Particle physics
8 min read
Why the proton isn’t a hard sphere
Firing protons
If you fire a high-energy electron at a proton, you do not see a neat elastic “bounce” off a single object. Instead, above a certain energy you start producing a messy spray of hadrons: the proton has been broken up. That regime is deep inelastic scattering (DIS), and it is the cleanest experimental window into the proton’s internal, momentum-sharing structure.
The DIS picture in one diagram
In DIS, an incoming lepton exchanges a highly virtual photon (or Z) with the proton. The key kinematic variable is Q^2 ≡ −q^2, the (positive) squared momentum transfer carried by the virtual boson. Large Q^2 means short distance scales: you are “resolving” smaller structures.
The parton model idealisation is:
the virtual photon hits a single parton (quark or gluon) inside the proton,
that parton carries a fraction x of the proton’s longitudinal momentum,
after the hit, the struck parton hadronises into a jet-like final state.
Bjorken-x: what does it actually mean?
The variable that captures “how much of the proton” you hit is Bjorken-x: x≡Q^2/(2 p ⋅ q), where p is the proton four-momentum and q is the momentum transfer. In the simple parton model, this x is literally the momentum fraction of the struck parton.
So:
large x ∼ 0.3 − 1 probes rare configurations where one quark carries a big chunk of the proton momentum (the “valence” region),
small x ≪ 0.1 probes the sea of quark–antiquark pairs and (especially) gluons.
PDFs: turning the proton into probabilities
A parton distribution function f_i(x,Q^2) is (roughly) the probability density to find a parton of type i carrying momentum fraction x when probed at resolution Q^2. The dependence on Q^2 is crucial: the proton looks different when you zoom in.
Practically, PDFs are not computed from first principles for all x and Q^2 in collider phenomenology. They are extracted from data (DIS, Drell–Yan, jet production, W/Z production…) in large global fits.
Scaling and scaling violations
Early DIS data suggested Bjorken scaling: structure functions looked approximately like functions of x alone, not Q^2. That is exactly what you’d expect if the proton were made of point-like constituents.
But QCD predicts a controlled breaking of this: scaling violations.
Why? Because at higher Q^2 you resolve additional radiation:
a quark can radiate a gluon before being struck,
a gluon can split into a quark–antiquark pair,
cascades of splittings reshuffle momentum towards lower x.
This evolution is encoded in the DGLAP equations: PDFs “run” with Q^2. Observing that running was (and remains) a major success of QCD.
Why PDF uncertainties matter (especially at colliders)
Even if the partonic calculation cross section is good, you still need the f_i. Uncertainties in PDFs therefore become uncertainties in predicted rates and shapes.
This is most important in two places:
Extreme kinematics: very high masses or very forward jets require large x, where PDFs are less constrained.
Background-limited searches: small mismodellings in the high-energy tails can look like an excess. A “new physics” bump is only convincing if the Standard Model prediction, PDFs included, has been stress-tested.
DIS taught us that the proton is a fluctuating, scale-dependent many-body system, and PDFs are the quantitative bookkeeping. If you want to claim a discovery, you must know, not guess, what the proton was likely doing when you collided it.