zbMATH — the first resource for mathematics

Numerical contour integration for loop integrals. (English) Zbl 1196.81066
Summary: A fully numerical method to calculate loop integrals, a numerical contour-integration method, is proposed. Loop integrals can be interpreted as a contour integral in a complex plane for an integrand with multi-poles in the plane. Stable and efficient numerical integrations an along appropriate contour can be performed for scalar and tensor integrals appearing in loop calculations of the standard model. Examples of 3- and 4-point diagrams in 1-loop integrals and 2- and 3-point diagrams in 2-loop integrals with arbitrary masses are shown.Moreover it is shown that numerical evaluations of the Hypergeometric function, which often appears in the loop integrals, can be performed using the numerical contour-integration method.

81-08 Computational methods for problems pertaining to quantum theory
Full Text: DOI
[1] Oyanagi, Y.; Kaneko, T.; Sasaki, T.; Kawabata, S.; Shimizu, Y., How to calculate one-loop diagram, (), 369, KEK-Preprint 88-6
[2] Fujimoto, J.; Shimizu, Y.; Kato, K.; Oyanagi, Y., Numerical approach to one-loop integrals, (), 407
[3] Fujimoto, J.; Shimizu, Y.; Kato, K.; Oyanagi, Y., Prog. theor. phys., 87, 1233, (1992)
[4] Fujimoto, J.; Shimizu, Y.; Kato, K.; Oyanagi, Y., Numerical approach to loop integrals, (), 625
[5] J. Fujimoto, Y. Shimizu, K. Kato, Y. Oyanagi, Numerical approach to two-loop integrals, KEK-Preprint 92-213
[6] Fujimoto, J.; Shimizu, Y.; Kato, K.; Kaneko, T., Internat. J. modern phys. C, 6, 525, (1995)
[7] Sato, M.; Sato, M.; Shintani, T., On zeta functions associated with prehomogeneous vector space, Nagoya math. J., Proc. natl. acad. sci. USA, 69, 1081, (1972), (The English translation of Sato’s lecture from Shintani’s Note (1970)) · Zbl 0249.10034
[8] Bernstein, L.N., Funct. anal. appl., 6, 66, (1972)
[9] Tkachov, F.V., Nucl. instrum. methods A, 389, 309, (1997)
[10] Passarino, G.; Ferroglia, A.; Passera, M.; Passarino, G.; Uccirati, S., Nucl. phys. B, Nucl. phys. B, Nucl. phys. B, 680, 199, (2004)
[11] Passarino, G.; Uccirati, S., Nucl. phys. B, 629, 97, (2002)
[12] Weber, M.M., Acta phys. polonica B, 55, 2655, (2004)
[13] Uccirati, S., Acta phys. polonica B, 55, 2573, (2004)
[14] de Doncker, E.; Shimizu, Y.; Fujimoto, J.; Yuasa, F.; Kaugars, K.; Cucos, L.; Van Voorst, J.; de Doncker, E.; Shimizu, Y.; Fujimoto, J.; Yuasa, F., Nucl. instrum. methods A, Comput. phys. comm., 159, 145, (2004)
[15] del Aguila, F.; Pittau, R., J. high energy phys. (JHEP), 0407, 017, (2004)
[16] van Hameren, A.; Vollinga, J.; Weinzierl, S.
[17] Giele, W.T.; Glover, E.W.N., J. high energy phys. (JHEP), 0404, 029, (2004)
[18] Binoth, T.; Guillet, J.Ph.; Heinrich, G.; Duplanžić, G.; Nižić, B.; Binoth, T.; Heinrich, G.; Kauer, N., Nucl. phys. B, Eur. phys. J. C, Nucl. phys. B, 654, 277, (2003)
[19] Bélanger, G.; Boudjema, F.; Fujimoto, J.; Ishikawa, T.; Kaneko, T.; Kato, K.; Shimizu, Y., (Section 5)
[20] Sloan, I.H.; Joe, S., Lattice methods for multiple integration, (1994), Oxford Univ. Press Oxford, For example, see · Zbl 0855.65013
[21] Fujimoto, J.; Igarashi, M.; Nakazawa, N.; Shimizu, Y.; Tobimatsu, K., Prog. theor. phys., 100, Suppl., 1, (1990)
[22] van Oldenborgh, G.J., Comput. phys. comm., 58, 1, (1991)
[23] Kawabata, S., Comput. phys. comm., Comput. phys. comm., 88, 309, (1995)
[24] Kreimer, D., Phys. lett. B, 273, 277, (1991), Numbers are obtained though a private communication with D. Kreimer. His calculation is performed based on the method described in:
[25] Bauberger, S.; Böhm, M., Nucl. phys. B, 445, 25, (1995)
[26] Kreimer, D.; Frink, A.; Kilian, U.; Kreimer, D., Phys. lett. B, Nucl. phys. B, 488, 426, (1997)
[27] Kurihara, Y.; Fujimoto, J.; Ishikawa, T.; Kato, K.; Kawabata, S.; Munehisa, T.; Tanaka, H., Nucl. phys. B, 654, 301, (2003)
[28] Mathematica ver.5.0, Wolfram Research, Inc
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.