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 $Q_0^2$ are also positive. In addition, the evolution to $Q^2<Q_0^2$ may violate positivity, providing therefore a 'time arrow'. The positivity condition is just $|\Delta P_{ij}(z)|\le P_{ij}(z)$ for $z<1$ for all types of partons, while the ${}^{\prime }{+}^{\prime }$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.

11 pages, LateX, eps sty, 3 eps figures

References in corpus (1)

• Two-loop anomalous dimensions for the structure function h1

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