Abstract: In the first half of my talk, I will present a way to parametrize new sources of CP violation in the Standard Model EFT (SMEFT) using flavor invariants. New sources of CP violation can feed into the QCD theta angle, jeopardizing solutions to the strong CP problem such as the axion. By means of an instanton calculation, I will show that the CP-violating parameters of the SMEFT indeed enter the theta angle as flavor-invariant quantities. Turning these arguments around, I will also discuss how the structure of the invariants allows to make statements about the way particular SMEFT phases can enter in the instanton calculation.
The second part of my talk will concern EFTs of axion-like particles (ALPs), that are usually built assuming a shift symmetry for the ALP due to the global U(1) Peccei-Quinn symmetry. However, it is generally believed that global symmetries, in particular axion shift symmetries, can only be approximate. I will present flavor invariants which act as order parameters for shift symmetry breaking of the axion. In this description, shift-breaking couplings are characterized in an explicit and flavor-invariant way and it is straightforward to give different power countings to the shift-conserving and shift-breaking sectors of the EFT. I will discuss properties of the invariants, like their CP parities and RGEs, and show how they can be used to identify shift-breaking contributions in observables.