$J/\psi$ Photoproduction from Threshold to HERA: Leading-Twist Convolution, Small-$x$ Pathology, and Eikonal Unitarization

We revisit near-threshold $J/\psi$ photoproduction on the nucleon within the OPE sum-rule framework combined with vector-meson dominance and dispersion relations, using modern NNLO gluon distributions (ABMP16, MSHT20, CT18, NNPDF4.0). Two complementary pathologies are identified. The moment-based cross-section reconstruction fails near threshold: the small-$x$ singularity of modern PDFs distorts the Mellin moment hierarchy and drives the threshold exponent to $a\simeq 16$-$20$, compared to $a\simeq 1$-$2$ for the 1999 scaling parametrization. The direct convolution approach avoids this artefact and describes the threshold data (GlueX, Cornell) for all four PDF sets, but overshoots HERA measurements at $W\gtrsim 90$~GeV by a factor $7$-$12$ - an intrinsic feature of leading-twist convolution with any small-$x$-singular PDF, already noted in the 1999 analysis. A minimal eikonal unitarization of the amplitude, with an energy-dependent saturation scale fitted to HERA data, reconciles the convolution with the full $W$-range measurements while leaving the threshold description unchanged. Near threshold the dispersive real part dominates the cross section, anchored by the OPE subtraction constant $M_{\psi N}(0)\simeq 36$-$39$~GeV$^{-2}$.