We discuss the method of conformal mappings applied to perturbative QCD. The approach is based on the Borel-Laplace integral regulated with the principal value prescription and the expansion of the Borel transform in powers of the variable which performs the conformal mapping of the cut Borel plane onto the unit disk. We write down the expression of the conformal mapping for the most general location of the singularities of the Borel transform and review the properties of the corresponding expansions of the correlators. Unlike the standard perturbative expansions, which are divergent, the modified expansions have a tamed behaviour at large orders and may even converge under some conditions. On the other hand, the expansion functions exhibit nonperturbative features similar to those of the expanded function. Using these properties, it was suggested recently that the expansions based on the conformal mapping of the Borel plane may provide an alternative to the standard OPE. We briefly review the arguments in favour of this conjecture and discuss the application of the method to the Adler function for massless quarks and the static quark self-energy calculated in lattice QCD.

我们讨论了应用于微扰 QCD 的保形映射方法。该方法基于 Borel-Laplace 积分，该积分由主值规定和 Borel 变换在变量的幂中的展开进行调节，该变量将切割的 Borel 平面共形映射到单位圆盘上。我们写下波雷尔变换奇点最一般位置的共形映射表达式，并回顾相关器相应展开的性质。与发散的标准微扰展开不同，修改后的展开在大阶数下具有驯服的行为，甚至可能在某些条件下收敛。另一方面，扩展函数表现出与扩展函数类似的非微扰特征。使用这些属性，最近有人建议，基于 Borel 平面的共形映射的扩展可以提供标准 OPE 的替代方案。我们简要回顾了支持这一猜想的论点，并讨论了该方法在无质量夸克的阿德勒函数和晶格 QCD 中计算的静态夸克自能中的应用。