Anomaly matching condition

For example, consider the quantum chromodynamics with N_f massless quarks. This theory has the flavor symmetry ${\displaystyle SU(N_{f})_{L}\times SU(N_{f})_{R}\times U(1)_{V}}$[lower-alpha 2] This flavor symmetry ${\displaystyle SU(N_{f})_{L}\times SU(N_{f})_{R}\times U(1)_{V}}$ becomes anomalous when the background gauge field is introduced. One may use either the degrees of freedom at the far low energy limit (far “IR” [lower-alpha 1]) or the degrees of freedom at the far high energy limit (far “UV”[lower-alpha 1]) in order to calculate the anomaly. In the former case one should only consider massless fermions or Nambu–Goldstone bosons which may be composite particles, while in the latter case one should only consider the elementary fermions of the underlying short-distance theory. In both cases, the answer must be the same. Indeed, in the case of QCD, the chiral symmetry breaking occurs and the Wess–Zumino–Witten term for the Nambu–Goldstone bosons reproduces the anomaly.[3]

One proves this condition by the following procedure:[1] we may add to the theory a gauge field which couples to the current related with this symmetry, as well as chiral fermions which couple only to this gauge field, and cancel the anomaly (so that the gauge symmetry will remain non-anomalous, as needed for consistency).

In the limit where the coupling constants we have added go to zero, one gets back to the original theory, plus the fermions we have added; the latter remain good degrees of freedom at every energy scale, as they are free fermions at this limit. The gauge symmetry anomaly can be computed at any energy scale, and must always be zero, so that the theory is consistent. One may now get the anomaly of the symmetry in the original theory by subtracting the free fermions we have added, and the result is independent of the energy scale.

Another way to prove the anomaly matching for continuous symmetries is to use the anomaly inflow mechanism.[4] To be specific, we consider four-dimensional spacetime in the following.

For global continuous symmetries ${\displaystyle G}$, we introduce the background gauge field ${\displaystyle A}$ and compute the effective action ${\displaystyle \Gamma [A]}$. If there is a 't Hooft anomaly for ${\displaystyle G}$, the effective action ${\displaystyle \Gamma [A]}$ is not invariant under the ${\displaystyle G}$ gauge transformation on the background gauge field ${\displaystyle A}$ and it cannot be restored by adding any four-dimensional local counter terms of ${\displaystyle A}$. Wess–Zumino consistency condition[5] shows that we can make it gauge invariant by adding the five-dimensional Chern–Simons action.

With the extra dimension, we can now define the effective action ${\displaystyle \Gamma [A]}$ by using the low-energy effective theory that only contains the massless degrees of freedom by integrating out massive fields. Since it must be again gauge invariant by adding the same five-dimensional Chern–Simons term, the 't Hooft anomaly does not change by integrating out massive degrees of freedom.