Higher-order Corrections Research Articles

Exact polarization of particles of any spin at global equilibrium

The polarization of the Λ particle offers the unique opportunity to study the hydrodynamic gradients in the Quark-Gluon Plasma formed in heavyion collisions. However, the theoretical formula commonly used to calculate polarization is only a linear order expansion in thermal vorticity and neglects higher-order corrections. Here, I present an exact calculation to all orders in (constant) thermal vorticity at global equilibrium, obtaining the analytic form of the spin density matrix and the polarization vector for massive particles of any spin. Finally, I extend these results to local equilibrium and assess their phenomenological impact by numerically calculating the polarization vector in a 3+1 hydrodynamic simulation.

Ion-optical updates and performance analysis of High energy FRagment Separator (HFRS) at HIAF

The High energy FRagment Separator (HFRS), a new generation in-flight radioactive separator, is under construction at the high-intensity Heavy Ion Accelerator Facility (HIAF) in China. To further optimize the separator performance and consider the manufacturing technology, the specifications of some magnetic elements of the HFRS separator have been modified from the original design reported earlier, and prototypes of partial magnetic elements have been fabricated and tested. In this paper, updates of magnetic parameters, test results of prototypes, and development of ion-optical calculation are presented. The high-order correction of aberrations is demonstrated. The performance of the separator is studied using fission reactions and fragmentation reactions of relativistic 238U projectiles. This work will guide the forthcoming experiments in the future HFRS separator.

Higher-order asymptotic corrections and their application to the Gamma Variance Model

We present improved methods for calculating confidence intervals and p values in situations where standard asymptotic approaches fail due to small sample sizes. We apply these techniques to a specific class of statistical model that can incorporate uncertainties in parameters that themselves represent uncertainties (informally, “errors on errors”) called the Gamma Variance Model. This model contains fixed parameters, generically denoted by ε\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varepsilon $$\\end{document}, that represent the relative uncertainties in estimates of standard deviations of Gaussian distributed measurements. If the ε\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varepsilon $$\\end{document} parameters are small, one can construct confidence intervals and p values using standard asymptotic methods. This is formally similar to the familiar situation of a large data sample, in which estimators for all adjustable parameters have Gaussian distributions. Here we address the important case where the ε\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varepsilon $$\\end{document} parameters are not small and as a consequence the first-order asymptotic distributions do not represent a good approximation. We investigate improved test statistics based on the technology of higher-order asymptotics (modified likelihood root and Bartlett correction). The effective application of higher-order corrections removes an important computational barrier to the use of the Gamma Variance Model.

Non-thermal WIMP production from higher order moduli decay

In a non-standard cosmological scenario heavy, long-lived particles, which we call moduli, dominate the energy density prior to Big Bang Nucleosynthesis. Weakly Interacting Massive Particles (WIMPs) may be produced non-thermally from moduli decays. The final relic abundance then depends on additional parameters such as the branching ratio of moduli to WIMPs and the modulus mass. This is of interest for WIMP candidates, such as a bino-like neutralino, where thermal production in standard cosmology leads to an overdensity. Previous works have shown that the correct dark matter (DM) relic density can then still be obtained if the moduli, with mass less than 107 GeV, decay to WIMPs with a branching ratio of less than 10-4. This upper bound could easily be violated once higher order corrections, involving final states with more than two particles, are included. We compute the branching ratios of three- and four-body decays of a modulus into final states involving two DM particles for general couplings. We then apply these expressions to sparticle production within the Minimal Supersymmetric Standard Model (MSSM) with neutralino DM. We find that this upper bound on the branching ratio can be satisfied in simplified models through an appropriate choice of as yet undetermined couplings. However, in the MSSM, it requires sparticle masses to be very close to half the modulus mass, in contrast to the idea of weak-scale supersymmetry.

Quantitative Phase Field Model for Electrochemical Systems

Modeling microstructure evolution in electrochemical systems is vital for understanding the mechanism of various electrochemical processes. In this work, we propose a general phase field framework that is fully variational and thus guarantees that the energy decreases upon evolution in an isothermal system. The bulk and interface free energies are decoupled using a grand potential formulation to enhance numerical efficiency. The variational definition of the overpotential is used, and the reaction kinetics is incorporated into the evolution equation for the phase field to correctly capture capillary effects and eliminate additional model parameter calibrations. A higher-order kinetic correction is derived to accurately reproduce general reaction models such as the Butler-Volmer, Marcus, and Marcus-Hush-Chidsey models. Electrostatic potentials in the electrode and the electrolyte are considered separately as independent variables, providing additional freedom to capture the interfacial potential jump. To handle realistic materials and processing parameters for practical applications, a driving force extension method is used to enhance the grid size by three orders of magnitude. Finally, we comprehensively verify our phase field model using classical electrochemical theory.

Two-dimensional unconfined seepage flow toward a highway cut slope

The unconfined gravity-flow system near a free-outflow boundary such as a highway cut slope was investigated by using a higher-order numerical model. Unlike the Dupuit–Forchheimer equation, which is applicable mainly to a hydraulic flow problem, the proposed model accounts for the effects of the vertical component of the flow for a full treatment of the problem of plane phreatic flow. The model equations were numerically solved by means of a finite-difference scheme. Their accuracy was then verified using the solutions of the full two-dimensional potential-flow method and rigorous mathematical approaches, revealing that for a face slope flatter than 70º, the differences between the solutions of the model and the earlier approaches for the relative seepage-face height were nearly negligible. The comparison results also demonstrated the substantial effects of the slope of the downstream face on this height and the seepage discharge. Furthermore, the accuracies of the model predictions for the phreatic-surface profile and the distributions of the piezometric head at different vertical sections are much better than the earlier method, which approximates the trapezoidal-shaped aquifer by its equivalent rectangular one. Such a satisfactory performance may be attributed to the model’s higher-order correction factor for the effects of the phreatic-surface curvature and steep slope.

The O(αt+αλ+ακ)2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal{O}(\\alpha _t+\\alpha _\\lambda +\\alpha _\\kappa )^2$$\\end{document} correction to the ρ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\rho $$\\end{document} parameter and its effect on the W boson mass calculation in the complex NMSSM

We present the prediction of the electroweak ρ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\rho $$\\end{document} parameter and the W boson mass in the CP-violating Next-to-Minimal Supersymmetric extension of the Standard Model (NMSSM) at the two-loop order. The ρ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\rho $$\\end{document} parameter is calculated at the full one-loop and leading and sub-leading two-loop order O(α+αtαs+αt+αλ+ακ2)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {O}}(\\alpha + \\alpha _t\\alpha _s + \\left( \\alpha _t+\\alpha _\\lambda +\\alpha _\\kappa \\right) ^2)$$\\end{document}. The new Δρ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Delta \\rho $$\\end{document} prediction is incorporated into a prediction of MW\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$M_W$$\\end{document} via a full supersymmetric (SUSY) one-loop calculation of Δr\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Delta r$$\\end{document}. Furthermore, we include all known state-of-the-art SM higher-order corrections to Δr\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Delta r$$\\end{document}. By comparing results for Δρ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\Delta \\rho $$\\end{document} obtained using on-shell (OS) and DR¯\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\overline{\ ext {DR}}$$\\end{document} renormalization conditions in the top/stop sector, we find that the scheme uncertainty is reduced at one-loop order by 55%, at two-loop O(αsαt)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {O}}(\\alpha _s\\alpha _t)$$\\end{document} by 22%, and at two-loop O(αt+ακ+αλ)2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${\\mathcal {O}}(\\alpha _t+\\alpha _\\kappa +\\alpha _\\lambda )^2$$\\end{document} by 16%, respectively. The influence of the two-loop results on the MW\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$M_W$$\\end{document} mass prediction is found to be sub-leading. The new calculation is made public in the computer program NMSSMCALC. We perform an extensive comparison in the W-mass, Higgs boson mass and the muon anomalous magnetic moment prediction between our calculation and three other publicly available tools and find very good agreement provided that the input parameters and renormalization scales are treated in the same way. Finally, we study the impact of the CP-violating phases on the W-mass prediction which is found to be smaller than the overall size of the SUSY corrections.

Higher-Order Corrections to the Effective Field Theory of Low-Energy Axions

Dark matter (DM) can be composed of a collection of axions, or axion-like particles (ALPs), whose existence is due to the spontaneous breaking of the Peccei–Quinn U(1) symmetry, which is the most compelling solution of the strong CP-problem of quantum chromodynamics (QCD). Axions must be spin-0 particles with very small masses and extremely weak interactions with themselves as well as with the particles that constitute the Standard Model. In general, the physics of axions is detailed by a quantum field theory of a real scalar field, ϕ. Nevertheless, it is more convenient to implement a nonrelativistic effective field theory with a complex scalar field, ψ, to characterize the mentioned axions in the low-energy regime. A possible application of this equivalent description is for studying the collapse of cold dark matter into more complex structures. There have been a few derivations of effective Lagrangians for the complex field ψ, which were all equivalent after a nonlocal space transformation between ϕ and ψ was found, and some other corrections were introduced. Our contribution herein is to further provide higher-order corrections; in particular, we compute the effective field theory Lagrangian up to order (ψ*ψ)5, also incorporating the fast-oscillating field fluctuations into the dominant slowly varying nonrelativistic field.

Heat transport and flow morphology of geostrophic rotating Rayleigh–Bénard convection in the presence of boundary flow

Using direct numerical simulations, we investigate the heat transport in bulk and boundary flows separately in rotating Rayleigh–Bénard convection in cylindrical cells. In the bulk we observe a steep scaling relationship between the Nusselt number ( $Nu$ ) and the Rayleigh number ( $Ra$ ), which is consistent with the results from simulations using periodic boundary conditions. For the boundary flow, we observe a power law $Nu_{BF}\sim (Ra/Ra_w)^1$ at the leading order, where $Nu_{BF}$ is the local Nusselt number of the boundary flow and $Ra_w$ is the onset Rayleigh number of the wall mode. We develop a model using the boundary layer marginal stability theory to explain this power law, and further show that a more precise description of the data can be obtained if a higher-order correction is introduced. A striking finding of our study is the observation of a sharp transition in flow state, manifested by a sudden drop in $Nu_{BF}$ with a corresponding collapse of the boundary flow coherency. After the transition, the boundary flow breaks into vortices, leading to a reduction in flow coherency and heat transport efficiency. As the physical properties of the vortices should not depend on the aspect ratio, $Nu_{BF}$ for all aspect ratios collapse together after the transition. Moreover, the centrifugal force helps trigger the breakdown of the coherent boundary flow state. For this reason, $Nu_{BF}$ for the cases with non-zero centrifugal force collapse together. We further develop a method that enables us to separate the contributions from the bulk and boundary flows in the global Nusselt number using only the global $Nu$ and it does not require the centrifugal force to be absent.

A Very Tentative Calculation of Some Coupling Constants and Particle Masses

In this article, we provide tentative first-order calculations of some coupling constants λ and particle mass corrections δm based on the results of prior work. We show that it is possible to derive values which are very roughly within the ballpark of the empirical data, though higher-order corrections are vital for higher accuracy. This article thus may provide a tentative clue on the deeper underlying physics if it exists. The difficulty of interpreting the physicality (if any) of the results are emphasized, and error discussions are made since the inclusion of this is considered vital. Finally, the importance of further work regarding the rigor of the derivations and the accuracy of the physical interpretations are underlined.

Differential contracting homotopy in higher-spin theory

A new efficient approach to the analysis of nonlinear higher-spin equations, that treats democratically auxiliary spinor variables ZA and integration homotopy parameters in the non-linear vertices of the higher-spin theory, is developed. Being most general, the proposed approach is the same time far simpler than those available so far. In particular, it is free from the necessity to use the Schouten identity. Remarkably, the problem of reconstruction of higher-spin vertices is mapped to certain polyhedra cohomology in terms of homotopy parameters themselves. The new scheme provides a powerful tool for the study of higher-order corrections in higher-spin theory and, in particular, its spin-locality. It is illustrated by the analysis of the lower order vertices, reproducing not only the results obtained previously by the shifted homotopy approach but also projectively-compact vertices with the minimal number of derivatives, that were so far unreachable within that scheme.

Exploring the higher-order QED effects on the differential distributions of vacuum pair production in relativistic heavy-ion collisions

Extensive studies have been conducted in the past few decades to investigate potential signatures of higher-order QED effects in high-energy electromagnetic scattering processes. In our previous work, we have identified evidence of higher-order corrections in the total cross-section for the vacuum pair production in relativistic heavy-ion collisions. However, the presence of higher-order QED corrections cannot be unambiguously proven solely based on total cross-section measurements due to substantial experimental and theoretical uncertainties. The objective of this paper is to explore the sensitivity of specific differential observables in the vacuum pair production to higher-order QED effects in high-energy heavy-ion collisions. These investigations will provide guidance in determining the presence or absence of higher-order QED processes by conducting precise measurements in future experiments.

On the high-energy instability of quarkonium production

The perturbative instability of NLO collinear factorisation (CF) computations of pT-integrated cross sections of heavy quarkonium production at high hadronic or photon-hadron collision energy is discussed. We resolve this problem via the matching of NLO CF computation with the resummation of higher-order corrections ∝αsnlnn−1⁡(sˆ/M2) at high partonic center of mass energies sˆ≫M2. The resummation is performed in the Doubly-Logarithmic Approximation (DLA) of High-Energy Factorisation (HEF) formalism. We also report the results of the first computation of one-loop corrections to impact-factors involving heavy quark-antiquark pair in the intermediate states considered in the Non-Relativistic QCD (NRQCD) factorisation formalism for quarkonium production: QQ¯[S0[8]1] and QQ¯[S0[1]1]. These results are necessary for the extension of our resummation formalism beyond DLA.

Conservative Fourier spectral method for a class of modified Zakharov system with high-order space fractional quantum correction

In this paper, we consider the Fourier spectral method and numerical investigation for a class of modified Zakharov system with high-order space fractional quantum correction. First, the numerical scheme of the system is developed with periodic boundary condition based on the Crank–Nicolson/leap-frog methods in time and the Fourier spectral method in space. Moreover, it is shown that the scheme preserves simultaneously mass and energy conservation laws. Second, we analyze stability and convergence of the numerical scheme. Last, the numerical experiments are given, and the results show the correctness of theoretical results and the efficiency of the conservative scheme.

A novel spaceborne ISAR imaging approach for space target with high-order translational motion compensation and spatial variant MTRC correction

ABSTRACT For spaceborne ISAR imaging of space targets, due to the high-speed relative motion between the radar and targets, both the translational and rotational motion exhibit the high-order characteristics. Conventional translational motion compensation (TMC) methods are ineffective in handling such high-order translational motion. Moreover, the migration through resolution cell (MTRC) stems from the presence of the high-order spatial variant phase of the echo signal, resulting in severe defocusing of ISAR images. In this paper, a novel spaceborne ISAR imaging approach for space targets is proposed, which incorporates the high-order TMC and the high-order MTRC correction. First, the slant range model for spaceborne ISAR imaging is derived as a third-order polynomial form. Based on this model, a modified high-order TMC method is proposed, which utilizes the minimum entropy method combined with the Broyden – Fletcher – Goldfarb – Shanno (BFGS) algorithm. The proposed method leverages orbital model information to obtain the initial TMC coefficients, enabling TMC to be completed more accurately and effectively. Then, a modified polynomial keystone transform (PKT) is proposed to correct the high-order MTRC. The PKT-based method effectively addresses both the high-order and first-order MTRC corrections simultaneously. Finally, a spaceborne ISAR imaging approach is proposed to obtain well-focused ISAR images. Simulated and real data results validate the analysis of the spaceborne ISAR imaging model and demonstrate the effectiveness of the spaceborne imaging approach for space targets proposed in this paper.

Reservoir-induced stabilization of a periodically driven classical spin chain: Local versus global relaxation.

Floquet theory is an indispensable tool for analyzing periodically driven quantum many-body systems. Although it does not universally extend to classical systems, some of its methodologies can be adopted in the presence of well-separated timescales. Here we use these tools to investigate the stroboscopic behaviors of a classical spin chain that is driven by a periodic magnetic field and coupled to a thermal reservoir. We detail and expand our previous work: we investigate the significance of higher-order corrections to the classical Floquet-Magnus expansion in both the high- and low-frequency regimes; explicitly probe the evolution dynamics of the reservoir; and further explore how the driven system and the reservoir synchronize with the applied field at low frequencies. In line with our earlier results, we find that the high-frequency regime is characterized by a local Floquet-Gibbs ensemble with the reservoir acting as a nearly-reversible heat sink. At low frequencies, the driven system rapidly enters a synchronized state, which can only be fully described in a global picture accounting for the concurrent relaxation of the reservoir in a fictitious magnetic field arising from the drive. We highlight how the evolving nature of the reservoir may still be incorporated in a local picture by introducing an effective temperature. Finally, we show that generic local-dissipation models that account for the influence of the reservoir on the driven system phenomenologically through Markovian dissipative equationsof motion can generally not reproduce the rich behavior that our microscopic simulations reveal. In particular, such models prove insufficient to account for the suppression of overall energy absorption that is induced by the here observed synchronization between driven system and reservoir.

Nonlinear high-order harmonics correction for phase measuring profilometry

Gamma nonlinearity severely affects the accuracy of phase measuring profilometry (PMP). Through averaging two phase maps with inverse errors, conventional methods including double phase-shifting (DPS) and Hilbert transform (HT) only compensate the fundamental harmonic while leaving high-order harmonics. To address this issue, this paper presents an efficient high-order harmonics correction (HHC) method. The averaged phase is encoded into factitious fringes on the computer. Then we directly apply HT on these factitious fringes to generate transformed fringes, while the addition of the ideal phase and residual harmonics is converted to multiplication. Finally, the residual high-order harmonics are suppressed, and the corrected phase is calculated using the phase-shifting algorithm. In comparison to conventional DPS and HT methods, the proposed HHC method can efficiently correct the residual high-order harmonics, thereby enhancing the measuring accuracy.

Configuration entropy of ϒ(1S) state in strong coupling plasma

To better understand the effects of strong coupling and QCD at high temperature in QGP, by using holographic model, we investigate the dissociation effect of bottomonium under the higher-order curvature corrections to the supergravity action corresponding to the corrections of large N expansion of boundary CFT in the side of field theory. The results show that effective potential is not a good physical quantity to estimate the dissociation strength of bottomonium in the case of finite wave number and considering the higher-order curvature corrections. Therefore, we calculate the quasinormal spectra (QNMs) and the differential configuration entropy (DCE). It is found that the dissociation effect is stronger for the stronger coupling.

Solitary Vortex Dynamics of 2D Bose-Einstein Condensates with Higher-Order Nonlinear Interactions

For study of Continuous matter waves in Bose-Einstein condensates in nonlinear and quantum atom optics, the two-dimensional Gross-Pitaevskii equation (GPE) is chosen as the reliable model for studying the dynamics of vortices in the framework of mean-field theory. In related problems in several recent studies showing that higher-order interrelationships are an indispensable component of the GPE even at the mean-field level, by numerically estimating the vortex dynamics variables. In this paper, derive the vortex soliton solutions using the variational method and investigate the effect of higher-order nonlinear corrections on the behavior of the vortex dynamics, which are shown to have an important impact on the vortex dynamics behavior.

Codebase release 1.0 for GRIFFIN

This paper describes a modular framework for the description of electroweak scattering and decay processes, including but not limited to Z-resonance physics. The framework consistently combines a complex-pole expansion near an s-channel resonance with a regular fixed-order perturbative description away from the resonance in a manifestly gauge-invariant scheme. Leading vertex correction contributions are encapsulated in form factors that can be predicted or treated as numerical fit parameters. This framework has been implemented in the publicly available object-oriented C++ library GRIFFIN. Version 1.0 of this library provides Standard Model predictions for the IR-subtracted matrix elements for the process f\bar{f} \to f'\bar{f}'ff‾→f′f‾′ with full NNLO and leading higher-order contributions on the Z-resonance, and with NLO corrections off-resonance. The library can straightforwardly be extended to include higher-order corrections, should they become available, or predictions for new physics models. It can be interfaced with Monte-Carlo programs to account for QED and QCD initial-state and final-state radiation.