用微分方程求三圈夸克静势的最后一个缺失因子
Evaluating the last missing ingredient for the three-loop quark static potential by differential equations
摘要：
我们分析评估的三圈费曼积分，这是最后一个失踪的成分的三圈夸克静态势的分析评估。为了评估积分，我们引入了一个辅助参数$y$，它对应于一些HQET传播子中的剩余能量。我们构造了一个109个主积分依赖于$y$的微分系统，并从极限$y\to \infty$的渐近行为固定边界条件。从极限$y\to 0$恢复原始积分。为了解决这些线性微分方程，我们试图找到一个$ε$形式的微分系统。虽然这一步似乎是，严格地说，是不可能的，我们成功地找到了一个$ε$ -形式的所有不可约对角块，这是足够的解决微分系统的一个$ε$扩展。我们找到了一个解决方案的重量6在多个多项式，并获得所需的三圈费曼积分的分析结果，通过采取限制$y\to 0$。作为一个副产品，我们获得了一些典型的HQET费曼积分的分析结果。
英文摘要：
We analytically evaluate the three-loop Feynman integral which was the last missing ingredient for the analytical evaluation of the three-loop quark static potential. To evaluate the integral we introduce an auxiliary parameter $y$, which corresponds to the residual energy in some of the HQET propagators. We construct a differential system for 109 master integrals depending on $y$ and fix boundary conditions from the asymptotic behaviour in the limit $y\to \infty$. The original integral is recovered from the limit $y\to 0$. To solve these linear differential equations we try to find an $ε$-form of the differential system. Though this step appears to be, strictly speaking, not possible, we succeed to find an $ε$-form of all irreducible diagonal blocks, which is sufficient for solving the differential system in terms of an $ε$ expansion. We find a solution up to weight six in terms of multiple polylogarithms and obtain an analytical result for the required three-loop Feynman integral by taking the limit $y\to 0$. As a by-product, we obtain analytical results for some Feynman integrals typical for HQET.
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