Beyond the Standard Model Orders of Charge–Parity Violation

In this thesis, we use flavour invariants to systematically study solutions to problems of the Standard Model (SM) of particle physics with different effective field theories (EFTs). In Part I, we study Charge–Parity (CP) violation in the SM and SM EFT extended with light sterile neutrinos. We construct the generating set of flavour invariants in the νSM allowing us to express any observable as a polynomial of those invariants. In addition, the invariants enable us to express the conditions for CP violation in a flavour-invariant way. We extend the results to the EFT interactions with different scenarios for the neutrino masses. Here, the form of the EFT flavour invariants and their suppression with the scale of new physics changes drastically depending on the nature of the neutrino masses. In Part II, we study different aspects of symmetry breaking in the EFTs of axionlike particles (ALPs). An essential property of ALPs is their shift symmetry (ShS) due to their pseudo-Nambu–Goldstone nature. We formulate flavour-invariant order parameters of ShS, which allow us to properly impose the power counting of the leading order EFT in the presence of a softly broken ShS. Using the Hilbert series, we count the number of operators appearing in the ALP EFT with and without a ShS above and below the electroweak scale. We use this information to construct operator bases, generalise the relations imposing ShS to higher orders and construct the leading order CP-odd flavour invariants. The axion solution to the strong CP problem can be spoiled by new CP violation in the ultraviolet in the presence of small instantons. Parameterising the new CP violation in the SMEFT, we show that newly constructed CP-odd SMEFT flavour invariants, featuring the strong CP angle, explicitly appear in the instanton computations and vice-versa that they can be used to systematise the computations. Using these results, we derive bounds on different small instanton and SMEFT flavour scenarios.