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

By Vladimir A. Smirnov

The publication provides asymptotic expansions of Feynman integrals in a number of limits of momenta and much, and their functions to difficulties of actual curiosity. the matter of growth is systematically solved via formulating common prescriptions that categorical phrases of the growth utilizing the unique Feynman necessary with its integrand multiplied right into a Taylor sequence in applicable momenta and lots more and plenty. wisdom of the constitution of the asymptotic growth on the diagrammatic point is vital in realizing find out how 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 concept, and Non-Relativistic QCD.

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Sample text

As we discussed in Chap. 2, it is natural to set such integrals to zero. But here we do not need to refer to such ad hoc recipes, because the two pieces of the integral arise quite naturally when they are evaluated with their own values of ε which provide convergence.

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.

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. It should be stressed that these expectations are, in general, a matter of experience rather than a corollary of a mathematical theorem. However we shall see that, in some simpler limits, these statements can indeed be justified. 3) for 0 < x < ε. The above formulae are modified just a little when there are more than two parameters: we multiply each small parameter by a dimensionless variable x, and deal with the resulting function of x in the limit x → 0.

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