---
title: Deep Inelastic Scattering and the Parton Model
module: Quantum Chromodynamics
moduleNumber: 8
lessonNumber: 3
order: 803
summary: >
  Scattering electrons hard off a proton resolves pointlike constituents. This lesson
  sets up the deep-inelastic kinematics, defines the structure functions F1 and F2,
  and reads Bjorken scaling as the signature of free spin-half partons. The
  Callan-Gross relation fixes the parton spin, the structure function becomes a
  charge-weighted sum of parton distributions, and the slow logarithmic scaling
  violations expose the gluon through DGLAP evolution.
topics: [Quantum Chromodynamics]
draft: false
sources:
  - book: Halzen & Martin
    ref: "Ch. 8–9 (deep inelastic scattering, partons, scaling)"
  - book: Griffiths
    ref: "Ch. 8 §8.3–8.4 (electron-proton scattering, the parton model)"
  - book: Thomson
    ref: "Ch. 8 (deep inelastic scattering and the quark-parton model)"
  - book: Tong
    ref: "The Standard Model (Cambridge Part III), §3.3 (the parton model)"
---

Confinement hides quarks inside hadrons, yet [asymptotic
freedom](/particle-physics/qcd/asymptotic-freedom-confinement) 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](/particle-physics/quark-model/eightfold-way-su3) into dynamical
objects inside the proton.

Throughout, $\hbar = c = 1$ 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,

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

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

$$
W^2 = (p + q)^2 = M^2 + 2M\nu - Q^2 .
$$

- **Elastic scattering** leaves the proton intact, $W = M$, which forces
  $Q^2 = 2M\nu$. 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, $W \gg M$, reached at large
  $Q^2$ and large $\nu$. 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**

$$
x = \frac{Q^2}{2\,p\cdot q} = \frac{Q^2}{2M\nu},
\qquad 0 \le x \le 1,
$$

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.

$$
% caption: Deep inelastic scattering kinematics. The incoming electron of energy E
% radiates a virtual photon of spacelike virtuality Q-squared, scatters to energy
% E-prime, and the photon strikes a constituent of the proton. The hadronic system
% recoils with invariant mass W, well above the proton mass in the deep inelastic
% region. The Bjorken variable x is the constituent's momentum fraction.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  % electron in
  \draw[thick, ->] (-4.4,2.2) -- (-2.9,1.7);
  \draw[thick] (-2.9,1.7) -- (-2.0,1.4);
  \node[left, font=\scriptsize] at (-4.4,2.2) {electron, E};
  \fill (-2.0,1.4) circle (2.0pt);
  % electron out
  \draw[thick, ->] (-2.0,1.4) -- (-0.6,2.0);
  \draw[thick] (-0.6,2.0) -- (0.5,2.5);
  \node[right, font=\scriptsize] at (0.5,2.5) {scattered, E$'$};
  % virtual photon
  \draw[acc, thick] (-2.0,1.4)
    sin (-1.7,1.1) cos (-1.4,0.8) sin (-1.1,0.5) cos (-0.8,0.2) sin (-0.5,-0.1) cos (-0.2,-0.4);
  \node[acc, right, font=\scriptsize] at (-1.2,0.55) {photon, $Q^2$};
  % proton blob
  \draw[thick, fill=black!8] (-0.2,-0.9) ellipse (0.9 and 0.7);
  \node at (-0.2,-0.9) {p};
  % remnant + jet
  \draw[thick, ->] (0.6,-0.6) -- (2.0,-0.2);
  \draw[thick, ->] (0.6,-0.9) -- (2.1,-0.9);
  \draw[thick, ->] (0.6,-1.2) -- (2.0,-1.6);
  \node[black, font=\scriptsize, right] at (2.1,-0.9) {hadrons, W};
\end{tikzpicture}
$$

## 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

$$
\frac{\d^2\sigma}{\d x\,\d Q^2}
  = \frac{4\pi\alpha^2}{x\,Q^4}
    \left[\left(1 - y\right) F_2(x,Q^2)
      + x\,y^2\,F_1(x,Q^2)\right],
$$

with $y = \nu/E$ the fractional energy loss.[^hm-struct] 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:

$$
F_i(x, Q^2) \to F_i(x)
\qquad (\text{large } Q^2).
$$

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 $\xi$ of the proton
momentum. The struck-parton kinematics require $\xi = x$: the Bjorken variable is the
momentum fraction of the parton that absorbed the photon.

$$
% caption: The proton as a collection of partons. In a fast-moving frame each parton
% carries a fraction of the proton momentum; the virtual photon strikes a single
% parton elastically. Bjorken scaling means the interaction resolves pointlike
% constituents, so the structure functions depend on the momentum fraction x alone
% and not on the resolving power Q-squared.
\begin{tikzpicture}[>=stealth, font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  % proton envelope
  \draw[thick, fill=black!6] (-1.6,-1.0) rectangle (1.6,1.0);
  % partons carrying fractions
  \foreach \x/\yy in {-1.1/0.5, -0.4/0.7, 0.3/0.4, 1.0/0.6, -0.9/-0.5, 0.1/-0.6, 0.9/-0.4}
    \fill[black] (\x,\yy) circle (2.0pt);
  % the struck parton highlighted
  \fill[acc] (0.3,0.4) circle (2.0pt);
  \draw[acc, thick] (0.3,0.4) circle (5pt);
  % incoming photon
  \draw[thick] (-2.9,1.9)
    sin (-2.6,1.6) cos (-2.3,1.3) sin (-2.0,1.0) cos (-1.7,0.7) sin (-1.4,0.5) cos (0.3,0.4);
  \node[left, font=\scriptsize] at (-2.9,1.9) {photon};
  % proton momentum arrow
  \draw[thick, ->] (-2.0,-1.5) -- (2.0,-1.5);
  \node[below, font=\scriptsize] at (0,-1.6) {proton momentum};
  \node[acc, right, font=\scriptsize] at (1.7,0.4) {fraction $x$};
\end{tikzpicture}
$$

## 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-$\tfrac12$ quarks of charge $e_i$ gives the structure functions as charge-weighted
sums,

$$
F_2(x) = \sum_i e_i^2\,x\,f_i(x),
\qquad
2x\,F_1(x) = F_2(x).
$$

The second identity is the **Callan-Gross relation**. It holds because the photon
scatters off spin-$\tfrac12$ constituents: a spin-$\tfrac12$ 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

$$
\frac{2x F_1}{F_2} \approx 1
$$

across the scaling region is direct evidence that the charged partons carry
spin $\tfrac12$ — they are quarks, not scalars.[^grif-cg]

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 $\int (u - \bar u)\,\d x = 2$
and $\int (d - \bar d)\,\d x = 1$ recover two up quarks and one down. But the momentum
sum rule reveals something the quark model omitted:

$$
\sum_i \int_0^1 x\,f_i(x)\,\d x \approx 0.5 .
$$

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.

$$
% caption: Parton distributions of the proton against momentum fraction, weighted by
% x. The valence up and down quarks peak near x of one third; the sea quarks and
% especially the gluons dominate at small x. Integrating x f(x) shows the quarks carry
% about half the momentum and the gluon the other half.
\begin{tikzpicture}[font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,0) -- (8.4,0) node[right, font=\scriptsize] {x};
  \draw[->, black] (0,0) -- (0,4.0) node[above, font=\scriptsize] {$x f(x)$};
  \foreach \x/\lab in {0/0, 2.6/0.3, 5.2/0.6, 7.8/0.9}
    \node[below, black, font=\scriptsize] at (\x,-0.05) {\lab};
  % gluon: large at small x, falls steeply (dashed, muted)
  \draw[black, thick, densely dashed]
    (0.35,3.7) .. controls (1.2,1.6) and (2.0,0.8) .. (3.2,0.45)
    .. controls (4.6,0.25) and (6.5,0.12) .. (7.9,0.06);
  \node[black, font=\scriptsize] at (1.5,2.6) {gluon};
  % up valence: broad bump near x=1/3
  \draw[black, very thick]
    (0.3,0.5) .. controls (1.6,1.9) and (2.4,2.15) .. (3.0,2.0)
    .. controls (4.2,1.6) and (6.0,0.5) .. (7.9,0.05);
  \node[black, font=\scriptsize] at (3.3,2.35) {up valence};
  % down valence: smaller bump
  \draw[black, thick]
    (0.3,0.35) .. controls (1.5,1.15) and (2.3,1.3) .. (2.9,1.2)
    .. controls (4.0,0.95) and (6.0,0.3) .. (7.9,0.04);
  \node[black, font=\scriptsize] at (5.0,0.95) {down valence};
\end{tikzpicture}
$$

## 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 $q\bar q$ pair. The parton content therefore depends weakly on the
resolution scale, and the PDFs evolve according to the **DGLAP equations**,

$$
\frac{\partial f_i(x, Q^2)}{\partial\ln Q^2}
  = \frac{\alpha_s(Q^2)}{2\pi}
    \sum_j \int_x^1 \frac{\d z}{z}\,
      P_{ij}\!\left(\frac{x}{z}\right) f_j(z, Q^2),
$$

with splitting functions $P_{ij}$ giving the probability that parton $j$ radiates
parton $i$.[^hm-dglap] 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.

$$
% caption: Scaling and its violation. The structure function F2 against Q-squared at
% fixed x is nearly flat, the Bjorken scaling that revealed pointlike quarks. The
% residual logarithmic slope, rising at small x and falling at large x, is the QCD
% scaling violation predicted by the DGLAP evolution and is direct evidence for gluon
% radiation.
\begin{tikzpicture}[font=\footnotesize, scale=1.0]
  \definecolor{acc}{HTML}{4A6FA5}
  \draw[->, black] (0,0) -- (8.4,0) node[right, font=\scriptsize] {$Q^2$ (log)};
  \draw[->, black] (0,0) -- (0,3.8) node[above, font=\scriptsize] {$F_2$};
  % small-x: rises slightly
  \draw[black, very thick] (0.6,2.7) .. controls (3.0,2.95) and (5.5,3.2) .. (8.0,3.4);
  \node[black, right, font=\scriptsize] at (8.0,3.4) {small x};
  \fill[black] (1.5,2.75) circle (1.6pt); \fill[black] (3.5,2.98) circle (1.6pt);
  \fill[black] (5.5,3.2) circle (1.6pt); \fill[black] (7.3,3.35) circle (1.6pt);
  % mid-x: nearly flat
  \draw[black, very thick] (0.6,1.7) .. controls (3.0,1.68) and (5.5,1.62) .. (8.0,1.55);
  \node[black, right, font=\scriptsize] at (8.0,1.55) {mid x};
  % large-x: falls
  \draw[black, very thick] (0.6,0.9) .. controls (3.0,0.72) and (5.5,0.55) .. (8.0,0.42);
  \node[black, right, font=\scriptsize] at (8.0,0.42) {large x};
  \fill[black] (1.5,0.85) circle (1.6pt); \fill[black] (3.5,0.7) circle (1.6pt);
  \fill[black] (5.5,0.54) circle (1.6pt); \fill[black] (7.3,0.44) circle (1.6pt);
\end{tikzpicture}
$$

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](/particle-physics/qcd/jets-hadronization)
of the next lesson.

[^hm-struct]: The DIS cross section in terms of $F_1$ and $F_2$, the structure-function decomposition of the hadronic tensor, and the Bjorken variables $x$, $y$, $\nu$, $Q^2$ are developed in Halzen & Martin, _Quarks and Leptons_, Ch. 8, and Thomson, _Modern Particle Physics_, Ch. 8.

[^grif-cg]: The parton-model result $F_2 = \sum_i e_i^2\,x f_i(x)$ and the Callan-Gross relation $2x F_1 = F_2$ for spin-$\tfrac12$ 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](https://pdg.lbl.gov).

[^hm-dglap]: 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](https://pdg.lbl.gov).