Positivity and renormalization of parton densities

Abstract

There have been recent debates about whether $\overline{\mathrm{MS}}$ 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 [Candido et al., Can $\overline{\mathrm{MS}}$ parton distributions be negative?, J. High Energy Phys. 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 parton density (or distribution) function (pdfs) are always positive. We show that under the conditions used in the derivation, that statement fails. This is associated with the use of dimensional regularization for both UV divergences (space-time dimension $n<4$) and for collinear divergences, with $n>4$. Collinear divergences appear in massless partonic quantities convoluted with bare pdfs, in the approach used by these and other authors, which we call ``track B.'' Divergent UV contributions are regulated and are positive when $n<4$, but can and often do become negative after analytic continuation to $n>4$. We explore ramifications of this idea and provide some elementary calculations in a model QFT that show how this situation can generically arise in reality. We examine the connection with the origin of the track B method. Our examination pinpoints considerable difficulties with track B that render it either wrong or highly problematic and explain that a different approach, which appears in some literature and that we call track A, does not suffer from this set of problems. The issue of positivity highlights that track-B methods can lead to wrong results of phenomenological importance. From our analysis we identify the restricted situations in which positivity tends to be violated.