Standard Model Prediction for Paramagnetic Electric Dipole Moments

Standard model $CP$ violation associated with the phase of the Cabibbo-Kobayashi-Maskawa quark mixing matrix is known to give small answers for the electric dipole moment (EDM) observables. Moreover, predictions for the EDMs of neutrons and diamagnetic atoms suffer from considerable uncertainties. We point out that the $CP$-violating observables associated with the electron spin (paramagnetic EDMs) are dominated by the combination of the electroweak penguin diagrams and $\mathrm{\Delta }I=1/2$ weak transitions in the baryon sector, and are calculable within chiral perturbation theory. The predicted size of the semileptonic operator $C_S$ is $7\times {10}^{-16}$, which corresponds to the equivalent electron EDM $d_e^{\mathrm{eq}}=1.0\times {10}^{-35}e\mathrm{cm}$. While still far from the current observational limits, this result is 3 orders of magnitude larger than previously believed.

Figure 1

${\mathrm{EW}}^3$ order diagram that dominates in the chiral limit. The top vertex is the $CP$-odd, $P$-even $K_S\overline{e}i{\gamma }_{5}e$ generated in ${\mathrm{EW}}^2$ order, and the bottom vertex is $CP$-even, $P$-odd $K_S\overline{N}N$ coupling generated at ${\mathrm{EW}}^1$ order.

Figure 2

The baryon pole diagrams that contribute to $C_S$ at the NLO level in the chiral limit. The left vertex is the nucleon-hyperon mixing induced by Eq. (8), while the top vertex is induced by Eq. (6). The vertices without black dots are the strong interaction with the coupling constants $D$ and $F$. The diagrams with the nucleon-hyperon mixing on the right side give the same amount of contribution.