Renormalons and the Renormalization Scheme

According to the recent paper [1], existence or absence of renormalon singularities is related with the analiticity properties of the Gell-Mann – Low function β(g) (g is a coupling constant). Briefly, the results are as follows: (a) Renormalon singularities are absent, if β(g) has a proper behavior at infinity, β(g) ∼ g with α ≤ 1, and its singularities at finite points gc are sufficiently weak, so that 1/β(g) is not integrable at gc (i.e. β(g) ∼ (g − gc) γ with γ > 1). (b) Renormalon singularities exist, if at least one condition named in (a) is violated. It is well-known [2] that the Gell-Mann – Low function β(g) depends on the renormalization scheme, and only two coefficients β2 and β3 are universal in the expansion β(g) = β2g 2 + β3g 3 + . . . In essence, the change of the renormalization scheme is simply a change of variables g = f(g̃), transferring β(g) to β̃(g̃) = β(f(g̃))/f (g̃). Function f(g) is subjected to certain physical restrictions, such as f(g) = g + O(g); in fact, these restrictions are poorly investigated. The interesting possibility arises, if these restrictions do not forbid to transform the β function of type (a) to the β function of type (b). In this case, existence or absence of renormalon singularities is not the internal property of the specific field theory but depends on the renormalization scheme, i.e. on the way of description. The observable quantities do not depend on the renormalization scheme and the latter can be chosen from convenience. On the one hand, the scheme without renormalons can be used to formulate the welldefined theory with unique predictions [1, 4]. In such a theory, large orders of perturbation expansion are determined by the Lipatov method, the Borel integral is well-defined and constructive summation of the perturbation series is possible, giving the possibility to solve different strong coupling problems [4]. It was argued in [1, 4] that the MOM scheme in φ theory and the MS scheme in QED and QCD are renormalon-free. On the other hand, one can deliberately choose an ”extremely renormalon” scheme, in order to justify the renormalon heuristics, which is extensively used in different applications [5]. For example, power corrections in QCD are determined generally by the wide set of diagrams and can be calculated starting from the ”renormalon end” [5] or from the ”instanton end” [6]. When the β function of type (a) is used, the main contribution to power corrections is determined by instantons; when the β function is of type (b), this contribution