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

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 Staśto 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.

On the normalization of unintegrated parton densities

“…While we found Ref [15]. interesting, we do not fully agree with the claim that the only difference between the KMR[10,22] and WMR[11] approaches is the use of the cutoffs.…”

“…More recently, the authors of Ref. [15] showed that a good numerical agreement between the differential and integral definitions is obtained if one combines the conditions of Refs. [12] and [13], i.e., cutoffs-dependent PDFs, and Eq.…”

“…( 2) is not always obeyed by the KMRW UPDFs, in particular at large x, see also Ref. [15]. In a variable-flavor-number scheme, the two main contributions to D-meson production are given by the gg → cc and cg → cg processes.…”

“…( 3) and ( 8), are, in general, not equivalent. These problems have already been discussed in the literature [12][13][14][15].…”

“…The usual references for the KMR and WMR formalisms are [10,11]. In [15], the authors say that the only difference between these two formalisms is the implementation of the cutoff. This claim is not necessarily true since in [10] the Sudakov factor is given by…”