Abstract
Feynman parameters 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. Using alpha parameters we will first evaluate one and two-loop integrals with general complex powers of the propagators, within dimensional regularization, for which results can be written in terms of gamma functions for general values of the dimensional regularization parameter. We will show then how these formulae, together with simple algebraic manipulations, enable us to evaluate some classes of Feynmans integrals.
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Notes
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Thanks to A.G. Grozin for pointing out this possibility!
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Much more terms of the \(\varepsilon \)-expansion, up to \(\varepsilon ^4\), of this non-planar diagram were obtained in [11]. Moreover, an explicit analytic result at general \(\varepsilon \) written in terms of hypergeometric series, was presented.
References
M. Beneke, A. Signer, V.A. Smirnov, Phys. Rev. Lett. 80, 2535 (1998)
K.S. Bjoerkevoll, P. Osland, G. Faeldt, Nucl. Phys. B 386, 303 (1992)
G.T. Bodwin, E. Braaten, G.P. Lepage, Phys. Rev. D 51, 1125
G.T. Bodwin, E. Braaten, G.P. Lepage, Phys. Rev. D 55, 5853 (1997)
H. Cheng, T.T. Wu, Expanding Protons: Scattering at High Energies (MIT Press, Cambridge, 1987)
K.G. Chetyrkin, A.L. Kataev, F.V. Tkachov, Nucl. Phys. B 174, 345 (1980)
J.C. Collins, Renormalization (Cambridge University Press, Cambridge, 1984)
A. Czarnecki, K. Melnikov, Phys. Rev. Lett. 87, 013001 (2001)
A.I. Davydychev, Phys. Lett. B 263, 107 (1991)
A.I. Davydychev, R. Delbourgo, J. Math. Phys. 39, 4299 (1998)
T. Gehrmann, T. Huber, D. Maitre, Phys. Lett. B 622, 295 (2005)
R.J. Gonsalves, Phys. Rev. D 28, 1542 (1983)
A.G. Grozin, Heavy Quark Effective Theory (Springer, Heidelberg, 2004)
G. Kramer, B. Lampe, J. Math. Phys. 28, 945 (1987)
G.P. Lepage et al., Phys. Rev. D 46, 4052 (1992)
A.V. Manohar, M.B. Wise, Heavy Quark Physics (Cambridge University Press, Cambridge, 2000)
W.L. van Neerven, Nucl. Phys. B 268, 453 (1986)
M. Neubert, Phys. Rep. 245, 259 (1994)
A. Pak, J. Phys. Conf. Ser. 368, 012049 (2012)
M.E. Peskin, D.V. Schroeder, An Introduction to Quantum Field Theory (Perseus, Reading, 1995)
J.L. Rosner, Ann. Phys. 44, 11 (1967)
A.V. Smirnov, http://science.sander.su/Tools-UF.htm
O.V. Tarasov, Nucl. Phys. B 480, 397 (1996)
O.V. Tarasov, Phys. Rev. D 54, 6479 (1996)
B.A. Thacker, G.P. Lepage, Phys. Rev. D 43, 196 (1991)
S. Weinzierl, arXiv:1005.1855 [hep-ph]
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Smirnov, V.A. (2012). Evaluating by Alpha and Feynman Parameters. In: Analytic Tools for Feynman Integrals. Springer Tracts in Modern Physics, vol 250. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-34886-0_3
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