By Vladimir A. Smirnov

ISBN-10: 3540423346

ISBN-13: 9783540423348

The publication provides asymptotic expansions of Feynman integrals in numerous limits of momenta and much, and their functions to difficulties of actual curiosity. the matter of enlargement is systematically solved by way of formulating common prescriptions that categorical phrases of the growth utilizing the unique Feynman imperative with its integrand increased right into a Taylor sequence in applicable momenta and much. wisdom of the constitution of the asymptotic growth on the diagrammatic point is vital in figuring out find out how to practice expansions on the operator point. most common examples of those expansions are offered: the operator product growth, the large-mass enlargement, Heavy Quark potent conception, and Non-Relativistic QCD.

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**Download PDF by Vladimir A. Smirnov: Applied Asymptotic Expansions in Momenta and Masses**

The ebook offers asymptotic expansions of Feynman integrals in numerous limits of momenta and lots more and plenty, and their purposes to difficulties of actual curiosity. the matter of growth is systematically solved through formulating common prescriptions that specific phrases of the growth utilizing the unique Feynman fundamental with its integrand extended right into a Taylor sequence in acceptable momenta and much.

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**Additional info for Applied Asymptotic Expansions in Momenta and Masses (Springer Tracts in Modern Physics)**

**Sample text**

For a given subset, the corresponding contribution can certainly be made into an absolutely convergent integral over αl by choosing the real parts of the regularization parameters λl of the UV/IR lines suﬃciently large/small. Then each piece is deﬁned as an analytic function of the regularization parameters. Collecting all the pieces extended to the point (ε; 1, . . , 1) gives, by deﬁnition [61], the dimensionally regularized Feynman integral (still considered for Euclidean external momenta) for an arbitrary graph.

18) k+m q q 0 After this we rewrite the two contributions above in identical form by extending the integrations to inﬁnite limits and compensating this by subtracting the corresponding additional pieces: fsmall (q, m, Λ) = fsmall ∼ Fsmall − 1 q ∞ dk Λ Λ flarge ∼ Flarge − dk 0 1 k −ε − 2 k+m q k −ε−1 −m k+q ∞ dk Λ Λ dk 0 k −ε+1 + ... k+m k −ε−2 + ... 11), respectively. 19b) there exists a domain of the regularization parameter ε for which it is absolutely convergent. All these integrals can be deﬁned for general ε by analytic continuation.

Another new approach [122] applies so-called Lorentz invariance identities together with IBP relations. This method is primarily oriented towards Feynman integrals with four or more external lines and is based on the fact that when the total dimension of the denominator and numerator in the Feynman integrals associated with a given graph is increased the total number of IBP and Lorentz invariance equations grows faster than the number of independent Feynman integrals (labelled by the powers of propagators and the powers of independent scalar products in the numerators).

### Applied Asymptotic Expansions in Momenta and Masses (Springer Tracts in Modern Physics) by Vladimir A. Smirnov

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