标题： 多点共形积分在$D$维空间中。 第一部分：二分Mellin-Barnes表示和重构 显示英文标题

标题： Multipoint conformal integrals in $D$ dimensions. Part I: Bipartite Mellin-Barnes representation and reconstruction

摘要： 我们提出了一种系统的方法，用于计算$n$点一环参数共形积分在$D$维度中的值，我们称之为重构过程。 它依赖于将共形积分分解为基函数，这些基函数是通过作用于一组主函数上的循环群$\mathbb{Z}_n$生成的。 为了确定主函数，我们引入了一种二分Mellin-Barnes表示，方法是将给定的共形积分拆分为两个加法部分，其中一部分可以以多变量广义超几何级数的形式显式求解。 对于盒子和五边形积分（即$n=4,5$）我们表明，二分表示中的可计算部分包含所有主函数。 特别是，这使我们能够将参数五边形积分表示为由循环群$\mathbb{Z}_5$从两个主函数生成的十个基函数之和。 所得表达式可以通过两种方式进行测试。 首先，当其中一个传播子幂设置为零时，五边形积分简化为已知的盒子积分，该积分也通过重构过程重新推导出来。 其次，进入非参数情况，我们重现了以对数形式给出的五边形积分的已知表达式，该表达式先前在计算共形积分的几何方法中得出。 我们最后考虑了六边形积分（$n=6$），我们表明来自二分表示可计算部分的基函数是不够的，需要更多的基函数。 在我们项目的第二部分中，我们将描述一种在$n$点情况下构建完整主函数/基函数集的方法。 显示英文摘要

摘要： We propose a systematic approach to calculating $n$-point one-loop parametric conformal integrals in $D$ dimensions which we call the reconstruction procedure. It relies on decomposing a conformal integral over basis functions which are generated from a set of master functions by acting with the cyclic group $\mathbb{Z}_n$. In order to identify the master functions we introduce a bipartite Mellin-Barnes representation by means of splitting a given conformal integral into two additive parts, one of which can be evaluated explicitly in terms of multivariate generalized hypergeometric series. For the box and pentagon integrals (i.e. $n=4,5$) we show that a computable part of the bipartite representation contains all master functions. In particular, this allows us to evaluate the parametric pentagon integral as a sum of ten basis functions generated from two master functions by the cyclic group $\mathbb{Z}_5$. The resulting expression can be tested in two ways. First, when one of propagator powers is set to zero, the pentagon integral is reduced to the known box integral, which is also rederived through the reconstruction procedure. Second, going to the non-parametric case, we reproduce the known expression for the pentagon integral given in terms of logarithms derived earlier within the geometric approach to calculating conformal integrals. We conclude by considering the hexagon integral ($n=6$) for which we show that those basis functions which follow from the computable part of the bipartite representation are not enough and more basis functions are required. In the second part of our project we will describe a method of constructing a complete set of master/basis functions in the $n$-point case.