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Higher order terms in the condensate fraction of a homogeneous and dilute Bose gas

The condensate fraction of a homogeneous and dilute Bose gas is expanded as a power series of $\sqrt{n a^3}$ as $N_0/N = 1 -c_1 (n a^3)^{1/2} -c_2 (n a^3) - c_3 (n a^3)^{3/2}\hdots.$ The coefficient $c_1$ is well-known as $8/3 \sqrtπ$, but the others are unknown yet. Considering two-body contact interactions and applying a canonical transformation method twice we developed the method to obtain the higher order coefficients analytically. An iteration method is applied to make up a cutoff in a fluctuation term. The coefficients ares $c_2=2(π- 8/3)$ and $c_3=(4/\sqrtπ) (π-8/3)(10/3-π)$.