Characterize the full z-dependent positivity-preserving semigroup

Characterize the complete z-dependent semigroup of admissible DGLAP scheme transformations that preserve positivity, including one completely monotone Mellin-moment sequence per sector subject to the classification constraints at z=1 and z=2, in order to connect the general scheme classification with positivity-oriented factorization schemes.

Background

The paper classifies scheme transformations that preserve the DGLAP form, charge-conjugation and flavor symmetries, and the momentum and valence-number sum rules. It then shows that positivity is more restrictive: positivity-preserving transformations form a semigroup rather than a group, and their x-space kernels must satisfy entrywise nonnegativity conditions.

For general moment-dependent transformations, positivity cannot be characterized merely by requiring positive Mellin moments. Each relevant moment sequence must instead be completely monotone, as characterized by Hausdorff’s moment theorem. The paper gives the necessary moment conditions and fully analyzes the moment-independent case, but leaves open a complete description of the z-dependent semigroup across all sectors while simultaneously enforcing the sum-rule constraints.