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arXiv · hep-ph/0301103

The Kinetic Interpretation of the DGLAP Equation, its Kramers-Moyal Expansion and Positivity of Helicity Distributions

Alessandro Cafarella·Claudio Corianó
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Abstract

According to a rederivation - due to Collins and Qiu - the DGLAP equation can be reinterpreted (in leading order) in a probabilistic way. This form of the equation has been used indirectly to prove the bound $|Δf(x,Q)| < f(x,Q)$ between polarized and unpolarized distributions, or positivity of the helicity distributions, for any $Q$. We reanalize this issue by performing a detailed numerical study of the positivity bounds of the helicity distributions. To obtain the numerical solution we implement an x-space based algorithm for polarized and unpolarized distributions to next-to-leading order in $α_s$, which we illustrate. We also elaborate on some of the formal properties of the Collins-Qiu form and comment on the underlying regularization, introduce a Kramers-Moyal expansion of the equation and briefly analize its Fokker-Planck approximation. These follow quite naturally once the master version is given. We illustrate this expansion both for the valence quark distribution $q_V$ and for the transverse spin distribution $h_1$.

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Alessandro Cafarella, Claudio Corianó. 2003-02-06. The Kinetic Interpretation of the DGLAP Equation, its Kramers-Moyal Expansion and Positivity of Helicity Distributions. https://doi.org/10.1142/s0217751x05020951

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