Small-$x$ Asymptotics of the Leading-Twist Flavor Singlet Quark TMDs

TL;DR

This work addresses the small-$x$ behavior of flavor-singlet, leading-twist quark TMDs by applying the Light-Cone Operator Treatment (LCOT) to express TMD operators through polarized dipole amplitudes. In the large-$N_c$ limit under linearized Double-Logarithmic Approximation (DLA), the authors derive and solve the evolution equations for all six flavor-singlet TMDs, obtaining explicit small-$x$ asymptotics: the Sivers and g1T grow as $(1/x)^{2.9\sqrt{\alpha_s N_c/(2\pi)}}$, transversity and pretzelosity share NS-like behavior, and Boer-Mulders and helicity worm-gear display near-sub-eikonal scaling. A notable finding is the oscillatory behavior of certain polarized dipoles, yet the overarching asymptotics remain robust and mirror flavor-non-singlet patterns in several channels. The results supply a complete, testable framework for small-$x$ TMD phenomenology, with implications for future Electron-Ion Collider measurements, lattice checks, and matching to CSS and twist-three evolution in broader QCD factorization schemes.

Abstract

In this paper, we investigate the small-$x$ behavior of the flavor-singlet, leading-twist quark Transverse Momentum Dependent parton distribution functions (TMDs) using the Light-Cone Operator Treatment. This formalism allows us to express TMD operators at small $x$ in terms of polarized dipole amplitudes, enabling a systematic approach to their small-$x$ evolution. We derive the evolution equations for these TMDs and solve them within the large-$N_c$ approximation under the linearized, Double-Logarithmic Approximation (DLA), where $N_c$ represents the number of quark colors. Expanding on previous work on unpolarized and helicity TMDs, we present the small-$x$ asymptotics for a comprehensive set of TMDs, including the Sivers function, helicity worm-gear, transversity, pretzelosity, Boer-Mulders, and transversity worm-gear distributions. Our results provide a complete picture of the small-$x$ asymptotic behavior for all leading-twist flavor-singlet quark TMDs. We also discuss the implications of our findings for phenomenological applications and outline potential avenues for further research, particularly in understanding non-linear effects and extending beyond the DLA and large-$N_c$ approximations.

Figures (6)

• Figure 1: Example of the class of diagrams which give the leading sub-eikonal and sub-sub-eikonal corrections for the TMDs considered here. The anti-quark propagates from the position $\zeta$ to $\underline{w}$ with momentum $k_1$, undergoes a sub-eikonal interaction with the proton which changes its transverse position from $\underline{w}$ on the left of the shock wave to $\underline{z}$ on the right of the shock wave. The anti-quark then propagates from $\underline{z}$ to the position $\xi$ with momentum $k_2$. The shock wave is denoted by the blue (grey) rectangle, while the sub-(sub-) eikonal interaction with the shock wave is denoted by the white box. The double line represents the eikonal Wilson line encoding the interactions of the quark in the dipole.
• Figure 2: The plots of logarithms of the absolute values of polarized dipole amplitudes $F^S_A$, $F^S_B$ and $F^{S~\rm{mag}}$ versus $s_{10}$ and $\eta$, for the range $-\eta_{\max}\leq s_{10}\leq \eta_{\max}$ and $0\leq\eta\leq\eta_{\max}$ with $\eta_{\max}=70$. The amplitudes are computed numerically using step size $\Delta = 0.12$.
• Figure 3: Typical classes of diagrams that appear in the evolution of polarized dipole amplitudes at large-$N_c$.
• Figure 4: Classes of diagrams contributing to the evolution of polarized dipole amplitudes where only polarized quark emission (class $\alpha$) and eikonal gluon emissions (remaining classes) are allowed.
• Figure 5: Plots illustrating $H^{1T,S}(s_{10},\eta)$ that results from initial condition \ref{['ICs10']}.