Deep-Inelastic Scattering and the Parton Model

Deep-inelastic lepton–hadron scattering converts an unobserved hadronic final state into measurable structure functions. In the Bjorken limit those functions depend primarily on the momentum fraction $x$, and the spin-$\tfrac12$ parton model relates them to quark distributions. QCD predicts the logarithmic violations of this scaling and organizes target-mass and higher-twist corrections.

Required background. Inclusive annihilation and the emergence of jets supplies inclusive real–virtual reasoning and the interpretation of quark charges and color.

Helpful background. The free-field OPE preview supplies the short-distance operator logic behind moments of structure functions.

DIS kinematics and the hadronic tensor

For $\ell(k)+H(P)\to\ell(k')+X$, define

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

The inclusive hadronic mass is

$$W^2=(P+q)^2=M^2+Q^2\left(\frac1x-1\right).$$

Thus genuinely deep-inelastic kinematics require both $Q^2$ and $W^2$ to be large compared with hadronic scales. Large $Q^2$ at $x\to1$ can leave $W^2$ small and enter the resonance or endpoint region.

For electromagnetic scattering on an unpolarized target, all hadronic dynamics is in

$$W^{\mu\nu}(P,q)= \frac{1}{4\pi}\sum_X(2\pi)^4\delta^{(4)}(P+q-p_X) \langle P|J^\mu(0)|X\rangle \langle X|J^\nu(0)|P\rangle.$$

Lorentz symmetry, parity, and current conservation give

$$W^{\mu\nu}= \left(-g^{\mu\nu}+\frac{q^\mu q^\nu}{q^2}\right)F_1 +\frac{1}{P\!\cdot q} \left(P^\mu-\frac{P\!\cdot q}{q^2}q^\mu\right) \left(P^\nu-\frac{P\!\cdot q}{q^2}q^\nu\right)F_2,$$

where $F_{1,2}=F_{1,2}(x,Q^2)$ in this convention. Contracting with the massless leptonic tensor, neglecting target-mass and electroweak corrections, yields

$$\frac{d^2\sigma}{dx\,dy} =\frac{4\pi\alpha^2s}{Q^4} \left[(1-y)F_2(x,Q^2)+y^2xF_1(x,Q^2)\right].$$

Equivalently, with $F_L=F_2-2xF_1$, the bracket is $\tfrac12\{[1+(1-y)^2]F_2-y^2F_L\}$. This equality is a useful normalization check.

The parton-model derivation

In a frame where the fast hadron has large momentum, suppose the photon scatters incoherently from a nearly collinear parton carrying momentum $p=\xi P$. For a massless struck quark, the final quark is on shell:

$$(\xi P+q)^2\simeq 2\xi P\!\cdot q-Q^2=0,$$

so $\xi=x$. The measured Bjorken variable therefore selects the struck parton’s longitudinal momentum fraction.

At leading order in the quark–parton model,

$$F_2(x)=x\sum_q e_q^2\,[q(x)+\bar q(x)], \qquad F_2(x)=2xF_1(x), \qquad F_L(x)=0.$$

The first equation weighs each quark or antiquark density by the square of its electromagnetic charge. The second is the Callan–Gross relation and follows from scattering on effectively massless spin-$\tfrac12$ constituents. The kinematic construction and leading parton-model result are derived in Schwartz 2014, §32.1, pp. 668–76.

“Bjorken scaling” means approximate independence of the dimensionless functions at fixed $x$ as $Q^2$ changes. It is not exact: QCD radiation changes the resolution of the hadron and creates logarithmic $Q^2$ dependence. Longitudinal structure also begins perturbatively beyond the naive parton model.

From probabilities to QCD factorization

The probability language is a useful leading approximation, but QCD needs renormalized operators and coefficient functions. At leading power,

$$F_a(x,Q^2) =\sum_i\left[C_{a,i}\!\left(\frac{Q}{\mu_F},\alpha_s(\mu_R)\right) \otimes f_{i/H}(\mu_F)\right](x) +O\!\left(\frac{M^2}{Q^2},\frac{\Lambda_{\mathrm{QCD}}^2}{Q^2}\right),$$

with

$$[C\otimes f](x)=\int_x^1\frac{dz}{z}\,C(z)f\!\left(\frac{x}{z}\right).$$

The separation depends on the factorization scheme and scale; the physical structure function does not, through the calculated order. The operator meaning of $f_{i/H}$ and this cancellation are developed on the next two routes.

The same framework produces checks from conserved quantum numbers. For a proton, for example, integrals of $q-\bar q$ count net flavor quantum numbers, while the momentum-weighted sum over all partons equals the hadron momentum. Which experimental structure-function integral realizes a particular sum rule depends on the current, flavor combination, perturbative coefficient, and possible power corrections.

Checks and limitations

Tensor check. Contract the decomposition with $q_\mu$; electromagnetic current conservation requires $q_\mu W^{\mu\nu}=0$.

Support check. In the leading parton picture, $0<x<1$. Values reconstructed outside that range signal finite resolution, nuclear motion, a convention mismatch, or unphysical approximations rather than ordinary single-hadron PDFs.

Spin check. At leading order $F_L=0$. A nonzero QCD contribution does not refute partons; it records radiation and coefficient functions beyond the naive model.

Kinematic check. Quote cuts or a domain in both $Q^2$ and $W^2$. Target-mass effects scale with $M^2/Q^2$, higher twist with additional inverse powers, and the $x\to1$ endpoint enhances both logarithms and power corrections.

Nuclear check. A nuclear target has additional binding, Fermi-motion, shadowing, and other many-body effects. The single-proton formula is not a complete nuclear analysis.

Common pitfalls

Calling a PDF a directly measured probability. A PDF is a renormalized, scheme-dependent matrix element inferred within a factorization framework. Only the complete structure function or cross section is observable.

Equating scaling with exact constancy. The parton model supplies the leading pattern; DGLAP evolution predicts systematic logarithmic scaling violation.

Taking $Q^2$ large while ignoring $W^2$. At large $x$, the final hadronic state can remain at low invariant mass. Both invariants are part of the domain statement.

Handoff

A factorization calculation should receive

$$\left\{x,Q^2,y,W^2,\ J^\mu,\ F_a\text{ convention},\ \text{target},\ \text{cuts},\ \text{leading-power remainder}\right\}.$$

Continue to collinear factorization and operator-defined PDFs to replace the probability model by a renormalized QCD statement, then to DGLAP evolution for the predicted scaling violation.

References

• Schwartz, Matthew D. Quantum Field Theory and the Standard Model. Cambridge University Press, 2014, §32.1, pp. 668–76. DOI.