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).
Highlights
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Using mathematical F-positivity constraints on the limit r → 0 behavior of the dipole amplitudes, we identify the common origin of the violation of F-positivity for various, phenomenologically convenient, dipole models
We have discussed the constraints on QCD dipole models arising from F -positivity
Summary
The introduction of the dipole formalism [1] in QCD predictions for high-energy deep-inelastic scattering (DIS) and other processes at “small x” has prompted a lot of phenomenological activity with good success during the recent years. Within this approximation, (1) relate, through a Fourier-Bessel transformation, two physical observables which are both positive This was our original motivation [4] to discuss Fourier-positivity properties in general, hoping to get constraints on the QCD dipole models. At present time and in a more general setting, it can be proven that QCD observables at small x like N (Y, r) and N (Y, k) are expected to be conjugated positive functions through a Bessel transformation. This characteristic feature of positive Fourier partners is more general in the dipole formalism and its various developments than the one that we originally considered [4].
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