By Vladimir A. Smirnov
The e-book offers asymptotic expansions of Feynman integrals in quite a few limits of momenta and much, and their functions to difficulties of actual curiosity. the matter of enlargement is systematically solved by way of formulating common prescriptions that specific phrases of the growth utilizing the unique Feynman imperative 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 vital in figuring out the way to practice expansions on the operator point. most common examples of those expansions are provided: the operator product growth, the large-mass growth, Heavy Quark powerful idea, and Non-Relativistic QCD.
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Extra info for Applied Asymptotic Expansions in Momenta and Masses (Springer Tracts in Modern Physics)
9). On the other hand, it can also be obtained from the unrenormalized S-matrix by replacing all the bare parameters in the Lagrangian by their renormalized values. 91) 48 2 Feynman Integrals: a Brief Review where the factor µε makes the dimension of the coupling constant independent of the parameter ε of dimensional regularization. 92) where znj are dimensionless constants. 84), can be implemented by modifying parameters of the Lagrangian looks (almost) obvious at the one-loop level. 93) where LR I is the renormalized Lagrangian, which consists of the interaction part of the bare Lagrangian, LB I , and counterterms.
9) (so that the Wick theorem is applied again after the action of this operation). 11). 84), where the sum runs only over complete subgraphs. , ). Here is a sketch of the proof. 84) is applied to the resulting diagrams. From the order-N expansion term of the exponent, diagrams with N vertices arise. 84), into subsets consisting of a diﬀerent number of vertices, and the number N is represented as N1 + 2N2 + . . + kNk + . . )Nk ] distinct ways. 94). 95) is generated by a polynomial j(x) composed of the ﬁelds present in the theory.
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. Note that if we proceed naively and begin to expand the corresponding integrand in a Taylor series in our small parameter m2 , we immediately obtain, in the leading order of the expansion, an IR divergence, which comes from the factor 1/(k 2 )2 .