Fourier-positivity constraints on QCD dipole models

TL;DR

This paper explores how Fourier-positivity constraints can be applied to QCD dipole models, revealing fundamental conflicts with current models that include a running coupling constant, impacting their physical consistency.

Contribution

It introduces Fourier-positivity as a constraint on QCD dipole models and identifies its violation in models with a running coupling, linking mathematical properties to physical assumptions.

Findings

Fourier-positivity constrains the behavior of dipole amplitudes at small r.

Violations of Fourier-positivity are linked to softer-than-color transparency behavior.

Current dipole models with running coupling conflict with Fourier-positivity constraints.

Abstract

Fourier-positivity, i.e. the mathematical property that a function has a positive Fourier transform, can be used as a constraint on the parametrization of QCD dipole-target cross-sections or Wilson line correlators in transverse position (r) space. They are Bessel transforms of positive transverse momentum dependent gluon distributions. Using mathematical Fourier-positivity constraints on the limit r -> 0 behavior of the dipole amplitudes, we identify the common origin of the violation of Fourier-positivity for various, however phenomenologically convenient, dipole models. It is due to the behavior r^{2+epsilon}, epsilon>0, softer, even slightly, than color transparency. Fourier-positivity seems thus to conflict with the present dipole formalism when it includes a QCD running coupling constant alpha(r).