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,[1] and independently by Roman Jackiw and Claudio Rebbi.
Classical ground states of this theory have a vanishing field strength tensor which corresponds to pure gauge configurations
[3] 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
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
, 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.
can be deformed to the constant map and so there is a single connected vacuum state.
, one can always calculate its winding number from a volume integral which in the temporal gauge is given by where
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.
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
In other words, the Hilbert space decomposes into superselection sectors since expectation values of gauge invariant operators between two different θ-vacua vanish
[6] 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
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
, 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[8] Here
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 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,[9] requiring the fine tuning of
If massless fermions are present in the theory then the vacuum angle becomes unobservable because the fermions suppress the instanton tunnelling between topological vacua.
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.
[11] 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.