Quantum Corrections to the Ground State Energy Density of a Homogeneous Bose Gas

Quantum corrections to the properties of a homogeneous interacting Bose gas at zero temperature can be calculated as a low-density expansion in powers of scattering length. We calculate the ground state energy density to second order in $\sqrt{\rho a^3}$. The coefficient of the $\rho a^3$ correction has a logarithmic term that was calculated in 1959. We present the first calculation of the constant under the logarithm. The constant depends not only on $a$, but also on an extra parameter that describes the low energy $3\to 3$ scattering of the bosons. In the case of alkali atoms whose scattering length $a$ is much larger than their size, we argue that the dependence on the extra parameter can be eliminated in favor of a logarithmic dependence on the size of the atom.