利用生成函数求解任意单圈约化
Solving arbitrary one-loop reduction via generating function
AI总结：
本文针对单圈约化中尚未充分解决的高阶极点部分，建立其生成函数的偏微分方程，导出并迭代求解积分递推关系，从而完整解决任意张量秩和高阶极点贡献的单圈约化问题，并为高效生成约化系数提供工具。
摘要：
近年来，生成函数的概念已被应用于单圈约化中。对于包含任意张量秩和高阶极点贡献的单圈积分，生成函数可以分解为张量部分和高阶极点部分。虽然张量部分在近期研究中已得到充分处理，但关于高阶极点部分仍缺乏令人满意的研究。在本工作中，我们彻底解决了这一问题。我们首先建立了控制高阶极点生成函数的偏微分方程。基于这些方程，我们导出了一个积分递推关系，并对其进行迭代求解。该方法使我们能够探索高阶极点约化的解析结构，并为高效生成约化系数提供了一种有价值的工具。
英文摘要：
Recently, the concept of generating function has been employed in one-loop reduction. For one-loop integrals encompassing arbitrary tensor ranks and higher-pole contributions, the generating function can be decomposed into a tensor part and a higher-pole part. While the tensor component has been thoroughly addressed in recent studies, there remains a lack of satisfactory investigations regarding the higher-pole part. In this work, we completely solve the problem. We first establish the partial differential equations governing the higher-pole generating function. Based on these equations, we derive an integration recursion relation and solve it iteratively. This approach enables us to explore the analytical structure of higher-pole reduction and provides a valuable tool for generating reduction coefficients efficiently.
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