Viscosity sum rules at large scattering lengths

We use the operator product expansion (OPE) and dispersion relations to obtain model-independent ``Borel-resummed'' sum rules for both shear and bulk viscosity of many-body systems of spin-1/2 fermions with predominantly short-range S-wave interactions. These sum rules relate Gaussian weights of the frequency-dependent viscosities to the Tan contact parameter C(a). Our results are valid for arbitrary values of the scattering length a, but receive small corrections from operators of dimension Δ>5 in the OPE and can be used to study transport properties in the vicinity of the a→∞ fixed point. In particular, we find that the exact dependence of the shear viscosity sum rule on scattering length is controlled by the function C(a). The sum rules that we obtain depend on a frequency scale ω₀ that can be optimized to maximize their overlap with low-energy data.