The $Q^2$ evolution of Soffer inequality

Abstract

DGLAP evolution equations may be presented in a form completely analogous to the Boltzmann equation. This provides a natural proof of the positivity of the spin-dependent parton distributions, provided the initial distributions at $Q^2_0$ are also positive. In addition, the evolution to $Q^2 < Q^2_0$ may violate positivity, providing therefore a 'time arrow'. The positivity condition is just $|\Delta P_{ij} (z)| \leq P_{ij} (z) $ for $z < 1$ for all types of partons, while the $'+'$prescription and terms containing $\delta(1-z)$ do not affect positivity. This method allows one to complete immediately the existing proof of Soffer inequality at leading and next-to-leading order.