Existence of a positivity-preserving symmetric-scheme resummation

Establish whether a resummation of the symmetric next-to-leading BFKL kernel admits a non-negative Lévy step measure and therefore a probabilistic evolution law while retaining the required perturbative behavior.

Background

The paper proves that the fixed-order next-to-leading logarithmic BFKL exponent in the symmetric scheme cannot generate a non-negative Lévy measure: its cubic collinear pole violates necessary positivity and convexity conditions at every positive coupling. The authors emphasize that this obstruction concerns the fixed-order truncation rather than all possible resummations.

Several resummation prescriptions are examined. The pure and matched all-poles forms fail positivity analytically, the full prescription fails at the tested couplings, and the finite-rapidity improved Green function fails the tested positivity criteria under a contour assumption. A positive radial completion can reproduce the computed perturbative order, but it is not derived from the full kernel. Consequently, whether any genuine resummation of the symmetric next-to-leading kernel restores positivity remains unresolved.