Applied Asymptotic Expansions in Momenta and Masses by Vladimir A. Smirnov

By Vladimir A. Smirnov

The booklet offers asymptotic expansions of Feynman integrals in numerous limits of momenta and lots more and plenty, and their functions to difficulties of actual curiosity. the matter of growth is systematically solved by means of formulating common prescriptions that categorical phrases of the growth utilizing the unique Feynman crucial with its integrand accelerated right into a Taylor sequence in acceptable momenta and much. wisdom of the constitution of the asymptotic growth on the diagrammatic point is essential in figuring out how you can practice expansions on the operator point. most common examples of those expansions are provided: the operator product enlargement, the large-mass growth, Heavy Quark powerful thought, and Non-Relativistic QCD.

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11), respectively. 19b) there exists a domain of the regularization parameter ε for which it is absolutely convergent. All these integrals can be defined for general ε by analytic continuation. 19b) into Taylor series in m and k, respectively. The resulting integrals are easily evaluated. 20b) respectively. It should be stressed that the above two integrals are initially defined in different domains of the parameter ε, namely Re ε > 0 and Re ε < 0, where they are absolutely convergent and from where they are analytically continued to general values of ε.

8, we have the following forests: {∅}, {γ}, {Γ } and {γ, Γ }, where γ is the one-loop subgraph. The corresponding R-operation takes the form R = 1 − Mγ − MΓ + MΓ Mγ ≡ (1 − MΓ ) (1 − Mγ ) . 88) Moreover, for the two-loop QED graph shown in Fig. 9, with two logarithmically divergent partially overlapping one-loop vertex subgraphs γ1 and γ2 , the complete set of forests is {∅}, {γ1 }, {γ2 }, {Γ }, {γ1 , Γ }, {γ2 , Γ }. The corresponding R-operation is R = 1 − M γ1 − M γ2 − M Γ + M Γ M γ1 + M Γ M γ2 ≡ (1 − MΓ ) (1 − Mγ1 − Mγ2 ) .

80), respectively, and obtain a zero value. This fact again exemplifies the consistency of our rules. Thus we are going to systematically apply the properties of dimensionally regularized Feynman integrals in any situation, no matter where the external momenta are considered to be. 7 How They Are Renormalized Although a regularization makes it possible to deal with divergent Feynman integrals, it does not actually remove UV divergences, because this operation is of an auxiliary character so that sooner or later it will be necessary to switch off the regularization.

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