= Structure-function scaling versus elastic suppression
An ideal <Bjorken scaling> model has $F_2(x,Q^2)=f(x)$: varying $Q^2$ at fixed <Bjorken x> leaves this dimensionless <deep-inelastic structure function> unchanged. The comparison requires extracting a <deep-inelastic structure function>, rather than plotting raw event counts. <Quantum chromodynamics> predicts departures from exact <Bjorken scaling>; finite <hadron> masses also limit the idealization.
For an original shape comparison, choose $f(x)=x(1-x)^3$ and define $S(Q^2)=F_2(0.3,Q^2)/F_2(0.3,Q_0^2)$ with $Q_0^2=2\,\mathrm{GeV}^2$. The ideal model has $S=1$. To display a contrasting size-dependent suppression, define an illustrative elastic-form-factor shape $G(Q^2)=(1+Q^2/\Lambda^2)^{-2}$ with the freely chosen $\Lambda^2=1\,\mathrm{GeV}^2$. Its normalized squared magnitude is
$$
A(Q^2)=\left|\frac{G(Q^2)}{G(Q_0^2)}\right|^2=\left(\frac{1+Q_0^2/\Lambda^2}{1+Q^2/\Lambda^2}\right)^4.
$$
Thus $A(5\,\mathrm{GeV}^2)=1/16$, whereas $S(5\,\mathrm{GeV}^2)=1$. The elastic-form-factor shape is a comparison of suppression mechanisms, not an elastic <differential scattering cross-section> assigned to $x=0.3$: actual <elastic scattering> from a <proton> has $W=M$ and $x=1$. Neither curve is measured data or a fit to the SLAC-MIT experiment.
\Image[/deep-inelastic-fixed-x-scaling-model.png]
{title=Fixed-x scaling model and illustrative elastic-form-factor suppression}
{description=Original calculated curves. The left panel compares normalized response shapes; the right checks physical outgoing <Electron> <energies> at the same <Bjorken x>. The flat curve is an ideal <Bjorken scaling> model, not an assertion of exact scaling in <Quantum chromodynamics>.}
The experimental interpretation required more than a flat curve. The angular dependence and small longitudinal response supported <spin one-half> <partons>, expressed at leading order by the <Callan-Gross relation> $F_2=2xF_1$. Identification with fractionally charged <quarks> also drew on later <Electron>, <muon> and <neutrino> comparisons. A scaling plot alone does not determine the individual <electric charges> of constituents.<Deep Inelastic Scattering: Comparisons with the Quark Model by Jerome Isaac Friedman>[1990 Nobel lecture, identification of the constituents and early neutrino results; 1991 reprint p. 623]{ref}
The leading <parton model> formula already shows the ambiguity: $F_2=x\sum_f Q_f^2(q_f+\bar q_f)$ combines squared <electric charges> with unknown <parton distribution functions>. One measured function cannot independently fix all these quantities. <Confinement> also explains why an inclusive final <hadron> state need not contain an isolated outgoing <quark>.
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