and use

$$ \langle [L_n, O_i O_j O_k] \rangle = 0, \quad \langle [\bar{L}_n, O_i O_j O_k] \rangle = 0, \qquad (D.18) $$

namely the vacuum is invariant under global conformal transformation. We will denote

$$ \mathbb{P}_3 = \frac{1}{z_{12}^{h_1+h_2-h_3} \bar{z}_{12}^{\bar{h}_1+\bar{h}_2-\bar{h}_3} z_{13}^{h_1+h_3-h_2} \bar{z}_{13}^{\bar{h}_1+\bar{h}_3-\bar{h}_2} z_{23}^{h_2+h_3-h_1} \bar{z}_{23}^{\bar{h}_2+\bar{h}_3-\bar{h}_1}}, \qquad (D.19) $$

$$ \tau_1 = \ln \frac{z_{23} \bar{z}_{23}}{z_{12} \bar{z}_{12} z_{13} \bar{z}_{13}}, \quad \tau_2 = \ln \frac{z_{13} \bar{z}_{13}}{z_{12} \bar{z}_{12} z_{23} \bar{z}_{23}}, \quad \tau_3 = \ln \frac{z_{12} \bar{z}_{12}}{z_{13} \bar{z}_{13} z_{23} \bar{z}_{23}}. \qquad (D.20) $$

For three-point functions that involve two non-logarithmic operators and a rank-2 Jordan block $(t, T)$ we have

$$ \langle \sigma(z_1, \bar{z}_1) \sigma(z_2, \bar{z}_2) T(z_3) \rangle = C_{\sigma\sigma T} \mathbb{P}_3, \qquad (D.21a) $$

$$ \langle \sigma(z_1, \bar{z}_1) \sigma(z_2, \bar{z}_2) t(z_3, \bar{z}_3) \rangle = (C_{\sigma\sigma t} + C_{\sigma\sigma T} \tau_3) \mathbb{P}_3, \qquad (D.21b) $$

and with a rank-3 Jordan block $(\Psi_2, \Psi_1, \Psi_0)$, we have

$$ \langle \sigma(z_1, \bar{z}_1) \sigma(z_2, \bar{z}_2) \Psi_0(z_3, \bar{z}_3) \rangle = C_{\sigma\sigma\Psi_0} \mathbb{P}_3, \qquad (D.22a) $$

$$ \langle \sigma(z_1, \bar{z}_1) \sigma(z_2, \bar{z}_2) \Psi_1(z_3, \bar{z}_3) \rangle = (C_{\sigma\sigma\Psi_1} + C_{\sigma\sigma\Psi_0} \tau_3) \mathbb{P}_3, \qquad (D.22b) $$

$$ \langle \sigma(z_1, \bar{z}_1) \sigma(z_2, \bar{z}_2) \Psi_2(z_3, \bar{z}_3) \rangle = (C_{\sigma\sigma\Psi_2} + C_{\sigma\sigma\Psi_1} \tau_3 + \frac{1}{2} C_{\sigma\sigma\Psi_0} \tau_3^2) \mathbb{P}_3. \qquad (D.22c) $$

Further consider the three-point functions involving two operators in a rank-2 Jordan block $(\tilde{\epsilon}, \epsilon)$ and a rank-2 Jordan block $(t, T)$, we have the following three-point functions:

$$ \langle \epsilon(z_1, \bar{z}_1) \epsilon(z_2, \bar{z}_2) T(z_3) \rangle = C_{\epsilon\epsilon T} \mathbb{P}_3, \qquad (D.23a) $$

$$ \langle \epsilon(z_1, \bar{z}_1) \epsilon(z_2, \bar{z}_2) t(z_3, \bar{z}_3) \rangle = (C_{\epsilon\epsilon t} + C_{\epsilon\epsilon T} \tau_3) \mathbb{P}_3, \qquad (D.23b) $$

and

$$ \langle \tilde{\epsilon}(z_1, \bar{z}_1) \epsilon(z_2, \bar{z}_2) T(z_3) \rangle = (C_{\tilde{\epsilon}\epsilon T} + C_{\epsilon\epsilon T} \tau_1) \mathbb{P}_3, \qquad (D.24a) $$

$$ \langle \tilde{\epsilon}(z_1, \bar{z}_1) \epsilon(z_2, \bar{z}_2) t(z_3, \bar{z}_3) \rangle = (C_{\tilde{\epsilon}\epsilon t} + C_{\tilde{\epsilon}\epsilon T} \tau_3 + C_{\epsilon\epsilon t} \tau_1 + C_{\epsilon\epsilon T} \tau_1 \tau_3) \mathbb{P}_3, \qquad (D.24b) $$

and

$$ \langle \tilde{\epsilon}(z_1, \bar{z}_1) \tilde{\epsilon}(z_2, \bar{z}_2) T(z_3) \rangle = (C_{\tilde{\epsilon}\tilde{\epsilon}T} + C_{\epsilon\epsilon\tilde{\epsilon}T} \tau_1 + C_{\tilde{\epsilon}\tilde{\epsilon}T} \tau_2 + C_{\epsilon\epsilon T} \tau_1 \tau_2) \mathbb{P}_3, \qquad (D.25a) $$

$$ \langle \tilde{\epsilon}(z_1, \bar{z}_1) \tilde{\epsilon}(z_2, \bar{z}_2) t(z_3, \bar{z}_3) \rangle = (C_{\tilde{\epsilon}\tilde{\epsilon}t} + C_{\epsilon\epsilon\tilde{\epsilon}t} \tau_1 + C_{\tilde{\epsilon}\tilde{\epsilon}T} \tau_3 + C_{\epsilon\epsilon t} \tau_2 + C_{\epsilon\epsilon\tilde{\epsilon}T} \tau_1 \tau_3 + C_{\tilde{\epsilon}\tilde{\epsilon}T} \tau_1 \tau_2 + C_{\epsilon\epsilon T} \tau_1 \tau_2 \tau_3) \mathbb{P}_3. \qquad (D.25b) $$

Note that Bose symmetry requires for example $C_{\epsilon\epsilon\tilde{\epsilon}T} = C_{\tilde{\epsilon}\tilde{\epsilon}T}$, etc.

Finally consider the rank-3 Jordan block $(\Psi_2, \Psi_1, \Psi_0)$, solving the conformal Ward identities gives:

$$ \langle \epsilon(z_1, \bar{z}_1) \epsilon(z_2, \bar{z}_2) \Psi_0(z_3, \bar{z}_3) \rangle = C_{\epsilon\epsilon\Psi_0} \mathbb{P}_3, \qquad (D.26a) $$

$$ \langle \epsilon(z_1, \bar{z}_1) \epsilon(z_2, \bar{z}_2) \Psi_1(z_3, \bar{z}_3) \rangle = (C_{\epsilon\epsilon\Psi_1} + C_{\epsilon\epsilon\Psi_0} \tau_3) \mathbb{P}_3, \qquad (D.26b) $$

$$ \langle \epsilon(z_1, \bar{z}_1) \epsilon(z_2, \bar{z}_2) \Psi_2(z_3, \bar{z}_3) \rangle = (C_{\epsilon\epsilon\Psi_2} + C_{\epsilon\epsilon\Psi_1} \tau_3 + \frac{1}{2} C_{\epsilon\epsilon\Psi_0} \tau_3^2) \mathbb{P}_3, \qquad (D.26c) $$
and

$$ \langle \tilde{\epsilon}(z_1, \tilde{z}_1) \epsilon(z_2, \tilde{z}_2) \Psi_0(z_3, \tilde{z}_3) \rangle = (C_{\tilde{\epsilon}\tilde{\epsilon}}\Psi_0 + C_{\epsilon\epsilon}\Psi_0\tau_1)\mathbb{P}_3, \quad (D.27a) $$

$$ \langle \tilde{\epsilon}(z_1, \tilde{z}_1) \epsilon(z_2, \tilde{z}_2) \Psi_1(z_3, \tilde{z}_3) \rangle = (C_{\tilde{\epsilon}\tilde{\epsilon}}\Psi_1 + C_{\epsilon\epsilon}\Psi_0\tau_3 + C_{\epsilon\epsilon}\Psi_1\tau_1 + C_{\epsilon\epsilon}\Psi_0\tau_1\tau_3)\mathbb{P}_3, \quad (D.27b) $$

$$ \langle \tilde{\epsilon}(z_1, \tilde{z}_1) \epsilon(z_2, \tilde{z}_2) \Psi_2(z_3, \tilde{z}_3) \rangle = (C_{\tilde{\epsilon}\tilde{\epsilon}}\Psi_2 + C_{\epsilon\epsilon}\Psi_1\tau_3 + C_{\epsilon\epsilon}\Psi_2\tau_1 + \frac{1}{2}C_{\tilde{\epsilon}\tilde{\epsilon}}\Psi_0\tau_1^2\tau_3 + C_{\epsilon\epsilon}\Psi_1\tau_1\tau_3 + \frac{1}{2}C_{\epsilon\epsilon}\Psi_0\tau_1\tau_3^2)\mathbb{P}_3, \quad (D.27c) $$

and

$$ \langle \tilde{\epsilon}(z_1, \tilde{z}_1) \tilde{\epsilon}(z_2, \tilde{z}_2) \Psi_0(z_3, \tilde{z}_3) \rangle = (C_{\tilde{\epsilon}\tilde{\epsilon}}\Psi_0 + C_{\epsilon\epsilon}\Psi_0\tau_1 + C_{\tilde{\epsilon}\tilde{\epsilon}}\Psi_0\tau_2 + C_{\epsilon\epsilon}\Psi_0\tau_1\tau_2)\mathbb{P}_3, \quad (D.28a) $$

$$ \langle \tilde{\epsilon}(z_1, \tilde{z}_1) \tilde{\epsilon}(z_2, \tilde{z}_2) \Psi_1(z_3, \tilde{z}_3) \rangle = (C_{\tilde{\epsilon}\tilde{\epsilon}}\Psi_1 + C_{\tilde{\epsilon}\tilde{\epsilon}}\Psi_0\tau_3 + C_{\epsilon\epsilon}\Psi_1\tau_1 + C_{\tilde{\epsilon}\tilde{\epsilon}}\Psi_1\tau_2 + C_{\epsilon\epsilon}\Psi_0\tau_1\tau_3 + C_{\epsilon\epsilon}\Psi_1\tau_1\tau_2 + C_{\epsilon\epsilon}\Psi_0\tau_1\tau_2\tau_3)\mathbb{P}_3, \quad (D.28b) $$

$$ \langle \tilde{\epsilon}(z_1, \tilde{z}_1) \tilde{\epsilon}(z_2, \tilde{z}_2) \Psi_2(z_3, \tilde{z}_3) \rangle = (C_{\tilde{\epsilon}\tilde{\epsilon}}\Psi_2 + C_{\epsilon\epsilon}\Psi_2\tau_1 + C_{\tilde{\epsilon}\tilde{\epsilon}}\Psi_2\tau_2 + C_{\tilde{\epsilon}\tilde{\epsilon}}\Psi_1\tau_3 + C_{\epsilon\epsilon}\Psi_2\tau_1\tau_2 + C_{\epsilon\epsilon}\Psi_0\tau_1\tau_2\tau_3 + \frac{1}{2}C_{\tilde{\epsilon}\tilde{\epsilon}}\Psi_0\tau_1^2\tau_2\tau_3 + \frac{1}{2}C_{\epsilon\epsilon}\Psi_0\tau_1\tau_2\tau_3^2 + \frac{1}{2}C_{\epsilon\epsilon}\Psi_0\tau_1\tau_2\tau_3^2)\mathbb{P}_3. \quad (D.28c) $$

Recall the possible change of basis (D.5) and (D.6) for the Jordan blocks. For the three-point functions (D.21) and (D.22), under (D.5) and (D.6) for the rank-2 and rank-3 Jordan blocks, the three-point constants change as

$$ C_{\sigma\sigma T} \rightarrow C_{\sigma\sigma T}, \quad C_{\sigma\sigma t} \rightarrow C_{\sigma\sigma t} + r_0 C_{\sigma\sigma T}, \quad (D.29) $$

and

$$ C_{\sigma\sigma\Psi_0} \rightarrow C_{\sigma\sigma\Psi_0}, \quad C_{\sigma\sigma\Psi_1} \rightarrow C_{\sigma\sigma\Psi_1} + r_1 C_{\sigma\sigma\Psi_0}, \quad C_{\sigma\sigma\Psi_2} \rightarrow C_{\sigma\sigma\Psi_2} + r_1 C_{\sigma\sigma\Psi_1} + r_0 C_{\sigma\sigma\Psi_0}. \quad (D.30) $$

Therefore, the three-point constants $C_{\sigma\sigma T}$ and $C_{\sigma\sigma\Psi_0}$ are “intrinsic” similar to the logarithmic couplings $b_0, a_0$ in the sense that they are independent of the basis and we have determined them in the main text. The three-point constants in (D.23) – (D.28) are similar. For example, under the change of basis for $(t, T)$ and $(\Psi_2, \Psi_1, \Psi_0)$, one has

$$ C_{\epsilon\epsilon T} \rightarrow C_{\epsilon\epsilon T}, \quad C_{\epsilon\epsilon t} \rightarrow C_{\epsilon\epsilon t} + r_0 C_{\epsilon\epsilon T}, \quad \dots \quad (D.31a) $$

$$ C_{\epsilon\epsilon\Psi_0} \rightarrow C_{\epsilon\epsilon\Psi_0}, \quad C_{\epsilon\epsilon\Psi_1} \rightarrow C_{\epsilon\epsilon\Psi_1} + r_1 C_{\epsilon\epsilon\Psi_0}, \quad \dots \quad (D.31b) $$

However, for $C_{\epsilon\epsilon T} = C_{\epsilon\epsilon\Psi_0} = 0$ as we have obtained in the main text for the percolation and SAW energy operator, the three-point constants

$$ C_{\epsilon\epsilon t} \rightarrow C_{\epsilon\epsilon t}, \quad C_{\epsilon\epsilon\Psi_1} \rightarrow C_{\epsilon\epsilon\Psi_1}, \quad (D.32) $$

are invariant under the change of basis. Similarly, under a change of basis for $(\tilde{\epsilon}, \epsilon)$, we find

$$ C_{\epsilon\tilde{\epsilon}T} \rightarrow C_{\epsilon\tilde{\epsilon}T}, \quad C_{\epsilon\tilde{\epsilon}\Psi_0} \rightarrow C_{\epsilon\tilde{\epsilon}\Psi_0}. \quad (D.33) $$

In this sense, these three-point constants become “intrinsic” due to the vanishing $C_{\epsilon\epsilon T}, C_{\epsilon\epsilon\Psi_0}$ and these are what we have computed in the main text. It would be very interesting to understand what this means physically.
### D.3 Logarithmic OPEs

Given the two- and three-point functions, we can write down the OPE for logarithmic operators. Let us first consider the simple case of logarithmic operators appearing in the OPE of non-logarithmic operators. For this, what we have in mind is OPE of (59) which was previously considered in [21]. We use this case to explain the generic method described in [55] and apply it further below for the more complicated cases.

Consider the OPE of non-logarithmic operators $\Phi$. We want to know the OPE structure involving the rank-2 Jordan block ($\psi_1, \psi_0$)

$$ \Phi(z_1, \bar{z}_1)\Phi(z_2, \bar{z}_2) \sim \psi_1(z_2, \bar{z}_2) + \psi_0(z_2, \bar{z}_2), \qquad (D.34) $$

or the rank-3 Jordan block

$$ \Phi(z_1, \bar{z}_1)\Phi(z_2, \bar{z}_2) \sim \Psi_2(z_2, \bar{z}_2) + \Psi_1(z_2, \bar{z}_2) + \Psi_0(z_2, \bar{z}_2). \qquad (D.35) $$

To do this, write down the matrix of

$$ (G_{\Phi}^{(3)})_k = \lim_{z_1 \to z_2, \bar{z}_1 \to \bar{z}_2} \langle \Phi(z_1, \bar{z}_1) \Phi(z_2, \bar{z}_2) \mathcal{O}_k(z_3, \bar{z}_3) \rangle, \qquad (D.36) $$

where $\mathcal{O}_k$ refers to the logarithmic multiplets. In the case that $\Phi$ is not logarithmic, we have only one row, and the number of columns correspond to the rank of the Jordan block $\mathcal{O}_k$. More concretely, for rank-2 $\psi_k$ we have

$$ G_{\Phi}^{(3)} = \left[ \lim_{z_1 \to z_2, \bar{z}_1 \to \bar{z}_2} \langle \Phi(z_1, \bar{z}_1) \Phi(\bar{z}_2, \bar{z}_2) \psi_1(z_3, \bar{z}_3) \rangle \lim_{z_1 \to z_2, \bar{z}_1 \to \bar{z}_2} \langle \Phi(z_1, \bar{z}_1) \Phi(z_2, \bar{z}_2) \psi_0(z_3, \bar{z}_3) \rangle \right]. \qquad (D.37) $$

Then we define the two-point function matrix (in the case of rank-2 $\psi_k$)

$$ G_{\psi}^{(2)} = \begin{bmatrix} \langle \psi_1(z_2, \bar{z}_2) \psi_1(z_3, \bar{z}_3) \rangle & \langle \psi_1(z_2, \bar{z}_2) \psi_0(z_3, \bar{z}_3) \rangle \\ \langle \psi_0(z_2, \bar{z}_2) \psi_1(z_3, \bar{z}_3) \rangle & \langle \psi_0(z_2, \bar{z}_2) \psi_0(z_3, \bar{z}_3) \rangle \end{bmatrix}. \qquad (D.38) $$

The OPE structure is then given by the matrix

$$ G_{\Phi} = G_{\Phi}^{(3)} (G_{\psi}^{(2)})^{-1}. \qquad (D.39) $$

In the case of rank-2 Jordan block with two-point functions

$$ \langle \psi_1(z_2, \bar{z}_2) \psi_1(z_3, \bar{z}_3) \rangle = \frac{b_1 - 2b_0 \ln(z_{23} \bar{z}_{23})}{z_{23}^{2h} \bar{z}_{23}^{2\bar{h}}}, \qquad (D.40a) $$

$$ \langle \psi_1(z_2, \bar{z}_2) \psi_0(z_3, \bar{z}_3) \rangle = \frac{b_0}{z_{23}^{2h} \bar{z}_{23}^{2\bar{h}}}, \qquad (D.40b) $$

$$ \langle \psi_0(z_2, \bar{z}_2) \psi_0(z_3, \bar{z}_3) \rangle = 0, \qquad (D.40c) $$

and three-point functions

$$ \langle \Phi(z_1, \bar{z}_1) \Phi(z_2, \bar{z}_2) \psi_0(z_3, \bar{z}_3) \rangle = \frac{C_{\Phi\Phi\psi_0}}{z_{12}^{2\Delta-h} \bar{z}_{12}^{2\Delta-h} z_{23}^{h} \bar{z}_{23}^{\bar{h}} z_{13}^{h} \bar{z}_{13}^{\bar{h}}}, \qquad (D.41a) $$

$$ \langle \Phi(z_1, \bar{z}_1) \Phi(z_2, \bar{z}_2) \psi_1(z_3, \bar{z}_3) \rangle = \frac{C_{\Phi\Phi\psi_1} + C_{\Phi\Phi\psi_0} \ln \frac{z_{12}\bar{z}_{12}}{z_{13}\bar{z}_{13}z_{23}\bar{z}_{23}}}{z_{12}^{2\Delta-h} \bar{z}_{12}^{2\Delta-h} z_{23}^{h} \bar{z}_{23}^{\bar{h}} z_{13}^{h} \bar{z}_{13}^{\bar{h}}}, \qquad (D.41b) $$
where $(\Delta, \Delta)$ indicate the dimension of $\Phi$ (taken to be diagonal here) and $(h, \bar{h})$ the dimension of $(\psi_1, \psi_0)$. Using (D.39) we can extract

$$ \Phi(z_1, \bar{z}_1)\Phi(z_2, \bar{z}_2) = (z_{12}\bar{z}_{12})^{-2\Delta} z_{12}^h \bar{z}_{12}^{\bar{h}} \left( \frac{C_{\Phi\Phi\psi_0}}{b_0} \psi_1 + \frac{C_{\Phi\Phi\psi_0}}{b_0} \ln(z_{12}\bar{z}_{12})\psi_0 + \frac{b_0 C_{\Phi\Phi\psi_1} - b_1 C_{\Phi\Phi\psi_0}}{b_0^2} \psi_0 + \dots \right). \quad (D.42) $$

Similarly in the case of rank-3 Jordan block we find

$$ \Phi(z_1, \bar{z}_1)\Phi(z_2, \bar{z}_2) = (z_{12}\bar{z}_{12})^{-2\Delta+h} \left( \begin{aligned} & \frac{C_{\Phi\Phi\psi_0}}{a_0}\psi_2 + \frac{C_{\Phi\Phi\psi_0}}{a_0}\ln(z_{12}\bar{z}_{12})\psi_1 + \frac{a_0 C_{\Phi\Phi\psi_1} - a_1 C_{\Phi\Phi\psi_0}}{a_0^2}\psi_1 \\ & + \frac{C_{\Phi\Phi\psi_0}}{2a_0}\ln^2(z_{12}\bar{z}_{12})\psi_0 + \frac{a_0 C_{\Phi\Phi\psi_1} - a_1 C_{\Phi\Phi\psi_0}}{a_0^2}\ln(z_{12}\bar{z}_{12})\psi_0 \\ & + \frac{a_0^2 C_{\Phi\Phi\psi_2} - a_2 a_0 C_{\Phi\Phi\psi_0} - a_1 a_0 C_{\Phi\Phi\psi_1} + a_1^2 C_{\Phi\Phi\psi_0}}{a_0^3}\psi_0 + \dots \end{aligned} \right). \quad (D.43) $$

Recall that the Jordan blocks are defined up to a change of basis (D.5) and (D.6) where for the rank-2 Jordan block there is one parameter $r_0$ and for the rank-3 Jordan block, there are two.<fnref n="22" /> This freedom allows us to reduce the logarithmic OPE to a simple form, by setting

$$ \begin{aligned} & b_0 C_{\Phi\Phi\psi_1} - b_1 C_{\Phi\Phi\psi_0} = 0, \\ & a_0 C_{\Phi\Phi\psi_1} - a_1 C_{\Phi\Phi\psi_0} = 0, \\ & a_0^2 C_{\Phi\Phi\psi_2} - a_2 a_0 C_{\Phi\Phi\psi_0} - a_1 a_0 C_{\Phi\Phi\psi_1} + a_1^2 C_{\Phi\Phi\psi_0} = 0, \end{aligned} \quad (D.44) $$

solved by

$$ C_{\Phi\Phi\psi_1} = \frac{b_1 C_{\Phi\Phi\psi_0}}{b_0}, \quad C_{\Phi\Phi\psi_1} = \frac{a_1 C_{\Phi\Phi\psi_0}}{a_0}, \quad C_{\Phi\Phi\psi_2} = \frac{a_2 C_{\Phi\Phi\psi_0}}{a_0}. \quad (D.45) $$

This simplifies the OPEs (D.42) and (D.43) to

$$ \Phi(z_1, \bar{z}_1)\Phi(z_2, \bar{z}_2) = (z_{12}\bar{z}_{12})^{-2\Delta} z_{12}^h \bar{z}_{12}^{\bar{h}} \frac{C_{\Phi\Phi\psi_0}}{b_0} (\psi_1 + \ln(z_{12}\bar{z}_{12})\psi_0) + \dots, \quad (D.46) $$

and

$$ \Phi(z_1, \bar{z}_1)\Phi(z_2, \bar{z}_2) = (z_{12}\bar{z}_{12})^{-2\Delta+h} \frac{C_{\Phi\Phi\psi_0}}{a_0} (\Psi_2 + \ln(z_{12}\bar{z}_{12})\Psi_1 + \frac{1}{2}\ln^2(z_{12}\bar{z}_{12})\Psi_0) + \dots, \quad (D.47) $$

as we have done in e.g. (59) in the main text.

The case of OPE of logarithmic operators is a direct generalization. Consider the logarithmic pair $(\tilde{\epsilon}, \epsilon)$. In this case, the matrix (D.37) has multiple rows:

$$ G_{\tilde{\epsilon}, \epsilon}^{(3)} = \begin{bmatrix} \lim_{z_1 \to \bar{z}_2, \tilde{z}_1 \to \tilde{z}_2} \langle \tilde{\epsilon}(z_1, \bar{z}_1) \tilde{\epsilon}(z_2, \bar{z}_2) \psi_1(z_3, \bar{z}_3) \rangle & \lim_{z_1 \to \bar{z}_2, \tilde{z}_1 \to \tilde{z}_2} \langle \tilde{\epsilon}(z_1, \bar{z}_1) \tilde{\epsilon}(z_2, \bar{z}_2) \psi_0(z_3, \bar{z}_3) \rangle \\ \lim_{z_1 \to \bar{z}_2, \tilde{z}_1 \to \tilde{z}_2} \langle \tilde{\epsilon}(z_1, \bar{z}_1) \epsilon(z_2, \bar{z}_2) \psi_1(z_3, \bar{z}_3) \rangle & \lim_{z_1 \to \bar{z}_2, \tilde{z}_1 \to \tilde{z}_2} \langle \tilde{\epsilon}(z_1, \bar{z}_1) \epsilon(z_2, \bar{z}_2) \psi_0(z_3, \bar{z}_3) \rangle \\ \lim_{z_1 \to \bar{z}_2, \tilde{z}_1 \to \tilde{z}_2} \langle \epsilon(z_1, \bar{z}_1) \epsilon(z_2, \bar{z}_2) \psi_1(z_3, \bar{z}_3) \rangle & \lim_{z_1 \to \bar{z}_2, \tilde{z}_1 \to \tilde{z}_2} \langle \epsilon(z_1, \bar{z}_1) \epsilon(z_2, \bar{z}_2) \psi_0(z_3, \bar{z}_3) \rangle \end{bmatrix}, \quad (D.48) $$

and similarly for the rank-3 Jordan block where there are three columns corresponding to $(\Psi_2, \Psi_1, \Psi_0)$. The OPE structure can be obtained similar to (D.39) through:

$$ G_{\tilde{\epsilon}, \epsilon} = G_{\tilde{\epsilon}, \epsilon}^{(3)} (G_{\Psi}^{(2)})^{-1}. \quad (D.49) $$

<footnote n="22">²²In [51], this was used to bring the two-point functions of the Jordan blocks into a canonical form. See appendix A in [51].</footnote>
The full expression in the generic case is quite long and tedious. However, for the energy logarithmic pair $(\tilde{\epsilon}, \epsilon)$, we find in the main text the special condition

$$C_{\epsilon\epsilon T} = 0, \quad C_{\epsilon\epsilon\Psi_0} = 0, \qquad (D.50)$$

which greatly simplifies the form of the OPE. Taking $(\psi_1, \psi_0)$ to be $(t, T)$ and $(\Psi_2, \Psi_1, \Psi_0)$ the rank-3 Jordan block, we can write down directly (e.g. for $\epsilon^{\text{perco}}$):

$$\epsilon(z_1, \bar{z}_1)\epsilon(z_2, \bar{z}_2) = (z_{12}\bar{z}_{12})^{-2h_{\epsilon}} \left\{ z_{12}^2 \frac{C_{\epsilon\epsilon t}}{b_0} T + \text{c.c.} + \dots + (z_{12}\bar{z}_{12})^2 \times \left( \frac{C_{\epsilon\epsilon\Psi_1}}{a_0}\Psi_1 + \frac{C_{\epsilon\epsilon\Psi_1}}{a_0}\ln(z_{12}\bar{z}_{12})\Psi_0 + \frac{a_0C_{\epsilon\epsilon\Psi_2} - a_1C_{\epsilon\epsilon\Psi_1}}{a_0^2}\Psi_0 \right) + \dots \right\}, \qquad (D.51)$$

$$\tilde{\epsilon}(z_1, \bar{z}_1)\epsilon(z_2, \bar{z}_2) = (z_{12}\bar{z}_{12})^{-2h_{\epsilon}} \left\{ z_{12}^2 \left( \frac{C_{\tilde{\epsilon}\tilde{\epsilon}T}}{b_0} t + \frac{b_0 C_{\tilde{\epsilon}\tilde{\epsilon}T} - b_1 C_{\tilde{\epsilon}\tilde{\epsilon}T}}{b_0^2} T + \frac{C_{\tilde{\epsilon}\tilde{\epsilon}T} - C_{\epsilon\epsilon t}}{b_0} \ln(z_{12}\bar{z}_{12})T \right) + \text{c.c.} + \dots \right. \\ \left. + (z_{12}\bar{z}_{12})^2 \left( \frac{C_{\tilde{\epsilon}\tilde{\epsilon}\Psi_0}}{a_0}\Psi_2 + \frac{a_0 C_{\tilde{\epsilon}\tilde{\epsilon}\Psi_1} - a_1 C_{\tilde{\epsilon}\tilde{\epsilon}\Psi_0}}{a_0^2}\Psi_1 \right. \right. \\ \left. + \frac{C_{\tilde{\epsilon}\tilde{\epsilon}\Psi_0} - C_{\epsilon\epsilon\Psi_1}}{a_0} \ln(z_{12}\bar{z}_{12})\Psi_1 \right. \\ \left. + \frac{a_0^2 C_{\tilde{\epsilon}\tilde{\epsilon}\Psi_2} - a_2 a_0 C_{\tilde{\epsilon}\tilde{\epsilon}\Psi_0} + a_1^2 C_{\tilde{\epsilon}\tilde{\epsilon}\Psi_0} - a_1 a_0 C_{\tilde{\epsilon}\tilde{\epsilon}\Psi_1}}{a_0^3}\Psi_0 \right. \\ \left. + \frac{a_1(C_{\epsilon\epsilon\Psi_1} - C_{\tilde{\epsilon}\tilde{\epsilon}\Psi_0}) - a_0(C_{\epsilon\epsilon\Psi_2} + C_{\tilde{\epsilon}\tilde{\epsilon}\Psi_1} - 2C_{\tilde{\epsilon}\tilde{\epsilon}\Psi_1})}{a_0^2}\Psi_0 \right. \\ \left. \times \ln(z_{12}\bar{z}_{12})\Psi_0 + \frac{C_{\tilde{\epsilon}\tilde{\epsilon}\Psi_0} - 2C_{\epsilon\epsilon\Psi_1}}{2a_0}\ln^2(z_{12}\bar{z}_{12})\Psi_0 \right) + \dots \right\}, \qquad (D.52)$$

$$\tilde{\epsilon}(z_1, \bar{z}_1)\tilde{\epsilon}(z_2, \bar{z}_2) = (z_{12}\bar{z}_{12})^{-2h_{\epsilon}} \left\{ z_{12}^2 \left( \frac{C_{\tilde{\epsilon}\tilde{\epsilon}T}}{b_0} t - \frac{2C_{\tilde{\epsilon}\tilde{\epsilon}T}}{b_0} \ln(z_{12}\bar{z}_{12})t + \frac{b_0 C_{\tilde{\epsilon}\tilde{\epsilon}t} - b_1 C_{\tilde{\epsilon}\tilde{\epsilon}T}}{b_0^2} T \right. \right. \\ \left. + \frac{b_0 C_{\tilde{\epsilon}\tilde{\epsilon}T} + 2b_1 C_{\tilde{\epsilon}\tilde{\epsilon}T} - 2b_0 C_{\tilde{\epsilon}\tilde{\epsilon}t}}{b_0^2} \ln(z_{12}\bar{z}_{12})T \right. \\ \left. + \frac{C_{\epsilon\epsilon t} - 2C_{\tilde{\epsilon}\tilde{\epsilon}T}}{b_0} \ln^2(z_{12}\bar{z}_{12})T \right) + \text{c.c.} + \dots \\ + (z_{12}\bar{z}_{12})^2 \left( \frac{C_{\tilde{\epsilon}\tilde{\epsilon}\Psi_0}}{a_0}\Psi_2 - \frac{2C_{\tilde{\epsilon}\tilde{\epsilon}\Psi_0}}{a_0} \ln(z_{12}\bar{z}_{12})\Psi_2 + \frac{a_0 C_{\tilde{\epsilon}\tilde{\epsilon}\Psi_1} - a_1 C_{\tilde{\epsilon}\tilde{\epsilon}\Psi_0}}{a_0^2}\Psi_1 \right. \\ \left. + \frac{a_0(C_{\tilde{\epsilon}\tilde{\epsilon}\Psi_0} - 2C_{\tilde{\epsilon}\tilde{\epsilon}\Psi_1}) + 2a_1 C_{\tilde{\epsilon}\tilde{\epsilon}\Psi_0}}{a_0^2} \ln(z_{12}\bar{z}_{12})\Psi_1 \right. \\ \left. + \frac{C_{\epsilon\epsilon\Psi_1} - 2C_{\tilde{\epsilon}\tilde{\epsilon}\Psi_0}}{a_0} \ln^2(z_{12}\bar{z}_{12})\Psi_1 \right) \\ + \frac{a_0^2 C_{\tilde{\epsilon}\tilde{\epsilon}\Psi_2} - a_2 a_0 C_{\tilde{\epsilon}\tilde{\epsilon}\Psi_0} + a_1^2 C_{\tilde{\epsilon}\tilde{\epsilon}\Psi_0} - a_1 a_0 C_{\tilde{\epsilon}\tilde{\epsilon}\Psi_1}}{a_0^3}\Psi_0 \\ + \frac{a_0^2(2C_{\tilde{\epsilon}\tilde{\epsilon}\Psi_1} - 4C_{\tilde{\epsilon}\tilde{\epsilon}\Psi_2}) - 2a_0(a_1(C_{\tilde{\epsilon}\tilde{\epsilon}\Psi_0} - 2C_{\tilde{\epsilon}\tilde{\epsilon}\Psi_1}) - 2a_2 C_{\tilde{\epsilon}\tilde{\epsilon}\Psi_0}) - 4a_1^2 C_{\tilde{\epsilon}\tilde{\epsilon}\Psi_0}}{2a_0^3} \ln(z_{12}\bar{z}_{12})\Psi_0 \\ + \frac{a_0^2(2C_{\epsilon\epsilon\Psi_2} - 4C_{\tilde{\epsilon}\tilde{\epsilon}\Psi_1} + C_{\tilde{\epsilon}\tilde{\epsilon}\Psi_0}) - 2a_0 a_1(C_{\epsilon\epsilon\Psi_1} - 2C_{\tilde{\epsilon}\tilde{\epsilon}\Psi_0})}{2a_0^3} \ln^2(z_{12}\bar{z}_{12})\Psi_0 \\ \left. + \frac{C_{\epsilon\epsilon\Psi_1} - C_{\tilde{\epsilon}\tilde{\epsilon}\Psi_0}}{a_0} \ln^3(z_{12}\bar{z}_{12})\Psi_0 \right) + \dots \right\}.$$
## D.4 Four-point functions

From the logarithmic conformal data, using the cluster decomposition properties, we can now assemble four-point functions involving logarithmic operators.

Consider first the s-channel limit of four-point function of a non-logarithmic operator $\Phi$ with dimension $(\Delta, \Delta)$. As usual, we place the operators $\Phi$ at the most convenient configurations $(\infty, 1, z, 0)$

$$ \begin{aligned} \langle \Phi(\infty, \infty) \Phi(1, 1) \Phi(z, \bar{z}) \Phi(0, 0) \rangle &= \lim_{z_1, \bar{z}_1 \to \infty} (z_1 \bar{z}_1)^{2\Delta} \langle \Phi(z_1, \bar{z}_1) \Phi(1, 1) \Phi(z, \bar{z}) \Phi(0, 0) \rangle \\ &= \sum_{\{\psi\}} \langle \Phi | \Phi(1, 1) | \psi \rangle G^{-1} \langle \psi | \Phi(z, \bar{z}) | \Phi \rangle, \end{aligned} \quad (D.54) $$

where $G^{-1}$ is the inverse of Gram matrix, and

$$ \langle \Phi | \Phi(1, 1) | \psi \rangle = \lim_{z_1, \bar{z}_1 \to \infty} \langle \Phi(z_1, \bar{z}_1) \Phi(1, 1) \psi(0, 0) \rangle, \quad (D.55) $$

$$ \langle \psi | \Phi(z, \bar{z}) | \Phi \rangle = \lim_{w, \bar{w} \to 0} \langle \tilde{\psi}(w, \bar{w}) \Phi(z, \bar{z}) \Phi(0, 0) \rangle, \quad (D.56) $$

where $\tilde{\psi}(w, \bar{w})$ is the transformation of the field $\psi(z, \bar{z})$ under inversion

$$ w = \frac{1}{z}, \qquad \bar{w} = \frac{1}{\bar{z}}. \quad (D.57) $$

Up to dimensions (2, 2), we have identified the following set of operators in the $c = 0$ percolation and SAW bulk CFTs:<fnref n="23" />

$$ \{\psi\} : \mathbb{I}, \varepsilon, \tilde{\varepsilon}, T, \bar{T}, t, \bar{t}, \partial \bar{t}, \partial t, \partial^2 \bar{t}, \partial^2 t, \Psi_0, \Psi_1, \Psi_2, \dots, \quad (D.58) $$

where $\varepsilon, \tilde{\varepsilon}$ refer to the energy operator logarithmic pair (see section 4.2). The Gram matrix for these states are given by:

$$ \begin{pmatrix} \langle \tilde{\varepsilon} | \tilde{\varepsilon} \rangle & \langle \varepsilon | \varepsilon \rangle \\ \langle \varepsilon | \tilde{\varepsilon} \rangle & \langle \tilde{\varepsilon} | \varepsilon \rangle \end{pmatrix} = \begin{pmatrix} \theta_1 & \gamma \\ \gamma & 0 \end{pmatrix}, \quad \begin{pmatrix} \langle t | t \rangle & \langle t | T \rangle \\ \langle T | t \rangle & \langle T | T \rangle \end{pmatrix} = \begin{pmatrix} \theta & b \\ b & 0 \end{pmatrix}, \quad \begin{pmatrix} \langle \bar{\varepsilon} t | \bar{\varepsilon} t \rangle & \langle \bar{\varepsilon} t | \partial \bar{t} \rangle \\ \langle \partial \bar{t} | \bar{\varepsilon} t \rangle & \langle \partial \bar{t} | \partial t \rangle \end{pmatrix} = \begin{pmatrix} 2b & 0 \\ 0 & 2b \end{pmatrix}, \quad (D.59) $$

and for the states with dimension (2, 2):

$$ \begin{pmatrix} \langle \Psi_2 | \Psi_2 \rangle & \langle \Psi_2 | \Psi_1 \rangle & \langle \Psi_2 | \Psi_0 \rangle & \langle \Psi_2 | \partial^2 t \rangle & \langle \Psi_2 | \partial^2 \bar{t} \rangle \\ \langle \Psi_1 | \Psi_2 \rangle & \langle \Psi_1 | \Psi_1 \rangle & \langle \Psi_1 | \Psi_0 \rangle & \langle \Psi_1 | \partial^2 t \rangle & \langle \Psi_1 | \partial^2 \bar{t} \rangle \\ \langle \Psi_0 | \Psi_2 \rangle & \langle \Psi_0 | \Psi_1 \rangle & \langle \Psi_0 | \Psi_0 \rangle & \langle \Psi_0 | \partial^2 t \rangle & \langle \Psi_0 | \partial^2 \bar{t} \rangle \\ \langle \partial^2 t | \Psi_2 \rangle & \langle \partial^2 t | \Psi_1 \rangle & \langle \partial^2 t | \Psi_0 \rangle & \langle \partial^2 t | \partial^2 t \rangle & \langle \partial^2 t | \partial^2 \bar{t} \rangle \\ \langle \partial^2 \bar{t} | \Psi_2 \rangle & \langle \partial^2 \bar{t} | \Psi_1 \rangle & \langle \partial^2 \bar{t} | \Psi_0 \rangle & \langle \partial^2 \bar{t} | \partial^2 t \rangle & \langle \partial^2 \bar{t} | \partial^2 \bar{t} \rangle \end{pmatrix} = \begin{pmatrix} a_2 & a_1 & a & 0 & 0 \\ a_1 & a & 0 & 0 & 0 \\ a & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 4b & 0 \\ 0 & 0 & 0 & 0 & 4b \end{pmatrix}. \quad (D.60) $$

In writing the third equation of (D.59), we have used

$$ \langle t | \bar{L}_1 \bar{L}_{-1} | t \rangle = \langle t | [\bar{L}_1, \bar{L}_{-1}] | t \rangle = 2 \langle t | \bar{L}_0 | t \rangle = 2 \langle t | T \rangle = 2b, \quad (D.61) $$

with $\bar{L}_1 | t \rangle = 0$. Moreover, for (D.60), we have used

$$ \langle t | \bar{L}_{-1}^2 \bar{L}_{-1}^2 | t \rangle = \langle t | 4\bar{L}_0 + 8\bar{L}_0^2 + 8\bar{L}_0 \bar{L}_{-1} \bar{L}_1 | t \rangle = 4\langle t | T \rangle = 4b. \quad (D.62) $$

The s-channel limit of the four-point function of the operator $\Phi$ is then given by: (the normalization

<footnote n="23">²³We are not concerned with the descendants of $\varepsilon, \tilde{\varepsilon}$ here.</footnote>
of $\Phi$ taken to be 1)

$$ \langle \Phi(\infty)\Phi(1)\Phi(z, \bar{z})\Phi(0) \rangle = (z\bar{z})^{-2\Delta} \left\{ \begin{aligned} & 1 + (z^2 + \bar{z}^2) \left[ \frac{C_{\Phi\Phi T}(2bC_{\Phi\Phi t} - C_{\Phi\Phi T}\theta)}{b^2} + \frac{C_{\Phi\Phi T}^2}{b} \ln(z\bar{z}) \right] \\ & + (z\bar{z}^2 + z^2\bar{z}) \frac{C_{\Phi\Phi T}^2}{2b} + (z\bar{z})^2 \frac{C_{\Phi\Phi T}^2}{2b} \\ & + (z\bar{z})^{h_\epsilon} \left[ \frac{C_{\Phi\Phi\epsilon}(2\gamma C_{\Phi\Phi\bar{\epsilon}} - C_{\Phi\Phi\epsilon}\theta_1)}{\gamma^2} + \frac{C_{\Phi\Phi\epsilon}^2}{\gamma} \ln(z\bar{z}) \right] \\ & + (z\bar{z})^2 \left[ \frac{a_1^2 C_{\Phi\Phi\Psi_0}^2 - aa_2 C_{\Phi\Phi\Psi_0}^2 - 2aa_1 C_{\Phi\Phi\Psi_0} C_{\Phi\Phi\Psi_1} + a^2 C_{\Phi\Phi\Psi_1}^2 + 2a^2 C_{\Phi\Phi\Psi_0} C_{\Phi\Phi\Psi_2}}{a^3} \right. \\ & \qquad \left. + \frac{C_{\Phi\Phi\Psi_0}(2aC_{\Phi\Phi\Psi_1} - a_1 C_{\Phi\Phi\Psi_0})}{a^2} \ln(z\bar{z}) + \frac{C_{\Phi\Phi\Psi_0}^2}{2a} \ln^2(z\bar{z}) \right] + \dots \end{aligned} \right\} \quad (D.63) $$

Clearly the expression of (D.63) is invariant upon changing the basis of the Jordan blocks (see (D.15), (D.16), (D.29) and (D.30)). The expression can be simplified using the fixed basis (D.45) and becomes

$$ \langle \Phi(\infty)\Phi(1)\Phi(z, \bar{z})\Phi(0) \rangle = (z\bar{z})^{-2\Delta} \left\{ \begin{aligned} & 1 + (z^2 + \bar{z}^2) \frac{C_{\Phi\Phi T}^2}{b^2} (\theta + b \ln(z\bar{z})) + (z\bar{z}^2 + z^2\bar{z}) \frac{C_{\Phi\Phi T}^2}{2b} \\ & + (z\bar{z})^2 \frac{C_{\Phi\Phi T}^2}{2b} + (z\bar{z})^{h_\epsilon} \frac{C_{\Phi\Phi\epsilon}^2}{\gamma^2} (\theta_1 + \gamma \ln(z\bar{z})) \\ & + (z\bar{z})^2 \frac{C_{\Phi\Phi\Psi_0}^2}{a_0^2} \left( \frac{a}{2} \ln^2(z\bar{z}) + a_1 \ln(z\bar{z}) + a_2 \right) + \dots \end{aligned} \right\} \quad (D.64) $$

Applying to the spin operator in percolation $\Phi \sim \sigma^{\text{perco}}$, we get the s-channel expansion of four-point function $\langle \sigma\sigma\sigma\sigma \rangle^{\text{perco}}$ in (143). The same expression (D.64) applied to the spin operator $\Phi \sim \sigma^{\text{hull}}$ for percolation hull in section 2.3 results in the correlation (72).

Let us now consider the bottom field $\epsilon$ of the energy rank-2 Jordan block in percolation and SAW. The previous expression (D.63) essentially applies but since the bottom field $\epsilon$ has zero norm, the contribution from identity operator drops out. We find

$$ \langle \epsilon(\infty)\epsilon(1)\epsilon(z, \bar{z})\epsilon(0) \rangle = (z\bar{z})^{-2h_\epsilon} \left\{ \begin{aligned} & (z^2 + \bar{z}^2) \left[ \frac{C_{\epsilon\epsilon T}(2bC_{\epsilon\epsilon t} - C_{\epsilon\epsilon T}\theta)}{b^2} + \frac{C_{\epsilon\epsilon T}^2}{b} \ln(z\bar{z}) \right] \\ & + (z\bar{z}^2 + z^2\bar{z}) \frac{C_{\epsilon\epsilon T}^2}{2b} + (z\bar{z})^2 \frac{C_{\epsilon\epsilon T}^2}{2b} \\ & + (z\bar{z})^{h_\epsilon} \left[ \frac{C_{\epsilon\epsilon\epsilon}(2\gamma C_{\epsilon\epsilon\bar{\epsilon}} - C_{\epsilon\epsilon\epsilon}\theta_1)}{\gamma^2} + \frac{C_{\epsilon\epsilon\epsilon}^2}{\gamma} \ln(z\bar{z}) \right] \\ & + (z\bar{z})^2 \left[ \frac{a_1^2 C_{\epsilon\epsilon\Psi_0}^2 - aa_2 C_{\epsilon\epsilon\Psi_0}^2 - 2aa_1 C_{\epsilon\epsilon\Psi_0} C_{\epsilon\epsilon\Psi_1} + a^2 C_{\epsilon\epsilon\Psi_1}^2 + 2a^2 C_{\epsilon\epsilon\Psi_0} C_{\epsilon\epsilon\Psi_2}}{a^3} \right. \\ & \qquad \left. + \frac{C_{\epsilon\epsilon\Psi_0}(2aC_{\epsilon\epsilon\Psi_1} - a_1 C_{\epsilon\epsilon\Psi_0})}{a^2} \ln(z\bar{z}) + \frac{C_{\epsilon\epsilon\Psi_0}^2}{2a} \ln^2(z\bar{z}) \right] + \dots \end{aligned} \right\} \quad (D.65) $$

For both $\epsilon^{\text{perco}}$ and $\epsilon^{\text{SAW}}$, since $C_{\epsilon\epsilon\epsilon} = C_{\epsilon\epsilon T} = C_{\epsilon\epsilon\Psi_0} = 0$, there turns out to be no logarithmic terms, but the four-point functions also do not vanish. One has

$$ \langle \epsilon(\infty)\epsilon(1)\epsilon(z, \bar{z})\epsilon(0) \rangle = (z\bar{z})^{-2h_\epsilon} \frac{C_{\epsilon\epsilon\Psi_1}^2}{a} (z\bar{z})^2 + \dots, \quad (D.66) $$

as we have seen in the main text eqs. (224) and note that this non-trivial four-point function is due to the non-vanishing coupling between the field $\epsilon$ and the middle field $\Psi_1$ in the rank-3 Jordan block.
Although we do not analyze this in this paper, it is also straightforward to write down the s-channel limit of the four-point function involving one top field $\tilde{\epsilon}$:

$$ \langle \epsilon(\infty) \tilde{\epsilon}(1) \epsilon(x) \epsilon(0) \rangle = (z\bar{z})^{-2h_{\tilde{\epsilon}}} \left\{ (z^2 + \bar{z}^2) \left[ \frac{b C_{\tilde{\epsilon}\tilde{\epsilon}T} C_{\tilde{\epsilon}\tilde{\epsilon}T} + b C_{\tilde{\epsilon}\tilde{\epsilon}T} C_{\tilde{\epsilon}\tilde{\epsilon}T} - C_{\tilde{\epsilon}\tilde{\epsilon}T} C_{\tilde{\epsilon}\tilde{\epsilon}T} \theta}{b^2} + \frac{C_{\tilde{\epsilon}\tilde{\epsilon}T} C_{\tilde{\epsilon}\tilde{\epsilon}T}}{b} \ln(z\bar{z}) \right] + \dots \right. \\ \left. + (z\bar{z})^2 (A + B \ln(z\bar{z}) + C \ln^2(z\bar{z})) + \dots \right\}, \quad (D.67) $$

where

$$ A = \frac{1}{a^3} \left( 2a_1^2 C_{\tilde{\epsilon}\tilde{\epsilon}\Psi_0} C_{\tilde{\epsilon}\tilde{\epsilon}\Psi_0} - 2aa_1(C_{\tilde{\epsilon}\tilde{\epsilon}\Psi_0} C_{\tilde{\epsilon}\tilde{\epsilon}\Psi_1} + C_{\tilde{\epsilon}\tilde{\epsilon}\Psi_0} C_{\tilde{\epsilon}\tilde{\epsilon}\Psi_1}) \right. \\ \left. + a(-2a_2 C_{\tilde{\epsilon}\tilde{\epsilon}\Psi_0} C_{\tilde{\epsilon}\tilde{\epsilon}\Psi_0} + 2a(C_{\tilde{\epsilon}\tilde{\epsilon}\Psi_1} C_{\tilde{\epsilon}\tilde{\epsilon}\Psi_1} + C_{\tilde{\epsilon}\tilde{\epsilon}\Psi_0} C_{\tilde{\epsilon}\tilde{\epsilon}\Psi_2} + C_{\tilde{\epsilon}\tilde{\epsilon}\Psi_0} C_{\tilde{\epsilon}\tilde{\epsilon}\Psi_2})) \right), \quad (D.68a) $$

$$ B = \frac{a C_{\tilde{\epsilon}\tilde{\epsilon}\Psi_0} C_{\tilde{\epsilon}\tilde{\epsilon}\Psi_1} + a C_{\tilde{\epsilon}\tilde{\epsilon}\Psi_0} C_{\tilde{\epsilon}\tilde{\epsilon}\Psi_1} - a_1 C_{\tilde{\epsilon}\tilde{\epsilon}\Psi_0} C_{\tilde{\epsilon}\tilde{\epsilon}\Psi_0}}{a^2}, \quad (D.68b) $$

$$ C = \frac{C_{\tilde{\epsilon}\tilde{\epsilon}\Psi_0} C_{\tilde{\epsilon}\tilde{\epsilon}\Psi_0}}{2a}. \quad (D.68c) $$

Again, the expressions are invariant under a change of basis in the Jordan blocks. The expression (D.67) again simplifies for $C_{\tilde{\epsilon}\tilde{\epsilon}T} = C_{\tilde{\epsilon}\tilde{\epsilon}\Psi_0} = 0$:

$$ \langle \epsilon(\infty) \tilde{\epsilon}(1) \epsilon(x) \epsilon(0) \rangle = (z\bar{z})^{-2h_i} \left( (z^2 + \bar{z}^2) \frac{C_{\tilde{\epsilon}\tilde{\epsilon}T} C_{\tilde{\epsilon}\tilde{\epsilon}T}}{b} + (z\bar{z})^2 \left( \frac{a C_{\tilde{\epsilon}\tilde{\epsilon}\Psi_0} C_{\tilde{\epsilon}\tilde{\epsilon}\Psi_2} + a C_{\tilde{\epsilon}\tilde{\epsilon}\Psi_1} C_{\tilde{\epsilon}\tilde{\epsilon}\Psi_1} - a_1 C_{\tilde{\epsilon}\tilde{\epsilon}\Psi_0} C_{\tilde{\epsilon}\tilde{\epsilon}\Psi_1}}{a^2} \right. \right. \\ \left. \left. + \frac{C_{\tilde{\epsilon}\tilde{\epsilon}\Psi_0} C_{\tilde{\epsilon}\tilde{\epsilon}\Psi_1}}{a} \ln(z\bar{z}) \right) + \dots \right). \quad (D.69) $$

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