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Bootstrapping pentagon functions

A preprint version of the article is available at arXiv.


In Phys. Rev. Lett. 116 (2016) 062001, the space of planar pentagon functions that describes all two-loop on-shell five-particle scattering amplitudes was introduced. In the present paper we present a natural extension of this space to non-planar pentagon functions. This provides the basis for our pentagon bootstrap program. We classify the relevant functions up to weight four, which is relevant for two-loop scattering amplitudes. We constrain the first entry of the symbol of the functions using information on branch cuts. Drawing on an analogy from the planar case, we introduce a conjectural second-entry condition on the symbol. We then show that the information on the function space, when complemented with some additional insights, can be used to efficiently bootstrap individual Feynman integrals. The extra information is read off of Mellin-Barnes representations of the integrals, either by evaluating simple asymptotic limits, or by taking discontinuities in the kinematic variables. We use this method to evaluate the symbols of two non-trivial non-planar five-particle integrals, up to and including the finite part.


  1. R.J. Eden, P.V. Landshoff, D.I. Olive and J.C. Polkinghorne, The analytic S-matrix, Cambridge University Press, (1966).

  2. Z. Bern, L.J. Dixon, D.C. Dunbar and D.A. Kosower, One loop n point gauge theory amplitudes, unitarity and collinear limits, Nucl. Phys. B 425 (1994) 217 [hep-ph/9403226] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  3. 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 

  4. 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  Google Scholar 

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

    ADS  Article  Google Scholar 

  6. L.J. Dixon, J.M. Drummond and J.M. Henn, Bootstrapping the three-loop hexagon, JHEP 11 (2011) 023 [arXiv:1108.4461] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  7. L.J. Dixon, J.M. Drummond and J.M. Henn, Analytic result for the two-loop six-point NMHV amplitude in N = 4 super Yang-Mills theory, JHEP 01 (2012) 024 [arXiv:1111.1704] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  8. L.J. Dixon, J.M. Drummond, C. Duhr and J. Pennington, The four-loop remainder function and multi-Regge behavior at NNLLA in planar N = 4 super-Yang-Mills theory, JHEP 06 (2014) 116 [arXiv:1402.3300] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  9. L.J. Dixon, M. von Hippel and A.J. McLeod, The four-loop six-gluon NMHV ratio function, JHEP 01 (2016) 053 [arXiv:1509.08127] [INSPIRE].

    Article  Google Scholar 

  10. S. Caron-Huot, L.J. Dixon, A. McLeod and M. von Hippel, Bootstrapping a Five-Loop Amplitude Using Steinmann Relations, Phys. Rev. Lett. 117 (2016) 241601 [arXiv:1609.00669] [INSPIRE].

    ADS  Article  Google Scholar 

  11. J.M. Drummond, G. Papathanasiou and M. Spradlin, A Symbol of Uniqueness: The Cluster Bootstrap for the 3-Loop MHV Heptagon, JHEP 03 (2015) 072 [arXiv:1412.3763] [INSPIRE].

    ADS  Article  Google Scholar 

  12. L.J. Dixon, J. Drummond, T. Harrington, A.J. McLeod, G. Papathanasiou and M. Spradlin, Heptagons from the Steinmann Cluster Bootstrap, JHEP 02 (2017) 137 [arXiv:1612.08976] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  13. J. Golden, A.B. Goncharov, M. Spradlin, C. Vergu and A. Volovich, Motivic Amplitudes and Cluster Coordinates, JHEP 01 (2014) 091 [arXiv:1305.1617] [INSPIRE].

    ADS  Article  Google Scholar 

  14. T. Dennen, M. Spradlin and A. Volovich, Landau Singularities and Symbology: One- and Two-loop MHV Amplitudes in SYM Theory, JHEP 03 (2016) 069 [arXiv:1512.07909] [INSPIRE].

    ADS  Article  Google Scholar 

  15. 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].

  16. 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 

  17. S. Badger, G. Mogull and T. Peraro, Local integrands for two-loop all-plus Yang-Mills amplitudes, JHEP 08 (2016) 063 [arXiv:1606.02244] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  18. 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  Google Scholar 

  19. S. Abreu, F. Febres Cordero, H. Ita, B. Page and M. Zeng, Planar Two-Loop Five-Gluon Amplitudes from Numerical Unitarity, arXiv:1712.03946 [INSPIRE].

  20. J. Bartels, L.N. Lipatov and A. Sabio Vera, BFKL Pomeron, Reggeized gluons and Bern-Dixon-Smirnov amplitudes, Phys. Rev. D 80 (2009) 045002 [arXiv:0802.2065] [INSPIRE].

    ADS  Google Scholar 

  21. J. Gluza, K. Kajda and T. Riemann, AMBRE: A Mathematica package for the construction of Mellin-Barnes representations for Feynman integrals, Comput. Phys. Commun. 177 (2007) 879 [arXiv:0704.2423] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  22. J. Blümlein et al., Non-planar Feynman integrals, Mellin-Barnes representations, multiple sums, PoS(LL2014)052 [arXiv:1407.7832] [INSPIRE].

  23. I. Dubovyk, J. Gluza and T. Riemann, Non-planar Feynman diagrams and Mellin-Barnes representations with AM BRE 3.0, J. Phys. Conf. Ser. 608 (2015) 012070 [INSPIRE].

    Google Scholar 

  24. M. Ochman and T. Riemann, MBsumsa Mathematica package for the representation of Mellin-Barnes integrals by multiple sums, Acta Phys. Polon. B 46 (2015) 2117 [arXiv:1511.01323] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  25. 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  Google Scholar 

  26. 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].

    ADS  Article  Google Scholar 

  27. J. Maldacena, D. Simmons-Duffin and A. Zhiboedov, Looking for a bulk point, JHEP 01 (2017) 013 [arXiv:1509.03612] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  28. V.A. Smirnov, Analytic tools for Feynman integrals, Springer Tracts Mod. Phys. 250 (2012) 1.

    MathSciNet  Article  Google Scholar 

  29. M. Czakon, Automatized analytic continuation of Mellin-Barnes integrals, Comput. Phys. Commun. 175 (2006) 559 [hep-ph/0511200] [INSPIRE].

    ADS  Article  Google Scholar 

  30. J.B. Tausk, Nonplanar massless two loop Feynman diagrams with four on-shell legs, Phys. Lett. B 469 (1999) 225 [hep-ph/9909506] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  31. Z. Bern, E. Herrmann, S. Litsey, J. Stankowicz and J. Trnka, Evidence for a Nonplanar Amplituhedron, JHEP 06 (2016) 098 [arXiv:1512.08591] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

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

    ADS  MathSciNet  Article  Google Scholar 

  33. S. Moch and P. Uwer, XSummer: Transcendental functions and symbolic summation in form, Comput. Phys. Commun. 174 (2006) 759 [math-ph/0508008] [INSPIRE].

    ADS  Article  Google Scholar 

  34. C. Schneider, Symbolic Summation Assists Combinatorics, Séminaire Lotharingien de Combinatoire 56 (2007) article B56b.

  35. D. Maître, HPL, a mathematica implementation of the harmonic polylogarithms, Comput. Phys. Commun. 174 (2006) 222 [hep-ph/0507152] [INSPIRE].

    ADS  Article  Google Scholar 

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

    ADS  MathSciNet  Article  Google Scholar 

  37. T. Gehrmann, J. Henn and A. Lo Presti, work in progress.

  38. L.J. Dixon, J.M. Drummond and J.M. Henn, The one-loop six-dimensional hexagon integral and its relation to MHV amplitudes in N = 4 SYM, JHEP 06 (2011) 100 [arXiv:1104.2787] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  39. J.M. Drummond and J.M. Henn, Simple loop integrals and amplitudes in N = 4 SYM, JHEP 05 (2011) 105 [arXiv:1008.2965] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  40. 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].

  41. D. Zagier, The Dilogarithm Function, in Proceedings, Les Houches School of Physics: Frontiers in Number Theory, Physics and Geometry II: On Conformal Field Theories, Discrete Groups and Renormalization: Les Houches, France, March 9–21, 2003, pp. 3–65.

  42. L. Lewin, Polylogarithms and Associated Functions, North-Holland, New York, U.S.A. (1981).

    MATH  Google Scholar 

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Correspondence to Dmitry Chicherin.

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Chicherin, D., Henn, J. & Mitev, V. Bootstrapping pentagon functions. J. High Energ. Phys. 2018, 164 (2018).

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  • Scattering Amplitudes
  • Perturbative QCD