Small-$x$ Asymptotics of the Gluon Helicity Distribution

TL;DR

This work establishes the small-x behavior of the proton's gluon helicity distribution at leading order in perturbative QCD in the large-N_c limit. It introduces and analyzes new polarized dipole operators that govern the dipole and Weizsäcker-Williams gluon helicity TMDs, deriving their double-logarithmic evolution and solving for the gluon helicity intercept α_h^G = (13)/(4√3)√(α_s N_c/(2π)) ≈ 1.88√(α_s N_c/(2π)). The results show α_h^G is about 20% smaller than the quark intercept α_h^q, due to the coupled quark-gluon helicity evolution and operator structure. Phenomenologically, the small-x gluon spin S_G may increase by 5–10% when extrapolated using the new intercept, emphasizing the need to incorporate these results into helicity PDF fits and future EIC analyses. The work also highlights fundamental differences between quark and gluon helicity evolution and sets the stage for further refinements including running coupling and finite-N_c effects.

Abstract

We determine the small-$x$ asymptotics of the gluon helicity distribution in a proton at leading order in perturbative QCD at large $N_c$. To achieve this, we begin by evaluating the dipole gluon helicity TMD at small $x$. In the process we obtain an interesting new result: in contrast to the unpolarized dipole gluon TMD case, the operator governing the small-$x$ behavior of the dipole gluon helicity TMD is different from the operator corresponding to the polarized dipole scattering amplitude (used in our previous work to determine the small-$x$ asymptotics of the quark helicity distribution). We then construct and solve novel small-$x$ large-$N_c$ evolution equations for the operator related to the dipole gluon helicity TMD. Our main result is the small-$x$ asymptotics for the gluon helicity distribution: $ΔG \sim \left( \tfrac{1}{x} \right)^{α_h^G}$ with $α_h^G = \tfrac{13}{4 \sqrt{3}} \, \sqrt{\tfrac{α_s \, N_c}{2 π}} \approx 1.88 \, \sqrt{\tfrac{α_s \, N_c}{2 π}}$. We note that the power $α_h^G$ is approximately 20$\%$ lower than the corresponding power $α_h^q$ for the small-$x$ asymptotics of the quark helicity distribution defined by $Δq \sim \left( \tfrac{1}{x} \right)^{α_h^q}$ with $α_h^q = \tfrac{4}{\sqrt{3}} \, \sqrt{\tfrac{α_s \, N_c}{2 π}} \approx 2.31 \, \sqrt{\tfrac{α_s \, N_c}{2 π}}$ found in our earlier work.

Figures (9)

• Figure 1: The polarized Wilson line \ref{['M:Vpol1']} in the quasi-classical approximation in $A^-=0$ gauge. The filled circles denote the spin-dependent sub-eikonal scattering.
• Figure 2: Diagrams illustrating contractions \ref{['e:qcontractions']} contributing to the evolution of the polarized dipole amplitude \ref{['eq:Gdef2']} for the quark helicity distribution. The blue band represents the classical fields (shock wave), the black vertex represents the sub-eikonal operator insertion \ref{['eq:noneik2']}, and the gray box represents the polarized Wilson line.
• Figure 3: Diagrams illustrating contractions contributing the evolution of the polarized dipole amplitude for the dipole gluon helicity distribution. The blue band represents the classical gluon fields (shock wave), the vertex $(i)$ denotes the sub-eikonal operator insertion, and the gray box represents the polarized Wilson line.
• Figure 4: Diagrams contributing the initial conditions for $G^i$ and $\Gamma^i$ in Eq. \ref{['Ginit']}.
• Figure 5: Linearized large-$N_c$ evolution of the new dipole function $G^i$ as written in Eq. (\ref{['M:evol4_1']}). Because there is no universal DLA parameter for the various terms, we have no a priori constraint on the relative sizes of $x_{20}$ and $x_{21}$, which makes enforcing lifetime ordering in these dipoles more subtle. We must distinguish between the ladder emission of polarized gluons (top line), which are constrained by the lifetime of dipole $21$ only, and the non-ladder emission of polarized gluons (middle line), which are constrained by the lifetimes of both dipoles $20$ and $21$. The "+ c.c." stands for adding mirror-reflected diagrams as well as the true complex conjugates in which line $0$ becomes a polarized quark line.