Deep Inelastic Scattering and the Parton Model

Confinement hides quarks inside hadrons, yet asymptotic freedom guarantees that a sufficiently hard probe sees them as nearly free. Firing high-energy electrons at a proton and measuring how they scatter is the cleanest such probe: the electron couples through a single virtual photon whose calculable QED vertex factors out, leaving the proton's internal structure encoded in a pair of measurable functions. The 1968 SLAC experiments found those functions nearly independent of the probe's resolution — Bjorken scaling — the hallmark of pointlike constituents, which a smooth charge distribution forbids. This lesson develops the kinematics, the structure functions, and the parton-model reading that turned the abstract quarks of the eightfold way into dynamical objects inside the proton.

Throughout, $ℏ = c = 1$ and the proton mass is $M$ .

Elastic versus deep inelastic scattering

An electron of four-momentum $k$ scatters to $k^{'}$ off a proton of momentum $p$ , exchanging a virtual photon of momentum $q = k - k^{'}$ . Two invariants describe the photon: its spacelike virtuality and the energy it transfers,

Q^{2} \equiv - q^{2} > 0, ν \equiv \frac{p \cdot q}{M} = E - E^{'},

the second equality holding in the proton rest frame with $E, E^{'}$ the incident and scattered electron energies. The invariant mass of the struck hadronic system is

W^{2} = (p + q)^{2} = M^{2} + 2 M ν - Q^{2} .

Elastic scattering leaves the proton intact, $W = M$ , which forces $Q^{2} = 2 M ν$ . The recoiling proton is a single particle, and the cross section falls steeply with $Q^{2}$ through the proton form factors, the mark of an extended charge distribution.
Deep inelastic scattering (DIS) shatters the proton, $W ≫ M$ , reached at large $Q^{2}$ and large $ν$ . Here the cross section falls far more slowly than the elastic form factors predict — the signature that the photon is striking something pointlike inside.

The dimensionless Bjorken variable

x = \frac{Q ^{2}}{2 p \cdot q} = \frac{Q ^{2}}{2 M ν}, 0 \leq x \leq 1,

measures the inelasticity: $x = 1$ is elastic scattering, and $x < 1$ opens the inelastic region. Its parton interpretation, developed below, is the fraction of the proton momentum carried by the struck constituent.

Deep inelastic scattering kinematics. The incoming electron of energy E radiates a virtual photon of spacelike virtuality Q-squared, scatters to energy E-prime, and the photon strikes a constituent of the proton. The hadronic system recoils with invariant mass W, well above the proton mass in the deep inelastic region. The Bjorken variable x is the constituent's momentum fraction.

Structure functions

Because the electron vertex is pure QED, the DIS cross section factorizes into a calculable leptonic tensor and an unknown hadronic tensor. Lorentz invariance and current conservation restrict the hadronic part to two scalar structure functions $F_{1}$ and $F_{2}$ , each a function of the two invariants $x$ and $Q^{2}$ . The double-differential cross section in the proton rest frame is

\frac{d ^{2} σ}{d x d Q ^{2}} = \frac{4 π α ^{2}}{x Q ^{4}} [(1 - y) F_{2} (x, Q^{2}) + x y^{2} F_{1} (x, Q^{2})],

with $y = ν / E$ the fractional energy loss.¹ Everything not known from QED is packed into $F_{1}$ and $F_{2}$ ; measuring them across the $(x, Q^{2})$ plane is the experimental program. For a structureless point charge the analogous functions would depend on $Q^{2}$ strongly; for the proton, the question is how they actually behave.

Bjorken scaling

The SLAC result was that at large $Q^{2}$ the structure functions depend only on $x$ , not on $Q^{2}$ separately:

F_{i} (x, Q^{2}) \to F_{i} (x) (large Q^{2}) .

This is Bjorken scaling. A form factor that depends on $Q^{2}$ signals a characteristic size; independence from $Q^{2}$ signals no size at all — the photon scatters off pointlike constituents. Bjorken and Feynman read the result as elastic scattering off free partons, later identified with the quarks. In the frame where the proton moves fast, time dilation freezes the partons during the brief interaction, so the photon strikes one free parton carrying a fraction $ξ$ of the proton momentum. The struck-parton kinematics require $ξ = x$ : the Bjorken variable is the momentum fraction of the parton that absorbed the photon.

The proton as a collection of partons. In a fast-moving frame each parton carries a fraction of the proton momentum; the virtual photon strikes a single parton elastically. Bjorken scaling means the interaction resolves pointlike constituents, so the structure functions depend on the momentum fraction x alone and not on the resolving power Q-squared.

The parton distributions and the Callan-Gross relation

Let $f_{i} (x) d x$ be the number of partons of type $i$ carrying momentum fraction in $[x, x + d x]$ — the parton distribution functions (PDFs). Scattering off free spin- $\frac{1}{2}$ quarks of charge $e_{i}$ gives the structure functions as charge-weighted sums,

F_{2} (x) = i \sum e_{i}^{2} x f_{i} (x), 2 x F_{1} (x) = F_{2} (x) .

The second identity is the Callan-Gross relation. It holds because the photon scatters off spin- $\frac{1}{2}$ constituents: a spin- $\frac{1}{2}$ parton has a nonzero magnetic coupling that ties the transverse structure function $F_{1}$ to the longitudinal $F_{2}$ . Spin- $0$ partons would give $F_{1} = 0$ instead. The measured ratio

\frac{2 x F _{1}}{F _{2}} \approx 1

across the scaling region is direct evidence that the charged partons carry spin $\frac{1}{2}$ — they are quarks, not scalars.²

For the proton, treating $u$ and $d$ as the valence content and integrating the PDFs reproduces the quark-model quantum numbers: the number sum rules $\int (u - \overset{u}{ˉ}) d x = 2$ and $\int (d - \overset{ˉ}{d}) d x = 1$ recover two up quarks and one down. But the momentum sum rule reveals something the quark model omitted:

i \sum 0 \int 1 x f_{i} (x) d x \approx 0.5.

The charged quarks carry only about half the proton's momentum. The missing half is carried by electrically neutral constituents invisible to the photon — the gluons. DIS measures the quark momentum and, by subtraction, discovers that the gluon field carries the rest.

Parton distributions of the proton against momentum fraction, weighted by x. The valence up and down quarks peak near x of one third; the sea quarks and especially the gluons dominate at small x. Integrating x f(x) shows the quarks carry about half the momentum and the gluon the other half.

Scaling violations and the discovery of gluons

Bjorken scaling is only approximate. Measured over a wide range of $Q^{2}$ , the structure functions drift logarithmically: at fixed $x$ , $F_{2}$ rises with $Q^{2}$ at small $x$ and falls at large $x$ . QCD predicts precisely this drift. Increasing $Q^{2}$ sharpens the probe, resolving each quark into a quark plus a radiated gluon, and each gluon into a $q \overset{q}{ˉ}$ pair. The parton content therefore depends weakly on the resolution scale, and the PDFs evolve according to the DGLAP equations,

\frac{\partial f _{i} ( x , Q ^{2} )}{\partial ln Q ^{2}} = \frac{α _{s} ( Q ^{2} )}{2 π} j \sum x \int 1 \frac{d z}{z} P_{ij} (\frac{x}{z}) f_{j} (z, Q^{2}),

with splitting functions $P_{ij}$ giving the probability that parton $j$ radiates parton $i$ .³ The equations do not predict the PDFs at any one scale — those are measured — but they predict how the PDFs change with $Q^{2}$ , and the data follow the prediction precisely. The scaling violations are the clearest DIS evidence for the gluon: a proton of quarks alone would scale exactly, and only the quark-gluon coupling produces the observed logarithmic drift.

Scaling and its violation. The structure function F2 against Q-squared at fixed x is nearly flat, the Bjorken scaling that revealed pointlike quarks. The residual logarithmic slope, rising at small x and falling at large x, is the QCD scaling violation predicted by the DGLAP evolution and is direct evidence for gluon radiation.

The parton model, corrected by DGLAP evolution, turns the proton into a calculable object: a set of measured PDFs plus perturbative QCD evolution predicts cross sections at every collider. The same partons reappear as the initial states of hadron-collider reactions and the final states of the jets of the next lesson.

The DIS cross section in terms of $F_{1}$ and $F_{2}$ , the structure-function decomposition of the hadronic tensor, and the Bjorken variables $x$ , $y$ , $ν$ , $Q^{2}$ are developed in Halzen & Martin, Quarks and Leptons, Ch. 8, and Thomson, Modern Particle Physics, Ch. 8. ↩
The parton-model result $F_{2} = \sum_{i} e_{i}^{2} x f_{i} (x)$ and the Callan-Gross relation $2 x F_{1} = F_{2}$ for spin- $\frac{1}{2}$ partons are derived in Halzen & Martin, Ch. 9, and Griffiths, Introduction to Elementary Particles, §8.4. The momentum sum rule and the inference that gluons carry about half the proton momentum are in the same chapters; numerical PDF values from the Particle Data Group, pdg.lbl.gov. ↩
The DGLAP evolution equations and splitting functions, and the interpretation of scaling violations as evidence for gluon radiation, are given in Halzen & Martin, Ch. 9, and Thomson, Ch. 8. Modern parton distributions and their $Q^{2}$ evolution are compiled by the Particle Data Group, pdg.lbl.gov. ↩

Elastic versus deep inelastic scattering

Structure functions

Bjorken scaling

The parton distributions and the Callan-Gross relation

Scaling violations and the discovery of gluons

Footnotes