Consider free scalar field in two dimensions with the standard action written in complex coordinates $S=\int d^2z\, \partial \phi\bar{\partial}\phi$. The two-point correlation function is known to be $$\left\langle\phi(z,\bar{z})\phi(w,\bar{w})\right\rangle = \alpha\ln|z-w|^2$$ here $\alpha$ is some unimportant constant.
On the other hand, one expects from the conformal symmetry that the correlator must be invariant under any holomorphic variable charge. Apparentely, it is not.
I am aware that the conformal symmetry is in some sense anomalous -- the symmetry algebra of the QFT is not the Witt algebra, but its central extension, the Virasoro algebra.
The question is whether this anomaly is seen right at the simplest example of the two-point correlation function for free scalar field? Or am I confused about something?
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