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: $$ \varDelta G\sim {\left(\frac{1}{x}\right)}^{\alpha_h^G} $$ with $$ {\alpha}_h^G=\frac{13}{4\sqrt{3}}\sqrt{\frac{\alpha_s{N}_c}{2\pi }}\approx 1.88\sqrt{\frac{\alpha_s{N}_c}{2\pi }} $$ . We note that the power α is approximately 20% lower than the corresponding power α for the small-x asymptotics of the quark helicity distribution defined by $$ \varDelta q\sim {\left(\frac{1}{x}\right)}^{\alpha_h^q} $$ with $$ {\alpha}_h^q=\frac{4}{\sqrt{3}}\sqrt{\frac{\alpha_s{N}_c}{2\pi }}\approx 2.31\sqrt{\frac{\alpha_s{N}_c}{2\pi }} $$ found in our earlier work.