By Vladimir A. Smirnov
The publication offers asymptotic expansions of Feynman integrals in numerous limits of momenta and much, and their purposes to difficulties of actual curiosity. the matter of growth is systematically solved by means of formulating common prescriptions that specific phrases of the growth utilizing the unique Feynman indispensable with its integrand improved right into a Taylor sequence in applicable momenta and much. wisdom of the constitution of the asymptotic growth on the diagrammatic point is essential in figuring out the way to practice expansions on the operator point. commonest examples of those expansions are offered: the operator product growth, the large-mass growth, Heavy Quark potent idea, and Non-Relativistic QCD.
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Extra info for Applied Asymptotic Expansions in Momenta and Masses
In any case, it is often necessary to know not only the pole and the ﬁnite part of a diagram but also some initial terms of its Laurent expansion in ε. g. 52)) so that we have obtained a result in the form of a series in q 2 /m2 . A lot of one-loop integration formulae can be derived by use of the alpha representation and subsequent straightforward integration. 1. 54). Consider, for example, the two-loop diagram shown in Fig. 4. 56) and is eﬀectively replaced by a line with index ε. Then the sequence of two lines with indices 1 and ε is replaced by one line with index 1 + ε, and the one-loop diagram so obtained, which has indices 2 and 1 + ε, is 36 2 Feynman Integrals: a Brief Review Fig.
8). Sometimes it is reasonable to convert general monomials to traceless monomials or vice versa. 40b) are of use. 45) can be taken into account by shifting the space–time dimension d and indices al of a given diagram, because any factor that arises after the diﬀerentiation with respect to the auxiliary parameters ul is a sum of products of positive integer powers of the α-parameters and negative integer powers of the function D. In particular, the factor 1/Dn is taken into account by the shift d → d + 2n.
63) where Ai = m2i − p2i . For the evaluation of diagrams with a small number of loops, the choice of applying either alpha or Feynman parameters is usually just a matter of taste. 61). 1. g. 8). Sometimes it is reasonable to convert general monomials to traceless monomials or vice versa. 40b) are of use. 45) can be taken into account by shifting the space–time dimension d and indices al of a given diagram, because any factor that arises after the diﬀerentiation with respect to the auxiliary parameters ul is a sum of products of positive integer powers of the α-parameters and negative integer powers of the function D.