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Pentagon functions for massless planar scattering amplitudes

Journal of High Energy Physics volume 2018, Article number: 103 (2018) Cite this article

A preprint version of the article is available at arXiv.

Abstract

Loop amplitudes for massless five particle scattering processes contain Feynman integrals depending on the external momentum invariants: pentagon functions. We perform a detailed study of the analyticity properties and cut structure of these functions up to two loops in the planar case, where we classify and identify the minimal set of basis functions. They are computed from the canonical form of their differential equations and expressed in terms of generalized polylogarithms, or alternatively as one-dimensional integrals. We present analytical expressions and numerical evaluation routines for these pentagon functions, in all kinematical configurations relevant to five-particle scattering processes.

References

  1. R.K. Ellis, Z. Kunszt, K. Melnikov and G. Zanderighi, One-loop calculations in quantum field theory: from Feynman diagrams to unitarity cuts, Phys. Rept. 518 (2012) 141 [arXiv:1105.4319] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  2. H. Ita, Two-loop integrand decomposition into master integrals and surface terms, Phys. Rev. D 94 (2016) 116015 [arXiv:1510.05626] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  3. S. Abreu et al., Subleading poles in the numerical unitarity method at two loops, Phys. Rev. D 95 (2017) 096011 [arXiv:1703.05255] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  4. S. Badger, C. Brønnum-Hansen, H.B. Hartanto and T. Peraro, First look at two-loop five-gluon scattering in QCD, Phys. Rev. Lett. 120 (2018) 092001 [arXiv:1712.02229] [INSPIRE].

    ADS  Article  MATH  Google Scholar 

  5. J. Gluza, K. Kajda and D.A. Kosower, Towards a basis for planar two-loop integrals, Phys. Rev. D 83 (2011) 045012 [arXiv:1009.0472] [INSPIRE].

    ADS  Google Scholar 

  6. R.M. Schabinger, A new algorithm for the generation of unitarity-compatible integration by parts relations, JHEP 01 (2012) 077 [arXiv:1111.4220] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  7. A. von Manteuffel and R.M. Schabinger, A novel approach to integration by parts reduction, Phys. Lett. B 744 (2015) 101 [arXiv:1406.4513] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  8. K.J. Larsen and Y. Zhang, Integration-by-parts reductions from unitarity cuts and algebraic geometry, Phys. Rev. D 93 (2016) 041701 [arXiv:1511.01071] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  9. T. Peraro, Scattering amplitudes over finite fields and multivariate functional reconstruction, JHEP 12 (2016) 030 [arXiv:1608.01902] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  10. D.A. Kosower, Direct solution of integration-by-parts systems, Phys. Rev. D 98 (2018) 025008 [arXiv:1804.00131] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  11. J. Böhm et al., Complete integration-by-parts reductions of the non-planar hexagon-box via module intersections, JHEP 09 (2018) 024 [arXiv:1805.01873] [INSPIRE].

    MathSciNet  Article  Google Scholar 

  12. H.A. Chawdhry, M.A. Lim and A. Mitov, Two-loop five-point massless QCD amplitudes within the IBP approach, arXiv:1805.09182 [INSPIRE].

  13. S. Badger, H. Frellesvig and Y. Zhang, A two-loop five-gluon helicity amplitude in QCD, JHEP 12 (2013) 045 [arXiv:1310.1051] [INSPIRE].

    ADS  Article  Google Scholar 

  14. T. Gehrmann, J.M. Henn and N.A. Lo Presti, Analytic form of the two-loop planar five-gluon all-plus-helicity amplitude in QCD, Phys. Rev. Lett. 116 (2016) 062001 [Erratum ibid. 116 (2016) 189903] [arXiv:1511.05409] [INSPIRE].

  15. S. Badger, G. Mogull, A. Ochirov and D. O’Connell, A complete two-loop, five-gluon helicity amplitude in Yang-Mills theory, JHEP 10 (2015) 064 [arXiv:1507.08797] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  16. D.C. Dunbar and W.B. Perkins, Two-loop five-point all plus helicity Yang-Mills amplitude, Phys. Rev. D 93 (2016) 085029 [arXiv:1603.07514] [INSPIRE].

    ADS  MathSciNet  Google Scholar 

  17. S. Abreu, F. Febres Cordero, H. Ita, B. Page and M. Zeng, Planar two-loop five-gluon amplitudes from numerical unitarity, Phys. Rev. D 97 (2018) 116014 [arXiv:1712.03946] [INSPIRE].

    ADS  Google Scholar 

  18. C.G. Papadopoulos, D. Tommasini and C. Wever, The pentabox master integrals with the simplified differential equations approach, JHEP 04 (2016) 078 [arXiv:1511.09404] [INSPIRE].

    ADS  Google Scholar 

  19. T. Gehrmann and E. Remiddi, Two loop master integrals for γ * → 3 jets: the planar topologies, Nucl. Phys. B 601 (2001) 248 [hep-ph/0008287] [INSPIRE].

  20. T. Gehrmann and E. Remiddi, Two loop master integrals for γ * → 3 jets: the nonplanar topologies, Nucl. Phys. B 601 (2001) 287 [hep-ph/0101124] [INSPIRE].

  21. T. Gehrmann and E. Remiddi, Analytic continuation of massless two loop four point functions, Nucl. Phys. B 640 (2002) 379 [hep-ph/0207020] [INSPIRE].

  22. T. Gehrmann and E. Remiddi, Differential equations for two loop four point functions, Nucl. Phys. B 580 (2000) 485 [hep-ph/9912329] [INSPIRE].

  23. J.M. Henn, Multiloop integrals in dimensional regularization made simple, Phys. Rev. Lett. 110 (2013) 251601 [arXiv:1304.1806] [INSPIRE].

    ADS  Article  Google Scholar 

  24. E. Remiddi and J.A.M. Vermaseren, Harmonic polylogarithms, Int. J. Mod. Phys. A 15 (2000) 725 [hep-ph/9905237] [INSPIRE].

  25. A.B. Goncharov, Multiple polylogarithms and mixed Tate motives, math/0103059 [INSPIRE].

  26. K.-T. Chen, Iterated path integrals, Bull. Am. Math. Soc. 83 (1977) 831.

    MathSciNet  Article  MATH  Google Scholar 

  27. A.B. Goncharov, M. Spradlin, C. Vergu and A. Volovich, Classical polylogarithms for amplitudes and Wilson loops, Phys. Rev. Lett. 105 (2010) 151605 [arXiv:1006.5703] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  28. F. Brown, Iterated integrals in quantum field theory, in the proceedings of the 6th Summer School on Geometric and Topological Methods for Quantum Field Theory, July 6–23, Villa de Leyva, Colombia (2009) [INSPIRE].

  29. C. Duhr, H. Gangl and J.R. Rhodes, From polygons and symbols to polylogarithmic functions, JHEP 10 (2012) 075 [arXiv:1110.0458] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  30. S. Caron-Huot and J.M. Henn, Iterative structure of finite loop integrals, JHEP 06 (2014) 114 [arXiv:1404.2922] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  31. S. Laporta, High precision calculation of multiloop Feynman integrals by difference equations, Int. J. Mod. Phys. A 15 (2000) 5087 [hep-ph/0102033] [INSPIRE].

  32. A.V. Smirnov, Algorithm FIRE — Feynman Integral REduction, JHEP 10 (2008) 107 [arXiv:0807.3243] [INSPIRE].

    ADS  Article  MATH  Google Scholar 

  33. A. von Manteuffel and C. Studerus, Reduze 2 — Distributed Feynman integral reduction, arXiv:1201.4330 [INSPIRE].

  34. T. Gehrmann and E. Remiddi, Numerical evaluation of harmonic polylogarithms, Comput. Phys. Commun. 141 (2001) 296 [hep-ph/0107173] [INSPIRE].

  35. T. Gehrmann and E. Remiddi, Numerical evaluation of two-dimensional harmonic polylogarithms, Comput. Phys. Commun. 144 (2002) 200 [hep-ph/0111255] [INSPIRE].

  36. J. Vollinga and S. Weinzierl, Numerical evaluation of multiple polylogarithms, Comput. Phys. Commun. 167 (2005) 177 [hep-ph/0410259] [INSPIRE].

  37. N. Arkani-Hamed, J.L. Bourjaily, F. Cachazo and J. Trnka, Local integrals for planar scattering amplitudes, JHEP 06 (2012) 125 [arXiv:1012.6032] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  38. P. Wasser, Analytic properties of Feynman integrals for scattering amplitudes, M.Sc. thesis, Johannes Gutenberg-Universität Mainz, Mainz, Germany (2016).

  39. D. Chicherin et al., Analytic result for the nonplanar hexa-box integrals, arXiv:1809.06240 [INSPIRE].

  40. J.M. Henn, A.V. Smirnov and V.A. Smirnov, Analytic results for planar three-loop four-point integrals from a Knizhnik-Zamolodchikov equation, JHEP 07 (2013) 128 [arXiv:1306.2799] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  41. J.M. Henn, Lectures on differential equations for Feynman integrals, J. Phys. A 48 (2015) 153001 [arXiv:1412.2296] [INSPIRE].

    ADS  MathSciNet  MATH  Google Scholar 

  42. D. Chicherin, J. Henn and V. Mitev, Bootstrapping pentagon functions, JHEP 05 (2018) 164 [arXiv:1712.09610] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  43. J. Henn, E. Herrmann and J. Parra-Martinez, Bootstrapping two-loop Feynman integrals for planar N = 4 SYM, arXiv:1806.06072 [INSPIRE].

  44. R. Britto, F. Cachazo, B. Feng and E. Witten, Direct proof of tree-level recursion relation in Yang-Mills theory, Phys. Rev. Lett. 94 (2005) 181602 [hep-th/0501052] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  45. Z. Bern, L.J. Dixon and D.A. Kosower, One loop corrections to five gluon amplitudes, Phys. Rev. Lett. 70 (1993) 2677 [hep-ph/9302280] [INSPIRE].

  46. J.M. Henn, K. Melnikov and V.A. Smirnov, Two-loop planar master integrals for the production of off-shell vector bosons in hadron collisions, JHEP 05 (2014) 090 [arXiv:1402.7078] [INSPIRE].

    ADS  Article  Google Scholar 

  47. A.V. Smirnov, FIESTA4: Optimized Feynman integral calculations with GPU support, Comput. Phys. Commun. 204 (2016) 189 [arXiv:1511.03614] [INSPIRE].

    ADS  Article  MATH  Google Scholar 

  48. S. Badger et al., Applications of integrand reduction to two-loop five-point scattering amplitudes in QCD, PoS(LL2018)006.

  49. S. Catani, The singular behavior of QCD amplitudes at two loop order, Phys. Lett. B 427 (1998) 161 [hep-ph/9802439] [INSPIRE].

  50. S.M. Aybat, L.J. Dixon and G.F. Sterman, The two-loop soft anomalous dimension matrix and resummation at next-to-next-to leading pole, Phys. Rev. D 74 (2006) 074004 [hep-ph/0607309] [INSPIRE].

  51. S. Borowka et al., SecDec-3.0: numerical evaluation of multi-scale integrals beyond one loop, Comput. Phys. Commun. 196 (2015) 470 [arXiv:1502.06595] [INSPIRE].

  52. C.W. Bauer, A. Frink and R. Kreckel, Introduction to the GiNaC framework for symbolic computation within the C++ programming language, J. Symb. Comput. 33 (2000) 1 [cs/0004015] [INSPIRE].

  53. B. Gough, GNU Scientific Library Reference Manual, Network Theory Ltd., U.K. (2009).

    Google Scholar 

  54. R. Brüser, S. Caron-Huot and J.M. Henn, Subleading Regge limit from a soft anomalous dimension, JHEP 04 (2018) 047 [arXiv:1802.02524] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  55. D. Chicherin, J.M. Henn and E. Sokatchev, Amplitudes from superconformal ward identities, Phys. Rev. Lett. 121 (2018) 021602 [arXiv:1804.03571] [INSPIRE].

    ADS  Article  Google Scholar 

  56. D. Chicherin, J.M. Henn and E. Sokatchev, Implications of nonplanar dual conformal symmetry, JHEP 09 (2018) 012 [arXiv:1807.06321] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  57. A. Hodges, Eliminating spurious poles from gauge-theoretic amplitudes, JHEP 05 (2013) 135 [arXiv:0905.1473] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  58. L.J. Mason and D. Skinner, Dual superconformal invariance, momentum twistors and grassmannians, JHEP 11 (2009) 045 [arXiv:0909.0250] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  59. F. Cachazo, S. He and E.Y. Yuan, Scattering of massless particles in arbitrary dimensions, Phys. Rev. Lett. 113 (2014) 171601 [arXiv:1307.2199] [INSPIRE].

    ADS  Article  Google Scholar 

  60. S. Weinzierl, On the solutions of the scattering equations, JHEP 04 (2014) 092 [arXiv:1402.2516] [INSPIRE].

    ADS  Article  Google Scholar 

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Authors and Affiliations

  1. Physik-Institut, Universität Zürich, Wintherturerstrasse 190, CH-8057, Zürich, Switzerland

    T. Gehrmann

  2. PRISMA Cluster of Excellence, Institute of Physics, Johannes Gutenberg University, D-55099, Mainz, Germany

    J. M. Henn

  3. MPI für Physik, Werner-Heisenberg-Institut, München, Germany

    J. M. Henn

  4. Institute for Particle Physics Phenomenology, Durham University, Durham, DH1 3LE, U.K.

    N. A. Lo Presti

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  1. T. Gehrmann
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Correspondence to T. Gehrmann.

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Gehrmann, T., Henn, J.M. & Lo Presti, N.A. Pentagon functions for massless planar scattering amplitudes. J. High Energ. Phys. 2018, 103 (2018). https://doi.org/10.1007/JHEP10(2018)103

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Keywords

  • Perturbative QCD
  • Scattering Amplitudes