从Schwinger表示中的$\mathcal{A}$-超几何系统导出费曼积分的正则微分方程
Canonical differential equations for Feynman integrals from $\mathcal{A}$-hypergeometric systems in the Schwinger representation
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中文总结 AI 辅助
本文提出从Schwinger表示中的GKZ超几何系统构造Feynman积分正则微分方程的方法，利用广义Euler积分和Frobenius基，所得正则基维数不超过master积分数量，实现微分系统的约化。
中文摘要 AI 辅助
我们采用$D$-模方法，直接从Feynman积分的Gel'fand-Kapranov-Zelevinsky（GKZ）超几何系统构造其正则微分方程。从Schwinger表示出发，我们识别出相关的广义Euler积分及其GKZ系统，作为Lee-Pomeransky表示的替代方案。广义Euler公式允许直接识别与Newton多面体的面相关的约化，从而减少所得微分系统中的变量数量。我们使用Frobenius基构造相关的Pfaffian系统，这也提供了对奇异轨迹的直接访问。经过适当的合理化变量变换后，系统被转化为正则的$\epsilon$-形式。该构造的一个关键特征是，正则基的维数由GKZ系统的完整秩决定，并且对于所考虑的示例，它小于或等于通过分部积分获得的master积分的数量，从而提供了微分系统的约化描述。
英文摘要
We follow a $D$-module approach to the construction of canonical differential equations for Feynman integrals directly from their Gel'fand-Kapranov-Zelevinsky (GKZ) hypergeometric systems. Starting from the Schwinger representation, we identify the associated generalized Euler integral and its GKZ system, as an alternative to the Lee-Pomeransky representation. The generalized Euler formulation allows reductions associated with facets of the Newton polytope to be identified directly, reducing the number of variables in the resulting differential systems. We construct the associated Pfaffian systems using Frobenius bases, which also provide direct access to the singular loci. After a suitable rationalizing change of variables, the systems are brought into canonical $ε$-form. A key feature of the construction is that the dimension of the canonical basis is determined by the holonomic rank of the GKZ system and, for the examples considered, it is smaller than or equal to the number of master integrals obtained from integration-by-parts, thus providing a reduced description of the differential system.
发表机构
- Universidad Nacional de Colombia Sede Medellín（哥伦比亚国立大学麦德林校区）
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