# Exact treatment of dispersion relations in pp and p\=p elastic scattering

## TL;DR

This paper introduces exact closed-form dispersion relations for proton-proton and proton-antiproton elastic scattering, enabling precise analysis of forward scattering data across all energies.

## Contribution

It provides novel exact expressions for the real parts and derivatives of scattering amplitudes based on Lerch's transcendent, improving the analysis of scattering data.

## Key findings

- Exact formulas for real parts of amplitudes at t=0
- Precise representations for derivatives of amplitudes
- Application-ready expressions for all energies

## Abstract

Based on a study of the properties of the Lerch's transcendent, exact closed forms of dispersion relations for amplitudes and for derivatives of amplitudes in pp and p\=p scattering are introduced. Exact and complete expressions are written for the real parts and for their derivatives at $t=0$ based on given inputs for the energy dependence of the total cross sections and of the slopes of the imaginary parts. The results are prepared for application in the analysis of forward scattering data of the pp and p\=p systems at all energies, where exact and precise representations can be written.