Regularization schemes and the multiplicative anomaly

Elizalde, Vanzo, and Zerbini have shown that the effective action of two free Euclidean scalar fields in flat space contains a `multiplicative anomalyʹ when ζ-function regularization is used. This is related to the Wodzicki residue. I show that there is no anomaly when using a wide range of other regularization schemes and that the anomaly can be removed by an unusual choice of renormalization scales. I define new types of anomalies and show that they have similar properties. Thus multiplicative anomalies encode no novel physics. They merely illustrate some dangerous aspects of ζ-function and Schwinger proper time regularization schemes.