Analytic tools for Feynman integrals by Vladimir A. Smirnov

By Vladimir A. Smirnov

Introduction.- Feynman Integrals: simple Definitions and Tools.-Evaluating by way of Alpha and Feynman Parameters.- zone Decompositions.- comparing by way of Mellin-Barnes Representation.- Integration by way of components and aid to grasp Integrals.- review through Differential Equations.- comparing grasp Integrals via Dimensional Recurrence and Analyticity.- Asymptotic Expansions in Momenta and Masses.- Tables.- a few certain features- Summation Formulae.- desk of MB Integrals.- a short assessment of a few different tools

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N is symmetric in its indices and is composed of the metric tensor and the vectors qi . 13). 40 3 Evaluating by Alpha and Feynman Parameters Let us now present a simple one-loop example and illustrate the trick with turning to integrals without numerators. Consider the Feynman integral corresponding to Fig. 14) where l is a momentum not related to p1 and p2 . 15) where U = α1 + α2 + α3 , V = q 2 α1 α2 . 16) l2 F(a1 , a2 , a3 ; 0; d + 2) 2π 1 + 2 a1 (a1 + 1)(l · p1 )2 F(a1 + 2, a2 , a3 ; 0; d + 4) π + 2a1 a2 (l · p1 )(l · p2 )F(a1 + 1, a2 + 1, a3 ; 0; d + 4) F(a1 , a2 , a3 ; 2; d) = +a2 (a2 + 1)(l · p2 )2 F(a1 , a2 + 2, a3 ; 0; d + 4) .

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On the other hand, from the pure theoretical and mathematical point of view, such a position is beneath criticism. 4 Regularization 25 variables, the analysis of convergence reduces to power counting in one-dimensional integrals. e. 38) i∈I this analysis is described in Sect. 4. As a result of this analysis, any Feynman integral at Euclidean external momenta is defined as meromorphic function of d with series of UV and IR poles [9, 27, 30, 33, 34, 36]. e. massless subdiagrams with zero external momenta.

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