Advertisement

Transformation, Reduction and Extrapolation Techniques for Feynman Loop Integrals

  • Elise de Doncker
  • Junpei Fujimoto
  • Nobuyuki Hamaguchi
  • Tadashi Ishikawa
  • Yoshimasa Kurihara
  • Yoshimitsu Shimizu
  • Fukuko Yuasa
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6017)

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.

Keywords

Feynman Diagram Extrapolation Method Loop Integral Extrapolation Technique Multivariate Integration 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Anastasiou, C., Beerli, S., Daleo, A.: Evaluating multi-loop Feynman diagrams with infrared and threshold singularities numerically. JHEP 0705, 71 (2005)Google Scholar
  2. 2.
    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)CrossRefGoogle Scholar
  3. 3.
    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)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Binoth, T., Heinrich, G.: An automized algorithm to compute infrared divergent multi-loop integrals. Nuclear Physics B 585, 741–759 (2000)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Bollini, C.G., Giambiagi, J.J.: Dimensional renormalization: the number of dimensions as a regularizing parameter. Nuovo Cimento B 12 20 (1972)Google Scholar
  6. 6.
    Brezinski, C.: A general extrapolation algorithm. Numerische Mathematik 35, 175–187 (1980)zbMATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    Buras, A.J., Czarnecki, A., Misiak, M., Urban, J.: Two-loop matrix element of the current-current operator in the decay BX s γ. Nuclear Physics B(611), 488–502 (2001)Google Scholar
  8. 8.
    Czarnecki, A., Marciano, W.J.: Electroweak radiative corrections to bs γ. Phys. Rev. Lett. 81(2), 277–280 (1998)CrossRefGoogle Scholar
  9. 9.
    Davis, P.J., Rabinowitz, P.: Methods of Numerical Integration. Academic Press, New York (1984)zbMATHGoogle Scholar
  10. 10.
    de Doncker, E.: Numerical Integration and Asymptotic Expansions. Ph.D. thesis, Katholieke Universiteit Leuven (1980)Google Scholar
  11. 11.
    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)Google Scholar
  12. 12.
    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
  13. 13.
    de Doncker, E., Shimizu, Y., Fujimoto, J., Yuasa, F.: Computation of loop integrals using extrapolation. Computer Physics Communications 159, 145–156 (2004)CrossRefGoogle Scholar
  14. 14.
    de Doncker, E., Shimizu, Y., Fujimoto, J., Yuasa, F.: Computation of Feynman loop integrals. PAMM - Wiley InterScience Journal 7(1) (2007)Google Scholar
  15. 15.
    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)CrossRefGoogle Scholar
  16. 16.
    Ferroglia, A., Passarino, G., Passera, M., Uccirati, S.: All-purpose numerical evaluation of one-loop multi-leg Feynman diagrams. Tech. rep., hep-ph/0209219Google Scholar
  17. 17.
    Ferroglia, A., Passera, M., Passarino, G., Uccirati, S.: Two-loop vertices in quantum field theory: Infrared convergent scalar configurations (2003), hep-ph/0311186Google Scholar
  18. 18.
    Fleischer, J., Tarasov, O.V.: Calculation of Feynman diagrams from their small momentum expansion. Zeitschrift für Physik C 64, 413–425 (1994)CrossRefGoogle Scholar
  19. 19.
    Ford, W., Sidi, A.: An algorithm for the generalization of the Richardson extrapolation process. SIAM Journal on Numerical Analysis 24, 1212–1232 (1987)zbMATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Fujimoto, J., Shimizu, Y., Kato, K., Oyanagi, Y.: Numerical approach to one-loop integrals. Progress of Theoretical Physics 87(5), 1233–1247 (1992)CrossRefGoogle Scholar
  21. 21.
    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)Google Scholar
  22. 22.
    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]Google Scholar
  23. 23.
    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
  24. 24.
    Genz, A.: The Approximate Calculation of Multidimensional Integrals using Extrapolation Methods. Ph.D. thesis, Univ. of Kent at Canterbury (1975)Google Scholar
  25. 25.
    Genz, A., Malik, A.: An imbedded family of multidimensional integration rules. SIAM J. Numer. Anal 20, 580–588 (1983)zbMATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    Hävie, T.: Generalized Neville-type extrapolation schemes. BIT 19, 204–213 (1979)CrossRefMathSciNetGoogle Scholar
  27. 27.
    HMLIB: Nucl. Instr. and Meth. A 559, 269 (2006)Google Scholar
  28. 28.
    Hooft, G., Veltman, M.: Regularization and renormalization of gauge fields. Nucl. Phys. B 44, 189 (1972)CrossRefGoogle Scholar
  29. 29.
    Hurth, T.: Present status of inclusive rare B decays (2003), hep-ph/0212304, CERN-TH/2002-264, SLAC-PUB-9604Google Scholar
  30. 30.
    Kawabata, S.: A new version of the multi-dimensional integration and event generation package bases/spring. Computer Physics Communications 88, 309–326 (1995)zbMATHCrossRefGoogle Scholar
  31. 31.
    Kurihara, Y.: Dimensionally regularized one-loop tensor integrals with massless internal particles (2005), hep-ph/0504251 v3Google Scholar
  32. 32.
    Kurihara, Y., Kaneko, T.: Numerical contour integration for loop integrals. Computer Physics Communications 174(7), 530–539 (2006)CrossRefGoogle Scholar
  33. 33.
    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)zbMATHCrossRefMathSciNetGoogle Scholar
  34. 34.
    Li, S.: Online Support for Multivariate Integration. PhD dissertation, Western Michigan University (December 2005)Google Scholar
  35. 35.
    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)CrossRefMathSciNetGoogle Scholar
  36. 36.
    Lyness, J.N., de Doncker, E.: On quadrature error expansions part I. Journal of Computational and Applied Mathematics 17, 131–149 (1987)zbMATHCrossRefMathSciNetGoogle Scholar
  37. 37.
    Lyness, J.N., de Doncker, E.: On quadrature error expansions II. The full corner singularity. Numerische Mathematik 64, 355–370 (1993)zbMATHCrossRefGoogle Scholar
  38. 38.
    Neubert, M.: Renormalization-group improved calculation of the Bx s γ branching ratio. hep-ph 1(16) (2004), 0408179, CLNS 04/1885Google Scholar
  39. 39.
    Passarino, G.: An approach toward the numerical evaluation of multiloop Feynman diagrams. Nucl. Phys. B 619, 257 (2001)zbMATHCrossRefMathSciNetGoogle Scholar
  40. 40.
    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)zbMATHGoogle Scholar
  41. 41.
    Shanks, D.: Non-linear transformations of divergent and slowly convergent sequences. J. Math. and Phys. 34, 1–42 (1955)zbMATHMathSciNetGoogle Scholar
  42. 42.
    Sidi, A.: Convergence properties of some nonlinear sequence transformations. Math. Comp. 33, 315–326 (1979)zbMATHCrossRefMathSciNetGoogle Scholar
  43. 43.
    Tarasov, O.V.: An algorithm for small momentum expansion of Feynman diagrams (1995); hep-ph/9505277Google Scholar
  44. 44.
    Tkachov, F.V.: Algebraic algorithms for multiloop calculations: The first 15 years. What’s next? Nucl. Phys. B 389, 309 (1997)Google Scholar
  45. 45.
    Vermaseren, J.A.M.: New features of FORM (2000), math-ph/0010025Google Scholar
  46. 46.
    Wynn, P.: On a device for computing the e m(s n) transformation. Mathematical Tables and Aids to Computing 10, 91–96 (1956)zbMATHCrossRefMathSciNetGoogle Scholar
  47. 47.
    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]Google Scholar
  48. 48.
    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]Google Scholar
  49. 49.
    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:0904.2823Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Elise de Doncker
    • 1
  • Junpei Fujimoto
    • 2
  • Nobuyuki Hamaguchi
    • 4
  • Tadashi Ishikawa
    • 2
  • Yoshimasa Kurihara
    • 2
  • Yoshimitsu Shimizu
    • 3
  • Fukuko Yuasa
    • 2
  1. 1.Department of Computer ScienceWestern Michigan UniversityU.S.A.
  2. 2.High Energy Accelerator Research Organization (KEK)Japan
  3. 3.Graduate University for Advanced Studies, Hayama, Miura-gunKanagawaJapan
  4. 4.Hitachi, Ltd., Software Division, Totsuka-kuYokohamaJapan

Personalised recommendations