论文标题
与任意质量的一环Feynman积分的功能降低
Functional reduction of one-loop Feynman integrals with arbitrary masses
论文作者
论文摘要
详细描述了一种具有大规模繁殖器的尺寸正规化的单环积分的功能降低方法。该方法基于作者提出的功能关系的重复应用。给出了明确的公式，用于将一环标量积分简化为简单的公式，其参数的参数是质量和运动不变式中多项式的比率。我们表明，根据$ n（n+1）/2 $通用质量和运动变量的一般标量$ n $ point积分不可或缺的积分，可以表示为仅取决于$ n $变量的积分的线性组合。后一个积分由$（n-1）$无量纲变量的高几何功能明确给出。还通过求解尺寸复发关系获得了2-，3和4点积分的分析表达，这些表达式取决于最小的变量。这些积分的结果表达式是根据高斯的超几何函数$ _2F_1 $，appell函数$ f_1 $和超几何lauricella -saran函数$ f_s $给出的。考虑了对运动变量的某些特殊值的功能还原过程的修改。
A method of functional reduction for the dimensionally regularized one-loop Feynman integrals with massive propagators is described in detail. The method is based on a repeated application of the functional relations proposed by the author. Explicit formulae are given for reducing one-loop scalar integrals to a simpler ones, the arguments of which are the ratios of polynomials in the masses and kinematic invariants. We show that a general scalar $n$-point integral, depending on $n(n+1)/2$ generic masses and kinematic variables, can be expressed as a linear combination of integrals depending only on $n$ variables. The latter integrals are given explicitly in terms of hypergeometric functions of $(n-1)$ dimensionless variables. Analytic expressions for the 2-, 3- and 4-point integrals, that depend on the minimal number of variables, were also obtained by solving the dimensional recurrence relations. The resulting expressions for these integrals are given in terms of Gauss' hypergeometric function $_2F_1$, the Appell function $F_1$ and the hypergeometric Lauricella - Saran function $F_S$. A modification of the functional reduction procedure for some special values of kinematical variables is considered.
