Gauge anomaly

In

All gauge anomalies must cancel out. Anomalies in gauge symmetries[2] lead to an inconsistency, since a gauge symmetry is required in order to cancel degrees of freedom with a negative norm which are unphysical (such as a photon polarized in the time direction). Indeed, cancellation occurs in the Standard Model.

The term gauge anomaly is usually used for vector gauge anomalies. Another type of gauge anomaly is the

.

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

attached to the loop where ${\displaystyle n=1+D/2}$ where ${\displaystyle D}$ is the spacetime dimension.

Let us look at the (semi)effective action we get after integrating over the

. If there is a gauge anomaly, the resulting action will not be gauge invariant. If we denote by ${\displaystyle \delta _{\epsilon }}$ the operator corresponding to an infinitesimal gauge transformation by ε, then the Frobenius consistency condition requires that

${\displaystyle \left[\delta _{\epsilon _{1}},\delta _{\epsilon _{2}}\right]{\mathcal {F}}=\delta _{\left[\epsilon _{1},\epsilon _{2}\right]}{\mathcal {F}}}$

for any functional ${\displaystyle {\mathcal {F}}}$, including the (semi)effective action S where [,] is the Lie bracket. As ${\displaystyle \delta _{\epsilon }S}$ is linear in ε, we can write

${\displaystyle \delta _{\epsilon }S=\int _{M^{d}}\Omega ^{(d)}(\epsilon )}$

where Ω^(d) is

(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

${\displaystyle \delta _{\epsilon }S=\int _{M^{d+1}}d\Omega ^{(d)}(\epsilon ).}$

The Frobenius consistency condition now becomes

${\displaystyle \left[\delta _{\epsilon _{1}},\delta _{\epsilon _{2}}\right]S=\int _{M^{d+1}}\left[\delta _{\epsilon _{1}}d\Omega ^{(d)}(\epsilon _{2})-\delta _{\epsilon _{2}}d\Omega ^{(d)}(\epsilon _{1})\right]=\int _{M^{d+1}}d\Omega ^{(d)}(\left[\epsilon _{1},\epsilon _{2}\right]).}$

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

${\displaystyle \delta _{\epsilon _{1}}d\Omega ^{(d)}(\epsilon _{2})-\delta _{\epsilon _{2}}d\Omega ^{(d)}(\epsilon _{1})=d\Omega ^{(d)}(\left[\epsilon _{1},\epsilon _{2}\right]).}$

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

${\displaystyle \delta _{\epsilon }\Omega ^{(d+1)}=d\Omega ^{(d)}(\epsilon ).}$

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

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

in d+2 dimensions. Ω^(d+2) is gauge invariant:

${\displaystyle \delta _{\epsilon }\Omega ^{(d+2)}=d\delta _{\epsilon }\Omega ^{(d+1)}=d^{2}\Omega ^{(d)}(\epsilon )=0}$

as d and δ_ε commute.

See also

• Chiral gauge theory
• Anomaly matching condition
• Green–Schwarz mechanism
• Mixed anomaly

References