the two-point function and reflect a single vertex operator, leading to $$\begin{aligned}
&& \left\langle \Phi^{j_1\omega_1}_{J_1}(x_1|z_1) \Phi^{j_2\omega_2}_{J_2}(x_2|z_2)\right\rangle= B^{j_1\omega_1}_{J_1}\int_{\mathbb C}d^2x_1'd^2z_1' \left|\left(x_1-x'_1\right)^{2J_1}(z_1-z'_1)^{\alpha_1}\right|^2 \times \nonumber \\
&& ~~~~~ ~~~~~ ~~~~~ ~~~~~ \left\langle \Phi^{-1-j_1,\omega_1}_{-1-J_1}(x'_1|z'_1)\Phi^{j_2,\omega_2}_{J_2}(x_2|z_2) \right\rangle.\nonumber\end{aligned}$$ For $J_1\sim -1-J_2$, the two-point function in the integral can be replaced by the bulk term written in (\[regular\]), so that $$\begin{aligned}
 \left\langle \Phi^{j_1\omega_1}_{J_1}(x_1|z_1) \Phi^{j_2\omega_2}_{J_2}(x_2|z_2)\right\rangle= -\frac{|1+2J_1|^2}{\pi^2}\delta_{\omega_1\omega_2}\delta^2(J_{12}^+)B^{j_1\omega_1}_{J_1}\left[\delta\left(j_{12}\right)+Y^{-1-j_1,\omega_1}_{-1-J_1}\delta\left(j_{12}^+\right)\right] \times \nonumber \\
 \int_{\mathbb C}d^2x_1'd^2z_1' \left|\left(x_1-x'_1\right)^{2J_1}(z_1-z'_1)^{-1+\omega_1(1+2J_1)}\left(x_1'-x_2\right)^{-2-2J_1}\left(z_1'-z_2\right)^{-2\Delta_1-2\omega_1(1+2J_1)}\right|^2.\nonumber\end{aligned}$$ After integrating over $x'$ and $z'$ we obtain $$\begin{aligned}
&& \left\langle \Phi^{j_1\omega_1}_{J_1}(x_1|z_1) \Phi^{j_2\omega_2}_{J_2}(x_2|z_2)\right\rangle= \delta_{\omega_1\omega_2}\delta^2(J_{12}^+)B^{j_1\omega_1}_{J_1}\left[\delta\left(j_{12}\right)+Y^{-1-j_1,\omega_1}_{-1-J_1}\delta\left(j_{12}^+\right)\right] \times \nonumber \\
&& ~~~~~ ~~~~~ ~~~~~ \pi \frac{\gamma(2\Delta_1+\omega_1(1+2J_1))\gamma(\omega_1(1+2J_1))}{\gamma(2\Delta_1+2\omega_1(1+2J_1))}\delta\left(x_{12}\right)\left|z_{12}^{-2\Delta_1-\omega_1(1+2J_1)}\right|^2, \nonumber\end{aligned}$$ or, more explicitly, $$\begin{aligned}
\label{contact4}
 \left\langle \Phi^{j_1\omega_1}_{J_1}(x_1|z_1) \Phi^{j_2\omega_2}_{J_2}(x_2|z_2)\right\rangle=\frac{i|1+2J_1|}{\pi} \delta_{\omega_1\omega_2}\delta^2(J_{12}^+)\left[\delta\left(j_{12}^+\right)+Y^{j_1\omega_1}_{-1-J_1}\delta\left(j_{12}\right)\right] \times \nonumber \\
 \sqrt{\frac{\gamma(2\Delta_1+\omega_1(1+2J_1))^2\gamma_{\omega_1}(1+j_1+J_1)\gamma_{\omega_1}(-1-j_1-J_1)}{\gamma(2\Delta_1)\gamma(2\Delta_1+2\omega_1(1+2J_1))\gamma_{\omega_1}(j_1-J_1)\gamma_{\omega_1}(-j_1+J_1)}}\delta\left(x_{12}\right)\left|z_{12}^{-2\Delta_1-\omega_1(1+2J_1)}\right|^2.\end{aligned}$$ The full expression of the propagator for the SL$(2,\mathbb R)$-WZNW model is obtained by adding (\[contact4\]) to (\[regular\]). A well suited parametrization of the two-point function can be obtained by considering the following [*ansatz*]{}: $$\begin{aligned}
\label{2ptdef}
 &&\left\langle \Phi^{j_1\omega_1}_{J_1}(x_1|z_1) \Phi^{j_2\omega_2}_{J_2}(x_2|z_2)\right\rangle = S_{J_1}^{j_1\omega_1}S_{J_2}^{j_2\omega_2}\delta_{\omega_1\omega_2}\left[\delta\left(j_{12}^+\right)+L^{j_1\omega_1}_{J_1}\delta\left(j_{12}\right)\right] \times \nonumber \\
 && ~~~~~ ~~~~~ ~~~~~ ~~~~~ ~~~~~ ~~~~~ ~~~~~ \left[\delta^2\left(J_{12}^+\right)\delta\left(x_{12}\right)+M^{j_1\omega_1}_{J_1}\delta^2\left(J_{12}\right)\left|x_{12}^{2J_1}\right|^2 \right]\left|z_{12}^{-\Delta_1-\Delta_2}\right|^2.\end{aligned}$$ Notice that the factorization of the overall constant is a highly nontrivial proposal that, if fulfilled, would allow us to absorb this factor through a proper redefinition of the spectral flowed vertex fields. Let us show that this is indeed the case.

It can be seen from 