- Gauge anomaly
In

theoretical physics , a**gauge anomaly**is an example of an anomaly: it is an effect ofquantum mechanics —usually aone-loop diagram —that invalidates thegauge symmetry of aquantum field theory ; i.e. of agauge 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

photon polarized in the time direction). Therefore all gauge anomalies must cancel out. This indeed happens in theStandard Model .The term

**gauge anomaly**is usually used for vector gauge anomalies. Another type of gauge anomaly is thegravitational anomaly , because reparametrization is a gauge symmetry ingravitation .**Calculation of the anomaly**In vector gauge anomalies (in gauge symmetries whose

gauge boson is a vector), the anomaly is achiral anomaly , and can be calculated exactly at one loop level, via aFeynman diagram with a chiralfermion running in the loop (a polygon) with "n" externalgauge boson s attached to the loop where $n=1+D/2$ where $D$ is thespacetime dimension. 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 fermion s. If there is a gauge anomaly, the resulting action will not be gauge invariant. If we denote by $delta\_epsilon$ the operator corresponding to an infinitesimal gauge transformation by ε, then the Frobenius consistency condition requires that:$left\; [delta\_\{epsilon\_1\},delta\_\{epsilon\_2\}\; ight]\; mathcal\{F\}=delta\_\{left\; [epsilon\_1,epsilon\_2\; ight]\; \}mathcal\{F\}$

for any functional $mathcal\{F\}$, including the (semi)effective action S where [,] is the

Lie bracket . As $delta\_epsilon\; S$ is linear in ε, we can write:$delta\_epsilon\; S=int\_\{M^d\}\; Omega^\{(d)\}(epsilon)$

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 theexterior product of p-forms. If the spacetime M^{d}is closed (i.e. without boundary) and oriented, then it is the boundary of some d+1 dimensional oriented manifold M^{d+1}. If we then arbitrarily extend the fields (including ε) as defined on M^{d}to M^{d+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:$delta\_epsilon\; S=int\_\{M^\{d+1\; dOmega^\{(d)\}(epsilon).$

The Frobenius consistency condition now becomes

:$left\; [delta\_\{epsilon\_1\},delta\_\{epsilon\_2\}\; ight]\; S=int\_\{M^\{d+1left\; [delta\_\{epsilon\_1\}dOmega^\{(d)\}(epsilon\_2)-delta\_\{epsilon\_2\}dOmega^\{(d)\}(epsilon\_1)\; ight]\; =int\_\{M^\{d+1dOmega^\{(d)\}(left\; [epsilon\_1,epsilon\_2\; ight]\; ).$

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

:$delta\_\{epsilon\_1\}dOmega^\{(d)\}(epsilon\_2)-delta\_\{epsilon\_2\}dOmega^\{(d)\}(epsilon\_1)=dOmega^\{(d)\}(left\; [epsilon\_1,epsilon\_2\; ight]\; ).$

Because of the Frobenius consistency condition, this means that there exists a d+1-form Ω

^{d+1}(not depending upon ε) defined over M^{d+1}satisfying:$delta\_epsilon\; Omega^\{(d+1)\}=dOmega^\{(d)\}(\; epsilon\; ).$

Ω

^{d+1}is often called aChern-Simons form .Once again, if we assume Ω

^{d+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:$Omega^\{(d+2)\}=dOmega^\{(d+1)\}$

in d+2 dimensions. Ω

^{d+2}is gauge invariant::$delta\_epsilon\; Omega^\{d+2\}=ddelta\_epsilon\; Omega^\{(d+1)\}=d^2Omega^\{(d)\}(epsilon)=0$

as d and δ

_{ε}commute.**See also***

chiral gauge theory

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