Exact treatment of dispersion relations in $pp$ and $p\overline{p}$ elastic scattering

Proofs are given of exact analytical solutions for general principal value integrations of the dispersion relations of $pp$ and $p\overline{p}$ scattering amplitudes. The proofs are based on recent developments in the study of properties of the Lerch's transcendent of the mathematical literature. Dispersion relations for the slopes of the amplitudes are mathematically defined and also solved exactly. The results are explicitly expanded, providing an important basis for improvement in the phenomenology of the hadronic interactions.

Figure 1

Energy dependence of $\rho \left(pp\right)=\mathrm{Re}{F}_{pp}(x,0)/\mathrm{Im}{F}_{pp}(x,0)$ and $\rho \left(p\overline{p}\right)=\mathrm{Re}{F}_{p\overline{p}}(x,0)/\mathrm{Im}{F}_{p\overline{p}}(x,0)$ predicted by DRs, with illustrative values 0 and $-300$ for the subtraction constant $K$.

Figure 2

With $\sigma$ and $B_I$ (for $pp$ and $p\overline{p}$ systems) given as inputs, the combinations of parameters $\rho B_R/2-\mu$ (for $pp$ and $p\overline{p})$ are predicted by the DRS, offering efficient control in the analysis of $d\sigma /dt$ data.