The instanton

The BPST instanton has a nontrivial winding number, which can be visualised as a non-trivial mapping of the circle on itself.

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:

${\displaystyle {\mathcal {L}}=-{\frac {1}{4}}F_{\mu \nu }^{a}F_{\mu \nu }^{a}}$

with F_μν^a = ∂_μA_ν^a – ∂_νA_μ^a + gε^abcA_μ^bA_ν^c the field strength. The instanton is a solution with finite action, so that F_μν must go to zero at space-time infinity, meaning that A_μ goes to a pure gauge configuration. Space-time infinity of our four-dimensional world is S³. The gauge group SU(2) has exactly the same structure, so the solutions with A_μ pure gauge at infinity are mappings from S³ onto itself.^[1] These mappings can be labelled by an integer number q, the Pontryagin index (or winding number). Instantons have q = 1 and thus correspond (at infinity) to gauge transformations which cannot be continuously deformed to unity.^[3] The BPST solution is thus topologically stable.

It can be shown that self-dual configurations obeying the relation F_μν^a = ± ½ ε^μναβ F_αβ^a minimize the action.^[4] Solutions with a plus sign are called instantons, those with the minus sign are anti-instantons.

Instantons and anti-instantons can be shown to minimise the action locally as follows:

${\displaystyle {\tilde {F}}_{\mu \nu }{\tilde {F}}^{\mu \nu }=F_{\mu \nu }F^{\mu \nu }}$, where ${\displaystyle {\tilde {F}}_{\mu \nu }={\frac {1}{2}}\epsilon _{\mu \nu }^{\rho \sigma }F_{\rho \sigma }}$.

${\displaystyle S=\int dx^{4}{\frac {1}{4}}F^{2}=\int dx^{4}{\frac {1}{8}}(F\pm {\tilde {F}})^{2}\mp \int dx^{4}{\frac {1}{4}}F{\tilde {F}}}$

The first term is minimised by self-dual or anti-self-dual configurations, whereas the last term is a total derivative and therefore depends only on the boundary (i.e. ${\displaystyle x\rightarrow \infty }$) of the solution; it is therefore a topological invariant and can be shown to be an integer number times some constant (the constant here is ${\displaystyle {\frac {8\pi ^{2}}{g^{2}}}}$). The integer is called instanton number (see Homotopy group).

Explicitly the instanton solution is given by^[5]

${\displaystyle A_{\mu }^{a}(x)={\frac {2}{g}}{\frac {\eta _{\mu \nu }^{a}(x-z)_{\nu }}{(x-z)^{2}+\rho ^{2}}}}$

with z_μ the center and ρ the scale of the instanton. η^a_μν is the 't Hooft symbol:

${\displaystyle \eta _{\mu \nu }^{a}={\begin{cases}\epsilon ^{a\mu \nu }&\mu ,\nu =1,2,3\\-\delta ^{a\nu }&\mu =4\\\delta ^{a\mu }&\nu =4\\0&\mu =\nu =4\end{cases}}.}$

For large x², ρ becomes negligible and the gauge field approaches that of the pure gauge transformation: ${\displaystyle {\frac {x^{0}+i\mathbf {x} \cdot \mathbf {\sigma } }{\sqrt {x^{2}}}}}$. Indeed, the field strength is:

${\displaystyle {\frac {1}{2}}\epsilon _{ijk}{F^{a}}_{jk}={F^{a}}_{0i}={\frac {4{\rho }^{2}\delta _{ai}}{g(x^{2}+\rho ^{2})^{2}}}}$

and approaches zero as fast as r⁻⁴ at infinity.

An anti-instanton is described by a similar expression, but with the 't Hooft symbol replaced by the anti-'t Hooft symbol ${\displaystyle {\bar {\eta }}_{\mu \nu }^{a}}$, which is equal to the ordinary 't Hooft symbol, except that the components with one of the Lorentz indices equal to four have opposite sign.

The BPST solution has many symmetries.^[6] Translations and dilations transform a solution into other solutions. Coordinate inversion (x^μ → x^μ/x²) transforms an instanton of size ρ into an anti-instanton with size 1/ρ and vice versa. Rotations in Euclidean four-space and special conformal transformations leave the solution invariant (up to a gauge transformation).

The classical action of an instanton equals^[4]

${\displaystyle S={\frac {8\pi ^{2}}{g^{2}}}.}$

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.

Other gauges

The expression for the BPST instanton given above is in the so-called regular Landau gauge. Another form exists, which is gauge-equivalent with the expression given above, in the singular Landau gauge. In both these gauges, the expression satisfies ∂_μA^μ = 0. In singular gauge the instanton is

${\displaystyle A_{\mu }^{a}(x)={\frac {2}{g}}{\frac {\rho ^{2}}{(x-z)^{2}}}{\frac {{\bar {\eta }}_{\mu \nu }^{a}(x-z)_{\nu }}{(x-z)^{2}+\rho ^{2}}}.}$

In singular gauge, the expression has a singularity in the center of the instanton, but goes to zero more swiftly for x to infinity.

When working in other gauges than the Landau gauge, similar expressions can be found in the literature.

In QCD

It is expected that BPST-like instantons play an important role in the vacuum structure of QCD. Instantons are indeed found in lattice calculations. The first computations performed with instantons used the dilute gas approximation. 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.^[8]

The dilute instanton gas model departs from the supposition that the QCD vacuum consists of a gas of BPST instantons. Although only the solutions with one or few instantons (or anti-instantons) are known exactly, a dilute gas of instantons and anti-instantons can be approximated by considering a superposition of one-instanton solutions at great distances from one another. '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.

Later, an instanton liquid model was studied. This model starts from the assumption that an ensemble of instantons cannot be described by a mere sum of separate instantons. Various models have been proposed, introducing interactions between instantons or using variational methods (like the "valley approximation") endeavouring to approximate the exact multi-instanton solution as closely as possible. Many phenomenological successes have been reached.^[8] Confinement seems to be the biggest issue in Yang–Mills theory for which instantons have no answer whatsoever.

In electroweak theory

The weak interaction is described by SU(2), so that instantons can be expected to play a role there as well. If so, they would induce baryon number violation. Due to the Higgs mechanism, instantons are not exact solutions anymore, but approximations can be used instead. One of the conclusions is that the presence of a gauge boson mass suppresses large instantons, so that the instanton gas approximation is consistent.

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.