Contact-interaction Bose-gas field integral ($S_E=\int [\bar{\psi }(\hslash {\partial }_{\tau }-{\hslash }^2{\nabla }^2/(2m)-\mu )\psi +g|\psi {|}^4/2]$)

For a uniform dilute repulsive gas, the periodic complex-field integral has uniform saddle $n_0=\mu /g$ for $\mu >0$. The paired Gaussian fluctuation kernel has determinant $(\hslash {\omega }_n{)}^2+{\epsilon }_k({\epsilon }_k+2g{n}_0)$, giving the Bogoliubov spectrum. In three dimensions the physical low-energy coupling is $g=4\pi {\hslash }^2{a}_s/m$ and the weak-depletion parameter is $n{a}_s^3\ll 1$. Contact coupling beyond mean field needs ultraviolet matching to the scattering length.

The regulated bare coupling $g_0$ is matched to the physical two-body scattering length. In the uniform one-loop grand potential this matching supplies the subtraction that makes $\frac{1}{2}\int d^{3}k/(2\pi {)}^3[E_k-{\epsilon }_k-\mu +{\mu }^2/(2{\epsilon }_k)]$ ultraviolet finite. Merely using the physical coupling in an unregulated zero-point integral would retain a divergent answer. The contact description and its cutoff cannot be extended to arbitrary momenta independently of that matching.