4-point function in CFT

Question

Im currently trying to understand the form of the 4 point function in CFT, i.e. how to derive equation 4.62 in Di Francesco et al. In particular, the coefficients of the $x_{ij}=|x_i-x_j|$. For four scalars $O_i(x_i)$ with dimensions $\Delta_i$, I know the general form should be:

$$<O_1(x_1)O_2(x_2)O_3(x_3)O_4(x_4)>=\frac{f(u,v)}{x_{12}^a x_{13}^b x_{14}^c x_{23}^d x_{24}^e x_{34}^f}$$

with $u$ and $v$ the crossing ratios, based on rotation and translation invariance. Scale invariance gives $$a+b+c+d+e+f=\Delta=\Sigma_{i=1}^4 \Delta_i,$$ then special conformal invariance gives me 4 more equations similar to those in 4.59 of Di Francesco et al for the 3 point function, leaving me with 5 equations on 6 unknowns that i can't solve uniquely. Am i missing something?

Answer 1

The problem is that there is too much freedom in your parametrization of the 4-point function:

If we take the cross ratios to be (different conventions give rise to similar arguments) $$ u = \frac{x_{12} x_{34}}{x_{14} x_{23}}, \qquad v = \frac{x_{13} x_{24}}{x_{14} x_{23}}, $$ then you can rewrite $$\frac{f(u,v)}{x_{12}^a x_{13}^b x_{14}^c x_{23}^d x_{24}^e x_{34}^f} = \frac{f'(u,v)}{x_{12}^{a-f} x_{13}^{b-e} x_{14}^{c+e+f} x_{23}^{d+e+f}}$$ where $$ f'(u,v) = u^{-f} v^{-e} f(u,v) $$ Now you can apply the logic that you were mentioning in the question: special conformal invariance gives you 3 equations that fix completely the coefficients $(a-f)$, $(b-e)$, $(c+e+f)$ and $(d+e+f)$, and you find that the 4-point function is fixed up to an unknown function of the two cross-ratios $u$ and $v$.