Double-Logarithmic Small-x Evolution for T-Odd Weizsäcker-Williams TMDs
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At small Bjorken x, we study the gluon Weizsäcker-Williams (WW) transverse-momentum-dependent parton distribution functions (TMDs) that are odd under time reversal. Specifically, this includes the Sivers function, $f_{1T}^{\perp}$, the linearly polarized gluon, $h_{1T}$, and the gluonic pretzelosity, $h_{1T}^{\perp}$. Their operator definitions can be expanded in powers of Bjorken x, whose first non-vanishing terms yield a trace of fundamental Wilson lines decorated by sub-eikonal insertions. These traces form sub-eikonal dipole amplitudes that contain small-x evolution resumming double logarithms, $\alpha_s\ln^2(1/x)$. The evolution equations become closed upon taking the large-$N_c$ limit, allowing us to determine that they scale asymptotically at small Bjorken $x$ with $x^{-2.9\sqrt{\alpha_sN_c/4\pi}}$. Despite the power-law growth, these asymptotic behaviors are suppressed by a power of $x$ due to the sub-eikonal nature of the WW TMDs, in contrast to the small-x asymptotic behavior of T-odd dipole gluon TMDs that contain eikonal contributions driven mainly by the spin-dependent odderon.