基于Schläfli的微分公式递归构造标量一阶积分
Recursive construction of scalar one-loop integrals in dimensional regularisation
AI总结：
本文提出基于Schläfli微分公式递归构造标量一阶积分的方法，证明其Laurent展开系数可由多重polylogarithms表示，并给出标量六边形的闭合表达式。
AI中文摘要：
我们基于Schläfli对双曲单纯形的微分公式，推导出一种新的递归结构，用于维数正规化标量一阶费曼积分。该递归关系将维数调节器ε的N点积分的Laurent系数与具有额外外部腿的低阶积分的系数联系起来。该构造以ε=0贡献为基础，这些贡献可以几何上解释为双曲空间中的单纯形体积，并以多重polylogarithms（MPLs）的形式已知。迭代该递归因此提供了一种构造任意阶ε展开的算法，用于计算具有任意质量和动力学的标量一阶积分，同时仍完全保留在MPLs的类别中。特别地，这证明了所有维数正规化标量一阶积分的Laurent展开系数都可以用MPLs表示。作为现有结果之外的第一个应用，我们获得了一个显式的闭合MPL表达式，用于标量六边形的O(ε)系数，具有任意质量和非壳欧几里得动力学。
英文摘要：
We derive a novel recursive structure for dimensionally regularised scalar one-loop Feynman integrals based on Schläfli's differential formula for hyperbolic simplices. The recursion relates the Laurent coefficients in the dimensional regulator $\varepsilon$ of an $N$-point integral to lower-order coefficients of integrals with additional external legs. The construction is seeded by the $\varepsilon=0$ contributions, which admit a geometric interpretation as volumes of simplices in hyperbolic space and are known in terms of multiple polylogarithms (MPLs). Iterating the recursion therefore provides a constructive algorithm for computing arbitrary orders in the $\varepsilon$-expansion of scalar one-loop integrals with arbitrary masses and kinematics, while remaining entirely within the class of MPLs. In particular, this establishes that all coefficients in the Laurent expansion of dimensionally regularised scalar one-loop integrals can be expressed in terms of MPLs. As a first application beyond existing results, we obtain an explicit closed MPL expression for the $\mathcal{O}(\varepsilon)$ coefficient of the scalar hexagon with arbitrary masses and off-shell Euclidean kinematics.
