Constraints to new physics models for the proton charge radius puzzle from the decay $K^+ \rightarrow \mu^+ +\nu + e^- + e^+$

A possible explanation for the discrepancy between electronic and muonic hydrogen measurements of the proton charge radius are new, lepton-universality violating interactions. Several new couplings and particles have been suggested that account for this discrepancy. At present, these explanations are poorly constrained. Experiments such as the upcoming kaon decay experiment at JPARC may constrain or eliminate some explanations by sensitivity to the decay channel $K^+ \rightarrow \mu^+ +\nu + e^- + e^+$. We calculate the predicted contributions of the various explanations to this channel. The predicted signals, if present, should be large enough to be resolved in the experiment.