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Viscosity spectral functions of resonating fermions in the quantum virial expansion

We consider two-component fermions with a zero-range interaction both in two and three dimensions and study their spectral functions of bulk and shear viscosities for an arbitrary scattering length. Here the Kubo formulas are systematically evaluated up to the second order in the quantum virial expansion applicable to the high-temperature regime. In particular, our computation of the bulk viscosity spectral function is facilitated by expressing it with the contact-contact response function, which can be measured experimentally under the periodic modulation of the scattering length. The obtained formulas are fully consistent with the known constraints on high-frequency tail and sum rule. Although our static shear viscosity agrees with that derived from the kinetic theory, our static bulk viscosity disagrees. Furthermore, the latter for three dimensions exhibits an unexpected non-analyticity of $ζ\sim(\ln a^2)/a^2$ in the unitarity limit $a\to\infty$, which thus challenges the "crossover" hypothesis.