Neutrino force at all length scales

TL;DR

This work derives the complete, gauge-invariant SM neutrino force valid at all length scales by incorporating self-energy, penguin, and box diagrams within the full electroweak theory and employing dispersion methods to obtain $V_0(r)$. The force transitions from the familiar long-range $\sim G_F^2/r^5$ behavior to a short-range $\sim 1/r$ form with $\ln^2(m_W r)$ enhancements, driven by electroweak dynamics rather than the four-Fermi approximation. The authors then embed this force into atomic parity violation (APV) calculations, showing that the neutrino force can contribute a few percent to APV in muonium/positronium and up to $\sim 0.3\%$ in heavy atoms, with potential implications for extracting $\sin^2\theta_W$ from APV data. They also assess parity-violating contributions from other SM fermions and compare their results to existing literature, arguing that neutrino-forced PV effects are non-negligible at one loop and should be included for precision SM tests. Overall, the work establishes a framework for evaluating long-range quantum forces in the SM and highlights measurable APV consequences that could probe the neutrino sector and weak mixing angle measurements.

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.

Figures (11)

• Figure 1: The two-neutrino mediated force in the framework of four-Fermi effective theory, which scales as $G_F^2/r^5$. The four-Fermi effective vertex is only valid at long distances when $r \gg \sqrt{G_F}$.
• Figure 2: The three possible ways to open up the four-Fermi effective vertex at short distances ($r \lesssim \sqrt{G_F}$) for the Standard Model neutrino force: (a) self-energy diagram, (b)+(c) penguin diagram, and (d) box diagram.
• Figure 3: $V_0(r)/m_Z$ as a function of the radial distance for the different diagrams that contribute to the neutrino force. Note that the self-energy and penguin potentials change sign while the box potential does not. For a comparison of all three potentials in the same diagram, see Fig. \ref{['fig:V0_all']}.
• Figure 4: Magnitudes of the neutrino force from self-energy (SE), penguin (PG), and box diagrams plotted as a function of the radial distance. Labels 'A' and 'R' represent the sign of the potential that renders it either attractive or repulsive. Note that the plot is made on a log-log scale and so the cusps in the potential occur when the potential changes sign. In reality there are no cusps in the potential.
• Figure 5: Accumulated charges $Q(c)$ for APV matrix elements as a function of the length scale $c\equiv m_Z r$. Upper left: $s$-wave $Z$ force and neutrino forces (SE, PG, and Box). Upper right and lower left: $p$-wave neutrino forces at short and long distances. Lower right: Comparison of $p$-wave and $d$-wave neutrino forces across atomic length scales, where the dashed line corresponds to the Bohr radius.