Helicity Evolution at Small $x$: Flavor Singlet and Non-Singlet Observables

Abstract

We extend our earlier results for the quark helicity evolution at small $x$ to derive the small-$x$ asymptotics of the flavor singlet and flavor non-singlet quark helicity TMDs and PDFs and of the $g_1$ structure function. In the flavor singlet case we re-derive the evolution equations obtained in our previous paper on the subject, performing additional cross-checks of our results. In the flavor non-singlet case we construct new small-$x$ evolution equations by employing the large-$N_c$ limit. All evolution equations resum double-logarithmic powers of ${\alpha }_s{\ln }^2(1/x)$ in the polarization-dependent evolution along with the single-logarithmic powers of ${\alpha }_s\ln (1/x)$ in the unpolarized evolution which includes saturation effects. We solve the linearized flavor non-singlet equation analytically, obtaining an intercept which agrees with the one calculated earlier by Bartels, Ermolaev and Ryskin using the infra-red evolution equations. Our numerical solution of the linearized large-$N_c$ evolution equations for the flavor singlet case is presented in the accompanying Letter and is further discussed here.

30 pages, 12 figures

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• Helicity at Small $x$: Oscillations Generated by Bringing Back the Quarks
• T-odd Leading-Twist Quark TMDs at Small $x$
• Gluon distributions and their applications to Ioffe-time distributions
• Accessing the nucleon transverse structure in inclusive deep inelastic scattering
• Multiparticle production in proton-nucleus collisions beyond eikonal accuracy
• JAM small-$x$ helicity phenomenology
• Can sea quark asymmetry shed light on the orbital angular momentum of the proton?