Improving the understanding and the applicability of regularization by dimensional reduction for ultraviolet and infrared divergences

正则化是量子场论重整化的一个基本步骤。它用定义良好的表达式取代了发散积分，从而允许量子场论在数学上的一致定义。在实际应用中，我们希望一个正则化方案尽可能多地保留基本性质（如洛伦兹不变性或其他对称性），但理想的方案并不存在。我们考虑了降维正则化（DRED）。特别是对于超对称理论，DRED通常是最好的选择。然而，几十年来人们已经知道了DRED的几个问题。尽管最近取得了进展，但这些问题再次被SPA（“超对称参数分析”）合作报告所强调，SPA是一个由100多名作者组成的全球合作项目。我们的研究项目正是集中在DRED最迫切的两个问题上：与超对称的一致性和与红外散度分解的一致性。我们将通过在几种情况下证明DRED的超对称性，识别和描述DRED违反超对称性的情况，以及在双环水平上证明与分解的一致性来给出明确的答案。此外，我们将提供DRED和维度正则化（DREG）之间易于使用的转换规则。这样，我们将显著提高对DRED的理论认识和实践适用性，完成SPA报告提出的七项任务之一。

Regularization is a fundamental step in the renormalization of quantum field theories. It replaces divergent integrals by well-defined expressions and thus allows a mathematically consistent definition of quantum field theories. For practical purposes it is desirable that a regularization scheme preserves as many fundamental properties (such as Lorentz invariance or other symmetries) as possible, but no ideal scheme exists. We consider regularization by dimensional reduction (DRED). Particularly for supersymmetric theories DRED is often the best choice. However, several problems of DRED have been known for decades. In spite of recent progress, these problems have again been highlighted by the report of the SPA (“Supersymmetry Parameter Analysis”) collaboration, a world-wide collaboration with more than 100 authors. Our research project is precisely focused on the two most pressing problems of DRED: consistency with supersymmetry and consistency with factorization of infrared divergences. We will give definite answers by proving supersymmetry of DRED in several cases, identifying and describing cases where DRED violates supersymmetry, and by proving consistency with factorization at the two-loop level. Furthermore, we will provide easy-to-use transition rules between DRED and dimensional regularization (DREG). In this way we will significantly improve the theoretical understanding and the practical applicability of DRED and complete one of the seven tasks put forward by the SPA report.