Counterterm technique and Wilson expansions

Asymptotic expansions in limits of large momenta and masses

“…Some time ago, the situation for operator asymptotic expansions was reversed in its character. For example, coefficient functions of Wilson expansion [45] TMx + Qξι)Ji{x + ρξ 2 ) e~°X QiQξuQξiWάx) (1.2) were naturally expressed by Zimmermann [50], and by Anikin and Zavialov [2,3,46], in terms of Green functions of composite operators J\ and Jh (here [Θi] is a basis of composite operators). However this expansion is not in powers and logarithms of the expansion parameter ρ because the coefficient functions QiQζuQζi) depend on it nontrivially.…”

“…The main purpose of this paper is to prove asymptotic expansions in limits of large momenta and/or masses with both above mentioned properties: the expansions are in powers and logarithms, and coefficient functions are written as renormalized Feynman amplitudes (at the diagrammatic level) or Green functions of composite operators (at the operator level). We shall apply a straightforward generalization of the method of Zimmermann, Anikin and Zavialov [2,3,46,50] that is based on pre-subtractions in certain subgraphs. This procedure removes the ultraviolet (UY) divergences which are generated in the limit under consideration.…”

“…To derive the asymptotic expansion of a Feynman amplitude in the given limit let us apply the method [2,3,46,50] that reduces to the construction of the remainder of the expansion. This remainder is nothing but the initial Feynman amplitude renormalized in a special subtraction procedure.…”

Renormalization and Operator Product Expansion in Theories with Massless Particles

“…1) A generalization of the light-cone expansion of ref. [13] for theories including mashless particles can easily be achieved by replacing the operator $@) introduced in [13] by cc 5 h 2) It is worth-while to derive the explicit formulae for the OPE in the MS-scheme…”

“…While analyzing different contributions to Gd it is useful to make the Fourier transform with respect to some variablesp, i.e. to apply the mixed representation [15] which, np to a nonessential coefficient has the form (I, 13) 13) exp ( -fH7(Z, p , ( I , 13).…”