2D Ghost CFT and two-point functions 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?
 A: 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.
A: The answer seems to be that, technically at least, the two point function $\langle b(z) c(w) \rangle$ does vanish on the sphere.  In the context of the standard $bc$ ghost system that shows up in bosonic string theory, the simplest nonzero correlation function on the sphere that involves both $b$ and $c$ is 
$$
\langle c(z_1) c(z_2) c(z_3) c(z_4) b(w) \rangle.
$$
This correlation function is a special case of (6.3.5) in Polchinski volume 1 and evaluates to 
$$
\frac{(z_1-z_2)(z_1-z_3)(z_2-z_3)(z_1-z_4)(z_2-z_4)(z_3-z_4)}{(z_1-w)(z_2-w)(z_3-w)(z_4-w)}.
$$ 
The Kronecker delta function $\delta_{h_i h_j}$ that shows up in the result for $\langle \phi_i(z) \phi_j(w) \rangle$ described in OP's first formula can be argued for based on the transformations of the fields under inversion $z \to 1/z$.  While $\langle b(z) c(w) \rangle$ cannot transform properly, as OP has noted, it would be reassuring to see that this five-point function does transform in the right way.  Indeed, the conformal factor
$$
\frac{z_1^2 z_2^2 z_3^2 z_4^2}{w^4}
$$
takes precisely the right form to convert this correlation function into one involving only $1/z_i$ and $1/w$.
(Note I am using the $SL(2, R)$ invariant vacuum state to compute these correlation functions.  There is also the option of using so-called Q-vacua, some of which are obtained by acting with $c$ operators at the poles of the sphere.  In this case, translation invariance is lost, and the correlation function $\langle \phi_i(z) \phi_j(w) \rangle$ can depend on a further dimensionless ratio $z/w$.)
