Extracting the Odderon from $pp$ and $p\overline{p}$ scattering data

Starting from a simple empirical parametrization of the scattering amplitude, successfully describing the dip-bump structure of elastic $pp$ scattering in $t$ at fixed values of $s$, we construct a toy model interpolating between missing energy intervals to extract the Odderon contribution from the difference between $\overline{p}p$ and $pp$ elastic and total cross sections. The model is fitted to data from $\sqrt{s}=23.5\mathrm{GeV}$ to 7 TeV and used to extract the Odderon and its ratio to the Pomeron. From our fits, a unit intercept Odderon follows, as predicted by J. Bartels, L. N. Lipatov, and G. P. Vacca, on the basis of perturbative quantum chromodynamics.

Figure 1

Timeline of proton and antiproton elastic scattering measurements. New accelerators are run first at the maximum available energies; however, at the start of the $\mathrm{S}p\overline{p}\mathrm{S}$ accelerator, the $pp$ and the $p\overline{p}$ elastic scattering data were measured at the same $\sqrt{s}=31$, 53, and 62 GeV.

Figure 2

The PB model fitted to the pp and $\overline{p}p$ data at discrete energy values. The $d\sigma /dt$ data are from Refs. [16, 17, 18, 19, 20, 21]. The data on ${\sigma }_{\text{tot}}$ are from the Particle Data Group database [22].

Figure 3

Energy-dependent values of the parameters extracted from a fit to $pp$ and $p\overline{p}$ data. The data on ${\sigma }_{\text{tot}}$ are from the Particle Data Group database [22].

Figure 4

Energy-dependent values of the parameters from a fit to $pp$ and $p\overline{p}$ data, constrained by ${\sigma }_t^{p\overline{p}}={\sigma }_t^{pp}$ as $\sqrt{s}\to \infty$. The lowest right icon shows a cross-check for the (asymptotically converting) total cross sections.

Figure 5

Odd (Odderon) and even (Pomeron) parts of the $d\sigma /dt$ cross sections calculated from the difference $|\mathcal{A}{|}_{\overline{p}p}^2-|\mathcal{A}{|}_{pp}^2={\mathrm{\Delta }}_{\text{Odd}}$ and from the sum $|\mathcal{A}{|}_{\overline{p}p}^2+|\mathcal{A}{|}_{pp}^2={\mathrm{\Sigma }}_{\text{Pom}}$ fitted to the data.

Figure 6

Real and imaginary parts of the difference ${\mathcal{A}}^{\overline{p}p}-{\mathcal{A}}^{pp}=\mathrm{\Delta }\mathcal{A}$ calculated from the present model fitted to the data. As seen from the figure, this difference tends to a (small) constant, corresponding to a unit intercept Odderon [12].