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 G²F=r₅, where GF is the Fermi constant. However, as r ≲ √GF, the four-Fermi theory breaks down and the neutrino force no longer has the 1=r5 scaling. For the first time, we derive a complete expression for the neutrino force that is valid at all distances. For r ≫ √GF, the result reduces to the known G²F=r₅; for r √GF p, 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 wave functions. 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.