Fourier-positivity constraints on QCD dipole models

Fourier-positivity (FF-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 space r  . They are Bessel transforms of positive transverse momentum dependent gluon distributions. Using mathematical FF-positivity constraints on the limit r→0r→0 behavior of the dipole amplitudes, we identify the common origin of the violation of FF-positivity for various, however phenomenologically convenient, dipole models. It is due to the behavior r2+ϵr2+ϵ, ϵ>0ϵ>0 softer, even slightly, than color transparency. FF-positivity seems thus to conflict with the present dipole formalism when it includes a QCD running coupling constant α(r)α(r).