  For a system with the ion beam co-propagating with the electron beam, such as
a traditional electron cooler or a Coherent electron Cooler (CeC), the
recombination rate is an important observable for matching the energy of the
electrons with the ions. In this work, we have developed the analytical
expressions to investigate how the recombination rate depends on the energy
difference of the two beams, with the influences from the transverse motions of
the particles being considered. The analytical results are then used to analyze
the measured recombination data collected during the CeC experiment in run 21.
