On the short-range behavior of neutrino forces beyond the Standard Model: from $1/r^5$ to $1/r^4$, $1/r^2$, and $1/r$

TL;DR

This paper explores how neutrino-mediated forces change at short distances, showing a transition from a $1/r^5$ behavior to Coulomb-like and other power-law potentials due to mediator effects beyond the Standard Model.

Contribution

It derives a unified formula for neutrino potentials across all regimes considering mediator scenarios, extending beyond the standard long-range $1/r^5$ form.

Findings

Potential transitions from $1/r^5$ to $1/r$, $1/r^4$, and $1/r^2$ at short distances.

Short-range behaviors depend on the type of mediator (t-channel or s-channel).

Derived a general formula valid for all regimes with non-relativistic particles.

Abstract

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.