Evaluation of the general three-loop vacuum Feynman integral

We discuss the systematic evaluation of 3-loop vacuum integrals with arbitrary masses. Using integration by parts, the general integral of this type can be reduced algebraically to a few basis integrals. We define a set of modified finite basis integrals that are particularly convenient for expressing renormalized quantities. The basis integrals can be computed numerically by solving coupled first-order differential equations, using as boundary conditions the analytically known special cases that depend on only one mass scale. We provide the results necessary to carry this out, and introduce an implementation in the form of a public software package called 3VIL (3-loop Vacuum Integral Library), which efficiently computes the numerical values of the basis integrals for any specified masses. 3VIL is written in C, and can be linked from C, C++, or FORTRAN code.

我们讨论对具有任意质量的三圈真空积分的系统评估。通过分部积分，这种类型的一般积分可以通过代数方法简化为几个基积分。我们定义了一组修正的有限基积分，它们对于表示重整化量特别方便。这些基积分可以通过求解耦合的一阶微分方程进行数值计算，将仅依赖于一个质量标度的已知解析特殊情况用作边界条件。我们提供了进行此操作所需的结果，并引入了一个名为3VIL（三圈真空积分库）的公共软件包形式的实现，它能针对任何指定的质量高效计算基积分的数值。3VIL是用C语言编写的，可以从C、C++或FORTRAN代码中链接。