Lévy structure of the forward fixed-coupling BFKL kernel and a fixed-order obstruction to positivity in the symmetric scheme
Abstract
We establish a Levy interpretation of normalized forward BFKL small-x evolution at leading order and fixed coupling, and exclude a probability law at fixed-order next-to-leading accuracy in the symmetric scheme. After the Marchesini-Onofri conjugation and growth subtraction, one positive step measure in closed form covers all conformal spins, on the cylinder of logarithmic transverse momentum and azimuth. The process is pure jump, and the Pomeron intercept is the relaxation rate of its first azimuthal harmonic. With s0=q1q2 and the coupling at that argument, the negative cubic collinear pole of the truncated eigenvalue excludes a probability law at every positive coupling and rapidity. Its leading coefficient is independent of Nc and nf. The pole dominates the leading-order simple pole within √(α̅/2) of the edge in the leading collinear approximation. Translation-preserving multiplicative conjugations cannot restore positivity. No non-negative step measure generates that truncated evolution, while a positive radial completion at zero conformal spin matches the computed order, so the obstruction is a property of the fixed-order truncation. The tested symmetric-scheme resummations also fail positivity, as shown in closed form for the pure and matched all-poles forms, except for a degenerate zero process, and by computation at the examined couplings for the full prescription. The improved finite-rapidity Green function fails at the displayed parameters under the contour assumption. A resummed kernel in another rapidity scheme has a non-negative step measure at tested couplings. An unweighted transverse walk describes the leading-order evolution with no approximation, but a probabilistic resummation of the symmetric next-to-leading kernel must establish positivity beyond perturbative matching, and whether one exists is left open.
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BibTeX
@misc{prygarin2026lévy,
title = {{Lévy structure of the forward fixed-coupling BFKL kernel and a fixed-order obstruction to positivity in the symmetric scheme}},
author = {Alex Prygarin and Claudelle Capasia Madjuogang Sandeu and Karam Shekh Yusuf},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.18269},
eprint = {2609.18269},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.18269},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Lévy structure of the forward fixed-coupling BFKL kernel and a fixed-order obstruction to positivity in the symmetric scheme
AU - Alex Prygarin
AU - Claudelle Capasia Madjuogang Sandeu
AU - Karam Shekh Yusuf
PY - 2026
DA - 2026/09/16
PB - arXiv
DO - 10.48550/arxiv.2609.18269
UR - https://arxiv.org/abs/2609.18269
AB - We establish a Levy interpretation of normalized forward BFKL small-x evolution at leading order and fixed coupling, and exclude a probability law at fixed-order next-to-leading accuracy in the symmetric scheme. After the Marchesini-Onofri conjugation and growth subtraction, one positive step measure in closed form covers all conformal spins, on the cylinder of logarithmic transverse momentum and azimuth. The process is pure jump, and the Pomeron intercept is the relaxation rate of its first azimuthal harmonic. With s_0=q_1q_2 and the coupling at that argument, the negative cubic collinear pole of the truncated eigenvalue excludes a probability law at every positive coupling and rapidity. Its leading coefficient is independent of N_c and n_f. The pole dominates the leading-order simple pole within √(α̅/2) of the edge in the leading collinear approximation. Translation-preserving multiplicative conjugations cannot restore positivity. No non-negative step measure generates that truncated evolution, while a positive radial completion at zero conformal spin matches the computed order, so the obstruction is a property of the fixed-order truncation. The tested symmetric-scheme resummations also fail positivity, as shown in closed form for the pure and matched all-poles forms, except for a degenerate zero process, and by computation at the examined couplings for the full prescription. The improved finite-rapidity Green function fails at the displayed parameters under the contour assumption. A resummed kernel in another rapidity scheme has a non-negative step measure at tested couplings. An unweighted transverse walk describes the leading-order evolution with no approximation, but a probabilistic resummation of the symmetric next-to-leading kernel must establish positivity beyond perturbative matching, and whether one exists is left open.
ER -