Transition temperature of a dilute homogeneous imperfect Bose gas

The leading-order effect of interactions on a homogeneous Bose gas is theoretically predicted to shift the critical temperature by an amount \Delta\Tc = # a_{scatt} n^{1/3} T_0 from the ideal gas result T_0, where a_{scatt} is the scattering length and n is the density. There have been several different theoretical estimates for the numerical coefficient #. We claim to settle the issue by measuring the numerical coefficient in a lattice simulation of O(2) phi^4 field theory in three dimensions---an effective theory which, as observed previously in the literature, can be systematically matched to the dilute Bose gas problem to reproduce non-universal quantities such as the critical temperature. We find # = 1.32 +- 0.02.