  We prove the following formula for the ground state energy density of a
dilute Bose gas with density $\rho$ in $2$ dimensions in the thermodynamic
limit \begin{align*} e^{\rm{2D}}(\rho) = 4\pi \rho^2 Y\left(1 - Y \vert \log Y
\vert + \left( 2\Gamma + \frac{1}{2} + \log(\pi) \right) Y \right) + o(\rho^2
Y^{2}). \end{align*} Here $Y= |\log(\rho a^2)|^{-1}$ and $a$ is the scattering
length of the two-body potential. This result in $2$ dimensions corresponds to
the famous Lee-Huang-Yang formula in $3$ dimensions. The proof is valid for
essentially all positive potentials with finite scattering length, in
particular it covers the crucial case of the hard core potential.
