By Vladimir A. Smirnov

The e-book provides asymptotic expansions of Feynman integrals in numerous limits of momenta and much, and their purposes to difficulties of actual curiosity. the matter of enlargement is systematically solved through formulating common prescriptions that specific phrases of the growth utilizing the unique Feynman imperative with its integrand accelerated right into a Taylor sequence in applicable momenta and much. wisdom of the constitution of the asymptotic enlargement on the diagrammatic point is essential in knowing easy methods to practice expansions on the operator point. commonest examples of those expansions are offered: the operator product enlargement, the large-mass growth, Heavy Quark powerful thought, and Non-Relativistic QCD.

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**Extra resources for Applied asymptotic expansions in momenta and masses**

**Sample text**

11). 84), where the sum runs only over complete subgraphs. , [147]). 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. As will be explained shortly (and, in detail, in Chap.

This appearance of divergences is characteristic (well, almost) of all expansions in momenta and masses and is a negligible price paid in order to obtain explicit formulae for individual terms that are as simple as possible in the general order of the expansion. 59). Since the argument of the hypergeometric functions is q 2 /m2 , this result is naturally written as an expansion in powers of this ratio. 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 .

6 with general indices. This ‘triangle’ rule takes the form 1= 1 a1 1+ 3− − p21 + a2 2+ 3− − p22 d − a1 − a2 − 2a3 . 69) When applied to the left triangle in Fig. 66). Generally, one tries to use all possible IBP relations. For example, for a two-loop Feynman integral over the loop momenta k and l depending on n external momenta qi , all possible IBP relations for derivatives (∂/∂kµ )pµ and (∂/∂lµ )pµ are used, where p = k, l, qi . e. reduction of a Feynman integral from a given family to master integrals, can be a rather non-trivial problem even at the two-loop level.