Analytic Continuation and Reciprocity Relation for Collinear Splitting in QCD

Abstract

It is well-known that direct analytic continuation of DGLAP evolution kernel (splitting functions) from space-like to time-like kinematics breaks down at three loops. We identify the origin of this breakdown as splitting functions are not analytic function of external momenta. However, splitting functions can be constructed from square of (generalized) splitting amplitudes. We establish the rule of analytic continuation for splitting amplitudes, and use them to determine the analytic continuation of certain holomorphic and anti-holomorphic part of splitting functions and transverse-momentum dependent distributions. In this way we derive the time-like splitting functions at three loops without ambiguity. We also propose a reciprocity relation for singlet splitting functions, and provide non-trivial evidence that it holds in QCD at least through three loops.

arXiv commentv1:6 pages, 5 figures; v2:references added

References in corpus (59)

• An effective field theory for collinear and soft gluons: heavy to light decays
• Soft-Collinear Factorization in Effective Field Theory
• The Three-Loop Splitting Functions in QCD: The Non-Singlet Case
• Summing Sudakov logarithms in B -> X_s + gamma in effective field theory
• The Three-Loop Splitting Functions in QCD: The Singlet Case
• Invariant Operators in Collinear Effective Theory
• Hard Scattering Factorization from Effective Field Theory
• QCD Factorization for Semi-Inclusive Deep-Inelastic Scattering at Low Transverse Momentum
• Transverse-momentum resummation and the spectrum of the Higgs boson at the LHC
• A Formalism for the Systematic Treatment of Rapidity Logarithms in Quantum Field Theory
• Drell-Yan production at small q_T, transverse parton distributions and the collinear anomaly
• QCD Factorization for Spin-Dependent Cross Sections in DIS and Drell-Yan Processes at Low Transverse Momentum
• Infrared factorization of tree-level QCD amplitudes at the next-to-next-to-leading order and beyond
• The Zero-Bin and Mode Factorization in Quantum Field Theory
• Parton-to-Pion Fragmentation Reloaded
• Double Unresolved Approximations to Multiparton Scattering Amplitudes
• Bootstrapping rapidity anomalous dimension for transverse-momentum resummation
• Two-Loop g -> gg Splitting Amplitudes in QCD
• The Structure of the Proton in the LHC Precision Era
• Unpolarized Transverse Momentum Dependent Parton Distribution and Fragmentation Functions at next-to-next-to-leading order
• Revisiting parton evolution and the large-x limit
• Calculation of the transverse parton distribution functions at next-to-next-to-leading order
• Vector boson production at hadron colliders: hard-collinear coefficients at the NNLO
• Reparameterization Invariance for Collinear Operators
• The Collinear Limit of the Energy-Energy Correlator
• Anomalous dimensions of high-spin operators beyond the leading order
• Model Independent Evolution of Transverse Momentum Dependent Distribution Functions (TMDs) at NNLL
• Four-Loop Non-Singlet Splitting Functions in the Planar Limit and Beyond
• The semi-inclusive jet function in SCET and small radius resummation for inclusive jet production
• Next-to-Next-to-Leading Order Evolution of Non-Singlet Fragmentation Functions
• Transverse parton distribution functions at next-to-next-to-leading order: the quark-to-quark case
• Quark Transverse Parton Distribution at the Next-to-Next-to-Next-to-Leading Order
• Two-loop splitting functions in QCD
• On Third-Order Timelike Splitting Functions and Top-Mediated Higgs Decay into Hadrons
• On the Next-to-Next-to-Leading Order Evolution of Flavour-Singlet Fragmentation Functions
• Simplicity from Recoil: The Three-Loop Soft Function and Factorization for the Energy-Energy Correlation
• On the Drell-Levy-Yan Relation to $O({\alpha }_s^2)$
• Exposing the QCD Splitting Function with CMS Open Data
• Next-to-Leading Order Evolution of Polarized and Unpolarized Fragmentation Functions
• Fragmentation in Jets at NNLO
• Transverse Parton Distribution and Fragmentation Functions at NNLO: the Quark Case
• Renormalization, Wilson lines, and transverse-momentum dependent parton distribution functions
• N=4 SUSY Yang--Mills: three loops made simple(r)
• Jet substructure using semi-inclusive jet functions within SCET
• Transverse momentum dependent distributions with jets
• Two-loop splitting amplitudes and the single-real contribution to inclusive Higgs production at N3LO
• On timelike and spacelike hard exclusive reactions
• Transverse momentum dependent distributions in $e^{+}{e}^{-}$ and semi-inclusive deep-inelastic scattering using jets
• Inclusive production of small radius jets in heavy-ion collisions
• Linearly polarized gluons at next-to-next-to leading order and the Higgs transverse momentum distribution
• Fragmentation Functions at Next-to-Next-to-Leading Order Accuracy
• Collinear factorization violation and effective field theory
• Tree-level splitting amplitudes for a quark into four collinear partons
• Soft Fragmentation on the Celestial Sphere
• Towards three-loop QCD corrections to the time-like splitting functions
• Cutting massless four-loop propagators
• Master integrals for splitting functions from differential equations in QCD
• Five-Particle Phase-Space Integrals in QCD
• Removing phase-space restrictions in factorized cross sections

Cited by in corpus (23)

• Collider Physics at the Precision Frontier
• The Four Loop QCD Rapidity Anomalous Dimension
• Extending Precision Perturbative QCD with Track Functions
• Spinning Gluons from the QCD Light-Ray OPE
• The Energy-Energy Correlation in the back-to-back limit at N${}^3$LO and N${}^3$LL${}^{\prime }$
• Energy-energy correlators in Deep Inelastic Scattering
• TMD Fragmentation Functions at N${}^3$LO
• Renormalization Group Flows for Track Function Moments
• Conformal Colliders Meet the LHC
• Scaling violation in power corrections to energy correlators from the light-ray OPE
• Global analysis of fragmentation functions to charged hadrons with high-precision data from the LHC
• Nonperturbative Effects in Energy Correlators: From Characterizing Confinement Transition to Improving ${\alpha }_s$ Extraction
• Unpolarized Quark and Gluon TMD PDFs and FFs at N${}^3$LO
• Resummation of small-x double logarithms in QCD: inclusive deep-inelastic scattering
• Next-to-Next-to-Leading Order QCD Corrections to Polarized Semi-Inclusive Deep-Inelastic Scattering
• NNLO QCD corrections to unpolarized and polarized SIDIS
• Automated calculation of Jet fragmentation at NLO in QCD
• Fragmentation Functions of Charged Hadrons at Next-to-Next-to-Leading Order and Constraints on the Proton Parton Distribution Functions
• Probing cold nuclear matter with energy correlators
• Single-inclusive hadron production in electron-positron annihilation at next-to-next-to-next-to-leading order in QCD
• Global analysis of fragmentation functions to light neutral hadrons
• Four-loop anomalous dimension of flavor non-singlet quark operator of twist two and Lorentz spin N for general gauge group: transcendental part
• Parton Fragmentation Functions