On the short-range behavior of neutrino forces beyond the Standard
  Model: from $1/r^5$ to $1/r^4$, $1/r^2$, and $1/r$
  The exchange of a pair of neutrinos between two objects, seperated by a
distance $r$, leads to a long-range effective potential proportional to
$1/r_{}^5$, assuming massless neutrinos and four-fermion contact interactions.
In this paper, we investigate how this known form of neutrino-mediated
potentials might be altered if the distance $r$ is sufficiently short,
corresponding to a sufficiently large momentum transfer which could invalidate
the contact interactions. We consider two possible scenarios to open up the
contact interactions by introducing a $t$-channel or an $s$-channel mediator.
We derive a general formula that is valid to describe the potential in all
regimes as long as the external particles remain non-relativistic. In both
scenarios, the potential decreases as $1/r_{}^5$ in the long-range limit as
expected. In the short-range limit, the $t$-channel potential exhibits the
Coulomb-like behavior (i.e. proportional to $1/r$), while the $s$-channel
potential exhibits $1/r_{}^4$ and $1/r_{}^2$ behaviors.
