Scattering in QM Consider a beam of particles scattering in potential V(r): NOTE: natural units The scattering rate is characterized by the interaction cross-section Use Fermi’s Golden Rule to get the transition rate where Mfi is the matrix element and r(Ef) is the density of final states. Number of particles scattered per unit time Incident flux

1st Order Perturbation Theory using plane wave solutions of form Require: Wave-function normalization Matrix element in perturbation theory Expression for flux Expression for density of states. Normalization: Normalize wave-functions to one particle in a box of side L:

Matrix Element: This contains the physics of the interaction where Incident Flux: Consider a “target” of area A and a beam of particles travelling at velocity vi towards the target. Any incident particle within a volume viA will cross the target area every second. Flux (F) = number of incident particles crossing unit area per second: where n is the number density of incident particles = 1 per L3

Density of States: for a box of side L states are given by the periodic boundary conditions: Each state occupies a volume in p space. Number of states between in p and p+dp in solid angle dW Density of states in energy For relativistic scattering: (Ep)

Putting all the separate bits together: Relativistic scattering vi= c = 1 and p  E

Breit-Wigner Cross-section Some particle interactions proceed via an intermediate RESONANT state which then decays Two stages: (Bohr Model) Formation Occurs when the collision energy s  the natural frequency of a resonant state. Decay The decay of the resonance Z is independent of the mode of formation and depends only on the properties of the Z.

The RESONANCE CROSS-SECTION is given by with The matrix element Mif is given by 2nd Order Perturbation Theory where the sum is over all intermediate states. ( Factors of L3 cancel as before )

Consider 1 intermediate state described by This describes a state with energy = Rate of decay of Z: Rate of formation of Z:

Hence, pi is the C.O.M. momentum natural units BREIT-WIGNER CROSS-SECTION

The factor g takes into account the SPIN and is the ratio of the number of spin states for the resonant state to the total number of spin states for the a+b system. Notes Total cross-section Replace Gf by G in the Breit-Wigner formula Elastic cross-section so Gf = Gi On peak of resonance (E=E0) Thus, From the measurement of stot and sel can infer g and hence spin of resonant state.

Example: (GeV) s (pb) GeV s)

Section IV - Feynman Diagrams

Theoretical Framework Macroscopic Microscopic Slow Classical Mechanics Quantum Mechanics Fast Special Relativity Quantum Field Theory ELECTROMAGNETISM: QUANTUM ELECTRODYNAMICS (QED) 1948 Feynman, Schwinger, Tomonaga (1965 Nobel Prize) ELECTROMAGNETISM: ELECTROWEAK UNIFICATION +WEAK 1968 Glashow, Weinberg, Salam (1979 Nobel Prize) STRONG: QUANTUM CHROMODYNAMICS (QCD) 1974 Politzer, Wilczek, Gross (2004 Nobel Prize) CMK : 2008 Nobel prize

Feynman Diagrams Results of calculations based on a single process in Time-Ordered Perturbation Theory depend on the reference frame. The sum of all time orderings is not frame dependent and provides the basis for our relativistic theory of Quantum Mechanics. The sum of all time orderings are represented by FEYNMAN DIAGRAMS. Space Time Time + Space = FEYNMAN DIAGRAM

Antimatter The negative energy solution is equivalent to a negative energy particle state travelling backwards in time  Interpret as a positive energy antiparticle travelling forwards in time. Then all solutions describe physical states with positive energy, going forward in time. Examples: e+e- annihilation pair creation g e+ e+ g <------- e- backwards in time = -------> e+ forwards in time but put e+ anyway !! arrows show charge flow All quantum numbers carried into vertex by e+, same as if viewed as outgoing e-.

Virtual Particles Forces arise due to the exchange of unobservable VIRTUAL particles. The mass of the virtual particle, q2, is given by and is not the physical mass m, i.e. it is OFF MASS-SHELL. The mass of a virtual particle can by +ve, -ve or imaginary. A virtual particle which is off-mass shell by amount Dm can only exist for time and range If q2 = m2, then the particle is real and can be observed. ħ =c=1 natural units

For virtual particle exchange, expect a contribution to the matrix element of where Qualitatively: the propagator is inversely proportional to how far the particle is off-shell. The further off-shell, the smaller the probability of producing such a virtual state. For m  0 ; e.g. single g exchange q2 0, very low energy transfer EM scattering COUPLING CONSTANT STRENGTH OF INTERACTION PHYSICAL (On-shell) mass VIRTUAL (Off-shell) mass PROPAGATOR

Feynman diagrams represent a term in the perturbation theory expansion of the matrix element for an interaction. Normally, a matrix element contains an infinite number of Feynman diagrams. But each vertex gives a factor of g, so if g is small (i.e. the perturbation is small) only need the first few. Example: QED Total amplitude Fermi’s Golden Rule Total rate

External Lines (visible particles) Internal lines (propagators) Spin ½ Particle Incoming Outgoing Antiparticle Spin 1 Spin ½ Particle (antiparticle) Spin 1 g, W and Z0 g Each propagator gives a factor of Each vertex gives a factor of the coupling constant, g.

q q e e p p u u n d d p d u e Examples: ELECTROMAGNETIC STRONG Electron-proton scattering Quark-antiquark annihilation WEAK u u n d d p d u e Neutron decay

SECTION V - QED

The Electromagnetic Vertex All electromagnetic interactions can be described by the photon propagator and the EM vertex: The coupling constant, g, is proportional to the fermion charge. Energy, momentum, angular momentum and charge always conserved. QED vertex NEVER changes particle type or flavour i.e. but not QED vertex always conserves PARITY

Pure QED Processes Compton Scattering (ge-ge-) e- g g e- Bremsstrahlung (e-e-g) Pair Production (g e+e-) e- g e- g e- e- g g e- Nucleus The processes e-e-g and g e+e-cannot occur for real e, g due to energy, momentum conservation. Ze Nucleus

p0 u c J/Y e+e- Annihilation p0 Decay J/+- The coupling strength determines “order of magnitude” of the matrix element. For particles interacting/decaying via EM interaction: typical values for cross-sections/lifetimes sem ~ 10-2 mb tem ~ 10-20 s u p0 c J/Y

Scattering in QED e e p p Examples: Calculate the “spin-less” cross-sections for the two processes: (1) (2) Fermi’s Golden rule and Born Approximation For both processes write the SAME matrix element is the strength of the interaction. measures the probability that the photon carries 4-momentum i.e. smaller probability for higher mass. e e p p Electron-proton scattering Electron-positron annihilation

“Spin-less” e-p Scattering q2 is the four-momentum transfer: Neglecting electron mass: i.e. and Therefore for ELASTIC scattering p p e e RUTHERFORD SCATTERING

Discovery of Quarks Virtual g carries 4-momentum Large q  High q wave-function oscillates rapidly in space and time  probes short distances and short time. Rutherford Scattering q2 small E = 8 GeV Excited states q2 increases E q2 large l<< size of proton q2 > 1 (GeV)2 Elastic scattering from quarks in proton

“Spin-less” e+e- Annihilation Same formula, but different 4-momentum transfer: Assuming we are in the centre-of-mass system Integrating gives total cross-section:

This is not quite correct, because we have neglected spin This is not quite correct, because we have neglected spin. The actual cross-section (using the Dirac equation) is Example: Cross-section at GeV (i.e. 11 GeV electrons colliding with 11 GeV positrons)

Experimental Tests of QED QED is an extremely successful theory tested to very high precision. Example: Magnetic moments of e,  : For a point-like spin ½ particle: Dirac Equation However, higher order terms introduce an anomalous magnetic moment i.e. g not quite 2. v v     O(1) O(a) O(a4) 12672 diagrams

O(a3) Agreement at the level of 1 in 108. QED provides a remarkable precise description of the electromagnetic interaction ! Experiment Theory

+     +     Higher Orders So far only considered lowest order term in the perturbation series. Higher order terms also contribute Second order suppressed by a2 relative to first order. Provided a is small, i.e. perturbation is small, lowest order dominates. Lowest Order +     Second Order Third Order +    

Running of  specifies the strength of the interaction between an electron and a photon. BUT a is NOT a constant. Consider an electric charge in a dielectric medium. Charge Q appears screened by a halo of +ve charges. Only see full value of charge Q at small distance. Consider a free electron. The same effect can happen due to quantum fluctuations that lead to a cloud of virtual e+e- pairs The vacuum acts like a dielectric medium The virtual e+e- pairs are polarised At large distances the bare electron charge is screened. At shorter distances, screening effect reduced and see a larger effective charge i.e. a. -Q + - e+ e- g

a increases with increasing q2 (i.e. closer to the bare charge) Measure a(q2) from etc a increases with increasing q2 (i.e. closer to the bare charge) At q2=0: a=1/137 At q2=(100 GeV)2: a=1/128 Can sum perturbation series and describe total interaction with an effective three level coupling - infinite series of Feynman diagrams with fixed EM is equivalent to single diagram with effective running EM(q2)