Quantum Corrections to the 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 $\sqrt{\rho a^3}$, where $\rho$ is the number density and $a$ is the S-wave 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, we argue that the second order quantum correction is dominated by the logarithmic term, where the argument of the logarithm is $\rho a \ell_V^2$, and $\ell_V$ is the length scale set by the van der Waals potential.