Asymptotic Expansions in Momenta and Masses and Calculation of Feynman Diagrams

General results on asymptotic expansions of Feynman diagrams in momenta and/or masses are reviewed. It is shown how they are applied for calculation of massive diagrams. Talk given at the workshop “Advances in non-perturbative and perturbative techniques”, Ringberg Castle, November 13–19, 1994. Permanent address: Nuclear Physics Institute of Moscow State University, Moscow 119899, Russia; E-mail: [email protected] 1. Earlier results. The problem of asymptotic expansions of Feynman amplitudes in momenta and/or masses is rather old. A limit of large momenta and masses is characterized by a subdivision of the set of all external momenta and internal masses of a diagram Γ into large Q ≡ {Q1, . . .Qi, . . .}, M ≡ {M1, . . .Mi, . . .}, and small q ≡ {q1, . . . qi, . . .}, m ≡ {m1, . . .mi, . . .} ones. The problem is to analyze the behavior of the corresponding Feynman integral FΓ(Q, q,M,m) in the limit FΓ(Q/ρ, q,M/ρ,m) as ρ → 0. For brevity, let us denote this limit as Q,M → ∞. Let us imply that the external momenta are not fixed at a mass shell. In 1960 Weinberg [1] described the leading large momentum behavior. Logarithmic corrections were characterized in [2]. Later it was proved that the large momentum asymptotic expansions are always performed in powers and logarithms of the expansion parameter [3]. In [4, 5, 6, 7, 8, 9, 10, 11] asymptotic expansions in various limit of large momenta and masses were obtained. A typical result is the expansion of the form FΓ(Q/ρ, q,m) ρ → 0 ∼ ∑ k,l Ck,l ρ k log ρ. (1) However the coefficient functions in these expansions are cumbersome. They are expressed in terms of numerous parametric integrals or in terms of Mellin integrals. Thus, the first of the following two natural properties of asymptotic expansions does not hold: (i) The coefficient functions Ck,l are expressed in a simple way through renormalized and/or regularized Feynman amplitudes; (ii) The expansion is in powers and logarithms. 2. Results in the simplest form for the large momentum limit. To write down the asymptotic expansion with these two properties it is worthwhile to introduce dimensional (with d = 4 − 2ε) regularization even in case the original diagram is ultravioletly finite. The following proposition is valid. In the large momentum limit, FΓ(Q/ρ, q,m; ε) Q → ∞ ∼ ∑ γ FΓ/γ(q,m; ε) ◦ Tqγ ,mγFγ(Q, q , m ; ε), (2) where the sum is over subgraphs γ of Γ such that each γ (a) contains all the vertices with the large external momenta and (b) is 1PI after contraction of these vertices. The term ‘asymptotic’ implies that the corresponding remainder satisfies necessary estimates and one knows nothing about the radius of convergence. In contrast to expansions in coupling constants which typically have zero radii of convergence, the large mass/momentum expansions of Feynman diagrams seem to have always non-zero radii of convergence. The results that are presented here hold both for Minkowski and Euclidean spaces. For Minkowski space, it is in fact sufficient to imply that the large external momenta are space-like. However, for the limit of large masses when all the momenta are small, there are no restriction on momenta. Another possible variant is to consider Feynman diagrams as distributions in momenta.