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.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
D.J. Gross and F. Wilczek, Asymptotically free gauge theories. 1, Phys. Rev. D 8 (1973) 3633 [INSPIRE].
H. Georgi and H.D. Politzer, Electroproduction scaling in an asymptotically free theory of strong interactions, Phys. Rev. D 9 (1974) 416 [INSPIRE].
G. Altarelli and G. Parisi, Asymptotic freedom in parton language, Nucl. Phys. B 126 (1977) 298 [INSPIRE].
K.J. Kim and K. Schilcher, Scaling violation in the infinite momentum frame, Phys. Rev. D 17 (1978) 2800 [INSPIRE].
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].
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].
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].
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].
G. Curci, W. Furmanski and R. Petronzio, Evolution of parton densities beyond leading order: the nonsinglet case, Nucl. Phys. B 175 (1980) 27 [INSPIRE].
W. Furmanski and R. Petronzio, Singlet parton densities beyond leading order, Phys. Lett. 97B (1980) 437 [INSPIRE].
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].
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].
R.K. Ellis and W. Vogelsang, The evolution of parton distributions beyond leading order: the singlet case, hep-ph/9602356 [INSPIRE].
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].
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].
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].
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].
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].
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].
A. Accardi et al., A critical appraisal and evaluation of modern PDFs, Eur. Phys. J. C 76 (2016) 471 [arXiv:1603.08906] [INSPIRE].
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].
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].
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].
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].
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.
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].
P.A. Baikov and K.G. Chetyrkin, New four loop results in QCD, Nucl. Phys. Proc. Suppl. 160 (2006) 76 [INSPIRE].
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].
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].
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].
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].
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].
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].
J.A.M. Vermaseren, New features of FORM, math-ph/0010025 [INSPIRE].
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].
M. Tentyukov and J.A.M. Vermaseren, The multithreaded version of FORM, Comput. Phys. Commun. 181 (2010) 1419 [hep-ph/0702279] [INSPIRE].
J.A.M. Vermaseren, Harmonic sums, Mellin transforms and integrals, Int. J. Mod. Phys. A 14 (1999) 2037 [hep-ph/9806280] [INSPIRE].
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].
A.K. Lenstra, H.W. Lenstra and L. Lovász, Factoring polynomials with rational coefficients, Math. Ann. 261 (1982) 515.
K. Matthews, Solving ax = b using the Hermite normal form, unpublished.
J.H. Silverman, The Xedni calculus and the elliptic curve discrete logarithm problem, Designs, Codes Crypt. 20 (2000) 5.
CALC webpage, http://www.numbertheory.org/calc/krm_calc.html.
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].
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].
G.P. Korchemsky, Asymptotics of the Altarelli-Parisi-Lipatov evolution kernels of parton distributions, Mod. Phys. Lett. A 4 (1989) 1257 [INSPIRE].
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].
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].
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].
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].
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].
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].
J. Kalinowski, K. Konishi and T.R. Taylor, Jet calculus beyond leading logarithms, Nucl. Phys. B 181 (1981) 221 [INSPIRE].
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].
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].
O. Gituliar, Master integrals for splitting functions from differential equations in QCD, JHEP 02 (2016) 017 [arXiv:1512.02045] [INSPIRE].
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].
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].
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].
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].
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].
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].
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].
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].
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].
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].
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].
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].
G. ’t Hooft, Dimensional regularization and the renormalization group, Nucl. Phys. B 61 (1973) 455 [INSPIRE].
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].
C.G. Bollini and J.J. Giambiagi, Dimensional renormalization: the number of dimensions as a regularizing parameter, Nuovo Cim. B 12 (1972) 20 [INSPIRE].
G. ’t Hooft and M.J.G. Veltman, Regularization and renormalization of gauge fields, Nucl. Phys. B 44 (1972) 189 [INSPIRE].
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].
P. Nogueira, Automatic Feynman graph generation, J. Comput. Phys. 105 (1993) 279.
B. Ruijl, T. Ueda and J. Vermaseren, FORM version 4.2, arXiv:1707.06453 [INSPIRE].
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].
F. Herzog, B. Ruijl, T. Ueda, J.A.M. Vermaseren and A. Vogt, FORM, diagrams and topologies, PoS(LL2016)073 [arXiv:1608.01834] [INSPIRE].
J.A.M. Vermaseren, The Minos database facility webpage, https://www.nikhef.nl/~form/maindir/others/minos/minos.html.
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].
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].
K.G. Chetyrkin and F.V. Tkachov, Infrared R operation and ultraviolet counterterms in the MS scheme, Phys. Lett. B 114 (1982) 340 [INSPIRE].
K.G. Chetyrkin and V.A. Smirnov, R * operation corrected, Phys. Lett. B 144 (1984) 419 [INSPIRE].
F. Herzog and B. Ruijl, The R * -operation for Feynman graphs with generic numerators, JHEP 05 (2017) 037 [arXiv:1703.03776] [INSPIRE].
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].
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].
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].
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].
E. Remiddi and J.A.M. Vermaseren, Harmonic polylogarithms, Int. J. Mod. Phys. A 15 (2000) 725 [hep-ph/9905237] [INSPIRE].
S. Moch and J.A.M. Vermaseren, Deep inelastic structure functions at two loops, Nucl. Phys. B 573 (2000) 853 [hep-ph/9912355] [INSPIRE].
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].
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/.
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].
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].
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].
J. Davies, C.H. Kom and A. Vogt, Resummation of small-x double logarithms in QCD: inclusive deep-inelastic scattering, to appear.
A. Vogt, Resummation of small-x double logarithms in QCD: semi-inclusive electron-positron annihilation, JHEP 10 (2011) 025 [arXiv:1108.2993] [INSPIRE].
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].
V.N. Velizhanin, Generalised double-logarithmic equation in QCD, arXiv:1412.7143 [INSPIRE].
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].
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].
V. Ravindran, Higher-order threshold effects to inclusive processes in QCD, Nucl. Phys. B 752 (2006) 173 [hep-ph/0603041] [INSPIRE].
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].
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].
B. Ruijl, Towards five loop calculations in QCD, http://www.physik.uzh.ch/en/seminars/ttpseminar/HS2016.html, seminar of 6 December 2016.
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].
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].
A. Vogt, Efficient evolution of unpolarized and polarized parton distributions with QCD-PEGASUS, Comput. Phys. Commun. 170 (2005) 65 [hep-ph/0408244] [INSPIRE].
T. Gehrmann and E. Remiddi, Numerical evaluation of harmonic polylogarithms, Comput. Phys. Commun. 141 (2001) 296 [hep-ph/0107173] [INSPIRE].
J. Ablinger, J. Blümlein, M. Round and C. Schneider, Algebraic and numeric representations of harmonic polylogarithms, their generalizations and special numbers, DESY-13-064.
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].
E. Gardi and L. Magnea, Factorization constraints for soft anomalous dimensions in QCD scattering amplitudes, JHEP 03 (2009) 079 [arXiv:0901.1091] [INSPIRE].
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].
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].
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].
R.H. Boels, T. Huber and G. Yang, The four-loop non-planar cusp anomalous dimension in N = 4 SYM, arXiv:1705.03444 [INSPIRE].
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].
M. Czakon, The four-loop QCD β-function and anomalous dimensions, Nucl. Phys. B 710 (2005) 485 [hep-ph/0411261] [INSPIRE].
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].
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].
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].
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].
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].
J.A.M. Vermaseren, Axodraw, Comput. Phys. Commun. 83 (1994) 45 [INSPIRE].
D. Binosi and L. Theussl, JaxoDraw: a graphical user interface for drawing Feynman diagrams, Comput. Phys. Commun. 161 (2004) 76 [hep-ph/0309015] [INSPIRE].
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
Corresponding author
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.
About this article
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
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP10(2017)041