Solve the closed system of sub-eikonal high-energy evolution equations for Q1, Q5, and related operators

Solve the closed system of rapidity-evolution equations that couple the quark dipole-type operators Q1(x_perp, x_B=0) and Q5(x_perp, x_B=0) with the auxiliary operators \tilde{Q}_1, \tilde{Q}_5, Q_1(x_perp,y_perp), and Q_5(x_perp,y_perp), including the flavor-singlet mixing with the gluon dipole operator, to determine their small-x energy evolution beyond the preliminary analyses presented here.

Background

The paper derives evolution equations for the sub-eikonal quark operators Q1 and Q5 at x_B=0 and rewrites them in a dipole-type operator basis that exposes the leading small-dipole behavior. These equations are not closed: they involve additional operators (\tilde{Q}_1, \tilde{Q}_5, Q_1(x_perp,y_perp), Q_5(x_perp,y_perp)) and, in the flavor-singlet channel, mixing with a gluon dipole operator.

While the needed additional evolution equations for some of these operators are known from prior work, the author explicitly does not tackle the full solution of the resulting coupled system here, identifying it as a task for future work. Solving this closed system would provide the complete high-energy evolution of the sub-eikonal quark operators relevant to the small-x behavior of DIS observables.

References

In this work, however, we will not consider the full problem of solving the closed system of evolution equations, and we leave it for future work.

— From Sub-eikonal DIS to Quark Distributions and their High-Energy Evolution  (2603.30000 - Chirilli, 31 Mar 2026) in Section 4 (Evolution equation), after Eqs. (evolutionQ1b)–(evolutionQ5NSb)

A helicity-dependent generalization of the JIMWLK equation is, however, not yet known, and even at large $N_c$ the non-linear terms enter at single-logarithmic accuracy, where linear contributions of the same order remain to be computed.

— Transverse momentum dependent parton distributions at small $x$  (2610.26004 - Caucal et al., 19 Oct 2026) in Section 4.1 and Section 4.2

Several questions remain open. In particular, it is necessary to understand how to model, within the CGC framework, the correlator of such quark background fields, as well as how the corresponding operator evolves at high energy.

— Transverse momentum dependent parton distributions at small $x$  (2610.26004 - Caucal et al., 19 Oct 2026) in Section 4.3, paragraph beginning “Several questions remain open”