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.Comment: 11 pages, 2 figures, 1 tabl

Froissart bound and self-similarity based models of proton structure functions

“…It needs at least three magnification factors not two as compared to earlier work in Ref. [45]. The ranges of validity for four different self-similarity based models with three magnification factors are: Thus the above analysis shows the Froissart saturated self-similarity based structure function has smaller validity ranges as compared to that of structure function without Froissart condition having power law growth in 1 x and log Q 2 .…”

“…. It was noted in [45] that if the magnification factor M 2 is changed to log 1 x , then it is possible to get a Froissart saturation like behavior in structure function. However, we observe that it is true only for PDF but not for the structure function.…”

“…Specifically in Ref [45], such possibility was first suggested. The present work is a generalization and improvement of it in the sense that the improved models can incorporate linear rise in log Q 2 instead of Q 2 and make them closer to data and QCD expectation.…”

Momentum fractions carried by quarks and gluons in models of proton structure functions at small x

“…Ref [39]. In a recent study [40], it is suggested that the logarithmic rise in Q 2 of structure function is also possible, in an improved singularity free self-similar based model with a wider kinematical range in x and Q 2 but even then it is far short off accounting the entire x − Q 2 range explored in HERA unlike perturbative QCD [28].…”

An Improved Singularity Free Self-Similar Model of Proton Structure Function

“…Self-similarity is a possible feature of multi-partons inside a proton at small x suggested by Lastovicka [1] and pursued by us in recent years [2,3,4,5,6]. More recently, we have examined its consequences in [7] double parton distribution function(dPDF) to be tested at LHC, in Froissart saturation [8] and Longitudinal structure function [9].…”