  We perform a complete next-to-leading order calculation of the non-Abelian
electric field correlator in a SU($N_c$) plasma, which encodes properties of
the plasma relevant for heavy particle bound state formation and dissociation,
and is different from the correlator for the heavy quark diffusion coefficient.
The calculation is carried out in the real-time formalism of thermal field
theory and includes both vacuum and finite temperature contributions. By
working in the $R_\xi$ gauge, we explicitly show the results are gauge
independent, infrared and collinear safe. The renormalization group equation of
this electric field correlator is determined by that of the strong coupling
constant. Our next-to-leading order calculation can be directly applied to any
dipole singlet-adjoint transition of heavy particle pairs. For example, it can
be used to describe dissociation and (re)generation of heavy quarkonia inside
the quark-gluon plasma well below the melting temperature, as well as heavy
dark matter pairs (or charged co-annihilating partners) in the early universe.
