Sub-eikonal vanishing of the WW-type h1L-perpendicular gluon TMD

Prove whether the Weizsäcker–Williams-type h_{1L}^{\perp} gluon TMD vanishes at sub-eikonal order in the small-x expansion.

Background

The WW-type h_{1L}{\perp} distribution vanishes at eikonal order, and the known sub-eikonal classical field does not appear to couple to its operator. The paper identifies the still-unproved possibility that the distribution also vanishes at the next order.

References

The sub-eikonal classical field does not appear to couple to the corresponding operator either, and it has been suggested that the WW-type $h_{1L}\perp$ may vanish at sub-eikonal order as well, which remains to be proven.

— Transverse momentum dependent parton distributions at small $x$  (2610.26004 - Caucal et al., 19 Oct 2026) in Section 4.1, paragraph beginning “The TMD h1L”