  We obtain the equation of state of interacting alpha matter and the critical
temperature of Bose-Einstein condensation of alpha particles within an
effective scalar field theory. We start from a non-relativistic model of
uniform alpha matter interacting with attractive two-body and repulsive
three-body potentials and reformulate this model as a O(2) symmetric scalar
\phi^6 field theory with negative quartic and positive sextic interactions.
Upon restricting the Matsubara sums, near the temperature of Bose-Einstein
condensation, to the zeroth order modes we further obtain an effective
classical theory in three spatial dimensions. The phase diagram of the alpha
matter is obtained from simulations of this effective field theory on a lattice
using local Monte-Carlo algorithms.
