the parton distribution function

A proton is not a tiny solid ball. It is a churning bag of quarks and gluons — collectively called partons — sharing the proton's total energy in constantly shifting proportions. When you collide protons, the real collision is between one parton from each proton, and you never get to choose which parton takes part or how much of the proton's energy it carries. The parton distribution function is the rulebook for this lottery: it tells you the probability that, when you reach into a proton, you grab a particular kind of parton carrying a particular fraction of the proton's momentum.

More precisely, a parton distribution function (PDF) gives, for each type of parton (up quark, down quark, gluon, and so on), the probability density of finding it carrying a fraction x of the proton's momentum. At small x the proton is dominated by gluons and a sea of quark-antiquark pairs; at large x the three 'valence' quarks (two up, one down) carry most of the momentum. Because the strong force cannot be calculated from first principles at these scales, PDFs are extracted from experimental data — chiefly from deep inelastic scattering — and then evolved to other energies using equations from quantum chromodynamics.

PDFs are unavoidable at hadron colliders like the LHC, because the colliding objects are protons, not pointlike particles. To predict the rate of any process, you must fold the parton-level cross section together with the PDFs of both protons. This makes PDFs a central ingredient in every Standard Model prediction and a leading source of theoretical uncertainty: imperfect knowledge of the gluon distribution, for instance, directly limits how precisely the Higgs production rate can be predicted. The honest point is that a PDF is not a fundamental constant of nature you can look up; it is a fitted, energy-dependent function carrying its own uncertainties, which then propagate into nearly every collider result.