Factorization in deep inelastic scattering at Björken limit: Reduction to (1+1)D integrable models
We propose a Sine-Gordon description of the longitudinal sector probed in small-$x$ deep inelastic scattering on an effective string world sheet. Integrability gives nonperturbative access to the leading small-$x$ behavior through exact form factors. We test the proposal against combined HERA data with their published correlated uncertainties. In the window $Q^2>1\,\mathrm{GeV}^2$, $x\le0.01$, $y<0.3$, local powers describe the final reduced cross sections with $χ^2/\mathrm{dof}=71.811/77$ across 19 virtuality slices. Matching the measured exponent $γ(Q^2)$ to the exact Sine-Gordon power defines the scale-dependent effective coupling through $β^2(Q^2)/π=16/[3-γ(Q^2)]$. A bounded crossover with illustrative limit $C=0.431$ is compatible with the data; the compact HERA-I $F_2$ surface fit gives $123.608/106$. We also evaluate the fixed-coupling three-particle normalization, obtaining $g_3(\infty)=0.589320245$, verify numerical convergence independently, and derive its monotonicity and asymptotic approach. The proposed QCD correspondence is distinct from these exact-model and truncated-spectral results: identifying the collective world-sheet field, electromagnetic-current matching, and scale evolution remains an open problem.