Can $$ \overline{\mathrm{MS}} $$ parton distributions be negative?

Abstract

It is common lore that Parton Distribution Functions (PDFs) in the $$ \overline{\mathrm{MS}} $$ MS ¯ factorization scheme can become negative beyond leading order due to the collinear subtraction which is needed in order to define partonic cross sections. We show that this is in fact not the case and next-to-leading order (NLO) $$ \overline{\mathrm{MS}} $$ MS ¯ PDFs are actually positive in the perturbative regime. In order to prove this, we modify the subtraction prescription, and perform the collinear subtraction in such a way that partonic cross sections remain positive. This defines a factorization scheme in which PDFs are positive. We then show that positivity of the PDFs is preserved when transforming from this scheme to $$ \overline{\mathrm{MS}} $$ MS ¯ , provided only the strong coupling is in the perturbative regime, such that the NLO scheme change is smaller than the LO term.

Curing the unphysical behaviour of NLO quarkonium production at the LHC and its relevance to constrain the gluon PDF at low scales

“…What we wish to argue here is that, since this subtraction is the only possible source of negative numbers at z = 1, if A a (z) happens to be negative in a given scheme where PDFs are supposedly positive (see [55] for MS), this signals that the AP CT have likely over-subtracted some collinear contributions from the real-emission contributions, and this can yield the observed negative hadronic cross sections. This is indeed what happens for quarkonia since the NLO threshold contributions (ŝ = M 2 ) are found to be positive-definite for η Q and several other states at least for μ F = μ R .…”

“…Understanding the possible connection with a recent study of the positivity of the PDFs in the MS scheme[55] and the collinear factorisation scheme[56] is left for a future study 13. Our choice has mainly been driven by technical reasons.…”

An open-source machine learning framework for global analyses of parton distributions

“…The n3fit library allows for a flexible specification of the neural network model adopted to parametrise the PDFs, whose settings can be selected automatically via the built-in hyperoptimisation tooling [30]. These include the neural network type and architecture, the activation functions, and the initialisation strategy; the choice of optimiser and of its corresponding parameters; and hyperparameters related to the implementation in the fit of theoretical constraints such as PDF positivity [31] and integrability. The settings for a PDF fit are input via a declarative run card.…”

Double parton distributions out of bounds in colour space

“…At given y one can hence always achieve a negative 8 J by taking the scales μ 1 and μ 2 sufficiently high.…”

“…Whether positivity bounds on parton distributions actually hold turns out to be a non-trivial question. At leading order (LO) accuracy, the positivity of PDFs can quite directly be deduced from the positivity of cross sections, whilst the situation is more involved at next-to-leading order (NLO) in the strong coupling [7,8]. When derived along such lines, positivity holds for renormalisation scales μ that are high enough for the approximations of leading-twist dominance and of the perturbative expansion to be valid.…”