The Three-Loop Splitting Functions in QCD: The Singlet Case

Abstract

We compute the next-to-next-to-leading order (NNLO) contributions to the splitting functions governing the evolution of the unpolarized flavour-singlet parton densities in perturbative QCD. The exact expressions are presented in both Mellin-N and Bjorken-x space. We also provide accurate parametrizations for practical applications. Our results agree with all partial results available in the literature. As in the non-singlet case, the correct leading logarithmic predictions for small momentum fractions x do not provide good estimates of the respective complete splitting functions. We investigate the size of the corrections and the stability of the NNLO evolution under variation of the renormalization scale. The perturbative expansion appears to converge rapidly at x >~ 10^-3. Relatively large third-order corrections are found at smaller values of x.

41 pages, Latex, 12 figures

References in corpus (1)

• The Three-Loop Splitting Functions in QCD: The Non-Singlet Case

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• What did HERA teach us about the structure of the proton?
• Rapidity distribution at soft-virtual and beyond for n-colorless particles to N${}^4$LO in QCD
• The first-order factorizable contributions to the three-loop massive operator matrix elements $A_{Qg}^{(3)}$ and $\Delta A_{Qg}^{(3)}$
• Charm Production and QCD Analysis at HERA and LHC
• A Variable Flavour Number Scheme for Heavy Quark Production at Small x
• Drell-Yan Transverse-Momentum Spectra at N${}^3$LL${}^{\prime }$ and Approximate N${}^4$LL with SCETlib
• Mixed QCD-electroweak corrections to Higgs production via gluon fusion in the small mass approximation
• NNLO QCD corrections to unpolarized and polarized SIDIS
• An Evaluation of The Proton Structure Functions $F_2$ and $F_L$ at Small $x$
• The SAGEX Review on Scattering Amplitudes, Chapter 4: Multi-loop Feynman Integrals
• The Fully-Differential Gluon Beam Function at NNLO
• Matching next-to-leading-order and high-energy-resummed calculations of heavy-quarkonium-hadroproduction cross sections
• Analytical approach for the approximate solution of the longitudinal structure function with respect to the GLR-MQ equation at small x
• Beam functions for $N$-jettiness at N${}^3$LO in perturbative QCD
• Threshold Improved QCD Corrections for Stop-Antistop production at Hadron colliders
• The non-planar contribution to the four-loop anomalous dimension of twist-2 operators: first moments in N=4 SYM and non-singlet QCD
• IR-Improved DGLAP Theory: Kernels, Parton Distributions, Reduced Cross Sections
• Resummed Higgs boson cross section at next-to SV to $\mathrm{NNLO}+\overline{\mathrm{NNLL}}$
• Non-planar universal anomalous dimension of twist-two operators with general Lorentz spin at four loops in N=4 SYM theory
• Two loop QCD corrections for the process Pseudo-scalar Higgs $\to 3$ partons
• New Approach to Parton Shower MC's for Precision QCD Theory: HERWIRI1.0(31)
• HERWIRI1.0: MC Realization of IR-Improved DGLAP-CS Parton Showers
• Third-order QCD results on form factors and coefficient functions
• Two-Loop QCD Helicity Amplitudes for (2+1)-Jet Production in Deep Inelastic Scattering
• Triple-collinear splittings with massive particles
• Heavy-flavor PDF evolution and variable-flavor number scheme uncertainties in deep-inelastic scattering
• Precision QCD Physics at the LHC
• Nonlinear correction to the longitudinal structure function at small x
• NNLO Logarithmic Expansions and Precise Determinations of the Neutral Currents near the Z Resonance at the LHC: The Drell-Yan case
• Digging for Top Squarks from Higgs data: from signal strengths to differential distributions
• Dissecting the collinear structure of quark splitting at NNLL
• Generalized threshold resummation in inclusive DIS and semi-inclusive electron-positron annihilation
• IR-Improved DGLAP Theory
• The Two-Loop Massless Off-Shell QCD Operator Matrix Elements to Finite Terms
• Review of AdS/CFT Integrability, Chapter III.4: Twist states and the cusp anomalous dimension
• Leading twist moments of the neutron structure function F2n
• Master integrals for splitting functions from differential equations in QCD
• Four-loop splitting functions at small $x$
• Going beyond Soft plus Virtual
• Determination of neutron fracture functions from a global QCD analysis of leading neutron production at HERA
• Double Higgs production at NNLO interfaced to parton showers in GENEVA
• Stairway to discovery: a report on the CMS programme of cross section measurements from millibarns to femtobarns
• Phenomenology of leading nucleon production in $ep$ collisions at HERA in the framework of fracture functions
• On the discrepancy of the low-x evolution kernels
• Impact of Inclusive Electron Ion Collider Data on Collinear Parton Distributions
• Two-loop evolution equations for flavor-singlet light-ray operators
• Next to SV resummed prediction for pseudoscalar Higgs boson production at NNLO$+\overline{\text{NNLL}}$
• Parton distribution functions and constraints on the intrinsic charm content of the proton using BHPS approach
• Large spin expansion of the long-range Baxter equation in the sl(2) sector of N=4 SYM
• Three loop form factors of a massive spin-2 particle with nonuniversal coupling
• Comparison of NNLO DIS scheme splitting functions with results from exact gluon kinematics at small x
• On twist-two operators in N=4 SYM
• Heavy Quark Fragmentation in $e^{+}{e}^{-}$ Collisions to NNLO+NNLL Accuracy in Perturbative QCD
• The behavior of the structure function by using the effective exponent at low-x
• High Energy Photoproduction
• Threshold enhanced cross sections for colorless productions
• Zero-jettiness soft function to third order in perturbative QCD
• The Color Dipole Picture and extracting the ratio of structure functions at small $x$
• The QCD Splitting Functions at Three Loops: Methods and Results
• The NNLO quark beam function for jet-veto resummation
• Twist-four gravitational form factor at NNLO QCD from trace anomaly constraints
• A Phenomenological Solution Small $x$ to the Longitudinal Structure Function Dynamical Behavior
• The Virtual Photon Structure to the Next-to-next-to-leading Order in QCD
• Resummation Effects in the Search of SM Higgs Boson at Hadron Colliders
• Renormalization of twist-two operators in covariant gauge to three loops in QCD
• Refining the GENEVA method for Higgs boson production via gluon fusion
• Resummed inclusive cross-section in ADD model at N${}^3$LL+NNLO
• Large spin behavior of anomalous dimensions and short-long strings duality
• NNLO QCD corrections to production of a spin-2 particle with non-universal couplings in the DY process
• The dark side of the proton
• Yangians in Integrable Field Theories, Spin Chains and Gauge-String Dualities
• First $O({\alpha }_s^3)$ heavy flavor contributions to deeply inelastic scattering
• Longitudinal structure function from the parton parameterization
• Photon structure function revisited
• Resummed next-to-soft corrections to rapidity distribution of Higgs Boson to $\text{NNLO}+\overline{\text{NNLL}}$
• Vector boson production in joint resummation
• Analytical solution of the longitudinal structure function $F_L$ in the leading and next-to-leading-order analysis at low x with respect to Laguerre polynomials method
• Ratio of the structure functions and the color dipole model bound
• Heavy quark impact factor in kT-factorization
• Fully NLO Parton Shower in QCD
• Next-to-soft-virtual resummed rapidity distribution for Drell-Yan process to $\text{NNLO}+\overline{\text{NNLL}}$
• Higher order approximations to the longitudinal structure function $F_L$ from the parametrization of $F_2$ based on the Laplace transformation
• Determination of ${\alpha }_S$ beyond NNLO using event shape averages
• Three loop MSbar transversity operator anomalous dimensions for fixed moment n <= 8
• Evolution of the longitudinal structure function at small x
• Nonlinear corrections on the parameterization methods
• Large x Resummation in Q^2 Evolution
• Three-loop off-forward evolution kernel for axial-vector operators in Larin's scheme
• Three-loop helicity amplitudes for photon+jet production
• Hard- Pomeron behavior of the Longitudinal Structure Function $F_L$ in the Next- to- Leading- Order at low $x$
• Single-inclusive hadron production in electron-positron annihilation at next-to-next-to-next-to-leading order in QCD
• Large-Spin Expansions of GKP Strings
• Improved analysis of moments of F3 in neutrino-nucleon scattering using the Bernstein polynomial method
• Small-x analysis on the effect of gluon recombinations inside hadrons in light of the GLR-MQ-ZRS equation
• Higgs boson rapidity distribution in bottom annihilation at NNLL and beyond
• On phenomenological study of the solution of nonlinear GLR-MQ evolution equation beyond leading order
• Analytical solution to DGLAP integro-differential equation in a simple toy-model with a fixed gauge coupling
• RI'/SMOM scheme amplitudes for deep inelastic scattering operators at one loop in QCD
• Evolution and interpolation of double parton distributions using Chebyshev grids
• Thrust distribution in Higgs decays up to the fifth logarithmic order
• Conformal constraints for anomalous dimensions of leading twist operators
• On higher-order flavour-singlet splitting and coefficient functions at large x
• Second order splitting functions and infrared safe cross sections in $\mathcal{N}=4$ SYM theory
• Target Mass Corrections for the Virtual Photon Structure Functions to the Next-to-next-to-leading Order in QCD
• Spin-2 Form Factors at Three Loop in QCD
• disorder: Deep inelastic scattering at high orders
• Semiclassical Strings in AdS_5 x S^5 and Automorphic Functions
• NNLO anomalous dimension matrix for twist-two flavor-singlet operators
• Analytical solution to DGLAP integro-differential equation via complex maps in domains of contour integrals
• Summary of ICHEP 2004
• Structure Functions and Parton Distributions
• The study of the gluon distribution function and reduced cross section behavior using the proton structure function
• Color dipole cross section in the DGLAP improved saturation model
• Hard QCD at hadron colliders
• NNLO QCD corrections to the resonant sneutrino/slepton production at Hadron Colliders
• Monte Carlo modelling of NLO DGLAP QCD evolution in the fully unintegrated form
• New constraints on the up-quark valence distribution in the proton
• Two-loop evolution kernels for colour dependent double parton distributions
• String-based axial and helicity-flip GPDs: a comparison to lattice QCD
• Framing energetic top-quark pair production at the LHC
• Benchmark of deep-inelastic-scattering structure functions at $\mathcal{O}({\alpha }_s^3)$
• NNLO electron structure functions (PDFs) from SCET
• A New Method for Calculating Differential Distributions Directly in Mellin Space
• Three-loop soft function for energetic electroweak boson production at hadron colliders
• Studies of Quantum Chromodynamics at the LHC
• PartonDensity.jl: a novel parton density determination code
• Global QCD fit from $Q^2=0$ to $Q^2=30000$ GeV${}^2$ with Regge-compatible initial condition
• An analysis of polarized parton distribution functions with nonlinear QCD evolution equations
• Highest states in light-cone $Ad{S}_5\times S^5$ superstring
• Nonlinear corrections for the nuclear gluon distribution in $eA$ processes
• Heavy quark effects on parton distribution functions in the unpolarized virtual photon up to the next-to-leading order in QCD
• Symbolic Summation and Higher Orders in Perturbation Theory
• On threshold resummation of singlet structure and fragmentation functions
• QED corrections to parton distributions and Altarelli-Parisi splitting functions in the polarized case
• Entanglement entropy, Monte Carlo event generators, and soft gluons DIScovery
• Structure constants of twist-two light-ray operators in the triple Regge limit
• O(α_s^3) contributions to F_L^{Q\bar{Q}}(x,Q^2) for large virtualities
• Universal structure of subleading infrared poles at strong coupling
• Resummation of large logarithms in the heavy quark effects on the parton distributions inside the virtual photon
• High-energy resummation at the LHC: the case of Drell-Yan processes
• Progress in Small x Resummation
• Dynamical parton distribution functions
• Algorithm to find an all-order in the running coupling solution to an equation of the DGLAP type
• Soft gluon resummation for gluon fusion $ZH$ production
• Single-valued representation of unpolarized and polarized semi-inclusive deep inelastic scattering at next-to-next-to-leading order
• Three-loop helicity amplitudes for four-quark scattering in massless QCD
• The Virtual Photon-Gluon Impact Factor with Massive Quarks and Exact Gluon Kinematics
• Unitarity methods for Mellin moments of Drell-Yan cross sections
• Sudakov Resummations at Higher Orders
• Analytical perturbation theory and Nucleon structure function in infrared region
• NNLO splitting and coefficient functions with time-like kinematics
• Anomalous dimensions at small spins
• QCD at hadron colliders
• Parton distributions in the virtual photon target up to NNLO in QCD
• The QCD Coupling and Parton Distributions at High Precision
• Threshold resummation for $Z$-boson pair production at NNLO+NNLL
• Markovian MC simulation of QCD evolution at NLO level with minimum k_T
• QCD Theory
• QCD analysis of $x{F}_3$ structure functions in deep-inelastic scattering: Mellin transform by Gegenbauer polynomial up to N${}^3$LO approximation
• Jet Physics from Static Charges in AdS
• Four-loop anomalous dimension of flavor non-singlet quark operator of twist two and Lorentz spin N for general gauge group: transcendental part
• Evolution of the color dipole cross section
• A semi-analytical $x$-space solution for parton evolution -- Application to non-singlet and singlet DGLAP equation
• Pseudoscalar Higgs boson decay to three parton amplitudes at NNLO to higher orders in the dimensional regulator
• Virtual Photon Structure Functions to NNLO in QCD
• Evolution of entropy at small $x$
• Three-loop corrections to the quark and gluon decomposition of the QCD trace anomaly and their applications
• Renormalization of gluonic leading-twist Operators in covariant Gauges
• High Energy Resummation of Jet Observables
• Soft Integrals and Soft Anomalous Dimensions at N${}^3$LO and Beyond
• QCD evolution equations from conformal symmetry
• Threshold resummation for $W$-boson pair production at NNLO+NNLL
• QCD analysis of valence structure functions using deep inelastic lepton-nucleon scattering