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.
Categories: hep-ph