Editing Gauge anomaly (section)

==Calculation of the anomaly==

Anomalies occur only in even spacetime dimensions. For example, the anomalies in the usual 4 spacetime dimensions arise from triangle Feynman diagrams.

===Vector gauge anomalies===

In [[Vector (geometric)#Vectors in particle physics|vector]] gauge anomalies (in [[gauge symmetry|gauge symmetries]] whose [[gauge boson]] is a vector), the anomaly is a [[chiral anomaly]], and can be calculated exactly at one loop level, via a [[Feynman diagram]] with a [[chirality (physics)|chiral]] [[fermion]] running in the loop with ''n'' external [[gauge boson]]s attached to the loop where <math>n=1+D/2</math> where <math>D</math> is the [[spacetime]] dimension.

[[Image:Triangle diagram.svg]]

Let us look at the (semi)[[effective action]] we get after integrating over the [[chiral fermion]]s. If there is a gauge anomaly, the resulting action will not be gauge invariant. If we denote by <math>\delta_\epsilon</math> the operator corresponding to an [[infinitesimal]] gauge transformation by ε, then the [[Frobenius theorem (differential topology)|Frobenius consistency condition]] requires that

:<math>\left[\delta_{\epsilon_1},\delta_{\epsilon_2}\right]\mathcal{F}=\delta_{\left[\epsilon_1,\epsilon_2\right]}\mathcal{F}</math>

for any functional <math>\mathcal{F}</math>, including the (semi)effective action S where [,] is the [[Lie bracket of vector fields|Lie bracket]]. As <math>\delta_\epsilon S</math> is linear in ε, we can write

:<math>\delta_\epsilon S=\int_{M^d} \Omega^{(d)}(\epsilon)</math>

where Ω<sup>(d)</sup> is [[differential form|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<sup>(d)</sup>(x) only depends upon the values of the fields and their derivatives at x) and that it can be expressed as the [[exterior product]] of p-forms. If the spacetime M<sup>d</sup> is [[closed manifold|closed]] (i.e. without boundary) and oriented, then it is the boundary of some d+1 dimensional oriented manifold M<sup>d+1</sup>. If we then arbitrarily extend the fields (including ε) as defined on M<sup>d</sup> to M<sup>d+1</sup> with the only condition being they match on the boundaries and the expression Ω<sup>(d)</sup>, being the exterior product of p-forms, can be extended and defined in the interior, then

:<math>\delta_\epsilon S=\int_{M^{d+1}} d\Omega^{(d)}(\epsilon).</math>

The Frobenius consistency condition now becomes

:<math>\left[\delta_{\epsilon_1},\delta_{\epsilon_2}\right]S=\int_{M^{d+1}}\left[\delta_{\epsilon_1}d\Omega^{(d)}(\epsilon_2)-\delta_{\epsilon_2}d\Omega^{(d)}(\epsilon_1)\right]=\int_{M^{d+1}}d\Omega^{(d)}(\left[\epsilon_1,\epsilon_2\right]).</math>

As the previous equation is valid for ''any'' arbitrary extension of the fields into the interior,

:<math>\delta_{\epsilon_1}d\Omega^{(d)}(\epsilon_2)-\delta_{\epsilon_2}d\Omega^{(d)}(\epsilon_1)=d\Omega^{(d)}(\left[\epsilon_1,\epsilon_2\right]).</math>

Because of the Frobenius consistency condition, this means that there exists a d+1-form Ω<sup>(d+1)</sup> (not depending upon ε) defined over M<sup>d+1</sup> satisfying

:<math>\delta_\epsilon \Omega^{(d+1)}=d\Omega^{(d)}( \epsilon ).</math>

Ω<sup>(d+1)</sup> is often called a [[Chern–Simons form]].

Once again, if we assume Ω<sup>(d+1)</sup> 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

:<math>\Omega^{(d+2)}=d\Omega^{(d+1)}</math>

in d+2 dimensions. Ω<sup>(d+2)</sup> is gauge invariant:

:<math>\delta_\epsilon \Omega^{(d+2)}=d\delta_\epsilon \Omega^{(d+1)}=d^2\Omega^{(d)}(\epsilon)=0</math>

as d and δ<sub>ε</sub> commute.