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Connection between Feynman integrals having different values of the space-time dimension. (English) Zbl 0925.81121
Summary: A systematic algorithm for obtaining recurrence relations for dimensionally regularized Feynman integrals with respect to the space-time dimension $$d$$ is proposed. The relation between $$d$$- and $$(d-2)$$-dimensional integrals is given in terms of a differential operator for which an explicit formula can be obtained for each Feynman diagram. We show how the method works for one-, two-, and three-loop integrals. The new recurrence relations with respect to $$d$$ are complementary to the recurrence relations which derive from the method of integration by parts. We find that the problem of the irreducible numerators in Feynman integrals can be naturally solved in the framework of the proposed generalized recurrence relations.

##### MSC:
 81T18 Feynman diagrams
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##### References:
 [1] J. Zinn-Justin, in: Quantum Field Theory and Critical Phenomena (1993) [2] A. M. Tsvelik, in: Quantum Field Theory in Condensed Matter Physics (1995) · Zbl 0968.82001 [3] D. J. Amit, in: Field Theory, the Renormalization Group and Critical Phenomena (1978) [4] G. ’t Hooft, Nucl. Phys. B44 pp 189– (1972) [5] G. G. Bollini, Nuovo Cimento B 12 pp 20– (1972) [6] C. M. Bender, Phys. Rev. D 46 pp 5557– (1992) [7] K. G. Chetyrkin, Nucl. Phys. B192 pp 159– (1981) [8] F. V. Tkachov, Phys. Lett. 100B pp 65– (1981) [9] V. Alfaro, in: High Energy Physics and Elementary Particles (1965) [10] E. E. Boos, Teor. Mat. Fiz. 89 pp 56– (1991) [11] D. J. Broadhurst, Z. Phys. C 60 pp 287– (1993) [12] S. Bauberger, Nucl. Phys. B434 pp 383– (1995) [13] O. V. Tarasov, Nucl. Phys. B [14] C. Itzykson, in: Quantum Field Theory (1980) [15] E. R. Speer, J. Math. Phys. (N.Y.) 15 pp 1– (1974) [16] M. C. Bergère, Commun. Math. Phys. 39 pp 1– (1974) [17] N. N. Bogoliubov, in: Introduction to the Theory of Quantized Fields (1980) [18] A. I. Davydychev, Phys. Lett. B 263 pp 107– (1991) [19] A. I. Davydychev, Phys. Rev. D 53 pp 7381– (1996) [20] D. J. Broadhurst, Z. Phys. C 54 pp 599– (1992) [21] L. Avdeev, Phys. Lett. B 336 pp 560– (1994) [22] L. Avdeev, Phys. Lett. B 349 pp 597– (1995) [23] L. M. Brown, Nuovo Cimento 22 pp 178– (1961) [24] D. B. Melrose, Nuovo Cimento A 40 pp 181– (1965) [25] F. R. Halpern, Phys. Rev. Lett. 10 pp 310– (1963) [26] Z. Bern, Nucl. Phys. B412 pp 751– (1994) · Zbl 1007.81512 [27] A. J. Macfarlane, Nucl. Phys. B77 pp 91– (1974) [28] A. J. Macfarlane, Nucl. Phys. B86 pp 548– (1975) [29] A. V. Efremov, Nuovo Cimento A 76 pp 122– (1983)
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