Outline of Topics discussed in the Lecture for Graduate Nuclear Physics 
(PHYS722/822 Fall 2003)

Nucleon Form Factors

Quasi-elastic scattering

• Impulse approximation (incoherent scattering of single nucleons inside nuclei)
• Scatter off proton moving inside nucleus A with initial momentum p, leading to a final state nucleus (A-1) moving with momentum -p and a final state proton moving with momentum p+q.
• Nonrelativistic approximation: Energy transfer n = [m_A-1 + E_exitation + T_kin(A-1) + m_p - m_A] + p²/2m_p + q²/2m_p + p^.q/m_p
• The first term is the missing energy E_miss (sum of separation energy, excitation energy and kinetic energy for the residual nucleus), typically of the order of 10's of MeV
• The second term is the part of the kinetic energy of the knocked-out proton (again from zero to 30 MeV)
• The third term is the result for free scattering from an unbound, stationary proton (more properly Q²/2m_p).
• The last term depends on the initial momentum of the proton and its direction relative to q. It can assume fairly large values, leading to a "broadening" of the quasi-elastic peak of several 100's of MeV at high energy.
• Relativistic (=correct) approach: Use 4-vectors for incoming electron, outgoing electron, and 4-momentum exchanged (= 4-momentum of virtual photon q = (n,q)). Initial 4-momentum of target at rest in lab (use whole nucleus!)  P = (M_A,0,0,0). Final state has 2 particles (other than electron): struck nucleon (proton) with momentum P₁ and residual A-1 nucleus with P_A-1. Write down 4-momentum conservation for initial (q + P) = final (P₁ + P_A-1) momenta.
  
• Cross section becomes ds/dW/dE' = ds/dW(free)*P(E'), where the probability distribution P(E') describes the likelihood of finding a proton with initial momentum p leading to a particular value of E' = E - n  according to above formulae. The width of this distribution can be used to extract an "experimental" value for the Fermi-momentum (largest possible momentum) for a given nucleus, and the shift from the free value Q²/2m_p can be interpreted as average missing energy E_miss. Early results from quasi-elastic inclusive scattering show good agreement with Fermi model and Shell model expectations.

Quasi-elastic scattering cont'd

• Measure ejected proton together with electron in final state (exclusive instead of inclusive):
• Completely determine initial momentum p, and therefore all terms except E_exitation in the formula for n (see last lecture)
• Can solve for E_exitation or E_miss , so we have complete kinematic information on initial momentum and energy of struck proton (this is unfortunately spoiled in the real world due to final state interactions - the proton does not escape the nucleus completely unperturbed).
• E_miss spectra show which "shell" the proton got knocked out of (sharp peaks for uppermost occupied shells, broad bumps for lower shells)
• Broadening due to correlations within nuclei: The total A-nucleon wave function is NOT a simple product of singe-nucleon energy eigenstates, but a superposition.
• Cut on different shells, study momentum distribution for protons in those shells. Findings: s-shell has maximum likelihood for zero momentum, while p-shell has zero likelihood for zero momentum (because of parity symmetry - wave function inverts sign under reflections).

Inelastic scattering

• Range of possible final states (ground state, excited states/resonances, break-up states) leading to range of possible final state energies for scattered electron => cross section ds/dW/dE'
• Isolated resonances: ds/dW/dE' = ds/dW*Sum[d(E'-E_f(R))] . E_f(R) is the final state energy of the electron, calculated using 4-momentum conservation, for the proper specific value of the resonance energy for a given resonance R.
• In reality: distribution can be broader if resonance has finite lifetime and therefore (Uncertainty principle) finite width.
• New kinematic variable: W = invariant mass (energy) of final state of the struck object (nucleon, nucleus) in its own center-of-mass frame.
• Formula: W² = m_A² + 2*m_A*n - Q² (if struck object had mass m_A)
• Elastic scattering ->  W = m_A -> n = Q²/2m_A
• Higher excitations -> larger W
• Quasi-elastic scattering -> full range of E' (or W) centered on (but slightly shifted from) elastic position for nucleon: Q²/2m_p
• New cross section formula: ds/dW/dE' = s_Mott[W₂(n,Q²) + 2tan²q/2 W₁(n,Q²)] (no recoil factor - included in W₂ and  W₁)
• Structure function W₁ contains magnetic/transverse excitation strength (think of wiggling a magnet with another magnet)
• Structure function W₂ contains both transverse and longitudinal (Coulomb) excitation strength

Inelastic scattering on the nucleon

• Same cross section formula
• W² = m_p² + 2*m_p*n - Q²
• W₂ and  W₁ are delta-functions at W = m_p and zero up to pion production threshold (no low-lying excitations known)
• At W = m_p + m_p (pion threshold; m_p = 0.14 GeV), W₂ and  W₁ begin to rise.
• Maxima at several resonance peaks (3 clearly visible in electron scattering)
• At even higher energy transfer (W > 2 GeV), smooth "scaling" curve ("quasielastic scattering from quarks")
• In the scaling region (W > 2 GeV, Q² > 1 GeV²), structure functions become functions of one variable only:
• W₁(n,Q²) = 1/m_p F₁(x)
• W₂(n,Q²) = 1/n F₂(x)
• x = Q²/2m_pn (Bjorken scaling variable)

(Deep) inelastic scattering

• Formalism, variable transformation to DQ², Dn
• W₂ and  W₁ as functions of invariant final state energy W: Elastic peak (delta-function), zero up to pion threshold (1078 MeV), meson production, resonances, deep inelastic regime
• Production of pions (p -> n + p⁺ , p -> p + p⁰ , n -> p + p^- , n -> n + p⁰ )
• Pions have I=1 (I₃ = -1,0,1 for p⁺ , p⁰ , p^- ), mass 133-139 MeV, spin 0, parity -. Lowest mass example of a class of subatomic particles called "mesons". All mesons have integer spin and interact strongly with protons and neutrons. Other examples: Kaons (K⁺, K^-, K⁰_L, K⁰_S), etas (h, h'), rhos (r), omegas (w), .... (all the way throught the Greek alphabeth).
• Protons in initial state can also be converted into other particles in the final state: neutrons, lambdas (L), sigmas (S),... (upper case Greek letters), in accordance with conserved quantum numbers. These particles (including the proton and its resonances) are called "baryons". All baryons have HALF-integer spins (1/2, 3/2, ...) and interact strongly with protons. Baryons are typically heavier than mesons, but there are some pretty heavy mesons, too.
• Resonant states are excited on top of "smooth" particle production spectrum: The lowest lying one is the Delta (D), which is a baryon with spin 3/2 and 4 different charge states: -,0,+,++ (isospin 3/2), and mass 1232 MeV. Higher lying states (P₁₁, S₁₁, D₁₃, F₁₅, ...) are broad and overlapping, leading to only 2 distinctive additional bumps.
• All resonance have "form factors" that fall off like the elastic proton form factors (leading to a decrease of W₂ and  W₁ as functions of Q² at fixed W).
• Deep inelastic scattering - smooth curve (sum of all resonance, multiple meson and baryon productions at W>2, independent of Q²  at Q² >1...3 GeV²)
• Better interpreted as quasi-elastic scattering from point-like dirac particles inside the proton (quarks)
• Interpretation: x = Q²/2mn is the fraction of the proton momentum carried by the struck quark, as measured in the Breit frame (where the virtual photon carries no energy)
• F₁(x) = 1/2 z_q² P_q(x), summed over all different kinds of quarks. P_q(x) is the probability of finding a quark of type q carrying momentum fraction x.
• F₂(x) = 2x F₁(x) (Callan-Gross relationship).
• For more details, see my writeup for the HUGS lecture series I gave in 1997.