Large Long-Distance Contributions to the Electric Dipole Moments of Charged Leptons in the Standard Model

We reevaluate the electric dipole moment (EDM) of charged leptons in the standard model using hadron effective models. We find unexpectedly large EDM generated by the hadron level long-distance effect, $d_e=5.8\times {10}^{-40}$, $d_{\mu }=1.4\times {10}^{-38}$, and $d_{\tau }=-7.3\times {10}^{-38}e\mathrm{cm}$, with an error bar of 70%, exceeding the conventionally known four-loop level elementary contribution by several orders of magnitude.

Figure 1

Example of four-loop level diagram contributing to the EDM of charged leptons generated by the CKM matrix at the quark level.

Figure 2

Long-distance contribution to the charged lepton EDM generated in the SM, with the (a) strong and (b) weak three-vector meson interactions. Other symmetric diagrams are not displayed. The $|\mathrm{\Delta }S|=1$ semileptonic interaction (gray blob) and the $|\mathrm{\Delta }S|=1$ hadronic interaction (black blob) are chosen so as to form the Jarlskog invariant.

Figure 3

Short-distance contribution to the $K^{*}$-charged lepton interaction, with $l=e$, $\mu$, $\tau$. The diagrams (a), (b) and (c) are the contributions of the box diagram with the two $W$ exchanges and the penguin diagrams with the quark-$Z$ and $W\text{−}Z$ vertices, respectively.

Figure 4

Factorization of the $|\mathrm{\Delta }S|=1$ vector meson vertices ($|\mathrm{\Delta }S|=1$ meson transition), with the (a) two-quark process, (b) one-quark process, and (c) three-meson interaction. The double crosses with “$⟨\overline{q}q⟩$” denote the chiral condensate $⟨0|\overline{q}q|0⟩$ ($q=d$, $s$). The black blob denotes the $|\mathrm{\Delta }S|=1$ four-quark interaction. There are similar diagrams with the $\rho$ meson replaced by $\omega$ or $\varphi$ mesons.

Figure 5

Plot of the SM predictions (quark and hadron levels) of the electron EDM compared with the experimental upper limits updated from the 1960s [6, 10, 12, 14, 19, 20, 22, 23, 24, 25]. The prospective experimental sensitivity [30] is also shown.