en>Wcherowi: revised page to properly reflect the definition

2014-08-06T05:38:11Z

revised page to properly reflect the definition

en>ChrisGualtieri: TypoScan Project / General Fixes, typos fixed: , → , using AWB

2012-05-01T03:43:36Z

TypoScan Project / General Fixes, typos fixed: , → , using AWB

New page

{{unreferenced|date=December 2010}}

In [[theoretical physics]], a '''gauge anomaly''' is an example of an [[anomaly (physics)|anomaly]]: it is an effect of [[quantum mechanics]]—usually a [[one-loop diagram]]—that invalidates the [[gauge symmetry]] of 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 [[photon]] polarized 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 (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 (a polygon) 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. Anomalies occur only in even spacetime dimensions. For example, the anomalies in the usual 4 spacetime dimensions arise from triangle Feynman diagrams.

[[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(d) 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(d)(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 Md is [[closed manifold|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

:<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(d+1) (not depending upon ε) defined over Md+1 satisfying

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

Ω(d+1) is often called a [[Chern-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

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

in d+2 dimensions. Ω(d+2) 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 δε commute.

==See also==
*[[chiral gauge theory]]

{{DEFAULTSORT:Gauge Anomaly}}
[[Category:Anomalies in physics]]

Amenable number - Revision history

en>Wcherowi: revised page to properly reflect the definition

en>ChrisGualtieri: TypoScan Project / General Fixes, typos fixed: , → , using AWB