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S. Moch

Publications and source records attributed to S. Moch.

At least 19 recordsLinked to original sources
arXiv2026

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

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.

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arXiv2026

Precision QCD with the Electron-Ion Collider

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.

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arXiv2025

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

We have extended our previous computations, performed analytically for a general gauge group, of the even-$N$ moments $\gamma_{\rm ik}^{\,(3)}(N)$ 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 ($\zeta$-function) contributions to $\gamma_{\rm ik}^{\,(3)}(N)$: only the $n_f^{0} \zeta_3$ parts of the quark-to-gluon (gq) and the $n_f^{1} \zeta_3$ 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 $n_f^{}$.

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arXiv2025

Determination of the strong coupling from high-energy data

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 $\alpha_s(m_Z)$ 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 $\alpha_s(m_Z,N_f=5)=0.1152 \pm 0.008$ 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.

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arXiv2025

Alien operators for PDF evolution

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.

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arXiv2025

Anomalous dimensions at small spins

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.

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arXiv2025

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

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.

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arXiv2024

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

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.

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arXiv2024

Constraints for twist-two alien operators in QCD

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.

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arXiv2024

Three-loop evolution kernel for transversity operator

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.

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arXiv2024

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

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.

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arXiv2024

Charm total cross sections with nonuniversal fragmentation treatment

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 $x$.

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arXiv2024

Status of QCD precision predictions for Drell-Yan rapidity distributions

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.

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arXiv2024

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

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.

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arXiv2023

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

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.

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arXiv2023

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

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.

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arXiv2023

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

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 $\alpha_s = 0.2$.

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