Neutrino force at all length scales

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.