Doctor of Philosophy, The Ohio State University, 2022, Physics
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 unknown pieces of the puzzle are the contributions from quark and gluon spins at small Bjorken $x$. This dissertation fills the gap by constructing the evolution of these quantities into the small-$x$ region, through their relations with the polarized dipole amplitudes. 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. (Here, $N_c$ and $N_f$ represents the number of quark colors and flavors, respectively.) 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 following small-$x$ asymptotics for $N_f \leq 5$:
$$g_1(x,Q^2)\sim \Delta\Sigma(x,Q^2) &\sim \Delta G(x,Q^2) \sim \left(\frac{1}{x}\right)^{\alpha_h\sqrt{\frac{\alpha_sN_c}{2\pi}}}\,.$$
with the intercept, $\alpha_h$, decreasing with $N_f$. In particular, in the large-$N_c$ limit, we have $\alpha_h = 3.66$, which agrees with the earlier work by Bartels, Ermolaev and Ryskin. Furthermore, at $N_f=6$, the asymptotic form becomes
$$g_1(x,Q^2)\sim \Delta\Sigma(x,Q^2) &\sim \Delta G(x,Q^2) \sim \left(\frac{1}{x}\right)^{\alpha_h\sqrt{\frac{\alpha_sN_c}{2\pi}}} \cos\left(\omega_h\sqrt{\frac{\alpha_sN_c}{2\pi}}\,\ln\frac{1}{x} + \varphi_h\right),$$
where the parameters, $\alpha_h$, $\omega_h$ and $\varphi_h$, are calculated and listed in the text. The emerging oscillation has a period spanning several units of rapidity. Finally, parts of the (open full item for complete abstract)
Committee: Yuri Kovchegov (Advisor); Michael Lisa (Committee Member); Christopher Hirata (Committee Member); Richard Furnstahl (Committee Member)
Subjects: Nuclear Physics; Physics