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Self-energy Feynman Diagrams with Four Loops and 11 Internal Lines

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Computational Science and Its Applications – ICCSA 2021 (ICCSA 2021)

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Abstract

We consider 4-loop Feynman diagrams with 11 internal lines. The associated 10-dimensional loop integrals are calculated for four diagrams with massive internal lines, and we further handle the massless case of the diagram referenced in the literature as M61. The computations are performed with double exponential (DE), Quasi-Monte Carlo (lattice and embedded lattice rules) and adaptive integration algorithms, which do not require any user input regarding the integrand behavior. The lattice rule methods are combined with a transformation to help alleviate boundary singularities. The embedded lattice rules are implemented in CUDA C and their execution is accelerated using an NVIDIA Quadro GV100 GPU, whereas DE is parallelized over MPI and executed on an AMD cluster. Adaptive integration is performed with the ParInt multivariate integration package, which is also layered over MPI. For the massless M61 diagram we use a dimensional regularization approach and extrapolation. The results will be compared with respect to accuracy and efficiency, and verified with pySecDec.

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Acknowledgments

We acknowledge the support of the Grant-in-Aid for Scientific Research (JP17K05428 and JP20K03941) from JSPS KAKENHI, as well as the National Science Foundation Award Number 1126438 that funded the cluster used for the computations with ParInt in this paper. We also thank our colleagues, Dr. T. Ishikawa, Dr. H. Daisaka, Dr. N. Nakasato and Dr. J. Kapenga, for creating and maintaining computing environments supporting our Feynman integration work.

Author information

Authors and Affiliations

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

    Elise de Doncker

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

    Fukuko Yuasa

Authors
  1. Elise de Doncker
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  2. Fukuko Yuasa
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Corresponding author

Correspondence to Elise de Doncker .

Editor information

Editors and Affiliations

  1. University of Perugia, Perugia, Italy

    Osvaldo Gervasi

  2. University of Basilicata, Potenza, Potenza, Italy

    Beniamino Murgante

  3. Covenant University, Ota, Nigeria

    Sanjay Misra

  4. University of Cagliari, Cagliari, Italy

    Chiara Garau

  5. University of Cagliari, Cagliari, Italy

    Ivan Blečić

  6. Monash University, Clayton, VIC, Australia

    David Taniar

  7. Kyushu Sangyo University, Fukuoka, Japan

    Bernady O. Apduhan

  8. University of Minho, Braga, Portugal

    Ana Maria A. C. Rocha

  9. Polytechnic University of Bari, Bari, Italy

    Eufemia Tarantino

  10. Polytechnic University of Bari, Bari, Italy

    Carmelo Maria Torre

A C and D functions

A C and D functions

The C and D functions for the diagrams of Fig. 1 are given in C code form below. We denote

$$\begin{aligned}&x_{i_1\ldots i_k} = \sum _{j = i_1}^{i_k} x_j \end{aligned}$$
(20)

We use \(x_a = x_{10}\) and \(x_b = x_{11}.\) Obvious notations also include \(xksq = x_k^2.\)

1.1 A.1 M61,  Fig. 1(a).

figure a

1.2 A.2 M62,  Fig. 1(b).

figure b

1.3 A.3 M63,  Fig. 1(c).

figure c

1.4 A.4 BH, Fig. 1(d).

figure d

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de Doncker, E., Yuasa, F. (2021). Self-energy Feynman Diagrams with Four Loops and 11 Internal Lines. In: Gervasi, O., et al. Computational Science and Its Applications – ICCSA 2021. ICCSA 2021. Lecture Notes in Computer Science(), vol 12953. Springer, Cham. https://doi.org/10.1007/978-3-030-86976-2_11

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  • DOI: https://doi.org/10.1007/978-3-030-86976-2_11

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