前向固定耦合BFKL核的Lévy结构及对称方案中正定性的固定阶障碍
Lévy structure of the forward fixed-coupling BFKL kernel and a fixed-order obstruction to positivity in the symmetric scheme
- Ariel University（阿里埃勒大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文证明前向固定耦合BFKL核在领头阶具有Lévy结构，但在对称方案的次领头固定阶截断中存在负三次共线极点，排除了概率律，且重求和无法恢复正定性。
AI中文摘要：
我们建立了在领头阶和固定耦合下归一化前向BFKL小-$x$演化的Lévy解释，并排除了对称方案中固定阶次领头精度下的概率律。在Marchesini-Onofri共轭和增长扣除之后，一个闭式正步长测度覆盖了所有共形自旋，位于对数横向动量与方位角的圆柱上。该过程是纯跳跃的，Pomeron截距是其第一方位角谐波的弛豫率。以$s_0=q_1q_2$及该参数处的耦合，截断特征值的负三次共线极点排除了所有正耦合和快度下的概率律。其首项系数与$N_c$和$n_f$无关。在领头共线近似中，该极点在边缘的$\sqrt{\bar\alpha/2}$范围内主导领头阶简单极点。保持平移的乘法共轭无法恢复正定性。没有非负步长测度能生成该截断演化，而零共形自旋处的正径向补全与计算阶数匹配，因此该障碍是固定阶截断的性质。所测试的对称方案重求和也未能满足正定性，如纯及匹配全极点形式的闭式所示，以及所检查耦合下完整处方的计算所示。改进的有限快度格林函数在轮廓假设下于所示参数处失效。另一快度方案中的重求和核在测试耦合下具有非负步长测度。一个未加权的横向行走无近似地描述了领头阶演化，但对称次领头核的概率性重求和必须在微扰匹配之外建立正定性，其是否存在仍未解决。
英文摘要：
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 $\sqrt{\barα/2}$ 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.
