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Large Scale Automatic Computations for Feynman Diagrams with up to Five Loops

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Computational Science and Its Applications – ICCSA 2020 (ICCSA 2020)

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Abstract

We give results by automatic integration methods for finite and UV-divergent 4-loop diagrams and a finite 5-loop case with massless internal lines. Non-adaptive methods include DE (Double Exponential), and Quasi-Monte Carlo (QMC) techniques. The latter are based on optimal lattice rules, implemented in Cuda-C for GPUs; or, for execution on PEZY/ Exascaler, the host program is written in C++ and the kernel is generated using the Goose compiler interface. DE is executed on similar hardware as QMC, with or without parallel libraries for MPI. Transformations are incorporated to alleviate or smoothen singularities on the boundaries of the domain. For adaptive integration we use the ParInt package layered over MPI on a cluster, as well as a new adaptive scheme that performs GPU evaluations of the cubature rules. For the UV-divergent diagram we apply a nonlinear extrapolation on a sequence of integral approximations generated for dimensional regularization. Some results are verified using computationally intensive symbolic/numerical evaluations with pySecDec.

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Notes

  1. 1.

    For those interested in English literature, “42” is the answer to the Ultimate Question of Life, the Universe and Everything, cf., Douglas Adams, “The Hitchhiker’s Guide to the Galaxy”  [1].

References

  1. Adams, D.: The Hitchhiker’s Guide to the Galaxy. Pocket Books, Simon & Shuster, Inc. (1981). ISBN 0-671-52721-5

    Google Scholar 

  2. Berntsen, J., Espelid, T.O., Genz, A.: An adaptive algorithm for the approximate calculation of multiple integrals. ACM Trans. Math. Softw. 17, 437–451 (1991)

    Article  MathSciNet  Google Scholar 

  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)

    Article  MathSciNet  Google Scholar 

  4. Binoth, T., Heinrich, G.: Numerical evaluation of multi-loop integrals by sector decomposition. Nucl. Phys. B 680, 375 (2004). hep-ph/0305234v1

    Article  Google Scholar 

  5. Borowka, S., Heinrich, G., Jahn, S., Jones, S.P., Kerner, M., Schlenk, J.: A GPU compatible quasi-Monte Carlo integrator interfaced to pySecDec. Comput. Phys. Commun. 240, 120–137 (2019). Preprint: arXiv:1811.11720v1 [hep-ph]. https://arxiv.org/abs/1811.11720. https://doi.org/10.1016/j.cpc.2019.02.015

  6. Cranley, R., Patterson, T.N.L.: Randomization of number theoretic methods for multiple integration. SIAM J. Numer. Anal. 13, 904–914 (1976)

    Article  MathSciNet  Google Scholar 

  7. De Ridder, L., Van Dooren, P.: An adaptive algorithm for numerical integration over an n-dimensional cube. J. Comput. Appl. Math. 2(3), 207–210 (1976)

    Article  Google Scholar 

  8. de Doncker, E., Almulihi, A., Yuasa, F.: High speed evaluation of loop integrals using lattice rules. J. Phys.: Conf. Series (JPCS) IOP Series 1085, 052005 (2018). http://iopscience.iop.org/article/10.1088/1742-6596/1085/5/052005

  9. de Doncker, E., Almulihi, A., Yuasa, F.: Transformed lattice rules for Feynman loop integrals. J. Phys.: Conf. Series (JPCS) IOP Series 1136, 012002 (2018). https://doi.org/10.1088/1742-6596/1136/1/012002

  10. de Doncker, E., Genz, A., Gupta, A., Zanny, R.: Tools for distributed adaptive multivariate integration on NOW’s: PARINT1.0 release. In: Supercomputing 1998 (1998)

    Google Scholar 

  11. de Doncker, E., Yuasa, F., Almulihi, A.: Efficient GPU integration for multi-loop Feynman diagrams with massless internal lines. In: Okada, H., Atluri, S.N. (eds.) ICCES 2019. MMS, vol. 75, pp. 737–747. Springer, Cham (2020). https://doi.org/10.1007/978-3-030-27053-7_62

    Chapter  Google Scholar 

  12. de Doncker, E., Yuasa, F., Almulihi, A., Nakasato, N., Daisaka, H., Ishikawa, T.: Numerical multi-loop integration on heterogeneous many-core processors. In: The Journal of Physics: Conference Series (JPCS), IOP Series (ACAT 2019) (2019, to appear)

    Google Scholar 

  13. de Doncker, E., Yuasa, F., Kato, K., Ishikawa, T., Kapenga, J., Olagbemi, O.: Regularization with numerical extrapolation for finite and UV-divergent multi-loop integrals. Comput. Phys. Commun. 224, 164–185 (2018). https://doi.org/10.1016/j.cpc.2017.11.001

    Article  MathSciNet  Google Scholar 

  14. Genz, A., Malik, A.: An adaptive algorithm for numerical integration over an n-dimensional rectangular region. J. Comput. Appl. Math. 6, 295–302 (1980)

    Article  Google Scholar 

  15. Goose - GRAPE9-MPx - for goose version 1.5.0: K & F Computing Research Co. (2014). (in Japanese)

    Google Scholar 

  16. Hahn, T.: Cuba - a library for multidimensional numerical integration. Comput. Phys. Commun. 176, 712–713 (2007). https://doi.org/10.1016/j.cpc.2007.03.006

    Article  MATH  Google Scholar 

  17. Hahn, T., Pérez-Victoria, M.: Automated one-loop calculations in four and D dimensions. Comput. Phys. Commun. 118(2–3), 153–165 (1999). hep-ph/9807565

    Article  Google Scholar 

  18. ’t Hooft, G., Veltman, M.: Scalar one-loop integrals. Nucl. Phys. B 153, 365–401 (1979)

    Google Scholar 

  19. Ishikawa, T., Nakazawa, N., Yasui, Y.: Numerical calculation of the full two-loop electroweak corrections to muon (g-2). Phys. Rev. D 99, 073004 (2019)

    Google Scholar 

  20. Korobov, N.M.: The approximate computation of multiple integrals. Dokl. Akad. Nauk SSSR 124, 1207–1210 (1959). (Russian)

    MathSciNet  MATH  Google Scholar 

  21. Korobov, N.M.: Properties and calculation of optimal coefficients. Doklady Akademii Nauk SSSR 132, 1009–1012 (1960). Russ. Eng. trans. Soviet Math. Doklady 1, 696–700

    Google Scholar 

  22. L’Equyer, P., Munger, D.: Algorithm 958: lattice builder: a general software tool for constructing rank-1 lattice rules. ACM Trans. Math. Softw. 42(2), 15:1–15:30 (2016)

    Google Scholar 

  23. MPI. http://www-unix.mcs.anl.gov/mpi/index.html

  24. Niederreiter, H.: Existence of good lattice points in the sense of hlawka. Monatshefte für Mathematik 86, 203–219 (1978)

    Article  MathSciNet  Google Scholar 

  25. Nuyens, D., Cools, R.: Fast algorithms for component-by-component construction of rank-1 lattice rules in shift-invariant reproducing kernel Hilbert spaces. Math. Comput. 75, 903–920 (2006)

    Article  MathSciNet  Google Scholar 

  26. Nuyens, D., Cools, R.: Fast component-by-component construction of rank-1 lattice rules with a non-prime number of points. J. Complex. 22, 4–28 (2006)

    Article  MathSciNet  Google Scholar 

  27. Olagbemi, O.E.: Scalable algorithms and hybrid parallelization strategies for multivariate integration with ParAdapt and CUDA. Ph.D. thesis, Western Michigan University (2019)

    Google Scholar 

  28. Olagbemi, O.E., de Doncker, E.: Scalable algorithms for multivariate integration with ParAdapt and CUDA. In: Proceedings of 2019 International Conference on Computational Science and Computational Intelligence. IEEE Computer Society (2019). https://american-cse.org/csci2019/pdfs/CSCI2019-14dQVW1stBtXVEInMQPd3t/558400a481/558400a481.pdf

  29. PEZY Computing/Exascaler Inc. http://www.exascaler.co.jp/

  30. Piessens, R., de Doncker, E., Überhuber, C.W., Kahaner, D.K.: QUADPACK. A Subroutine Package for Automatic Integration. Springer Series in Computational Mathematics, vol. 1. Springer, Heidelberg (1983). https://doi.org/10.1007/978-3-642-61786-7

    Book  MATH  Google Scholar 

  31. Rice, J.R.: A metalgorithm for adaptive quadrature. J. Assoc. Comput. Mach. 22, 61–82 (1975)

    Article  MathSciNet  Google Scholar 

  32. Ruijl, B., Herzog, F., Ueda, T., Vermaseren, J.A.M., Vogt, A.: The \({R}^*\)-operation and five loops calculations. In: 13th International Symposium on Radiative Corrections (2017)

    Google Scholar 

  33. Ruijl, B., Herzog, F., Ueda, T., Vermaseren, J.A.M., Vogt, A.: The \({R}^*\)-operation and combinatorial challenges at five loops. J. Phys.: Conf. Series (JPCS) IOP Series 1085, 052006 (2018). https://doi.org/10.1088/1742-6596/1085/5/052006

  34. Ruijl, B., Ueda, T., Vermaseren, J.A.M.: Forcer, a FORM program for the parametric reduction of four-loop massless propagator diagrams. Technical report, 2017-019, Nikhef (2017). arXiv:1704.06650 [hep-ph]

  35. Sanders, J., Kandrot, E.: CUDA by Example: An Introduction to General-Purpose GPU Programming. Pearson, London (2010)

    Google Scholar 

  36. Sidi, A.: A new variable transformation for numerical integration. Int. Ser. Numer. Math. 112, 359–373 (1993)

    MathSciNet  MATH  Google Scholar 

  37. Sloan, I., Joe, S.: Lattice Methods for Multiple Integration. Oxford University Press, Oxford (1994)

    MATH  Google Scholar 

  38. Sugihara, M.: Optimality of the double exponential formula - functional analysis approach. Numer. Math. 75(3), 379–395 (1997)

    Article  MathSciNet  Google Scholar 

  39. Takahasi, H., Mori, M.: Double exponential formulas for numerical integration. Publ. Res. Inst. Math. Sci. 9(3), 721–741 (1974)

    Article  MathSciNet  Google Scholar 

  40. Torii, S., Ishikawa, H.: ZettaScaler: liquid immersion cooling manycore based supercomputer. Technical report, ExaScaler Inc., PEZY Computing K. K. (2017)

    Google Scholar 

  41. Wynn, P.: On a device for computing the \(e_m(s_n)\) transformation. Math. Tables Aids Comput. 10, 91–96 (1956)

    Article  MathSciNet  Google Scholar 

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Acknowledgments

We acknowledge the support from the National Science Foundation under Award Number 1126438 that funded the cluster used for the computations with ParInt and ParAdapt in this paper. Furthermore we rely on the Grant-in-Aid for Scientific Research (17K05428) of JSPS, and on partial support by the Large Scale Computational Sciences with Heterogeneous Many-Core Computers, Grant-in-Aid for High Performance Computing with General Purpose Computers from MEXT (Ministry of Education, Culture, Sports, Science and Technology-Japan). We also sincerely thank the reviewers of this paper for their valuable comments.

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Authors and Affiliations

  1. Western Michigan University, Kalamazoo, MI, 49008, USA

    E. de Doncker & O. Olagbemi

  2. High Energy Accelerator Research Organization (KEK), 1-1 OHO, Tsukuba, Ibaraki, 305-0801, Japan

    F. Yuasa & T. Ishikawa

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  1. E. de Doncker
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  2. F. Yuasa
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  3. O. Olagbemi
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  4. T. Ishikawa
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Correspondence to E. de Doncker .

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  1. University of Perugia, Perugia, Italy

    Osvaldo Gervasi

  2. University of Basilicata, Potenza, Potenza, Italy

    Beniamino Murgante

  3. Chair- Center of ICT/ICE, Covenant University, Ota, Nigeria

    Sanjay Misra

  4. University of Cagliari, Cagliari, Italy

    Chiara Garau

  5. University of Cagliari, Cagliari, Italy

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  6. Clayton School of Information Technology, Monash University, Clayton, VIC, Australia

    David Taniar

  7. Department of Information Science, Kyushu Sangyo University, Fukuoka, Japan

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  8. University of Minho, Braga, Portugal

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  9. Polytechnic University of Bari, Bari, Italy

    Eufemia Tarantino

  10. Polytechnic University of Bari, Bari, Italy

    Carmelo Maria Torre

  11. Department of Neurology, University of Massachusetts Medical School, Worcester, MA, USA

    Yeliz Karaca

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de Doncker, E., Yuasa, F., Olagbemi, O., Ishikawa, T. (2020). Large Scale Automatic Computations for Feynman Diagrams with up to Five Loops. In: Gervasi, O., et al. Computational Science and Its Applications – ICCSA 2020. ICCSA 2020. Lecture Notes in Computer Science(), vol 12253. Springer, Cham. https://doi.org/10.1007/978-3-030-58814-4_11

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  • DOI: https://doi.org/10.1007/978-3-030-58814-4_11

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