奇异点展开求解Feynman积分微分方程的策略

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"通过奇异点附近的展开来求解Feynman积分的微分方程" 在粒子物理和量子场论中，Feynman积分是计算复杂物理过程的关键工具。这些积分通常涉及到高维度和非平凡的数学结构，使得直接求解变得极具挑战性。本文提出了一种新的策略，专注于通过奇异点附近的幂级数展开来解决与Feynman积分相关的微分方程，从而能够获取主积分的高精度结果。 Feynman积分涉及多尺度问题，当积分依赖于一个或多个变量时，其复杂性显著增加。在这个特定的研究中，研究者关注的是具有两个尺度的Feynman积分，即那些非平凡地取决于一个变量的积分。这种情况下的积分通常伴随着微分方程，但这些方程可能无法简化到规范形式。在这种情况下，传统的求解方法可能会遇到困难。为了应对这一挑战，作者开发了一种算法，专门针对微分方程无法达到规范形式的情形。他们利用幂级数展开的方法，即在奇异点附近对微分方程进行展开，然后逐步求解每一项，从而得到主积分的精确表达。这种方法能够有效地处理积分的复杂性和奇异性。为了证明这种方法的有效性，研究团队提供了一个计算机代码示例，该代码应用于一个具体的四环广义日落积分问题。这个例子包含三个非零质量和两个零质量的粒子，它展示了如何应用所提出的算法。该代码可以在指定的扩展阶数和正则化参数下，为实轴上的任意点计算出所需的轴精度的主积分值。此外，这篇研究发表在JHEP（Journal of High Energy Physics）上，是一篇开放访问的文章，意味着任何感兴趣的人都可以免费阅读和使用其中的方法。文章的接收日期为2017年11月26日，经过修订和接受后，最终于2018年3月5日发表。作者包括Roman N. Lee、Alexander V. Smirnov和Vladimir A. Smirnov，他们分别来自俄罗斯的布德克尔核物理研究所、莫斯科国立大学计算研究中心和斯克利贝尔茨基核物理研究所。这项工作为处理复杂的Feynman积分问题提供了新的视角，通过奇异点附近的幂级数展开策略，使得高精度计算成为可能，对于理论物理和粒子物理学的研究具有重要意义。

JHEP03(2018)008

transformations from this extended class it is not always possible to obtain an -form.

The simplest example where an -form is impossible is given by the two-loop propagator

sunset diagram with three identical masses. In this example, as well in other known

examples without -form, DE can still be reduced to the form where the right-hand side of

the diﬀerential system is a linear function of .

However, ‘integrating out’ the constant term in such a form of DE appears to be an

essentially more complicated problem. This can be seen in the known examples where

results are expressed in terms of elliptic functions. In practice, it can happen that such

‘elliptic’ master integrals appear only in a small number of sectors. (A sector is speciﬁed by

a distribution of the set of indices (powers of propagators) into positive and non-positive

values.) A ﬁrst example of a calculation of a full set of the master integrals with ‘elliptic

sectors’ can be found in ref. [21]. A recent example of a calculation relevant to a physical

problem is ref. [22], where elliptic functions appear only in two sectors and ﬁnal results

are expressed either in terms of multiple polylogarithms or, for the elliptic sectors and for

some sectors involving complicated letters with square roots, in terms of two and three-fold

iterated integrals suitable for numerical evaluation. Moreover, in refs. [23, 24] a strategy to

obtain parametric representations for master integrals applicable also in situations without

-form was described and illustrated through one-, two- and three-loop examples.

Other examples of calculations of individual Feynman integrals in situations where

-form is impossible can be found in [25, 26] (see also references therein), where results are

expressed in terms of elliptic generalizations of polylogarithms [25] or iterated integrals of

modular forms [26]. One more class of elliptic generalization of multiple polylogarithms

was recently introduced in ref. [27]. In particular, it includes functions appearing in the

-expansion of the imaginary part of the two-loop massive sunset diagram. However, these

new functions do not have the same status as harmonic polylogarithms and multiple poly-

logarithms, at least in the practical sense, i.e. there are no codes to evaluate them at a

given point with a desired precision. Anyway, it looks like we are very far, even in lower

loops orders, from answering the following question: ‘What is the class of functions which

can appear in results for Feynman integrals in situations where -form is impossible’?

On the other hand, thinking positively, we may say that knowing a diﬀerential system

and the corresponding boundary conditions gives almost as much information about Feyn-

man integrals as knowing their explicit expressions in terms of some class of functions. In

fact, some properties of the integrals are even more accessible via DE. In particular, singu-

larities of DE provide a way to examine the branching properties of integrals. Numerical

values of the integrals can be obtained from a numerical solution

of the diﬀerential system.

Many computer algebra systems contain tools to solve this task (e.g. NDSolve procedure in

Mathematica system). However, there is one complication that does not allow to use these

tools immediately. Namely, we would like to keep  as a variable and evaluate solutions of

DE as series expansions in .

Recently, a strict criterion of the existence of an -form was presented in ref. [20].

Examples of solving DE for Feynman integrals numerically can be found in refs. [28, 29].

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