Showing 1–50 of 277 results for author: Moch, S

Simultaneous extraction of top quark mass, strong coupling, effective mixing angle, and proton PDFs using inclusive DIS and proton-proton collision data

Authors: Mikel Mendizabal Morentin, Katerina Lipka, Giovanni Limatola, Efstathios Agrafiotis, Oleksandr Zenaiev, XiaoMin Shen, Sven Olaf Moch

Abstract: We present the first simultaneous determination of the proton parton distribution functions (PDFs) together with the strong coupling, the top quark pole mass, and the effective weak mixing angle. The analysis is performed at next-to-next-to-leading order in QCD using the xFitter framework, based on inclusive deep-inelastic scattering measurements at HERA and precise measurements of jet, electrowea… ▽ More We present the first simultaneous determination of the proton parton distribution functions (PDFs) together with the strong coupling, the top quark pole mass, and the effective weak mixing angle. The analysis is performed at next-to-next-to-leading order in QCD using the xFitter framework, based on inclusive deep-inelastic scattering measurements at HERA and precise measurements of jet, electroweak boson, and top quark-antiquark pair production in proton-proton collisions at the LHC. Non-relativistic QCD effects in top quark-antiquark pair production are considered at next-to-leading order, and combined with next-to-leading logarithmic soft-gluon resummation. Simultaneously with PDFs and Standard Model (SM) parameters, relevant top quark SM effective-field-theory (SMEFT) operators are constrained. The precision of the obtained SM parameters is competitive with the state-of-the art individual extractions and accounts for the correlations among those parameters and PDFs. This work paves the way for future global interpretation of the LHC data in terms of QCD, SM and SMEFT. △ Less

Submitted 12 August, 2026; v1 submitted 24 July, 2026; originally announced July 2026.

Report number: DESY-26-099

HyperFORM -- a FORM package for parametric integration with hyperlogarithms

Authors: Adam Kardos, Sven-Olaf Moch, Oliver Schnetz

Abstract: HyperFORM brings the parametric integration of hyperlogarithms, weighted by rational prefactors, into the symbolic-manipulation system FORM. It ports the capabilities of Erik Panzer's Maple package HyperInt, capitalizing on FORM's speed with bulky algebraic input and on its ability to spread a single calculation across many processor cores. We keep the description of the method brief and concentra… ▽ More HyperFORM brings the parametric integration of hyperlogarithms, weighted by rational prefactors, into the symbolic-manipulation system FORM. It ports the capabilities of Erik Panzer's Maple package HyperInt, capitalizing on FORM's speed with bulky algebraic input and on its ability to spread a single calculation across many processor cores. We keep the description of the method brief and concentrate instead on how the package is organized and driven: a fully self-contained program for the three-loop zigzag period serves as a worked illustration, and timing measurements for zigzags through six loops gauge its present reach. HyperFORM is released openly and applies to a broad class of problems, the evaluation of Feynman integrals prominently among them. △ Less

Submitted 1 July, 2026; originally announced July 2026.

Comments: 9 pages, 4 figures, 1 table; contribution to the proceedings of the Loops and Legs in Quantum Field Theory conference (LL2026), April 12-17, 2026, Bayreuth, Germany

Heavy-quark pair-production in DIS at NLO QCD matched to a parton shower

Authors: Santiago Castro, Clara Del Pio, Adam Kardos, Sven-Olaf Moch, Aris Spourdalakis

Abstract: We present theoretical predictions for heavy-quark pair-production in deep-inelastic scattering (DIS) at next-to-leading order (NLO) in quantum chromodynamics (QCD), matched to a parton shower in the POWHEG framework. We revisit the NLO heavy-quark pair-production cross section and implement a consistent matching to parton-shower evolution, with careful treatment of heavy-quark mass effects and th… ▽ More We present theoretical predictions for heavy-quark pair-production in deep-inelastic scattering (DIS) at next-to-leading order (NLO) in quantum chromodynamics (QCD), matched to a parton shower in the POWHEG framework. We revisit the NLO heavy-quark pair-production cross section and implement a consistent matching to parton-shower evolution, with careful treatment of heavy-quark mass effects and the avoidance of double counting between fixed-order and parton-shower radiation. In addition, we compare the virtual NLO corrections available in the literature to one-loop amplitudes obtained through massification in the small-mass limit. This provides an independent validation of the virtual contributions. The study is presently restricted to the gluon-initiated channel, which dominates the kinematic region of interest at the HERA collider and remains important for the future Electron-Ion Collider. △ Less

Submitted 19 June, 2026; originally announced June 2026.

Comments: 14 pages, 5 figures, 1 table

Properties and implications of the four-loop non-singlet splitting functions in QCD

Authors: S. Moch, A. Vogt

Abstract: We have studied the recently completed analytic all-N expressions for the four-loop anomalous dimensions corresponding to the next-to-next-to-next-to-leading order splitting functions for the non-singlet quark distribution in perturbative QCD. The results agree with fixed-N values beyond those published so far. Their structural consistency with theoretical requirements is established. They are use… ▽ More We have studied the recently completed analytic all-N expressions for the four-loop anomalous dimensions corresponding to the next-to-next-to-next-to-leading order splitting functions for the non-singlet quark distribution in perturbative QCD. The results agree with fixed-N values beyond those published so far. Their structural consistency with theoretical requirements is established. They are used to cast the four-loop gluon virtual anomalous dimension and the next-to-next-to-next-to-next-to-leading logarithmic threshold-resummation coefficients for lepton-pair and Higgs production in hadron-hadron collisions and deep-inelastic scattering into their final analytical forms. Further properties and consequences of the new results are addressed, in particular a new structure seen most clearly in the small-x logarithms occurring in a quartic-Casimir contribution. △ Less

Submitted 5 May, 2026; originally announced May 2026.

Comments: 37 pages, LaTeX, 6 ancillary FORM files with results and useful relations

Report number: PUBDB-2026-01424, LTH 1416

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

Authors: B. A. Kniehl, S. -O. Moch, V. N. Velizhanin, A. Vogt

Abstract: We consider the anomalous dimension $γ_{gg}^{(3)}(N)$ 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, estab… ▽ More We consider the anomalous dimension $γ_{gg}^{(3)}(N)$ 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 $\mathcal{N}=1,4$ supersymmetric Yang-Mills theories. We also present the contribution to the rational part of $γ_{gg}^{(3)}(N)$ with color factor $C_F^2n_f^2$. Exact contributions to the four-loop splitting function $P_{gg}^{(3)}(x)$ hence resulting via inverse Mellin transformation help us to reduce theoretical uncertainties in scaling violations of parton distribution functions in QCD. △ Less

Submitted 10 August, 2026; v1 submitted 28 April, 2026; originally announced April 2026.

Comments: 11 pages, minor corrections included, text and references added, Supplemental Material extended, accepted for publication in PRL

Report number: PUBDB-2026-01395, LTH 1415

Graphical Functions by Examples

Authors: Mrigankamauli Chakraborty, Marco Klann, Sven-Olaf Moch, Pooja Mukherjee, Tobias Porsche, Oliver Schnetz, Leonid A. Shumilov

Abstract: Graphical functions have emerged as a powerful framework for evaluating multi-loop Feynman integrals in perturbative quantum field theory. Defined as massless three-point position-space integrals, they reveal rich analytic structures and have enabled major advances, including the highest-loop results currently known in several quantum field theories. Their role extends to conformal field theory, a… ▽ More Graphical functions have emerged as a powerful framework for evaluating multi-loop Feynman integrals in perturbative quantum field theory. Defined as massless three-point position-space integrals, they reveal rich analytic structures and have enabled major advances, including the highest-loop results currently known in several quantum field theories. Their role extends to conformal field theory, and recent algorithmic developments now allow many graphical functions to be computed automatically. This review, based on graduate-level lectures held by O.S. in 2025/26 at the University of Hamburg, introduces the central ideas behind graphical functions, covering periods, Feynman residues, and the treatment of regular and singular cases in both integer and non-integer dimensions. It also discusses connections to momentum space and self-duality, and provides guidance for further study, offering a coherent entry point into a topic not addressed in standard textbooks. △ Less

Submitted 10 August, 2026; v1 submitted 28 April, 2026; originally announced April 2026.

Report number: PUBDB-2026-01399

Threshold Top-Quark Pair-Production: Cross Sections and Key Uncertainties

Authors: Maria Vittoria Garzelli, Giovanni Limatola, Sven-Olaf Moch, Matthias Steinhauser, Oleksandr Zenaiev

Abstract: We study theoretical uncertainties in predicting top-quark pair-production near threshold at the LHC using the non-relativistic QCD framework. We include variations in the top-quark mass and width, the strong coupling $α_s$, renormalization and factorization scales, and parton distribution functions, as well as uncertainties from the color-singlet and octet Green's functions that describe quasi-bo… ▽ More We study theoretical uncertainties in predicting top-quark pair-production near threshold at the LHC using the non-relativistic QCD framework. We include variations in the top-quark mass and width, the strong coupling $α_s$, renormalization and factorization scales, and parton distribution functions, as well as uncertainties from the color-singlet and octet Green's functions that describe quasi-bound toponium formation. These uncertainties are compared with those from standard fixed-order QCD predictions, and implications for ATLAS and CMS analyses are discussed. For the LHC at 13 TeV center-of-mass energy, the integral of the top-quark pair invariant-mass distribution from 340 to 350 GeV is 11.67 pb with ${}^{+1.43}_{-1.47}$ pb uncertainty. The corresponding excess after subtracting the POWHEG-BOX result is 4.15 pb with the same uncertainties. △ Less

Submitted 10 April, 2026; originally announced April 2026.

Precision QCD with the Electron-Ion Collider

Authors: C. Alexandrou, M. Arratia, E. C. Aschenauer, A. Avkhadiev, P. V. Balachandran, V. Bertone, I. Borsa, M. Cerutti, X. Chu, W. Cosyn, D. de Florian, A. Dumitru, M. Engelhardt, R. Fatemi, S. Forte, Y. Fu, L. Gamberg, H. Gao, T. Gehrmann, A. Gehrmann-De Ridder, Y. Go, Y. Guo, Y. Hatta, J. Haug, T. J. Hobbs , et al. (44 additional authors not shown)

Abstract: This document summarizes the discussions at the program "Precision QCD with the Electron Ion Collider", held from May to June 2025 at the Institute for Nuclear Theory (INT) at the University of Washington. The program was co-sponsored by the INT and by the Center for Frontiers in Nuclear Science (CFNS, Stony Brook University). Over its five-week duration it brought together about 70 theorists, exp… ▽ More This document summarizes the discussions at the program "Precision QCD with the Electron Ion Collider", held from May to June 2025 at the Institute for Nuclear Theory (INT) at the University of Washington. The program was co-sponsored by the INT and by the Center for Frontiers in Nuclear Science (CFNS, Stony Brook University). Over its five-week duration it brought together about 70 theorists, experimentalists and computer scientists all interested in the physics program at the future Electron Ion Collider in preparation at Brookhaven National Laboratory. Key topics at the program were: higher-order perturbative-QCD calculations and techniques; nuclear structure and tomography; comparisons of phenomenological and lattice determinations of parton distribution functions; identification of signature observables for saturated gluons; assessment of the importance of AI techniques for EIC studies and detector development. △ Less

Submitted 6 April, 2026; originally announced April 2026.

Comments: Summary of the 2025 joint CFNS-INT program: Precision QCD with the Electron-Ion Collider. 165 pages, 35 figures

Report number: INT-PUB-26-011

NNLO QCD corrections to unpolarized and polarized electroweak structure functions in semi-inclusive deep-inelastic scattering

Authors: Saurav Goyal, Sven-Olaf Moch, Vaibhav Pathak, V. Ravindran

Abstract: We present results for unpolarized and polarized semi-inclusive deep-inelastic scattering mediated by electroweak gauge bosons at next-to-next-to-leading order (NNLO) in perturbative quantum chromodynamics. The results include all relevant structure functions arising from both neutral current (NC) and charged current (CC) interactions, incorporating contributions from all partonic channels with fu… ▽ More We present results for unpolarized and polarized semi-inclusive deep-inelastic scattering mediated by electroweak gauge bosons at next-to-next-to-leading order (NNLO) in perturbative quantum chromodynamics. The results include all relevant structure functions arising from both neutral current (NC) and charged current (CC) interactions, incorporating contributions from all partonic channels with full flavor dependence. These corrections are crucial for improving the theoretical precision. A detailed numerical analysis of the NNLO corrections demonstrates their phenomenological importance, revealing sizable effects and a significant reduction in residual scale dependence in the kinematic range probed by the future Electron-Ion-Collider. These results will serve as a critical input for future global extractions of parton distributions functions and fragmentation functions. △ Less

Submitted 31 March, 2026; originally announced March 2026.

Comments: 15 pages, 4 figures

Resummation of small-spin singularities in anomalous dimensions of twist-two operators

Authors: Alexander N. Manashov, Sven-Olaf Moch, Leonid A. Shumilov

Abstract: Anomalous dimensions of leading-twist operators in QCD play an important role in precision predictions for high-energy processes, since they govern the scale evolution of parton distributions. Their analytic structure as a function of spin is particularly important due to the complexity of higher-loop computations. In these proceedings, we discuss the resummation of the certain type of such singul… ▽ More Anomalous dimensions of leading-twist operators in QCD play an important role in precision predictions for high-energy processes, since they govern the scale evolution of parton distributions. Their analytic structure as a function of spin is particularly important due to the complexity of higher-loop computations. In these proceedings, we discuss the resummation of the certain type of such singularities that share common features with those appearing in the quark flavor-nonsinglet sector of QCD. Our main focus is on the interplay between Gross-Neveu-Yukawa model in $ε$ expansion and Gross-Neveu in $1/N$ expansion. Such resummation allows one to predict the higher-loop singular behavior and reveals connections with the conformal Regge theory and recent studies of detector operators in QCD and various conformal field theories. △ Less

Submitted 15 February, 2026; originally announced February 2026.

Comments: 10 pages. RADCOR 2025 proceedings

Heavy-quark production in deep-inelastic scattering -- Mellin moments of structure functions

Authors: Marco Klann, Sven-Olaf Moch, Kay Schönwald

Abstract: We compute Mellin moments of the heavy-quark structure functions in deep-inelastic scattering at next-to-leading order in quantum chromodynamics, retaining their full dependence on the heavy-quark mass. Using the optical theorem and the operator product expansion, we derive analytic results for fixed Mellin moments $N = 2$ to $22$ of the structure functions $F_2$ and $F_L$. Our results reproduce t… ▽ More We compute Mellin moments of the heavy-quark structure functions in deep-inelastic scattering at next-to-leading order in quantum chromodynamics, retaining their full dependence on the heavy-quark mass. Using the optical theorem and the operator product expansion, we derive analytic results for fixed Mellin moments $N = 2$ to $22$ of the structure functions $F_2$ and $F_L$. Our results reproduce the known expressions in the relevant asymptotic limits, in particular for virtualities of the exchanged photon $Q^2$ much larger than the heavy-quark mass squared $m^2$, and are in agreement with existing parametrisations of the next-to-leading-order coefficient functions. The computational set-up developed in this work also provides a direct pathway toward extending these calculations to next-to-next-to-leading order. △ Less

Submitted 5 February, 2026; v1 submitted 4 February, 2026; originally announced February 2026.

Soft and Jet functions for SCET at four loops in QCD

Authors: Saurav Goyal, Sven-Olaf Moch, Vaibhav Pathak, V. Ravindran

Abstract: Soft-Collinear Effective Theory is a framework for systematically organizing and resumming the logarithmic contributions that occur in high-energy reactions. It provides a factorized description of cross sections in terms of hard, jet, soft, and beam functions. As the latter are universal, they can be obtained from the well-known perturbative results in quantum chromodynamics (QCD) for deep-inelas… ▽ More Soft-Collinear Effective Theory is a framework for systematically organizing and resumming the logarithmic contributions that occur in high-energy reactions. It provides a factorized description of cross sections in terms of hard, jet, soft, and beam functions. As the latter are universal, they can be obtained from the well-known perturbative results in quantum chromodynamics (QCD) for deep-inelastic scattering, Drell-Yan and Higgs boson productions. Using the recent results ~\cite{Kniehl:2025ttz} on four-loop eikonal $(f^I)$ and collinear anomalous dimensions $(B^I)$ for quarks and gluons, $I=q,g$, as well as perturbative results from previous orders, we present four-loop predictions for the quark and gluon soft and jet functions. They constitute an important component of the $N$-jettiness subtraction method at $\rm{N^4LO}$ accuracy in QCD, which eventually may enable the calculation of fully-differential cross sections at higher orders. △ Less

Submitted 29 December, 2025; originally announced December 2025.

Comments: 25 pages, 4 ancillary files

Additional results on the four-loop flavour-singlet splitting functions in QCD

Authors: G. Falcioni, F. Herzog, S. Moch, A. Pelloni, A. Vogt

Abstract: We have extended our previous computations, performed analytically for a general gauge group, of the even-$N$ moments $γ_{\rm ik}^{\,(3)}(N)$ of the four-loop flavour-singlet splitting functions $P_{\rm ik}^{\,(3)}(x)$ to $N = 22$. The numerical QCD results perfectly agree with all predictions resulting from the approximations for $P_{\rm ik}^{\,(3)}(x)$ that we obtained before from the moments… ▽ More We have extended our previous computations, performed analytically for a general gauge group, of the even-$N$ moments $γ_{\rm ik}^{\,(3)}(N)$ of the four-loop flavour-singlet splitting functions $P_{\rm ik}^{\,(3)}(x)$ to $N = 22$. The numerical QCD results perfectly agree with all predictions resulting from the approximations for $P_{\rm ik}^{\,(3)}(x)$ that we obtained before from the moments $N \leq 20 $ and endpoint constraints, confirming their reliability for collider-physics applications. Due to the additional analytical constraints provided by our new $N = 22$ results, we are now closing in on determining the all-$N$ forms of all non-rational ($ζ$-function) contributions to $γ_{\rm ik}^{\,(3)}(N)$: only the $n_f^{0} ζ_3$ parts of the quark-to-gluon (gq) and the $n_f^{1} ζ_3$ parts of the gluon-to-gluon (qg) cases still need to be completed. Finally we extend our approximations for all $P_{\rm ik}^{\,(3)}(x)$ to $n_f^{} = 6$ light flavours and update those for $P_{\rm gq}^{\,(3)}(x)$ at lower $n_f^{}$. △ Less

Submitted 11 December, 2025; originally announced December 2025.

Comments: 13 pages, LaTeX, 8 ancillary files (FORM files with results and FORTRAN subroutines)

Report number: DESY-25-138, LTH 1410

HyperFORM -- a FORM package for parametric integration with hyperlogarithms

Authors: Adam Kardos, Sven-Olaf Moch, Oliver Schnetz

Abstract: We present an implementation of algorithms for the symbolic integration of hyperlogarithms multiplied by rational functions in the computer algebra system FORM. This implementation encompasses cases where hyperlogarithms have rational letters or a rational argument. It complements the previous implementation, HyperInt, in MAPLE by Erik Panzer, utilizing the advantages of FORM in the efficient hand… ▽ More We present an implementation of algorithms for the symbolic integration of hyperlogarithms multiplied by rational functions in the computer algebra system FORM. This implementation encompasses cases where hyperlogarithms have rational letters or a rational argument. It complements the previous implementation, HyperInt, in MAPLE by Erik Panzer, utilizing the advantages of FORM in the efficient handling of large symbolic expressions. Among a wide range of applications, this approach enables the computation of many Feynman integrals. △ Less

Submitted 20 March, 2026; v1 submitted 25 November, 2025; originally announced November 2025.

Comments: 33 pages, 6 figures and 2 tables, version accepted for publication

Determination of the strong coupling from high-energy data

Authors: S. Alekhin, M. V. Garzelli, S. Moch, O. Zenaiev

Abstract: We determine the strong coupling from high-energy data for the Drell-Yan (DY) process and top-quark hadro-production collected at the Large Hadron Collider and the Tevatron combined with the world data on deep-inelastic scattering (DIS) and fixed-target DY data. The theory description uses results at next-to-next-to-leading order in perturbative QCD in the $\overline{\mathrm{MS}}$-scheme together… ▽ More We determine the strong coupling from high-energy data for the Drell-Yan (DY) process and top-quark hadro-production collected at the Large Hadron Collider and the Tevatron combined with the world data on deep-inelastic scattering (DIS) and fixed-target DY data. The theory description uses results at next-to-next-to-leading order in perturbative QCD in the $\overline{\mathrm{MS}}$-scheme together with leading order QED evolution. The DIS data are subject to stringent kinematic cuts to suppress the contribution of power corrections. We apply a cut on the hadronic invariant mass squared $W^2 \geq 12.5~$GeV$^2$ together with a series of cuts on momentum transfer squared $Q^2$. Discarding higher-twist terms we find that the value of strong coupling $α_s(m_Z)$ preferred by the data stabilizes at large enough cuts on $Q^2$, when low-$Q^2$ DIS data sensitive to power corrections are effectively removed. In particular, we extract the value of $α_s(m_Z,N_f=5)=0.1152 \pm 0.008$ for $N_f=5$ light flavors with the cut $Q^2>10~$GeV$^2$. In the absence of higher-twist terms less tight cuts on $Q^2$ show a clear deterioration of the fit and lead to rising values of the strong coupling, shifted upwards by about two standard deviations. △ Less

Submitted 24 October, 2025; originally announced October 2025.

Comments: 9 pages LaTeX, 2 figures

Report number: DESY 25--137

NNLO QCD$\otimes$QED corrections to unpolarized and polarized SIDIS

Authors: Saurav Goyal, Roman N. Lee, Sven-Olaf Moch, Vaibhav Pathak, V. Ravindran

Abstract: We present the first computation of next-to-next-to-leading order (NNLO) pure QED and mixed QCD$\otimes$QED corrections to unpolarized and polarized semi-inclusive deep-inelastic scattering (SIDIS). Building on our previous NNLO QCD results, these corrections are crucial for improving the theoretical precision. The coefficient functions are derived within the QCD factorization framework using dime… ▽ More We present the first computation of next-to-next-to-leading order (NNLO) pure QED and mixed QCD$\otimes$QED corrections to unpolarized and polarized semi-inclusive deep-inelastic scattering (SIDIS). Building on our previous NNLO QCD results, these corrections are crucial for improving the theoretical precision. The coefficient functions are derived within the QCD factorization framework using dimensional regularization, with consistent renormalization and mass factorization. A detailed phenomenological analysis shows that the NNLO QED and QCD$\otimes$QED terms enhance perturbative stability and reduce scale uncertainties. These results are essential for high-precision SIDIS predictions at future facilities such as the Electron-Ion Collider. △ Less

Submitted 21 October, 2025; originally announced October 2025.

Comments: 3 Tables, 2 figures

The inclusive Higgs boson cross-section in gluon-gluon fusion in soft-virtual approximation at fourth order in QCD

Authors: Goutam Das, Sven-Olaf Moch

Abstract: We present precise results for the inclusive Higgs boson cross-section in gluon-gluon fusion at the LHC considering state-of-the-art fourth-order results in perturbative QCD arising from the dominant soft and virtual gluon emissions. Utilizing four-loop QCD results for the gluon-form factor, the splitting function and related anomalous dimensions, we study the effects of threshold enhanced soft gl… ▽ More We present precise results for the inclusive Higgs boson cross-section in gluon-gluon fusion at the LHC considering state-of-the-art fourth-order results in perturbative QCD arising from the dominant soft and virtual gluon emissions. Utilizing four-loop QCD results for the gluon-form factor, the splitting function and related anomalous dimensions, we study the effects of threshold enhanced soft gluon emissions and estimate their impact on the total cross-section at the fourth order. Our study highlights the role of these higher-order contributions in improving the perturbative convergence and in significantly reducing the renormalization and factorization scale uncertainties. The results provide strong evidence for the perturbative stability and reliability of Higgs boson cross-section predictions at the LHC, thereby reinforcing the robustness of theoretical inputs in precision Higgs phenomenology. We also provide cross-section predictions using a large set of available parton distribution functions and show that, together with the value of the strong coupling $α_s(m_Z)$, they cause the largest residual uncertainty for the Higgs boson cross-section in gluon-gluon fusion. △ Less

Submitted 27 January, 2026; v1 submitted 20 October, 2025; originally announced October 2025.

Comments: Published Version, 8 pages, 2 tables, Contribution to the LHC Higgs Working Group Report 5

Report number: TTK-25-34, P3H-25-080

Journal ref: SciPost Phys. Comm. Rep. 17 (2026)

Alien operators for PDF evolution

Authors: S. Van Thurenhout, G. Falcioni, F. Herzog, S. Moch

Abstract: Understanding the scale dependence of parton distribution functions is vital for precision physics at hadron colliders. The well-known DGLAP evolution equation relates this scale dependence to the QCD splitting functions, which can be calculated perturbatively in terms of the anomalous dimensions of leading-twist gauge-invariant operators. The computation of the latter in general requires one to t… ▽ More Understanding the scale dependence of parton distribution functions is vital for precision physics at hadron colliders. The well-known DGLAP evolution equation relates this scale dependence to the QCD splitting functions, which can be calculated perturbatively in terms of the anomalous dimensions of leading-twist gauge-invariant operators. The computation of the latter in general requires one to take into account contributions of gauge-variant (or alien) operators. In this talk, we discuss the systematic study of these alien operators at arbitrary spin. Specifically, using generalized BRST symmetry relations, we derive the one-loop couplings and Feynman rules of the aliens necessary to perform the operator renormalization up to four loops in QCD. This provides an important step towards the determination of the four-loop splitting functions which will be of significant phenomenological importance at future colliders. △ Less

Submitted 2 September, 2025; originally announced September 2025.

Comments: 5 pages, LaTex, Contribution to proceedings of the European Physical Society Conference on High Energy Physics (EPS-HEP2025)

Soft and virtual corrections to semi-inclusive DIS up to four loops in QCD

Authors: Saurav Goyal, Sven-Olaf Moch, Vaibhav Pathak, Narayan Rana, V. Ravindran

Abstract: We apply the threshold resummation formalism for semi-inclusive deep-inelastic scattering (SIDIS) to derive the soft and virtual corrections for the SIDIS cross section up to four loops in QCD. Using the recently computed next-to-next-to-leading order QCD corrections for the SIDIS cross section together with known results for the form factor and splitting functions in QCD up to four loops, we deri… ▽ More We apply the threshold resummation formalism for semi-inclusive deep-inelastic scattering (SIDIS) to derive the soft and virtual corrections for the SIDIS cross section up to four loops in QCD. Using the recently computed next-to-next-to-leading order QCD corrections for the SIDIS cross section together with known results for the form factor and splitting functions in QCD up to four loops, we derive the complete soft and collinear contributions to the SIDIS coefficient functions at four-loop order. We also include systematically the next-to-leading power corrections, which are suppressed near threshold. The numerical analysis of the new four-loop corrections shows a small effect on the cross section underpinning the very good perturbative stability of the SIDIS process at that order in perturbation theory, including the reduced dependence on the renormalization and factorization scales $μ_R$ and $μ_F$. △ Less

Submitted 30 June, 2025; originally announced June 2025.

Comments: 54 pages, 7 figures, 5 tables

Report number: DESY-25-079

A novel phenomenological approach to total charm cross-section measurements at the LHC

Authors: Yewon Yang, Achim Geiser, Sven-Olaf Moch, Oleksandr Zenaiev

Abstract: We propose a novel, data-driven method for determining total charm cross sections in proton-proton collisions by extrapolating measured fiducial cross sections without assuming any particular fragmentation model. The recently observed charm fragmentation non-universality at the LHC experimentally establishes strongly increased baryon production fractions and correspondingly decreased meson product… ▽ More We propose a novel, data-driven method for determining total charm cross sections in proton-proton collisions by extrapolating measured fiducial cross sections without assuming any particular fragmentation model. The recently observed charm fragmentation non-universality at the LHC experimentally establishes strongly increased baryon production fractions and correspondingly decreased meson production fractions compared to electron-positron collisions, with a very significant $p_T$ dependence. The novel method accounts for this non-universality and its $p_T$-dependence through a data-driven extrapolation function called ddFONLL. Applied to $D^0$ production at 5 and 13 TeV, this approach yields total charm cross sections that fully incorporate the fragmentation non-universality and increase significantly compared to the previous measurements still based on fragmentation universality. The results are consistent with NNLO QCD predictions and enable direct comparisons free from fragmentation assumptions. We use this to evaluate the sensitivity of total cross-section measurements to parton distribution functions and the charm-quark mass. An outlook is given on the potential of further expanding the use of the ddFONLL method. △ Less

Submitted 14 May, 2026; v1 submitted 27 June, 2025; originally announced June 2025.

Comments: 34 pages, 18 figures, published in EPJC

Report number: DESY-25-091

Journal ref: Eur. Phys. J. C 86, 225 (2026)

Anomalous dimensions at small spins

Authors: A. N. Manashov, S. Moch, L. A. Shumilov

Abstract: In perturbation theory, the anomalous dimensions of twist-two operators have poles at negative or small positive integer values of spin and therefore must be resummed at these points. It was observed earlier that a certain quadratic combination of the anomalous dimensions remains finite at the right-most singularities, providing an efficient tool for resummation. In this paper, we analyze the smal… ▽ More In perturbation theory, the anomalous dimensions of twist-two operators have poles at negative or small positive integer values of spin and therefore must be resummed at these points. It was observed earlier that a certain quadratic combination of the anomalous dimensions remains finite at the right-most singularities, providing an efficient tool for resummation. In this paper, we analyze the small-spin behavior of the anomalous dimensions for all types of twist-two operators in the $O(N)$-symmetric $\varphi^4$ model at the four-loop level, in the complex $\varphi^3$ model at the three-loop level, and the Gross-Neveu-Yukawa model at the two-loop level. We find that the behavior of the anomalous dimensions at singular points is consistent with theoretical expectations, and we present expressions for the resummed anomalous dimensions. △ Less

Submitted 10 July, 2026; v1 submitted 5 June, 2025; originally announced June 2025.

Comments: Published version: 31 pages, 3 figures. Typos corrected, references added

Report number: DESY-25-077

Journal ref: JHEP 09 (2025) 106

Flavour Non-Singlet Splitting Functions at Four Loops in QCD -- The Fermionic Contributions

Authors: B. A. Kniehl, S. Moch, V. N. Velizhanin, A. Vogt

Abstract: We have determined the fourth-order $n_f$ contributions to the two splitting functions governing the evolution of all flavor differences of quark distributions of hadrons in perturbative quantum chromodynamics with $n_f$ light flavors. The analytic forms of these functions are presented in both Mellin $N$-space and momentum-fraction $x$-space for a general gauge group. In the small-$x$ limit doubl… ▽ More We have determined the fourth-order $n_f$ contributions to the two splitting functions governing the evolution of all flavor differences of quark distributions of hadrons in perturbative quantum chromodynamics with $n_f$ light flavors. The analytic forms of these functions are presented in both Mellin $N$-space and momentum-fraction $x$-space for a general gauge group. In the small-$x$ limit double logarithms occur, but the small-$x$ rise of both splitting functions is confined to extremely small $x$-values, $x \lesssim 10^{-6}$. The large-$x$ limit includes the $n_f$-part of the four-loop quark virtual anomalous dimension. Using this result we obtain also the $n_f$ contributions to the corresponding gluonic quantity and the complete threshold-enhanced logarithms from soft-gluon emission for a large class of inclusive observables, including Higgs boson production in gluon-gluon fusion. △ Less

Submitted 14 May, 2025; originally announced May 2025.

Comments: 21 pages latex, 2 figures; 1 ancillary file (FORM file)

Report number: DESY 25-067, LTH 1399

Top-quark pole mass extraction at NNLO accuracy

Authors: S. Alekhin, M. V. Garzelli, J. Mazzitelli, S. -O. Moch, O. Zenaiev

Abstract: We describe our recent NNLO QCD extraction of the top-quark pole mass from fits to experimental data on total inclusive and normalized (multi)-differential cross sections for $t\bar{t} + X$ hadroproduction, using as input various modern PDF + $α_s(M_Z)$ sets. We find top-quark mass values compatible among each other and with the PDG 2024 preferred value. We describe our recent NNLO QCD extraction of the top-quark pole mass from fits to experimental data on total inclusive and normalized (multi)-differential cross sections for $t\bar{t} + X$ hadroproduction, using as input various modern PDF + $α_s(M_Z)$ sets. We find top-quark mass values compatible among each other and with the PDG 2024 preferred value. △ Less

Submitted 28 December, 2024; originally announced December 2024.

Comments: 6 pages, 2 figures, invited contribution to the Proceedings of the 12th Large Hadron Collider Physics Conference (LHCP2024), Boston, USA, 3-7 June 2024

Report number: CERN-TH-2024-225

Journal ref: PoS LHCP2024 (2025) 320

NNLO QCD corrections to unpolarized and polarized SIDIS

Authors: Saurav Goyal, Roman N. Lee, Sven-Olaf Moch, Vaibhav Pathak, Narayan Rana, V. Ravindran

Abstract: The semi-inclusive deep-inelastic scattering (SIDIS) process requires the presence of an identified hadron H$'$ in the final state, which arises from the scattering of a lepton with an initial hadron P. By employing factorization in quantum chromodynamics (QCD), SIDIS provides essential knowledge on the hadron structure, enabling the exploration of parton distribution functions (PDFs) and fragment… ▽ More The semi-inclusive deep-inelastic scattering (SIDIS) process requires the presence of an identified hadron H$'$ in the final state, which arises from the scattering of a lepton with an initial hadron P. By employing factorization in quantum chromodynamics (QCD), SIDIS provides essential knowledge on the hadron structure, enabling the exploration of parton distribution functions (PDFs) and fragmentation functions (FFs). The coefficient functions for SIDIS can be calculated in perturbative QCD and are currently known to the next-to-next-to-leading order (NNLO) for the cases, where the incoming lepton and the hadron P are either both polarized or unpolarized. We present a detailed description of these NNLO computations, including a thorough discussion of all the partonic channels, the calculation of the amplitudes and master integrals for the phase-space integration as well as the renormalization of ultraviolet divergences and mass factorization of infrared divergences in dimensional regularization through NNLO. We provide an extensive phenomenological analysis of the effects of NNLO corrections on SIDIS cross sections for different PDFs and FFs and various kinematics, including those of the future Electron-Ion Collider (EIC). We find that these corrections are not only significant but also crucial for reducing the dependence on the renormalization and factorization scales $μ_R$ and $μ_F$ to obtain stable predictions. △ Less

Submitted 31 October, 2025; v1 submitted 26 December, 2024; originally announced December 2024.

Comments: ancillary files updated;

Journal ref: Phys.Rev.D 111 (2025) 9, 094007

Updated predictions for toponium production at the LHC

Authors: M. V. Garzelli, G. Limatola, S. -O. Moch, M. Steinhauser, O. Zenaiev

Abstract: We provide an update on QCD predictions for top-quark pair production close to threshold including bound state effects at the Large Hadron Collider. We compute the top-quark pair invariant mass distribution $dσ/dM_{t\bar{t}}$, including Coulomb resummation for bound-state effects, as well as threshold resummation for emissions of soft and collinear gluons. We discuss uncertainty estimates and pres… ▽ More We provide an update on QCD predictions for top-quark pair production close to threshold including bound state effects at the Large Hadron Collider. We compute the top-quark pair invariant mass distribution $dσ/dM_{t\bar{t}}$, including Coulomb resummation for bound-state effects, as well as threshold resummation for emissions of soft and collinear gluons. We discuss uncertainty estimates and present a proposal for the use of these predictions in experimental analyses. △ Less

Submitted 15 May, 2025; v1 submitted 21 December, 2024; originally announced December 2024.

Comments: 10 pages LaTeX, 4 figures; 2 ancillary files with cross section predictions (Journal version)

Report number: CERN-TH-2024-222, DESY 24-207, P3H-24-104, TTP24-048

NNLO phase-space integrals for semi-inclusive deep-inelastic scattering

Authors: Taushif Ahmed, Saurav Goyal, Syed Mehedi Hasan, Roman N. Lee, Sven-Olaf Moch, Vaibhav Pathak, Narayan Rana, Andreas Rapakoulias, V. Ravindran

Abstract: We evaluate the phase-space integrals that arise in double real emission diagrams for semi-inclusive deep-inelastic scattering at next-to-next-to-leading order (NNLO) in QCD. Utilizing the reverse unitarity technique, we convert these integrals into loop integrals, allowing us to employ integration-by-parts identities and reduce them to a set of master integrals. The master integrals are then solv… ▽ More We evaluate the phase-space integrals that arise in double real emission diagrams for semi-inclusive deep-inelastic scattering at next-to-next-to-leading order (NNLO) in QCD. Utilizing the reverse unitarity technique, we convert these integrals into loop integrals, allowing us to employ integration-by-parts identities and reduce them to a set of master integrals. The master integrals are then solved using the method of differential equations and expressed in terms of Goncharov polylogarithms. By examining the series expansion in the dimensional regulator, we discover additional relations among some of the master integrals. As an alternative approach, we solve the master integrals by decomposing them into angular and radial components. The angular parts are evaluated using Mellin-Barnes representation, while special attention is given to the singular structures of the radial integrals to handle them accurately. Here the results are provided in terms of one-fold integrals over classical polylogarithms. This approach provides a clearer understanding of the origin of soft and collinear singularities. △ Less

Submitted 28 March, 2025; v1 submitted 21 December, 2024; originally announced December 2024.

Comments: Changed style to revtex, added one reference

NNLO fits of top-quark mass using total, single-differential and double-differential $t\bar{t}+X$ cross-section data

Authors: S. Alekhin, M. V. Garzelli, J. Mazzitelli, S. -O. Moch, O. Zenaiev

Abstract: We describe the fits of the top-quark mass value at NNLO using as input the double-differential distributions in rapidity and invariant mass of $t\bar{t}$ pairs obtained by the ATLAS and CMS collaborations from unfolding of their experimental data to the parton level, compared to NNLO theory predictions. We consider different state-of-the-art PDF sets, finding results of the fits compatible among… ▽ More We describe the fits of the top-quark mass value at NNLO using as input the double-differential distributions in rapidity and invariant mass of $t\bar{t}$ pairs obtained by the ATLAS and CMS collaborations from unfolding of their experimental data to the parton level, compared to NNLO theory predictions. We consider different state-of-the-art PDF sets, finding results of the fits compatible among each other within uncertainties. On the other hand, we observe some tension among the fits to different datasets. △ Less

Submitted 18 December, 2024; originally announced December 2024.

Comments: Contribution to proceedings of the 42nd International Conference on High Energy Physics (ICHEP 2024), 18-24 July 2024

Report number: CERN-TH-2024-220

Journal ref: PoS ICHEP2024 (2025) 393

Four-loop splitting functions in QCD -- The gluon-gluon case --

Authors: G. Falcioni, F. Herzog, S. Moch, A. Pelloni, A. Vogt

Abstract: We have computed the even-N moments N =< 20 of the gluon-gluon splitting function P_{gg} at the fourth order of perturbative QCD via the renormalization of off-shell operator matrix elements. Our results, derived analytically for a general compact simple gauge group, agree with all results obtained for this function so far, in particular with the lowest five moments obtained via structure function… ▽ More We have computed the even-N moments N =< 20 of the gluon-gluon splitting function P_{gg} at the fourth order of perturbative QCD via the renormalization of off-shell operator matrix elements. Our results, derived analytically for a general compact simple gauge group, agree with all results obtained for this function so far, in particular with the lowest five moments obtained via structure functions in deep-inelastic scattering. Using our new moments and all available endpoint constraints, we construct improved approximations for the four-loop P_{gg}(x) that should be sufficient for a wide range of collider-physics applications. The N^3LO contributions to the scale derivative of the gluon distribution, resulting from these and the corresponding quark-to-gluon splitting functions, amount to 1% or less at x >~ 10^{-4} at a standard reference scale with alpha_s = 0.2. △ Less

Submitted 10 October, 2024; originally announced October 2024.

Comments: 15 pages, LaTeX, 3 figures, 2 ancillary files (FORM file with results and FORTRAN subroutine). arXiv admin note: text overlap with arXiv:2404.09701

Report number: ZU-TH 47/24, DESY-24-144, LTH 1384

Constraints for twist-two alien operators in QCD

Authors: G. Falcioni, F. Herzog, S. Moch, S. Van Thurenhout

Abstract: Parton evolution equations in QCD are controlled by the anomalous dimensions of gauge-invariant twist-two spin-$N$ quark and gluon operators. Under renormalization, these mix with gauge-variant operators of the same quantum numbers, referred to as alien operators. Our work addresses the systematic study of these alien operators at arbitrary spin $N$, using generalized BRST symmetry relations to de… ▽ More Parton evolution equations in QCD are controlled by the anomalous dimensions of gauge-invariant twist-two spin-$N$ quark and gluon operators. Under renormalization, these mix with gauge-variant operators of the same quantum numbers, referred to as alien operators. Our work addresses the systematic study of these alien operators at arbitrary spin $N$, using generalized BRST symmetry relations to derive their couplings and Feynman rules at all values of $N$. We observe how the all-$N$ structure of the generalized (anti-)BRST constraints relates the couplings of alien operators with $n+1$ gluons to those with $n$ gluons. Realizing a bootstrap, we present all one-loop results necessary for performing the operator renormalization up to four loops in QCD. △ Less

Submitted 10 December, 2024; v1 submitted 4 September, 2024; originally announced September 2024.

Comments: 28 pages, LaTex, 4 figures, all-N results in ancillary Mathematica file, journal version

Report number: ZU-TH 43/24, DESY-24-108

Journal ref: JHEP 11 (2024) 080

Three-loop evolution kernel for transversity operator

Authors: A. N. Manashov, S. Moch, L. A. Shumilov

Abstract: We calculate quantum corrections to the symmetry generators for the transversity operators in quantum chromodynamics (QCD) in the two-loop approximation. Using this result, we obtain the evolution kernel for the corresponding operators at three loops. The explicit expression for the anomalous dimension matrix in the Gegenbauer basis is given for the first few operators. We calculate quantum corrections to the symmetry generators for the transversity operators in quantum chromodynamics (QCD) in the two-loop approximation. Using this result, we obtain the evolution kernel for the corresponding operators at three loops. The explicit expression for the anomalous dimension matrix in the Gegenbauer basis is given for the first few operators. △ Less

Submitted 23 September, 2024; v1 submitted 17 July, 2024; originally announced July 2024.

Comments: 28 pages, 2 figures, v2: references added, typos corrected

Analysis of $(n+1)$ and $n$-parton contributions for computing QCD jet cross sections in the local analytic subtraction scheme

Authors: Bakar Chargeishvili, Giuseppe Bevilacqua, Adam Kardos, Sven-Olaf Moch, Zoltán Trócsányi

Abstract: We analyze and implement the Local Analytic Sector Subtraction (LASS) scheme for handling infrared singularities in next-to-next-to-leading order (NNLO) calculations in perturbative QCD. We examine the key aspects of the scheme including sector function construction, singular limit parametrization, subtraction counterterm derivation, and integration techniques. As a proof-of-concept, we numericall… ▽ More We analyze and implement the Local Analytic Sector Subtraction (LASS) scheme for handling infrared singularities in next-to-next-to-leading order (NNLO) calculations in perturbative QCD. We examine the key aspects of the scheme including sector function construction, singular limit parametrization, subtraction counterterm derivation, and integration techniques. As a proof-of-concept, we numerically implement LASS for the process $e^+e^- \rightarrow 3$ jets. In this study we examine the limiting behavior of subtraction terms for real-virtual contribution and explicitly demonstrate the pole cancellation of the double-virtual contribution. Differential cross sections are computed for several event shape observables, showing the stability and efficiency of LASS scheme. This work lays the foundation for developing an automated tool for NNLO QCD calculations using this promising scheme. △ Less

Submitted 2 July, 2024; originally announced July 2024.

Comments: 10 pages, 4 figures, contribution to the proceedings of Loops and Legs in Quantum Field Theory (LL2024), Wittenberg, Germany

LASS, the numerics

Authors: Adam Kardos, Giuseppe Bevilacqua, Bakar Chargeishvili, Sven-Olaf Moch, Zoltan Trocsanyi

Abstract: We summarize our efforts to create a numerical implementation of the Local Analytical Subtraction Scheme (LASS) for obtaining NNLO QCD predictions in electron-positron collisions. We focus on the regularization of double-real radiation contributions, which are typically the most computationally intensive part of NNLO calculations. We describe the structure of the subtraction terms in LASS and our… ▽ More We summarize our efforts to create a numerical implementation of the Local Analytical Subtraction Scheme (LASS) for obtaining NNLO QCD predictions in electron-positron collisions. We focus on the regularization of double-real radiation contributions, which are typically the most computationally intensive part of NNLO calculations. We describe the structure of the subtraction terms in LASS and our approach to numerically validate them. Using arbitrary precision arithmetic, we demonstrate the proper convergence of individual subtraction terms, overlap removal terms, and spurious singularity cancellations in various singular limits. We show results for the specific case of three-jet production in $e^+e^-$ collisions. Our results validate the correctness of both the LASS formalism and our numerical implementation, setting the stage for full NNLO calculations using this method. △ Less

Submitted 2 July, 2024; originally announced July 2024.

Comments: 11 pages, 4 figures, contribution to the proceedings of Loops and Legs in Quantum Field Theory (LL2024), Wittenberg, Germany

NNLO PDFs driven by top-quark data

Authors: S. Alekhin, M. V. Garzelli, S. -O. Moch, O. Zenaiev

Abstract: We study the impact of state-of-the-art top-quark data collected at the Large Hadron Collider on parton distribution functions (PDFs). Following the ABMP methodology, the fit extracts simultaneously proton PDFs, the strong coupling $α_s(M_Z)$ and heavy-quark masses at next-to-next-to-leading order (NNLO) accuracy in QCD. It includes recent high-statistics data on absolute total inclusive cross sec… ▽ More We study the impact of state-of-the-art top-quark data collected at the Large Hadron Collider on parton distribution functions (PDFs). Following the ABMP methodology, the fit extracts simultaneously proton PDFs, the strong coupling $α_s(M_Z)$ and heavy-quark masses at next-to-next-to-leading order (NNLO) accuracy in QCD. It includes recent high-statistics data on absolute total inclusive cross sections for $t\bar{t}+X$, the sum of $(t + X)$ and $(\bar{t} + X)$ hadroproduction, and normalized inclusive data double-differential in the invariant mass and rapidity of the $t\bar{t}$ pair at $\sqrt{S}=13$ TeV. The gluon PDF at large $x$ and the top-quark mass value derived from these data are well compatible with the previous ABMP16 results, but with significantly smaller uncertainties, reduced by up to a factor of two. At NNLO in QCD we obtain for the strong coupling the value $α_s^{(n_f=5)}(M_Z)= 0.1150 \pm 0.0009$ and for the top-quark mass in the ${\overline{\mbox{MS}}}$-scheme $m_t(m_t) = 160.6 \pm 0.6$ GeV, corresponding to $m_t^{\rm pole} = 170.2 \pm 0.7$ GeV in the on-shell scheme. The new fit, dubbed ABMPtt, is publicly released in grids in LHAPDF format. △ Less

Submitted 22 January, 2025; v1 submitted 29 June, 2024; originally announced July 2024.

Comments: 33 pages, 22 figures, 4 tables, interpolation tables for top-antitop production available at https://ploughshare.web.cern.ch v2: minor revision accounting for the comments of the Referee, to be published in EPJC

Report number: DESY 24-092

Journal ref: Eur. Phys. J. C 85 (2025) 162, 1 - 25

A Benchmarking of QCD Evolution at Approximate $N^3LO$

Authors: A. Cooper-Sarkar, T. Cridge, F. Giuli, L. A. Harland-Lang, F. Hekhorn, J. Huston, G. Magni, S. Moch, R. S. Thorne

Abstract: We present a detailed benchmarking of different treatments of the QCD evolution of unpolarized PDFs at approximate $N^3LO$ ($aN^3LO$) order in the QCD coupling. Namely, the implementations in the public $aN^3LO$ releases of the MSHT and NNPDF global PDF fitters, as well as that of the theoretical FHMRUVV collaboration are compared. This follows the same procedure as in previous benchmarking exerci… ▽ More We present a detailed benchmarking of different treatments of the QCD evolution of unpolarized PDFs at approximate $N^3LO$ ($aN^3LO$) order in the QCD coupling. Namely, the implementations in the public $aN^3LO$ releases of the MSHT and NNPDF global PDF fitters, as well as that of the theoretical FHMRUVV collaboration are compared. This follows the same procedure as in previous benchmarking exercises at lower order, that is by considering the impact of this evolution on a set of simple toy PDFs. △ Less

Submitted 23 June, 2024; originally announced June 2024.

Charm total cross sections with nonuniversal fragmentation treatment

Authors: A. Geiser, Y. Yang, S. Moch, O. Zenaiev

Abstract: Total charm-pair cross sections in $pp$ collisions are interesting because they can be calculated to NNLO in QCD without any reference to fragmentation effects. On the other hand, the fiducial differential charm cross sections from which the total cross sections must be extrapolated are currently known to NLO+NLL at most (e.g. FONLL), and must be treated for known effects of nonuniversal charm fra… ▽ More Total charm-pair cross sections in $pp$ collisions are interesting because they can be calculated to NNLO in QCD without any reference to fragmentation effects. On the other hand, the fiducial differential charm cross sections from which the total cross sections must be extrapolated are currently known to NLO+NLL at most (e.g. FONLL), and must be treated for known effects of nonuniversal charm fragmentation. A new procedure using the FONLL framework as input for an empirical parametrization of the data in both shape and normalization, with all its parameters actually fitted to data, is used to derive so-called data-driven FONLL (ddFONLL) parametrizations which can be used to extrapolate the differential cross sections to total cross sections with minimal bias. This includes an empirical treatment of all known non-universal charm fragmentation effects, in particular for the baryon-to-meson ratio as a function of transverse momentum. The total charm-pair cross sections obtained in this way, which supersede all previous ones obtained using the assumption of charm-fragmentation universality, are consistent with NNLO predictions, and allow first studies of their sensitivity e.g. to the charm-quark mass and/or the NNLO gluon PDF at very low transverse momentum fractions $x$. △ Less

Submitted 5 June, 2024; originally announced June 2024.

Comments: 6 pages, 6 Figures, contribution to the 2024 QCD session of the 58th Rencontres de Moriond

Les Houches 2023: Physics at TeV Colliders: Standard Model Working Group Report

Authors: J. Andersen, B. Assi, K. Asteriadis, P. Azzurri, G. Barone, A. Behring, A. Benecke, S. Bhattacharya, E. Bothmann, S. Caletti, X. Chen, M. Chiesa, A. Cooper-Sarkar, T. Cridge, A. Cueto Gomez, S. Datta, P. K. Dhani, M. Donega, T. Engel, S. Ferrario Ravasio, S. Forte, P. Francavilla, M. V. Garzelli, A. Ghira, A. Ghosh , et al. (59 additional authors not shown)

Abstract: This report presents a short summary of the activities of the "Standard Model" working group for the "Physics at TeV Colliders" workshop (Les Houches, France, 12-30 June, 2023). This report presents a short summary of the activities of the "Standard Model" working group for the "Physics at TeV Colliders" workshop (Les Houches, France, 12-30 June, 2023). △ Less

Submitted 2 June, 2024; originally announced June 2024.

Comments: Proceedings of the Standard Model Working Group of the 2023 Les Houches Workshop, Physics at TeV Colliders, Les Houches 12-30 June 2023. 48 pages

Report number: DESY-24-076

Status of QCD precision predictions for Drell-Yan rapidity distributions

Authors: S. Alekhin, S. Amoroso, L. Buonocore, A. Huss, S. Kallweit, A. Kardos, J. Michel, S. Moch, F. Petriello, L. Rottoli, Z. Trócsányi, M. Wiesemann

Abstract: We compute differential distributions for Drell-Yan processes at the LHC and the Tevatron colliders at next-to-next-to-leading order in perturbative QCD, including fiducial cuts on the decay leptons in the final state. The comparison of predictions obtained with four different codes shows excellent agreement, once linear power corrections from the fiducial cuts are included in those codes that rel… ▽ More We compute differential distributions for Drell-Yan processes at the LHC and the Tevatron colliders at next-to-next-to-leading order in perturbative QCD, including fiducial cuts on the decay leptons in the final state. The comparison of predictions obtained with four different codes shows excellent agreement, once linear power corrections from the fiducial cuts are included in those codes that rely on phase-space slicing subtraction schemes. For Z-boson production we perform a detailed study of the symmetric cuts on the transverse momenta of the decay leptons. Predictions at fixed order in perturbative QCD for those symmetric cuts, typically imposed in experiments, suffer from an instability. We show how this can be remedied by an all-order resummation of the fiducial transverse momentum spectrum, and we comment on the choice of cuts for future experimental analyses. △ Less

Submitted 8 April, 2025; v1 submitted 30 May, 2024; originally announced May 2024.

Comments: 41 pages LaTeX, 19 figures, 6 tables (journal version EPJC)

Report number: DESY 24-073, CERN-TH-2024-068, ZU-TH 26/24, MPP-2024-106

Next-to-Next-to-Leading Order QCD Corrections to Polarized Semi-Inclusive Deep-Inelastic Scattering

Authors: Saurav Goyal, Roman N. Lee, Sven-Olaf Moch, Vaibhav Pathak, Narayan Rana, V. Ravindran

Abstract: Polarized semi-inclusive deep-inelastic scattering (SIDIS) is a key process in the quest for a resolution of the proton spin puzzle. We present the complete results for the polarized SIDIS process at next-to-next-to-leading order (NNLO) in perturbative quantum chromodynamics. Our analytical results include all partonic channels for the scattering of polarized leptons off hadrons and a spin-average… ▽ More Polarized semi-inclusive deep-inelastic scattering (SIDIS) is a key process in the quest for a resolution of the proton spin puzzle. We present the complete results for the polarized SIDIS process at next-to-next-to-leading order (NNLO) in perturbative quantum chromodynamics. Our analytical results include all partonic channels for the scattering of polarized leptons off hadrons and a spin-averaged hadron identified in the final state. A numerical analysis of the NNLO corrections illustrates their significance and the reduced residual scale dependence in the kinematic range probed by the future Electron-Ion-Collider EIC. △ Less

Submitted 31 October, 2025; v1 submitted 15 April, 2024; originally announced April 2024.

Comments: 6 pages, 3 figures; ancillary files updated;

Journal ref: Phys. Rev. Lett. 133, 211905 (2024)

Four-loop splitting functions in QCD -- The quark-to-gluon case

Authors: G. Falcioni, F. Herzog, S. Moch, A. Pelloni, A. Vogt

Abstract: We present the even-N moments N =< 20 of the fourth-order (N^3LO) contribution P_{gq}^(3)(x) to the quark-to-gluon splitting function in perturbative QCD. These moments, obtained by analytically computing off-shell operator matrix elements for a general gauge group, agree with all known results, in particular with the moments N =< 10 derived before from structure functions in deep-inelastic scatte… ▽ More We present the even-N moments N =< 20 of the fourth-order (N^3LO) contribution P_{gq}^(3)(x) to the quark-to-gluon splitting function in perturbative QCD. These moments, obtained by analytically computing off-shell operator matrix elements for a general gauge group, agree with all known results, in particular with the moments N =< 10 derived before from structure functions in deep-inelastic scattering. Using the new moments and the available endpoint constraints, we construct approximations for P_{gq}^(3)(x) which improve upon those obtained from the lowest five even moments. The remaining uncertainties of this function are now practically irrelevant at momentum fractions x > 0.1. The resulting errors of the convolution of P_{gq} at N^3LO with a typical quark distribution are small at x >~ 10^{-3} and exceed 1% only at x ~< 10^{-4} for a strong coupling alpha_s = 0.2. The present results for P_{gq}^(3)(x) should thus be sufficient for most collider-physics applications. △ Less

Submitted 15 April, 2024; originally announced April 2024.

Comments: 15 pages, LaTex, 2 figures, 2 ancillary files (FORM file with results and FORTRAN subroutine)

Report number: ZU-TH 20/24, DESY-24-053, LTH 1367

NNLO QCD corrections to semi-inclusive DIS

Authors: Saurav Goyal, Sven-Olaf Moch, Vaibhav Pathak, Narayan Rana, V. Ravindran

Abstract: We present the first results for the next-to-next-to leading order (NNLO) corrections to the semi-inclusive deep-inelastic scattering process in perturbative quantum chromodynamics. We consider the quark initiated flavor non-singlet process and obtain the complete contributions analytically at leading color. All relevant virtual and real emission Feynman diagrams have been computed using integrati… ▽ More We present the first results for the next-to-next-to leading order (NNLO) corrections to the semi-inclusive deep-inelastic scattering process in perturbative quantum chromodynamics. We consider the quark initiated flavor non-singlet process and obtain the complete contributions analytically at leading color. All relevant virtual and real emission Feynman diagrams have been computed using integration-by-parts reduction to master integrals and two approaches for their subsequent evaluation (parametric phase-space integration and method of differential equations). The numerical analysis demonstrates the significance of the NNLO corrections and their great impact on the reduction of the residual scale dependence. △ Less

Submitted 20 March, 2024; v1 submitted 29 December, 2023; originally announced December 2023.

Comments: 6 pages, 1 figure; Ancillary file updated

Impact of SeaQuest data on sea-quark PDFs at large $x$

Authors: S. Alekhin, M. V. Garzelli, S. Kulagin, S. -O. Moch

Abstract: We investigate the impact of the recently released FNAL-E906 (SeaQuest) data on the ratio of proton-deuteron and proton-proton Drell-Yan production cross-sections on the sea quark PDFs. We find that they have constraining power on the light-quark sea isospin asymmetry $(\bar{d}-\bar{u})(x)$ and on the $(\bar{d}/\bar{u})(x)$ ratio at large longitudinal momentum fraction $x$ values, and that their c… ▽ More We investigate the impact of the recently released FNAL-E906 (SeaQuest) data on the ratio of proton-deuteron and proton-proton Drell-Yan production cross-sections on the sea quark PDFs. We find that they have constraining power on the light-quark sea isospin asymmetry $(\bar{d}-\bar{u})(x)$ and on the $(\bar{d}/\bar{u})(x)$ ratio at large longitudinal momentum fraction $x$ values, and that their constraints turn out to be compatible with those from Drell-Yan data in collider experiments (Tevatron and Large Hadron Collider) and in the old fixed-target experiment by the FNAL-E866 collaboration. We study the impact of nuclear corrections due to the deuteron target, finding them within 1% in most of the kinematic region covered by SeaQuest. We perform a new proton PDF fit, including SeaQuest data, using the ABMP16 methodology and we compare it to the ABMP16 baseline. △ Less

Submitted 19 December, 2023; originally announced December 2023.

Comments: 6 pages, 2 figures, 2 tables; talk at "The European Physical Society Conference on High Energy Physics (EPS-HEP2023)", 21-25 August 2023, Hamburg, Germany; submitted to PoS - Proceedings of Science

Compatibility between theoretical predictions and experimental data for top-antitop hadroproduction at NNLO QCD accuracy

Authors: Maria Vittoria Garzelli, Javier Mazzitelli, Sven-Olaf Moch, Oleksandr Zenaiev

Abstract: We compare double-differential normalized production cross sections for top-antitop $+ X$ hadroproduction at NNLO QCD accuracy, as obtained through a customized version of the MATRIX framework interfaced to PineAPPL, with recent data by the ATLAS and CMS collaborations. We take into account theory uncertainties due to scale variation and we see how predictions vary as a function of parton distribu… ▽ More We compare double-differential normalized production cross sections for top-antitop $+ X$ hadroproduction at NNLO QCD accuracy, as obtained through a customized version of the MATRIX framework interfaced to PineAPPL, with recent data by the ATLAS and CMS collaborations. We take into account theory uncertainties due to scale variation and we see how predictions vary as a function of parton distribution function (PDF) choice and top-quark pole mass value, considering different state-of-the-art PDF fits with their uncertainties. Notwithstanding the overall reasonable good agreement, we observe discrepancies at the level of a few $σ$'s between data and theoretical predictions in some kinematical regions, which can be alleviated by refitting the top-quark mass value, and/or the PDFs and/or $α_s(M_Z)$, considering the correlations between these three quantities. In a fit of top-quark mass standalone, we notice that, for all considered PDF + $α_s(M_Z)$ sets used as input, some datasets point towards top-quark pole mass values lower by about $2\,σ$'s than those emerging from fitting other datasets, suggesting a possible tension between experimental measurements using different decay channels, and/or the need of better estimating uncertainties on the latter. △ Less

Submitted 5 December, 2023; v1 submitted 16 November, 2023; originally announced November 2023.

Comments: 6 pages, 2 figures, talk at "The European Physical Society Conference on High Energy Physics (EPS-HEP2023)", 21-25 August 2023, Hamburg, Germany; submitted to PoS - Proceedings of Science

Top-quark pole mass extraction at NNLO accuracy, from total, single- and double-differential cross sections for $t\bar{t}+X$ production at the LHC

Authors: M. V. Garzelli, J. Mazzitelli, S. -O. Moch, O. Zenaiev

Abstract: We extract the top-quark mass value in the on-shell renormalization scheme from the comparison of theoretical predictions for $pp \rightarrow t\bar{t} + X$ at next-to-next-to-leading order (NNLO) QCD accuracy with experimental data collected by the ATLAS and CMS collaborations for absolute total, normalized single-differential and double-differential cross-sections during Run 1, Run 2 and the ongo… ▽ More We extract the top-quark mass value in the on-shell renormalization scheme from the comparison of theoretical predictions for $pp \rightarrow t\bar{t} + X$ at next-to-next-to-leading order (NNLO) QCD accuracy with experimental data collected by the ATLAS and CMS collaborations for absolute total, normalized single-differential and double-differential cross-sections during Run 1, Run 2 and the ongoing Run 3 at the Large Hadron Collider (LHC). For the theory computations of heavy-quark pair-production we use the MATRIX framework, interfaced to PineAPPL for the generation of grids of theory predictions, which can be efficiently used a-posteriori during the fit, performed within xFitter. We take several state-of-the-art parton distribution functions (PDFs) as input for the fit and evaluate their associated uncertainties, as well as the uncertainties arising from renormalization and factorization scale variation. Fit uncertainties related to the datasets are also part of the extracted uncertainty of the top-quark mass and turn out to be of similar size as the combined scale and PDF uncertainty. Fit results from different PDF sets agree among each other within 1$σ$ uncertainty, whereas some datasets related to $t\bar{t}$ decay in different channels (dileptonic vs. semileptonic) point towards top-quark mass values in slight tension among each other, although still compatible within $2.5 σ$ accuracy. Our results are compatible with the PDG 2022 top-quark pole-mass value. Our work opens the road towards more complex simultaneous NNLO fits of PDFs, the strong coupling $α_s(M_Z)$ and the top-quark mass, using the currently most precise experimental data on $t\bar{t} + X$ total and multi-differential cross-sections from the LHC. △ Less

Submitted 29 May, 2024; v1 submitted 9 November, 2023; originally announced November 2023.

Report number: DESY-23-179, PSI-PR-23-42

Journal ref: JHEP05(2024)321

Exploring SMEFT Couplings Using the Forward-Backward Asymmetry in Neutral Current Drell-Yan Production at the LHC

Authors: Andrii Anataichuk, Sven-Olaf Moch, Hamed Abdolmaleki, Simone Amoroso, Daniel Britzger, Filippo Dattola, Juri Fiaschi, Francesco Giuli, Alexander Glazov, Francesco Hautmann, Agnieszka Luszczak, Sara Taheri Monfared, Fred Olness, Federico Vazzoler, Oleksandr Zenaiev

Abstract: Neutral current Drell-Yan (DY) lepton-pair production is considered in the framework of the Standard Model Effective Field Theory (SMEFT). Using the open-source fit platform xFitter, we investigate the impact of high-statistics measurements of the neutral current DY (NCDY) forward-backward asymmetry $A_{\rm{FB}}$ near the weak boson mass scale in the present and forthcoming stages of the Large Had… ▽ More Neutral current Drell-Yan (DY) lepton-pair production is considered in the framework of the Standard Model Effective Field Theory (SMEFT). Using the open-source fit platform xFitter, we investigate the impact of high-statistics measurements of the neutral current DY (NCDY) forward-backward asymmetry $A_{\rm{FB}}$ near the weak boson mass scale in the present and forthcoming stages of the Large Hadron Collider (LHC). Besides recovering earlier results on the $A_{\rm{FB}}$ sensitivity to parton distribution functions, we analyze the precision determination of $Z$-boson couplings to left-handed and right-handed $u$-quarks and $d$-quarks, and explore Beyond-Standard-Model contributions using the SMEFT framework. We perform a sensitivity study and comment on the role of the $A_{\rm{FB}}$ asymmetry for the electroweak SMEFT fit and precision $Z$-boson physics at the LHC and high-luminosity HL-LHC. △ Less

Submitted 10 December, 2024; v1 submitted 30 October, 2023; originally announced October 2023.

Comments: Published version

Report number: DESY-23-160

Journal ref: Eur. Phys. J. C 84 (2024) 1277

Additional moments and x-space approximations of four-loop splitting functions in QCD

Authors: S. Moch, B. Ruijl, T. Ueda, J. Vermaseren, A. Vogt

Abstract: We have extended our previous computations of the even-N moments of the flavour-singlet four-loop splitting functions to N = 12 for the pure-singlet quark case and N = 10 for all other cases. These results, obtained using physical quantities in inclusive deep-inelastic scattering, have been and will be used to validate conceptionally much more challenging determinations of these splitting function… ▽ More We have extended our previous computations of the even-N moments of the flavour-singlet four-loop splitting functions to N = 12 for the pure-singlet quark case and N = 10 for all other cases. These results, obtained using physical quantities in inclusive deep-inelastic scattering, have been and will be used to validate conceptionally much more challenging determinations of these splitting functions from off-shell operator matrix elements (OMEs). For the quark-gluon and gluon-gluon splitting functions, which have yet to be computed to higher N using OMEs, we construct approximations based on our moments and endpoint constraints, where we present new large-x results for the gluon-gluon case. These approximations facilitate an approximate N^3LO evolution of parton distributions which are sufficiently accurate outside the region of small momentum fractions x. △ Less

Submitted 9 October, 2023; originally announced October 2023.

Comments: 14 pages, 3 figures. Analytical and numerical results in ancillary FORM and Fortran files

Report number: DESY-23-150, Nikhef 23-016, LTH 1354

The double fermionic contribution to the four-loop quark-to-gluon splitting function

Authors: G. Falcioni, F. Herzog, S. Moch, J. Vermaseren, A. Vogt

Abstract: We have computed the first 30 even-N moments for the double fermionic (nf^2) part of the quark-to-gluon splitting function P_{gq} at the fourth order of perturbative QCD via the renormalization of off-shell operator matrix elements. From these results we have determined the all-N form, and hence the exact x-space expression, using systems of Diophantine equations for its coefficients. The dominant… ▽ More We have computed the first 30 even-N moments for the double fermionic (nf^2) part of the quark-to-gluon splitting function P_{gq} at the fourth order of perturbative QCD via the renormalization of off-shell operator matrix elements. From these results we have determined the all-N form, and hence the exact x-space expression, using systems of Diophantine equations for its coefficients. The dominant and subdominant leading small-x nf^2 contributions to P_{gq}^(3)(x) are of the form 1/x ln x and ln^4 x, respectively; the leading large-x term is ln^4 (1-x). The coefficient of the first of these is new, the other two agree with results obtained before and thus provide checks of our results. △ Less

Submitted 2 October, 2023; originally announced October 2023.

Comments: 10 pages, LaTex, 1 eps-based figure. Analytical results in ancillary FORM files

Report number: ZU-TH 62/23, DESY 23-146, Nikhef 2023-015, LTH 1353

Four-loop splitting functions in QCD -- The gluon-to-quark case

Authors: G. Falcioni, F. Herzog, S. Moch, A. Vogt

Abstract: We have computed the even-$N$ moments $N \leq 20$ of the gluon-to-quark splitting function $P_{\rm qg}$ at the fourth order of perturbative QCD via the renormalization of off-shell operator matrix elements. Our results, derived analytically for a general gauge group, agree with all results obtained for this function so far, in particular with the lowest five moments obtained via physical cross sec… ▽ More We have computed the even-$N$ moments $N \leq 20$ of the gluon-to-quark splitting function $P_{\rm qg}$ at the fourth order of perturbative QCD via the renormalization of off-shell operator matrix elements. Our results, derived analytically for a general gauge group, agree with all results obtained for this function so far, in particular with the lowest five moments obtained via physical cross sections. Using our new moments and all available endpoint constraints, we construct approximations for the four-loop $P_{\rm qg}(x)$ that should be sufficient for a wide range of collider-physics applications. The N$^3$LO corrections resulting from these and the corresponding quark-quark splitting functions lead to a marked improvement of the perturbative accuracy for the scale derivative of the singlet quark distribution, with effects of 1% or less at $x \gtrsim 10^{\,-4}$ at a standard reference scale with $α_s = 0.2$. △ Less

Submitted 15 April, 2024; v1 submitted 9 July, 2023; originally announced July 2023.

Comments: journal version (v2 typos corrected); 17 pages latex, 3 figures, 2 ancillary files (v2 FORM file with results and FORTRAN subroutine)

Report number: DESY 23-096, LTH 1345

Journal ref: Phys.Lett.B 846 (2023) 138215

On evolution kernels of twist-two operators

Authors: Yao Ji, Alexander Manashov, Sven-Olaf Moch

Abstract: The evolution kernels that govern the scale dependence of the generalized parton distributions are invariant under transformations of the $\mathrm{SL}(2,\mathrm R)$ collinear subgroup of the conformal group. Beyond one loop the symmetry generators, due to quantum effects, differ from the canonical ones. We construct the transformation which brings the {\it full} symmetry generators back to their c… ▽ More The evolution kernels that govern the scale dependence of the generalized parton distributions are invariant under transformations of the $\mathrm{SL}(2,\mathrm R)$ collinear subgroup of the conformal group. Beyond one loop the symmetry generators, due to quantum effects, differ from the canonical ones. We construct the transformation which brings the {\it full} symmetry generators back to their canonical form and show that the eigenvalues (anomalous dimensions) of the new, canonically invariant, evolution kernel coincide with the so-called parity respecting anomalous dimensions. We develop an efficient method that allows one to restore an invariant kernel from the corresponding anomalous dimensions. As an example, the explicit expressions for NNLO invariant kernels for the twist two flavor-nonsinglet operators in QCD and for the planar part of the universal anomalous dimension in $ N=4$ SYM are presented. △ Less

Submitted 29 March, 2024; v1 submitted 4 July, 2023; originally announced July 2023.

Comments: 12 pages, two figures; The three-loop QCD kernel in electronic form is given in the ancillary file. V2: published version; reference added; typo corrected

Report number: TUM--HEP--1461/23, MPP--2023--137, DESY--23--091

Impact of SeaQuest data on PDF fits at large $x$

Authors: S. Alekhin, M. V. Garzelli, S. Kulagin, S. -O. Moch

Abstract: We evaluate the impact of recent SeaQuest (E906 experiment) data on dimuon production in proton-deuteron and proton-proton collisions on parton distribution functions (PDFs). We find these data in a good agreement with the QCD predictions based on PDFs fitted to the Tevatron and LHC data on forward production of $W$ and $Z$ bosons. As a basis for this study we use the ABMP16 PDF fits and show that… ▽ More We evaluate the impact of recent SeaQuest (E906 experiment) data on dimuon production in proton-deuteron and proton-proton collisions on parton distribution functions (PDFs). We find these data in a good agreement with the QCD predictions based on PDFs fitted to the Tevatron and LHC data on forward production of $W$ and $Z$ bosons. As a basis for this study we use the ABMP16 PDF fits and show that they turn out to be compatible with the SeaQuest data, and that these data have constraining power, allowing to reduce the uncertainties on the isospin asymmetry of the light-sea-quark distribution at large longitudinal momentum fraction $x$. We discuss the nuclear corrections needed to describe the deuteron and show that they affect the theoretical description of the proton-deuteron Drell-Yan cross section at the level of $\mathcal{O}(0.5 - 1)$\%. We also comment on the compatibility of the SeaQuest results with other state-of-the-art PDF fits and show that these data are in clear disagreement with models proposing an SU(3)-flavor symmetric quark sea. Finally, we perform a comparison between the second Mellin moments of the light-quark PDFs and recent results from various lattice QCD computations, which demonstrates good compatibility, albeit limited by the uncertainties inherent in current lattice QCD simulations. △ Less

Submitted 19 July, 2023; v1 submitted 2 June, 2023; originally announced June 2023.

Comments: 23 pages, 7 figures, 3 tables, v2: report number updated, references added, typos corrected, discussion slightly revised and extended. Further small modifications still to come in a next version

Report number: INR-TH-2023-012

Four-loop splitting functions in QCD -- The quark-quark case

Authors: G. Falcioni, F. Herzog, S. Moch, A. Vogt

Abstract: We have computed the even-$N$ moments $N\leq 20$ of the pure-singlet quark splitting function $P_{\,\rm ps}$ at the fourth order of perturbative QCD via the anomalous dimensions of off-shell flavour-singlet operator matrix elements. Our results, derived analytically for a general gauge group, agree with all results obtained for this function so far, in particular with the lowest six even moments o… ▽ More We have computed the even-$N$ moments $N\leq 20$ of the pure-singlet quark splitting function $P_{\,\rm ps}$ at the fourth order of perturbative QCD via the anomalous dimensions of off-shell flavour-singlet operator matrix elements. Our results, derived analytically for a general gauge group, agree with all results obtained for this function so far, in particular with the lowest six even moments obtained via physical cross sections. Using these results and all available endpoint constraints, we construct approximations for $P_{\rm ps}$ at four loops that should be sufficient for most collider-physics applications. Together with the known results for the non-singlet splitting function $P_{\rm ns}^{\,+}$ at this order, this effectively completes the quark-quark contribution for the evolution of parton distribution at N$^{\:\!3}$LO accuracy. Our new results thus provide a major step towards fully consistent N$^{\:\!3}$LO calculations at the LHC and the reduction of the residual uncertainty in the parton evolution to the percent level. △ Less

Submitted 15 February, 2023; originally announced February 2023.

Comments: 17 pages latex, 2 figures, 2 ancillary files (FORM file with results and FORTRAN subroutine)

Report number: DESY 23--022, LTH 1333