费曼积分的Stieltjes自举方法
The Stieltjes Bootstrap for Feynman Integrals
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中文总结 AI 辅助
本文证明标量费曼积分在满足特定指数条件时为Stieltjes函数，利用该性质结合微分方程与半定优化，在欧几里得区域高效计算主积分，并成功应用于三圈电弱修正中的未知积分。
中文摘要 AI 辅助
我们证明了标量费曼积分在每个运动学变量$-p_i\cdot p_j$以及粒子质量中都是Stieltjes函数，适用于任意圈阶、任意数量的外腿以及任意内部质量，只要满足某个指数条件。该证明基于第二Symanzik多项式的规范分解，其系数由跨越双森林索引且显式非负。我们针对运动学变量满足关系的情况进行了专门分析，这种情况在物理应用中经常发生。我们认为，分部积分关系允许选择主积分基，其中每个元素都是Stieltjes函数。Stieltjes性质保证了Padé逼近的收敛性，并提供了对主积分泰勒系数的行列式（Hankel）约束，这些约束超越了先前已知的完全单调性界限。我们开发了一种数值策略，将这些Stieltjes约束与微分方程相结合，通过半定优化在欧几里得区域确定主积分。我们在六维有质量盒积分的教学示例上演示了这种方法，然后将其应用于计算先前未知的三圈双质量积分，这些积分是$Z$玻色子和希格斯玻色子自能的三圈电弱修正所需的。
英文摘要
We prove that scalar Feynman integrals are Stieltjes functions in each kinematic variable $-p_i\cdot p_j$, as well as in the particle masses, for arbitrary loop order, arbitrary numbers of external legs, and arbitrary internal masses, whenever a certain exponent condition is satisfied. The proof rests on a canonical decomposition of the second Symanzik polynomial with manifestly non-negative coefficients indexed by spanning two-forests. We present a dedicated analysis of the case in which the kinematic variables satisfy relations, as frequently happens in physical applications. We argue that integration-by-parts relations allow the selection of master-integral bases in which every element is a Stieltjes function. The Stieltjes property guarantees the convergence of Padé approximants and provides determinantal (Hankel) constraints on the Taylor coefficients of master integrals that go beyond previously known complete-monotonicity bounds. We develop a numerical strategy that combines these Stieltjes constraints with differential equations to determine the master integrals in the Euclidean region via semidefinite optimization. We demonstrate this approach on the pedagogical example of the six-dimensional massive box integral, and then apply it to the calculation of previously unknown three-loop two-mass integrals needed for three-loop electroweak corrections to the $Z$- and Higgs-boson self-energies.
发表机构
- Max-Planck-Institut für Physik, Werner-Heisenberg-Institut（马克斯·普朗克物理研究所）
- Max Planck Institute for Mathematics in the Sciences（马克斯·普朗克数理科学研究所）
- Leung Center for Cosmology and Particle Astrophysics（梁林宇宙学与粒子天体物理中心）
- INFN, Sezione di Milano（意大利国家核物理学院米兰分部）
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