All-order survival of small-x factorization beyond leading order

Determine whether the leading-order small-x factorization framework based on process-dependent Wilson-line correlators survives at next-to-leading order and beyond.

Background

At leading order, process-dependent small-x cross sections can be expressed in terms of a finite set of Wilson-line correlators, even when standard moderate-x TMD factorization is violated. The unresolved issue is whether this operator-based predictivity persists after radiative corrections and at higher perturbative orders.

References

This holds at leading order for a particular class of processes, and whether it survives at NLO and beyond is, to our knowledge, open.

— Transverse momentum dependent parton distributions at small $x$  (2610.26004 - Caucal et al., 19 Oct 2026) in Section 2, paragraph beginning “At moderate x”

Although this generically remains an open problem, in particular for gluon TMDs with gauge-link structures more complicated than those of the WW TMD, there is increasing evidence that the resummation can be performed using an evolution equation formally equivalent to the Collins-Soper-Sterman equation.

— Transverse momentum dependent parton distributions at small $x$  (2610.26004 - Caucal et al., 19 Oct 2026) in Section 3.2, subsection “Resummation of Sudakov logarithms”

How these additional contributions should be resummed, and in particular whether they exponentiate in the same way as in the CSS solution, remains an open question (for the single logarithm depending on the jet radius R, the resummation should be implemented at the level of the jet function).

— Transverse momentum dependent parton distributions at small $x$  (2610.26004 - Caucal et al., 19 Oct 2026) in Section 3.2.2, paragraph beginning “The coefficients A_i and B_i”

Second, for processes with colored partons in both the initial and the final state, such as forward dihadron production in $pA$ collisions, it is not known whether the finite operator basis found at leading order survives at next-to-leading order, nor whether CSS resummation applied to each distribution can be matched to a next-to-leading-order small-$x$ calculation (Secs.~\ref{sec:tmds} and~\ref{subsub:Sudakovresum}).

— Transverse momentum dependent parton distributions at small $x$  (2610.26004 - Caucal et al., 19 Oct 2026) in Section 7, paragraph beginning “While these results place”