Advertisement

A non-planar two-loop three-point function beyond multiple polylogarithms

  • Andreas von Manteuffel
  • Lorenzo TancrediEmail author
Open Access
Regular Article - Theoretical Physics

Abstract

We consider the analytic calculation of a two-loop non-planar three-point function which contributes to the two-loop amplitudes for \( t\overline{t} \) production and γγ production in gluon fusion through a massive top-quark loop. All subtopology integrals can be written in terms of multiple polylogarithms over an irrational alphabet and we employ a new method for the integration of the differential equations which does not rely on the rationalization of the latter. The top topology integrals, instead, in spite of the absence of a massive three-particle cut, cannot be evaluated in terms of multiple polylogarithms and require the introduction of integrals over complete elliptic integrals and polylogarithms. We provide one-fold integral representations for the solutions and continue them analytically to all relevant regions of the phase space in terms of real functions, extracting all imaginary parts explicitly. The numerical evaluation of our expressions becomes straightforward in this way.

Keywords

NLO Computations QCD Phenomenology 

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.

References

  1. [1]
    F.V. Tkachov, A Theorem on Analytical Calculability of Four Loop Renormalization Group Functions, Phys. Lett. B 100 (1981) 65 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  2. [2]
    K.G. Chetyrkin and F.V. Tkachov, Integration by Parts: The Algorithm to Calculate β-functions in 4 Loops, Nucl. Phys. B 192 (1981) 159 [INSPIRE].ADSCrossRefGoogle Scholar
  3. [3]
    S. Laporta, High precision calculation of multiloop Feynman integrals by difference equations, Int. J. Mod. Phys. A 15 (2000) 5087 [hep-ph/0102033] [INSPIRE].
  4. [4]
    A.V. Kotikov, Differential equations method: New technique for massive Feynman diagrams calculation, Phys. Lett. B 254 (1991) 158 [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  5. [5]
    E. Remiddi, Differential equations for Feynman graph amplitudes, Nuovo Cim. A 110 (1997) 1435 [hep-th/9711188] [INSPIRE].ADSGoogle Scholar
  6. [6]
    T. Gehrmann and E. Remiddi, Differential equations for two loop four point functions, Nucl. Phys. B 580 (2000) 485 [hep-ph/9912329] [INSPIRE].
  7. [7]
    A.V. Kotikov, The property of maximal transcendentality in the N = 4 Supersymmetric Yang-Mills, arXiv:1005.5029 [INSPIRE].
  8. [8]
    J.M. Henn, Multiloop integrals in dimensional regularization made simple, Phys. Rev. Lett. 110 (2013) 251601 [arXiv:1304.1806] [INSPIRE].ADSCrossRefGoogle Scholar
  9. [9]
    A.B. Goncharov, Geometry of configurations, polylogarithms, and motivic cohomology, Adv. Math. 114 (1995) 197.MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    A.B. Goncharov, Multiple polylogarithms, cyclotomy and modular complexes, Math. Res. Lett. 5 (1998) 497 [arXiv:1105.2076] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  11. [11]
    E. Remiddi and J.A.M. Vermaseren, Harmonic polylogarithms, Int. J. Mod. Phys. A 15 (2000) 725 [hep-ph/9905237] [INSPIRE].
  12. [12]
    T. Gehrmann and E. Remiddi, Two loop master integrals for gamma * → 3 jets: The planar topologies, Nucl. Phys. B 601 (2001) 248 [hep-ph/0008287] [INSPIRE].
  13. [13]
    A.B. Goncharov, Galois symmetries of fundamental groupoids and noncommutative geometry, Duke Math. J. 128 (2005) 209 [math/0208144] [INSPIRE].
  14. [14]
    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].ADSMathSciNetCrossRefGoogle Scholar
  15. [15]
    F. Brown, On the decomposition of motivic multiple zeta values, arXiv:1102.1310 [INSPIRE].
  16. [16]
    C. Duhr, H. Gangl and J.R. Rhodes, From polygons and symbols to polylogarithmic functions, JHEP 10 (2012) 075 [arXiv:1110.0458] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  17. [17]
    C. Duhr, Hopf algebras, coproducts and symbols: an application to Higgs boson amplitudes, JHEP 08 (2012) 043 [arXiv:1203.0454] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  18. [18]
    T. Gehrmann and E. Remiddi, Numerical evaluation of harmonic polylogarithms, Comput. Phys. Commun. 141 (2001) 296 [hep-ph/0107173] [INSPIRE].
  19. [19]
    T. Gehrmann and E. Remiddi, Numerical evaluation of two-dimensional harmonic polylogarithms, Comput. Phys. Commun. 144 (2002) 200 [hep-ph/0111255] [INSPIRE].
  20. [20]
    J. Vollinga and S. Weinzierl, Numerical evaluation of multiple polylogarithms, Comput. Phys. Commun. 167 (2005) 177 [hep-ph/0410259] [INSPIRE].
  21. [21]
    A. Sabry, Fourth order spectral functions for the electron propagator, Nucl. Phys. 33 (1962) 401.MathSciNetCrossRefzbMATHGoogle Scholar
  22. [22]
    D.J. Broadhurst, The Master Two Loop Diagram With Masses, Z. Phys. C 47 (1990) 115 [INSPIRE].Google Scholar
  23. [23]
    S. Bauberger, F.A. Berends, M. Böhm and M. Buza, Analytical and numerical methods for massive two loop selfenergy diagrams, Nucl. Phys. B 434 (1995) 383 [hep-ph/9409388] [INSPIRE].
  24. [24]
    S. Bauberger and M. Böhm, Simple one-dimensional integral representations for two loop selfenergies: The master diagram, Nucl. Phys. B 445 (1995) 25 [hep-ph/9501201] [INSPIRE].
  25. [25]
    M. Caffo, H. Czyz, S. Laporta and E. Remiddi, The master differential equations for the two loop sunrise selfmass amplitudes, Nuovo Cim. A 111 (1998) 365 [hep-th/9805118] [INSPIRE].ADSGoogle Scholar
  26. [26]
    S. Laporta and E. Remiddi, Analytic treatment of the two loop equal mass sunrise graph, Nucl. Phys. B 704 (2005) 349 [hep-ph/0406160] [INSPIRE].
  27. [27]
    S. Bloch and P. Vanhove, The elliptic dilogarithm for the sunset graph, J. Number Theor. 148 (2015) 328 [arXiv:1309.5865] [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  28. [28]
    E. Remiddi and L. Tancredi, Schouten identities for Feynman graph amplitudes; The Master Integrals for the two-loop massive sunrise graph, Nucl. Phys. B 880 (2014) 343 [arXiv:1311.3342] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  29. [29]
    L. Adams, C. Bogner and S. Weinzierl, The two-loop sunrise graph with arbitrary masses, J. Math. Phys. 54 (2013) 052303 [arXiv:1302.7004] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  30. [30]
    L. Adams, C. Bogner and S. Weinzierl, The two-loop sunrise graph in two space-time dimensions with arbitrary masses in terms of elliptic dilogarithms, J. Math. Phys. 55 (2014) 102301 [arXiv:1405.5640] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  31. [31]
    L. Adams, C. Bogner and S. Weinzierl, The two-loop sunrise integral around four space-time dimensions and generalisations of the Clausen and Glaisher functions towards the elliptic case, J. Math. Phys. 56 (2015) 072303 [arXiv:1504.03255] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  32. [32]
    L. Adams, C. Bogner and S. Weinzierl, The iterated structure of the all-order result for the two-loop sunrise integral, J. Math. Phys. 57 (2016) 032304 [arXiv:1512.05630] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  33. [33]
    S. Bloch, M. Kerr and P. Vanhove, Local mirror symmetry and the sunset Feynman integral, arXiv:1601.08181 [INSPIRE].
  34. [34]
    E. Remiddi and L. Tancredi, Differential equations and dispersion relations for Feynman amplitudes. The two-loop massive sunrise and the kite integral, Nucl. Phys. B 907 (2016) 400 [arXiv:1602.01481] [INSPIRE].
  35. [35]
    K.-T. Chen, Iterated path integrals, Bull. Am. Math. Soc. 83 (1977) 831 [INSPIRE].MathSciNetCrossRefzbMATHGoogle Scholar
  36. [36]
    A. Primo and L. Tancredi, On the maximal cut of Feynman integrals and the solution of their differential equations, Nucl. Phys. B 916 (2017) 94 [arXiv:1610.08397] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  37. [37]
    L. Adams, C. Bogner, A. Schweitzer and S. Weinzierl, The kite integral to all orders in terms of elliptic polylogarithms, J. Math. Phys. 57 (2016) 122302 [arXiv:1607.01571] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  38. [38]
    U. Aglietti, R. Bonciani, L. Grassi and E. Remiddi, The two loop crossed ladder vertex diagram with two massive exchanges, Nucl. Phys. B 789 (2008) 45 [arXiv:0705.2616] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  39. [39]
    S. Caron-Huot and K.J. Larsen, Uniqueness of two-loop master contours, JHEP 10 (2012) 026 [arXiv:1205.0801] [INSPIRE].ADSCrossRefGoogle Scholar
  40. [40]
    R. Bonciani, V. Del Duca, H. Frellesvig, J.M. Henn, F. Moriello and V.A. Smirnov, Two-loop planar master integrals for Higgs→ 3 partons with full heavy-quark mass dependence, JHEP 12 (2016) 096 [arXiv:1609.06685] [INSPIRE].ADSCrossRefGoogle Scholar
  41. [41]
    A. von Manteuffel and C. Studerus, Reduze 2 — Distributed Feynman Integral Reduction, arXiv:1201.4330 [INSPIRE].
  42. [42]
    C. Studerus, Reduze-Feynman Integral Reduction in C++, Comput. Phys. Commun. 181 (2010) 1293 [arXiv:0912.2546] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  43. [43]
    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].
  44. [44]
    R.H. Lewis, Computer Algebra System Fermat, http://home.bway.net/lewis.
  45. [45]
    T. Gehrmann, A. von Manteuffel, L. Tancredi and E. Weihs, The two-loop master integrals for \( q\overline{q}\to VV \), JHEP 06 (2014) 032 [arXiv:1404.4853] [INSPIRE].ADSCrossRefGoogle Scholar
  46. [46]
    R.N. Lee and A.A. Pomeransky, Critical points and number of master integrals, JHEP 11 (2013) 165 [arXiv:1308.6676] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  47. [47]
    M. Argeri et al., Magnus and Dyson Series for Master Integrals, JHEP 03 (2014) 082 [arXiv:1401.2979] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  48. [48]
    R.N. Lee, Reducing differential equations for multiloop master integrals, JHEP 04 (2015) 108 [arXiv:1411.0911] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  49. [49]
    J. Ablinger, A. Behring, J. Blümlein, A. De Freitas, A. von Manteuffel and C. Schneider, Calculating Three Loop Ladder and V-Topologies for Massive Operator Matrix Elements by Computer Algebra, Comput. Phys. Commun. 202 (2016) 33 [arXiv:1509.08324] [INSPIRE].ADSMathSciNetCrossRefzbMATHGoogle Scholar
  50. [50]
    L. Tancredi, Integration by parts identities in integer numbers of dimensions. A criterion for decoupling systems of differential equations, Nucl. Phys. B 901 (2015) 282 [arXiv:1509.03330] [INSPIRE].
  51. [51]
    H. Ita, Two-loop Integrand Decomposition into Master Integrals and Surface Terms, Phys. Rev. D 94 (2016) 116015 [arXiv:1510.05626] [INSPIRE].ADSGoogle Scholar
  52. [52]
    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
  53. [53]
    C. Meyer, Transforming differential equations of multi-loop Feynman integrals into canonical form, JHEP 04 (2017) 006 [arXiv:1611.01087] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar
  54. [54]
    A. Georgoudis, K.J. Larsen and Y. Zhang, Azurite: An algebraic geometry based package for finding bases of loop integrals, arXiv:1612.04252 [INSPIRE].
  55. [55]
    M. Prausa, epsilon: A tool to find a canonical basis of master integrals, arXiv:1701.00725 [INSPIRE].
  56. [56]
    O. Gituliar and V. Magerya, Fuchsia: a tool for reducing differential equations for Feynman master integrals to epsilon form, arXiv:1701.04269 [INSPIRE].
  57. [57]
    E. Weihs, private communication.Google Scholar
  58. [58]
    H.j. Lenstra, A. Lenstra and L. Lovász, Factoring polynomials with rational coefficients, Math. Ann. 261 (1982) 515.Google Scholar
  59. [59]
    PARI Group, PARI/GP version 2.7.2, Université Bordeaux, available from http://pari.math.u-bordeaux.fr/.
  60. [60]
    A. von Manteuffel and R.M. Schabinger, in preparation.Google Scholar
  61. [61]
    E. Panzer, On hyperlogarithms and Feynman integrals with divergences and many scales, JHEP 03 (2014) 071 [arXiv:1401.4361] [INSPIRE].ADSCrossRefGoogle Scholar
  62. [62]
    A. von Manteuffel, E. Panzer and R.M. Schabinger, A quasi-finite basis for multi-loop Feynman integrals, JHEP 02 (2015) 120 [arXiv:1411.7392] [INSPIRE].ADSMathSciNetCrossRefGoogle Scholar

Copyright information

© The Author(s) 2017

Authors and Affiliations

  1. 1.Department of Physics and AstronomyMichigan State UniversityEast Lansing, MIU.S.A.
  2. 2.PRISMA Cluster of ExcellenceJohannes Gutenberg UniversityMainzGermany
  3. 3.Institut für Theoretische TeilchenphysikKarlsruhe Institute of Technology (KIT)KarlsruheGermany

Personalised recommendations