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 by using the degrees of freedom of the theory at some energy scale, must not depend on what scale is chosen for the calculation. It is also known as 't Hooft UV-IR anomaly matching condition, where UV stand for the high-energy limit, and IR for the low-energy limit of the theory.

Terminology - 't Hooft anomaly

There are two closely related but different notions in quantum field theory that are both called anomalies: Adler-Bell-Jackiw anomaly and 't Hooft anomaly.

If we say that the symmetry of the theory has an 't Hooft anomaly, it means that the symmetry is exact as a global symmetry of the quantum theory but there is an obstruction in gauging it.^[2]

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

Anomaly matching for continuous symmetry

't Hooft's anomaly matching condition claims that an 't Hooft anomaly of continuous symmetry can be computed both in the UV and IR degrees of freedom and they give the same answer.

Example

For example, let us consider the quantum chromodynamics with Nf massless quarks. This theory has the flavor symmetry SU(Nf)_L×SU(Nf)_R×U(1)_V (the axial U(1) symmetry is broken by the chiral anomaly or instantons so we do not include it). This flavor symmetry SU(Nf)_L×SU(Nf)_R×U(1)_V becomes anomalous when the background gauge field is introduced. One may use either the degrees of freedom at the far IR (i.e. low energy limit) or the degrees of freedom at the far UV (i.e. high energy limit) 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 tern for the Nambu-Goldstone bosons reproduces the anomaly.

Proof

One proves this condition by the following procedure:^[3] 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 $G$ , we introduce the background gauge field $A$ and compute the effective action $\Gamma [A]$ . If there is an 't Hooft anomaly for $G$ , the effective action $\Gamma [A]$ is not invariant under the $G$ gauge transformation on the background gauge field $A$ and it cannot be restored by adding any four-dimensional local counter terms of $A$ . Wess-Zumino consistency condition^[5] shows that we can make it gauge invariant by adding the five-dimensional Chern-Simons action. We can now define the effective action $\Gamma [A]$ 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.

References

Citations

↑ Naturalness, Chiral Symmetry and Spontaneous Chiral Symmetry Breaking by G.'t Hooft.
↑ A. Kapustin, R. Thorngren, Anomalous Discrete Symmetries in Three Dimensions and Group Cohomology, Phys. Rev. Lett. 112, 231602, arXiv:1404.3230 [hep-th]
↑ Naturalness, Chiral Symmetry and Spontaneous Chiral Symmetry Breaking by G.'t Hooft.
↑ C.G. CALLAN, Jr. and J. A. HARVEY, ANOMALIES AND FERMION ZERO MODES ON STRINGS AND DOMAIN WALLS, Nuclear Physics B250 (1985) 427-436
↑ J. Wess, B. Zumino, Consequences of anomalous ward identities, Phys. Lett. B 37 (1971) 95

't Hooft, G. (1980). "Naturalness, Chiral Symmetry and Spontaneous Chiral Symmetry Breaking". In 't Hooft, G. Recent Developments in Gauge Theories. Plenum Press. ISBN 978-0-306-40479-5.
Y. Frishman; A. Scwimmer; T. Banks; S. Yankielowicz (1981). "The Axial Anomaly and the Bound State Spectrum in Confining Theories". Nucl. Phys. B. 177: 157–171. Bibcode:1981NuPhB.177..157F. doi:10.1016/0550-3213(81)90268-6.

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