arXiv 1901.094422019-01-27hep-ph

Feynman积分的泛函约化

Functional reduction of Feynman integrals

Tarasov O.

备注：: 37 pages

摘要：

本文提出了一种将费曼积分化为一组较少变量的积分的方法。该方法是基于作者提出的函数方程的迭代应用。本文详细地描述了用无质量内传播子将单圈标量三角积分和盒积分简化为简单积分的方法。依赖于三个变量的三角积分表示为依赖于两个变量的三个积分的和。通过求解这些积分的维数递推关系，得到了用$_2F_1$高斯超几何函数和对数函数表示的解析表达式。利用该函数方程，将依赖于6个运动学变量的无质量内传播子的单圈盒积分表示为12项之和。这些项与同一积分成比例，仅取决于每项不同的三个变量。对于这个三元积分，通过求解关于时空维数$d$的递推关系，得到了用$F_1$ Appell和$_2F_1$ Gauss超几何函数表示的解析结果.考虑了某些运动变量为零的箱积分的简化方程。

英文摘要：

A method for reducing Feynman integrals, depending on several kinematic variables and masses, to a combination of integrals with fewer variables is proposed. The method is based on iterative application of functional equations proposed by the author. The reduction of the one-loop scalar triangle and box integrals with massless internal propagators to simpler integrals is described in detail. The triangle integral depending on three variables is represented as a sum over three integrals depending on two variables. By solving the dimensional recurrence relations for these integrals, an analytic expression in terms of the $_2F_1$ Gauss hypergeometric function and the logarithmic function was derived. By using the functional equations, the one-loop box integral with massless internal propagators, which depends on six kinematic variables, was expressed as a sum of 12 terms. These terms are proportional to the same integral depending only on three variables different for each term. For this integral with three variables, an analytic result in terms of the $F_1$ Appell and $_2F_1$ Gauss hypergeometric functions was derived by solving the recurrence relation with respect to the spacetime dimension $d$. The reduction equations for the box integral with some kinematic variables equal to zero are considered.

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