J. Abate and P. Valkó. Multi‐precision laplace transform inversion. International Journal for Numerical Methods in Engineering, 60:979 – 993, 06 2004. doi:10.1002/nme.995.


Rabah Abdul Khalek and others. Parton Distributions with Theory Uncertainties: General Formalism and First Phenomenological Studies. Eur. Phys. J. C, 79(11):931, 2019. arXiv:1906.10698, doi:10.1140/epjc/s10052-019-7401-4.


J. Ablinger, A. Behring, J. Blümlein, A. De Freitas, A. Goedicke, A. von Manteuffel, C. Schneider, and K. Schönwald. The unpolarized and polarized single-mass three-loop heavy flavor operator matrix elements A$_gg,Q$ and \ensuremath \Delta A$_gg,Q$. JHEP, 12:134, 2022. arXiv:2211.05462, doi:10.1007/JHEP12(2022)134.


J. Ablinger, A. Behring, J. Blümlein, A. De Freitas, A. von Manteuffel, C. Schneider, and K. Schönwald. The non-first-order-factorizable contributions to the three-loop single-mass operator matrix elements $A_Qg^(3)$ and $\Delta A_Qg^(3)$. 3 2024. arXiv:2403.00513.


J. Ablinger, A. Behring, J. Blümlein, A. De Freitas, A. Hasselhuhn, A. von Manteuffel, M. Round, C. Schneider, and F. Wißbrock. 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, 886:733–823, 2014. arXiv:1406.4654, doi:10.1016/j.nuclphysb.2014.07.010.


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 f2(x,q2) and the anomalous dimension. Nuclear Physics B, 890:48–151, Jan 2015. URL:, doi:10.1016/j.nuclphysb.2014.10.008.


J. Ablinger, J. Blümlein, A. De Freitas, A. Hasselhuhn, A. von Manteuffel, M. Round, and C. Schneider. The $O(\alpha _s^3 T_F^2)$ Contributions to the Gluonic Operator Matrix Element. Nucl. Phys. B, 885:280–317, 2014. arXiv:1405.4259, doi:10.1016/j.nuclphysb.2014.05.028.


J. Ablinger, J. Blümlein, A. De Freitas, A. Hasselhuhn, A. von Manteuffel, M. Round, C. Schneider, and F. Wißbrock. The transition matrix element a_gq(n) of the variable flavor number scheme at o(α_s^3). Nuclear Physics B, 882:263–288, May 2014. URL:, doi:10.1016/j.nuclphysb.2014.02.007.


J. Ablinger, J. Blümlein, S. Klein, C. Schneider, and F. Wissbrock. The $O(\alpha _s^3)$ Massive Operator Matrix Elements of $O(n_f)$ for the Structure Function $F_2(x,Q^2)$ and Transversity. Nucl. Phys. B, 844:26–54, 2011. arXiv:1008.3347, doi:10.1016/j.nuclphysb.2010.10.021.


Jakob Ablinger. Computer Algebra Algorithms for Special Functions in Particle Physics. PhD thesis, Linz U., 4 2012. arXiv:1305.0687.


Simon Albino and Richard D. Ball. Soft resummation of quark anomalous dimensions and coefficient functions in MS-bar factorization. Phys. Lett. B, 513:93–102, 2001. arXiv:hep-ph/0011133, doi:10.1016/S0370-2693(01)00742-0.


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, 854:133–152, 2012. arXiv:1107.2263, doi:10.1016/j.nuclphysb.2011.08.028.


Guido Altarelli and G. Parisi. Asymptotic Freedom in Parton Language. Nucl. Phys., B126:298–318, 1977. doi:10.1016/0550-3213(77)90384-4.


P. A. Baikov, K. G. Chetyrkin, and J. H. Kühn. Five-Loop Running of the QCD coupling constant. Phys. Rev. Lett., 118(8):082002, 2017. arXiv:1606.08659, doi:10.1103/PhysRevLett.118.082002.


Richard D. Ball, Valerio Bertone, Marco Bonvini, Stefano Forte, Patrick Groth Merrild, Juan Rojo, and Luca Rottoli. Intrinsic charm in a matched general-mass scheme. Physics Letters B, 754:49–58, Mar 2016. URL:, doi:10.1016/j.physletb.2015.12.077.


Richard D. Ball, Marco Bonvini, and Luca Rottoli. Charm in Deep-Inelastic Scattering. JHEP, 11:122, 2015. arXiv:1510.02491, doi:10.1007/JHEP11(2015)122.


A. Behring, I. Bierenbaum, J. Blümlein, A. De Freitas, S. Klein, and F. Wißbrock. The logarithmic contributions to the $O(\alpha ^3_s)$ asymptotic massive Wilson coefficients and operator matrix elements in deeply inelastic scattering. Eur. Phys. J. C, 74(9):3033, 2014. arXiv:1403.6356, doi:10.1140/epjc/s10052-014-3033-x.


Valerio Bertone, Stefano Carrazza, and Juan Rojo. APFEL: A PDF Evolution Library with QED corrections. Comput. Phys. Commun., 185:1647–1668, 2014. arXiv:1310.1394, doi:10.1016/j.cpc.2014.03.007.


I. Bierenbaum, J. Blümlein, A. De Freitas, A. Goedicke, S. Klein, and K. Schönwald. O(\ensuremath \alpha s2) polarized heavy flavor corrections to deep-inelastic scattering at Q2 \ensuremath \gg m2. Nucl. Phys. B, 988:116114, 2023. arXiv:2211.15337, doi:10.1016/j.nuclphysb.2023.116114.


Isabella Bierenbaum, Johannes Blümlein, and Sebastian Klein. Mellin Moments of the O(alpha**3(s)) Heavy Flavor Contributions to unpolarized Deep-Inelastic Scattering at Q**2 \ensuremath >\ensuremath > m**2 and Anomalous Dimensions. Nucl. Phys. B, 820:417–482, 2009. arXiv:0904.3563, doi:10.1016/j.nuclphysb.2009.06.005.


Isabella Bierenbaum, Johannes Blümlein, and Sebastian Klein. The Gluonic Operator Matrix Elements at O(alpha(s)**2) for DIS Heavy Flavor Production. Phys. Lett. B, 672:401–406, 2009. arXiv:0901.0669, doi:10.1016/j.physletb.2009.01.057.


Johannes Blümlein. Analytic continuation of mellin transforms up to two-loop order. Computer Physics Communications, 133(1):76–104, Dec 2000. URL:, doi:10.1016/s0010-4655(00)00156-9.


Johannes Blümlein. Structural Relations of Harmonic Sums and Mellin Transforms up to Weight w = 5. Comput. Phys. Commun., 180:2218–2249, 2009. arXiv:0901.3106, doi:10.1016/j.cpc.2009.07.004.


Johannes Blümlein, Jakob Ablinger, Arnd Behring, Abilio De Freitas, Andreas von Manteuffel, Carsten Schneider, and C. Schneider. Heavy Flavor Wilson Coefficients in Deep-Inelastic Scattering: Recent Results. PoS, QCDEV2017:031, 2017. arXiv:1711.07957, doi:10.22323/1.308.0031.


Johannes Blümlein and Stefan Kurth. Harmonic sums and Mellin transforms up to two loop order. Phys. Rev. D, 60:014018, 1999. arXiv:hep-ph/9810241, doi:10.1103/PhysRevD.60.014018.


Marco Bonvini. Resummation of soft and hard gluon radiation in perturbative QCD. PhD thesis, Genoa U., 2012. arXiv:1212.0480.


Marco Bonvini and Simone Marzani. Four-loop splitting functions at small $x$. JHEP, 06:145, 2018. arXiv:1805.06460, doi:10.1007/JHEP06(2018)145.


Andy Buckley, James Ferrando, Stephen Lloyd, Karl Nordström, Ben Page, Martin Rüfenacht, Marek Schönherr, and Graeme Watt. LHAPDF6: parton density access in the LHC precision era. Eur. Phys. J. C, 75:132, 2015. arXiv:1412.7420, doi:10.1140/epjc/s10052-015-3318-8.


M. Buza, Y. Matiounine, J. Smith, and W. L. van Neerven. Charm electroproduction viewed in the variable-flavour number scheme versus fixed-order perturbation theory. The European Physical Journal C, 1(1-2):301–320, Mar 1998. URL:, doi:10.1007/bf01245820.


Matteo Cacciari, Paolo Nason, and Carlo Oleari. Crossing heavy-flavor thresholds in fragmentation functions. JHEP, 10:034, 2005. arXiv:hep-ph/0504192, doi:10.1088/1126-6708/2005/10/034.


Alessandro Cafarella, Claudio Corianò, and Marco Guzzi. Precision studies of the nnlo dglap evolution at the lhc with candia. Computer Physics Communications, 179(9):665–684, Nov 2008. URL:, doi:10.1016/j.cpc.2008.06.004.


Alessandro Candido, Stefano Forte, and Felix Hekhorn. Can $ \overline \mathrm MS $ parton distributions be negative? JHEP, 11:129, 2020. arXiv:2006.07377, doi:10.1007/JHEP11(2020)129.


S. Carrazza, E. R. Nocera, C. Schwan, and M. Zaro. PineAPPL: combining EW and QCD corrections for fast evaluation of LHC processes. JHEP, 12:108, 2020. arXiv:2008.12789, doi:10.1007/JHEP12(2020)108.


Stefano Carrazza. Parton distribution functions with QED corrections. PhD thesis, Milan U., 2015. arXiv:1509.00209, doi:10.13130/carrazza-stefano_phd2015-07-06.


K. G. Chetyrkin, G. Falcioni, F. Herzog, and J. A. M. Vermaseren. Five-loop renormalisation of QCD in covariant gauges. JHEP, 10:179, 2017. [Addendum: JHEP 12, 006 (2017)]. arXiv:1709.08541, doi:10.1007/JHEP10(2017)179.


K.G. Chetyrkin, Johann H. Kuhn, and Christian Sturm. QCD decoupling at four loops. Nucl. Phys. B, 744:121–135, 2006. arXiv:hep-ph/0512060, doi:10.1016/j.nuclphysb.2006.03.020.


J. Davies, C. -H. Kom, S. Moch, and A. Vogt. Resummation of small-x double logarithms in QCD: inclusive deep-inelastic scattering. 2 2022. arXiv:2202.10362.


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, 915:335–362, 2017. arXiv:1610.07477, doi:10.1016/j.nuclphysb.2016.12.012.


Daniel de Florian, Germán F. R. Sborlini, and Germán Rodrigo. QED corrections to the Altarelli–Parisi splitting functions. Eur. Phys. J. C, 76(5):282, 2016. arXiv:1512.00612, doi:10.1140/epjc/s10052-016-4131-8.


Daniel de Florian, Germán F. R. Sborlini, and Germán Rodrigo. Two-loop QED corrections to the Altarelli-Parisi splitting functions. JHEP, 10:056, 2016. arXiv:1606.02887, doi:10.1007/JHEP10(2016)056.


M. Dittmar and others. Working Group I: Parton distributions: Summary report for the HERA LHC Workshop Proceedings. WGI, 2005. arXiv:hep-ph/0511119.


Yu. L. Dokshitzer, G. Marchesini, and G. P. Salam. Revisiting parton evolution and the large-x limit. Phys. Lett. B, 634:504–507, 2006. arXiv:hep-ph/0511302, doi:10.1016/j.physletb.2006.02.023.


Yuri L. Dokshitzer. Calculation of the Structure Functions for Deep Inelastic Scattering and e+ e- Annihilation by Perturbation Theory in Quantum Chromodynamics. Sov. Phys. JETP, 46:641–653, 1977. [Zh. Eksp. Teor. Fiz.73,1216(1977)].


Claude Duhr, Bernhard Mistlberger, and Gherardo Vita. Soft integrals and soft anomalous dimensions at N$^3$LO and beyond. JHEP, 09:155, 2022. arXiv:2205.04493, doi:10.1007/JHEP09(2022)155.


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. arXiv:2310.01245, doi:10.1016/j.physletb.2023.138351.


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. arXiv:2307.04158, doi:10.1016/j.physletb.2023.138215.


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. arXiv:2302.07593, doi:10.1016/j.physletb.2023.137944.


E. G. Floratos, C. Kounnas, and R. Lacaze. Higher Order QCD Effects in Inclusive Annihilation and Deep Inelastic Scattering. Nucl. Phys. B, 192:417–462, 1981. doi:10.1016/0550-3213(81)90434-X.


Stefano Forte, Eric Laenen, Paolo Nason, and Juan Rojo. Heavy quarks in deep-inelastic scattering. Nucl. Phys. B, 834:116–162, 2010. arXiv:1001.2312, doi:10.1016/j.nuclphysb.2010.03.014.


Thomas Gehrmann, Andreas von Manteuffel, Vasily Sotnikov, and Tong-Zhi Yang. Complete $ N_f^2 $ contributions to four-loop pure-singlet splitting functions. JHEP, 01:029, 2024. arXiv:2308.07958, doi:10.1007/JHEP01(2024)029.


W. Giele and others. The QCD / SM working group: Summary report. In 2nd Les Houches Workshop on Physics at TeV Colliders, 275–426. 4 2002. arXiv:hep-ph/0204316.


M. Gluck, E. Reya, M. Stratmann, and W. Vogelsang. Next-to-leading order radiative parton model analysis of polarized deep inelastic lepton - nucleon scattering. Phys. Rev. D, 53:4775–4786, 1996. arXiv:hep-ph/9508347, doi:10.1103/PhysRevD.53.4775.


M. Gluck, E. Reya, and A. Vogt. Parton fragmentation into photons beyond the leading order. Phys. Rev. D, 48:116, 1993. [Erratum: Phys.Rev.D 51, 1427 (1995)]. doi:10.1103/PhysRevD.51.1427.


M. Glück, E. Reya, and A. Vogt. Radiatively generated parton distributions for high-energy collisions. Z. Phys. C, 48:471–482, 1990. doi:10.1007/BF01572029.


V. N. Gribov and L. N. Lipatov. Deep inelastic e p scattering in perturbation theory. Sov. J. Nucl. Phys., 15:438–450, 1972. [Yad. Fiz.15,781(1972)].


Thomas Hahn, Sebastian Paßehr, and Christian Schappacher. FormCalc 9 and Extensions. PoS, LL2016:068, 2016. arXiv:1604.04611, doi:10.1088/1742-6596/762/1/012065.


Johannes M. Henn, Gregory P. Korchemsky, and Bernhard Mistlberger. The full four-loop cusp anomalous dimension in $\mathcal N=4$ super Yang-Mills and QCD. JHEP, 04:018, 2020. arXiv:1911.10174, doi:10.1007/JHEP04(2020)018.


F. Herzog, B. Ruijl, T. Ueda, J. A. M. Vermaseren, and A. Vogt. The five-loop beta function of Yang-Mills theory with fermions. JHEP, 02:090, 2017. arXiv:1701.01404, doi:10.1007/JHEP02(2017)090.


H. Kawamura, N. A. Lo Presti, S. Moch, and A. Vogt. On the next-to-next-to-leading order QCD corrections to heavy-quark production in deep-inelastic scattering. Nucl. Phys. B, 864:399–468, 2012. arXiv:1205.5727, doi:10.1016/j.nuclphysb.2012.07.001.


K.S. Kölbig. Programs for computing the logarithm of the gamma function, and the digamma function, for complex argument. Computer Physics Communications, 4(2):221 – 226, 1972. URL:, doi:


Tao Liu and Matthias Steinhauser. Decoupling of heavy quarks at four loops and effective Higgs-fermion coupling. Phys. Lett. B, 746:330–334, 2015. arXiv:1502.04719, doi:10.1016/j.physletb.2015.05.023.


Thomas Luthe, Andreas Maier, Peter Marquard, and York Schroder. The five-loop Beta function for a general gauge group and anomalous dimensions beyond Feynman gauge. JHEP, 10:166, 2017. arXiv:1709.07718, doi:10.1007/JHEP10(2017)166.


Thomas Luthe, Andreas Maier, Peter Marquard, and York Schröder. Towards the five-loop Beta function for a general gauge group. JHEP, 07:127, 2016. arXiv:1606.08662, doi:10.1007/JHEP07(2016)127.


A. Mitov, S. Moch, and A. Vogt. Next-to-Next-to-Leading Order Evolution of Non-Singlet Fragmentation Functions. Phys. Lett. B, 638:61–67, 2006. arXiv:hep-ph/0604053, doi:10.1016/j.physletb.2006.05.005.


Alexander Mitov and Sven-Olaf Moch. QCD Corrections to Semi-Inclusive Hadron Production in Electron-Positron Annihilation at Two Loops. Nucl. Phys. B, 751:18–52, 2006. arXiv:hep-ph/0604160, doi:10.1016/j.nuclphysb.2006.05.018.


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. arXiv:2310.05744, doi:10.1016/j.physletb.2024.138468.


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. JHEP, 10:041, 2017. arXiv:1707.08315, doi:10.1007/JHEP10(2017)041.


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. arXiv:2111.15561, doi:10.1016/j.physletb.2021.136853.


S. Moch, J. A. M. Vermaseren, and A. Vogt. The Three loop splitting functions in QCD: The Nonsinglet case. Nucl. Phys., B688:101–134, 2004. arXiv:hep-ph/0403192, doi:10.1016/j.nuclphysb.2004.03.030.


S. Moch, J. A. M. Vermaseren, and A. Vogt. The Three-Loop Splitting Functions in QCD: The Helicity-Dependent Case. Nucl. Phys. B, 889:351–400, 2014. arXiv:1409.5131, doi:10.1016/j.nuclphysb.2014.10.016.


S. Moch, J. A. M. Vermaseren, and A. Vogt. On \ensuremath \gamma 5 in higher-order QCD calculations and the NNLO evolution of the polarized valence distribution. Phys. Lett. B, 748:432–438, 2015. arXiv:1506.04517, doi:10.1016/j.physletb.2015.07.027.


S. Moch and A. Vogt. On third-order timelike splitting functions and top-mediated Higgs decay into hadrons. Phys. Lett. B, 659:290–296, 2008. arXiv:0709.3899, doi:10.1016/j.physletb.2007.10.069.


Claudio Muselli. Resummations of Transverse Momentum Distributions. PhD thesis, Università degli studi di Milano, Dipartimento di Fisica, 2017. doi:10.13130/muselli-claudio_phd2017-12-01.


Luca Rottoli. private communication.


E. Ruiz Arriola. NLO evolution for large scale distances, positivity constraints and the low-energy model of the nucleon. Nucl. Phys. A, 641:461–475, 1998. doi:10.1016/S0375-9474(98)00489-8.


Y. Schroder and M. Steinhauser. Four-loop decoupling relations for the strong coupling. JHEP, 01:051, 2006. arXiv:hep-ph/0512058, doi:10.1088/1126-6708/2006/01/051.


G. Soar, S. Moch, J. A. M. Vermaseren, and A. Vogt. On Higgs-exchange DIS, physical evolution kernels and fourth-order splitting functions at large x. Nucl. Phys. B, 832:152–227, 2010. arXiv:0912.0369, doi:10.1016/j.nuclphysb.2010.02.003.


Levan R. Surguladze. O(alpha**n alpha-s**m) corrections in e+ e- annihilation and tau decay. 9 1996. arXiv:hep-ph/9803211.


J. A. M. Vermaseren, S. A. Larin, and T. van Ritbergen. The four loop quark mass anomalous dimension and the invariant quark mass. Phys. Lett. B, 405:327–333, 1997. arXiv:hep-ph/9703284, doi:10.1016/S0370-2693(97)00660-6.


A. Vogt. Efficient evolution of unpolarized and polarized parton distributions with QCD-PEGASUS. Comput. Phys. Commun., 170:65–92, 2005. arXiv:hep-ph/0408244, doi:10.1016/j.cpc.2005.03.103.


A. Vogt, S. Moch, and J. A. M. Vermaseren. The Three-loop splitting functions in QCD: The Singlet case. Nucl. Phys., B691:129–181, 2004. arXiv:hep-ph/0404111, doi:10.1016/j.nuclphysb.2004.04.024.