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 . Classical ground states of this theory have a vanishing field strength tensor which corresponds to pure gauge configurations , where at each point in spacetime is some gauge transformation belonging to the non-abelian gauge group . To ensure that the action is finite, approaches some fixed value as . Since all points at spatial infinity now behave as a single new point, the spatial manifold behaves as a 3-sphere so that every pure gauge choice for the gauge field is described by a mapping . 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 . For example, the gauge group has an underlying manifold of so that the mapping is , which has a homotopy group of . 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 is mapped onto the group , with negative windings occurring due to a flipped orientation. Only mappings with the same