Soft Gluon Wave Function and Evolution Operator in the CGC at Next-to-Leading Order

We construct the soft gluon light cone wave function of a fast moving hadron and the associated unitary evolution operator $\Omega$ up to $\mathcal{O}(g^{2})$ in pure Yang Mills theory $(N_{f} = 0)$, within the Color Glass Condensate (CGC) framework. Working in light cone gauge, we perform a Born Oppenheimer separation between fast valence and soft modes and implement the eikonal approximation, which allows $\Omega$ to be written as a fully normal ordered series in soft gluon creation and annihilation operators. The expansion coefficients are functionals of the non-commuting valence color charge density operators $\rho$. We fix the coefficients by using the unitarity and explicit diagrammatic calculations within light-cone perturbation theory (LCPT). As an application, we diagonalize the pure Yang Mills soft Hamiltonian through orders $g$ and $g^{2}$ and show that the off diagonal mixing elements between Fock sectors cancel leading to the coherent background field energy proportional to $\rho^{2}$.