Small-$x$ Asymptotics of the Gluon Helicity Distribution

We determine the small-$x$ asymptotics of the gluon helicity distribution in a proton at leading order in perturbative QCD at large $N_c$. To achieve this, we begin by evaluating the dipole gluon helicity TMD at small $x$. In the process we obtain an interesting new result: in contrast to the unpolarized dipole gluon TMD case, the operator governing the small-$x$ behavior of the dipole gluon helicity TMD is different from the operator corresponding to the polarized dipole scattering amplitude (used in our previous work to determine the small-$x$ asymptotics of the quark helicity distribution). We then construct and solve novel small-$x$ large-$N_c$ evolution equations for the operator related to the dipole gluon helicity TMD. Our main result is the small-$x$ asymptotics for the gluon helicity distribution: $\Delta G \sim \left( \tfrac{1}{x} \right)^{\alpha_h^G}$ with $\alpha_h^G = \tfrac{13}{4 \sqrt{3}} \, \sqrt{\tfrac{\alpha_s \, N_c}{2 \pi}} \approx 1.88 \, \sqrt{\tfrac{\alpha_s \, N_c}{2 \pi}}$. We note that the power $\alpha_h^G$ is approximately 20$\%$ lower than the corresponding power $\alpha_h^q$ for the small-$x$ asymptotics of the quark helicity distribution defined by $\Delta q \sim \left( \tfrac{1}{x} \right)^{\alpha_h^q}$ with $\alpha_h^q = \tfrac{4}{\sqrt{3}} \, \sqrt{\tfrac{\alpha_s \, N_c}{2 \pi}} \approx 2.31 \, \sqrt{\tfrac{\alpha_s \, N_c}{2 \pi}}$ found in our earlier work.