Analytic continuation and perturbative expansions in QCD

Starting from the divergence pattern of perturbative quantum chromodynamics, we propose a novel, non-power series replacing the standard expansion in powers of the renormalized coupling constant $a$. The coefficients of the new expansion are calculable at each finite order from the Feynman diagrams, while the expansion functions, denoted as $W_n(a)$, are defined by analytic continuation in the Borel complex plane. The infrared ambiguity of perturbation theory is manifest in the prescription dependence of the $W_n(a)$. We prove that the functions $W_n(a)$ have branch point and essential singularities at the origin $a=0$ of the complex $a$-plane and their perturbative expansions in powers of $a$ are divergent, while the expansion of the correlators in terms of the $W_n(a)$ set is convergent under quite loose conditions