Opportunistic CP Violation

In the electroweak sector of the Standard Model, CP violation arises through a very particular interplay between the three quark generations, as described by the Cabibbo--Kobayashi--Maskawa (CKM) mechanism and the single Jarlskog invariant $J_4$. Once generalized to the Standard Model Effective Field Theory (SMEFT), this peculiar pattern gets modified by higher-dimensional operators, whose associated Wilson coefficients are usually split into CP-even and odd parts. However, CP violation at dimension four, i.e., at the lowest order in the EFT expansion, blurs this distinction: any Wilson coefficient can interfere with $J_4$ and mediate CP violation. In this paper, we study such interferences at first order in the SMEFT expansion, $\mathcal{O}(1/\Lambda^2)$, and we capture their associated parameter space via a set of 1551 linear CP-odd flavor invariants. This construction describes both new, genuinely CP-violating quantities as well as the interference between $J_4$ and CP-conserving ones. We call this latter possibility \textit{opportunistic CP violation}. Relying on an appropriate extension of the matrix rank to Taylor expansions, which we dub \emph{Taylor rank}, we define a procedure to organize the invariants in terms of their magnitude, so as to retain only the relevant ones at a given precision. We explore how this characterization changes when different assumptions are made on the flavor structure of the SMEFT coefficients. Interestingly, some of the CP-odd invariants turn out to be less suppressed than $J_4$, even when they capture opportunistic CPV, demonstrating that CP-violation in the SM, at dimension 4, is \textit{accidentally small.}