Abstract
We address the computation of Feynman loop integrals, which are required for perturbation calculations in high energy physics, as they contribute corrections to the scattering amplitude for the collision of elementary particles. Results in this field can be used in the verification of theoretical models, compared with data measured at colliders.
We made a numerical computation feasible for various types of one and two-loop Feynman integrals, by parametrizing the integral to be computed and extrapolating to the limit as the parameter introduced in the denominator of the integrand tends to zero. In order to handle additional singularities at the boundaries of the integration domain, the extrapolation can be preceded by a transformation and/or by a sector decomposition. With the goal of demonstrating the applicability of the combined integration and extrapolation methods to a wide range of problems, we give a survey of earlier work and present additional applications with new results. We aim for an automatic or semi-automatic approach, in order to greatly reduce the amount of analytic manipulation required before the numeric approximation.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Anastasiou, C., Beerli, S., Daleo, A.: Evaluating multi-loop Feynman diagrams with infrared and threshold singularities numerically. JHEP 0705, 71 (2005)
Bélanger, G., Boudjema, F., Fujimoto, J., Ishikawa, T., Kaneko, T., Kato, K., Shimizu, Y.: Automatic calculations in high energy physics and GRACE at one-loop. Physics Reports 430, 117–209 (2006)
Berntsen, J., Espelid, T.O., Genz, A.: Algorithm 698: DCUHRE-an adaptive multidimensional integration routine for a vector of integrals. ACM Trans. Math. Softw. 17, 452–456 (1991)
Binoth, T., Heinrich, G.: An automized algorithm to compute infrared divergent multi-loop integrals. Nuclear Physics B 585, 741–759 (2000)
Bollini, C.G., Giambiagi, J.J.: Dimensional renormalization: the number of dimensions as a regularizing parameter. Nuovo Cimento B 12 20 (1972)
Brezinski, C.: A general extrapolation algorithm. Numerische Mathematik 35, 175–187 (1980)
Buras, A.J., Czarnecki, A., Misiak, M., Urban, J.: Two-loop matrix element of the current-current operator in the decay B →X s γ. Nuclear Physics B(611), 488–502 (2001)
Czarnecki, A., Marciano, W.J.: Electroweak radiative corrections to b →s γ. Phys. Rev. Lett. 81(2), 277–280 (1998)
Davis, P.J., Rabinowitz, P.: Methods of Numerical Integration. Academic Press, New York (1984)
de Doncker, E.: Numerical Integration and Asymptotic Expansions. Ph.D. thesis, Katholieke Universiteit Leuven (1980)
de Doncker, E., Li, S., Fujimoto, J., Shimizu, Y., Yuasa, F.: Regularization and extrapolation methods for infrared divergent loop integrals. In: Sunderam, V.S., van Albada, G.D., Sloot, P.M.A., Dongarra, J. (eds.) ICCS 2005. LNCS, vol. 3514, pp. 165–171. Springer, Heidelberg (2005)
de Doncker, E., Li, S., Shimizu, Y., Fujimoto, J., Yuasa, F.: Numerical computation of a non-planar two-loop vertex diagram. In: LoopFest, V. (ed.) Stanford Linear Accelerator Center (2006), http://www.conf.slac.stanford.edu/loopfestv/proc/present/DEDONCKER.pdf
de Doncker, E., Shimizu, Y., Fujimoto, J., Yuasa, F.: Computation of loop integrals using extrapolation. Computer Physics Communications 159, 145–156 (2004)
de Doncker, E., Shimizu, Y., Fujimoto, J., Yuasa, F.: Computation of Feynman loop integrals. PAMM - Wiley InterScience Journal 7(1) (2007)
de Doncker, E., Shimizu, Y., Fujimoto, J., Yuasa, F., Cucos, L., Van Voorst, J.: Loop integration results using numerical extrapolation for a non-scalar integral. Nuclear Instuments and Methods in Physics Research Section A 539, 269–273 (2004)
Ferroglia, A., Passarino, G., Passera, M., Uccirati, S.: All-purpose numerical evaluation of one-loop multi-leg Feynman diagrams. Tech. rep., hep-ph/0209219
Ferroglia, A., Passera, M., Passarino, G., Uccirati, S.: Two-loop vertices in quantum field theory: Infrared convergent scalar configurations (2003), hep-ph/0311186
Fleischer, J., Tarasov, O.V.: Calculation of Feynman diagrams from their small momentum expansion. Zeitschrift für Physik C 64, 413–425 (1994)
Ford, W., Sidi, A.: An algorithm for the generalization of the Richardson extrapolation process. SIAM Journal on Numerical Analysis 24, 1212–1232 (1987)
Fujimoto, J., Shimizu, Y., Kato, K., Oyanagi, Y.: Numerical approach to one-loop integrals. Progress of Theoretical Physics 87(5), 1233–1247 (1992)
Fujimoto, J., Shimizu, Y., Kato, K., Oyanagi, Y.: Numerical approach to two-loop integrals. In: Proc. of the VIIth Workshop on High Energy Physics and Quantum Field Theory (1992)
Fujimoto, J., Ueda, T.: New implementation of the sector decomposition on FORM. In: XII Advanced Computing and Analysis Techniques in Physics Research) poS (ACAT 2008), vol. 120 (2009), ArXiv:0902.2656v1 [hep-ph]
Fujimoto, J., Ueda, T.: New implementation of the sector decomposition on FORM, aCAT08 talk slides (2008), http://indico.cern.ch/conferenceOtherViews.py?confId=34666&view=static&showDate=all&showSession=all&detailLevel=contribution
Genz, A.: The Approximate Calculation of Multidimensional Integrals using Extrapolation Methods. Ph.D. thesis, Univ. of Kent at Canterbury (1975)
Genz, A., Malik, A.: An imbedded family of multidimensional integration rules. SIAM J. Numer. Anal 20, 580–588 (1983)
Hävie, T.: Generalized Neville-type extrapolation schemes. BIT 19, 204–213 (1979)
HMLIB: Nucl. Instr. and Meth. A 559, 269 (2006)
Hooft, G., Veltman, M.: Regularization and renormalization of gauge fields. Nucl. Phys. B 44, 189 (1972)
Hurth, T.: Present status of inclusive rare B decays (2003), hep-ph/0212304, CERN-TH/2002-264, SLAC-PUB-9604
Kawabata, S.: A new version of the multi-dimensional integration and event generation package bases/spring. Computer Physics Communications 88, 309–326 (1995)
Kurihara, Y.: Dimensionally regularized one-loop tensor integrals with massless internal particles (2005), hep-ph/0504251 v3
Kurihara, Y., Kaneko, T.: Numerical contour integration for loop integrals. Computer Physics Communications 174(7), 530–539 (2006)
Levin, D., Sidi, A.: Two classes of non-linear transformations for accelerating the convergence of infinite integrals and series. Appl. Math. Comp. 9, 175–215 (1981)
Li, S.: Online Support for Multivariate Integration. PhD dissertation, Western Michigan University (December 2005)
Lyness, J.N.: Applications of extrapolation techniques to multidimensional quadrature of some integrand functions with a singularity. Journal of Computational Physics 20, 346–364 (1976)
Lyness, J.N., de Doncker, E.: On quadrature error expansions part I. Journal of Computational and Applied Mathematics 17, 131–149 (1987)
Lyness, J.N., de Doncker, E.: On quadrature error expansions II. The full corner singularity. Numerische Mathematik 64, 355–370 (1993)
Neubert, M.: Renormalization-group improved calculation of the B→x s γ branching ratio. hep-ph 1(16) (2004), 0408179, CLNS 04/1885
Passarino, G.: An approach toward the numerical evaluation of multiloop Feynman diagrams. Nucl. Phys. B 619, 257 (2001)
Piessens, R., de Doncker, E., Überhuber, C.W., Kahaner, D.K.: QUADPACK, A Subroutine Package for Automatic Integration. Series in Computational Mathematics. Springer, Heidelberg (1983)
Shanks, D.: Non-linear transformations of divergent and slowly convergent sequences. J. Math. and Phys. 34, 1–42 (1955)
Sidi, A.: Convergence properties of some nonlinear sequence transformations. Math. Comp. 33, 315–326 (1979)
Tarasov, O.V.: An algorithm for small momentum expansion of Feynman diagrams (1995); hep-ph/9505277
Tkachov, F.V.: Algebraic algorithms for multiloop calculations: The first 15 years. What’s next? Nucl. Phys. B 389, 309 (1997)
Vermaseren, J.A.M.: New features of FORM (2000), math-ph/0010025
Wynn, P.: On a device for computing the e m (s n ) transformation. Mathematical Tables and Aids to Computing 10, 91–96 (1956)
Yasui, Y., Ueda, T., de Doncker, E., Fujimoto, J., Hamaguchi, N., Ishikawa, T., Shimizu, Y., Yuasa, F.: Status reports from the grace group. In: International Colliders Workshop LCWS/ILC (2007), arXiv:0710.2957v1 [hep-ph]
Yuasa, F., de Doncker, E., Fujimoto, J., Hamaguchi, N., Ishikawa, T., Shimizu, Y.: Precise numerical results of IR-vertex and box integration with extrapolation. In: Proc. of the XI ACAT workshop, Advanced Computing and Analysis Techniques in physics research (2007), arXiv:0709.0777v2 [hep-ph]
Yuasa, F., Ishikawa, T., Fujimoto, J., Hamaguchi, N., de Doncker, E., Shimizu, Y.: Numerical evaluation of Feynman integrals by a direct computation method. In: Proc. of the XII ACAT workshop, Advanced Computing and Analysis Techniques in physics research (2008), arXiv:0904.2823
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2010 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
de Doncker, E. et al. (2010). Transformation, Reduction and Extrapolation Techniques for Feynman Loop Integrals. In: Taniar, D., Gervasi, O., Murgante, B., Pardede, E., Apduhan, B.O. (eds) Computational Science and Its Applications – ICCSA 2010. ICCSA 2010. Lecture Notes in Computer Science, vol 6017. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-12165-4_12
Download citation
DOI: https://doi.org/10.1007/978-3-642-12165-4_12
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-12164-7
Online ISBN: 978-3-642-12165-4
eBook Packages: Computer ScienceComputer Science (R0)