Calculation of Feynman diagrams by the “Uniqueness” method

A method of calculating massless Feynman integrals on the basis of a “uniqueness” relation is developed and applications are discussed. The method involves several algebraic steps but not the calculation of integrals of elementary or special functions or expansion in infinite series and summation of them. The simplicity and effectiveness of the method are illustrated by the calculation of two- and three-loop dimensionally regularized integrals. The technique is applied to the analytic calculation of the last diagrams that complete the five-loop renormalization-group calculations in the $\varphi^4$ theory.