论文标题
紫外线的理性术语在两个循环中
Rational Terms of UV Origin at Two Loops
论文作者
论文摘要
用于构建单循环幅度的有效数值算法的出现在NLO计算的自动化中起着至关重要的作用，并且在两个循环下开发相似的算法是NNLO自动化的自然策略。在数值框架内，循环积分的分子通常在四个维度中构造，而缺少的理性术语是由$（d-4）$ - 尺寸 - $ 1/（d-4）$ poles in $ d $ dimensions的相互作用产生的。在一个循环中，这种合理术语仅来自紫外线差异，并且可以通过与过程无关的局部反应来恢复。在本文中，我们调查了两个循环中紫外线原点的理性术语的行为。主要结果是一个通用公式，该公式将紫外线的减法与两个循环的相关有理部件的重建相结合。该公式具有与R型的结构相同的结构，并且所有极点和有理零件都通过一组有限的与过程无关的局部反应来描述。我们还提出了一个一般公式，用于计算基于单尺度t t t t的任何可统一的理论中所有相关的两循环合理反对物。作为第一个应用程序，我们得出了$r_ξ$ -Gauge中QED的全套两循环有理反对。
The advent of efficient numerical algorithms for the construction of one-loop amplitudes has played a crucial role in the automation of NLO calculations, and the development of similar algorithms at two loops is a natural strategy for NNLO automation. Within a numerical framework the numerator of loop integrals is usually constructed in four dimensions, and the missing rational terms, which arise from the interplay of the $(D-4)$-dimensional parts of the loop numerator with $1/(D-4)$ poles in $D$ dimensions, are reconstructed separately. At one loop, such rational terms arise only from UV divergences and can be restored through process-independent local counterterms. In this paper we investigate the behaviour of rational terms of UV origin at two loops. The main result is a general formula that combines the subtraction of UV poles with the reconstruction of the associated rational parts at two loops. This formula has the same structure as the R-operation, and all poles and rational parts are described through a finite set of process-independent local counterterms. We also present a general formula for the calculation of all relevant two-loop rational counterterms in any renormalisable theory based on one-scale tadpole integrals. As a first application, we derive the full set of two-loop rational counterterms for QED in the $R_ξ$-gauge.
