- Gauge anomaly
theoretical physics, a gauge anomaly is an example of an anomaly: it is an effect of quantum mechanics—usually a one-loop diagram—that invalidates the gauge symmetryof a quantum field theory; i.e. of a gauge theory.
Anomalies in gauge symmetries lead to an inconsistency, since a gauge symmetry is required in order to cancel unphysical degrees of freedom with a negative norm (such as a
photonpolarized in the time direction). Therefore all gauge anomalies must cancel out. This indeed happens in the Standard Model.
The term gauge anomaly is usually used for vector gauge anomalies. Another type of gauge anomaly is the
gravitational anomaly, because reparametrization is a gauge symmetry in gravitation.
Calculation of the anomaly
In vector gauge anomalies (in gauge symmetries whose
gauge bosonis a vector), the anomaly is a chiral anomaly, and can be calculated exactly at one loop level, via a Feynman diagramwith a chiral fermionrunning in the loop (a polygon) with "n" external gauge bosons attached to the loop where where is the spacetimedimension. Anomalies occur only in even spacetime dimensions. For example, the anomalies in the usual 4 spacetime dimensions arise from triangle Feynman diagrams.
Let us look at the (semi)effective action we get after integrating over the
chiral fermions. If there is a gauge anomaly, the resulting action will not be gauge invariant. If we denote by the operator corresponding to an infinitesimal gauge transformation by ε, then the Frobenius consistency condition requires that
for any functional , including the (semi)effective action S where [,] is the
Lie bracket. As is linear in ε, we can write
where Ω(4) is d-form as a functional of the nonintegrated fields and is linear in ε. Let us make the further assumption (which turns out to be valid in all the cases of interest) that this functional is local (i.e. Ω(d)(x) only depends upon the values of the fields and their derivatives at x) and that it can be expressed as the
exterior productof p-forms. If the spacetime Md is closed (i.e. without boundary) and oriented, then it is the boundary of some d+1 dimensional oriented manifold Md+1. If we then arbitrarily extend the fields (including ε) as defined on Md to Md+1 with the only condition being they match on the boundaries and the expression Ω(d), being the exterior product of p-forms, can be extended and defined in the interior, then
The Frobenius consistency condition now becomes
As the previous equation is valid for "any" arbitrary extension of the fields into the interior,
Because of the Frobenius consistency condition, this means that there exists a d+1-form Ωd+1 (not depending upon ε) defined over Md+1 satisfying
Ωd+1 is often called a
Once again, if we assume Ωd+1 can be expressed as an exterior product and that it can be extended into a d+1 -form in a d+2 dimensional oriented manifold, we can define
in d+2 dimensions. Ωd+2 is gauge invariant:
as d and δε commute.
chiral gauge theory
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