Using the Fourier transform (A.2), one can rewrite the combinations of propagators appearing in (D.2) in the following way

$$ \begin{aligned} \frac{(n \cdot y_{12})^S}{(y_{12}^2)^{2-\frac{\Delta-S}{2}}} &= \frac{(-1)^S}{4^{2-\frac{\Delta}{2}} \pi^2} \frac{\Gamma(\frac{\Delta+S}{2})}{\Gamma(2-\frac{\Delta-S}{2})} (n \cdot \partial_{y_1})^S \int d^4 k \frac{e^{ik \cdot y_{12}}}{(k^2)^{\frac{\Delta+S}{2}}} \\ \frac{(n \cdot y_{12})^{S-1}}{(y_{12}^2)^{2-\frac{\Delta-S}{2}}} &= \frac{(-1)^{S-1}}{4^{\frac{5-\Delta}{2}} \pi^2} \frac{\Gamma(\frac{\Delta+S}{2}-1)}{\Gamma(2-\frac{\Delta-S}{2})} (n \cdot \partial_{y_1})^{S-1} \int d^4 k \frac{e^{ik \cdot y_{12}}}{(k^2)^{\frac{\Delta+S}{2}-1}}. \end{aligned} \quad (D.3) $$

Substituting (D.2) together with (D.3) into (D.1) and transforming the integrals in momentum space by means of (A.3), one can compute the trivial integrations obtaining

$$ \begin{aligned} h_{t\Delta,S} = c_F^3 \frac{(-i)^{S+1} \pi^2}{4^{2-\frac{\Delta}{2}}} & \left[ \frac{\Gamma(\frac{\Delta+S}{2})}{\Gamma(3-\frac{\Delta-S}{2})} \int d^4 k e^{ik \cdot x_{21}} \frac{\operatorname{tr}(\sigma_\eta \bar{\sigma}_\mu \sigma_\nu \bar{\sigma}_\rho) x_{12}^\eta k^\mu k^\nu k^\rho}{(k^2)^{2+\frac{\Delta+S}{2}}} (n \cdot k)^S \right. \\ & \left. - \frac{S}{2} \frac{\Gamma(\frac{\Delta+S}{2}-1)}{\Gamma(3-\frac{\Delta-S}{2})} \int d^4 k e^{ik \cdot x_{21}} \frac{\operatorname{tr}(\sigma_\eta \bar{\sigma}_\mu \sigma_\nu \bar{\sigma}_\rho) x_{12}^\eta k^\mu n^\nu k^\rho}{(k^2)^{1+\frac{\Delta+S}{2}}} (n \cdot k)^{S-1} \right]. \end{aligned} \quad (D.4) $$

Let's focus on the integral in the first line of (D.4). Using the trace of four $\sigma$-matrices given in (A.11) and transforming the integral back to position space by means of (A.2), we have

$$ \begin{aligned} & \int d^4 k e^{ik \cdot x_{21}} \frac{\operatorname{tr}(\sigma_\eta \bar{\sigma}_\mu \sigma_\nu \bar{\sigma}_\rho) x_{12}^\eta k^\mu k^\nu k^\rho}{(k^2)^{2+\frac{\Delta+S}{2}}} (n \cdot k)^S \\ & = i^{-S-1} 4^{\frac{3}{2}-\frac{\Delta+S}{2}} \pi^2 \frac{\Gamma(1-\frac{\Delta+S}{2})}{\Gamma(1+\frac{\Delta+S}{2})} (x_{12} \cdot \partial_{y_2}) (n \cdot \partial_{y_2})^S \frac{1}{(x_{12}^2)^{1-\frac{\Delta+S}{2}}}. \end{aligned} \quad (D.5) $$

Repeating the same procedure for the second line of (D.4) we obtain

$$ \begin{aligned} & \int d^4 k e^{ik \cdot x_{21}} \frac{\operatorname{tr}(\sigma_\eta \bar{\sigma}_\mu \sigma_\nu \bar{\sigma}_\rho) x_{12}^\eta k^\mu n^\nu k^\rho}{(k^2)^{1+\frac{\Delta+S}{2}}} (n \cdot k)^{S-1} = \frac{4^{2-\frac{\Delta+S}{2}} \pi^2}{i^{S+1}} \left[ \frac{\Gamma(1-\frac{\Delta+S}{2})}{\Gamma(1+\frac{\Delta+S}{2})} \right. \\ & \left. \times (x_{12} \cdot \partial_{y_2}) (n \cdot \partial_{y_2})^S \frac{1}{(x_{12}^2)^{1-\frac{\Delta+S}{2}}} + 2 \frac{\Gamma(2-\frac{\Delta+S}{2})}{\Gamma(\frac{\Delta+S}{2})} (n \cdot \partial_{y_2})^{S-1} \frac{n \cdot x_{12}}{(x_{12}^2)^{2-\frac{\Delta+S}{2}}} \right]. \end{aligned} \quad (D.6) $$

Finally, plugging the integrals (D.5) and (D.6) into (D.4) and performing the derivatives we arrive at the following expression for the eigenvalue

$$ h_{t\Delta,S} = 4\pi^4 c_F^3 \frac{(\Delta - S)^2 + S(S+2)}{(\Delta + S)(\Delta - S - 4)(\Delta - S - 2)(\Delta + S - 2)}, \quad (D.7) $$

where we set $x_{12}^2 = (n \cdot x_{12}) = 1$. Notice that setting $S = 0$ we find (5.27), as expected.

## E Cancellation of the spurious poles

In order to confirm the validity of equation (3.27) we should show that the physical poles given by the zeroes of the spectral equation (3.25) are the only contributions to the four-point correlators under study. This fact, well known for the bi-scalar reduction of our
theory (see appendix B in [24]), needs a proof for the full $\chi$CFT. It appears that additional possible contributions could come from the extra poles in $g_{\Delta,S}(u, v)$ and the measure factor $1/c_2(\Delta, S)$. In this appendix we will show that these contributions cancel each other thanks to a symmetry relation fulfilled by the eigenvalues $h_i_{\Delta,S}$ of the Bethe-Salpeter kernels.

The conformal block $g_{\Delta,S}(u, v)$ has simple poles at $\Delta_{S-n} = S + 3 - n$ (with $n = 1, 2, \dots, S$), namely $2i\nu_n = S + 1 - n$. Its residue at the pole $\nu = \nu_n$ is given by $r_n g_{S+3,S-n}(u, v)$ where (see for example appendix B in [60]):

$$r_n = (-1)^n \frac{in\Gamma^2\left(\frac{1}{2}(n - \Delta_1 + \Delta_2 + 1)\right)}{2\Gamma(n + 1)^2\Gamma^2\left(\frac{1}{2}(-n - \Delta_1 + \Delta_2 + 1)\right)}. \qquad (E.1)$$

This results in the following extra contribution to (3.27):

$$R_{S,m}^g = \left( \frac{r_m}{c_2(\Delta_{S-m}, S)} \frac{h_{b\Delta_{S-m},S}}{1 - \chi_b h_{b\Delta_{S-m},S} - \chi_f h_{f\Delta_{S-m},S}} \right) g_{S+3,S-m}(u, v), \quad 1 \le n \le S < \infty. \qquad (E.2)$$

In addition to that, the measure factor $1/c_2(\Delta, S)$ develops poles at $\Delta = S + 3 + k$, $k = 0, 1, 2, \dots$. The corresponding contribution can be expressed as

$$R_{S,k}^{c_2} = - \left( \frac{r_k}{c_2(\Delta_S, S+k)} \frac{h_{b\Delta_{S+k},S}}{1 - \chi_b h_{b\Delta_{S+k},S} - \chi_f h_{f\Delta_{S+k},S}} \right) g_{S+3+k,S}(u, v), \quad 0 \le S, k < \infty. \qquad (E.3)$$

The overall contribution of these terms is the sum over all non-negative integers $S, k$ of the generic term

$$R_{S,k}^{c_2} + R_{S+k,k}^{g} = - \left( \frac{r_k}{c_2(\Delta_S, S+k)} g_{S+3+k,S}(u, v) \right) \times \left[ \frac{h_{b\Delta_{S+k},S}}{1 - \chi_b h_{b\Delta_{S+k},S} - \chi_f h_{f\Delta_{S+k},S}} - \frac{h_{b\Delta_S,S+k}}{1 - \chi_b h_{b\Delta_S,S+k} - \chi_f h_{f\Delta_S,S+k}} \right]. \qquad (E.4)$$

A possible vanishing condition for the full contribution is then

$$r_k[h_{b\Delta_{S+k},S}(1 - \chi_b h_{b\Delta_{S+k},S} - \chi_f h_{f\Delta_{S+k},S}) - h_{b\Delta_{S+k},S}(1 - \chi_b h_{b\Delta_{S+k},S} - \chi_f h_{f\Delta_{S+k},S})] = 0 \qquad (E.5)$$

for any $k \in \mathbb{N}$. We can actually verify in both sectors under study that the following set of stronger conditions is fulfilled

$$r_k(h_{b3+S+k,S} - h_{b3+S,S+k}) = 0, \quad k = 0, 1, 2, \dots \qquad (E.6)$$

$$r_k(h_{f3+S+k,S} - h_{f3+S,S+k}) = 0, \quad k = 0, 1, 2, \dots. \qquad (E.7)$$

It is easy to check that plugging (E.6) into (E.5), one is left with the condition (E.7), which means that (E.6) together with (E.7) are a sufficient condition for (E.5). To prove these equations hold, we notice first of all that at $\Delta_1 = \Delta_2 = 1$ (E.1) vanishes at odd $n$, so it would be sufficient to prove (E.6), (E.7) at even $k \in 2\mathbb{N}$. Moreover, equation (E.6) has been checked in [24], where it was enough to state the cancelation of spurious poles in $\text{Tr}[\phi_1^2]$ sector. Let us verify the second condition (E.7) at even integer $k$. Starting from the sector $\text{Tr}[\phi_1^2]$ it is equivalent to

$$\tilde{h}_{f3+S+k,S} - \tilde{h}_{f3+S,S+k} = 0, \qquad (E.8)$$
where we recalled the definition of $\tilde{h}_{f\Delta,S}$ (5.31). Equation (E.8) actually coincides with the vanishing condition for spurious contribution in the “one-magnon” $\text{Tr}[\phi_1^2\phi_2]$ sector of bi-scalar theory, and is verified in [24]. We can finally check (E.7) for the sector $\text{Tr}[\phi_1\phi_2^\dagger]$. Recalling that in this case

$$h_{F\Delta,S} = -\frac{c_F^4[(\Delta-2)^2 + S(S+2)]h_{B\Delta,S}^2}{2c_B^6 - 8c_B^3c_F^2\pi^2[(\Delta-2)^2 + S(S+2)]\xi_j\xi_{j+}h_{B\Delta,S}}, \quad (E.9)$$

and that $h_{B3+S+k,S} = h_{B3+S,S+k}$, as we already know from [24], we are left to verify that

$$[(\Delta - 2)^2 + S(S + 2)]_{\Delta=3+s+k, S=s} = [(\Delta - 2)^2 + S(S + 2)]_{\Delta=3+s, S=s+k}, \quad (E.10)$$

which is trivially true for any integer $k$.

## F Operator mixing and logarithmic multiplet

In both sectors $\text{Tr}[\phi_1^2]$ and $\text{Tr}[\phi_1\phi_2^\dagger]$ of our theory the exchanged physical operators in the OPE s-channel of the 4-point correlators under analysis present mixing. Namely, due to the wide matter content of the theory, the renormalized operators are not just rescaled and normal-ordered monomials of elementary fields and derivatives, but linear coupling-dependant combination of several such terms which share the same symmetries. Concretely, in our theory we deal with single trace primary operators as

$$\mathcal{O}_1(x) = \text{tr}[\chi_{i_1}\chi_{i_2}\cdots\chi_{i_L}](x),$$

made up of elementary fields of the theory $\chi_{i_k}(x)$ eventually dressed by tensor structures and derivatives. Given the quantum numbers of such a term $\mathcal{O}_1$, that is Cartan's $U(1)^{\otimes 3}$ charge, twist and tensor rank $S$, it is usually possible to write a few other conformal primaries with the same numbers, say $\{\mathcal{O}_2, \mathcal{O}_3, \dots\}$. This allows in general some of the two-point functions $\langle \mathcal{O}_i(x)\mathcal{O}_j(0)^\dagger \rangle$ to not vanish at $i \neq j$, that is to have transitions $\mathcal{O}_i \to \mathcal{O}_j$. We define the anomalous dimension matrix $\gamma_{ij}$ as

$$-\mu \frac{d}{d\mu} Z_{\mathcal{O}_i} = \gamma_{ij} Z_{\mathcal{O}_j}, \quad (F.1)$$

being $Z_{\mathcal{O}_i}$ the renormalization of operator $\mathcal{O}_i$ and $\mu$ the scale. In absence of transitions, namely $\gamma_{ij} = \delta_{ij}\gamma_i$, mixing does not happen and each operator $\mathcal{O}_i$ has anomalous dimension $\gamma_i$. Otherwise, one has to bring the mixing matrix $\gamma_{ij}$ into diagonal form via a rotation over the basis of local primaries $\{\mathcal{O}_1, \mathcal{O}_2, \dots\}$. The operators of the new basis are linear combinations of the kind

$$\mathcal{O}'_i(x) = c_{1,i}(\xi) \mathcal{O}_1 + c_{2,i}(\xi) \mathcal{O}_2 + \dots \quad (F.2)$$

and they do not mix among each other. The anomalous dimension of $\mathcal{O}'_i(x)$ is the corresponding eigenvalue of the matrix, namely $\gamma'_i$. The existence of a basis of eigenvectors for $\gamma_{ij}$-matrix is ensured by its hermiticity in unitary theories. The absence of invariance under
hermitian conjugation of (2.2) prevent to come to similar conclusions for $\chi$CFT theory. In particular, performing the planar limit can lead to “one-way” transitions

$$ \langle \mathcal{O}_i(x) \mathcal{O}_j(0)^{\dagger} \rangle \neq 0 \qquad \langle \mathcal{O}_i(x) \mathcal{O}_j(0)^{\dagger} \rangle = 0, \qquad (F.3) $$

and the correspondent mixing matrix can be only brought into Jordan canonical form, e.g. for the mixing of four primaries:

$$ \gamma_{ij} \rightarrow (S\gamma S^{-1})_{ij} = \begin{bmatrix} 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & \gamma'_3 & 0 \\ 0 & 0 & 0 & \gamma'_4 \end{bmatrix}. \qquad (F.4) $$

The matrix (F.4) contains a $2 \times 2$ Jordan block, together with two diagonal terms $\gamma'_3$ and $\gamma'_4$, corresponding to two renormalized operators with such anomalous dimensions. The physical interpretation of Jordan blocks leads to the formulation of logarithmic CFT (see [31, 61]). In the example (F.4) the block corresponds to a rank-2 logarithmic multiplet. This means that the corresponding operators of the new basis, $O'_1, O'_2$ show 2-point functions of the kind

$$ \langle \mathcal{O}'_1(x) \mathcal{O}'_1^{\dagger}(0) \rangle = \frac{k \ln(\mu^2 x^2)}{(x)^{2\Delta_0}} \qquad \langle \mathcal{O}'_1(x) \mathcal{O}'_2^{\dagger}(0) \rangle = \frac{k}{(x)^{2\Delta_0}} $$

$$ \langle \mathcal{O}'_2(x) \mathcal{O}'_1^{\dagger}(0) \rangle = \frac{k}{(x)^{2\Delta_0}} \qquad \langle \mathcal{O}'_2(x) \mathcal{O}'_2^{\dagger}(0) \rangle = 0, $$

where $\Delta_0$ is the bare dimension of $O_i$ operators, and $\mu$ the energy scale. This phenomenon, the presence of log-multiplets in $\chi$CFT has first be noticed by J.Caetano [32] for its bi-scalar reduction (2.6) and some examples of its occurrence in the context of Fishnet CFT have been presented in [33]. Despite such logarithmic operators appear in our theory, we are mostly interested in selecting the non-logarithmic ones: indeed these are the only exchanged in the OPE of the correlators under study, as the solutions of spectral equations (4.26) and (5.32) correspond to non-protected operators $\Delta(\xi) \neq \Delta_0$.

## F.1 tr[$\phi_1^2$] sector

This first sector is characterized by the Cartan R-charge of two $\phi_1$ fields (2, 0, 0). The equation (4.26) shows physical solutions for every even twist. In particular there is only one solution at twist-2 (4.29), and two at twist-4 (4.30) and higher (4.31) both for spin $S = 0$ and $S > 0$. The twist-2 solution is easily interpreted as the scaling dimension of

$$ \mathrm{tr}[\phi_1(n \cdot \partial)^S \phi_1] + \text{permutations} \quad S = 0, 2, \dots, \qquad (F.5) $$

indeed for any $S$ there is no other twist-2 conformal primary with charge (2, 0, 0). On the other hand for $\Delta_0 - S = t \ge 4$ we can list several primaries with the right set of Cartan's charges. Let us concentrate on the scalar case $S = 0$ of twist four; we find 9 scalar conformal primaries which have the right set of charges

$$ \begin{aligned} \mathcal{O}_1 &= \mathrm{tr}[\phi_1^3 \phi_1^\dagger] & \mathcal{O}_j &= \mathrm{tr}[\phi_1^2 \phi_j \phi_j^\dagger] & \mathcal{O}_{2+j} &= \mathrm{tr}[\phi_1^2 \phi_j^\dagger \phi_j] & \mathcal{O}_{4+j} &= \mathrm{tr}[\phi_1 \phi_j \phi_1 \phi_j^\dagger] \\ \mathcal{O}_8 &= \mathrm{tr}[\bar{\psi}_2 \bar{\psi}_3 \phi_1] & \mathcal{O}_9 &= \mathrm{tr}[\bar{\psi}_3 \bar{\psi}_2 \phi_1], & j &= 2, 3. \end{aligned} $$
As also the structure of (4.1) shows, this sector is fully described in terms of the $\chi_0$ CFT, thus the mixing transitions are realized by the vertices of (2.4). At any coupling $O_1$ shows no planar transitions, and we deal with a set of 8 conformal primaries which at non-zero couplings $\xi_2, \xi_3$ mix among themselves. This fact is apparently in contrast with the presence of only two twist-4 exchanged operators in the OPE expansion of section 4.5, and can be explained with the arising of logarithmic multiplets of operators with $\Delta = \Delta_0 = 4$, not being solutions of (4.26). Indeed, for instance, the following planar transitions

$$ \mathcal{O}_{4+j} \rightarrow \mathcal{O}_2, \mathcal{O}_5 \qquad \mathcal{O}_8 \rightarrow \mathcal{O}_4, \mathcal{O}_3, \qquad (F.6) $$

can happen respectively starting from order $\xi^2$ and $\xi^3$, while they lack the hermitian conjugate due to chirality of (2.4). One can actually check that there is no conjugate transition to (F.6) at any order. This suggest that matrix $\gamma_{ij}$ won't be diagonalizable and presents Jordan blocks in its canonical form, i.e. logarithmic operators.

## F.2 $\text{tr}[\phi_1\phi_2^\dagger]$ sector

In this sector the Cartan's $U(1)^{\otimes 3}$ charge is the difference between those of $\phi_1$ and $\phi_2$, that is $(1, -1, 0)$. The solutions of equation (5.32) are three in the scalar case $S = 0$, one of twist-2 (5.38) and two of twist-4 (5.39). Interestingly, for tensors $S > 0$ there is an additional solution of twist-2 (5.35). The explanation of such difference is that there is a unique way to realize a scalar $\Delta_0 = 2$ operator with $(1, -1, 0)$ charge, that is $\text{tr}[\phi_1\phi_2^\dagger]$, while for higher rank tensors $S > 0$ two operators can mix up. Starting from $S = 1$ we find that in addition to $\text{Tr}[\phi_1(n \cdot \partial)\phi_2^\dagger]$, the operator $\text{Tr}[\psi_1(n \cdot \sigma)\bar{\psi}_2] = -\text{Tr}[\bar{\psi}_2(n \cdot \bar{\sigma})\psi_1]$ satisfy the constraint of quantum numbers and behave as an $SO(4)$ vector. Therefore operator mixing happens among two conformal primaries

$$ \mathcal{O}_1 = \text{tr}[\phi_1(n \cdot \partial)^S \phi_2^\dagger] \quad \text{and} \quad \mathcal{O}_2 = \text{tr}[\psi_1(n \cdot \sigma)(n \cdot \partial)^{S-1} \bar{\psi}_2], \qquad (F.7) $$

at any $S \in 2\mathbb{N}_+$. Indeed, being interested in symmetric traceless tensors (type I [44]), no other bilinear covariant made of fermions can enter the game at higher $S$. The allowed planar transitions are:

$$ \mathcal{O}_1 \rightarrow \mathcal{O}_2 \qquad \mathcal{O}_2 \rightarrow \mathcal{O}_1, \qquad (F.8) $$

together with the “diagonal” terms $\langle \mathcal{O}_i(x) \mathcal{O}_i^\dagger(0) \rangle$, for $i = 1, 2$. Dealing with single trace operators made of two elementary fields the effect of chiral interaction is cancelled by the trace cyclicity. The resulting transitions are symmetric and the anomalous dimension matrix is diagonalizable. Recalling the couplings redefinition (5.33) and (5.37), its first perturbative order $\xi^2$ should coincide with:

$$ \gamma'_{ij} = (S\gamma S^{-1})_{ij} = \begin{pmatrix} -\frac{\lambda}{2} \left[ \lambda - \frac{\zeta}{\sqrt{S(S+1)}} \right] & 0 \\ 0 & \frac{\lambda}{2} \left[ \lambda + \frac{\zeta}{\sqrt{S(S+1)}} \right] \end{pmatrix}. \qquad (F.9) $$
As a result the two physical operators exchanged in the OPE of 5 have the form

$$ \mathcal{O}'_{\pm} = \pm c_{1,S}(\mu, \lambda) \mathcal{O}_1 + \left( \frac{1}{2} \pm c_{2,S}(\mu, \lambda) \right) \mathcal{O}_2, \qquad (\text{F.10}) $$

and correspond to solutions (5.35).

At twist-4, there are already for the scalar case $S = 0$ several local operators satisfying the constraints of bare dimension $\Delta_0 = 4$ and Cartan's charge $(1, -1, 0)$. We can compactly list them

$$ \mathcal{O}_j = \text{tr}[\phi_1 \phi_2^\dagger \phi_j \phi_j^\dagger] \qquad \mathcal{O}_{j+3} = \text{tr}[\phi_1 \phi_2^\dagger \phi_j^\dagger \phi_j] \qquad \mathcal{O}_{j+6} = \text{tr}[\phi_2^\dagger \phi_1 \phi_j \phi_j^\dagger] \qquad j = 1, 2, 3 $$

$$ \mathcal{O}_{10} = \text{tr}[\phi_1 \phi_3 \phi_2^\dagger \phi_3^\dagger] \qquad \mathcal{O}_{11} = \text{tr}[\phi_1 \phi_3^\dagger \phi_2^\dagger \phi_3] \qquad \mathcal{O}_{12} = \text{tr}[\phi_2^\dagger \phi_1 \phi_3^\dagger \phi_3] $$

$$ \mathcal{O}_{13} = \text{tr}[\bar{\psi}_2 \bar{\psi}_3 \phi_1^\dagger] \qquad \mathcal{O}_{14} = \text{tr}[\bar{\psi}_3 \bar{\psi}_2 \phi_1^\dagger] \qquad \mathcal{O}_{15} = \text{tr}[\psi_1 \psi_3 \phi_1] \qquad \mathcal{O}_{16} = \text{tr}[\psi_3 \psi_1 \phi_1]. $$

At non-zero couplings {$\xi_1, \xi_2, \xi_3$} these 16 operators mix into linear combinations obtained by the analysis of their transitons. As it happens in the first sector F.1, one can check that the effect of chiral interaction breaks the hermiticity of $\gamma_{ij}$ matrix, preventing its diagonalization. Its associated Jordan canonical form $\gamma'_{ij}$ will contain several log-multiplets of various rank, together with two diagonal elements, the anomalous dimensions (5.39).

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