Scattering Cross Sections (light projectile off heavy target) a + b  c + d counting number of final states counting number of initial states “flux” from (E)=dN/dE Note: If a and b are unpolarized (randomly polarized) the experiment cannot distinguish between or separate out the contributions from different possibilities, but measures the scattered total of all possible spin combinations: averaging over all possible (equally probable) initial states

mfi2 or m2fi2 fi fi or fi fi g fi fj fk We will find that everything is in fact derivable from a comprehensive Lagrangian (better yet a Lagrangian density, where . As a preview assume L describes some generic fields: fi, fj, fk wave functions…of matter fields …or interaction fields (potentials) Invariance principles (symmetries) will guarantee that schematically we will find 3 basic type of terms: mfi2 or m2fi2 1 mass terms: Fermion Boson 2 kinetic energy terms: fi fi or fi fi 3 interaction terms: g fi fj fk may also contain derivatives or >3 fields…or both

g Cross sections or decay rates are theoretically computed/predicted from the Matrix Elements (transition probabilities) pick out the relevant terms in L can be expanded in a series of approximations [the coefficients of each term being powers of a coupling constant, g] Griffiths outlines the Feynmann rules that translate L terms into M factors propagator mass & kinetic terms vertex g interaction terms g

Vertices get hooked together with propagators, with each vertex contributing one power of coupling to the calculation of the matrix element for the process. g g

You know from Quantum Mechanics: Amplitudes are, in general, COMPLEX NUMBERS In e+e- collisions, can’t distinguish All diagrams with the same initial and final states must be added, then squared. e- e- g e+ e+ + e- e- The cross terms introduced by squaring describe interference between the diagrams (sometimes suppressing rates!) g e+ e+

kinematic constraints d = [flux] × | M |2 × (E) × 4 initial state properties statistical factor counting the number of ways final state produced kinematic constraints on 4-momentum will be a Lorentz invariant phase space dp3 2E Matrix elements get squared electromagnetic 1 137 e e2  Basic vertex of any interaction introduces a coupling factor, g g weak g g2 GF10-5GeV-2 g2 is the minimum factor associated with any process…usually the expansion parameter (coefficient) of any series approximation for the matrix element W± strong gs gs2  s  0.1 g

Schrödinger’s equation For “free” particles (unbounded in the “continuum”) the solutions to Schrödinger’s equation with no potential Sorry!…this V is a volume appearing for normalization V

q q pi q = ki - kf =(pi-pf )/ħ pi momentum transfer the momentum given up (lost) by the scattered particle pi q = ki - kf =(pi-pf )/ħ pi

Proton-proton (strong interaction) cross sections The strong force has a very short effective range (unlike the coulomb force) If assume a simple “black disk” model with fixed geometric cross section: typical hadron size rp~1fm = 10-13cm (1 barn=10-24 cm2) This ignores the dependence on E or resonances, but from 1 to several 1000 GeV of beam energy its approximately correct!

35-40 mb pp collisions pp collisions Note: Elastic scattering 1/E dependence

Letting one cat out of the bag: Protons, anti-protons, neutrons are each composed of 3 quarks The (lighter) mesons (+, 0, -, K+, K0, K-, …) … 2 quarks Might predict: ~38mb ~42mb and ±p~ 25mb K±p~ 20mb

n  p + e- + e -  e- + e +  Ne*  Ne +  N  C + e + + e The transition rate, W (the “Golden Rule”) of initialfinal is also invoked to understand ab+c (+  ) decays Some observed decays n  p + e- + e -  e- + e +  Fundamental particle decays Ne*  Ne +  N  C + e + + e Pu  U +  20 10 20 10 13 7 13 6 Nuclear decays 236 94 232 92 How do you calculate an “overlap” between ???

J conserved. Any decay that’s possible will happen! It almost seems a self-evident statement: Any decay that’s possible will happen! What makes it possible? What sort of conditions must be satisfied? Total charge q conserved. J conserved.

probability of surviving to time t mean lifetime  = 1/ For any free particle (separation of space-time components) Such an expression CANNOT describe an unstable particle since Instead mathematically introduce the exponential factor:

a decaying probability then a decaying probability of surviving Note: G=għ Also notice: effectively introduces an imaginary part to E

Applying a Fourier transform: What’s this represent? E distribution of the unstable state still complex!

Expect some constant Breit-Wigner Resonance Curve

1.0  MAX 0.5  = FWHM E Eo When SPIN of the resonant state is included:

130-eV neutron resonances Transmission 130-eV neutron resonances scattering from 59Co -ray yield for neutron radiative capture

+p elastic scattering cross-section in the region of the Δ++ resonance. The central mass is 1232 MeV with a width =120 MeV

e+e- anything near the Z0 resonance plotted against cms energy Cross-section for the reaction e+e- anything near the Z0 resonance plotted against cms energy

In general: cross sections for free body decays (not resonances) are built exactly the same way as scattering cross sections. except for how the “flux” factor has to be defined DECAYS (2-body example) (2-body) SCATTERING enforces conservation of energy/momentum when integrating over final states in C.O.M. in Lab frame: Now the relativistic invariant phase space of both recoiling target and scattered projectile

Number scattered per unit time = (FLUX) × N × total density (a rate) of targets (a rate) /cm2·sec A concentration focused into a small spot and small time interval size of each target Notice: is a function of flux!