Abstract
The electron, positron, and photon Parton Distribution Functions (PDFs) of the unpolarised electron have recently been computed at the next-to-leading logarithmic accuracy in QED, by adopting the overline{mathrm{MS}} factorisation scheme. We present here analogous results, obtained by working in a different framework that is inspired by the so-called DIS scheme. We derive analytical solutions relevant to the large-z region, where we show that the behaviour of the PDFs depends in a dramatic way on whether running-α effects are included to all orders, as opposed to being truncated to some fixed order. By means of suitable initial and evolution conditions, next-to-leading logarithmic accurate PDFs are obtained whose large-z functional forms are identical to those of their leading logarithmic counterparts.
Highlights
Accuracy [12, 13].1 Among other things, this extension has a strong phenomenological motivation, in view of the precision targets relevant to future colliders
The electron, positron, and photon Parton Distribution Functions (PDFs) of the unpolarised electron have recently been computed at the next-to-leading logarithmic accuracy in QED, by adopting the MS factorisation scheme
We have studied analytically the behaviour of the NLL-accurate electron and photon PDFs of the electron in the z → 1 region, by considering a DIS-like factorisation scheme, called ∆
Summary
We write the evolution equations for the PDFs [18,19,20,21] in the same way as it has been done in ref. [13], namely by introducing a PDF evolution operator, and by defining it so as it works in a generic factorisation scheme. We point out that, owing to eq (2.13), at the NLO only the first column of the K matrix gives a non-null contribution; in other words, only the Kαiαj elements with αj equal to the particle identity (i.e. α1) are relevant This is in keeping with the elementary nature of that particle, so that at the NLO it is the branchings α1 → αi + X that are sufficient to control the factorisation scheme. At this level of precision, the choice of the Kαiαj elements with αj = α1 is essentially arbitrary, since they are associated with NNLO contributions Note that this is not true for the partonic short distance cross sections, where those functions may contribute at the NLO as well; this is irrelevant as far as PDF evolution is concerned, which is our goal here. We shall show in the following that this is justified: while the difference between these two equations is of NNLO in both QCD and in QED, in the latter case it leads to dramatically different behaviours in the z → 1 region; this does not happen in QCD
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