Probing $\mu e\gamma \gamma$ contact interactions with $\mu \to e$ conversion

Contact interactions of a muon, an electron and two photons can contribute to the decay $\mu \to e\gamma \gamma$, but also to the conversion of a muon into an electron in the electric field of a nucleus. We calculate the $\mu \to e$ conversion rate, and show that for the coefficients of operators involving the combination $FF\propto |\overrightarrow{E}{|}^2$ (as opposed to $F\tilde{F}\propto \overrightarrow{E}·\overrightarrow{B}$), the current bound on $\mu \to e$ conversion is more sensitive than the bound on $\mu \to e\gamma \gamma$.

Figure 1

Matching of scalar heavy fermion operators ${\mathcal{O}}_{S,XY}^{\psi \psi }$, ${\mathcal{O}}_{S,XX}^{\psi \psi }$ onto the two-photon operator ${\mathcal{O}}_{FF,X}$.

Figure 2

$|F_A|$ as a function of atomic number ($Z$) for the target nucleus. Nuclear distributions are 3pG for $Z=16$, 28, 38, 40, 42, 50, 56, and 83 (blue circle), and 2pF for other nuclei (red diamond).

Figure 3

Integrand of $F_A$ for $Z=13$, 38, and 79. The horizontal axis shows the dimensionless distance from a nuclear center. The amplitude for $Z=13$ is multiplied by a factor 5.

Figure 4

Diagrams for two photons from the $FF$ operator interacting with the nucleus $A$.

Figure 5

Branching ratios for $\mu \to e$ conversion normalized to that for aluminum. The solid line shows the result for ${\mathcal{O}}_{FF,X}=\overline{e}{P}_X\mu F^{\alpha \beta }{F}_{\alpha \beta }$, where we consider two contributions: one is the interaction with the classical electric field of the nucleus, and the other is the effect of loop mixing of ${\mathcal{O}}_{FF,X}$ into ${\mathcal{O}}_{S,X}=(\overline{e}{P}_X\mu )(\overline{p}p)$ due to two-photon exchange. If neglecting the loop contribution, we obtain the dashed line. For comparison, the dotted line gives the normalized branching ratio for only the scalar operator ${\mathcal{O}}_{S,X}$.