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

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By Vladimir A. Smirnov

The booklet 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 specific phrases of the growth utilizing the unique Feynman vital with its integrand accelerated 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 tips to practice expansions on the operator point. most common examples of those expansions are offered: the operator product enlargement, the large-mass growth, Heavy Quark potent thought, and Non-Relativistic QCD.

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K 2 −i0. 54)). We also prefer to use al for integer and λl for general complex indices. e. FΓ (q; λ1 , λ2 , d) = iπ d/2 Γ (λ1 + λ2 + ε − 2) 1 × 0 dξ ξ λ1 (1 − ξ)λ2 λ1 +λ2 +ε−2 [m21 ξ + m22 (1 − ξ) − q 2 ξ(1 − ξ) − i0] . 53) Suppose that the masses are zero. 54) where G(λ1 , λ2 ) = Γ (λ1 + λ2 + ε − 2)Γ (2 − ε − λ1 )Γ (2 − ε − λ2 ) . 5 How They Are Evaluated Γ (ε)Γ (1 − ε)2 dd k = iπ d/2 . g. 56). Let us present another example of evaluation of Feynman diagrams by means of alpha parameters: consider Fig.

So, it is a decomposition of the given set of momenta and masses into these two subsets that determines a limit of momenta and masses. For simplicity, let us first consider an h-loop Feynman integral FΓ corresponding to a graph Γ with two external vertices and depending on two parameters. e. m2 −q 2 . We have 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 .

As we shall see, the remainder of the expansion is obtained from the original diagram by the action of an operation given by the forest formula with an appropriate (pre-)subtraction operator. 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.

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