Analytic components for the hadronic total cross-section: Fractional calculus and Mellin transform

In high-energy hadron-hadron collisions, the dependence of the total cross-section ($\sigma_{tot}$) with the energy still constitutes an open problem for QCD. Phenomenological analyses usually relies on analytic parameterizations provided by the Regge-Gribov formalism and fits to the experimental data. In this framework, the singularities of the scattering amplitude in the complex angular momentum plane determine the asymptotic behavior of $\sigma_{tot}$ in terms of the energy. Usual applications connect simple and triple pole singularities with asymptotic power and logarithmic-squared functions of the energy, respectively. More restrict applications have considered as a leading component for $\sigma_{tot}$ an empirical function consisting of a logarithmic raised to a real exponent, which is treated as a free fit parameter. With this function, data reductions lead to good descriptions of the experimental data and real (not integer) values of the exponent. In this paper, making use of two independent formalisms (fractional calculus and Mellin transform), we first show that the singularity associated with this empirical function is a branch point and then we explore the mathematical consequences of the result and possible physical interpretations. After reviewing the determination of the singularities from asymptotic forms of interest through the Mellin transform, we demonstrate that the same analytic results can be obtained by means of the Caputo fractional derivative, leading, therefore, to fractional calculus interpretations. Besides correlating Mellin transform, non-local fractional derivatives and exploring the generalization from integer to real exponents and derivative orders, this fractional calculus result may provide insights for physical interpretations on the asymptotic rise of $\sigma_{tot}$.