# Higher point function in CFT The 2-point function in CFT can be fixed by a choice of basis. The 3-point function is fixed by [[0028 Conformal symmetry|conformal symmetry]] up to a constant. The 4-point function is more interesting and is a non-trivial function of two cross-ratios. At higher points, there are more cross-ratios, which makes the life harder. ## Refs - [[2020#Fortin, Ma, Skiba]] discusses different channels of higher point functions in CFT ## 4-point - $\left\langle\mathcal{O}_{\Delta_{1}}\left(\boldsymbol{x}_{1}\right) \mathcal{O}_{\Delta_{2}}\left(\boldsymbol{x}_{2}\right) \mathcal{O}_{\Delta_{3}}\left(\boldsymbol{x}_{3}\right) \mathcal{O}_{\Delta_{4}}\left(\boldsymbol{x}_{4}\right)\right\rangle=f(u, v) \prod_{1 \leq i<j \leq 4} x_{i j}^{2 \delta_{i j}}$ - $2 \delta_{i j}=\frac{\Delta_{t}}{3}-\Delta_{i}-\Delta_{j}, \quad \Delta_{t}=\sum_{i=1}^{4} \Delta_{i}$ - known since [[Polyakov1970]] - conformal cross ratios: $u=\frac{x_{13}^{2} x_{24}^{2}}{x_{14}^{2} x_{23}^{2}}, \quad v=\frac{x_{12}^{2} x_{34}^{2}}{x_{13}^{2} x_{24}^{2}}$ ## 5-point - $L_{5 \mid \mathrm{comb}}^{\left(\Delta_{i_{3}}, \Delta_{i_{4}}, \Delta_{i_{2}}, \Delta_{i_{5}}, \Delta_{i_{1}}\right)}=\left(\frac{\eta_{14}}{\eta_{12} \eta_{24}}\right)^{\frac{\Delta_{i_{2}}}{2}}\left(\frac{\eta_{24}}{\eta_{23} \eta_{34}}\right)^{\frac{\Delta_{i_{3}}}{2}}\left(\frac{\eta_{23}}{\eta_{24} \eta_{34}}\right)^{\frac{\Delta_{i_{4}}}{2}}\left(\frac{\eta_{14}}{\eta_{15} \eta_{45}}\right)^{\frac{\Delta_{i_{5}}}{2}}\left(\frac{\eta_{45}}{\eta_{14} \eta_{15}}\right)^{\frac{\Delta_{i_{1}}}{2}}$ - $u_{1}^{5}=\frac{\eta_{12} \eta_{34}}{\eta_{14} \eta_{23}}, \quad u_{2}^{5}=\frac{\eta_{15} \eta_{24}}{\eta_{12} \eta_{45}}, \quad v_{11}^{5}=\frac{\eta_{13} \eta_{24}}{\eta_{14} \eta_{23}}, \quad v_{12}^{5}=\frac{\eta_{14} \eta_{25}}{\eta_{12} \eta_{45}}, \quad v_{22}^{5}=\frac{\eta_{24} \eta_{35}}{\eta_{23} \eta_{45}}$ ## $n$-point - there are $n(n-3)/2$ such cross ratios - see e.g. [[NikolovTodorov2000]][](https://arxiv.org/abs/hep-th/0009004) - [[Rosenhaus2018]] gives a general form of a *block* in the n-point function, but the leg factors have the same form as a n-point correlation function