By Vladimir A. Smirnov
The e-book 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 enlargement is systematically solved by means of formulating common prescriptions that specific phrases of the growth utilizing the unique Feynman fundamental with its integrand multiplied 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 knowing how one can practice expansions on the operator point. commonest examples of those expansions are offered: the operator product growth, the large-mass growth, Heavy Quark potent thought, and Non-Relativistic QCD.
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Additional info for Applied Asymptotic Expansions in Momenta and Masses (Springer Tracts in Modern Physics)
In this chapter, the limits we are going to study are characterized and the form of the asymptotic expansion we are aiming at is described. An example of a one-loop Feynman integral in the large-momentum limit and a related toy one-dimensional example are then used to formulate two basic strategies for expanding Feynman integrals that we are going to apply. The ﬁrst of them, expansion by regions, is physically motivated and turns out to be more general. In this strategy, one divides the whole integration domain into various regions, then performs some simpliﬁcations and eventually obtains an expansion as a sum of contributions from the regions that can be handled in much easier way than the initial diagram.
To be speciﬁc, let us choose the small parameter to be the square of a mass and the −q 2 . e. m2 2 2 chosen q to be Euclidean so that q = −Q , with Q > 0. Experience tells us that, in all limits, we obtain expansions of Feynman integrals in powers and logarithms. 1) n=n0 j=0 where x = m2 /Q2 and ω is the degree of divergence of the graph Γ . The sum over n runs from some minimal value. The index n can generally take, in some limits, not only integer but also half-integer values. The second index, j, is bounded, for any n, by twice the number of loops.
50). 50) and obtain a set of relations between Feynman integrals. e. with the lowest powers of propagators, in addition to integrals that are easily evaluated, for example, in terms of gamma functions; (b) to evaluate the master integrals. If both parts of this programme are fulﬁlled then the evaluation of the given family of integrals reduces to algebraic manipulations and substitutions of values of the master and ‘boundary’ integrals. Let us illustrate how the IBP method works using the massless diagram of Fig.
Applied Asymptotic Expansions in Momenta and Masses (Springer Tracts in Modern Physics) by Vladimir A. Smirnov