BPST instanton

The BPST instanton is an essentially non-perturbative classical solution of the Yang–Mills field equations.

It is found when minimizing the Yang–Mills SU(2) Lagrangian density: with Fμνa = ∂μAνa – ∂νAμa + gεabcAμbAνc the field strength.

The instanton is a solution with finite action, so that Fμν must go to zero at space-time infinity, meaning that Aμ goes to a pure gauge configuration.

Instantons have q = 1 and thus correspond (at infinity) to gauge transformations which cannot be continuously deformed to unity.

It can be shown that self-dual configurations obeying the relation Fμνa = ± ⁠1/2⁠ εμναβ Fαβa minimize the action.

The classical action of an instanton equals[4] Since this quantity comes in an exponential in the path integral formalism this is an essentially non-perturbative effect, as the function e−1/x^2 has vanishing Taylor series at the origin, despite being nonzero elsewhere.

At finite temperature the BPST instanton generalizes to what is called a caloron.

When turning to a Yang–Mills theory with spontaneous symmetry breaking due to the Higgs mechanism, one finds that BPST instantons are not exact solutions to the field equations anymore.

In order to find approximate solutions, the formalism of constrained instantons can be used.

[7] It is expected that BPST-like instantons play an important role in the vacuum structure of QCD.

The results obtained did not solve the infrared problem of QCD, making many physicists turn away from instanton physics.

Later, though, an instanton liquid model was proposed, turning out to be a more promising approach.

't Hooft calculated the effective action for such an ensemble,[5] and he found an infrared divergence for big instantons, meaning that an infinite amount of infinitely big instantons would populate the vacuum.

The weak interaction is described by SU(2), so that instantons can be expected to play a role there as well.

Due to the Higgs mechanism, instantons are not exact solutions anymore, but approximations can be used instead.

Due to the non-perturbative nature of instantons, all their effects are suppressed by a factor of e−16π2/g2, which, in electroweak theory, is of the order 10−179.

Instantons are also closely related to merons,[10] singular non-dual solutions of the Euclidean Yang–Mills field equations of topological charge 1/2.