The gamma5-problem of dimensional regularization - the HVBM scheme for the electroweak Standard Model at the 2-loop level

违反宇称（本质上是左右对称）的电弱相互作用存在的本质基本事实。它的技术描述使用了所谓的gamma5矩阵。自1972年t Hooft和Veltman的开创性工作以来，维度正则化（DREG）为基本粒子理论的高精度提供了一个数学上一致且非常成功的框架。然而，Hooft和Veltman已经指出了伽玛5的一个问题。这个问题本质上意味着DREG不允许同时在数学上一致和计算上简单的处理gamma5。在许多处理伽玛5的建议中，只有一个已经建立了严格的证明和数学一致性：由't Hooft/Veltman和Breitenlohner/Maison （HVBM方案）提出的原始方案。然而，HVBM方案在实际应用中很难应用于普遍存在γ - 5的电弱相互作用中，并且HVBM打破了重要的规范不变性。因此，从业者传统上采用替代方案，这些方案在技术上更容易使用，但有其他缺点，如有限（有时不清楚）的有效性范围和/或缺乏数学一致性（可能导致错误或模糊的结果）。未来对精确预测的需求增加了人们对γ - 5问题的兴趣。最近所有的讨论都集中在HVBM的替代办法上，并证实了替代办法的不令人满意的状况和局限性，突出表明需要进一步的工作和进展。因此，直接关注HVBM方案的互补、不妥协的研究似乎是有希望的、及时的和有价值的。在此，我们提出这样一项研究。我们将使用HVBM方案，并将其应用于双环水平的电弱标准模型，大大超出了当前的状态。目的是建立如何在现实的理论基础上进行hvb -scheme的多回路计算，并为今后这类的二回路和三回路计算提供便利。研究计划由系统的五个步骤组成，首先应用于单环和双环水平的简单模型，然后应用于全电弱标准模型。一个主要的成果是确定所谓的规范不变性-恢复和消失的反项-这些是通用的构建块，将作为计算机代码发布，并允许以直接的方式进行未来的HVBM计算。此外，该项目将回答几个关键的概念问题，这些问题涉及基于HVBM的计算与更传统的计算之间的关系，以及HVBM方案中各种选择的优缺点。通过这种方式，该项目还将对传统方案提供重要的反馈，这些反馈可以通过与严格的HVBM结果进行比较来验证和改进。

A fundamental fact of nature of the existence of electroweak interactions which violate parity (essentially left-right symmetry). Its technical description uses the so-called gamma5-matrix. Since the seminal work of 't Hooft and Veltman in 1972, dimensional regularization (DREG) provides a mathematically consistent and extremely successful framework for high-precision in elementary particle theory. However, already 't Hooft and Veltman pointed out a problem with gamma5. The problem means in essence that DREG allows no treatment of gamma5 which is simultaneously mathematically consistent and computationally straightforward. Among the many proposals to treat gamma5, there is only one for which rigorous proofs and mathematical consistency have been established: the original one by 't Hooft/Veltman and Breitenlohner/Maison (HVBM scheme). However the HVBM scheme is difficult to apply in practice to the electroweak interactions where gamma5 is ubiquitous and where HVBM breaks an important gauge invariance. Therefore, practioners have traditionally applied alternative schemes which are technically easier to use but have other drawbacks such as limited (sometimes unclear) range of validity and/or lack of mathematical consistency (possibly leading to wrong or ambiguous results). The future need for precise predictions has increased the interest in the gamma5-problem. All recent discussions have focused on alternatives to HVBM and have confirmed the unsatisfactory status and the limitations of alternative schemes, highlighting the need for further work and progress. Hence a complementary, no-compromise study focusing directly on the HVBM scheme seems promising, timely and valuable. Here we propose such a study. We will use the HVBM scheme and apply it to the electroweak Standard Model at the two-loop level, significantly beyond the current status. The goal is to establish how to do HVBM-scheme multi-loop computations in realistic theories and facilitate future two- and three-loop computations of this kind. The research plan consists of a systematic sequence of five steps which are applied at the one-loop and two-loop level first to simpler models and then to the full electroweak Standard Model. A major deliverable is the determination of so-called gauge invariance-restoring and evanescent counterterms --- these are universal building blocks which will be published as computer codes and which will allow to carry out future HVBM computations in a straightforward way. In addition the project will answer several crucial conceptual questions, related to the relationship between HVBM-based and more traditional calculations and related to the pros and cons of various options within the HVBM scheme. In this way the project will also provide important feedback on the traditional schemes, which can be verified and improved by comparing with the rigorous HVBM results.