Neutrino force at all length scales

TL;DR

This paper derives a comprehensive expression for the neutrino force valid at all distances, revealing its behavior at short and long ranges, and explores its significant effects on atomic parity violation experiments and potential for new physics insights.

Contribution

It provides the first complete formula for the neutrino force across all length scales, extending beyond the traditional long-range approximation.

Findings

Neutrino force scales as 1/r at short distances and as G_F^2/r^5 at long distances.

Neutrino force contributes 4% in muonium and 16% in positronium relative to Z-exchange.

Implications for detecting neutrino force and measuring the weak mixing angle in APV experiments.

Abstract

The Standard Model predicts a long-range force mediated by a pair of neutrinos, known as ``the neutrino force". It scales as $G_F^2/r^5$, where $G_F$ is the Fermi constant. However, as $r \lesssim \sqrt{G_F}$, the four-Fermi theory breaks down and the neutrino force no longer has the $1/r^5$ scaling. For the first time, we derive a complete expression for the neutrino force that is valid at all distances. For $r \gg \sqrt{G_F}$, the result reduces to the known $G_F^2/r^5$; for $r \ll \sqrt{G_F}$, it scales as $1/r$. We explore the implications of this result for atomic parity violation (APV) experiments. A key feature of the neutrino force is that it is a long-range effect compared to the atomic length scale. Thus, in general, it cannot be simply treated as a correction to the tree-level $Z$-exchange diagram without considering the atomic wavefunctions. We calculate the effects in muonium and positronium, finding that the neutrino force contributes about 4\% and 16\%, respectively, compared to the leading $ Z$ exchange. This indicates a significant impact on APV, with important implications for detecting the neutrino force and measuring the weak mixing angle in APV experiments.