Testing $WW\gamma$ vertex in radiative muon decay

Large numbers of muons will be produced at facilities developed to probe the lepton-flavor-violating process $\mu \to e\gamma$. We show that by constructing a suitable asymmetry, radiative muon decay $\mu \to e\gamma {\nu }_{\mu }{\overline{\nu }}_e$ can also be used to test the $WW\gamma$ vertex at such facilities. The process has two missing neutrinos in the final state, and upon integrating their momenta the partial differential decay rate shows no radiation-amplitude zero. However, we establish that an easily separable part of the normalized differential decay rate that is odd under the exchange of photon and electron energies does have a zero in the case of the standard model (SM). This new type of zero has hitherto not been studied in the 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 $\mathcal{O}({10}^{-2})$ with a satisfactory significance level.

Figure 1

Feynman diagrams for radiative muon decay.

Figure 2

Feynman rule for the effective $WW\gamma$ vertex.

Figure 3

The variations of the functions $f(x_p,y_p,q_p^2)$ and $F_o(x_p,y_p,q_p^2)$ are shown in the $x_p\text{−}{y}_p$ plane in the left and right panels, respectively, where $q_p^2=0.01$. The blue line in both panels indicates one boundary of the phase space with $\cos \theta =-1$ or $(q_p^4-q_p^2+x_p^2-y_p^2)=0$. In the left panel, the blue region indicates a negative-valued $f(x_p,y_p,q_p^2)$, the brown region indicates a positive-valued $f(x_p,y_p,q_p^2)$, and the black curve indicates $f(x_p,y_p,q_p^2)=0$. In the right panel, the yellow region indicates a negative-valued $F_o(x_p,y_p,q_p^2)$, the green region indicates a positive-valued $F_o(x_p,y_p,q_p^2)$, and the red curve indicates $F_o(x_p,y_p,q_p^2)=0$.

Figure 4

The variation of $F_o(x_p,y_p,q_p^2)$ for different $q_p^2$ in the $x_p\text{−}{y}_p$ plane. Each green dot represents a bin according to the experimental resolutions of the photon energy, electron energy, and the angle between them. The red dots stand for the bins with $\delta |A_{\eta }|/|A_{\eta }|\le 10$ in that bin. The purple curve indicates $F_o=0$ in a different $q_p^2$ plane. Our numerical analysis includes the bins corresponding to the red dots only. This results in an optimal sensitivity to ${\eta }_{\gamma }$.