arXiv subjects

F. Herzog

Publications and source records attributed to F. Herzog.

17 recordsLinked to original sources

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 $γ_{\rm ik}^{\,(3)}(N)$ 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 $n_f^{}$.

hep-ph↗

arXiv2025

Five-loop beta function for gauge theories: computations, results and consequences

At the end of 2016, we computed the five-loop (N$^4$LO) contributions to the beta function in perturbative Quantum Chromodynamics (QCD), its generalization to non-Abelian gauge theories with a simple compact Lie group, and for Quantum Electrodynamics (QED). Here we recall main tools used in and specifically developed for this computation and its main analytic and numerical results. The development work carried out for this project facilitated further even more involved analytic five-loop computations. We briefly summarize also their numerical QCD results for Higgs-boson decay to hadrons in the heavy-top limit and for two N$^4$LO splitting functions for the evolution of quark distributions of hadrons. The latter lead to a first realistic estimate of the five-loop contribution to another important quantity in perturbative QCD, the quark cusp anomalous dimension.

hep-ph↗

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.

hep-ph↗

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.

hep-ph↗

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.

hep-ph↗

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.

hep-ph↗

arXiv2024

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 $α_s = 0.2$.

hep-ph↗

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.

hep-ph↗

arXiv2023

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

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.

hep-ph↗

arXiv2019

Five-loop contributions to low-N non-singlet anomalous dimensions in QCD

We present the first calculations of next-to-next-to-next-to-next-to-leading order (N^4LO) contributions to anomalous dimensions of spin-N twist-2 operators in perturbative QCD. Specifically, we have obtained the respective non-singlet quark-quark anomalous dimensions at N=2 and N=3 to the fifth order in the strong coupling alpha_s. These results set the scale for the N^4LO contributions to the evolution of the non-singlet quark distributions of hadrons outside the small-x region, and facilitate a first approximate determination of the five-loop cusp anomalous dimension. While the N^4LO coefficients are larger than expected from the lower-order results, their inclusion stabilizes the perturbative expansions for three or more light flavours at a sub-percent accuracy for alpha_s < 0.3.

hep-ph↗

arXiv2018

Anomalous dimensions and splitting functions beyond the next-to-next-to-leading order

We report on recent progress on the splitting functions for the evolution of parton distributions and related quantities, the (lightlike) cusp anomalous dimensions, in perturbative QCD. New results are presented for the four-loop (next-to-next-to-next-to-leading order, N^3LO) contributions to the flavour-singlet splitting functions and the gluon cusp anomalous dimension. We present first results, the moments N=2 and N=3, for the five-loop (N^4LO) non-singlet splitting functions.

hep-ph↗

arXiv2018

Five-loop renormalisation of QCD in covariant gauges

We present the complete set of vertex, wave function and charge renormalisation constants in QCD in a general simple gauge group and with the complete dependence on the covariant gauge parameter $ξ$ in the minimal subtraction scheme of conventional dimensional regularisation. Our results confirm all already known results, which were obtained in the Feynman gauge, and allow the extraction of other useful gauges such as the Landau gauge. We use these results to extract the Landau gauge five-loop anomalous dimensions of the composite operator $A^2$ as well as the Landau gauge scheme independent gluon, ghost and fermion propagators at five loops.

hep-ph↗

arXiv2018

R*-operation and five-loop calculations

We sketch how the R*-operation can be used to compute the pole terms of Feynman diagrams. We identify computational difficulties when performing five-loop calculations, and provide four solutions that drastically reduce the number of terms that are generated. Using these methods, we have computed the beta function for Yang-Mills theory with fermions, the R-ratio in electron-positron annihilation, and Higgs decays to quarks and gluons at five-loop accuracy. The results for the beta function and Higgs decay width to gluons in the heavy-top limit are briefly discussed. There is no need for six-loop extensions of these calculations in the near future.

hep-ph↗

arXiv2018

The method of global R* and its applications

The global R* operation is a powerful method for computing renormalisation group functions. This technique, based on the principle of infrared rearrangement, allows to express all the ultraviolet counterterms in terms of massless propagator integrals. In this talk we present the main features of global R* and its application to the renormalisation of QCD. By combining this approach with the use of the program Forcer for the evaluation of the relevant Feynman integrals, we renormalise for the first time QCD at five loops in covariant gauges.

hep-ph↗

arXiv2017

On Higgs decays to hadrons and the R-ratio at N^4LO

We present the first determination of Higgs-boson decay to hadrons at the next-to-next-to-next-to-next-to-leading order of perturbative QCD in the limit of a heavy top quark and massless light flavours. This result has been obtained by computing the absorptive parts of the relevant five-loop self-energy for a general gauge group and combining the outcome with the corresponding coefficient function already known to this order in QCD. Our new result reduces the uncertainty due to the truncation of the perturbation series to a fraction of the uncertainty due to the present error of the strong coupling constant. We have also performed the corresponding but technically simpler computations for direct Higgs decay to bottom quarks and for the electromagnetic R-ratio in e^+ e^- -> hadrons, thus verifying important fifth-order results obtained so far only by one group.

hep-ph↗

arXiv2017

Experimental determination of Rashba and Dresselhaus parameters and $g^*$-factor anisotropy via Shubnikov-de Haas oscillations

The spin splitting of conduction band electrons in inversion-asymmetric InGaAs/InP quantum wells is studied by Shubnikov-de Haas measurements combining the analysis of beating patterns and coincidence measurements in doubly tilted magnetic fields. The method allows us to determine the absolute values of the Rashba and linear Dresselhaus spin-orbit interaction coefficients, their relative sign and the full Landé g-tensor. This is achieved by analyzing the anisotropy of the beat node positions with respect to both polar and azimuthal angles between the magnetic field direction and the quantum well normal. We show that the spin-orbit interaction is dominated by a large Rashba coefficient together with a linear Dresselhaus coefficient that is 10 $\%$ of the Rashba coefficient. Their relative sign is found to be positive. The g-tensor is found to have a marked out-of-plane anisotropy and a smaller but distinct in-plane anisotropy due to spin-orbit interaction.

cond-mat.mes-hall↗

arXiv2017

The five-loop beta function of Yang-Mills theory with fermions

We have computed the five-loop corrections to the scale dependence of the renormalized coupling constant for Quantum Chromodynamics (QCD), its generalization to non-Abelian gauge theories with a simple compact Lie group, and for Quantum Electrodynamics (QED). Our analytical result, obtained using the background field method, infrared rearrangement via a new diagram-by-diagram implementation of the R* operation and the Forcer program for massless four-loop propagators, confirms the QCD and QED results obtained by only one group before. The numerical size of the five-loop corrections is briefly discussed in the standard MSbar scheme for QCD with n_f flavours and for pure SU(N) Yang-Mills theory. Their effect in QCD is much smaller than the four-loop contributions, even at rather low scales.

hep-ph↗