even under such field redefinitions, the combination remains unchanged. For example, the CP violation due to the mass term can be eliminated by picking , but then all the CP violation goes to the θ-term which is now proportional to . If instead the θ-term is eliminated through a chiral transformation, then there will be a CP violating complex mass with a phase . Practically, it is usually useful to put all the CP violation into the θ-term and thus only deal with real masses. In the Standard Model where one deals with six quarks whose masses are described by the Yukawa matrices and , the physical CP violating angle is . Since the θ-term has no contributions to perturbation theory, all effects from strong CP violation is entirely non-perturbative. Notably, it gives rise to a neutron electric dipole moment Current experimental upper bounds on the dipole moment give an upper bound of cm, which requires . The angle can take any value between zero and , so it taking on such a particularly small value is a fine-tuning problem called the strong CP problem. Proposed solutions The strong CP problem is solved automatically if one of the quarks is massless. In that case one can perform a set of chiral transformations on all the massive quark fields to get rid of their complex mass phases and then perform another chiral transformation on the massless quark field to eliminate the residual θ-term without also introducing a complex mass term for that field. This then gets rid of all CP violating terms in the theory. The problem with this solution is that all quarks are known to be massive from experimental matching with lattice calculations. Even if one of the quarks was essentially massless to solve the problem, this would in itself just be another fine-tuning problem since there is nothing requiring a quark mass to take on such a small value. The most popular solution to the problem is through the Peccei–Quinn mechanism. This introduces a new global anomalous symmetry which is then spontaneously broken at low energies, giving rise to a pseudo-Goldstone boson called an axion. The axion ground state dynamically forces the theory to be CP-symmetric by setting . Axions are also considered viable candidates for dark matter and axion-like particles are also predicted by string theory. Other less popular proposed solutions exist such as Nelson–Barr models. These