Helicity of Quarks and Gluons at Small Bjorken $x$

Abstract

The proton spin puzzle is a longstanding problem in high-energy nuclear physics: how the proton spin distributes among the spin and orbital angular momenta of the quarks and gluons inside. Two of the unresolved pieces of the puzzle are the contributions to quark and gluon spins from the region of small Bjorken $x$. This dissertation fills the gap by constructing the evolution of these quantities into the small-$x$ region using a modified dipole formalism. The dominant contributions to the evolution equations resum powers of $\alpha_s\ln^2(1/x)$, where $\alpha_s$ is the strong coupling constant. In general, these evolution equations do not close. However, once the large-$N_c$ or large-$N_c\& N_f$ limit is taken, they turn into a system of linear integral equations that can be solved iteratively. At large $N_c$, the evolution equations are shown to be consistent with the gluon sector of the polarized DGLAP evolution in the small-$x$ limit. We numerically solve the equations in the large-$N_c$ and large-$N_c\& N_f$ limits and obtain the exponential growth in $\ln(1/x)$ for $N_f \leq 5$, with the intercept decreasing with $N_f$. For the large-$N_c$ limit, we have $\alpha_h = 3.66$, which agrees up to the uncertainty with the earlier work by Bartels, Ermolaev and Ryskin. Furthermore, at $N_f=6$, the asymptotic form attains an oscillation in $\ln(1/x)$ on top of the exponential growth, with the period spanning several units of rapidity. Finally, parts of the single-logarithmic corrections to the small-$x$ helicity evolution is also derived, resumming powers of $\alpha_s\ln(1/x)$. There, the effects of the unpolarized small-$x$ evolution and the running coupling are also included for consistency. The complete single-logarithmic corrections can be derived based on the framework established here. Altogether, these equations will provide the most precise small-$x$ helicity evolution to date.