Self-similarity and the Froissart bound

Abstract

The Froissart bound implies that the total cross section (or, equivalently, the structure function) cannot rise faster than the logarithmic growth of ${\ln }^2\left(\frac{1}{x}\right)$. In this work, we show that such a slow growth is not compatible with the notion of self-similarity. As a result, it calls for the modification of the defining transverse-momentum-dependent parton density function (TMD PDF) of a self-similarity based proton structure function $F_2\left(x,Q^2\right)$ at small \textit{x}. Using plausible assumptions, we obtain the Froissart saturation condition on this TMD PDF.

Comments: 11 pages, 2 figures, 1 table