  We study QCD with two colors and quarks in the fundamental representation at
finite baryon density in the limit of light quark masses. In this limit the
free energy of this theory reduces to the free energy of a chiral Lagrangian
which is based on the symmetries of the microscopic theory. In earlier work
this Lagrangian was analyzed at the mean field level and a phase transition to
a phase of condensed diquarks was found at a chemical potential of half the
diquark mass (which is equal to the pion mass). In this article we analyze this
theory at next-to-leading order in chiral perturbation theory. We show that the
theory is renormalizable and calculate the next-to-leading order free energy in
both phases of the theory. By deriving a Landau-Ginzburg theory for the order
parameter we show that the finite one-loop contribution and the next-to-leading
order terms in the chiral Lagrangian do not qualitatively change the phase
transition. In particular, the critical chemical potential is equal to half the
next-to-leading order pion mass, and the phase transition is second order.
