Quantum Electrodynamics and the Higgs Mechanism

Transcription

1 Quantum Electrodynamics and the Higgs Mechanism Jakob Jark Jørgensen 4. januar 009

2 QED and the Higgs Mechanism INDHOLD Indhold 1 Introduction Quantum Electrodynamics 3.1 Obtaining a Gauge Theory The S-Matrix Expansion The Interaction Term Identifying Processes in the S-matrix expansion Transition Amplitudes The Higgs Mechanism Spontaneous Symmetry Breaking The Local U1 Invariant Model Applying the Higgs Mechanism The Physical Local SU U1 Invariant Model The SU Group and SU U1 Invariance Applying the Higgs Mechanism Conclusion 33 5 Appendices Appendix A Appendix B Appendix B Appendix B Appendix B Appendix B Appendix B Bibliography 43 Jakob Jark Jørgensen 1

3 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 interactions between electrons, positrons and photons. We shall obtain the electromagnetic interaction by requiring invariance of the Lagrangian density for the electron-positron field under local U1 phase transformations. We shall then develop a perturbative theory known as the S-matrix expansion, and use this to derive transition amplitudes for elementary electromagnetic processes. We will see that interactions can be interpreted as being brought about by exchange of virtual intermediate particles. In the second part of the report we focus on the Higgs mechanism. We start out by explaining the concept of spontaneous symmetry breaking using a simple model. This simple model will be gauged and we shall apply the Higgs mechanism thereby making the gauge bosons massive, while preserving the gauge symmetry of the theory. In the final section we shall see that the W ± and Z 0 bosons of the electroweak theory become massice by application of the Higgs mechanism. We shall obtain expressions for the invariant masses of the W ±, Z 0 and Higgs bosons. Jakob Jark Jørgensen

4 QED and the Higgs Mechanism. Quantum Electrodynamics Quantum Electrodynamics In this first part of the report we will consider quantum electrodynamics which deals with the interactions between electrons, positrons and photons. We will develop a perturbative theory known as the S-matrix expansion, which to a given order can be used to derive transition amplitudes for elementary processes of QED. We will see that this perturbative theory leads one to interpret interactions as being brought about by exchange of virtual intermediate particles..1 Obtaining a Gauge Theory We start out by considering the classical Lagrangian density for the electron-positron field: L x = ψx iγ µ µ m ψx..1 The notation is as follows: The four-spinor fields ψx and ψx = ψx γ 0 satisfy the Dirac equation and the adjoint Dirac equation respectively. These equations of motion can be derived from the Lagrangian density by the Euler Lagrange equations. In the quantized theory, the quanta of the fields will be electrons and positrons, both of mass m. γ µ = γ 0, γ 1, γ, γ 3 is a vector consisting of the Dirac 4 4 gamma matrices. µ = x µ is the covariant four gradient operator. x µ = t, x 1 is the contravariant space-time fourvector and will be denoted as x. Repeated indices implies summation. We will consider the symmetry of the field theory specified by.1. We first note that the Lagrangian density is invariant under global U1 transformations 3 : ψx ψ x = e iα ψx, ψ x ψ x = e iα ψ x.. Invariance under. allows one to change the phase of the fields by the same amount at each point in space. This seems very restrictive for a field theory and we shall demand invariance under the more general local U1 phase transformations. Doing so we obtain what is called a gauge theory 4 Under local U1 gauge transformations the fields transform according to., with α = αx, ie. the transformation is dependent on the space-time position..1 transform according to: L L = ψe iα iγ µ µ m e iα ψ = ψe iα iγ µ [e iα µ ψ + i µ αe iα ψ] me iα ψ = L ψγ µ µ αψ..3 1 We use natural units, ie. = c = 1. The metric used to define the covariant fourvector x µ from the contravariant x µ ie. x µ = g µνg ν is specified by g 00 = g 11 = g = g 33 = 1, g µν = 0 if µ ν. 3 This symmetry implies current conservation so that a conserved charge can be associated with the field 4 Gauge theories have the advantage of being for example renomalizable, these points will not be discussed here. Jakob Jark Jørgensen 3

5 QED and the Higgs Mechanism.1 Obtaining a Gauge Theory The Lagrangian density is not invariant under U1 gauge transformation. Gauge invariance can be obtained by coupling a gauge field A µ to the matter field. This coupling is made by replacing the ordinary differential µ in the Lagrangian density by the covariant differential D µ : D µ = µ iea µ x..4 A µ x is a real vector field, and will be called the gauge field. e is a constant specifying the coupling. Under U1 transformation the covariant derivative of ψ transform according to: D µ ψ D µ ψ = µ iea µ ψ..5 We see that the theory will be gauge invariant if the covariant derivative transform as the fields themselves. This will be demanded: D µ ψ D µ ψ = e iα D µ ψ. This requirement determines how the gauge field must transform under local U1 transformations. From.5 we obtain: Dµ ψ = µ iea µ e iα = e iα µ ie A µ 1 e µα ψ. From this and.4 we see that the gauge field must transform according to: A µ 1 e µα = A µ A µ = A µ + 1 e 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 Lagrangian density. Replacing µ in.1 by D µ we obtain: L = ψ iγ µ D µ m ψ = ψ iγ µ ] [ µ iea µ m ψ ψ iγ µ µ m ψ + e ψ /Aψ..6 }{{}}{{} Free e + /e field. Interaction. 5.6 contains the free electron-positron field Lagrangian density and a term which is interpreted 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 free gauge field. This term must of course be U1 gauge invariant as well. We note that A µ has 4 components and is a real field. In the quantized theory this implies the quanta of the gauge field being neutral spin-one bosons. Concerning the mass of these gauge bosons, we note that a Lorentz invariant mass term would have the form 1 m γa µ A µ. However, such a term is not invariant under local U1 transformations: 1 m γa µ A µ 1 m γa µa µ = 1 m γ 5 We use the notation /A = γ µ A µ. Aµ + µα e A µ + µ α 1 e m γa µ A µ..7 Jakob Jark Jørgensen 4

6 QED and the Higgs Mechanism.1 Obtaining a Gauge Theory Therefore we require the gauge bosons to be massless. Since the gauge field describes massless spin-1 bosons, we shall add the Lagrangian density of the free photon field: 1 4 F µνxf µν x where F µν = ν A µ µ A ν..8 This term is seen to be gauge invariant: ν A µ µ A ν ν A µ µ A ν = ν Aµ + 1 e µα µ Aν + 1 e να = ν A µ µ A ν + 1 e ν µ α 1 e µ ν α = ν A µ µ A ν. The full local U1 Lagrangian density then reads: L = 1 4 F µνf µν }{{} Free photon field. + ψ iγ µ µ m ψ }{{} Free e + /e field. + e ψ /Aψ }{{}..9 Interaction..9 is the Lagrangian density describing the interacting electron-positron and photon fields of quantum electrodynamics QED 6. It contains terms describing the free electronpositron field, the free electromagnetic field and the electromagnetic interaction. The same Lagrangian density will be obtained by making the minimal substitution of nonrelativistic quantum mechanics: i t i t ea 0, i i ea j..10 We see that this exactly corresponds to replacing the ordinary differential by the covariant differential D µ. To summarize: By requiring local U1 gauge invariance of the Lagrangian density.1 we are led to introduce a gauge field coupled to the matter field. This gauge field can be interpreted 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 one cannot derive interactions by requiring gauge invariance. One merely developes gauge invariant forms of the theories which may or may not be confirmed by experiments. For example were the W ± and Z 0 bosons predicted to exist and their masses determined by such arguments, before they were observed, as discussed in [1, chapter 1, page 63]. 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.9. This term will be treated as a perturbation of the free fields and we shall show how to derive transition amplitudes for elementary processes using perturbation theory. The application of perturbation theory is particularly successful for QED where the coupling between photons and electrons, measured by the fine structure constant α x µ, is particularly small. 6 In this context.9 describes the fields before second quantization is applied. Upon second quantization these fields become the field operators of QED. The canonical quantization procedure will not be described in this report, and it will in general not be specified whether we are working with the fields before or after second quantization. Jakob Jark Jørgensen 5

7 QED and the Higgs Mechanism. The S-Matrix Expansion. The S-Matrix Expansion In this section we will develop the perturbation theory which can be used to derive probability transition amplitudes for elementary processes. We shall be working in the interaction picture of quantum mechanics. The Hamiltonian of the system is split into the free-field Hamiltonian H 0 and an interaction Hamiltonian H I. The time developement of state vectors Φt are governed by: i d dt Φt = H It Φt..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 a definite number of well separated particles such that the do not interact with definite momenta and spin/polarization properties long before the scattering. For example: i = e + r p; γ s k, 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.11 i evolves into a final state Φ in which particles are again far apart and non interacting turning of the interacting leaves the state vectors constant in time as is seen from.11. The S-matrix is defined to relate the initial and final states: Φ = S Φ = S i..1 The final state Φ does not specify a definite number of particles with definite momenta and spin/polarization properties as the initial state does. Instead the final state contains all possible outcomes of the scattering. Φ can be expanded in a complete 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 = f S i. To work out the corresponding probability transition amplitudes f S i, we need an expression for the S-matrix. For the initial condition i = Φ.11 can be written as the integral eqution: t Φt = i + i dt 1 H I t 1 Φt We set: Φt 1 = i + i t1 dt H I t Φt, and so on for Φt, Φt 3... Jakob Jark Jørgensen 6

8 QED and the Higgs Mechanism.3 The Interaction Term And solve.13 iteratively: t Φt = i + i dt 1 H I t 1 i + i [ Φt = 1 + i t dt 1 H I t 1 + i t t dt H I t i + i dt 3 H I t 3... t1 ] dt 1 dt H I t 1 H I t +... i. In the limit T this gives us an expression for the S-matrix, the so called S-matrix expansion: S = 1 + i S = i n n=0 n! t1 t1 dt 1 H I t 1 + i dt 1 dt H I t 1 H I t d 4 x 1 d 4 x... d 4 x n T {H I x 1 H I x... H I x n }..15 In.15 the Hamiltonian density H I x is introduced and a more compact summation notation is used. The equivalence of the two forms.14 and.15 where time ordering is introduced 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 H I will only be useful if the interaction energy is small. As mentioned before this is the case for QED and the method developed above can be applied with succes, as discussed in [1, chapter 5, page ]. In the next sections we shall se how to derive transition amplitudes in second order of this perturbation theory..3 The Interaction Term We will now consider the interaction term L I = e ψ /Aψ of the Lagrangian density.9. This term is going to be the input of.15. The Hamiltonian density of the interaction term is obtained using general formulas: H x = π r x φ r x L φ r, µ φ r, π r x = L φ r Noting that L I does not depend on time derivatives of any of the fields we simply obtain: H I x = 0 L I = e ψx /Axψx..17 As mentioned before we are working in the interaction picture. The fields in.17 are upon second quantization the interacting fields in the interaction picture. However we will se below that formulas for the free ie. non interacting fields in the Heisenberg picture is also valid for the fields in.17. We first consider formulas for the free fields. The free fields are written as Fourier expansions in the complete sets of plane wave 7 r labels the fields of the system, for A µ for example It labels the components. π r is the conjugate momenta of the fields. Jakob Jark Jørgensen 7

9 QED and the Higgs Mechanism.3 The Interaction Term solutions to the equations of motion for the fields. For ψx ψx the equation of motion is the adjoint Dirac equation. For A µ x the equation of motion is the covariant formulation of Maxwell s equations. These equations can be derived from the free field Lagrangian density terms in.9, by means of the Euler Lagrange equations. The expansions are, from [1, pages 67 and 84]: A µ x = A + µ x + A µ x = 1 ɛ rµ a r ke ikx + ɛ rµ a rke ikx..18 V ω k r,k The k summation is over k allowed by imposing periodic boundary conditions on a cubic enclosure of volume V. k is the wave four-vector with k 0 = w k. The r summation, r = 0 to r = 3 is over four polarization states, described by ɛ rµ. ψx = ψ + x + ψ x = m c r pu r pe ipx + d rpv r pe ipx..19 V E r,p p ψx = ψ + x + ψ x = r,p m V E p d r p v r pe ipx + c rpū r pe ipx..0 Here p is the energy momentum four-vector and the p summation is over p allowed by imposing periodic boundary conditions on a cubic enclosure of volume V. The r summation, r = 1, is over spin states described by the spinors u r p and v r p. E p stems from normalization of u r p and v r p 8 9. Upon second quantization the fields become operators. The Fourier expansion coefficients become creation and annihilation operators of particles with definite momentum and spin/polarization. The state e + r p; γ s k is for example created from the vacuum by: e + r p; γ s k = d rpa sk 0. Vice verca e + r p; γ s k becomes the vacuum state by: 0 = d r pa s k e + r p; γ s k. In the notation used above A + x A x contains only annihilation creation operators. Therefore A + x A x is interpreted as annihilating creating a photon of definite position x. Correspondingly ψ + x ψ x annihilates creates an electron at x, 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 mechanics, and the fields represents Heisenberg operators. As mentioned before we shall be using interacting fields in the interaction picture. It so happens that the solutions for the free fields and the interpretation in terms of raising and lowering operators in the Heisenberg picture, is also valid for the interaction fields in the interaction picture. This follows from two points: 8 The dagger notation is used since the expansion coefficients become operators upon second quantization. 9 The expansions of the free fields will not be discussed in any more detail in this report. We shall be using these and other results derived in [1, chapters -5] 10 The procedure of second quantization, an the interpretation in terms of creation and annihilation operators will not be discussed in this report. Jakob Jark Jørgensen 8

10 QED and the Higgs Mechanism.4 Identifying Processes in the S-matrix expansion 1. In the interaction picture operators satisfies the equation of motion: i d dt OI t = [O I t, H 0 ]. Ie. only involving the free field Hamiltionian H 0. The free fields satisfy the same equation of motion in the Heisenberg picture 11.. Since the interaction term of.9 does not involve any derivatives, the fields conjugate to the interacting fields will be the same as the fields conjugate to the free fields. Since the Heisenberg picture and the interaction picture are related by a unitary transformation, the interacting fields and the free fields therefore satisfy the same commutation relations. It follows from these two points that the expansions of the free fields and the interpretation in terms of creation and annihilation operators in the Heisenberg picture, is valid for the interacting fields in the interaction picture. The propagators will also have the same form in the interaction picture as in the Heisenberg picture. The input for.15 is then: H I x = en[ ψx /Axψx] = en[ ψ+ + ψ /A + + /A ψ + + ψ ] x..1 Here normal ordering is introduced. In a normal product all annihilation operators stand to 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.1 consists of the interacting fields, each expanded in terms of raising and lowering operators..15 with.1 as the input will therefore describe a large number of different processes. In the next section we will se how to pick out from the S-matrix expansion the terms which contribute to a given process. Afterwards we will se how to derive transition amplitudes f S i for the processes..4 Identifying Processes in the S-matrix expansion It is straightforward to pick out terms from the S-matrix expansion which contribute to a given transition i f. The terms must contain the right annihilation operators to annihilate the particles present in i and the right creation operators to create the particles present in f. In this section we will see how to pick out the terms contributing to a given process, and we will describe the processes by Feynman diagrams. We start out by considering the first order term S 1 of.15: S 1 = ie d 4 x 1 T {N[ ψ /Aψ] x1 }. 11 In the Heisenberg picture the equation of motion is: i d dt OH t = [O H t, H]+i OH. For the operators t we consider the last term drop out cf. [1, page 3] Jakob Jark Jørgensen 9

11 QED and the Higgs Mechanism.4 Identifying Processes in the S-matrix expansion a b Figur 1: Figure a is the Feynman diagram for the process corresponding to the term N[ 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 interpreted as follows: A line entering from the left-hand side represents a particle present initially. A line leaving the diagram on the right-hand side represents a particle present finally. Curvy lines represent photons and straight lines represent fermions. The arrows on fermion lines distinguish 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 normal products, into a sum of normal products. In this way we obtain processes with no intermediate particles 1. For S 1 we just get: S 1 = ie d 4 x 1 N[ ψ+ + ψ /A + + /A ψ + + ψ ] x1... gives rise to eight different processes. The term d4 x 1 N[ ψ + /A + ψ + ] x1 for example corresponds to the annihilation of an initial electron positron pair and the annihilation of an initial photon. Notice that we integrate over all space, corresponding to the process happening at any point in space-time. This process is represented by the Feynman diagram in figure 1 a. For real particles we have p = m for electrons and positrons and k = 0 for photons. Energy and momentum cannot be conserved for real particles for any of the processes arising from S 1, and the processes are therefore not physical processes. This is illustrated by the following simple example. We consider the process arising from the term N[ ψ /A + ψ + ] x1. The process is represented by the Feynman diagram in figure 1 b. An initial electron and photon are annihilated and a final electron is created, ie. a photon is absorbed by an electron. Energy-momentum conservation implies p = k + p 1. From this we obtain: p = k + p 1 + k µ p 1µ = m + k 0 p 10 k p 1..3 In the reference frame of the incoming electron p 1 = 0 such that p m for the outgoing electron. Ie. for real particles, energy and momentum cannot be conserved in 1 A detailed discussion of Wicks theorem is found fx. in [1, chapter 6]. It will not be discussed in this report. Jakob Jark Jørgensen 10

12 QED and the Higgs Mechanism.4 Identifying Processes in the S-matrix expansion this process. The processes arising from S 1 are referred to as the basic vertex parts of QED. In fact all other QED Feynman diagrams can be built up by combining such basic vertexes. The processes arising form the second order term S can for example be constructed by combining two vertexes. We will see an example of this below. The second order term S is: S = e d 4 x 1 d 4 x T {N[ ψ /Aψ] x1 N[ ψ /Aψ] x }..4 Expanding this by use of Wicks theorem we obtain 13 : S = e d 4 x 1 d 4 x N[ ψ /Aψ x 1 ψ /Aψ x ] e d 4 x 1 d 4 x N[ ψ /A ψ x 1 ψ /Aψ x }{{} ] e d 4 x 1 d 4 x N[ ψ /Aψ x 1 ψ /A }{{} ψ x ].5 e d 4 x 1 d 4 x N[ ψ /Aψ x 1 ψ /Aψ x }{{} ] e d 4 x 1 d 4 x N[ ψ /A ψ x 1 ψ /A ψ x }{{} ] }{{} e d 4 x 1 d 4 x N[ ψ /A ψ x 1 ψ /Aψ x }{{} ] e d 4 x 1 d 4 x N[ ψ /Aψ x 1 ψ /A }{{} ψ x ] }{{}}{{} 4 e x 1 d 4 x N[ ψ /A ψ x1 ψ /A ψ x }{{} ]. }{{}}{{} The braces represents nonvanishing contractions. These contractions are Feynman propagators and can be interpreted as the propagation of a virtual intermediate particle between two space-time points. Eg. ψx 1 ψx represents a virtual intermediate fermion propagating between x 1 and x 14. By the term virtual we mean that the particle }{{} is not a physical particle for which p = m fermion or k = 0 photon. For the processes arising from S energy and momentum can be conserved for initial and final physical particles, and the terms in S describe real physical processes. As we shall see below, an interaction between physical particles is represented as an exchange of virtual particles. As an example we will consider the processes arising from.5. This term contains four uncontracted fermion operators. These correspond to creation and annihilation of fermions at x 1 and x such that.5 will represent fermion-fermion scattering processes: e e, e + e + and e + + e e + + e. The photon-photon contraction can be 13 Integration boundaries will be omitted in the following, and we just write: R R d4 x 1d 4 x = R d 4 x 1d 4 x to ease notation. 14 An introduction to covariant commutation relations from which the propagators arise will not be given 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. Jakob Jark Jørgensen 11

13 QED and the Higgs Mechanism.4 Identifying Processes in the S-matrix expansion interpreted as mediating the interaction between the fermions at x 1 and x by exchange of virtual intermediate photons. Notice that all values of x 1 and x are integrated over, corresponding to the processes happening at any two space-time points. Let us now pick out the terms from.5 which contribute to the process e + + e e + +e, known as Bhabha scattering. As mentioned before we need just the right combination of annihilation and creation operators to annihilate the particles present initially and to create the particles present finally. In this case we need ψ + and ψ + to annihilate the initial positron and electron respectively and ψ and ψ to create the final positron and electron..5 consists of four terms wich has this combination of operators: S e + + e e + + e = e d 4 x 1 d 4 x { N [ ψ+ γ µ ψ ψ x 1 γ ν ψ + ] Aµ x x 1 A ν x }{{} +N[ ψ γ µ ψ + ψ+ x 1 γ ν ψ ] Aµ x x 1 A ν x }{{}.6 N [ ψ+ γ µ ψ + x 1 ψ γ ν ψ x ] Aµ x 1 A ν x }{{} +N[ ψ γ µ ψ x 1 ψ+ γ ν ψ + x ] Aµ x 1 A ν x }{{} }..7 The rest of the terms correspond to different processes. By permuting the fermion operators and changing the integration variables, it is seen that the two terms in.6 are equal as are two terms in.7. For example for the first term in.6: ψ + α x 1 γ µ αβ ψ β x 1 ψ δ x γ ν δγ ψ+ γ x A µ x 1 A ν x }{{} = 1 4 ψ δ x γ ν δγ ψ+ γ x ψ + α x 1 γ µ αβ ψ β x 1 A ν x A µ x 1 }{{}. The spinor indices have been written out explicitly and a minus sign is introduced for each time two fermion operators are interchanged. Interchanging the integration variables we se that the two terms in.6 are equal. Therefore: S e + + e e + + e = e d 4 x 1 d 4 x { N [ ψ γ µ ψ + x 1 ψ+ γ ν ψ x ] Aµ x 1 A ν x }{{} +.8 N [ ψ γ µ ψ ψ+ x 1 γ ν ψ + ] Aµ x x 1 A ν x }..9 }{{} This will be the term which to second order in the s-matrix expansion will contribute to Bhabha scattering. The processes.8 and.9 are represented by the Feynman diagrams in figure. Notice that these Feynman diagrams can be built up from basic vertexes such as those in figure 1. As mentioned before this is the case for all QED processes. The curvy line combining the two vertexes represents the exchange of virtual photons bringing about the scattering process. Concerning energy and momentum conservation, energy and momentum can only be conserved at each vertex for initial and final physical particles if the intermediate photon is virtual. Jakob Jark Jørgensen 1

14 QED and the Higgs Mechanism.5 Transition Amplitudes a b Figur : a corresponds to.8. It is the scattering process contributing to Bhabha scattering. An initial electron is annihilated and a final electron is created at a point x 1. An initial positron is annihilated and a final positron is created at a point x. b corresponds to.9. It is the electron-positron pair annihilation-creation process contributing to Bhabha scattering. An initial electron-positron pair is annihilated at one point x, and a final electron-positron is created at another point x 1. Both processes contributes to Bhabha scattering. For both processes a virtual intermediate photon propagates between x 1 and x, and is interpreted as bringing about the interaction. As done for Bhabha scattering we can pick out from.5 the processes corresponding to electron-electron scattering and positron-positron scattering. In a similar way we can describe the processes rising from the other terms in S..5 Transition Amplitudes In this section we will se how to derive transition amplitudes. As mentioned before the initial state i and the final state f specifies a definite number of particles with definite momenta spin and polarization labels will be omitted in what follows. Ie. they are momentum eigenstates of the particles present and we shall work in momentum space when deriving transition amplitudes. We shall use the Fourier expansions of the fields.19 and.0 and the Fourier transform of the photon propagater into momentum space. The latter is given by: A µ x 1 A ν x }{{} = id µνx 1 x = 1 π 4 d 4 kid µν ke ikx 1 x..30 The functional form of the propagator id µν will not be discussed here. From the expansions.19,.0 and.18 we note that 15 : m m ψ + x e p = 0 upe ipx, V E ψ+ x e + p = 0 vpe ipx,.31 p V E p 1 A + µ x γk = 0 ɛ µ ke ikx..3 V ω k 15 Spin/polarization labels will be omitted in what follows to ease notation. Jakob Jark Jørgensen 13

15 QED and the Higgs Mechanism.5 Transition Amplitudes ψ x 0 = p m e p V E p ūpe ipx, ψ x 0 = p m e + p vpe ipx.33 V E p A µ x 0 = k 1 γk ɛ µ ke ikx..34 V ω k 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 term from the S-matrix expansion contributing to this process is: S p 1 ie d 4 x 1 N[ ψ /A + ψ + ] x1. As noted before energy and momentum cannot be conserved in this process for physical particles. We will now see that the transition amplitude is explicitly zero for physical particles. The transition amplitude is given by: f S p 1 i = ie e p d 4 x ψ 1 x 1 /A + x 1 ψ + x 1 e p; γk Using.31,.3 and.33 we obtain: m f S p 1 i = ie e p d 4 x ψ 1 x 1 γ µ A + µ x 1 γk upe ipx 1 V E p m m 1 = ie e p d 4 x 1 e p e ix 1 p k p ūp /ɛkup. V E p p V E p V ω k Noting that the set of states is orthonormal 16, and doing the x 1 integration we obtain: f S 1 e + γ e i = π 4 δ 4 p k p m m 1 M,.35 V E p V E p V ω k M = ūp /ɛk = p pup. M is called the Feynman amplitude and is treated with care when deriving cross sections. Notice how the delta function in.35 implies momentum conservation at the vertex. If all three particles are physical we cannot have p k p = 0 and we see that the transition amplitude is explicitly zero for physical particles. Ie. the transition does not represent a physical 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 of this below when we consider Bhabha scattering. To obtain physical processes we must go at least to second order in the S-matrix expansion. As an example we will consider Bhabha scattering: i = e p 1 ; e + p f = e p 1; e + p. Ie. the transition amplitude is to second order perturbation theory given by: f s i = e p 1; e + p S e + + e e + + e e p 1 ; e + p As mentioned before.11 is a unitary transformation and therefore preserves scalarproducts. Jakob Jark Jørgensen 14

16 QED and the Higgs Mechanism.5 Transition Amplitudes With the S-matrix term given by.8 and.9. We first write.8 and.9 into the same 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 a N [ ψ γ µ ψ + x 1 ψ+ γ ν ψ x ] N[c 1 c1dd ],.37 S b N [ ψ γ µ ψ x 1 ψ+ γ ν ψ + x ] N[c 1 d dc1]..38 Here we have labeled the initial final electron with momentum p 1 p 1 by 1 1, and analogously for the initial and final positron. We only need to consider the terms with the appropriate creation and annihilation operators since all other terms arising from the expansions of ψ and ψ will become zero for the transition amplitude.36. Arranging.37 and.38 in the same normal order gives: N[c 1 c1dd ] = 1 c 1 d c1d, N[c 1 d dc1] = 1c 1 d c1d. Ie. we see that there is a relative sign factor of 1 between the two contributions which we are now denoting as S a and S b. It is important to note that the spinor and gamma matrix structure of.37 and.38 remains intact. We are only interchanging the expansion coefficients, not spinors and gamma matrices. Using.31,.33 and inserting.30 we obtain the matrix element for S a : e p 1; e + p S a e p 1 ; e + p = e e p 1; e + p d 4 x 1 d 4 x {N [ ψ γ µ ψ + ψ+ x 1 γ ν ψ ] Aµ x x 1 A ν x }{{} e p 1 ; e + p = m m m m ie e p 1; e + p d 4 x 1 d 4 x e p ; e + p V E p,p p1 V E p1 V E p V E p e ip 1 x 1 e ip 1x 1 e ip x e ip xūp γ µ 1 up 1 π 4 vp γ ν vp d 4 kd µν ke ikx 1 x. Noting that the set of states is orthonormal and doing the x 1 and x integration we find: ie m 1 V E p p π 4 d 4 kπ 4 δ 4 p 1 k p 1 π 4 δ 4 p + k p }{{ } δ 4 p 1 +p p 1 p π 4 δ p 4 +k p ūp 1γ µ up 1 vp γ ν vp D µν k = π 4 δ 4 p 1 + p m p 1 p M a,.39 V E p M a = ie ūp 1γ µ up 1 vp γ ν vp D µν k = p p p.39 Is our final result. It gives the transition amplitude for the scattering process contributing to Bhabha scattering. Notice that the delta functions explicitly imply energymomentum conservation at both vertexes. This fixes the momentum of the intermediate Jakob Jark Jørgensen 15

17 QED and the Higgs Mechanism.5 Transition Amplitudes photon: k = p 1 p 1 = p p. As mentioned before energy-momentum cannot be conserved for three physical particles at a vertex, implying the photon being virtual. By the same procedure we obtain for S i S b f = ie p m 1 V E p π 4 b : d 4 kπ 4 δ 4 p 1 + p k π 4 δ 4 k p 1 p }{{} δ 4 p 1 +p p 1 p π 4 δ 4 k p 1 p ūp 1γ µ vp vp γ ν up 1 D µν k = π 4 δ 4 p 1 + p m p 1 p M b,.40 V E p M b = ie ūp 1γ µ vp vp γ ν up 1 D µν k = p 1 + p p.40 is the transition amplitude for the pair annihilation-creation process contributing to Bhabha scattering. As before the delta functions imply energy-momentum conservation at each vertex and the momentum of the intermediate virtual photon is fixed: k = p 1 + p = p 1 + p. As for.39,.40 explicitly gives vanishing transition amplitudes for processes for which the initial and final particles does not conserve energy and momentum..39 and.40 are the final results of this first part of the report. Starting out by requiring gauge invariance for the electron-positron field theory, we obtained the electromagnetic interaction. We treated the interaction as a perturbation of the free fields and developed a perturbation theory, the S-matrix expansion, which can be used to derive transition amplitudes for the elementary electromagnetic processes. We have seen how to extract terms from the S-matrix expansion which contribute to a given process, and finally we have seen how to derive transition amplitudes for real physical processes of QED. Using transition amplitudes one can derive cross sections and differential cross sections which are the physical observable quantities. This will not be considered in this report, instead we will now tend to spontaneous symmetry breaking and the Higgs mechanism. Jakob Jark Jørgensen 16

18 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 of the electron-positron field, thereby obtaining the electromagnetic interaction. In this second part of the report we shall be focusing on symmetries and on the consequences of breaking them. We shall develop the concept of spontaneous symmetry breaking, leading to the Higgs Mechanism. The Higgs mechanism will be used to obtain invariant masses for the W ± and Z 0 bosons of the standard electroweak symmetry, while preserving the gauge invariance of the theory. 3.1 Spontaneous Symmetry Breaking To explain the idea of spontaneous symmetry breaking we consider a classical field theory whose Lagrangian density is invariant under global U1 phase transformations. The system is specified by a complex scalar field H: Hx = RH + IH = φ 1 + iφ, φ 1, φ R. 3.1 Under global U1 phase transformation H transforms according to: H H = e iα H, α [0, π]. The transformation changes the phase angle of the field, corresponding to a rotation in the complex plane. The phase angle is changed by the same amount at each point in space, therefore the term global phase transformation. We consider a simple model with potential and kinetic energy density terms: V x = m H H + λ H H. K x = µ H µ H. 3. Giving a Lagrangian density which obviously has global U1 symmetry: L x = µ H µ H m H λ H m and λ are real parameters 17. Treating H and H as independent fields the fields conjugate to H and H are given by: πx = L Ḣ = 1 c H, And the hamiltonian density is obtained: π x = L H = 1 c Ḣ. H x = πḣ + π H L = 1 c H Ḣ µh µ H + V = 0 Hx 0 Hx + Hx Hx + V H The first parameter is called m to explicitly illustrate the fourdimensionality of the Lagrangian density. Jakob Jark Jørgensen 17

19 QED and the Higgs Mechanism 3.1 Spontaneous Symmetry Breaking a m > 0 b m < 0 Figur 3: Figure a and b illustrates the appearence of the potential energy density V φ 1, φ = m φ 1 + φ + λ φ 1 + φ for the cases m > 0 and m < 0. V is plottet on the vertical axis and φ 1 and φ on either of the horizontal axis. For m > 0 V has an absolute minumum for φ 1 = φ = 0, ie. Hx = 0 specifies a uniqe ground state. The ground state possesses the same global U1 symmetry as the Lagrangian density, therefore spontaneous symmetry breaking cannot occur. For m < 0 V possesses a whole circle of absoslute minima corresponding to φ 1 + φ = m λ, ie. Hx = m λ eiθ, 0 θ < π corresponds to minimum field energy. The ground state is not uniqe. Spontaneous symmetry breaking occurs if we choose one particular value og θ to represent the ground state, since such a ground state does not posses global U1 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 the first to terms in 3.4 are positive definite and vanish for constant Hx. Minimizing H x therefore corresponds to finding that constant Hx which minimizes V H. Ie. to find the ground state we only need to consider the potential term 3.. For the potential term we have: V H = V φ 1, φ = m φ 1 + φ + λ φ 1 + φ. 3.5 To very different situations occur depending on the sign of m, these situations are illustrated in figure 3. For m > 0 V φ 1, φ has an absolute minimum corresponding to φ 1 x = φ x = 0, ie. Hx = 0 determines an unique ground state. The ground state has global U1 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 upon 18 The 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 8-83]. Jakob Jark Jørgensen 18

20 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 m < 0 the situation is very different. Extrema are found by requiring: V = m φ 1 + 4λφ 1 φ φ 1 + φ V = 0 = m φ + 4λφ φ 1 φ 1 + φ = 0 m φ 1 + φ + 4λ φ1 + φ φ 1 + φ = 0 φ1 + φ m + 4λφ 1 + φ = 0. In this case we see that V φ 1, φ has a local maximum at φ 1 = φ = 0, ie. Hx = 0. Concerning the ground state we note that V φ 1, φ has absolute mimima for φ 1 + φ = m λ, ie. V H possesses a whole circle of absolute minima in the complex plane given by: m H 0 = λ eiθ, 0 θ < π. 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 state with less symmetry than the Lagrangian density, ie. spontaneous symmetry breaking can occur. Global U1 transformation which leaves L invariant, will rotate the vector representing H 0 around the origin. degenerate. If we choose one particular θ to represent the ground state, the ground state will not be invariant under global U1 transformation as the Lagrangian density is. Rotation in the complex plane will instead transform it into another state lying on the circle of absolute minima See figure 4. The ground state therefore has less symmetry than the Lagrangian density has. Such a theory is called a theory with spontaneous symmetry breaking. We notice that for m > 0, spontaneous symmetry breaking cannot occur since the uniqe ground state possesses the same symmetry as the Lagrangian density. We will now consider the consequences of spontaneous symmetry breaking. We first note Jakob Jark Jørgensen 19

21 QED and the Higgs Mechanism 3. The Local U1 Invariant Model that since L is invariant under global U1 transformation, it doesn t matter which θ we choose to represent the ground state. We choose θ = 0 such that: m H 0 = λ 1 v. 3.7 We now introduce two real fields σx and ηx: Hx = 1 v + σx + iηx, 3.8 Thereby letting σx and ηx represent deviations from the ground state configuration Hx = H 0. The Lagrangian density 3.3 then becomes Derivation is found in section 5. of the appendix: L x = 1 µσx µ σx 1 λv σx µηx µ ηx 3.10 λvσx σx + ηx λ 4 σx + ηx 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 σx and ηx are two real Klein Gordon fields. Upon second quantization these fields leads to neutral spin 0 particles. The quanta of the σ-boson field have mass v λ = m 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 a pertubative theory with massive scalar bosons and massless Goldstone bosons. In the next sections the mechanism of spontaneous symmetry breaking will be extended to create massive vector bosons in a gauge invariant theory. This is called the Higgs mechanism, and we shall call Hx the Higgs field and the scalar bosons associated with the σx field will be called Higgs bosons. 3. The Local U1 Invariant Model We have so far been considering a model whose Lagrangian density was constructed to have global U1 symmetry. We shall now extend our model by requiring invariance under local U1 transformation, in this way we obtain a gauge theory. We will apply the mechanism of spontaneous symmetry breaking to the theory and we will see that no goldstone bosons are obtained. Instead the degree of freedom associated with the ηx field will somehow be transferred to the gauge field, which becomes massive in the process. The procedure of gauging the theory resembles the one used for the electron positron Jakob Jark Jørgensen 0

22 QED and the Higgs Mechanism 3. The Local U1 Invariant Model field. However, it will be done here before we tend to the mechanism of spontaneous symmetry breaking. As before we will se that requiring invariance under local U1 transformations leads one to introduce a gauge field coupled to the matter field. For U1 transformations the fields transform according to: H H = e iαx H, αx R. 3.1 α is now dependent on the space-time coordinate, therefore the term local phase transformation. The potential term 3. of the Lagrangian density is obviously invariant with respect to such transformations. Considering the kinetic term we note that: µ H = µ e iαx H = e iα µ + i µ α H. Implying that the kinetic term is not invariant under local U1 transformations. Gauge invariance can be restored by introducing a gauge field A µ coupled to the matter field. This coupling is made as before by replacing the ordinary differential by the covariant differential D µ : D µ = µ + iga µ x A µ is a real vector field and g is the coupling constant. The covariant differential of H transform according to: D µ H D µ H = µ + iga µ H = µ + iga µ e iα H = e iα µ + ig A µ + µα H. g 3.14 The kinetic term which is now D µ H D µ H would be invariant if D µ H transforms like: D µ H D µ H = e iα D µ H. From 3.14 we note that this is the case, if the gauge field transforms according to: A µ = A µ + µα g ie. A µ x A µx = A µ x µαx g To summarize our approach: Demanding invariance of the theory under local U1 transformations we are led to couple to the matter field a gauge field A µ x which transform according to The coupling is made by replacing the ordinary differential in the Lagrangian density 3.3 by the covariant differential. Doing so we obtain: L = D µ H D µ H V = µ iga µ H µ + iga µ H V = µ H µ H V + ig µ H A µ H iga µ H µ H + g A µ A µ HH = µ H µ H V + iga µ H µ H H µ H + g A µ A µ HH }{{}}{{} free matter field lagrangian interaction terms Jakob Jark Jørgensen 1

23 QED and the Higgs Mechanism 3. The Local U1 Invariant Model The first underbraced terms are the free matter field lagrangian. The remaining terms are interpreted as terms describing interactions between the free matter field and the gauge field. These can in principle be treated by the perturbation theory we developed in section.. 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, which does not break the symmetry. As for the electron-positron field, we add the Lagrangian density of the free electromagnetic field.8. The the full Lagrangian density then reads: L = µ H µ H V 1 }{{} 4 F µνf µν + iea µ H µ H H µ H + e A µ A µ HH }{{}}{{} free matter field free gauge field interaction terms 3..1 Applying the Higgs Mechanism We now consider the symmetry of the ground state for the cases m > 0 and m < 0. For the ground state we must have A µ x = 0 in both cases. The analysis therefore parallels that for the global U1 symmetric model. For m > 0 the ground state corresponds to A µ x = 0 and Hx = 0. The ground state is uniqe and possesses the same symmetry as the Lagrangian density, thus spontaneous symmetry breaking does not occur. For m < 0 the ground state will be degenerate as before leading to spontaneous symmetry breaking. 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 σx and ηx represent deviations from the ground state according to 3.8. In terms of these new coordinates the Lagrangian density becomes derivation is found in section 5.3 of the appendix: L x = 1 µσx µ σx 1 λv σ µηx µ ηx 3.19 F µν F µν + 1 gv A µ A µ gva µ µ η + interaction terms. 3.1 As before we interpret 3.18 as a Lagrangian density for the free real scalar σ field, which is seen to obey the Klein Gordon equation of motion. Ie. the quanta of the field are spin 0 bosons with mass v λ = m looks as the Lagrangian density for the Goldstone boson field. Concerning the term 3.0 we see something interesting. This term looks like the free gauge field Lagrangian density plus an invariant mass term. It appers that the gauge bosons somehow have aquired a mass in the process! However we need to be careful with these interpretations. The first term in 3.1 cannot be regarded as an interaction term, since it is quadratic in the fields as the free field terms are. Therefore η and A µ cannot be regarded as independent coordinates changing η will affect a term containing A µ and vice verca. Counting degrees of freedom we come to the conclusion that something needs to be fixed. Jakob Jark Jørgensen

24 QED and the Higgs Mechanism3.3 The Physical Local SU U1 Invariant Model For 3.17 we count 4 degrees of freedom. One for H, one for H and to for A µ 19. For 3.18, 3.19, 3.0, 3.1 we count 5 degrees of freedom. One for σ one for η and three for A µ, 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 the transformed Lagrangian density must contain an unphysical field. This unphysical field is the η field. As discussed in [1, page 85-86] it so happens that a local U1 gauge transformation of the form 3.1, which transform H into H = 1 v + σ, always can be found. Choosing this so called unitary gauge, absorbs the extra degree of freedom. This corresponds to remove the η field from the transformed Lagrangian density, ie.: L x = 1 µσx µ σx 1 λv σ F µν F µν + 1 gv A µ A µ + interaction terms.. 3. 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 free field Lagrangian density of a real massive vector field A µ, which quanta has the invariant mass gv. We see that the Higgs mechanism does not generate Goldstone bosons. Instead the degree of freedom associated with the ηx field has been absorbed, giving the gauge field an extra degree of freedom and an invariant mass term. Referring to the discussion on.7, adding an invariant mass for the vector field A µ would destroy the gauge invariance of However by applying the Higgs mechanism, the gauge bosons acquire mass while the gauge invariance of the theory is preserved. 3.3 The Physical Local SU U1 Invariant Model We will now extend the model further by requiring invariance under SU U1 gauge transformations. In this way we will obtain SU U1 gauge bosons which will be the W ± and Z 0 bosons of weak interactions, and photons. When formulating weak interactions as a gauge theory, one has to assume that all leptons and the W ± and Z 0 gauge bosons are massless. Adding mass terms to the Lagrangian density will simply destroy the gauge invariance of the theory, in analogy with what we have seen for the theory However, the Higgs mechanism can be used to introduce nonzero mass terms for the gauge bosons and leptons, while preserving the symmetry. In this section we will see how the W ± and Z 0 bosons become massive when coupled to the Higgs field, and the Higgs mechanism is applied. We start out by constructing a global SU invariant Higgs field theory, and then require local SU U1 gauge invariance. 19 From the study of the free photon field, one knows that of the four degrees of freedom A µ appers to have one is removed by the subsidiary condition: [a 3k a 0k] Ψ = 0 Where a 0k and a 3k are lowering operators of scalar and longitudinal photons respectively and Ψ is the state. The other can be removed by gauge transformation leaving one with two transverse photons. If A µ is massive, only the scalar degree of freedom can be removed. As discussed in fx. [1, chapter 5]. Jakob Jark Jørgensen 3