Anomaly matching condition

In quantum field theory, the anomaly matching condition[1] by Gerard 't Hooft states that the calculation of any chiral anomaly for the flavor symmetry must not depend on what scale is chosen for the calculation if it is done by using the degrees of freedom of the theory at some energy scale. It is also known as the 't Hooft condition and the 't Hooft UV-IR anomaly matching condition.[lower-alpha 1]

't Hooft anomalies

There are two closely related but different types of obstructions to formulating a quantum field theory that are both called anomalies: chiral, or Adler-Bell-Jackiw anomalies, and 't Hooft anomalies.

If we say that the symmetry of the theory has a 't Hooft anomaly, it means that the symmetry is exact as a global symmetry of the quantum theory, but there is some impediment to using it as a gauge in the theory.[2]

As an example of a 't Hooft anomaly, we consider quantum chromodynamics with ${\displaystyle N_{f}}$ massless fermions: This is the ${\displaystyle SU(N_{c})}$ gauge theory with ${\displaystyle N_{f}}$ massless Dirac fermions. This theory has the global symmetry ${\displaystyle SU(N_{f})_{L}\times SU(N_{f})_{R}\times U(1)_{V}}$, which is often called the flavor symmetry, and this has a 't Hooft anomaly.

Anomaly matching for continuous symmetry

The anomaly matching condition by G. 't Hooft proposes that a 't Hooft anomaly of continuous symmetry can be computed both in the high-energy and low-energy degrees of freedom (“UV” and “IR”[lower-alpha 1]) and give the same answer.

Example

For example, consider the quantum chromodynamics with N_f massless quarks. This theory has the flavor symmetry SU(N_f)_L×SU(N_f)_R×U(1)_V[lower-alpha 2] This flavor symmetry SU(N_f)_L×SU(N_f)_R×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]

Proof

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.

Alternative proof

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.

Notes

1. In the context of quantum field theory, “UV” actually means the high-energy limit of a theory, and “IR” means the low-energy limit, by analogy to the upper and lower peripheries of visible light, but not actually meaning either light or those particular energies.
2. The axial U(1) symmetry is broken by the chiral anomaly or instantons so is not included in the example.

References

1. 't Hooft, G. (1980). "Naturalness, Chiral Symmetry, and Spontaneous Chiral Symmetry Breaking". In 't Hooft, G. (ed.). Recent Developments in Gauge Theories. Plenum Press. ISBN 978-0-306-40479-5.
2. Kapustin, A.; Thorngren, R. (2014). "Anomalous discrete symmetries in three dimensions and group cohomology". Physical Review Letters. 112 (23): 231602. arXiv:1403.0617. Bibcode:2014PhRvL.112w1602K. doi:10.1103/PhysRevLett.112.231602.
3. Frishman, Y.; Scwimmer, A.; Banks, T.; Yankielowicz, S. (1981). "The axial anomaly and the bound state spectrum in confining theories". Nuclear Physics B. 177 (1): 157–171. Bibcode:1981NuPhB.177..157F. doi:10.1016/0550-3213(81)90268-6.
4. Callan, Jr., C.G.; Harvey, J.A. (1985). "Anomalies and fermion zero modes on strings and domain walls". Nuclear Physics B. 250 (1–4): 427–436. Bibcode:1985NuPhB.250..427C. doi:10.1016/0550-3213(85)90489-4.
5. Wess, J.; Zumino, B. (1971). "Consequences of anomalous ward identities". Physics Letters B. 37 (1): 95. Bibcode:1971PhLB...37...95W. doi:10.1016/0370-2693(71)90582-X.