Positivity and renormalization of parton densities

TL;DR

This paper critically examines recent claims about the positivity of parton densities in quantum field theory, revealing flaws in the proofs and clarifying conditions under which positivity may be violated, with implications for phenomenology.

Contribution

The paper identifies errors in recent positivity proofs of parton densities and clarifies the conditions under which positivity can be violated, challenging previous assumptions.

Findings

Recent positivity proofs are flawed due to incorrect assumptions about bare PDFs.

Positivity of PDFs can fail under certain conditions in quantum field theory.

Methodological issues in factorization derivations can lead to incorrect phenomenological conclusions.

Abstract

There have been recent debates about whether $\overline{\text{MS}}$ parton densities exactly obey positivity bounds (including the Soffer bound), and whether the bounds should be applied as a constraint on global fits to parton densities and on nonperturbative calculations. A recent paper (JHEP ${\bf \text{11}}$ (2020) 129) appears to provide a proof of positivity in contradiction with earlier work by other authors. We examine their derivation and find that its primary failure is in the apparently uncontroversial statement that bare pdfs are always positive. We show that under the conditions used in the derivation, that statement fails. We provide some elementary calculations in a model QFT that show how this situation can generically arise in reality. In addition, we observe that the methods used in the derivation are in common with much, but not all, other work where factorization is derived. Our examination pinpoints considerable difficulties with these methods that render them either wrong or highly problematic. The issue of positivity highlights that these methods can lead to wrong results of phenomenological importance. From our analysis we identify the restricted situations in which positivity can be violated.