Eta Decay and Muonic Puzzles

New physics motivated by muonic puzzles (proton radius and muon $g-2$ discrepancies) is studied. Using a light scalar boson $\phi$, assuming Yukawa interactions, accounts for these muonic puzzles simultaneously. Our previous work limits the existence of such a scalar boson's mass $m_\phi$ from about 160 keV to 60 MeV. We improve this result by including the influence of all of the possible particles that couple to the $\phi$ in computing the decay rate. Doing this involves including the strong interaction physics, involving quarks, necessary to compute the $\eta\pi\phi$ vertex function. The Nambu-Jona-Lasinio model, which accounts for the spontaneous symmetry breaking that yields the constituent mass is employed to represent the relevant strong-interaction physics. We use the $\eta\pi\phi$ vertex function to reanalyze the electron beam dump experiments. The result is that the allowed range of $m_\phi$ lies between about 160 keV and 3.5 MeV. This narrow range represents an inviting target for ruling out or discovering this scalar boson. A possible UV completion of our phenomenological model is discussed.