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.
Highlights
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 Unintegrated parton distribution functions (UPDFs) model, i.e., the integral (I-MRW) and differential (D-MRW) MRW UPDFs
In this work we show that both of the solutions suggested in the Refs. [19,20] are incomplete, and the true equality between the I-MRW and D-MRW UPDFs derivations can only be obtained if the cutoff-dependent PDFs and the additional term at the same time be included into the formalism
In order to understand the roots of these problems, we show numerically the derivation steps of reaching to the I-MRW UPDF from the D-MRW UPDF, i.e., the Eqs. (12) and (13)
Summary
The MRW formalism at the leading order (LO) level, can be written in two alternative forms, i.e., integral (I-MRW) and differential (D-MRW) UPDFs derivations. [19] suggested that only in the cutoff-dependent parton distribution functions (PDFs) can solve this discrepancy. [20] contradicts the above idea [19] and claims that there is no need for the cutoff-dependent PDFs, if one introduces another term to the D-MRW UPDF. [19,20] are incomplete, and the true equality between the I-MRW and D-MRW UPDFs derivations can only be obtained if the cutoff-dependent PDFs and the additional term at the same time be included into the formalism. The structure of the paper is as follows: In the Sect. 2, the integral and differential forms of the MRW UPDFs are in detail explained.
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