In deep inelastic electron-proton scattering, the structure function F_2(x) measures the momentum distribution of charged partons inside the proton. If the proton contained only three free quarks (uud), what would F_2(x) look like?
AThree delta functions at x = 1/3, each carrying one-third of the proton momentum
BA smooth distribution peaked at x = 1/3, because the quarks share momentum equally
CA broad distribution peaked at moderate x, because the quarks are bound and exchange momentum through gluon interactions — but the integral of F_2 over x would equal the sum of e_i^2 times the momentum fractions, which for three valence quarks would account for all the proton's momentum
DA flat distribution from x = 0 to x = 1
If quarks were non-interacting, each would carry exactly 1/3 of the proton momentum and F_2(x) would have delta-function peaks. In reality, gluon exchange smears the momentum distribution. The valence quarks produce a broad distribution peaked around x ~ 0.15-0.3. Crucially, integrating x*f(x) for all quarks gives only about 50% of the proton momentum -- the other 50% is carried by gluons, which are electrically neutral and invisible to photon exchange. This 'momentum sum rule' violation was key evidence for gluons.
Question 2 True / False
Bjorken scaling states that the proton structure functions F_1(x,Q^2) and F_2(x,Q^2) depend only on x and not on Q^2 at high Q^2. This scaling is exact in QCD.
TTrue
FFalse
Answer: False
Bjorken scaling holds at leading order in the parton model (non-interacting point-like quarks), but QCD corrections produce logarithmic violations: the structure functions depend weakly on Q^2 through terms proportional to alpha_s * ln(Q^2/mu^2). As Q^2 increases, gluon radiation produces more sea quarks at low x and depletes quarks at high x. These scaling violations are described by the DGLAP evolution equations and are one of the most precise tests of QCD. The violations were observed experimentally and their agreement with QCD predictions earned Gross, Politzer, and Wilczek the 2004 Nobel Prize.
Question 3 Short Answer
DIS experiments measure the ratio R = sigma_L/sigma_T of longitudinal to transverse virtual photon cross sections. The Callan-Gross relation predicts R = 0 for spin-1/2 partons. Why?
Think about your answer, then reveal below.
Model answer: A massless spin-1/2 particle conserves helicity, so it cannot absorb a longitudinally polarized virtual photon (which would require a helicity flip). This gives sigma_L = 0 and hence R = 0, or equivalently F_2 = 2xF_1 (the Callan-Gross relation). Experimentally, R is small but nonzero (a few percent), consistent with QCD corrections from gluon radiation and nonzero quark masses. If partons were spin-0 (scalar), sigma_T would vanish instead. The measurement of R ~ 0 at SLAC was direct evidence that partons have spin 1/2 and are therefore quarks.
The Callan-Gross relation connects the spin of the partons to a measurable ratio of cross sections. Its approximate validity was among the first confirmations that the point-like partons observed in DIS were indeed quarks.
Question 4 Multiple Choice
The kinematic variable x in DIS is often called 'Bjorken x.' In the infinite-momentum frame, x has a simple physical interpretation. What is it?
AThe scattering angle of the electron
BThe fraction of the proton's momentum carried by the struck parton
CThe energy of the virtual photon divided by the proton mass
DThe number of quarks participating in the interaction
In a frame where the proton has very large momentum P, a parton carrying momentum fraction x has 4-momentum xP. The virtual photon with 4-momentum q strikes this parton elastically, and the kinematics require x = Q^2/(2P*q) = Q^2/(2M*nu). This identification of x as the parton momentum fraction is exact in the parton model and receives small corrections in QCD. The distribution of partons as a function of x is encoded in the parton distribution functions f_i(x).