Sub-eikonal stress and model dependence of the small-$x$ gluon D-term

Abstract

The leading-eikonal small-$x$ dipole gives a compact representation of the gluon momentum form factor $A_g(t)$. We show that the same information is not sufficient to determine the gluon stress form factor $C_g(t)$, and hence the gluon D-term. The reason is kinematic and operatorial: in a Drell-Yan frame $A_g(t)$ is projected by $T_g^{++}$, whereas $C_g(t)$ is projected by the symmetric-traceless transverse stress $T_g^{ij}$. This stress projection first appears through next-to-eikonal fields and is represented by gauge-invariant stress-decorated Wilson lines containing $F^{i-}$, $F^{ij}$, or equivalent sub-eikonal target fields. We construct this operator and match it to the local energy-momentum tensor at tree level, obtaining an operator-level no-go statement: the ordinary dipole $S_x(b_\perp,r_\perp)$, or a saturation profile $Q_s^2(x,b)$, does not by itself determine the sign of the small-$x$ gluon D-term. We then give a finite-correlation response model in which a positive kernel generates an anti-aligned response $F^{i-}=-\epsilon R_{\rm NE}F^{i+}$, so that $\Lambda_{\rm NE}>0$ and $D_g(0)<0$ within that response class. A Gaussian benchmark and numerical scans over Gaussian, Woods-Saxon, power-edge, and McLerran-Venugopalan (MV)-inspired profiles show a stable negative forward D-term for this anti-aligned model, together with the expected core-shell pressure pattern. The gluon D-term is therefore a next-to-eikonal stress probe, not a universal leading-eikonal saturation observable.