High orders of perturbation theory. Are renormalons significant?

Abstract

According to Lipatov, the high orders of perturbation theory are determined by saddle-point configurations (instantons) of the corresponding functional integrals. According to t'Hooft, some individual large diagrams, renormalons, are also significant and they are not contained in the Lipatov contribution. The history of the conception of renormalons is presented, and the arguments in favor of and against their significance are discussed. The analytic properties of the Borel transforms of functional integrals, Green functions, vertex parts, and scaling functions are investigated in the case of \phi^4 theory. Their analyticity in a complex plane with a cut from the first instanton singularity to infinity (the Le Guillou - Zinn-Justin hypothesis) is proved. It rules out the existence of the renormalon singularities pointed out by t'Hooft and demonstrates the nonconstructiveness of the conception of renormalons as a whole. The results can be interpreted as an indication of the internal consistency of \phi^4 theory.