Divergent Integrals Treatment in Quantum Field Theory

This paper presents a brief review on regularization methods and shows that the combination of two techniques could preserve symmetries in all orders of the Perturbation Theory. We will see, with two simple examples in the one-loop-order, what a regularization method needs to preserve symmetries. Furthermore, it will be shown that the problems with symmetry violations by regularization methods are related to the consequences of not being able to simply shift a variable in these divergent integrals. To facilitate the analysis between methods and provide analytical expressions for the finite parts of amplitudes, it was systematized in terms of a set of special functions. In the Appendix we introduce these functions and then cases of specific interest, functions continuity and some useful limits will be presented.

本文对正则化方法（regularization method）进行了简要综述，并证明两种技术的结合可在微扰论（Perturbation Theory）的所有阶次中保持对称性。我们将通过单圈阶（one-loop-order）下的两个简单示例，阐释正则化方法为保持对称性所需满足的条件。此外，本文还将证明：正则化方法引发的对称性破缺问题，与无法在这类发散积分（divergent integrals）中直接进行变量平移所带来的后果密切相关。为便于不同方法间的对比分析，并为振幅的有限部分提供解析表达式，本文已基于一组特殊函数（special functions）完成了体系化整理。在附录（Appendix）中，我们将先介绍这些特殊函数，随后给出若干具有特定研究价值的案例、函数的连续性及若干实用极限的相关内容。