Download Applied asymptotic expansions in momenta and masses by Vladimir A. Smirnov PDF

By Vladimir A. Smirnov

The e-book provides asymptotic expansions of Feynman integrals in a variety of limits of momenta and lots more and plenty, and their purposes to difficulties of actual curiosity. the matter of growth is systematically solved by way of formulating common prescriptions that categorical phrases of the growth utilizing the unique Feynman indispensable with its integrand elevated right into a Taylor sequence in acceptable momenta and lots more and plenty. wisdom of the constitution of the asymptotic growth on the diagrammatic point is vital in knowing how one can practice expansions on the operator point. most common examples of those expansions are offered: the operator product enlargement, the large-mass enlargement, Heavy Quark powerful conception, and Non-Relativistic QCD.

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86) (and can be chosen as the BPHZ subtraction or some other operator that provides the necessary properties of the renormalization). e. families of non-overlapping subgraphs), consisting of 1PI divergent subgraphs. By definition, the empty forest is also implied in the sum, with the corresponding term equal to unity. For example, for the one-loop graph Γ of Fig. 1, there are two forests, ∅ and Γ , so that the R-operation takes the form R = 1 − M . For the two-loop scalar graph shown in Fig. 8, we have the following forests: {∅}, {γ}, {Γ } and {γ, Γ }, where γ is the one-loop subgraph.

However, we want an expansion in the opposite variable. 54) that relates the hypergeometric functions with arguments z and 1/z and obtain FΓ (q 2 , m2 ; d) = iπ d/2 + (ε − 1)m2 2 2 2 F1 1, ε; 1 − ε; m /q εq 2 Γ (2 − ε)Γ (ε) Γ (1 − 2ε) − m2 q2 1+ε 1− m2 q2 −2ε . 76), closing the integration contour to the right and taking two series of residues, at the points z1,n = −ε + n and z2,n = n, where n = 0, 1, . . But we want to arrive at the terms of this expansion in an essentially easier way, and by following some general prescriptions which will avoid the need for us to evaluate this full result or to perform an analysis of MB integrals oriented towards concrete expressions for the given diagram.

9. Two-loop QED diagram Although dimensional renormalization can be also described through an appropriate subtraction operator [72], this is achieved by a cumbersome procedure, at the cost of introducing different space–time dimensions for 1PI subgraphs, so that it is more reasonable to apply the MS and MS schemes within the recursive definition. However, we shall certainly need the forest formula for another purpose: it happens that the structure of the R-operation represented by the forest formula arises when one solves the problem of asymptotic expansions in off-shell limits of momenta and masses.

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