Explicit solutions for effective four- and five-loop QCD running coupling

We start with the explicit solution, in terms of the Lambert W function, of the renormalization group equation (RGE) for the gauge coupling in the supersymmetric Yang-Mills theory described by the well-known NSVZ beta-function. We then construct a class of beta-functions for which the RGE can be solved in terms of the Lambert W function. These beta-functions are expressed in terms of a function which is a truncated Laurent series in the inverse u of the gauge coupling a equivalent to alpha/pi. The parameters in the Laurent series can be adjusted so that the first coefficients of the Taylor expansion of the beta-function in the gauge coupling a reproduce the four-loop or five-loop QCD (or SQCD) beta-function.