The Universal Transverse Momentum Dependent Soft Function at NNLO

Abstract

All (un)polarized transverse momentum dependent functions (TMDs), both distribution and fragmentation functions, are defined with the same universal soft function, which cancels spurious rapidity divergences within an individual TMD and renders them well-defined hadronic quantities. Moreover, it is independent of the kinematics, whether it is Drell-Yan, deep inelastic scattering or $e^{+}{e}^{-}\to 2$ hadrons. In this paper we provide this soft function at next-to-next-to-leading order (NNLO), necessary for the calculation of all TMDs at the same order, and to perform the resummation of large logarithms at next-to-next-to-next-to-leading-logarithmic accuracy. From the results we obtain the $D$ function at NNLO, which governs the evolution of all TMDs. This work represents the first independent and direct calculation of this quantity. Given the all order relation through a Casimir scaling between the soft function relevant for gluon TMDs and the one for quark TMDs, we also obtain the first at NNLO. The used regularization method to deal with the rapidity divergences is discussed as well.

12 pages, 2 figures. v2: minor rewording, accepted for publication in PRD

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Cited by in corpus (89)

• Large-Momentum Effective Theory
• Unpolarized Transverse Momentum Dependent Parton Distribution and Fragmentation Functions at next-to-next-to-leading order
• Parton Fragmentation Functions
• Analysis of vector boson production within TMD factorization
• Non-perturbative structure of semi-inclusive deep-inelastic and Drell-Yan scattering at small transverse momentum
• Soft-/rapidity- anomalous dimensions correspondence
• Quark Transverse Parton Distribution at the Next-to-Next-to-Next-to-Leading Order
• Subleading Power Rapidity Divergences and Power Corrections for $q_T$
• Lattice-QCD Calculations of TMD Soft Function Through Large-Momentum Effective Theory
• Towards Quasi-Transverse Momentum Dependent PDFs Computable on the Lattice
• Resummation of Transverse Momentum Distributions in Distribution Space
• Transverse momentum dependent PDFs at N${}^3$LO
• Rapidity renormalized TMD soft and beam functions at two loops
• Drell-Yan $q_T$ Resummation of Fiducial Power Corrections at N${}^3$LL
• Transverse momentum dependent distributions with jets
• TMD Soft Function from Large-Momentum Effective Theory
• Transverse momentum dependent distributions in $e^{+}{e}^{-}$ and semi-inclusive deep-inelastic scattering using jets
• Lattice QCD Study of Transverse-Momentum Dependent Soft Function
• The Four Loop QCD Rapidity Anomalous Dimension
• Self-contained definition of the Collins-Soper kernel
• Transverse momentum dependent factorization for lattice observables
• Linearly polarized gluons at next-to-next-to leading order and the Higgs transverse momentum distribution
• Extraction of the Sivers function from SIDIS, Drell-Yan, and $W^{\pm }/Z$ boson production data with TMD evolution
• Transverse Parton Distribution and Fragmentation Functions at NNLO: the Gluon Case
• Probing Transverse-Momentum Distributions With Groomed Jets
• PDF bias and flavor dependence in TMD distributions
• Twist-2 matching of transverse momentum dependent distributions
• A Toolbox for $q_T$ and $0$-Jettiness Subtractions at N${}^3$LO
• Computing Light-Front Wave Functions Without Light-Front Quantization: A Large-Momentum Effective Theory Approach
• Proper definition and evolution of generalized transverse momentum distributions
• Collinear matching for Sivers function at next-to-leading order
• Transverse momentum dependent operator expansion at next-to-leading power
• Two-loop anomalous dimensions of generic dijet soft functions
• Recoil-free azimuthal angle for precision boson-jet correlation
• Computing one-loop corrections to effective vertices with two scales in the EFT for Multi-Regge processes in QCD
• Quantum Simulation of Light-Front Parton Correlators
• Generic dijet soft functions at two-loop order: correlated emissions
• Joint transverse momentum and threshold resummation beyond NLL
• Renormalization and Matching for the Collins-Soper Kernel from Lattice QCD
• TMD factorization for dijet and heavy-meson pair in DIS
• Matching of transverse momentum dependent distributions at twist-3
• Transverse-momentum-dependent wave functions and Soft functions at one-loop in Large Momentum Effective Theory
• Threshold resummation for computing large-$x$ parton distribution through large-momentum effective theory
• Unpolarized Quark and Gluon TMD PDFs and FFs at N${}^3$LO
• Structure of the nucleon at leading and subleading twist in the covariant parton model
• Precision boson-jet azimuthal decorrelation at hadron colliders
• Transverse $\Lambda$ Polarization in $e^{+}{e}^{-}$ collisions
• Effective Field Theory for jet substructure in heavy ion collisions
• Transverse momentum dependent distributions in dijet and heavy hadron pair production at EIC
• Soft factors for double parton scattering at NNLO
• Muon-electron backward scattering: a prime example for endpoint singularities in SCET
• Gauge-invariant TMD factorization for Drell-Yan hadronic tensor at small x
• Collinear expansion for color singlet cross sections
• Definition and evolution of transverse momentum dependent distribution of twist-three
• Prospects for quarkonium studies at the high-luminosity LHC
• Drell-Yan Transverse-Momentum Spectra at N${}^3$LL${}^{\prime }$ and Approximate N${}^4$LL with SCETlib
• A fast and accurate method for perturbative resummation of transverse momentum-dependent observables
• Quarkonium TMD fragmentation functions in NRQCD
• Quasi Transverse Momentum Dependent Distributions at Next-to-Next-to-Leading order
• Structure of rapidity divergences in soft factors
• Transverse momentum moments
• Conformal invariance of TMD rapidity evolution
• Twist-4 T-even proton TMDs in the light-front quark-diquark model
• Gluon Transverse-Momentum-Dependent Distributions from Large-Momentum Effective Theory
• Quark Wigner Distributions and GTMDs of Pion in the Light-Front Holographic Model
• Sivers, Boer-Mulders and worm-gear distributions at next-to-leading order
• Nonperturbative renormalization of asymmetric staple-shaped operators in twisted mass lattice QCD
• GTMDs and the factorization of exclusive double Drell-Yan
• QCD$\times$QED evolution of TMDs
• Two-loop splitting in double parton distributions: the colour non-singlet case
• Matching generalised transverse-momentum-dependent distributions onto generalised parton distributions at one loop
• Gluon TMD fragmentation function into quarkonium
• Automated Calculation of Dijet Soft Functions in Soft-Collinear Effective Theory
• Fracture Functions in Different Kinematic Regions and Their Factorizations
• Renormalization of the next-to-leading-power $\gamma \gamma \to h$ and $gg\to h$ soft quark functions
• Exploring slicing variables for jet processes
• Disentangling observable dependence in SCETI and SCETII anomalous dimensions: angularities at two loops
• A formalism for the resummation of non-factorizable observables in SCET
• Light-Cone Singularities and Transverse-Momentum-Dependent Factorization at Twist-3
• Two-loop evolution kernels for colour dependent double parton distributions
• W-boson production in TMD factorization
• Nucleon tomography. What can we do better today than Rutherford 100 years ago?
• $q_T$-slicing with multiple jets
• SUSY, Casimir scaling and probabilistic properties of gluon and quark jets evolution
• Perturbative Corrections to Quark TMDPDFs in the Background-Field Method: Gauge Invariance, Equations of Motion, and Multiple Interactions
• QCD Beyond Leading Power
• Revisiting the time-ordering issue of TMD soft factors: causality, coordinate space analyticity and new equalities
• Connecting baryon light-front wave functions to quasi-transverse-momentum-dependent correlators in lattice QCD
• Soft background fields at next-to-leading power in transverse momentum dependent SIDIS with jets