Phenomenology of Particle Physics NIU Spring 2002 PHYS586 Lecture Notes Copyright (C) 2002 Stephen P. Martin Physics Department Northern Illinois University DeKalb IL 60115 [email protected] corrections and updates: http://zippy.physics.niu.edu/phys586.html June 23, 2005 Diligent efforts have been made to eliminate all misteaks. – Anonymous Contents 1 Special Relativity and Lorentz Transformations 4 2 Relativistic quantum mechanics of single particles 10 2.1 Klein-GordonandDiracequations . .......... 10 2.2 SolutionsoftheDiracEquation . ........... 16 2.3 TheWeylequation ................................. ...... 23 3 Maxwell’s equations and electromagnetism 26 4 Field Theory and Lagrangians 28 4.1 The field concept and Lagrangian dynamics . ............ 28 4.2 Quantization of free scalar field theory . .............. 35 4.3 Quantization of free Dirac fermion field theory . ............... 40 5 Interacting scalar field theories 43 5.1 Scalar field with φ4 coupling.................................. 43 5.2 Scattering processes and cross sections . ............... 48 5.3 Scalar field with φ3 coupling.................................. 55 5.4 Feynmanrules .................................... ..... 62 1 6 Quantum Electro-Dynamics (QED) 69 6.1 QEDLagrangianandFeynmanrules . ......... 69 6.2 e e+ µ µ+ .......................................... 77 − → − 6.3 e e+ ff........................................... 82 − −→ 6.4 Helicities in e e+ µ µ+ .................................. 86 − → − 6.5 Bhabha scattering (e e+ e e+) .............................. 92 − → − 6.6 Crossingsymmetry................................ ....... 98 6.7 e µ+ e µ+ .......................................... 99 − → − 6.8 Moller scattering (e e e e )...............................100 − − → − − 6.9 Gauge invariance in Feynman diagrams . ...........101 6.10 Compton scattering (γe γe )...............................103 − → − 6.11 e+e γγ ...........................................110 − → 7 Decay processes 113 7.1 Decayratesandpartialwidths . ..........113 7.2 Two-bodydecays................................... 114 7.3 Scalar decays to fermion-antifermion pairs; Higgs decay ..................116 7.4 Three-bodydecays ................................. 119 8 Fermi theory of weak interactions 122 8.1 Weaknucleardecays ............................... .......122 8.2 Muondecay....................................... 124 8.3 Correctionstomuondecay . ........134 8.4 Inverse muon decay (e ν ν µ )..............................136 − µ → e − 8.5 e ν µ ν ..........................................138 − e → − µ 8.6 Charged currents and π± decay................................139 8.7 Unitarity, renormalizability, and the W boson........................145 9 Gauge theories 150 9.1 Groupsandrepresentations . ..........150 9.2 The Yang-Mills Lagrangian and Feynman rules . ............160 10 Quantum Chromo-Dynamics (QCD) 167 10.1 QCDLagrangianandFeynmanrules . ..........167 10.2 Quark-quark scattering (qq qq)...............................169 → 10.3 Renormalization ................................ ........171 10.4 Gluon-gluon scattering (gg gg)...............................180 → 10.5 The parton model, parton distribution functions, and hadron-hadron scattering . 182 10.6 Top-antitop production in pp collisions............................191 2 10.7 Top-antitop production in pp collisions............................193 10.8 Kinematics in hadron-hadron scattering . ..............194 + 10.9 Drell-Yan scattering (` `− production in hadron collisions) . 196 11 Spontaneous symmetry breaking 199 11.1 Globalsymmetrybreaking. ..........199 11.2 Local symmetry breaking and the Higgs mechanism . ..............202 11.3 Goldstone’s Theorem and the Higgs mechanism in general .................206 12 The standard electroweak model 208 12.1 SU(2) U(1) representationsandLagrangian . 208 L × Y 12.2 The Standard Model Higgs mechanism . ...........212 12.3 Fermion masses and Cabibbo-Kobayashi-Maskawa mixing . ...............217 Homework Problems 225 Appendix A: Natural units and conversions 241 Appendix B: Dirac Spinor Formulas 242 Index 243 3 1 Special Relativity and Lorentz Transformations A successful description of elementary particles must be consistent with the two pillars of modern physics: special relativity and quantum mechanics. Let us begin by establishing some important features of special relativity. Spacetime has four dimensions. For any given event (for example, a firecracker explodes, or a particle decays to two other particles) one can assign a four-vector position: (ct,x,y,z) = (x0,x1,x2,x3)= xµ (1.1) The Greek indices µ,ν,ρ,... run over the values 0, 1, 2, 3, and c is the speed of light in vacuum. As a matter of terminology, xµ is an example of a contravariant four-vector. It is often useful to change our coordinate system (or “inertial reference frame”) according to µ µ µ ν x x0 = L x . (1.2) → ν Such a change of coordinates is called a Lorentz transformation. Here Lµ is a 4 4 real matrix which ν × parameterizes the Lorentz transformation. It is not arbitrary, however, as we will soon see. The laws of physics should not depend on what coordinate system we use; this is a guiding principle in making a sensible theory. As a simple example of a Lorentz transformation, suppose we rotate our coordinate system about the z-axis by an angle α. Then in the new coordinate system: µ x0 = (ct0,x0, y0,z0) (1.3) where ct0 = ct x0 = x cos α + y sin α y0 = x sin α + y cos α − z0 = z. (1.4) Alternatively, we could go to a frame moving with respect to the original frame with velocity v along the z direction, with the origins of the two frames coinciding at time t = t0 = 0. Then: ct0 = γ(ct βz) − x0 = x y0 = y z0 = γ(z βct). (1.5) − where β = v/c; γ = 1/ 1 β2 (1.6) − q 4 Another way of rewriting this is to define the rapidity η by β = tanh η, so that γ = cosh η and βγ = sinh η. Then we can rewrite eq. (1.5), 0 0 3 x0 = x cosh η x sinh η − 1 1 x0 = x 2 2 x0 = x 3 0 3 x0 = x sinh η + x cosh η. (1.7) − This is an example of a boost. Another example of a contravariant four-vector is given by the 4-momentum formed from the energy E and spatial momentum ~p of a particle: pµ = (E/c, ~p). (1.8) All contravariant four-vectors transform the same way under a Lorentz transformation: µ µ ν a0 = L νa . (1.9) A key property of special relativity is that for any two events one can define a proper interval which is independent of the Lorentz frame, and which tells us how far apart the two events are in a coordinate-independent sense. So, consider two events occurring at xµ and xµ + dµ, where dµ is some four-vector displacement. The proper interval between the events is (∆τ)2 = (d0)2 (d1)2 (d2)2 (d3)2 = g dµdν (1.10) − − − µν where 10 0 0 0 1 0 0 gµν = − (1.11) 0 0 1 0 − 0 0 0 1 − is known as the metric tensor. Here, and from now on, we adopt the Einstein summation convention, in which repeated indices µ,ν,... are taken to be summed over. It is an assumption of special relativity that gµν is the same in every inertial reference frame. The existence of the metric tensor allows us to define covariant four-vectors by lowering an index: x = g xν = (ct, x, y, z), (1.12) µ µν − − − p = g pν = (E/c, p , p , p ). (1.13) µ µν − x − y − z Furthermore, one can define an inverse metric gµν so that µν µ g gνρ = δρ , (1.14) 5 µ where δν =1 if µ = ν, and otherwise = 0. It follows that 10 0 0 0 1 0 0 gµν = − . (1.15) 0 0 1 0 − 0 0 0 1 − Then one has, for any vector aµ, ν µ µν aµ = gµνa ; a = g aν. (1.16) It follows that covariant four-vectors transform as ν aµ0 = Lµ aν (1.17) where (note the positions of the indices!) ν νσ ρ Lµ = gµρg L σ. (1.18) Because one can always use the metric to go between contravariant and covariant four-vectors, people often use a harmlessly sloppy terminology and neglect the distinction, simply referring to them as four-vectors. If aµ and bµ are any four-vectors, then aµbνg = a b gµν = a bµ = aµb a b (1.19) µν µ ν µ µ ≡ · is a scalar quantity. For example, if pµ and qµ are the four-momenta of any two particles, then p q is a · Lorentz-invariant; it does not depend on which inertial reference frame it is measured in. In particular, a particle with mass m satisfies the on-shell condition p2 = pµp = E2/c2 ~p2 = m2c2. (1.20) µ − This equation, plus the conservation of four-momentum, is enough to solve most problems in relativistic kinematics. The Lorentz-invariance of equation (1.19) implies that, if aµ and bµ are constant four-vectors, then µ ν µ ν gµνa0 b0 = gµνa b , (1.21) so that µ ν ρ σ ρ σ gµνL ρL σa b = gρσa b . (1.22) Since aµ and bν are arbitrary, it must be that: µ ν gµνL ρL σ = gρσ. (1.23) 6 This is the fundamental constraint that a Lorentz transformation matrix must satisfy. In matrix form, it could be written as LT gL = g. If we contract eq. (1.23) with gρκ, we obtain κ ν κ Lν L σ = δσ (1.24) Applying this to eqs. (1.2) and (1.17), we find that the inverse Lorentz transformation of any four-vector is ν µ ν a = a0 Lµ (1.25) µ aν = aµ0 L ν (1.26) Let us now consider some particular Lorentz transformations. To begin, we note that as a matrix, det(L)= 1. (See homework problem.) An example of a “large” Lorentz transformation with det(L)= ± 1 is: − 1 0 0 0 − 0 100 Lµ = . (1.27) ν 0 010 0 001 This just flips the sign of the time coordinate, and is therefore known as time reversal: 0 0 1 1 2 2 3 3 x0 = x x0 = x x0 = x x0 = x . (1.28) − Another “large” Lorentz transformation is parity, or space inversion: 10 0 0 µ 0 1 0 0 L ν = − , (1.29) 0 0 1 0 − 0 0 0 1 − so that: 0 0 1 1 2 2 3 3 x0 = x x0 = x x0 = x x0 = x .