did not include the integration of the Liouville (or Weyl) mode $\sigma$. We will turn this point in the next section. Our starting point  and  are the same as the one in \[\].

We also note the above integration measure is invariant under the conformal transformation as noted in \[\]: $$\delta\phi=V\partial\phi+h\left(\partial V\right)\phi,\quad
  \delta\overline\phi=V\partial\overline\phi,
\ee$$ where $h$ is the conformal weight of the generic field $\phi$.

In the path integral formulation \[\], the anomaly is generally ascribed to a non-invariance of the integration measure and the Jacobian factor associated with the anomalous transformation gives rise to the anomaly. The Jacobian factor of general conformal fields in a conformally flat background, is analyzed in \[\]. According to \[\], under an infinitesimal change of the integration variable, $\tilde\phi\rightarrow\tilde\phi+\varepsilon(x)\tilde\phi$, a logarithm of the Jacobian factor $J$ is given by $$\ln J=\pm{1\over2\pi}\int d^2x\varepsilon(x)
  \left[\left(a-b\over3\right)\partial\partialbar\sigma
  +M^2e^{-(2a+b)\sigma}\right],
\eqn\twoeight$$ and, for $\tilde\phi\rightarrow\tilde\phi+\varepsilon(x)\partial\tilde\phi$, $$\eqalign{
  &\ln J
  =\pm{1\over24\pi}\int d^2x\varepsilon(x)
  \biggl[\left(b^2-4ab\right)\partial\sigma\partialbar\partial\sigma
  +\left(2a-3b\right)\partialbar\partial^2\sigma
\cr
  &\qquad\qquad\qquad\qquad\qquad\qquad\qquad\qquad
  -12M^2\left(a+b\right)\partial\sigma e^{-(2a+b)\sigma}\biggr].
\cr
}
\eqn\twonine$$ In  and , the double sign ($\pm$) corresponds to the statistics of the field $\phi$. The trace operation that is necessary to evaluate the above Jacobian factors is regularized by using an exponential type damping factor $e^{-H/M^2}$ with the regulator $H$, which 