Anomaly matching condition

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In quantum field theory, the anomaly matching condition by Gerard 't Hooft states that the calculation of any chiral anomaly 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.

For example, one may calculate the anomaly of a global U(1) current by considering which fermionic degrees of freedom the theory has. 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 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.

One proves this condition by the following procedure: 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.