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Four-Loop Gluon Anomalous Dimension of General Lorentz Spin: Transcendental Part

B. A. Kniehl ^1,*, S.-O. Moch ^1,†, V. N. Velizhanin ^1,‡, and A. Vogt ^1,2,§

¹II. Institute for Theoretical Physics, Hamburg University, D-22761 Hamburg, Germany
²Department of Mathematical Sciences, University of Liverpool, Liverpool L69 3BX, United Kingdom

^*Contact author: kniehl@desy.de
^†Contact author: sven-olaf.moch@desy.de
^‡Contact author: vitaly.velizhanin@desy.de
^§Contact author: andreas.vogt@liverpool.ac.uk

Phys. Rev. Lett. 137, 081901 – Published 17 August, 2026

DOI: https://doi.org/10.1103/g4l4-t4cv

Abstract

We consider the anomalous dimension $γ_{g g}^{(3)} (N)$ of the twist-two gluon operator of arbitrary Lorentz spin $N$ in the quark flavor singlet sector of a general gauge theory at four loops and construct its contribution proportional to $ζ (3)$ in analytic form by applying the Lenstra-Lenstra-Lovász algorithm to the available low- $N$ moments. We exploit generalized Gribov-Liptov reciprocity, establish new self-tuning relations for the anomalous dimension matrix of the singlet sector, and inject information from $N = 1$ , 4 supersymmetric Yang-Mills theories. We also present the contribution to the rational part of $γ_{g g}^{(3)} (N)$ with color factor $C_{F}^{2} n_{f}^{2}$ . Exact contributions to the four-loop splitting function $P_{g g}^{(3)} (x)$ hence resulting via inverse Mellin transformation help us to reduce theoretical uncertainties in scaling violations of parton distribution functions in QCD.

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References (103)

J. D. Bjorken and E. A. Paschos, Inelastic electron-proton and $γ$ -proton scattering and the structure of the nucleon, Phys. Rev. 185, 1975 (1969).
H. Fritzsch, M. Gell-Mann, and H. Leutwyler, Advantages of the color octet gluon picture, Phys. Lett. 47B, 365 (1973).
V. N. Gribov and L. N. Lipatov, Deep inelastic $e p$ scattering in perturbation theory, Yad. Fiz. 15, 781 (1972) [Sov. J. Nucl. Phys. 15, 438 (1972)].
V. N. Gribov and L. N. Lipatov, $e^{+} e^{-}$ -pair annihilation and deep inelastic $e p$ scattering in perturbation theory, Yad. Fiz. 15, 1218 (1972) [Sov. J. Nucl. Phys. 15, 675 (1972)].
G. Altarelli and G. Parisi, Asymptotic freedom in parton language, Nucl. Phys. B126, 298 (1977).
Yu. L. Dokshitzer, Calculation of structure functions of deep-inelastic scattering and $e^{+} e^{-}$ annihilation by perturbation theory in quantum chromodynamics, Zh. Eksp. Teor. Fiz. 73, 1216 (1977) [Sov. Phys. JETP 46, 641 (1977)].
J. C. Collins, D. E. Soper, and G. Sterman, Factorization of hard processes in QCD, Adv. Ser. Dir. High Energy Phys. 5, 1 (1989).
S. Alekhin, M. V. Garzelli, S. Moch, and O. Zenaiev, Determination of the strong coupling from high-energy data, Eur. Phys. J. C 86, 161 (2026).
S. Alekhin and S. Moch, Heavy-quark deep-inelastic scattering with a running mass, Phys. Lett. B 699, 345 (2011).
S. Alekhin, K. Daum, K. Lipka, and S. Moch, Determination of the charm-quark mass in the $\bar{MS}$ scheme using charm production data from deep inelastic scattering at HERA, Phys. Lett. B 718, 550 (2012).
D. J. Gross and F. Wilczek, Asymptotically free gauge theories. I, Phys. Rev. D 8, 3633 (1973).
D. J. Gross and F. Wilczek, Asymptotically free gauge theories. II, Phys. Rev. D 9, 980 (1974).
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. B129, 66 (1977); B139, 545(E) (1978).
A. González-Arroyo, C. López, and F. J. Ynduráin, Second-order contributions to the structure functions in deep inelastic scattering (I). Theoretical calculations, Nucl. Phys. B153, 161 (1979).
A. González-Arroyo, C. López, and F. J. Ynduráin, Testing QCD predictions beyond the leading order for non-singlet structure functions in neutrino scattering, Nucl. Phys. B174, 474 (1980).
G. Curci, W. Furmanski, and R. Petronzio, Evolution of parton densities beyond leading order: The non-singlet case, Nucl. Phys. B175, 27 (1980).
W. Furmanski and R. Petronzio, Singlet parton densities beyond leading order, Phys. Lett. 97B, 437 (1980).
S. A. Larin, T. van Ritbergen, and J. A. M. Vermaseren, The next-next-to-leading QCD approximation for non-singlet moments of deep inelastic structure functions, Nucl. Phys. B427, 41 (1994).
S. A. Larin, P. Nogueira, T. van Ritbergen, and J. A. M. Vermaseren, The 3-loop QCD calculation of the moments of deep inelastic structure functions, Nucl. Phys. B492, 338 (1997).
S. Moch, J. A. M. Vermaseren, and A. Vogt, The three-loop splitting functions in QCD: The non-singlet case, Nucl. Phys. B688, 101 (2004).
A. Vogt, S. Moch, and J. A. M. Vermaseren, The three-loop splitting functions in QCD: The singlet case, Nucl. Phys. B691, 129 (2004).
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. B890, 48 (2014).
J. Blümlein, P. Marquard, C. Schneider, and K. Schönwald, The three-loop unpolarized and polarized non-singlet anomalous dimensions from off shell operator matrix elements, Nucl. Phys. B971, 115542 (2021).
J. Blümlein, P. Marquard, C. Schneider, and K. Schönwald, The massless three-loop Wilson coefficients for the deep-inelastic structure functions $F_{2}$ , $F_{L}$ , $x F_{3}$ and $g_{1}$ , J. High Energy Phys. 11 (2022) 156.
T. Gehrmann, A. von Manteuffel, and T. Z. Yang, Renormalization of twist-two operators in covariant gauge to three loops in QCD, J. High Energy Phys. 04 (2023) 041.
J. A. Gracey, Anomalous dimension of non-singlet Wilson operators at $O (1 / N_{f})$ in deep inelastic scattering, Phys. Lett. B 322, 141 (1994).
V. N. Velizhanin, Four loop anomalous dimension of the second moment of the non-singlet twist-2 operator in QCD, Nucl. Phys. B860, 288 (2012).
V. N. Velizhanin, Four-loop anomalous dimension of the third and fourth moments of the nonsinglet twist-2 operator in QCD, Int. J. Mod. Phys. A 35, 2050199 (2020).
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. B915, 335 (2017).
S. Moch, B. Ruijl, T. Ueda, J. A. M. Vermaseren, and A. Vogt, Four-loop non-singlet splitting functions in the planar limit and beyond, J. High Energy Phys. 10 (2017) 041.
J. Davies and A. Vogt, Absence of $π^{2}$ terms in physical anomalous dimensions in DIS: Verification and resulting predictions, Phys. Lett. B 776, 189 (2018).
S. Moch, B. Ruijl, T. Ueda, J. A. M. Vermaseren, and A. Vogt, On quartic colour factors in splitting functions and the gluon cusp anomalous dimension, Phys. Lett. B 782, 627 (2018).
G. Das, S. Moch, and A. Vogt, Approximate four-loop QCD corrections to the Higgs-boson production cross section, Phys. Lett. B 807, 135546 (2020).
S. Moch, B. Ruijl, T. Ueda, J. A. M. Vermaseren, and A. Vogt, Low moments of the four-loop splitting functions in QCD, Phys. Lett. B 825, 136853 (2022).
G. Falcioni, F. Herzog, S. Moch, and A. Vogt, Four-loop splitting functions in QCD—The quark-quark case, Phys. Lett. B 842, 137944 (2023).
T. Gehrmann, A. von Manteuffel, V. Sotnikov, and T.-Z. Yang, Complete $N_{f}^{2}$ contributions to four-loop pure-singlet splitting functions, J. High Energy Phys. 01 (2024) 029.
G. Falcioni, F. Herzog, S. Moch, and A. Vogt, Four-loop splitting functions in QCD—The gluon-to-quark case, Phys. Lett. B 846, 138215 (2023).
G. Falcioni, F. Herzog, S. Moch, J. Vermaseren, and A. Vogt, The double fermionic contribution to the four-loop quark-to-gluon splitting function, Phys. Lett. B 848, 138351 (2024).
S. Moch, B. Ruijl, T. Ueda, J. Vermaseren, and A. Vogt, Additional moments and $x$ -space approximations of four-loop splitting functions in QCD, Phys. Lett. B 849, 138468 (2024).
T. Gehrmann, A. von Manteuffel, V. Sotnikov, and T.-Z. Yang, The $N_{f} C_{F}^{3}$ contribution to the non-singlet splitting function at four-loop order, Phys. Lett. B 849, 138427 (2024).
G. Falcioni, F. Herzog, S. Moch, A. Pelloni, and A. Vogt, Four-loop splitting functions in QCD—The quark-to-gluon case, Phys. Lett. B 856, 138906 (2024).
G. Falcioni, F. Herzog, S. Moch, and S. Van Thurenhout, Constraints for twist-two alien operators in QCD, J. High Energy Phys. 11 (2024) 080.
G. Falcioni, F. Herzog, S. Moch, A. Pelloni, and A. Vogt, Four-loop splitting functions in QCD—the gluon-gluon case, Phys. Lett. B 860, 139194 (2025).
B. A. Kniehl and V. N. Velizhanin, Four-loop anomalous dimension of flavor nonsinglet twist-two operator of general lorentz spin in QCD: $ζ (3)$ term, Phys. Rev. Lett. 134, 131901 (2025).
B. A. Kniehl, S. Moch, V. N. Velizhanin, and A. Vogt, Flavor nonsinglet splitting functions at four loops in QCD: Fermionic contributions, Phys. Rev. Lett. 135, 071902 (2025).
G. Falcioni, F. Herzog, S. Moch, A. Pelloni, and A. Vogt, Additional results on the four-loop flavour-singlet splitting functions in QCD, Phys. Lett. B 875, 140278 (2026).
B. A. Kniehl and V. N. Velizhanin, Four-loop anomalous dimension of flavor non-singlet quark operator of twist two and Lorentz spin $N$ for general gauge group: Transcendental part, Nucl. Phys. B1028, 117492 (2026).
T. Gehrmann, A. von Manteuffel, V. Sotnikov, and T. Yang, The four-loop non-singlet splitting functions in QCD, arXiv:2604.09534.
M. Jamin and R. Miravitllas, Absence of even-integer $ζ$ -function values in Euclidean physical quantities in QCD, Phys. Lett. B 779, 452 (2018).
P. A. Baikov and K. G. Chetyrkin, The structure of generic anomalous dimensions and no- $π$ theorem for massless propagators, J. High Energy Phys. 06 (2018) 141.
J. A. M. Vermaseren, Harmonic sums, Mellin transforms and integrals, Int. J. Mod. Phys. A 14, 2037 (1999).
A. K. Lenstra, H. W. Lenstra, and L. Lovász, Factoring polynomials with rational coefficients, Math. Ann. 261, 515 (1982).
M. Albrech, D. Cadé, X. Pujol, and D. Stehlé (The fplll development team), fplll, a lattice reduction library, 2016, available at https://github.com/fplll/fplll.
See Supplemental Material at http://link.aps.org/supplemental/10.1103/g4l4-t4cv for a typical application of fplll and the all- $N$ results for $γ_{g g, ζ_{k}}^{(3)}$ with $k = 3$ , 4, 5.
V. N. Velizhanin, Six-loop anomalous dimension of twist-three operators in $N = 4$ SYM, J. High Energy Phys. 11 (2010) 129.
V. N. Velizhanin, Twist-2 at five loops: Wrapping corrections without wrapping computations, J. High Energy Phys. 06 (2014) 108.
C. Marboe, V. Velizhanin, and D. Volin, Six-loop anomalous dimension of twist-two operators in planar $N = 4$ SYM theory, J. High Energy Phys. 07 (2015) 084.
C. Marboe and V. Velizhanin, Twist-2 at seven loops in planar $N = 4$ SYM theory: Full result and analytic properties, J. High Energy Phys. 11 (2016) 013.
B. A. Kniehl and V. N. Velizhanin, Nonplanar cusp and transcendental anomalous dimension at four loops in $N = 4$ supersymmetric Yang-Mills theory, Phys. Rev. Lett. 126, 061603 (2021).
B. A. Kniehl and V. N. Velizhanin, Non-planar universal anomalous dimension of twist-two operators with general Lorentz spin at four loops in $N = 4$ SYM theory, Nucl. Phys. B968, 115429 (2021).
V. N. Velizhanin, NNNLLA BFKL pomeron eigenvalue in the planar $N = 4$ SYM theory, arXiv:2106.06527.
B. A. Kniehl and V. N. Velizhanin, Anomalous dimensions of twist-two operators in extended $N = 2$ and $N = 4$ super Yang-Mills theories, Nucl. Phys. B1001, 116511 (2024).
B. A. Kniehl and V. N. Velizhanin, Nonplanar four-loop anomalous dimensions of twist-two operators in $N = 4$ supersymmetric Yang-Mills theory: Higher moment, general result, and cusp anomalous dimension, Phys. Rev. D 111, L061902 (2025).
V. N. Velizhanin, Three loop anomalous dimension of the non-singlet transversity operator in QCD, Nucl. Phys. B864, 113 (2012).
Sun Tzu Suan Ching, 3rd to 5th century AD, edited by D. E. Knuth, The Art of Computer Programming, Vol. 1: Fundamental Algorithms, 3rd ed. (Adison-Wesley, Reading, MA, 1997).
S. Moch, J. A. M. Vermaseren, and A. Vogt, The three-loop splitting functions in QCD: The helicity-dependent case, Nucl. Phys. B889, 351 (2014).
S. D. Drell, D. J. Levy, and T.-M. Yan, Theory of deep-inelastic lepton-nucleon scattering and lepton-pair annihilation processes. I, Phys. Rev. 187, 2159 (1969).
J. Blümlein, V. Ravindran, and W. L. van Neerven, On the Drell-Levy-Yan relation to $O (α_{s}^{2})$ , Nucl. Phys. B586, 349 (2000).
Yu. L. Dokshitzer, G. Marchesini, and G. P. Salam, Revisiting parton evolution and the large- $x$ limit, Phys. Lett. B 634, 504 (2006).
Yu. L. Dokshitzer and G. Marchesini, $N = 4$ SUSY Yang-Mills: Three loops made simple(r), Phys. Lett. B 646, 189 (2007).
B. Basso and G. P. Korchemsky, Anomalous dimensions of high-spin operators beyond the leading order, Nucl. Phys. B775, 1 (2007).
D. J. Gross and F. Wilczek, Ultraviolet behavior of non-Abelian gauge theories, Phys. Rev. Lett. 30, 1343 (1973).
H. D. Politzer, Reliable perturbative results for strong interactions?, Phys. Rev. Lett. 30, 1346 (1973).
D. R. T. Jones, Two-loop diagrams in Yang-Mills theory, Nucl. Phys. B75, 531 (1974).
W. E. Caswell, Asymptotic behavior of non-Abelian gauge theories to two-loop order, Phys. Rev. Lett. 33, 244 (1974).
O. V. Tarasov, A. A. Vladimirov, and A. Yu. Zharkov, The Gell-Mann-Low function of QCD in the three-loop approximation, Phys. Lett. B 93, 429 (1980).
S. A. Larin and J. A. M. Vermaseren, The three-loop QCD $β$ -function and anomalous dimensions, Phys. Lett. B 303, 334 (1993).
T. van Ritbergen, J. A. M. Vermaseren, and S. A. Larin, The four-loop $β$ -function in quantum chromodynamics, Phys. Lett. B 400, 379 (1997).
K. G. Chetyrkin, Four-loop renormalization of QCD: Full set of renormalization constants and anomalous dimensions, Nucl. Phys. B710, 499 (2005).
M. Czakon, The four-loop QCD $β$ -function and anomalous dimensions, Nucl. Phys. B710, 485 (2005).
T. Luthe, A. Maier, P. Marquard, and Y. Schröder, Towards the five-loop Beta function for a general gauge group, J. High Energy Phys. 07 (2016) 127.
P. A. Baikov, K. G. Chetyrkin, and J. H. Kühn, Five-loop running of the QCD coupling constant, Phys. Rev. Lett. 118, 082002 (2017).
F. Herzog, B. Ruijl, T. Ueda, J. A. M. Vermaseren, and A. Vogt, The five-loop beta function of Yang-Mills theory with fermions, J. High Energy Phys. 02 (2017) 090.
T. Luthe, A. Maier, P. Marquard, and Y. Schröder, The five-loop Beta function for a general gauge group and anomalous dimensions beyond Feynman gauge, J. High Energy Phys. 10 (2017) 166.
L. F. Alday, A. Bissi, and T. Łukowski, Large spin systematics in CFT, J. High Energy Phys. 11 (2015) 101.
H. Chen, T.-Z. Yang, H.-X. Zhu, and Y.-J. Zhu, Analytic continuation and reciprocity relation for collinear splitting in QCD, Chin. Phys. C 45, 043101 (2021).
S.-O. Moch and L. A. Shumilov, Analytic properties of anomalous dimensions of twist-two composite operators (to be published).
A. V. Kotikov and L. N. Lipatov, DGLAP and BFKL equations in the $N = 4$ supersymmetric gauge theory, Nucl. Phys. B661, 19 (2003); B685, 405(E) (2004).
A. V. Kotikov and L. N. Lipatov, On the highest transcendentality in $N = 4$ SUSY, Nucl. Phys. B769, 217 (2007).
N. Beisert, B. Eden, and M. Staudacher, Transcendentality and crossing, J. Stat. Mech. (2007) P01021.
Z. Bajnok, R. A. Janik, and T. Łukowski, Four loop twist two, BFKL, wrapping and strings, Nucl. Phys. B816, 376 (2009).
W. L. van Neerven and E. B. Zijlstra, Order $α_{s}^{2}$ contributions to the deep inelastic Wilson coefficient, Phys. Lett. B 272, 127 (1991).
S. Moch and J. A. M. Vermaseren, Deep-inelastic structure functions at two loops, Nucl. Phys. B573, 853 (2000).
J. A. M. Vermaseren, A. Vogt, and S. Moch, The third-order QCD corrections to deep-inelastic scattering by photon exchange, Nucl. Phys. B724, 3 (2005).
S. Moch, J. A. M. Vermaseren, and A. Vogt, Third-order QCD corrections to the charged-current structure function $F_{3}$ , Nucl. Phys. B813, 220 (2009).
G. P. Korchemsky, Asymptotics of the Altarelli-Parisi-Lipatov evolution kernels of parton distributions, Mod. Phys. Lett. A 4, 1257 (1989).
A. P. Bukhvostov, É. A. Kuraev, L. N. Lipatov, and G. V. Frolov, Supersymmetry and anomalous dimensionalities of quasiparton operators in quantum chromodynamics, Pis’ma Zh. Eksp. Teor. Fiz. 41, 77 (1985) [JETP Lett. 41, 92 (1985)].
A. P. Bukhvostov, G. V. Frolov, L. N. Lipatov, and E. A. Kuraev, Evolution equations for quasi-partonic operators, Nucl. Phys. B258, 601 (1985).
A. V. Belitsky, D. Müller, and A. Schäfer, Implications of $N = 1$ supersymmetry for QCD conformal operators, Phys. Lett. B 450, 126 (1999).
B. Ruijl, T. Ueda, and J. A. M. Vermaseren, FORCER, a FORM program for the parametric reduction of four-loop massless propagator diagrams, Comput. Phys. Commun. 253, 107198 (2020).
J. A. M. Vermaseren, New features of FORM, arXiv:math-ph/0010025.
J. Kuipers, T. Ueda, J. A. M. Vermaseren, and J. Vollinga, FORM version 4.0, Comput. Phys. Commun. 184, 1453 (2013).
B. Ruijl, T. Ueda, and J. A. M Vermaseren, FORM version 4.2, arXiv:1707.06453.

Phys. Rev. Lett. 137, 081901– Published 17 August, 2026

Vol. 137, Iss. 8 — 21 August 2026

Received 28 April 2026
Revised 9 July 2026
Accepted 23 July 2026

DOI: https://doi.org/10.1103/g4l4-t4cv

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Four-Loop Gluon Anomalous Dimension of General Lorentz Spin: Transcendental Part

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