Spelling Out Leptonic CP Violation in the Language of Invariant Theory

In terms of flavor invariants, we establish the intimate connection between leptonic CP violation in the canonical seesaw model for neutrino masses and that in the seesaw effective field theory (SEFT). For the first time, we calculate the Hilbert series and explicitly construct the primary flavor invariants in the SEFT by considering both the dimension-five Weinberg operator ${\cal O}^{\alpha \beta}_5 = \overline{\ell^{}_{\alpha \rm L}} \widetilde{H} \widetilde{H}^{\rm T} \ell^{\rm C}_{\beta \rm L}$ and the dimension-six operator ${\cal O}^{\alpha \beta}_6 = \overline{\ell^{}_{\alpha \rm L}} \widetilde{H} {\rm i} \not\!\partial \widetilde{H}^\dagger \ell^{}_{\beta \rm L}$ at the tree-level matching. The inclusion of only the Wilson coefficients $C^{\alpha \beta}_5$ and $C^{\alpha \beta}_6$ already enables the SEFT to incorporate all physical information about the full seesaw model. Moreover, the minimal sufficient and necessary conditions for CP conservation both in the SEFT and in the full theory are clarified, and the matching between the flavor invariants in both theories is accomplished. Through the matching of flavor invariants, the CP asymmetries necessary for successful leptogenesis are directly linked to those in neutrino-neutrino and neutrino-antineutrino oscillations at low energies. Surprisingly, it is revealed that the precise measurements of $C^{\alpha \beta}_5$ and $C^{\alpha \beta}_6$ in low-energy experiments are powerful enough to probe the full seesaw model, including CP violation for cosmological matter-antimatter asymmetry.