Upper Bound for the Ground State Energy of a Dilute Bose Gas of Hard Spheres

We consider a gas of bosons interacting through a hard-sphere potential with radius $a$ in the thermodynamic limit. We derive an upper bound for the ground state energy per particle at low density. Our bound captures the leading term $4\pi \rho a$ and shows that corrections are smaller than $C\rho a{\left(\rho a^3\right)}^{1/2}$ , for a sufficiently large constant $C>0$ . In combination with a known lower bound, our result implies that the first sub-leading term to the ground state energy of a dilute gas of hard spheres is, in fact, of the order $\rho a{\left(\rho a^3\right)}^{1/2}$ , in agreement with the Lee-Huang-Yang prediction.