By Vladimir A. Smirnov
Introduction.- Feynman Integrals: easy Definitions and Tools.-Evaluating via Alpha and Feynman Parameters.- region Decompositions.- comparing by way of Mellin-Barnes Representation.- Integration by means of components and aid to grasp Integrals.- evaluate through Differential Equations.- comparing grasp Integrals via Dimensional Recurrence and Analyticity.- Asymptotic Expansions in Momenta and Masses.- Tables.- a few precise capabilities- Summation Formulae.- desk of MB Integrals.- a short assessment of a few different equipment
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17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. M. A. Smirnov, Nucl. Phys. B 522, 321 (1998) T. Binoth, G. Heinrich, Nucl. Phys. B 585, 741 (2000) T. Binoth, G. Heinrich, Nucl. Phys. T. Bodwin, E. P. Lepage, Phys. Rev. T. Bodwin, E. P. Lepage, Phys. Rev. D 55, 5853 (1997) C. Bogner, S. Weinzierl, Feynman graph polynomials. Int. J. Mod. Phys. N. V. Shirkov, Introduction to Theory of Quantized Fields, 3rd edn. G. J. Giambiagi, Nuovo Cim. B 12, 20 (1972) P. Breitenlohner, D. Maison, Commun.
M . e. 36) and take the limit where some indices ai tend to negative integers. After this, the integration is taken only over the parameters corresponding to positive indices and one obtains in the integrand a polynomial due to the differentiation in the parameters corresponding to negative indices and setting these parameters to zero. Let us now present the alpha representation of scalar dimensionally regularized integrals in a modified form by making the change of variables αl = ηαl , where αl = 1.
G. S. Wightman (Reidel, Dodrecht, 1976) p. 25. 36. R. Speer, Ann. Inst. H. Poincaré 23, 1 (1977) 37. A. P. Lepage, Phys. Rev. D 43, 196 (1991) 38. G. Wilson, Phys. Rev. D 7, 2911 (1973) 39. com 40. I. Zavialov, Renormalized Quantum Field Theory (Kluwer Academic Publishers, Dodrecht, 1990) Chapter 3 Evaluating by Alpha and Feynman Parameters Feynman parameters1 are very well known and often used in practical calculations. They are closely related to alpha parameters introduced in Chap. 2. The use of both kinds of parameters enables us to transform Feynman integrals over loop momenta into parametric integrals where Lorentz invariance becomes manifest.
Analytic tools for Feynman integrals by Vladimir A. Smirnov
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