Initial momentum anisotropies in the kT-factorization of the CGC I: Gradient Expansion

Abstract

We revisit single-inclusive gluon production in the dilute--dilute limit of the Color Glass Condensate and show that $k_\perp$-factorization is only the zeroth order of a systematic gradient expansion in the transverse positions of the two colliding sources, organized in powers of $1/(Q_sR)$, with $Q_s$ the saturation scale and $R$ the size over which the sources vary. Collecting terms order by order builds a ladder of scalar and tensor structures from the local distributions and their transverse derivatives, yielding an extended $k_\perp$-factorized formula that retains the spatial dependence of the sources and, through kinetic moments of the spectrum, the energy-momentum tensor. Repeating the expansion from a real-time computation of the classical Glasma fields in the future light cone reproduces the momentum-space result in the eikonal limit, but additionally retains coherent interference between amplitude and conjugate. This adds terms to the ladder (flow responses, chromo-electric--magnetic interference, and a longitudinal energy flux) further suppressed by powers of the large outgoing momentum $|\pp|\gg Q_s$. The hierarchy thus reveals momentum-space anisotropies present in the initial state before any hydrodynamic evolution, which leading $k_\perp$-factorization misses by construction. Since each order is a local operator on the same unintegrated distributions, these corrections can be added directly to existing saturation-based initial-state models, giving a dynamically generated initial momentum anisotropy and a more faithful early-time energy-momentum tensor without abandoning the tractable factorized form. Set by gradients of the transverse density profiles, they grow towards more dilute, fluctuation-dominated systems, e.g. light-ion collisions. Phenomenological consequences will be developed in a companion paper.