  We calculate the electron-ion temperature equilibration rate in a fully
ionized, weakly to moderately coupled plasma, using an exact treatment of the
Fermi-Dirac electrons. The temperature is sufficiently high so that the
quantum-mechanical Born approximation to the scattering is valid. At the heart
of this calculation lies the method of dimensional continuation, a technique
that we borrow from quantum field theory and use in a novel fashion to regulate
the kinetic equations in a consistent manner. We can then perform a systematic
perturbation expansion and thereby obtain a finite first-principles result to
leading and next-to-leading order. Unlike model building, this systematic
calculation yields an estimate of its own error and thus prescribes its domain
of applicability. The calculational error is small for a weakly to moderately
coupled plasma, for which our result is nearly exact. It should also be
emphasized that our calculation becomes unreliable for a strongly coupled
plasma, where the perturbative expansion that we employ breaks down, and one
must then utilize model building and computer simulations. Besides providing
new and potentially useful results, we use this calculation as an opportunity
to explain the method of dimensional continuation in a pedagogical fashion.
Interestingly, in the regime of relevance for many inertial confinement fusion
experiments, the degeneracy corrections are comparable in size to the
subleading quantum correction below the Born approximation. For consistency, we
therefore present this subleading quantum-to-classical transition correction in
addition to the degeneracy correction.
