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, O �(1/?(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 opportunistic CP violation. Relying on an appropriate extension of the matrix rank to Taylor expansions, which we dub 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 accidentally small.