$\displaystyle-\frac{\Gamma\left(J+\Delta_{\psi}\right)\Gamma\left(J+\Delta_{\phi}-\frac{1}{2}\right)\Gamma\left(J+\Delta_{\phi}+\Delta_{\psi}-1\right)}{\Gamma\left(J+\frac{1}{2}\right)\Gamma\left(\Delta_{\psi}+\frac{1}{2}\right)\Gamma\left(\Delta_{\phi}\right)\Gamma\left(2J+\Delta_{\phi}+\Delta_{\psi}-\frac{3}{2}\right)}\;,$
(4.4.8a) $\displaystyle P_{22}^{(s)}\left([\phi\partial\psi]_{0,J}\right)=$
$\displaystyle-\frac{(2J+1)2^{3-2\Delta_{\phi}}\Gamma(J+1)\Gamma\left(\Delta_{\psi}\right)\Gamma\left(J+\Delta_{\psi}\right)\left(\Delta_{\psi}+\Delta_{\phi}+J-1\right)}{\sqrt{\pi}\left(2\Delta_{\psi}-1\right)\Gamma(2J+3)\Gamma\left(2\Delta_{\psi}-2\right)\Gamma\left(\Delta_{\phi}\right)\Gamma\left(J+\Delta_{\phi}+\Delta_{\psi}-\frac{1}{2}\right)}$
$\displaystyle\times\frac{\Gamma\left(J+\Delta_{\phi}+\frac{1}{2}\right)\Gamma\left(2J+2\Delta_{\phi}+2\Delta_{\psi}-3\right)}{\Gamma\left(2J+\Delta_{\phi}+\Delta_{\psi}-\frac{1}{2}\right)}\;.$
(4.4.8b)

where we define shorthand notation232323The reason for the difference between
eqn. (4.4.9) and
$P_{\mathcal{O}}^{a,b}=(-1)^{\ell}f_{\psi_{1}\psi_{2}\mathcal{O}}^{a}f_{\psi_{3}\psi_{4}\mathcal{O}}^{b}$
as given in eqn. (3.2.41) is the difference in the conformal block
normalization used in this chapter and chapter 3 — also see footnote 11.

$\displaystyle
P^{(s)}_{ab}(\mathcal{O})\coloneqq\lambda^{a}_{\mathcal{O}_{1}\mathcal{O}_{2}\mathcal{O}}\lambda^{b}_{\mathcal{O}_{3}\mathcal{O}_{4}\mathcal{O}}\;.$
(4.4.9)

At large $J$, the leading behavior is

$\displaystyle P_{11}^{(s)}\left([\phi\psi]_{0,J}\right)\sim$
$\displaystyle-\frac{\sqrt{\pi}2^{-\Delta_{\psi}-\Delta_{\phi}-2J+\frac{5}{2}}J^{\Delta_{\psi}+\Delta_{\phi}-1}}{\Gamma\left(\Delta_{\psi}+\frac{1}{2}\right)\Gamma(\Delta_{\phi})}\;,$
(4.4.10a) $\displaystyle
P_{22}^{(s)}\left([\phi\partial\psi]_{0,J}\right)\sim$
$\displaystyle-\frac{\sqrt{\pi}\left(\Delta_{\psi}-1\right)2^{-\Delta_{\psi}-\Delta_{\phi}-2J+\frac{3}{2}}J^{\Delta_{\psi}+\Delta_{\phi}-2}}{\Gamma\left(\Delta_{\psi}+\frac{1}{2}\right)\Gamma\left(\Delta_{\phi}\right)}\;.$
(4.4.10b)

As we are working with three-point structures such that
$\lambda_{\phi\psi\mathcal{O},a}=(-1)^{J-\frac{1}{2}}\lambda_{\psi\phi\mathcal{O},a}$,
we see that

$\displaystyle\left(\lambda_{\psi\phi\mathcal{O},1}\right)^{2}\sim$
$\displaystyle-(-1)^{J-\frac{1}{2}}\frac{\sqrt{\pi}2^{-\Delta_{\psi}-\Delta_{\phi}-2J+\frac{5}{2}}J^{\Delta_{\psi}+\Delta_{\phi}-1}}{\Gamma\left(\Delta_{\psi}+\frac{1}{2}\right)\Gamma(\Delta_{\phi})}\;,$
(4.4.11a) $\displaystyle\left(\lambda_{\psi\phi\mathcal{O},2}\right)^{2}\sim$
$\displaystyle-(-1)^{J-\frac{1}{2}}\frac{\sqrt{\pi}\left(\Delta_{\psi}-1\right)2^{-\Delta_{\psi}-\Delta_{\phi}-2J+\frac{3}{2}}J^{\Delta_{\psi}+\Delta_{\phi}-2}}{\Gamma\left(\Delta_{\psi}+\frac{1}{2}\right)\Gamma\left(\Delta_{\phi}\right)}\;.$
(4.4.11b)

In the free theory limit with
$\\{\Delta_{\phi},\Delta_{\psi}\\}\rightarrow\\{\frac{1}{2},1\\}$, these
become

$\displaystyle\left(\lambda_{\psi\phi\mathcal{O},1}\right)^{2}\sim-(-1)^{J-\frac{1}{2}}4^{1-J}\sqrt{\frac{J}{\pi}}\;,\quad\left(\lambda_{\psi\phi\mathcal{O},2}\right)^{2}\sim
0\;.$ (4.4.12)

We see that this matches the results of [148] once we take the normalizations
into account. Specifically, they normalize their operators as

$\displaystyle\expectationvalue{\mathcal{O}_{\Delta,\ell}(X_{1},S_{1})\mathcal{O}_{\Delta,\ell}(X_{2},S_{2})}$
$\displaystyle=c^{\text{there}}_{\mathcal{O}}i^{2\ell}\frac{\expectationvalue{S_{1}S_{2}}^{2\ell}}{X_{12}^{2\Delta+2\ell}}\;,$
(4.4.13) $\displaystyle c^{\text{there}}_{\mathcal{O}}$
$\displaystyle=\frac{4^{\Delta-1}(-1)^{\frac{1}{2}-J}\Gamma\left(J+\frac{3}{2}\right)}{\left(\frac{1}{2}\right)_{J+\frac{1}{2}}}\;,$
(4.4.14)

while we take $c^{\text{here}}_{\mathcal{O}}=1$. Thus we have

$\displaystyle
c_{\mathcal{O}}^{\text{there}}\left(\lambda_{\psi\phi\mathcal{O},1}\right)^{2}\sim-4J\;,$
(4.4.15)

which can be compared with Eq. (4.4) of [148].

####  4.4.2 $\expectationvalue{\psi_{1}\psi_{2}\psi_{2}\psi_{1}}$

Now we turn to correlators containing four fermions. For a correlator of the
form $\expectationvalue{\psi_{1}\psi_{2}\psi_{2}\psi_{1}}$, the identity
operator only appears in the $14\rightarrow 23$ channel, hence eqn. (4.4.4)
becomes

$\displaystyle\rho_{ab}^{(s)}(\mathcal{O})=\eta^{\mathcal{O}}_{(ac)(bd)}S^{d}_{e}(\widetilde{\psi}_{2}[\widetilde{\psi}_{1}]\mathcal{O})S^{e}_{f}([\widetilde{\psi}_{2}]\psi_{1}\mathcal{O})\left(\expectationvalue{\widetilde{\psi}_{1}\widetilde{\psi}_{2}\widetilde{\mathcal{O}}}^{c},\expectationvalue{\psi_{2}\psi_{1}\mathcal{O}}^{f}\right)\;.$
(4.4.16)

When we calculate $\rho$, which we will not reproduce here, we see that it is
block-diagonal, with the upper $2\times 2$ block having poles at
$\Delta=\Delta_{\psi_{1}}+\Delta_{\psi_{2}}+J-1+2n$, and with the lower
$2\times 2$ block having poles at
$\Delta=\Delta_{\psi_{1}}+\Delta_{\psi_{2}}+J+2n$. These correspond to the
double-twist families $[\psi_{1\alpha}\psi_{2\beta}]_{n,J}$ and
$[\psi_{1\alpha}\psi_{2}^{\alpha}]_{n,J}$ respectively. We then can read off
the explicit results for OPE coefficients.

For the leading tower ($n=0$) we obtain:

$\displaystyle
P^{(s)}_{[\psi_{1\alpha}\psi_{2\beta}]_{0,J}}=-\frac{\Gamma\left(J+\Delta_{1}-\frac{1}{2}\right)\Gamma\left(J+\Delta_{2}-\frac{1}{2}\right)\Gamma\left(J+\Delta_{1}+\Delta_{2}-1\right)}{\Gamma\left(\Delta_{1}+\frac{1}{2}\right)\Gamma\left(\Delta_{2}+\frac{1}{2}\right)\Gamma(J)\Gamma\left(2J+\Delta_{1}+\Delta_{2}-2\right)}\begin{pmatrix}0&0\\\
0&1\end{pmatrix},$ (4.4.17) $\displaystyle
P^{(s)}_{[\psi_{1\alpha}\psi_{2}^{\alpha}]_{0,J}}=-\frac{\left(\Delta_{1}+\Delta_{2}+2J-1\right)\Gamma\left(J+\Delta_{1}-\frac{1}{2}\right)\Gamma\left(J+\Delta_{2}-\frac{1}{2}\right)\Gamma\left(J+\Delta_{1}+\Delta_{2}-2\right)}{4\Gamma\left(\Delta_{1}+\frac{1}{2}\right)\Gamma\left(\Delta_{2}+\frac{1}{2}\right)\Gamma(J+2)\Gamma\left(2J+\Delta_{1}+\Delta_{2}\right)}$
$\displaystyle\times\begin{pmatrix}\left(\Delta_{1}+\Delta_{2}-2\right)\left(\Delta_{1}+J-\frac{1}{2}\right)\left(\Delta_{2}+J-\frac{1}{2}\right)+c_{1}&\left(\Delta_{1}-\Delta_{2}\right)J(J+1)\left(\Delta_{1}+\Delta_{2}+J-\frac{3}{2}\right)\\\
-\left(\Delta_{1}-\Delta_{2}\right)J(J+1)\left(\Delta_{1}+\Delta_{2}+J-\frac{3}{2}\right)&\left(\Delta_{1}+\Delta_{2}-2\right)\left(\Delta_{1}+J-\frac{1}{2}\right)\left(\Delta_{2}+J-\frac{1}{2}\right)-c_{1}\end{pmatrix}\;,$

with

$c_{1}=\frac{1}{4}\left(2\Delta_{1}-1\right)\left(\Delta_{1}+\Delta_{2}-2\right)\left(2\Delta_{2}-1\right)\\\
+\left(\Delta_{1}+\Delta_{2}-2\right)J^{3}+\left(\Delta_{1}^{2}+\left(4\Delta_{2}-\frac{9}{2}\right)\Delta_{1}+\Delta_{2}^{2}-\frac{9\Delta_{2}}{2}+3\right)J^{2}\\\
+\left(\Delta_{1}+\Delta_{2}-1\right)\left(\Delta_{1}\left(2\Delta_{2}-1\right)-\Delta_{2}\right)J\;.$
(4.4.18)

Results for higher-twist towers are given in section 4.4.4.

####  4.4.3 $\expectationvalue{\psi\psi\psi\psi}$

For identical fermions $\expectationvalue{\psi\psi\psi\psi}$, both the t- and
u-channels contribute, hence we have

$\displaystyle\rho_{ab}^{(s)}(\mathcal{O})=$
$\displaystyle\eta^{\mathcal{O}}_{(ac)(bd)}S^{d}_{e}(\widetilde{\psi}[\widetilde{\psi}]\mathcal{O})S^{e}_{f}([\widetilde{\psi}]\psi\mathcal{O})\left(\expectationvalue{\widetilde{\psi}\widetilde{\psi}\widetilde{\mathcal{O}}}^{c},\expectationvalue{\psi\psi\mathcal{O}}^{f}\right)$
$\displaystyle-\eta^{\mathcal{O}}_{(ac)(bd)}S^{d}_{e}([\widetilde{\psi}]\widetilde{\psi}\mathcal{O})S^{e}_{f}(\psi[\widetilde{\psi}]\mathcal{O})\left(\expectationvalue{\widetilde{\psi}\widetilde{\psi}\widetilde{\mathcal{O}}}^{c},\expectationvalue{\psi\psi\mathcal{O}}^{f}\right).$

For the leading ($n=0$) tower, the explicit expressions for the OPE
coefficients are as follows:

$\displaystyle P^{(s)}_{[\psi_{\alpha}\psi_{\beta}]_{n,J}}=$
$\displaystyle\frac{2(-1)^{J+1}\left(\Delta_{\psi}+J-1\right)\Gamma\left(J+\Delta_{\psi}-\frac{1}{2}\right){}^{2}\Gamma\left(J+2\Delta_{\psi}-1\right)}{\Gamma(J)\Gamma\left(\Delta_{\psi}+\frac{1}{2}\right){}^{2}\Gamma\left(2J+2\Delta_{\psi}-1\right)}\begin{pmatrix}0&0\\\
0&1+(-1)^{J}\end{pmatrix},$ (4.4.19) $\displaystyle
P^{(s)}_{[\psi_{\alpha}\psi^{\alpha}]_{n,J}}=$
$\displaystyle\frac{(-1)^{J+1}2^{-2J}J\Gamma\left(\Delta_{\psi}\right)\Gamma\left(J+\Delta_{\psi}+\frac{1}{2}\right)\Gamma\left(J+2\Delta_{\psi}\right)}{\left(1-2\Delta_{\psi}\right){}^{2}\Gamma(J+1)\Gamma\left(\Delta_{\psi}-\frac{1}{2}\right)\Gamma\left(2\Delta_{\psi}-2\right)\Gamma\left(J+\Delta_{\psi}\right)}\left(\begin{array}[]{cc}\frac{1+(-1)^{J}}{J\left(J+2\Delta_{\psi}-2\right)}&0\\\
0&\frac{1-(-1)^{J}}{(J+1)\left(J+2\Delta_{\psi}-1\right)}\\\
\end{array}\right).$

After an appropriate change of basis, these results match perfectly to those
calculated using the lightcone bootstrap at large $J$ [1].

####  4.4.4 MFT Coefficients for Higher Twist Towers

In the subsections above we presented the MFT coefficients for the leading
twist towers; here we will present the results for higher twist towers.

###### For $\expectationvalue{\phi\psi\psi\phi}:$

$\displaystyle P_{11}^{(s)}\left([\phi\psi]_{n,J}\right)=$
$\displaystyle\,\frac{(2J+1)\Gamma(J+1)\cos\left(\pi\left(\Delta_{\psi}+\Delta_{\phi}\right)\right)\Gamma\left(n+\Delta_{\psi}-1\right)\Gamma\left(n+\Delta_{\phi}-\frac{1}{2}\right)\Gamma\left(J+n+\Delta_{\psi}\right)}{2^{4-2\Delta_{\phi}}\pi^{3/2}n!\Gamma\left(J+\frac{3}{2}\right)\Gamma\left(\Delta_{\psi}-1\right)\Gamma\left(\Delta_{\psi}+\frac{1}{2}\right)\Gamma\left(2\Delta_{\phi}-1\right)\Gamma(J+n+1)}$
$\displaystyle\times\frac{\left(2\Delta_{\psi}+2\Delta_{\phi}+4J+4n-3\right)\Gamma\left(J+n+\Delta_{\phi}-\frac{1}{2}\right)\Gamma\left(-2n-\Delta_{\phi}-\Delta_{\psi}+\frac{7}{2}\right)}{\Gamma\left(2J+2n+\Delta_{\phi}+\Delta_{\psi}-\frac{1}{2}\right)}$
$\displaystyle\times\frac{\Gamma\left(n+\Delta_{\phi}+\Delta_{\psi}-\frac{5}{2}\right)\Gamma\left(J+n+\Delta_{\phi}+\Delta_{\psi}-\frac{3}{2}\right)\Gamma\left(J+2n+\Delta_{\phi}+\Delta_{\psi}-1\right)}{\Gamma\left(J+2n+\Delta_{\phi}+\Delta_{\psi}-\frac{3}{2}\right)},$
(4.4.20a) $\displaystyle P_{22}^{(s)}\left([\phi\partial\psi]_{n,J}\right)=$
$\displaystyle\,P_{11}^{(s)}\left([\phi\psi]_{n,J}\right)\frac{\left(\Delta_{\psi}+n-1\right)\left(\Delta_{\phi}+J+n-\frac{1}{2}\right)\left(\Delta_{\psi}+\Delta_{\phi}+n-\frac{5}{2}\right)}{2(J+n+1)\left(\frac{1}{4}\left(2\Delta_{\psi}+2\Delta_{\phi}-5\right)+n\right)\left(\Delta_{\psi}+\Delta_{\phi}+J+2n-\frac{3}{2}\right)}$
$\displaystyle\times\frac{\left(\frac{1}{2}\left(\Delta_{\psi}+\Delta_{\phi}+J-1\right)+n\right)}{\left(J+\frac{1}{4}\left(2\Delta_{\psi}+2\Delta_{\phi}+4n-3\right)\right)}\;.$
(4.4.20b)

###### For $\expectationvalue{\psi_{1}\psi_{2}\psi_{2}\psi_{1}}:$

$P^{(s)}_{[\psi_{1\alpha}\psi_{2\beta}]_{n,J}}=-\begin{pmatrix}\frac{n\left(\Delta_{1}+\Delta_{2}+2J+2n-2\right)}{4J}c_{1}&c_{2}\\\
c_{2}&\left(\Delta_{1}+\Delta_{2}+J+2n-2\right)\left(\Delta_{1}+\Delta_{2}+2J+2n-2\right)c_{3}\end{pmatrix}\\\
\times\frac{\Gamma\left(J+n+\Delta_{2}-\frac{1}{2}\right)\Gamma\left(J+n+\Delta_{1}+\Delta_{2}-\frac{5}{2}\right)\Gamma\left(J+2n+\Delta_{1}+\Delta_{2}-3\right)}{\Gamma\left(2n+\Delta_{1}+\Delta_{2}-3\right)\Gamma\left(J+2n+\Delta_{1}+\Delta_{2}-\frac{5}{2}\right)\Gamma\left(2J+2n+\Delta_{1}+\Delta_{2}-1\right)}\\\
\times\frac{\Gamma\left(J+\frac{3}{2}\right)\Gamma\left(n+\Delta_{1}-1\right)\Gamma\left(n+\Delta_{2}-1\right)\Gamma\left(n+\Delta_{1}+\Delta_{2}-3\right)\Gamma\left(J+n+\Delta_{1}-\frac{1}{2}\right)}{(J+1)n!\Gamma\left(\Delta_{1}-1\right)\Gamma\left(\Delta_{1}+\frac{1}{2}\right)\Gamma\left(\Delta_{2}-1\right)\Gamma\left(\Delta_{2}+\frac{1}{2}\right)\Gamma(J)\Gamma\left(J+n+\frac{3}{2}\right)},$
(4.4.21)

and

$P^{(s)}_{[\psi_{1\alpha}\psi_{2}^{\alpha}]_{n,J}}=-2^{2\Delta_{1}+2\Delta_{2}-6}\begin{pmatrix}\frac{1+J}{2}d_{1}&d_{2}\\\
-d_{2}&-\frac{J}{2}d_{3}\end{pmatrix}\\\
\times\frac{\Gamma\left(J+\frac{3}{2}\right)\left(\Delta_{1}+\Delta_{2}+2J+2n-1\right)\Gamma\left(n+\Delta_{1}-1\right)\Gamma\left(n+\Delta_{2}-1\right)\Gamma\left(n+\Delta_{1}+\Delta_{2}-2\right)}{\pi\left(2\Delta_{1}-1\right)\left(2\Delta_{2}-1\right)(J+1)n!\Gamma\left(2\Delta_{1}-2\right)\Gamma\left(2\Delta_{2}-2\right)\Gamma(J+1)\Gamma\left(J+n+\frac{3}{2}\right)}\\\
\times\frac{\Gamma\left(J+n+\Delta_{1}-\frac{1}{2}\right)\Gamma\left(J+n+\Delta_{2}-\frac{1}{2}\right)\Gamma\left(J+n+\Delta_{1}+\Delta_{2}-\frac{3}{2}\right)\Gamma\left(J+2n+\Delta_{1}+\Delta_{2}-2\right)}{\Gamma\left(2n+\Delta_{1}+\Delta_{2}-2\right)\Gamma\left(J+2n+\Delta_{1}+\Delta_{2}-\frac{3}{2}\right)\Gamma\left(2J+2n+\Delta_{1}+\Delta_{2}\right)},$
(4.4.22)

where

$\displaystyle c_{1}=$ $\displaystyle
4J^{2}\left(\Delta_{1}+\Delta_{2}+n-3\right)+4J\left(\Delta_{1}+\Delta_{2}+n-3\right)\left(\Delta_{1}+\Delta_{2}+2n-2\right)$
(4.4.23)
$\displaystyle+\left(\Delta_{1}+\Delta_{2}+2n-3\right)\left(2\Delta_{1}+2\Delta_{2}+2n-5\right),$
$\displaystyle c_{2}=$ $\displaystyle
4n\left(\Delta_{1}+\Delta_{2}+n-\frac{7}{2}\right)\left(\frac{1}{2}\left(\Delta_{1}+\Delta_{2}+J-2\right)+n\right)\left(J+\frac{1}{2}\left(\Delta_{1}+\Delta_{2}+2n-2\right)\right),$
$\displaystyle c_{3}=$ $\displaystyle
J^{2}+J\left(\Delta_{1}+\Delta_{2}+2n-2\right)+(2n+1)\left(\Delta_{1}+\Delta_{2}+n-3\right),$
$\displaystyle d_{1}=$ $\displaystyle
2J^{2}\left(\Delta_{1}+\Delta_{2}+2n-2\right)+\left(2\Delta_{1}+2n-1\right)\left(\Delta_{1}+\Delta_{2}+2n-2\right)\left(2\Delta_{2}+2n-1\right)$
$\displaystyle+J\left(-9\Delta_{1}-9\Delta_{2}+2\left(\Delta_{1}^{2}+\Delta_{2}^{2}+4\Delta_{1}\Delta_{2}+6n^{2}+\left(6\Delta_{1}+6\Delta_{2}-9\right)n\right)+6\right),$
$\displaystyle d_{2}=$
$\displaystyle\left(\Delta_{1}-\Delta_{2}\right)J(J+1)\left(\Delta_{1}+\Delta_{2}+J+2n-\frac{3}{2}\right),$
$\displaystyle d_{3}=$ $\displaystyle
J\left(-11\Delta_{1}-11\Delta_{2}+2\left(\Delta_{1}^{2}+\Delta_{2}^{2}+4\Delta_{1}\Delta_{2}+6n^{2}+\left(6\Delta_{1}+6\Delta_{2}-11\right)n\right)+10\right)$
$\displaystyle+4\left(\Delta_{1}+n-1\right)\left(\Delta_{2}+n-1\right)\left(\Delta_{1}+\Delta_{2}+2n-1\right)+2J^{2}\left(\Delta_{1}+\Delta_{2}+2n-2\right)\;.$

###### For $\expectationvalue{\psi\psi\psi\psi}:$

$P^{(s)}_{[\psi_{\alpha}\psi_{\beta}]_{n,J}}=n\frac{(-1)^{J+1}\left((-1)^{J}+1\right)2^{-2J-4n+1}\Gamma\left(J+\frac{3}{2}\right)\Gamma\left(n+\Delta_{\psi}-1\right)\Gamma\left(n+2\Delta_{\psi}-3\right)}{\left(1-2\Delta_{\psi}\right)^{2}\Gamma(J+2)\Gamma(n+1)\Gamma\left(2\left(\Delta_{\psi}-1\right)\right){}^{2}\Gamma\left(J+n+\frac{3}{2}\right)}\\\
\times\frac{\Gamma\left(J+n+\Delta_{\psi}-\frac{1}{2}\right)\Gamma\left(J+n+2\Delta_{\psi}-\frac{5}{2}\right)\Gamma\left(J+2n+2\Delta_{\psi}-3\right)}{\Gamma\left(n+\Delta_{\psi}-\frac{3}{2}\right)\Gamma\left(J+n+\Delta_{\psi}-1\right)\Gamma\left(J+2n+2\Delta_{\psi}-\frac{5}{2}\right)}\begin{pmatrix}c_{1}&c_{2}\\\
c_{2}&\frac{1}{n}c_{3}\end{pmatrix},$ (4.4.24)

and

$P^{(s)}_{[\psi_{\alpha}\psi^{\alpha}]_{n,J}}=\frac{(-1)^{J}2^{-2J-4n}\Gamma\left(J+\frac{3}{2}\right)\Gamma\left(n+\Delta_{\psi}\right)\Gamma\left(n+2\Delta_{\psi}-2\right)\Gamma\left(J+2n+2\Delta_{\psi}\right)}{\left(1-2\Delta_{\psi}\right){}^{2}\Gamma(J+2)\Gamma(n+1)\Gamma\left(2\Delta_{\psi}-2\right){}^{2}\Gamma\left(J+n+\frac{3}{2}\right)\Gamma\left(J+2n+2\Delta_{\psi}-\frac{3}{2}\right)}\\\
\times\frac{\Gamma\left(J+n+\Delta_{\psi}+\frac{1}{2}\right)\Gamma\left(J+n+2\Delta_{\psi}-\frac{3}{2}\right)}{\Gamma\left(n+\Delta_{\psi}-\frac{1}{2}\right)\Gamma\left(J+n+\Delta_{\psi}\right)}\left(\begin{array}[]{cc}-\frac{\left(1+(-1)^{J}\right)(J+1)}{J+2\left(n+\Delta_{\psi}-1\right)}&0\\\
0&\frac{\left(-1+(-1)^{J}\right)J}{J+2n+2\Delta_{\psi}-1}\\\
\end{array}\right),$ (4.4.25)

where

$\displaystyle c_{1}=$ $\displaystyle
4J^{2}\left(2\Delta_{\psi}+n-3\right)+8J\left(\Delta_{\psi}+n-1\right)\left(2\Delta_{\psi}+n-3\right)+\left(2\Delta_{\psi}+2n-3\right)\left(4\Delta_{\psi}+2n-5\right),$
(4.4.26) $\displaystyle c_{2}=$ $\displaystyle
2J\left(4\Delta_{\psi}+2n-7\right)\left(J+2\left(\Delta_{\psi}+n-1\right)\right),$
$\displaystyle c_{3}=$ $\displaystyle
4J\left(J+2\left(\Delta_{\psi}+n-1\right)\right)\left(J^{2}+2J\left(\Delta_{\psi}+n-1\right)+(2n+1)\left(2\Delta_{\psi}+n-3\right)\right).$

###  4.5 CFT data of a general spectrum

In this section, we will use the techniques we developed in previous section
to calculate the anomalous dimensions of the double-twist operators exchanged
in the $\expectationvalue{\psi\phi\phi\psi}$ and
$\expectationvalue{\psi\psi\psi\psi}$ correlators. For this, we will primarily
focus on using the coefficient of the double poles in eqn. (A.2.7) to obtain
$\delta hP$, and we will divide it by $P^{\text{MFT}}$ to obtain $\delta
h=\gamma/2$.242424One can further improve these results by considering the
residues of eqn. (A.2.7) to obtain $\delta P$ as in (A.2.12) or the example in
4.5.2, with which one schematically has $\delta h=\frac{\delta
hP}{P^{\text{MFT}}+\delta P}$ up to possible mixing between different twist
towers. Necessary $\mathcal{K}$ coefficients needed for these computations are
provided in [2]. We also give an example of calculating OPE coefficient
corrections in section 4.5.2.

####  4.5.1 Applications in the computation of anomalous dimensions

As we reviewed in section 4.3.5, $6j$ symbols develop double poles when
certain relations are satisfied; in the case of eqn. (A.2.7), we have double
poles in $\Delta^{\prime}$ at
$\Delta^{\prime}=\Delta_{1}+\Delta_{2}+J^{\prime}$ if
$\Delta_{1}+\Delta_{2}=\Delta_{3}+\Delta_{4}$. Here these correspond to
double-twist operators $[\phi_{1}\phi_{2}]$. We will stick to decomposition
coefficients $\mathcal{K}$ with $\mathcal{O}_{6}^{\prime}=\mathcal{O}_{6}$, so
taking the coefficient of double poles of eqn. (A.2.7) roughly translates into
relating $\delta
hP_{[\mathcal{O}_{1}\mathcal{O}_{2}]}\evaluated{}_{\mathcal{O}_{6}}$ to
$\delta hP_{[\phi_{1}\phi_{2}]}\evaluated{}_{\mathcal{O}_{6}}$, where here
$\phi_{i}$ are some fictitious scalar operators. For later convenience, we
define

$\displaystyle\mathfrak{dp}_{1}^{J,n}(\phi_{1},\phi_{2},\mathcal{O}_{6})\equiv$
$\displaystyle\lim\limits_{\Delta\rightarrow\Delta_{1}+\Delta_{2}+J+2n}\left(\Delta-\Delta_{1}-\Delta_{2}-J-2n\right)^{2}\frac{S(\phi_{3}\phi_{4}[\widetilde{\mathcal{O}}_{5}])}{\eta^{(s)}_{\mathcal{O}_{5}}}\left(\begin{array}[]{ccc}\phi_{1}&\phi_{2}&\mathcal{O}_{6}\\\
\phi_{2}&\phi_{1}&\mathcal{O}_{\Delta,J}\end{array}\right),$ (4.5.1c)
$\displaystyle\mathfrak{dp}_{2}^{J,n}(\phi_{1},\phi_{2},\mathcal{O}_{6})\equiv$
$\displaystyle\lim\limits_{\Delta\rightarrow\Delta_{1}+\Delta_{2}+J+2n}\left(\Delta-\Delta_{1}-\Delta_{2}-J-2n\right)^{2}\frac{S(\phi_{3}\phi_{4}[\widetilde{\mathcal{O}}_{5}])}{\eta^{(s)}_{\mathcal{O}_{5}}}\left(\begin{array}[]{ccc}\phi_{1}&\phi_{2}&\mathcal{O}_{6}\\\
\phi_{1}&\phi_{2}&\mathcal{O}_{\Delta,J}\end{array}\right),$ (4.5.1f)

which describe the corrections $\delta hP$ with external scalars after the OPE
coefficients are stripped off:

$\displaystyle\delta
hP_{[\phi_{1}\phi_{2}]}\evaluated{}_{\mathcal{O}_{6}\in\phi_{1}\times\phi_{1}}=$
$\displaystyle\lambda_{\phi_{1}\phi_{1}\mathcal{O}_{6}}\lambda_{\phi_{2}\phi_{2}\mathcal{O}_{6}}\mathfrak{dp}_{1}^{J,n}(\phi_{1},\phi_{2},\mathcal{O}_{6})\;,$
(4.5.2) $\displaystyle\delta
hP_{[\phi_{1}\phi_{2}]}\evaluated{}_{\mathcal{O}_{6}\in\phi_{1}\times\phi_{2}}=$
$\displaystyle\lambda_{\phi_{1}\phi_{2}\mathcal{O}_{6}}^{2}\mathfrak{dp}_{2}^{J,n}(\phi_{1},\phi_{2},\mathcal{O}_{6})\;.$

As a reminder, in these formulas $\eta$ is the normalization for the scalar
partial wave.

The particular decomposition we will be using in eqn. (A.2.7) will be the one
with $\mathcal{O}_{6}$ kept constant, hence by taking the coefficients of
double poles on both sides, we obtain

$(\delta
hP)_{ab}([\mathcal{O}_{1}\mathcal{O}_{2}]_{n,J})\evaluated{}_{G^{(t),fg}_{\mathcal{O}_{6}}}=\\\
-\lambda_{326,f}\lambda_{146,g}\sum\limits_{\phi_{i},J^{\prime},n^{\prime}}\mathcal{K}^{fg}_{ab}\footnotesize\left(\begin{array}[]{cccccc}\mathcal{O}_{1}&\mathcal{O}_{2}&\mathcal{O}_{3}&\mathcal{O}_{4}&[\mathcal{O}_{1}\mathcal{O}_{2}]_{n,J}&\mathcal{O}_{6}\\\
\phi_{1}&\phi_{2}&\phi_{2}&\phi_{1}&[\phi_{1}\phi_{2}]_{n^{\prime},J^{\prime}}&\mathcal{O}_{6}\end{array}\right)\normalsize\mathfrak{dp}_{1}^{J^{\prime},n^{\prime}}(\phi_{1},\phi_{2},\mathcal{O}_{6})\\\
-\lambda_{326,f}\lambda_{146,g}\sum\limits_{\phi_{i},J^{\prime},n^{\prime}}\mathcal{K}^{fg}_{ab}\footnotesize\left(\begin{array}[]{cccccc}\mathcal{O}_{1}&\mathcal{O}_{2}&\mathcal{O}_{3}&\mathcal{O}_{4}&[\mathcal{O}_{1}\mathcal{O}_{2}]_{n,J}&\mathcal{O}_{6}\\\
\phi_{1}&\phi_{2}&\phi_{1}&\phi_{2}&[\phi_{1}\phi_{2}]_{n^{\prime},J^{\prime}}&\mathcal{O}_{6}\end{array}\right)\normalsize\mathfrak{dp}_{2}^{J^{\prime},n^{\prime}}(\phi_{1},\phi_{2},\mathcal{O}_{6})\;.$
(4.5.3)

For example, for the leading parity-even tower in the s-channel of
$\expectationvalue{\psi\phi\phi\psi}$, we find252525In this section, we use
the following shorthand notation for brevity:
$\phi^{a}_{i}\equiv\phi_{\Delta_{i}+a}$,
$\widetilde{\phi}^{a}_{i}\equiv\phi_{3-\Delta_{i}+a}$,
$\mathcal{O}_{i}^{a,b}\equiv\mathcal{O}_{\Delta_{i}+a,\ell_{i}+b}$, and
$\widetilde{\mathcal{O}}_{i}^{a,b}\equiv\mathcal{O}_{3-\Delta_{i}+a,\ell_{i}+b}$.

$(\delta
hP)_{11}([\phi\psi]^{+}_{0,J})\evaluated{}_{G^{(t)}_{\mathcal{O}_{6}}}=-i\lambda_{\phi\phi\mathcal{O}_{6}}\lambda_{\psi\psi\mathcal{O}_{6}}^{1}\mathfrak{dp}_{1}^{J-\frac{1}{2},0}(\psi^{\frac{1}{2}},\phi,\mathcal{O}_{6})\\\
-i\lambda_{\phi\phi\mathcal{O}_{6}}\lambda_{\psi\psi\mathcal{O}_{6}}^{2}(1-\delta_{0,\ell_{6}})\bigg{(}\frac{\left(J+\frac{1}{2}\right)(J+1)}{\left(\Delta_{6}-1\right)\ell_{6}}\mathfrak{dp}_{1}^{J+\frac{1}{2},0}(\psi^{-\frac{1}{2}},\phi,\mathcal{O}_{6})\\\
+\frac{2\left(\Delta_{\psi}+J-1\right){}^{2}\left(\Delta_{\psi}+\Delta_{\phi}+J-2\right)\left(2\Delta_{\psi}+2\Delta_{\phi}+2J-5\right)}{\left(\Delta_{6}-1\right)\ell_{6}\left(2\Delta_{\psi}+2\Delta_{\phi}+4J-5\right)\left(2\Delta_{\psi}+2\Delta_{\phi}+4J-3\right)}\mathfrak{dp}_{1}^{J-\frac{1}{2},0}(\psi^{-\frac{1}{2}},\phi,\mathcal{O}_{6})\\\
+\frac{\left(-2\Delta_{\psi}-\Delta_{6}+\ell_{6}+4\right)\left(2\Delta_{\psi}-\Delta_{6}+\ell_{6}-1\right)}{4\left(\Delta_{6}-1\right)\ell_{6}}\mathfrak{dp}_{1}^{J-\frac{1}{2},0}(\psi^{\frac{1}{2}},\phi,\mathcal{O}_{6})\bigg{)}\;.$
(4.5.4)

For the parity-odd tower, we instead have

$(\delta
hP)_{22}\left([\phi\psi]^{-}_{0,J}\right)\evaluated{}_{G^{(t)}_{\mathcal{O}_{6}}}=i\lambda_{\phi\phi\mathcal{O}_{6}}\lambda_{\psi\psi\mathcal{O}_{6}}^{1}\bigg{(}\frac{J+\frac{1}{2}}{J+1}\mathfrak{dp}_{1}^{J+\frac{1}{2},0}(\psi^{\frac{1}{2}},\phi,\mathcal{O}_{6})\\\
+\frac{2\left(2\Delta_{\phi}+2J-1\right)^{2}\left(\Delta_{\psi}+\Delta_{\phi}+J-1\right)}{\left(2\Delta_{\psi}+2\Delta_{\phi}+2J-3\right)\left(2\Delta_{\psi}+2\Delta_{\phi}+4J-3\right)\left(2\Delta_{\psi}+2\Delta_{\phi}+4J-1\right)}\mathfrak{dp}_{1}^{J-\frac{1}{2},0}(\psi^{\frac{1}{2}},\phi,\mathcal{O}_{6})\bigg{)}\\\
+i\lambda_{\phi\phi\mathcal{O}_{6}}\lambda_{\psi\psi\mathcal{O}_{6}}^{2}(1-\delta_{0,\ell_{6}})\bigg{(}\frac{1}{\left(\Delta_{6}-1\right)\ell_{6}}\mathfrak{dp}_{1}^{J-\frac{1}{2},1}(\psi^{-\frac{1}{2}},\phi,\mathcal{O}_{6})\\\
+\frac{(2J+1)\left(-2\Delta_{\psi}-\Delta_{6}+\ell_{6}+4\right)\left(2\Delta_{\psi}-\Delta_{6}+\ell_{6}-1\right)}{8\left(\Delta_{6}-1\right)(J+1)\ell_{6}}\mathfrak{dp}_{1}^{J+\frac{1}{2},0}(\psi^{\frac{1}{2}},\phi,\mathcal{O}_{6})\\\
+\frac{\left(J+\frac{1}{2}\right)\left(\Delta_{\psi}-1\right){}^{2}\left(\Delta_{\psi}+\Delta_{\phi}-\frac{5}{2}\right)\left(\Delta_{\psi}+\Delta_{\phi}+J-1\right)}{\left(\Delta_{6}-1\right)(J+1)\ell_{6}\left(\Delta_{\psi}+\Delta_{\phi}-\frac{3}{2}\right)\left(\Delta_{\psi}+\Delta_{\phi}+J-\frac{3}{2}\right)}\mathfrak{dp}_{1}^{J-\frac{1}{2},0}(\psi^{\frac{1}{2}},\phi,\mathcal{O}_{6})\\\
+\frac{\left(2\Delta_{\phi}+2J-1\right){}^{2}\left(-2\Delta_{\psi}-\Delta_{6}+\ell_{6}+4\right)\left(2\Delta_{\psi}-\Delta_{6}+\ell_{6}-1\right)\left(\Delta_{\psi}+\Delta_{\phi}+J-1\right)}{2\left(\Delta_{6}-1\right)\ell_{6}\left(2\Delta_{\psi}+2\Delta_{\phi}+2J-3\right)\left(2\Delta_{\psi}+2\Delta_{\phi}+4J-3\right)\left(2\Delta_{\psi}+2\Delta_{\phi}+4J-1\right)}\mathfrak{dp}_{1}^{J-\frac{1}{2},0}(\psi^{\frac{1}{2}},\phi,\mathcal{O}_{6})\bigg{)}\;.$
(4.5.5)

We would like to point out the appearance of
$\mathfrak{dp}_{1}^{J-\frac{1}{2},1}(\psi^{-\frac{1}{2}},\phi,\mathcal{O}_{6})$
which indicates that we may need to extract the data of _non-leading twist_
scalar towers from the scalar $6j$ symbol in order to obtain the _leading
twist_ spinning towers. This happens for the cases where the scaling
dimensions of $\mathcal{O}_{1,2}$ and $\ell_{5}$ are shifted downwards while
the scaling dimension of $\mathcal{O}_{5}$ is shifted upwards. We also note
that the absence of $\lambda_{\psi\psi\mathcal{O}_{6}}^{3,4}$ follows from our
parity-definite choice of three-point structures.262626The OPE coefficients
$\lambda_{\psi\psi\mathcal{O}_{6}}^{3,4}$ can only appear through the product
of $3-$point structures
$\expectationvalue{\phi\phi\mathcal{O}_{6}}\expectationvalue{\psi\psi\mathcal{O}_{6}}^{3,4}$,
which are parity-odd under $X\rightarrow-X$ in embedding space (see footnote 5
as well). In the s-channel this can only match to the non-diagonal pieces
$\expectationvalue{\psi\phi\mathcal{O}_{5}}^{1,2}\expectationvalue{\psi\phi\mathcal{O}_{5}}^{2,1}$
for which no double pole appears: $(\delta
hP)_{12}\left([\phi\psi]_{0,J}\right)=(\delta
hP)_{21}\left([\phi\psi]_{0,J}\right)=0$.

Using the $\mathcal{K}$ coefficients given explicitly in the attached
Mathematica notebook of [2], one can similarly obtain all cases for four
fermions as well. For brevity, we reproduce here a few cases for $\ell_{6}=0$:

$(\delta
hP)_{22}\left([\psi\psi]^{+}_{0,J}\right)\evaluated{}_{G^{(t)}_{\phi_{6}}}=\lambda_{\psi\psi\mathcal{O}_{6}}^{1}\lambda_{\psi\psi\mathcal{O}_{6}}^{1}\mathfrak{dp}_{1}^{J-1,0}(\psi^{\frac{1}{2}},\psi^{\frac{1}{2}},\phi_{6})\\\
+\lambda_{\psi\psi\mathcal{O}_{6}}^{3}\lambda_{\psi\psi\phi_{6}}^{3}\bigg{(}\frac{2J}{\left(\Delta_{6}-1\right)^{2}}\mathfrak{dp}_{2}^{J,0}(\psi^{-\frac{1}{2}},\psi^{\frac{1}{2}},\phi_{6})\\\
+\frac{\left(\Delta_{\psi}+\frac{J-2}{2}\right)\left(\Delta_{\psi}+J-\frac{3}{2}\right)}{\left(\Delta_{6}-1\right){}^{2}\left(\Delta_{\psi}+J-1\right)}\mathfrak{dp}_{2}^{J-1,0}(\psi^{-\frac{1}{2}},\psi^{\frac{1}{2}},\phi_{6})\bigg{)}\;,$
(4.5.6)

and

$\displaystyle(\delta hP)_{11}\left([\psi\psi]^{+}_{0,J}\right)=(\delta
hP)_{12}\left([\psi\psi]^{+}_{0,J}\right)=(\delta
hP)_{21}\left([\psi\psi]^{+}_{0,J}\right)=0\;.$ (4.5.7)

Unlike the parity-even case, parity-odd families have non-zero off-diagonal
components:

$\displaystyle(\delta
hP)_{34}\left([\psi\psi]^{+}_{0,J}\right)\evaluated{}_{G^{(t)}_{\phi_{6}}}=\frac{\lambda_{\psi\psi\phi_{6}}^{3}\lambda_{\psi\psi\phi_{6}}^{3}}{(\Delta_{6}-1)^{2}}\bigg{(}-\mathfrak{dp}_{2}^{J-1,1}(\psi^{-\frac{1}{2}},\psi^{\frac{1}{2}},\phi_{6})+\frac{J\left(J+1\right)\left(2J+3\right)}{2J+1}\mathfrak{dp}_{2}^{J+1,0}(\psi^{-\frac{1}{2}},\psi^{\frac{1}{2}},\phi_{6})$
(4.5.8)
$\displaystyle\qquad+\frac{\left(2\Delta_{\psi}+J-2\right)\left(2\Delta_{\psi}+J-1\right)\left(2\Delta_{\psi}+2J-3\right)\left(2\Delta_{\psi}+2J-1\right)\left(4\Delta_{\psi}+2J-5\right)}{64\left(\Delta_{\psi}+J-1\right){}^{2}\left(4\Delta_{\psi}+2J-3\right)}\mathfrak{dp}_{2}^{J-1,0}(\psi^{-\frac{1}{2}},\psi^{\frac{1}{2}},\phi_{6})$
$\displaystyle\qquad+\frac{J\left(2\Delta_{\psi}+J-1\right)\left(8\Delta_{\psi}^{3}+2\left(4J^{2}-8J+3\right)\Delta_{\psi}-4J^{2}+4(4J-3)\Delta_{\psi}^{2}+4J-3\right)}{8\left(2\Delta_{\psi}-1\right)\left(\Delta_{\psi}+J-1\right)\left(\Delta_{\psi}+J\right)}\mathfrak{dp}_{2}^{J,0}(\psi^{-\frac{1}{2}},\psi^{\frac{1}{2}},\phi_{6})\bigg{)},$
$\displaystyle(\delta
hP)_{43}\left([\psi\psi]^{+}_{0,J}\right)\evaluated{}_{G^{(t)}_{\phi_{6}}}=(\delta
hP)_{34}\left([\psi\psi]^{+}_{0,J}\right)\evaluated{}_{G^{(t)}_{\phi_{6}}}\,,$

where one can write down $(\delta
hP)_{33}\left([\psi\psi]^{+}_{0,J}\right)\neq 0$ and $(\delta
hP)_{44}\left([\psi\psi]^{+}_{0,J}\right)\neq 0$ as well. However, all non-
block-diagonal terms such as $(\delta
hP)_{13}\left([\psi\psi]^{+}_{0,J}\right)$ are zero as the corresponding
structures do not develop double poles. This is why we do not have terms with
the mixed coefficient
$\lambda_{\psi\psi\mathcal{O}_{6}}^{1}\lambda_{\psi\psi\mathcal{O}_{6}}^{3}$.

To calculate the scalar coefficients $\mathfrak{dp}_{i}^{J,0}$ we will use the
Lorentzian inversion formula [127, 133, 1], combined with either dimensional
reduction of the 3d block [215, 1] or resummations of the lightcone expansion
[232]. Using dimensional reduction we find for the $n=0$ double-twist
operators:

$\displaystyle\mathfrak{dp}_{1}^{J,0}(\phi_{1},\phi_{2},\mathcal{O}_{6})$
$\displaystyle=-\sum\limits_{p=0}^{\infty}\sum\limits_{q=\text{max}(-p,p-2J_{6})}^{p}2\kappa^{0,0}_{2\bar{h}}\sin(\pi(h_{6}-2h_{1}))\sin(\pi(h_{6}-2h_{2}))$
$\displaystyle\hskip
113.81102pt\times\mathcal{A}^{0,0}_{p,q}\Omega^{h_{1}h_{2}h_{2}h_{1}}_{\bar{h},h_{6}+p,2h_{2}}\frac{\Gamma(2(\bar{h}+q))}{\Gamma^{2}(\bar{h}+q)}\bigg{|}_{\bar{h}=h_{1}+h_{2}+J},$
(4.5.9)
$\displaystyle\mathfrak{dp}_{2}^{J,0}(\phi_{1},\phi_{2},\mathcal{O}_{6})$
$\displaystyle=-\sum\limits_{p=0}^{\infty}\sum\limits_{q=\text{max}(-p,p-2J_{6})}^{p}2\kappa^{h_{21},h_{12}}_{2\bar{h}}\sin^{2}(\pi(h_{6}-h_{1}-h_{2}))$
$\displaystyle\times\mathcal{A}^{h_{21},h_{12}}_{p,q}\Omega^{h_{1}h_{2}h_{1}h_{2}}_{\bar{h},h_{6}+p,h_{1}+h_{2}}\frac{\Gamma(2(\bar{h}+q))}{\Gamma(h_{12}+\bar{h}+q)\Gamma(h_{21}+\bar{h}+q)}\bigg{|}_{\bar{h}=h_{1}+h_{2}+J}.$
(4.5.10)

Here we have defined $h=\frac{1}{2}(\Delta-J)$,
$\bar{h}=\frac{1}{2}(\Delta+J)$, $h_{ij}=h_{i}-h_{j}$, and

$\displaystyle\kappa^{a,b}_{2\bar{h}}=\frac{\Gamma(\bar{h}+a)\Gamma(\bar{h}-a)\Gamma(\bar{h}+b)\Gamma(\bar{h}-b)}{2\pi^{2}\Gamma(2\bar{h})\Gamma(2\bar{h}-1)}\,.$
(4.5.11)

The coefficients $\mathcal{A}^{a,b}_{p,q}$ come from performing dimensional
reduction for 3d blocks in terms of the chiral, 2d blocks and were found for
$a=b=0$ in [215].272727In comparison to [215] we use
$\mathcal{A}^{\text{here}}_{p,q}=\mathcal{A}^{\text{there}}_{\frac{p_{q}}{2},\bar{h}-h+q-p}$,
and for Eq. (2.35) there we use
$c^{(d)}_{\ell}=\frac{(d-2)_{\ell}}{\left(\frac{d-2}{2}\right)_{\ell}}$. For
general $a$ and $b$ we can compute $\mathcal{A}^{a,b}_{h,\bar{h}}$ recursively
using the Casimir equation. For explicit results, we will mainly be interested
in the large spin asymptotics, in which case we can restrict to $p=q=0$ and
use $\mathcal{A}^{a,b}_{0,0}=1$. Finally, the function $\Omega$ is given by
[133]:

$\displaystyle\Omega^{h_{1},h_{2},h_{3},h_{4}}_{h_{5},h_{6},p}$
$\displaystyle=\frac{\Gamma(2h_{5})\Gamma(h_{6}-p+1)\Gamma(h_{5}+h_{12}-h_{6}+p-1)\Gamma(-h_{12}+h_{34}+h_{6}-p+1)}{\Gamma(h_{5}+h_{12})\Gamma(h_{5}+h_{34})\Gamma(h_{5}-h_{12}+h_{6}-p+1)}$
$\displaystyle\hskip
28.45274pt{}_{4}F_{3}\biggl{[}\genfrac{.}{.}{0.0pt}{}{h_{23}+h_{6}\mskip
8.0muh_{6}-h_{14}\mskip 8.0mu-h_{12}+h_{34}+h_{6}-p+1\mskip
8.0muh_{6}-p+1}{2h_{6}\mskip 8.0muh_{5}-h_{12}+h_{6}-p+1\mskip 8.0mu-
h_{5}-h_{12}+h_{6}-p+2};1\biggr{]}$
$\displaystyle+\frac{\Gamma(2h_{6})\Gamma(h_{5}+h_{13}+p-1)\Gamma(h_{5}+h_{42}+p-1)\Gamma(-h_{5}-h_{12}+h_{6}-p+1)}{\Gamma(h_{6}-h_{14})\Gamma(h_{23}+h_{6})\Gamma(h_{5}+h_{12}+h_{6}+p-1)}$
$\displaystyle\hskip
28.45274pt{}_{4}F_{3}\biggl{[}\genfrac{.}{.}{0.0pt}{}{h_{5}+h_{13}+p-1\mskip
8.0muh_{5}+h_{42}+p-1\mskip 8.0muh_{5}+h_{34}\mskip
8.0muh_{5}+h_{12}}{h_{5}+h_{12}+h_{6}+p-1\mskip 8.0mu2h_{5}\mskip
8.0muh_{5}+h_{12}-h_{6}+p};1\biggr{]}\,.$ (4.5.12)

When we study double-twist operators with large spin, or equivalently large
$\bar{h}$, the first ${}_{4}F_{3}$ hypergeometric yields the asymptotic,
large-spin prediction while the second ${}_{4}F_{3}$ gives effects which are
exponentially suppressed.

By inserting eqn. (4.5.1) and eqn. (4.5.1) into eqn. (4.5.3), we can obtain
$\delta hP$ for various double-twist operators of fermions. Below we will
consider some examples.

#####  $\delta hP$ of double-twist towers
$[\phi\psi_{\alpha}]^{\pm}_{0,\ell_{5}}$ due to scalar exchange:

From eqn. (4.5.4) and eqn. (4.5.5), we find that

$(\delta
hP)_{11}^{(p)}=-\lambda_{\phi\phi\phi_{6}}\lambda_{\psi\psi\phi_{6}}^{1}\frac{(-1)^{\ell_{5}+1}\Gamma\left(\Delta_{6}\right)\left(\sin\left(\pi\left(-\Delta_{\psi}-\Delta_{\phi}+\Delta_{6}\right)\right)+\sin\left(\pi\left(\Delta_{\psi}-\Delta_{\phi}\right)\right)\right)}{\Gamma\left(\frac{\Delta_{6}}{2}\right)}\\\
\times\Omega^{\frac{2\Delta_{\psi}+1}{4},\frac{\Delta_{\phi}}{2},\frac{\Delta_{\phi}}{2},\frac{2\Delta_{\psi}+1}{4}}_{\frac{2\Delta_{\psi}+2\Delta_{\phi}+4\ell_{5}-1}{4},\frac{\Delta_{6}}{2},\Delta_{\phi}}$
(4.5.13a) and $(\delta
hP)_{22}^{(p)}=-\lambda_{\phi\phi\phi_{6}}\lambda_{\psi\psi\phi_{6}}^{1}\frac{(-1)^{\ell_{5}}\Gamma\left(\Delta_{6}\right)\cos\left(\pi\left(\Delta_{\psi}-\frac{\Delta_{6}}{2}\right)\right)\sin\left(\pi\left(\Delta_{\phi}-\frac{\Delta_{6}}{2}\right)\right)}{\Gamma\left(\frac{\Delta_{6}}{2}\right)^{2}}\\\
\times\Bigg{(}\frac{\left(2\ell_{5}+1\right)}{\ell_{5}+1}\Omega^{\frac{2\Delta_{\psi}+1}{4},\frac{\Delta_{\phi}}{2},\frac{\Delta_{\phi}}{2},\frac{2\Delta_{\psi}+1}{4}}_{\frac{2\Delta_{\psi}+2\Delta_{\phi}+4\ell_{5}+3}{4},\frac{\Delta_{6}}{2},\Delta_{\phi}}\\\
-\frac{4\left(2\Delta_{\phi}+2\ell_{5}-1\right){}^{2}\left(\Delta_{\psi}+\Delta_{\phi}+\ell_{5}-1\right)}{\left(2\Delta_{\psi}+2\Delta_{\phi}+2\ell_{5}-3\right)\left(2\Delta_{\psi}+2\Delta_{\phi}+4\ell_{5}-3\right)\left(2\Delta_{\psi}+2\Delta_{\phi}+4\ell_{5}-1\right)}\Omega^{\frac{2\Delta_{\psi}+1}{4},\frac{\Delta_{\phi}}{2},\frac{\Delta_{\phi}}{2},\frac{2\Delta_{\psi}+1}{4}}_{\frac{2\Delta_{\psi}+2\Delta_{\phi}+4\ell_{5}-1}{4},\frac{\Delta_{6}}{2},\Delta_{\phi}}\Bigg{)}\,.$
(4.5.13b)

By expanding the perturbative terms at large $\ell$, we can obtain the leading
order behavior

$\displaystyle(\delta
hP)_{11}([\phi\psi]^{+}_{0,\ell_{5}})\evaluated{}_{G^{(t)}_{\phi_{6}}}\sim{}$
$\displaystyle{}\lambda_{\phi\phi\phi_{6}}\lambda_{\psi\psi\phi_{6}}^{1}\frac{\sqrt{\pi}(-1)^{\ell_{5}+1}\Gamma\left(\Delta_{6}\right)2^{-\Delta_{\psi}-\Delta_{\phi}-2\ell_{5}+\frac{5}{2}}\ell_{5}^{\Delta_{\psi}+\Delta_{\phi}-\Delta_{6}-1}}{\Gamma\left(\frac{\Delta_{6}}{2}\right){}^{2}\Gamma\left(-\frac{\Delta_{6}}{2}+\Delta_{\psi}+\frac{1}{2}\right)\Gamma\left(\Delta_{\phi}-\frac{\Delta_{6}}{2}\right)},$
(4.5.14a) $\displaystyle(\delta
hP)_{22}([\phi\psi]^{-}_{0,\ell_{5}})\evaluated{}_{G^{(t)}_{\phi_{6}}}\sim{}$
$\displaystyle{}\frac{\lambda_{\phi\phi\phi_{6}}\lambda_{\psi\psi\phi_{6}}^{1}(-1)^{\ell_{5}}\left(-2\Delta_{\psi}+\Delta_{6}+2\right)\Gamma\left(\frac{1}{2}\left(\Delta_{6}+1\right)\right)\ell_{5}^{\Delta_{\psi}+\Delta_{\phi}-\Delta_{6}-2}}{2^{\Delta_{\psi}+\Delta_{\phi}-\Delta_{6}+2\ell_{5}+\frac{1}{2}}\Gamma\left(\frac{\Delta_{6}}{2}\right)\Gamma\left(-\frac{\Delta_{6}}{2}+\Delta_{\psi}+\frac{1}{2}\right)\Gamma\left(\Delta_{\phi}-\frac{\Delta_{6}}{2}\right)}.$
(4.5.14b)

By dividing these by the MFT coefficients at large $\ell$ give in eqn.
(4.4.10), we obtain the anomalous dimensions at leading order:

$\displaystyle\gamma_{[\phi\psi]^{+}_{0,\ell_{5}}}\evaluated{}_{G^{(t)}_{\phi_{6}}}={}$
$\displaystyle{}\frac{2}{P_{11}^{(s)}}(\delta
hP)_{11}\evaluated{}_{G^{(t)}_{\phi_{6}}}=\frac{i\lambda_{\phi\phi\phi_{6}}\lambda_{\psi\psi\phi_{6}}^{1}}{\ell_{5}^{\Delta_{6}}}\frac{2^{\Delta_{6}}\Gamma\left(\frac{1}{2}\left(\Delta_{6}+1\right)\right)\Gamma\left(\Delta_{\psi}+\frac{1}{2}\right)\Gamma\left(\Delta_{\phi}\right)}{\sqrt{\pi}\Gamma\left(\frac{\Delta_{6}}{2}\right)\Gamma\left(\frac{1}{2}\left(-\Delta_{6}+2\Delta_{\psi}+1\right)\right)\Gamma\left(\frac{1}{2}\left(2\Delta_{\phi}-\Delta_{6}\right)\right)},$
(4.5.15a)
$\displaystyle\gamma_{[\phi\psi]^{-}_{0,\ell_{5}}}\evaluated{}_{G^{(t)}_{\phi_{6}}}={}$
$\displaystyle{}\frac{2}{P_{22}^{(s)}}(\delta
hP)_{22}\evaluated{}_{G^{(t)}_{\phi_{6}}}=-\frac{i\lambda_{\phi\phi\phi_{6}}\lambda_{\psi\psi\phi_{6}}^{1}}{\ell_{5}^{\Delta_{6}}}\frac{2^{\Delta_{6}-1}\left(\frac{\Delta_{6}}{2}-\Delta_{\psi}+1\right)\Gamma\left(\frac{\Delta_{6}+1}{2}\right)\Gamma\left(\Delta_{\psi}+\frac{1}{2}\right)\Gamma\left(\Delta_{\phi}\right)}{\sqrt{\pi}\left(\Delta_{\psi}-1\right)\Gamma\left(\frac{\Delta_{6}}{2}\right)\Gamma\left(\frac{-\Delta_{6}+2\Delta_{\psi}+1}{2}\right)\Gamma\left(\frac{2\Delta_{\phi}-\Delta_{6}}{2}\right)}.$
(4.5.15b)

#####  $\delta hP$ of parity-even double-twist tower
$[\phi\psi_{\alpha}]^{+}_{0,\ell_{5}}$ due to stress tensor exchange:

By inserting eqn. (4.5.1) and eqn. (4.5.1) into eqn. (4.5.4), we obtain

$(\delta
hP)_{11}([\phi\psi]^{+}_{0,\ell_{5}})\evaluated{}_{G^{(t)}_{T}}=\frac{3\Delta_{\phi}\sin\left(\pi\ell_{5}\right)\Gamma\left(\ell_{5}+\Delta_{\psi}\right)\Gamma\left(\ell_{5}+\Delta_{\phi}-\frac{1}{2}\right)\Gamma\left(\ell_{5}+\Delta_{\phi}+\Delta_{\psi}-\frac{3}{2}\right)}{2\sqrt{2}\pi^{2}c_{T}\Gamma\left(\ell_{5}+1\right)\Gamma\left(\Delta_{\psi}-1\right)\Gamma\left(\Delta_{\phi}-\frac{1}{2}\right)\Gamma\left(2\ell_{5}+\Delta_{\phi}+\Delta_{\psi}-\frac{1}{2}\right)}\\\
\times\Bigg{(}\left(2\Delta_{\psi}+1\right)\left(2\Delta_{\psi}+2\Delta_{\phi}+4\ell_{5}-3\right){}_{4}F_{3}\biggl{[}\genfrac{.}{.}{0.0pt}{}{\frac{1}{2}\mskip
8.0mu\frac{1}{2}\mskip 8.0mu\frac{3}{2}-\Delta_{\phi}\mskip
8.0mu1-\Delta_{\psi}}{1\mskip 8.0mu\ell_{5}+1\mskip
8.0mu-\ell_{5}-\Delta_{\phi}-\Delta_{\psi}+\frac{5}{2}};1\biggr{]}\\\
-4\left(\Delta_{\psi}+\ell_{5}-1\right)\left(\Delta_{\psi}+\Delta_{\phi}+\ell_{5}-2\right){}_{4}F_{3}\biggl{[}\genfrac{.}{.}{0.0pt}{}{\frac{1}{2}\mskip
8.0mu\frac{1}{2}\mskip 8.0mu\frac{3}{2}-\Delta_{\phi}\mskip
8.0mu2-\Delta_{\psi}}{1\mskip 8.0mu\ell_{5}+1\mskip
8.0mu-\ell_{5}-\Delta_{\phi}-\Delta_{\psi}+\frac{7}{2}};1\biggr{]}\\\
+4\left(\ell_{5}+\frac{1}{2}\right)\left(\Delta_{\phi}+\ell_{5}-\frac{1}{2}\right){}_{4}F_{3}\biggl{[}\genfrac{.}{.}{0.0pt}{}{\frac{1}{2}\mskip
8.0mu\frac{1}{2}\mskip 8.0mu\frac{3}{2}-\Delta_{\phi}\mskip
8.0mu2-\Delta_{\psi}}{1\mskip 8.0mu\ell_{5}+2\mskip
8.0mu-\ell_{5}-\Delta_{\phi}-\Delta_{\psi}+\frac{5}{2}};1\biggr{]}\Bigg{)}\\\
+\text{ (non-perturbative terms)},$ (4.5.16)

where we set

$\displaystyle\ell_{6}=2\;,\quad\Delta_{6}=3\;,\quad\lambda_{\phi\phi
T}=-\frac{3\Delta_{\phi}\Gamma\left(\frac{3}{2}\right)}{2(2\pi)^{3/2}\sqrt{c_{T}}}\;,\quad\lambda_{\psi\psi
T}^{1}=\frac{3i(\Delta_{\psi}-1)}{4\sqrt{c_{T}}}\;,\quad\lambda_{\psi\psi
T}^{2}=-\frac{3i}{2\sqrt{c_{T}}},$ (4.5.17)

where $c_{T}$ is the central charge, which here is defined as the
normalization of the stress tensor two-point function. At large spin, the
leading order term is then

$\displaystyle(\delta
hP)_{11}([\phi\psi]^{+}_{0,\ell_{5}})\evaluated{}_{G^{(t)}_{T}}\sim\frac{3i(-1)^{\ell_{5}}\Delta_{\phi}2^{-\Delta_{\psi}-\Delta_{\phi}-2\ell_{5}+4}\ell_{5}^{\Delta_{\psi}+\Delta_{\phi}-2}}{\pi^{3/2}c_{T}\Gamma\left(\Delta_{\psi}-1\right)\Gamma\left(\Delta_{\phi}-\frac{1}{2}\right)},$
(4.5.18)

and we obtain the anomalous dimension via eqn. (4.4.10):

$\displaystyle\gamma_{[\phi\psi]^{+}_{0,\ell_{5}}}\evaluated{}_{G^{(t)}_{T}}=\frac{2}{P_{11}^{(s)}}(\delta
hP)_{11}\evaluated{}_{G^{(t)}_{T}}=\frac{1}{\ell_{5}}\frac{3\sqrt{2}\Gamma\left(\Delta_{\psi}+\frac{1}{2}\right)\Gamma\left(\Delta_{\phi}+1\right)}{\pi^{2}c_{T}\Gamma\left(\Delta_{\psi}-1\right)\Gamma\left(\Delta_{\phi}-\frac{1}{2}\right)}.$
(4.5.19)

#####  $\delta hP$ of parity-even double-twist tower
$[\psi\psi]^{+}_{0,\ell_{5}}$ due to the exchange of a generic parity-even
operator or parity-odd scalar:

Similar to the previous examples, we obtain the relevant $\delta hP$ by
inserting eqn. (4.5.1) and eqn. (4.5.1) into eqn. (4.5.6). As the expressions
are quite lengthy, we will not reproduce the full results but instead present
their asymptotic forms at large spin. We find that

$\displaystyle(\delta
hP)_{11}([\psi\psi]^{+}_{0,\ell_{5}})\evaluated{}_{G^{(t)}_{\mathcal{O}_{6}}}={}$
$\displaystyle{}(\delta
hP)_{12}([\psi\psi]^{+}_{0,\ell_{5}})\evaluated{}_{G^{(t)}_{\mathcal{O}_{6}}}=(\delta
hP)_{21}([\psi\psi]^{+}_{0,\ell_{5}})\evaluated{}_{G^{(t)}_{\mathcal{O}_{6}}}=0,$
(4.5.20) $\displaystyle(\delta
hP)_{22}([\psi\psi]^{+}_{0,\ell_{5}})\evaluated{}_{G^{(t)}_{\mathcal{O}_{6}}}\sim{}$
$\displaystyle{}\left(\lambda_{\psi\psi\mathcal{O}_{6}}^{1}\right)^{2}\frac{(-1)^{\ell_{5}+1}2^{-2\Delta_{\psi}+\Delta_{6}-2\ell_{5}+\ell_{6}+2}\ell_{5}^{2\Delta_{\psi}-\Delta_{6}+\ell_{6}-\frac{1}{2}}\Gamma\left(\frac{1}{2}\left(\ell_{6}+\Delta_{6}+1\right)\right)}{\Gamma\left(\frac{1}{2}\left(\ell_{6}+\Delta_{6}\right)\right)\Gamma\left(\frac{1}{2}\left(\ell_{6}-\Delta_{6}+1\right)+\Delta_{\psi}\right)^{2}},$

where contributions due to $\lambda_{\psi\psi\mathcal{O}_{6}}^{2,3,4}$ come at
subleading order.282828We see that this result matches the one calculated
using lightcone bootstap methods in [1], once the change of basis and the
difference in conformal block normalization is taken into account; compare
eqn. (4.5.20) here with (3.30c) there. By dividing by the MFT coefficients
given in eqn. (4.4.19), we obtain the anomalous dimension at large spin:

$\displaystyle\gamma_{[\psi\psi]^{+}_{0,\ell_{5}}}\evaluated{}_{G^{(t)}_{\mathcal{O}_{6}}}=\frac{2}{P_{22}^{(s)}}(\delta
hP)_{22}\evaluated{}_{G^{(t)}_{\mathcal{O}_{6}}}=\frac{\left(\lambda_{\psi\psi\mathcal{O}_{6}}^{1}\right)^{2}}{\ell_{5}^{\Delta_{6}-\ell_{6}}}\frac{2^{\Delta_{6}+\ell_{6}}\Gamma\left(\Delta_{\psi}+\frac{1}{2}\right){}^{2}\Gamma\left(\frac{1}{2}\left(\ell_{6}+\Delta_{6}+1\right)\right)}{\sqrt{\pi}\Gamma\left(\frac{1}{2}\left(\ell_{6}+\Delta_{6}\right)\right)\Gamma\left(\frac{1}{2}\left(\ell_{6}-\Delta_{6}+1\right)+\Delta_{\psi}\right)^{2}}\;.$
(4.5.21)

For example, for stress tensor exchange we can impose eqn. (4.5.17) which
yields

$\displaystyle\gamma_{[\psi\psi]^{+}_{0,\ell_{5}}}\evaluated{}_{G^{(t)}_{T}}=-\frac{1}{\ell_{5}}\frac{48\Gamma\left(\Delta_{\psi}+\frac{1}{2}\right)^{2}}{\pi
c_{T}\Gamma\left(\Delta_{\psi}-1\right)^{2}}\;,$ (4.5.22)

whereas for parity-even scalar exchange it becomes

$\displaystyle\gamma_{[\psi\psi]^{+}_{0,\ell_{5}}}\evaluated{}_{G^{(t)}_{\phi_{6}}}=\frac{\left(\lambda_{\psi\psi\mathcal{O}_{6}}^{1}\right)^{2}}{\ell_{5}^{\Delta_{6}}}\frac{2^{\Delta_{6}}\Gamma\left(\frac{\Delta_{6}+1}{2}\right)\Gamma\left(\Delta_{\psi}+\frac{1}{2}\right)^{2}}{\sqrt{\pi}\Gamma\left(\frac{\Delta_{6}}{2}\right)\Gamma\left(-\frac{\Delta_{6}}{2}+\Delta_{\psi}+\frac{1}{2}\right)^{2}}\;.$
(4.5.23)

For the exchange of a parity-odd scalar in the crossed channel, we still have
$(\delta hP)_{ij}=0$ unless $i=j=2$, which now becomes

$\displaystyle(\delta
hP)_{22}([\psi\psi]^{+}_{0,\ell_{5}})\evaluated{}_{G^{(t)}_{\phi_{6}}}\sim{}$
$\displaystyle{}\left(\lambda_{\psi\psi\phi_{6}}^{3}\right)^{2}\frac{(-1)^{\ell_{5}}\Gamma\left(\frac{\Delta_{6}}{2}\right)2^{-2\Delta_{\psi}+\Delta_{6}-2\ell_{5}+1}\ell_{5}^{2\Delta_{\psi}-\Delta_{6}-\frac{3}{2}}}{\Gamma\left(\frac{1}{2}\left(\Delta_{6}+1\right)\right)\Gamma\left(\Delta_{\psi}-\frac{\Delta_{6}}{2}\right){}^{2}}\;,$
(4.5.24)

from which we can extract the anomalous dimension as

$\displaystyle\gamma_{[\psi\psi]^{+}_{0,\ell_{5}}}\evaluated{}_{G^{(t)}_{\phi_{6}}}=\frac{2}{P_{22}^{(s)}}(\delta
hP)_{22}\evaluated{}_{G^{(t)}_{\phi_{6}}}=\frac{\left(\lambda_{\psi\psi\phi_{6}}^{3}\right)^{2}}{\ell_{5}^{\Delta_{6}+1}}\frac{2^{\Delta_{6}-1}\Gamma\left(\frac{\Delta_{6}}{2}\right)\Gamma\left(\Delta_{\psi}+\frac{1}{2}\right){}^{2}}{\sqrt{\pi}\Gamma\left(\frac{\Delta_{6}+1}{2}\right)\Gamma\left(\Delta_{\psi}-\frac{\Delta_{6}}{2}\right)^{2}}\;.$
(4.5.25)

We see that the anomalous dimensions in eqn. (4.5.22), eqn. (4.5.23), and eqn.
(4.5.25) match precisely to the results computed using large-spin expansions
in [1].

####  4.5.2 Computation of corrections to OPE Coefficients: a working example

In eqn. (A.2.7) we relate the OPE function of spinning operators to the scalar
$6j$ symbols, which we reproduce for reader’s convenience:

$\rho_{ac}^{(s)}(\mathcal{O})S^{c}_{b}(\mathcal{O}_{3}\mathcal{O}_{4}[\widetilde{\mathcal{O}}])\evaluated{}_{G^{(t),fg}_{\mathcal{O}_{6}}}=\lambda_{326,f}\lambda_{146,g}\sum\limits_{\phi_{i},\mathcal{O}^{\prime},\mathcal{O}^{\prime}_{6}}\mathcal{K}^{fg}_{ab}\footnotesize\left(\begin{array}[]{cccccc}\mathcal{O}_{1}&\mathcal{O}_{2}&\mathcal{O}_{3}&\mathcal{O}_{4}&\mathcal{O}&\mathcal{O}_{6}\\\
\phi_{1}&\phi_{2}&\phi_{3}&\phi_{4}&\mathcal{O}^{\prime}&\mathcal{O}^{\prime}_{6}\end{array}\right)\normalsize\\\
\times\qquad\frac{S(\phi_{3}\phi_{4}[\widetilde{\mathcal{O}^{\prime}}])}{\eta^{(s)}_{\mathcal{O}^{\prime}}}\left(\begin{array}[]{ccc}\phi_{1}&\phi_{2}&\mathcal{O}^{\prime}_{6}\\\
\phi_{3}&\phi_{4}&\mathcal{O}^{\prime}\end{array}\right).$ (4.5.26)

By taking the double poles in $\Delta$ on both sides, we can extract $\delta
hP$ for spinning operators in terms of scalar data, which we detailed and
illustrated in section 4.5.1. In this section, we will use this equation to
extract correction to OPE coefficients for double twist operators
$[\phi\psi]^{+}_{0}$ due to an exchange of a scalar in the crossed channel.

We see in eqn. (4.5.4) that the $\delta hP$ for $[\phi\psi]^{+}_{0,J}$ reads
as

$\displaystyle(\delta
hP)_{11}([\phi\psi]^{+}_{0,J})\evaluated{}_{G^{(t)}_{\phi_{6}}}=$
$\displaystyle-i\lambda_{\phi\phi\phi_{6}}\lambda_{\psi\psi\phi_{6}}^{1}\mathfrak{dp}_{1}^{J-\frac{1}{2},0}(\psi^{\frac{1}{2}},\phi,\phi_{6})\;,$
(4.5.27) $\displaystyle(\delta
hP)_{12}([\phi\psi]^{+}_{0,J})\evaluated{}_{G^{(t)}_{\phi_{6}}}=$
$\displaystyle(\delta
hP)_{21}([\phi\psi]^{+}_{0,J})\evaluated{}_{G^{(t)}_{\phi_{6}}}=(\delta
hP)_{22}([\phi\psi]^{+}_{0,J})\evaluated{}_{G^{(t)}_{\phi_{6}}}=0\;,$

as only the first term in eqn. (A.2.14) contributes. For $(\delta P)$ on the
other hand, we do not need double poles (single poles are sufficient) and
there are also cross terms, hence we have

$\displaystyle(\delta
P)_{11}([\phi\psi]^{+}_{0,J})\evaluated{}_{G^{(t)}_{\phi_{6}}}=$
$\displaystyle-i\lambda_{\phi\phi\phi_{6}}\lambda_{\psi\psi\phi_{6}}^{1}\mathfrak{p}_{1,+}^{J-\frac{1}{2},0}\;,$
(4.5.28) $\displaystyle(\delta
P)_{12}([\phi\psi]^{+}_{0,J})\evaluated{}_{G^{(t)}_{\phi_{6}}}=$
$\displaystyle-\lambda_{\phi\phi\phi_{6}}\lambda_{\psi\psi\phi_{6}}^{3}\bigg{(}\frac{i\left(2\ell_{5}+1\right)}{\Delta_{6}-1}\mathfrak{p}_{1,-}^{J+\frac{1}{2},0}$
$\displaystyle\;\;+\frac{8i\left(\Delta_{\psi}+\ell_{5}-1\right)\left(\Delta_{\phi}+\ell_{5}-1\right)\left(\Delta_{\psi}+\Delta_{\phi}+\ell_{5}-2\right)}{\left(\Delta_{6}-1\right)\left(2\Delta_{\psi}+2\Delta_{\phi}+4\ell_{5}-5\right)\left(2\Delta_{\psi}+2\Delta_{\phi}+4\ell_{5}-3\right)}\mathfrak{p}_{1,-}^{J-\frac{1}{2},0}\bigg{)}\;,$
$\displaystyle(\delta
P)_{12}([\phi\psi]^{+}_{0,J})\evaluated{}_{G^{(t)}_{\phi_{6}}}=$
$\displaystyle
i\lambda_{\phi\phi\phi_{6}}\lambda_{\psi\psi\phi_{6}}^{3}\frac{1}{\Delta_{6}-1}\mathfrak{p}_{2,-}^{J-\frac{1}{2},0}\;,$
$\displaystyle(\delta
P)_{22}([\phi\psi]^{+}_{0,J})\evaluated{}_{G^{(t)}_{\phi_{6}}}=$
$\displaystyle 0\;,$

where we define the shorthand notation

$\displaystyle\mathfrak{p}_{i,\pm}^{J,n}\equiv\mathfrak{p}_{i}^{J,n}(\psi^{\pm},\phi,\phi,\psi^{\frac{1}{2}},\phi_{6})$
(4.5.29)

for

$\displaystyle\mathfrak{p}_{1}^{J,n}(\phi_{1},\phi_{2},\phi_{3},\phi_{4},\mathcal{O}_{6})\equiv$
$\displaystyle\lim\limits_{\Delta\rightarrow\Delta_{1}+\Delta_{2}+J+2n}\left(\Delta-\Delta_{1}-\Delta_{2}-J-2n\right)\frac{S(\phi_{3}\phi_{4}[\widetilde{\mathcal{O}}_{\Delta,J}])}{\eta^{(s)}_{\mathcal{O}_{\Delta,J}}}\left(\begin{array}[]{ccc}\phi_{1}&\phi_{2}&\mathcal{O}_{6}\\\
\phi_{3}&\phi_{4}&\mathcal{O}_{\Delta,J}\end{array}\right),$ (4.5.30c)
$\displaystyle\mathfrak{p}_{2}^{J,n}(\phi_{1},\phi_{2},\phi_{3},\phi_{4},\mathcal{O}_{6})\equiv$
$\displaystyle\lim\limits_{\Delta\rightarrow\Delta_{3}+\Delta_{4}+J+2n}\left(\Delta-\Delta_{3}-\Delta_{4}-J-2n\right)\frac{S(\phi_{3}\phi_{4}[\widetilde{\mathcal{O}}_{\Delta,J}])}{\eta^{(s)}_{\mathcal{O}_{\Delta,J}}}\left(\begin{array}[]{ccc}\phi_{1}&\phi_{2}&\mathcal{O}_{6}\\\
\phi_{3}&\phi_{4}&\mathcal{O}_{\Delta,J}\end{array}\right).$ (4.5.30f)

We can compute $\mathfrak{p}$ similar to $\mathfrak{dp}$ and include both
perturbative and nonperturbative corrections to OPE coefficients. For brevity,
we only reproduce the leading piece of the perturbative correction at large
$\ell$:

$(\delta
P)([\phi\psi]^{+}_{0,\ell_{5}})\evaluated{}_{G^{(t)}_{\phi_{6}}}=\frac{i\sqrt{\pi}(-1)^{\ell_{5}-\frac{1}{2}}2^{-\Delta_{\psi}-\Delta_{\phi}+\Delta_{6}-2\ell_{5}+\frac{5}{2}}\ell_{5}^{\Delta_{\psi}+\Delta_{\phi}-\Delta_{6}-1}}{\Gamma\left(-\frac{\Delta_{6}}{2}+\Delta_{\psi}+\frac{1}{2}\right)\Gamma\left(\Delta_{\phi}-\frac{\Delta_{6}}{2}\right)}\\\
\times\begin{pmatrix}-\lambda_{\phi\phi\phi_{6}}\lambda_{\psi\psi\phi_{6}}^{1}\frac{\Gamma\left(\frac{\Delta_{6}+1}{2}\right)H_{\frac{\Delta_{6}-2}{2}}}{\sqrt{\pi}\Gamma\left(\frac{\Delta_{6}}{2}\right)}&\lambda_{\phi\phi\phi_{6}}\lambda_{\psi\psi\phi_{6}}^{3}\frac{\Gamma\left(-\frac{\Delta_{6}}{2}+\Delta_{\psi}+\frac{1}{2}\right)}{2\sqrt{\ell_{5}}\Gamma\left(\Delta_{\psi}-\frac{\Delta_{6}}{2}\right)}\\\
\lambda_{\phi\phi\phi_{6}}\lambda_{\psi\psi\phi_{6}}^{3}\frac{\Gamma\left(-\frac{\Delta_{6}}{2}+\Delta_{\psi}+\frac{1}{2}\right)}{2\sqrt{\ell_{5}}\Gamma\left(\Delta_{\psi}-\frac{\Delta_{6}}{2}\right)}&0\end{pmatrix},$
(4.5.31)

where $H_{a}$ is the Harmonic number. As a consistency check, we see that
setting

$\displaystyle\Delta_{6}\rightarrow
0\;,\quad\lambda_{\phi\phi\phi_{6}}\rightarrow
1\;,\quad\lambda_{\psi\psi\phi_{6}}^{1}\rightarrow
i\;,\quad\lambda_{\psi\psi\phi_{6}}^{3}\rightarrow 0$ (4.5.32)

reduces the result to the MFT coefficient eqn. (4.4.11a).

##  Chapter 5 Discussion

In this thesis we reviewed various analytic tools that have been developed in
recent years and used them to extend the progress of the analytic conformal
bootstrap program to fermionic conformal field theories in three dimensional
spacetimes.

In the first part of the thesis, we used the $\mathrm{SL}(2,\mathbb{R})$
expansion of the conformal blocks and the known formulae for the summation of
the _Casimir-irregular_ terms to derive the relation between the CFT data of
different channels in the lightcone bootstrap. This old-style lightcone
bootstrap has been used for scalars and is sufficient to derive the large spin
behavior of the double twist families in the spectrum, although the results
surprisingly match the numerics even at low spins for scalar theories such as
the $3d$ Ising and $\mathrm{O}(N)$ models. We extended this technique to
fermions and derived the OPE coefficients and the anomalous dimensions for the
large spin spectrum of fermionic CFTs. In particular, we provided both leading
and next-to-leading order terms in the large spin expansion.

In the second part of the thesis, we provided a different approach for the
analytic analysis of CFTs with spinning operators (such as fermions). This
approach relies on deriving relations between the $6j$ symbols of the
conformal group and using the known $6j$ symbols for scalars. The relations
between the $6j$ symbols are derived using the conformally invariant pairings
of the three point structures and the differential operators that can be used
to shift the conformal weights of a given operator. When combined, these
technologies allow one to compute the CFT data of fermionic operators _non-
perturbatively_ , that is by including both the perturbative and exponentially
suppressed terms in spin. The second piece is inaccessible with the
traditional lightcone bootstrap that we used in the first part of the thesis,
and it ensures that the result is analytic all the way down to low spins,
yielding expressions far more compatible with the numerical results. We first
demonstrated the importance of these non-perturbative pieces in the $3d$ Ising
and $\mathrm{O}(N)$ models, then extended the calculations to generic unitary
fermionic CFTs by deriving the analytic expressions for their double twist
spectrum.

While extending the application of the Lorentzian inversion formula to
spinning correlators, we derived relations between the $6j$ symbols of
different representations of the conformal group.111The reader can refer
toeqn. (A.1.27) for this relation and the definition of
$\mathcal{J}-$coefficients. Aside from their employment in the construction of
the OPE function (hence the derivation of the CFT data), the
$\mathcal{J}-$coefficients that we have constructed for this relation may be
useful on their own. As $6j$ symbols are kinematic objects of the conformal
group without any dynamic data, the relations between these objects are valid
in any system with the conformal symmetry, and we believe that our results for
these coefficients may have pure mathematical value, considering our
derivation (and the resultant $J-$coefficients) are completely exact without
any sort of approximation. We hope that further mathematical insight can be
obtained from these objects in the future.

There are several open questions left that can be considered by the tools and
results of this thesis. For instance, our demonstration for the improvement of
the analytic predictions for $3d$ Ising and $\mathrm{O}(N)$ models with the
inclusion of non-perturbative effects can be extended to concrete fermionic
CFTs, such as GNY models. Would we then expect new precise analytic
predictions that match the results of the numerical bootstrap [81, 226]? It is
also natural to ask if we can predict analytic trajectories that cannot be
accessed using scalar correlators in $3d$ $\mathcal{N}=1$ SCFTs such as the
supersymmetric Ising [233, 234] and Wess-Zumino [235] models. This is an
interesting question as the implications of imposing analytic bootstrap
constraints for all external operators in the same supermultiplet has not yet
been fully understood, so our results for external fermionic operators can be
utilized to address this question in SCFTs.

Moreover, it is also straightforward to go to higher spin now that we
understand how to spin down a fermionic $6j$ symbol in $3d$. As a simple
example, our results could then also be used to study correlation functions of
conserved currents $J_{\mu}$ in the $O(N)$ vector model. There are many
physically relevant observables, such $\expectationvalue{JJT}$, which are only
accessible with spinning correlators. Based on [126, 127, 1] we now know that
the current and stress tensor lie on the double-twist trajectories composed of
the fundamental scalars, $\phi$, so these correlators are now within reach of
analytic methods.

Finally, we note that our results are directly applicable to the study of
Witten diagrams with external fermionic operators. For example, by studying
the contribution of the stress tensor $T^{\mu\nu}$ to a fermionic correlator,
e.g. $\expectationvalue{\phi\psi\psi\phi}$, we can derive the binding energy
for a two-particle state dual to $[\phi\psi]_{n}$, due to tree-level graviton
exchange. The anomalous dimension, or corresponding 6j symbol, can then be
used to bootstrap a graviton loop in AdS4 [236, 133]. Likewise, by extending
our computations to external currents, we can derive similar results for gauge
interactions in AdS4, which would provide valuable data that can be cross-
checked with perturbative bulk computations.222As part of his PhD work not
included in this thesis [237, 238, 239, 240, 241, 242, 243], the author
studied AdS4 gluons in momentum space, showing that written in a particular
gauge any tree-level gluon Witten diagram can be purely computed algebraically
(without any bulk-integration) which is demonstrated by the computation of the
explicit results for several higher point amplitudes [237, 238]. It would be
interesting to connect to these bulk computations from a boundary perspective.
Lastly, if one wants to study an AdS theory with fermions, we need to
understand the tree-level fermionic correlators to fully determine a one-loop
scalar four point function. We therefore hope the results presented here are
useful in the wider study of AdS4 correlators.

##  Appendix A $6j$ Symbols

In section 4.3.5 we defined and reviewed $6j$ symbols. In this appendix, we
will further discuss $6j$ symbols and detail our method of deriving relations
between $6j$ symbols for operators of different spins.

###  A.1 Spinning Down the $6j$ Symbol

We first draw the attention of the reader to the problem of inverting a single
partial wave (or block) with external spinning operators. This allows us to
compute corrections to the anomalous dimensions of double twist operators. As
a reminder, the general form of the $6j$ symbol is given by

$\displaystyle\left\\{\begin{array}[]{ccc}\mathcal{O}_{1}&\mathcal{O}_{2}&\mathcal{O}_{6}\\\
\mathcal{O}_{3}&\mathcal{O}_{4}&\mathcal{O}_{5}\end{array}\right\\}^{abcd}$
$\displaystyle=\left(\widetilde{\Psi}^{(s),ab}_{\mathcal{O}_{5}},\Psi^{(t),cd}_{\mathcal{O}_{6}}\right)$
(A.1.3) $\displaystyle=\int
d^{d}x_{1}...d^{d}x_{6}\expectationvalue{\widetilde{\mathcal{O}}_{1}\widetilde{\mathcal{O}}_{2}\widetilde{\mathcal{O}}_{5}}\expectationvalue{\widetilde{\mathcal{O}}_{3}\widetilde{\mathcal{O}}_{4}\mathcal{O}_{5}}\expectationvalue{\mathcal{O}_{3}\mathcal{O}_{2}\mathcal{O}_{6}}\expectationvalue{\mathcal{O}_{1}\mathcal{O}_{4}\widetilde{\mathcal{O}}_{6}}.$
(A.1.4)

Our strategy will in a way be the reverse of the scalar case [133]. Instead of
using the Lorentzian inversion formula, we will follow the strategy outlined
in [152] and use weight-shifting operators to calculate the $6j$ symbol for
external fermions in terms of the $6j$ symbol for external scalars. We then
use the expression (4.3.73), which splits the scalar $6j$ symbol into two
pieces from inverting the physical block and its shadow to find the
corresponding split for the fermionic $6j$ symbol.

To start, we use the results of [152] to write the t-channel spinning partial
wave as a differential operator acting on the partial wave for external
scalars:

$\displaystyle\Psi^{(t),ab}_{\mathcal{O}}(x_{i})=\mathfrak{D}^{ab}_{t}\Psi^{(t),\text{scalar}}_{\mathcal{O}^{\prime}}(x_{i})\;,$
(A.1.5)

where $\Psi^{(t),\text{scalar}}_{\mathcal{O}^{\prime}}(x_{i})$ is a partial
wave for four external scalars,
$\expectationvalue{\phi_{3}\phi_{2}\phi_{1}\phi_{4}}$. We are being very
schematic here and it should be understood that $\mathfrak{D}^{ab}_{t}$ is a
sum of multiple weight-shifting operators. For each term in the sum we may
have to choose a different shifted operator $\mathcal{O}^{\prime}$, with
scaling dimension and spin shifted from $\mathcal{O}$, as well as different
external scaling dimensions $\Delta_{i}$ of the scalar partial wave.

Given this expression we can simplify the spinning $6j$ symbol by taking the
adjoint of this operator:

$\displaystyle\left(\widetilde{\Psi}^{(s),ab}_{\mathcal{O}_{5}},\Psi^{(t),ab}_{\mathcal{O}_{6}}\right)$
$\displaystyle=\left(\mathfrak{D}^{ab}_{s}\widetilde{\Psi}^{(s),\text{scalar}}_{\mathcal{O}^{\prime}_{5}},\mathfrak{D}^{ab}_{t}\Psi^{(t),\text{scalar}}_{\mathcal{O}^{\prime}_{6}}\right)$
$\displaystyle=\left(\widetilde{\Psi}^{(s),\text{scalar}}_{\mathcal{O}^{\prime}_{5}},\mathfrak{D}^{*,ab}_{s}\mathfrak{D}^{ab}_{t}\Psi^{(t),\text{scalar}}_{\mathcal{O}^{\prime}_{6}}\right).$
(A.1.6)

The two weight-shifting operators acting on the t-channel conformal partial
wave then give a linear combination of undifferentiated t-channel conformal
partial waves, at the price of more shifts for the internal and external
labels. In the end we are left with an equality of the form

$\displaystyle\left\\{\begin{array}[]{ccc}\mathcal{O}_{1}&\mathcal{O}_{2}&\mathcal{O}_{6}\\\
\mathcal{O}_{3}&\mathcal{O}_{4}&\mathcal{O}_{5}\end{array}\right\\}^{abcd}=\sum\limits_{\phi_{i},\mathcal{O}^{\prime}_{5,6}}\mathcal{J}^{abcd}\footnotesize\left(\begin{array}[]{cccccc}\mathcal{O}_{1}&\mathcal{O}_{2}&\mathcal{O}_{3}&\mathcal{O}_{4}&\mathcal{O}_{5}&\mathcal{O}_{6}\\\
\phi_{1}&\phi_{2}&\phi_{3}&\phi_{4}&\mathcal{O}^{\prime}_{5}&\mathcal{O}^{\prime}_{6}\end{array}\right)\normalsize\left\\{\begin{array}[]{ccc}\phi_{1}&\phi_{2}&\mathcal{O}^{\prime}_{6}\\\
\phi_{3}&\phi_{4}&\mathcal{O}^{\prime}_{5}\end{array}\right\\},$ (A.1.13)

where the sum runs over some set of scaling dimensions for the fictitious
external scalars $\phi_{i}$, whose dimensions are related to $\mathcal{O}_{i}$
by some (half-)integer shift, and over both scaling dimensions and spins for
$\mathcal{O}^{\prime}_{5,6}$, which are again related to the
$\mathcal{O}_{5,6}$ by (half-)integer shifts in both labels. We now turn to
how to compute these coefficients.

The general strategy to compute the decomposition factors $J^{abcd}$ in eqn.
(A.1.13) can be systematized by the following procedure:

  * •

Write a three-point structure in terms of weight-shifting operators acting on
three-point structures of lower spins.

  * •

Use integration by parts and crossing symmetry of covariant three-point
structures to move the weight-shifting operators such that they act on the
same operator.

  * •

By using irreducibility of the representations, the weight-shifting operators
become multiples of the identity.

  * •

Repeat until all three-point structures are of the form
$\expectationvalue{\phi\phi\mathcal{O}}$, i.e. we are left with three-point
functions involving two scalars.

Below, we will unpack this procedure further by detailing each step in the
explicit decomposition of the $6j$ symbol of two external scalars and two
external fermions.

In our conventions, the $6j$-symbol for the correlator
$\expectationvalue{\phi\phi\psi\psi}$ reads as

$\displaystyle\left\\{\begin{matrix}\phi_{1}&\phi_{2}&\mathcal{O}_{6}\\\
\psi_{3}&\psi_{4}&\mathcal{O}_{5}\end{matrix}\right\\}^{\raisebox{-2.1097pt}{\scalebox{1.8}{$\cdot$}}s_{2}t_{1}t_{2}}=\int
d^{d}x_{1}...d^{d}x_{6}\expectationvalue{\widetilde{\phi}_{1}\widetilde{\phi}_{2}\widetilde{\mathcal{O}}_{5}}\expectationvalue{\widetilde{\psi}_{3}\widetilde{\psi}_{4}\mathcal{O}_{5}}^{s_{2}}\expectationvalue{\psi_{3}\phi_{2}\mathcal{O}_{6}}^{t_{1}}\expectationvalue{\phi_{1}\psi_{4}\widetilde{\mathcal{O}}_{6}}^{t_{2}},$
(A.1.14)

where the first three-point structure is already in the appropriate form for a
$6j$ symbol of four external scalars, so all we need to do is to massage the
remaining structures. As the first step, we use eqn. (4.3.112) to rewrite
$\expectationvalue{\widetilde{\psi}_{3}\widetilde{\psi}_{4}\mathcal{O}_{5}}^{s_{2}}$:111We
use the same shorthand notation as is used in section 4.5.1:
$\phi^{a}_{i}\equiv\phi_{\Delta_{i}+a}$,
$\widetilde{\phi}^{a}_{i}\equiv\phi_{3-\Delta_{i}+a}$,
$\mathcal{O}_{i}^{a,b}\equiv\mathcal{O}_{\Delta_{i}+a,\ell_{i}+b}$, and
$\widetilde{\mathcal{O}}_{i}^{a,b}\equiv\mathcal{O}_{3-\Delta_{i}+a,\ell_{i}+b}$.

$\int
d^{d}x_{3}d^{d}x_{4}d^{d}x_{6}\expectationvalue{\widetilde{\psi}_{3}\widetilde{\psi}_{4}\mathcal{O}_{5}}^{s_{2}}\expectationvalue{\psi_{3}\phi_{2}\mathcal{O}_{6}}^{t_{1}}\expectationvalue{\phi_{1}\psi_{4}\widetilde{\mathcal{O}}_{6}}^{t_{2}}\\\
=\int
d^{d}x_{3}d^{d}x_{4}d^{d}x_{6}\sum\limits_{a,b}\kappa^{s_{2}}_{3,ab}(\widetilde{\psi}_{3}\widetilde{\psi}_{4}\mathcal{O}_{5})\mathcal{D}_{1}^{-a,+}\mathcal{D}_{2}^{-b,+}\expectationvalue{\widetilde{\phi}_{3}^{a}\widetilde{\phi}_{4}^{b}\mathcal{O}_{5}}\expectationvalue{\psi_{3}\phi_{2}\mathcal{O}_{6}}^{t_{1}}\expectationvalue{\phi_{1}\psi_{4}\widetilde{\mathcal{O}}_{6}}^{t_{2}}\;.$
(A.1.15)

After that, we integrate by parts to obtain

$\int
d^{d}x_{3}d^{d}x_{4}d^{d}x_{6}\expectationvalue{\widetilde{\psi}_{3}\widetilde{\psi}_{4}\mathcal{O}_{5}}^{s_{2}}\expectationvalue{\psi_{3}\phi_{2}\mathcal{O}_{6}}^{t_{1}}\expectationvalue{\phi_{1}\psi_{4}\widetilde{\mathcal{O}}_{6}}^{t_{2}}=\int
d^{d}x_{3}d^{d}x_{4}d^{d}x_{6}\\\
\sum\limits_{a,b}\kappa^{s_{2}}_{3,ab}(\widetilde{\psi}_{3}\widetilde{\psi}_{4}\mathcal{O}_{5})\expectationvalue{\widetilde{\phi}_{3}^{a}\widetilde{\phi}_{4}^{b}\mathcal{O}_{5}}\left[\left(\mathcal{D}_{1}^{-a,+}\right)^{*}_{A}\expectationvalue{\psi_{3}\phi_{2}\mathcal{O}_{6}}^{t_{1}}\right]\left[\left(\mathcal{D}_{2}^{-b,+}\right)^{A*}\expectationvalue{\phi_{1}\psi_{4}\widetilde{\mathcal{O}}_{6}}^{t_{2}}\right]\;,$
(A.1.16)

where we are showing the spinor indices of weight-shifting operators
explicitly. Witheqn. (4.3.102), the equation further reduces to

$\displaystyle\int d^{d}x_{3}$ $\displaystyle
d^{d}x_{4}d^{d}x_{6}\expectationvalue{\widetilde{\psi}_{3}\widetilde{\psi}_{4}\mathcal{O}_{5}}^{s_{2}}\expectationvalue{\psi_{3}\phi_{2}\mathcal{O}_{6}}^{t_{1}}\expectationvalue{\phi_{1}\psi_{4}\widetilde{\mathcal{O}}_{6}}^{t_{2}}$
$\displaystyle=$ $\displaystyle\int
d^{d}x_{3}d^{d}x_{4}d^{d}x_{6}\sum\limits_{a,b}\kappa^{s_{2}}_{3,ab}(\widetilde{\psi}_{3}\widetilde{\psi}_{4}\mathcal{O}_{5})\zeta_{0}^{-a,+}\zeta_{0}^{-b,+}\expectationvalue{\widetilde{\phi}_{3}^{a}\widetilde{\phi}_{4}^{b}\mathcal{O}_{5}}$
$\displaystyle\times\left[\left(\mathcal{D}_{1}^{-a,-}\right)_{A}\expectationvalue{\psi_{3}\phi_{2}\mathcal{O}_{6}}^{t_{1}}\right]\left[E_{\phi\psi\mathcal{O}\rightarrow\mathcal{O}\phi\psi}^{t_{2}t_{2}^{\prime}}\left(\mathcal{D}_{3}^{-b,-}\right)^{A}\expectationvalue{\widetilde{\mathcal{O}}_{6}\phi_{1}\psi_{4}}^{t_{2}^{\prime}}\right],$
(A.1.17)

where we defined the exchange matrix $E$

$\displaystyle\expectationvalue{\mathcal{O}_{2}\mathcal{O}_{1}\mathcal{O}}^{a}=E^{ab}_{21\mathcal{O}\to
12\mathcal{O}}\expectationvalue{\mathcal{O}_{1}\mathcal{O}_{2}\mathcal{O}}^{b}$
(A.1.18)

to reorder the last tensor structure so that the weight-shifting operator acts
on the third operator.222We do this because we will use the convention of
[152] for finite-dimensional representations, where the weight-shifting
operator acts on the first (third) operator in the $s$ ($t$) channel. We can
then use the crossing for conformally covariant three-point structures as
derived in [152], which reads as

$\displaystyle\left(\mathcal{D}^{-a,-b}_{3}\right)^{A}\expectationvalue{\mathcal{O}_{1}\mathcal{O}_{2}\mathcal{O}_{3}^{a,b}}^{m}=\sum_{c,d,n}\left\\{\begin{matrix}\mathcal{O}_{1}&\mathcal{O}_{2}&\mathcal{O}_{1}^{c,d}\\\
\mathcal{O}_{3}&S&\mathcal{O}_{3}^{a,b}\end{matrix}\right\\}^{mn}\left(\mathcal{D}^{-c,-d}_{1}\right)^{A}\expectationvalue{\mathcal{O}_{1}^{c,d}\mathcal{O}_{2}\mathcal{O}_{3}}^{n}$
(A.1.19)

for the fermionic representation of weight-shifting operators in $3d$. Thus we
obtain

$\displaystyle\int d^{d}x_{3}$ $\displaystyle
d^{d}x_{4}d^{d}x_{6}\expectationvalue{\widetilde{\psi}_{3}\widetilde{\psi}_{4}\mathcal{O}_{5}}^{s_{2}}\expectationvalue{\psi_{3}\phi_{2}\mathcal{O}_{6}}^{t_{1}}\expectationvalue{\phi_{1}\psi_{4}\widetilde{\mathcal{O}}_{6}}^{t_{2}}$
$\displaystyle=$ $\displaystyle\int
d^{d}x_{3}d^{d}x_{4}d^{d}x_{6}\sum\limits_{a,b}\kappa^{s_{2}}_{3,ab}(\widetilde{\psi}_{3}\widetilde{\psi}_{4}\mathcal{O}_{5})\zeta_{0}^{-a,+}\zeta_{0}^{-b,+}\expectationvalue{\widetilde{\phi}_{3}^{a}\widetilde{\phi}_{4}^{b}\mathcal{O}_{5}}$
$\displaystyle\times$
$\displaystyle\left[\left(\mathcal{D}_{1}^{-a,-}\right)_{A}\expectationvalue{\psi_{3}\phi_{2}\mathcal{O}_{6}}^{t_{1}}\right]\left[E_{\phi\psi\mathcal{O}\rightarrow\mathcal{O}\phi\psi}^{t_{2}t_{2}^{\prime}}\sum_{c,d}\left\\{\begin{matrix}\widetilde{\mathcal{O}}_{6}&\phi_{1}&\widetilde{\mathcal{O}}_{6}^{c,d}\\\
\phi_{4}^{-b}&S&\psi_{4}\end{matrix}\right\\}^{t_{2}^{\prime}\raisebox{-1.89873pt}{\scalebox{1.8}{$\cdot$}}}\left(\mathcal{D}^{-c,-d}_{1}\right)^{A}\expectationvalue{\widetilde{\mathcal{O}}_{6}\phi_{1}\phi_{4}^{-b}}\right].$
(A.1.20)

We can integrate by parts and use the crossing again to get both weight-
shifting operators to act on the same operator:

$\displaystyle\int$ $\displaystyle
d^{d}x_{3}d^{d}x_{4}d^{d}x_{6}\expectationvalue{\widetilde{\psi}_{3}\widetilde{\psi}_{4}\mathcal{O}_{5}}^{s_{2}}\expectationvalue{\psi_{3}\phi_{2}\mathcal{O}_{6}}^{t_{1}}\expectationvalue{\phi_{1}\psi_{4}\widetilde{\mathcal{O}}_{6}}^{t_{2}}$
$\displaystyle=$ $\displaystyle\int
d^{d}x_{3}d^{d}x_{4}d^{d}x_{6}\sum\limits_{a,b,c,d}\kappa^{s_{2}}_{3,ab}(\widetilde{\psi}_{3}\widetilde{\psi}_{4}\mathcal{O}_{5})\zeta_{0}^{-a,+}\zeta_{0}^{-b,+}\zeta_{\ell_{t}+d}^{-c,-d}\left\\{\begin{matrix}\widetilde{\mathcal{O}}_{6}&\phi_{1}&\widetilde{\mathcal{O}}_{6}^{c,d}\\\
\phi_{4}^{-b}&S&\psi_{4}\end{matrix}\right\\}^{t_{2}^{\prime}\raisebox{-1.89873pt}{\scalebox{1.8}{$\cdot$}}}E_{\phi\psi\mathcal{O}\rightarrow\mathcal{O}\phi\psi}^{t_{2}t_{2}^{\prime}}$
$\displaystyle\times\expectationvalue{\widetilde{\phi}_{3}^{a}\widetilde{\phi}_{4}^{b}\mathcal{O}_{5}}\Bigg{[}\sum_{e,f,n}\left\\{\begin{matrix}\psi_{3}&\phi_{2}&\mathcal{O}_{3}^{e,f}\\\
\mathcal{O}_{6}^{-c,d}&S&\mathcal{O}_{6}\end{matrix}\right\\}^{t_{1}n}\left(\mathcal{D}_{1}^{-a,-}\right)_{A}\left(\mathcal{D}^{-e,-f}_{1}\right)^{A}\expectationvalue{\mathcal{O}_{3}^{e,f}\phi_{2}\mathcal{O}_{6}^{-c,d}}^{n}\Bigg{]}\expectationvalue{\widetilde{\mathcal{O}}_{6}\phi_{1}\phi_{4}^{-b}}.$
(A.1.21)

By the irreducibility of the representations, we have

$\displaystyle\left(\mathcal{D}^{-a,-b}\right)_{A}\left(\mathcal{D}^{c,d}\right)^{A}\mathcal{O}_{\Delta,\ell}=\delta^{ac}\delta^{bd}\beta^{\Delta,\ell}_{ab}\mathcal{O}_{\Delta,\ell}$
(A.1.22)

with which we finally obtain333One can derive $\beta$ by acting on the two
point function with weight shifting operators in embedding space. In our
conventions, we have  $\displaystyle\beta_{a,b}^{\Delta,\ell}\equiv
b(a+2\Delta-3)(b+2\ell+1)(2ab\Delta+a(2b+1)(a+b-2)+2\ell)\quad\text{ for
}a,b=\pm 1\cdot$ (A.1.23)

$\displaystyle\int$ $\displaystyle
d^{d}x_{3}d^{d}x_{4}d^{d}x_{6}\expectationvalue{\widetilde{\psi}_{3}\widetilde{\psi}_{4}\mathcal{O}_{5}}^{s_{2}}\expectationvalue{\psi_{3}\phi_{2}\mathcal{O}_{6}}^{t_{1}}\expectationvalue{\phi_{1}\psi_{4}\widetilde{\mathcal{O}}_{6}}^{t_{2}}$
$\displaystyle=d^{d}x_{3}d^{d}x_{4}d^{d}x_{6}\sum\limits_{a,b,c,d}\beta_{a,+}^{\Delta_{3}-a,0}\kappa^{s_{2}}_{3,ab}(\widetilde{\psi}_{3}\widetilde{\psi}_{4}\mathcal{O}_{5})\zeta_{0}^{-a,+}\zeta_{0}^{-b,+}\zeta_{\ell_{t}+d}^{-c,-d}E_{\phi\psi\mathcal{O}\rightarrow\mathcal{O}\phi\psi}^{t_{2}t_{2}^{\prime}}$
$\displaystyle\times\left\\{\begin{matrix}\widetilde{\mathcal{O}}_{6}&\phi_{1}&\widetilde{\mathcal{O}}_{6}^{c,d}\\\
\phi_{4}^{-b}&S&\psi_{4}\end{matrix}\right\\}^{t_{2}^{\prime}\raisebox{-2.1097pt}{\scalebox{1.8}{$\cdot$}}}\left\\{\begin{matrix}\psi_{3}&\phi_{2}&\phi_{3}^{-a}\\\
\mathcal{O}_{6}^{-c,d}&S&\mathcal{O}_{6}\end{matrix}\right\\}^{t_{1}\raisebox{-2.1097pt}{\scalebox{1.8}{$\cdot$}}}\expectationvalue{\widetilde{\phi}_{3}^{a}\widetilde{\phi}_{4}^{b}\mathcal{O}_{5}}\expectationvalue{\phi_{3}^{-a}\phi_{2}\mathcal{O}_{6}^{-c,d}}^{n}\expectationvalue{\phi_{1}\phi_{4}^{-b}\widetilde{\mathcal{O}}_{6}^{c,d}}.$
(A.1.24)

Inserting this into eqn. (A.1.14), we get

$\left\\{\begin{matrix}\phi_{1}&\phi_{2}&\mathcal{O}_{6}\\\
\psi_{3}&\psi_{4}&\mathcal{O}_{5}\end{matrix}\right\\}^{\raisebox{-2.1097pt}{\scalebox{1.8}{$\cdot$}}s_{2}t_{1}t_{2}}=\sum\limits_{a,b,c,d}\mathcal{J}^{\raisebox{-2.1097pt}{\scalebox{1.8}{$\cdot$}}s_{2}t_{1}t_{2}}\footnotesize\left(\begin{array}[]{cccccc}\phi_{1}&\phi_{2}&\psi_{3}&\psi_{4}&\mathcal{O}_{5}&\mathcal{O}_{6}\\\
\phi_{1}&\phi_{2}&\phi_{3}^{-a}&\phi_{4}^{-b}&\mathcal{O}_{5}&\mathcal{O}_{6}^{-c,d}\end{array}\right)\normalsize\\\
\times\left\\{\begin{matrix}\phi_{1}&\phi_{2}&\mathcal{O}_{6}^{-c,d}\\\
\phi_{3}^{-a}&\phi_{4}^{-b}&\mathcal{O}_{5}\end{matrix}\right\\}\;,$ (A.1.25)

where

$\mathcal{J}^{\raisebox{-2.1097pt}{\scalebox{1.8}{$\cdot$}}s_{2}t_{1}t_{2}}\footnotesize\left(\begin{array}[]{cccccc}\phi_{1}&\phi_{2}&\psi_{3}&\psi_{4}&\mathcal{O}_{5}&\mathcal{O}_{6}\\\
\phi_{1}&\phi_{2}&\phi_{3}^{-a}&\phi_{4}^{-b}&\mathcal{O}_{5}&\mathcal{O}_{6}^{-c,d}\end{array}\right)\normalsize=\beta_{a,+}^{\Delta_{3}-a,0}\kappa^{s_{2}}_{3,ab}(\widetilde{\psi}_{3}\widetilde{\psi}_{4}\mathcal{O}_{5})\zeta_{0}^{-a,+}\zeta_{0}^{-b,+}\zeta_{\ell_{t}+d}^{-c,-d}\\\
\times
E_{\phi\psi\mathcal{O}\rightarrow\mathcal{O}\phi\psi}^{t_{2}t_{2}^{\prime}}\left\\{\begin{matrix}\widetilde{\mathcal{O}}_{6}&\phi_{1}&\widetilde{\mathcal{O}}_{6}^{c,d}\\\
\phi_{4}^{-b}&S&\psi_{4}\end{matrix}\right\\}^{t_{2}^{\prime}\raisebox{-2.1097pt}{\scalebox{1.8}{$\cdot$}}}\left\\{\begin{matrix}\psi_{3}&\phi_{2}&\phi_{3}^{-a}\\\
\mathcal{O}_{6}^{-c,d}&S&\mathcal{O}_{6}\end{matrix}\right\\}^{t_{1}\raisebox{-2.1097pt}{\scalebox{1.8}{$\cdot$}}}.$
(A.1.26)

Note that if $\ell_{5}=0$, there are only 2 independent structures for
$\expectationvalue{\psi_{3}\psi_{4}\phi_{5}}^{s_{2}}$, hence we need to take a
$2\times 2$ invertible submatrix of $\kappa_{3}$. We can do this by
restricting to the structures
$\expectationvalue{\psi_{3}\psi_{4}\phi_{5}}^{1,3}$ in eqn. (3.1.15), and
fixing $b=\frac{1}{2}$ in eqn. (A.1.25) instead of summing over
$b=\pm\frac{1}{2}$.

Despite the complicated and lengthy expressions, the procedure is actually
quite straightforward and most easily tractable in the diagrammatic notation,
see figure A.1 for a summary of the decomposition above. However, one should
only use the diagrammatic expressions as a guide, as there are ambiguities in
their meaning, most notably sign ambiguities as they do not carry the
information of the order of operators in the equations.

$\Rightarrow$ $\Rightarrow$ $\Rightarrow$ $\Rightarrow$ $\Rightarrow$

Figure A.1:  Step by step diagrammatic illustration for the decomposition of
the $6j$ symbol for $\expectationvalue{\phi\psi\psi\phi}$ into the $6j$ symbol
for $\expectationvalue{\phi\phi\phi\phi}$. The idea is as follows: one re-
expresses a fermionic three-point structure,
$\expectationvalue{\psi\psi\mathcal{O}}$, in terms of weight-shifting
operators acting on a scalar three-point structure,
$\expectationvalue{\phi\phi\mathcal{O}}$. The weight-shifting operators are
then moved inside the diagrammatic loop until they act on the same leg. By the
irreducibility of the representations, i.e.
$\left(\mathcal{D}^{a,b}\right)_{A}\left(\mathcal{D}^{-c,-d}\right)^{A}\sim\delta^{ac}\delta^{bd}$,
the diagram reduces to that of a scalar $6j$ symbol. To be able to move around
the weight-shifting operators, one either integrates by parts or uses the
crossing relation for covariant three-point structures as explained in the
main text. The diagrams above correspond to the equations eqn. (A.1.14), eqn.
(A.1.15), eqn. (A.1), eqn. (A.1), eqn. (A.1.21), and eqn. (A.1.24)
respectively.

For spinning correlators beyond $\expectationvalue{\phi\phi\psi\psi}$ we can
repeat the procedure above and recursively spin-down the $6j$ symbol. For
this, we first define a generalized form of eqn. (A.1.13):

$\displaystyle\left\\{\begin{array}[]{ccc}\mathcal{O}_{1}&\mathcal{O}_{2}&\mathcal{O}_{6}\\\
\mathcal{O}_{3}&\mathcal{O}_{4}&\mathcal{O}_{5}\end{array}\right\\}^{abcd}=\sum\limits_{\phi_{i},\mathcal{O}^{\prime}_{5,6}}\mathcal{J}^{abcd}_{efgh}\footnotesize\left(\begin{array}[]{cccccc}\mathcal{O}_{1}&\mathcal{O}_{2}&\mathcal{O}_{3}&\mathcal{O}_{4}&\mathcal{O}_{5}&\mathcal{O}_{6}\\\
\mathcal{O}^{\prime}_{1}&\mathcal{O}^{\prime}_{2}&\mathcal{O}^{\prime}_{3}&\mathcal{O}^{\prime}_{4}&\mathcal{O}^{\prime}_{5}&\mathcal{O}^{\prime}_{6}\end{array}\right)\normalsize\left\\{\begin{array}[]{ccc}\mathcal{O}^{\prime}_{1}&\mathcal{O}^{\prime}_{2}&\mathcal{O}^{\prime}_{6}\\\
\mathcal{O}^{\prime}_{3}&\mathcal{O}^{\prime}_{4}&\mathcal{O}^{\prime}_{5}\end{array}\right\\}^{efgh},$
(A.1.27)

where we obtain the ultimate result $J^{abcd}$ of eqn. (A.1.13) by summing
over these intermediate factors $J^{abcd}_{efgh}$.

For four external fermions, we only need to repeat this process twice, where
in the first step we reduce from four external fermions to two external
fermions and two external scalars, and then in the second step we reduce from
two external fermions and two external scalars to four external scalars. The
second step is already what we derived above, so the only new ingredient is
the first step:

$\left\\{\begin{array}[]{ccc}\psi_{1}&\psi_{2}&\mathcal{O}_{6}\\\
\psi_{3}&\psi_{4}&\mathcal{O}_{5}\end{array}\right\\}^{s_{1}s_{2}t_{1}t_{2}}=\sum\limits_{a,b,c,d,t_{1}^{\prime},t_{2}^{\prime}}\mathcal{J}^{s_{1}s_{2}t_{1}t_{2}}_{\raisebox{-2.1097pt}{\scalebox{1.8}{$\cdot$}}s_{2}t_{1}^{\prime}t_{2}^{\prime}}\footnotesize\left(\begin{array}[]{cccccc}\psi_{1}&\psi_{2}&\psi_{3}&\psi_{4}&\mathcal{O}_{5}&\mathcal{O}_{6}\\\
\phi_{1}^{-a}&\phi_{2}^{-b}&\psi_{3}&\psi_{4}&\mathcal{O}_{5}&\mathcal{O}_{6}^{-c,d}\end{array}\right)\normalsize\\\
\times\left\\{\begin{array}[]{ccc}\phi_{1}^{-a}&\phi_{2}^{-b}&\mathcal{O}_{6}^{-c,d}\\\
\psi_{3}&\psi_{4}&\mathcal{O}_{5}\end{array}\right\\}^{\raisebox{-2.1097pt}{\scalebox{1.8}{$\cdot$}}s_{2}t_{1}^{\prime}t_{2}^{\prime}}.$
(A.1.28)

Combining this with eqn. (A.1.25), we obtain the final result

$\left\\{\begin{array}[]{ccc}\psi_{1}&\psi_{2}&\mathcal{O}_{6}\\\
\psi_{3}&\psi_{4}&\mathcal{O}_{5}\end{array}\right\\}^{s_{1}s_{2}t_{1}t_{2}}=\sum\limits_{a,\dots,h}\mathcal{J}^{s_{1}s_{2}t_{1}t_{2}}\footnotesize\left(\begin{array}[]{cccccc}\psi_{1}&\psi_{2}&\psi_{3}&\psi_{4}&\mathcal{O}_{5}&\mathcal{O}_{6}\\\
\phi_{1}^{-a}&\phi_{2}^{-b}&\phi_{3}^{-e}&\phi_{4}^{-f}&\mathcal{O}_{5}&\mathcal{O}_{6}^{-c-g,d+h}\end{array}\right)\normalsize\\\
\times\left\\{\begin{array}[]{ccc}\phi_{1}^{-a}&\phi_{2}^{-b}&\mathcal{O}_{6}^{-c-g,d+h}\\\
\phi_{3}^{-e}&\phi_{4}^{-f}&\mathcal{O}_{5}\end{array}\right\\},$ (A.1.29)

where

$\mathcal{J}^{s_{1}s_{2}t_{1}t_{2}}\footnotesize\left(\begin{array}[]{cccccc}\psi_{1}&\psi_{2}&\psi_{3}&\psi_{4}&\mathcal{O}_{5}&\mathcal{O}_{6}\\\
\phi_{1}^{-a}&\phi_{2}^{-b}&\phi_{3}^{-e}&\phi_{4}^{-f}&\mathcal{O}_{5}&\mathcal{O}_{6}^{-c-g,d+h}\end{array}\right)\normalsize\\\
=\sum\limits_{t_{1}^{\prime},t_{2}^{\prime}}\mathcal{J}^{s_{1}s_{2}t_{1}t_{2}}_{\raisebox{-2.1097pt}{\scalebox{1.8}{$\cdot$}}s_{2}t_{1}^{\prime}t_{2}^{\prime}}\footnotesize\left(\begin{array}[]{cccccc}\psi_{1}&\psi_{2}&\psi_{3}&\psi_{4}&\mathcal{O}_{5}&\mathcal{O}_{6}\\\
\phi_{1}^{-a}&\phi_{2}^{-b}&\psi_{3}&\psi_{4}&\mathcal{O}_{5}&\mathcal{O}_{6}^{-c,d}\end{array}\right)\normalsize\\\
\times\mathcal{J}^{\raisebox{-2.1097pt}{\scalebox{1.8}{$\cdot$}}s_{2}t_{1}^{\prime}t_{2}^{\prime}}\footnotesize\left(\begin{array}[]{cccccc}\phi_{1}^{-a}&\phi_{2}^{-b}&\psi_{3}&\psi_{4}&\mathcal{O}_{5}&\mathcal{O}_{6}^{-c,d}\\\
\phi_{1}^{-a}&\phi_{2}^{-b}&\phi_{3}^{-e}&\phi_{4}^{-f}&\mathcal{O}_{5}&\mathcal{O}_{6}^{-c-g,d+h}\end{array}\right)\normalsize.$
(A.1.30)

We can derive
$J^{s_{1}s_{2}t_{1}t_{2}}_{\raisebox{-2.1097pt}{\scalebox{1.8}{$\cdot$}}s_{2}t_{1}^{\prime}t_{2}^{\prime}}$
in a similar manner to how we derived
$J^{\raisebox{-2.1097pt}{\scalebox{1.8}{$\cdot$}}s_{2}t_{1}t_{2}}$. For
brevity we skip the intermediate steps, illustrated diagrammatically in figure
A.2, and only present the final result here:

$\mathcal{J}^{s_{1}s_{2}t_{1}t_{2}}_{\raisebox{-2.1097pt}{\scalebox{1.8}{$\cdot$}}s_{2}t_{1}^{\prime}t_{2}^{\prime}}\footnotesize\left(\begin{array}[]{cccccc}\psi_{1}&\psi_{2}&\psi_{3}&\psi_{4}&\mathcal{O}_{5}&\mathcal{O}_{6}\\\
\phi_{1}^{-a}&\phi_{2}^{-b}&\psi_{3}&\psi_{4}&\mathcal{O}_{5}&\mathcal{O}_{6}^{-c,d}\end{array}\right)\normalsize=\sum\limits_{u_{1},u_{2},u_{1}^{\prime},u_{2}^{\prime}}\kappa^{s_{1}}_{3,ab}(\widetilde{\psi}_{1}\widetilde{\psi}_{2}\widetilde{\mathcal{O}}_{5})\zeta_{0}^{-b,+}\zeta_{0}^{-a,+}\zeta_{\ell_{t}+d}^{-c,-d}\\\
\times\beta_{b,+}^{\Delta_{2}-b,0}E_{\psi_{1}\psi_{2}\mathcal{O}\rightarrow\psi_{2}\psi_{1}\mathcal{O}}^{t_{1}u_{1}}\left\\{\begin{matrix}\psi_{2}&\psi_{3}&\phi_{2}^{-b}\\\
\mathcal{O}_{6}^{-c,d}&S&\mathcal{O}_{6}\end{matrix}\right\\}^{u_{1}u_{1}^{\prime}}E_{\phi\psi\mathcal{O}\rightarrow\psi\phi\mathcal{O}}^{u_{1}^{\prime}t_{1}^{\prime}}E_{\psi_{1}\psi_{2}\mathcal{O}\rightarrow\mathcal{O}\psi_{2}\psi_{1}}^{t_{2}u_{2}}\\\
\times\left\\{\begin{matrix}\widetilde{\mathcal{O}}_{6}&\psi_{4}&\widetilde{\mathcal{O}}_{6}^{c,d}\\\
\phi_{1}^{-a}&S&\psi_{1}\end{matrix}\right\\}^{u_{2}u_{2}^{\prime}}E_{\mathcal{O}\psi\phi\rightarrow\phi\psi\mathcal{O}}^{u_{2}^{\prime}t_{2}^{\prime}}\;.$
(A.1.31)

$\Rightarrow$ $\Rightarrow$ $\Rightarrow$ $\Rightarrow$ $\Rightarrow$

Figure A.2:  Step by step diagrammatic illustration for the decomposition of
$6j$ symbol of four external fermions, $\expectationvalue{\psi\psi\psi\psi}$,
in terms of $6j$ symbols of two external fermions and two external scalars.

###  A.2 OPE Function and its Decomposition

In section 4.3.5 we discussed how $6j$ symbols are related to OPE
coefficients. We reproduce eqn. (4.3.97) for convenience:

$\displaystyle\lambda_{125,a}\lambda_{345,b}\bigg{|}_{G^{(t),fg}_{\mathcal{O}_{6}}}$
$\displaystyle=-\Res_{\Delta=\Delta_{5}}\rho_{ac}^{(s)}(\mathcal{O})S^{c}_{b}(\mathcal{O}_{3}\mathcal{O}_{4}[\widetilde{\mathcal{O}}])\bigg{|}_{G^{(t),fg}_{\mathcal{O}_{6}}}$
$\displaystyle=(-1)^{1+\Sigma_{55}+\Sigma_{12}+\Sigma_{13}+\Sigma_{23}}\lambda_{326,f}\lambda_{146,g}$
$\displaystyle\quad\times\Res_{\Delta=\Delta_{5}}\eta_{(ad)(ce)}^{\Delta,J}\left(\begin{array}[]{ccc}\mathcal{O}_{1}&\mathcal{O}_{2}&\mathcal{O}_{6}\\\
\mathcal{O}_{3}&\mathcal{O}_{4}&\mathcal{O}_{\Delta,J}\end{array}\right)^{defg}S^{c}_{b}(\mathcal{O}_{3}\mathcal{O}_{4}[\widetilde{\mathcal{O}}])\bigg{|}_{J=J_{5}}.$
(A.2.3)

We aim to relate the inversion of a single block for external fermions to the
inversion of a single block for a scalar four-point function. For that, by
comparing (4.3.73), (4.3.92), and (A.1.13) we write down

$\left(\begin{array}[]{ccc}\mathcal{O}_{1}&\mathcal{O}_{2}&\mathcal{O}_{6}\\\
\mathcal{O}_{3}&\mathcal{O}_{4}&\mathcal{O}_{5}\end{array}\right)^{abcd}=(-1)^{\Sigma_{66}}\left(S^{-1}(\mathcal{O}_{1}\mathcal{O}_{4}[\widetilde{\mathcal{O}_{6}}])\right)^{d}_{e}\\\
\times\sum\limits_{\phi_{i},\mathcal{O}^{\prime}_{5,6}}\mathcal{J}^{abce}\footnotesize\left(\begin{array}[]{cccccc}\mathcal{O}_{1}&\mathcal{O}_{2}&\mathcal{O}_{3}&\mathcal{O}_{4}&\mathcal{O}_{5}&\mathcal{O}_{6}\\\
\phi_{1}&\phi_{2}&\phi_{3}&\phi_{4}&\mathcal{O}^{\prime}_{5}&\mathcal{O}^{\prime}_{6}\end{array}\right)\normalsize
K^{\phi_{1}\phi_{4}}_{\widetilde{\mathcal{O}}^{\prime}_{6}}\left(\begin{array}[]{ccc}\phi_{1}&\phi_{2}&\mathcal{O}^{\prime}_{6}\\\
\phi_{3}&\phi_{4}&\mathcal{O}^{\prime}_{5}\end{array}\right).$ (A.2.4)

We then find the following expression for the OPE function:

$\displaystyle\rho_{ac}^{(s)}(\mathcal{O})S^{c}_{b}(\mathcal{O}_{3}\mathcal{O}_{4}[\widetilde{\mathcal{O}}])\evaluated{}_{G^{(t),fg}_{\mathcal{O}_{6}}}=\lambda_{326,f}\lambda_{146,g}\sum\limits_{\phi_{i},\mathcal{O}^{\prime},\mathcal{O}^{\prime}_{6}}$
$\displaystyle\mathcal{K}^{fg}_{ab}\footnotesize\left(\begin{array}[]{cccccc}\mathcal{O}_{1}&\mathcal{O}_{2}&\mathcal{O}_{3}&\mathcal{O}_{4}&\mathcal{O}&\mathcal{O}_{6}\\\
\phi_{1}&\phi_{2}&\phi_{3}&\phi_{4}&\mathcal{O}^{\prime}&\mathcal{O}^{\prime}_{6}\end{array}\right)\normalsize$
(A.2.7)
$\displaystyle\qquad\frac{S(\phi_{3}\phi_{4}[\widetilde{\mathcal{O}^{\prime}}])}{\eta^{(s)}_{\mathcal{O}^{\prime}}}\left(\begin{array}[]{ccc}\phi_{1}&\phi_{2}&\mathcal{O}^{\prime}_{6}\\\
\phi_{3}&\phi_{4}&\mathcal{O}^{\prime}\end{array}\right),$ (A.2.10)

where $\eta^{(s)}_{\mathcal{O}_{5}}$ is the normalization of the scalar
partial wave for $\expectationvalue{\phi_{1}\phi_{2}\phi_{3}\phi_{4}}$ and we
have defined

$\mathcal{K}^{fg}_{ab}\footnotesize\left(\begin{array}[]{cccccc}\mathcal{O}_{1}&\mathcal{O}_{2}&\mathcal{O}_{3}&\mathcal{O}_{4}&\mathcal{O}&\mathcal{O}_{6}\\\
\phi_{1}&\phi_{2}&\phi_{3}&\phi_{4}&\mathcal{O}^{\prime}&\mathcal{O}^{\prime}_{6}\end{array}\right)\normalsize\equiv(-1)^{\Sigma_{55}+\Sigma_{66}+\Sigma_{12}+\Sigma_{13}+\Sigma_{23}}K^{\phi_{1}\phi_{4}}_{\widetilde{\mathcal{O}}^{\prime}_{6}}\frac{S^{c}_{b}(\mathcal{O}_{3}\mathcal{O}_{4}[\widetilde{\mathcal{O}}])}{S(\phi_{3}\phi_{4}[\widetilde{\mathcal{O}}^{\prime}])}\\\
\times\left(S^{-1}(\mathcal{O}_{1}\mathcal{O}_{4}[\widetilde{\mathcal{O}_{6}}])\right)^{g}_{h}\eta_{\mathcal{O}^{\prime}}^{(s)}\eta_{(ad)(ce)}^{(s)\mathcal{O}}\mathcal{J}^{defh}\footnotesize\left(\begin{array}[]{cccccc}\mathcal{O}_{1}&\mathcal{O}_{2}&\mathcal{O}_{3}&\mathcal{O}_{4}&\mathcal{O}&\mathcal{O}_{6}\\\
\phi_{1}&\phi_{2}&\phi_{3}&\phi_{4}&\mathcal{O}^{\prime}&\mathcal{O}^{\prime}_{6}\end{array}\right)\normalsize.$
(A.2.11)

We expect the physical poles for inverting a fermionic block to come from the
physical poles from inverting a scalar block. We then have for example

$\lambda_{125,a}\lambda_{345,b}\evaluated{}_{G^{(t),fg}_{\mathcal{O}_{6}}}=-\lambda_{326,f}\lambda_{146,g}\sum\limits_{\phi_{i},\mathcal{O}^{\prime},\mathcal{O}^{\prime}_{6}}\mathcal{K}^{fg}_{ab}\footnotesize\left(\begin{array}[]{cccccc}\mathcal{O}_{1}&\mathcal{O}_{2}&\mathcal{O}_{3}&\mathcal{O}_{4}&\mathcal{O}_{5}&\mathcal{O}_{6}\\\
\phi_{1}&\phi_{2}&\phi_{3}&\phi_{4}&\mathcal{O}^{\prime}_{5}&\mathcal{O}^{\prime}_{6}\end{array}\right)\normalsize\\\
\times\Res_{\Delta=\Delta_{5}}\frac{S(\phi_{3}\phi_{4}[\widetilde{\mathcal{O}^{\prime}_{\Delta,J}}])}{\eta^{(s)}_{\mathcal{O}^{\prime}_{\Delta,J}}}\left(\begin{array}[]{ccc}\phi_{1}&\phi_{2}&\mathcal{O}^{\prime}_{6}\\\
\phi_{3}&\phi_{4}&\mathcal{O}_{\Delta,J}^{\prime}\end{array}\right)\evaluated{}_{J=J_{5}}.$
(A.2.12)

Here we have assumed the inversion of a single block just has single poles. In
general when studying a correlator
$\expectationvalue{\mathcal{O}_{1}\mathcal{O}_{2}\mathcal{O}_{3}\mathcal{O}_{4}}$
with $\Delta_{1}+\Delta_{2}=\Delta_{3}+\Delta_{4}$ we find both single and
double poles. The double poles give the OPE coefficients times the anomalous
dimensions while the single poles gives the OPE coefficients themselves [127].

Equation A.2.7 is the main result of this approach: by using weight-shifting
operators successively, we can express CFT data of spinning operators in terms
of $6j$ symbols of external scalars and the decomposition coefficients
$\mathcal{K}$. The former can be calculated efficiently using the Lorentzian
inversion formula whereas the latter is given in terms of partial wave
normalization factors, shadow matrices, and $6j$ symbol decomposition
coefficients $\mathcal{J}$, each of which we have computed explicitly.

We presented the most general form in eqn. (A.2.7), however one can in fact
choose either $\mathcal{O}_{5}$ or $\mathcal{O}_{6}$ to stay the same by
moving the weight-shifting operators through the other leg only. Indeed, we
kept $\mathcal{O}_{5}$ the same in the calculation of the $\mathcal{J}$
coefficients both for $\expectationvalue{\phi\phi\psi\psi}$ and for
$\expectationvalue{\psi\psi\psi\psi}$, as we can observe in eqn. (A.1.25) and
eqn. (A.1.29). One can similarly compute $\mathcal{J}$ while keeping
$\mathcal{O}_{6}$ constant, though a separate calculation is not necessary:
there are several identities between various $\mathcal{J}$ coefficients, which
follow from the symmetries of the $6j$ symbols that we have summarized in
appendix A.4. In particular, via eqn. (A.4.5), we have

$\mathcal{J}^{abcd}_{efgh}\footnotesize\left(\begin{array}[]{cccccc}\mathcal{O}_{1}&\mathcal{O}_{2}&\mathcal{O}_{3}&\mathcal{O}_{4}&\mathcal{O}_{5}&\mathcal{O}_{6}\\\
\mathcal{O}_{1}^{\delta_{1},\epsilon_{1}}&\mathcal{O}_{2}^{\delta_{2},\epsilon_{2}}&\mathcal{O}_{3}^{\delta_{3},\epsilon_{3}}&\mathcal{O}_{4}^{\delta_{4},\epsilon_{4}}&\mathcal{O}_{5}^{\delta_{5},\epsilon_{5}}&\mathcal{O}_{6}^{\delta_{6},\epsilon_{6}}\end{array}\right)\normalsize=(-1)^{\Sigma_{55}+\Sigma_{5^{\prime}5^{\prime}}+\Sigma_{66}+\Sigma_{6^{\prime}6^{\prime}}}\\\
\times\mathcal{J}^{dcba}_{hgfe}\footnotesize\left(\begin{array}[]{cccccc}\widetilde{\mathcal{O}}_{1}&\widetilde{\mathcal{O}}_{4}&\widetilde{\mathcal{O}}_{3}&\widetilde{\mathcal{O}}_{2}&\mathcal{O}_{6}&\mathcal{O}_{5}\\\
\widetilde{\mathcal{O}}_{1}^{-\delta_{1},\epsilon_{1}}&\widetilde{\mathcal{O}}_{4}^{-\delta_{4},\epsilon_{4}}&\widetilde{\mathcal{O}}_{3}^{-\delta_{3},\epsilon_{3}}&\widetilde{\mathcal{O}}_{2}^{-\delta_{2},\epsilon_{2}}&\mathcal{O}_{6}^{\delta_{6},\epsilon_{6}}&\mathcal{O}_{5}^{\delta_{5},\epsilon_{5}}\end{array}\right)\normalsize.$
(A.2.13)

Let us now turn to the explicit results for the $\mathcal{K}^{fg}_{ab}$
coefficients. Despite the complicated intermediate steps, the final form they
take is quite simple as they are relatively short meromorphic functions in
scaling dimensions and spins. For example, the only nonzero $\mathcal{K}$
coefficients for $\expectationvalue{\psi\phi\phi\psi}$ arising from the
exchange of a scalar $\phi_{6}$ in the t-channel are

$\displaystyle\mathcal{K}^{\raisebox{-1.89873pt}{\scalebox{1.8}{$\cdot$}}1}_{11}\footnotesize\left(\begin{array}[]{cccccc}\psi_{1}&\phi_{2}&\phi_{2}&\psi_{1}&\mathcal{O}_{5}&\phi_{6}\\\
\phi_{1}^{\frac{1}{2}}&\phi_{2}&\phi_{2}&\phi_{1}^{\frac{1}{2}}&\mathcal{O}_{5}^{\frac{1}{2},-\frac{1}{2}}&\phi_{6}\end{array}\right)\normalsize={}$
$\displaystyle{}i,$ (A.2.14c)
$\displaystyle\mathcal{K}^{\raisebox{-1.89873pt}{\scalebox{1.8}{$\cdot$}}1}_{11}\footnotesize\left(\begin{array}[]{cccccc}\psi_{1}&\phi_{2}&\phi_{2}&\psi_{1}&\mathcal{O}_{5}&\phi_{6}\\\
\phi_{1}^{\frac{1}{2}}&\phi_{2}&\phi_{2}&\phi_{1}^{\frac{1}{2}}&\mathcal{O}_{5}^{-\frac{1}{2},\frac{1}{2}}&\phi_{6}\end{array}\right)\normalsize={}$
$\displaystyle{}\frac{i\left(\Delta_{5}-\frac{3}{2}\right)\left(\ell_{5}+\frac{1}{2}\right)\left(\ell_{5}-\Delta_{251}+\frac{5}{2}\right)^{2}}{4\left(\Delta_{5}-2\right)\left(\ell_{5}+1\right)\left(\ell_{5}-\Delta_{5}+2\right)\left(\ell_{5}-\Delta_{5}+3\right)},$
(A.2.14f)
$\displaystyle\mathcal{K}^{\raisebox{-1.89873pt}{\scalebox{1.8}{$\cdot$}}1}_{22}\footnotesize\left(\begin{array}[]{cccccc}\psi_{1}&\phi_{2}&\phi_{2}&\psi_{1}&\mathcal{O}_{5}&\phi_{6}\\\
\phi_{1}^{\frac{1}{2}}&\phi_{2}&\phi_{2}&\phi_{1}^{\frac{1}{2}}&\mathcal{O}_{5}^{\frac{1}{2},\frac{1}{2}}&\phi_{6}\end{array}\right)\normalsize={}$
$\displaystyle{}-\frac{i\left(\ell_{5}+\frac{1}{2}\right)}{\ell_{5}+1},$
(A.2.14i)
$\displaystyle\mathcal{K}^{\raisebox{-1.89873pt}{\scalebox{1.8}{$\cdot$}}1}_{22}\footnotesize\left(\begin{array}[]{cccccc}\psi_{1}&\phi_{2}&\phi_{2}&\psi_{1}&\mathcal{O}_{5}&\phi_{6}\\\
\phi_{1}^{\frac{1}{2}}&\phi_{2}&\phi_{2}&\phi_{1}^{\frac{1}{2}}&\mathcal{O}_{5}^{-\frac{1}{2},-\frac{1}{2}}&\phi_{6}\end{array}\right)\normalsize={}$
$\displaystyle{}-\frac{i\left(\Delta_{5}-\frac{3}{2}\right)\left(\Delta_{251}+\ell_{5}-\frac{3}{2}\right)^{2}}{4\left(\Delta_{5}-2\right)\left(\Delta_{5}+\ell_{5}-2\right)\left(\Delta_{5}+\ell_{5}-1\right)},$
(A.2.14l)
$\displaystyle\mathcal{K}^{\raisebox{-1.89873pt}{\scalebox{1.8}{$\cdot$}}3}_{12}\footnotesize\left(\begin{array}[]{cccccc}\psi_{1}&\phi_{2}&\phi_{2}&\psi_{1}&\mathcal{O}_{5}&\phi_{6}\\\
\phi_{1}^{-\frac{1}{2}}&\phi_{2}&\phi_{2}&\phi_{1}^{\frac{1}{2}}&\mathcal{O}_{5}^{\frac{1}{2},\frac{1}{2}}&\phi_{6}\end{array}\right)\normalsize={}$
$\displaystyle{}\frac{i\left(\ell_{5}+\frac{1}{2}\right)\left(\ell_{5}-\Delta_{125}+\frac{5}{2}\right)}{\left(\Delta_{6}-1\right)\left(\ell_{5}+1\right)},$
(A.2.14o)
$\displaystyle\mathcal{K}^{\raisebox{-1.89873pt}{\scalebox{1.8}{$\cdot$}}3}_{12}\footnotesize\left(\begin{array}[]{cccccc}\psi_{1}&\phi_{2}&\phi_{2}&\psi_{1}&\mathcal{O}_{5}&\phi_{6}\\\
\phi_{1}^{-\frac{1}{2}}&\phi_{2}&\phi_{2}&\phi_{1}^{\frac{1}{2}}&\mathcal{O}_{5}^{-\frac{1}{2},-\frac{1}{2}}&\phi_{6}\end{array}\right)\normalsize={}$
$\displaystyle{}\frac{i\left(\Delta_{152}+\ell_{5}-\frac{3}{2}\right)\left(\Delta_{251}+\ell_{5}-\frac{3}{2}\right)}{4\left(\Delta_{5}-\frac{3}{2}\right)^{-1}\left(\Delta_{5}-2\right)\left(\Delta_{6}-1\right)\left(\Delta_{5}+\ell_{5}-2\right)}$
(A.2.14r)
$\displaystyle\qquad\times\frac{\left(\Delta_{1}+\Delta_{2}+\Delta_{5}+\ell_{5}-\frac{9}{2}\right)}{\Delta_{5}+\ell_{5}-1}\;,$
(A.2.14s)
$\displaystyle\mathcal{K}^{\raisebox{-1.89873pt}{\scalebox{1.8}{$\cdot$}}3}_{21}\footnotesize\left(\begin{array}[]{cccccc}\psi_{1}&\phi_{2}&\phi_{2}&\psi_{1}&\mathcal{O}_{5}&\phi_{6}\\\
\phi_{1}^{-\frac{1}{2}}&\phi_{2}&\phi_{2}&\phi_{1}^{\frac{1}{2}}&\mathcal{O}_{5}^{\frac{1}{2},\frac{1}{2}}&\phi_{6}\end{array}\right)\normalsize={}$
$\displaystyle{}\frac{i\left(\Delta_{125}+\ell_{5}-\frac{3}{2}\right)}{\Delta_{6}-1},$
(A.2.14v)
$\displaystyle\mathcal{K}^{\raisebox{-1.89873pt}{\scalebox{1.8}{$\cdot$}}3}_{21}\footnotesize\left(\begin{array}[]{cccccc}\psi_{1}&\phi_{2}&\phi_{2}&\psi_{1}&\mathcal{O}_{5}&\phi_{6}\\\
\phi_{1}^{-\frac{1}{2}}&\phi_{2}&\phi_{2}&\phi_{1}^{\frac{1}{2}}&\mathcal{O}_{5}^{-\frac{1}{2},\frac{1}{2}}&\phi_{6}\end{array}\right)\normalsize={}$
$\displaystyle{}\frac{i\left(\Delta_{5}-\frac{3}{2}\right)\left(\ell_{5}+\frac{1}{2}\right)\left(\ell_{5}-\Delta_{1}-\Delta_{2}-\Delta_{5}+\frac{11}{2}\right)}{4\left(\Delta_{5}-2\right)\left(\Delta_{6}-1\right)\left(\ell_{5}+1\right)\left(\ell_{5}-\Delta_{5}+2\right)}$
(A.2.14y)
$\displaystyle\qquad\times\frac{\left(\ell_{5}-\Delta_{152}+\frac{5}{2}\right)\left(\ell_{5}-\Delta_{251}+\frac{5}{2}\right)}{-\Delta_{5}+\ell_{5}+3},$
(A.2.14z)

We would like to emphasize two points. Firstly, as there is not a unique way
to write eqn. (A.2.12), the statement that the $\mathcal{K}$ in eqn. (A.2.14)
are the only nonzero coefficients for $\expectationvalue{\psi\phi\phi\psi}$
with an exchange of a scalar $\phi_{6}$ in the t-channel should be understood
for a particularly chosen decomposition in eqn. (A.2.12). One can of course
change the decomposition, which would then require a new set of $\mathcal{K}$
coefficients. For example, we used a set of $\mathcal{K}$ coefficients with
$\mathcal{O}_{6}$ held constant in eqn. (A.2.14); another set with
$\mathcal{O}_{5}$ held constant instead can be immediately obtained via eqn.
(A.2.13).444It should be noted that not all different decompositions are
related to each other via symmetries. For example,
$\mathcal{O}_{4}^{\prime}=\phi_{1}^{\frac{1}{2}}$ whereas
$\mathcal{O}_{1}^{\prime}=\phi_{1}^{\pm\frac{1}{2}}$ in eqn. (A.2.14): this
follows from fixing $b=\frac{1}{2}$ in eqn. (A.1.26) as we noted after the
equation. If we were to fix $b=-\frac{1}{2}$ instead, we would then have a set
of $\mathcal{K}$ coefficients with
$\mathcal{O}_{4}^{\prime}=\phi_{1}^{-\frac{1}{2}}$ and
$\mathcal{O}_{1}^{\prime}=\phi_{1}^{\pm\frac{1}{2}}$, and these new
coefficients are not related to eqn. (A.2.14) in any manifestly symmetric way.
Secondly, we note that the absence of
$\mathcal{K}^{\raisebox{-2.1097pt}{\scalebox{1.8}{$\cdot$}}1}_{12}$,
$\mathcal{K}^{\raisebox{-2.1097pt}{\scalebox{1.8}{$\cdot$}}1}_{21}$,
$\mathcal{K}^{\raisebox{-2.1097pt}{\scalebox{1.8}{$\cdot$}}3}_{11}$, and
$\mathcal{K}^{\raisebox{-2.1097pt}{\scalebox{1.8}{$\cdot$}}3}_{22}$ is not
coincidental: they are forbidden by the parity symmetry as we work in a
parity-definite basis.555We would like to caution the reader that this
statement follows from the symmetries of three-point structures
$\expectationvalue{\mathcal{O}_{1}\mathcal{O}_{2}\mathcal{O}_{3}}^{a}$ under
the transformation $X\rightarrow-X$ in embedding space, hence it is true
whether the relevant physical theory has parity symmetry or not, i.e.
$\expectationvalue{\mathcal{O}_{1}\mathcal{O}_{2}\mathcal{O}_{3}}^{a}$ are
merely formal entities and should not be thought of as physical three-point
structures.

###  A.3 $\mathcal{K}$ Coefficients

In this appendix, we present the explicit expression for the $\mathcal{K}$
coefficients defined in eqn. (A.2.11) for
$\expectationvalue{\psi\phi\phi\psi}$. As there are a different number of
three-point tensor structures depending on whether $\ell_{5,6}=0$, the minimal
complete set of nonzero $\mathcal{K}$ coefficients differs for each case. We
already presented the results for $\expectationvalue{\psi\phi\phi\psi}$ with
$\ell_{6}=0$ in eqn. (A.2.14), so we will detail the
$\expectationvalue{\psi\phi\phi\psi}$ with $\ell_{6}\neq 0$ below. For
$\expectationvalue{\psi_{1}\psi_{2}\psi_{2}\psi_{1}}$ and
$\expectationvalue{\psi\psi\psi\psi}$, the coefficients become quite lengthy
so we do not reproduce them here; please see the Mathematica file of [2] for
their explicit expressions.

For the correlator $\expectationvalue{\psi\phi\phi\psi}$, the list below
constitutes a sufficient set of nonzero $\mathcal{K}$ coefficients if the
exchanged operator in t-channel is not a scalar.666For scalar exchange in the
t-channel, see eqn. (A.2.14). For convenience, we will use a different
shorthand notation in this section, i.e.

$\displaystyle\Delta_{abc}=\Delta_{a}+\Delta_{b}+\Delta_{c}\;,\quad\Delta_{ab}^{c}=\Delta_{a}+\Delta_{b}-\Delta_{c}.$
(A.3.1)

The coefficients are:

$\begin{array}[]{cc}\scalebox{0.8}{
$\mathcal{K}^{\raisebox{-1.68776pt}{\scalebox{1.8}{$\cdot$}}2}_{11}\footnotesize\left(\begin{array}[]{cccccc}\psi_{1}&\phi_{2}&\phi_{2}&\psi_{1}&\mathcal{O}_{5}&\mathcal{O}_{6}\\\
\phi_{1}^{-\frac{1}{2}}&\phi_{2}&\phi_{2}&\phi_{1}^{-\frac{1}{2}}&\mathcal{O}_{5}^{-\frac{1}{2},-\frac{1}{2}}&\mathcal{O}_{6}\end{array}\right)\normalsize$
}&\frac{i\left(\Delta_{5}-\frac{3}{2}\right)\left(\Delta_{125}+\ell_{5}-\frac{9}{2}\right){}^{2}\left(\Delta_{15}^{2}+\ell_{5}-\frac{3}{2}\right){}^{2}}{16\left(\Delta_{5}-2\right)\left(\Delta_{6}-1\right)\ell_{6}\left(\Delta_{5}+\ell_{5}-2\right)\left(\Delta_{5}+\ell_{5}-1\right)}\\\
\scalebox{0.8}{
$\mathcal{K}^{\raisebox{-1.68776pt}{\scalebox{1.8}{$\cdot$}}2}_{22}\footnotesize\left(\begin{array}[]{cccccc}\psi_{1}&\phi_{2}&\phi_{2}&\psi_{1}&\mathcal{O}_{5}&\mathcal{O}_{6}\\\
\phi_{1}^{-\frac{1}{2}}&\phi_{2}&\phi_{2}&\phi_{1}^{-\frac{1}{2}}&\mathcal{O}_{5}^{-\frac{1}{2},\frac{1}{2}}&\mathcal{O}_{6}\end{array}\right)\normalsize$
}&-\frac{i\left(\Delta_{5}-\frac{3}{2}\right)\left(\ell_{5}+\frac{1}{2}\right)\left(-\Delta_{125}+\ell_{5}+\frac{11}{2}\right){}^{2}\left(-\Delta_{15}^{2}+\ell_{5}+\frac{5}{2}\right){}^{2}}{16\left(\Delta_{5}-2\right)\left(\Delta_{6}-1\right)\left(\ell_{5}+1\right)\ell_{6}\left(-\Delta_{5}+\ell_{5}+2\right)\left(-\Delta_{5}+\ell_{5}+3\right)}\\\
\scalebox{0.8}{
$\mathcal{K}^{\raisebox{-1.68776pt}{\scalebox{1.8}{$\cdot$}}2}_{22}\footnotesize\left(\begin{array}[]{cccccc}\psi_{1}&\phi_{2}&\phi_{2}&\psi_{1}&\mathcal{O}_{5}&\mathcal{O}_{6}\\\
\phi_{1}^{-\frac{1}{2}}&\phi_{2}&\phi_{2}&\phi_{1}^{-\frac{1}{2}}&\mathcal{O}_{5}^{\frac{1}{2},-\frac{1}{2}}&\mathcal{O}_{6}\end{array}\right)\normalsize$
}&-\frac{i\left(\Delta_{12}^{5}+\ell_{5}-\frac{3}{2}\right){}^{2}}{4\left(\Delta_{6}-1\right)\ell_{6}}\\\
\scalebox{0.8}{
$\mathcal{K}^{\raisebox{-1.68776pt}{\scalebox{1.8}{$\cdot$}}2}_{11}\footnotesize\left(\begin{array}[]{cccccc}\psi_{1}&\phi_{2}&\phi_{2}&\psi_{1}&\mathcal{O}_{5}&\mathcal{O}_{6}\\\
\phi_{1}^{-\frac{1}{2}}&\phi_{2}&\phi_{2}&\phi_{1}^{-\frac{1}{2}}&\mathcal{O}_{5}^{\frac{1}{2},\frac{1}{2}}&\mathcal{O}_{6}\end{array}\right)\normalsize$
}&\frac{i\left(\ell_{5}+\frac{1}{2}\right)\left(-\Delta_{12}^{5}+\ell_{5}+\frac{5}{2}\right){}^{2}}{4\left(\Delta_{6}-1\right)\left(\ell_{5}+1\right)\ell_{6}}\\\
\scalebox{0.8}{
$\mathcal{K}^{\raisebox{-1.68776pt}{\scalebox{1.8}{$\cdot$}}3}_{12}\footnotesize\left(\begin{array}[]{cccccc}\psi_{1}&\phi_{2}&\phi_{2}&\psi_{1}&\mathcal{O}_{5}&\mathcal{O}_{6}\\\
\phi_{1}^{-\frac{1}{2}}&\phi_{2}&\phi_{2}&\phi_{1}^{\frac{1}{2}}&\mathcal{O}_{5}^{-\frac{1}{2},-\frac{1}{2}}&\mathcal{O}_{6}\end{array}\right)\normalsize$
}&\frac{i\left(\Delta_{5}-\frac{3}{2}\right)\left(\Delta_{125}+\ell_{5}-\frac{9}{2}\right)\left(\Delta_{15}^{2}+\ell_{5}-\frac{3}{2}\right)\left(\Delta_{25}^{1}+\ell_{5}-\frac{3}{2}\right)}{8\left(\Delta_{5}-2\right)\left(\Delta_{6}-1\right)\left(\Delta_{5}+\ell_{5}-2\right)\left(\Delta_{5}+\ell_{5}-1\right)}\\\
\scalebox{0.8}{
$\mathcal{K}^{\raisebox{-1.68776pt}{\scalebox{1.8}{$\cdot$}}4}_{12}\footnotesize\left(\begin{array}[]{cccccc}\psi_{1}&\phi_{2}&\phi_{2}&\psi_{1}&\mathcal{O}_{5}&\mathcal{O}_{6}\\\
\phi_{1}^{-\frac{1}{2}}&\phi_{2}&\phi_{2}&\phi_{1}^{\frac{1}{2}}&\mathcal{O}_{5}^{-\frac{1}{2},-\frac{1}{2}}&\mathcal{O}_{6}\end{array}\right)\normalsize$
}&\frac{i\left(\Delta_{5}-\frac{3}{2}\right)\left(\Delta_{125}+\ell_{5}-\frac{9}{2}\right)\left(\Delta_{15}^{2}+\ell_{5}-\frac{3}{2}\right)\left(\Delta_{25}^{1}+\ell_{5}-\frac{3}{2}\right)}{8\left(\Delta_{5}-2\right)\ell_{6}\left(\Delta_{5}+\ell_{5}-2\right)\left(\Delta_{5}+\ell_{5}-1\right)}\\\
\scalebox{0.8}{
$\mathcal{K}^{\raisebox{-1.68776pt}{\scalebox{1.8}{$\cdot$}}3}_{21}\footnotesize\left(\begin{array}[]{cccccc}\psi_{1}&\phi_{2}&\phi_{2}&\psi_{1}&\mathcal{O}_{5}&\mathcal{O}_{6}\\\
\phi_{1}^{-\frac{1}{2}}&\phi_{2}&\phi_{2}&\phi_{1}^{\frac{1}{2}}&\mathcal{O}_{5}^{-\frac{1}{2},\frac{1}{2}}&\mathcal{O}_{6}\end{array}\right)\normalsize$
}&\frac{i\left(\Delta_{5}-\frac{3}{2}\right)\left(\ell_{5}+\frac{1}{2}\right)\left(-\Delta_{125}+\ell_{5}+\frac{11}{2}\right)\left(-\Delta_{15}^{2}+\ell_{5}+\frac{5}{2}\right)\left(-\Delta_{25}^{1}+\ell_{5}+\frac{5}{2}\right)}{8\left(\Delta_{5}-2\right)\left(\Delta_{6}-1\right)\left(\ell_{5}+1\right)\left(-\Delta_{5}+\ell_{5}+2\right)\left(-\Delta_{5}+\ell_{5}+3\right)}\\\
\scalebox{0.8}{
$\mathcal{K}^{\raisebox{-1.68776pt}{\scalebox{1.8}{$\cdot$}}4}_{21}\footnotesize\left(\begin{array}[]{cccccc}\psi_{1}&\phi_{2}&\phi_{2}&\psi_{1}&\mathcal{O}_{5}&\mathcal{O}_{6}\\\
\phi_{1}^{-\frac{1}{2}}&\phi_{2}&\phi_{2}&\phi_{1}^{\frac{1}{2}}&\mathcal{O}_{5}^{-\frac{1}{2},\frac{1}{2}}&\mathcal{O}_{6}\end{array}\right)\normalsize$
}&\frac{i\left(\Delta_{5}-\frac{3}{2}\right)\left(\ell_{5}+\frac{1}{2}\right)\left(-\Delta_{125}+\ell_{5}+\frac{11}{2}\right)\left(-\Delta_{15}^{2}+\ell_{5}+\frac{5}{2}\right)\left(-\Delta_{25}^{1}+\ell_{5}+\frac{5}{2}\right)}{8\left(\Delta_{5}-2\right)\left(\ell_{5}+1\right)\ell_{6}\left(-\Delta_{5}+\ell_{5}+2\right)\left(-\Delta_{5}+\ell_{5}+3\right)}\\\
\scalebox{0.8}{
$\mathcal{K}^{\raisebox{-1.68776pt}{\scalebox{1.8}{$\cdot$}}3}_{21}\footnotesize\left(\begin{array}[]{cccccc}\psi_{1}&\phi_{2}&\phi_{2}&\psi_{1}&\mathcal{O}_{5}&\mathcal{O}_{6}\\\
\phi_{1}^{-\frac{1}{2}}&\phi_{2}&\phi_{2}&\phi_{1}^{\frac{1}{2}}&\mathcal{O}_{5}^{\frac{1}{2},-\frac{1}{2}}&\mathcal{O}_{6}\end{array}\right)\normalsize$
}&\frac{i\left(\Delta_{12}^{5}+\ell_{5}-\frac{3}{2}\right)}{2\left(\Delta_{6}-1\right)}\\\
\scalebox{0.8}{
$\mathcal{K}^{\raisebox{-1.68776pt}{\scalebox{1.8}{$\cdot$}}4}_{21}\footnotesize\left(\begin{array}[]{cccccc}\psi_{1}&\phi_{2}&\phi_{2}&\psi_{1}&\mathcal{O}_{5}&\mathcal{O}_{6}\\\
\phi_{1}^{-\frac{1}{2}}&\phi_{2}&\phi_{2}&\phi_{1}^{\frac{1}{2}}&\mathcal{O}_{5}^{\frac{1}{2},-\frac{1}{2}}&\mathcal{O}_{6}\end{array}\right)\normalsize$
}&\frac{i\left(\Delta_{12}^{5}+\ell_{5}-\frac{3}{2}\right)}{2\ell_{6}}\\\
\scalebox{0.8}{
$\mathcal{K}^{\raisebox{-1.68776pt}{\scalebox{1.8}{$\cdot$}}3}_{12}\footnotesize\left(\begin{array}[]{cccccc}\psi_{1}&\phi_{2}&\phi_{2}&\psi_{1}&\mathcal{O}_{5}&\mathcal{O}_{6}\\\
\phi_{1}^{-\frac{1}{2}}&\phi_{2}&\phi_{2}&\phi_{1}^{\frac{1}{2}}&\mathcal{O}_{5}^{\frac{1}{2},\frac{1}{2}}&\mathcal{O}_{6}\end{array}\right)\normalsize$
}&\frac{i\left(\ell_{5}+\frac{1}{2}\right)\left(-\Delta_{12}^{5}+\ell_{5}+\frac{5}{2}\right)}{2\left(\Delta_{6}-1\right)\left(\ell_{5}+1\right)}\\\
\scalebox{0.8}{
$\mathcal{K}^{\raisebox{-1.68776pt}{\scalebox{1.8}{$\cdot$}}4}_{12}\footnotesize\left(\begin{array}[]{cccccc}\psi_{1}&\phi_{2}&\phi_{2}&\psi_{1}&\mathcal{O}_{5}&\mathcal{O}_{6}\\\
\phi_{1}^{-\frac{1}{2}}&\phi_{2}&\phi_{2}&\phi_{1}^{\frac{1}{2}}&\mathcal{O}_{5}^{\frac{1}{2},\frac{1}{2}}&\mathcal{O}_{6}\end{array}\right)\normalsize$
}&\frac{i\left(\ell_{5}+\frac{1}{2}\right)\left(-\Delta_{12}^{5}+\ell_{5}+\frac{5}{2}\right)}{2\left(\ell_{5}+1\right)\ell_{6}}\\\
\scalebox{0.8}{
$\mathcal{K}^{\raisebox{-1.68776pt}{\scalebox{1.8}{$\cdot$}}3}_{21}\footnotesize\left(\begin{array}[]{cccccc}\psi_{1}&\phi_{2}&\phi_{2}&\psi_{1}&\mathcal{O}_{5}&\mathcal{O}_{6}\\\
\phi_{1}^{\frac{1}{2}}&\phi_{2}&\phi_{2}&\phi_{1}^{-\frac{1}{2}}&\mathcal{O}_{5}^{-\frac{1}{2},-\frac{1}{2}}&\mathcal{O}_{6}\end{array}\right)\normalsize$
}&\frac{i\left(\Delta_{5}-\frac{3}{2}\right)\left(\Delta_{125}+\ell_{5}-\frac{9}{2}\right)\left(\Delta_{15}^{2}+\ell_{5}-\frac{3}{2}\right)\left(\Delta_{25}^{1}+\ell_{5}-\frac{3}{2}\right)}{8\left(\Delta_{5}-2\right)\left(\Delta_{6}-1\right)\left(\Delta_{5}+\ell_{5}-2\right)\left(\Delta_{5}+\ell_{5}-1\right)}\\\
\scalebox{0.8}{
$\mathcal{K}^{\raisebox{-1.68776pt}{\scalebox{1.8}{$\cdot$}}4}_{21}\footnotesize\left(\begin{array}[]{cccccc}\psi_{1}&\phi_{2}&\phi_{2}&\psi_{1}&\mathcal{O}_{5}&\mathcal{O}_{6}\\\
\phi_{1}^{\frac{1}{2}}&\phi_{2}&\phi_{2}&\phi_{1}^{-\frac{1}{2}}&\mathcal{O}_{5}^{-\frac{1}{2},-\frac{1}{2}}&\mathcal{O}_{6}\end{array}\right)\normalsize$
}&\frac{i\left(\Delta_{5}-\frac{3}{2}\right)\left(1-\Delta_{6}\right)\left(\Delta_{125}+\ell_{5}-\frac{9}{2}\right)\left(\Delta_{15}^{2}+\ell_{5}-\frac{3}{2}\right)\left(\Delta_{25}^{1}+\ell_{5}-\frac{3}{2}\right)}{8\left(\Delta_{5}-2\right)\left(\Delta_{6}-1\right)\ell_{6}\left(\Delta_{5}+\ell_{5}-2\right)\left(\Delta_{5}+\ell_{5}-1\right)}\\\
\scalebox{0.8}{
$\mathcal{K}^{\raisebox{-1.68776pt}{\scalebox{1.8}{$\cdot$}}3}_{12}\footnotesize\left(\begin{array}[]{cccccc}\psi_{1}&\phi_{2}&\phi_{2}&\psi_{1}&\mathcal{O}_{5}&\mathcal{O}_{6}\\\
\phi_{1}^{\frac{1}{2}}&\phi_{2}&\phi_{2}&\phi_{1}^{-\frac{1}{2}}&\mathcal{O}_{5}^{-\frac{1}{2},\frac{1}{2}}&\mathcal{O}_{6}\end{array}\right)\normalsize$
}&\frac{i\left(\Delta_{5}-\frac{3}{2}\right)\left(\ell_{5}+\frac{1}{2}\right)\left(-\Delta_{125}+\ell_{5}+\frac{11}{2}\right)\left(-\Delta_{15}^{2}+\ell_{5}+\frac{5}{2}\right)\left(-\Delta_{25}^{1}+\ell_{5}+\frac{5}{2}\right)}{8\left(\Delta_{5}-2\right)\left(\Delta_{6}-1\right)\left(\ell_{5}+1\right)\left(-\Delta_{5}+\ell_{5}+2\right)\left(-\Delta_{5}+\ell_{5}+3\right)}\\\
\scalebox{0.8}{
$\mathcal{K}^{\raisebox{-1.68776pt}{\scalebox{1.8}{$\cdot$}}4}_{12}\footnotesize\left(\begin{array}[]{cccccc}\psi_{1}&\phi_{2}&\phi_{2}&\psi_{1}&\mathcal{O}_{5}&\mathcal{O}_{6}\\\
\phi_{1}^{\frac{1}{2}}&\phi_{2}&\phi_{2}&\phi_{1}^{-\frac{1}{2}}&\mathcal{O}_{5}^{-\frac{1}{2},\frac{1}{2}}&\mathcal{O}_{6}\end{array}\right)\normalsize$
}&\frac{i\left(\Delta_{5}-\frac{3}{2}\right)\left(1-\Delta_{6}\right)\left(\ell_{5}+\frac{1}{2}\right)\left(-\Delta_{125}+\ell_{5}+\frac{11}{2}\right)\left(-\Delta_{15}^{2}+\ell_{5}+\frac{5}{2}\right)\left(-\Delta_{25}^{1}+\ell_{5}+\frac{5}{2}\right)}{8\left(\Delta_{5}-2\right)\left(\Delta_{6}-1\right)\left(\ell_{5}+1\right)\ell_{6}\left(-\Delta_{5}+\ell_{5}+2\right)\left(-\Delta_{5}+\ell_{5}+3\right)}\\\
\scalebox{0.8}{
$\mathcal{K}^{\raisebox{-1.68776pt}{\scalebox{1.8}{$\cdot$}}3}_{12}\footnotesize\left(\begin{array}[]{cccccc}\psi_{1}&\phi_{2}&\phi_{2}&\psi_{1}&\mathcal{O}_{5}&\mathcal{O}_{6}\\\
\phi_{1}^{\frac{1}{2}}&\phi_{2}&\phi_{2}&\phi_{1}^{-\frac{1}{2}}&\mathcal{O}_{5}^{\frac{1}{2},-\frac{1}{2}}&\mathcal{O}_{6}\end{array}\right)\normalsize$
}&\frac{i\left(\Delta_{12}^{5}+\ell_{5}-\frac{3}{2}\right)}{2\left(\Delta_{6}-1\right)}\\\
\scalebox{0.8}{
$\mathcal{K}^{\raisebox{-1.68776pt}{\scalebox{1.8}{$\cdot$}}4}_{12}\footnotesize\left(\begin{array}[]{cccccc}\psi_{1}&\phi_{2}&\phi_{2}&\psi_{1}&\mathcal{O}_{5}&\mathcal{O}_{6}\\\
\phi_{1}^{\frac{1}{2}}&\phi_{2}&\phi_{2}&\phi_{1}^{-\frac{1}{2}}&\mathcal{O}_{5}^{\frac{1}{2},-\frac{1}{2}}&\mathcal{O}_{6}\end{array}\right)\normalsize$
}&\frac{i\left(1-\Delta_{6}\right)\left(\Delta_{12}^{5}+\ell_{5}-\frac{3}{2}\right)}{2\left(\Delta_{6}-1\right)\ell_{6}}\\\
\scalebox{0.8}{
$\mathcal{K}^{\raisebox{-1.68776pt}{\scalebox{1.8}{$\cdot$}}3}_{21}\footnotesize\left(\begin{array}[]{cccccc}\psi_{1}&\phi_{2}&\phi_{2}&\psi_{1}&\mathcal{O}_{5}&\mathcal{O}_{6}\\\
\phi_{1}^{\frac{1}{2}}&\phi_{2}&\phi_{2}&\phi_{1}^{-\frac{1}{2}}&\mathcal{O}_{5}^{\frac{1}{2},\frac{1}{2}}&\mathcal{O}_{6}\end{array}\right)\normalsize$
}&\frac{i\left(\ell_{5}+\frac{1}{2}\right)\left(-\Delta_{12}^{5}+\ell_{5}+\frac{5}{2}\right)}{2\left(\Delta_{6}-1\right)\left(\ell_{5}+1\right)}\\\
\scalebox{0.8}{
$\mathcal{K}^{\raisebox{-1.68776pt}{\scalebox{1.8}{$\cdot$}}4}_{21}\footnotesize\left(\begin{array}[]{cccccc}\psi_{1}&\phi_{2}&\phi_{2}&\psi_{1}&\mathcal{O}_{5}&\mathcal{O}_{6}\\\
\phi_{1}^{\frac{1}{2}}&\phi_{2}&\phi_{2}&\phi_{1}^{-\frac{1}{2}}&\mathcal{O}_{5}^{\frac{1}{2},\frac{1}{2}}&\mathcal{O}_{6}\end{array}\right)\normalsize$
}&\frac{i\left(1-\Delta_{6}\right)\left(\ell_{5}+\frac{1}{2}\right)\left(-\Delta_{12}^{5}+\ell_{5}+\frac{5}{2}\right)}{2\left(\Delta_{6}-1\right)\left(\ell_{5}+1\right)\ell_{6}}\\\
\scalebox{0.8}{
$\mathcal{K}^{\raisebox{-1.68776pt}{\scalebox{1.8}{$\cdot$}}1}_{22}\footnotesize\left(\begin{array}[]{cccccc}\psi_{1}&\phi_{2}&\phi_{2}&\psi_{1}&\mathcal{O}_{5}&\mathcal{O}_{6}\\\
\phi_{1}^{\frac{1}{2}}&\phi_{2}&\phi_{2}&\phi_{1}^{\frac{1}{2}}&\mathcal{O}_{5}^{-\frac{1}{2},-\frac{1}{2}}&\mathcal{O}_{6}\end{array}\right)\normalsize$
}&-\frac{i\left(\Delta_{5}-\frac{3}{2}\right)\left(\Delta_{25}^{1}+\ell_{5}-\frac{3}{2}\right){}^{2}}{4\left(\Delta_{5}-2\right)\left(\Delta_{5}+\ell_{5}-2\right)\left(\Delta_{5}+\ell_{5}-1\right)}\end{array}$
$\begin{array}[]{cc}\scalebox{0.8}{
$\mathcal{K}^{\raisebox{-1.68776pt}{\scalebox{1.8}{$\cdot$}}2}_{22}\footnotesize\left(\begin{array}[]{cccccc}\psi_{1}&\phi_{2}&\phi_{2}&\psi_{1}&\mathcal{O}_{5}&\mathcal{O}_{6}\\\
\phi_{1}^{\frac{1}{2}}&\phi_{2}&\phi_{2}&\phi_{1}^{\frac{1}{2}}&\mathcal{O}_{5}^{-\frac{1}{2},-\frac{1}{2}}&\mathcal{O}_{6}\end{array}\right)\normalsize$
}&\frac{i\left(\Delta_{5}-\frac{3}{2}\right)\left(\Delta_{1}+\frac{1}{2}\left(-\Delta_{6}+\ell_{6}-1\right)\right)\left(\Delta_{1}+\frac{1}{2}\left(\Delta_{6}-\ell_{6}-4\right)\right)\left(\Delta_{25}^{1}+\ell_{5}-\frac{3}{2}\right){}^{2}}{4\left(\Delta_{5}-2\right)\left(\Delta_{6}-1\right)\ell_{6}\left(\Delta_{5}+\ell_{5}-2\right)\left(\Delta_{5}+\ell_{5}-1\right)}\\\
\scalebox{0.8}{
$\mathcal{K}^{\raisebox{-1.68776pt}{\scalebox{1.8}{$\cdot$}}1}_{11}\footnotesize\left(\begin{array}[]{cccccc}\psi_{1}&\phi_{2}&\phi_{2}&\psi_{1}&\mathcal{O}_{5}&\mathcal{O}_{6}\\\
\phi_{1}^{\frac{1}{2}}&\phi_{2}&\phi_{2}&\phi_{1}^{\frac{1}{2}}&\mathcal{O}_{5}^{-\frac{1}{2},\frac{1}{2}}&\mathcal{O}_{6}\end{array}\right)\normalsize$
}&\frac{i\left(\Delta_{5}-\frac{3}{2}\right)\left(\ell_{5}+\frac{1}{2}\right)\left(-\Delta_{25}^{1}+\ell_{5}+\frac{5}{2}\right){}^{2}}{4\left(\Delta_{5}-2\right)\left(\ell_{5}+1\right)\left(-\Delta_{5}+\ell_{5}+2\right)\left(-\Delta_{5}+\ell_{5}+3\right)}\\\
\scalebox{0.8}{
$\mathcal{K}^{\raisebox{-1.68776pt}{\scalebox{1.8}{$\cdot$}}2}_{11}\footnotesize\left(\begin{array}[]{cccccc}\psi_{1}&\phi_{2}&\phi_{2}&\psi_{1}&\mathcal{O}_{5}&\mathcal{O}_{6}\\\
\phi_{1}^{\frac{1}{2}}&\phi_{2}&\phi_{2}&\phi_{1}^{\frac{1}{2}}&\mathcal{O}_{5}^{-\frac{1}{2},\frac{1}{2}}&\mathcal{O}_{6}\end{array}\right)\normalsize$
}&-\frac{i\left(\Delta_{5}-\frac{3}{2}\right)\left(\ell_{5}+\frac{1}{2}\right)\left(\Delta_{1}+\frac{1}{2}\left(-\Delta_{6}+\ell_{6}-1\right)\right)\left(\Delta_{1}+\frac{1}{2}\left(\Delta_{6}-\ell_{6}-4\right)\right)\left(-\Delta_{25}^{1}+\ell_{5}+\frac{5}{2}\right){}^{2}}{4\left(\Delta_{5}-2\right)\left(\Delta_{6}-1\right)\left(\ell_{5}+1\right)\ell_{6}\left(-\Delta_{5}+\ell_{5}+2\right)\left(-\Delta_{5}+\ell_{5}+3\right)}\\\
\scalebox{0.8}{
$\mathcal{K}^{\raisebox{-1.68776pt}{\scalebox{1.8}{$\cdot$}}1}_{11}\footnotesize\left(\begin{array}[]{cccccc}\psi_{1}&\phi_{2}&\phi_{2}&\psi_{1}&\mathcal{O}_{5}&\mathcal{O}_{6}\\\
\phi_{1}^{\frac{1}{2}}&\phi_{2}&\phi_{2}&\phi_{1}^{\frac{1}{2}}&\mathcal{O}_{5}^{\frac{1}{2},-\frac{1}{2}}&\mathcal{O}_{6}\end{array}\right)\normalsize$
}&i\\\ \scalebox{0.8}{
$\mathcal{K}^{\raisebox{-1.68776pt}{\scalebox{1.8}{$\cdot$}}2}_{11}\footnotesize\left(\begin{array}[]{cccccc}\psi_{1}&\phi_{2}&\phi_{2}&\psi_{1}&\mathcal{O}_{5}&\mathcal{O}_{6}\\\
\phi_{1}^{\frac{1}{2}}&\phi_{2}&\phi_{2}&\phi_{1}^{\frac{1}{2}}&\mathcal{O}_{5}^{\frac{1}{2},-\frac{1}{2}}&\mathcal{O}_{6}\end{array}\right)\normalsize$
}&-\frac{i\left(\Delta_{1}+\frac{1}{2}\left(-\Delta_{6}+\ell_{6}-1\right)\right)\left(\Delta_{1}+\frac{1}{2}\left(\Delta_{6}-\ell_{6}-4\right)\right)}{\left(\Delta_{6}-1\right)\ell_{6}}\\\
\scalebox{0.8}{
$\mathcal{K}^{\raisebox{-1.68776pt}{\scalebox{1.8}{$\cdot$}}1}_{22}\footnotesize\left(\begin{array}[]{cccccc}\psi_{1}&\phi_{2}&\phi_{2}&\psi_{1}&\mathcal{O}_{5}&\mathcal{O}_{6}\\\
\phi_{1}^{\frac{1}{2}}&\phi_{2}&\phi_{2}&\phi_{1}^{\frac{1}{2}}&\mathcal{O}_{5}^{\frac{1}{2},\frac{1}{2}}&\mathcal{O}_{6}\end{array}\right)\normalsize$
}&-\frac{i\left(\ell_{5}+\frac{1}{2}\right)}{\ell_{5}+1}\\\ \scalebox{0.8}{
$\mathcal{K}^{\raisebox{-1.68776pt}{\scalebox{1.8}{$\cdot$}}2}_{22}\footnotesize\left(\begin{array}[]{cccccc}\psi_{1}&\phi_{2}&\phi_{2}&\psi_{1}&\mathcal{O}_{5}&\mathcal{O}_{6}\\\
\phi_{1}^{\frac{1}{2}}&\phi_{2}&\phi_{2}&\phi_{1}^{\frac{1}{2}}&\mathcal{O}_{5}^{\frac{1}{2},\frac{1}{2}}&\mathcal{O}_{6}\end{array}\right)\normalsize$
}&\frac{i\left(\ell_{5}+\frac{1}{2}\right)\left(\Delta_{1}+\frac{1}{2}\left(-\Delta_{6}+\ell_{6}-1\right)\right)\left(\Delta_{1}+\frac{1}{2}\left(\Delta_{6}-\ell_{6}-4\right)\right)}{\left(\Delta_{6}-1\right)\left(\ell_{5}+1\right)\ell_{6}}\end{array}$
(A.3.2)

###  A.4 Symmetries of $6j$ Symbols

Figure A.3:  Diagrammatic form of the $6j$ symbol
$\left(\Psi^{\widetilde{(s)}cd}_{\widetilde{\mathcal{O}}_{5}},\Psi^{(t)ab}_{\mathcal{O}_{6}}\right)$
as a tetrahedron.

By representing the $6j$ symbol as a tetrahedron as in figure A.3, we can
reveal its symmetries, as was done in [133]. Explicitly, we can consider the
three transformations as the generators of the symmetry group:

  1.   2. S1

Rotation around the axis that passes through the vertex $a$ and the center of
the triangle $\Delta bcd$, generated by the permutation
$(1\widetilde{5}\widetilde{4})(\widetilde{1}54)(236)(\widetilde{2}\widetilde{3}\widetilde{6})(bcd)$:

$\displaystyle\left\\{\begin{aligned}
\mathcal{O}_{1}\mathcal{O}_{2}\mathcal{O}_{6}\\\
\mathcal{O}_{3}\mathcal{O}_{4}\mathcal{O}_{5}\end{aligned}\right\\}^{cdab}=(-1)^{2\ell_{\mathcal{O}_{6}}}\left\\{\begin{aligned}
\widetilde{\mathcal{O}}_{5}\mathcal{O}_{3}\mathcal{O}_{2}\\\
\mathcal{O}_{6}\widetilde{\mathcal{O}}_{1}\mathcal{O}_{4}\end{aligned}\right\\}^{dbac}\;.$
(A.4.1a)

  3. S2

Rotation around the axis that passes through the vertex $c$ and the center of
the triangle $\Delta abd$, generated by the permutation
$(125)(\widetilde{1}\widetilde{2}\widetilde{5})(\widetilde{3}46)(3\widetilde{4}\widetilde{6})(bad)$:

$\displaystyle\left\\{\begin{aligned}
\mathcal{O}_{1}\mathcal{O}_{2}\mathcal{O}_{6}\\\
\mathcal{O}_{3}\mathcal{O}_{4}\mathcal{O}_{5}\end{aligned}\right\\}^{cdab}=(-1)^{2\ell_{\mathcal{O}_{1}}}\left\\{\begin{aligned}
\mathcal{O}_{2}\mathcal{O}_{5}\widetilde{\mathcal{O}}_{3}\\\
\widetilde{\mathcal{O}}_{4}\mathcal{O}_{6}\mathcal{O}_{1}\end{aligned}\right\\}^{cbda}\;.$
(A.4.1b)

  4. S3

Reflection with respect to the plane that passes through the points $c$, $d$,
and the midpoint of the line segment $\overline{ab}$, generated by the
permutation
$(12)(\widetilde{1}\widetilde{2})(34)(\widetilde{3}\widetilde{4})(6\widetilde{6})(ba)$:

$\displaystyle\left\\{\begin{aligned}
\mathcal{O}_{1}\mathcal{O}_{2}\mathcal{O}_{6}\\\
\mathcal{O}_{3}\mathcal{O}_{4}\mathcal{O}_{5}\end{aligned}\right\\}^{cdab}=(-1)^{2\ell_{\mathcal{O}_{6}}}\left\\{\begin{aligned}
\mathcal{O}_{2}\mathcal{O}_{1}\widetilde{\mathcal{O}}_{6}\\\
\mathcal{O}_{4}\mathcal{O}_{3}\mathcal{O}_{5}\end{aligned}\right\\}^{cdba}\;.$
(A.4.1c)

The overall phases in the front follow from the fermionic nature of the
correlators and can be checked explicitly.

The validity of eqn. (A.4.1) depends on the choice of three point basis. For
example, the first two equalities require us to work in a basis which respects
the cyclic permutations; i.e., we should have
$\expectationvalue{\mathcal{O}_{1}\mathcal{O}_{2}\mathcal{O}_{3}}^{a}=\expectationvalue{\mathcal{O}_{2}\mathcal{O}_{3}\mathcal{O}_{1}}^{a}=\expectationvalue{\mathcal{O}_{2}\mathcal{O}_{3}\mathcal{O}_{1}}^{a}$.
Generically, we can always find a basis which respects this property.

The equality eqn. (A.4.1c) on the other hand requires the basis to respect
inversions, i.e. we should have
$\expectationvalue{\mathcal{O}_{1}\mathcal{O}_{2}\mathcal{O}_{3}}^{a}=\expectationvalue{\mathcal{O}_{2}\mathcal{O}_{1}\mathcal{O}_{3}}^{a}$.
We can always choose a basis to respect this _unless_ we have
$\ell_{\mathcal{O}_{1}}=\ell_{\mathcal{O}_{2}}$. In that case, we can no
longer choose two independent bases
$\expectationvalue{\mathcal{O}_{1}\mathcal{O}_{2}\mathcal{O}_{3}}^{a}$ and
$\expectationvalue{\mathcal{O}_{2}\mathcal{O}_{1}\mathcal{O}_{3}}^{a}$ to
satisfy the required equality; we need the same basis to satisfy this
condition. However, if we work in a parity definite basis, all _nonzero_ $6j$
symbols will have an even number of parity odd three-point structures,
therefore the equality holds. Assuming we are in such a basis, we can use
following relations to derive all permutations:

$\displaystyle\textbf{S1}^{3}=\textbf{S2}^{3}=\textbf{S3}^{2}=\textbf{E}\text{
with
}(\textbf{S2}*\textbf{S1})^{2}=(\textbf{S3}*\textbf{S1})^{4}=(\textbf{S3}*\textbf{S2})^{2}=\textbf{E}$
(A.4.2)

where E is the identity transformation.

In summary, we can derive eqn. (A.4.1) and similar identities by considering
the inversions and rotations of the tetrahedron and are valid in a parity
definite basis with a cyclic property. These conditions are trivially
satisfied for external scalars as there is only one three-point structure.

An interesting set of transformations is the one that does not move the edges
$5,6$. There are only three such permutations:

$\displaystyle(12)(\widetilde{1}\widetilde{2})(34)(\widetilde{3}\widetilde{4})(6\widetilde{6})(ab),$
(A.4.3)
$\displaystyle(5\widetilde{5})(23)(\widetilde{2}\widetilde{3})(14)(\widetilde{1}\widetilde{4})(cd),$
$\displaystyle(13)(\widetilde{1}\widetilde{3})(24)(\widetilde{2}\widetilde{4})(5\widetilde{5})(6\widetilde{6})(ab)(cd),$

which yields

$\left\\{\begin{aligned} \mathcal{O}_{1}\mathcal{O}_{2}\mathcal{O}_{6}\\\
\mathcal{O}_{3}\mathcal{O}_{4}\mathcal{O}_{5}\end{aligned}\right\\}^{cdab}=(-1)^{2\ell_{\mathcal{O}_{6}}}\left\\{\begin{aligned}
\mathcal{O}_{2}\mathcal{O}_{1}\widetilde{\mathcal{O}}_{6}\\\
\mathcal{O}_{4}\mathcal{O}_{3}\mathcal{O}_{5}\end{aligned}\right\\}^{cdba}=(-1)^{2\ell_{\mathcal{O}_{5}}}\left\\{\begin{aligned}
\mathcal{O}_{4}\mathcal{O}_{3}\mathcal{O}_{6}\\\
\mathcal{O}_{2}\mathcal{O}_{1}\widetilde{\mathcal{O}}_{5}\end{aligned}\right\\}^{dcab}\\\
=(-1)^{2\ell_{\mathcal{O}_{5}}+2\ell_{\mathcal{O}_{6}}}\left\\{\begin{aligned}
\mathcal{O}_{3}\mathcal{O}_{4}\widetilde{\mathcal{O}}_{6}\\\
\mathcal{O}_{1}\mathcal{O}_{2}\widetilde{\mathcal{O}}_{5}\end{aligned}\right\\}^{dcba}\;.$
(A.4.4)

But we also know how to relate $\left\\{\begin{aligned}
\mathcal{O}_{1}\mathcal{O}_{2}\mathcal{O}_{6}\\\
\mathcal{O}_{3}\mathcal{O}_{4}\widetilde{\mathcal{O}}_{5}\end{aligned}\right\\}$
to $\left\\{\begin{aligned} \mathcal{O}_{1}\mathcal{O}_{2}\mathcal{O}_{6}\\\
\mathcal{O}_{3}\mathcal{O}_{4}\mathcal{O}_{5}\end{aligned}\right\\}$, and
likewise for $\mathcal{O}_{6}$, due to shadow symmetry of the partial waves.

We may also be interested in interchanging $\mathcal{O}_{5,6}$ in the $6j$
symbol, and this can be achieved with the transformation
$\mathfrak{T}=(1\widetilde{1})(2\widetilde{4})(3\widetilde{3})(56)(ad)(bc)$:

$\displaystyle\left\\{\begin{aligned}
\mathcal{O}_{1}\mathcal{O}_{2}\mathcal{O}_{6}\\\
\mathcal{O}_{3}\mathcal{O}_{4}\mathcal{O}_{5}\end{aligned}\right\\}^{cdab}=(-1)^{2\left(\ell_{\mathcal{O}_{5}}+\ell_{\mathcal{O}_{6}}\right)}\left\\{\begin{aligned}
\widetilde{\mathcal{O}}_{1}\widetilde{\mathcal{O}}_{4}\mathcal{O}_{5}\\\
\widetilde{\mathcal{O}}_{3}\widetilde{\mathcal{O}}_{2}\mathcal{O}_{6}\end{aligned}\right\\}^{badc}.$
(A.4.5)

Likewise, with the transformation
$\mathfrak{T}=(1\widetilde{3})(2\widetilde{2})(4\widetilde{4})(5\widetilde{6})(ac)(bd)$,
we get

$\displaystyle\left\\{\begin{aligned}
\mathcal{O}_{1}\mathcal{O}_{2}\mathcal{O}_{6}\\\
\mathcal{O}_{3}\mathcal{O}_{4}\mathcal{O}_{5}\end{aligned}\right\\}^{cdab}=\left\\{\begin{aligned}
\widetilde{\mathcal{O}}_{3}\widetilde{\mathcal{O}}_{2}\widetilde{\mathcal{O}}_{5}\\\
\widetilde{\mathcal{O}}_{1}\widetilde{\mathcal{O}}_{4}\widetilde{\mathcal{O}}_{6}\end{aligned}\right\\}^{abcd}.$
(A.4.6)

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