In a 2D CFT, is the free boson $X$ a primary field? A primary field $\mathcal{O}(w,\bar{w})$ with weight $(h,\bar{h})$ is defined by having the following OPEs with the stress tensor:
$$T(z)\mathcal{O}(w,\bar{w})=\frac{h\mathcal{O}}{(z-w)^2}+\frac{\partial \mathcal{O}}{z-w},\qquad \bar{T}(\bar{z})\mathcal{O}(w,\bar{w})=\frac{\bar{h}\mathcal{O}}{(\bar{z}-\bar{w})^2}+\frac{\bar{\partial} \mathcal{O}}{\bar{z}-\bar{w}}.$$
Let us consider a theory of a free boson defined by:
$$S = \int d^2 z\ \partial X\bar{\partial}X.$$
Then $\partial X$ is a $(1,0)$ primary field with and $\bar{\partial}X$ is a $(0,1)$ primary field (see e.g. Tong's string theory notes, section 4.3.3).
The field $X$ itself has the OPE
$$T(z)X(w,\bar{w})=\frac{\partial X}{z-w},\qquad\bar{T}(\bar{z})X(w,\bar{w})=\frac{\bar{\partial} X}{\bar{z}-\bar{w}}.$$
(See e.g. Polchiski eq. 2.4.6). So, by the definition above, it should be a primary field with weight $h=\bar{h}=0$.
However, I've looked through many textbooks and lecture notes (e.g. Di Francesco, Blumenhagen, Polchinski, Tong...) and it is never explicitly said that $X$ is primary. So... Is it primary or not?

*

*If it is primary, then how come its descendants $\partial X$ and $\bar{\partial} X$ are also primary?


*If it is not primary, then how come its OPE is exactly the correct OPE for a primary field of weight $(0,0)$?
 A: I do not agree that $X$ is a primary field. Primary field is defined by its transformation properties under the conformal group (see e.g. yellow book). In particular, under scaling transformation, a correlation function involving primary operators, transforms as
$$
\langle \mathcal{O}_1(\lambda x_1)\ldots\mathcal{O}_n(\lambda x_n)\rangle=\lambda^{-\Delta_1-\ldots-\Delta_n}\langle \mathcal{O}_1(x_1)\ldots\mathcal{O}_n(x_n)\rangle,
$$
where $\Delta_i=h_i+\bar{h}_i$. It follows that for a primary field of dimension $(0,0)$ its two-point function must be constant. This is not true for $X$.
In fact, $X$ is not a local field in the sense of operator-state correspondence. The set of Virasoro primary operators in free boson CFT is $1,\partial X,\bar\partial X$ and the continuos family $e^{i\alpha X}$ for $\alpha\neq 0$.
A: To see why the descendants are primary, you can use 
$$
\partial\left(T(z)X(w,\overline{w})\right) = T(z)\partial X(w,\overline{w}) = \frac{\partial^2 X}{z-w} + \frac{\partial X}{(z-w)^2}
$$
And see that it is a primary field of weight $h = 1$, $\overline{h} = 0$, and similarly for the other field....
