Quark and Gluon Helicity Evolution at Small $x$: Revised and Updated

Abstract

We revisit the problem of small Bjorken-$x$ evolution of the gluon and flavor-singlet quark helicity distributions in the shock wave ($s$-channel) formalism. Earlier works on the subject in the same framework resulted in an evolution equation for the gluon field-strength $F^{12}$ and quark "axial current" $\bar{\psi }{\gamma }^{+}{\gamma }^5\psi$ operators (sandwiched between the appropriate light-cone Wilson lines) in the double-logarithmic approximation (DLA: summing powers of ${\alpha }_s{\ln }^2(1/x)$ with ${\alpha }_s$ the strong coupling constant). In this work, we observe that an important mixing of the above operators with another gluon operator, ${\overleftarrow{D}}^i{D}^i$, also sandwiched between the light-cone Wilson lines (with the repeated index $i=1,2$ summed over), was missing in the previous works. This operator has the physical meaning of the sub-eikonal (covariant) phase: its contribution to helicity evolution is shown to be proportional to another sub-eikonal operator, $D^i-{\overleftarrow{D}}^i$, which is related to the Jaffe-Manohar polarized gluon distribution. In this work we include this operator into small-$x$ helicity evolution, and construct a novel evolution mixing all three operators ($D^i-{\overleftarrow{D}}^i$, $F^{12}$, and $\bar{\psi }{\gamma }^{+}{\gamma }^5\psi$), generalizing the previous results. We also construct closed DLA evolution equations in the large-$N_c$ and large-$N_c\&N_f$ limits, with $N_c$ and $N_f$ the numbers of quark colors and flavors, respectively. Solving the large-$N_c$ equations numerically we obtain the following small-$x$ asymptotics of the quark and gluon helicity distributions $\Delta \Sigma$ and $\Delta G$, along with the $g_1$ structure function, \[ΔΣ(x,Q^2)\simΔG(x,Q^2)\sim g_1(x,Q^2)\sim\left(\frac{1}{x}\right)^{3.66\,\sqrt{\frac{α_s\,N_c}{2π}}},\] in complete agreement with the earlier work by Bartels, Ermolaev and Ryskin.

65 pages, 8 figures; v2, v3: typos corrected, comments added; v4: minor corrections in intermediate steps without changes to the final results

References in corpus (83)

• 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
• An effective field theory for collinear and soft gluons: heavy to light decays
• Quantum Chromodynamics and Other Field Theories on the Light Cone
• Gluon Distribution Functions for Very Large Nuclei at Small Transverse Momentum
• Small-x F_2 Structure Function of a Nucleus Including Multiple Pomeron Exchanges
• Soft-Collinear Factorization in Effective Field Theory
• 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
• Nonlinear Gluon Evolution in the Color Glass Condensate: II
• Unitarization of the BFKL Pomeron on a Nucleus
• The Renormalization Group Equation for the Color Glass Condensate
• Soft-collinear effective theory and heavy-to-light currents beyond leading power
• The Wilson renormalization group for low x physics: Gluon evolution at finite parton density
• Unitarity at small Bjorken x
• Single and Double BFKL Pomeron Exchange and a Dipole Picture of High Energy Hard Processes
• The Spin Structure of the Nucleon
• Unitarity and the BFKL Pomeron
• 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
• Gluon saturation and initial conditions for relativistic heavy ion collisions
• First simultaneous extraction of spin-dependent parton distributions and fragmentation functions from a global QCD analysis
• The Calculation of the Two-Loop Spin Splitting Functions $P_{ij}^{(1)}(x)$
• Spin Physics and Polarized Structure Functions
• A New Odderon Solution in Perturbative QCD
• The Construction of Gauge-Links in Arbitrary Hard Processes
• The Color Glass Condensate and High Energy Scattering in QCD
• Odderon in the Color Glass Condensate
• The Three-Loop Splitting Functions in QCD: The Helicity-Dependent Case
• Mining for Gluon Saturation at Colliders
• 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
• On the Behaviour of Non--Singlet Structure Functions at Small $x$
• Solution of the Odderon Problem
• Perturbative Odderon in the Dipole Model
• Quark and Gluon Orbital Angular Momentum and Spin in Hard Processes
• Gluon orbital angular momentum at small-$x$
• Gluon TMD in particle production from low to moderate x
• Helicity Evolution at Small $x$: Flavor Singlet and Non-Singlet Observables
• Sub-eikonal corrections to scattering amplitudes at high energy
• Small-$x$ Helicity Evolution: an Operator Treatment
• On Orbital Angular Momentum in Deep Inelastic Scattering
• Twist analysis of the nucleon spin in QCD
• Wilson lines and orbital angular momentum
• A classical Odderon in QCD at high energies
• Small-$x$ Asymptotics of the Gluon Helicity Distribution
• Baryon Stopping and the Valence Quark Distribution at Small x
• Helicity-dependent generalization of the JIMWLK evolution
• The Singlet Contribution to the Structure Function $g_1(x,Q^2)$ at Small $x$
• Next-to-next-to-eikonal corrections in the CGC
• The three-loop polarized singlet anomalous dimensions from off-shell operator matrix elements
• Small-$x$ Asymptotics of the Quark Helicity Distribution: Analytic Results
• Odderon and seven Pomerons: QCD Reggeon field theory from JIMWLK evolution
• Probing Parton Orbital Angular Momentum in Longitudinally Polarized Nucleon
• Regge Asymptotics of Scattering with Flavour Exchange in QCD
• Proton Spin Structure at Small-$x$
• Power corrections to TMD factorization for Z-boson production
• Evolution Equations for Higher Moments of Angular Momentum Distributions
• Quarks at next-to-eikonal accuracy in the CGC I: Forward quark-nucleus scattering
• Solution of the multi-reggeon compound state problem in multi-colour QCD
• High-energy Operator Product Expansion at sub-eikonal level
• pT-Broadening and production processes versus dipole/quadrupole amplitudes at next-to-leading order
• Helicity-dependent extension of the McLerran-Venugopalan model
• Quark Sivers Function at Small $x$: Spin-Dependent Odderon and the Sub-Eikonal Evolution
• Higher-twist corrections to gluon TMD factorization
• 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 $p_t$
• The role of the chiral anomaly in polarized deeply inelastic scattering I: Finding the triangle graph inside the box diagram in Bjorken and Regge asymptotics
• 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
• Orbital Angular Momentum at Small $x$
• Reggeon Interactions in Perturbative QCD
• Studying the Perturbative Reggeon
• The role of the chiral anomaly in polarized deeply inelastic scattering II: Topological screening and transitions from emergent axion-like dynamics
• Spin-dependent Pomeron and Odderon in elastic proton-proton scattering
• $CP$-odd gluonic operators in QCD spin physics
• Parton distribution function for topological charge at one loop

Cited by in corpus (4)

• 50 Years of Quantum Chromodynamics
• Quark Sivers Function at Small $x$: Spin-Dependent Odderon and the Sub-Eikonal Evolution
• DIS dijet production at next-to-eikonal accuracy in the CGC
• Small-x Evolution of the Gluon Generalized Parton Distribution Eg of the Nucleon