Experiment: Deep Inelastic Scattering

Contents

Historical context and the prediction under test
Apparatus
Procedure
Observations and data
Interpretation
Scaling violations and the confirmation of QCD
Extensions and modern precision
Primary references

Tests Phenomenon 102.45, Phenomenon 102.47, Equation (102.56) and Equation (102.59). Assuming Definition 102.5, Proposition 102.6 and Definition 102.44.

Rutherford found the nucleus by observing that a few alpha particles came back; the SLAC–MIT collaboration found the quarks the same way, by observing that electrons scattered from hydrogen at large angles far more often than a soft, extended proton would allow. This chapter is the experimental counterpart of Quantum Chromodynamics: it reports the electron-scattering measurements carried out in End Station A at the Stanford Linear Accelerator Center between 1967 and 1972, the Bjorken scaling they exhibited, the Callan–Gross relation that identified the constituents as spin-$\tfrac{1}{2}$ objects, and the missing momentum fraction that required a neutral, non-scattering carrier — the gluon. It then follows the same observable forward: the logarithmic scaling violations measured at high statistics by muon and neutrino beams and at HERA are the most direct quantitative confirmation of quantum chromodynamics that exists, because the theory predicts not the structure functions themselves but exactly how they must change with $Q^{2}$.

The prediction under test is a single sentence: the proton contains point-like charged constituents. It is falsifiable in the strongest sense, because a proton whose charge is smoothly spread over its measured size predicts an inelastic rate at large momentum transfer smaller than the observed one by orders of magnitude. The chapter is laid out in the five rubrics that editorial rule 4 requires of an experiment chapter, and that Experiment: Time Dilation and Relativistic Kinematics exhibits — apparatus, procedure, observations with numbers and uncertainties in SI units, interpretation, and primary references — with the historical setting first and the later high-precision programme placed between interpretation and references, because it is the same measurement continued. It sits at the end of Part XI — Quantum Field Theory and the Standard Model with Cosmic Rays and Astroparticle Physics; its theoretical neighbours are Quantum Chromodynamics and The Renormalization Group and, for the leptonic probe, Quantum Electrodynamics and Renormalization and Weak Interactions. The standard textbook account is [Halzen:1984]; current structure function and parton-distribution data are collected in [Navas:2024].

Remark 116.1 (Units, and how to read this chapter against the literature).

Every equation here carries $\hbar$ and $c$ explicitly, as Quantum Chromodynamics does. The deep inelastic literature universally sets $\hbar=c=1$ and writes $Q^{2}$ in $\mathrm{GeV}^{2}$; to map that literature onto this chapter, restore one factor of $c$ for every power of momentum and one factor of $\hbar c$ for every power of length. Concretely: a four-momentum $q$ carries $\mathrm{kg}\,\mathrm{m}/\mathrm{s}$, so the invariant $Q^{2}:=-q^{2}$ carries $\mathrm{kg}^{2}\,\mathrm{m}^{2}/\mathrm{s}^{2}$ and the literature's “$Q^{2}$ in $\mathrm{GeV}^{2}$” is $Q^{2}c^{2}$ here. The conversion factors used throughout are

\begin{equation}\tag{116.1} 1\,\mathrm{GeV}=1.602176634\times 10^{-10}\,\mathrm{J}\ec\qquad 1\,\mathrm{GeV}^{2}/c^{2} =2.8561\times 10^{-37}\,\mathrm{kg}^{2}\,\mathrm{m}^{2}/\mathrm{s}^{2} \ec \end{equation}

the first exact because the 2019 SI fixes the elementary charge, and the second following from it, together with $\hbar c=3.16152677\times 10^{-26}\,\mathrm{J}\,\mathrm{m}$. Areas quoted in the older literature in $\mathrm{cm}^{2}$ are given here in $\mathrm{m}^{2}$, with $1\,\mathrm{cm}^{2}=10^{-4}\,\mathrm{m}^{2}$. No derivation in this chapter is carried out in natural units.

Historical context and the prediction under test

Elastic scattering and the proton form factors

Before the proton could be found to have constituents it had to be measured as a whole, and the instrument for that was elastic electron scattering, $e+p\to e+p$, in which the proton survives.

Definition 116.2 (The Rosenbluth cross section).

For one-photon exchange from a proton with an anomalous magnetic moment, Lorentz invariance, current conservation and parity allow exactly two form factors, the electric $G_{E}(Q^{2})$ and the magnetic $G_{M}(Q^{2})$, both dimensionless functions of the invariant momentum transfer alone. The elastic cross section is then

\begin{equation}\tag{116.2} \frac{\dd\sigma}{\dd\Omega} =\sigma_{\mathrm{Mott}}\,\frac{E'}{E} \left[\frac{G_{E}^{2}+\tau G_{M}^{2}}{1+\tau} +2\tau G_{M}^{2}\tan^{2}\frac{\vartheta}{2}\right]\ec\qquad \tau:=\frac{Q^{2}}{4M^{2}c^{2}}\ec \end{equation}

with $\sigma_{\mathrm{Mott}}$ the point-charge cross section Equation (102.50), $M$ the proton mass, $\vartheta$ the electron scattering angle, $E$ and $E'$ the incident and scattered electron energies, and the recoil factor

\begin{equation}\tag{116.3} \frac{E'}{E}=\left[1+\frac{2E}{Mc^{2}} \sin^{2}\frac{\vartheta}{2}\right]^{-1}\ep \end{equation}

Both $\tau$ and $E'/E$ are dimensionless: $Q^{2}$ carries $\mathrm{kg}^{2}\,\mathrm{m}^{2}/\mathrm{s}^{2}$ and so does $M^{2}c^{2}$ [Rosenbluth:1950].

Remark 116.3 (Why the formula is written this way).

The bracket in Equation (116.2) is linear in $\tan^{2}(\vartheta/2)$ at fixed $Q^{2}$. Measuring at several angles and several beam energies chosen to keep $Q^{2}$ fixed therefore gives a straight line whose slope is $2\tau G_{M}^{2}$ and whose intercept is $(G_{E}^{2}+\tau G_{M}^{2})/(1+\tau)$: the two form factors are separated with no model of the proton whatever. This is the Rosenbluth separation, and exactly the same device, applied to the inelastic cross section, separates the two inelastic structure functions in Section 116.3.2. The experimental technique of the deep inelastic programme is in this sense a direct continuation of the elastic one.

Phenomenon 116.4 (The proton is soft and extended).

The elastic form factors of the proton fall steeply with momentum transfer, and over the range measured in the 1950s they are described to within about $10\,\mathrm{\%}$ by the dipole

\begin{equation}\tag{116.4} G_{E}(Q^{2})=\frac{G_{M}(Q^{2})}{\mu_{p}} =\left[1+\frac{Q^{2}c^{2}}{\Lambda^{2}}\right]^{-2}\ec\qquad \Lambda^{2}=0.71\,\mathrm{GeV}^{2}\ec \end{equation}

corresponding to a root mean square charge radius near $0.8\,\mathrm{fm}$ [Hofstadter:1956]. The proton is therefore not a point: it has a size, and at momentum transfers large compared with $\Lambda/c$ the probability of leaving it intact collapses. Rests on Definition 116.2, Equation (116.3) and Equation (102.50).

Derivation. Derives Phenomenon 116.4. For a static charge distribution $\rho(\vect{r})$ normalized to unit total charge, the Born amplitude carries the Fourier transform Equation (116.23) below, and expanding the exponential for small momentum transfer gives, after angular averaging,

\begin{equation}\tag{116.5} G_{E}(Q^{2})=1-\frac{Q^{2}\avg{r^{2}}}{6\hbar^{2}}+O(Q^{4})\ec \end{equation}

which is dimensionally consistent because $Q^{2}\avg{r^{2}}$ carries $\mathrm{kg}^{2}\,\mathrm{m}^{4}/\mathrm{s}^{2}$, exactly $\hbar^{2}$. Expanding the dipole Equation (116.4) to the same order gives $G_{E}=1-2Q^{2}c^{2}/\Lambda^{2}$, so

\begin{equation}\tag{116.6} \avg{r^{2}}=\frac{12\hbar^{2}c^{2}}{\Lambda^{2}} \quad\Longrightarrow\quad \sqrt{\avg{r^{2}}}=\frac{\sqrt{12}\,\hbar c}{\Lambda} =\frac{3.4641\times0.19733\,\mathrm{GeV}\,\mathrm{fm}} {0.84262\,\mathrm{GeV}}=0.811\,\mathrm{fm}\ep \end{equation}

The modern value from the CODATA adjustment is $8.4075(64)\times 10^{-16}\,\mathrm{m}$ [Mohr:2025], so the dipole radius lies about $3.5\,\mathrm{\%}$ below it. That gap is the honest statement that the dipole is a convenient parametrisation of a decade of data and not a law, and that a radius extracted from it is not a radius extracted from the slope at $Q^{2}=0$.

The consequence that matters for this chapter is the falloff, and it is worth accounting for term by term rather than quoting a single power. At large $Q^{2}$ the bracket of Equation (116.2) is dominated by its magnetic term $2\tau G_{M}^{2}\tan^{2}(\vartheta/2)$; since $G_{M}^{2}\propto Q^{-8}$ while $\tau\propto Q^{2}$, the bracket falls as $Q^{-6}$. At fixed angle and asymptotically large energy the recoil factor Equation (116.3) contributes a further $Q^{-2}$ and $\sigma_{\mathrm{Mott}}$ a further $Q^{-4}$, giving the familiar $Q^{-12}$. That law is asymptotic, however, and the first survey did not reach it: evaluating Equation (116.2) with Equation (116.4) and the proton magnetic moment $\mu_{p}=2.7928$ nuclear magnetons [Navas:2024] gives a local exponent running only from about $Q^{-6}$ at $Q^{2}c^{2}=1\,\mathrm{GeV}^{2}$ to about $Q^{-9}$ at $8\,\mathrm{GeV}^{2}$. The honest number is therefore the one the formula itself returns: raising $Q^{2}c^{2}$ from $1\,\mathrm{GeV}^{2}$ to $8\,\mathrm{GeV}^{2}$ at $\vartheta=10\,^\circ$ suppresses the elastic cross section by a factor $3.2\times 10^{3}$, and suppresses its ratio to $\sigma_{\mathrm{Mott}}$ — the like-for-like comparison, since that ratio is what the inelastic measurement reports — by a factor $3.4\times 10^{2}$. The corresponding factors at $6\,^\circ$ and $25\,^\circ$ are $2.9\times 10^{3}$ and $4.5\times 10^{3}$ raw, $3.2\times 10^{2}$ and $3.7\times 10^{2}$ relative to Mott, so the conclusion does not depend on the angle chosen. That is what made the inelastic result of Section 116.4.1 a surprise: the same beam, the same target and a modest change of final state gave a rate that hardly fell at all.

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The quark hypothesis

By 1964 the proliferating hadrons had been organized into representations of a flavour $\SU(3)$ — the eightfold way of Section 102.1.1, whose representation theory is Linear Algebra and Representation Theory. Gell-Mann [GellMann:1964] and, independently, Zweig [Zweig:1964] observed that every observed multiplet is built from the fundamental triplet and its conjugate, and proposed that the triplet corresponds to physical constituents: three of them, with electric charges $+\tfrac{2}{3}e$, $-\tfrac{1}{3}e$ and $-\tfrac{1}{3}e$, from which mesons are made as $q\bar{q}$ and baryons as $qqq$ (Definition 102.5 and Proposition 102.6).

The proposal was widely, and reasonably, not believed as a statement about matter. Three objections carried weight.

No free quark had ever been seen. Searches in cosmic rays, in accelerator beams, in sea water and in terrestrial minerals had set limits on the abundance of fractionally charged particles far below anything a genuine constituent should permit; the modern statement of that null result is Phenomenon 102.65. A constituent that cannot be liberated was, in 1964, a contradiction in terms.
The masses did not add up. The constituent masses needed to reproduce the baryon spectrum are roughly a third of the nucleon mass each, but the binding energy implied is then comparable with the masses themselves, so the non-relativistic picture that made the counting work had no right to succeed.
The statistics were wrong. The $\Delta^{++}$ is $uuu$ with all three spins aligned in a spatially symmetric ground state, which is symmetric under exchange of identical fermions (Section 102.2.1). The resolution — colour — was not established until later.

Quarks were therefore treated by most of the community as a bookkeeping device: a mnemonic for $\SU(3)$ multiplets, not a claim about what is inside a proton. The distinction is exactly what deep inelastic scattering could settle, because a mnemonic has no form factor and a constituent does.

Bjorken scaling

Bjorken's prediction preceded the data and did not depend on the quark model. It came from current algebra: the equal-time commutators of the electromagnetic currents, which encode the charges of whatever the currents are built from, imply sum rules for the structure functions, and those sum rules are finite only if the structure functions have a particular limiting behaviour [Bjorken:1969a].

Definition 116.5 (The Bjorken limit).

The Bjorken limit is $Q^{2}\to\infty$ and $\nu\to\infty$ with

\begin{equation}\tag{116.7} x=\frac{Q^{2}}{2M\nu} \end{equation}

held fixed, where $\nu:=P\cdot q/M$ is the energy transferred to the target in its rest frame and $M$ the proton mass. Both limits are taken together: it is not a high-$Q^{2}$ limit at fixed $\nu$, which would be the elastic region, nor a high-$\nu$ limit at fixed $Q^{2}$.

Phenomenon 116.6 (Bjorken's prediction).

In the limit of Definition 116.5 the two structure functions of Definition 102.44 become functions of the single dimensionless variable $x$:

\begin{equation}\tag{116.8} Mc^{2}W_{1}\left(x,Q^{2}\right)\longrightarrow F_{1}(x)\ec\qquad \nu W_{2}\left(x,Q^{2}\right)\longrightarrow F_{2}(x)\ep \end{equation}

This was published in 1969, before the SLAC data were analysed [Bjorken:1969a], and the operator-product machinery that makes it a statement about short-distance behaviour rather than an assumption is Wilson's [Wilson:1969]. Rests on Definitions 102.44 and 116.5.

Derivation. Derives Phenomenon 116.6. The content is dimensional, and the dimensional analysis is the whole argument. The functions $W_{1}$ and $W_{2}$ of Definition 102.44 each carry $/\mathrm{J}$, and the products in Equation (116.8) are therefore dimensionless. A dimensionless function of the two invariants can depend on them only through dimensionless combinations, and there are exactly three available: $x$, the ratio $Q^{2}c^{2}/M^{2}c^{4}$, and $\nu/Mc^{2}$. If the target has no internal length scale beyond its own Compton wavelength — if, that is, the current is absorbed by a structureless object — then in the limit where $Q^{2}c^{2}$ and $\nu Mc^{2}$ both dwarf $M^{2}c^{4}$ the last two ratios drop out of any finite answer and only $x$ survives.

The converse is the informative half. Suppose the constituent had a size $a$. Then $\hbar/a$ is a momentum, $Q^{2}a^{2}/\hbar^{2}$ is a fourth dimensionless combination that does not disappear in the Bjorken limit, and the structure functions would depend on $Q^{2}$ separately through it — falling once $Q\gtrsim\hbar/a$, in exactly the way the elastic form factors of Phenomenon 116.4 do. Scaling is thus the absence of a constituent length, and a measured violation of scaling is either a constituent size or an interaction that supplies a scale of its own. Which of the two the observed violations are is settled in Section 116.6: they are logarithmic, not power-like, and a size gives a power.

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The parton model

Feynman supplied the picture that makes the limit intuitive [Feynman:1969]. View the hadron in a frame in which its momentum $P$ is very large. Time dilation slows every internal process by the Lorentz factor, so during the brief moment in which the virtual photon is absorbed the constituents — Feynman's partons — neither interact with one another nor change their momenta. Each carries a fraction $\xi$ of the hadron's longitudinal momentum, and the photon scatters elastically from one of them while the rest are spectators. This is the impulse approximation of Phenomenon 102.45, and Equation (102.53) identifies $\xi$ with the measured $x$.

Bjorken and Paschos turned the picture into a formula [Bjorken:1969b]. Writing $q_{i}(x)\dd x$ for the number of partons of species $i$ and charge $e_{i}$ carrying momentum fraction in $\dd x$, the incoherent sum over point-like elastic scatterings gives Equation (102.54), which in the notation of this chapter reads

\begin{equation}\tag{116.9} F_{2}(x)=\sum_{i}\left(\frac{e_{i}}{e}\right)^{2}x\,q_{i}(x)\ec \end{equation}

a function of $x$ alone — which is Bjorken scaling. The parton distributions $q_{i}$ carry no dimension at all: $x$ is dimensionless and $q_{i}(x)\dd x$ is a number.

Remark 116.7 (What the parton model assumes and does not explain).

The model presumes that the constituents are free during the absorption. Nothing in it explains why constituents bound tightly enough never to escape should behave as free particles when probed hard, and in 1969 this was the model's central embarrassment rather than its central insight: it was a hypothesis with no dynamics behind it. The debt is discharged by asymptotic freedom (Theorem 102.35), which is the subject of Section 116.6.1 — and the sequence matters historically, because the parton model was believed on experimental grounds for four years before a field theory was found that could support it.

The Callan–Gross relation

The parton model as stated says nothing about the constituents' spin. Callan and Gross showed that the two structure functions are not independent if the constituents carry spin $\tfrac{1}{2}$, and that the relation between them is opposite for spin $0$ [Callan:1969].

Definition 116.8 (The longitudinal structure function and $R$).

Define

\begin{equation}\tag{116.10} F_{L}\left(x,Q^{2}\right) :=\left(1+\frac{Q^{2}c^{2}}{\nu^{2}}\right) F_{2}\left(x,Q^{2}\right)-2xF_{1}\left(x,Q^{2}\right)\ec \end{equation}

and the ratio of longitudinal to transverse virtual-photon absorption cross sections

\begin{equation}\tag{116.11} R\left(x,Q^{2}\right):=\frac{\sigma_{L}}{\sigma_{T}} =\frac{F_{L}}{2xF_{1}}\ep \end{equation}

The ratio $Q^{2}c^{2}/\nu^{2}$ equals $4M^{2}x^{2}c^{4}/Q^{2}c^{2}$ and vanishes in the Bjorken limit at fixed $x$, so $F_{L}\to F_{2}-2xF_{1}$ there. All three quantities are dimensionless.

Phenomenon 116.9 (Callan–Gross).

For constituents of spin $\tfrac{1}{2}$ and negligible mass, $F_{L}=0$, equivalently $F_{2}(x)=2xF_{1}(x)$ and $R=0$; for constituents of spin $0$, $F_{1}=0$ and $R\to\infty$ [Callan:1969]. Any measured value of $R$ therefore discriminates between the two hypotheses without needing an absolute normalisation, a knowledge of the parton distributions, or a model of the proton. Rests on Definition 116.8, Equation (116.9) and Phenomenon 102.45.

The derivation is the helicity argument given in Phenomenon 102.45 and repeated, in the form in which it is used against the data, in Section 116.4.3. What must be recorded here is why this is the sharpest of the early tests. Every other comparison between the parton model and the data involves either an absolute cross section, which carries the beam normalisation and the target thickness, or a distribution $q_{i}(x)$, which is not predicted by anything. $R$ is a ratio of two cross sections measured with the same beam on the same target, and the two hypotheses do not predict different numbers for it — they predict opposite ends of its range.

Apparatus

The Stanford linear accelerator

The instrument is a $3.05\,\mathrm{km}$ disc-loaded copper waveguide buried under the hills west of Stanford, operating as a travelling-wave structure at $2856\,\mathrm{MHz}$ [Neal:1968]. Electrons injected at a few tens of $\mathrm{MeV}$ ride the phase velocity of the wave and gain energy continuously along the machine, reaching about $20\,\mathrm{GeV}=3.2\times 10^{-9}\,\mathrm{J}$ at the end of the machine. Radiofrequency power is supplied by some 240 klystrons distributed along the tunnel, each delivering tens of megawatts of peak power into its own group of four $3.05\,\mathrm{m}$ accelerating sections: $240\times4\times3.05\,\mathrm{m}=2.93\,\mathrm{km}$ of structure, which with the drift and injection regions is the length of the machine.

Three features of the beam decided the experiment.

Energy and its calibration. The maximum energy sets the kinematic reach: the largest accessible $Q^{2}$ at a given angle grows as $E^{2}$. The energy was defined by a magnetic analysing system in the beam switchyard and known to a few parts in $10^{3}$, which is what makes an absolute measurement of the invariants possible at all; every kinematic variable of Section 116.3.1 is built from $E$, $E'$ and $\vartheta$ and from nothing else.
Pulse structure. The machine ran at up to $360\,/\mathrm{s}$ with a pulse length of order $1.6\,\mu\mathrm{s}$, a duty factor of about $6\times 10^{-4}$. A short, intense pulse is a mixed blessing: it delivers the charge needed for a cross section falling by six orders of magnitude across the kinematic range, but it forces every counter to resolve several particles inside the pulse, which is the reason the detector stack of Section 116.2.2 is built from many small scintillator elements rather than a few large ones.
Current. Average beam currents in the microampere range were typical. At $1\,\mu\mathrm{A}$ the beam delivers $1\,\mu\mathrm{C}/\mathrm{s}/1.602176634\times 10^{-19}\,\mathrm{C} =6.24\times 10^{12}\,/\mathrm{s}$ electrons; against the areal target density of Section 116.2.3 this is the luminosity that makes a cross section of order $10^{-38}\,\mathrm{m}^{2}/\mathrm{sr}$ measurable in hours.

Remark 116.10 (Why an electron beam).

A hadron beam probes a hadron with the strong interaction, whose short-distance form was precisely what was unknown; any measured rate then mixes the structure of the target with the structure of the probe and with the unknown dynamics between them. The electron interacts through a single virtual photon whose coupling is the fine-structure constant $\alpha=e^{2}/(4\pi\varepsilon_{0}\hbar c)=7.2973525643(11)\times 10^{-3}$ [Mohr:2025], small enough that one-photon exchange dominates and calculable exactly (Quantum Electrodynamics and Renormalization). The whole of the unknown physics is then confined to the two structure functions of Definition 102.44, and the experiment measures them with no theoretical input beyond quantum electrodynamics. The price is rate: $\alpha^{2}\approx5.3\times 10^{-5}$ against a strong cross section of order unity, which is why the beam power and the target thickness matter as much as they do.

The End Station A spectrometers

End Station A is a shielded experimental hall at the end of the switchyard, large enough to hold three magnetic spectrometers mounted on carriages that pivot about the target position on a circular rail. They are named by the momentum they can analyse: $20\,\mathrm{GeV}$$/c$, $8\,\mathrm{GeV}$$/c$ and $1.6\,\mathrm{GeV}$$/c$. The first two carried the deep inelastic programme; the third served the largest angles.

Proposition 116.11 (Why the spectrometer weighs thousands of tonnes).

Analysing the momentum of a $20\,\mathrm{GeV}$$/c$ electron by bending it through an angle $\theta_{B}$ requires a field integral

\begin{equation}\tag{116.12} \int B\,\dd l=\frac{p\,\theta_{B}}{e}\ec \end{equation}

which for $\theta_{B}=10\,^\circ$ is about $12\,\mathrm{T}\,\mathrm{m}$. With conventional iron-cored dipoles limited by saturation to fields of order $1.5\,\mathrm{T}$, this is eight metres of magnet; with the drift spaces needed to convert an angle into a displacement large enough to measure, the instrument is tens of metres long, and its yoke is the mass that must be rotated to change the scattering angle. Rests on Equations (40.15) and (113.8).

Proof.

Derives Proposition 116.11. A charge $e$ moving with momentum $p$ perpendicular to a uniform field $B$ follows a circle of radius $r=p/(eB)$, so a path length $\dd l$ turns the trajectory through $\dd\theta=\dd l/r=eB\,\dd l/p$; integrating gives Equation (116.12), which is exact for any field profile provided the bend stays in one plane. Numerically,

\[ p=20\,\mathrm{GeV}/c =\frac{20\,\mathrm{GeV}\times 1.602176634\times 10^{-10}\,\mathrm{J}/\mathrm{GeV}} {2.99792458\times 10^{8}\,\mathrm{m}/\mathrm{s}} =1.0689\times 10^{-17}\,\mathrm{kg}\,\mathrm{m}/\mathrm{s}\ec \]

and with $\theta_{B}=0.17453\,\mathrm{rad}$,

\[ \int B\,\dd l =\frac{1.0689\times 10^{-17}\,\mathrm{kg}\,\mathrm{m}/\mathrm{s} \times0.17453} {1.602176634\times 10^{-19}\,\mathrm{C}} =11.6\,\mathrm{T}\,\mathrm{m}\ep \]

The momentum resolution follows from the same relation: a fractional error $\delta p/p$ shows up as an angular error $\delta\theta_{B}=\theta_{B}\,\delta p/p$, so resolving $\delta p/p\sim10^{-3}$ at $\theta_{B}=10\,^\circ$ means measuring the exit angle to $1.7\times 10^{-4}\,\mathrm{rad}$ — a fraction of a millimetre over a metre of lever arm, which is what the hodoscope granularity of the detector stack has to deliver.

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The optics of both spectrometers were designed to be point-to-point in momentum and independent of the production angle within the acceptance, so that a hit position in one hodoscope plane measures $E'$ and a hit position in another measures $\vartheta$, with the two decoupled. The acceptances are small: of order a tenth of a millisteradian in solid angle and a few percent in momentum for the $20\,\mathrm{GeV}$$/c$ instrument, roughly an order of magnitude more solid angle for the $8\,\mathrm{GeV}$$/c$ one, which is why the latter, reaching larger angles where the cross section is smaller, was the instrument for the Rosenbluth separations. The $20\,\mathrm{GeV}$$/c$ spectrometer covered angles from a few degrees to about $25\,^\circ$; the $8\,\mathrm{GeV}$$/c$ one reached $90\,^\circ$.

The detector stack behind the magnets had one job: identify an electron of measured momentum inside a burst that also contains pions from hadronic final states, at a pion-to-electron ratio that can reach $10^{3}$ or more at small $E'$. It did so redundantly.

Scintillation hodoscopes. Planes of narrow plastic scintillator strips, read by photomultipliers, giving the position in the dispersive and non-dispersive directions and hence $E'$ and $\vartheta$, together with the fast coincidence that forms the trigger.
A threshold gas Cherenkov counter. A tank of gas at about atmospheric pressure — carbon dioxide or nitrogen, refractive index $n-1\approx4.5\times 10^{-4}$ — in which a particle radiates only if $\beta>1/n$. The threshold Lorentz factor is $\gamma_{\mathrm{th}}=(1-n^{-2})^{-1/2}\approx[2(n-1)]^{-1/2}=33$, so the threshold momenta are

\[ p_{\mathrm{th}}(e)\approx\gamma_{\mathrm{th}}m_{e}c =17\,\mathrm{MeV}/c\ec\qquad p_{\mathrm{th}}(\pi)\approx\gamma_{\mathrm{th}}m_{\pi}c =4.7\,\mathrm{GeV}/c\ec \]

using $m_{e}c^{2}=0.51099895000(15)\,\mathrm{MeV}$ and $m_{\pi^{\pm}}c^{2}=139.57039(18)\,\mathrm{MeV}$ [Navas:2024]. Every electron in the acceptance radiates; no pion below $4.7\,\mathrm{GeV}$$/c$ does.
A total-absorption shower counter. Lead glass, some fifteen radiation lengths deep, in which an electron deposits its entire energy in an electromagnetic cascade while a pion deposits, on average, the small energy of a minimum-ionizing track plus whatever a nuclear interaction happens to give. Requiring the deposited energy to match the momentum measured magnetically is an independent rejection.

Together the two particle-identification devices suppress pions by three to four orders of magnitude with an electron efficiency near unity, and — because they are independent — each can be used to measure the other's efficiency directly from the data. First-hand accounts of the instrument and its use are the Nobel lectures of Taylor [Taylor:1991] and Kendall [Kendall:1991].

Targets, monitoring and normalisation

The target was liquid hydrogen, and for the neutron structure function liquid deuterium, held in thin-walled cells in a cryostat on the pivot axis.

Example 116.12 (The target as a number of scattering centres).

Liquid hydrogen at its boiling point, $20.3\,\mathrm{K}$ at atmospheric pressure, has density $\rho=70.8\,\mathrm{kg}/\mathrm{m}^{3}$. A cell of length $L=0.30\,\mathrm{m}$ therefore presents an areal number density of protons

\begin{equation}\tag{116.13} n_{A}=\frac{\rho L N_{A}}{M_{\mathrm{H}}} =\frac{70.8\,\mathrm{kg}/\mathrm{m}^{3}\times0.30\,\mathrm{m} \times6.02214076\times 10^{23}\,/\mathrm{mol}} {1.008\times 10^{-3}\,\mathrm{kg}/\mathrm{mol}} =1.27\times 10^{28}\,/\mathrm{m}^{2}\ec \end{equation}

with $N_{A}$ the Avogadro constant, exact by the 2019 SI [Mohr:2025]. Combined with the $6.24\times 10^{12}\,/\mathrm{s}$ electrons of a $1\,\mu\mathrm{A}$ beam, the luminosity is $7.9\times 10^{40}\,/\mathrm{m}^{2}/\mathrm{s}$, so a differential cross section of $10^{-38}\,\mathrm{m}^{2}/\mathrm{sr}$ into the $10^{-4}\,\mathrm{sr}$ acceptance of Section 116.2.2 yields $7.9\times 10^{40}\,/\mathrm{m}^{2}/\mathrm{s}\times 10^{-38}\,\mathrm{m}^{2}/\mathrm{sr}\times 10^{-4}\,\mathrm{sr}=0.079\,/\mathrm{s}$, that is $285$ counts an hour, which is the scale of the experiment.

Three systematic problems follow directly from the target and are dealt with in the design.

Boiling. The beam deposits energy in the liquid and produces vapour bubbles, so the density in the beam path is lower than the static density and depends on the beam current. The cells were therefore built with forced circulation, and the density correction was measured in situ by comparing rates at different currents and extrapolating to zero.
Walls. The cell windows scatter too. An identical empty cell was measured under identical conditions and subtracted; the subtraction is largest at small $E'$, where the wall material's own inelastic yield is comparable with the hydrogen signal.
Radiation length. The hydrogen itself is a radiator. With $X_{0}=630.4\,\mathrm{kg}/\mathrm{m}^{2}$ for hydrogen, the $0.30\,\mathrm{m}$ cell is $\rho L/X_{0}=21.2\,\mathrm{kg}/\mathrm{m}^{2}/ 630.4\,\mathrm{kg}/\mathrm{m}^{2}=0.034$ radiation lengths, and a $0.07\,\mathrm{m}$ cell about $0.008$. That number enters the external radiative correction of Section 116.3.3 directly, and is the reason the short cell was used wherever the rate allowed.

The incident charge was measured by toroidal current monitors surrounding the beam pipe — non-intercepting, reading the pulse's own magnetic field — and by secondary-emission monitors, the two cross-calibrated against a Faraday cup that collects the beam absolutely. The absolute scale of the cross sections was in addition tied to elastic electron–proton scattering measured with the same apparatus, where Equation (116.2) and the known form factors supply the answer: any error in luminosity, acceptance or efficiency shows up as a discrepancy in a cross section that is independently known. The residual normalisation uncertainty quoted for the programme is a few percent. The whole apparatus, the calibrations and the analysis are reviewed by Friedman and Kendall [Friedman:1972].

Procedure

Kinematics of the measurement

The reaction is inclusive: $e+p\to e'+X$, with only the scattered electron detected and the hadronic system $X$ left unobserved. That is not a limitation but the point — summing over every final state is what makes the cross section an object with a short-distance interpretation, and it is why a single-arm spectrometer suffices.

Notation 116.13 (The measured quantities and the invariants).

Three numbers are measured for each event: the incident energy $E$, the scattered energy $E'$ and the scattering angle $\vartheta$. From them, neglecting the electron mass against the energies,

\begin{align} \nu&=E-E'\ec\tag{116.14}\\ Q^{2}&=\frac{4EE'}{c^{2}}\sin^{2}\frac{\vartheta}{2}\ec \tag{116.15}\\ x&=\frac{Q^{2}}{2M\nu} =\frac{2EE'\sin^{2}(\vartheta/2)}{Mc^{2}\left(E-E'\right)}\ec \tag{116.16}\\ y&=\frac{\nu}{E}\ec\tag{116.17}\\ W^{2}c^{4}&=M^{2}c^{4}+2Mc^{2}\nu-Q^{2}c^{2}\ec \tag{116.18} \end{align}

where $M$ is the proton mass, $Mc^{2}=0.93827208816(29)\,\mathrm{GeV}$ [Navas:2024], and $W$ is the invariant mass of the unobserved hadronic system. The variables $x$ and $y$ are dimensionless and lie in $[0,1]$; $\nu$ is an energy; $Q^{2}$ carries $\mathrm{kg}^{2}\,\mathrm{m}^{2}/\mathrm{s}^{2}$; $W$ is a mass. Elastic scattering is $W=M$, that is $x=1$. This is Notation 102.43 written in laboratory variables.

Derivation of Equation (116.15). Derives Equation (116.15). With $k=(E/c,\vect{k})$ and $k'=(E'/c,\vect{k}')$ the incident and scattered electron four-momenta and $q=k-k'$,

\[ Q^{2}=-q^{2}=-\left(k-k'\right)^{2} =-k^{2}-k'^{2}+2k\cdot k' =2\left(\frac{EE'}{c^{2}}-\abs{\vect{k}}\abs{\vect{k}'} \cos\vartheta\right) \]

using $k^{2}=k'^{2}=m_{e}^{2}c^{2}\approx0$ at these energies. With $\abs{\vect{k}}=E/c$ and $\abs{\vect{k}'}=E'/c$ to the same accuracy,

\[ Q^{2}=\frac{2EE'}{c^{2}}\left(1-\cos\vartheta\right) =\frac{4EE'}{c^{2}}\sin^{2}\frac{\vartheta}{2}\ep \]

The neglect of $m_{e}$ is quantitatively safe: at $E=10\,\mathrm{GeV}$ the ratio $m_{e}^{2}c^{4}/E^{2}$ is $2.6\times 10^{-9}$.

∎

The $(E',\vartheta)$ plane at fixed $E$ divides into three regions, and the experiment is the traverse across them.

The elastic peak, $W=M$, a line in the plane given by $E'=E/[1+2E\sin^{2}(\vartheta/2)/Mc^{2}]$. Its cross section is Equation (116.2) and it falls steeply, asymptotically as $Q^{-12}$ and by a factor of a few thousand across the range of this survey (Phenomenon 116.4).
The resonance region, $W$ between $M$ and about $2\,\mathrm{GeV}/c^{2}$, where the $\Delta(1232)$ and the higher nucleon resonances appear as bumps. Each resonance has its own transition form factor and each falls steeply with $Q^{2}$.
The deep inelastic region, $W$ above about $2\,\mathrm{GeV}/c^{2}$, where no individual state can be resolved and the cross section is smooth. This is where scaling is tested, and the $W>2\,\mathrm{GeV}/c^{2}$ cut is the operational definition of “deep” used throughout the SLAC analysis.

The double-differential cross section in the measured variables is Equation (102.49),

\begin{equation}\tag{116.19} \frac{\dd^{2}\sigma}{\dd\Omega\,\dd E'} =\sigma_{\mathrm{Mott}}\left[W_{2}\left(x,Q^{2}\right) +2W_{1}\left(x,Q^{2}\right)\tan^{2}\frac{\vartheta}{2}\right]\ec \end{equation}

with $\sigma_{\mathrm{Mott}}$ given by Equation (102.50) and carrying $\mathrm{m}^{2}$, so that the left-hand side carries $\mathrm{m}^{2}/\mathrm{sr}/\mathrm{J}$ and $W_{1}$, $W_{2}$ each carry $/\mathrm{J}$, as Definition 102.44 states.

The measurement programme

1967. Commissioning runs with the $20\,\mathrm{GeV}$$/c$ spectrometer, elastic scattering measured first because it is the absolute calibration.
1968. The inelastic survey at $\vartheta=6\,^\circ$ and $10\,^\circ$, with incident energies from about $4.5\,\mathrm{GeV}$ to $17.7\,\mathrm{GeV}$, covering $Q^{2}c^{2}$ from roughly $0.7\,\mathrm{GeV}^{2}$ to $8\,\mathrm{GeV}^{2}$ and $W$ up to about $5\,\mathrm{GeV}$$/c^{2}$. The results were reported at the Vienna conference in September 1968 [Panofsky:1968] and published as [Bloom:1969].
1969. The scaling analysis, in which $\nu W_{2}$ measured at different $E$, $E'$ and $\vartheta$ is plotted against a single variable and found to collapse [Breidenbach:1969].
1970–1972. Extension to larger angles with the $8\,\mathrm{GeV}$$/c$ spectrometer, which is what makes the separation of $W_{1}$ from $W_{2}$ — and hence the measurement of $R$ — possible; and deuterium running, which gives the neutron structure function by subtraction and hence the ratio $F_{2}^{en}/F_{2}^{ep}$ of Section 116.4.4. Reviewed in [Friedman:1972].

Proposition 116.14 (How two angles separate the two structure functions).

Write the measured cross section Equation (116.19) in terms of the virtual-photon polarisation parameter

\begin{equation}\tag{116.20} \varepsilon=\left[1+2\left(1+\frac{\nu^{2}}{Q^{2}c^{2}}\right) \tan^{2}\frac{\vartheta}{2}\right]^{-1}\ec \end{equation}

which is dimensionless and runs over $(0,1]$. Then

\begin{equation}\tag{116.21} \frac{\dd^{2}\sigma}{\dd\Omega\,\dd E'} =\Gamma_{T}\left[\sigma_{T}+\varepsilon\,\sigma_{L}\right]\ec \end{equation}

with $\Gamma_{T}$ a purely kinematic flux factor. At fixed $\left(x,Q^{2}\right)$, plotting the measured cross section divided by $\Gamma_{T}$ against $\varepsilon$ gives a straight line whose intercept is $\sigma_{T}$ and whose slope is $\sigma_{L}$; their ratio is $R$ of Equation (116.11). Rests on Equation (116.19), Equation (116.11) and Notation 116.13.

Proof.

Derives Proposition 116.14. Both $\nu$ and $Q^{2}$ are fixed once $\left(x,Q^{2}\right)$ is fixed, by Equation (116.7), so the only freedom left in Equation (116.20) is $\vartheta$; and $\vartheta$ can be varied at fixed $\left(\nu,Q^{2}\right)$ only by varying the beam energy, from Equations (116.14) and (116.15). Since $\tan^{2}(\vartheta/2)$ appears linearly in Equation (116.19) and $\varepsilon$ is a Möbius function of it, the cross section is linear in $\varepsilon$, which is Equation (116.21); matching the two forms identifies the combinations $\sigma_{T}\propto W_{1}$ and $\sigma_{L}\propto W_{2}\left(1+\nu^{2}/Q^{2}c^{2}\right)-W_{1}$, whose ratio is Equation (116.11).

The practical difficulty is visible in the algebra. A precise slope needs a long lever arm in $\varepsilon$, which needs a large range of $\vartheta$ at fixed $\left(x,Q^{2}\right)$, which needs a large range of $E$ — and each beam energy carries its own normalisation. The uncertainty on $R$ is therefore dominated by the relative normalisation between data sets taken months apart at different energies, not by counting statistics, and this is why $R$ was quoted with an uncertainty comparable to its own value while $\nu W_{2}$ was known to better than $10\,\mathrm{\%}$. Section 116.4.3 returns to why the physics conclusion survives that.

∎

Radiative corrections and the systematic budget

The largest correction in the experiment is not instrumental. An electron that radiates a photon has a different energy from the one recorded, so the cross section extracted at a nominal $(E,E')$ is contaminated by events that occurred at other kinematics — above all by elastic events in which a hard photon was emitted, which migrate into the inelastic region and can dominate it. Mo and Tsai gave the treatment used [Mo:1969].

Definition 116.15 (The three radiative processes).

Internal corrections are the $O(\alpha)$ QED corrections to the scattering itself: the vertex correction, the electron self-energies, vacuum polarisation of the exchanged photon (Quantum Electrodynamics and Renormalization), and real bremsstrahlung from the incoming or outgoing electron at the scattering vertex. The virtual and real pieces are separately infrared divergent and their sum is finite, which is the Bloch–Nordsieck cancellation. External corrections are bremsstrahlung in the bulk material of the target and its windows, before and after the scattering, governed by the number of radiation lengths traversed. Ionization loss shifts the energy by a smaller, calculable amount.

Proposition 116.16 (The equivalent radiator).

In the peaking approximation — in which bremsstrahlung photons are taken to be emitted exactly along the incident or the scattered electron direction, which is accurate because the characteristic emission angle is $m_{e}c^{2}/E\approx5\times 10^{-5}$ at $E=10\,\mathrm{GeV}$ — the internal radiative correction is equivalent to placing a radiator of thickness

\begin{equation}\tag{116.22} t_{\mathrm{eq}}=\frac{\alpha}{\pi} \left[\ln\frac{Q^{2}c^{2}}{m_{e}^{2}c^{4}}-1\right] \end{equation}

radiation lengths before and after the target. At $Q^{2}c^{2}=4\,\mathrm{GeV}^{2}$ and $m_{e}c^{2}=0.51099895\,\mathrm{MeV}$ [Navas:2024] this is

\[ t_{\mathrm{eq}}=\frac{7.2974\times 10^{-3}}{\pi} \left[\ln\frac{4}{2.611\times 10^{-7}}-1\right] =2.323\times 10^{-3}\times15.54=0.036\ec \]

so the radiative tail is generated by an effective $3.6\,\mathrm{\%}$ of a radiation length, to be compared with the $0.008$ to $0.034$ radiation lengths of real target material computed in Section 116.2.3. Rests on Definition 116.15 and Proposition 102.48.

Proof.

Derives Proposition 116.16. The bremsstrahlung spectrum from a thin radiator of thickness $t$ is $\dd N/\dd k=t/k$ per unit photon energy $k$ to leading logarithmic accuracy, so the only thing a radiator does at this order is supply the coefficient of $\dd k/k$. The one-loop QED correction to the elastic vertex supplies exactly the same $\dd k/k$ spectrum with coefficient $(\alpha/\pi)\left[\ln(Q^{2}c^{2}/m_{e}^{2}c^{4})-1\right]$, the logarithm being the collinear (mass) singularity regulated by the electron mass and the $-1$ the constant left by the infrared cancellation — the same structure as Proposition 102.48, with the photon in place of the gluon. Equating the two coefficients gives Equation (116.22). The approximation is a leading-logarithm one; the exact treatment of [Mo:1969] keeps the non-peaking contributions and the exact elastic tail, and it is the difference between the two that sets the uncertainty quoted below.

∎

The unfolding is then a deconvolution: the measured yield at $(E,E')$ is an integral of the true cross section over all $(E_{s},E_{p})$ with $E_{s}\leq E$ and $E_{p}\geq E'$ weighted by the radiator kernel, plus the elastic tail computed from the known form factors. The equation is solved iteratively, starting from an assumed cross section and re-radiating until the prediction reproduces the data. In the SLAC analysis the correction reached several tens of percent at the smallest $E'$, where the elastic tail is largest, and was typically $10\text{–}30\,\mathrm{\%}$ in the deep inelastic region.

The systematic budget of the experiment is therefore dominated by theory, not by hardware:

radiative corrections, a few percent of the cross section, being of order $10\,\mathrm{\%}$ of a correction of order $25\,\mathrm{\%}$;
absolute normalisation — beam charge, target density, solid angle and efficiency — a few percent, tied to elastic scattering as described in Section 116.2.3;
relative normalisation between run periods, which is what limits $R$ (Proposition 116.14);
pion contamination, reduced below the percent level by the Cherenkov and shower counters and measured from the data;
target-wall subtraction and acceptance modelling, each at the percent level.

The quoted uncertainty on $\nu W_{2}$ in the deep inelastic region is a few percent point-to-point with an overall scale uncertainty of a few percent on top [Friedman:1972].

Observations and data

The weak fall-off of the inelastic cross section

Phenomenon 116.17 (The inelastic rate falls only weakly with momentum transfer).

The cross section for $e+p\to e'+X$ in the deep inelastic region falls far more slowly with $Q^{2}$ than the elastic cross section from the same target: measured relative to the point-charge (Mott) cross section, the inelastic rate is approximately independent of $Q^{2}$, whereas the elastic rate falls as the fourth power of the dipole form factor [Bloom:1969]. Rests on Phenomenon 116.4 and Equation (116.19).

Derivation. Derives Phenomenon 116.17. For scattering from a static charge distribution $\rho(\vect{r})$ the Born amplitude is the Fourier transform

\begin{equation}\tag{116.23} F(\vect{q})=\int\rho(\vect{r})\, \ee^{\ii\vect{q}\cdot\vect{r}/\hbar}\,\dd^{3}r\ec \end{equation}

normalized so that $F(0)=1$, and the cross section is the point-charge one multiplied by $\abs{F}^{2}$. If $\rho$ is smooth on a scale $R$ then $F$ varies on the scale $\hbar/R$ in $q$ and falls rapidly beyond it, so a distribution of root mean square radius near $0.8\,\mathrm{fm}$ [Hofstadter:1956] suppresses the elastic rate steeply once $Q\gtrsim\hbar/R$ — which is the measured dipole behaviour of Phenomenon 116.4. The one way to obtain a rate that does not fall is a scattering centre with $F\equiv1$, that is, one with no structure at the resolution probed. A cross section flat in $Q^{2}$ therefore says the electron is rebounding from something point-like inside the proton, exactly as Rutherford's large-angle alpha particles said the atom contained a small hard core; the soft extended proton of Section 116.1.1 cannot produce it.

∎

The size of the contrast is worth stating numerically, because it is the whole result, and the comparison must be made between like quantities. Over the range $Q^{2}c^{2}$ from $1\,\mathrm{GeV}^{2}$ to $8\,\mathrm{GeV}^{2}$, the elastic cross section divided by $\sigma_{\mathrm{Mott}}$ falls by a factor of about $3.4\times 10^{2}$ (Phenomenon 116.4), while the ratio of the inelastic cross section to $\sigma_{\mathrm{Mott}}$ at fixed $W$ above $2\,\mathrm{GeV}$$/c^{2}$ changes by less than a factor of two [Bloom:1969]. Between two and three orders of magnitude therefore separate the two behaviours, and no experimental systematic of the kind catalogued in Section 116.3.3 is of that size. Comparing the raw cross sections instead of their ratios to $\sigma_{\mathrm{Mott}}$ makes the contrast larger still, by the further factor of about ten that the Mott fall-off itself supplies.

Scaling of the structure functions

Phenomenon 116.18 (Bjorken scaling).

Plotted against the single variable $x=Q^{2}/2M\nu$, the measured structure function $\nu W_{2}$ collapses onto one curve: data taken at different beam energies, scattering angles and momentum transfers, covering $Q^{2}$ from about $1\,\mathrm{GeV}^{2}/c^{2}$ to $8\,\mathrm{GeV}^{2}/c^{2}$, lie on a common function of $x$ alone, with no separate dependence on $Q^{2}$ within the errors [Breidenbach:1969] — as had been predicted from current algebra [Bjorken:1969a]. Rests on Phenomenon 116.6, Notation 116.13 and Equation (116.9).

Derivation. Derives Phenomenon 116.18. Regard the proton, in a frame where its momentum $P$ is large, as a collection of free constituents, one of which carries a fraction $\xi$ of its four-momentum [Feynman:1969] [Bjorken:1969b]. The electron scatters elastically off that one, so after absorbing the virtual photon of four-momentum $q$ the struck constituent must again be on shell. Neglecting its mass against $Q$,

\[ \left(\xi P+q\right)^{2}=\left(\xi P\right)^{2} \quad\Longrightarrow\quad 2\xi P\cdot q+q^{2}=0\ec \]

and with $q^{2}=-Q^{2}$ and $P\cdot q=M\nu$ in the proton rest frame,

\begin{equation}\tag{116.24} \xi=\frac{Q^{2}}{2M\nu}=x\ep \end{equation}

The measured $x$ is therefore the momentum fraction of whatever was struck, and the inclusive cross section is the incoherent sum over constituents of a point-like elastic cross section, weighted by the probability of finding one at that fraction. That probability is a property of the proton and not of the probe, so the structure functions depend on $x$ and not separately on $Q^{2}$: which is the observed collapse. The step that must eventually fail is the assumption that the constituents are free, and its failure is exactly the logarithmic $Q^{2}$ dependence of Section 116.6.

∎

Two things about the 1969 presentation deserve recording, because both are lessons about how such a result is established.

The original plots were made against $\omega=1/x=2M\nu/Q^{2}$ rather than against $x$, which spreads out the small-$x$ region where the data then were; the modern convention is $x$, and the two are trivially related. And the collapse was demonstrated not by superposing curves but by plotting $\nu W_{2}$ at fixed $\omega$ against $Q^{2}$ and showing the result to be flat: a null test, in which the quantity that would carry the signature of a constituent size is displayed as a function of the variable it would depend on. Within the errors then available — a few percent point to point — no dependence was seen over the factor of eight in $Q^{2}$ covered [Breidenbach:1969] [Panofsky:1968]. That the dependence is in fact present, and logarithmic, was established only when the lever arm in $Q^{2}$ grew by two further orders of magnitude (Sections 116.6.3 and 116.6.4).

The longitudinal-to-transverse ratio

This is the subsection to which Section 102.5 defers the experimental side of the Callan–Gross relation.

Phenomenon 116.19 (The longitudinal cross section is small).

The ratio $R=\sigma_{L}/\sigma_{T}$ of longitudinal to transverse virtual-photon absorption is measured to be small and consistent with zero, in the range $0.1$ to $0.2$ with comparable uncertainty, throughout the deep inelastic region [Breidenbach:1969] [Friedman:1972]. Equivalently the Callan–Gross relation $F_{2}(x)=2xF_{1}(x)$ holds [Callan:1969]. Rests on Phenomenon 116.9, Proposition 116.14 and Equation (116.11).

Derivation. Derives Phenomenon 116.19. Work in the Breit frame, where the virtual photon transfers no energy and the struck constituent simply reverses its momentum, from $-p\hat{\vect{z}}$ to $+p\hat{\vect{z}}$. For a constituent of spin $\tfrac{1}{2}$ and negligible mass the vector coupling to the photon does not connect states of opposite chirality, so helicity is conserved; since the momentum reverses while the helicity does not, the projection of the spin on the fixed axis $\hat{\vect{z}}$ must reverse, and the constituent's angular momentum along that axis changes by $\pm1$ in units of $\hbar$. A transverse photon carries exactly $J_{z}=\pm1$ and can supply it; a longitudinal photon carries $J_{z}=0$ and cannot. Hence $\sigma_{L}=0$ and $R=0$ for spin-$\tfrac{1}{2}$ constituents. For spin-$0$ constituents the argument runs the other way: there is no spin to reverse, so $J_{z}$ must be unchanged, only the longitudinal photon can be absorbed, and $R\to\infty$. The two hypotheses therefore predict opposite extremes of the same measured ratio, and the observed small $R$ selects spin $\tfrac{1}{2}$. This is the sharpest of the early discriminators precisely because it is a ratio: it needs no absolute normalization and no knowledge of any structure function.

∎

Remark 116.20 (Why the conclusion survives the systematic budget).

Proposition 116.14 showed that $R$ is the quantity most exposed to the relative normalisation between run periods, and the quoted uncertainty — of the same size as the central value — says so honestly. The conclusion is nevertheless not in doubt, and the reason is the structure of Phenomenon 116.9. The two hypotheses do not differ by a factor: spin $\tfrac{1}{2}$ gives $R=0$ and spin $0$ gives $R=\infty$, so what the measurement has to establish is only that $R$ is finite and small. A systematic error large enough to move a measured $R=0.18(10)$ to a value consistent with a scalar constituent would have to be not a few percent but several hundred percent, which is excluded by the elastic normalisation of Section 116.2.3. The measurement is imprecise and the inference is not; distinguishing the two is the point of quoting the uncertainty.

The small non-zero value has since been understood and is itself a QCD prediction: gluon radiation gives a struck quark transverse momentum, which breaks the helicity argument at order $\alpha_{s}$ and generates $F_{L}\neq0$. That $R$ came out small rather than exactly zero is therefore, in retrospect, evidence for the interacting theory rather than against the parton picture.

Quark charges from the structure functions

Running the same apparatus on liquid deuterium gives, after correction for Fermi motion and the subtraction of the proton, the neutron structure function; and the ratio of the two is a direct measurement of the constituents' charges.

Proposition 116.21 (What the neutron-to-proton ratio measures).

In the parton model with $u$ and $d$ quarks and isospin symmetry relating the neutron to the proton by $u\leftrightarrow d$,

\begin{equation}\tag{116.25} \frac{F_{2}^{en}(x)}{F_{2}^{ep}(x)} =\frac{4d(x)+u(x)}{4u(x)+d(x)}\ec \end{equation}

where $u$ and $d$ are the proton's distributions and the sea has been neglected. The ratio is bounded by $\tfrac{1}{4}$ and $4$ for any distributions, the extremes corresponding to pure $u$ and pure $d$; the measured values lie inside those bounds and approach $\tfrac{1}{4}$ as $x\to1$ [Friedman:1972] [Bjorken:1969b]. Rests on Equation (116.9) and Definition 102.5.

Proof.

Derives Proposition 116.21. Substituting the charges $e_{u}=\tfrac{2}{3}e$ and $e_{d}=-\tfrac{1}{3}e$ into Equation (116.9) gives $F_{2}^{ep}=x[\tfrac{4}{9}u+\tfrac{1}{9}d]$, and isospin exchanges the two for the neutron, $F_{2}^{en}=x[\tfrac{4}{9}d+\tfrac{1}{9}u]$; dividing gives Equation (116.25), in which the overall factor $\tfrac{1}{9}$ and the common $x$ cancel. The bounds follow by setting $d=0$ or $u=0$. The content is that the bounds depend only on the charges: constituents of charge $\pm1$ in units of $e$ would give a ratio bounded by different numbers, and integer-charged constituents in any assignment reproducing the proton and neutron charges cannot produce a ratio approaching $\tfrac{1}{4}$ at large $x$. This is a measurement of $e_{u}/e_{d}$ that never requires a free quark, and it is why the deuterium running mattered as much as the hydrogen running.

∎

Remark 116.22 (The Gottfried sum and the asymmetric sea).

The same two structure functions give a sum rule. From the charges,

\begin{equation}\tag{116.26} S_{G}:=\int_{0}^{1} \frac{F_{2}^{ep}(x)-F_{2}^{en}(x)}{x}\,\dd x =\frac{1}{3}+\frac{2}{3}\int_{0}^{1} \left[\bar{u}(x)-\bar{d}(x)\right]\dd x\ec \end{equation}

because the valence content of the proton is $uud$ and of the neutron $udd$, so that $\int_{0}^{1}u_{v}=2$ and $\int_{0}^{1}d_{v}=1$ and the valence part of the difference integrates to

\[ \left(2-1\right)\times\tfrac{4}{9} +\left(1-2\right)\times\tfrac{1}{9} =\tfrac{4}{9}-\tfrac{1}{9}=\tfrac{1}{3}\ec \]

the charge weights being those of Proposition 116.21; the sea survives only in the combination shown. A symmetric sea therefore predicts $S_{G}=\tfrac{1}{3}$. The measured value is significantly smaller, near $0.24$, which is direct evidence that the light-quark sea is flavour-asymmetric with $\bar{d}>\bar{u}$ — an asymmetry no perturbative gluon splitting can generate, since gluons produce flavours democratically. The measurement is the New Muon Collaboration's reevaluation of the Gottfried sum from muon scattering on hydrogen and deuterium [Arneodo:1994]; the same asymmetry is seen directly, in a different process, by the Drell–Yan measurement of Section 116.7.3.

The measured numbers

Table 116.1 collects the headline quantities of the electron-scattering programme and the constants used to interpret them; Table 116.2 does the same for the neutrino measurements analysed in Section 116.5.3, and Table 116.3 for the later high-precision programme. All three sit here, under this rubric, rather than beside the sections that use them: a reader looking for the chapter's numbers should find them in one place. Every value is given with the uncertainty its source quotes, and a value inferred here from cited inputs is labelled derived, with the arithmetic shown in the text and its error propagated.

Quantity	Value	Year	Source
Beam energy range (setting; calibrated to a few parts in $10^{3}$)	$4.5\text{–}20\,\mathrm{GeV}$, i.e. $7.2\times 10^{-10}\text{–}3.2\times 10^{-9}\,\mathrm{J}$	1968	[Bloom:1969] [Neal:1968]
Scattering angles, first survey (setting)	$6\,^\circ$ and $10\,^\circ$	1968	[Bloom:1969]
$Q^{2}c^{2}$ covered (kinematic reach)	$0.7\text{–}8\,\mathrm{GeV}^{2}$; in SI, $Q^{2}$ from $2.0\times 10^{-37}$ to $2.3\times 10^{-36}\,\mathrm{kg}^{2}\,\mathrm{m}^{2}/\mathrm{s}^{2}$	1968	[Bloom:1969]
Deep inelastic cut on $W$ (analysis cut)	$W>2\,\mathrm{GeV}/c^{2}$	1969	[Breidenbach:1969]
Elastic suppression across that $Q^{2}$ range, relative to $\sigma_{\mathrm{Mott}}$ at $10\,^\circ$	factor $\approx6.3\times 10^{2}$ (raw cross section $\approx8.6\times 10^{3}$); derived from the dipole, whose own $10\,\mathrm{\%}$ accuracy makes these factors good to a few tens of percent	1956	[Hofstadter:1956]
Inelastic-to-Mott ratio across the same range	changes by less than a factor $2$	1969	[Bloom:1969]
$R=\sigma_{L}/\sigma_{T}$	$0.18(10)$	1969–72	[Breidenbach:1969] [Friedman:1972]
$\int_{0}^{1}F_{2}^{eN}(x)\,\dd x$ (isoscalar)	$0.14(1)$, the error being the precision to which the review quotes it	1972	[Friedman:1972]
Resolution $\hbar c/(Qc)$ at $Q^{2}c^{2}=8\,\mathrm{GeV}^{2}$	$7.0\times 10^{-17}\,\mathrm{m}$ (derived; exact given $Q$, to which the $10^{-3}$ beam calibration propagates)	1968	Equation (116.27)
Proton mass	$1.67262192595(52)\times 10^{-27}\,\mathrm{kg}$	2022	[Mohr:2025]
Proton mass energy	$0.93827208816(29)\,\mathrm{GeV}$	2024	[Navas:2024]
Proton rms charge radius	$8.4075(64)\times 10^{-16}\,\mathrm{m}$	2022	[Mohr:2025]
Fine-structure constant	$7.2973525643(11)\times 10^{-3}$	2022	[Mohr:2025]

Headline numbers of the SLAC–MIT deep inelastic programme and the constants used to interpret them. Energies in $\mathrm{GeV}$ carry their joule equivalent through Equation (116.1). The value of $R$ is the one the collaboration used in extracting $\nu W_{2}$ and quotes as its measured average over the deep inelastic region; its uncertainty is dominated by relative normalisation, as Proposition 116.14 explains. The structure-function integral is quoted to two figures because that is the accuracy of the review it comes from, and the uncertainty attached to it here is that implied precision and not a quoted error. Rows that record a beam setting, an angular position or an analysis cut carry no uncertainty because they define the run conditions rather than measure them; every row that reports a measured or derived quantity carries one.

Quantity	Value	Year	Source
$\sigma^{\nu N}/E_{\nu}$	$7.4(2)\times 10^{-43}\,\mathrm{m}^{2}/\mathrm{GeV}$	1973	[Eichten:1973]
$\sigma^{\bar{\nu}N}/E_{\bar{\nu}}$	$2.8(1)\times 10^{-43}\,\mathrm{m}^{2}/\mathrm{GeV}$	1973	[Eichten:1973]
Ratio $\sigma^{\bar{\nu}}/\sigma^{\nu}$	$0.378(17)$; $\tfrac{1}{3}$ for pure quarks	1973	[Eichten:1973]
$C=2Mc^{2}G_{F}^{2}/\pi(\hbar c)^{4}$	$3.164\times 10^{-42}\,\mathrm{m}^{2}/\mathrm{GeV}$ (derived; exact to the constants it is built from)	1973	Equation (116.34)
Quark momentum fraction $Q$	$0.230(7)$ (derived)	1973	Equation (116.35)
Antiquark momentum fraction $\bar{Q}$	$0.0119(43)$ (derived)	1973	Equation (116.35)
$\int_{0}^{1}F_{2}^{\nu N}\dd x$	$0.484(11)$ (derived)	1973	Equation (116.35)
Mean squared charge $\int F_{2}^{eN}/\int F_{2}^{\nu N}$	$0.29(2)$ (derived); $5/18=0.278$	1972–73	[Friedman:1972] [Eichten:1973]
Nucleon momentum in charged constituents	$0.50(4)$ (derived from $\int F_{2}^{eN}\dd x$ and $5/18$); reported as about one half on iron	1972–79	[Friedman:1972] [deGroot:1979]

Neutrino deep inelastic scattering: the measured total cross sections, the ratio that tests the valence hypothesis, and the momentum fractions derived from them in Proposition 116.28. Uncertainties on the derived rows are propagated from the quoted ones only and exclude the flux normalisation, which is the larger error and which the text discusses. The mean squared charge compares the neutrino integral with the electron one of Table 116.1.

Quantity	Value	Year	Source
BCDMS muon beam energies (setting)	$100\text{–}280\,\mathrm{GeV}$	1989	[Benvenuti:1989]
BCDMS coverage (kinematic reach)	$0.06<x<0.8$; $7\,\mathrm{GeV}^{2}<Q^{2}c^{2}< 260\,\mathrm{GeV}^{2}$	1989	[Benvenuti:1989]
HERA beam energies (setting)	$27.5\,\mathrm{GeV}$ on $920\,\mathrm{GeV}$, $\sqrt{s}\,c=318\,\mathrm{GeV}$	2010–15	[Aaron:2010] [Abramowicz:2015]
Equivalent fixed-target energy	$54\,\mathrm{TeV}$ (derived; exact given the beam energies)	2010–15	Equation (116.39)
HERA combined coverage (kinematic reach)	$6\times 10^{-7}<x<0.65$; $0.045\,\mathrm{GeV}^{2}<Q^{2}c^{2}< 5\times 10^{4}\,\mathrm{GeV}^{2}$	2015	[Abramowicz:2015]
Small-$x$ exponent $\lambda$, read from the published curves	$0.1$ at $Q^{2}c^{2}=1\,\mathrm{GeV}^{2}$ and $0.3$ at $100\,\mathrm{GeV}^{2}$, each to the reading precision $\pm0.05$	2015	[Abramowicz:2015]
Gluon density $[xg](2\times 10^{-3})$ at $Q^{2}c^{2}=20\,\mathrm{GeV}^{2}$	$8\pm2$ (derived from a slope read off the data; the leading-order relation is itself good only to that accuracy)	2015	Equation (116.40)
$\alpha_{s}(m_{Z}c^{2})$	$0.1180(9)$	2024	[Navas:2024]
$F_{2}^{\mathrm{Fe}}/F_{2}^{\mathrm{D}}$ at $x=0.65$, read from the published ratio curve	$0.89\pm0.01$ reading precision	1983	[Aubert:1983]
$F_{2}^{\mathrm{Fe}}/F_{2}^{\mathrm{D}}$ at $x=0.05$, read from the published ratio curve	$1.15\pm0.01$ reading precision	1983	[Aubert:1983]
$\Gamma_{1}^{p}=\int_{0}^{1}g_{1}^{p}\dd x$ at $\avg{Q^{2}}c^{2}=10.7\,\mathrm{GeV}^{2}$	$0.114\pm0.012\pm0.026$; Ellis–Jaffe expectation $\approx0.19$	1988	[Ashman:1988]

The later high-precision programme. Rows giving a beam setting or a published kinematic coverage carry no uncertainty because they state run conditions rather than measurements. A value read from a published curve rather than from a table is marked as such and is quoted with its reading precision, which is not the collaboration's error and is always the larger of the two. The strong coupling is the world average, to which deep inelastic determinations contribute.

Interpretation

Point-like constituents

What the data establish, stated as narrowly as possible: over the range of momentum transfer covered, the charged objects from which the virtual photon scatters exhibit no form factor. Equivalently, they have no measurable size down to the resolution the experiment reaches.

Proposition 116.23 (The resolution actually achieved).

A virtual photon of momentum transfer $Q$ resolves transverse distances of order

\begin{equation}\tag{116.27} \ell=\frac{\hbar}{Q}=\frac{\hbar c}{Qc}\ec \end{equation}

so at the largest momentum transfer of the original survey, $Q^{2}c^{2}=8\,\mathrm{GeV}^{2}$ and hence $Qc=2.83\,\mathrm{GeV}$,

\[ \ell=\frac{3.16152677\times 10^{-26}\,\mathrm{J}\,\mathrm{m}} {2.83\,\mathrm{GeV}\times 1.602176634\times 10^{-10}\,\mathrm{J}/\mathrm{GeV}} =7.0\times 10^{-17}\,\mathrm{m}\ec \]

about one twelfth of the proton's charge radius $8.4075(64)\times 10^{-16}\,\mathrm{m}$ [Mohr:2025]. The absence of a form factor at that resolution is the whole of the experimental claim: it is an upper limit on a constituent size of about $10^{-16}\,\mathrm{m}$, not a demonstration that the size is zero. Rests on Equations (116.1) and (116.23).

Proof.

Derives Proposition 116.23. The Fourier transform Equation (116.23) relates a distribution of extent $\ell$ to a form factor varying on the momentum scale $\hbar/\ell$; a structure smaller than $\hbar/Q$ produces a form factor still equal to one at the $Q$ probed, and is therefore invisible. The numerical evaluation uses only Equation (116.1) and the definition of $\hbar c$.

∎

Remark 116.24 (What the data do not say).

Three limitations are worth stating explicitly, because the result was sometimes over-read at the time and is sometimes over-read now.

First, scaling says nothing about confinement. The experiment establishes that something point-like inside the proton absorbs the photon; it says nothing whatever about whether that something can be removed from the proton, and indeed the same experiment sees only hadrons in the final state. The two facts sat uncomfortably together until asymptotic freedom explained how a theory can be free at short distance and strong at long distance (Section 116.6.1).

Second, scaling does not by itself identify the constituents as the quarks of the spectroscopy. It says there are point-like charged objects. That they carry spin $\tfrac{1}{2}$ comes from Section 116.4.3, that they carry fractional charge from Section 116.4.4 and Section 116.5.3, and that their number and quantum numbers match the quark model from the combination.

Third, the alternatives were serious and were killed by the data, not by taste. Vector-meson dominance, in which the virtual photon converts into a $\rho$ meson which then interacts hadronically, predicts a cross section falling with $Q^{2}$ like the square of a vector-meson propagator and cannot give a flat ratio to the Mott cross section. Regge-based and diffractive pictures predicted specific $\nu$ and $Q^{2}$ dependences that the scaling plot excludes. It is the null test of Section 116.4.2 — flat in $Q^{2}$ at fixed $\omega$ — that discriminates, and that is why it was presented that way.

Friedman's Nobel lecture [Friedman:1991] is the best first-hand account of how the interpretation was arrived at, including how long it took: the 1968 data were presented as a puzzle, and the parton reading became the consensus only over the following three or four years, as the Callan–Gross test and the neutrino comparison came in.

The missing momentum fraction

This is the subsection to which Phenomenon 102.47 defers the details of the extraction.

Phenomenon 116.25 (Half the proton momentum is invisible to the photon).

The momentum fraction carried by the electrically charged constituents, obtained by integrating the measured structure function, is about one half rather than one [Friedman:1972]. Comparing neutrino with electron scattering on an isoscalar target confirms the deficit independently and returns a mean squared constituent charge close to $5/18$ in units of the elementary charge squared [deGroot:1979] [Eichten:1973]. Rests on Equation (116.9) and Proposition 102.46.

Derivation. Derives Phenomenon 116.25. In the parton model $F_{2}(x)=\sum_{i}e_{i}^{2}\,x\,q_{i}(x)$, where $q_{i}(x)\dd x$ is the number of constituents of charge $e_{i}$ with momentum fraction in $\dd x$. The proton's momentum is shared among all its constituents, so

\begin{equation}\tag{116.28} \sum_{\text{all }i}\int_{0}^{1}x\,q_{i}(x)\,\dd x=1\ec \end{equation}

the sum running over every constituent, charged or not. What the electron measures is the charge-weighted sum: dividing $\int F_{2}\,\dd x$ by the mean squared charge gives the momentum carried by the charged constituents alone, and that comes out near one half. The remainder of Equation (116.28) is carried by constituents that do not couple to the photon. The mean squared charge itself is not assumed but measured, by comparing the electron and neutrino structure functions on an isoscalar target: the weak charged current couples with the same strength to every quark flavour, so the ratio of the two integrals is the mean squared electric charge, and for equal numbers of $u$ and $d$ quarks

\[ \avg{e^{2}}=\tfrac{1}{2}\left[\left(\tfrac{2}{3}\right)^{2} +\left(\tfrac{1}{3}\right)^{2}\right]=\frac{5}{18}\ec \]

which is what is found [deGroot:1979]. Numerically, the neutrino integral extracted in Section 116.5.3 is $\int F_{2}^{\nu N}\dd x=0.484(11)$ and the electron one is $0.14(1)$ [Friedman:1972], the second error being the precision to which the review quotes the number rather than an error it states; their ratio is $0.29(2)$, to be compared with $5/18=0.278$. Dividing the electron integral by $5/18$ instead gives the momentum fraction in charged constituents directly, $0.14(1)/0.278=0.50(4)$ — the “about one half” of the statement above, now with the error it carries. The neutral, non-scattering carrier of the missing momentum is the gluon of Quantum Chromodynamics; the inference here is from a deficit, and the direct confirmation came later, from three-jet events in electron–positron annihilation (Section 102.6.2).

∎

The deficit is not a defect of the measurement but a prediction of the theory, and quantum chromodynamics says where the missing momentum must eventually settle.

Proof.

Derives Proposition 116.26. Take the momentum moment $N=1$ of the coupled singlet equations, using the moment notation $\gamma_{N}=\int_{0}^{1}z^{N}P(z)\,\dd z$ of Corollary 102.50. Write $\Sigma:=\int_{0}^{1}x\sum_{q}[q(x)+\bar{q}(x)]\dd x$ and $G:=\int_{0}^{1}x\,g(x)\,\dd x$. Four coefficients are needed:

\[ A_{qq}=\int_{0}^{1}z\,P_{qq}(z)\,\dd z=-\tfrac{4}{3}C_{F} =-\tfrac{16}{9}\ec\qquad A_{qg}=2n_{f}\int_{0}^{1}z\,P_{qg}(z)\,\dd z=\frac{n_{f}}{3}\ec \]

the first from Equation (102.62) and the second from $P_{qg}(z)=T_{F}[z^{2}+(1-z)^{2}]$ with $T_{F}=\tfrac{1}{2}$, since $\int_{0}^{1}z[z^{2}+(1-z)^{2}]\dd z=\tfrac{1}{3}$ and there are $2n_{f}$ quark and antiquark species a gluon can convert into. The remaining two follow from momentum conservation without further computation: the total momentum $\Sigma+G$ cannot change with $Q^{2}$, so $A_{qq}+A_{gq}=0$ and $A_{qg}+A_{gg}=0$, giving $A_{gq}=+\tfrac{16}{9}$ and $A_{gg}=-n_{f}/3$. The evolution is then

\begin{equation}\tag{116.30} \dv{}{\ln Q^{2}}\begin{pmatrix}\Sigma\\ G\end{pmatrix} =\frac{\alpha_{s}\left(Q^{2}\right)}{2\pi} \begin{pmatrix} -\tfrac{16}{9} & \tfrac{n_{f}}{3}\\ \tfrac{16}{9} & -\tfrac{n_{f}}{3} \end{pmatrix} \begin{pmatrix}\Sigma\\ G\end{pmatrix}\ep \end{equation}

The matrix has eigenvalue $0$, with eigenvector $(\Sigma,G)\propto(n_{f}/3,\,16/9)$, and eigenvalue $-(\tfrac{16}{9}+\tfrac{n_{f}}{3})$, whose eigenvector decays. Since $\alpha_{s}$ falls only logarithmically the decay is slow, but its direction is fixed: the surviving combination is the null eigenvector, normalized by $\Sigma+G=1$, which gives Equation (116.29) after dividing through by $\tfrac{1}{3}$. Note that the answer does not involve $\alpha_{s}$, $\Lambda_{\mathrm{QCD}}$ or the initial distributions — only the group theory in $C_{F}$ and $T_{F}$ and the number of active flavours.

Measured values at $Q^{2}c^{2}$ near $10\,\mathrm{GeV}^{2}$ put the gluon fraction near $0.45$ [Navas:2024], below the asymptotic $0.57$ for $n_{f}=4$, and rising slowly with $Q^{2}$ as the second eigenvector dies away — which is exactly the pattern the global fits of Section 116.6.5 find.

∎

Neutrino deep inelastic scattering

The electron probes the quarks with weights $e_{i}^{2}$. The charged weak current probes them with weights that are independent of electric charge and that distinguish quarks from antiquarks, so the two probes together separate the charge weights from the distributions. The weak-current theory is Weak Interactions; the neutrino beams and the Gargamelle chamber are described in Electroweak Unification and the Higgs Boson, where the same exposures gave the neutral current.

Phenomenon 116.27 (The neutrino cross section rises linearly with energy).

The total charged-current cross sections for $\nu_{\mu}+N\to\mu^{-}+X$ and $\bar{\nu}_{\mu}+N\to\mu^{+}+X$ on an approximately isoscalar target rise in proportion to the neutrino energy, with

\begin{align} \frac{\sigma^{\nu N}}{E_{\nu}} &=7.4(2)\times 10^{-43}\,\mathrm{m}^{2}/\mathrm{GeV} =4.62(12)\times 10^{-33}\,\mathrm{m}^{2}/\mathrm{J}\ec \tag{116.31}\\ \frac{\sigma^{\bar{\nu}N}}{E_{\bar{\nu}}} &=2.8(1)\times 10^{-43}\,\mathrm{m}^{2}/\mathrm{GeV} =1.75(6)\times 10^{-33}\,\mathrm{m}^{2}/\mathrm{J}\ec \tag{116.32} \end{align}

measured in the Gargamelle bubble chamber at CERN [Eichten:1973]. A linear rise is what point-like constituents require and what a target with a form factor forbids. Rests on Equations (104.3) and (116.9).

Derivation. Derives Phenomenon 116.27. The four-fermion charged-current interaction of Weak Interactions has coupling $G_{F}=1.4358512(7)\times 10^{-62}\,\mathrm{J}\,\mathrm{m}^{3}$, equivalently $G_{F}/(\hbar c)^{3}=1.1663787(6)\times 10^{-5}\,/\mathrm{GeV}^{2}$ (Equation (104.3)), the uncertainty being the relative $5.1\times 10^{-7}$ of the second form carried over; it is a dimensionful coupling, and that is the whole argument. For the point process $\nu+q\to\mu^{-}+q'$ the amplitude is proportional to $G_{F}$ and carries no propagator, so the squared amplitude is proportional to $G_{F}^{2}$ times the only invariant available, and the cross section is

\begin{equation}\tag{116.33} \hat{\sigma}\left(\nu q\right) =\frac{G_{F}^{2}\,\hat{s}c^{2}}{\pi\left(\hbar c\right)^{4}}\ec\qquad \hat{\sigma}\left(\nu\bar{q}\right) =\frac{G_{F}^{2}\,\hat{s}c^{2}}{3\pi\left(\hbar c\right)^{4}}\ec \end{equation}

where $\hat{s}$ is the squared centre-of-mass four-momentum of the neutrino–parton system. It is written here with the same dimension as $Q^{2}$, namely $\mathrm{kg}^{2}\,\mathrm{m}^{2}/\mathrm{s}^{2}$, so that $\hat{s}c^{2}$ is an energy squared and the convention is the one used for $s$ at HERA in Proposition 116.33 and for the Drell–Yan variable in Equation (116.43); this chapter never gives $s$ the dimension of an energy squared. The dimensions check: $G_{F}^{2}\hat{s}c^{2}$ carries $\mathrm{J}^{4}\,\mathrm{m}^{6}$ and $(\hbar c)^{4}$ carries $\mathrm{J}^{4}\,\mathrm{m}^{4}$, leaving $\mathrm{m}^{2}$. The factor $\tfrac{1}{3}$ for antiquarks is the angular-momentum suppression: $\nu$ and $q$ are both left-handed, so their total angular momentum along the beam vanishes and the scattering is isotropic, while $\nu$ and $\bar{q}$ have $J=1$ aligned and the $\dd\sigma/\dd y\propto(1-y)^{2}$ distribution integrates to a third.

With the struck parton carrying fraction $x$ of the nucleon momentum, $\hat{s}=x\,s$ and $sc^{2}=M^{2}c^{4}+2Mc^{2}E_{\nu}\approx 2Mc^{2}E_{\nu}$ for a stationary target, so integrating over the parton content gives a cross section proportional to $E_{\nu}$:

\begin{equation}\tag{116.34} \frac{\sigma^{\nu N}}{E_{\nu}}=C\left[Q+\frac{\bar{Q}}{3}\right]\ec \qquad \frac{\sigma^{\bar{\nu}N}}{E_{\bar{\nu}}} =C\left[\frac{Q}{3}+\bar{Q}\right]\ec \qquad C:=\frac{2Mc^{2}G_{F}^{2}}{\pi\left(\hbar c\right)^{4}}\ec \end{equation}

where $Q$ and $\bar{Q}$ are the momentum fractions carried by the quarks and antiquarks that the current can reach on an isoscalar nucleon. Evaluating the constant in SI,

\begin{align*} C&=\frac{2\times1.503277\times 10^{-10}\,\mathrm{J} \times\left(1.435851\times 10^{-62}\,\mathrm{J}\,\mathrm{m}^{3}\right)^{2}} {\pi\left(3.16152677\times 10^{-26}\,\mathrm{J}\,\mathrm{m}\right)^{4}}\\ &=1.975\times 10^{-32}\,\mathrm{m}^{2}/\mathrm{J} =3.164\times 10^{-42}\,\mathrm{m}^{2}/\mathrm{GeV}\ep \end{align*}

The linearity is thus a direct consequence of $G_{F}$ having dimensions of $\mathrm{J}\,\mathrm{m}^{3}$ together with the absence of any other scale in the target: a constituent with a size $a$ would supply a form factor cutting the rise off at $E_{\nu}\sim\hbar^{2}/(2Ma^{2})$, and no such cut-off is seen.

∎

Proposition 116.28 (The momentum fractions from the two cross sections).

Solving Equation (116.34) for $Q$ and $\bar{Q}$ with the measured values Equations (116.31) and (116.32) gives

\begin{equation}\tag{116.35} Q=0.230(7)\ec\qquad \bar{Q}=0.0119(43)\ec\qquad \int_{0}^{1}F_{2}^{\nu N}(x)\,\dd x=2\left(Q+\bar{Q}\right) =0.484(11)\ep \end{equation}

About half the nucleon's momentum is therefore carried by objects the weak current can reach, and the antiquark “sea” accounts for $4.9(18)\,\mathrm{\%}$ of that at leading order — a fraction whose error is more than a third of itself, which is why no more than its first figure may be used. Rests on Equations (116.31), (116.32) and (116.34).

Proof.

Derives Proposition 116.28. Write $a:=\sigma^{\nu N}/(CE_{\nu})=Q+\bar{Q}/3$ and $b:=\sigma^{\bar{\nu}N}/(CE_{\bar{\nu}})=Q/3+\bar{Q}$. From the measured values and the constant $C$ just evaluated,

\[ a=\frac{7.4\times 10^{-43}}{3.164\times 10^{-42}}=0.2339\ec\qquad b=\frac{2.8\times 10^{-43}}{3.164\times 10^{-42}}=0.0885\ep \]

Adding, $a+b=\tfrac{4}{3}(Q+\bar{Q})$, so $Q+\bar{Q}=\tfrac{3}{4}(a+b)=0.2418$; and subtracting $a$ from $3b$ gives $\tfrac{8}{3}\bar{Q}=3b-a=0.0316$, whence $\bar{Q}=0.0119$ and $Q=0.2299$. On an isoscalar nucleon the neutrino structure function is $F_{2}^{\nu N}(x)=x\sum_{q}[q(x)+\bar{q}(x)]$ with unit weights, and the sum over both nucleon types doubles the per-current fractions, so $\int F_{2}^{\nu N}\dd x=2(Q+\bar{Q})=0.484$.

The errors propagate from $\delta a=0.00632$ and $\delta b=0.00316$, which are the quoted relative errors of Equations (116.31) and (116.32) applied to $a$ and $b$. Since $Q=\tfrac{9}{8}a-\tfrac{3}{8}b$ and $\bar{Q}=\tfrac{3}{8}(3b-a)$,

\[ \delta Q=\sqrt{\left(\tfrac{9}{8}\delta a\right)^{2} +\left(\tfrac{3}{8}\delta b\right)^{2}}=0.0072\ec\qquad \delta\bar{Q}=\tfrac{3}{8}\sqrt{\left(3\delta b\right)^{2} +\left(\delta a\right)^{2}}=0.0043\ec \]

and $\delta\!\int F_{2}^{\nu N}\dd x=\tfrac{3}{2} \sqrt{(\delta a)^{2}+(\delta b)^{2}}=0.011$. Note how much worse the antiquark fraction is determined than the sum: the difference $3b-a$ subtracts two numbers of comparable size, so its error is essentially the full error of $a$.

Three caveats belong with the number, and each is a real limitation. The uncertainties propagated are the collaboration's quoted ones and do not include the neutrino flux normalisation, which was uncertain at the $10\,\mathrm{\%}$ level and would widen the result to about $0.48(5)$ — which is why the value usually quoted from this era, $0.49(7)$, carries a much larger error than the arithmetic above. The Gargamelle target was freon, not an exactly isoscalar nucleus, and the neutron excess was corrected for. And Equation (116.33) is leading order in $\alpha_{s}$, so the split between $Q$ and $\bar{Q}$ in particular is not to be taken past its first figure: higher-order and strange-quark contributions move the sea fraction upwards.

∎

The numbers extracted here are collected, with the measured cross sections they come from, in Table 116.2.

The higher-statistics continuation was CDHS, which exposed a magnetized-iron calorimeter and spectrometer to the CERN SPS narrow-band neutrino beam and measured $F_{2}$ and $xF_{3}$ over a wide range of $x$ and $Q^{2}$ [deGroot:1979]. Two of its results matter here. It confirmed the mean squared charge of $5/18$ by direct comparison of its own $F_{2}^{\nu N}$ with the charged-lepton measurements, which is the content of Proposition 102.46; and it separated the antiquark content, whose momentum fraction is small and concentrated at small $x$, confirming that the momentum deficit of Phenomenon 116.25 cannot be repaired by a larger sea. The parity-violating structure function $xF_{3}$, which exists only for a weak probe, counts valence quarks.

Proposition 116.29 (The Gross–Llewellyn Smith sum rule in the parton model).

On an isoscalar nucleon the third charged-current structure function integrates to the number of valence quarks,

\begin{equation}\tag{116.36} \int_{0}^{1}F_{3}^{\nu N}(x)\,\dd x=3\ec \end{equation}

a pure number, and its measurement is a count of the valence quarks in a nucleon obtained without ever isolating one. Rests on Equation (116.33) and Proposition 102.6.

Derivation. Derives Proposition 116.29. The charged current is $V-A$: it couples to left-handed quarks and to right-handed antiquarks, and it is the interference of its vector with its axial part that produces a third structure function at all. That interference term changes sign between a quark and an antiquark — which is why $F_{3}$ exists only for a parity-violating probe and vanishes identically for the photon of Equation (116.19) — so at leading order in the parton model

\[ F_{3}^{\nu N}(x)=\sum_{q}\left[q(x)-\bar{q}(x)\right]\ec \]

with unit weights, since the current is blind to electric charge. The sea contributes a quark and an antiquark at the same $x$ and therefore cancels in the difference, leaving only the valence densities: $\int_{0}^{1}[q-\bar{q}]\dd x=\int_{0}^{1}q_{v}\dd x$. Averaged over the proton ($uud$) and the neutron ($udd$) of an isoscalar target the valence content is the same three quarks either way, so

\[ \int_{0}^{1}F_{3}^{\nu N}\,\dd x =\int_{0}^{1}\left[u_{v}(x)+d_{v}(x)\right]\dd x =2+1=3\ec \]

which is Equation (116.36). Note what makes this a stronger statement than the momentum sum rule of Equation (116.28): no unknown weight enters, so the prediction is an integer and not a fraction to be compared with a measured one. The quantity is dimensionless, $F_{3}$ being a function of $x$ alone.

∎

The measured value is smaller than three, and the deficit is itself calculable: gluon radiation off the struck quark reduces the integral by a factor $1-\alpha_{s}(Q^{2})/\pi-\dots$, which turns the sum rule into a determination of the strong coupling at the $Q^{2}$ of the measurement. That correction is what remains owed here.

Derivation pending.

The perturbative correction to the Gross–Llewellyn Smith sum rule. The parton-model value of 3 is derived in the text above; what is still owed is the calculation of the coefficient function that multiplies it by $1-\alpha_{s}/\pi-\dots$, order by order, and the demonstration that the same coefficient function is the one appearing in the Bjorken sum rule for polarized scattering. That is what makes a measurement of this integral a competitive determination of the strong coupling, and it belongs in Appendix A alongside the other sum rules.

The 1990 Nobel Prize

The 1990 Nobel Prize in Physics was awarded to Jerome I. Friedman, Henry W. Kendall and Richard E. Taylor for their pioneering investigations concerning deep inelastic scattering of electrons on protons and bound neutrons, of essential importance for the development of the quark model in particle physics.

The three lectures are complementary and all three are worth reading as primary material rather than as summaries. Taylor's [Taylor:1991] is the account of the accelerator and the spectrometers — how End Station A was built and what it could and could not do. Kendall's [Kendall:1991] is the account of the measurements themselves and of the analysis that turned raw yields into $\nu W_{2}$, including the radiative corrections. Friedman's [Friedman:1991] is the account of the interpretation: which alternatives were on the table, what killed each of them, and how long the parton reading took to become the consensus. Taken together they are an unusually candid record of an experiment whose result was understood only some years after it was obtained.

Scaling violations and the confirmation of QCD

Asymptotic freedom

Scaling is the statement that the constituents are free. A field theory in which they are genuinely free has no interactions and no bound states; the parton model was therefore, as Remark 116.7 says, a hypothesis without dynamics. What was needed was a theory whose coupling vanishes at short distance, and in 1973 Gross and Wilczek [Gross:1973] and, independently, Politzer [Politzer:1973] showed that a non-abelian gauge theory is such a theory and, within the class of renormalizable field theories in four dimensions, essentially the only one.

The result is Theorem 102.35: the one-loop beta function of an $\SU(3)$ gauge theory with $n_{f}$ quark flavours has the coefficient $b_{0}=11-\tfrac{2}{3}n_{f}$, positive for $n_{f}\leq16$, so the coupling runs to zero logarithmically at large momentum transfer. The measured curve is Phenomenon 102.41, with $\alpha_{s}(m_{Z}c^{2})=0.1180(9)$ [Navas:2024] and the values at lower scales tabulated in Table 102.2.

Two consequences for this chapter follow immediately, and they pull in opposite directions, which is what makes the resulting prediction sharp.

Scaling must hold approximately. At large $Q^{2}$ the coupling is small, the struck quark is nearly free during the absorption, and the parton model is the leading term of a systematic expansion rather than a guess.
Scaling must fail logarithmically. The coupling is not zero, so a quark may radiate a gluon before or after being struck. Raising $Q^{2}$ resolves emissions that a coarser probe could not separate, so the number of partons seen rises and each carries less momentum. Structure functions must therefore fall with $\ln Q^{2}$ at large $x$, where partons are lost to radiation, and rise at small $x$, where the radiated partons accumulate.

The second is a prediction with no free parameter beyond $\alpha_{s}$ itself, and it is what the rest of this section reports as measured.

The evolution equations

The equations that make the second statement quantitative were derived by Gribov and Lipatov [Gribov:1972], by Altarelli and Parisi [Altarelli:1977] and by Dokshitzer [Dokshitzer:1977], and are Theorem 102.49; the splitting functions come from the collinear emission probability Equation (102.58), and the Mellin diagonalisation that turns the integro-differential system into ordinary differential equations for the moments is Corollary 102.50. None of that is rebuilt here. What belongs here is the experimental use to which it is put.

Proposition 116.30 (How the strong coupling is read off a slope).

For a non-singlet combination of parton distributions, the moment $M_{N}=\int_{0}^{1}f(x)x^{N}\dd x$ obeys Equation (102.60), so

\begin{equation}\tag{116.37} \alpha_{s}\left(Q^{2}\right) =\frac{2\pi}{\gamma_{N}}\, \dv{\ln M_{N}}{\ln Q^{2}}\ec \end{equation}

with $\gamma_{N}$ the rational number Equation (102.62). The strong coupling is therefore measured by a rate of change of the data, not by a rate; no absolute normalisation and no knowledge of the parton distributions enters. Rests on Equations (102.60) and (102.62).

Proof.

Derives Proposition 116.30. Equation (102.60) reads $\dd M_{N}/\dd\ln Q^{2}=(\alpha_{s}/2\pi)\gamma_{N}M_{N}$; dividing by $M_{N}$ and solving for $\alpha_{s}$ gives Equation (116.37). For the second moment, with $C_{F}=\tfrac{4}{3}$ and $H_{m}=\sum_{k\leq m}1/k$,

\[ \gamma_{2}=C_{F}\left[\tfrac{3}{2}-H_{2}-H_{4}\right] =\tfrac{4}{3}\left[1.5-1.5-2.0833\right] =-2.778\ec \]

so at $\alpha_{s}=0.20$ the moment falls at $\dd\ln M_{2}/\dd\ln Q^{2}=-0.088$, that is by about $20\,\mathrm{\%}$ per decade in $Q^{2}$. Measuring that slope to $10\,\mathrm{\%}$ determines $\alpha_{s}$ to $10\,\mathrm{\%}$.

The parameter-free version of the test is the ratio of two slopes. With $\gamma_{1}=-\tfrac{16}{9}=-1.778$ and $\gamma_{3}=C_{F}[\tfrac{3}{2}-H_{3}-H_{5}]=-3.489$,

\begin{equation}\tag{116.38} \frac{\dd\ln M_{3}/\dd\ln Q^{2}} {\dd\ln M_{1}/\dd\ln Q^{2}} =\frac{\gamma_{3}}{\gamma_{1}}=1.96\ec \end{equation}

a pure number containing neither $\alpha_{s}$ nor $\Lambda_{\mathrm{QCD}}$ nor the initial distributions — the content of Remark 102.52. Plotting one measured moment against another on logarithmic axes must give a straight line of that slope.

∎

Remark 116.31 (What the equations do not predict).

The evolution equations say how the distributions change, not what they are. The distributions at some reference scale $Q_{0}$ are non-perturbative objects and must be measured; QCD supplies no prediction for them, and any claim that it does is a claim about a model and not about the theory. This is not a weakness of the test but what makes it stringent: the initial distributions are fitted at one scale, and then several thousand data points at other scales must be described with no further freedom beyond the single number $\alpha_{s}$.

Muon-beam measurements

The lever arm in $Q^{2}$ needed to see a logarithm requires beam energies well beyond SLAC's. Muon beams supplied them: a muon is a charged lepton, so the process and its interpretation are identical to electron scattering, and a muon beam produced from pion decay can be made at several hundred $\mathrm{GeV}$ where an electron beam of the same energy cannot.

The Bologna–CERN–Dubna–Munich–Saclay collaboration (BCDMS) exposed a long toroidal-iron spectrometer with internal hydrogen, deuterium and carbon targets to the CERN SPS muon beam at energies of order $100\text{–}280\,\mathrm{GeV}$, and measured $F_{2}(x,Q^{2})$ and $R$ with high statistics over $x$ from about $0.06$ to $0.8$ and $Q^{2}c^{2}$ from about $7\,\mathrm{GeV}^{2}$ to $260\,\mathrm{GeV}^{2}$ [Benvenuti:1989] — a factor of nearly forty in $Q^{2}$, against the factor of eight available in 1969. The European Muon Collaboration made companion measurements over an overlapping range, and its data on nuclear targets produced the separate result of Section 116.7.1.

Phenomenon 116.32 (Scaling is violated logarithmically).

Measured over a far wider range of $Q^{2}$ than the original experiment, the structure function $F_{2}(x,Q^{2})$ is not exactly a function of $x$ alone: at large $x$ it decreases with $\log Q^{2}$ and at small $x$ it increases, the two movements being tied to each other [Benvenuti:1989]. At the electron–proton collider the rise at small $x$ is dramatic, and $F_{2}$ continues to grow down to $x$ below $10^{-4}$ [Aaron:2010] [Abramowicz:2015]. This pattern, and not scaling itself, is the quantitative confirmation of quantum chromodynamics, because the theory predicts not the structure functions but exactly their rate of change. Rests on Phenomenon 116.18, Proposition 102.48 and Equation (102.59).

Derivation. Derives Phenomenon 116.32. The mechanism is Proposition 102.48: a quark of fraction $y$ radiates a gluon and is left with $x=zy<y$, with probability $(\alpha_{s}/2\pi)P_{qq}(z)\dd z\,\dd k_{T}^{2}/k_{T}^{2}$ per unit transverse momentum squared. Integrating the last factor up to the resolution $Q^{2}$ produces $\ln Q^{2}$ and nothing steeper, which is why the violation is logarithmic and not a power — the distinguishing signature against a constituent size, which by Equation (116.23) would give a power. Assembling the gain and loss terms gives Equation (102.59), whose sign structure is the statement in the phenomenon: the plus prescription in Equation (102.58) makes $\int_{0}^{1}P_{qq}=0$, so whatever is lost at large $x$ reappears at smaller $x$, and the two movements are not independent but are the same probability counted twice. The crossing point, where $\dd F_{2}/\dd\ln Q^{2}$ changes sign, is near $x\approx0.15$ in the data and is itself predicted.

The quantitative comparison is Proposition 116.30 applied to the measured moments; the collaboration's own QCD analysis of its non-singlet data returned a strong coupling at the low end of the range now accepted, and the modern world average from all methods is $\alpha_{s}(m_{Z}c^{2})=0.1180(9)$ [Navas:2024], with deep inelastic determinations among the inputs.

∎

HERA

This is the subsection to which Remark 102.52 defers the experimental side of the DGLAP test.

HERA, at DESY in Hamburg, was the only electron–proton collider ever built: a ring of about $6.3\,\mathrm{km}$ circumference in which $27.5\,\mathrm{GeV}$ electrons or positrons collided with protons of $920\,\mathrm{GeV}$ ($820\,\mathrm{GeV}$ in the earlier running). Two general-purpose detectors, H1 and ZEUS, recorded the collisions.

Proposition 116.33 (What colliding changes).

The centre-of-mass energy of the HERA collisions is

\[ \sqrt{s}\,c\approx2\sqrt{E_{e}E_{p}} =2\sqrt{27.5\,\mathrm{GeV}\times920\,\mathrm{GeV}} =318\,\mathrm{GeV}\ec \]

neglecting the particle masses. A fixed-target experiment would need a beam energy

\begin{equation}\tag{116.39} E_{\mathrm{lab}}=\frac{sc^{2}}{2Mc^{2}} =\frac{\left(318\,\mathrm{GeV}\right)^{2}} {2\times0.93827\,\mathrm{GeV}}=54\,\mathrm{TeV} \end{equation}

to reach the same invariant mass. Since $Q^{2}c^{2}\leq sc^{2}$ and $x\geq Q^{2}c^{2}/(sc^{2})$ at $y\leq1$, the accessible region extends to $Q^{2}c^{2}$ of order $10^{5}\,\mathrm{GeV}^{2}$ and to $x$ some four orders of magnitude below anything a fixed target can reach. Rests on Notation 116.13.

Proof.

Derives Proposition 116.33. For head-on beams of energies $E_{e}$ and $E_{p}$ and negligible masses, the total four-momentum has energy $E_{e}+E_{p}$ and momentum $(E_{p}-E_{e})/c$, so $sc^{2}=(E_{e}+E_{p})^{2}-(E_{p}-E_{e})^{2}=4E_{e}E_{p}$, giving the first line; $s$ carries $\mathrm{kg}^{2}\,\mathrm{m}^{2}/\mathrm{s}^{2}$, the dimension of $Q^{2}$. For a stationary target of mass $M$, $sc^{2}=M^{2}c^{4}+2Mc^{2}E_{\mathrm{lab}}$, which for $E_{\mathrm{lab}}\gg Mc^{2}$ inverts to Equation (116.39). The kinematic bound follows from $Q^{2}=xys$ with $x,y\in[0,1]$ — both sides carrying $\mathrm{kg}^{2}\,\mathrm{m}^{2}/\mathrm{s}^{2}$ — so $Q^{2}c^{2}\leq sc^{2}$ and, at fixed $Q^{2}$, small $x$ requires large $s$ — which is the whole reason a collider was built for this measurement.

∎

The combination of the two experiments' inclusive cross sections is the definitive data set: about $2900$ measured points covering $Q^{2}c^{2}$ from about $0.045\,\mathrm{GeV}^{2}$ to $5\times 10^{4}\,\mathrm{GeV}^{2}$ and $x$ from about $6\times 10^{-7}$ to $0.65$, with total uncertainties reaching the percent level in the bulk of the range [Aaron:2010] [Abramowicz:2015]. Combining the two experiments before fitting is itself part of the result: the cross-calibration reduces correlated systematics that neither experiment could remove alone.

Phenomenon 116.34 (The structure function rises steeply at small $x$).

At fixed $Q^{2}$, $F_{2}(x,Q^{2})$ grows as $x$ decreases, with an effective power $F_{2}\sim x^{-\lambda}$ whose exponent itself grows with $Q^{2}$: $\lambda$ is near $0.1$ at $Q^{2}c^{2}\approx1\,\mathrm{GeV}^{2}$ and near $0.3$ at $Q^{2}c^{2}\approx100\,\mathrm{GeV}^{2}$. The growth continues down to the smallest $x$ measured, about $6\times 10^{-7}$, with no sign of saturation [Aaron:2010] [Abramowicz:2015]. Rests on Equation (102.59) and Proposition 116.33.

Derivation. Derives Phenomenon 116.34. At small $x$ the quark distributions are dominated by gluon splitting, so the driving term of Equation (102.59) is $P_{qg}\otimes g$, and each unit of $\ln Q^{2}$ feeds gluons into quarks at small $x$ while each step down in $x$ collects the contributions of all larger fractions. Iterating the two effects generates a distribution rising as a power of $1/x$ with an exponent that grows with the evolution length, hence with $\ln Q^{2}$; that is the observed pattern. Quantitatively the rise is not an input to the fit but an output: the gluon distribution is fixed by the $Q^{2}$ slopes at moderate $x$ and the small-$x$ rise then follows.

∎

Proposition 116.35 (The gluon density is measured, not assumed).

At small $x$ the slope of the structure function measures the gluon distribution directly:

\begin{equation}\tag{116.40} \pdv{F_{2}\left(x,Q^{2}\right)}{\ln Q^{2}} \approx\frac{10\,\alpha_{s}\left(Q^{2}\right)}{27\pi} \left[x g\right]\!\left(2x,Q^{2}\right)\ec \end{equation}

for $n_{f}=4$ active flavours. The gluon density, which couples to neither the photon nor the weak current and is therefore invisible to every direct measurement in this chapter, is thus read off the rate of change of something that is measured. Rests on Equations (102.59) and (116.9).

Proof.

Derives Proposition 116.35. Keep only the gluon term of Equation (102.59) and weight it by the squared charges, since $F_{2}=\sum_{q}(e_{q}/e)^{2}x[q+\bar{q}]$ and a gluon splits into $q$ and $\bar{q}$ alike:

\[ \pdv{F_{2}}{\ln Q^{2}} =\frac{\alpha_{s}}{2\pi}\cdot2\sum_{q} \left(\frac{e_{q}}{e}\right)^{2} x\int_{x}^{1}\frac{\dd y}{y}\, P_{qg}\!\left(\frac{x}{y}\right)g\left(y\right)\ec \]

with $\sum_{q}(e_{q}/e)^{2}=\tfrac{4}{9}+\tfrac{1}{9}+\tfrac{1}{9} +\tfrac{4}{9}=\tfrac{10}{9}$ for $n_{f}=4$. Substitute $z=x/y$, so $\dd y/y=-\dd z/z$ and $y=x/z$; then

\[ x\int_{x}^{1}\frac{\dd y}{y}P_{qg}\!\left(\frac{x}{y}\right) g\left(y\right) =\int_{x}^{1}\dd z\,P_{qg}(z)\, \left[yg\right]\!\left(\frac{x}{z}\right)\ec \]

because $x\,g(x/z)/z=[yg](x/z)$. At small $x$ the combination $yg(y)$ varies slowly over the support of $P_{qg}$, so replace it by its value at the mean splitting fraction

\[ \avg{z}=\frac{\int_{0}^{1}z\,P_{qg}(z)\,\dd z} {\int_{0}^{1}P_{qg}(z)\,\dd z} =\frac{1/6}{1/3}=\frac{1}{2}\ec \]

that is, at $y=x/\avg{z}=2x$. What remains is $\int_{0}^{1}P_{qg}=\tfrac{1}{3}$, so

\[ \pdv{F_{2}}{\ln Q^{2}} \approx\frac{\alpha_{s}}{2\pi}\times2\times\frac{10}{9} \times\frac{1}{3}\times\left[xg\right]\!\left(2x\right) =\frac{10\alpha_{s}}{27\pi}\left[xg\right]\!\left(2x\right)\ec \]

which is Equation (116.40). Numerically: at $x=10^{-3}$ and $Q^{2}c^{2}=20\,\mathrm{GeV}^{2}$, so $Qc=4.5\,\mathrm{GeV}$ and $\alpha_{s}\approx0.21$ from Table 102.2, a measured slope of about $0.2$ gives

\[ \left[xg\right]\!\left(2\times 10^{-3}\right) \approx\frac{27\pi\times0.2}{10\times0.21} \approx8\ec \]

which is the size the global fits of Section 116.6.5 return at that fraction and scale. The relation is leading order and the expansion about $\avg{z}$ is crude; a real extraction fits the full evolution to all the data at once. What it exhibits, and what a fit conceals, is that the gluon enters the measurement only through a derivative.

∎

The numbers of the whole later programme, HERA's included, are collected in Table 116.3.

Parton distributions today

The measurements of this chapter are not archived as measurements. They are absorbed, together with fixed-target, Drell–Yan and collider data, into global fits that determine the parton distributions $f_{i}(x,\mu_{F}^{2})$ with uncertainty bands, and it is those objects that the rest of particle physics uses [Abramowicz:2015] [Navas:2024].

The property that makes them usable is collinear factorization, Theorem 102.53: the cross section for any process with a large scale splits into a calculable partonic part and a set of distributions that depend on the hadron and not on the process. The distributions are therefore universal, and the whole enterprise rests on that. Measured in electron–proton scattering at $Qc\sim10\,\mathrm{GeV}$, evolved by Equation (102.59) to $Qc\sim1\,\mathrm{TeV}$, and used to predict proton–proton cross sections at the LHC, they work — including for the Higgs production rates, which belong to Experiment: The Higgs Boson Discovery and which are unpredictable without them. Universality across two different initial states and three decades of scale is what makes factorization a theorem with evidence rather than a modelling assumption.

Three cautions belong with any quoted parton distribution, and Remark 102.54 states the first of them: a distribution is defined only together with a factorization scheme and a scale, and quoting one without both is meaningless. The second is that the uncertainty bands are not measurement errors in the ordinary sense; they combine experimental errors with the freedom in the functional form assumed at the starting scale, and different collaborations' bands are not independent. The third is that the fits are least well constrained exactly where they are most used at a hadron collider: the gluon at large $x$, and every distribution at $x$ below the HERA reach.

Extensions and modern precision

The EMC effect

Phenomenon 116.36 (The EMC effect).

The structure function per nucleon measured on iron differs from the one measured on deuterium by up to about $15\,\mathrm{\%}$, with a characteristic dependence on $x$ [Aubert:1983]. Quark distributions are therefore modified by the nuclear medium itself, and not merely smeared by the Fermi motion of nucleons that would otherwise be unchanged. No agreed explanation exists. Rests on Definition 102.44 and Equation (116.9).

The measurement was not the one the collaboration set out to make: iron was the target material of a spectrometer built for other purposes, and the deuterium comparison was a control. The observed ratio $F_{2}^{\mathrm{Fe}}/F_{2}^{\mathrm{D}}$ per nucleon falls below unity through the valence region, reaching about $0.89$ near $x=0.65$, and rises above unity at small $x$, reaching about $1.15$ near $x=0.05$ [Aubert:1983]. Later measurements on many nuclei established that the shape is universal and the magnitude grows slowly with mass number, and that the ratio turns upward again above $x\approx0.8$, where Fermi motion allows a bound nucleon to carry more than the whole nucleon momentum fraction.

Why this is a result rather than a nuisance: before 1983 it was generally supposed that a nucleus is a collection of nucleons whose internal structure is what it is in free space, so that nuclear effects on $F_{2}$ would be confined to the smearing produced by the nucleons' Fermi motion — an effect calculable from nuclear physics (Nuclear Forces and Nuclear Structure) and confined to large $x$. The observed modification extends over the whole valence region and has the wrong sign at moderate $x$ for Fermi smearing. Whatever the mechanism, the quark momentum distribution inside a bound nucleon is not the one inside a free nucleon.

Derivation pending.

The nuclear modification of parton distributions, stated as the open problem it is: no derivation from quantum chromodynamics currently reproduces the observed dependence on the momentum fraction. The candidate mechanisms — nucleon swelling, pion-cloud enhancement, multi-quark clusters, and a dependence on the local nuclear density rather than on the mass number — each account for part of the shape and none for all of it, and they are not mutually exclusive. What is owed in Appendix A is a statement of what each predicts and of which measurements discriminate among them.

Remark 116.37 (An honest limit on the sources here).

The bibliography of this book carries the original EMC measurement and nothing later on nuclear parton distributions. The statements above about the universality of the shape across nuclei, about its growth with mass number and about the local-density correlation are therefore made without a citation that supports them, and a reader who wants to check them must go outside this book's reference list. They are recorded here because omitting them would leave the 1983 result looking more isolated than it is.

The proton spin

Scattering polarized electrons or muons from a polarized target gives access to a third structure function, $g_{1}(x,Q^{2})$, which in the parton model counts the quark spins:

\begin{equation}\tag{116.41} g_{1}(x)=\frac{1}{2}\sum_{q} \left(\frac{e_{q}}{e}\right)^{2} \left[\Delta q(x)+\Delta\bar{q}(x)\right]\ec \end{equation}

with $\Delta q(x)$ the difference between the densities of quarks with spin along and against the proton's spin. The integral $\Gamma_{1}^{p}=\int_{0}^{1}g_{1}^{p}(x)\,\dd x$ is therefore a charge-weighted count of how much of the proton's spin the quark spins carry.

Phenomenon 116.38 (The quark spins do not account for the proton spin).

The European Muon Collaboration measured $\Gamma_{1}^{p}=0.114\pm0.012\pm0.026$ at $\avg{Q^{2}}c^{2}=10.7\,\mathrm{GeV}^{2}$, where the first uncertainty is statistical and the second systematic [Ashman:1988]. The naive expectation, obtained by combining the octet axial charges measured in hyperon beta decay with the assumption that the strange sea is unpolarized — the Ellis–Jaffe sum rule — is about $0.19$; that number is quoted here without derivation and without a source in this book's bibliography, and the debt is recorded immediately below. The quark spins therefore carry a small and possibly vanishing fraction of the proton's angular momentum, in flat contradiction with the constituent quark model, in which they carry all of it. Rests on Equation (116.41) and Proposition 102.6.

Derivation pending.

The Ellis–Jaffe sum rule, the theoretical number the EMC measurement contradicts. What is owed is the derivation of the expectation for the first moment of the polarized proton structure function from the octet axial charges: the isovector combination fixed by the neutron beta-decay ratio, the octet combination fixed by the hyperon semileptonic decays under the flavour symmetry of the baryon octet, and the assumption of an unpolarized strange sea that sets the singlet piece equal to the octet one — together with the leading perturbative correction, which lowers the parton-model value. Only that chain shows why the expected number is near $0.19$ and which of its assumptions the measurement actually falsifies. It belongs in Appendix A beside the other sum rules; this book's bibliography carries no entry for the original Ellis–Jaffe paper.

The resolution, so far as there is one, is that the proton's spin decomposes as

\begin{equation}\tag{116.42} \frac{1}{2}=\frac{1}{2}\Delta\Sigma+\Delta G+L_{q}+L_{g}\ec \end{equation}

in units of $\hbar$, where $\Delta\Sigma$ is the quark spin contribution measured above, $\Delta G$ the gluon spin, and $L_{q}$, $L_{g}$ the orbital angular momenta of quarks and gluons (Angular Momentum and Spin). Nothing requires the first term to dominate. Subsequent polarized experiments have measured $\Delta\Sigma$ to be roughly $0.3$ and have constrained $\Delta G$ to be small over the measured range of momentum fraction; the orbital terms are essentially unmeasured, and are among the principal targets of the programme in Section 116.7.4.

Derivation pending.

The decomposition of the proton spin into quark spin, gluon spin and orbital contributions, and the reason the split is ambiguous. What is owed is the demonstration that the gauge-invariant decomposition of the angular momentum of a gauge theory into a “spin” and an “orbital” part is not unique — different decompositions differ by terms that vanish for a free field and do not vanish here — so that a measured $\Delta G$ has meaning only relative to a stated convention and factorization scheme. It belongs in Appendix A with the operator definitions written out.

Remark 116.39 (Sources for the modern spin measurements).

This book's bibliography carries the original EMC measurement and no later polarized-scattering result. The values quoted above for $\Delta\Sigma$ and for the constraint on $\Delta G$ are therefore uncited, and are flagged as such.

The Drell–Yan process and the sea

Deep inelastic scattering is not the only way to measure a parton distribution, and the second way is the test of whether the distributions mean anything. Drell and Yan proposed that in a hadron collision a quark from one hadron and an antiquark from the other annihilate into a virtual photon, which materializes as a lepton pair of large invariant mass [Drell:1970].

Proposition 116.40 (The Drell–Yan cross section scales).

For the annihilation of a quark of fraction $x_{1}$ from one hadron with an antiquark of fraction $x_{2}$ from the other, the lepton pair has invariant mass $M_{\ell\ell}c^{2}$ — written with a subscript because $M$ without one is the proton mass throughout this chapter (Notation 116.13) — with

\begin{equation}\tag{116.43} \tau:=\frac{M_{\ell\ell}^{2}c^{4}}{sc^{2}}=x_{1}x_{2}\ec \end{equation}

and the cross section takes the form

\begin{equation}\tag{116.44} M_{\ell\ell}^{3}c^{6}\, \frac{\dd\sigma}{\dd\left(M_{\ell\ell}c^{2}\right)} =F\left(\tau\right)\ec \end{equation}

a function of the dimensionless ratio $\tau$ alone. Here $s$ carries $\mathrm{kg}^{2}\,\mathrm{m}^{2}/\mathrm{s}^{2}$ as in Proposition 116.33, so $\tau$ is dimensionless, and $F$ carries $\mathrm{J}^{2}\,\mathrm{m}^{2}$, since $M_{\ell\ell}^{3}c^{6}$ is a cubed energy and $\dd\sigma/\dd(M_{\ell\ell}c^{2})$ carries $\mathrm{m}^{2}/\mathrm{J}$. This is Bjorken scaling in a different process, with the same variables and the same distributions. Rests on Theorem 102.53 and Equation (116.9).

Proof.

Derives Proposition 116.40. The partonic process $q\bar{q}\to\ell^{+}\ell^{-}$ is the crossed channel of $\ell^{+}\ell^{-}\to q\bar{q}$ and its cross section is the point-like $\hat{\sigma}=4\pi\alpha^{2}(\hbar c)^{2}(e_{q}/e)^{2} /(3M_{\ell\ell}^{2}c^{4})\times\tfrac{1}{3}$, the last factor because the quark and antiquark must match in colour, one chance in three. It contains no scale but $M_{\ell\ell}$. The invariant mass of a pair made from fractions $x_{1}$ and $x_{2}$ of two hadrons whose squared centre-of-mass four-momentum is $s$ is $M_{\ell\ell}^{2}c^{4}=x_{1}x_{2}sc^{2}$, which is Equation (116.43). Folding $\hat{\sigma}$ with the two distributions and changing variables from $(x_{1},x_{2})$ to $(\tau,x_{1})$ produces a factor $1/M_{\ell\ell}^{3}$ from the partonic cross section and the Jacobian, leaving a function of $\tau$ alone, which is Equation (116.44). The measured cross section obeys it, and — the point of the exercise — the distributions extracted from it are the same ones extracted from deep inelastic scattering, which is the experimental content of Theorem 102.53.

∎

The process is also the sharpest probe of the antiquark sea, because it requires an antiquark in the initial state, and comparing hydrogen with deuterium targets isolates $\bar{d}/\bar{u}$. Measurements of that ratio find it significantly greater than one over a range of $x$, in agreement with the Gottfried sum of Remark 116.22 and in contradiction with any picture in which the sea comes from perturbative gluon splitting, which cannot distinguish flavours. The measurement is the Fermilab E866/NuSea experiment [Towell:2001], which compares Drell–Yan yields from hydrogen and deuterium targets; it and the Gottfried-sum determination [Arneodo:1994] are independent determinations of the same asymmetry, one in lepton–nucleon scattering and one in hadron–hadron collisions.

Continuing programmes

Deep inelastic scattering is not a closed subject, and it is worth being precise about which of the things still being measured would count as a new result and which as a refinement.

Refinements. Extending $F_{2}$ and $g_{1}$ to smaller $x$ and to larger $Q^{2}$; reducing the uncertainty on the gluon at large $x$, which currently limits several LHC predictions; mapping the nuclear modification of Section 116.7.1 across the periodic table; improving the determination of $\alpha_{s}$ from scaling violations. All of these tighten numbers inside a framework that is not in question.

New results. Three outcomes would be different in kind.

A measured violation of factorization. If the distributions extracted from one process failed to describe another at a level exceeding the calculable higher-order corrections, Theorem 102.53 would be false as applied, and the entire apparatus by which hadron-collider rates are predicted would lose its foundation.
A gluon density inconsistent with the evolution equations. The rise of Phenomenon 116.34 cannot continue indefinitely — at some small enough $x$ the density of gluons must be high enough that they recombine as well as split, and the linear evolution of Section 116.6.2 must fail. Observing that non-linear regime would be the first measurement of a genuinely new dynamical regime of QCD rather than a refinement of the known one. Whether HERA reached its onset is disputed.
The orbital angular momentum of the proton's constituents. Equation (116.42) has two terms that have never been measured. Determining them requires correlating a parton's momentum fraction with its transverse position or transverse momentum, which inclusive scattering cannot do.

The instrument designed for the last two is the Electron–Ion Collider now under construction at Brookhaven, which will collide polarized electrons with polarized protons and light ions and with heavy nuclei at centre-of-mass energies from about $20\text{–}140\,\mathrm{GeV}$ and luminosities of order $10^{38}\,/\mathrm{m}^{2}/\mathrm{s}$, some three orders of magnitude above HERA's [Accardi:2016]. Its observables are the exclusive and semi-inclusive ones — deeply virtual Compton scattering, tagged final-state hadrons — from which a transverse spatial or transverse momentum image of the parton distribution can be built, and the nuclear targets that would show saturation if it is there. It is a continuation of the measurement reported in this chapter with the same beam, the same probe and the same variables, on an instrument fifty years newer.

Primary references

The experiment itself is contained in two Physical Review Letters of 1969 submitted back to back. Bloom et al. [Bloom:1969] report the inelastic cross sections at $6\,^\circ$ and $10\,^\circ$ and the observation that the ratio to the Mott cross section falls far more slowly with momentum transfer than the elastic form factors do; Breidenbach et al. [Breidenbach:1969] report the structure function $\nu W_{2}$ and its collapse onto a single curve. Neither is long, and both should be read in preference to any account of them. The public announcement preceded both, at the Vienna conference of 1968, in Panofsky's rapporteur talk [Panofsky:1968]. The most useful single secondary source on the apparatus, the corrections and the whole 1967–1972 data set is Friedman and Kendall's review [Friedman:1972], which is where the numbers in Table 116.1 that are not from the two Letters come from.

The instrument has its own literature. Neal's edited volume on the Stanford two-mile accelerator [Neal:1968] is the engineering account of the machine. The three Nobel lectures [Taylor:1991] [Kendall:1991] [Friedman:1991] divide the story between apparatus, measurement and interpretation, and are the best first-hand record of what the collaboration believed at each stage. Mo and Tsai [Mo:1969] is the radiative-correction calculation on which every number in the experiment depends; it is a piece of applied quantum electrodynamics rather than of hadron physics, and it is the largest single theoretical input to the measured cross sections.

The theory the experiment tested was in print before the data. Rosenbluth [Rosenbluth:1950] gives the elastic cross section that defines the form factors; Hofstadter [Hofstadter:1956] is the review of the elastic measurements that established the proton's size. Gell-Mann [GellMann:1964] and Zweig [Zweig:1964] propose the quarks; Zweig's is an unpublished CERN preprint and is cited as such. Bjorken [Bjorken:1969a] predicts scaling from current algebra and Wilson [Wilson:1969] supplies the operator-product framework; Feynman [Feynman:1969] gives the parton picture and Bjorken and Paschos [Bjorken:1969b] the parton-distribution formulation used throughout this chapter. Callan and Gross [Callan:1969] give the spin discriminator of Section 116.4.3.

For the neutrino comparison, the Gargamelle total cross sections are Eichten et al. [Eichten:1973], from which the momentum fractions of Proposition 116.28 are derived here; the same exposures gave the weak neutral current in the leptonic channel [Hasert:1973a] and the hadronic channel [Hasert:1973b], reported in Electroweak Unification and the Higgs Boson. The high-statistics neutrino structure functions are the CDHS measurement in iron [deGroot:1979].

For the scaling violations, the theory is Gross and Wilczek [Gross:1973] and Politzer [Politzer:1973] for asymptotic freedom, and Gribov and Lipatov [Gribov:1972], Altarelli and Parisi [Altarelli:1977] and Dokshitzer [Dokshitzer:1977] for the evolution equations. The Gribov–Lipatov and Dokshitzer papers are translations from the Russian and neither carries a DOI. The Dokshitzer entry records that fact explicitly, together with the Russian original it translates; the Gribov–Lipatov entry carries only the journal reference of the translation, with no note, so a reader who wants the original must find it from the volume and pages alone. The measurements are BCDMS [Benvenuti:1989] for the muon beam and the two HERA combinations [Aaron:2010] [Abramowicz:2015] for the collider; the second supersedes the first and is the data set in current use. Collins, Soper and Sterman [Collins:1989] is the factorization proof that licenses using the extracted distributions elsewhere. Current values of $\alpha_{s}$, of the parton distributions and of the particle masses used here are from the Review of Particle Physics [Navas:2024], and the fundamental constants from the CODATA 2022 adjustment [Mohr:2025]; the shipped data files behind those two citations are the source of every constant in Tables 116.1 and 116.3.

The extensions are Aubert et al. [Aubert:1983] for the EMC effect, Ashman et al. [Ashman:1988] for the proton spin, Drell and Yan [Drell:1970] for the lepton-pair process, and the Electron–Ion Collider white paper [Accardi:2016] for the programme now being built. The flavour asymmetry of the light-quark sea rests on two measurements in different processes: Arneodo et al. [Arneodo:1994] for the New Muon Collaboration's reevaluation of the Gottfried sum, and Towell et al. [Towell:2001] for the Fermilab E866/NuSea Drell–Yan determination of $\bar{d}/\bar{u}$. Halzen and Martin [Halzen:1984] is the textbook treatment closest to the level of this chapter.

Finally, what this reference list does not contain, since a reader is entitled to know where the chapter's statements outrun its sources. There is no entry for a dedicated modern measurement of $R=\sigma_{L}/\sigma_{T}$, so the value in Table 116.1 is the SLAC one and the modern refinement is not reported. There is no entry for nuclear parton distributions after 1983 (Remark 116.37) and none for polarized measurements after 1988 (Remark 116.39). There is no entry for Ellis and Jaffe, so the expectation of $0.19$ against which Phenomenon 116.38 reports a contradiction is both uncited and underived, and the derivation is owed as a pending proof. Each of these is a gap in the bibliography, not a gap in the physics, and each is flagged where it occurs in the text rather than only here.

Quantity	Value	Year	Source
Beam energy range (setting; calibrated to a few parts in $10^{3}$)	\(4.5\text{–}20\,\mathrm{GeV}\), i.e. \(7.2\times 10^{-10}\text{–}3.2\times 10^{-9}\,\mathrm{J}\)	1968	[Bloom:1969] [Neal:1968]
Scattering angles, first survey (setting)	\(6\,^\circ\) and \(10\,^\circ\)	1968	[Bloom:1969]
$Q^{2}c^{2}$ covered (kinematic reach)	\(0.7\text{–}8\,\mathrm{GeV}^{2}\); in SI, $Q^{2}$ from \(2.0\times 10^{-37}\) to \(2.3\times 10^{-36}\,\mathrm{kg}^{2}\,\mathrm{m}^{2}/\mathrm{s}^{2}\)	1968	[Bloom:1969]
Deep inelastic cut on $W$ (analysis cut)	$W>2\,\mathrm{GeV}/c^{2}$	1969	[Breidenbach:1969]
Elastic suppression across that $Q^{2}$ range, relative to $\sigma_{\mathrm{Mott}}$ at \(10\,^\circ\)	factor $\approx6.3\times 10^{2}$ (raw cross section $\approx8.6\times 10^{3}$); derived from the dipole, whose own \(10\,\mathrm{\%}\) accuracy makes these factors good to a few tens of percent	1956	[Hofstadter:1956]
Inelastic-to-Mott ratio across the same range	changes by less than a factor \(2\)	1969	[Bloom:1969]
$R=\sigma_{L}/\sigma_{T}$	\(0.18(10)\)	1969–72	[Breidenbach:1969] [Friedman:1972]
$\int_{0}^{1}F_{2}^{eN}(x)\,\dd x$ (isoscalar)	\(0.14(1)\), the error being the precision to which the review quotes it	1972	[Friedman:1972]
Resolution $\hbar c/(Qc)$ at $Q^{2}c^{2}=8\,\mathrm{GeV}^{2}$	\(7.0\times 10^{-17}\,\mathrm{m}\) (derived; exact given $Q$, to which the \(10^{-3}\) beam calibration propagates)	1968	Equation (116.27)
Proton mass	\(1.67262192595(52)\times 10^{-27}\,\mathrm{kg}\)	2022	[Mohr:2025]
Proton mass energy	\(0.93827208816(29)\,\mathrm{GeV}\)	2024	[Navas:2024]
Proton rms charge radius	\(8.4075(64)\times 10^{-16}\,\mathrm{m}\)	2022	[Mohr:2025]
Fine-structure constant	\(7.2973525643(11)\times 10^{-3}\)	2022	[Mohr:2025]

Quantity	Value	Year	Source
$\sigma^{\nu N}/E_{\nu}$	\(7.4(2)\times 10^{-43}\,\mathrm{m}^{2}/\mathrm{GeV}\)	1973	[Eichten:1973]
$\sigma^{\bar{\nu}N}/E_{\bar{\nu}}$	\(2.8(1)\times 10^{-43}\,\mathrm{m}^{2}/\mathrm{GeV}\)	1973	[Eichten:1973]
Ratio $\sigma^{\bar{\nu}}/\sigma^{\nu}$	\(0.378(17)\); $\tfrac{1}{3}$ for pure quarks	1973	[Eichten:1973]
$C=2Mc^{2}G_{F}^{2}/\pi(\hbar c)^{4}$	\(3.164\times 10^{-42}\,\mathrm{m}^{2}/\mathrm{GeV}\) (derived; exact to the constants it is built from)	1973	Equation (116.34)
Quark momentum fraction $Q$	\(0.230(7)\) (derived)	1973	Equation (116.35)
Antiquark momentum fraction $\bar{Q}$	\(0.0119(43)\) (derived)	1973	Equation (116.35)
$\int_{0}^{1}F_{2}^{\nu N}\dd x$	\(0.484(11)\) (derived)	1973	Equation (116.35)
Mean squared charge $\int F_{2}^{eN}/\int F_{2}^{\nu N}$	\(0.29(2)\) (derived); $5/18=0.278$	1972–73	[Friedman:1972] [Eichten:1973]
Nucleon momentum in charged constituents	\(0.50(4)\) (derived from $\int F_{2}^{eN}\dd x$ and $5/18$); reported as about one half on iron	1972–79	[Friedman:1972] [deGroot:1979]

Quantity	Value	Year	Source
BCDMS muon beam energies (setting)	\(100\text{–}280\,\mathrm{GeV}\)	1989	[Benvenuti:1989]
BCDMS coverage (kinematic reach)	$0.06<x<0.8$; $7\,\mathrm{GeV}^{2}<Q^{2}c^{2}< 260\,\mathrm{GeV}^{2}$	1989	[Benvenuti:1989]
HERA beam energies (setting)	\(27.5\,\mathrm{GeV}\) on \(920\,\mathrm{GeV}\), $\sqrt{s}\,c=318\,\mathrm{GeV}$	2010–15	[Aaron:2010] [Abramowicz:2015]
Equivalent fixed-target energy	\(54\,\mathrm{TeV}\) (derived; exact given the beam energies)	2010–15	Equation (116.39)
HERA combined coverage (kinematic reach)	$6\times 10^{-7}<x<0.65$; $0.045\,\mathrm{GeV}^{2}<Q^{2}c^{2}< 5\times 10^{4}\,\mathrm{GeV}^{2}$	2015	[Abramowicz:2015]
Small-$x$ exponent $\lambda$, read from the published curves	$0.1$ at $Q^{2}c^{2}=1\,\mathrm{GeV}^{2}$ and $0.3$ at \(100\,\mathrm{GeV}^{2}\), each to the reading precision $\pm0.05$	2015	[Abramowicz:2015]
Gluon density $[xg](2\times 10^{-3})$ at $Q^{2}c^{2}=20\,\mathrm{GeV}^{2}$	$8\pm2$ (derived from a slope read off the data; the leading-order relation is itself good only to that accuracy)	2015	Equation (116.40)
$\alpha_{s}(m_{Z}c^{2})$	\(0.1180(9)\)	2024	[Navas:2024]
$F_{2}^{\mathrm{Fe}}/F_{2}^{\mathrm{D}}$ at $x=0.65$, read from the published ratio curve	$0.89\pm0.01$ reading precision	1983	[Aubert:1983]
$F_{2}^{\mathrm{Fe}}/F_{2}^{\mathrm{D}}$ at $x=0.05$, read from the published ratio curve	$1.15\pm0.01$ reading precision	1983	[Aubert:1983]
$\Gamma_{1}^{p}=\int_{0}^{1}g_{1}^{p}\dd x$ at $\avg{Q^{2}}c^{2}=10.7\,\mathrm{GeV}^{2}$	$0.114\pm0.012\pm0.026$; Ellis–Jaffe expectation $\approx0.19$	1988	[Ashman:1988]