Scale evolution of T-odd gluon TMDs at small $x$

We study the transverse momentum dependent (TMD) evolution of the three leading twist dipole type T-odd gluon TMDs inside a transversely polarized nucleon, all of which at small $x$ dynamically originate from the spin dependent odderon. Their energy dependence presents a unique opportunity to study the polarization dependent TMD evolution in the small-$x$ region, where the distributions are identical up to a normalization constant at tree level. We further propose to study their evolution via azimuthal asymmetries in virtual photon-jet production in polarized proton-proton collisions at RHIC. We present model predictions for the asymmetries as functions of the large jet or photon transverse momentum and $Q^2$ which set the hard scales in this process.

Figure 1

The considered gluon TMDs at different scales at $x=0.01$, using the diquark model result as the initial condition.

Figure 2

The ratios ${\mathcal{R}}_1$ and ${\mathcal{R}}_2$ evolved to the scale $\mu =10\mathrm{GeV}$ for $x=0.01$ in comparison to the tree level model input that satisfies ${\mathcal{R}}_1={\mathcal{R}}_2$.

Figure 3

Estimates of the azimuthal asymmetries as a function of $k_{\perp }$ for $P_{\perp }=10\mathrm{GeV}$ (left panel), and a function of $P_{\perp }$ for $k_{\perp }=2.5\mathrm{GeV}$ (right panel) at $\sqrt{s}=500\mathrm{GeV}$. The jet and virtual photon rapidities $y_J$, $y_{\gamma }^{*}$ are integrated over the regions $[-1.5,-0.5],[-3,-2]$, respectively. The invariant mass of the virtual photon $Q^2$ is integrated over the region [30, 80] ${\mathrm{GeV}}^2$ in the left figure, and set to be $\frac{1}{2}{P}_{\perp }^2$ in the right one.