PDFs depend on both x (parton momentum fraction) and Q^2 (resolution scale). The x dependence must be measured, but the Q^2 dependence is predicted by the DGLAP evolution equations. What physical process drives the Q^2 evolution?
AQuarks accelerate as the proton moves faster
BAt higher Q^2, the virtual photon resolves shorter distances and sees the quarks splitting into quark-gluon pairs and gluons splitting into quark-antiquark pairs — this QCD radiation redistributes momentum from high-x to low-x partons as Q^2 increases
CHigher Q^2 means more energy is available to create new quarks
DThe strong coupling constant changes with Q^2, making the proton expand
DGLAP evolution is driven by the splitting functions P_{ab}(z), which give the probability of parton a emitting parton b carrying fraction z of its momentum. A quark can radiate a gluon (P_{qg}), a gluon can split into a quark-antiquark pair (P_{gq}), and a gluon can radiate a gluon (P_{gg}). At higher Q^2, more of these splittings are resolved, populating the low-x region and depleting high-x partons. The splitting functions are calculable in perturbative QCD, so the Q^2 evolution is a firm prediction.
Question 2 Short Answer
The gluon PDF g(x, Q^2) is the dominant parton distribution at small x. At the LHC (Q^2 ~ 10^4 GeV^2), more than 50% of the proton's momentum is carried by gluons. Yet gluons cannot be directly probed by virtual photons in DIS. How is the gluon PDF determined?
Think about your answer, then reveal below.
Model answer: The gluon PDF is determined indirectly from several sources: (1) the momentum sum rule — the gluon carries whatever momentum fraction is not accounted for by quarks; (2) scaling violations in F_2 — the Q^2 dependence of the quark distributions is driven by gluon splitting, so measuring dF_2/d(ln Q^2) constrains g(x); (3) the longitudinal structure function F_L, which is directly proportional to g(x) at leading order; (4) jet production cross sections in pp and ppbar collisions, which are dominated by gluon-gluon and quark-gluon scattering; and (5) direct photon production and heavy quark production. Modern PDF fits combine all these data in global QCD analyses.
The indirect nature of gluon PDF extraction means it carries larger uncertainties than the quark PDFs, especially at very small and very large x. This is a major source of systematic uncertainty for LHC cross section predictions.
Question 3 Multiple Choice
Two major collaborations (CT, MSHT, NNPDF) provide independent PDF sets. These are critical inputs for predicting cross sections at the LHC. Why do different PDF sets give different predictions, and how are the differences handled?
AThey use different experimental data and so are measuring different things
BThey use similar data but different methodological choices — functional forms for the x dependence, treatment of experimental uncertainties, perturbative order, heavy quark schemes — so their central values and uncertainty bands differ. Cross section predictions quote a 'PDF uncertainty' by comparing results across sets or using the error sets provided by each group
CThe differences are negligible and purely cosmetic
DOnly one collaboration is correct; the others are deprecated
PDF extraction is a complex inference problem: parameterize the x dependence at a starting scale Q_0^2, evolve to all other scales using DGLAP, and fit to thousands of data points. Different groups make different choices about parameterization flexibility (NNPDF uses neural networks with minimal assumptions; CT/MSHT use fixed functional forms), data selection, and uncertainty propagation. The resulting differences are real and propagated to LHC predictions as 'PDF uncertainties,' which are often the dominant theoretical uncertainty for precision cross sections like Higgs production via gluon fusion.