Can electron and muon $g-2$ anomalies be jointly explained in SUSY?

The FNAL+BNL measurements for muon $g-2$ is $4.2\sigma$ above the SM prediction, and the Berkeley $^{133}$Cs measurement for the fine-structure constant $\alpha_{\rm em}$ leads to the SM prediction for electron $g-2$ which is $2.4\sigma$ above the experimental value. Hence, a joint explanation of both anomalies requires a positive contribution to muon $g-2$ and a negative contribution to electron $g-2$, which is rather challenging. In this work we explore the possibility of such a joint explanation in the minimal supersymmetric standard model (MSSM). Assuming no universality between smuon and selectron soft masses, we find out a part of parameter space for a joint explanation at $2\sigma$ level, i.e., $\mu M_1,\mu M_2<0$, $m_{L1}, m_{E2}<200$ GeV, $m_{L2}$ being much larger than the soft masses of other sleptons, $|M_1|<125$ GeV and $\mu<400$ GeV. This part of parameter space can survive LHC and LEP constraints, but gives an over-abundance for dark matter if the bino-like lightest neutralino is assumed to be the dark matter candidate. With the assumption that the dark matter candidate is a superWIMP (say a pseudo-goldstino in multi-sector SUSY breaking scenarios, whose mass can be as light as GeV and produced from the late-decay of the thermally freeze-out lightest neutralino), the dark matter problem can be avoided. So, we conclude that the MSSM may give a joint explanation for the muon and electron $g-2$ anomalies at $2\sigma$ level (the muon $g-2$ anomaly can be even ameliorated to $1\sigma$).