Skip to main content
Download PDF

Four-loop non-singlet splitting functions in the planar limit and beyond

Journal of High Energy Physics volume 2017, Article number: 41 (2017) Cite this article

A preprint version of the article is available at arXiv.

Abstract

We present the next-to-next-to-next-to-leading order (N3LO) contributions to the non-singlet splitting functions for both parton distribution and fragmentation functions in perturbative QCD. The exact expressions are derived for the terms contributing in the limit of a large number of colours. For the remaining contributions, approximations are provided that are sufficient for all collider-physics applications. From their threshold limits we derive analytical and high-accuracy numerical results, respectively, for all contributions to the four-loop cusp anomalous dimension for quarks, including the terms proportional to quartic Casimir operators. We briefly illustrate the numerical size of the four-loop corrections, and the remarkable renormalization-scale stability of the N3LO results, for the evolution of the non-singlet parton distribution and the fragmentation functions. Our results appear to provide a first point of contact of four-loop QCD calculations and the so-called wrapping corrections to anomalous dimensions in \( \mathcal{N}=4 \) super Yang-Mills theory.

References

  1. D.J. Gross and F. Wilczek, Asymptotically free gauge theories. 1, Phys. Rev. D 8 (1973) 3633 [INSPIRE].

  2. H. Georgi and H.D. Politzer, Electroproduction scaling in an asymptotically free theory of strong interactions, Phys. Rev. D 9 (1974) 416 [INSPIRE].

    ADS  Google Scholar 

  3. G. Altarelli and G. Parisi, Asymptotic freedom in parton language, Nucl. Phys. B 126 (1977) 298 [INSPIRE].

    ADS  Article  Google Scholar 

  4. K.J. Kim and K. Schilcher, Scaling violation in the infinite momentum frame, Phys. Rev. D 17 (1978) 2800 [INSPIRE].

    ADS  Google Scholar 

  5. E.G. Floratos, D.A. Ross and C.T. Sachrajda, Higher order effects in asymptotically free gauge theories: the anomalous dimensions of Wilson operators, Nucl. Phys. B 129 (1977) 66 [Erratum ibid. B 139 (1978) 545] [INSPIRE].

  6. E.G. Floratos, D.A. Ross and C.T. Sachrajda, Higher order effects in asymptotically free gauge theories: 2. Flavor singlet Wilson operators and coefficient functions, Nucl. Phys. B 152 (1979) 493 [INSPIRE].

  7. A. Gonzalez-Arroyo, C. Lopez and F.J. Yndurain, Second order contributions to the structure functions in deep inelastic scattering. 1. Theoretical calculations, Nucl. Phys. B 153 (1979) 161 [INSPIRE].

  8. A. Gonzalez-Arroyo and C. Lopez, Second order contributions to the structure functions in deep inelastic scattering. 3. The singlet case, Nucl. Phys. B 166 (1980) 429 [INSPIRE].

  9. G. Curci, W. Furmanski and R. Petronzio, Evolution of parton densities beyond leading order: the nonsinglet case, Nucl. Phys. B 175 (1980) 27 [INSPIRE].

    ADS  Article  Google Scholar 

  10. W. Furmanski and R. Petronzio, Singlet parton densities beyond leading order, Phys. Lett. 97B (1980) 437 [INSPIRE].

    ADS  Article  Google Scholar 

  11. E.G. Floratos, C. Kounnas and R. Lacaze, Higher order QCD effects in inclusive annihilation and deep inelastic scattering, Nucl. Phys. B 192 (1981) 417 [INSPIRE].

    ADS  Article  Google Scholar 

  12. R. Hamberg and W.L. van Neerven, The correct renormalization of the gluon operator in a covariant gauge, Nucl. Phys. B 379 (1992) 143 [INSPIRE].

    ADS  Article  Google Scholar 

  13. R.K. Ellis and W. Vogelsang, The evolution of parton distributions beyond leading order: the singlet case, hep-ph/9602356 [INSPIRE].

  14. S. Moch, J.A.M. Vermaseren and A. Vogt, The three loop splitting functions in QCD: the nonsinglet case, Nucl. Phys. B 688 (2004) 101 [hep-ph/0403192] [INSPIRE].

  15. A. Vogt, S. Moch and J.A.M. Vermaseren, The three-loop splitting functions in QCD: the singlet case, Nucl. Phys. B 691 (2004) 129 [hep-ph/0404111] [INSPIRE].

  16. J. Ablinger, J. Blümlein, S. Klein, C. Schneider and F. Wissbrock, The O(α 3 s ) massive operator matrix elements of O(n f ) for the structure function F 2(x, Q 2) and transversity, Nucl. Phys. B 844 (2011) 26 [arXiv:1008.3347] [INSPIRE].

    ADS  Article  MATH  Google Scholar 

  17. J. Ablinger et al., The 3-loop non-singlet heavy flavor contributions and anomalous dimensions for the structure function F 2(x, Q 2) and transversity, Nucl. Phys. B 886 (2014) 733 [arXiv:1406.4654] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  18. J. Ablinger, A. Behring, J. Blümlein, A. De Freitas, A. von Manteuffel and C. Schneider, The 3-loop pure singlet heavy flavor contributions to the structure function F 2(x, Q 2) and the anomalous dimension, Nucl. Phys. B 890 (2014) 48 [arXiv:1409.1135] [INSPIRE].

    ADS  MATH  Google Scholar 

  19. J. Ablinger, A. Behring, J. Blümlein, A. De Freitas, A. von Manteuffel and C. Schneider, The three-loop splitting functions \( {P}_{qg}^{(2)}\; and\;{P}_{gg}^{\left(2,{N}_F\right)} \), Nucl. Phys. B 922 (2017) 1 [arXiv:1705.01508] [INSPIRE].

    ADS  Article  MATH  Google Scholar 

  20. A. Accardi et al., A critical appraisal and evaluation of modern PDFs, Eur. Phys. J. C 76 (2016) 471 [arXiv:1603.08906] [INSPIRE].

    ADS  Article  Google Scholar 

  21. C. Anastasiou, C. Duhr, F. Dulat, F. Herzog and B. Mistlberger, Higgs boson gluon-fusion production in QCD at three loops, Phys. Rev. Lett. 114 (2015) 212001 [arXiv:1503.06056] [INSPIRE].

    ADS  Article  Google Scholar 

  22. J.A.M. Vermaseren, A. Vogt and S. Moch, The third-order QCD corrections to deep-inelastic scattering by photon exchange, Nucl. Phys. B 724 (2005) 3 [hep-ph/0504242] [INSPIRE].

  23. S. Moch, J.A.M. Vermaseren and A. Vogt, Third-order QCD corrections to the charged-current structure function F 3, Nucl. Phys. B 813 (2009) 220 [arXiv:0812.4168] [INSPIRE].

    ADS  Article  MATH  Google Scholar 

  24. J. Davies, A. Vogt, S. Moch and J.A.M. Vermaseren, Non-singlet coefficient functions for charged-current deep-inelastic scattering to the third order in QCD, PoS(DIS2016)059 [arXiv:1606.08907] [INSPIRE].

  25. J. Davies, S. Moch, J.A.M. Vermaseren and A. Vogt, Third-order QCD corrections to charged-current and polarized structure function in DIS, to appear.

  26. F.A. Dreyer and A. Karlberg, Vector-boson fusion Higgs production at three loops in QCD, Phys. Rev. Lett. 117 (2016) 072001 [arXiv:1606.00840] [INSPIRE].

    ADS  Article  Google Scholar 

  27. P.A. Baikov and K.G. Chetyrkin, New four loop results in QCD, Nucl. Phys. Proc. Suppl. 160 (2006) 76 [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  28. V.N. Velizhanin, Four loop anomalous dimension of the second moment of the non-singlet twist-2 operator in QCD, Nucl. Phys. B 860 (2012) 288 [arXiv:1112.3954] [INSPIRE].

    ADS  Article  MATH  Google Scholar 

  29. V.N. Velizhanin, Four loop anomalous dimension of the third and fourth moments of the non-singlet twist-2 operator in QCD, arXiv:1411.1331 [INSPIRE].

  30. P.A. Baikov, K.G. Chetyrkin and J.H. Kühn, Massless propagators, R(s) and multiloop QCD, Nucl. Part. Phys. Proc. 261-262 (2015) 3 [arXiv:1501.06739] [INSPIRE].

  31. B. Ruijl, T. Ueda, J.A.M. Vermaseren, J. Davies and A. Vogt, First Forcer results on deep-inelastic scattering and related quantities, PoS(LL2016)071 [arXiv:1605.08408] [INSPIRE].

  32. J. Davies, A. Vogt, B. Ruijl, T. Ueda and J.A.M. Vermaseren, Large-N f contributions to the four-loop splitting functions in QCD, Nucl. Phys. B 915 (2017) 335 [arXiv:1610.07477] [INSPIRE].

    ADS  Article  MATH  Google Scholar 

  33. B. Ruijl, T. Ueda and J.A.M. Vermaseren, Forcer, a FORM program for the parametric reduction of four-loop massless propagator diagrams, arXiv:1704.06650 [INSPIRE].

  34. J.A.M. Vermaseren, New features of FORM, math-ph/0010025 [INSPIRE].

  35. J. Kuipers, T. Ueda, J.A.M. Vermaseren and J. Vollinga, FORM version 4.0, Comput. Phys. Commun. 184 (2013) 1453 [arXiv:1203.6543] [INSPIRE].

  36. M. Tentyukov and J.A.M. Vermaseren, The multithreaded version of FORM, Comput. Phys. Commun. 181 (2010) 1419 [hep-ph/0702279] [INSPIRE].

  37. J.A.M. Vermaseren, Harmonic sums, Mellin transforms and integrals, Int. J. Mod. Phys. A 14 (1999) 2037 [hep-ph/9806280] [INSPIRE].

  38. J. Blümlein and S. Kurth, Harmonic sums and Mellin transforms up to two loop order, Phys. Rev. D 60 (1999) 014018 [hep-ph/9810241] [INSPIRE].

  39. A.K. Lenstra, H.W. Lenstra and L. Lovász, Factoring polynomials with rational coefficients, Math. Ann. 261 (1982) 515.

    MathSciNet  Article  MATH  Google Scholar 

  40. K. Matthews, Solving ax = b using the Hermite normal form, unpublished.

  41. J.H. Silverman, The Xedni calculus and the elliptic curve discrete logarithm problem, Designs, Codes Crypt. 20 (2000) 5.

  42. CALC webpage, http://www.numbertheory.org/calc/krm_calc.html.

  43. V.N. Velizhanin, Three loop anomalous dimension of the non-singlet transversity operator in QCD, Nucl. Phys. B 864 (2012) 113 [arXiv:1203.1022] [INSPIRE].

    ADS  Article  MATH  Google Scholar 

  44. S. Moch, J.A.M. Vermaseren and A. Vogt, The three-loop splitting functions in QCD: the helicity-dependent case, Nucl. Phys. B 889 (2014) 351 [arXiv:1409.5131] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  45. G.P. Korchemsky, Asymptotics of the Altarelli-Parisi-Lipatov evolution kernels of parton distributions, Mod. Phys. Lett. A 4 (1989) 1257 [INSPIRE].

    ADS  Article  Google Scholar 

  46. J.M. Henn, A.V. Smirnov, V.A. Smirnov and M. Steinhauser, A planar four-loop form factor and cusp anomalous dimension in QCD, JHEP 05 (2016) 066 [arXiv:1604.03126] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  47. J. Henn, A.V. Smirnov, V.A. Smirnov, M. Steinhauser and R.N. Lee, Four-loop photon quark form factor and cusp anomalous dimension in the large-N c limit of QCD, JHEP 03 (2017) 139 [arXiv:1612.04389] [INSPIRE].

    ADS  Article  Google Scholar 

  48. W.L. van Neerven and A. Vogt, NNLO evolution of deep inelastic structure functions: the nonsinglet case, Nucl. Phys. B 568 (2000) 263 [hep-ph/9907472] [INSPIRE].

  49. W.L. van Neerven and A. Vogt, NNLO evolution of deep inelastic structure functions: the singlet case, Nucl. Phys. B 588 (2000) 345 [hep-ph/0006154] [INSPIRE].

  50. W.L. van Neerven and A. Vogt, Improved approximations for the three loop splitting functions in QCD, Phys. Lett. B 490 (2000) 111 [hep-ph/0007362] [INSPIRE].

  51. J. Kalinowski, K. Konishi, P.N. Scharbach and T.R. Taylor, Resolving QCD jets beyond leading order: quark decay probabilities, Nucl. Phys. B 181 (1981) 253 [INSPIRE].

    ADS  Article  Google Scholar 

  52. J. Kalinowski, K. Konishi and T.R. Taylor, Jet calculus beyond leading logarithms, Nucl. Phys. B 181 (1981) 221 [INSPIRE].

    ADS  Article  Google Scholar 

  53. T. Munehisa, H. Okada, K. Kudoh and K. Kitani, Two loop anomalous dimensions of timelike cut vertices and scaling violation of fragmentation functions in QCD, Prog. Theor. Phys. 67 (1982) 609 [INSPIRE].

    ADS  Article  Google Scholar 

  54. A. Mitov and S.-O. Moch, QCD corrections to semi-inclusive hadron production in electron-positron annihilation at two loops, Nucl. Phys. B 751 (2006) 18 [hep-ph/0604160] [INSPIRE].

  55. O. Gituliar, Master integrals for splitting functions from differential equations in QCD, JHEP 02 (2016) 017 [arXiv:1512.02045] [INSPIRE].

    ADS  Article  Google Scholar 

  56. A. Mitov, S. Moch and A. Vogt, Next-to-next-to-leading order evolution of non-singlet fragmentation functions, Phys. Lett. B 638 (2006) 61 [hep-ph/0604053] [INSPIRE].

  57. S. Moch and A. Vogt, On third-order timelike splitting functions and top-mediated Higgs decay into hadrons, Phys. Lett. B 659 (2008) 290 [arXiv:0709.3899] [INSPIRE].

    ADS  Article  Google Scholar 

  58. A.A. Almasy, S. Moch and A. Vogt, On the next-to-next-to-leading order evolution of flavour-singlet fragmentation functions, Nucl. Phys. B 854 (2012) 133 [arXiv:1107.2263] [INSPIRE].

    ADS  Article  MATH  Google Scholar 

  59. D.P. Anderle, F. Ringer and M. Stratmann, Fragmentation functions at next-to-next-to-leading order accuracy, Phys. Rev. D 92 (2015) 114017 [arXiv:1510.05845] [INSPIRE].

    ADS  Google Scholar 

  60. NNPDF collaboration, V. Bertone, S. Carrazza, N.P. Hartland, E.R. Nocera and J. Rojo, A determination of the fragmentation functions of pions, kaons and protons with faithful uncertainties, Eur. Phys. J. C 77 (2017) 516 [arXiv:1706.07049] [INSPIRE].

  61. V.N. Gribov and L.N. Lipatov, Deep inelastic ep scattering in perturbation theory, Sov. J. Nucl. Phys. 15 (1972) 438 [Yad. Fiz. 15 (1972) 781] [INSPIRE].

  62. V.N. Gribov and L.N. Lipatov, e + e − pair annihilation and deep inelastic ep scattering in perturbation theory, Sov. J. Nucl. Phys. 15 (1972) 675 [Yad. Fiz. 15 (1972) 1218] [INSPIRE].

  63. M. Stratmann and W. Vogelsang, Next-to-leading order evolution of polarized and unpolarized fragmentation functions, Nucl. Phys. B 496 (1997) 41 [hep-ph/9612250] [INSPIRE].

  64. J. Blümlein, V. Ravindran and W.L. van Neerven, On the Drell-Levy-Yan relation to O(α 2 s ), Nucl. Phys. B 586 (2000) 349 [hep-ph/0004172] [INSPIRE].

  65. Yu. L. Dokshitzer, G. Marchesini and G.P. Salam, Revisiting parton evolution and the large-x limit, Phys. Lett. B 634 (2006) 504 [hep-ph/0511302] [INSPIRE].

  66. Yu. L. Dokshitzer and G. Marchesini, N = 4 SUSY Yang-Mills: three loops made simple(r), Phys. Lett. B 646 (2007) 189 [hep-th/0612248] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  67. B. Basso and G.P. Korchemsky, Anomalous dimensions of high-spin operators beyond the leading order, Nucl. Phys. B 775 (2007) 1 [hep-th/0612247] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  68. G. ’t Hooft, Dimensional regularization and the renormalization group, Nucl. Phys. B 61 (1973) 455 [INSPIRE].

  69. W.A. Bardeen, A.J. Buras, D.W. Duke and T. Muta, Deep inelastic scattering beyond the leading order in asymptotically free gauge theories, Phys. Rev. D 18 (1978) 3998 [INSPIRE].

    ADS  Google Scholar 

  70. C.G. Bollini and J.J. Giambiagi, Dimensional renormalization: the number of dimensions as a regularizing parameter, Nuovo Cim. B 12 (1972) 20 [INSPIRE].

    Google Scholar 

  71. G. ’t Hooft and M.J.G. Veltman, Regularization and renormalization of gauge fields, Nucl. Phys. B 44 (1972) 189 [INSPIRE].

  72. I. Bierenbaum, J. Blümlein and S. Klein, Mellin moments of the O(α 3 s ) heavy flavor contributions to unpolarized deep-inelastic scattering at Q 2 ≫ m 2 and anomalous dimensions, Nucl. Phys. B 820 (2009) 417 [arXiv:0904.3563] [INSPIRE].

    ADS  Article  MATH  Google Scholar 

  73. P. Nogueira, Automatic Feynman graph generation, J. Comput. Phys. 105 (1993) 279.

    ADS  MathSciNet  Article  MATH  Google Scholar 

  74. B. Ruijl, T. Ueda and J. Vermaseren, FORM version 4.2, arXiv:1707.06453 [INSPIRE].

  75. T. van Ritbergen, A.N. Schellekens and J.A.M. Vermaseren, Group theory factors for Feynman diagrams, Int. J. Mod. Phys. A 14 (1999) 41 [hep-ph/9802376] [INSPIRE].

  76. F. Herzog, B. Ruijl, T. Ueda, J.A.M. Vermaseren and A. Vogt, FORM, diagrams and topologies, PoS(LL2016)073 [arXiv:1608.01834] [INSPIRE].

  77. J.A.M. Vermaseren, The Minos database facility webpage, https://www.nikhef.nl/~form/maindir/others/minos/minos.html.

  78. S. Moch, J.A.M. Vermaseren and A. Vogt, On γ 5 in higher-order QCD calculations and the NNLO evolution of the polarized valence distribution, Phys. Lett. B 748 (2015) 432 [arXiv:1506.04517] [INSPIRE].

    ADS  Article  MATH  Google Scholar 

  79. B. Ruijl, T. Ueda, J.A.M. Vermaseren and A. Vogt, Four-loop QCD propagators and vertices with one vanishing external momentum, JHEP 06 (2017) 040 [arXiv:1703.08532] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  80. K.G. Chetyrkin and F.V. Tkachov, Infrared R operation and ultraviolet counterterms in the MS scheme, Phys. Lett. B 114 (1982) 340 [INSPIRE].

    ADS  Article  Google Scholar 

  81. K.G. Chetyrkin and V.A. Smirnov, R * operation corrected, Phys. Lett. B 144 (1984) 419 [INSPIRE].

    ADS  Article  Google Scholar 

  82. F. Herzog and B. Ruijl, The R * -operation for Feynman graphs with generic numerators, JHEP 05 (2017) 037 [arXiv:1703.03776] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  83. F. Herzog, B. Ruijl, T. Ueda, J.A.M. Vermaseren and A. Vogt, The five-loop β-function of Yang-Mills theory with fermions, JHEP 02 (2017) 090 [arXiv:1701.01404] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  84. F. Herzog, B. Ruijl, T. Ueda, J.A.M. Vermaseren and A. Vogt, On Higgs decays to hadrons and the R-ratio at N 4 LO, JHEP 08 (2017) 113 [arXiv:1707.01044] [INSPIRE].

    ADS  Article  Google Scholar 

  85. D.J. Broadhurst, A.L. Kataev and C.J. Maxwell, Comparison of the Gottfried and Adler sum rules within the large-N c expansion, Phys. Lett. B 590 (2004) 76 [hep-ph/0403037] [INSPIRE].

  86. V.M. Braun, A.N. Manashov, S. Moch and M. Strohmaier, Three-loop evolution equation for flavor-nonsinglet operators in off-forward kinematics, JHEP 06 (2017) 037 [arXiv:1703.09532] [INSPIRE].

    ADS  Article  Google Scholar 

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

  88. S. Moch and J.A.M. Vermaseren, Deep inelastic structure functions at two loops, Nucl. Phys. B 573 (2000) 853 [hep-ph/9912355] [INSPIRE].

  89. T. Lukowski, A. Rej and V.N. Velizhanin, Five-loop anomalous dimension of twist-two operators, Nucl. Phys. B 831 (2010) 105 [arXiv:0912.1624] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  90. V.N. Velizhanin, Results related with the calculations of the full five-loop anomalous dimension of twist-two operators in the planar N = 4 SYM theory, webpage, http://thd.pnpi.spb.ru/~velizh/5loop/.

  91. R. Kirschner and L.N. Lipatov, Double logarithmic asymptotics and Regge singularities of quark amplitudes with flavor exchange, Nucl. Phys. B 213 (1983) 122 [INSPIRE].

    ADS  Article  Google Scholar 

  92. J. Blümlein and A. Vogt, On the behavior of nonsinglet structure functions at small x, Phys. Lett. B 370 (1996) 149 [hep-ph/9510410] [INSPIRE].

  93. A. Vogt et al., Progress on double-logarithmic large-x and small-x resummations for (semi-)inclusive hard processes, PoS(LL2012)004 [arXiv:1212.2932] [INSPIRE].

  94. J. Davies, C.H. Kom and A. Vogt, Resummation of small-x double logarithms in QCD: inclusive deep-inelastic scattering, to appear.

  95. A. Vogt, Resummation of small-x double logarithms in QCD: semi-inclusive electron-positron annihilation, JHEP 10 (2011) 025 [arXiv:1108.2993] [INSPIRE].

    ADS  Article  MATH  Google Scholar 

  96. C.H. Kom, A. Vogt and K. Yeats, Resummed small-x and first-moment evolution of fragmentation functions in perturbative QCD, JHEP 10 (2012) 033 [arXiv:1207.5631] [INSPIRE].

    ADS  Article  Google Scholar 

  97. V.N. Velizhanin, Generalised double-logarithmic equation in QCD, arXiv:1412.7143 [INSPIRE].

  98. S. Moch, J.A.M. Vermaseren and A. Vogt, Higher-order corrections in threshold resummation, Nucl. Phys. B 726 (2005) 317 [hep-ph/0506288] [INSPIRE].

  99. S. Moch and A. Vogt, Higher-order soft corrections to lepton pair and Higgs boson production, Phys. Lett. B 631 (2005) 48 [hep-ph/0508265] [INSPIRE].

  100. V. Ravindran, Higher-order threshold effects to inclusive processes in QCD, Nucl. Phys. B 752 (2006) 173 [hep-ph/0603041] [INSPIRE].

  101. T. Ahmed, M. Mahakhud, N. Rana and V. Ravindran, Drell-Yan production at threshold to third order in QCD, Phys. Rev. Lett. 113 (2014) 112002 [arXiv:1404.0366] [INSPIRE].

    ADS  Article  Google Scholar 

  102. J.A. Gracey, Anomalous dimension of nonsinglet Wilson operators at O(1/N f ) in deep inelastic scattering, Phys. Lett. B 322 (1994) 141 [hep-ph/9401214] [INSPIRE].

  103. B. Ruijl, Towards five loop calculations in QCD, http://www.physik.uzh.ch/en/seminars/ttpseminar/HS2016.html, seminar of 6 December 2016.

  104. V. Ravindran, J. Smith and W.L. van Neerven, Two-loop corrections to Higgs boson production, Nucl. Phys. B 704 (2005) 332 [hep-ph/0408315] [INSPIRE].

  105. L.J. Dixon, L. Magnea and G.F. Sterman, Universal structure of subleading infrared poles in gauge theory amplitudes, JHEP 08 (2008) 022 [arXiv:0805.3515] [INSPIRE].

    ADS  Article  Google Scholar 

  106. A. Vogt, Efficient evolution of unpolarized and polarized parton distributions with QCD-PEGASUS, Comput. Phys. Commun. 170 (2005) 65 [hep-ph/0408244] [INSPIRE].

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

  108. J. Ablinger, J. Blümlein, M. Round and C. Schneider, Algebraic and numeric representations of harmonic polylogarithms, their generalizations and special numbers, DESY-13-064.

  109. T. van Ritbergen, J.A.M. Vermaseren and S.A. Larin, The four loop β-function in quantum chromodynamics, Phys. Lett. B 400 (1997) 379 [hep-ph/9701390] [INSPIRE].

  110. E. Gardi and L. Magnea, Factorization constraints for soft anomalous dimensions in QCD scattering amplitudes, JHEP 03 (2009) 079 [arXiv:0901.1091] [INSPIRE].

    ADS  Article  Google Scholar 

  111. T. Becher and M. Neubert, On the structure of infrared singularities of gauge-theory amplitudes, JHEP 06 (2009) 081 [Erratum ibid. 11 (2013) 024] [arXiv:0903.1126] [INSPIRE].

  112. E. Gardi and L. Magnea, Infrared singularities in QCD amplitudes, Nuovo Cim. C32N5-6 (2009) 137 [Frascati Phys. Ser. 50 (2010)] [arXiv:0908.3273] [INSPIRE].

  113. V. Ahrens, M. Neubert and L. Vernazza, Structure of infrared singularities of gauge-theory amplitudes at three and four loops, JHEP 09 (2012) 138 [arXiv:1208.4847] [INSPIRE].

    ADS  Article  Google Scholar 

  114. R.H. Boels, T. Huber and G. Yang, The four-loop non-planar cusp anomalous dimension in N = 4 SYM, arXiv:1705.03444 [INSPIRE].

  115. A. Grozin, J.M. Henn, G.P. Korchemsky and P. Marquard, The three-loop cusp anomalous dimension in QCD and its supersymmetric extensions, JHEP 01 (2016) 140 [arXiv:1510.07803] [INSPIRE].

    ADS  MathSciNet  Article  Google Scholar 

  116. M. Czakon, The four-loop QCD β-function and anomalous dimensions, Nucl. Phys. B 710 (2005) 485 [hep-ph/0411261] [INSPIRE].

  117. A. Grozin, Leading and next-to-leading large-N f terms in the cusp anomalous dimension and quark-antiquark potential, PoS(LL2016)053 [arXiv:1605.03886] [INSPIRE].

  118. R.N. Lee, A.V. Smirnov, V.A. Smirnov and M. Steinhauser, The n 2 f contributions to fermionic four-loop form factors, Phys. Rev. D 96 (2017) 014008 [arXiv:1705.06862] [INSPIRE].

    ADS  Google Scholar 

  119. J.C. Collins and R.J. Scalise, The renormalization of composite operators in Yang-Mills theories using general covariant gauge, Phys. Rev. D 50 (1994) 4117 [hep-ph/9403231] [INSPIRE].

  120. Z. Bajnok, R.A. Janik and T. Lukowski, Four loop twist two, BFKL, wrapping and strings, Nucl. Phys. B 816 (2009) 376 [arXiv:0811.4448] [INSPIRE].

    ADS  MathSciNet  Article  MATH  Google Scholar 

  121. A.V. Kotikov, L.N. Lipatov, A. Rej, M. Staudacher and V.N. Velizhanin, Dressing and wrapping, J. Stat. Mech. 10 (2007) P10003 [arXiv:0704.3586] [INSPIRE].

    Article  Google Scholar 

  122. J.A.M. Vermaseren, Axodraw, Comput. Phys. Commun. 83 (1994) 45 [INSPIRE].

    ADS  Article  MATH  Google Scholar 

  123. D. Binosi and L. Theussl, JaxoDraw: a graphical user interface for drawing Feynman diagrams, Comput. Phys. Commun. 161 (2004) 76 [hep-ph/0309015] [INSPIRE].

Download references

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.

Author information

Authors and Affiliations

  1. II. Institute for Theoretical Physics, Hamburg University, Luruper Chaussee 149, D-22761, Hamburg, Germany

    S. Moch

  2. Nikhef Theory Group, Science Park 105, 1098 XG, Amsterdam, The Netherlands

    B. Ruijl, T. Ueda & J. A. M. Vermaseren

  3. Leiden Centre of Data Science, Leiden University, Niels Bohrweg 1, 2333 CA, Leiden, The Netherlands

    B. Ruijl

  4. Department of Mathematical Sciences, University of Liverpool, Liverpool, L69 3BX, U.K.

    A. Vogt

Authors
  1. S. Moch
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. B. Ruijl
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. T. Ueda
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. J. A. M. Vermaseren
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. A. Vogt
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to S. Moch.

Additional information

ArXiv ePrint: 1707.08315

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Moch, S., Ruijl, B., Ueda, T. et al. Four-loop non-singlet splitting functions in the planar limit and beyond. J. High Energ. Phys. 2017, 41 (2017). https://doi.org/10.1007/JHEP10(2017)041

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/JHEP10(2017)041

Keywords

  • Perturbative QCD
  • Resummation