Theta vacuum

In quantum field theory, the theta vacuum is the semi-classical vacuum state of non-abelian Yang–Mills theories specified by the vacuum angle "θ" that arises when the state is written as a superposition of an infinite set of topologically distinct vacuum states. The dynamical effects of the vacuum are captured in the Lagrangian formalism through the presence of a "θ"-term which in quantum chromodynamics leads to the fine tuning problem known as the strong CP problem. It was discovered in 1976 by Curtis Callan, Roger Dashen, and David Gross, and independently by Roman Jackiw and Claudio Rebbi.
Yang–Mills vacuum.
Topological vacua.
The semi-classical vacuum structure of non-abelian Yang–Mills theories is often investigated in Euclidean spacetime in some fixed gauge such as the temporal gauge formula_1. Classical ground states of this theory have a vanishing field strength tensor which corresponds to pure gauge configurations formula_2, where at each point in spacetime formula_3 is some gauge transformation belonging to the non-abelian gauge group formula_4. To ensure that the action is finite, formula_3 approaches some fixed value formula_6 as formula_7. Since all points at spatial infinity now behave as a single new point, the spatial manifold formula_8 behaves as a 3-sphere formula_9 so that every pure gauge choice for the gauge field is described by a mapping formula_10.
When every ground state configuration can be smoothly transformed into every other ground state configuration through a smooth gauge transformation then the theory has a single vacuum state, but if there are topologically distinct configurations then it has multiple vacua. This is because if there are two different configurations that are not smoothly connected, then to transform one into the other one must pass through a configuration with non-vanishing field strength tensor, which will have non-zero energy. This means that there is an energy barrier between the two vacua, making them distinct.
The question of whether two gauge configurations can be smoothly deformed into each other is formally described by the homotopy group of the mapping formula_10. For example, the gauge group formula_12 has an underlying manifold of formula_13 so that the mapping is formula_14, which has a homotopy group of formula_15. This means that every mapping has some integer associated with it called its winding number, also known as its Pontryagin index, with it roughly describing to how many times the spatial formula_13 is mapped onto the group formula_13, with negative windings occurring due to a flipped orientation. Only mappings with the same winding number can be smoothly deformed into each other and are said to belong to the same homotopy class. Gauge transformations which preserve the winding number are called small gauge transformations while ones that change the winding number are called large gauge transformations.
For other non-abelian gauge groups formula_4 it is sufficient to focus on one of their formula_19 subgroups, ensuring that formula_20. This is because every mapping of formula_13 onto formula_4 can be continuously deformed into a mapping onto an formula_19 subgroup of formula_4, a result that follows from Botts theorem. This is in contrast to abelian gauge groups where every mapping formula_25 can be deformed to the constant map and so there is a single connected vacuum state. For a gauge field configuration formula_26, one can always calculate its winding number from a volume integral which in the temporal gauge is given by
where formula_28 is the coupling constant. The different classes of vacuum states with different winding numbers formula_29 are referred to as topological vacua. 
Theta vacua.
Topological vacua are not candidate vacuum states of Yang–Mills theories since they are not eigenstates of large gauge transformations and so aren't gauge invariant. Instead acting on the state formula_29 with a large gauge transformation formula_31 with winding number formula_32 will map it to a different topological vacuum formula_33. The true vacuum has to be an eigenstate of both small and large gauge transformations. Similarly to the form that eigenstates take in periodic potentials according to Bloch's theorem, the vacuum state is a coherent sum of topological vacua
This set of states indexed by the angular variable formula_35 are known as "θ"-vacua. They are eigenstates of both types of gauge transformations since now formula_36. In pure Yang–Mills, each value of formula_37 will give a different ground state on which excited states are built, leading to different physics. In other words, the Hilbert space decomposes into superselection sectors since expectation values of gauge invariant operators between two different "θ"-vacua vanish formula_38 if formula_39.
Yang–Mills theories exhibit finite action solutions to their equations of motion called instantons. They are responsible for tunnelling between different topological vacua with an instanton with winding number formula_40 being responsible for a tunnelling from a topological vacuum formula_41 to formula_42. Instantons with formula_43 are known as BPST instantons. Without any tunnelling the different "θ"-vacua would be degenerate, however instantons lift the degeneracy, making the various different "θ"-vacua physically distinct from each other. The ground state energy of the different vacua is split to take the form formula_44, 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 formula_46 is the Hamiltonian, formula_47 the Yang–Mills action, and formula_48 is a new CP violating contribution to the Lagrangian called the "θ"-term
where formula_50 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 formula_51. In contrast to other total derivatives that can be added to the Lagrangian, this one has physical consequences in non-perturbative physics because formula_52 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 formula_37 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 formula_54. 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.