Infinite number of primaries in CFT I want to prove the fact that there are infinite number of primary operators in CFT by Conformal bootstrap.
However, for that I need to show that the crossed conformal blocks $g_{\Delta,\ell}(1-z,1-\bar{z})$ behave like $\log z$ in the limit $z \to 0$ and $z = \bar{z}$.
In 2D and 4D, rather in any even dimensions, using the explicit expression due to Dolan and Osborn I managed to prove this.
However I am clueless about how to do it for general dimensions.
It would be extremely helpful if someone could shed some light into this matter.
 A: You don't need to set $z=\bar{z}$ for the $\log z$ to appear, you can leave $\bar{z}$ arbitrary. Then, take the limit $\bar{z} \to 0$ and the block in any dimension becomes
$$ g_{\Delta,\ell}(z,\bar{z}) \to \bar{z}^{\frac{\Delta-\ell}2} k_{\Delta+\ell}(z) $$
where
$$ k_\beta(z) = z^{\beta/2} {}_2F_1(\beta/2,\beta/2,\beta,z) $$
is the $SL_2(\mathbb{R})$-block. You can see this from the Casimir operator,
$$ C_2 = D_z + D_{\bar{z}} + (d-2) \frac{z\bar{z}}{z-\bar{z}}[(1-z)\partial_z-(1-\bar{z})\partial_{\bar{z}})] $$
where $D$ is some differential operator. The third term vanishes like $\mathcal{O}(\bar{z})$ and hence the conformal block factorizes. Now take the limit $z \to 1$ and the $\log 1-z$ ($\log z$ in the t-channel) appears.
A: I am hereby writing an answer to my own question. Please actively comment on your opinions and let me know if my answer works.
From DSD's notes, we have the following expression for conformal blocks using the "rho-configuration" of the 4-pt functions:
$g_{\Delta_{O},\ell_{O}}(z,\bar{z}) = \sum_{n = 0,2,\cdots,j}B_{n,j}r^{\Delta + n}C_{j}^{\frac{d-2}{2}}(\cos\Theta)$
Now, restricting to $z = \bar{z} \implies \Theta = 0$.
So that means we are looking at $C_{j}^{\frac{d-2}{2}}(1)$. But lets, do it in a different way.
Lets, look at $C_{j}^{\frac{d-2}{2}}(1-x)$ where $x \to 0$ also let us look at only the $n=0$ order term. From, wikipedia: Any n-th order coefficient can be written as:
$C_{n}^{(\alpha)}(x) = \frac{(2\alpha)_{n}}{n!}{}_{2}F_{1}(-n,2\alpha + n;\alpha +1/2;\frac{1-x}{2})$
So, we get:
$ C_{0}^{(\alpha)}(1-x) \propto \log x$ when $x \to 0$.
Thus, we get a logarithmic dependence in the crossed blocks in any dimensions.
