Odderon Form Factors in Reggeized Spin-2 Pomeron and Spin-3 Odderon Exchange in $pp$ and $p\bar p$ Elastic Scattering

Abstract

We investigate the form-factor dependence of Reggeized tensor Pomeron and Odderon exchanges in high-energy elastic $pp$ and $p\bar p$ scattering. The spin structure is implemented through explicit covariant spin-2 and spin-3 projectors, kept factorized from the Reggeized scalar kernels, so that vertex effects can be separated from trajectory dynamics. Seven Odderon--proton form-factor parametrizations are tested against a global dataset including TOTEM $pp$ data at $\sqrt{s}=2.76$, $7$, $8$, and $13$~TeV and Tevatron $p\bar p$ data at $\sqrt{s}=1.80$ and $1.96$~TeV. A clear hierarchy is found. Six dipole, polynomial, Gaussian, and hybrid parametrizations give comparable fit qualities, $χ^2_{\rm red}\simeq 1.44$--$1.48$, whereas a one-parameter exponential form, $F_{\mathbb O}(t)=\exp[-B|t|/2]$, yields $χ^2_{\rm red}=0.98$ for 138 degrees of freedom. The fitted couplings and Regge slopes remain comparatively stable across the form-factor choices, indicating that the improvement is driven mainly by the Odderon--proton vertex rather than by large compensating shifts in trajectory parameters. The exponential form admits an impact-parameter interpretation as a Gaussian transverse profile, with an effective radius $\sqrt{\langle b^2\rangle}=\sqrt{2B}\,\hbar c$. The extracted radii are of hadronic size and suggest a peripheral soft Odderon interaction. The shrinking $t$-range over which the single-Regge-exchange description remains accurate at increasing energy indicates the onset of absorptive and unitarity corrections. These results provide a compact phenomenological framework for connecting the $pp/p\bar p$ dip--bump difference with the transverse structure of $C$-odd color-singlet exchange.