Holographic Open/Closed Exchange in Double Deeply Virtual Compton Scattering: Fixed-$j$ Structural Matching to the $\pm$-Basis Wilson Kernels

Abstract

We show that fixed--$j$ holographic double deeply virtual Compton scattering (DDVCS), and its DVCS limit, give a fixed-scale structural match to the singlet conformal-OPE Wilson-kernel family of QCD in the leading-twist unpolarized singlet vector channel. The open-string $C_1(\delta,\vartheta)$ hypergeometric kernel was derived by Nishio--Watari; the new closed-string result is that the BPST upper Witten vertex gives the same family with a different near-boundary power count, because $\Delta_c(j)$ is fixed by the BPST trajectory rather than by open-Regge data. The Mellin exponent is therefore derived as $\delta_c(j)=j+\Delta_c(j)-2=2j+\gamma_c(j)$ rather than inserted by hand. The even open branch gives the parallel unprotected counterpart in the projected singlet vector amplitude. At $Q=\mu=\mu_0=\mu_\ast$, the point is not merely that equal-endpoint evolution is trivial; it is that the ultraviolet Witten vertex has already produced the fixed--$j$ conformal kernel before the lower conformal moment is matched. In the conformal partial-wave/CS representation, beta-proportional conformal-anomaly terms and scheme transformations are coefficient/evolution bookkeeping away from the matching point, not replacements for the projected fixed-scale hypergeometric $\eta/\xi$ Wilson-kernel family. The protected/unprotected $j=2$ split identifies the protected closed branch with the $(-)$ conformal partial wave and the even open branch with the unprotected $(+)$ counterpart; these are mixed singlet eigenchannels, not literal unmixed quark/gluon operators at finite $N_c$. Thus, once the lower Witten vertex is matched to constrained conformal moments, the DVCS/DDVCS deconvolution problem is organized in a physical operator basis rather than in arbitrary $x$-space profiles.