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By Vladimir A. Smirnov
Introduction.- Feynman Integrals: uncomplicated Definitions and Tools.-Evaluating by way of Alpha and Feynman Parameters.- quarter Decompositions.- comparing via Mellin-Barnes Representation.- Integration via components and aid to grasp Integrals.- review through Differential Equations.- comparing grasp Integrals by way of Dimensional Recurrence and Analyticity.- Asymptotic Expansions in Momenta and Masses.- Tables.- a few targeted services- Summation Formulae.- desk of MB Integrals.- a short evaluate of a few different tools
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Example text
An obvious way of regularizing Feynman integrals is to introduce a cut-off at large values of the loop momenta. Another well-known regularization procedure is the Pauli–Villars regularization [26], which is described by the replacement p2 1 1 1 → 2 − 2 2 2 −m p −m p − M2 22 2 Feynman Integrals: Basic Definitions and Tools and its generalizations. For finite values of the regularization parameter M, this procedure clearly improves the UV asymptotics of the integrand. Here the limiting value of the regularization parameter is M = ∞.
43) This integral is divergent, in contrast to the original Feynman integral defined for general q 2 . Thus on-shell or threshold dimensionally regularized Feynman integrals are defined by the alpha representation or by integrals over the loop momenta with restriction of some kinematical invariants to appropriate values in the corresponding integrands. In this sense, these regularized integrals are ‘formal’ values of general Feynman integrals at the chosen variables. Note that the products of the free fields in the Lagrangian are not required to be normal-ordered, so that products of fields of the same sort at the same point are allowed.
3): 6 These arguments can be found, for example, in [20], and, ironically, even in a pure mathematical book [16]. 28 2 Feynman Integrals: Basic Definitions and Tools dd k Γ (ε) = iπ d/2 . 43) This integral is divergent, in contrast to the original Feynman integral defined for general q 2 . Thus on-shell or threshold dimensionally regularized Feynman integrals are defined by the alpha representation or by integrals over the loop momenta with restriction of some kinematical invariants to appropriate values in the corresponding integrands.