Renormalization of a three-dimensional dilute-gas contact coupling ($\frac{1}{g_0}=\frac{m}{4\pi {\hslash }^2{a}_s}-{\int }^{\Lambda }\frac{d^{3}k}{(2\pi {)}^3}\frac{1}{2{\epsilon }_k}$)

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.