Dijet production in DIS off a large nucleus at next-to-eikonal accuracy in a Gaussian model within the CGC framework

We develop a Gaussian model to evaluate the decorated dipole and quadrupole operators that arise beyond the eikonal approximation in the Color Glass Condensate framework. While the method is general and applicable to arbitrary beyond-eikonal Wilson line structures, we employ it for dijet production in deep inelastic scattering at next-to-eikonal accuracy. After validating the model at the eikonal level, we compute all next-to-eikonal operator structures entering the dijet cross section. We show that some of them do not contribute to this observable, while others vanish identically. Therefore, in the Gaussian model next-to-eikonal corrections to dijet production in deep inelastic scattering originate solely from a given type of operators and from next-to-eikonal three-point correlators. The resulting expressions are provided in a form suitable for numerical implementation.

💡 Research Summary

The paper presents a systematic approach to incorporate next‑to‑eikonal (NEik) corrections into the Color Glass Condensate (CGC) description of deep‑inelastic scattering (DIS) dijet production off a large nucleus. While the eikonal approximation—retaining only the leading terms in the high‑energy limit—has been the work‑horse of CGC phenomenology, it neglects sub‑leading effects of order 1/γ (the Lorentz boost factor of the target). Recent theoretical developments have identified a set of “decorated” Wilson‑line operators, i.e. standard Wilson lines with insertions of transverse gauge fields, covariant derivatives, or field‑strength tensors, that encode these NEik effects. However, existing models such as the McLerran‑Venugopalan (MV) or Golec‑Biernat‑Wüsthoff (GBW) parameterisations only provide the standard dipole ⟨Tr U(x)U†(y)⟩ and quadrupole ⟨Tr U(x)U†(y)U(z)U†(w)⟩ correlators; they cannot be directly applied to the decorated operators needed at NEik order.

To fill this gap, the authors construct a Gaussian model for the background color charge density ρ of the target. The model assumes a Gaussian probability distribution for ρ with a two‑point correlator ⟨ρ^a(x) ρ^b(y)⟩ = δ^{ab} μ^2(x⊥) δ^{(2)}(x⊥ − y⊥), while higher‑order correlators vanish. This is essentially a reformulation of the MV model that is analytically tractable for any Wilson‑line functional, including those with insertions. The longitudinal extent of the target is taken to infinity (L⁺→∞), which is justified because the background field is a pure gauge outside the support region.

The paper proceeds in several stages:

1. Eikonal baseline – The authors first recapitulate the standard DIS dijet cross‑section at eikonal order, expressed in terms of the dipole d(v,w) and quadrupole Q(w′,v′,v,w) operators. They then validate the Gaussian model by reproducing the known MV/GBW results for these correlators, showing that the model correctly captures saturation physics through the saturation scale Q_s encoded in μ^2.

2. Definition of decorated operators – Three families of NEik Wilson lines are introduced:

   • Type‑1: a single transverse gauge field insertion A_j.
   • Type‑2: two covariant derivatives acting on the Wilson line.
   • Type‑3: a field‑strength tensor insertion F_{ij}. Corresponding decorated dipoles d^{(1)}j, d^{(2)}, d^{(3)}{ij} and quadrupoles Q^{(1)}j, Q^{(2)}, Q^{(3)}{ij} are defined, with a star indicating the longitudinal position of the insertion.

3. Gaussian averaging of decorated operators – By expanding the Wilson lines in powers of the background field and using Wick’s theorem for the Gaussian distribution, the authors reduce all higher‑order correlators to products of the two‑point function. Importantly, three‑point correlators vanish, which dramatically simplifies the algebra. The resulting expressions for each decorated operator are written as the eikonal dipole/quadrupole multiplied by explicit transverse derivatives of the two‑point correlator G(x⊥−y⊥) and by factors of μ^2. Color algebra is handled in Appendix A, confirming that many potential contributions cancel.

4. Selection of physically relevant terms – Substituting the decorated operators into the NEik dijet cross‑section (originally derived in Refs.