关于括号法
On the Method of Brackets
摘要：
括号法（MoB）是一种计算定积分的方法，起源于负维积分法。它最初提出的费曼积分的评价，适用时，它给出的结果（多）系列的组合。在这里，我们专注于一些MoB的局限性，并通过研究梅林-巴恩斯（MB）表示技术来解决这些问题。最近，由于基于圆锥壳的新的计算方法的发展，在后者的研究中出现了重要的进展（参见Phys. Rev. Lett. 127，151601（2021））。这两种方法之间的比较有助于理解MoB的局限性，特别是当术语发散序列出现时。因此，MB技术被发现优于MoB，主要有两个原因：1。在MB方法中，形成给定积分的级数表示的级数集合的选择遵循圆锥壳的特定交叉，与MoB相反，其不需要对所涉及的级数进行任何收敛分析，以及2. MB可以用来评估共振（即对数）的情况下，MoB失败，由于外观的termwise发散系列。此外，我们表明，最近增加的规则5的MoB自然出现的结果，在MB的背景下的剩余定理。
英文摘要：
The Method of Brackets (MoB) is a technique used to compute definite integrals, that has its origin in the negative dimensional integration method. It was originally proposed for the evaluation of Feynman integrals for which, when applicable, it gives the results in terms of combinations of (multiple) series. We focus here on some of the limitations of MoB and address them by studying the Mellin-Barnes (MB) representation technique. There has been significant process recently in the study of the latter due to the development of a new computational approach based on conic hulls (see Phys. Rev. Lett. 127, 151601 (2021)). The comparison between the two methods helps to understand the limitations of the MoB, in particular when termwise divergent series appear. As a consequence, the MB technique is found to be superior over MoB for two major reasons: 1. the selection of the sets of series that form a series representation for a given integral follows, in the MB approach, from specific intersections of conic hulls, which, in contrast to MoB, does not need any convergence analysis of the involved series, and 2. MB can be used to evaluate resonant (i.e. logarithmic) cases where MoB fails due to the appearance of termwise divergent series. Furthermore, we show that the recently added Rule 5 of MoB naturally emerges as a consequence of the residue theorem in the context of MB.
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