  The very-low temperature thermal effective mass m* of paramagnetic and
ferromagnetic electrons in a uniform electron fluid in two dimensions is
studied. Analytical and numerical evaluations are used to meaningfully define
an m*, even in the the Hartree-Fock approximation. The Hartree-Fock m*
decreases linearly with the electron-disk radius r_s. Correlation effects lead
to strong cancellations between exchange and correlation. Thus the effective
mass is enhanced with increasing r_s for the unpolarized fluid, while m*
decreases with the r_s of the polarized fluid. The effective mass is calculated
from the coefficient of the quadratic temperature dependence of
exchange-correlation free energy F_{xc}. This is calculated in a physically
transparent manner using a new formula for the effective mass. This uses the
T=0 pair-distribution functions of Gori-Giorgi et al., and the temperature
derivative of a quantum analogue of the potential of mean-force well known in
the statistical mechanics of classical fluids. The results are compared with
recent quantum Monte-Carlo simulations at T=0, as well as with other available
experimental and theoretical data for the effective mass.
