Chiral anomaly
Such events are expected to be prohibited according to classical conservation laws, but it is known there must be ways they can be broken, because we have evidence of charge–parity non-conservation ("CP violation").
Many physicists suspect that the fact that the observable universe contains more matter than antimatter is caused by a chiral anomaly.
These calculations suggested that the decay of the pion was suppressed, clearly contradicting experimental results.
The nature of the anomalous calculations was first explained in 1969 by Stephen L. Adler[2] and John Stewart Bell & Roman Jackiw.
These are famously divergent, and require a regularization to be applied, to obtain the renormalized amplitudes.
In order for the renormalization to be meaningful, coherent and consistent, the regularized diagrams must obey the same symmetries as the zero-loop (classical) amplitudes.
At the time that the Adler–Bell–Jackiw anomaly was being explored in physics, there were related developments in differential geometry that appeared to involve the same kinds of expressions.
These were not in any way related to quantum corrections of any sort, but rather were the exploration of the global structure of fiber bundles, and specifically, of the Dirac operators on spin structures having curvature forms resembling that of the electromagnetic tensor, both in four and three dimensions (the Chern–Simons theory).
After considerable back and forth, it became clear that the structure of the anomaly could be described with bundles with a non-trivial homotopy group, or, in physics lingo, in terms of instantons.
Instantons are a form of topological soliton; they are a solution to the classical field theory, having the property that they are stable and cannot decay (into plane waves, for example).
Put differently: conventional field theory is built on the idea of a vacuum – roughly speaking, a flat empty space.
These non-trivial configurations are also candidates for the vacuum, for empty space; yet they are no longer flat or trivial; they contain a twist, the instanton.
In mathematics, non-trivial configurations are found during the study of Dirac operators in their fully generalized setting, namely, on Riemannian manifolds in arbitrary dimensions.
Roughly speaking, the symmetries of Minkowski spacetime, Lorentz invariance, Laplacians, Dirac operators and the U(1)xSU(2)xSU(3) fiber bundles can be taken to be a special case of a far more general setting in differential geometry; the exploration of the various possibilities accounts for much of the excitement in theories such as string theory; the richness of possibilities accounts for a certain perception of lack of progress.
The neutral pion itself was discovered in the 1940s; its decay rate (width) was correctly estimated by J. Steinberger in 1949.
[6] The correct form of the anomalous divergence of the axial current is obtained by Schwinger in 1951 in a 2D model of electromagnetism and massless fermions.
[7] That the decay of the neutral pion is suppressed in the current algebra analysis of the chiral model is obtained by Sutherland and Veltman in 1967.
[8][9] An analysis and resolution of this anomalous result is provided by Adler[2] and Bell & Jackiw[3] in 1969.
However, the quantum numbers, including parity and angular momentum, taken to be conserved, prohibit the decay of the pion, at least in the zero-loop calculations (quite simply, the amplitudes vanish.)
If the quarks are assumed to be massive, not massless, then a chirality-violating decay is allowed; however, it is not of the correct size.
The contribution of the mass is given by the Sutherland and Veltman result; it is termed "PCAC", the partially conserved axial current.)
Current day research is focused on similar phenomena in different settings, including non-trivial topological configurations of the electroweak theory, that is, the sphalerons.
, but not of the measure μ and therefore not of the generating functional of the quantized theory (ℏ is Planck's action-quantum divided by 2π).
The anomaly is proportional to the instanton number of a gauge field to which the fermions are coupled.
(Note that the gauge symmetry is always non-anomalous and is exactly respected, as is required for the theory to be consistent.)
The amplitude for this process can be calculated directly from the change in the measure of the fermionic fields under the chiral transformation.
The Standard Model of electroweak interactions has all the necessary ingredients for successful baryogenesis, although these interactions have never been observed[11] and may be insufficient to explain the total baryon number of the observed universe if the initial baryon number of the universe at the time of the Big Bang is zero.
Baryons are not conserved by the usual electroweak interactions due to quantum chiral anomaly.
is conserved: However, quantum corrections known as the sphaleron destroy this conservation law: instead of zero in the right hand side of this equation, there is a non-vanishing quantum term, where C is a numerical constant vanishing for ℏ =0, and the gauge field strength
An important fact is that the anomalous current non-conservation is proportional to the total derivative of a vector operator,
