Deep Inelastic Scattering and Parton Distributions

Deep inelastic scattering (DIS) is the process that revealed quarks inside the proton and remains the cleanest window on the proton's partonic structure. It is the QCD analogue of the worked QED calculations (Compton, hydrogen): a concrete computation that turns the QCD Lagrangian into measured numbers — the structure functions, Bjorken scaling, its logarithmic violation (DGLAP), and the parton distribution functions that make hadron colliders predictive.

Conventions: $\hslash =c=1$, mostly-minus metric.

Kinematics

DIS is lepton–nucleon scattering $\ell (k)+N(P)\to \ell (k^{\prime })+X$, where $X$ is any hadronic final state (the nucleon is smashed, hence inelastic). A single virtual photon (or $W/Z$) of momentum $q=k-k^{\prime }$ probes the target. The invariants are

$$Q^2=-q^2>0,x=\frac{Q^2}{2P\cdot q},y=\frac{P\cdot q}{P\cdot k},$$

with Bjorken $x$ ($0<x\le 1$) the key variable. "Deep" means $Q^2\gg M_N^2$; "inelastic" means the final hadronic mass $W^2=(P+q{)}^2\gg M_N^2$.

Structure functions

The cross section factorizes into a calculable leptonic tensor and an unknown hadronic tensor $W^{\mu \nu }$, parameterized by two (for electromagnetic DIS) structure functions $F_1(x,Q^2)$ and $F_2(x,Q^2)$:

$$\frac{d^2\sigma }{dxd{Q}^2}=\frac{4\pi {\alpha }^2}{x{Q}^4}\left[\right(1-y)F_2(x,Q^2)+x{y}^2{F}_1(x,Q^2)].$$

The structure functions encode everything about the target's internal structure; QCD's job is to predict them.

The parton model and Bjorken scaling

Bjorken and Feynman's parton model: at large $Q^2$ the virtual photon scatters elastically off a single point-like constituent (a parton) carrying a fraction $\xi$ of the proton momentum. Elastic kinematics on the parton force $\xi =x$ — the struck parton carries exactly the Bjorken-$x$ fraction. Summing incoherently over partons of charge $Q_i$ with number density $f_i(x)$ gives

$$\boxed{F_2(x)=\sum _i{Q}_i^{2}x{f}_i(x),F_2=2x{F}_1(\text{Callan–Gross}).}$$

Two sharp predictions, both confirmed at SLAC (1968–69):

• Bjorken scaling — $F_2$ depends on $x$ alone, not on $Q^2$. Scaling means the partons are point-like; a target with intrinsic size would show $Q^2$-dependence at $Q^2\sim 1/{\text{size}}^2$. This was the discovery that quarks are real, point-like constituents.
• Callan–Gross relation $F_2=2x{F}_1$ — follows from the partons being spin-$\frac{1}{2}$ Dirac fermions (spin-0 partons would give $F_1=0$). Confirmed, establishing quark spin.

The parton distribution functions $f_i(x)$ — the probability of finding parton $i$ with momentum fraction $x$ — are the physical content extracted from $F_2$.

Scaling violations: DGLAP evolution

QCD corrects exact scaling. The struck quark can radiate a gluon before being hit (asymptotic freedom makes this calculable), so the PDFs acquire a slow, logarithmic $Q^2$-dependence — scaling violation. The evolution is governed by the DGLAP equations (Dokshitzer–Gribov–Lipatov– Altarelli–Parisi):

$$\boxed{\frac{\partial f_i(x,Q^2)}{\partial \ln Q^2}=\frac{{\alpha }_s(Q^2)}{2\pi }\sum _j{\int }_x^1\frac{d\xi }{\xi }{P}_{ij}\left(\frac{x}{\xi }\right)f_j(\xi ,Q^2),}$$

with the splitting functions $P_{ij}(z)$ giving the probability that parton $j$ radiates to become parton $i$ carrying fraction $z$ (e.g. $P_{qq},P_{gq},P_{qg},P_{gg}$). DGLAP is the renormalization-group equation for the PDFs — it resums the collinear logarithms $\ln Q^2$ exactly as the running coupling resums UV logarithms. Its prediction for how $F_2$ drifts with $Q^2$ (rising at small $x$, falling at large $x$) is confirmed across orders of magnitude in $Q^2$ at HERA and fixed-target experiments — one of the most stringent tests of perturbative QCD.

Parton distribution functions (PDFs)

The $f_i(x,Q^2)$ are nonperturbative (they describe the proton's bound-state structure, beyond perturbation theory) and are extracted from global fits to DIS and collider data. They are not computable from Feynman diagrams — but they are universal: the same PDF appears in every process. Key facts:

• Sum rules constrain them: momentum conservation ${\sum }_i{\int }_0^{1}x{f}_{i}dx=1$ (and famously, quarks carry only $\sim 50\%$ of the proton momentum — the rest is gluons, an early sign of the dynamical gluon field), and valence-number rules ${\int }_0^1(u-\bar{u})dx=2$, ${\int }_0^1(d-\bar{d})dx=1$ for the proton.
• Flavor content: valence $uud$, plus a sea of $q\bar{q}$ pairs and gluons.

Factorization: making hadron colliders predictive

The universality of PDFs is codified in the factorization theorem, which separates the (nonperturbative, universal) PDFs from the (perturbative, process- specific) partonic cross section — the master relation quoted in QCD/from-postulates §Factorization:

$${\sigma }_{AB\to X}=\sum _{i,j}\int d{x}_{1}d{x}_2{f}_{i/A}(x_1,{\mu }_F)f_{j/B}(x_2,{\mu }_F){\hat{\sigma }}_{ij\to X}(x_1{x}_{2}s,{\alpha }_s({\mu }_R),{\mu }_F).$$

• $f_{i/A}$ — PDFs, measured once (e.g. in DIS), used everywhere.
• ${\hat{\sigma }}_{ij\to X}$ — the partonic cross section, computed perturbatively with the Feynman rules and cross-section formulas.
• ${\mu }_F$ — the factorization scale; its dependence is governed by DGLAP and cancels between the PDF and $\hat{\sigma }$ order by order.

This is precisely what lets LHC predictions proceed despite confinement: the incalculable long-distance physics is packaged into universal PDFs, and everything short-distance is perturbative.

Summary

• DIS kinematics: $Q^2=-q^2$, Bjorken $x=Q^2/2P\cdot q$.
• Parton model: $F_2={\sum }_i{Q}_i^{2}x{f}_i(x)$, with Bjorken scaling (point-like partons) and Callan–Gross $F_2=2x{F}_1$ (spin-$\frac{1}{2}$).
• DGLAP evolution gives calculable scaling violations — the RG equation for PDFs.
• PDFs are nonperturbative but universal; factorization turns them into predictions for any collider process.

Where this fits

• The observables this computes: QCD observables §3.
• The companion QCD calculation: Jets and the Running Coupling.
• The theory: QCD from postulates, asymptotic freedom.

References

• Halzen & Martin, Quarks and Leptons, Ch. 8–10.
• Ellis, Stirling & Webber, QCD and Collider Physics.
• Peskin & Schroeder, An Introduction to Quantum Field Theory, Ch. 17–18.