Advertisement

Analytic result for the nonplanar hexa-box integrals

  • D. ChicherinEmail author
  • T. Gehrmann
  • J. M. Henn
  • N. A. Lo Presti
  • V. Mitev
  • P. Wasser
Open Access
Regular Article - Theoretical Physics

Abstract

In this paper, we analytically compute all master integrals for one of the two non-planar integral families for five-particle massless scattering at two loops. We first derive an integral basis of 73 integrals with constant leading singularities. We then construct the system of differential equations satisfied by them, and find that it is in canonical form. The solution space is in agreement with a recent conjecture for the non-planar pentagon alphabet. We fix the boundary constants of the differential equations by exploiting constraints from the absence of unphysical singularities. The solution of the differential equations in the Euclidean region is expressed in terms of iterated integrals. We cross-check the latter against previously known results in the literature, as well as with independent Mellin-Barnes calculations.

Keywords

Perturbative QCD Scattering Amplitudes 

Notes

Open Access

This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.

Supplementary material

References

  1. [1]
    S. Badger, H. Frellesvig and Y. Zhang, A two-loop five-gluon helicity amplitude in QCD, JHEP 12 (2013) 045 [arXiv:1310.1051] [INSPIRE].ADSCrossRefGoogle Scholar
  2. [2]
    H. Ita, Two-loop integrand decomposition into master integrals and surface terms, Phys. Rev. D 94 (2016) 116015 [arXiv:1510.05626] [INSPIRE].ADSMathSciNetGoogle Scholar
  3. [3]
    S. Abreu, F. Febres Cordero, H. Ita, M. Jaquier and B. Page, Subleading poles in the numerical unitarity method at two loops, Phys. Rev. D 95 (2017) 096011 [arXiv:1703.05255] [INSPIRE].ADSMathSciNetGoogle Scholar
  4. [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].ADSCrossRefGoogle Scholar
  5. [5]
    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].
  6. [6]
    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].ADSGoogle Scholar
  7. [7]
    T. Gehrmann, J.M. Henn and N.A. Lo Presti, Pentagon functions for massless planar scattering amplitudes, JHEP 10 (2018) 103 [arXiv:1807.09812] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  8. [8]
    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].ADSGoogle Scholar
  9. [9]
    R.M. Schabinger, A new algorithm for the generation of unitarity-compatible integration by parts relations, JHEP 01 (2012) 077 [arXiv:1111.4220] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    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].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    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].ADSMathSciNetGoogle Scholar
  12. [12]
    T. Peraro, Scattering amplitudes over finite fields and multivariate functional reconstruction, JHEP 12 (2016) 030 [arXiv:1608.01902] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  13. [13]
    D.A. Kosower, Direct solution of integration-by-parts systems, Phys. Rev. D 98 (2018) 025008 [arXiv:1804.00131] [INSPIRE].ADSGoogle Scholar
  14. [14]
    J. Böhm, A. Georgoudis, K.J. Larsen, M. Schulze and Y. Zhang, Complete sets of logarithmic vector fields for integration-by-parts identities of Feynman integrals, Phys. Rev. D 98 (2018) 025023 [arXiv:1712.09737] [INSPIRE].ADSGoogle Scholar
  15. [15]
    J. Böhm, A. Georgoudis, K.J. Larsen, H. Schönemann and Y. Zhang, Complete integration-by-parts reductions of the non-planar hexagon-box via module intersections, JHEP 09 (2018) 024 [arXiv:1805.01873] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  16. [16]
    H.A. Chawdhry, M.A. Lim and A. Mitov, Two-loop five-point massless QCD amplitudes within the IBP approach, arXiv:1805.09182 [INSPIRE].
  17. [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].ADSGoogle Scholar
  18. [18]
    D. Chicherin, J. Henn and V. Mitev, Bootstrapping pentagon functions, JHEP 05 (2018) 164 [arXiv:1712.09610] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  19. [19]
    D. Chicherin, J.M. Henn and E. Sokatchev, Scattering amplitudes from superconformal Ward identities, Phys. Rev. Lett. 121 (2018) 021602 [arXiv:1804.03571] [INSPIRE].ADSCrossRefGoogle Scholar
  20. [20]
    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].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  21. [21]
    J.M. Henn, Multiloop integrals in dimensional regularization made simple, Phys. Rev. Lett. 110 (2013) 251601 [arXiv:1304.1806] [INSPIRE].ADSCrossRefGoogle Scholar
  22. [22]
    Z. Bern, E. Herrmann, S. Litsey, J. Stankowicz and J. Trnka, Evidence for a nonplanar amplituhedron, JHEP 06 (2016) 098 [arXiv:1512.08591] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  23. [23]
    P. Wasser, Analytic properties of Feynman integrals for scattering amplitudes, M.Sc. thesis, Mainz Univ., Mainz, Germany (2016).Google Scholar
  24. [24]
    S. Laporta, High precision calculation of multiloop Feynman integrals by difference equations, Int. J. Mod. Phys. A 15 (2000) 5087 [hep-ph/0102033] [INSPIRE].
  25. [25]
    A. von Manteuffel and C. Studerus, Reduze 2 — distributed Feynman integral reduction, arXiv:1201.4330 [INSPIRE].
  26. [26]
    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].
  27. [27]
    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].
  28. [28]
    T. Gehrmann and E. Remiddi, Analytic continuation of massless two loop four point functions, Nucl. Phys. B 640 (2002) 379 [hep-ph/0207020] [INSPIRE].
  29. [29]
    E. Remiddi and J.A.M. Vermaseren, Harmonic polylogarithms, Int. J. Mod. Phys. A 15 (2000) 725 [hep-ph/9905237] [INSPIRE].
  30. [30]
    A.B. Goncharov, Multiple polylogarithms and mixed Tate motives, math.AG/0103059 [INSPIRE].
  31. [31]
    T. Gehrmann and E. Remiddi, Numerical evaluation of harmonic polylogarithms, Comput. Phys. Commun. 141 (2001) 296 [hep-ph/0107173] [INSPIRE].
  32. [32]
    T. Gehrmann and E. Remiddi, Numerical evaluation of two-dimensional harmonic polylogarithms, Comput. Phys. Commun. 144 (2002) 200 [hep-ph/0111255] [INSPIRE].
  33. [33]
    J. Vollinga and S. Weinzierl, Numerical evaluation of multiple polylogarithms, Comput. Phys. Commun. 167 (2005) 177 [hep-ph/0410259] [INSPIRE].
  34. [34]
    S. Abreu, B. Page and M. Zeng, Differential equations from unitarity cuts: nonplanar hexa-box integrals, JHEP 01 (2019) 006 [arXiv:1807.11522] [INSPIRE].ADSCrossRefGoogle Scholar
  35. [35]
    A.V. Kotikov, Differential equations method: new technique for massive Feynman diagrams calculation, Phys. Lett. B 254 (1991) 158 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  36. [36]
    E. Remiddi, Differential equations for Feynman graph amplitudes, Nuovo Cim. A 110 (1997) 1435 [hep-th/9711188] [INSPIRE].ADSGoogle Scholar
  37. [37]
    T. Gehrmann and E. Remiddi, Differential equations for two loop four point functions, Nucl. Phys. B 580 (2000) 485 [hep-ph/9912329] [INSPIRE].
  38. [38]
    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].
  39. [39]
    J.M. Henn, A.V. Smirnov and V.A. Smirnov, Evaluating single-scale and/or non-planar diagrams by differential equations, JHEP 03 (2014) 088 [arXiv:1312.2588] [INSPIRE].ADSCrossRefGoogle Scholar
  40. [40]
    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].ADSCrossRefGoogle Scholar
  41. [41]
    C. Duhr, H. Gangl and J.R. Rhodes, From polygons and symbols to polylogarithmic functions, JHEP 10 (2012) 075 [arXiv:1110.0458] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  42. [42]
    A.V. Smirnov, FIESTA4: optimized Feynman integral calculations with GPU support, Comput. Phys. Commun. 204 (2016) 189 [arXiv:1511.03614] [INSPIRE].ADSCrossRefzbMATHGoogle Scholar
  43. [43]
    S. Badger, C. Brønnum-Hansen, H.B. Hartanto and T. Peraro, Analytic helicity amplitudes for two-loop five-gluon scattering: the single-minus case, JHEP 01 (2019) 186 [arXiv:1811.11699] [INSPIRE].ADSCrossRefGoogle Scholar
  44. [44]
    S. Abreu, J. Dormans, F. Febres Cordero, H. Ita and B. Page, Analytic form of the planar two-loop five-gluon scattering amplitudes in QCD, Phys. Rev. Lett. 122 (2019) 082002 [arXiv:1812.04586] [INSPIRE].CrossRefGoogle Scholar
  45. [45]
    S. Abreu, L.J. Dixon, E. Herrmann, B. Page and M. Zeng, The two-loop five-point amplitude in N = 4 super-Yang-Mills theory, arXiv:1812.08941 [INSPIRE].
  46. [46]
    D. Chicherin, J.M. Henn, P. Wasser, T. Gehrmann, Y. Zhang and S. Zoia, Analytic result for a two-loop five-particle amplitude, arXiv:1812.11057 [INSPIRE].
  47. [47]
    D. Chicherin, T. Gehrmann, J.M. Henn, P. Wasser, Y. Zhang and S. Zoia, All master integrals for three-jet production at NNLO, arXiv:1812.11160 [INSPIRE].

Copyright information

© The Author(s) 2019

Authors and Affiliations

  1. 1.PRISMA Cluster of Excellence, Institute of PhysicsJohannes Gutenberg UniversityMainzGermany
  2. 2.Physik-InstitutUniversität ZürichZürichSwitzerland
  3. 3.MPI für PhysikWerner-Heisenberg-InstitutMünchenGermany
  4. 4.Institute for Particle Physics PhenomenologyDurham UniversityDurhamU.K.

Personalised recommendations