large mediator masses cannot be applied beyond their domains of applicability.

An effective theory description is appropriate when $M \gtrsim \sqrt{s}$ and $g \lesssim 1$. Then these parameters can be characterized in a combination analogous to the Fermi constant for low-energy weak interactions, i.e., a dimensionful coupling $$G \equiv \frac{1}{\sqrt{4\pi}} \frac{g^{2}}{M^{2}} \,.$$ Constraints on $\nu$SI are sometimes quoted using just $G$. This does not provide as much information as a region in the $g$-$M$ plane, because of the degeneracy in $g/M$ and the unspecified limits of applicability. In Fig. \[fig:constraint\], we plot some diagonal contours of constant $G$. The fact that the line given by $G\sim G_F \sim 10^{-5} \; {\rm GeV}^{-2}$ is not the same as the single point ($M \sim 100$ GeV, $g \sim 1$) that defines the weak interactions illustrates our caution about characterizing $\nu$SI with only $G$.

One general framework for directly testing $\nu$SI is to use neutrinos from sources that travel a long distance through the C$\nu$B. The only possibly relevant standard model process is the Z-burst scenario [@Weiler:1982qy; @Weiler:1997sh], where a high energy neutrino interacts with the C$\nu$B through a Z-boson resonance. However, the required neutrino energy is extremely high, $\sim 10^{14}\,{\rm GeV}$, and 