Lecture 2 - Feynman Diagrams & Experimental Measurements From drawing diagrams of interactions to calculating scattering cross-sections In this lecture only for electromagnetic interactions between (hypothetical) spinless charged elementary particles 15/1/10 Particle Physics Lecture 2 Steve Playfer

Particle Physics Lecture 2 Steve Playfer Two-Body Collisions p3 (scattered) LAB Frame p2= 0 “Fixed target” experiments q y p1 (beam) p4 (recoil) p3=pf* p1=pi* q* CM Frame “Collider” experiments p2= -pi* p4= -pf* Elastic scattering has pi*= pf* 15/1/10 Particle Physics Lecture 2 Steve Playfer

Drawing Feynman Diagrams “virtual” particles are exchanged In the middle Initial state particles are on the left Final state particles are on the right Each interaction vertex has a coupling constant Warning – do not interpret diagrams literally as time (x) and space (y) coordinates! fermions antifermions photons gluons W, Z, H bosons 15/1/10 Particle Physics Lecture 2 Steve Playfer

Particle Physics Lecture 2 Steve Playfer The Feynman Rules Initial and final state particles have wavefunctions: Spin 0 bosons are plane waves Spin 1/2 fermions have Dirac spinors Spin 1 bosons have polarization vectors em Vertices have dimensionless coupling constants: Electromagnetism has √(4pa) = e Strong interaction has √as = gs Weak interactions have gL and gR (or cA and cV) Virtual particles have propagators: Virtual photon propagator is 1/q2 Virtual W boson propagator is 1/(q2 –MW2) Virtual fermion propagator is (gmqm + m)/(q2 – m2) (see Lecture 3 for definition of gamma matrices gm) 15/1/10 Particle Physics Lecture 2 Steve Playfer

Relativistic Kinematics Four momenta of initial and final state particles: pm = [ E, p ] with m2 = E2 – |p|2 Four momenta of virtual particles are “off the mass shell” qm = [ q0, q ] with q2 = q02 – |q|2 ≠ m2 Each component of four momentum is conserved Four momentum is conserved at each vertex of a Feynman diagram Lorentz transformations between LAB and CM frames p’x = g(px – bE) E’ = g(E – bpx) with b = p*/E* and g2= 1/(1–b2) Apply same Lorentz transformation to all initial and final state particles Amplitudes must be Lorentz invariant. All products of two four vectors are Lorentz invariants. Examples on this page are q2 and m2 15/1/10 Particle Physics Lecture 2 Steve Playfer

Particle Physics Lecture 2 Steve Playfer Mandelstam Variables The variables s,t and u are Lorentz invariants p1 s-channel p3 p1 t-channel p3 t = (p1- p3)2 s = (p1+ p2)2 p2 p4 p2 p4 For highly relativistic elastic scattering where p~E, m << E: s = 4 p*2 t = -2p*2 (1-cosq*) u = -2p*2 (1+cosq*) where p* is the CM momentum of the particles, and q* is the CM scattering angle p1 u-channel p4 u = (p1- p4)2 p2 p3 15/1/10 Particle Physics Lecture 2 Steve Playfer

Particle Wavefunctions A free particle is a plane wave: y = N e –ip.x where p.x = pmxm = h(k.x – wt) (this wavefunction contains no spin, charge or colour information) Probability density: r = i [ y* dy - y dy* ] dt dt Normalisation of density of states in a box of Volume V: ∫V r d3x = 2E (usually take V=1, N=1) Probability current: jm = i [ y* dmy - y dmy* ] For a particle transition p1 ® p3: jm = (p1 + p3 ) e –i(p3-p1).x 15/1/10 Particle Physics Lecture 2 Steve Playfer

Scattering Cross-sections A cross-section describes the rate of scattering during a collision between two particles. Units are barns (1 b = 10-28 m2). Cross-section: ds = Wfi = 2p |Mfi|2 rf ri ri where Wfi is the transition rate and Mfi is the matrix element ri is an initial state flux factor (number of states and relative velocity) rf = dNf/dE is the final state phase space factor (density of final states) Total cross-section s=∫ds is a Lorentz invariant physical observable 15/1/10 Particle Physics Lecture 2 Steve Playfer

Initial & Final State Factors Each initial state particle has 2Ei states. Flux factor also requires relative velocity between particles vi ri = 2EA 2EB vi Lorentz invariant final state phase space has a density per final state particle of: 1 d3p rf = 1 Pj=1,n d3pj (2p)3 2E (2p)(3n-4) 2Ej For a relativistic two-body scattering process A+B ® C+D in the CM system the three momenta are pA = -pB = pi pC = -pD = pf ri = 4pi√s rf = 1 pf dW 16p2 √s where √s is the total CM energy 15/1/10 Particle Physics Lecture 2 Steve Playfer

Differential Cross-section The differential cross-section for A+B ® C+D in the CM system is: ds = 1 1 pf |Mfi |2 dW 64p2 s pi N.B. Solid angle element dW = d(cos q) df depends on frame of reference because cos q* ≠ cos qlab (see Dynamics & Relativity lectures) Angular dependence of the differential cross-section is contained in the matrix element Mfi For elastic scattering note that pi = pf = p* For inelastic scattering usually have pf < pi 15/1/10 Particle Physics Lecture 2 Steve Playfer

Matrix Element for Spinless Scattering Hypothetical interaction in which two spinless charged particles exchange a virtual photon (and we ignore higher order diagrams) q = p1– p3 = p4 – p2 √a Vertex Couplings p2 p4 M = a (p1 + p3) (p2 + p4) d4(p1 + p2– p3 – p4) q2 Photon Propagator Four momentum conservation Plane Waves M = a (s – u) (in terms of Mandelstam variables) t 15/1/10 Particle Physics Lecture 2 Steve Playfer

Spinless differential cross-section The amplitude for a transition from an initial state to a final state is given by the sum of all possible Feynman diagrams. Spinless matrix element for two non-identical particles: M = a (u - s) (t-channel) t but for two identical particles it becomes: M = a (u - s) + a (t – s) t u (t-channel + u-channel) because you can’t tell which final state particle is which! For elastic scattering of identical spinless particles: ds = a2 (u - s) + (t - s) 2 dW 64p2s t u [ ] 15/1/10 Particle Physics Lecture 2 Steve Playfer

Including Polarization If the two-body scattering process A+B ® C+D involves fermions, the Matrix element has to include the initial and final spin states: Mfi ( ) Mfi ( ) Mfi ( ) Mfi ( ) These are not necessarily the same! If fermion spins are observed we measure polarized cross-sections If spins are not known the cross-section is unpolarized To calculate unpolarized cross-sections: Initial spin states are averaged over Final spin states are summed over 15/1/10 Particle Physics Lecture 2 Steve Playfer

Particle Physics Lecture 2 Steve Playfer Particle Decays m1 M m2 -p* p* Two-Body M > m1 + m2 p*2 = (M2 – m12 – m22)2 – 4m12m22 4M2 p2 m2 m1 M p1 Three-Body m3 p3 M > m1 + m2 + m3 -p1 = p2 + p3 15/1/10 Particle Physics Lecture 2 Steve Playfer

Decay Rates & Lifetimes Partial decay width of a particle mass M to a final state f: Gf = h Wfi = 2p |Mfi |2 rf For a decay to a two-body final state (m1, m2) : rf = 1 pf with E1 = √(m12 + pf2) = M2 + m12 – m22 16p2 M2 2M For several possible decay modes the total decay width and branching fractions are: G = Sf Gf B (i®f) = Gf / G The proper lifetime in the rest frame is t = h/G and the decay length in a moving frame is L = gbct 15/1/10 Particle Physics Lecture 2 Steve Playfer

Particle Physics Lecture 2 Steve Playfer Three-body Decays The final state phase space is given by: rf = 1 Pj=1,3 d3pj (2p)5 2Ej The allowed kinematic region is determined by the available energy Q : Q = M - Sj=1,3 mj = Sj=1,3 Tj (where Tj are the Kinetic energies of the final state particles) Using the centre of mass energy/momentum constraints: √s = M = √(M122 + p122 ) + √(m32 + p122 ) (and similarly for p13 and p23) where M122 = (E1 + E2)2 - p122 and p12 = (p1 + p2) From this it can be shown that the invariant mass squared of 1 & 2 is: M122 = s + m32 – 2√s E3 It is usual to represent the kinematic region by a “Dalitz plot” The axes of these plots are either M122 and M232 or T3 and T1 15/1/10 Particle Physics Lecture 2 Steve Playfer

Particle Physics Lecture 2 Steve Playfer Dalitz Plots T1 T1 Non-relativistic limit T=p2/2m Relativistic limit T=E Allowed regions for three identical final state particles T2 T3 T2 T3 B ® ppp In general the Matrix element depends on the position in the Dalitz plot (M122, M232) or (T3,T1) These areas removed because of background 15/1/10 Particle Physics Lecture 2 Steve Playfer