Interplay of the CGC and TMD frameworks to all orders in kinematic twist

A framework for an improved TMD (iTMD) factorization scheme at small x, involving off-shell perturbative subamplitudes, was recently developed as an interpolation between the TMD k$_{t}$ ≪ Q regime and the BFKL k$_{t}$ ∼ Q regime. In this article, we study the relation between CGC and iTMD amplitudes. We first show how the dipole-size expansion of CGC amplitudes resembles the twist expansion of a TMD amplitude. Then, by isolating kinematic twists, we prove that iTMD amplitudes are obtained with infinite kinematic twist accuracy by simply getting rid of all genuine twist contributions in a CGC amplitude. Finally we compare the amplitudes obtained via a proper kinematic twist expansion to those obtained via a more standard dilute expansion to show the relation between the iTMD framework and the dilute low x framework. This allows to reinterpret the dilute expansion as a Wandzura-Wilczek approximation in the large k$_{t}$ regime.