Combined explanations of $\mathbf{(}g-2{\mathbf{)}}_{\mu ,e}$ and implications for a large muon EDM

With the long-standing tension between experiment and standard-model (SM) prediction in the anomalous magnetic moment of the muon, $a_{\mu }=(g-2{)}_{\mu }/2$, at the level of $3–4\sigma$, it is natural to ask if there could be a sizable effect in the electric dipole moment (EDM) $d_{\mu }$ as well. In this context it has often been argued that in UV complete models the electron EDM, which is very precisely measured, excludes a large effect in $d_{\mu }$. However, the recently observed $2.5\sigma$ tension in $a_e=(g-2{)}_e/2$, if confirmed, requires that the muon and electron sectors effectively decouple to avoid constraints from $\mu \to e\gamma$. We briefly discuss UV complete models that possess such a decoupling, which can be enforced by an Abelian flavor symmetry $L_{\mu }-L_{\tau }$. We show that, in such scenarios, there is no reason to expect a correlation between the electron and muon EDM, so that the latter can be sizable. New limits on $d_{\mu }$ improved by up to two orders of magnitude are expected from the upcoming $(g-2{)}_{\mu }$ experiments at Fermilab and J-PARC. Beyond, a proposed dedicated muon EDM experiment at PSI could further advance the limit. In this way, future improved measurements of $a_e$, $a_{\mu }$, as well as the fine-structure constant $\alpha$ are not only set to provide exciting precision tests of the SM, but, in combination with EDMs, to reveal crucial insights into the flavor structure of physics beyond the SM.

Figure 1

Generic 1-loop diagram contributing to the dipole operator with fermion $L$ and scalar or vector $\varphi$ or $V$, respectively.

Figure 2

Generic diagrams contributing to the dipole operator in Model I.

Figure 3

Allowed regions of $a_{\ell }$ in the ${\lambda }_E={\lambda }_L–M_E=M_L$ plane for ${\kappa }_L=0$ and ${\kappa }_E=\mp 1$ for muon (left) and electron (right). The bounds are derived from $\sigma (h\to {\mu }^{+}{\mu }^{-})/\sigma (h\to {\mu }^{+}{\mu }^{-}{)}_{\mathrm{SM}}=0\pm 1.3$ [85, 86, 87], $\sigma (h\to e^{+}{e}^{-})/\sigma (h\to e^{+}{e}^{-}{)}_{\mathrm{SM}}<3.7\times {10}^5$ [88], $Z\to \ell \ell$ [85, 89], and direct searches for new heavy charged leptons [90]. The $h\to \ell \ell$ limits are implemented at $2\sigma$, the ones for $Z\to \ell \ell$ at $3\sigma$, as explained in the main text.

Figure 4

Three-loop diagram that produces a contribution to the electron EDM by an insertion of the muon EDM operator indicated by the cross. The other diagrams with insertions at the remaining muon–photon vertices as well as the permutations at the electron line are not shown.

Figure 5

Present and future direct limits on $|d_{\mu }|$ from BNL [26] (dark blue), see (3), Fermilab/J-PARC (blue), and the proposed PSI experiment (light blue). The dark red and light red regions refer to the ACME 2013 [27] and ACME 2018 [28] limits on $|d_e|$, respectively, where the latter provides an indirect bound on $|d_{\mu }|$ slightly better than the BNL direct bound. The blue dashed lines indicate the limits on $|d_e|$ that would be required to match the anticipated direct limits from Fermilab/J-PARC and PSI. The black line defines the relation (36), with the upper-left half referring to limits on $|d_{\mu }|$ and the lower-right to limits on $|d_e|$.

Figure 6

Sketch (a) and cross section (b) of the compact storage ring setup to search for a muon EDM (not to scale). (a) Polarized muons ($P\approx 0.9$) from in-flight decays of pions with a momentum of $200\mathrm{MeV}/\mathrm{c}$ from target E (not shown) of the high intensity proton accelerator of PSI arrive every 19.75 ns at the beam telescope. A dipole magnet with $B=1.5\mathrm{T}$ with a circular electrode system forms the magnetic storage ring. Once the beam telescope identifies a muon within the acceptance of the storage ring, the inflector, an air coil perturbing the otherwise homogeneous field, will be synchronously ramped down using a $1/2$ integer resonant injection scheme. After about twenty turns (orange orbits), the muon will stay on the stable (red) orbit until it decays. A positron tracker around the orbit will count the decay positrons relative to time. In general only one muon at a time will be stored. A veto sends the muon beam on a beam dump (not shown) until the decay is registered or a time-out occurs. Then the next cycle starts with ramping of the inflector field. (b) The muon orbit (red spot) is surrounded by a positron tracker system (blue) inside a vacuum chamber (gray outer lines). An electrode system encapsulated in a ground ring creates the electric field.

Figure 7

Contour lines defining the muon EDM (in units of $e\mathrm{cm}$) as a function of $\mathrm{\Delta }{a}_{\mu }$ and the phase of the Wilson coefficient $c_R^{\mu \mu }$. The red regions are currently preferred by the measurement of $a_{\mu }$ and the blue regions are the expected sensitivity of the Fermilab/J-PARC (dark blue) and the proposed PSI experiment (light blue). Note that since a chirally enhanced effect is preferred, $\arg [c_R^{\mu \mu }]$ is a free phase of the theory and in general expected to fulfill $\tan (\arg [c_R^{\mu \mu }])=\mathcal{O}(1)$. The limit on the phase derived from the current limit for $|d_{\mu }|$ is so close to 90° that it is not visible in the plot.