Testing $WW\gamma$ vertex in radiative muon decay

Large numbers of muons will be produced at facilities developed to probe lepton flavor violating process $\mu\to e\gamma$. We show that by constructing a suitable asymmetry, radiative muon decay $\mu\to e \gamma\nu_\mu\bar{\nu}_e$ can also be used to test the $WW\gamma$ vertex at such facilities. The process has two missing neutrinos in the final state and on integrating their momenta, the partial differential decay rate shows no radiation-amplitude-zero. We establish, however, that an easily separable part of the normalized differential decay rate, odd under the exchange of photon and electron energies, does have a zero in the case of standard model (SM). This "new type of zero" has hitherto not been studied in literature. A suitably constructed asymmetry using this fact, enables a sensitive probe for the $WW\gamma$ vertex beyond the SM. With a simplistic analysis, we find that the $C$ and $P$ conserving dimension four $WW\gamma$ vertex can be probed at ${\cal O}(10^{-2})$ with satisfactory significance level.