= Renormalization of a three-dimensional dilute-gas contact coupling
{title2=$\frac1{g_0}=\frac m{4\pi\hbar^2a_s}-\int^\Lambda\frac{d^3k}{(2\pi)^3}\frac1{2\varepsilon_k}$}
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 $\tfrac12\int d^3k/(2\pi)^3[E_k-\varepsilon_k-\mu+\mu^2/(2\varepsilon_k)]$ 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.
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