Diffractive deep inelastic scattering in the dipole picture: the $q\bar{q}g$ contribution in exact kinematics

We compute the $q\bar{q}g$ contribution to the diffractive structure functions in high-energy deep inelastic scattering. The obtained result corresponds to a finite part of the next-to-leading-order contribution to the diffractive cross section. Previous phenomenological applications have included this contribution only in the high-$Q^2$ or high-$M_X^2$ limits in the case of a soft gluon, and we numerically demonstrate that these existing estimates do not provide a good approximation for the full $q\bar{q}g$ contribution. Furthermore, we demonstrate that in addition to the soft gluon contribution, there is an equally important soft quark contribution to the diffractive structure functions at high $Q^2$.