the form , where the constant of proportionality will depend on how strong the instanton tunnelling is. The complicated structure of the θ-vacuum can be directly incorporated into the Yang–Mills Lagrangian by considering the vacuum-vacuum transitions in the path integral formalism Here is the Hamiltonian, the Yang–Mills action, and is a new CP violating contribution to the Lagrangian called the θ-term where is the dual field strength tensor and the trace is over the group generators. This term is a total derivative meaning that it can be written in the form . In contrast to other total derivatives that can be added to the Lagrangian, this one has physical consequences in non-perturbative physics because is not gauge invariant. In quantum chromodynamics the presence of this term leads to the strong CP problem since it gives rise to a neutron electric dipole moment which has not yet been observed, requiring the fine tuning of to be very small. Modification due to fermions If massless fermions are present in the theory then the vacuum angle becomes unobservable because the fermions suppress the instanton tunnelling between topological vacua. This can be seen by considering a Yang–Mills theory with a single massless fermion . In the path integral formalism the tunnelling by an instanton between two topological vacua takes the form This differs from the pure Yang–Mills result by the fermion determinant acquired after integrating over the fermionic fields. The determinant vanishes because the Dirac operator with massless fermions has at least one zero eigenvalue for any instanton configuration. While instantons no longer contribute to tunnelling between topological vacua, they instead play a role in violating axial charge and thus give rise to the chiral condensate. If instead the theory has very light fermions then the θ-term is still present, but its effects are heavily suppressed as they must be proportional to the fermion masses. See also Instanton Strong CP problem References Gauge theories Quantum chromodynamics