Fourier-positivity constraints on QCD dipole models

Abstract : Fourier-positivity ($\mathscr{F}$-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 $\mathscr{F}$-positivity constraints on the limit $r$ $\rightarrow$ 0 behavior of the dipole amplitudes, we identify the common origin of the violation of $\mathscr{F}$-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. $\mathscr{F}$-positivity seems thus to conflict with the present dipole formalism when it includes a QCD running coupling constant $\alpha$($r$).