On the ambiguity between differential and integral forms of the Martin-Ryskin-Watt unintegrated parton distribution function model

TL;DR

This paper investigates the differences between integral and differential forms of the Martin-Ryskin-Watt unintegrated parton distribution functions, explaining the reasons for their discrepancy and proposing a combined approach for consistency.

Contribution

It provides a detailed analysis of the non-equivalence between the integral and differential MRW UPDFs and suggests a combined prescription to reconcile both forms.

Findings

Identified the discrepancy between I-MRW and D-MRW UPDFs.

Numerically derived the I-MRW from D-MRW to understand their relation.

Proposed a combined approach to achieve consistent UPDF structures.

Abstract

In this work, we study the structure of the leading order Martin-Ryskin-Watt (MRW) unintegrated parton distribution function (UPDF) and explain in detail why there exists discrepancy between the two different definitions of this UPDF model, i.e., the integral (I-MRW) and differential (D-MRW) MRW UPDFs. We perform this investigation with both angular and strong ordering cutoffs. The derivation footsteps of obtaining the I-MRW UPDF from the D-MRW ones are numerically performed, and the reason of such non-equivalency between the two forms is clearly explained. We show and find out that both methods suggested in the papers by Golec-Biernat and Stasto as well as that of Guiot have shortcomings, and only the combination of their prescriptions can give us the same UPDF structure from both of these two different versions of the MRW UPDF, namely I-MRW and the D-MRW UPDFs.