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

By Vladimir A. Smirnov

The e-book provides asymptotic expansions of Feynman integrals in quite a few limits of momenta and lots more and plenty, and their functions to difficulties of actual curiosity. the matter of growth is systematically solved via formulating common prescriptions that specific phrases of the growth utilizing the unique Feynman crucial 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 essential in realizing easy methods 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 powerful concept, and Non-Relativistic QCD.

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A less trivial property is that 32 2 Feynman Integrals: a Brief Review • a derivative of an integral with respect to a mass or momentum equals the corresponding integral of the derivative. This is also a consequence (see [61, 208]) of the above definition. To prove this statement, one uses standard algebraic relations between the functions entering the alpha representation [181, 35]. ) A corollary of the last property is the possibility of integrating by parts and always neglecting surface terms: • dd k1 .

The modified MS scheme [13] (MS) is obtained from the MS scheme by the replacement µ2 → µ2 eγE /(4π) for the massive parameter of dimensional regularization that enters through the factors of µ−2ε per loop. Note that the most important part of the basic theorem about the Roperation in the framework of dimensional renormalization [35] is just the above polynomial dependence of the diagrammatic counterterms Pγ (q, m) on the masses and external momenta. Although a requirement for this polynomial dependence on the masses [71] is not obligatory from the Lagrangian point of view, it turns out to be rather natural.

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) dξ ξ λ1 (1 − ξ)λ2 1 × 0 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 2 dd k d/2 Γ (ε)Γ (1 − ε) = iπ . g. 56). Let us present another example of evaluation of Feynman diagrams by means of alpha parameters: consider Fig.

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