Topics covered: Particle Physics 2006
- Lecture 1 :
- Lecture 2:
- Basics of QM: operators, observables, time evolution,
symmetries and conservation laws
- A short review of angular momentum algebra
- Parity operator P: formal properties, determination of
P eigenvalues of baryons, pions, kaons
- Lecture 3:
- Charge conjugation operator C: action on mesons and baryons,
C eigenvalues of photon, pi0, eta
- Time reversal operator T
- Action of T and P on operators like r, p, sigma, E, B and
their combinations
- Isospin symmetry, application for calculating ratios of
cross sections
- Assignment of isospin quantum numbers to various particles
- G parity and its eigenvalue for pions
- Lecture 4:
- Flavour SU(3), mesons and baryons, operators U+-, V+-, I+-
- Pesudoscalar octet: wavefunctions in terms of quarks,
eta-eta' mixing
- Vector meson octet: phi-omega mixing
- Baryon octet: wavefunctions in terms of quarks
- Baryon decuplet: colour quantum number
- Lecture 5:
- Applications of flavour SU(3)
- Magnetic moments of baryons
- Mass splittings through Hyperfine terms
- Predictions of hadron-hadron cross sections through quark-quark
cross sections
- Confirming quark charges through decays of
[rho, omega, phi] -> e+ e- and the ratio of
[pi- C] and [pi+ C] cross sections
- Lecture 6:
- Modifying Schrodinger's equation to make it invariant under local
U(1) gauge symmetry
- Starting with U(1) gauge symmetry to get equations of electromagnetism
- QM electromagnetism: equations of motion through [x,H] and
[dx/dt,H]
- Lecture 7:
- Maxwell's equations in terms of the vector potential A
- Equation of motion for A, simplification under Lorentz gauge
- Solution for A, getting rid of unphysical degrees of
freedom to get two polarizations for photon
- Lagrangian for EM field without referring to Maxwell,
using only U(1) symmetry, equations of motion for A
- Lecture 8:
- Klein-Gordan equation as a relativistic generalization of
Schrodinger's equation
- KG current, discussion of antiparticles a la Dirac,
Pauli-Weisskopf, Feynman-Stuckelberg
- Lagrangian that can give rise to KG equation
- Lagrangian for a photon as a special case obeying [KG + U(1)],
masslessness of photon through U(1) symmetry
- Interaction Lagrangian of photon and charged KG field
- Lecture 9:
- Nonrelativistic perturbation theory a la Halzen and Martin
- First order PT leading to Fermi' golden rule
- Second order PT for understanding of intermediate states
- Electrodynamics of spinless particles: Transition amplitude
T_{f i}
- Lecture 10:
- Electrodynamics of spinless particles: initial flux, final state
phase space, cross section, matrix element M
- Explicit calculation for ``spinless'' electron-muon scattering
- Crossing symmetries for M
- Additional terms for identical particles: [e- e-] and [e- e+]
scattering
- Mandelstam variables: s, t, u and their properties
- M and cross sections in terms of s, t, u
- Angular dependence of scattering
- Interpretation of propagator as time-ordered product
- Lecture 11:
- A detour through quantum field theory a la Veltman (Diagrammatica)
- Fields as "local" combinations of Fourier transforms of
creation and annihilation operators
- Scattering theory: S-matrix, T-matrix, in and out states
- U(t) as the evolution operator, theorem that gives dU/dt in
terms of H_in
- Lecture 12:
- Solving the evolution equation for U(t): Dyson series with
time-ordering, explicitly till second order
- pi-pi scattering upto second order terms, by using
H = g pi pi sigma
- Obtaining Feynman propagator from the second order term
in Dyson equation
- Feynman rules and diagrams for pi-pi scattering
- Lecture 13:
- Dirac equation from an attempt to find a linear relativistic
generalization of Schrodinger's equation
- Dirac equation as four coupled differential equations
- Basic properties of gamma matrices
- Conjugate equation with psi.bar instead of psi.dagger
- Dirac current: Is it a 4-vector ? (left for exercise)
- Solution of Dirac equation in rest frame, positive and
negative energy solutions, interpretation
- Solution of Dirac equation for any momentum p
- Helicity operator, quantum numbers for all 4 solutions
of the Dirac equation
- Lecture 14:
- Dirac equation describes spin-1/2 particles: through calculation of
[H,L] and [H,Sigma] where H = alpha p + beta m
- u and v as particle-antiparticle solutions of Dirac equation
- Defining charge conjugation by adding EM interaction term,
association of v(1) as charge conjugate of u(1) etc
- Properties of u and v spinors: the equations they and their
conjugates satisfy, orthogonality [u.dagger(s) u(s)]
- Useful relations: Sum[u(s) u.bar(s)] = p.slash + m, etc
- Lecture 15:
- Traces of gamma matrices
- Electrodynamics of spin-1/2 particles a la Halzen and Martin
- Electron-muon scattering: summation over final state spins,
averaging over initial spins, converting |M|^2 expressions
in terms of traces
- calculating cross sections using fluxes, final state phase
space etc
- Crossing symmetries
- Lecture 16:
- Identical particles and spin-1/2 scattering
- Definition of fermion fields, consistency of Fermion fields
anticommutation with H commutation
- Motivation of a negative sign under fermion interchange
using the time ordered product a la Veltman
- Fermion vertex as "spinless" vetrex plus magnetic moment interaction
- Propagators for spinless particles, spin-1/2 fermions,
massless and massive spin-1 bosons
- Bilinear invariants for general interactions
- Lecture 17:
- Weak interactions: examples, lifetimes, cross sections
- Differences from EM interactions: decay and scattering rates,
angular distributions, s-dependence
- Coupling constant G has to have dimensions E^-2
- Weak charged current involves only left-handed particles
(V-A interaction)
- Properties of left and right projevtion operators P_L and P_R
- Interpretation of G in terms of W propagator
- Lecture 18:
- Calculation of decay rate in terms of the matrix element,
initial flux and phase space
- Muon decay: complete calculation to get the decay rate and
energy spectrum of decay products
- Lecture 19:
- Beta decay: endpoint spectrum as a function of neutrino mass
- Pi+ decay: decay constant as a parameter, ratio of decay to electron vs muon as a test of V-A theory
- Lecture 20:
- Understanding symmetries like P, C, Isospin and their breaking
explicitly in terms of the Lagrangian, for EM and weak charged current
- Allowed vertices in weak charged current: magnitudes of CKM matrix
elements (CP violation untouched)
- Drawing all possible Feynman diagrams involving EM and weak charged
current for various processes, relationships between cross sections /decay
rates using Feynman diagrams and CKM elements
- Lecture 21:
- Weak neutral current: examples of allowed processes, no FCNC
- Form of the current J_NC: c_V and c_A
- Postulation of SU(2)_L for weak interactions: the presence of
J(+/-)_weak suggests J_3(weak), which is left-handed and hence cannot
reproduce the c_V and c_A values
- J_3(weak) combined with J_em can give the combinations
J_NC and J_Y (Hypercharge)
- Determinations of Y eigenvalues through J_Y = 2 (J_em - J_3)
- Mixing of SU(2)_L and U(1)_Y with coupling constants g and g'
to get J_NC and J_em, relations between g, g', e, theta_W (Weinberg angle)
- Reproducing c_V and c_A interms of the Weinberg angle
- rho parameter \approx 1, prediction awaits Weinberg-Salam model
- Lecture 22:
- Review of weak CC and NC processes
- neutrino-electron and antineutrino-electron cross sections and
angular dependences for the CC part, relating to neutrino-quark cross
sections
- CC-NC interference in neutrino-electron scattering
- EM-NC interference in e+ e- -> mu+ mu-
- Calculating the total width of Z -> f fbar, determining the number of neutrino species through invisible Z width
- Lecture 23:
- Gauge symmetries as a means of restricting terms in the Lagrangian
- Obtaining Lagrangians from equations of motion for free particles
described by: Klein-Gordan, Dirac, Maxwell equations
- Interactions predicted through the requirement of gauge invariance
- Mass terms for the gauge bosons forbidden with gauge symmetries
- Non-abelian gauge symmetries: infinitesimal transformations,
determination of covariant derivatives D_mu, transformation properties
of the gauge field under the symmetry, the gauge invariant
``kinetic energy'' combination
- Lecture 24:
- The problems with massive gauge bosons: gauge symmetry broken, and
more importantly, renormaliation not possible (only an illustration)
- Solution: spontaneous symmetry breaking (SSB) that breaks the
gauge symmetry in ``initial conditions''
- SSB for a 1-D potential (parity symmetry) and U(1) global:
massive Higgs scalar and massless Goldstone boson
- SSB for local U(1): massive Higgs scalar, massless Goldstone
boson degree of freedom (dof) converting to the longitudinal
polarization dof for the massless gauge bosons, giving it mass
- High and low scale Feynman diagrams and couplings
- SSB for SU(2) gauge symmetry: 3 Goldstone bosons getting
absorbed into longitudinal dofs of 3 W bosons, which become
massive, mass proportional to the vacuum expectation value (vev)
of the Higgs
- Lecture 25:
- The standard model: SU(3) x SU(2)_L x U(1)_Y
- The general Lagrangian that satisfies the gauge symmetries,
interactions and currents
- SSB through adding a Higgs that is an SU(2) doublet, has Y=1
[implies Phi = (phi+ phi0)^T] and vev of Phi = (0 v)^T/sqrt[2]
- Masses of gauge bosons by expanding the relevant term in the
lagrangian: M_W = g v /2, M_Z = g v / (2 cos[theta_W]), M_A =0.
Prediction of rho parameter to be unity at the leading order
- Fermion masses through Yukawa couplings through the Higgs and
its conjugate, masses proportional to coupling strength as a test
of Higgs mechanism
- A long list of particle physics topics not covered in this course:
- QCD, confinement, perturbative QCD, hadronic structure functions,
deep inelastic scattering
- Renormalization, anomalies
- Flavour physics: B physics, CP violation, neutrinos
- Physics Beyond the Standard Model: SUSY, unification, extra dimensions
- etc.