Helicity Evolution at Small x

We construct small-$x$ evolution equations which can be used to calculate quark and anti-quark helicity TMDs and PDFs, along with the $g_1$ structure function. These evolution equations resum powers of $\as \, \ln^2 (1/x)$ in the polarization-dependent evolution along with the powers of $\as \, \ln (1/x)$ in the unpolarized evolution which includes saturation effects. The equations are written in an operator form in terms of polarization-dependent Wilson line-like operators. While the equations do not close in general, they become closed and self-contained systems of non-linear equations in the large-$N_c$ and large-$N_c \, \& \, N_f$ limits. After solving the large-$N_c$ equations numerically we obtain the following small-$x$ asymptotics for the flavor-singlet $g_1$ structure function along with quarks hPDFs and helicity TMDs (in absence of saturation effects): $g_1^S (x, Q^2) \sim \Delta q^S (x, Q^2) \sim g_{1L}^S (x, k_T^2) \sim \left( \frac{1}{x} \right)^{\alpha_{h}} \approx \left( \frac{1}{x} \right)^{2.31 \: {\sqrt{\frac{\alpha_s N_c}{2 \pi}}}}.$ \notag We also give an estimate of how much of the proton's spin may be at small $x$ and what impact this has on the so-called ``spin crisis.''