Standard model contribution to the electric dipole moment of the deuteron, $^3$H, and $^3$He nuclei

We calculate for the first time the electric dipole moment (EDM) of the deuteron, $^3$H, and $^3$He nuclei generated by the one-meson exchange CP-odd nuclear force in the standard model. The effective $|\Delta S| = 1$ four-quark operators are matched to the $|\Delta S| = 1$ standard model processes involving the CP phase of the Cabibbo-Kobayashi-Maskawa matrix at the electroweak scale and run down to the hadronic scale $\mu = 1$ GeV according to the renormalization group evolution in the next-to-leading logarithmic order. At the hadronic scale, the hadron matrix elements are modeled in the factorization approach. We then obtain the one-meson (pion, eta meson, and kaon) exchange CP-odd nuclear force, which is the combination of the $|\Delta S| = 1$ meson-baryon vertices which issue from the penguin operator and the hyperon-nucleon transition. From this CP-odd nuclear force, the nuclear EDM is calculated with the realistic Argonne $v18$ interaction and the CP-odd nuclear force using the Gaussian expansion method. It is found that the EDMs of light nuclear systems are of order $O(10^{-31}) e $ cm. We also estimate the standard model contribution to other hadronic CP violating observables such as the EDMs of $^6$Li, $^9$Be nuclei, and the atomic EDMs of $^{129}$Xe, $^{199}$Hg, $^{211}$Rn, and $^{225}$Ra generated through the nuclear Schiff moment. We then analyze the source of theoretical uncertainties and show some possible ways to overcome them.