发表机构
Ariel University（阿里埃尔大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究给出平面N = 4超杨-米尔斯在奇数共形自旋n下三圈BFKL本征值的封闭形式，从积分项提取原子表，分析系数结构，发现算术障碍，本征值是Kotikov - Lipatov形式的无积情况，还验证了相关函数。
AI 中文摘要
我们给出了平面N = 4超杨-米尔斯在奇数共形自旋n下的三圈（次下领先阶）BFKL本征值的封闭形式，它是连续嵌套调和和与z = (|n| - 1)/2 + iν在整数平移阶梯上的有理函数的有限组合。对于每个奇数自旋，原子表通过有理算术从Caron - Huot - Herranen三圈积分项中精确提取；一个命令可为任何奇数n >= 3重新生成它，n = 1是给定的边界块。未使用推测能在每个自旋处重现这些积分项的重整化主生成函数（通过n = 17以及m = 9、11、13、15、21、31验证）来生成每个自旋的表。在ν = 0时，截距通过n = 91逐个自旋地重现独立的量子谱曲线值。系数是1、π²和ζ₃的有理倍数；在固定的基本基中，47个槽中的40个以两个阶梯距离的调和和形式闭合（所有自旋形式为推测）。剩余的五层内核包括一个服从完整递推的网格；一旦调和和分离，系数加权阶梯坍缩为两个深度为一的多伽马超越函数。我们报告了一个算术障碍：在计算范围内，异常网格系数在每个n = p + 2（n - 2为素数）处招募一个新的分母素数，我们证明并推测这是一个有限规律且会持续；在该延续上，它们的生成数据不能是代数的或全局有界的。推测偶数自旋包含在同一个主函数中。在每个奇数自旋处，本征值是Kotikov - Lipatov厄米可分离形式的无积情况。一篇配套快报宣布了该结果。
英文摘要
We give the colour-singlet eigenvalue of the three-loop BFKL kernel of planar N=4 super Yang-Mills in closed form at odd conformal spin n. It splits into a holomorphic block and its conjugate, chi_2(n,nu)=(1/4)[F_n(z)+F_n(zbar)] with z=(|n|-1)/2+i nu, and the block is a finite combination of nested harmonic sums of one variable, rational functions and transcendental constants, of uniform transcendental weight five. At the two lower orders the sums are evaluated at the two conjugate points only. At three loops they are evaluated at every point of the integer ladder running from the reflected endpoint -zbar up to z, and one step beyond it, and for all but seven of the slots the coefficient at each rung is fixed by the two distances to the ends. Summing over the one-variable functions before summing along the ladder replaces the rules of those slots by one function of one variable under each ladder weight. The remaining seven are carried by a sum along the ladder whose summand moves with the summation index. All forty-seven coefficient slots follow from rules uniform in the conformal spin at every odd n>=3, with no per-spin coefficient tabulated in the definition; the n=1 boundary block is supplied separately. Those rules are a reconstruction from the computed spins, exact at every one of them, verified over the range n<=99 and at the holdout spin n=101, and not proved at arbitrary odd n. Along nu=0 the intercepts agree with the Quantum Spectral Curve values at every computed odd spin through n=91. That comparison tests the three-loop integrand and the reduction performed here together, at one point of each spin, and for most of those spins it is the first comparison of its kind.
Comments53 pages, 1 figure, 5 tables. The result is announced in arXiv:2607.17608. Ancillary files include the per-spin atom tables for odd n <= 101, the coefficient catalogue, the closed-form rules, an evaluator, and the programs that rebuild every coefficient from the rules, check the printed equations against the tables, and reproduce the holdout spin n=101
