Using neural networks for efficient evaluation of high multiplicity scattering amplitudes

Abstract

Precision theoretical predictions for high multiplicity scattering rely on the evaluation of increasingly complicated scattering amplitudes which come with an extremely high CPU cost. For state-of-the-art processes this can cause technical bottlenecks in the production of fully differential distributions. In this article we explore the possibility of using neural networks to approximate multi-variable scattering amplitudes and provide efficient inputs for Monte Carlo integration. We focus on QCD corrections to e+e jets up to one-loop and up to five jets. We demonstrate reliable interpolation when a series of networks are trained to amplitudes that have been divided into sectors defined by their infrared singularity structure. Complete simulations for one-loop distributions show speed improvements of at least an order of magnitude over a standard approach.

A preprint version of the article is available at ArXiv.

References

  1. [1]

    M. Czakon, Tops from light quarks: full mass dependence at two-loops in QCD, Phys. Lett.B 664 (2008) 307 [arXiv:0803.1400] [INSPIRE].

    ADS  Article  Google Scholar 

  2. [2]

    S. Borowka et al., Higgs boson pair production in gluon fusion at next-to-leading order with full top-quark mass dependence, Phys. Rev. Lett.117 (2016) 012001 [Erratum ibid.117 (2016) 079901] [arXiv:1604.06447] [INSPIRE].

  3. [3]

    G. Heinrich et al., NLO predictions for Higgs boson pair production with full top quark mass dependence matched to parton showers, JHEP08 (2017) 088 [arXiv:1703.09252] [INSPIRE].

    ADS  Article  Google Scholar 

  4. [4]

    S.P. Jones, M. Kerner and G. Luisoni, Next-to-leading-order QCD corrections to Higgs boson plus jet production with full top-quark mass dependence, Phys. Rev. Lett.120 (2018) 162001 [arXiv:1802.00349] [INSPIRE].

    ADS  Article  Google Scholar 

  5. [5]

    G. Heinrich et al., Probing the trilinear Higgs boson coupling in di-Higgs production at NLO QCD including parton shower effects, JHEP06 (2019) 066 [arXiv:1903.08137] [INSPIRE].

    ADS  Article  Google Scholar 

  6. [6]

    G. Cybenko, Approximation by superpositions of a sigmoidal function, Math. Control. Sign. Syst.2 (1989) 303.

    MathSciNet  Article  Google Scholar 

  7. [7]

    S. Badger, B. Biedermann, P. Uwer and V. Yundin, Numerical evaluation of virtual corrections to multi-jet production in massless QCD, Comput. Phys. Commun.184 (2013) 1981 [arXiv:1209.0100] [INSPIRE].

    ADS  Article  Google Scholar 

  8. [8]

    J. Bendavid, Efficient Monte Carlo integration using boosted decision trees and generative deep neural networks, arXiv:1707.00028 [INSPIRE].

  9. [9]

    M.D. Klimek and M. Perelstein, Neural network-based approach to phase space integration, arXiv:1810.11509 [INSPIRE].

  10. [10]

    E. Bothmann et al., Exploring phase space with neural importance sampling, SciPost Phys.8 (2020) 069 [arXiv:2001.05478] [INSPIRE].

    ADS  Article  Google Scholar 

  11. [11]

    C. Gao et al., Event generation with normalizing flows, Phys. Rev.D 101 (2020) 076002 [arXiv:2001.10028] [INSPIRE].

    ADS  Google Scholar 

  12. [12]

    L. Dinh, D. Krueger and Y. Bengio, NICE: Non-linear independent components estimation, in the proceedings of 3rdInternational Conference on Learning Representations (ICLR 2015), May 7–9, San Diega, U.S.A. (2015).

  13. [13]

    S. Otten et al., DeepXS: Fast approximation of MSSM electroweak cross sections at NLO, Eur. Phys. J.C 80 (2020) 12 [arXiv:1810.08312] [INSPIRE].

    ADS  Article  Google Scholar 

  14. [14]

    I. Goodfellow et al., Generative adversarial nets, in Advances in neural information processing systems 27 , Z. Ghahramani et al. eds., Curran Associates Inc., U.S.A. (2014),

  15. [15]

    S. Otten et al., Event generation and statistical sampling for physics with deep generative models and a density information buffer, arXiv:1901.00875 [INSPIRE].

  16. [16]

    B. Hashemi et al., LHC analysis-specific datasets with generative adversarial networks, arXiv:1901.05282 [INSPIRE].

  17. [17]

    R. Di Sipio, M. Faucci Giannelli, S. Ketabchi Haghighat and S. Palazzo, DijetGAN: a generative-adversarial network approach for the simulation of QCD dijet events at the LHC, JHEP08 (2020) 110 [arXiv:1903.02433] [INSPIRE].

    Google Scholar 

  18. [18]

    A. Butter, T. Plehn and R. Winterhalder, How to GAN LHC events, SciPost Phys.7 (2019) 075 [arXiv:1907.03764] [INSPIRE].

    ADS  Article  Google Scholar 

  19. [19]

    A. Butter, T. Plehn and R. Winterhalder, How to GAN event subtraction, arXiv:1912.08824 [INSPIRE].

  20. [20]

    S. Carrazza and F.A. Dreyer, Lund jet images from generative and cycle-consistent adversarial networks, Eur. Phys. J.C 79 (2019) 979 [arXiv:1909.01359] [INSPIRE].

    ADS  Article  Google Scholar 

  21. [21]

    SHiP collaboration, Fast simulation of muons produced at the SHiP experiment using generative adversarial networks, 2019 JINST14 P11028 [arXiv:1909.04451] [INSPIRE].

  22. [22]

    F. Bishara and M. Montull, (Machine) learning amplitudes for faster event generation, arXiv:1912.11055 [INSPIRE].

  23. [23]

    J. Bullock, n3jet, https://github.com/JosephPB/n3jet, (2020).

  24. [24]

    G. Ossola, C.G. Papadopoulos and R. Pittau, Reducing full one-loop amplitudes to scalar integrals at the integrand level, Nucl. Phys.B 763 (2007) 147 [hep-ph/0609007] [INSPIRE].

  25. [25]

    Z. Bern, L.J. Dixon, D.C. Dunbar and D.A. Kosower, Fusing gauge theory tree amplitudes into loop amplitudes, Nucl. Phys.B 435 (1995) 59 [hep-ph/9409265] [INSPIRE].

  26. [26]

    R. Britto, F. Cachazo and B. Feng, Generalized unitarity and one-loop amplitudes in N = 4 super-Yang-Mills, Nucl. Phys.B 725 (2005) 275 [hep-th/0412103] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  27. [27]

    R.K. Ellis, W.T. Giele and Z. Kunszt, A numerical unitarity formalism for evaluating one-loop amplitudes, JHEP03 (2008) 003 [arXiv:0708.2398] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  28. [28]

    W.T. Giele, Z. Kunszt and K. Melnikov, Full one-loop amplitudes from tree amplitudes, JHEP04 (2008) 049 [arXiv:0801.2237] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  29. [29]

    D. Forde, Direct extraction of one-loop integral coefficients, Phys. Rev.D 75 (2007) 125019 [arXiv:0704.1835] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  30. [30]

    C.F. Berger et al., An automated implementation of on-shell methods for one-loop amplitudes, Phys. Rev.D 78 (2008) 036003 [arXiv:0803.4180] [INSPIRE].

    ADS  Google Scholar 

  31. [31]

    S.D. Badger, Direct extraction of one loop rational terms, JHEP01 (2009) 049 [arXiv:0806.4600] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  32. [32]

    F.A. Berends and W.T. Giele, Recursive calculations for processes with n gluons, Nucl. Phys.B 306 (1988) 759 [INSPIRE].

    ADS  Article  Google Scholar 

  33. [33]

    T. Binoth et al., A proposal for a standard interface between Monte Carlo tools and one-loop programs, Comput. Phys. Commun.181 (2010) 1612 [arXiv:1001.1307] [INSPIRE].

    ADS  Article  Google Scholar 

  34. [34]

    S. Frixione, Z. Kunszt and A. Signer, Three jet cross-sections to next-to-leading order, Nucl. Phys.B 467 (1996) 399 [hep-ph/9512328] [INSPIRE].

  35. [35]

    R. Frederix, S. Frixione, F. Maltoni and T. Stelzer, Automation of next-to-leading order computations in QCD: the FKS subtraction, JHEP10 (2009) 003 [arXiv:0908.4272] [INSPIRE].

    ADS  Article  Google Scholar 

  36. [36]

    M. Czakon and D. Heymes, Four-dimensional formulation of the sector-improved residue subtraction scheme, Nucl. Phys.B 890 (2014) 152 [arXiv:1408.2500] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  37. [37]

    JADE collaboration, Experimental studies on multi-jet production in e+eannihilation at PETRA energies, Z. Phys.C 33 (1986) 23 [INSPIRE].

  38. [38]

    R. Kleiss, W.J. Stirling and S.D. Ellis, A new Monte Carlo treatment of multiparticle phase space at high-energies, Comput. Phys. Commun.40 (1986) 359 [INSPIRE].

    ADS  Article  Google Scholar 

  39. [39]

    J.H. Friedman, M.H. Wright, An adaptive importance sampling procedure, Stanford University, U.S.A. (1981).

    Google Scholar 

  40. [40]

    G.P. Lepage, A new algorithm for adaptive multidimensional integration, J. Comput. Phys.27 (1978) 192 [INSPIRE].

    ADS  Article  Google Scholar 

  41. [41]

    G.P. Lepage, VEGAS: an adaptive multidimensional integration program, CLNS-80/447 (1980).

  42. [42]

    W.H. Press and G.R. Farrar, Recursive stratified sampling for multidimensional Monte Carlo integration, Comp. Phys.190 (1990) 4.

  43. [43]

    T. Ohl, Vegas revisited: adaptive Monte Carlo integration beyond factorization, Comput. Phys. Commun.120 (1999) 13 [hep-ph/9806432] [INSPIRE].

  44. [44]

    S. Jadach, Foam: a general purpose cellular Monte Carlo event generator, Comput. Phys. Commun.152 (2003) 55 [physics/0203033] [INSPIRE].

  45. [45]

    K. Kroeninger, S. Schumann and B. Willenberg, (M C )3– a Multi-Channel Markov Chain Monte Carlo algorithm for phase-space sampling, Comput. Phys. Commun.186 (2015) 1 [arXiv:1404.4328] [INSPIRE].

  46. [46]

    P.D. Draggiotis, A. van Hameren and R. Kleiss, SARGE: an algorithm for generating QCD antennas, Phys. Lett.B 483 (2000) 124 [hep-ph/0004047] [INSPIRE].

  47. [47]

    A. van Hameren and C.G. Papadopoulos, A hierarchical phase space generator for QCD antenna structures, Eur. Phys. J.C 25 (2002) 563 [hep-ph/0204055] [INSPIRE].

  48. [48]

    R. Frederix, S. Frixione, K. Melnikov and G. Zanderighi, NLO QCD corrections to five-jet production at LEP and the extraction of αs (MZ), JHEP11 (2010) 050 [arXiv:1008.5313] [INSPIRE].

    ADS  Article  Google Scholar 

  49. [49]

    F. Chollet et al., Keras, https://github.com/fchollet/keras (2015).

  50. [50]

    M. Abadi et al., TensorFlow: large-scale machine learning on heterogeneous systems, https://www.tensorflow.org/ (2015).

  51. [51]

    D.P. Kingma and J. Ba, Adam: a method for stochastic optimization, arXiv:1412.6980 [INSPIRE].

  52. [52]

    I. Goodfellow, Y. Bengio and A. Courville, Deep learning. MIT Press, U.S.A. (2016).

  53. [53]

    N. Tagasovska and D. Lopez-Paz, Single-model uncertainties for deep learning, NeurlPS (2019) [arXiv:1811.00908].

  54. [54]

    Y. Gal, Uncertainty in deep learning, Ph.D. thesis, University of Cambridge, Cambridge U.K. (2016).

  55. [55]

    B. Nachman, A guide for deploying Deep Learning in LHC searches: how to achieve optimality and account for uncertainty, arXiv:1909.03081 [INSPIRE].

  56. [56]

    B. Nachman and C. Shimmin, AI safety for high energy physics, arXiv:1910.08606 [INSPIRE].

  57. [57]

    S. Bollweg et al., Deep-learning jets with uncertainties and more, SciPost Phys.8 (2020) 006 [arXiv:1904.10004] [INSPIRE].

    ADS  Article  Google Scholar 

  58. [58]

    C. Englert, P. Galler, P. Harris and M. Spannowsky, Machine learning uncertainties with adversarial neural networks, Eur. Phys. J.C 79 (2019) 4 [arXiv:1807.08763] [INSPIRE].

    ADS  Article  Google Scholar 

  59. [59]

    K. Cranmer, J. Pavez and G. Louppe, Approximating likelihood ratios with calibrated discriminative classifiers, arXiv:1506.02169 [INSPIRE].

  60. [60]

    S. Frixione, E. Laenen, P. Motylinski and B.R. Webber, Single-top production in MC@NLO, JHEP03 (2006) 092 [hep-ph/0512250] [INSPIRE].

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Badger, S., Bullock, J. Using neural networks for efficient evaluation of high multiplicity scattering amplitudes. J. High Energ. Phys. 2020, 114 (2020). https://doi.org/10.1007/JHEP06(2020)114

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Keywords

  • NLO Computations
  • QCD Phenomenology