量子场论中幂次发散问题的圈正规化研究

The divergences in Quantum Field Theories (QFT) can be classified into logarithmic divergences and power-law divergences. The logarithmic divergences are well understood and well treated by perturbative renormalization under the viewpoint of Wilsonian Exact Renormalization Group Equations and Effective Field Theories (EFT). The resulting running coupling constants and running masses can characterize the dependency of the higher order corrections on the scattering energy scale in scattering processes, which can be used to improve perturbation theory by solving the Renormalization Group Equations (RGE). But, the power-law divergences are not well understood and well addressed. The most commonly used regularization method in the literature is the Dimensional Regularization which discards the power-law divergences. To study the power-law divergence one need a regularization which can preserve various symmetries and the original power-law divergent behaviour. It turns out to be the Loop Regularization. In this project we will use the Loop Regularization to study the power-law divergences in QFT. This includes investigating whether the quadratic divergence in the Standard Model (SM) Higgs mass correction has observable effect, whether the bare mass parameter is a physical parameter, whether the fine-tuning problem has been misinterpreted, how to subtract the power-law divergences in EFT, how the bare parameters and renormalized parameters run under Exact RGE and perturbative RGE, whether the power-law running coupling constant is a useful concept. The innovation of this project is the use of Loop Regularization which can preserve symmetries and power-law divergent behaviour, and trying to provide a new viewing angle to treat the fine-tuning problem, the renormalization of EFT, and the power-law running problem.

量子场论包含的发散问题可分为对数发散和幂次发散。对数发散在Wilson的严格重整化群和有效场论观点下可通过微扰重整化很好地理解和处理，相应的跑动耦合常数等概念刻画了散射过程中高阶修正对散射能标的依赖关系，通过重整化群可用这种依赖关系改进微扰论。但幂次发散则未得到很好的理解和处理。文献最常用的维数正规化丢弃了幂次发散。为研究幂次发散需用保持对称性和幂次发散行为的正规化，即圈正规化。本项目将研究圈正规化对场论中幂次发散的处理和理解。这包括检查标准模型中黑格斯质量的平方发散是否具有可观测效应，裸质量参数是否是一个物理的参数，精细调节问题是否被错误解读，有效场论中幂次发散应该如何减除，裸参量和重整化参量各自的重整化群跑动如何，幂次跑动的耦合常数是否为一个有用的概念。项目的创新之处在于使用了保持对称性和幂次发散行为的圈正规化来研究，并尝试对精细调节问题、有效场论的重整化、幂次跑动等提供新的视角。