高斯超几何函数的Feynman积分与递归分析

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本文是核物理B期刊上的一篇研究论文，由Georg Puhlfürst和Stephan Stieberger共同撰写，发表于2016年。该研究关注的是费曼积分的递归关系及其与高斯超几何函数的关系，特别是在半整数值附近的展开。文章的核心是通过解析求解递归关系来确定高斯超几何函数在半整数值周围的ϵ展开的封闭表达式和紧凑表达式。这些函数通常在量子场论和数学物理中出现，特别是在处理多维和复杂积分时。高斯超几何函数是一种特殊的超几何函数，具有广泛的应用，包括在解决特殊类型的微分方程时。文章指出，这种ϵ展开可以与具有四个正则奇异点的基础Fuchs系统（一种线性微分方程系统）的规范化解相对应。Fuchs系统在理解复分析中的解析延拓和局部行为方面至关重要。通过计算这个系统的正则化ζ级数，研究人员能够找到两个独立的关联项，它们的比例给出了特定值下的ϵ展开。此外，作者还利用已知的一圈有质量的泡泡图积分作为示例，展示如何获得费曼积分的全阶ϵ展开，并讨论如何构建费曼积分的表示。费曼积分是量子场论中的基本工具，用于计算粒子相互作用的概率。在这一点上，文章提供了一种方法，使得对这些复杂积分的解析处理成为可能，这对于精确计算物理过程至关重要。这篇论文为理解和计算与费曼积分相关的复杂数学结构提供了新的洞察，特别是涉及到高斯超几何函数和Fuchs系统的展开。这种方法对于物理学家和数学家在处理量子场论中的计算问题时可能非常有用，尤其是在高精度计算和理论预测的背景下。

172 G. Puhlfürst, S. Stieberger / Nuclear Physics B 906 (2016) 168–193

g(0;y) =

,g(1;y) =

1 −y

,g(−1;y) =

1 +y

. (2.7)

Weight w = 1 functions are simple logarithms:

H(0;y) = ln(y) , H(1;y) =−ln(1 −y) , H(−1;y) = ln(1 +y) . (2.8)

In the next subsection we will encounter differential equations satisﬁed by functions for which

we know their boundary conditions at y = 1. Therefore, it is advantageous to introduce the fol-

lowing integral operators

J(0)f (y) =



f(t),

J(1)f (y) =



1 −t

f(t),

J(−1)f (y) =



1 +t

f(t), (2.9)

with their integrations starting at the point 1. We shall use a shorter notation for products of these

operators:

J(m

,...,m

) ≡J(m

)J (m

)...J(m

), m

∈{0, 1, −1,θ} ,J(θ)≡y

(2.10)

Products of integral operators (2.9) acting on functions independent of y, like the constant func-

tion

1, can recursively be written as HPLs using

J(m

) 1 = H(m

;y) −H(m

;1),

J(m

)H (m

,...,m

;y) =H(m

,...,m

;y) −H(m

,...,m

;1), (2.11)

with m

∈{0, 1, −1}, e.g.:

J(0,...,0



 

) 1 = H(0,...,0



 

;y) ,

J(1, 0) 1 = H(1, 0;y) −H(1, 0;1),

J(0, 1, 0) 1 = H(0, 1, 0;y) −H(0, 1, 0;1) −H(0;y)H(1, 0;1),

J(0, −1, 0) 1 = H(0, −1, 0;y) −H(0, −1, 0;1) −H(0;y)H(−1, 0;1). (2.12)

Since HPLs with argument 1 can have divergences, the second terms on the r.h.s. of eqs. (2.11)

reveal that some products of integral operators are ill-deﬁned. This happens when J(1) is the

rightmost operator of a product acting on a constant. It can be seen in the following sections

that these cases ne

ver appear in our results. Divergent HPLs also arise for other operator prod-

ucts, however, they can be removed using the underlying product algebra. Consider for example

J(1, 1, 0) and its representation in terms HPLs following from eqs. (2.11):

In the sequel we will not write this 1.

G. Puhlfürst, S. Stieberger / Nuclear Physics B 906 (2016) 168–193 173

J(1, 1, 0) =H(1, 1, 0;y) −H(1, 0;1)H (1;y) − H(1, 1, 0;1) +H(1, 0;1)H (1;1).

(2.13)

According to the identity

H(1, 0;1)H (1;1) =−H(0, 1, 1;1) +H(1, 1, 0;1) (2.14)

the divergences in the last two terms of (2.13) cancel each other.

The essential dif

ference between the integral operators (2.9) and the integrations in the deﬁ-

nition (2.6) of HPLs is the lower bound. See also eq. (2.37) or Ref. [15] for polylogarithms with

lower integration limits not chosen to be zero.

Products of inte

gral operators (2.9) are closely related to the ‘symbol’ of the underlying func-

tion, together with the prescription that the integration path goes from 1 to z. Indeed, a sequence

of integral operators immediately determines a ‘symbol’, which when integrated against this

path turns it into the Chen iterated integral representation of the function (see e.g. Ref. [16] for

an introduction for ph

ysicists).

Furthermore we introduce the generalized operator product

,...,c

} , (2.15)

as the sum of all the





α=1

,...,j



(2.16)

possible distinct permutations of non-commutative factors c

, each one appearing j

times (i =

1, 2, ..., n). We will call the non-negative integers j

indices and the factors c

arguments of the

generalized operator product (2.15). Some examples are:

}=c

,...,c

}={c

,...,c

} ,

}=c

}=1 . (2.17)

A recursive deﬁnition and basic properties of the generalized operator product can be found

in [5]. The object (2.15) is useful to handle non-commutative quantities, such as the integral

operators (2.9).

2.2. Differential equations for hypergeometric functions

In this section we describe some of the achievements originally developed in [13] for the

calculation of expansions (2.5). Some of the formulas derived here are the foundation of the

method we use in the next section to obtain the all-order expansion.

Applying the differential operator θ :=z

on the series (2.1), it is easy to show, that hyper-

geometric functions satisfy:

(

θ +a

)



a,b





1 +a,b



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