Better understanding of CP violationFlavour-Invariant Contributions to the Strong CP Problem

28 May 2025

Photo: Christoph Grojean

The potential of the axion is shifted by new sources of CP violation. The contribution of generic new sources of CP violation can be captured by flavour invariants of the coefficients in the effective field theory (bottom). These allow us to directly estimate the contribution using a diagrammatic approach (top right).

The postulation of an axion poses an elegant solution to the strong CP problem of the Standard Model. The construction of such a solution could be jeopardized by new sources of CP violation at high energies. Using an effective field theory approach to describe these new sources of CP violation, a group of Quantum Universe researchers from DESY together with scientists from the University of Minnesota has shown how to compute these corrections systematically, leading to a better understanding of theories with axions in the presence of CP violation.

The Standard Model (SM) of particle physics is extremely successful at describing Nature as we observe it around us. However, there are still some phenomena in Nature and some peculiarities in the theory that we do not fully understand yet. One of these problems of the SM is the so-called strong CP problem. Here, CP refers to the Charge-Parity symmetry, which if sufficiently violated can explain why there is more matter than antimatter in the Universe. At the heart of the strong CP problem is a term in the defining equations of the SM, sometimes referred to as the θ-term. This term is not forbidden by any fundamental principles of Nature, but violates CP. By measuring how neutrons react to external electric fields, a property referred to as their electric dipole moment, it can be inferred that the coefficient of the θ-term – called θ, hence the name – has to be extremely small of the order of θ~10^-11. This is in stark contrast with our expectations: as the term is not forbidden by any fundamental principle, we would expect θ to be a number close to 1. In other words, in Nature the strong interactions preserve CP, while in our best theory of the strong interactions a violation of CP – even with a large magnitude – is not forbidden. This is the strong CP problem, which calls for an explanation.

A popular solution to this problem is the axion solution postulating the existence of a new particle – the axion –, which turns the constant parameter θ into a dynamical field, similar to an electric field that can take different fields at different points in space-time. In this extension of the SM, the dynamical relaxation of the θ parameter to zero is ensured by the form of the potential of the axion, which is simply a periodic function of the axion field itself. This potential has to be minimised at zero field values of the axion for the mechanism to work, as the value of the axion field at the minimum of the potential determines the residual size of the θ parameter, which is strongly constrained by experiment as mentioned above. In many realisations of the axion mechanism this is indeed the case, because the potential for the axion is generated by the strong interactions, which are known to preserve CP. Moving the potential minimum away from zero is indeed only possible with new sources of sizable CP violation appearing in the theory. In general, new sources of sizable CP violation are expected to appear in extensions of the SM, because they are necessary to explain another shortcoming of the Standard Model: the baryon asymmetry of the Universe. Therefore, it is important to quantify how much CP violation can be allowed in the high-energy completion of the SM, and in which sector of the theory it can be introduced, in order not to spoil the axion solution at the current experimental sensitivity.

A well-motivated class of models beyond the SM includes heavy new particles that we have not seen in our experiments yet, because they are too massive to be directly produced. In this case, the effect of the heavy new physics on the SM particles can be captured in an effective field theory (EFT). These effective theories can be used whenever experiments take place at energies far below the mass of the heavy new particles, where the direct effects of the new physics on the SM particles can no longer be resolved. This is similar to how we can use Ohm’s law without understanding everything that goes on in a conductor at the microscopic level. At the microscopic level some more fundamental physics plays together in such a way that, at the macroscopic level, the material can effectively be described as a conductor following Ohm’s law. In the same way, the interactions between the heavy new particles and the SM particles can be captured by effective interactions among only the SM particles without specifying details about the more fundamental physics taking place at higher energy scales. 

Employing these EFT techniques, we have computed the shift of the axion potential. Here, CP violation manifests itself as an imaginary coefficient in the EFT interactions which due to the structure of the EFT have to appear in a very specific way in the computations of the shifted minimum, namely as so-called flavour invariants (cf. bottom expression in the figure). 

The computations to obtain the shift of the potential in a given quantum field theory turn out to be complicated. The main idea for starting the research of the paper was to find out whether using the very particular and constrained form of the flavour invariants could predict the form of the computations, which can be organised using diagrams as shown in the figure above. This indeed turned out to be true. Together with some counting rules, scientists showed that one can bypass most of the full computation and get a good estimate of the full result. This procedure significantly reduces the computational effort required and provides a systematic understanding of how CP-violating heavy new physics affects the axion potential. 

Researchers used their results to constrain certain high-energy completions of the Standard Model that feature heavy CP-violating new physics changing the minimum of the axion potential. This will lead to a better understanding of the possible realisations of the axion mechanism and its limitations, which will hopefully also facilitate experimental searches for axions, like the ALPSII experiment at DESY.