Standard Model of Particle Physics

Transcription

1 Standard Model of Particle Physics SU3 C SU L U1 Y by Dennis Dunn Version date: Thursday, 15 May :0

2 Copyright c 001 Dennis Dunn. USEFUL REFERENCES Lagrangian field theory & symmetry, D Dunn Lecture notes for module PH4A0 An introduction to the Standard Model of particle physics W N Cottingham and D A Greenwood 1998 Cambridge University Press ISBN Particle Physics B R Martin and G Shaw 001 Wiley ISBN Quantum Field Theory L H Ryder 1985 Cambridge University Press ISBN X Quantum Field Theory F Mandl and G Shaw 1996 Wiley ISBN

3 Contents 1 INTRODUCTION Aims Summary Range of interactions Strength of interactions List of Symbols Dimensions and Dimensional Analysis BUILDING A FIELD THEORY 9.1 Left-handed spinor fields Positive k t Negative k t Orthogonality of eigenvectors Normalization Helicity General solution Energy Right-handed spinor fields FIELDS AND PARTICLES Creation and annihilation operators properties Anti-commutation relations Number operators Zero particle or Vacuum state Particle states Particle energy Normal ordering CONSTRUCTING THE STANDARD MODEL I BASIC INGREDIENTS SU L U1 Y SU L U1 Y SYMMETRIES AND CONSTRUCTION OF LAGRANGIAN

4 4 CONTENTS 5 CONSTRUCTING THE STANDARD MODEL II LOCAL GAUGE SYMMETRIES AND INTERACTIONS Local SUN Gauge Invariance Dyadics and the gauge field Lagrangians Summary CONSTRUCTING THE STANDARD MODEL II MASS HIGGS FIELDS AND ELECTROMAGNETISM The electromagnetic field PREDICTIONS OF THE STANDARD MODEL W-mass Z-Interactions Experimental Data W-Lepton Interactions Summary of assumptions and prescribed parameters of the Standard Model Eigenvalue structure APPENDIX 1: SPACE-TIME SYMMETRIES NOETHER S THEOREM Infinitesimal Symmetry Transformations Current Densities Time translation invariance & energy conservation Space-Time Translations Eigenvalues and eigenfunctions Rotations Generalizations of the rotation generators Spinor representation Lorentz Transformations Basic Fields Pauli Spinor Fields Dirac spinor Index 56

5 Chapter 1 INTRODUCTION 1.1 Aims To provide an introduction to the basic aspects of the Standard Model of Particle Physics SU3 C SU L U1 Y. 1. Summary Particle physics is well described within the Standard Model SM. SM provides an elegant theoretical framework and has successfully passed many precise experimental tests. It does however leave unanswered many questions and it is not nearly as predictive as we would like. I aim to study elementary particles. By elementary particles I those which we believe are not composed of other particles. These are of two types of particles, the basic building blocks of matter known as matter particles and the particles which mediate the interactions between these matter particles. The matter particles are fermions and are described by spinor fields with spin 1. These are classified into two groups: leptons and quarks. The sets of leptons and quarks occur in 3 generations. The generations have identical properties except for the particle masses. The first generation consists of: electron charge -1; electron neutrino charge 0; up quark charge 3 ; down quark charge 1 3 ; Each of the particles has a corresponding anti-particle with the opposite charge. Each quark particle occurs in three colours variously called red, white and blue; red, green and blue or whatever combination takes your fancy! The second generation consists of: muon charge -1; muon neutrino charge 0;

6 6 INTRODUCTION charm quark charge 3 ; strange quark charge 1 3 ; and the third generation tauon charge -1; tauon neutrino charge 0; top quark charge 3 ; bottom quark charge 1 3 ; I shall denote the anti-particles by a bar over the symbol: e = e +, µ = µ +, τ = τ + ; ν e, ν µ, ν τ ; u, d, c, s, t, b. Quarks are not seen in nature and therefore the elementary quarks must be combined into the experimentally observed matter particles, the hadrons. These composite particles are classified into baryons and mesons. The baryons are fermions made of three quarks, qqq. For example the proton, p is uud, and the neutron, n is ddu. The mesons are bosons made of one quark and one antiquark as for instance the pions, π + is ud and π is du. Ignoring the gravitational interaction because no-one knows how to incorporate this!, all the relevant interactions in Particle Physics are mediated by the exchange of a Bose particle which is described by a vector field. That is a spinor field with spin s = 1. The photon, γ, is the exchanged particle in the electromagnetic interaction,; the eight gluons g α ; α = 1,..., 8 mediate the strong interaction between quarks; and the three particles W ± and Z are the corresponding bosons mediating the weak interactions. SM is a quantum field theory that is based on the gauge symmetry group SU3 C SU L U1 Y. The significance of the subscripts will be explained later. This gauge group includes the symmetry group of the strong interactions, SU3 C, and the symmetry group of the electro-weak interactions, SU L U1 Y. The group symmetry of the electromagnetic interactions, U1 EM, appears in SM as a subgroup of SU L U1 Y and so the weak and electromagnetic interactions are unified. SUN originates from the group of N N Unitary matrices. Such matrices satisfy the relations U U = I = U U where I is the N N unit matrix. From this it follows that the magnitude of the determinant of U is 1. The S in the SUN indicates that we have chosen the particular case in which the determinant and not just its magnitude is 1. If we allowed the determinant to have an arbitrary phase the group would be UN. The gauge fields of SM arise from local symmetries. The eight gluons arise from the local SU3 C symmetry. In general a local SUN symmetry has N 1 symmetry generators and each generator corresponds to a separate gauge field. The group UN would have N generators. Hence in SU3 there are 3 1 = 8 gauge fields. The γ, W ± and Z particles are the four gauge bosons of SU L U1 Y : SU L has 1 = 3 generators and U1 Y has 1. The main physical properties of these gauge particles are:

7 1. Summary 7 The 8 gluons are massless and electrically neutral. They interact with the quarks and also with themselves. However they do not interact with the leptons. The weak fields, W ± and Z represent massive particles which interact with quarks and leptons; and are also self-interacting. The Z is electrically neutral. The W ± are charged with Q = ±1 and hence also interact with the photon field. The photon is massless and has zero charge and hence does not interact with itself. It interacts with all charged particles Range of interactions The infinite range of the electromagnetic interaction corresponds to the mediation by the massless photon. By the Uncertainty Principle, a particle of mass M can exist as part of an intermediate state for a time Mc, and in this time can travel a distance less than. The short range of the weak interactions Mc of about m arises from the exchange of a massive gauge particle with a mass of the order of M V 100GeV. Finally, the strong interactions whose range is, in principle, infinite, as we should expect because of the exchange of a massless gluon. However the nature of the effective potential produces some surprising results. The potential corresponding to the exchange of photons for two particles with equal and opposite charge Q is V r = Q 4πɛ 0 r The effective potential is an approximate concept and is, in some sense, the time-average of the actual interaction. This potential has a maximum value of zero and so a state with positive energy corresponds to free particles. The corresponding effective potential due to the exchange of gluons is V r = A r + B + Cr The presence of the term proportional to r implies that V r has no maximum value. This means there are no free states and so the particles are always bound or confined. A harmonic oscillator potential has similar properties. The above effective potential was determined by a computational method known as Lattice Chromodynamics. 1.. Strength of interactions The strength of the electromagnetic interaction is governed by the size of the electromagnetic coupling constant e or equivalently by α e = e 4πɛ 0 c = The weak and electromagnetic interactions are related and have essentially the same strength, α w α e and the strong interaction has α s List of Symbols Lagrangian density L

8 8 INTRODUCTION Energy H Space-time translation generators P Rotation generators J Lorentz transformation generators K Lorentz-rotation generators à = 1 J + i K B = 1 J i K Current density J 1..4 Dimensions and Dimensional Analysis The basic independent dimensional unit is length [L. Dependent dimensional units which I shall use sparingly are: Time [T. This is simply proportional to [L: [T = 1 c [L This in the same category as the relation between meters and feet. Mass [M. This is simply proportional to [L: [M = c [L Again this in the same category as the relation between meters and feet. The dimensions of some significant quantities are: Action A [L 0 Lagrangian Density L [L 4 Spinor field [L 3 Vector electromagnetic field à [L 1 Space-time translation generator P [L 1 Rotation generator J [L 0 Current density [L 3 Charge [L 0

9 Chapter BUILDING A FIELD THEORY Before I get to particles I am going to setup a field theory. The idea that a field theory could and should describe particles originated in quantum mechanics. The building blocks of this field theory are chosen to be the simple representations of the Lorentz- Rotation group. That is, I want the basic fields to represent the properties of space-time. The representatives of the Lorentz-Rotation group the details are given in Appendix 1 can be specified by a pair of indices a, b where both a and b can take the values 0, 1, 1, 3,.... I shall only make use of the simpler representations: 0, 0 0, 1 1, 0 1, 1 Scalar field single component Left-handed spinor field components Right-handed spinor field components Vector field 4 components The Scalar field is invariant under spatial rotations and Lorentz transformations. Such a field is thought to represent a Higgs particle. I represent an infinitesimal rotation by δθ and an infinitesimal Lorentz transformation by δφ. The magnitude of δθ is the angle of rotation and its direction is the axis of rotation. The magnitude of δφ is the speed of the Lorentz transformation divided by c and its direction is the direction of the transformation. The total change in a field is given by δf x, t = iδθ Jf + iδφ Kf.1 where J and K are the appropriate operators for the type of field f. J is the generator of rotations and K is the generator of Lorentz transformations. For a scalar field the operators J and K are both zero. For a right-handed spinor field the operators are J = 1 σ i K = σ.

10 10 BUILDING A FIELD THEORY and for a left-handed spinor they are J = 1 σ i K = σ.3 In these equations σ is the vector Pauli spinor matrix. That is σ is a vector in which each component is a matrix. These matrices are: σ z = [ Left-handed spinor fields [ 0 1 σ x = 1 0 [ 0 i σ y = i 0 I denote a left-handed spinor field by Ψ L x. This is a two-component field and is usually written as a column vector: Ψ L ψ1 =.5 ψ.4 The corresponding adjoint field is Ψ L x and this is usually written as row vector Ψ L = ψ1 ψ The Lagrangian for the left-handed spinor field is L L = 1 Ψ L Pt Ψ L + Ψ L P σ Ψ L + Pt Ψ L Ψ L + PΨ L σ Ψ L.6.7 and the equation satisfied by the spinor is i d ΨL d x t = i σ Ψ L.8 This Lagrangian is, of course, invariant under any space-time translation and there are corresponding current densities. If I consider the time-translation symmetry then Noether s theorem see the appendix equations 9.13 and 9.14 gives rise to the following space-time vector which is the energy current density. i Ψ L σ Ψ L Ψ L σψ L i Ψ L d Ψ L σ x ẽt ẽx d x t i Ψ L d Ψ L σ y d ΨL σ y Ψ L i Ψ L d Ψ L σ z ẽy d x t d x t ẽz d x t d ΨL d x t σ x Ψ L d ΨL σ z Ψ L d x t.9 Note: The overall sign can be changed without affecting the vector properties. In practice the sign is chosen so that the positive frequency eigenvectors give a positive energy. If I use equation.8 this energy current density vector can also be written as i L d ΨL Ψ d ΨL ẽt dx t dx t Ψ L d Ψ L σ y i ẽy d x t Ψ L d ΨL σ y Ψ L d x t i Ψ L d Ψ L σ x d ΨL ẽx d x t d x t i Ψ L d Ψ L σ z ẽz d x t σ x Ψ L d ΨL σ z Ψ L d x t.10

11 .1 Left-handed spinor fields 11 The integral of the time-like component of this current density vector is particularly important: This is the energy. H L = i = i dv Ψ L σ Ψ L Ψ L σψ L dv L d ΨL Ψ d.11 ΨL d x t d xt ΨL I can easily generate a solution of the equation.8 by considering a Fourier representation: k Ψ L x = Ψ L k exp i k x = Ψ L exp i k x k kt x t = Ψ L exp i k x ωt.1 I will simplify the problem by assuming that the universe is finite with volume V. In this case, the spacepart of k will be quantized. If space is a rectangular block with sides L x, L y, L z then the allowed values are k x = πn x L x, k y = πn y, k z = πn z L y L z where n x etc are integers. There is no restriction yet on the time-component. Inserting this representation into equation.8 generates an eigenvalue equation. The eigenvalues k t and the eigenvectors are those of the matrix [ k z k x + ik y k x + ik y k z ψ L = k t ψ L.13 There are two solutions: k t = ± k x + k y + k z = ± k. In both cases the wave travels at the speed of light Positive k t The eigenvector in this case can be written as ψ L + k = [ 1 kx + ik y k k + k z k + k z k z 0 ψ L + k = [ 1 k kz k k x ik y k k z k z < 0.15

12 1 BUILDING A FIELD THEORY.1. Negative k t The eigenvector in the case of negative k t is [ ψ L k 1 kx ik = y k k k z k k z k z 0 ψ L k = [ 1 k + kz k k x + ik y k + k z k z > Orthogonality of eigenvectors The eigenvectors satisfy the orthogonality relations ψ L k ψ L k + = 0 = ψ L + and they are normalized as ψ L + k ψ L k k ψ L k + = 1 = ψ L k ψ L k Normalization I need to justify the normalization of the eigenvectors used above. There is no clear cut choice for the normalization because no simple quadratic function of Ψ L and Ψ L which is a Lorentz invariant. However I can make use of the following space-time vector see equation??: i Ψ L σ Ψ L Ψ L σψ L i Ψ L Ψ L σ x i ẽt ẽx x t i Ψ L Ψ L σ y ΨL σ y Ψ L i Ψ L Ψ L σ z ẽy x t x t ẽz x t Ψ L x t σ x Ψ L ΨL σ z Ψ L x t which is the energy current density. In the case of the field.15 with positive frequency this space-time vector becomes: ẽ t k ψ L k + ψ L k + ẽ x k ψ L k + σ x ψ L k + ẽ y k ψ L k + σ y ψ L k + ẽ z k ψ L k + σ z ψ L + k.19 Hence the term k ψ L + k ψ L k + transforms like the time-component of a space-time vector. However k is already the time-component of a vector because it is equal to k t. Hence the remaining factor ψ L k + ψ L k + must be a scalar. Therefore I can choose this to be 1.

13 .1 Left-handed spinor fields 13 ψ L + k ψ L k + = 1.0 This is the choice embodied in equation.15. A similar choice will be made for the normalization of the negative frequency eigenvector..1.5 Helicity I define an operator called helicity ĥ = 1 σ k k.1 This gives the component of the spin σ in the direction of propagation of the wave k. Applying this operator to a positive frequency eigenvector gives a value 1. That is the helicity is 1. Hence a left-handed spinor field with positive frequency has a spin which is opposite to the direction of propagation. If I apply it to a negative frequency eigenvector I get a a value +1. That is the helicity is +1. Hence a left-handed spinor field with negative frequency has a spin which is parallel to the direction of propagation..1.6 General solution The general solution for the left-handed spinor field can be constructed by taking combinations of all eigenvectors. For convenience, I shall assume that the 3D space is large but finite with volume V. This has the advantage that I can use a Fourier series rather than a Fourier integral in the infinite volume case. The general solution can be written as a Fourier series Ψ L x = 1 e i k x k x t ψ L k + c L k + e i k x k x t ψ L k c L k V The complex constants c L k k and c L k are the arbitrary Fourier coefficients.. Notice that I use to denote complex conjugation. The overline, as in c, has at the moment no special significance here..1.7 Energy The conserved energy of this general solution can be evaluated by inserting the general solution into the expression for the energy.11. The resultant energy is H L = k c L k c L k c L k c L k k.3 This has a simple interpretation: The first term in the sum represents the energy of the positive frequency components of the wave and the second term that of the negative frequency components. This presents a problem. The energy has no lower bound: it can be arbitrarily large and negative.

14 14 BUILDING A FIELD THEORY. Right-handed spinor fields The right-handed spinor field can be treated in exactly the same way. The Lagrangian and equation of motion are L R = 1 Ψ R Pt Ψ R Ψ R P σ Ψ R + Pt Ψ R Ψ R PΨ R σ Ψ R.4 and the equation satisfied by the spinor is i d ΨR d x t = i σ Ψ R.5 Again there is a time-translational symmetry and this leads to a space-time vector which is the energy current density i Ψ R σ Ψ R Ψ R σψ R i Ψ R Ψ R σ x ẽt ẽx x t i Ψ R Ψ R σ y ΨR σ y Ψ R i Ψ R Ψ R σ z ẽy x t x t ẽz x t ΨR x t σ x Ψ R ΨR σ z Ψ R x t.6 Note: The overall sign can be changed without affecting the vector properties. In practice the sign is chosen so that the positive frequency eigenvectors give a positive energy. The energy associated with this field is H R = i and the general solution for the right-handed spinor field is k dv Ψ R σ Ψ R Ψ R σψ R.7 Ψ R x = 1 e i k x k x t ψ R k + c R k + e i k x k x t ψ R k c R k V.8 The eigenvectors are simply related to those of the left-handed spinors: ψ R k + = ψ L + k = ψ L k ψ R k = ψ L.9 k = ψ L k + The positive- and negative-frequency solutions have opposite helicities to the left-handed spinor. That is, a right-handed spinor field with positive frequency has a spin which is parallel to the direction of propagation. The energy of the general solution is H R = k c R k c R k c R k c R k.30 k Again this can be arbitrarily large and negative.

15 Chapter 3 FIELDS AND PARTICLES There are two significant problems relating to what we have done so far. The energy associated with a left-handed or right-handed spinor field can be arbitrarily large and negative. There is as yet no obvious relation between the field and particles. In this chapter I replace the fields of the previous chapters by field operators. These field operators describe the creation and annihilation of particles and the properties of these operators solve the negative energy problem. Consider the general solution of the left spinor field: Ψ L x = 1 e ik x k xt ψ L + k c L k + e ik x k xt ψ L k c L k V I replace this by a field operator k Ψ L x = 1 e ik x k xt ψ L + k ĉ L k + e ik x k xt ψ L k ĉ L k V k and I replace the adjoint field by Ψ L x = 1 e ik x k xt ψ L + k ĉ L k + eik x k xt ψ L k ĉ L k V k 3.3 The essential feature here is that the Fourier coefficients have been replaced by operators c L k, c L k, c L k, and c L k ĉ L k, ĉ L k, ĉ L k, and ĉ L k. The operators ĉ L k and ĉ L k describe the creation of particles; and the operators ĉ L k and ĉ L k describe the annihilation of particles.

16 16 FIELDS AND PARTICLES So the field operator Ψ L x is a mixture of operators which create left-handed spinor particles and which destroy left-handed spinor anti-particles. Similarly, the field operator Ψ L x is a mixture of operators which create left-handed spinor antiparticles and which destroy left-handed spinor particles. In principle I should start the whole process again from the beginning with the new operator-valued Lagrangians. However providing I take care over the order in which the operators occur everything proceeds as before. 3.1 Creation and annihilation operators properties The operators ĉ L k, and ĉ L k describe the creation of a left-handed spinor particle and of a lefthanded spinor anti-particle. The operators ĉ L k, and ĉ L k describe the annihilation of a left-handed spinor particle and of a left-handed spinor anti-particle Anti-commutation relations The operators for the particles are defined to have the following anti-commutation relations: Similarly for the anti-particle operators ĉ L k ĉ L k + ĉ L k ĉ L k = 0 ĉ L k ĉ L k + ĉ L k ĉ L k = 0 ĉ L k ĉ L k + ĉ L k ĉ L k = δ k,k 3.4 ĉ L k ĉ L k + ĉ L k ĉ L k = 0 ĉ L k ĉ L k + ĉ L k ĉ L k = ĉ L k ĉ L k + ĉ L k ĉ L k = δ k,k The form of these relations are chosen to produce fermions. That is, particles that obey the Pauli exclusion principle. 3. Number operators I want to show that the operator n L k = ĉ L k ĉ L k 3.6 has the properties of a number operator for the left-handed spinor with wave-vector k. What are the required properties of such an operator? In general a number operator should have eigenvalues 0, 1,,.... However, experimentally, a spin 1 particle should obey the Pauli exclusion principle: there can be no more than one particle in a given state. Hence I require the number operator to have eigenvalues 0 and 1. Consider the square of the operator written out in detail n L k = ĉ L k ĉ L k ĉ L k ĉ L k

17 3.3 Zero particle or Vacuum state 17 If I invert the middle two operators on the rhs, making use of the commutation relations 3.4 then this operator can be written as n L k = ĉ L k ĉ L k ĉ L k ĉ L k ĉ L k ĉ L k However, again from the commutation relations 3.4, the second term on the rhs is identically zero. Hence I have proved the relation n L k = n L k 3.7 You should be able to show that this implies that the only eigenvalues are, indeed, 0 and 1. The total number operator can be simply defined as n L = k n L k = k ĉ L k ĉ L k 3.8 The eigenvalues of this operator give the total number of left-handed spinor particles. I can similarly define number operators for left-handed spinor anti-particles; and for right-handed spinor particles and anti-particles: n L n R n R 3.3 Zero particle or Vacuum state I define a state Φ 0 in which there are no particles. This is called the zero-particle or vacuum state. By definition this is an eigenvector of every number operator with eigenvalue zero: n L k Φ 0 = 0 = n L k Φ 0 n R k Φ 0 = 0 = n R k Φ It follows from this definition I shall leave this as an exercise that Φ 0 satisfies the slightly simpler relations ĉ L k Φ 0 = 0 = ĉ L k Φ 0 ĉ R k Φ 0 = 0 = ĉ R k Φ That is, any annihilation operator applied to Φ 0 gives zero. This makes some sort of sense: You cannot annihilate what is not there! 3.4 Particle states I can construct states which do contain particles by operating on Φ 0 with the particle creation operators. For example ĉ L k Φ 0 ĉ L k Φ 0 describes a state with one left-spinor particle with wave-vector k and another state with one left-spinor anti-particle with wave-vector k. A more complicated example is Φ a = ĉ L k ĉ L k Φ 0

18 18 FIELDS AND PARTICLES This describes a state with two left-spinor particles, one with wave-vector k and one with with wavevector k. If these operators are applied in the opposite order Φ b = ĉ L k ĉ L k Φ 0 then we have the well-known anti-symmetry property of Fermi states: Φ a = Φ b. That is, interchanging two particles introduces a minus sign. If the two operators have the same wave-vector: Φ c = ĉ L k ĉ L k Φ 0 then the anti-commutation rules 3.4 ensures that Φ c = 0. So it is is impossible to have two particles of the same type with the same wave-vector. The anti-commutation rules therefore imply the Fermi exclusion principle. 3.5 Particle energy The energy for a left-handed spinor operator field can be determined exactly as for the simple spinor field providing we take care of the operator order. Ĥ L = k ĉ L k ĉ L k ĉ L k ĉ L k 3.11 k Using the anti-commutator relations the second term can be replaced by Ĥ L = k ĉ L k ĉ L k + ĉ L k ĉ L k 1 k 3.1 The final term is simply a constant and can be removed. This leaves an expression for the energy operator which has a simple and consistent interpretation: The first term is the number operator for the left-handed spinor particle multiplied by its positive energy; the second term is the number operator for the left-handed spinor anti-particle multiplied by its positive energy. The total energy eigenvalues are then clearly positive. 3.6 Normal ordering The satisfactory result for the energy operator is achieved by subtracting a simple but infinite constant. Adding or subtracting infinity is definitely something to be avoided. This can be done by replacing the original Lagrangian by one that is normally ordered. Normal ordering is the process of placing creation operators to the left and annihilation operators to the right whilst taking into account the anticommutation properties. I shall denote normal ordering by :... :. Simple examples are: : ĉ L k ĉ L k : = ĉ L k ĉ L k : ĉ L k ĉ L k : = ĉ L k ĉ L k This ensures a positive energy operator without the need for subtracting an infinite constant.

19 Chapter 4 CONSTRUCTING THE STANDARD MODEL I BASIC INGREDIENTS In this chapter I introduce the basic ingredients of the Standard Model and the symmetries that are the foundations of the theory. The basic ingredients are: Spinors These occur in three sets called generations with each generation containing 7 spinor fields ν el u L u R e L d L d R e R ν µl s L s R µ L c L c R µ R ν τl t L t R τ L b L b R τ R The subscripts L and R refer to left-handed and right-handed spinors. Clearly there is a built-in left-right asymmetry. Scalar fields Higgs Fields There are two complex scalar that is Lorentz-rotation invariant fields H + and H 0. The significance of the labels will become clear later. 4.1 SU L U1 Y In addition to Lorentz-rotation symmetry, the Lagrangian for the SM is required to be invariant under the gauge transformation groups SU L, left-isospin symmetry and U1 Y, hypercharge symmety.

20 0 CONSTRUCTING THE STANDARD MODEL I BASIC INGREDIENTS SU L This Lie symmetry group has three generators T 1, T and T 3. These are formally identical to the rotation generators Ĵx, Ĵy and Ĵz see Lagrangian Methods section 4.3 and satisfy the same commutation rules : T 1 T T T1 = i T 3 T T3 T 3 T = i T T 3 T1 T 1 T3 = i T Analysis is, of course, identical to the rotation case. We can use T + = T 1 + i T as a raising operator and T = T 1 i T as a lowering operator to construct a complete specification of the representations. These can be specified as common eigenvectors of T = T 1 + T + T 3 and of T 3. The eigenvalues have the following properties: T tt + 1 t = 0, 1, 1, 3,... T 3 m m = t, 1 t, t,..., t, t 1, t 4. The basic ingredients are assumed to belong to representations of SU L corresponding to either t = 0 or t = 1 First the t = 0 representations: These are the right-handed spinors. where φ R is any of the right-handed spinors. t = 0 T φ R = 0 T 3 φ R = 0 T + φ R = 0 T φ R = Next the t = 1 representations. t = 1 T φ L = 3 4 φ L T 3 ν el = 1 ν el T 3 u L = 1 u L T 3 H + = 1 H+ T 3 e L = 1 e L T 3 d L = 1 d L T 3 H 0 = 1 H0 T + ν el = 0 T+ u L = 0 T+ H + = 0 T + e L = ν el T+ d L = u L T+ H 0 = H T ν el = e L T u L = d L T H + = H 0 T e L = 0 T d L = 0 T H 0 = U1 Y This Lie symmetry group is simple and only has one generator Ŷ the hypercharge generator. This has the following properties:

21 4.1 SU L U1 Y 1 Ŷ ν el = 1 ν el Ŷ e L = 1 e L Ŷ e R = e R Ŷ u L = 1 6 u L Ŷ d L = 1 6 d L Ŷ u R = 3 u R Ŷ d R = 1 3 d R Ŷ H + = 1 H+ Ŷ H 0 = 1 H0 4.5 These rather strange properties are chosen to fit known experimental conservation laws left-isospin & hypercharge. I have only included one of the three generations in these relations: The other two give identical results SYMMETRIES AND CONSTRUCTION OF LAGRANGIAN The next step is to write down the complete transformation properties of each of the basic fields under Lorentz-rotation symmetry; SU L left-isospin symmetry and under U1 Y hypercharge symmetry. A general finite transformation under SU L U1 Y can be written as exp iβŷ exp i α 1 T1 + α T + α 3 T3 4.6 However it is simpler to consider only infinitesimal transformations. I denote the infinitesimal rotation by the vector θ; the Lorentz transformation by vector φ ; the SU rotation by vector α; and the U1 transformation by scalar β. The changes in the fields under these transformations are: ν el = i [ θ + i φ σν el + α 3 β ν el + α 1 i α e L e L = i [ θ + i φ σe L α 3 + β e L + α 1 + i α ν el e R = i [ θ i φ σ β e R u L = i 1 [ θ + i φ σul + α β u L + α 1 i α d L u R = i [ θ i φ σ β u R d L = i 1 [ θ + i φ σdl + α β d L + α 1 + i α u L d R = i [ θ i φ σ 3 β d R 4.7 H + = i [ α3 + β H + + α 1 i α H 0 H 0 = i [ α3 + β H 0 + α 1 + i α H + Then, in constructing the Lagrangian, I choose only the simplest completely invariant combinations of fields. The resulting Lagrangian is

22 CONSTRUCTING THE STANDARD MODEL I BASIC INGREDIENTS L = 1 ν P el t ν el + ν P el σ ν el + Pt ν el νel + PνeL σ νel + 1 e P L t e L + e P L σ e L + Pt e L el + PeL σ el + 1 e P R t e R e P R σ e R + Pt e R er PeR σ er + 1 u P L t u L + u P L σ u L + Pt u L ul + PuL σ ul + 1 u P R t u R u P R σ u R + Pt u R ur PuR σ ur + 1 d P L t d L + d P L σ d L + Pt d L dl + PdL σ dl d P R t d R d P R σ d R + Pt d R dr PdR σ dr PH + PH PH 0 PH 0 V H + H + + H 0 H 0 λ e ν el H+ + e L H0 e R + e R H + ν el + H 0 e L λ u u R H 0 u L H + d L + ul H 0 d L H + u R λ d d R H + u L + H 0 d L + ul H + + d L H 0 d R This basically consists of 7 massless spinor Lagrangians; Higgs Lagrangians which may have mass depending on the choice for V; and three terms which couple the spinors and the Higgs fields. It is these last three terms which ultimately give rise to the masses of the spinor particles. At present, V. is an arbitrary potential; and λ e, λ d and λ u are arbitrary coupling constants. Notice that for a Dirac spinor, ignoring the complications of SU and U1, it was shown earlier that mass could be incorporated simply by including in the Lagrangian a term µ Ψ R Ψ L + Ψ L Ψ R where the two spinors are any left- and right-spinors. However, if you check this, you will find that such a term is not SU L U 1 Y -invariant. Hence such a term cannot be included in the Lagrangian. The particle masses arise in a much more complicated way in this theory.

23 Chapter 5 CONSTRUCTING THE STANDARD MODEL II LOCAL GAUGE SYMMETRIES AND INTERACTIONS In this chapter I show that the concept of local gauge symmetries leads to introduction of new fields which interact with the basic matter fields. Gauge invariance is a symmetry property which just involves the fields: there is no transformation of the co-ordinates. In the previous chapter I have constructed a Lagrangian which is Lorentz invariant and also invariant under the global gauge transformations SU L U1 Y. The significance of the term global is that the parameters of the transformations α and β are assumed to be constant throughout all space and time. I now remove this restriction and allow α and β to vary. That is α α x β β x I then require that the Lagrangian be invariant under this local gauge transformation. However the terms that involve the derivatives of the fields cause problems: On making the transformation such terms involve derivatives of α and β and these derivatives do not cancel. 5.1 Local SUN Gauge Invariance I need to consider the local SU L U1 Y gauge invariance. However since later on I shall also need to investigate local SU3 invariance, I consider first a more general form of gauge invariance in which SU is replaced by SUN. In this case the fields are a considered composite objects with N field components. For example, in the SU case ν el and e L would be considered to be the two components of a composite SU field. In general the Lagrangian is a function of S, S, S and S where S is a column vector with N components and S is a row vector with N components. In this type of global gauge transformation

24 4 CONSTRUCTING THE STANDARD MODEL II LOCAL GAUGE SYMMETRIES AND INTERACTIONS these quantities change as S e iβŷ US S e iβŷ S U S e iβŷ US S e iβŷ S U 5.1 U is a unitary N N matrix and U is its hermitian conjugate. These matrices have determinant equal to 1 and satisfy the equations UU = I = U U 5. where I is the N N unit matrix. If this is to be a symmetry, these fields must appear in the Lagrangian in such a way that the terms involving β and the matrices U and U cancel. Strictly the SUN gauge transformation is just the part involving the N N unitary matrices this is what the notation means but I have also included the U1 symmetry. If, as stated above, I promote the transformation to be local β and the matrices are allowed to be functions of x. In this case the change in the variables becomes S e iŷ β x U x S S e iŷ β x S U x S e iβ xŷ U x + i β Ŷ + U U S 5.3 [ S e iβ xŷ U x + i β Ŷ + U U S The terms involving the derivatives do not cancel and I attempt to remedy this by introducing new fields. In this case the required fields are B, which is introduced to attempt to eliminate β, and W which is introduced to eliminate the matrix U U. This field W is rather complicated: It is both a space-time vector and an N N matrix. In detail there are 4 matrices W t, W x, W y and W z. These new fields are introduced into the Lagrangian by making the replacements S ig1 Ŷ B ig W [ S ig1 Ŷ B ig W S S 5.4 g 1 and g are at the moment unknown coupling constants. I choose the properties of the new fields so that under the local gauge transformation the changes in the new fields exactly cancel the changes in S. The required transformation properties are B x B x + 1 g 1 β x W U WU i 1 g U U 5.5 With the introduction of these new fields the quantities in the Lagrangian change under the local gauge transformation as S e iŷ β x U x S S e iŷ β x S U x ig1 Ŷ B ig W S e iβŷ U ig1 Ŷ B ig W S [ ig1 Ŷ B [ ig W S e iβŷ U ig1 Ŷ B ig W 5.6 S

25 5. Dyadics and the gauge field Lagrangians 5 The gradients of β and U have all been eliminated and the other factors all cancel because of the assumed global symmetry. The resulting Lagrangian can be written as L = L S + L SB + L SW 5.7 where L SB describes the interaction with the B field and L SW describes the interaction with the W fields. I need to add the Lagrangians of fields B and W themselves. 5. Dyadics and the gauge field Lagrangians I consider first the B field. The Lagrangian L B for the field B itself must be such that it does not change under the local gauge transformation. That is it must not change under the transformation B x B x + 1 g 1 β x. This can be done if L B contains only derivatives of B x in the anti-symmetric combinations The effect of the local gauge transformation on such a term is B µν = B µ x ν B ν x µ 5.8 B µ B ν B µ B ν 1 β β = B µ B ν 5.9 x ν x µ x ν x µ g 1 x ν x µ x µ x ν x ν x µ That is it has no effect! In order to form a Lorentz invariant Lagrangian I first make a dyadic out of B µν see Lagrangian notes page 8: B = µν ẽ µ B µν ẽ ν 5.10 I can define a dyadic scalar product by analogy with the normal scalar product for vectors as à B = µνµ ν This dyadic scalar product is, of course, Lorentz-rotation invariant. ẽµ ẽ µ ẽν ẽ ν A µν B µ ν 5.11 Hence if I take the dyadic scalar product of B with itself this is guaranteed to be Lorentz invariant. I can use this concept to construct a simple Lorentz invariant Lagrangian for the B gauge field: [ Bt L B = 1 4 B B = 1 + B c t B B 5.1 B t and B are the time and space components of B x. I now turn to the more complicated task of choosing L W. This has to be chosen in such a way that it is invariant under the change W U WU i U U g

26 6 CONSTRUCTING THE STANDARD MODEL II LOCAL GAUGE SYMMETRIES AND INTERACTIONS In order to see how to do this, I first look at the change, under the local gauge transformation, of the quantities which worked for the B field µ W ν ν W µ. This is µ W ν ν W µ U µ W ν ν W µ U +UW ν µ U UW µ ν U + µ U W ν U ν U W µ U +i 1 g µ U ν U ν U µ U 5.13 This does not look very hopeful! However if I look at the change in the quantity i W µ W ν W ν W µ I get i W µ W ν W ν W µ iu W µ W ν W ν W µ U + 1 g UWν µ U UW µ ν U + 1 g µ U W ν U ν U W µ U i 1 g µ U ν U ν U µ U it is clear that the last three lines of 5.14 are exactly equal to the last three line of 5.13 divided by g. Hence the transformation properties of the combination is much simpler: W µν = µ W ν ν W µ ig W µ W ν W ν W µ 5.15 µ W ν ν W µ ig W µ W ν W ν W µ U [ µ W ν ν W µ ig W µ W ν W ν W µ U 5.16 Apart from multiplication by the U matrices, it is unaltered. Again I can form a Lorentz invariant by first forming a dyadic and then taking the dyadic scalar product as before. The problem of the the pre- and post-multiplication by the U matrices can be solved by taking the trace of the resulting matrices. The trace of a matrix is the sum of the diagonal elements. Trace has the following property: T race [XYZ = T race [YZX = T race [ZXY 5.17 for any square matrices X, Y and Z. Hence, applying this to the current situation, [ T race UCU = T race [CU U = T race [C 5.18 and the matrices U and U cancel. The resulting Lagrangians for the Gauge fields are L B = 1 4 B B L W = 1 [ W T race W 5.19 The numerical pre-factors are arbitrary but these are the conventional values. Changing the pre-factor is equivalent to scaling the amplitude of the field. In SUN the matrix field W can be represented by N 1 independent space-time vector fields. There are two particularly important cases. SU3: Here the matrix field is represented by eight vector fields. These fields lead to the eight fields responsible for the strong interaction the gluons. I shall return to this case later. SU: In this case to which I now return, the matrix field is represented by three vector fields.

27 5. Dyadics and the gauge field Lagrangians 7 W = T 1 W1 + T W1 + T 3 W3 where T j are the three generators of the SU L gauge transformation. The transformation U is given by [ U x = exp i α 1 x T 1 + α x T + α 3 x T Although this is the obvious representation, a simpler representation can be chosen: Since T + and T are much simpler operators than T 1 and T, I choose to express the latter in terms of the former and I also introduce W = W 1 i W 5. W = W 1 + i W Note that these are complex vector fields and one is the adjoint of the other. Using this representation: and W = T + W + T W + T3 W3 [ U x = exp i α x T + + α x T + α 3 x T where α = α 1 + iα The transformation properties of the fields W, W and W 3 can be determined from the general relation 5.6. These are quite complicated for a finite transformation but for the infinitesimal case the resulting transformations, including B, are: B B 1 g 1 β W W + 1 g α i α W 3 + i α 3 W W W + 1 g α + i α W 3 i α 3 W W 3 W g α3 + i W α i W α 5.5 With these changes of notation each derivative in the original Lagrangian has to be replaced by Φ ig1 BŶ ig W T+ + W T + W 3 T3 Φ 5.6 Since both the matrix operators Ŷ and T 3 are diagonal whereas both T + and T are both non-diagonal, this change to the Lagrangian is better written as Φ ig1 BŶ ig 3 W T3 Φ ig W T+ + W T 5.7 Φ The general dyadic 5.15 can be expressed, in this case, as where the individual terms are W µν = T + W µν + T W µν + T 3 W 3 µν 5.8

28 8 CONSTRUCTING THE STANDARD MODEL II LOCAL GAUGE SYMMETRIES AND INTERACTIONS W µν = W ν x µ W µ x ν ig W 3 µ W ν W 3 νw µ W µν = Wν x µ Wµ x ν + ig W 3 µ W ν W 3 νw µ W 3 µν = W 3 ν x µ W 3 µ x ν + i g Wµ W ν W ν W µ 5.9 The resulting Lagrangian 5.19 for these gauge fields including the B field is L BW = 1 [ W T race W 1 4 B B L BW = [ W W + W W + B B 5.3 Summary It is important to emphasize what has been achieved. This procedure has shown that insisting on local gauge invariance forces the introduction of new gauge fields; specifies the form of interactions with the particle fields; and also specifies the Lagrangians for these gauge fields. Hence local gauge symmetry gives rise to interactions between particles via the gauge fields.

29 Chapter 6 CONSTRUCTING THE STANDARD MODEL II MASS HIGGS FIELDS AND ELECTROMAGNETISM In this chapter I show that massive fields can be produced from initially massless fields. The Higgs part of the total Lagrangian is [ + i g 1 B + i g W 3 H + + i g W H 0 [ i g 1 B i g W 3 H + i g W H [ + i g 1 B i g W 3 H 0 + i g W H + [ i g 1 B + i g W 3 H 0 i g W H + V The potential associated with the Higgs fields is chosen to be: V = m H 4v H + H + + H 0 H 0 v where m H and v are real, positive constants. This is unusual in that the minimum does not occur where the fields are zero. I refer the details of the Higgs mechanism to Ryder Chapter 8; Cottingham and Greenwood, Chapters 10 and 11; Mandl and Shaw chapter 13. The result of applying the Higgs mechanism is that I can choose a gauge transformation so that H + x 0 H 0 x v + h x where h x is a real field. With the application of these changes the Higgs Lagrangian becomes 1 h h m H h + h3 v + h4 4v 1 8 g 1 B g W 3 g 8 W W v + h g 1 B g W v + h 6.4

30 30 CONSTRUCTING THE STANDARD MODEL II MASS HIGGS FIELDS AND ELECTROMAGNETISM The first term corresponds to a scalar Higgs particle with mass m H and with some non-linear interactions. The second terms provides the mass for the W and W particles. The final term also provides mass but is somewhat more complicated. 6.1 The electromagnetic field Notice that the fields B and W 3 only occur in the combination g 1 B g W 3. Also from 5.7 it can be seen that these fields B and W 3 enter the gradient modifications in very similar ways. I can make use of this to introduce combinations of the fields which simplify the above Higgs Lagangian; introduce the familiar electromagnetic fields; and which introduce a more familiar conservation quantity which is just the sum of Ŷ and T 3. Q = Ŷ + T Q is the charge operator: Its eigenvalues are charge. New fields à and Z which are combinations of the B and W 3 fields can be defined by or, equivalently à = cos θ w B + sin θ w W 3 Z = sin θ w B + cos θ w W B = cos θ w à sin θ w Z W 3 = sin θ w à + cos θ w Z 6.7 In terms of these new fields, the combination g 1 B g W 3 that appears in the Higgs Lagrangian becomes à g 1 cos θ w g sin θ w Z g 1 sin θ w + g cos θ w 6.8 I choose the angle θ w so that the term in à vanishes. g 1 cos θ w = g sin θ w = g A 6.9 where, at the moment, g A is not specified. With this choice the Higgs Lagrangian becomes 1 h h m H h + h3 v + h4 4v g W 8 W W v + h g Z 8 Z Z v + h 6.10 The coupling constants g W and g Z have been defined as g Z = g A 1 sinθwcosθw 1 g W = g A sinθw Notice that g W is just what I had previously called g. In the Higgs Lagrangian v is simply a constant and the terms 6.11

31 6.1 The electromagnetic field 31 g W8 v W W = m W W W g Z8 v Z Z = mz Z Z 6.1 introduce masses for the W and Z fields. There is no mass term for the A field and I identify this as the electromagnetic field. That is, the electromagnetic field has been identified as that gauge field which has zero mass. The coupling g A introduced above is precisely the coupling to the electromagnetic field and is identified as the electric charge in dimensionless units. In terms of conventional variables g A is given by e g A = = ε0 c Notice that g A is not predicted by the theory but is simply fitted to the known results of electromagnetism. With this choice the change to the general gradient of the spinor and scalar fields in the new Lagrangian is Φ iga Qà ig Z T3 sin θ w Q Z Φ ig W W T+ + W T 6.14 Φ It may also be convenient to make the Z-interaction look more like the electro-magnetic interaction by defining the Z-charge Q Z = T 3 sin θ w Q 6.15 In this case the modification to the gradient terms is given by Φ iga Qà ig Z QZ Z Φ ig W W T+ + W T Φ However there is a price to pay! Introducing à and Z complicates the gauge field Lagrangian I can express the gauge field Lagrangian in terms of the dyadics à and Z which are defined as: A µν = Aν x µ With these definitions the gauge field Lagrangian is L AZW = 1 4 where I have defined the components of Ω to be Aµ x ν Z µν = Zν x µ Zµ x ν 6.17 [ à à + Z Z + W W i 4 g A Ã Ω i 4 g Zcos θ w Z Ω g W Ω Ω 6.18 Ω µν = W µ W ν W ν W µ 6.19

32

33 Chapter 7 PREDICTIONS OF THE STANDARD MODEL I investigate some of the simpler predictions of the Standard model. There are some parameters of the standard model which are simply fitted to experimental results. The ones which are important for the weak-electromagnetic interactions are: g A This is fitted to the electric charge G F This is a parameter called the Fermi interaction constant. It was determined before the advent of the Standard Model by fitting the decay-rate of the µ particle. In the Standard Model it is given by gw mw 8 m Z This is fitted to the known Z mass These give rise to the values: g A = G F = GeV m Z = Gev sinθ W = ± sin θ W = ± cosθ W = ± cos θ W = ± g W = ± g Z = ± Note that you will find many different values for sin θ W in the literature. The reason is that the value used is an effective value which take into vertex corrections. Unfortunately this effective value depends on the energy of the particular process involved.

34 34 PREDICTIONS OF THE STANDARD MODEL 7.1 W-mass A very simple prediction based on the above fitted data is the mass of the W-particle. According to 6.11 the W-particle mass is given by m W = g W gz m Z = m Z cosθ w = m Z = 79.97GeV 7. The experimental value for m W is GeV. The small difference can be explained and calculated in terms of self-energy corrections. 7. Z-Interactions The various Z-interactions predicted by 6.16 are all of the form : g Z Q Z ψ L Z tψ L ψ L Z σψ L g Z Q Z ψ R Z tψ R + ψ R Z σψ 7.3 R where ψ L is any of the left-handed spinor fields and ψ R is any of the right-handed spinor fields. Q Z is the Z-charge of the particular field. It is important to note that the structure of the interactions only depends on the value of the Z-charge. These Z-charges are shown in the table. T 3 Q Q Z 1 1 ν L 0 e L 1-1 sin θ w 1 e R 0-1 sin θ w 1 u L sin θ w u R sin θ w d L d R sin θ w sin θ w Consider the processes depicted by the following figures: e ν Z e ν

35 7. Z-Interactions 35 e l Z e e l q Z e q These refer to processes in which an electron collides with a positron anti-electron; emits a Z-particle which then decays into either neutrino + anti-neutrino; or lepton + anti-lepton or quark + anti-quark. The neutrino can be any of the three types of neutrino; the lepton can be any of the three types of lepton electron, µ and τ; and the quark can be almost any of the quarks. The almost in the last sentence arises because the top quark is too massive to be created in this process. In each of the diagrams, time runs from left to right. So that particle lines starting on the left represent initial particles and particle lines ending on the right represent final particles. The arrows on the particle lines need some explanation: For incident particles, an arrow pointing towards the vertex refers to a particle and an arrow pointing away from the vertex refers to an anti-particle. For the final particles the convention is that an arrow pointing away from the vertex refers to a particle and an arrow pointing towards the vertex refers to an anti-particle. Don t expect me to defend this strange convention. Each particle line and gauge field line has an associated space-time wave-vector. These space-time vectors are conserved at each vertex: effectively conserving energy the time part and momentum the space parts. For the initial and final particle lines these wave-vectors satisfy: where m is the mass of the corresponding particle Experimental Data k k = k x + k y + k x k t = m 7.4 The experimental probabilities for the possible outcomes of the above process are, in the case where the initial electron-positron energy is about that required to produce a Z-particle: e + e Z all neutrinos ± e + e Z e + e ± e + e Z µ + µ ± e + e Z τ + τ ± e + e Z q q hadrons ± 0.00