Small-$x$ Helicity Evolution: an Operator Treatment

Abstract

We rederive the small-$x$ evolution equations governing quark helicity distribution in a proton using solely an operator-based approach. In our previous works on the subject, the evolution equations were derived using a mix of diagrammatic and operator-based methods. In this work, we re-derive the double-logarithmic small-$x$ evolution equations for quark helicity in terms of the "polarized Wilson lines", the operators consisting of light-cone Wilson lines with one or two non-eikonal local operator insertions which bring in helicity dependence. For the first time we give explicit and complete expressions for the quark and gluon polarized Wilson line operators, including insertions of both the gluon and quark sub-eikonal operators. We show that the double-logarithmic small-$x$ evolution of the "polarized dipole amplitude" operators, made out of regular light-cone Wilson lines along with the polarized ones constructed here, reproduces the equations derived in our earlier works. The method we present here can be used as a template for determining the small-$x$ asymptotics of any transverse momentum-dependent (TMD) quark (or gluon) parton distribution functions (PDFs), and is not limited to helicity.

arXiv comment31 pages, 11 figures

References in corpus (55)

• Computing Quark and Gluon Distribution Functions for Very Large Nuclei
• Gauge-Invariant Decomposition of Nucleon Spin and Its Spin-Off
• Operator expansion for high-energy scattering
• Gluon Distribution Functions for Very Large Nuclei at Small Transverse Momentum
• Small-x F_2 Structure Function of a Nucleus Including Multiple Pomeron Exchanges
• Nonlinear Gluon Evolution in the Color Glass Condensate: I
• The Color Glass Condensate
• The Wilson renormalization group for low x physics: towards the high density regime
• Green's Functions in the Color Field of a Large Nucleus
• Unitarization of the BFKL Pomeron on a Nucleus
• The Renormalization Group Equation for the Color Glass Condensate
• The complete tree-level result up to order $1/Q$ for polarized deep-inelastic leptoproduction
• The Intrinsic Glue Distribution at Very Small x
• The Wilson renormalization group for low x physics: Gluon evolution at finite parton density
• Non-Abelian Weizsacker-Williams field and a two-dimensional effective color charge density for a very large nucleus
• The angular momentum controversy: What's it all about and does it matter?
• Factorization and high-energy effective action
• Saturation Physics and Deuteron--Gold Collisions at RHIC
• Evolution at small x_bj: The Color Glass Condensate
• Inclusive Gluon Production in DIS at High Parton Density
• Gluon saturation and initial conditions for relativistic heavy ion collisions
• Quantum structure of the non-Abelian Weizsacker-Williams field for a very large nucleus
• Gluon and Wilson loop TMDs for hadrons of spin $\le$ 1
• Distribution of linearly polarized gluons inside a large nucleus
• Electron Ion Collider: The Next QCD Frontier - Understanding the glue that binds us all
• Odderon in the Color Glass Condensate
• Proton Spin Structure from Measurable Parton Distributions
• Flavor Singlet Contribution to the Structure Function $g_1$ at Small-x
• Next-to-eikonal corrections in the CGC: gluon production and spin asymmetries in pA collisions
• A Simple Derivation of the JIMWLK Equation
• Perturbative Odderon in the Dipole Model
• Gluon orbital angular momentum at small-$x$
• Gluon TMD in particle production from low to moderate x
• Linearly-polarized small-x gluons in forward heavy-quark pair production
• The RHIC SPIN Program: Achievements and Future Opportunities
• Sub-eikonal corrections to scattering amplitudes at high energy
• Helicity Evolution at Small $x$: Flavor Singlet and Non-Singlet Observables
• Helicity Evolution at Small-x
• Small-$x$ asymptotics of the quark helicity distribution
• A New Mechanism for Generating a Single Transverse Spin Asymmetry
• Small-$x$ Asymptotics of the Gluon Helicity Distribution
• Baryon Stopping and the Valence Quark Distribution at Small x
• Small-$x$ Asymptotics of the Quark Helicity Distribution: Analytic Results
• Anisotropy of the semi-classical gluon field of a large nucleus at high energy
• Regge Asymptotics of Scattering with Flavour Exchange in QCD
• Medium Induced Transverse Momentum Broadening in Hard Processes
• pT-Broadening and production processes versus dipole/quadrupole amplitudes at next-to-leading order
• Calculating TMDs of a Large Nucleus: Quasi-Classical Approximation and Quantum Evolution
• Elastic scattering of a quark from a color field: longitudinal momentum exchange
• Reggeon Interactions in Perturbative QCD
• Studying the Perturbative Reggeon
• Valence Quark Transversity at Small $x$
• Nonsinglet Contributions to the Structure Function g1 at Small-x
• Quark-Antiquark Exchange in ${\gamma }^{\ast }{\gamma }^{\ast }$ Scattering
• Classical Gluon Production Amplitude for Nucleus-Nucleus Collisions: First Saturation Correction in the Projectile

Cited by in corpus (39)

• Mining for Gluon Saturation at Colliders
• Snowmass 2021 whitepaper: Proton structure at the precision frontier
• Helicity-dependent generalization of the JIMWLK evolution
• Ab Initio Coupling of Jets to Collective Flow in the Opacity Expansion Approach
• Proton Spin Structure at Small-$x$
• High-energy Operator Product Expansion at sub-eikonal level
• Quarks at next-to-eikonal accuracy in the CGC I: Forward quark-nucleus scattering
• Helicity-dependent extension of the McLerran-Venugopalan model
• Quark Sivers Function at Small $x$: Spin-Dependent Odderon and the Sub-Eikonal Evolution
• Quark and Gluon Helicity Evolution at Small $x$: Revised and Updated
• Helicity at Small $x$: Oscillations Generated by Bringing Back the Quarks
• Quark and scalar propagators at next-to-eikonal accuracy in the CGC through a dynamical background gluon field
• Quark jets scattering from a gluon field: from saturation to high $p_t$
• Rapidity loss, spin and angular asymmetries in scattering of a quark from color field of a proton (nucleus)
• Helicity Evolution at Small $x$: the Single-Logarithmic Contribution
• First analysis of world polarized DIS data with small-$x$ helicity evolution
• The role of the chiral anomaly in polarized deeply inelastic scattering II: Topological screening and transitions from emergent axion-like dynamics
• DIS dijet production at next-to-eikonal accuracy in the CGC
• Single Inclusive Hadron Production in DIS at Small $x$: Next to Leading Order Corrections
• Analytic Solution for the Revised Helicity Evolution at Small $x$ and Large $N_c$: New Resummed Gluon-Gluon Polarized Anomalous Dimension and Intercept
• T-odd Leading-Twist Quark TMDs at Small $x$
• Structure of the nucleon at leading and subleading twist in the covariant parton model
• Valence Quark Transversity at Small $x$
• Ultrarelativistic quark-nucleus scattering in a light-front Hamiltonian approach
• Scattering and gluon emission in a color field: a light-front Hamiltonian approach
• Small $x$ Physics Beyond Eikonal Approximation: an Effective Hamiltonian Approach
• Gluon-mediated inclusive Deep Inelastic Scattering from Regge to Bjorken kinematics
• Multiparticle production in proton-nucleus collisions beyond eikonal accuracy
• Helicity Evolution at Small $x$: Revised Asymptotic Results at Large $N_c\&N_f$
• Lensing Mechanism Meets Small-$x$ Physics: Single Transverse Spin Asymmetry in $p^{\uparrow }+p$ and $p^{\uparrow }+A$ Collisions
• Structure functions at small x from world-lines. I: Unpolarized distributions
• Global analysis of polarized DIS & SIDIS data with improved small-$x$ helicity evolution
• Unifying approaches: derivation of Balitsky hierarchy from the Lipatov effective action
• Quark TMDs from back-to-back dijet production at forward rapidities in pA collisions beyond eikonal accuracy in the CGC
• Color Glass Condensate meets High Twist Expansion
• JAM small-$x$ helicity phenomenology
• Parton model contributions as next-to-eikonal corrections to the dipole factorization of DIS and SIDIS at low $x_{Bj}$
• Single-Logarithmic Corrections to Small-$x$ Helicity Evolution
• Proton spin from small-$x$ with constraints from the valence quark model