Realistic model for a fifth force explaining anomaly in {⁸Be^*} to {⁸Be} \;{e^+e^-} Decay

A $6.8\,\sigma$ anomaly has been reported in the opening angle and invariant mass distributions of $e^+e^-$ pairs produced in ${^8Be}$ nuclear transitions. It has been shown that a protophobic fifth force mediated by a $17\,\textrm{MeV}$ gauge boson $X$ with pure vector current interactions can explain the data through the decay of an excited state to the ground state, ${^8Be^*} \to {^8Be}\, X$, and then the followed saturating decay $X \to e^+e^-$. In this work we propose a renormalizable model to realize this fifth force. Although axial-vector current interactions also exist in our model, their contributions cancel out in the iso-scalar interaction for ${^8Be^*} \to {^8Be} \,X$. Within the allowed parameter space, this model can alleviate the $(g-2)_\mu$ anomaly problem and can be probed by the LHCb experiment. Several other implications are discussed.