field being neutral spin-one bosons. Concerning themass of these gauge bosons, we note that a Lorentz invariant mass term would have theform 1

2m2γAµA

µ. However, such a term is not invariant under local U(1) transformations:

12m2γAµA

µ → 12m2γA′µA′µ =

12m2γ

(Aµ +

∂µα

e

)(Aµ +

∂µα

e

)6= 1

2m2γAµA

µ. (2.7)

5We use the notation /A = γµAµ.


QED and the Higgs Mechanism 2.1 Obtaining a Gauge Theory

Therefore we require the gauge bosons to be massless. Since the gauge field describesmassless spin-1 bosons, we shall add the Lagrangian density of the free photon field:

−14Fµν(x)Fµν(x) where Fµν = ∂νAµ − ∂µAν . (2.8)

This term is seen to be gauge invariant:

∂νAµ − ∂µAν → ∂νA′µ − ∂µA′ν = ∂ν

(Aµ +

1e

(∂µα))− ∂µ

(Aν +

1e

(∂να))

= ∂νAµ − ∂µAν +1e∂ν∂µα−

1e∂µ∂να = ∂νAµ − ∂µAν .

The full local U(1) Lagrangian density then reads:

L = −14FµνF

µν︸︷︷︸Free photon field.

+ ψ(iγµ∂µ −m

)ψ︸︷︷︸

Free e+/e− field.

+ eψ /Aψ︸︷︷︸Interaction.

. (2.9)

2.9 is the Lagrangian density describing the interacting electron-positron and photon fi-elds of quantum electrodynamics (QED)6. It contains terms describing the free electron-positron field, the free electromagnetic field and the electromagnetic interaction. Thesame Lagrangian density will be obtained by making the minimal substitution of non-relativistic quantum mechanics:

i∂

∂t→ i

∂

∂t− eA0, −i∇ → −i∇− eAj . (2.10)

We see that this exactly corresponds to replacing the ordinary differential ∂∂xµ by the

covariant differential Dµ.To summarize: By requiring local U(1) gauge invariance of the Lagrangian density 2.1 weare led to introduce a gauge field coupled to the matter field. This gauge field can be in-terpreted as the photon field and in this way we obtain the electromagnetic interaction.However we have not derived the electromagnetic interaction, and in other areas onecannot derive interactions by requiring gauge invariance. One merely developes gaugeinvariant forms of the theories which may or may not be confirmed by experiments. Forexample were the W± and Z0 bosons predicted to exist and their masses determinedby such arguments, before they were observed, as discussed in [1, chapter 12, page 263].This case will be considered later when we tend to the Higgs mechanism.In the following sections we shall focus on the interaction term in 2.9. This term will betreated as a perturbation of the free fields and we shall show how to derive transitionamplitudes for elementary processes using perturbation theory. The application of per-turbation theory is particularly successful for QED where the coupling between photonsand electrons, measured by the fine structure constant α ≈ 1

137 , is particularly small.

6In this context 2.9 describes the fields before second quantization is applied. Upon second quantiza-tion these fields become the field operators of QED. The canonical quantization procedure will not bedescribed in this report, and it will in general not be specified whether we are working with the fields’before’ or ’after’ second quantization.


QED and the Higgs Mechanism 2.2 The S-Matrix Expansion

2.2 The S-Matrix Expansion

In this section we will develop the perturbation theory which can be used to deriveprobability transition amplitudes for elementary processes.We shall be working in the interaction picture of quantum mechanics. The Hamiltonianof the system is split into the free-field Hamiltonian H0 and an interaction HamiltonianHI . The time developement of state vectors |Φ(t)〉 are governed by:

id

dt|Φ(t)〉 = HI(t)|Φ(t)〉. (2.11)

This is a unitary transformation and scalar products are preserved.We shall focus on scattering processes. We let an intial state |i〉 = |Φ(−∞)〉 specify adefinite number of well separated particles (such that the do not interact) with definitemomenta and spin/polarization properties long before the scattering. For example: |i〉 =|e+r p; γsk〉, ie. containing one positron of momentum p with spin specified by r and one

photon of momentum k with polarization specified by s. By means of 2.11 |i〉 evolves intoa final state |Φ(∞)〉 in which particles are again far apart and non interacting (turningof the interacting leaves the state vectors constant in time as is seen from 2.11).The S-matrix is defined to relate the initial and final states:

|Φ(∞)〉 = S|Φ(−∞)〉 = S|i〉. (2.12)

The final state |Φ(∞)〉 does not specify a definite number of particles with definitemomenta and spin/polarization properties as the initial state does. Instead the finalstate contains all possible outcomes of the scattering. |Φ(∞)〉 can be expanded in acomplete set of definite final states |f〉 (which are analogous to |i〉):

|Φ(∞)〉 =∑f

|f〉〈f |Φ(∞)〉 =∑f

|f〉〈f |S|i〉.

The probability that a measurement after the collision will correspond to a final state|f〉 is then:

|〈f |Φ(∞)〉|2 = |〈f |S|i〉|2.

To work out the corresponding probability transition amplitudes 〈f |S|i〉, we need anexpression for the S-matrix. For the initial condition |i〉 = |Φ(−∞)〉 2.11 can be writtenas the integral eqution:

|Φ(t)〉 = |i〉+ (−i)∫ t

−∞dt1HI(t1)|Φ(t1)〉. (2.13)

We set:

|Φ(t1)〉 = |i〉+ (−i)∫ t1

−∞dt2HI(t2)|Φ(t2)〉, and so on for |Φ(t2)〉, |Φ(t3)〉 . . .


QED and the Higgs Mechanism 2.3 The Interaction Term

And solve 2.13 iteratively:

|Φ(t)〉 = |i〉+ (−i)∫ t

−∞dt1HI(t1)

(|i〉+ (−i)

∫ t1

−∞dt2HI(t2)

(|i〉+ (−i)

∫ t2

−∞dt3HI(t3) . . .⇒

|Φ(t)〉 =[1 + (−i)

∫ t

−∞dt1HI(t1) + (−i)2

∫ t

−∞dt1

∫ t1

−∞dt2HI(t1)HI(t2) + . . .

]|i〉.

In the limit T →∞ this gives us an expression for the S-matrix, the so called S-matrixexpansion:

S = 1 + (−i)∫ ∞−∞

dt1HI(t1) + (−i)2

∫ ∞−∞

dt1

∫ t1

−∞dt2HI(t1)HI(t2) + . . . (2.14)

S =∞∑n=0

(−i)n

n!

∫ ∞−∞

. . .

∫ ∞−∞

d4x1d4x2 . . . d

4xnT{HI(x1)HI(x2) . . .HI(xn)}. (2.15)

In 2.15 the Hamiltonian density HI(x) is introduced and a more compact summationnotation is used. The equivalence of the two forms 2.14 and 2.15 where time ordering isintroduced is shown to hold in the appendix section 5.1.This perturbation solution, where the S-matrix is expressed as a series in powers of HI

will only be useful if the interaction energy is small. As mentioned before this is thecase for QED and the method developed above can be applied with succes, as discussedin [1, chapter 5, page 99-101]. In the next sections we shall se how to derive transitionamplitudes in second order of this perturbation theory.

2.3 The Interaction Term

We will now consider the interaction term LI = eψ /Aψ of the Lagrangian density 2.9.This term is going to be the input of 2.15. The Hamiltonian density of the interactionterm is obtained using general formulas:

H (x) = πr(x)φr(x)−L (φr, ∂µφr), πr(x) =∂L

∂φr. (2.16)

7 Noting that LI does not depend on time derivatives of any of the fields we simplyobtain:

HI(x) = 0−LI = −eψ(x) /A(x)ψ(x). (2.17)

As mentioned before we are working in the interaction picture. The fields in 2.17 are(upon second quantization) the interacting fields in the interaction picture. However wewill se below that formulas for the free ie. non interacting fields in the Heisenberg pictureis also valid for the fields in 2.17. We first consider formulas for the free fields.The free fields are written as Fourier expansions in the complete sets of plane wave

7r labels the fields of the system, for Aµ for example It labels the components. πr is the conjugatemomenta of the fields.


QED and the Higgs Mechanism 2.3 The Interaction Term

solutions to the equations of motion for the fields. For (ψ(x)) ψ(x) the equation of mo-tion is the (adjoint) Dirac equation. For Aµ(x) the equation of motion is the covariantformulation of Maxwell’s equations. These equations can be derived from the free fi-eld Lagrangian density terms in 2.9, by means of the Euler Lagrange equations. Theexpansions are, from [1, pages 67 and 84]:

Aµ(x) = A+µ (x) +A−µ (x) =

∑r,k

√1

2V ωk

(εrµar(k)e−ikx + εrµa

†rke

ikx

). (2.18)

The k summation is over k allowed by imposing periodic boundary conditions on a cubicenclosure of volume V . k is the wave four-vector with k0 = wk. The r summation, r = 0to r = 3 is over four polarization states, described by εrµ.

ψ(x) = ψ+(x) + ψ−(x) =∑r,p

√m

V Ep

(cr(p)ur(p)e−ipx + d†r(p)vr(p)eipx

). (2.19)

ψ(x) = ψ+(x) + ψ−(x) =∑r,p

√m

V Ep

(dr(p)vr(p)e−ipx + c†r(p)ur(p)eipx

). (2.20)

Here p is the energy momentum four-vector and the p summation is over p allowed byimposing periodic boundary conditions on a cubic enclosure of volume V . The r sum-mation, r = 1, 2 is over spin states described by the spinors ur(p) and vr(p). Ep stemsfrom normalization of ur(p) and vr(p)8 9.Upon second quantization the fields become operators. The Fourier expansion coeffici-ents become creation and annihilation operators of particles with definite momentumand spin/polarization. The state |e+

r p; γsk〉 is for example created from the vacuumby: |e+

r p; γsk〉 = d†r(p)a†s(k)|0〉. Vice verca |e+r p; γsk〉 becomes the vacuum state by:

|0〉 = dr(p)as(k)|e+r p; γsk〉.

In the notation used above A+(x) (A−(x)) contains only annihilation (creation) ope-rators. Therefore A+(x) (A−(x)) is interpreted as annihilating (creating) a photon ofdefinite position x. Correspondingly ψ+(x) (ψ−(x)) annihilates (creates) an electron atx, and ψ+(x) (ψ−(x)) annihilates (creates) a positron at x. 10

Second quantization of the free fields is done in the Heisenberg picture of quantum me-chanics, and the fields represents Heisenberg operators. As mentioned before we shall beusing interacting fields in the interaction picture. It so happens that the solutions forthe free fields and the interpretation in terms of raising and lowering operators in theHeisenberg picture, is also valid for the interaction fields in the interaction picture. Thisfollows from two points:

8The ’dagger’ notation is used since the expansion coefficients become operators upon second quan-tization.

9The expansions of the free fields will not be discussed in any more detail in this report. We shall beusing these and other results derived in [1, chapters 2-5]

10The procedure of second quantization, an the interpretation in terms of creation and annihilationoperators will not be discussed in this report.


QED and the Higgs Mechanism 2.4 Identifying Processes in the S-matrix expansion

1. In the interaction picture operators satisfies the equation of motion:

id

dtOI(t) = [OI(t), H0].

Ie. only involving the free field Hamiltionian H0. The free fields satisfy the sameequation of motion in the Heisenberg picture11.

2. Since the interaction term of 2.9 does not involve any derivatives, the fields conju-gate to the interacting fields will be the same as the fields conjugate to the freefields. Since the Heisenberg picture and the interaction picture are related by aunitary transformation, the interacting fields and the free fields therefore satisfythe same commutation relations.

It follows from these two points that the expansions of the free fields and the interpre-tation in terms of creation and annihilation operators in the Heisenberg picture, is validfor the interacting fields in the interaction picture. The propagators will also have thesame form in the interaction picture as in the Heisenberg picture.The input for 2.15 is then:

HI(x) = −eN [ψ(x) /A(x)ψ(x)] = −eN [(ψ+ + ψ−

)(/A

+ + /A−)(

ψ+ + ψ−)]x. (2.21)

Here normal ordering is introduced. In a normal product all annihilation operators standto the right of all creation operators. It makes all vacuum expectation values vanish.When arranging a product of operators in normal order one treats the boson (fermion)operators as though all commutators (anti-commutators) vanish.The Hamiltonian density 2.21 consists of the interacting fields, each expanded in termsof raising and lowering operators. 2.15 with 2.21 as the input will therefore describe alarge number of different processes. In the next section we will se how to pick out fromthe S-matrix expansion the terms which contribute to a given process. Afterwards wewill se how to derive transition amplitudes 〈f |S|i〉 for the processes.

2.4 Identifying Processes in the S-matrix expansion

It is straightforward to pick out terms from the S-matrix expansion which contribute toa given transition |i〉 → |f〉. The terms must contain the right annihilation operatorsto annihilate the particles present in |i〉 and the right creation operators to create theparticles present in |f〉. In this section we will see how to pick out the terms contributingto a given process, and we will describe the processes by Feynman diagrams.We start out by considering the first order term S(1) of 2.15:

S(1) = ie

∫ ∞−∞

d4x1T{N [ψ /Aψ]x1}.

11In the Heisenberg picture the equation of motion is: i ddtOH(t) = [OH(t), H]+i ∂O

H

∂t. For the operators

we consider the last term drop out cf. [1, page 23]



(a) (b)

Figur 1: Figure (a) is the Feynman diagram for the process corresponding to the termN [ψ+ /A

+ψ+]x1 in S(1). An initial electron, positron and photon are annihilated. Figure

(b) is the Feynman diagram corresponding to N [ψ− /A+ψ+]x1 . An initial electron and

photon are annihilated and a final electron is created. The diagrams are to be interpretedas follows: A line entering from the left-hand side represents a particle present initially. Aline leaving the diagram on the right-hand side represents a particle present finally. Curvylines represent photons and straight lines represent fermions. The arrows on fermion linesdistinguish electrons (arrows point to the right) from positrons (arrows point to the left).

We shall use Wick’s theorem to expand a time ordered product whose factors are nor-mal products, into a sum of normal products. In this way we obtain processes with nointermediate particles12. For S(1) we just get:

S(1) = ie

∫ ∞−∞

d4x1N [(ψ+ + ψ−

)(/A

+ + /A−)(

ψ+ + ψ−)]x1 . (2.22)

2.22 gives rise to eight different processes. The term∫∞−∞ d

4x1N [ψ+ /A+ψ+]x1 for example

corresponds to the annihilation of an initial electron positron pair and the annihilationof an initial photon. Notice that we integrate over all space, corresponding to the pro-cess happening at any point in space-time. This process is represented by the Feynmandiagram in figure 1 (a).For real particles we have p2 = m2 for electrons and positrons and k2 = 0 for photons.

Energy and momentum cannot be conserved for real particles for any of the processesarising from S(1), and the processes are therefore not physical processes. This is illu-strated by the following simple example. We consider the process arising from the termN [ψ− /A+

ψ+]x1 . The process is represented by the Feynman diagram in figure 1 (b). Aninitial electron and photon are annihilated and a final electron is created, ie. a photonis absorbed by an electron. Energy-momentum conservation implies p2 = k + p1. Fromthis we obtain:

p22 = k2 + p2

1 + 2kµp1µ = m2 + 2k0p10 − 2~k · ~p1. (2.23)

In the reference frame of the incoming electron ~p1 = 0 such that p22 6= m2 for the

outgoing electron. Ie. for real particles, energy and momentum cannot be conserved in12A detailed discussion of Wicks theorem is found fx. in [1, chapter 6]. It will not be discussed in this

report.



this process.The processes arising from S(1) are referred to as the basic vertex parts of QED. In factall other QED Feynman diagrams can be built up by combining such basic vertexes.The processes arising form the second order term S(2) can for example be constructedby combining two vertexes. We will see an example of this below.The second order term S(2) is:

S(2) = −e2

2

∫ ∞−∞

∫ ∞−∞

d4x1d4x2T{N [ψ /Aψ]x1N [ψ /Aψ]x2}. (2.24)

Expanding this by use of Wicks theorem we obtain13:

S(2) =−e2

2

∫d4x1d

4x2N [(ψ /Aψ

)x1

(ψ /Aψ

)x2

]− e2

2

∫d4x1d

4x2N [(ψ /Aψ

)x1

(ψ︸︷︷︸ /Aψ)x2

]

−e2

2

∫d4x1d

4x2N [(ψ /Aψ

)x1

(ψ /A︸︷︷︸ψ)x2

] (2.25)

−e2

2

∫d4x1d

4x2N [(ψ /Aψ

)x1

(ψ /Aψ︸︷︷︸)x2

]− e2

2

∫d4x1d

4x2N [(ψ /Aψ

)x1

(ψ︸︷︷︸ /A︸︷︷︸ψ

)x2

]

−e2

2

∫d4x1d

4x2N [(ψ /Aψ

)x1

(ψ︸︷︷︸ /Aψ︸︷︷︸

)x2

]− e2

2

∫d4x1d

4x2N [(ψ /Aψ

)x1

(ψ /A︸︷︷︸ψ︸︷︷︸

)x2

]

−e2

2

∫ 4

x1d4x2N [

(ψ /Aψ

)x1

(ψ︸︷︷︸ /A︸︷︷︸ψ︸︷︷︸

)x2

].

The braces represents nonvanishing contractions. These contractions are Feynman pro-pagators and can be interpreted as the propagation of a virtual intermediate particlebetween two space-time points. Eg. ψ(x1)ψ(x2)︸︷︷︸ represents a virtual intermediate fer-

mion propagating between x1 and x214. By the term ’virtual’ we mean that the particle

is not a physical particle for which p2 = m2 (fermion) or k2 = 0 (photon).For the processes arising from S(2) energy and momentum can be conserved for initialand final physical particles, and the terms in S(2) describe real physical processes. As weshall see below, an interaction between physical particles is represented as an exchangeof virtual particles.As an example we will consider the processes arising from 2.25. This term contains fouruncontracted fermion operators. These correspond to creation and annihilation of fer-mions at x1 and x2 such that 2.25 will represent fermion-fermion scattering processes:2e− → 2e−, 2e+ → 2e+ and e+ + e− → e+ + e−. The photon-photon contraction can be

13Integration boundaries will be omitted in the following, and we just write:R∞−∞

R∞−∞ d

4x1d4x2 =R

d4x1d4x2 to ease notation.

14An introduction to covariant commutation relations from which the propagators arise will not begiven here. We shall be using the covariant formulation of the propagators as developed in for example[1, chapters 3-5]. When needed formulas will be citet.



interpreted as mediating the interaction between the fermions at x1 and x2 by exchangeof virtual intermediate photons. Notice that all values of x1 and x2 are integrated over,corresponding to the processes happening at any two space-time points.Let us now pick out the terms from 2.25 which contribute to the process e+ + e− →e+ +e−, known as Bhabha scattering. As mentioned before we need just the right combi-nation of annihilation and creation operators to annihilate the particles present initiallyand to create the particles present finally. In this case we need ψ+ and ψ+ to annihilatethe initial positron and electron respectively and ψ− and ψ− to create the final positronand electron. 2.25 consists of four terms wich has this combination of operators:

S(2)(e+ + e− → e+ + e−

)= −e

2

2

∫d4x1d

4x2{

N[(ψ+γµψ−

)x1

(ψ−γνψ+

)x2

]Aµ(x1)Aν(x2)︸︷︷︸+N

[(ψ−γµψ+

)x1

(ψ+γνψ−

)x2

]Aµ(x1)Aν(x2)︸︷︷︸

(2.26)

N[(ψ+γµψ+

)x1

(ψ−γνψ−

)x2

]Aµ(x1)Aν(x2)︸︷︷︸+N

[(ψ−γµψ−

)x1

(ψ+γνψ+

)x2

]Aµ(x1)Aν(x2)︸︷︷︸}.

(2.27)

The rest of the terms correspond to different processes. By permuting the fermion ope-rators and changing the integration variables, it is seen that the two terms in 2.26 areequal as are two terms in 2.27. For example for the first term in 2.26:

ψ+α (x1)γµαβψ

−β (x1)ψ−δ (x2)γνδγψ

+γ (x2)Aµ(x1)Aν(x2)︸︷︷︸ =

(−1)4ψ−δ (x2)γνδγψ+γ (x2)ψ+

α (x1)γµαβψ−β (x1)Aν(x2)Aµ(x1)︸︷︷︸ .

The spinor indices have been written out explicitly and a minus sign is introduced foreach time two fermion operators are interchanged. Interchanging the integration variableswe se that the two terms in 2.26 are equal. Therefore:

S(2)(e+ + e− → e+ + e−

)= −e2

∫d4x1d

4x2{

N[(ψ−γµψ+

)x1

(ψ+γνψ−

)x2

]Aµ(x1)Aν(x2)︸︷︷︸+ (2.28)

N[(ψ−γµψ−

)x1

(ψ+γνψ+

)x2

]Aµ(x1)Aν(x2)︸︷︷︸}. (2.29)

This will be the term which to second order in the s-matrix expansion will contribute toBhabha scattering.The processes 2.28 and 2.29 are represented by the Feynman diagrams in figure 2. Noticethat these Feynman diagrams can be built up from basic vertexes such as those in figure1. As mentioned before this is the case for all QED processes. The curvy line combiningthe two vertexes represents the exchange of virtual photons bringing about the scatte-ring process. Concerning energy and momentum conservation, energy and momentumcan only be conserved at each vertex (for initial and final physical particles) if the inter-mediate photon is virtual.


QED and the Higgs Mechanism 2.5 Transition Amplitudes

(a) (b)

Figur 2: (a) corresponds to 2.28. It is the scattering process contributing to Bhabhascattering. An initial electron is annihilated and a final electron is created at a point’x1’. An initial positron is annihilated and a final positron is created at a point ’x2’. (b)corresponds to 2.29. It is the electron-positron pair annihilation-creation process contri-buting to Bhabha scattering. An initial electron-positron pair is annihilated at one pointx2, and a final electron-positron is created at another point x1. Both processes contribu-tes to Bhabha scattering. For both processes a virtual intermediate photon propagatesbetween x1 and x2, and is interpreted as bringing about the interaction.

As done for Bhabha scattering we can pick out from 2.25 the processes correspondingto electron-electron scattering and positron-positron scattering. In a similar way we candescribe the processes rising from the other terms in S(2).

2.5 Transition Amplitudes

In this section we will se how to derive transition amplitudes. As mentioned beforethe initial state |i〉 and the final state |f〉 specifies a definite number of particles withdefinite momenta (spin and polarization labels will be omitted in what follows). Ie. theyare momentum eigenstates of the particles present and we shall work in momentumspace when deriving transition amplitudes. We shall use the Fourier expansions of thefields 2.19 and 2.20 and the Fourier transform of the photon propagater into momentumspace. The latter is given by:

Aµ(x1)Aν(x2)︸︷︷︸ = iDµν(x1 − x2) =1

(2π)4

∫ ∞−∞

d4kiDµν(k)e−ik(x1−x2). (2.30)

The functional form of the propagator iDµν will not be discussed here.From the expansions 2.19, 2.20 and 2.18 we note that15:

ψ+(x)|e−p〉 = |0〉√

m

V Epu(p)e−ipx, ψ+(x)|e+p〉 = |0〉

√m

V Epv(p)e−ipx, (2.31)

A+µ (x)|γk〉 = |0〉

√1

2V ωkεµ(k)e−ikx. (2.32)

15Spin/polarization labels will be omitted in what follows to ease notation.



ψ−(x)|0〉 =∑p

|e−p〉√

m

V Epu(p)eipx, ψ−(x)|0〉 =

∑p

|e+p〉√

m

V Epv(p)eipx (2.33)

A−µ (x)|0〉 =∑k

|γk〉√

12V ωk

εµ(k)eikx. (2.34)

As first example we derive the transition amplitude for |i〉 = |e−p; γk〉 → |f〉 = |e−p′〉.This process is represented by the Feynman diagram in figure 1 (b). The first order termfrom the S-matrix expansion contributing to this process is: S(1)

p ≡ ie∫d4x1N [ψ− /A+

ψ+]x1 .As noted before energy and momentum cannot be conserved in this process for physicalparticles. We will now see that the transition amplitude is explicitly zero for physicalparticles.The transition amplitude is given by:

〈f |S(1)p |i〉 = ie〈e−p′|

∫d4x1ψ

−(x1) /A+(x1)ψ+(x1)|e−p; γk〉

Using 2.31, 2.32 and 2.33 we obtain:

〈f |S(1)p |i〉 = ie〈e−p′|

∫d4x1ψ

−(x1)γµA+µ (x1)|γk〉

√m

V Epu(p)e−ipx1

= ie〈e−p′|∫d4x1

∑p′′

|e−p′′〉√

m

V Ep′′

√m

V Ep

√1

2V ωkeix1

(p′′−k−p

)u(p′′)/ε(k)u(p).

Noting that the set of states is orthonormal 16, and doing the x1 integration we obtain:

〈f |S(1)(e− + γ → e−

)|i〉 = (2π)4δ(4)

(p′ − k − p

)√ m

V Ep′

√m

V Ep

√1

2V ωkM , (2.35)

M = u(p′)/ε(k = p′ − p)u(p).

M is called the Feynman amplitude and is treated with care when deriving cross sections.Notice how the delta function in 2.35 implies momentum conservation at the vertex. If allthree particles are physical we cannot have p′− k− p = 0 and we see that the transitionamplitude is explicitly zero for physical particles. Ie. the transition does not represent aphysical process. If one of the particles is virtual energy-momentum can be conserved.However a virtual line must be connected to another vertex. We will se an example ofthis below when we consider Bhabha scattering.To obtain physical processes we must go at least to second order in the S-matrix expan-sion. As an example we will consider Bhabha scattering:

|i〉 = |e−p1; e+p2〉 → |f〉 = |e−p′1; e+p′2〉.

Ie. the transition amplitude is to second order perturbation theory given by:

〈f |s|i〉 = 〈e−p′1; e+p′2|S(2)(e+ + e− → e+ + e−

)|e−p1; e+p2〉 (2.36)

16As mentioned before 2.11 is a unitary transformation and therefore preserves scalarproducts.



With the S-matrix term given by 2.28 and 2.29. We first write 2.28 and 2.29 into thesame normal order of operators. We will see that there is a relative sign factor of

(− 1)

between the two contributions implicit in the normal product.Omitting the spinor and gamma matrix structure we have:

S(2)a ∼N

[(ψ−γµψ+

)x1

(ψ+γνψ−

)x2

]∼ N [c†(1′)c(1)d(2)d†(2′)], (2.37)

S(2)b ∼N

[(ψ−γµψ−

)x1

(ψ+γνψ+

)x2

]∼ N [c†(1′)d†(2′)d(2)c(1)]. (2.38)

Here we have labeled the initial (final) electron with momentum p1 (p′1) by 1 (1′), andanalogously for the initial and final positron. We only need to consider the terms withthe appropriate creation and annihilation operators since all other terms arising fromthe expansions of ψ and ψ will become zero for the transition amplitude 2.36. Arranging2.37 and 2.38 in the same normal order gives:

N [c†(1′)c(1)d(2)d†(2′)] = (−1)2c†(1′)d†(2′)c(1)d(2),

N [c†(1′)d†(2′)d(2)c(1)] = (−1)c†(1′)d†(2′)c(1)d(2).

Ie. we see that there is a relative sign factor of (−1) between the two contributionswhich we are now denoting as S(2)

a and S(2)b . It is important to note that the spinor and

gamma matrix structure of 2.37 and 2.38 remains intact. We are only interchanging theexpansion coefficients, not spinors and gamma matrices. Using 2.31, 2.33 and inserting2.30 we obtain the matrix element for S(2)

a :

〈e−p′1; e+p′2|S(2)a |e−p1; e+p2〉 =

−e2〈e−p′1; e+p′2|∫d4x1d

4x2{N[(ψ−γµψ+

)x1

(ψ+γνψ−

)x2

]Aµ(x1)Aν(x2)︸︷︷︸|e−p1; e+p2〉 =

−ie2〈e−p′1; e+p′2|∫d4x1d

4x2

∑p′,p′′

|e−p′; e+p′′〉√

m

V Ep1′

√m

V Ep1

√m

V Ep2

√m

V Ep′′·

eip′1x1e−ip1x1e−ip2x2eip

′′x2 u(p′)γµu(p1)1

(2π)4v(p2)γνv(p′′)

∫d4kDµν(k)e−ik(x1−x2).

Noting that the set of states is orthonormal and doing the x1 and x2 integration we find:

− ie2∏p

√m

V Ep

1(2π)4

∫d4k(2π)4 δ(4)

(p′1 − k − p1

)(2π)4δ(4)

(p′2 + k − p2

)︸︷︷︸δ(4)(p′1+p′2−p1−p2

)(2π)4δ(4)

(p′2+k−p2

)· u(p′1)γµu(p1)v(p2)γνv(p′2)Dµν(k)

= (2π)4δ(4)(p′1 + p′2 − p1 − p2

)∏p

√m

V EpMa, (2.39)

Ma = −ie2u(p′1)γµu(p1)v(p2)γνv(p′2)Dµν(k = p2 − p′2)

2.39 Is our final result. It gives the transition amplitude for the scattering process con-tributing to Bhabha scattering. Notice that the delta functions explicitly imply energy-momentum conservation at both vertexes. This fixes the momentum of the intermediate



photon: k = p′1 − p1 = p′2 − p2. As mentioned before energy-momentum cannot be con-served for three physical particles at a vertex, implying the photon being virtual.By the same procedure we obtain for S(2)

b :

〈i|S(2)b |f〉 = ie2

∏p

√m

V Ep

1(2π)4

∫d4k(2π)4 δ(4)

(p′1 + p′2 − k

)(2π)4δ(4)

(k − p1 − p2

)︸︷︷︸δ(4)(p′1+p′2−p1−p2

)(2π)4δ(4)

(k−p1−p2

)· u(p′1)γµv(p′2)v(p2)γνu(p1)Dµν(k)

= (2π)4δ(4)(p′1 + p′2 − p1 − p2

)∏p

√m

V EpMb, (2.40)

Mb = ie2u(p′1)γµv(p′2)v(p2)γνu(p1)Dµν(k = p1 + p2)

2.40 is the transition amplitude for the pair annihilation-creation process contributingto Bhabha scattering. As before the delta functions imply energy-momentum conser-vation at each vertex and the momentum of the intermediate virtual photon is fixed:k = p1 + p2 = p′1 + p′2. As for 2.39, 2.40 explicitly gives vanishing transition amplitudesfor processes for which the initial and final particles does not conserve energy and mo-mentum.2.39 and 2.40 are the final results of this first part of the report. Starting out by requiringgauge invariance for the electron-positron field theory, we obtained the electromagneticinteraction. We treated the interaction as a perturbation of the free fields and developeda perturbation theory, the S-matrix expansion, which can be used to derive transitionamplitudes for the elementary electromagnetic processes. We have seen how to extractterms from the S-matrix expansion which contribute to a given process, and finally wehave seen how to derive transition amplitudes for real physical processes of QED. Usingtransition amplitudes one can derive cross sections and differential cross sections whichare the physical observable quantities. This will not be considered in this report, insteadwe will now tend to spontaneous symmetry breaking and the Higgs mechanism.


QED and the Higgs Mechanism 3. The Higgs Mechanism

3 The Higgs Mechanism

We started this report by requiring a specific symmetry for the Lagrangian density ofthe electron-positron field, thereby obtaining the electromagnetic interaction. In thissecond part of the report we shall be focusing on symmetries and on the consequences ofbreaking them. We shall develop the concept of spontaneous symmetry breaking, leadingto the Higgs Mechanism. The Higgs mechanism will be used to obtain invariant massesfor the W± and Z0 bosons of the standard electroweak symmetry, while preserving thegauge invariance of the theory.

3.1 Spontaneous Symmetry Breaking

To explain the idea of spontaneous symmetry breaking we consider a classical field theorywhose Lagrangian density is invariant under global U(1) phase transformations.The system is specified by a complex scalar field H:

H(x) = <H + =H = φ1 + iφ2, φ1, φ2 ∈ R. (3.1)

Under global U(1) phase transformation H transforms according to:

H → H ′ = eiαH, α ∈ [0, 2π].

The transformation changes the phase angle of the field, corresponding to a rotation inthe complex plane. The phase angle is changed by the same amount at each point inspace, therefore the term global phase transformation.We consider a simple model with potential and kinetic energy density terms:

V (x) = m2H∗H + λ(H∗H

)2. K (x) = ∂µH

∗∂µH. (3.2)

Giving a Lagrangian density which obviously has global U(1) symmetry:

L (x) = ∂µH∗∂µH −m2|H|2 − λ|H|4. (3.3)

m2 and λ are real parameters 17.Treating H and H∗ as independent fields the fields conjugate to H and H∗ are given by:

π(x) =∂L

∂H=

1c2H∗, π∗(x) =

∂L

∂H∗=

1c2H.

And the hamiltonian density is obtained:

H (x) = πH + π∗H∗ −L

= 21c2H∗H − ∂µH∗∂µH + V = ∂0H(x)∗∂0H(x) +∇H(x)∗∇H(x) + V (H).

(3.4)

17The first parameter is called m2 to explicitly illustrate the fourdimensionality of the Lagrangiandensity.


QED and the Higgs Mechanism 3.1 Spontaneous Symmetry Breaking

(a) m2 > 0 (b) m2 < 0

Figur 3: Figure (a) and (b) illustrates the appearence of the potential energy densityV (φ1, φ2) = m2

(φ2

1 + φ22

)+ λ

(φ2

1 + φ22

)2 for the cases m > 0 and m < 0. V is plotteton the vertical axis and φ1 and φ2 on either of the horizontal axis. For m > 0 V hasan absolute minumum for φ1 = φ2 = 0, ie. H(x) = 0 specifies a uniqe ground state.The ground state possesses the same global U(1) symmetry as the Lagrangian density,therefore spontaneous symmetry breaking cannot occur. For m < 0 V possesses a whole

circle of absoslute minima corresponding to φ21 + φ2

2 = −m2

2λ , ie. H(x) =√−m2

2λ eiθ, 0 ≤θ < 2π corresponds to minimum field energy. The ground state is not uniqe. Spontaneoussymmetry breaking occurs if we choose one particular value og θ to represent the groundstate, since such a ground state does not posses global U(1) symmetry.

For the field energy to be bounded from below we set λ > 0.We will focus on the ground state, the state of lowest field energy. We notice that thefirst to terms in 3.4 are positive definite and vanish for constant H(x). Minimizing H (x)therefore corresponds to finding that constant H(x) which minimizes V (H). Ie. to findthe ground state we only need to consider the potential term 3.2.For the potential term we have:

V (H) = V (φ1, φ2) = m2(φ2

1 + φ22

)+ λ(φ2

1 + φ22

)2. (3.5)

To very different situations occur depending on the sign of m2, these situations areillustrated in figure 3. For m2 > 0 V (φ1, φ2) has an absolute minimum corresponding toφ1(x) = φ2(x) = 0, ie. H(x) = 0 determines an unique ground state. The ground statehas global U(1) symmetry as the Lagrangian density has. Omitting the term λ|H|4 18

we are left with the Lagrangian density for the complex Klein Gordon field, which upon18The term can be regarded as a perturbation of the other terms, and represents a selfinteraction of

the particles in the quantized theory. As discussed in [1, page 282-283].


QED and the Higgs Mechanism 3.1 Spontaneous Symmetry Breaking

second quantization gives rise to two oppositely charged spin 0 particles off mass m.For m2 < 0 the situation is very different. Extrema are found by requiring:

∂V

∂φ1= 2m2φ1 + 4λφ1

(φ2

1 + φ22

)= 0 ∧ ∂V

∂φ2= 2m2φ2 + 4λφ2

(φ2

1 + φ22

)= 0⇒

2m2(φ1 + φ2

)+ 4λ

(φ1 + φ2

)(φ2

1 + φ22

)= 0⇔

(φ1 + φ2

)(2m2 + 4λ(φ2

1 + φ22))

= 0.

In this case we see that V (φ1, φ2) has a local maximum at φ1 = φ2 = 0, ie. H(x) = 0.Concerning the ground state we note that V (φ1, φ2) has absolute mimima for φ2

1 +φ22 =

−m2

2λ , ie. V (H) possesses a whole circle of absolute minima in the complex plane givenby:

H0 =

√−m2

2λeiθ, 0 ≤ θ < 2π. (3.6)

In this case the ground state does not correspond to a uniqe value of H, but is instead

Figur 4: Circle of H’s in the complex plane corresponding to minimum field energy.Choosing a particular value of θ to represent the ground state leaves the ground statewith less symmetry than the Lagrangian density, ie. spontaneous symmetry breakingcan occur. Global U(1) transformation which leaves L invariant, will rotate the vectorrepresenting H0 around the origin.

degenerate. If we choose one particular θ to represent the ground state, the ground statewill not be invariant under global U(1) transformation as the Lagrangian density is.Rotation in the complex plane will instead transform it into another state lying on thecircle of absolute minima ( See figure 4). The ground state therefore has less symmetrythan the Lagrangian density has. Such a theory is called a theory with spontaneoussymmetry breaking. We notice that for m2 > 0, spontaneous symmetry breaking cannotoccur since the uniqe ground state possesses the same symmetry as the Lagrangiandensity.We will now consider the consequences of spontaneous symmetry breaking. We first note


QED and the Higgs Mechanism 3.2 The Local U(1) Invariant Model

that since L is invariant under global U(1) transformation, it doesn’t matter which θwe choose to represent the ground state. We choose θ = 0 such that:

H0 =

√−m2

2λ≡ 1√

2v. (3.7)

We now introduce two real fields σ(x) and η(x):

H(x) =1√2

(v + σ(x) + iη(x)

), (3.8)

Thereby letting σ(x) and η(x) represent deviations from the ground state configurationH(x) = H0. The Lagrangian density 3.3 then becomes (Derivation is found in section5.2 of the appendix):

L (x) =12∂µσ(x)∂µσ(x)− 1

2(2λv2)σ(x)2 (3.9)

+12∂µη(x)∂µη(x) (3.10)

− λvσ(x)(σ(x)2 + η(x)2

)− λ

4(σ(x)2 + η(x)2

)2. (3.11)

3.11 and 3.3 are the same Lagrangian density expressed in terms of different variables,and must lead to the same physical result.If wee treat 3.9 and 3.10 as free field Lagrangian densities and 3.11 as interaction terms,it is seen that that σ(x) and η(x) are two real Klein Gordon fields. Upon second quan-tization these fields leads to neutral spin 0 particles. The quanta of the σ-boson fieldhave mass v

√2λ =

√−2m2 while the quanta of the η-boson field are massless, the latter

are called Goldstone bosons.It is seen that we by the mechanism of spontaneous symmetry breaking have created apertubative theory with massive scalar bosons and massless Goldstone bosons. In the ne-xt sections the mechanism of spontaneous symmetry breaking will be extended to createmassive vector bosons in a gauge invariant theory. This is called the Higgs mechanism,and we shall call H(x) the Higgs field and the scalar bosons associated with the σ(x)field will be called Higgs bosons.

3.2 The Local U(1) Invariant Model

We have so far been considering a model whose Lagrangian density was constructedto have global U(1) symmetry. We shall now extend our model by requiring invarianceunder local U(1) transformation, in this way we obtain a gauge theory. We will applythe mechanism of spontaneous symmetry breaking to the theory and we will see thatno goldstone bosons are obtained. Instead the degree of freedom associated with theη(x) field will somehow be transferred to the gauge field, which becomes massive in theprocess.The procedure of ’gauging’ the theory resembles the one used for the electron positron



field. However, it will be done here before we tend to the mechanism of spontaneoussymmetry breaking. As before we will se that requiring invariance under local U(1)transformations leads one to introduce a gauge field coupled to the matter field.

For U(1) transformations the fields transform according to:

H → H ′ = eiα(x)H, α(x) ∈ R. (3.12)

α is now dependent on the space-time coordinate, therefore the term local phase trans-formation. The potential term 3.2 of the Lagrangian density is obviously invariant withrespect to such transformations. Considering the kinetic term we note that:

∂µH′ = ∂µ

(eiα(x)H

)= eiα

(∂µ + i(∂µα)

)H.

Implying that the kinetic term is not invariant under local U(1) transformations. Gaugeinvariance can be restored by introducing a gauge field Aµ coupled to the matter field.This coupling is made as before by replacing the ordinary differential by the covariantdifferential Dµ:

Dµ =(∂µ + igAµ(x)

). (3.13)

Aµ is a real vector field and g is the coupling constant. The covariant differential of Htransform according to:

DµH →(DµH

)′ = (∂µ + igA′µ)H ′ =

(∂µ + igA′µ

)eiαH = eiα

(∂µ + ig

(A′µ +

(∂µα)g

)H.

(3.14)

The kinetic term which is now(DµH

)∗(DµH)

would be invariant if DµH transformslike:

DµH →(DµH

)′ = eiαDµH.

From 3.14 we note that this is the case, if the gauge field transforms according to:

Aµ = A′µ +∂µα

gie. Aµ(x)→ A′µ(x) = Aµ(x)− ∂µα(x)

g. (3.15)

To summarize our approach: Demanding invariance of the theory under local U(1) trans-formations we are led to couple to the matter field a gauge field Aµ(x) which transformaccording to 3.15. The coupling is made by replacing the ordinary differential in theLagrangian density 3.3 by the covariant differential. Doing so we obtain:

L =(DµH

)∗(DµH

)− V

=(∂µ − igAµ

)H∗(∂µ + igAµ

)H − V

= ∂µH∗∂µH − V + ig∂µH

∗AµH − igAµH∗∂µH + g2AµAµHH∗

= ∂µH∗∂µH − V︸︷︷︸

free matter field lagrangian

+ igAµ(H∂µH∗ −H∗∂µH) + g2AµA

µHH∗︸︷︷︸interaction terms

. (3.16)



The first underbraced terms are the free matter field lagrangian. The remaining termsare interpreted as terms describing interactions between the free matter field and thegauge field. These can in principle be treated by the perturbation theory we developedin section 2.2. However that is not the aim of this part of the report.To obtain the full Lagrangian density we must add a term for the free gauge field, whichdoes not break the symmetry. As for the electron-positron field, we add the Lagrangiandensity of the free electromagnetic field 2.8. The the full Lagrangian density then reads:

L = ∂µH∗∂µH − V︸︷︷︸

free matter field

− 14FµνF

µν︸︷︷︸free gauge field

+ ieAµ(H∗∂µH −H∂µH∗) + e2AµA


. (3.17)

3.2.1 Applying the Higgs Mechanism

We now consider the symmetry of the ground state for the cases m2 > 0 and m2 < 0. Forthe ground state we must have Aµ(x) = 0 in both cases. The analysis therefore parallelsthat for the global U(1) symmetric model. For m2 > 0 the ground state corresponds toAµ(x) = 0 and H(x) = 0. The ground state is uniqe and possesses the same symmetryas the Lagrangian density, thus spontaneous symmetry breaking does not occur. Form2 < 0 the ground state will be degenerate as before leading to spontaneous symmetrybreaking. We again obtain the circle of H ′s (3.6) corresponding to minimum field energy.We shall choose θ = 0 to represent the ground state 3.7 and we let the fields σ(x) andη(x) represent deviations from the ground state according to 3.8. In terms of these newcoordinates the Lagrangian density becomes (derivation is found in section 5.3 of theappendix):

L (x) =12∂µσ(x)∂µσ(x)− 1

2(2λv2)σ2 (3.18)

+12∂µη(x)∂µη(x) (3.19)

− FµνFµν +12

(gv)2AµAµ (3.20)

+ gvAµ∂µη + interaction terms. (3.21)

As before we interpret 3.18 as a Lagrangian density for the free real scalar σ field, whichis seen to obey the Klein Gordon equation of motion. Ie. the quanta of the field arespin 0 bosons with mass v

√2λ =

√−2m2. 3.19 looks as the Lagrangian density for the

Goldstone boson field. Concerning the term 3.20 we see something interesting. This termlooks like the free gauge field Lagrangian density plus an invariant mass term. It appersthat the gauge bosons somehow have aquired a mass in the process! However we needto be careful with these interpretations. The first term in 3.21 cannot be regarded as aninteraction term, since it is quadratic in the fields as the free field terms are. Thereforeη and Aµ cannot be regarded as independent coordinates (changing η will affect a termcontaining Aµ and vice verca).Counting degrees of freedom we come to the conclusion that something needs to be fixed.


QED and the Higgs Mechanism3.3 The Physical Local SU(2)× U(1) Invariant Model

For 3.17 we count 4 degrees of freedom. One for H, one for H∗ and to for Aµ 19. For3.18, 3.19, 3.20, 3.21 we count 5 degrees of freedom. One for σ one for η and three forAµ, since this field appears to be massive, ie. it has a longitudinal degree of freedom.A coordinate change should not affect the number of degrees of freedom. Therefore thetransformed Lagrangian density must contain an unphysical field. This unphysical fieldis the η field. As discussed in [1, page 285-286] it so happens that a local U(1) gaugetransformation of the form 3.12, which transform H into H ′ = 1√

2

(v+σ

), always can be

found. Choosing this so called unitary gauge, absorbs the extra degree of freedom. Thiscorresponds to remove the η field from the transformed Lagrangian density, ie.:

L (x) =12∂µσ(x)∂µσ(x)− 1

2(2λv2)σ2

− FµνFµν +12

(gv)2AµAµ

+ interaction terms.. (3.22)

Again we interpred the first term as a free field Lagrangian density of the real Klein-Gordon field σ. The second term can now without problems be interpreted as the freefield Lagrangian density of a real massive vector field Aµ, which quanta has the invariantmass gv.We see that the Higgs mechanism does not generate Goldstone bosons. Instead the degreeof freedom associated with the η(x) field has been absorbed, giving the gauge field anextra degree of freedom and an invariant mass term. Referring to the discussion on 2.7,adding an invariant mass for the vector field Aµ would destroy the gauge invariance of3.17. However by applying the Higgs mechanism, the gauge bosons acquire mass whilethe gauge invariance of the theory is preserved.

3.3 The Physical Local SU(2)× U(1) Invariant Model

We will now extend the model further by requiring invariance under SU(2)×U(1) gaugetransformations. In this way we will obtain SU(2) × U(1) gauge bosons which will bethe W± and Z0 bosons of weak interactions, and photons. When formulating weakinteractions as a gauge theory, one has to assume that all leptons and the W± and Z0

gauge bosons are massless. Adding mass terms to the Lagrangian density will simplydestroy the gauge invariance of the theory, in analogy with what we have seen for thetheory 3.17. However, the Higgs mechanism can be used to introduce nonzero mass termsfor the gauge bosons and leptons, while preserving the symmetry. In this section we willsee how the W± and Z0 bosons become massive when coupled to the Higgs field, andthe Higgs mechanism is applied. We start out by constructing a global SU(2) invariantHiggs field theory, and then require local SU(2)× U(1) gauge invariance.

19From the study of the free photon field, one knows that of the four degrees of freedom Aµ appersto have one is removed by the subsidiary condition: [a3(k) − a0(k)]|Ψ〉 = 0 Where a0(k) and a3(k) arelowering operators of scalar and longitudinal photons respectively and |Ψ〉 is the state. The other canbe removed by gauge transformation leaving one with two transverse photons. If Aµ is massive, only the’scalar’ degree of freedom can be removed. As discussed in fx. [1, chapter 5].



3.3.1 The SU(2) Group and SU(2)× U(1) Invariance

The SU(2) group is the set of Special U nitary 2 × 2 matrices. Special means that thedeterminant of the group elements are unity, and unitary means that each group elementobeys U †U = 1. We note that any unitary matrix can be written in terms of a hermitianmatrix: U = eiH .The elements of SU(N) groups are labelled by continous parameters such as the Eulerangles for the rotation group SO(3). For SU(2) it can be shown that three independentparameters are needed to label a group element uniquely, and that the generators of thegroup are the hermitian pauli 2× 2 spin matrices. The group elements are then writtenas:

U(α) = exp(iτiαi

2

), (3.23)

Where:

τ1 =(

0 11 0

), τ2 =

(0 −ii 0

), τ3 =

(1 00 −1

), [τi, τj ] = 2iεijkτk,

Are the Pauli spin matrices wich are the generators of the group (εijk is the Levi-Civitasymbol), and:

α =(α1, α2, α3

), (3.24)

Uniquely labels the group elements. As before a constant α implies a global SU(2) trans-formation whereas an α dependent on spacetime, ie. α = α(x) implies a local SU(2)gauge transformation.

To extend the model 3.3, H must first of all be an isospinor:

H(x) =(φA(x)φB(x)

)=(<φA + i=φA<φB + i=φB

)=(φ1(x) + iφ2(x)φ3(x) + iφ4(x)

).

Instead of considering H and H∗ as before we shall consider H and H†. These fieldstransform according to:

H → H ′ = U(α)H = exp(iτiαi

2

)H. (3.25)

H† → H ′† =(U(α)H

)†= H†U(α)† = H† exp

(−iτiαi

2

). (3.26)

3.3 is then easily generalized to be global SU(2) invariant:

L (x) = ∂µH†∂µH −m2H†H − λ

(H†H

)2. (3.27)

Where H† and H are treated as independent fields if one wishes to derive the equationof motion for the fields.



We shall now require local SU(2) gauge invariance of the Lagrangian density. The pro-cedure resembles the one used earlier when requiring local U(1) invariance, but in thecase of local SU(2) we will be let to introduce three gauge fields coupled to the Higgsfield. The quanta of these gauge fields will be the W± and Z0 bosons known from theweak interactions.We first note, that the kinetic term is not invariant under local SU(2) transformationssince:

∂µH′ = ∂µ exp

(iτiαi(x)

2

)H =

(12iτi∂µαi(x)

)exp(iτiαi(x)

2

)H + exp

(iτiαi(x)

2

)∂µH

= exp(iτiαi(x)

2

)(∂µ +

12i∂µ(τiαi(x)

))H. (3.28)

In analogy with the local U(1) case, gauge invariance can be restored by coupling threegauge fields to the Higgs field. As before this coupling is made by replacing the ordinarydifferential by a covariant differential:

∂µH replaced by DµH =(∂µ +

12igτiWiµ

)H. (3.29)

Wiµ are real vector fields. g is the coupling constant. Notice that we need three gauge fieldin this case, since the SU(2) group elements are labeled by three independent functions,α1(x), α2(x), α3(x).To obtain gauge invariance we would like the covariant differential to transform accordingto:

DµH → (DµH)′ = U(α(x))DµH, (3.30)

Ie. as the fields themselves. The kinetic term will then be SU(2) gauge invariant. Forthis to be the case, we work out how the gauge fields must transform. We substituteW ′ = 1

2 iτiW′iµ for convenience and require:

DµH′ =U

(DµH

)=(∂µ + gW ′

)(UH

)⇔

U

((∂µ + gW

)H

)= ∂

(UH

)+ gW ′UH ⇔

U∂µH + gUWH =(∂µU

)H + U∂µH + gW ′UH ⇔(

gW ′U + ∂µU − gUW )H = 0⇒W ′U = UW − 1g∂µU ⇔

W ′ = UWU † − 1g

(∂µU

)U † = UWU † +

1gU∂µU

†. (3.31)

In the last step we used ∂µ(UU †

)= 0 ⇔ ∂µ

(U)U † = −U∂µU †. To obtain gauge in-

variance we require the gauge fields to transform according to 3.31. Notice that thisderivation generalizes to SU(N) cases. We also se that 3.31 immediately reduces to thetransformation law 3.15 for the U(1) case.



In section 5.4 of the appendix it is show that for infinitesimal transformations, the gaugefields must transform according to:

Wiµ →W ′iµ = Wiµ − ∂µαi −1gεijkαj(x)Wkµ(x). (3.32)

We want the theory to be local U(1) invariant as well. This is easily accomplished.For the earlier model we have seen that local U(1) invariance required introduction ofthe Aµ field through the covariant differential 3.13. To make the Lagrangian density3.27 SU(2) × U(1) invariant we simply add a term analogous to 3.13 in the covariantdifferential:

DµH =(∂µ +

12igτiWiµ + ig′Y Bµ

)H. (3.33)

We then define the fields Wiµ to be U(1) gauge invariant and the field Bµ 20 to be SU(2)gauge invariant. The Lagrangian density obtained by substituting 3.33 in 3.27 is thenSU(2)× U(1) invariant:

L H =(DµH

)†(DµH

)−m2H†H − λ

(H†H

)2. (3.34)

We note that for the theory to be SU(2)×U(1) invariant we had to couple 4 gauge fieldsto the Higgs field. One field for each function needed to label a given transformation.To obtain the full Lagrangian density we need to add terms for the free gauge fields. Wenote that the fields have four components and are real, this implies the quanta of thefields being spin-1 bosons. We also note that the fields cannot have an invariant massterm. Analogous to 2.7 such terms would destroy the SU(2)× U(1) gauge symmetry ofthe Lagrangian density. For the Bµ field we note that the gauge transformation law iscompletely analogous to that for the electromagnetic field Aµ, and we add the free gaugefield term:

−14BµνB

µν , Bµν = ∂νBµ − ∂µBν , Invariant under U(1) gauge transformations.

Since we defined Bµ to be SU(2) gauge invariant, this term will be SU(2)×U(1) gaugeinvariant.For the free Wiµ fields the analogous expressions would be:

−14FiµνF

µνi , Fiµν = ∂νWiν − ∂µWiν .

However, on account of the term gεijkαj(x)Wkµ(x) in the transformation law 3.32 for Wiµ

such a term would not be SU(2) gauge invariant. To restore gauge invariance additionalinteraction terms must be introduced. It can be shown that the following term is SU(2)invariant:

−14GiµνG

µνi , Giµν = Fiµν + gεijkWjµWkν (3.35)

20we choose a local U(1) transformation of the form exp(iY f(x)), where Y is the hypercharge. Thegauge field is denoted Bµ and its transformation law is: Bµ → B′µ = Bµ − 1

g′ ∂µf .



Derivation can be found in [1, chapter 12, appendix]. Since Wiµ is defined to be U(1)gauge invariant, the term will be SU(2) × U(1) invariant. The Lagrangian density forthe gauge bosons is then (derivation can be found in section 5.5 of the appendix):

L B = −14BµνB

µν − 14GiµνG

µνi ⇒

L B = −14BµνB

µν − 14FiµνF

µνi︸︷︷︸

free gauge field terms.

+ gεijkWiµWjν∂µW ν

k −14g2εijkεilmWjµWkνW

µl W

νm︸︷︷︸

gauge field interaction terms.

. (3.36)

And the full Lagrangian density reads:

L HB = L H + L B. (3.37)

In the next section we will express the gauge fields in terms of the fields W1µ, W2µ, Zµand Aµ (explanation follows). The quanta of these fields will be W±, Z0 bosons andphotons. We note that the free gauge field terms of 3.36 describe massless bosons. Ofcourse we would like the photons to remain massless, but we would like the W± andZ0 bosons to be massive. This could be achieved by adding the invariant mass terms12m

2WW1µW

µ1 + 1

2m2WW2µW

µ2 + 1

2m2ZZµZ

µ (This would correspond to adding analogousterms to 3.36). However doing so will destroy the SU(2)× U(1) gauge invariance of thetheory as mentioned above. In the next section we will see that by applying the Higgsmechanism, we can make the W± and Z0 bosons massive and let the photons remainmassless while keeping the SU(2)× U(1) gauge invariance of the theory.

3.3.2 Applying the Higgs Mechanism

As we have seen before two very different situations occurs depending on whether m2 > 0or m2 < 0 in 3.34. For m > 0 the classical energy density is a minimum for all fieldsvanishing, such that the gound state is unique as before. Spontaneous symmetry breakingcannot occur in this case and therefore the Higgs mechanism cannot be applied.For m2 < 0 the energy will be a minimum for a constant Higgs field and all otherfields vanishing. This is analogous to the previous cases considered, and we only need toconsider the potential term to specify the ground state:

V = −m2H†H − λ(H†H)2 = −m2(|φA|2 + |φB|2)− λ(|φA|2 + |φB|2)2

Minimizing this we obtain as before:

∂V

∂|φA|= 0 ∧ ∂V

∂|φB|= 0⇒

min. for: |φA|2 + |φB|2 = φ21 + φ2

2 + φ23 + φ2

4 =−m2

2λ. (3.38)



3.38 gives an expresseion for a n4 hypersphere of radius√−m2

2λ in a fourdimensionalspace spanned by φ1, φ2, φ3, φ4 axes. All points (φ1, φ2, φ3, φ4) located on the hyperspherecorresponds to minimum field energy. Again the ground state is degenerate and choosinga particular point to represent the ground state leads to spontaneous symmetry breaking.

To simplify computations we choose: φ1 = φ2 = φ4 = 0, φ3 =√−m2

2λ such that:

H0 =1√2

(0v

), v ≡

√−m2

λ.

The Higgs field is expressed as:

H(x) =1√2

(η1(x) + iη2(x)

v + σ(x) + iη3(x)

).

Where the fields σ, η1, η2, η3 represents deviations from the ground state configurationH0. By the mechanism of spontaneous symmetry breaking we have expressed the field interms of new coordinates. The Lagrangian density 3.34 expressed in terms of these newcoordinates will of course be equivalent to the Lagrangian density expressed in terms ofthe ’old’ coordinates.As in the previous case, 3.39 contains unphysical fields. This is so since the isospinor bya gauge transformation can be transformed into:

H(x) =1√2

(0

v + σ(x)

). (3.39)

This can be achieved by a SU(2) transformation which transforms 3.39 into a downspinor followed by a U(1) transformation which transforms this downspinor into a realquantity, as discussed in [1, page 298]. We work shall in this unitary gauge.The further analysis starts by substituting 3.39 into 3.37. We first consider the term(DµH

)†(DµH

):(

DµH)†(

DµH)

=(∂µH

† − 12igH†τiWiµ − ig′Y BµH†

)(∂µH +

12igτiW

µi H + ig′Y BµH

)=

12∂µσ∂

µσ +12g′2Y 2v2BµB

µ +18g2v2

(W1µW

µ1 +W2µW

µ2︸︷︷︸

Mass terms

+W3µWµ3

)− 1

2gg′Y v2W3µB

µ + Interaction terms of third and fourth order in the fields. (3.40)

Derivation is found in section 5.7 of the appendix. Notice that we have obtained invariantmass terms for the W1µ and W2µ fields. Concerning the W3µ and Bµ we note that theterm −1

2gg′Y v2W3µB

µ cannot be regarded as an interaction term since it is quadraticin the fields as the free field terms are. Therefore W3µ and Bµ cannot be regarded asindependent fields and the W3µW

µ3 and BµBµ terms cannot be regarded as mass terms.



This problem will now be solved by defining new variables.We first note that the ’mixing’ term:

12g′2Y 2v2BµB

µ +18g2v2W3µW

µ3 −

12gg′Y v2W3µB

µ (3.41)

Represents a quadratic form. We write 3.41 as:

(W3µ Bµ

)︸︷︷︸~xTµ

(12g′2Y 2v2 −1

4gg′Y v2

−14gg′Y v2 1

8g2v2

)︸︷︷︸

A

(Wµ

3

Bµ

)︸︷︷︸

~xµ

. (3.42)

Since A is symmetric it can be diagonalized by a rotation:

QTAQ = D, Q =(

cos θW sin θW− sin θW cos θW

).

The angle θW which specifies the rotation which diagonalizes A, is called the Weinbergangle. We now define new variables Zµ(x) and Aµ(x) by:(W3µ

Bµ

)︸︷︷︸

~xµ

=(

cos θW sin θW− sin θW cos θW

)︸︷︷︸

Q

(ZµAµ

)︸︷︷︸~x′µ

, ie.W3µ(x) = cos θWZµ(x) + sin θWAµ(x).Bµ(x) = − sin θWZµ(x) + cos θWAµ(x).

(3.43)

Substituting the new variables in 3.42 we obtain:

~xTµA~xµ =

(Q~x′µ

)TAQ~x′

µ= ~x

′Tµ Q

TAQ~x′µ = ~x′Tµ D~x

′µ. (3.44)

Since D is diagonal, it follows that all cross terms in the new variables (eg. cZµAµ)vanish from the expression.The mixing term is expressed in terms of the new variables by substituting 3.43 into3.41:

12g′2Y 2v2BµB

µ +18g2v2W3µW

µ3 −

12gg′Y v2W3µB

µ

=12g′2Y 2v2

(− sin θWZµ + cos θWAµ

)(− sin θWZµ + cos θWAµ

)+ . . .

= v2(1

2g′2Y 2 sin2 θW +

18g2 cos2 θW +

12gg′Y cos θW sin θW

)ZµZ

µ

+ v2(1

2g′2Y 2 cos2 θW +

18g2 sin2 θW −

12gg′Y sin θW cos θW

)AµA

µ

+ v2(1

4g2 cos θW sin θW − g′2Y 2 cos θW sin θW

− 12gg′Y cos2 θW +

12gg′Y sin2 θW

)ZµA

µ. (3.45)



Although we have not specified the relationship among the constants g, g′, Y we knowthat the coefficient of the crossterm ZµA

µ must vanish by definition. This fact will beused to specify the Weinberg angle21.We want the field Aµ to be the photon field. Since photons have zero invariant mass thecoefficient of the AµAµ term must vanish. As we will see these requirements (that theZµA

µ and AµAµ terms vanish) are compatible.

We set the coefficient to the ZµAµ term equal to zero and solve for g′Y :

−g′2Y 2 cos θW sin θW +12g(

sin2 θW − cos2 θW)g′Y +

14g2 cos θW sin θW = 0.

This is just an ordinary second ordor equation and the solutions are easily obtainedusing standard formulas:

g′Y = −12g cot θW ∧ g′Y =

12g tan θW .

We then set the coefficient to the AµAµ term equal to zero and solve for g′Y :

12g′2Y 2 cos2 θW −

12gg′Y sin θW cos θW +

18g2 sin2 θW = 0.

The solution is easily obtained:

g′Y =12g tan θW . (3.46)

Therefore we choose g′Y = 12 tan θW such that the Aµ field can be interpreted as the

photon field. Inserting this into 3.45 will explicitly remove the AµAµ and ZµAµ terms

from the expression and leave the ZµZµ term behind.It can be shown that one must set g′ = g tan θW to interpret Aµ as the photon field.This is discussed in [1, chapter 12] 22. 3.46 then requires Y = 1

2 , ie. we must assign aweak hypercharge of Y = 1

2 to the Higgs field, for photons to remain massless. InsertingY = 1

2 and g′ = g tan θW in 3.45 yields:

v2(1

2g′2Y 2 sin2 θW +

18g2 cos2 θW +

12gg′Y cos θW sin θW

)ZµZ

µ

= v2g2(1

8tan2 θW sin2 θW +

18

cos2 θW +14

tan θW cos θW sin θW)ZµZ

µ

=v2g2

8 cos2 θW

(sin4 θW + cos4 θW + 2 sin2 θW cos2 θW

)︸︷︷︸(sin2 θW+cos2 θW )2=1

ZµZµ =

v2g2

8 cos2 θWZµZ

µ. (3.47)

By considering the term(DµH

)†(DµH

)we have obtained invariant mass terms for the

W1µ, W2µ and Zµ gauge fields. The quanta of these fields are known as the W± and Z0

21We note that the Weinberg angle in fact can be determined by experiment. This will not be discussedhere.

22Derivation will not be repeated here, since it has little to do with the Higgs mechanism.



bosons. And the photons remain massless.We will now consider the rest of the Lagrangian density 3.37 (exept interaction terms).For the potential term of 3.34 we obtain the invariant mass term for the Higgs field byinserting 3.39:

−m2H†H − λ(H†H

)2 = −12m2(v + σ

)2 − λ(v + σ)4

= −12(v√

2λ)2σ2 + interaction terms. (3.48)

The derivation is analogous to the derivation done for the earlier model. This is shownin section 5.2 of the appendix.All that remains is to express the free gauge field terms of 3.36 in terms of the newvariables Zµ and Aµ. In section 5.6 of the appendix it is shown that:

−14BµνB

µν − 14FiµνF

µνi = −1

4F1µνF

1µν − 14F2µνF

2µν − 14FµνF

µν − 14ZµνZ

µν . (3.49)

−14F1µνF

1µν and −14F2µνF

2µν denotes the W1µ and W2µ fields respectively. −14FµνF

µν

denotes the photon field Aµ and −14ZµνZ

µν denotes the Zµ field. Substituting 3.40, 3.47for the mixing term, 3.48 and 3.49 in 3.37 and omitting interaction terms we obtain thefull Lagrangian density as a result of the Higgs mechanism:

L H + L B =− 14FµνF

µν

− 14F1µνF

1µν +12(1

2vg)2W1µW

µ1

− 14F2µνF

2µν +12(1

2vg)2W2µW

µ2

− 14ZµνZ

µν +12( vg

2 cos θW

)2ZµZ

µ

+12∂µσ∂

µσ − 12(v√

2λ)2σ2.

From this we read of the mass terms for the W±, Z0 and Higgs bosons:

mW± =12vg, mZ0 =

vg

2 cos θW=

mW±

cos θW, mσ = v

√2λ =

√−2m2. (3.50)

That does it. By applying the Higgs mechanism we have obtained invariant masses forthe W± and Z0 gauge bosons of the theory, while photons remain massless. These bosonswere coupled to the theory by requiring SU(2)×U(1) gauge invariance, and as we haveseen all these bosons were initially required to be massless. Otherwise the SU(2)×U(1)gauge invariance would be destroyed. However, by applying the Higgs mechanism theW± and Z0 gauge bosons become massive while preserving the SU(2) × U(1) gaugeinvariance of the theory.3.50 is the final result of this last part of the report. We have obtained invariant masses



for the W±, Z0 and Higgs bosons in terms of the basic parameters v, g, θW and λ, m2. Asdiscussed in [1, chapter 14] the masses of W± and Z0 bosons can be determined by threeexperimental well known quantities: The fine structure constant α, the Fermi couplingconstant G and the Weinberg angle θW . We will just cite the relations between theseexperimental constants and v, g, θW . Discussions can be found in [1, chapter 11-14].

α =e2

4π=

1137.04

, g sin θW = e, v =(

1√2G

)− 12

, G = 1.166 · 10−5 GeV−2,

sin2 θW = 0.235± 0.005. (3.51)

Inserting these relations in 3.50 one obtains the following masses for the W± and Z0

bosons [1, page 301]:

mW± = 76.9± 0.8 GeV, mZ0 = 87.9± 0.6 GeV. (3.52)

This agrees fairly well 23 with the experimental masses:

mW± = 80.22± 0.26 GeV mZ0 = 91.17± 0.02 GeV. (3.53)

As mentioned before, the above arguments were indeed used to determine the massesof the W± and Z0 bosons before the particles were detected. However, the parameter λand m2 cannot be measured and the mass of the Higgs boson cannot be predicted fromknown data. This make the search for the Higgs boson much harder.

23Radiative corrections are neglected, making the results worse.


QED and the Higgs Mechanism 4. Conclusion

4 Conclusion

Concerning the first part of the report we obtained the electromagnetic interaction byrequiring gauge invariance of the electron-positron field theory. We have treated theinteraction as a perturbation of the free fields and have developed a perturbative methodfor deriving transition amplitudes for elementary processes. This is known as the S-matrix expansion. We have seen how to extract from the S-matrix expansion termswhich to a given order contribute to a process. We have seen that the first oder termsdoes not describe physical processes and we have seen that the transition amplitudesare explicitly zero. To obtain physical processes we considered second order terms of theS-matrix expansion. As an example we picked out the terms contributing to Bhabhascattering and we derived the transition amplitudes 2.39 and 2.40 which represents thefinal result of the first part.In the second part of the report we explained the idea of spontaneous symmetry breakingby considering a simple global U(1) invariant model. We have seen that spontaneoussymmetry breaking occurs if the ground state is degenerate. We extended the model byrequiring local U(1) gauge invariance. This led us to couple a gauge field to the Higgsfield. By application of the Higgs mechanism the gauge bosons were seen to becomemassive while the gauge symmetry was preserved.We extended the model further by requiring local SU(2)×U(1) gauge invariance. This ledus to couple four gauge fields to the Higgs field. By application of the Higgs mechanismwe obtained invariant masses for the W± and Z0 gauge bosons, while photons remainedmassless and the gauge symmetry of the theory was preserved. The expressions for theinvariant masses of the W±, Z0 and Higgs bosons 3.50 represents the final result of thelast section.


QED and the Higgs Mechanism 5. Appendices

5 Appendices

5.1 Appendix A1

S = 1 + (−i)∫ ∞−∞

dt1HI(t1) + (−i)2

∫ ∞−∞

dt1

∫ t1

−∞dt2HI(t1)HI(t2) + . . . (5.1)

S =∞∑n=0

(−i)n

n!

∫ ∞−∞

. . .

∫ ∞−∞

d4x1d4x2 . . . d

4xnT{HI(x1)HI(x2) . . .HI(xn)}. (5.2)

The equivalence of the two terms 5.1 and 5.2 can be shown to hold for each term in theseries. Here we shall just consider the second terms in the series. The procedure appliesto the rest of the terms.For a time ordered product we have:

T{HI(t1)HI(t2)} =

{HI(t1)HI(t2) for t1 > t2.

HI(t2)HI(t1) for t2 > t1.

Substituting this into 5.2 we obtain for the second term (the hamiltonian densities whichare trivial to introduce will be omitted):

12

∫ ∞−∞

dt1

∫ ∞−∞

dt2T{HI(t1)HI(t2)} =

12

∫ ∞−∞

dt1

(∫ t1

−∞dt2HI(t1)HI(t2) +

∫ ∞t1

dt2HI(t2)HI(t1))

(5.3)

The two terms in 5.3 double up since:∫ ∞−∞

dt1

∫ ∞t1

dt2HI(t2)HI(t1) =∫ ∞−∞

dt2

∫ t2

−∞dt1HI(t2)HI(t1) =∫ ∞

−∞dt1

∫ t1

−∞dt2HI(t1)HI(t2).

In the first expression we integrate over the whole interval with t2 > t1. In the secondexpression we integrate over the whole interval with t1 < t2. This is of course the same.We just interchange variables to obtain the last expression. Ie. the two terms in 5.3double up, making 5.3 equal to the second term of the series 5.1.


QED and the Higgs Mechanism 5.2 Appendix B1

5.2 Appendix B1

We have:

L (x) = ∂µH∗∂µH︸︷︷︸

Kinetic term

−(m2|H|2 + λ|H|4

)︸︷︷︸Potential term

.

H0 =

√−m2

2λ≡ 1√

2v, m2 = −v2λ.

H(x) =1√2

(v + σ(x) + iη(x)

).

For the kinetic term of the lagrangian density we obtain:

∂µ1√2

(v + σ − iη

)· ∂µ 1√

2

(v + σ + iη

)=

12(∂µσ∂

µσ + i∂µσ∂µη − i∂µη∂µσ + ∂µη∂

µη)

=12∂µσ∂

µσ +12∂µη∂

µη.

For |H|2 and |H|4 we obtain:

|H|2 =12((v + σ)2 + η2

)=(v2 + σ2 + 2vσ + η2

).

|H|4 =14

((σ2 + η2

)2 +(v2 + 2vσ

)2 + 2(σ2 + η2

)(v2 + 2vσ

)︸︷︷︸v4+4v2σ2+4v3σ+2σ2v2+4vσ3+2η2v2+4vση2

).

We then obtain for the potential term:

m2|H|2 + λ|H|4 =

−v2λ

2(v2 + σ2 + 2vσ + η2

)+λ

4((σ2 + η2)2 + v4 + 4v2σ2 + 4v3σ + 2v2σ2 + 4vσ3 + 2η2v2 + 4vση2

)=

−λv4

2+λv4

4︸︷︷︸constant term

+λ

4(σ2 + η2

)2 + λvσ(σ2 + η2

)+ λv2σ2.

The constant term is of no significance in the Lagrangian density and is omittet. Com-bining the kinetic and potential terms we obtain the Lagrangian density:

L (x) =12∂µσ(x)∂µσ(x)− λv2σ(x)2

+12∂µη(x)∂µη(x)

−λvσ(x)(σ(x)2 + η(x)2

)− λ

4(σ(x)2 + η(x)2

)2.



5.3 Appendix B2

We have:

L = ∂µH∗∂µH − V︸︷︷︸

free matter field

− 14FµνF

µν︸︷︷︸free gauge field

+ igAµ(H∂µH∗ −H∗∂µH) + g2AµA


.

H =1√2

(v + σ + iη

).

For the terms in the free matter field Lagrangian we obviously get the same result asbefore. For the other terms we have:

− igAµ(H∂µH∗ −H∗∂µH

)=

− ig

2Aµ

((v + σ)∂µ(v + σ)− i(v + σ)∂µη + iη∂µ(v + σ) + η∂µη

−((v + σ)∂µ(v + σ) + i(v + σ)∂µη − iη∂µ(v + σ) + η∂µη

))=

− ig

2Aµ

(2i(v + σ)∂µη − 2iη∂µ(v + σ)

)=

gvAµ∂µη + terms cubic and quartic in the fields.

g2AµAµHH∗ =

12g2AµA

µ(v + σ + iη

)(v + σ − iη

)=

12

(gv)2AµAµ + terms cubic and quartic in the fields. (5.4)

Combining the terms we obtain:

L (x) =12∂µσ(x)∂µσ(x)− 1

2(2λv2)σ2 (5.5)

+12∂µη(x)∂µη(x) (5.6)

− FµνFµν +12

(gv)2AµAµ (5.7)

+ gvAµ∂µη + interaction terms. (5.8)



5.4 Appendix B3

It is a complicated business to derive the explicit transformation law for the gauge fieldsfor finite functions αi(x) so we only consider infinitesimal transformations.24

The gauge fields must undergo a transformation:

Wiµ →Wiµ + δWiµ.

Applying this to 3.30 for infinitesimal transformations we obtain:

DµH =(∂µ +

12igτiWiµ

)H →

(DµH

)′ =(∂µ +12igτiWiµ +

12igτiδWiµ

)(1 +

12iτiαi

)H.

(5.9)

To obtain gauge invariance we require:

U(α(x))DµH =(1 +

12iτiαi

)DµH

=(1 +

12igτiαi

)(∂µ +

12igτiWiµ

)H, (5.10)

Ie. we want to equate the right hand sites of 5.9 and 5.10. First we define the matrices:

α = τiαi, Wµ = τiWiµ (5.11)

From the commutation relations of the pauli matrices: [τi, τj ] = εijkτk we have:

[α,Wµ] = 2iεijkτiαjWµk. (5.12)

Equating the right hand sites of 5.9 and 5.10 we obtain:(∂µ +

12igWµ +

12igδWµ

)(1 +

12iα)H =

(1 +

12iα)(∂µ +

12igWµ

)H ⇔

12i∂µαH +

12iα∂µH+

(− 1

4gWµα+

12igδWµ −

neglected︷︸︸︷14gδWµα

)H =

12iα∂µH −

14gαWµH ⇔

12igδWµ = −1

2i∂µα+

14gWµα−

14gαWµ ⇔

δWµ = −1g∂µα+

12i[α,Wµ]. (5.13)

24Mandl.Shaw p 269



Substituting the commutation relation 5.12 yields:

δWµ = −1g∂µα− εijkτiαjWµk ie.

τiδWiµ = −1gτi∂µαi − τiεijkαjWµk

= τi(− 1g∂µαi − εijkαjWµk

)⇔

δWiµ = −1g∂µαi − εijkαjWµk. (5.14)

Which is the transformation law 3.32.



5.5 Appendix B4

The Lagrangian density for the gauge bosons are:

L B = −14BµνB

µν − 14GiµνG

µνi . (5.15)

Substituting Giµν = Fiµν + gεijkWjµWkν yields

L B = −14BµνB

µν − 14(Fiµν + gεijkWjµWkν

)(Fµνi + gεilmW

µl W

νm

)= −1

4BµνB

µν − 14FiµνF

µνi −

14g2εijkεilmWjµWkνW

µl W

νm

− 14gεijkWjµWkνF

µνi −

14gεilmFiµνW

µl W

νm. (5.16)

5.16 yields:

−14gεijkWjµWkνF

µνi −

14gεilmFiµνW

µl W

νm = −1

2gεijkF

µνi WjµWkν .

Substituting Fµνi = ∂νWµi − ∂µW ν

i yields:

− 12gεijkF

µνi WjµWkν = −1

2gεijk

(∂νWµ

i − ∂µW ν

i

)WjµWkν

= −12gεijk∂

νWµi WjµWkν +

12gεijk∂

µW νi WjµWkν

= −12gεijk∂

µW νi WjνWkµ +

12gεijkWiµWjν∂

µW νk

= +12gεijkWiνWjµ∂

µW νk +

12gεijkWiµWjν∂

µW νk = gεijkWiµWjν∂

µW νk (5.17)

Where the proporties of the Levi Civita symbol has been used. Ie.

L B = −14BµνB

µν − 14FiµνF

µνi −

14g2εijkεilmWjµWkνW

µl W

νm + gεijkWiµWjν∂

µW νk .

(5.18)

Which is 3.36.



5.6 Appendix B5

We whish to express:

L B0 = −1

4BµνB

µν − 14FiµνF

µνi . (5.19)

In terms of the fields Aµ, Wµ, Zµ. These fields are defined by:

Bµν = − sin θZµ + cos θAµ (5.20)W3µ = cos θZµ + sin θAµ (5.21)

Wµ =1√2

(Wiµ − iW2µ

), W †µ =

1√2

(Wiµ + iW2µ

)(5.22)

θ is the weak mixing angle.We now obtain:

−14BµνB

µν =14(

cos θ∂νAµ − sin θ∂νZµ + sin θ∂µZν − cos θ∂µAν)

·(

cos θ∂νAµ − sin θ∂νZµ + sin θ∂µZν − cos θ∂µAν)

=14

(cos θ

(∂νAµ − ∂µAν

)− sin θ

(∂νZµ − ∂µZµ

))·(

cos θ(∂νAµ − ∂µAν

)− sin θ

(∂νZµ − ∂µZµ

))=

14

(sin2 θZµνZ

µν + cos2 θFµνFµν − 2 sin θ cos θZµνFµν

). (5.23)

14FiµνF

iµν =14(F1µνF

1µν + F2µνF2µν + F3µνF

3µν).

We first consider:

14F3µνF

3µν =14(

cos θ∂νZµ + sin θ∂νAµ − cos θ∂µZν − sin θ∂µAν)

·(

cos θ∂νZµ + sin θ∂νAµ − cos θ∂µZν − sin θ∂µAν)

=14

(cos2 θZµνZ

µν + sin2 θFµνFµν + 2 sin θ cos θZµνFµν

). (5.24)

We then consider:

14(F1µνF

1µν + F2µνF2µν)

=14(∂νW1µ − ∂µW1ν

)(∂νWµ

1 − ∂µW ν

1

)+

14(∂νW2µ − ∂µW2ν

)(∂νWµ

2 − ∂µW ν

2

)=

12(∂νW1µ∂

νWµ1 − ∂νW1µ∂

µW ν1 + ∂νW2µ∂

νWµ2 − ∂νW2µ∂

µW ν2

). (5.25)



Differentiating 5.22 yiels:

∂vWµ =1√2

(∂vWiµ − i∂vW2µ

), W †µ =

1√2

(∂vWiµ + i∂vW2µ

)⇒

∂vWµ∂νWµ† =

12(∂νW1µ∂

νWµ1 + ∂νW2µ∂

νWµ2

)(5.26)

−∂vWµ∂µW ν† = −1

2(∂νW1µ∂

µW ν1 + ∂νW2µ∂

µW ν2

). (5.27)

Substituting 5.26 and 5.27 in 5.25 yields:

14(F1µνF

1µν + F2µνF2µν)

=12F †µνF

µν . (5.28)

Substituting 5.23, 5.24 and 5.28 in 5.19 yields:

L B0 = −1

4FµνF

µν − 12F †WµνF

µνW −

14ZµνZ

µν .



5.7 Appendix B6

We consider the term(DµH

)†(DµH

):(

DµH)†(

DµH)

=(∂µH

† − 12igH†τiWiµ − ig′Y BµH†

)(∂µH +

12igτiW

µi H + ig′Y BµH

)= ∂µH

†∂µH + g′2Y 2H†BµBµH +

14g2H†τiWiµτjW

µj H

+12ig∂µH

†τiWµi H −

12igH†τiWiµ∂

µH + ig′Y ∂µH†BµH − ig′Y H†Bµ∂µH

+12gg′Y H†τiWiµB

µH +12gg′H†BµτiW

µuH.

Substituting:

τiWiµ =(

0 11 0

)W1µ +

(0 −ii 0

)W2µ +

(1 00 −1

)W3µ,

And:

H(x) =1√2

(0

v + σ(x)

)And doing the matrix multiplications yields:

12∂µσ∂

µσ +12g′2Y 2(v + σ)2BµB

µ +18g2(v + σ)2

(W1µW

µ1 +W2µW

µ2 +W3µW

µ3

)− 1

4ig∂µσ(v + σ)2W3µ +

14ig(v + σ)∂µσW3µ

+12ig′Y ∂µσB

µ(v + σ)− 14ig′(v + σ)Bµ∂µσ

+14gg′Y (v + σ)2W3µB

µ +14gg′Y (v + σ)2W3µB

µ ⇒

(DµH

)†(DµH

)=

=12∂µσ∂

µσ +12g′2Y 2v2BµB

µ +18g2v2

(W1µW

µ1 +W2µW

µ2 +W3µW

µ3

)− 1

2gg′Y v2W3µB

µ + Third and fourth order terms in the fields.


QED and the Higgs Mechanism 6. Bibliography

6 Bibliography

Litteratur

[1] F. Mandl and G. Shaw. Quantum field theory. John Wiley and Sons, revised edition,reprinted 2008.

[2] Francesco Sannino. Lecture notes on strong interactions. Can be found on the website: http://hep.sdu.dk/Strong/Strong.pdf.

[3] Eugen Merzbacher. Quantum Mechanics. John Wiley and Sons, third edition 1998.

[4] David J. Griffiths. Introduction to Quantum Mechanics. Person Education Interna-tional, second edition, 2005

[5] J. J. Sakurai. Advanced Quantum Mechanics. Addison Wesley, 1967.

[6] Erwin Kreyszig. Advanced Engineering Mathematics. Wiley and sons, 9th edition.

[7] Herbet Goldstein. Classical Mechanics. Addison Wsley, second edition 1980.

[8] David Tong. Lectures on Quantum Field Theory. Can be found on the web site:http://www.damtp.cam.ac.uk/user/tong/qft.html.

[9] Higgs Physics at the LHC. The website is found on:http://www.hep.lu.se/atlas//thesis/egede/thesis-node6.html.


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Quantum Electrodynamics and the Higgs Mechanismcp3-origins.dk/content/thesis/2009-01-joergensen.pdf · QED and the Higgs Mechanism 1. Introduction 1 Introduction This report comprises