Regularization of the path-integral measure for anomalies

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As was shown for the first time by Fujikawa, the anomaly is fundamentally a variation of the functional integral measure under transformation. Fujikawa's original prescription of 1979 for the variation of the integral measure looks to be at first sight an artifact. In this paper we will show that it is not and that it is fully equivalent to the authentic field-theoretical treatment for a two-point function. To do this we first examine various ways of solving the factor A(x) in Fujikawa's expression for the functional integral measure. We define the anomaly as A(x)-${A}_{f}$(x), where ${A}_{f}$(x) is the Fujikawa factor for the free field. We propose a regulator which leads to a finite result for any anomaly. We show that A(x) can be defined in terms of the proper time through a splitting procedure. The original Fujikawa prescription for A(x) is shown to be closely related to the proper-time description of the anomaly, initiated by Schwinger. Its equivalence to the authentic field-theoretical treatment is proven as a consequence of these investigations. The \ensuremath{\zeta}-functional regularization for A(x) is also examined. We examine the way to deduce the anomaly from the effective potential by adopting the ${\ensuremath{\varphi}}^{4}$ model as an example. Comparison of the path-integral prescription with this procedure enables one to clarify the nature of divergence appearing in the original Fujikawa form of A(x). The renormalization-group equation for the effective potential is solved exactly to obtain the precise form of the \ensuremath{\beta} function in terms of which we reexpress the result obtained earlier for A(x). Finally we discuss the physical significance of the renormalization-group equation for the case of broken symmetry.