Implication of the hidden sub-GeV bosons for the $(g-2)_\mu$, $^8$Be-$^4$He anomaly, proton charge radius, EDM of fermions and dark axion portal

We discuss new physics phenomenology of hidden scalar ($S$), pseudoscalar ($P$), vector ($V$) and axial-vector ($A$) particles coupled to nucleons and leptons, which could give contributions to proton charge radius, $(g-2)_\mu$, $^8$Be-$^4$He anomaly and electric dipole moment (EDM) of Standard Model (SM) particles. In particular, we estimate sensitivity of NA64$\mu$ experiment to observe muon missing energy events involving hidden scalar and vector particles. That analysis is based on {\tt GEANT4} Monte Carlo simulation of the signal process of muon scattering off target nuclei $\mu N \to \mu N S(V)$ followed by invisible boson decay into Dark Matter (DM) particles, $S(V)\to \chi \chi$. The existence of light sub-GeV bosons could possibly explain the muon $(g-2)$ anomaly observed. We also summarize existing bounds on ATOMKI $X17(J^P=0^-, 1^\pm)$ boson coupling with neutron, proton and electron. We implement these constraints to estimate the contribution of $P$, $V$ and $A$ particles to proton charge radius via direct 1-loop calculation of Sachs form factors. The analysis reveals the corresponding contribution is negligible. We also calculate bounds on dark axion portal couplings of dimension-five operators, which contribute to the EDMs of leptons and neutron.