Constraints on muon-specific dark forces

The recent measurement of the Lamb shift in muonic hydrogen allows for the most precise extraction of the charge radius of the proton which is currently in conflict with other determinations based on $e-p$ scattering and hydrogen spectroscopy. This discrepancy could be the result of some new muon-specific force with O(1-100) MeV force carrier---in this paper we concentrate on vector mediators. Such an explanation faces challenges from the constraints imposed by the $g-2$ of the muon and electron as well as precision spectroscopy of muonic atoms. In this work we complement the family of constraints by calculating the contribution of hypothetical forces to the muonium hyperfine structure. We also compute the two-loop contribution to the electron parity violating amplitude due to a muon loop, which is sensitive to the muon axial-vector coupling. Overall, we find that the combination of low-energy constraints favors the mass of the mediator to be below 10 MeV, and that a certain degree of tuning is required between vector and axial-vector couplings of new vector particles to muons in order to satisfy constraints from muon $g-2$. However, we also observe that in the absence of a consistent standard model embedding, high energy weak-charged processes accompanied by the emission of new vector particles are strongly enhanced by $(E/m_V)^2$, with $E$ a characteristic energy scale and $m_V$ the mass of the mediator. In particular, leptonic $W$ decays impose the strongest constraints on such models completely disfavoring the remainder of the parameter space.