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We show that the shift in the transition temperature ofthe dilute homogeneous Bose gas is non-analytic in the scattering amplitude,$a$. The first correction beyond the positive linear shift in $a$ is negativeand of order $a^2\\ln a$. This non-universal non-analytic structure indicateshow the discrepancies between numerical calculations at finite $a$ can bereconciled with calculations of the limit $a \\to0$, since the linearity isapparent only for anomalously small $a$

Nonanalytic Dependence of the Transition Temperature of the Homogeneous Dilute Bose Gas on Scattering Length