  We implement a recursive Green function method to extract the Fock space (FS)
propagator and associated self-energy across the many-body localization (MBL)
transition, for one-dimensional interacting fermions in a random onsite
potential. We show that the typical value of the imaginary part of the local FS
self-energy, \Delta_t, related to the decay rate of an initially localized
state, acts as a probabilistic order parameter for the thermal to MBL phase
transition; and can be used to characterize critical properties of the
transition as well as the multifractal nature of MBL states as a function of
disorder strength W. In particular, we show that a fractal dimension D_s
extracted from \Delta_t jumps discontinuously across the transition, from D_s<1
in the MBL phase to D_s= 1 in the thermal phase. Moreover, \Delta_t follows an
asymmetrical finite-size scaling form across the thermal-MBL transition, where
a non-ergodic volume in the thermal phase diverges with a Kosterlitz-Thouless
like essential singularity at the critical point W_c, and controls the
continuous vanishing of \Delta_t as W_c is approached. In contrast, a
correlation length ({\xi}) extracted from \Delta_t exhibits a power-law
divergence on approaching W_c from the MBL phase.
