2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/167639Divergence exponents of the first-order quantum correction of a two-dimensional hard-sphere Bose atoms are obtained by an effective field theory method. The first-order correction to the ground-state energy density with respect to the zeroth-order is given by $\cale_1/\cale_0 \sim |D-2|^{-\al}|\lnγ|^{-\al'}$, where $D$ is the spatial dimension, and $γ$ is the gas parameter ($γ=n a^D$). As $D \to 2$, $\al =\al'=1$. We show that the first-order quantum correction of the energy density is not perturbative in low dimensions of $D < 2.2$ regardless of any gas parameter which is much less that 1.9 pages, 3 figures,Statistical Mechanicstext