For some reason I am suddenly confused over something which should be quit elementary.
In two-dimensional CFT's the two-point functions of quasi-primary fields are fixed by global $SL(2,\mathbb C)/\mathbb Z_2$ invariance to have the form
$$\langle \phi_i(z)\phi_j(w)\rangle = \frac{d_{ij}}{(z-w)^{2h_i}}\delta_{h_i,h_j}.$$
So a necessary requirement for a non-vanishing two-point function is $h_i = h_j$. Now consider the Ghost System which contains the two primary fields $b(z)$ and $c(z)$ with the OPE's
$$T(z)b(w)\sim \frac{\lambda}{(z-w)^2}b(w) + \frac 1{z-w}\partial b(w),$$
$$T(z)c(w)\sim \frac{1-\lambda}{(z-w)^2}c(w) + \frac 1{z-w}\partial c(w).$$
These primary fields clearly don't have the same conformal weight for generic $\lambda$, $h_b\neq h_c$. However their two-point function is
$$\langle c(z)b(w)\rangle = \frac 1{z-w}.$$
Why isn't this forced to be zero? Am I missing something very trivial, or are there any subtleties here?
Best Answer
1) Everything OP writes(v1) above his last equation is correct. The $bc$ OPE reads
$$ {\cal R}c(z)b(w) ~\sim~ \frac 1{z-w} ,$$
where ${\cal R}$ denotes radial ordering.
2) To calculate the two-point function
$$\langle c(z)b(w)\rangle $$
(which as OP writes must vanish if the conformal dimensions for $b$ and $c$ are different) is more subtle due to the presence of the ghost number anomaly, i.e. the vacuum should be prepared with certain modes of the $bc$ system, see e.g. Polchinski, String Theory, Vol. 1, Sections 2.5-2.7.