We investigate a large class of perturbative QCD (pQCD) renormalization schemes whose beta functions β(a) are meromorphic functions of the running coupling and give finite positive value of the coupling a(Q ²) in the infrared regime (‘freezing’), a(Q ²) → a ₀ for Q ² → 0. Such couplings automatically have no singularities on the positive axis of the squared momenta Q ² ( ≡ − q ²). Explicit integration of the renormalization group equation (RGE) leads to the implicit (inverted) solution for the coupling, of the form . An analysis of this solution leads us to an algebraic algorithm for the search of the Landau singularities of a(Q ²) on the first Riemann sheet of the complex Q ²-plane, i.e., poles and branching points (with cuts) outside the negative semiaxis. We present specific representative examples of the use of such algorithm, and compare the found Landau singularities with those seen after the 2-dimensional numerical integration of the RGE in the entire first Riemann sheet, where the latter approach is numerically demanding and may not always be precise. The specific examples suggest that the presented algebraic approach is useful to find out whether the running pQCD coupling has Landau singularities and, if yes, where precisely these singularities are.

我们研究了一大类微扰 QCD (pQCD) 重整化方案，其 β 函数β ( a ) 是运行耦合的亚纯函数，并在红外区域 (“冻结”) 中给出了耦合a ( Q ² ) 的有限正值，a ( Q ² ) → a ₀表示Q ² → 0。这样的耦合在平方动量Q ² ( ≡ − q ^{2 )}的正轴上自动没有奇点）。重整化群方程 (RGE) 的显式积分导致耦合的隐式（逆）解，形式为。对该解的分析使我们得到了一种代数算法，用于在复数Q ²的第一个黎曼表上搜索a ( Q ² )的朗道奇点 -平面，即负半轴外的极点和分支点（带切口）。我们给出了使用这种算法的具体代表性例子，并将发现的朗道奇点与在整个第一个黎曼表中对 RGE 进行二维数值积分后看到的那些进行比较，后者的方法在数值上要求很高，可能并不总是精确的。具体例子表明，所提出的代数方法有助于找出正在运行的 pQCD 耦合是否具有朗道奇点，如果是，则这些奇点的确切位置。