$\displaystyle\mathcal{M}^{\overline{\rm III}}$
$\displaystyle=\frac{4\pi^{2}m_{1}^{4}m_{2}^{4}(2\sigma^{2}-1)^{3}\log^{2}x}{3\left(\sigma^{2}-1\right)q^{2}}$
(181)
$\displaystyle\quad+\frac{4\pi^{2}m_{1}^{2}m_{2}^{2}}{15\epsilon}\Bigg{[}\left(m_{1}^{2}+m_{2}^{2}\right)\left(40\sigma^{4}-73\sigma^{2}+13\right)-20m_{1}m_{2}\sigma$
$\displaystyle\qquad\qquad\qquad\quad+\Big{(}\big{(}m_{1}^{2}+m_{2}^{2}\big{)}\sigma-
m_{1}m_{2}(6\sigma^{4}-5\sigma^{2}+1)\Big{)}\frac{10\log
x}{\sqrt{\sigma^{2}-1}}\Bigg{]}\,.$

####  6.1.2 $\rm IX$ and $\overline{\rm IX}$ sectors

The integral family for the IX sector is defined in (123) in terms of the full
matter propagators. The corresponding soft integral family is derived by
replacing the full matter propagators $\rho^{\rm IX}_{1}$ and $\rho^{\rm
IX}_{2}$ with the linear propagators $\tilde{\rho}^{\rm IX}_{1}$ and
$\tilde{\rho}^{\rm IX}_{2}$ respectively

$\displaystyle
j^{\text{IX}}_{i_{1},i_{2},\cdots,i_{9}}=\int_{\ell_{1},\ell_{2}}\frac{\big{(}\rho^{\rm
IX,8}_{g}\big{)}^{-i_{8}}\big{(}\rho^{\rm
IX,9}_{g}\big{)}^{-i_{9}}}{\big{(}\rho^{\rm
IX,1}_{g}\big{)}^{i_{1}}\big{(}\rho^{\rm
IX,2}_{g}\big{)}^{i_{2}}\big{(}\rho^{\rm
IX,3}_{g}\big{)}^{i_{3}}\big{(}\tilde{\rho}^{\rm
IX,4}_{1}\big{)}^{i_{4}}\big{(}\tilde{\rho}^{\rm
IX,5}_{1}\big{)}^{i_{5}}\big{(}\tilde{\rho}^{\rm
IX,6}_{2}\big{)}^{i_{6}}\big{(}\tilde{\rho}^{\rm IX,7}_{2}\big{)}^{i_{7}}}\,,$
(182)

where the set of propagators is given by

$\displaystyle\rho^{\rm IX,1}_{g}$ $\displaystyle=\ell_{1}^{2}\,,$
$\displaystyle\quad\rho^{\rm IX,2}_{g}$ $\displaystyle=\ell_{2}^{2}\,,$
$\displaystyle\quad\rho^{\rm IX,3}_{g}$
$\displaystyle=(\ell_{1}+\ell_{2}+q)^{2},$ (183)
$\displaystyle\tilde{\rho}^{\rm IX,4}_{1}$ $\displaystyle=2\ell_{1}\cdot
u_{1}\,,$ $\displaystyle\quad\tilde{\rho}^{\rm IX,5}_{1}$
$\displaystyle=-2\ell_{2}\cdot u_{1}\,,$ $\displaystyle\quad\tilde{\rho}^{\rm
IX,6}_{2}$ $\displaystyle=2\ell_{2}\cdot u_{2}\,,$
$\displaystyle\tilde{\rho}^{\rm IX,7}_{2}$
$\displaystyle=2(\ell_{1}+\ell_{2})\cdot u_{2}\,,$
$\displaystyle\quad\rho^{\rm IX,8}_{g}$ $\displaystyle=(\ell_{1}+q)^{2}\,,$
$\displaystyle\quad\rho^{\rm IX,9}_{g}$ $\displaystyle=(\ell_{2}+q)^{2}\,.$

There are 15 master integrals for the $j^{\rm IX}_{i_{1}\cdots i_{9}}$:

$\displaystyle f^{\text{IX}}_{1}$
$\displaystyle=-q^{2}\epsilon^{2}j^{\text{IX}}_{2,2,1,0,0,0,0,0,0}\,,$ (184)
$\displaystyle f^{\text{IX}}_{2}$
$\displaystyle=-\sqrt{y^{2}-1}\epsilon^{4}j^{\text{IX}}_{1,1,1,1,0,0,1,0,0}\,,$
$\displaystyle f^{\text{IX}}_{3}$
$\displaystyle=q^{2}\sqrt{y^{2}-1}\epsilon^{3}j^{\text{IX}}_{1,2,1,1,0,0,1,0,0}\,,$
$\displaystyle f^{\text{IX}}_{4}$
$\displaystyle=q^{2}y\epsilon^{3}j^{\text{IX}}_{1,2,1,1,0,0,1,0,0}-q^{2}\epsilon^{2}j^{\text{IX}}_{1,1,1,2,0,0,2,0,0}\,,$
$\displaystyle f^{\text{IX}}_{5}$
$\displaystyle=-\sqrt{y^{2}-1}\epsilon^{4}j^{\text{IX}}_{1,1,1,1,0,1,0,0,0}\,,$
$\displaystyle f^{\text{IX}}_{6}$
$\displaystyle=q^{2}\sqrt{y^{2}-1}\epsilon^{3}j^{\text{IX}}_{1,1,2,1,0,1,0,0,0}\,,$
$\displaystyle f^{\text{IX}}_{7}$
$\displaystyle=-q^{2}y\epsilon^{3}j^{\text{IX}}_{1,1,2,1,0,1,0,0,0}-q^{2}\epsilon^{2}j^{\text{IX}}_{1,1,1,2,0,2,0,0,0}\,,$
$\displaystyle f^{\text{IX}}_{8}$
$\displaystyle=-(1-6\epsilon)\epsilon^{3}j^{\text{IX}}_{1,1,1,1,1,0,0,0,0}\,,$
$\displaystyle f^{\text{IX}}_{9}$
$\displaystyle=q^{2}\sqrt{y^{2}-1}\epsilon^{3}j^{\text{IX}}_{1,1,2,0,1,1,0,0,0}\,,$
$\displaystyle f^{\text{IX}}_{10}$
$\displaystyle=-q^{2}\left(y^{2}-1\right)\epsilon^{4}j^{\text{IX}}_{1,1,1,1,1,1,1,0,0}\,,$
$\displaystyle f^{\text{IX}}_{11}$
$\displaystyle=-|q|\epsilon^{3}j^{\text{IX}}_{1,1,2,0,1,0,0,0,0}\,,$
$\displaystyle f^{\text{IX}}_{12}$
$\displaystyle=|q|\epsilon^{3}j^{\text{IX}}_{1,1,1,1,0,2,0,0,0}\,,$
$\displaystyle f^{\text{IX}}_{13}$
$\displaystyle=|q|\epsilon^{3}j^{\text{IX}}_{1,1,1,2,0,0,1,0,0}\,,$
$\displaystyle f^{\text{IX}}_{14}$
$\displaystyle=|q|\sqrt{y^{2}-1}\epsilon^{4}j^{\text{IX}}_{1,1,1,1,1,0,1,0,0}\,,$
$\displaystyle f^{\text{IX}}_{15}$
$\displaystyle=|q|\sqrt{y^{2}-1}\epsilon^{4}j^{\text{IX}}_{1,1,1,1,1,1,0,0,0}\,.$

Differentiating the master integrals with respect to $y$ and changing the
variable to $x$, we have a set of differential equations

$\frac{\mathrm{d}\vec{f}_{\rm IX}}{\mathrm{d}x}=\epsilon\bigg{(}\frac{A_{\rm
IX,0}}{x}+\frac{A_{\rm IX,1}}{x-1}+\frac{A_{\rm
IX,-1}}{x+1}\bigg{)}\vec{f}_{\rm IX}\,,$ (185)

where the $A_{\rm IX}$ are $15\times 15$ constant matrices decomposed as

$A_{\mathrm{IX},i}=\left(\begin{array}[]{cc}A_{\mathrm{IX},i}^{(\mathrm{e})}&0\\\
0&A_{\mathrm{IX},i}^{(\mathrm{o})}\end{array}\right)\,,$ (186)

and

$\displaystyle
A_{\mathrm{IX},0}^{(\mathrm{e})}=\left(\begin{array}[]{cccccccccc}0&0&0&0&0&0&0&0&0&0\\\
\frac{1}{2}&-6&0&-1&0&0&0&0&0&0\\\ \frac{3}{2}&0&2&-2&0&0&0&0&0&0\\\
0&12&2&0&0&0&0&0&0&0\\\ -\frac{1}{2}&0&0&0&-6&0&1&0&0&0\\\
-\frac{3}{2}&0&0&0&0&2&2&0&0&0\\\ 0&0&0&0&-12&-2&0&0&0&0\\\
0&0&0&0&0&0&0&0&0&0\\\ -\frac{3}{4}&0&0&0&0&0&0&0&0&0\\\
0&0&-\frac{1}{2}&0&0&-1&0&0&1&0\end{array}\right)\,,\qquad A_{\mathrm{IX},\pm
1}^{(\mathrm{e})}=\left(\begin{array}[]{cccccccccc}0&0&0&0&0&0&0&0&0&0\\\
0&6&0&0&0&0&0&0&0&0\\\ 0&0&-2&0&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0&0&0&0\\\
0&0&0&0&6&0&0&0&0&0\\\ 0&0&0&0&0&-2&0&0&0&0\\\ 0&0&0&0&0&0&0&0&0&0\\\
0&0&0&0&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0&0&0&0\\\
0&0&0&0&0&0&0&0&0&0\end{array}\right)$ (187)
$A_{\mathrm{IX},0}^{(\mathrm{o})}=\left(\begin{array}[]{ccccc}0&0&0&0&0\\\
0&-2&0&0&0\\\ 0&0&-2&0&0\\\ 0&-1&1&0&0\\\ 0&1&0&0&0\end{array}\right)\,,\quad
A_{\mathrm{IX},+1}^{(\mathrm{o})}=\left(\begin{array}[]{ccccc}0&0&0&0&0\\\
3&6&0&0&0\\\ -3&0&-2&0&0\\\ 0&0&0&0&0\\\ 0&0&0&0&0\end{array}\right)\,,\quad
A_{\mathrm{IX},-1}^{(\mathrm{o})}=\left(\begin{array}[]{ccccc}0&0&0&0&0\\\
-3&-2&0&0&0\\\ 3&0&6&0&0\\\ 0&0&0&0&0\\\ 0&0&0&0&0\\\
0&0&0&0&0\end{array}\right)\,.$ (188)

The solution of the differential equation is presented in (293).

After the tensor reduction for the integrands of the IX sector,
$\mathcal{M}^{\rm IX}$ is represented by a linear combination of the master
integrals

$\mathcal{M}^{\rm IX}=\sum_{i=1}^{15}c^{\rm IX}_{i}f^{\rm
IX}_{i}\Big{|}_{\text{non-analytic}}\,,$ (189)

and the explicit form of the coefficients is presented in Appendix D.1.3.
Substituting all the results into (189), we have

$\displaystyle\mathcal{M}^{\rm IX}$
$\displaystyle=\frac{10\pi^{2}m_{1}^{4}m_{2}^{4}\left(2\sigma^{2}-1\right)^{3}\log^{2}x}{3q^{2}\left(\sigma^{2}-1\right)}$
(190)
$\displaystyle\quad+\frac{2i\pi^{3}m_{1}^{3}m_{2}^{3}\left(m_{1}+m_{2}\right)(2\sigma^{2}-1)^{2}}{|q|\sqrt{\sigma^{2}-1}}\Bigg{[}-\frac{2\sigma}{\epsilon}+4\sigma\log
x+\frac{2\sigma^{2}-1}{\sigma+1}\Bigg{]}$
$\displaystyle\quad+\frac{2\pi^{2}m_{1}^{2}m_{2}^{2}}{3\epsilon}\Bigg{[}6m_{1}m_{2}\sigma^{2}\left(\frac{3\sigma-4\sigma^{3}}{\sigma^{2}-1}+\frac{5\left(2\sigma^{4}-3\sigma^{2}+1\right)\log
x}{\left(\sigma^{2}-1\right)^{\frac{3}{2}}}\right)$
$\displaystyle\qquad\qquad\qquad\quad-(m_{1}^{2}+m_{1}^{2})\left(\frac{62\sigma^{6}-91\sigma^{4}+34\sigma^{2}-2}{\sigma^{2}-1}-\frac{\sigma\log
x}{\sqrt{\sigma^{2}-1}}\right)\Bigg{]}\,.$

Let us consider the $\overline{\rm IX}$ sector. As the $\overline{\rm III}$
sector, the master integrals and the differential equations can be derived by
the crossing relation (107). The classical amplitude from the $\overline{\rm
IX}$ sector $\mathcal{M}^{\overline{\rm IX}}$ is expanded as

$\mathcal{M}^{\overline{\rm IX}}=\sum_{i=1}^{15}c^{\overline{\rm
IX}}_{i}f^{\overline{\rm IX}}_{i}\Big{|}_{\rm non-analytic}\,,$ (191)

and the explicit form of the coefficients and values of the master integrals
are denoted in Appendix D.1.4 and (294). Again, the coefficients in the IX and
$\overline{\rm IX}$ sectors are related by the crossing relation (107)

$c^{\rm IX}_{i}=c^{\overline{\rm IX}}_{i}\Big{|}_{\rm crossing}\,.$ (192)

As before, this relation provides a consistency check for the solutions of the
off-shell recursions, and we showed that our result satisfies the relation.

Substituting the results into (191), we have

$\displaystyle\mathcal{M}^{\overline{{\rm IX}}}$
$\displaystyle=-\frac{8\pi^{2}m_{1}^{4}m_{2}^{4}\left(2\sigma^{2}-1\right)^{3}\log^{2}x}{3\left(\sigma^{2}-1\right)q^{2}}$
(193)
$\displaystyle\quad+\frac{2\pi^{2}m_{1}^{2}m_{2}^{2}}{3\epsilon}\Bigg{[}\big{(}m_{1}^{2}+m_{1}^{2}\big{)}\left(28\sigma^{4}-16\sigma^{2}+1\right)-6m_{2}m_{1}\sigma$
$\displaystyle\qquad\qquad\qquad\quad+\left(\big{(}m_{1}^{2}+m_{2}^{2}\big{)}-6m_{2}m_{1}\left(8\sigma^{4}-6\sigma^{2}+1\right)\sigma\right)\frac{\log
x}{\sqrt{\sigma^{2}-1}}\Bigg{]}\,.$

####  6.1.3 Total amplitude from the $\Box\\!\Box$ class

Adding up the results for each sector, (178), (181), (190) and (193), we have
the total two-loop amplitude

$\displaystyle\mathcal{M}^{\Box\\!\Box}$
$\displaystyle=\frac{2\pi^{4}m_{1}^{4}m_{2}^{4}\left(2\sigma^{2}-1\right)^{3}}{3q^{2}\left(\sigma^{2}-1\right)}\Bigg{[}1-\frac{6}{\epsilon^{2}}\Bigg{]}$
(194)
$\displaystyle\quad+\frac{4i\pi^{3}m_{1}^{3}m_{2}^{3}\left(m_{1}+m_{2}\right)\left(2\sigma^{2}-1\right)}{\sqrt{\sigma^{2}-1}|q|}\Bigg{[}\frac{1}{\epsilon}-2\log
2-\frac{2(2\sigma^{2}-1)^{2}}{\sigma^{2}-1}\Bigg{]}$
$\displaystyle\quad+\frac{\pi^{2}m_{1}^{3}m_{2}^{3}}{\epsilon}\Bigg{[}\frac{2\sigma\left(8\sigma^{2}\left(\sigma^{4}-2\sigma^{2}+2\right)-7\right)}{\left(\sigma^{2}-1\right)^{2}}-16\left(2\sigma^{2}-1\right)\sqrt{\sigma^{2}-1}\log
x$
$\displaystyle\qquad\qquad\qquad\quad+\frac{(m_{1}^{2}+m_{2}^{2})\left(-88\sigma^{6}+244\sigma^{4}-184\sigma^{2}+33\right)}{5m_{1}m_{2}\left(\sigma^{2}-1\right)^{2}}\Bigg{]}\,.$

###  6.2 $\mathbf{H}$ class

The $\mathbf{H}$ class consists of the two sectors: H and $\overline{\rm H}$.
The total amplitude contribution from the $\mathbf{H}$ class is given by

$\mathcal{M}^{\mathbf{H}}=\mathcal{M}^{\rm H}+\mathcal{M}^{\overline{\rm
H}}\,.$ (195)

For the H-sector, the integral family is defined in (127), using the full
matter propagators $\rho^{\rm H}_{1,2}$. The corresponding soft integral
family that employs the linearized matter propagators $\tilde{\rho}^{\rm
H}_{1,2}$ is given by

$\displaystyle j^{\text{\rm
H}}_{i_{1},i_{2},\cdots,i_{9}}=\int_{\ell_{1},\ell_{2}}\frac{\big{(}\tilde{\rho}^{\rm
H,8}_{1}\big{)}^{-i_{8}}\big{(}\tilde{\rho}^{\rm
H,9}_{2}\big{)}^{-i_{9}}}{\big{(}\tilde{\rho}^{\rm
H,1}_{g}\big{)}^{i_{1}}\big{(}\tilde{\rho}^{\rm
H,2}_{g}\big{)}^{i_{2}}\big{(}\tilde{\rho}^{\rm
H,3}_{g}\big{)}^{i_{3}}\big{(}\tilde{\rho}^{\rm
H,4}_{g}\big{)}^{i_{4}}\big{(}\tilde{\rho}^{\rm
H,5}_{g}\big{)}^{i_{5}}\big{(}\tilde{\rho}^{\rm
H,6}_{1}\big{)}^{i_{6}}\big{(}\tilde{\rho}^{\rm H,7}_{2}\big{)}^{i_{7}}}\,,$
(196)

where the propagators are defined by

$\displaystyle\rho^{\rm H,1}_{g}$ $\displaystyle=(\ell_{1}+q)^{2}\,,$
$\displaystyle\qquad\rho^{\rm H,2}_{g}$ $\displaystyle=(\ell_{2}+q)^{2}\,,$
$\displaystyle\qquad\rho^{\rm H,3}_{g}$ $\displaystyle=\ell_{1}^{2}\,,$ (197)
$\displaystyle\rho^{\rm H,4}_{g}$ $\displaystyle=\ell_{2}^{2}\,,$
$\displaystyle\qquad\rho^{\rm H,5}_{g}$
$\displaystyle=(\ell_{1}+\ell_{2}+q)^{2}\,,$
$\displaystyle\qquad\tilde{\rho}^{\rm H,6}_{1}$ $\displaystyle=2\ell_{1}\cdot
u_{1}\,,$ $\displaystyle\tilde{\rho}^{\rm H,7}_{2}$
$\displaystyle=2\ell_{2}\cdot u_{2}\,,$ $\displaystyle\qquad\tilde{\rho}^{\rm
H,8}_{1}$ $\displaystyle=-2\ell_{2}\cdot u_{1}\,,$
$\displaystyle\qquad\tilde{\rho}^{\rm H,9}_{2}$ $\displaystyle=-2\ell_{1}\cdot
u_{2}\,.$

Here $\tilde{\rho}^{\rm H,8}_{1}$ and $\tilde{\rho}^{\rm H,9}_{2}$ are the
auxiliary propagators, which are not included in the propagator set of the
H-diagram.

Taking the IBP reduction, we identify 16 master integrals for $j^{\text{\rm
H}}_{i_{1},i_{2},\cdots,i_{9}}$. We may divide the master integrals into the
even and odd powers in $|q|$ terms. Although the H-sector amplitude at the
potential region requires the even terms only, we present all the master
integrals for completeness. Note that, as we will see later, the odd $|q|$
terms appear in the IY class, even though these are irrelevant to the total
amplitude. The master integrals corresponding to the even $|q|$ terms are as
follows:

$\displaystyle f^{\text{H}}_{1}$
$\displaystyle=-q^{2}\epsilon^{2}j^{\text{H}}_{2,2,0,0,1,0,0,0,0}\,,$ (198)
$\displaystyle f^{\text{H}}_{2}$
$\displaystyle=-(1-4\epsilon)\epsilon^{2}j^{\text{H}}_{0,1,0,1,1,2,0,0,0}\,,$
$\displaystyle f^{\text{H}}_{3}$
$\displaystyle=q^{4}\epsilon^{2}j^{\text{H}}_{2,1,1,2,0,0,0,0,0}\,,$
$\displaystyle f^{\text{H}}_{4}$
$\displaystyle=q^{2}\epsilon^{4}j^{\text{H}}_{1,1,1,1,0,1,1,0,0}\,,$
$\displaystyle f^{\text{H}}_{5}$
$\displaystyle=-\sqrt{y^{2}-1}\epsilon^{4}j^{\text{H}}_{1,1,0,0,1,1,1,0,0}\,,$
$\displaystyle f^{\text{H}}_{6}$
$\displaystyle=q^{2}\sqrt{y^{2}-1}\epsilon^{3}j^{\text{H}}_{1,1,0,0,2,1,1,0,0}\,,$
$\displaystyle f^{\text{H}}_{7}$
$\displaystyle=q^{2}y\epsilon^{3}j^{\text{H}}_{1,1,0,0,2,1,1,0,0}+q^{2}\epsilon^{2}j^{\text{H}}_{1,1,0,0,1,2,2,0,0}\,,$
$\displaystyle f^{\text{H}}_{8}$
$\displaystyle=-\frac{\epsilon^{2}(4\epsilon-1)}{\sqrt{y^{2}-1}}\left(yj^{\text{H}}_{1,0,1,0,1,0,2,0,0}+(2\epsilon-1)j^{\text{H}}_{1,0,1,0,1,1,1,0,0}\right)\,,$
$\displaystyle f^{\text{H}}_{9}$
$\displaystyle=-q^{4}\sqrt{y^{2}-1}\epsilon^{4}j^{\text{H}}_{1,1,1,1,1,1,1,0,0}\,,$
$\displaystyle f^{\text{H}}_{10}$
$\displaystyle=2q^{2}y\epsilon^{4}j^{\text{H}}_{1,1,1,1,0,1,1,0,0}+q^{2}\epsilon^{4}j^{\text{H}}_{1,1,1,1,1,1,1,-1,-1}$
$\displaystyle\quad-\frac{(3\epsilon-2)(3\epsilon-1)\epsilon
j^{\text{H}}_{0,0,1,1,1,0,0,0,0}}{q^{2}}+\frac{1}{2}(2\epsilon-1)\epsilon^{2}j^{\text{H}}_{1,1,1,1,0,0,0,0,0}\,,$

and the odd $|q|$ terms are

$\displaystyle f^{\rm H}_{11}$
$\displaystyle=\frac{\epsilon^{2}(3\epsilon-1)(2\epsilon-1)j^{\rm
H}_{1,1,0,0,1,1,0,0,0}}{|q|}\,,$ (199) $\displaystyle f^{\rm H}_{12}$
$\displaystyle=\epsilon^{3}|q|j^{\rm H}_{1,1,0,0,1,1,2,0,0}\,,$ $\displaystyle
f^{\rm H}_{13}$ $\displaystyle=\epsilon^{3}(2\epsilon-1)|q|j^{\rm
H}_{1,1,1,1,0,1,0,0,0}\,,$ $\displaystyle f^{\rm H}_{14}$
$\displaystyle=\frac{(1-2\epsilon)^{2}\epsilon^{2}j^{\rm
H}_{,0,1,0,1,1,1,0,0,0}}{|q|}\,,$ $\displaystyle f^{\rm H}_{15}$
$\displaystyle=\epsilon^{3}|q|j^{\rm H}_{1,0,1,0,1,1,2,0,0}\,,$ $\displaystyle
f^{\rm H}_{16}$ $\displaystyle=\frac{16\epsilon^{4}|q|j^{\rm
H}_{1,0,1,0,1,1,2,0,0}}{(2\epsilon+1)(y+1)}-\frac{4(y+1)\epsilon^{4}|q|j^{\rm
H}_{1,1,0,0,1,1,2,0,0}}{2\epsilon+1}$
$\displaystyle\quad+(y-1)\epsilon^{3}|q|^{5/2}j^{\rm
H}_{1,1,1,1,1,1,2,0,0}+\frac{16y\epsilon^{3}(1-2\epsilon)^{2}j^{\rm
H}_{0,1,0,1,1,1,0,0,0}}{(y+1)(2\epsilon+1)|q|}$
$\displaystyle\quad+\frac{12\epsilon^{3}(3\epsilon-1)(2\epsilon-1)j^{\rm
H}_{1,1,0,0,1,1,0,0,0}}{(2\epsilon+1)|q|}\,.$

Acting on the differential operator (147), we can obtain a set of differential
equations

$\frac{\mathrm{d}\vec{f}_{\rm H}}{\mathrm{d}x}=\epsilon\bigg{(}\frac{A_{\rm
H,0}}{x}+\frac{A_{\rm H,1}}{x-1}+\frac{A_{\rm H,-1}}{x+1}\bigg{)}\vec{f}_{\rm
H}\,,$ (200)

where the constant matrices $A_{\mathrm{H},i}$ are decomposed into the even
and the odd terms

$A_{\mathrm{H},i}=\left(\begin{array}[]{cc}A_{\mathrm{H},i}^{(\mathrm{e})}&0\\\
0&A_{\mathrm{H},i}^{(\mathrm{o})}\end{array}\right)\,,$ (201)

and the explicit expressions of the even terms are given by

$A_{\mathrm{H},0}^{(\mathrm{e})}=\left(\begin{array}[]{cccccccccc}0&0&0&0&0&0&0&0&0&0\\\
0&0&0&0&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0&0&0&0\\\
-\frac{1}{2}&0&0&0&-6&0&-1&0&0&0\\\ -\frac{3}{2}&0&0&0&0&2&-2&0&0&0\\\
0&0&0&0&12&2&0&0&0&0\\\ 0&2&0&0&0&0&0&2&0&0\\\ 2&-4&0&0&0&4&2&4&2&-2\\\
-1&0&-1&0&12&8&0&8&2&-2\end{array}\right)\,,\qquad A_{\mathrm{H},\pm
1}^{(\mathrm{e})}=\left(\begin{array}[]{cccccccccc}0&0&0&0&0&0&0&0&0&0\\\
0&0&0&0&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0&0&0&0\\\ 0&0&0&0&0&0&0&0&0&0\\\
0&0&0&0&6&0&0&0&0&0\\\ 0&0&0&0&0&-2&0&0&0&0\\\ 0&0&0&0&0&0&0&0&0&0\\\
0&0&0&0&0&0&0&-2&0&0\\\ 0&0&0&0&0&-4&0&-4&-2&0\\\ 1&0&1&\pm
4&0&0&0&0&0&2\end{array}\right)\,.$ (202)

We omit the odd terms since these do not contribute to the amplitude from the
$\mathbf{H}$ class.

As before, we can represent $\mathcal{M}^{\rm H}$ in terms of the master
integrals through the IBP reduction to the H sector

$\mathcal{M}^{\rm H}=\sum_{i=1}^{10}c^{\rm H}_{i}f^{\rm H}_{i}\,.$ (203)

The explicit form of the coefficients $c^{\rm H}_{i}$ and the solutions of the
differential equation are listed in Appendix D.2.1 and (295). If we substitute
the values of the master integrals, we have the amplitude

$\displaystyle\mathcal{M}^{\rm H}$
$\displaystyle=-\frac{8\pi^{2}m_{1}^{3}m_{2}^{3}}{3\epsilon}\bigg{(}8\sigma+\frac{12\sigma^{4}-8\sigma^{2}+9}{\sqrt{\sigma^{2}-1}}\log
x\bigg{)}\,.$ (204)

Similarly, we derive the master integral and its master integrals for the
$\overline{\rm H}$ sector using the crossing relation (196) and (198). The
amplitude from the $\overline{\rm H}$ sector $\mathcal{M}^{\overline{\rm H}}$
is expanded as

$\mathcal{M}^{\overline{\rm H}}=\sum_{i=1}^{10}c^{\overline{\rm
H}}_{i}f^{\overline{\rm H}}_{i}\,,$ (205)

and the explicit form of the coefficients and values of the master integrals
are denoted in Appendix D.2.2 and (296). Combining all the results, we have

$\displaystyle\mathcal{M}^{\overline{\rm H}}$
$\displaystyle=-\frac{4\pi^{2}m_{1}^{3}m_{2}^{3}}{3\epsilon}\bigg{(}8\sigma+\frac{12\sigma^{4}-8\sigma^{2}+9}{\sqrt{\sigma^{2}-1}}\log
x\bigg{)}\,.$ (206)

Finally, we derive the amplitudes from the total $\mathbf{H}$ class by adding
the results of the H and $\overline{\rm H}$ sectors in (204) and (206)
respectively

$\displaystyle\mathcal{M}^{\mathbf{H}}$
$\displaystyle=-\frac{4\pi^{2}m_{1}^{3}m_{2}^{3}}{\epsilon}\bigg{(}8\sigma+\frac{12\sigma^{4}-8\sigma^{2}+9}{\sqrt{\sigma^{2}-1}}\log
x\bigg{)}\,.$ (207)

###  6.3  $\mathbf{IY}$ and inverted $\mathbf{IY}$ classes

Finally, let us consider the IY and  IY classes. The $\mathbf{IY}$ class
consists of the 6 sectors, $\rm IY$, $\rm YI$, $\rm/\\!\\!\\!Y$,
$\overline{\rm IY}$, $\overline{\rm YI}$, and $\overline{\rm/\\!\\!\\!Y}$.
Within this class, one can find three pairs under the horizontal mirror duals,
which are connected through the dual mapping in (108),

${\rm IY}\iff{\rm YI}\,,\qquad\overline{\rm IY}\iff\overline{\rm
YI}\,,\qquad{\rm/\\!\\!\\!Y}\iff\overline{\rm/\\!\\!\\!Y}\,.$ (208)

Given these relations, it suffices to consider the IY, $\overline{\rm IY}$ and
$\rm/\\!\\!\\!Y$ sectors alone. Furthermore, the  IY class exhibits vertical
mirror duality to the IY class. This allows us to derive the corresponding
results through the simple mapping in (108), without additional computations.
As a result, our ensuing discussion will focus on IBP reduction and the
amplitudes for the IY, $\overline{\rm IY}$ and $\rm/\\!\\!\\!Y$ sectors. The
amplitudes from the $\mathbf{IY}$ and  IY classes are

$\displaystyle\mathcal{M}^{\textbf{IY}}$ $\displaystyle=2\mathcal{M}^{\rm
IY}+2\mathcal{M}^{\rm IY}+2\mathcal{M}^{\text{/\\!\\!\\!\rm{Y}}}\,,$ (209)
$\displaystyle\mathcal{M}^{\textbf{\scalebox{-1.0}[1.0]{\rotatebox[origin={c}]{180.0}{IY}}}}$
$\displaystyle=\mathcal{M}^{\textbf{IY}}\big{|}_{m_{1}\leftrightarrow m_{2}}$

One remarkable property of the master integrals in IY class is that these can
be represented solely by master integrals in the other classes. This means
that solving the differential equations arising in the $\mathbf{IY}$ and  IY
classes is unnecessary.

####  6.3.1 IY and $\overline{\rm IY}$ sectors

The soft integral family for the IY sector is derived by replacing the full
matter propagators $\rho^{\rm IY}_{1}$ and $\rho^{\rm IY}_{2}$ in (130) with
their linearized counterparts $\tilde{\rho}^{\rm IY}_{1}$ and
$\tilde{\rho}^{\rm IY}_{2}$ respectively

$\displaystyle j^{\text{\rm
IY}}_{i_{1},i_{2},\cdots,i_{9}}=\int_{\ell_{1},\ell_{2}}\frac{\big{(}\tilde{\rho}^{\rm
IY,8}_{2}\big{)}^{-i_{8}}\big{(}\rho^{\rm
IY,9}_{g}\big{)}^{-i_{9}}}{\big{(}\rho^{\rm
IY,1}_{g}\big{)}^{i_{1}}\big{(}\rho^{\rm
IY,2}_{g}\big{)}^{i_{2}}\big{(}\rho^{\rm
IY,3}_{g}\big{)}^{i_{3}}\big{(}\rho^{\rm
IY,4}_{g}\big{)}^{i_{4}}\big{(}\tilde{\rho}^{\rm
IY,5}_{1}\big{)}^{i_{5}}\big{(}\tilde{\rho}^{\rm
IY,6}_{1}\big{)}^{i_{6}}\big{(}\tilde{\rho}^{\rm IY,7}_{2}\big{)}^{i_{7}}}\,,$
(210)

where the set of the propagators are defined by

$\displaystyle\rho^{\rm IY,1}_{g}$ $\displaystyle=(\ell_{1}+q)^{2}\,,$
$\displaystyle\quad\rho^{\rm IY,2}_{g}$ $\displaystyle=\ell_{1}^{2}\,,$
$\displaystyle\quad\rho^{\rm IY,3}_{g}$ $\displaystyle=\ell_{2}^{2}\,,$ (211)
$\displaystyle\rho^{\rm IY,4}_{g}$
$\displaystyle=(\ell_{1}+\ell_{2}+q)^{2}\,,$
$\displaystyle\quad\tilde{\rho}^{\rm IY,5}_{1}$ $\displaystyle=2\ell_{2}\cdot
u_{1}\,,$ $\displaystyle\quad\tilde{\rho}^{\rm IY,6}_{1}$
$\displaystyle=-2\ell_{1}\cdot u_{1}\,,$ $\displaystyle\tilde{\rho}^{\rm
IY,7}_{2}$ $\displaystyle=2\ell_{1}\cdot u_{2}\,,$
$\displaystyle\quad\tilde{\rho}^{\rm IY,8}_{2}$ $\displaystyle=-2\ell_{2}\cdot
u_{2}\;,$ $\displaystyle\quad\rho^{\rm IY,9}_{g}$
$\displaystyle=(\ell_{2}+q)^{2}\,.$

Here $\tilde{\rho}^{\rm IY,8}_{2}$ and $\rho^{\rm IY,9}_{g}$ are the auxiliary
propagators.

Interestingly, the soft integral family for the IY sector can be represented
by the integral families for the III and H sectors,

$\displaystyle j^{\text{IY}}_{0,i_{2},i_{3},i_{4},i_{5},i_{6},i_{7},0,0}$
$\displaystyle=j^{\text{III}}_{i_{3},i_{2},i_{4},i_{5},i_{6},0,i_{7},0,0}\,,$
(212) $\displaystyle j^{\text{IY}}_{i_{1},i_{2},0,i_{4},i_{5},0,0,0,0}$
$\displaystyle=j^{\text{H}}_{0,i_{1},0,i_{2},i_{4},i_{5},0,0,0}\,,$
$\displaystyle j^{\text{IY}}_{i_{1},i_{2},0,i_{4},i_{5},0,i_{7},0,0}$
$\displaystyle=j^{\text{H}}_{0,i_{1},0,i_{2},i_{4},i_{5},i_{7},0,0}\,.$

This implies that we do not have to define the master integrals for the IY
sector; we exploit the master integrals defined in the III and H sectors.
There are a total of 13 master integrals in this sector:

$\displaystyle\text{III}:\quad f^{\rm IY}_{1}$ $\displaystyle=f^{\rm
III}_{1}\,,$ $\displaystyle\quad f^{\rm IY}_{2}$ $\displaystyle=f^{\rm
III}_{2}\,,$ $\displaystyle\quad f^{\rm IY}_{3}$ $\displaystyle=f^{\rm
III}_{3}\,,$ $\displaystyle\quad f^{\rm IY}_{4}$ $\displaystyle=f^{\rm
III}_{4}\,,$ (213) $\displaystyle f^{\rm IY}_{5}$ $\displaystyle=f^{\rm
III}_{5}\,,$ $\displaystyle\quad f^{\rm IY}_{6}$ $\displaystyle=f^{\rm
III}_{6}\,,$ $\displaystyle\quad f^{\rm IY}_{7}$ $\displaystyle=f^{\rm
III}_{8}\,,$ $\displaystyle\quad f^{\rm IY}_{8}$ $\displaystyle=f^{\rm
III}_{9}\,,$ $\displaystyle f^{\rm IY}_{9}$ $\displaystyle=f^{\rm
III}_{10}\,,$ $\displaystyle\text{H even}:\quad f^{\rm IY}_{10}$
$\displaystyle=f^{\rm H}_{2}\,,$ $\displaystyle\quad f^{\rm IY}_{11}$
$\displaystyle=f^{\rm H}_{8}\,,$ $\displaystyle\text{H odd}:\quad f^{\rm
IY}_{12}$ $\displaystyle=f^{\rm H}_{14}\,,$ $\displaystyle\quad f^{\rm
IY}_{13}$ $\displaystyle=f^{\rm H}_{15}\,.$

Consequently, we do not have to solve the differential equations to determine
these master integrals – we can use the previous results.

After the IBP reduction, the amplitude from the IY sector is represented by a
linear combination of the master integrals

$\mathcal{M}^{\rm IY}=\sum_{i=1}^{13}c^{\rm IY}_{i}f^{\rm IY}_{i}\Big{|}_{\rm
non-analytic}\,.$ (214)

Again, we keep the non-analytic terms only for the same reason in the other
classes. The explicit form of the coefficients $c^{\rm IY}_{i}$ are listed in
Appendix D.3.1. If we substitute the values of the master integrals, we have
the amplitude

$\displaystyle\mathcal{M}^{\rm IY}$
$\displaystyle=-\frac{i\pi^{3}m_{1}^{4}m_{2}^{3}\left(2\sigma^{2}-1\right)}{2\sqrt{\sigma^{2}-1}|q|}\Bigg{[}\sigma^{2}-1+2\left(15\sigma^{2}-7\right)\log
2+\frac{7-15\sigma^{2}}{\epsilon}\Bigg{]}$ (215) $\displaystyle\
+\frac{\pi^{2}m_{1}^{3}m_{2}^{2}}{45\left(\sigma^{2}-1\right)\epsilon}\Bigg{[}6m_{1}\left(120\sigma^{6}-98\sigma^{4}-19\sigma^{2}+12\right)+5m_{2}\sigma\left(146\sigma^{4}-157\sigma^{2}+29\right)\Bigg{]}$
$\displaystyle\
-\frac{4\pi^{2}m_{1}^{3}m_{2}^{2}\left(2\sigma^{2}-1\right)}{3\epsilon\sqrt{\sigma^{2}-1}}\left(4m_{1}\sigma+m_{2}\left(2\sigma^{2}+3\right)\right)\log
x$ $\displaystyle\
-\frac{4m_{1}^{4}m_{2}^{3}\sigma\left(2\sigma^{2}-1\right)}{\epsilon^{2}|q|}\bigg{[}\bigg{(}13-\frac{15}{\epsilon}+\frac{4}{\epsilon^{2}}\bigg{)}\frac{f^{\rm
H}_{14}}{(1-2\epsilon)^{2}}+\bigg{(}\frac{6\sigma^{2}-1}{\epsilon}+\frac{1-2\sigma^{2}}{\epsilon^{2}}\bigg{)}f^{\rm
H}_{15}\bigg{]}\,.$

We now turn to the crossed one. The integral family and its master integrals
for the $\overline{\rm IY}$ sector can be derived by the crossing relation
from (210) and (216). The master integrals are given by

$\displaystyle\text{III}:\quad f^{\overline{\rm IY}}_{1}$
$\displaystyle=f^{\overline{\rm III}}_{1}\,,$ $\displaystyle\quad
f^{\overline{\rm IY}}_{2}$ $\displaystyle=f^{\overline{\rm III}}_{2}\,,$
$\displaystyle\quad f^{\overline{\rm IY}}_{3}$ $\displaystyle=f^{\overline{\rm
III}}_{3}\,,$ $\displaystyle\quad f^{\overline{\rm IY}}_{4}$
$\displaystyle=f^{\overline{\rm III}}_{4}\,,$ (216) $\displaystyle
f^{\overline{\rm IY}}_{5}$ $\displaystyle=f^{\overline{\rm III}}_{5}\,,$
$\displaystyle\quad f^{\overline{\rm IY}}_{6}$ $\displaystyle=f^{\overline{\rm
III}}_{6}\,,$ $\displaystyle\quad f^{\overline{\rm IY}}_{7}$
$\displaystyle=f^{\overline{\rm III}}_{8}\,,$ $\displaystyle\quad
f^{\overline{\rm IY}}_{8}$ $\displaystyle=f^{\overline{\rm III}}_{9}\,,$
$\displaystyle f^{\overline{\rm IY}}_{9}$ $\displaystyle=f^{\overline{\rm
III}}_{10}\,,$ $\displaystyle\text{H even}:\quad f^{\overline{\rm IY}}_{10}$
$\displaystyle=f^{\overline{\rm H}}_{2}\,,$ $\displaystyle\quad
f^{\overline{\rm IY}}_{11}$ $\displaystyle=f^{\overline{\rm H}}_{8}\,,$
$\displaystyle\text{H odd}:\quad f^{\overline{\rm IY}}_{12}$
$\displaystyle=f^{\overline{\rm H}}_{14}\,,$ $\displaystyle\quad
f^{\overline{\rm IY}}_{13}$ $\displaystyle=f^{\overline{\rm H}}_{15}\,.$

Note that $f^{\rm H}_{2}=f^{\overline{\rm H}}_{2}$ and $f^{\rm
H}_{14}=f^{\overline{\rm H}}_{14}$ because these do not have the $u_{2}$
dependence. Applying the IBP reduction, the amplitude
$\mathcal{M}^{\overline{\rm IY}}$ is represented by a linear combination of
the above master integrals

$\mathcal{M}^{\overline{\rm IY}}=\sum_{i=1}^{13}c^{\overline{\rm
IY}}_{i}f^{\overline{\rm IY}}_{i}\Big{|}_{\rm non-analytic}\,,$ (217)

and the explicit form of the coefficients is denoted in Appendix D.3.2.
Combining all the results, we have

$\displaystyle\mathcal{M}^{\overline{\rm IY}}$
$\displaystyle=-\frac{4\pi^{2}m_{1}^{3}m_{2}^{2}}{45}\Bigg{[}\frac{3m_{1}\left(30\sigma^{4}-41\sigma^{2}+6\right)+5m_{2}\sigma\left(23\sigma^{2}-14\right)}{\epsilon}\Bigg{]}$
(218)
$\displaystyle\quad-\frac{2\pi^{2}m_{1}^{3}m_{2}^{2}\left(2\sigma^{2}-1\right)}{3\epsilon\sqrt{\sigma^{2}-1}}\left(m_{2}\left(2\sigma^{2}+3\right)-4m_{1}\sigma\right)\log
x$
$\displaystyle\quad+\frac{4m_{1}^{4}m_{2}^{3}\sigma\left(2\sigma^{2}-1\right)}{(1-2\epsilon)^{2}\epsilon^{2}|q|}\left(13-\frac{15}{\epsilon}+\frac{4}{\epsilon^{2}}\right)f^{\overline{\text{H}}}_{14}$
$\displaystyle\quad-\frac{4m_{1}^{4}m_{2}^{3}\left(2\sigma^{2}-1\right)}{\epsilon^{3}|q|}\left(6\sigma^{2}-1+\frac{1-2\sigma^{2}}{\epsilon}\right)f^{\overline{\text{\rm
H}}}_{15}\,.$

####  6.3.2 $/\\!\\!\\!\rm{Y}$ sector

The soft integral family associated with the $/\\!\\!\\!\rm{Y}$-sector is
defined from (132) by replacing the full matter propagators $\rho_{1,2}$ with
the linearized propagators $\tilde{\rho}_{1,2}$

$\displaystyle
j^{\text{/\\!\\!\\!\rm{Y}}}_{i_{1},i_{2},\cdots,i_{9}}=\int_{\ell_{1},\ell_{2}}\frac{\big{(}\tilde{\rho}^{\text{/\\!\\!\\!\rm{Y}},8}_{2}\big{)}^{-i_{8}}\big{(}\rho^{\text{/\\!\\!\\!\rm{Y}},9}_{g}\big{)}^{-i_{9}}}{\big{(}\rho^{\text{/\\!\\!\\!\rm{Y}},1}_{g}\big{)}^{i_{1}}\big{(}\rho^{\text{/\\!\\!\\!\rm{Y}},2}_{g}\big{)}^{i_{2}}\big{(}\rho^{\text{/\\!\\!\\!\rm{Y}},3}_{g}\big{)}^{i_{3}}\big{(}\rho^{\text{/\\!\\!\\!\rm{Y}},4}_{g}\big{)}^{i_{4}}\big{(}\tilde{\rho}^{\text{/\\!\\!\\!\rm{Y}},5}_{1}\big{)}^{i_{5}}\big{(}\tilde{\rho}^{\text{/\\!\\!\\!\rm{Y}},6}_{1}\big{)}^{i_{6}}\big{(}\tilde{\rho}^{\text{/\\!\\!\\!\rm{Y}},7}_{2}\big{)}^{i_{7}}}\,,$
(219)

where the propagators are defined by

$\displaystyle\rho^{\text{/\\!\\!\\!\rm{Y}},1}_{g}$
$\displaystyle=(\ell_{1}+q)^{2}\,,$
$\displaystyle\quad\rho^{\text{/\\!\\!\\!\rm{Y}},2}_{g}$
$\displaystyle=\ell_{1}^{2}\,,$
$\displaystyle\quad\rho^{\text{/\\!\\!\\!\rm{Y}},3}_{g}$
$\displaystyle=\ell_{2}^{2}\,,$ (220)
$\displaystyle\tilde{\rho}^{\text{/\\!\\!\\!\rm{Y}},5}_{1}$
$\displaystyle=2\ell_{2}\cdot u_{1}\,,$
$\displaystyle\quad\rho^{\text{/\\!\\!\\!\rm{Y}},4}_{g}$
$\displaystyle=(\ell_{1}+\ell_{2}+q)^{2}\,,$
$\displaystyle\quad\tilde{\rho}^{\text{/\\!\\!\\!\rm{Y}},6}_{1}$
$\displaystyle=2(\ell_{1}+\ell_{2})\cdot u_{1}\,,$
$\displaystyle\tilde{\rho}^{\text{/\\!\\!\\!\rm{Y}},7}_{2}$
$\displaystyle=2\ell_{1}\cdot u_{2}\,,$
$\displaystyle\quad\tilde{\rho}^{\text{/\\!\\!\\!\rm{Y}},8}_{2}$
$\displaystyle=-2\ell_{2}\cdot u_{2}\;,$
$\displaystyle\quad\rho^{\text{/\\!\\!\\!\rm{Y}},9}_{g}$
$\displaystyle=(\ell_{2}+q)^{2}\,.$

Here $\tilde{\rho}^{\text{/\\!\\!\\!\rm{Y}},8}_{2}$ and
$\rho^{\text{/\\!\\!\\!\rm{Y}},9}_{g}$ are auxiliary propagators for
consistent IBP reduction.

As for the IY sector, the soft integral family of the /Y class can be
represented by the soft integral families for the IX, H, and
$\overline{\text{H}}$ sectors as follows:

$\displaystyle
j^{\text{/\\!\\!\\!\rm{Y}}}_{0,i_{2},i_{3},i_{4},i_{5},i_{6},i_{7},0,0}$
$\displaystyle=j^{\text{IX}}_{i_{2},i_{3},i_{4},i_{7},0,i_{5},i_{6},0,0}\,,$
(221) $\displaystyle
j^{\text{/\\!\\!\\!\rm{Y}}}_{i_{1},i_{2},0,i_{4},i_{5},0,i_{7},0,0}$
$\displaystyle=j^{\text{H}}_{i_{1},0,i_{2},0,i_{4},i_{7},i_{5},0,0}\,,$
$\displaystyle
j^{\text{/\\!\\!\\!\rm{Y}}}_{i_{1},i_{2},i_{3},0,0,i_{6},i_{7},0,0}$
$\displaystyle=j^{\overline{\text{H}}}_{i_{1},0,i_{2},0,i_{3},i_{7},i_{6},0,0}\,.$

This implies that we can construct the master integrals for the /Y sector
by employing the previous results in the IX and H sectors. There are a total
of 18 master integrals in this sector:

$\displaystyle\text{IX}:\quad f^{\text{/\\!\\!\\!\rm{Y}}}_{1}$
$\displaystyle=f^{\rm IX}_{1}\,,$ $\displaystyle\quad
f^{\text{/\\!\\!\\!\rm{Y}}}_{2}$ $\displaystyle=f^{\rm IX}_{2}\,,$
$\displaystyle\quad f^{\text{/\\!\\!\\!\rm{Y}}}_{3}$ $\displaystyle=f^{\rm
IX}_{3}\,,$ $\displaystyle\quad f^{\text{/\\!\\!\\!\rm{Y}}}_{4}$
$\displaystyle=f^{\rm IX}_{4}\,,$ (222) $\displaystyle
f^{\text{/\\!\\!\\!\rm{Y}}}_{5}$ $\displaystyle=f^{\rm IX}_{5}\,,$
$\displaystyle\quad f^{\text{/\\!\\!\\!\rm{Y}}}_{6}$ $\displaystyle=f^{\rm
IX}_{6}\,,$ $\displaystyle\quad f^{\text{/\\!\\!\\!\rm{Y}}}_{7}$
$\displaystyle=f^{\rm IX}_{7}\,,$ $\displaystyle\quad
f^{\text{/\\!\\!\\!\rm{Y}}}_{8}$ $\displaystyle=f^{\rm IX}_{8}\,,$
$\displaystyle f^{\text{/\\!\\!\\!\rm{Y}}}_{9}$ $\displaystyle=f^{\rm
IX}_{11}\,,$ $\displaystyle\quad f^{\text{/\\!\\!\\!\rm{Y}}}_{10}$
$\displaystyle=f^{\rm IX}_{12}\,,$ $\displaystyle\quad
f^{\text{/\\!\\!\\!\rm{Y}}}_{11}$ $\displaystyle=f^{\rm IX}_{13}\,,$
$\displaystyle\quad f^{\text{/\\!\\!\\!\rm{Y}}}_{12}$ $\displaystyle=f^{\rm
IX}_{14}\,,$ $\displaystyle\text{H even}:\quad
f^{\text{/\\!\\!\\!\rm{Y}}}_{13}$ $\displaystyle=f^{\rm H}_{2}\,,$
$\displaystyle\quad f^{\text{/\\!\\!\\!\rm{Y}}}_{14}$ $\displaystyle=f^{\rm
H}_{8}\,,$ $\displaystyle\text{$\overline{\rm H}$ even}:\quad
f^{\text{/\\!\\!\\!\rm{Y}}}_{15}$ $\displaystyle=f^{\overline{\rm H}}_{8}\,,$
$\displaystyle\text{H odd}:\quad f^{\text{/\\!\\!\\!\rm{Y}}}_{16}$
$\displaystyle=f^{\rm H}_{14}\,,$ $\displaystyle\quad
f^{\text{/\\!\\!\\!\rm{Y}}}_{17}$ $\displaystyle=f^{\rm H}_{15}\,,$
$\displaystyle\text{$\overline{\rm H}$ odd}:\quad
f^{\text{/\\!\\!\\!\rm{Y}}}_{18}$ $\displaystyle=f^{\overline{\rm H}}_{15}\,.$

After the IBP reduction, $\mathcal{M}^{\text{/\\!\\!\\!\rm{Y}}}$ is
represented by the following linear combination

$\mathcal{M}^{\text{/\\!\\!\\!\rm{Y}}}=\sum_{i=1}^{18}c^{\text{/\\!\\!\\!\rm{Y}}}_{i}f^{\text{/\\!\\!\\!\rm{Y}}}_{i}\,,$
(223)

and the explicit form of the coefficients $c^{\text{/\\!\\!\\!\rm{Y}}}_{i}$
are listed in Appendix D.3.3.

If we substitute the values of the master integrals for the IX, H and
$\overline{\rm H}$ sectors, we have

$\displaystyle\mathcal{M}^{\text{/\\!\\!\\!\rm{Y}}}$
$\displaystyle=-\frac{2\pi^{2}m_{1}^{3}m_{2}^{2}}{225}\Bigg{[}\frac{15\left(m_{1}\left(30\sigma^{4}-41\sigma^{2}+6\right)-5m_{2}\sigma\left(\sigma^{2}-1\right)\right)}{\epsilon}$
(224)
$\displaystyle\qquad\qquad\qquad\quad+m_{1}\left(600\sigma^{4}-2353\sigma^{2}+353\right)+25m_{2}\sigma\left(29-50\sigma^{2}\right)\Bigg{]}$
$\displaystyle\
+\frac{4\pi^{2}m_{1}^{3}m_{2}^{2}}{\sqrt{\sigma^{2}-1}}\Bigg{[}\frac{\left(2\sigma^{2}-1\right)}{3\epsilon}\left(2m_{1}\sigma+3m_{2}\left(4\sigma^{2}-1\right)\right)\log(x)\Bigg{]}$
$\displaystyle\
+\frac{4m_{1}^{4}m_{2}^{3}\left(2\sigma^{2}-1\right)}{\epsilon^{3}|q|}\left(6\sigma^{2}-1+\frac{1-2\sigma^{2}}{\epsilon}\right)\left(f^{\text{\rm
H}}_{15}+f^{\overline{\text{\rm H}}}_{15}\right)\,.$

Here we do not substitute the explicit expressions of $f^{\text{\rm H}}_{15}$
and $f^{\overline{\text{\rm H}}}_{15}$, because these do not contribute to the
total amplitude.

####  6.3.3 Total $\mathbf{IY}$

The total contribution from the $\mathbf{IY}$ class can be derived by
substituting the results in each sector, (215), (218) and (224), into (209)

$\displaystyle\mathcal{M}^{\mathbf{IY}}$
$\displaystyle=-\frac{i\pi^{3}m_{1}^{4}m_{2}^{3}\left(2\sigma^{2}-1\right)}{\sqrt{\sigma^{2}-1}|q|}\Bigg{[}\sigma^{2}-1-\big{(}15\sigma^{2}-7\big{)}\bigg{(}\frac{1}{\epsilon}-2\log
2\bigg{)}\Bigg{]}$ (225)
$\displaystyle\quad+\frac{2\pi^{2}m_{1}^{3}m_{2}^{2}}{15\left(\sigma^{2}-1\right)\epsilon}\Bigg{[}6m_{1}\left(62\sigma^{4}-69\sigma^{2}+12\right)+5m_{2}\sigma\left(22\sigma^{4}-11\sigma^{2}-5\right)\Bigg{]}$
$\displaystyle\quad+\frac{4\pi^{2}m_{1}^{3}m_{2}^{3}}{\epsilon(\sigma^{2}-1)^{\frac{3}{2}}}\big{(}12\sigma^{4}-16\sigma^{2}+5\big{)}\log
x\,.$

As we have mentioned, $f^{\text{\rm H}}_{14}$, $f^{\overline{\text{\rm
H}}}_{15}$, $f^{\text{\rm H}}_{15}$ and $f^{\overline{\text{\rm H}}}_{15}$ are
cancled as we expected.

###  6.4 Total two-loop amplitude

The total two-loop amplitude in the potential region is derived by
substituting the results for each class, (194), (207) and (225), into (168)

$\displaystyle\mathcal{M}_{\rm
pot}^{{\scriptscriptstyle(2)}}(k_{1},k_{2},k_{3},k_{4})$ (226)
$\displaystyle=\frac{4G^{3}\pi
m_{1}^{4}m_{2}^{4}\left(2\sigma^{2}-1\right)^{3}\left(\pi^{2}\epsilon^{2}-6\right)}{3\epsilon^{2}\left(\sigma^{2}-1\right)q^{2}}+\frac{6G^{3}i\pi^{2}m_{1}^{3}m_{2}^{3}(5\sigma^{2}-1)(2\sigma^{2}-1)(m_{1}+m_{2})}{\epsilon\sqrt{\sigma^{2}-1}|q|}$
$\displaystyle\quad-2G^{3}\log|q|^{2}\Bigg{[}m_{1}^{2}m_{2}^{2}\pi\left(m_{1}^{2}+m_{2}^{2}+2m_{1}m_{2}\sigma\right)\left[\frac{\left(2\sigma^{2}-1\right)^{3}}{(\sigma^{2}-1)^{2}}+\frac{3\left(5\sigma^{2}-1\right)\left(2\sigma^{2}-1\right)}{2(\sigma^{2}-1)}\right]$
$\displaystyle\qquad\qquad\qquad\quad+\frac{\pi
m_{1}^{2}m_{2}^{2}}{6}\bigg{[}\left(3-54\sigma^{2}\right)\left(m_{1}^{2}+m_{2}^{2}\right)+m_{1}m_{2}\sigma\left(4\sigma^{2}+206\right)\bigg{]}$
$\displaystyle\qquad\qquad\qquad\quad+\frac{8\pi
m_{1}^{3}m_{2}^{3}\left(4\sigma^{4}-12\sigma^{2}-3\right)}{\sqrt{\sigma^{2}-1}}\operatorname{arcsinh}\left(\sqrt{\frac{\sigma-1}{2}}\right)\Bigg{]}\,.$

This result exactly matches with the classical two-loop amplitude at potential
region Bern:2019crd .

##  7 Conclusion

This article introduces a novel calculation framework for the efficient and
systematic computations of the classical scattering amplitude. The focus is on
elucidating the conservative dynamics of non-rotating binary black hole
systems, especially during the inspiral phase absent of gravitational wave
emission. Our framework builds upon quantum off-shell recursion techniques and
offers a systematic method for classifying classical diagrams through specific
power-counting rules. On top of this classification, we employ IBP reduction
on the loop integrands, deriving the $2\to 2$ classical amplitude for massive
scalar particles within a potential region.

Initially, we introduce the field theory description that treats black holes
as scalar point particles. The action comprises the EH action, along with two
massive scalar theories interacting through the gravitational forces only.
Adopting Cheung and Remmen’s convention of metric perturbations, we developed
the perturbative Einstein equation and the scalar equations of motion.
Notably, the DS equation plays a crucial role in deriving the quantum off-
shell recursions, which serve as a quantum counterpart to the classical
equations.

Next, we derived the quantum off-shell recursion for the field theory. This
was achieved by substituting the quantum perturbiner expansions for the
graviton and scalar fields and their descendant fields into the DS equation up
to two-loop order. The perturbiner formalism directly relates the gap between
scattering amplitudes and solutions of EoMs or DS equations. We explicitly
derived the recursions and solved them up to the two loops.

We then introduced a new power-counting prescription for identifying the
classical diagrams. For optimizing the IBP reduction, we provided a systematic
classification scheme of the classical diagrams by introducing the classes and
sectors directly associated with integral families. For consistent truncation
of the quantum contributions in the loop integrands for the classical
diagrams, we applied the soft expansion established in Parra-Martinez:2020dzs
.

We provided a concrete example of our prescription by analyzing the classical
one-loop amplitude, demonstrating its consistency with known results. Applying
the classification scheme, we showed that one class and two sectors exist in
this class. We then performed the soft expansion and the IBP reduction for the
one-loop integrands. The classical one-loop integrand can be represented by
the linear combinations of the one-loop master integrals. Referring to the
differential equations and their solutions corresponding to the master
integrals, we derived the one-loop classical amplitude in the potential region
consistent with the known results.

Similarly, the classical two-loop diagrams are classified – the four classes,
$\Box\\!\Box$, H, IY and  IY , and their sectors. We identified the mirror
dual pairs for each sector and showed that the  IY -class can be obtained from
the results of the IY-class by the simple relations. Again, we performed the
soft expansion and the IBP reduction to the two-loop integrands for each
sector to represent the soft two-loop integrands in terms of linear
combinations of the master integrals. Interestingly, the entire master
integrals of the IY-class can be represented by the master integrals of the
$\Box\\!\Box$ and H classes. Thus, we do not have to derive and solve the
differential equations. Referring to the solutions of the differential
equations obtained by Parra-Martinez:2020dzs , we derived the two-loop
classical amplitudes in the potential region. The amplitude exactly agrees
with the known results in Bern:2019crd .

This paper applies our formalism to the 3PM amplitude for the non-rotating
binary black hole system. However, many directions remain open for future
research. One of the immediate challenges is to derive the 4PM results for the
same setup by three-loop computations Dlapa:2021npj ; Dlapa:2021vgp ;
Bern:2021yeh ; Bern:2021dqo ; Bern:2022jvn . The extension of our framework to
include rotating black holes Porto:2005ac ; Vaidya:2014kza ; Bini:2017xzy ;
Cachazo:2017jef ; Vines:2017hyw ; Guevara:2017csg ; Bini:2018ywr ;
Guevara:2018wpp ; Levi:2018nxp ; Vines:2018gqi ; Chung:2019duq ;
Damgaard:2019lfh ; Guevara:2019fsj ; Maybee:2019jus ; Aoude:2020onz ;
Bern:2020buy ; Guevara:2020xjx ; Levi:2020kvb ; Levi:2020uwu ;
Jakobsen:2022fcj ; Aoude:2023dui ; Aoude:2023vdk ; DiVecchia:2023frv ;
Heissenberg:2023uvo ; Jakobsen:2023ndj ; Jakobsen:2023hig  and the finite size
correction for describing other celestial systems, such as neutron star
binaries or black hole-neutron star systems, by incorporating the higher
dimensional operators Bini:2020flp ; Cheung:2020gbf ; Aoude:2020ygw ;
Bern:2020uwk ; Kalin:2020lmz ; Haddad:2020que ; Cheung:2020sdj ;
Mandal:2023hqa . Furthermore, it would be interesting to apply the KMOC
formulation Kosower:2018adc ; Maybee:2019jus ; Herrmann:2021lqe ;
Herrmann:2021tct ; Cristofoli:2021vyo ; Cristofoli:2021jas  for describing
gravitational wave radiation.

###### Acknowledgements.

We thank Jung-Wook Kim, Seok Kim, Sangmin Lee, and Mao Zeng for their useful
comments and discussion. We would like to express our special gratitude to
Kimyeong Lee and Jeong-Hyuck Park for suggesting to initiate this work and
encouraging us. This work is supported by appointment to the JRG Program at
the APCTP through the Science and Technology Promotion Fund and Lottery Fund
of the Korean Government. KL is also supported by the National Research
Foundation of Korea(NRF) grant funded by the Korean government(MSIT)
RS-2023-00249451 and the Korean Local Governments of Gyeongsangbuk-do Province
and Pohang City. KC is supported by the NRF grant funded by the Korean
government (MSIT) 2022R1F1A1068489.

##  Appendix A Equations of Motions and $\mathcal{I}$-tensors

The EoM for the field theory is substantially complicated due to the graviton
sector. This makes the derivation of the DS equation difficult. As we have
discussed in Section 2.1, it is efficient to introduce the
$\mathcal{I}$-tensors. Recall that the perturbed Einstein equation in (22) is
decomposed into two parts, $\mathcal{G}^{\mu\nu}$ and $\mathcal{T}^{\mu\nu}$.
The general form of the curvature perturbations $\mathcal{G}^{\mu\nu}_{n}$ is
given by

$\displaystyle\mathcal{G}^{\mu\nu}_{n}$
$\displaystyle=\mathcal{I}^{\mu\nu,M_{1},\cdots,M_{n-1},\kappa N,\lambda
P}_{n,1}h^{M_{1}}\cdots
h^{M_{n-1}}\partial_{\kappa}h^{N}\partial_{\lambda}h^{P}$ (227)
$\displaystyle\quad+\mathcal{I}^{\mu\nu,M_{1},\cdots,M_{n-1},N,\kappa\lambda
P}_{n,2}h^{M_{1}}\cdots
h^{M_{n-1}}h^{N}\partial_{\kappa}\partial_{\lambda}h^{P}\,.$

The general form of the $n$-th order of the energy-momentum perturbations
$\mathcal{T}^{\mu\nu}_{n}$ are given by

$\displaystyle\mathcal{T}^{\mu\nu}_{n}$
$\displaystyle=\frac{1}{2}\sum_{I=1}^{2}\left[\mathcal{I}_{\rm
EM}^{\mu\nu,N_{1},\cdots N_{n-1},I}h^{N_{1}}\cdots
h^{N_{n-1}}\varphi_{I}^{2}\right]\,,\qquad n>1\,.$ (228)

The EoM of the scalar fields are defined in (25). The general $n$-th order
term (for $n>1$) of the EoM is also denoted by

$\mathcal{I}^{I,M_{1},M_{2},\cdots,M_{n}}_{\phi}h^{M_{1}}h^{M_{2}}\cdots
h^{M_{n}}\phi^{I}\,.$ (229)

Here we denote the $\mathcal{I}$-tensors explicitly associated with the
classical two-loop amplitudes. The relevant $\mathcal{I}$-tensors are as
follows:

$\displaystyle\mathcal{I}_{1,1}^{\mu\nu,\rho_{1}\kappa_{1}\lambda_{1},\rho_{2}\kappa_{2}\lambda_{2}}\,,$
$\displaystyle\mathcal{I}_{1,2}^{\mu\nu,\kappa_{1}\lambda_{1},\rho_{1}\rho_{2}\kappa_{2}\lambda_{2}}\,,$
(230)
$\displaystyle\mathcal{I}^{\mu\nu,\kappa_{1}\lambda_{1},\rho_{1}\kappa_{2}\lambda_{2},\rho_{2}\kappa_{3}\lambda_{3}}_{2,1}\,,$
$\displaystyle\mathcal{I}^{\mu\nu,\kappa_{1}\lambda_{1},\kappa_{2}\lambda_{2},\rho_{1}\rho_{2}\kappa_{3}\lambda_{3}}_{2,2}\,,$
$\displaystyle\mathcal{I}^{\mu\nu,\kappa_{1}\lambda_{1},I}_{\rm EM}\,,$
$\displaystyle\mathcal{I}^{\mu\nu,\kappa_{1}\lambda_{1},\kappa_{2},\lambda_{2},I}_{\rm
EM}\,,$
$\displaystyle\mathcal{I}^{I,\kappa_{1}\lambda_{1},\kappa_{2}\lambda_{2}}_{\phi}\,,$
$\displaystyle\mathcal{I}^{I,\kappa_{1}\lambda_{1},\kappa_{2}\lambda_{2}}_{\phi}\,.$

###  A.1 Graviton sector

Let us consider the curvature tensor part, $\mathcal{G}^{\mu\nu}_{1}$ and
$\mathcal{G}^{\mu\nu}_{2}$ using the $\mathcal{I}$-tensors. First
$\mathcal{G}^{\mu\nu}_{1}$ is defined in (23) as

$\mathcal{G}_{1}^{\mu\nu}=\mathcal{I}_{1,1}^{\mu\nu,\rho_{1}\kappa_{1}\lambda_{1},\rho_{2}\kappa_{2}\lambda_{2}}\partial_{\rho_{1}}h_{\kappa_{1}\lambda_{1}}\partial_{\rho_{2}}h_{\kappa_{2}\lambda_{2}}+\mathcal{I}_{1,2}^{\mu\nu,\kappa_{1}\lambda_{1},\rho_{1}\rho_{2}\kappa_{2}\lambda_{2}}h_{\kappa_{1}\lambda_{1}}\partial_{\rho_{1}}\partial_{\rho_{2}}h_{\kappa_{2}\lambda_{2}}\,.$
(231)

We want to present the explicit form of the
$\mathcal{I}^{\mu\nu,\rho_{1}\kappa_{1}\lambda_{1},\rho_{2}\kappa_{2}\lambda_{2}}_{1,1}$
and
$\mathcal{I}^{\mu\nu,\kappa_{1}\lambda_{1},\rho_{1}\rho_{2}\kappa_{2}\lambda_{2}}_{1,2}$.
For simplicity, we decompose the $\mathcal{I}$-tensors according to the
following tensor structure:

$\displaystyle\mathcal{I}_{1,1}^{\mu\nu,\rho_{1}\kappa_{1}\lambda_{1},\rho_{2}\kappa_{2}\lambda_{2}}$
$\displaystyle=\bar{\mathcal{I}}_{1,1}^{\mu\nu,\rho_{1}\kappa_{1}\lambda_{1},\rho_{2}\kappa_{2}\lambda_{2}}+\bar{\mathcal{I}}_{1,1}^{\mu\nu,\rho_{2}\kappa_{2}\lambda_{2},\rho_{1}\kappa_{1}\lambda_{1}}\,,$
(232)
$\displaystyle\mathcal{I}_{1,2}^{\mu\nu,\kappa_{1}\lambda_{1},\rho_{1}\rho_{2}\kappa_{2}\lambda_{2}}$
$\displaystyle=\bar{\mathcal{I}}_{1,2}^{\mu\nu,\kappa_{1}\lambda_{1},\rho_{1}\rho_{2}\kappa_{2}\lambda_{2}}+\bar{\mathcal{I}}_{1,2}^{\nu\mu,\kappa_{1}\lambda_{1},\rho_{1}\rho_{2}\kappa_{2}\lambda_{2}}\,,$

where

$\displaystyle{\bar{\mathcal{I}}_{1,1}}^{\mu\nu,\rho_{1}\kappa_{1}\lambda_{1},\rho_{2}\kappa_{2}\lambda_{2}}$
(233)
$\displaystyle=\frac{1}{8}\eta^{\mu\nu}\eta^{\rho_{1}\rho_{2}}\eta^{\kappa_{1}(\kappa_{2}}\eta^{\lambda_{2})\lambda_{1}}+\frac{1}{16}\eta^{\rho_{1}(\mu}\eta^{\nu)\rho_{2}}\eta^{\kappa_{1}\lambda_{1}}\eta^{\kappa_{2}\lambda_{2}}-\frac{1}{8}\eta^{\mu\nu}\eta^{\rho_{1}(\kappa_{1}}\eta^{\lambda_{1})\rho_{2}}\eta^{\kappa_{2}\lambda_{2}}$
$\displaystyle\quad-\frac{1}{4}\eta^{\mu(\kappa_{1}}\eta^{\lambda_{1})(\kappa_{2}}\eta^{\lambda_{2})\nu}\eta^{\rho_{1}\rho_{2}}-\frac{1}{4}\eta^{\rho_{2}(\kappa_{1}}\eta^{\lambda_{1})(\mu}\eta^{\nu)(\kappa_{2}}\eta^{\lambda_{2})\rho_{1}}-\frac{1}{8}\eta^{\rho_{1}(\mu}\eta^{\nu)\rho_{2}}\eta^{\kappa_{1}(\kappa_{2}}\eta^{\lambda_{2})\lambda_{1}}$
$\displaystyle\quad+\frac{1}{4}\eta^{\mu(\kappa_{2}}\eta^{\lambda_{2})\nu}\eta^{\rho_{1}(\kappa_{1}}\eta^{\lambda_{1})\rho_{2}}+\frac{1}{2}\eta^{\rho_{1}(\mu}\eta^{\nu)({\kappa_{2}}}\eta^{{\lambda_{2}})({\kappa_{1}}}\eta^{{\lambda_{1}})\rho_{2}}\,,$

and

$\displaystyle\bar{\mathcal{I}}_{1,2}^{\mu\nu,\kappa_{1}\lambda_{1},\rho_{1}\rho_{2}\kappa_{2}\lambda_{2}}$
(234)
$\displaystyle=\eta^{\mu(\kappa_{1}}\eta^{\lambda_{1})\nu}\bigg{[}\frac{1}{8}\eta^{\rho_{1}\rho_{2}}\eta^{\kappa_{2}\lambda_{2}}+\frac{1}{4}\eta^{\rho_{1}({\kappa_{1}}}\eta^{{\lambda_{1}})\rho_{2}}\bigg{]}-\frac{\eta^{\mu\nu}}{8}\bigg{[}\eta^{\kappa_{2}\lambda_{2}}\eta^{\kappa_{1}(\rho_{1}}\eta^{\rho_{2})\lambda_{1}}-\eta^{\kappa_{1}(\kappa_{2}}\eta^{\lambda_{2})\lambda_{2}}\eta^{\rho_{1}\rho_{2}}\bigg{]}$
$\displaystyle\quad-\frac{1}{2}\eta^{\mu({\kappa_{1}}}\eta^{{\lambda_{1}})({\kappa_{2}}}\eta^{{\lambda_{2}})\nu}\eta^{\rho_{1}\rho_{2}}+\frac{1}{2}\eta^{\mu({\kappa_{1}}}\eta^{{\lambda_{1}})({\lambda_{2}}}\eta^{{\kappa_{2}})(\rho_{1}}\eta^{\rho_{2})\nu}\,.$

Similarly, the $\mathcal{G}^{\mu\nu}_{2}$ is defined in (23) as follows:

$\displaystyle\mathcal{G}^{\mu\nu}_{2}$
$\displaystyle=\mathcal{I}^{\mu\nu,\kappa_{1}\lambda_{1},\rho_{1}\kappa_{2}\lambda_{2},\rho_{2}\kappa_{3}\lambda_{3}}_{2,1}h^{\kappa_{1}\lambda_{1}}\partial_{\rho_{1}}h^{\kappa_{2}\lambda_{2}}\partial_{\rho_{2}}h^{\kappa_{3}\lambda_{3}}$
(235)
$\displaystyle\quad+\mathcal{I}^{\mu\nu,\kappa_{1}\lambda_{1},\kappa_{2}\lambda_{2},\rho_{1}\rho_{2}\kappa_{3}\lambda_{3}}_{2,2}h^{\kappa_{1}\lambda_{1}}h^{\kappa_{2}\lambda_{2}}\partial_{\rho_{1}}\partial_{\rho_{2}}h^{\kappa_{3}\lambda_{3}}\,.$

Again, we decompose the $\mathcal{I}$-tensors as the

$\displaystyle\mathcal{I}^{\mu\nu,\kappa_{1}\lambda_{1},\rho_{1}\kappa_{2}\lambda_{2},\rho_{2}\kappa_{3}\lambda_{3}}_{2,1}$
$\displaystyle=\bar{\mathcal{I}}^{\mu\nu,\kappa_{1}\lambda_{1},\rho_{1}\kappa_{2}\lambda_{2},\rho_{2}\kappa_{3}\lambda_{3}}_{2,1}+\bar{\mathcal{I}}^{\mu\nu,\kappa_{1}\lambda_{1},\rho_{2}\kappa_{3}\lambda_{3},\rho_{1}\kappa_{2}\lambda_{2}}_{2,1}$
(236)
$\displaystyle\mathcal{I}^{\mu\nu,\kappa_{1}\lambda_{1},\kappa_{2}\lambda_{2},\rho_{1}\rho_{2}\kappa_{3}\lambda_{3}}_{2,2}$
$\displaystyle=\bar{\mathcal{I}}^{\mu\nu,\kappa_{1}\lambda_{1},\kappa_{2}\lambda_{2},\rho_{1}\rho_{2}\kappa_{3}\lambda_{3}}_{2,2}+\bar{\mathcal{I}}^{\nu\mu,\kappa_{1}\lambda_{1},\kappa_{2}\lambda_{2},\rho_{1}\rho_{2}\kappa_{3}\lambda_{3}}_{2,2}\,,$

where

$\displaystyle\bar{\mathcal{I}}_{2,1}^{\mu\nu,\kappa_{1}\lambda_{1},\rho_{1}\lambda_{2}\kappa_{2},\rho_{2}\kappa_{3}\lambda_{3}}$
(237)
$\displaystyle=\frac{\eta^{\mu\nu}}{8}\Big{[}2\eta^{\rho_{1}\rho_{2}}\eta^{{\kappa_{1}}({\kappa_{2}}}\eta^{{\lambda_{2}})({\kappa_{3}}}\eta^{{\lambda_{3}}){\lambda_{1}}}-\eta^{\rho_{1}({\kappa_{1}}}\eta^{{\lambda_{1}})\rho_{2}}\eta^{{\kappa_{2}}({\kappa_{3}}}\eta^{{\lambda_{3}}){\lambda_{2}}}-\eta^{\rho_{1}({\kappa_{2}}}\eta^{{\lambda_{2}})\rho_{2}}\eta^{{\kappa_{1}}({\kappa_{3}}}\eta^{{\lambda_{3}}){\lambda_{1}}}\Big{]}$
$\displaystyle\quad-\frac{1}{8}\eta^{\rho_{1}(\mu}\eta^{\nu)\rho_{2}}\Big{[}2\eta^{{\kappa_{1}}({\kappa_{2}}}\eta^{{\lambda_{2}})({\kappa_{3}}}\eta^{{\lambda_{3}}){\lambda_{1}}}-\eta^{{\kappa_{1}}({\kappa_{2}}}\eta^{{\lambda_{2}}){\lambda_{1}}}\eta^{{\kappa_{3}}{\lambda_{3}}}\Big{]}$
$\displaystyle\quad+\frac{1}{8}\eta^{\mu({\kappa_{1}}}\eta^{{\lambda_{1}})\nu}\Big{[}\eta^{{\kappa_{2}}({\kappa_{3}}}\eta^{{\lambda_{3}}){\lambda_{2}}}\eta^{\rho_{1}\rho_{2}}+\eta^{\rho_{1}({\kappa_{3}}}\eta^{{\lambda_{3}})\rho_{2}}\eta^{{\kappa_{2}}{\lambda_{2}}}\Big{]}$
$\displaystyle\quad-\frac{\eta^{\rho_{1}\rho_{2}}}{4}\Big{[}\eta^{\mu({\kappa_{1}}}\eta^{{\lambda_{1}})({\kappa_{2}}}\eta^{{\lambda_{2}})({\kappa_{3}}}\eta^{{\lambda_{3}})\nu}+\eta^{\mu({\kappa_{2}}}\eta^{{\lambda_{2}})({\kappa_{3}}}\eta^{{\lambda_{3}})({\kappa_{1}}}\eta^{{\lambda_{1}})\nu}+\eta^{\mu({\kappa_{3}}}\eta^{{\lambda_{3}})({\kappa_{1}}}\eta^{{\lambda_{1}})({\kappa_{2}}}\eta^{{\lambda_{2}})\nu}\Big{]}$
$\displaystyle\quad+\frac{1}{4}\Big{[}\eta^{\mu({\kappa_{1}}}\eta^{{\lambda_{1}})({\kappa_{2}}}\eta^{{\lambda_{2}})\nu}+\eta^{\nu({\kappa_{1}}}\eta^{{\lambda_{1}})({\kappa_{2}}}\eta^{{\lambda_{2}})\mu}\Big{]}\eta^{\rho_{1}({\kappa_{3}}}\eta^{{\lambda_{3}})\rho_{2}}+\frac{1}{4}\eta^{\mu({\kappa_{2}}}\eta^{{\lambda_{2}})({\kappa_{3}}}\eta^{{\lambda_{3}})\nu}\eta^{\rho_{1}({\kappa_{1}}}\eta^{{\lambda_{1}})\rho_{2}}$
$\displaystyle\quad+\frac{1}{2}\eta^{\rho_{1}(\mu}\eta^{\nu)({\kappa_{3}}}\eta^{{\lambda_{3}})({\kappa_{1}}}\eta^{{\lambda_{1}})({\kappa_{2}}}\eta^{{\lambda_{2}})\rho_{2}}+\frac{1}{2}\eta^{\rho_{1}(\mu}\eta^{\nu)({\kappa_{1}}}\eta^{{\lambda_{1}})({\kappa_{3}}}\eta^{{\lambda_{3}})({\kappa_{2}}}\eta^{{\lambda_{2}})\rho_{2}}$
$\displaystyle\quad-\frac{1}{2}\eta^{\rho_{1}({\kappa_{3}}}\eta^{{\lambda_{3}})(\mu}\eta^{\nu)({\kappa_{1}}}\eta^{{\lambda_{1})}{(\kappa_{2}}}\eta^{{\lambda_{2}})\rho_{2}}\,,$

and

$\displaystyle\bar{\mathcal{I}}_{2,2}^{\mu\nu,\kappa_{1}\lambda_{1},\kappa_{2}\lambda_{2},\rho_{1}\rho_{2}\kappa_{3}\lambda_{3}}$
(238)
$\displaystyle=\frac{\eta^{\mu\nu}\eta^{\rho_{1}\rho_{2}}}{16}\Big{[}\eta^{{\kappa_{1}}({\kappa_{2}}}\eta^{{\lambda_{2})}{(\kappa_{3}}}\eta^{{\lambda_{3}}){\lambda_{1}}}+\eta^{\lambda_{1}({\kappa_{2}}}\eta^{{\lambda_{2})}{(\kappa_{3}}}\eta^{{\lambda_{3}}){{\kappa_{1}}}}\Big{]}+\frac{1}{8}\eta^{\mu({\kappa_{1}}}\eta^{{\lambda_{1}})({\kappa_{2}}}\eta^{{\lambda_{2}})\nu}\eta^{{\kappa_{3}}{\lambda_{3}}}\eta^{\rho_{1}\rho_{2}}$
$\displaystyle\quad-\frac{\eta^{\mu\nu}}{16}\Big{[}\eta^{\rho_{1}({\kappa_{2}}}\eta^{{\lambda_{2}})\rho_{2}}\eta^{{\kappa_{1}}({\kappa_{3}}}\eta^{{\lambda_{3}}){\lambda_{1}}}+\eta^{\rho_{1}({\kappa_{1}}}\eta^{{\lambda_{1}})\rho_{2}}\eta^{{\kappa_{2}}({\kappa_{3}}}\eta^{{\lambda_{3}}){\lambda_{2}}}\Big{]}$
$\displaystyle\quad+\frac{\eta^{\mu({\kappa_{1}}}\eta^{{\lambda_{1}})\nu}}{16}\Big{[}\eta^{{\kappa_{2}}({\kappa_{3}}}\eta^{{\lambda_{3}}){\lambda_{2}}}\eta^{\rho_{1}\rho_{2}}-\eta^{\rho_{1}({\kappa_{2}}}\eta^{{\lambda_{2}})\rho_{2}}\eta^{{\kappa_{3}}{\lambda_{3}}}\Big{]}$
$\displaystyle\quad+\frac{\eta^{\mu({\kappa_{2}}}\eta^{{\lambda_{2}})\nu}}{16}\Big{[}\eta^{{\kappa_{1}}({\kappa_{3}}}\eta^{{\lambda_{3}}){\lambda_{1}}}\eta^{\rho_{1}\rho_{2}}+\eta^{\rho_{1}({\kappa_{1}}}\eta^{{\lambda_{1}})\rho_{2}}\eta^{{\kappa_{3}}{\lambda_{3}}}\Big{]}$
$\displaystyle\quad-\frac{\eta^{\rho_{1}\rho_{2}}}{4}\Big{[}\eta^{\mu({\kappa_{1}}}\eta^{{\lambda_{1}})({\kappa_{2}}}\eta^{{\lambda_{2}})({\kappa_{3}}}\eta^{{\lambda_{3}})\nu}+\eta^{\mu({\kappa_{2}}}\eta^{{\lambda_{2}})({\kappa_{3}}}\eta^{{\lambda_{3}})({\kappa_{1}}}\eta^{{\lambda_{1}})\nu}+\eta^{\mu({\kappa_{3}}}\eta^{{\lambda_{3}})({\kappa_{1}}}\eta^{{\lambda_{1}})({\kappa_{2}}}\eta^{{\lambda_{2}})\nu}\Big{]}$
$\displaystyle\quad+\frac{1}{4}\eta^{\mu({\kappa_{1}}}\eta^{{\lambda_{1}})({\kappa_{3}}}\eta^{{\lambda_{3}})\nu}\eta^{\rho_{1}({\kappa_{2}}}\eta^{{\lambda_{2}})\rho_{2}}+\frac{1}{4}\eta^{\mu({\kappa_{2}}}\eta^{{\lambda_{2}})({\kappa_{3}}}\eta^{{\lambda_{3}})\nu}\eta^{\rho_{1}({\kappa_{1}}}\eta^{{\lambda_{1}})\rho_{2}}$
$\displaystyle\quad+\frac{1}{4}\eta^{\mu({\kappa_{2}}}\eta^{{\lambda_{2}})({\kappa_{1}}}\eta^{{\lambda_{1}})({\kappa_{3}}}\eta^{{\lambda_{3}})(\rho_{1}}\eta^{\rho_{2})\nu}+\frac{1}{8}\eta^{\mu(\rho_{1}}\eta^{\rho_{2})({\kappa_{3}}}\eta^{{\lambda_{3}})({\kappa_{2}}}\eta^{{\lambda_{2}})({\kappa_{1}}}\eta^{{\lambda_{1}})\nu}\,.$

Let us consider the energy-momentum tensor part. Since the
$\mathcal{T}^{M}_{1}$ does not involve the $\mathcal{I}$-tensor, we focus on
the $\mathcal{T}^{M}_{2}$ and $\mathcal{T}^{M}_{3}$,

$\mathcal{T}^{M}_{2}=\mathcal{I}_{\rm
EM}^{M,N}\Theta^{N}_{2}\,,\qquad\mathcal{T}^{M}_{3}=\mathcal{I}_{\rm
EM}^{M,N,P}\Theta^{N,P}_{3}\,,$ (239)

where

$\displaystyle\mathcal{I}^{\mu\nu,\kappa_{1}\lambda_{1}}_{\rm
EM}=\frac{1}{2}\eta^{\mu(\kappa_{1}}\eta^{\lambda_{1})\nu}-\frac{1}{4}\eta^{\mu\nu}\eta^{\kappa_{1}\lambda_{1}}\,,$
(240)

and

$\displaystyle\mathcal{I}^{\mu\nu,\kappa_{1}\lambda_{1},\kappa_{2}\lambda_{2}}_{\rm
EM}$
$\displaystyle=\frac{1}{16}\eta^{\mu\nu}\eta^{\kappa_{1}\lambda_{1}}\eta^{\kappa_{2}\lambda_{2}}-\frac{1}{8}\eta^{\mu\nu}\eta^{\kappa_{2}(\kappa_{1}}\eta^{\lambda_{1})\lambda_{2}}-\frac{1}{8}\eta^{\mu(\kappa_{1}}\eta^{\lambda_{1})\nu}\eta^{\kappa_{2}\lambda_{2}}$
(241)
$\displaystyle\quad-\frac{1}{8}\eta^{\mu(\kappa_{2}}\eta^{\lambda_{2})\nu}\eta^{\kappa_{1}\lambda_{1}}+\frac{1}{4}\eta^{\mu(\kappa_{1}}\eta^{\lambda_{1})(\kappa_{2}}\eta^{\lambda_{2})\nu}+\frac{1}{4}\eta^{\nu(\kappa_{1}}\eta^{\lambda_{1})(\kappa_{2}}\eta^{\lambda_{2})\mu}\,.$

Finally, $\mathcal{I}$-tensors that appear in the EoM of $\phi_{I}$ (25) are

$\displaystyle\mathcal{I}^{\kappa_{1}\lambda_{1},\kappa_{2}\lambda_{2}}_{\phi}$
$\displaystyle=\frac{1}{8}\eta^{\kappa_{1}\lambda_{1}}\eta^{\kappa_{2}\lambda_{2}}-\frac{1}{4}\eta^{\kappa_{1}(\kappa_{2}}\eta^{\lambda_{2})\lambda_{1}}\,,$
(242)
$\displaystyle\mathcal{I}^{\kappa_{1}\lambda_{1},\kappa_{2}\lambda_{2},\kappa_{3}\lambda_{3}}_{\phi}$
$\displaystyle=-\frac{1}{48}\eta^{\kappa_{1}\lambda_{1}}\eta^{\kappa_{2}\lambda_{2}}\eta^{\kappa_{3}\lambda_{3}}+\frac{1}{8}\eta^{\kappa_{1}\lambda_{1}}\eta^{\kappa_{2}(\kappa_{3}}\eta^{\lambda_{3})\lambda_{2}}$
$\displaystyle\quad+\frac{1}{8}\eta^{\kappa_{2}\lambda_{2}}\eta^{\kappa_{3}(\kappa_{1}}\eta^{\lambda_{1})\lambda_{3}}+\frac{1}{8}\eta^{\kappa_{3}\lambda_{3}}\eta^{\kappa_{1}(\kappa_{2}}\eta^{\lambda_{2})\lambda_{1}}$
$\displaystyle\quad-\frac{1}{12}\eta^{\kappa_{1}(\kappa_{2}}\eta^{\lambda_{2})(\kappa_{3}}\eta^{\lambda_{3})\lambda_{1}}-\frac{1}{12}\eta^{\lambda_{1}(\kappa_{2}}\eta^{\lambda_{2})(\kappa_{3}}\eta^{\lambda_{3})\kappa_{1}}\,.$

##  Appendix B Descendant equations

###  B.1 Deformation rule for the product of multiple fields

We explain the general structure of the deformation, which is a key step in
deriving the DS equation. We first consider arbitrary fields
$A_{x},B_{x},C_{x},D_{x}$ and their external sources $a_{x},b_{x},c_{x},d_{x}$
respectively. The deformation is defined as follows:

$\displaystyle A_{x}\mapsto A_{x}+\frac{\hbar}{i}\frac{\delta}{\delta
a_{x}}\,,\qquad B_{x}\mapsto B_{x}+\frac{\hbar}{i}\frac{\delta}{\delta
b_{x}}\,,$ (243) $\displaystyle C_{x}\mapsto
C_{x}+\frac{\hbar}{i}\frac{\delta}{\delta c_{x}}\,,\qquad D_{x}\mapsto
D_{x}+\frac{\hbar}{i}\frac{\delta}{\delta d_{x}}\,.$

We introduce a shorthand notation for the functional derivatives as

$\displaystyle\psi^{ab}_{xy}$ $\displaystyle=\frac{\delta A_{x}}{\delta
b_{y}}\,,$ $\displaystyle\quad\psi^{bc}_{yz}$ $\displaystyle=\frac{\delta
B_{y}}{\delta c_{z}}\,,$ $\displaystyle\quad\psi^{ca}_{zx}$
$\displaystyle=\frac{\delta C_{z}}{\delta a_{x}}\,,$ (244)
$\displaystyle\psi^{abc}_{xyz}$ $\displaystyle=\frac{\delta^{2}A_{x}}{\delta
b_{y}\delta c_{z}}\,,$ $\displaystyle\quad\psi^{bca}_{yzx}$
$\displaystyle=\frac{\delta^{2}B_{y}}{\delta c_{z}\delta a_{x}}\,,$
$\displaystyle\quad\psi^{cab}_{zxy}$
$\displaystyle=\frac{\delta^{2}C_{z}}{\delta a_{x}\delta b_{y}}\,,$

and their permutations as

$\displaystyle\hat{\psi}^{ab}_{xy}$
$\displaystyle=\frac{1}{2}\Big{(}\psi^{ab}_{xy}+\psi^{ba}_{yx}\Big{)}\,,$
(245) $\displaystyle\hat{\psi}^{abc}_{xyz}$
$\displaystyle=\frac{1}{3!}\Big{(}\psi_{xyz}^{abc}+\psi_{xzy}^{acb}+\psi_{yxz}^{bac}+\psi_{yzx}^{bca}+\psi_{zxy}^{cab}+\psi_{zyx}^{cba}\Big{)}\,.$

Under the deformation, we derive the following rules:

$\displaystyle A_{x}B_{y}\mapsto
A_{x}B_{y}+\frac{\hbar}{i}\hat{\psi}^{ab}_{x,y}\,,$ (246) $\displaystyle
A_{x}B_{y}C_{z}\mapsto
A_{x}B_{y}C_{z}+\frac{\hbar}{i}\left[A_{x}\hat{\psi}_{yz}^{bc}+B_{y}\hat{\psi}_{zx}^{ca}+C_{z}\hat{\psi}_{xy}^{ab}\right]+\Big{(}\frac{\hbar}{i}\Big{)}^{2}\hat{\psi}_{xyz}^{abc}\,,$
$\displaystyle A_{x}B_{y}C_{z}D_{w}\mapsto A_{x}B_{y}C_{z}D_{w}$
$\displaystyle\quad~{}+\frac{\hbar}{i}\bigg{[}\hat{\psi}^{ab}_{xy}C_{z}D_{w}+\hat{\psi}^{ac}_{xz}B_{y}D_{w}+\hat{\psi}^{ad}_{xw}B_{y}C_{z}+\hat{\psi}^{bc}_{yz}A_{x}D_{w}+\hat{\psi}^{bd}_{yw}A_{x}C_{z}+\hat{\psi}^{cd}_{zw}A_{x}B_{y}\bigg{]}$
$\displaystyle\quad~{}+\Big{(}\frac{\hbar}{i}\Big{)}^{2}\bigg{[}A_{x}\hat{\psi}_{yzw}^{bcd}+B_{y}\hat{\psi}_{xzw}^{acd}+C_{z}\hat{\psi}_{xyw}^{abd}+D_{w}\hat{\psi}_{xyz}^{abc}+\hat{\psi}^{ab}_{xy}\hat{\psi}^{cd}_{zw}+\hat{\psi}^{ac}_{xz}\hat{\psi}^{bd}_{yw}+\hat{\psi}^{ad}_{xw}\hat{\psi}^{bc}_{yz}\bigg{]}$
$\displaystyle\quad~{}+\Big{(}\frac{\hbar}{i}\Big{)}^{3}\hat{\psi}_{xyzw}^{abcd}\,.$

###  B.2 Explicit form of the DS equations

The DS equation for $\varphi^{I}_{x}$ is given by

$\displaystyle\left(\square-
m_{I}^{2}\right)\varphi^{I}_{x}=-\rho_{x}^{I}+\kappa\Omega^{I}_{1}+\kappa^{2}\mathcal{I}^{I,M,N}_{\phi}\Omega^{IMN}_{2}+\kappa^{3}\mathcal{I}^{I,M,N,P}_{\phi}\Omega^{IMNP}_{3}\,,$
(247)

where

$\displaystyle\Omega^{I}_{1}$
$\displaystyle=\partial_{\kappa}\big{(}h_{x}^{\kappa\lambda}\partial_{\lambda}\varphi_{x}^{I}\big{)}-\frac{m_{I}^{2}}{2}h^{\kappa\kappa}_{x}\varphi_{x}^{I}+\left(\partial_{\kappa}^{y}\partial_{\lambda}^{x}+\partial_{\kappa}^{x}\partial_{\lambda}^{x}\right)\hat{\psi}_{xy}^{I\kappa\lambda}\Big{|}_{y\to
x}-\frac{m_{I}^{2}}{2}\hat{\psi}^{\kappa\kappa,I}_{xx}\,,$ (248)
$\displaystyle\Omega^{IMN}_{2}$
$\displaystyle=h_{x}^{M}h_{x}^{N}\varphi_{x}^{I}+\hat{\psi}_{xx}^{MN}\varphi^{I}_{x}+2h^{(M}_{x}\hat{\psi}_{xx}^{N)I}+\hat{\psi}_{xxx}^{MNI}\,,$
$\displaystyle\Omega^{IMNP}_{3}$
$\displaystyle=h_{x}^{M}h_{x}^{N}h_{x}^{P}\varphi_{x}^{I}+3\Big{[}h^{(M}_{x}\hat{\psi}^{NP)}_{xx}\varphi^{I}_{x}+h^{(M}_{x}h^{N}_{x}\hat{\psi}^{P)I}_{x,x}\Big{]}$
$\displaystyle\quad+\Big{[}\hat{\psi}^{MNP}_{xxx}\varphi^{I}_{x}+3h^{(M}\hat{\psi}_{xxx}^{NP)I}+3\hat{\psi}^{(MN}_{xx}\hat{\psi}^{P)I}_{xx}\Big{]}\,,$

where

$\displaystyle\hat{\psi}^{AB}_{xy}$
$\displaystyle=\frac{1}{2}\frac{\hbar}{i}\left(\psi^{AB}_{xy}+\psi^{BA}_{yx}\right)\,,$
(249) $\displaystyle\hat{\psi}_{xyz}^{ABC}$
$\displaystyle=\frac{1}{3!}\bigg{(}\frac{\hbar}{i}\bigg{)}^{2}\left(\psi_{xyz}^{ABC}+\psi_{xzy}^{ACB}+\psi_{yzx}^{BCA}+\psi_{yxz}^{BAC}+\psi_{zxy}^{CAB}+\psi_{zyx}^{CBA}\right)\,.$

The DS equation for $h^{M}$ is given by

$\Box
h_{x}^{M}=2P^{M,N}\bigg{(}\sum_{n=1}^{2}\kappa^{n}\mathcal{G}^{N}_{n}-\sum_{n=1}^{3}\kappa^{n}\mathcal{T}^{N}_{n}-2j^{N}_{x}\bigg{)}\,,$
(250)

where $\mathcal{G}^{M}_{x}$ represents the curvature perturbations

$\displaystyle\mathcal{G}^{M}_{1}$ $\displaystyle=\mathcal{I}^{M,\rho N,\sigma
P}_{1,1}\Lambda^{\rho N,\sigma P}_{1,1}+\mathcal{I}^{M,N,\rho\sigma
P}_{1,2}\Lambda^{N,\rho\sigma P}_{1,2}\,,$ (251)
$\displaystyle\mathcal{G}^{M}_{2}$ $\displaystyle=\mathcal{I}^{M,N,\rho
P,\sigma Q}_{2,1}\Lambda^{N,\rho P,\sigma
Q}_{2,1}+\mathcal{I}^{M,N,P,\rho\sigma Q}_{2,2}\Lambda^{N,P,\rho\sigma
Q}_{2,2}\,,$

and

$\displaystyle\mathcal{T}^{M}_{1}=\Theta^{M}_{1}\,,\qquad\mathcal{T}^{M}_{2}=\mathcal{I}_{\rm
EM}^{M,N}\Theta^{N}_{2}\,,\qquad\mathcal{T}^{M}_{3}=\mathcal{I}_{\rm
EM}^{M,N,P}\Theta^{N,P}_{3}\,.$ (252)

The explicit structure of $\Lambda$ and $\Theta$ are given by

$\displaystyle\Lambda^{\rho N,\sigma P}_{1,1}$
$\displaystyle=\partial_{\rho}h^{N}_{x}\partial_{\sigma}h^{P}_{x}+\frac{\hbar}{i}\partial^{x}_{\rho}\partial^{y}_{\sigma}\hat{\psi}_{xy}^{NP}\Big{|}_{y\to
x}\,,$ (253) $\displaystyle\Lambda^{N,\rho\sigma P}_{1,2}$
$\displaystyle=h^{N}_{x}\partial_{\rho}\partial_{\sigma}h^{P}_{x}+\frac{\hbar}{i}\partial^{y}_{\rho}\partial^{y}_{\sigma}\hat{\psi}_{xy}^{NP}\Big{|}_{y\to
x}\,,$ $\displaystyle\Lambda^{N,\rho P,\sigma Q}_{2,1}$
$\displaystyle=h^{N}_{x}\partial_{\rho}h^{P}_{x}\partial_{\sigma}h^{Q}_{x}+\frac{\hbar}{i}\Big{[}\partial_{\rho}^{x}\partial_{\sigma}^{y}\hat{\psi}_{xy}^{PQ}h^{N}_{x}+\partial_{\rho}^{y}\hat{\psi}^{NP}_{xy}\partial^{x}_{\sigma}h_{x}^{Q}+\partial_{\sigma}^{y}\hat{\psi}^{NQ}_{xy}\partial^{x}_{\rho}h_{x}^{P}\Big{]}_{y\to
x}$
$\displaystyle\quad+\left(\frac{\hbar}{i}\right)^{2}\partial_{\rho}^{y}\partial_{\sigma}^{z}\hat{\psi}_{xyz}^{NPQ}\Big{|}_{y,z\to
x}$ $\displaystyle\Lambda^{N,P,\rho\sigma Q}_{2,2}$
$\displaystyle=h^{N}_{x}h^{P}_{x}\partial_{\rho}\partial_{\sigma}h^{Q}_{x}+\frac{\hbar}{i}\Big{[}\hat{\psi}_{xy}^{NP}\partial_{\rho}^{x}\partial_{\sigma}^{x}h_{x}^{Q}+h^{N}_{x}\partial_{\rho}^{y}\partial_{\sigma}^{y}\hat{\psi}_{xy}^{PQ}+h^{P}_{x}\partial_{\rho}^{x}\partial_{\sigma}^{x}\hat{\psi}_{xy}^{QN}\Big{]}_{y\to
x}$
$\displaystyle\quad+\left(\frac{\hbar}{i}\right)^{2}\partial_{\rho}^{z}\partial_{\sigma}^{z}\hat{\psi}_{xyz}^{NPQ}\Big{|}_{y,z\to
x}$

where

$\displaystyle\Theta^{\mu\nu}_{1}$
$\displaystyle=\sum_{I=1}^{2}\Bigg{[}\partial_{\mu}\varphi^{I}_{x}\partial_{\nu}\varphi^{I}_{x}+\frac{m_{I}^{2}}{2}\eta_{\mu\nu}\varphi^{I}_{x}\varphi^{I}_{x}+\frac{\hbar}{i}\bigg{(}\partial^{x}_{\mu}\partial^{y}_{\nu}+\frac{m_{I}^{2}}{2}\eta_{\mu\nu}\bigg{)}\hat{\psi}^{I,I}_{x,y}\Big{|}_{y\to
x}\Bigg{]}\,,$ (254) $\displaystyle\Theta^{N}_{2}$
$\displaystyle=\sum_{I=1}^{2}m^{2}_{I}\left[h^{N}_{x}(\varphi^{I}_{x})^{2}+\frac{\hbar}{i}\Big{(}2\hat{\psi}^{N,I}_{x,x}\varphi^{I}_{x}+h^{N}_{x}\hat{\psi}^{I,I}_{x,x}\Big{)}+\left(\frac{\hbar}{i}\right)^{2}\hat{\psi}^{N,I,I}_{x,x,x}\right]\,,$
$\displaystyle\Theta^{N,P}_{3}$
$\displaystyle=\sum_{I=1}^{2}m^{2}_{I}\left[h^{N}_{x}h^{P}_{x}(\varphi^{I}_{x})^{2}+\frac{\hbar}{i}\Big{(}\hat{\psi}_{x,x}^{N,P}(\varphi^{I}_{x})^{2}+h^{(N}_{x}h^{P)}_{x}\hat{\psi}^{I,I}_{x,x}+4h^{(N}_{x}\hat{\psi}^{P),I}_{x,x}\varphi^{I}_{x}\Big{)}\right.$
$\displaystyle\qquad\qquad\quad\left.+\left(\frac{\hbar}{i}\right)^{2}\Big{(}3\hat{\psi}^{(N,P}_{x,x}\hat{\psi}^{I),I}_{x,x}+2\hat{\psi}^{N,P,I}_{x,x,x}\varphi^{I}_{x}+2h^{(N}_{x}\hat{\psi}^{P),I,I}_{x,x,x}\Big{)}\right]\,.$

###  B.3 DS equations for the descendant fields

To solve the DS equations, we need to derive the equations for the descendant
fields. It is straightforward to achieve by acting the functional derivative
$\frac{\delta}{\delta\mathbf{j}^{A}_{x}}$ on the DS equations for $h^{\mu\nu}$
and $\varphi^{I}$.

Let us consider the scalar field sector. The DS equation for the first
descendant fields $\psi^{I,A}$ are given by

$\displaystyle\left(\square-m_{I}^{2}\right)\psi^{IA}_{xy}$
$\displaystyle=-\hat{\delta}_{xy}\delta^{IA}+\kappa\Delta^{A}_{y}\big{[}\Omega^{I}_{1}\big{]}+\kappa^{2}\mathcal{I}^{M,N}_{\varphi,I}\Delta^{A}_{y}\big{[}\Omega^{IMN}_{2}\big{]}+\kappa^{3}\mathcal{I}^{M,N,P}_{\varphi,I}\Delta^{A}_{y}\big{[}\Omega^{IMNP}_{3}\big{]}\,,$
(255)

where

$\displaystyle\Delta^{A}_{y}\big{[}\Omega^{I}_{1}\big{]}$
$\displaystyle=\partial^{x}_{\kappa}\Big{(}\psi^{\kappa\lambda
A}_{xy}\partial_{\lambda}\varphi_{x}^{I}+h^{\kappa\lambda}_{x}\partial^{x}_{\lambda}\psi^{IA}_{xy}\Big{)}-\frac{m^{2}_{I}}{2}\Big{(}\psi^{\lambda\lambda
A}_{xy}\varphi^{I}_{x}+h_{x}^{\lambda\lambda}\psi^{IA}_{xy}\Big{)}$ (256)
$\displaystyle\quad+\bigg{[}\left(\partial_{\kappa}^{x}\partial_{\lambda}^{x}+\partial_{\kappa}^{z}\partial_{\lambda}^{x}\right)\hat{\psi}_{xz|y}^{I\kappa\lambda|A}-\frac{m_{I}^{2}}{2}\hat{\psi}^{\lambda\lambda
I|A}_{xz|y}\bigg{]}_{z\to x}\,,$
$\displaystyle\Delta^{A}_{y}\big{[}\Omega^{IMN}_{2}\big{]}$
$\displaystyle=2h^{(M}_{x}\psi^{N)A}_{xy}\varphi^{I}_{x}+h^{M}_{x}h^{N}_{x}\psi^{I,A}_{x,y}$
$\displaystyle\quad+\Big{[}\hat{\psi}^{MN|A}_{xz|y}\varphi^{I}_{x}+\hat{\psi}^{MN}_{xz}\psi^{IA}_{xy}+2\psi^{(M|A|}_{xy}\hat{\psi}^{N)I}_{xz}+2h^{(M}_{x}\hat{\psi}^{N)I|A}_{xz|y}\Big{]}_{z\to
x}\,,$ $\displaystyle\Delta^{A}_{y}\big{[}\Omega^{IMNP}_{3}\big{]}$
$\displaystyle=3h^{(M}_{x}h^{N}_{x}\psi^{P)A}_{xy}\varphi^{I}_{x}+h^{M}_{x}h^{N}_{x}h^{P}_{x}\psi^{IA}_{xy}$
$\displaystyle\quad+3\Big{[}\psi^{(M|A|}_{xy}\hat{\psi}^{NP)}_{xz}\varphi^{I}_{x}+h^{(M}_{x}\hat{\psi}^{NP)|A}_{xz|y}\varphi^{I}_{x}+h^{(M}_{x}\hat{\psi}^{NP)}_{xz}\psi^{IA}_{xy}$
$\displaystyle\qquad\quad+2\psi^{(M|A|}_{xy}h^{N}_{x}\hat{\psi}^{P)I}_{xz}+h^{(M}_{x}h^{N}_{x}\hat{\psi}^{P)I|A}_{xz|y}\Big{]}_{z\to
x}\,,$

Here we denote the functional derivative to the first descendant fields as

$\displaystyle\hat{\psi}^{AB|C}_{xy|z}:=\frac{\delta\hat{\psi}^{AB}_{xy}}{\delta\mathbf{j}^{C}_{z}}=\frac{1}{2}\Big{(}\psi_{xyz}^{ABC}+\psi_{yxz}^{BAC}\Big{)}\,.$
(257)

Next, the DS equation for second descendant fields $\psi^{I,AB}_{x,y,z}$

$\displaystyle\left(\square-
m_{I}^{2}\right)\psi^{IAB}_{xyz}=\kappa\Delta^{A,B}_{y,z}\big{[}\Omega^{I}_{1}\big{]}+\kappa^{2}\mathcal{I}^{M,N}_{\varphi,I}\Delta^{A,B}_{y,z}\big{[}\Omega^{IMN}_{2}\big{]}+\kappa^{3}\mathcal{I}^{M,N,P}_{\varphi,I}\Delta^{A,B}_{y,z}\big{[}\Omega^{IMNP}_{3}\big{]}\,,$
(258)

where

$\displaystyle\Delta^{A,B}_{y,z}\big{[}\Omega^{I}_{1}\big{]}$
$\displaystyle=\partial_{\kappa}\left(\psi^{\kappa\lambda
AB}_{xyz}\partial_{\lambda}\varphi^{I}_{x}+\psi_{xy}^{\kappa\lambda
A}\partial_{\lambda}\psi^{IB}_{xz}+\psi_{xz}^{\kappa\lambda
B}\partial_{\lambda}\psi^{IA}_{xy}+h_{x}^{\kappa\lambda}\partial_{\lambda}\psi^{IAB}_{xyz}\right)$
(259) $\displaystyle\quad-\frac{m^{2}_{I}}{2}\Big{(}\psi^{\lambda\lambda
AB}_{xyz}\varphi^{I}_{x}+\psi^{\lambda\lambda
A}_{xy}\psi^{IB}_{xz}+\psi_{xz}^{\lambda\lambda
B}\psi^{IA}_{xy}+h_{x}^{\lambda\lambda}\psi^{IAB}_{xyz}\Big{)}\,,$
$\displaystyle\Delta^{A,B}_{y,z}\big{[}\Omega^{IMN}_{2}\big{]}$
$\displaystyle=2\Big{(}\psi^{(M|B|}_{x,z}\psi^{N)A}_{xy}+h^{(M}_{x}\psi^{N)AB}_{xyz}\Big{)}\varphi^{I}_{x}+2h^{(M}_{x}\psi^{N)A}_{xy}\psi^{IB}_{xz}$
$\displaystyle\quad+2h^{(M}_{x}\psi^{N)B}_{xz}\psi^{IA}_{xy}+h^{M}_{x}h^{N}_{x}\psi^{IAB}_{xyz}\,,$
$\displaystyle\Delta^{A,B}_{y,z}\big{[}\Omega^{IMNP}_{3}\big{]}$
$\displaystyle=6h^{(M}_{x}\psi^{N|B|}_{xz}\psi^{P)A}_{xy}\varphi^{I}_{x}+3h^{(M}_{x}h^{N}_{x}\psi^{P)AB}_{xyz}\varphi^{I}_{x}$
$\displaystyle\quad+3h^{(M}_{x}h^{N}_{x}\psi^{P)A}_{xy}\psi^{IB}_{xz}+3h^{(M}_{x}h^{N}_{x}\psi^{P)B}_{xz}\psi^{IA}_{xy}+h^{M}_{x}h^{N}_{x}h^{P}_{x}\psi^{IAB}_{xyz}\,.$

We then consider the graviton sector. The first descendent equation for the
EoM of $h^{\mu\nu}$ is given by a functional derivative with respect to
$\mathbf{j}^{A}$ to the EoM

$\Box\psi_{xy}^{MA}=2P^{M,N}\Big{(}\sum_{n=1}^{2}\kappa^{n}\Delta^{A}_{y}\big{[}\mathcal{G}^{N}_{n}\big{]}-\sum_{n=1}^{3}\kappa^{n}\Delta^{A}_{y}\big{[}{\mathcal{T}}^{N}_{n}\big{]}-2\hat{\delta}_{x,y}\delta^{N,A}\Big{)}\,,$
(260)

where the $\Delta$ operation on $\mathcal{G}$

$\displaystyle\Delta^{A}_{y}\big{[}\mathcal{G}^{M}_{1}\big{]}$
$\displaystyle=\mathcal{I}^{M,\rho N,\sigma
P}_{1,1}\Delta^{A}_{y}\big{[}\Lambda^{\rho N,\sigma
P}_{1,1}\big{]}+\mathcal{I}^{M,N,\rho\sigma
P}_{1,2}\Delta^{A}_{y}\big{[}\Lambda^{N,\rho\sigma P}_{1,2}\big{]}\,,$ (261)
$\displaystyle\Delta^{A}_{y}\big{[}\mathcal{G}^{M}_{2}\big{]}$
$\displaystyle=\mathcal{I}^{M,N,\rho P,\sigma
Q}_{2,1}\Delta^{A}_{y}\big{[}\Lambda^{N,\rho P,\sigma
Q}_{2,1}\big{]}+\mathcal{I}^{M,N,P,\rho\sigma
Q}_{2,2}\Delta^{A}_{y}\big{[}\Lambda^{N,P,\rho\sigma Q}_{2,2}\big{]}\,,$

and $\Delta$ operation on $\mathcal{T}$

$\displaystyle\Delta^{A}_{y}\big{[}\mathcal{T}^{M}_{1}\big{]}=\Delta^{A}_{y}\big{[}\Theta^{M}_{1}\big{]}\,,$
(262)
$\displaystyle\Delta^{A}_{y}\big{[}\mathcal{T}^{M}_{2}\big{]}=\mathcal{I}_{\rm
EM}^{M,N}\Delta^{A}_{y}\big{[}\Theta^{N}_{2}\big{]}\,,$
$\displaystyle\Delta^{A}_{y}\big{[}\mathcal{T}^{M}_{3}\big{]}=\mathcal{I}_{\rm
EM}^{M,N,P}\Delta^{A}_{y}\big{[}\Theta^{N,P}_{3}\big{]}\,.$

with

$\displaystyle\Delta^{A}_{y}\big{[}\Lambda^{\rho N,\sigma P}_{1,1}\big{]}$
$\displaystyle=\partial^{x}_{\rho}\psi^{NA}_{xy}\partial^{x}_{\sigma}h^{P}_{xy}+\partial^{x}_{\rho}h^{N}_{x}\partial^{x}_{\sigma}\psi^{PA}_{xy}+\partial^{x}_{\rho}\partial^{z}_{\sigma}\hat{\psi}_{xz|y}^{NP|A}\Big{|}_{z\to
x}\,,$ (263) $\displaystyle\Delta^{A}_{y}\big{[}\Lambda^{N,\rho\sigma
P}_{1,2}\big{]}$
$\displaystyle=\psi^{NA}_{xy}\partial_{\rho}\partial_{\sigma}h^{P}_{x}+h^{N}_{x}\partial^{x}_{\rho}\partial^{x}_{\sigma}\psi^{PA}_{xy}+\partial^{z}_{\rho}\partial^{z}_{\sigma}\hat{\psi}_{xz|y}^{NP|A}\Big{|}_{z\to
x}\,,$ $\displaystyle\Delta^{A}_{y}\big{[}\Lambda^{N,\rho P,\sigma
Q}_{2,1}\big{]}$
$\displaystyle=\psi^{NA}_{xy}\partial_{\rho}h^{P}_{x}\partial_{\sigma}h^{Q}_{x}+h^{N}_{x}\partial^{x}_{\rho}\psi^{PA}_{xy}\partial^{x}_{\sigma}h^{Q}_{x}+h^{N}_{x}\partial^{x}_{\rho}h^{P}_{x}\partial^{x}_{\sigma}\psi^{QA}_{xy}$
$\displaystyle\quad+\Big{[}\partial_{\rho}^{x}\partial_{\sigma}^{z}\hat{\psi}_{xz|y}^{PQ|A}h^{N}_{x}+\partial_{\rho}^{x}\partial_{\sigma}^{z}\hat{\psi}_{xz}^{PQ}\psi^{NA}_{xy}+\partial_{\rho}^{y}\hat{\psi}^{NP|A}_{xz|y}\partial^{x}_{\sigma}h_{x}^{Q}$
$\displaystyle\qquad\quad+\partial_{\rho}^{y}\hat{\psi}^{NP}_{xz}\partial^{x}_{\sigma}\psi_{xy}^{QA}+\partial_{\sigma}^{y}\hat{\psi}^{NQ|A}_{xz|y}\partial^{x}_{\rho}h_{x}^{P}+\partial_{\sigma}^{y}\hat{\psi}^{NQ}_{xy}\partial^{x}_{\rho}\psi_{xy}^{PA}\Big{]}_{z\to
x}\,,$ $\displaystyle\Delta^{A}_{y}\big{[}\Lambda^{N,P,\rho\sigma
Q}_{2,2}\big{]}$
$\displaystyle=\psi^{NA}_{xy}h^{P}_{x}\partial_{\rho}\partial_{\sigma}h^{Q}_{x}+h^{N}_{x}\psi^{PA}_{xy}\partial_{\rho}\partial_{\sigma}h^{Q}_{x}+h^{N}_{x}h^{P}_{x}\partial_{\rho}\partial_{\sigma}\psi^{QA}_{xy}$
$\displaystyle\quad+\Big{[}\hat{\psi}_{xzy}^{NPA}\partial_{\rho}^{x}\partial_{\sigma}^{x}h_{x}^{Q}+\hat{\psi}_{xz}^{NP}\partial_{\rho}^{x}\partial_{\sigma}^{x}\psi_{xy}^{QA}+\psi^{NA}_{xy}\partial_{\rho}^{z}\partial_{\sigma}^{z}\hat{\psi}_{xz}^{PQ}$
$\displaystyle\qquad\quad+h^{N}_{x}\partial_{\rho}^{z}\partial_{\sigma}^{z}\hat{\psi}_{xz|y}^{PQ|A}+\psi^{PA}_{xy}\partial_{\rho}^{x}\partial_{\sigma}^{x}\hat{\psi}_{xz}^{QN}+h^{P}_{x}\partial_{\rho}^{x}\partial_{\sigma}^{x}\hat{\psi}_{xz|y}^{QN|A}\Big{]}_{z\to
x}\,,$

and

$\displaystyle\Delta^{A}_{y}\big{[}\Theta^{\kappa\lambda}_{1}\big{]}$
$\displaystyle=\partial^{x}_{\kappa}\psi^{IA}_{xy}\partial^{x}_{\lambda}\varphi^{I}_{x}+\partial^{x}_{\kappa}\varphi^{I}_{x}\partial^{x}_{\lambda}\psi^{IA}_{xy}+m_{I}^{2}\eta_{\kappa\lambda}\psi^{IA}_{xy}\varphi^{I}_{x}+\Big{(}\partial^{x}_{\kappa}\partial^{y}_{\lambda}+\frac{m_{I}^{2}}{2}\eta_{\kappa\lambda}\Big{)}\hat{\psi}^{II|A}_{xz|y}\Big{|}_{z\to
x}\,,$ (264) $\displaystyle\Delta^{A}_{y}\big{[}\Theta^{N,I}_{2}\big{]}$
$\displaystyle=\psi^{NA}_{xy}(\varphi^{I}_{x})^{2}+2h^{N}_{x}\varphi^{I}_{x}\psi^{IA}_{xy}$
$\displaystyle\quad+\Big{[}2\hat{\psi}^{NI|A}_{xx|y}\varphi^{I}_{x}+2\hat{\psi}^{NI}_{xx}\varphi^{IA}_{xy}+\psi^{NA}_{xy}\hat{\psi}^{II}_{xx}+h^{N}_{x}\hat{\psi}^{II|A}_{xx|y}\Big{]}\,,$
$\displaystyle\Delta^{A}_{y}\big{[}\Theta^{N,P,I}_{3}\big{]}$
$\displaystyle=2\psi^{(N|A|}_{xy}h^{P)}_{x}(\varphi^{I}_{x})^{2}+2h^{N}_{x}h^{P}_{x}\varphi^{I}_{x}\psi^{IA}_{xy}$
$\displaystyle\quad+\Big{[}\hat{\psi}_{xx|y}^{NP|A}(\varphi^{I}_{x})^{2}+2\hat{\psi}_{xx}^{N,P}\varphi^{I}_{x}\psi^{IA}_{xy}+2\psi^{(N|A|}_{xy}h^{P)}_{x}\hat{\psi}^{II}_{xx}+h^{N}_{x}h^{P}_{x}\hat{\psi}^{II|A}_{xx|y}$
$\displaystyle\qquad\quad+4\psi^{(N|A|}_{xy}\hat{\psi}^{P)I}_{xx}\varphi^{I}_{x}+4h^{(N}_{x}\hat{\psi}^{P)I|A}_{xx|y}\varphi^{I}_{x}+4h^{(N}_{x}\hat{\psi}^{P)I}_{xx}\psi^{IA}_{x}\,.\Big{]}$

Next, the DS equation for the second descendent field up to $\hbar^{2}$ order
is given by

$\Box\psi_{xy}^{MAB}=2P^{M,N}\Big{(}\sum_{n=1}^{2}\kappa^{n}\Delta^{A,B}_{y,z}\big{[}\mathcal{G}^{N}_{n}\big{]}-\sum_{n=1}^{3}\kappa^{n}\Delta^{A,B}_{y,z}\big{[}{\mathcal{T}}^{N}_{n}\big{]}\Big{)}\,,$
(265)

where the $\Delta$ operations on $\mathcal{G}$

$\displaystyle\Delta^{A,B}_{y,z}\big{[}\mathcal{G}^{M}_{1}\big{]}$
$\displaystyle=\mathcal{I}^{M,\rho N,\sigma
P}_{1,1}\Delta^{A,B}_{y,z}\big{[}\Lambda^{\rho N,\sigma
P}_{1,1}\big{]}+\mathcal{I}^{M,N,\rho\sigma
P}_{1,2}\Delta^{A,B}_{y,z}\big{[}\Lambda^{N,\rho\sigma P}_{1,2}\big{]}\,,$
(266) $\displaystyle\Delta^{A,B}_{y,z}\big{[}\mathcal{G}^{M}_{2}\big{]}$
$\displaystyle=\mathcal{I}^{M,N,\rho P,\sigma
Q}_{2,1}\Delta^{A,B}_{y,z}\big{[}\Lambda^{N,\rho P,\sigma
Q}_{2,1}\big{]}+\mathcal{I}^{M,N,P,\rho\sigma
Q}_{2,2}\Delta^{A,B}_{y,z}\big{[}\Lambda^{N,P,\rho\sigma Q}_{2,2}\big{]}\,,$

and $\Delta$ operation on $\mathcal{T}$

$\displaystyle\Delta^{A,B}_{y,z}\big{[}\mathcal{T}^{M}_{1}\big{]}=\Delta^{A,B}_{y,z}\big{[}\Theta^{M}_{1}\big{]}\,,$
(267)
$\displaystyle\Delta^{A,B}_{y,z}\big{[}\mathcal{T}^{M}_{2}\big{]}=\mathcal{I}_{\rm
EM}^{M,N}\Delta^{A,B}_{y,z}\big{[}\Theta^{N}_{2}\big{]}\,,$
$\displaystyle\Delta^{A,B}_{y,z}\big{[}\mathcal{T}^{M}_{3}\big{]}=\mathcal{I}_{\rm
EM}^{M,N,P}\Delta^{A,B}_{y,z}\big{[}\Theta^{N,P}_{3}\big{]}\,.$

with

$\displaystyle\Delta^{A,B}_{y,z}\big{[}\Lambda^{\rho N,\sigma P}_{1,1}\big{]}$
$\displaystyle=\partial_{\rho}^{x}\psi_{xyz}^{NAB}\partial_{\sigma}^{x}h_{x}^{P}+2\partial_{\rho}^{x}\psi^{N(A}_{xy}\partial_{\sigma}^{x}\psi_{xz}^{|P|B)}+\partial_{\rho}^{x}h_{x}^{N}\partial_{\sigma}^{x}\psi_{xyz}^{PAB}\,,$
(268) $\displaystyle\Delta^{A,B}_{y,z}\big{[}\Lambda^{N,\rho\sigma
P}_{1,2}\big{]}$
$\displaystyle=\psi^{NAB}_{xyz}\partial_{\rho}^{x}\partial_{\sigma}^{x}h_{x}^{P}+\psi^{NA}_{xy}\partial_{\rho}^{x}\partial_{\sigma}^{x}\psi_{xz}^{PB}+\psi^{NB}_{xz}\partial^{x}_{\rho}\partial^{x}_{\sigma}\psi^{PA}_{xy}+h_{x}^{N}\partial^{x}_{\rho}\partial^{x}_{\sigma}\psi^{NAB}_{xyz}\,,$
$\displaystyle\Delta^{A,B}_{y,z}\big{[}\Lambda^{N,\rho P,\sigma
Q}_{2,1}\big{]}$
$\displaystyle=\psi^{NAB}_{xyz}\partial_{\rho}^{x}h_{x}^{P}\partial_{\sigma}^{x}h_{x}^{Q}+h_{x}^{N}\partial_{\rho}^{x}\psi^{PAB}_{xyz}\partial_{\sigma}^{x}h_{x}^{Q}+h_{x}^{N}\partial_{\rho}^{x}h_{x}^{P}\partial_{\sigma}^{x}\psi^{QAB}_{xyz}$
$\displaystyle+\psi^{NA}_{xy}(\partial_{\rho}^{x}\psi_{xz}^{PB}\partial_{\sigma}^{x}h_{x}^{Q}+\partial_{\rho}^{x}h_{x}^{P}\partial_{\sigma}^{x}\psi^{QB}_{xz})+\psi^{NB}_{xz}(\partial_{\rho}^{x}\psi^{PA}_{xy}\partial^{x}_{\sigma}h_{x}^{Q}+\partial_{\rho}^{x}h_{x}^{P}\partial_{\sigma}^{x}\psi^{QA}_{xy})$
$\displaystyle+h_{x}^{N}(\partial_{\rho}^{x}\psi^{PA}_{xy}\partial^{x}_{\sigma}\psi^{QB}_{xz}+\partial^{x}_{\rho}\psi^{PB}_{xz}\partial_{\sigma}^{x}\psi^{QA}_{xy})\,,$
$\displaystyle\Delta^{A,B}_{y,z}\big{[}\Lambda^{N,P,\rho\sigma
Q}_{2,2}\big{]}$
$\displaystyle=2h_{x}^{(N}\psi^{P)AB}_{xyz}\partial^{x}_{\rho}\partial^{x}_{\sigma}h_{x}^{Q}+h_{x}^{N}h_{x}^{P}\partial_{\rho}^{x}\partial_{\sigma}^{x}\psi_{xyz}^{QAB}+(\psi^{NA}_{xy}\psi^{PB}_{xz}+\psi^{NB}_{xz}\psi^{PA}_{xy})\partial_{\rho}^{x}\partial_{\sigma}^{x}h_{x}^{Q}$
$\displaystyle+2h_{x}^{(N}\psi^{P)A}_{xy}\partial_{\rho}^{x}\partial_{\sigma}^{x}\psi^{QB}_{xz}+2h_{x}^{(N}\psi^{P)B}_{xz}\partial_{\rho}^{x}\partial_{\sigma}^{x}\psi^{QA}_{xy}\,,$

and

$\displaystyle\Delta^{A,B}_{y,z}\big{[}\Theta^{\kappa\lambda}_{1}\big{]}$
$\displaystyle=2\partial^{x}_{(\kappa}\psi^{IAB}_{xyz}\partial^{x}_{\lambda)}\varphi^{I}_{x}+2\partial^{x}_{(\kappa}\psi^{IA}_{xy}\partial_{\lambda)}^{x}\psi^{IB}_{xz}+m_{I}^{2}\eta_{\kappa\lambda}(\psi^{IAB}_{xyz}\varphi^{I}_{x}+\psi^{IA}_{xy}\psi^{IB}_{xz})\,,$
(269) $\displaystyle\Delta^{A,B}_{y,z}\big{[}\Theta^{N,I}_{2}\big{]}$
$\displaystyle=\psi^{NAB}_{xyz}(\varphi^{I}_{x})^{2}+2\psi^{NA}_{xy}\psi^{IB}_{xz}\varphi_{x}^{I}+2\psi_{xz}^{NB}\psi^{IA}_{xy}\varphi_{x}^{I}+2h_{x}^{N}\psi^{IAB}_{xyz}\varphi_{x}^{I}+2h_{x}^{N}\psi^{IA}_{xy}\psi^{IB}_{xz}\,,$
$\displaystyle\Delta^{A,B}_{y,z}\big{[}\Theta^{N,P,I}_{3}\big{]}$
$\displaystyle=2h^{(N}_{x}\psi^{P)AB}_{xyz}(\varphi_{x}^{I})^{2}+2\psi^{(N|A|}_{xy}\psi^{P)B}_{xz}(\varphi_{x}^{I})^{2}+2\varphi_{x}^{N}\psi^{PA}_{xy}\psi^{IB}_{xz}\varphi_{x}^{I}+4\psi^{(N|A|}_{xy}\psi^{I)B}_{xz}h_{x}^{P}\varphi_{x}^{I}$
$\displaystyle\quad+2\psi_{xz}^{PB}\psi^{IA}_{xy}h^{N}_{x}\varphi_{x}^{I}+2h_{x}^{N}h_{x}^{P}\psi^{IAB}_{xyz}\varphi_{x}^{I}+2h_{x}^{N}h_{x}^{P}\psi^{IA}_{xy}\psi^{IB}_{xz}.$

##  Appendix C Explicit Form of the Quantum Recursion Relations

Here we collect the $\Omega,\Lambda,\Theta$ currents for the quantum recursion
relation of scalar field (75) and graviton (77). The $\Omega$ currents are

$\displaystyle\Omega^{{\scriptscriptstyle(n)}}\big{|}^{I}_{\hat{\mathcal{P}}}$
$\displaystyle=k^{\kappa}_{\hat{\mathcal{P}}}\Big{\lceil}J^{\kappa\lambda},k^{\lambda}\Phi^{I}\Big{\rfloor}^{{\scriptscriptstyle(n)}}_{\hat{\mathcal{P}}}+\frac{m_{I}^{2}}{2}\Big{\lceil}J^{\lambda\lambda},\Phi^{I}\Big{\rfloor}^{{\scriptscriptstyle(n)}}_{\hat{\mathcal{P}}}$
(270)
$\displaystyle\quad+\int_{\ell}\bigg{[}\frac{k^{\kappa}_{\hat{\mathcal{P}}}}{2}\Big{(}k_{\hat{\mathcal{P}},\ell}^{\lambda}\Psi^{{\scriptscriptstyle(n-1)}}\big{|}_{\ell|\hat{\mathcal{P}}}^{I,\kappa\lambda}+\ell^{\lambda}\Psi^{{\scriptscriptstyle(n-1)}}\big{|}_{\ell|\hat{\mathcal{P}}}^{\kappa\lambda,I}\Big{)}+\frac{m_{I}^{2}}{2}\hat{\Psi}^{{\scriptscriptstyle(n-1)}}\big{|}^{I,\lambda\lambda}_{\ell|\hat{\mathcal{P}}}\bigg{]}\,,$
$\displaystyle\Omega^{{\scriptscriptstyle(n)}}\big{|}^{IMN}_{\hat{\mathcal{P}}}$
$\displaystyle=\Big{\lceil}J^{M}{,}J^{N}{,}\Phi^{I}\Big{\rfloor}^{{\scriptscriptstyle(n)}}_{\hat{\mathcal{P}}}+\int_{\ell}\Big{[}2\big{\lceil}J^{M}{,}\hat{\Psi}_{\ell}^{IN}\big{\rfloor}^{{\scriptscriptstyle(n-1)}}_{\hat{\mathcal{P}}}+\big{\lceil}\hat{\Psi}_{\ell}^{MN}{,}\Phi^{I}\big{\rfloor}^{{\scriptscriptstyle(n-1)}}_{\hat{\mathcal{P}}}\Big{]}$
$\displaystyle\quad+\int_{\ell_{1}\ell_{2}}\hat{\Psi}^{{\scriptscriptstyle(n-2)}}\big{|}_{\ell_{1}\ell_{2}|\hat{\mathcal{P}}}^{MNI}\,,$
$\displaystyle\Omega^{{\scriptscriptstyle(n)}}\big{|}^{IMNP}_{\hat{\mathcal{P}}}$
$\displaystyle=\Big{\lceil}J^{M}{,}J^{N}{,}J^{P}{,}\Phi^{I}\Big{\rfloor}^{{\scriptscriptstyle(n)}}_{\hat{\mathcal{P}}}$
$\displaystyle\quad+\int_{\ell}\left[3\big{\lceil}J^{(M}{,}J^{N}{,}\hat{\Psi}^{P),I}_{\ell}\big{\rfloor}^{{\scriptscriptstyle(n-1)}}_{\hat{\mathcal{P}}}+\frac{3}{2}\Big{\lceil}\Delta^{(P}_{\ell}\big{\lceil}J^{M}{,}J^{N)}\big{\rfloor}{,}\Phi^{I}\Big{\rfloor}^{{\scriptscriptstyle(n-1)}}_{\hat{\mathcal{P}}}\right]$
$\displaystyle\quad+\int_{\ell_{1}\ell_{2}}\left[\big{\lceil}\hat{\Psi}_{\ell_{1}\ell_{2}}^{MNP},\Phi^{I}\big{\rfloor}^{{\scriptscriptstyle(n-2)}}_{\hat{\mathcal{P}}}+3\big{\lceil}\hat{\Psi}_{\ell_{1}\ell_{2}}^{I(MN},J^{P)}\big{\rfloor}^{{\scriptscriptstyle(n-2)}}_{\hat{\mathcal{P}}}+3\big{\lceil}\Psi^{(MN}_{\ell_{2}},\hat{\Psi}_{\ell_{1}}^{P)I}\big{\rfloor}^{{\scriptscriptstyle(n-2)}}_{\hat{\mathcal{P}}}\right]\,.$

The $\Lambda$ currents are

$\displaystyle\Lambda^{\scriptscriptstyle(n)}_{1,1}\big{|}^{\rho N,\sigma
P}_{\hat{\mathcal{P}}}$
$\displaystyle=\big{\lceil}k^{\rho}J^{N},k^{\sigma}J^{P}\big{\rfloor}^{\scriptscriptstyle(n)}_{\hat{\mathcal{P}}}+\int_{\ell}k^{\rho}_{\ell}\ell^{\sigma}\Psi^{\scriptscriptstyle(n-1)}\big{|}_{\ell|\hat{\mathcal{P}}}^{N,P}\,,$
(271)
$\displaystyle\Lambda^{\scriptscriptstyle(n)}_{1,2}\big{|}^{N,\rho\sigma,P}_{\hat{\mathcal{P}}}$
$\displaystyle=\big{\lceil}J^{N},k^{\rho}k^{\sigma}J^{P}\big{\rfloor}^{\scriptscriptstyle(n)}_{\hat{\mathcal{P}}}+\frac{1}{2}\int_{\ell}\Big{(}k^{\rho}_{\ell}k^{\sigma}_{\ell}\Psi^{\scriptscriptstyle(n-1)}\big{|}^{P,N}_{\ell|\hat{\mathcal{P}}}+\ell^{\rho}\ell^{\sigma}\Psi^{\scriptscriptstyle(n-1)}\big{|}^{N,P}_{\ell|\hat{\mathcal{P}}}\Big{)}\,,$
$\displaystyle\Lambda^{\scriptscriptstyle(n)}_{2,1}\big{|}^{N,\rho P,\sigma
Q}_{\hat{\mathcal{P}}}$
$\displaystyle=\big{\lceil}J^{N},k^{\rho}J^{P},k^{\sigma}J^{Q}\big{\rfloor}^{\scriptscriptstyle(n)}_{\hat{\mathcal{P}}}+\frac{1}{2}\int_{\ell}\Delta^{N}_{\ell}\big{\lceil}k^{\rho}J^{P},k^{\sigma}J^{Q}\big{\rfloor}^{{\scriptscriptstyle(n-1)}}_{\hat{\mathcal{P}}}$
$\displaystyle+\int_{\ell}\ell^{\sigma}\Big{[}\big{\lceil}k^{\rho}J^{P},\Psi_{\ell}^{NQ}\big{\rfloor}^{\scriptscriptstyle(n-1)}_{\hat{\mathcal{P}}}+\big{\lceil}J^{N},k^{\rho}_{\ell}\Psi_{\ell}^{PQ}\big{\rfloor}^{\scriptscriptstyle(n-1)}_{\hat{\mathcal{P}}}\Big{]}$
$\displaystyle+\frac{1}{3}\int_{\ell_{1}\ell_{2}}\bigg{[}k^{\rho}_{\hat{\mathcal{P}},\ell_{12}}\ell_{1}^{\sigma}\Psi^{\scriptscriptstyle(n-2)}\big{|}_{\ell_{1}\ell_{2}|\hat{\mathcal{P}}}^{PQN}+\ell_{2}^{\rho}k^{\sigma}_{\hat{\mathcal{P}},\ell_{12}}\Psi^{\scriptscriptstyle(n-2)}\big{|}_{\ell_{1}\ell_{2}|\hat{\mathcal{P}}}^{QNP}+\ell_{1}^{\rho}\ell_{2}^{\sigma}\Psi^{\scriptscriptstyle(n-2)}\big{|}^{NPQ}_{\ell_{1}\ell_{2}|\hat{\mathcal{P}}}\bigg{]}\,,$
$\displaystyle\Lambda^{\scriptscriptstyle(n)}_{2,2}\big{|}^{N,P,\rho\sigma,Q}_{\hat{\mathcal{P}}}$
$\displaystyle=\big{\lceil}J^{N},J^{P},k^{\rho}k^{\sigma}J^{Q}\big{\rfloor}^{\scriptscriptstyle(n)}_{\hat{\mathcal{P}}}$
$\displaystyle+\int_{\ell}\Big{[}\big{\lceil}k^{\rho}k^{\sigma}J^{Q},\Psi_{\ell}^{NP}\big{\rfloor}^{\scriptscriptstyle(n-1)}_{\hat{\mathcal{P}}}+\big{\lceil}J^{N},k^{\rho}_{\ell}k^{\sigma}_{\ell}\Psi_{\ell}^{QP}+\ell^{\rho}\ell^{\sigma}\Psi_{\ell}^{PQ}\big{\rfloor}^{\scriptscriptstyle(n-1)}_{\hat{\mathcal{P}}}\Big{]}$
$\displaystyle+\frac{1}{3}\int_{\ell_{1}\ell_{2}}\bigg{[}\ell_{1}^{\rho}\ell_{1}^{\sigma}\Psi^{\scriptscriptstyle(n-2)}\big{|}_{\ell_{1}\ell_{2}|\hat{\mathcal{P}}}^{PQN}+k^{\rho}_{\hat{\mathcal{P}},\ell_{12}}k^{\sigma}_{\hat{\mathcal{P}},\ell_{12}}\Psi^{\scriptscriptstyle(n-2)}\big{|}_{\ell_{1}\ell_{2}|\hat{\mathcal{P}}}^{QNP}+\ell_{2}^{\rho}\ell_{2}^{\sigma}\Psi^{\scriptscriptstyle(n-2)}\big{|}_{\ell_{1}\ell_{2}|\hat{\mathcal{P}}}^{NPQ}\bigg{]}\,.$

The $\Theta$ currents are

$\displaystyle\Theta^{\scriptscriptstyle(n)}_{1}\big{|}^{\rho\sigma}_{\hat{\mathcal{P}}}$
$\displaystyle=\sum_{I=1}^{2}\bigg{[}\big{\lceil}k^{\rho}\Phi^{I},k^{\sigma}\Phi^{I}\big{\rfloor}^{{\scriptscriptstyle(n)}}_{\hat{\mathcal{P}}}-\frac{m_{I}^{2}}{2}\eta^{\rho\sigma}\big{[}\Phi^{I},\Phi^{I}\big{]}^{{\scriptscriptstyle(n)}}_{\hat{\mathcal{P}}}+\int_{\ell}\Big{(}k^{(\rho}_{\ell}\ell^{\sigma)}-\frac{m_{I}^{2}}{2}\eta^{\rho\sigma}\Big{)}\Psi^{{\scriptscriptstyle(n)}}\big{|}_{\ell|\hat{\mathcal{P}}}^{II}\bigg{]}\,,$
(272)
$\displaystyle\Theta^{\scriptscriptstyle(n)}_{2}\big{|}^{N}_{\hat{\mathcal{P}}}$
$\displaystyle=\sum_{I=1}^{2}m_{I}^{2}\Bigg{[}\big{\lceil}J^{N},\Phi^{I},\Phi^{I}\big{\rfloor}^{{\scriptscriptstyle(n)}}_{\hat{\mathcal{P}}}+\int_{\ell}\left(\big{\lceil}J^{N}{,}\Psi^{II}_{\ell}\big{\rfloor}^{{\scriptscriptstyle(n-1)}}_{\hat{\mathcal{P}}}+2\big{\lceil}\Phi^{I}{,}\hat{\Psi}^{IN}_{\ell}\big{\rfloor}^{{\scriptscriptstyle(n-1)}}_{\hat{\mathcal{P}}}\right)$
$\displaystyle\qquad\qquad+\int_{\ell_{1}\ell_{2}}\hat{\Psi}^{{\scriptscriptstyle(n-2)}}_{\hat{\mathcal{P}}}\big{|}^{NII}_{\ell_{1}\ell_{2}}\Bigg{]}\,,$
$\displaystyle\Theta^{\scriptscriptstyle(n)}_{3}\big{|}^{N,P}_{\hat{\mathcal{P}}}$
$\displaystyle=\sum_{I=1}^{2}m_{I}^{2}\Bigg{[}\big{\lceil}\Phi^{I}{,}\Phi^{I}{,}J^{N}{,}J^{P}\big{\rfloor}^{{\scriptscriptstyle(n)}}_{\hat{\mathcal{P}}}+\int_{\ell}\Big{(}\big{\lceil}\Phi^{I}{,}\Phi^{I}{,}\Psi_{\ell}^{NP}\big{\rfloor}^{{\scriptscriptstyle(n-1)}}_{\hat{\mathcal{P}}}+\big{\lceil}J^{N}{,}J^{P}{,}\Psi^{II}_{\ell}\big{\rfloor}^{{\scriptscriptstyle(n-1)}}_{\hat{\mathcal{P}}}\Big{)}$
$\displaystyle\qquad\qquad+\int_{\ell}\left(\Big{\lceil}\Delta^{I}_{\ell}\big{\lceil}J^{N}{,}J^{P}\big{\rfloor},\Phi^{I}\Big{\rfloor}^{{\scriptscriptstyle(n-1)}}_{\hat{\mathcal{P}}}+\Big{\lceil}\Delta^{P}_{\ell}\big{[}\Phi^{I},\Phi^{I}\big{]},J^{N}\Big{\rfloor}^{{\scriptscriptstyle(n-1)}}_{\hat{\mathcal{P}}}\right)$
$\displaystyle\qquad\qquad+\int_{\ell_{1}\ell_{2}}\bigg{[}\big{\lceil}\Psi_{\ell_{1}}^{NP},\Psi_{\ell_{2}}^{II}\big{\rfloor}^{{\scriptscriptstyle(n-2)}}_{\hat{\mathcal{P}}}+2\big{\lceil}\hat{\Psi}_{\ell_{1}}^{PI},\hat{\Psi}_{\ell_{2}}^{NI}\big{\rfloor}^{{\scriptscriptstyle(n-2)}}_{\hat{\mathcal{P}}}$
$\displaystyle\qquad\qquad\qquad\quad+2\big{\lceil}\Phi^{I},\hat{\Psi}^{NPI}_{\ell_{1}\ell_{2}}\big{\rfloor}^{{\scriptscriptstyle(n-2)}}_{\hat{\mathcal{P}}}+2\big{\lceil}J^{N},\hat{\Psi}^{PII}_{\ell_{1}\ell_{2}}\big{\rfloor}^{{\scriptscriptstyle(n-2)}}_{\hat{\mathcal{P}}}\bigg{]}\Bigg{]}\,.$

##  Appendix D IBP reduction of the Two loop integrals

In Section 6, the classical amplitudes in each sector are represented by
linear combinations of the master integrals through IBP reduction. Here, we
denote the explicit forms of the IBP reduction of the two-loop integrands.

###  D.1 $\Box\\!\Box$ class

The IBP reduction for each $I^{\rm class}_{i}$

####  D.1.1 III sector

The coefficients are

$\displaystyle c^{\text{III}}_{1}$
$\displaystyle=\frac{m_{1}^{2}m_{2}^{2}\left(2\sigma^{2}-1\right)}{3\left(\sigma^{2}-1\right)^{2}\epsilon^{3}(2\epsilon-1)}\Big{[}c^{\rm
III}_{1,1}(m_{1}^{2}+m_{2}^{2})+c^{\rm III}_{1,2}m_{1}m_{2}\Big{]}\,,$ (273)
$\displaystyle c^{\text{III}}_{2}$
$\displaystyle=\frac{8m_{1}^{2}m_{2}^{2}}{\left(\sigma^{2}-1\right)^{3/2}\epsilon^{4}(2\epsilon-1)}\Big{[}c^{\rm
III}_{2,1}(m_{1}^{2}+m_{2}^{2})+c^{\rm III}_{2,2}m_{1}m_{2}.\Big{]}\,,$
$\displaystyle c^{\text{III}}_{3}$
$\displaystyle=\frac{m_{1}^{2}m_{2}^{2}\left(2\sigma^{2}-1\right)^{3}\left(\sigma\left(m_{1}^{2}+m_{2}^{2}\right)+2m_{1}m_{2}\right)}{\left(\sigma^{2}-1\right)^{3/2}\epsilon^{3}}$
$\displaystyle c^{\text{III}}_{4}$
$\displaystyle=\frac{2m_{1}^{2}m_{2}^{2}\left(2\sigma^{2}-1\right)}{3\left(\sigma.^{2}-1\right)^{2}\epsilon^{3}(2\epsilon-1)}\Big{[}c^{\rm
III}_{4,1}(m_{1}^{2}+m_{2}^{2})+c^{\rm III}_{4,2}m_{1}m_{2}\Big{]}\,,$
$\displaystyle c^{\text{III}}_{5}$
$\displaystyle=-\frac{2m_{1}^{2}m_{2}^{2}\left(2\sigma^{2}-1\right)}{3\left(\sigma^{2}-1\right)^{3/2}\epsilon^{3}(2\epsilon-1)}\Big{[}c^{\rm
III}_{5,1}(m_{1}^{2}+m_{2}^{2})+c^{\rm III}_{5,2}m_{1}m_{2}\Big{]}\,,$
$\displaystyle c^{\text{III}}_{6}$
$\displaystyle=\frac{2m_{1}^{2}m_{2}^{2}}{3\left(\sigma^{2}-1\right)^{2}\epsilon^{3}(6\epsilon-5)(6\epsilon-1)}\Big{[}c^{\rm
III}_{6,1}(m_{1}^{2}+m_{2}^{2})+c^{\rm III}_{6,2}m_{1}m_{2}\Big{]}\,,$
$\displaystyle c^{\text{III}}_{7}$
$\displaystyle=8m_{1}^{2}m_{2}^{2}\left(2\sigma^{2}-1\right)^{3}\left(\frac{m_{1}^{2}+2m_{1}m_{2}\sigma+m_{2}^{2}}{3\left(\sigma^{2}-1\right)^{2}\epsilon^{3}}-\frac{m_{1}^{2}m_{2}^{2}}{\left(\sigma^{2}-1\right)\epsilon^{4}q^{2}}\right)\,,$
$\displaystyle c^{\text{III}}_{8}$ $\displaystyle=0\,,$ $\displaystyle
c^{\text{III}}_{9}$ $\displaystyle=0\,,$ $\displaystyle c^{\text{III}}_{10}$
$\displaystyle=-\frac{16m_{1}^{3}m_{2}^{3}(2\sigma^{2}-1)(m_{1}+m_{2})\left((1-2\sigma^{2})^{2}\epsilon-(\sigma-1)(4\sigma^{3}-2\sigma+1)\right)}{(\sigma-1)^{\frac{3}{2}}\epsilon^{4}q}\,,$

where

$\displaystyle c^{\rm III}_{1,1}$
$\displaystyle=6\epsilon^{2}+12\sigma^{6}(\epsilon-1)(2\epsilon-3)+16\sigma^{4}(5\epsilon-4)+\sigma^{2}(34-\epsilon(18\epsilon+35))+11\epsilon-7\,,$
(274) $\displaystyle c^{\rm III}_{1,2}$
$\displaystyle=2\sigma(2\epsilon-1)\left(-20\sigma^{4}+26\sigma^{2}-5+6\left(1-2\sigma^{2}\right)^{2}\epsilon\right)\,,$
$\displaystyle c^{\rm III}_{2,1}$
$\displaystyle=\sigma\left(2(\sigma^{2}-1)+2(2\sigma^{2}-1)^{3}\epsilon^{3}-\left(1-2\sigma^{2}\right)^{2}\left(10\sigma^{2}-9\right)\epsilon^{2}+4\sigma^{2}\left(6\sigma^{4}-11\sigma^{2}+5\right)\epsilon\right)\,,$
$\displaystyle c^{\rm III}_{2,2}$
$\displaystyle=2\left(2\sigma^{2}-1\right)(2\epsilon-1)\left(\sigma^{4}\left(4\epsilon^{2}-3\right)-2\sigma^{2}\left(2\epsilon^{2}+\epsilon-2\right)+\epsilon^{2}+2\epsilon-1\right)\,,$
$\displaystyle c^{\rm III}_{4,1}$
$\displaystyle=6\epsilon^{2}+12\sigma^{6}(\epsilon-1)(2\epsilon-3)+16\sigma^{4}(5\epsilon-4)+\sigma^{2}(34-\epsilon(18\epsilon+35))+11\epsilon-7\,,$
$\displaystyle c^{\rm III}_{4,2}$
$\displaystyle=2\sigma(2\epsilon-1)\left(-20\sigma^{4}+26\sigma^{2}-5+6\left(1-2\sigma^{2}\right)^{2}\epsilon\right)\,,$
$\displaystyle c^{\rm III}_{5,1}$
$\displaystyle=4\sigma\left(\sigma^{4}(4(\epsilon-1)\epsilon+3)+\sigma^{2}(-4(\epsilon-2)\epsilon-5)+(\epsilon-3)\epsilon\right)+5\sigma\,,$
$\displaystyle c^{\rm III}_{5,2}$
$\displaystyle=2\left(20\sigma^{2}+4\sigma^{4}(4\epsilon(\epsilon+1)-5)-8\sigma^{2}\epsilon(2\epsilon+3)+4\epsilon(\epsilon+3)-3\right)\,,$
$\displaystyle c^{\rm III}_{6,1}$
$\displaystyle=-108\epsilon^{3}-144\epsilon^{2}+453\epsilon-191+3\sigma^{2}(\epsilon(36\epsilon(5\epsilon+7)-935)+476)$
$\displaystyle\quad-6\sigma^{4}(3\epsilon(12\epsilon(3\epsilon+11)-343)+508)+4\sigma^{6}(664-3\epsilon(12\epsilon(3\epsilon-29)+533))$
$\displaystyle\quad+24\sigma^{8}(\epsilon-1)(6\epsilon-7)(6\epsilon-5)\,,$
$\displaystyle c^{\rm III}_{6,2}$
$\displaystyle=2\sigma(6\epsilon-5)\Big{(}8\sigma^{6}(6\epsilon-5)(6\epsilon-1)-12\sigma^{2}(4\sigma^{2}-1)$
$\displaystyle\qquad\qquad\qquad-36\sigma^{2}(2\sigma^{2}-1)\epsilon(6\epsilon-7)-36(\epsilon-1)\epsilon+19\Big{)}\,.$

####  D.1.2 $\overline{\rm III}$

The coefficients are

$\displaystyle c^{\overline{\rm III}}_{1}$
$\displaystyle=\frac{m_{1}^{2}m_{2}^{2}\left(2\sigma^{2}-1\right)}{3\left(\sigma^{2}-1\right)^{2}\epsilon^{3}(2\epsilon-1)}\Big{[}c^{\overline{\rm
III}}_{1,1}(m_{1}^{2}+m_{2}^{2})+c^{\overline{\rm
III}}_{1,2}m_{1}m_{2}\Big{]}\,,$ (275) $\displaystyle c^{\overline{\rm
III}}_{2}$
$\displaystyle=-\frac{8m_{1}^{2}m_{2}^{2}}{\left(\sigma^{2}-1\right)^{3/2}\epsilon^{4}(2\epsilon-1)}\Big{[}c^{\overline{\rm
III}}_{2,1}(m_{1}^{2}+m_{2}^{2})+c^{\overline{\rm
III}}_{2,2}m_{1}m_{2}.\Big{]}\,,$ $\displaystyle c^{\overline{\rm III}}_{3}$
$\displaystyle=\frac{m_{1}^{2}m_{2}^{2}\left(2\sigma^{2}-1\right)^{3}\left(\sigma\left(m_{1}^{2}+m_{2}^{2}\right)-2m_{1}m_{2}\right)}{\left(\sigma^{2}-1\right)^{3/2}\epsilon^{3}}$
$\displaystyle c^{\overline{\rm III}}_{4}$
$\displaystyle=\frac{2m_{1}^{2}m_{2}^{2}\left(2\sigma^{2}-1\right)}{3\left(\sigma^{2}-1\right)^{2}\epsilon^{3}(2\epsilon-1)}\Big{[}c^{\overline{\rm
III}}_{4,1}(m_{1}^{2}+m_{2}^{2})+c^{\overline{\rm
III}}_{4,2}m_{1}m_{2}\Big{]}\,,$ $\displaystyle c^{\overline{\rm III}}_{5}$
$\displaystyle=\frac{2m_{1}^{2}m_{2}^{2}\left(2\sigma^{2}-1\right)}{3\left(\sigma^{2}-1\right)^{3/2}\epsilon^{3}(2\epsilon-1)}\Big{[}c^{\overline{\rm
III}}_{5,1}(m_{1}^{2}+m_{2}^{2})+c^{\overline{\rm
III}}_{5,2}m_{1}m_{2}\Big{]}\,,$ $\displaystyle c^{\overline{\rm III}}_{6}$
$\displaystyle=\frac{2m_{1}^{2}m_{2}^{2}}{3\left(\sigma^{2}-1\right)^{2}\epsilon^{3}(6\epsilon-5)(6\epsilon-1)}\Big{[}c^{\overline{\rm
III}}_{6,1}(m_{1}^{2}+m_{2}^{2})+c^{\overline{\rm
III}}_{6,2}m_{1}m_{2}\Big{]}\,,$ $\displaystyle c^{\overline{\rm III}}_{7}$
$\displaystyle=8m_{1}^{2}m_{2}^{2}\left(2\sigma^{2}-1\right)^{3}\left(\frac{m_{1}^{2}-2m_{1}m_{2}\sigma+m_{2}^{2}}{3\left(\sigma^{2}-1\right)^{2}\epsilon^{3}}-\frac{m_{1}^{2}m_{2}^{2}}{\left(\sigma^{2}-1\right)\epsilon^{4}q^{2}}\right)\,,$
$\displaystyle c^{\overline{\rm III}}_{8}$ $\displaystyle=0\,,$ $\displaystyle
c^{\overline{\rm III}}_{9}$ $\displaystyle=0\,,$ $\displaystyle
c^{\overline{\rm III}}_{10}$
$\displaystyle=\frac{16m_{1}^{3}m_{2}^{3}(2\sigma^{2}-1)(m_{1}+m_{2})\left((1-2\sigma^{2})^{2}\epsilon-(\sigma+1)(4\sigma^{3}-2\sigma-1)\right)}{(\sigma-1)^{\frac{3}{2}}\epsilon^{4}q}\,,$

where

$\displaystyle c^{\overline{\rm III}}_{1,1}$
$\displaystyle=6\left(4\sigma^{6}-3\sigma^{2}+1\right)\epsilon^{2}+\left(-60\sigma^{6}+80\sigma^{4}-35\sigma^{2}+11\right)\epsilon+36\sigma^{6}-64\sigma^{4}+34\sigma^{2}-7\,,$
(276) $\displaystyle c^{\overline{\rm III}}_{1,2}$
$\displaystyle=-2\sigma(2\epsilon-1)\left(6\left(1-2\sigma^{2}\right)^{2}\epsilon-20\sigma^{4}+26\sigma^{2}-5\right)\,,$
$\displaystyle c^{\overline{\rm III}}_{2,1}$
$\displaystyle=-\sigma\left[2(2\sigma^{2}{-}1)^{3}\epsilon^{3}-\left(2\sigma^{2}{-}1\right)^{2}\left(10\sigma^{2}{-}9\right)\epsilon^{2}+4\sigma^{2}\left(6\sigma^{4}-11\sigma^{2}+5\right)\epsilon+2(\sigma^{2}-1)\right]\,,$
$\displaystyle c^{\overline{\rm III}}_{2,2}$
$\displaystyle=2\left(2\sigma^{2}-1\right)(2\epsilon-1)\left(\left(1-2\sigma^{2}\right)^{2}\epsilon^{2}+\left(2-2\sigma^{2}\right)\epsilon-3\sigma^{4}+4\sigma^{2}-1\right)\,,$
$\displaystyle c^{\overline{\rm III}}_{4,1}$
$\displaystyle=6\left(4\sigma^{6}-3\sigma^{2}+1\right)\epsilon^{2}-\left(60\sigma^{6}-80\sigma^{4}+35\sigma^{2}-11\right)\epsilon+36\sigma^{6}-64\sigma^{4}+34\sigma^{2}-7\,,$
$\displaystyle c^{\overline{\rm III}}_{4,2}$
$\displaystyle=-2\sigma(2\epsilon-1)\left(6\left(1-2\sigma^{2}\right)^{2}\epsilon-20\sigma^{4}+26\sigma^{2}-5\right)\,,$
$\displaystyle c^{\overline{\rm III}}_{5,1}$
$\displaystyle=-\sigma\big{(}4\left(1-2\sigma^{2}\right)^{2}\epsilon^{2}-4\left(4\sigma^{4}-8\sigma^{2}+3\right)\epsilon+\left(12\sigma^{4}-20\sigma^{2}+5\right)\big{)}\,,$
$\displaystyle c^{\overline{\rm III}}_{5,2}$
$\displaystyle=8\left(1-2\sigma^{2}\right)^{2}\epsilon^{2}+8\left(4\sigma^{4}-6\sigma^{2}+3\right)\epsilon-40\sigma^{4}+40\sigma^{2}-6\,,$
$\displaystyle c^{\overline{\rm III}}_{6,1}$
$\displaystyle=108\left(\sigma^{2}+1\right)\left(2\sigma^{2}-1\right)^{3}\epsilon^{3}-36\left(72\sigma^{8}-116\sigma^{6}+66\sigma^{4}-21\sigma^{2}+4\right)\epsilon^{2}$
$\displaystyle\quad+3\left(856\sigma^{8}-2132\sigma^{6}+2058\sigma^{4}-935\sigma^{2}+151\right)\epsilon$
$\displaystyle\quad+4\sigma^{2}\left(-210\sigma^{6}+664\sigma^{4}-762\sigma^{2}+357\right)-191\,,$
$\displaystyle c^{\overline{\rm III}}_{6,2}$
$\displaystyle=-2\sigma(6\epsilon-5)\Big{[}36(2\sigma^{2}{-}1)^{3}\epsilon^{2}-36\left(8\sigma^{6}{-}14\sigma^{4}{+}7\sigma^{2}{-}1\right)\epsilon+40\sigma^{6}-48\sigma^{4}-12\sigma^{2}+19\Big{]}\,.$

####  D.1.3 IX

The coefficients are

$\displaystyle c^{\text{IX}}_{1}$
$\displaystyle=-\frac{m_{1}^{2}m_{2}^{2}\left(2\sigma^{2}-1\right)}{6\left(\sigma^{2}-1\right)^{2}\epsilon^{4}(2\epsilon-1)}\Big{[}c^{\rm
IX}_{1,1}(m_{1}^{2}+m_{2}^{2})+c^{\rm IX}_{1,2}m_{1}m_{2}\Big{]}\,,$ (277)
$\displaystyle c^{\text{IX}}_{2}$
$\displaystyle=\frac{2m_{1}^{2}m_{2}^{2}}{\left(\sigma^{2}-1\right)^{3/2}\epsilon^{4}(2\epsilon-1)}\Big{[}c^{\rm
IX}_{2,1}(m_{1}^{2}+m_{2}^{2})+c^{\rm IX}_{2,2}m_{1}m_{2}.\Big{]}\,,$
$\displaystyle c^{\text{IX}}_{3}$
$\displaystyle=-\frac{m_{1}^{2}m_{2}^{2}\left(2\sigma^{2}-1\right)}{2\left(\sigma^{2}-1\right)^{3/2}\epsilon^{3}(2\epsilon-1)}\Big{[}c^{\rm
IX}_{3,1}(m_{1}^{2}+m_{2}^{2})+c^{\rm IX}_{3,2}m_{1}m_{2}.\Big{]}$
$\displaystyle c^{\text{IX}}_{4}$
$\displaystyle=\frac{m_{1}^{2}m_{2}^{2}}{6\left(\sigma^{2}-1\right)^{2}\epsilon^{4}(2\epsilon-1)}\Big{[}c^{\rm
IX}_{4,1}(m_{1}^{2}+m_{2}^{2})+c^{\rm IX}_{4,2}m_{1}m_{2}\Big{]}\,,$
$\displaystyle c^{\text{IX}}_{5}$
$\displaystyle=\frac{2m_{1}^{2}m_{2}^{2}}{\left(\sigma^{2}-1\right)^{3/2}\epsilon^{4}(2\epsilon-1)}\Big{[}c^{\rm
IX}_{5,1}(m_{1}^{2}+m_{2}^{2})+c^{\rm IX}_{5,2}m_{1}m_{2}\Big{]}\,,$
$\displaystyle c^{\text{IX}}_{6}$
$\displaystyle=-\frac{m_{1}^{2}m_{2}^{2}\left(2\sigma^{2}-1\right)}{\left(\sigma^{2}-1\right)^{3/2}\epsilon^{3}(2\epsilon-1)}\Big{[}c^{\rm
IX}_{6,1}(m_{1}^{2}+m_{2}^{2})+c^{\rm IX}_{6,2}m_{1}m_{2}\Big{]}\,,$
$\displaystyle c^{\text{IX}}_{7}$
$\displaystyle=\frac{m_{1}^{2}m_{2}^{2}}{6\left(\sigma^{2}-1\right)^{2}\epsilon^{4}(2\epsilon-1)}\Big{[}c^{\rm
IX}_{7,1}(m_{1}^{2}+m_{2}^{2})+c^{\rm IX}_{7,2}m_{1}m_{2}\Big{]}\,,$
$\displaystyle c^{\text{IX}}_{8}$
$\displaystyle=-\frac{m_{1}^{2}m_{2}^{2}\left(2\sigma^{2}-1\right)}{6\left(\sigma^{2}-1\right)^{2}\epsilon^{4}(6\epsilon-1)}\Big{[}c^{\rm
IX}_{8,1}(m_{1}^{2}+m_{2}^{2})+c^{\rm IX}_{8,2}m_{1}m_{2}\Big{]}\,,$
$\displaystyle c^{\text{IX}}_{9}$
$\displaystyle=\frac{m_{1}^{2}m_{2}^{2}\left(2\sigma^{2}-1\right)}{3\left(\sigma^{2}-1\right)^{3/2}\epsilon^{3}(2\epsilon-1)}\Big{[}c^{\rm
IX}_{9,1}(m_{1}^{2}+m_{2}^{2})+c^{\rm IX}_{9,2}m_{1}m_{2}\Big{]}\,,$
$\displaystyle c^{\text{IX}}_{10}$
$\displaystyle=4m_{1}^{2}m_{2}^{2}\left(2\sigma^{2}-1\right)^{2}\Big{[}\frac{2m_{1}^{2}m_{2}^{2}\left(1-2\sigma^{2}\right)}{\left(\sigma^{2}-1\right)\epsilon^{4}q^{2}}+\frac{c^{\rm
IX}_{10,1}\left(m_{1}^{2}+m_{2}^{2}\right)+c^{\rm
IX}_{10,2}m_{1}m_{2}}{3\left(\sigma^{2}-1\right)^{2}\epsilon^{4}}\Big{]}\,,$
$\displaystyle c^{\text{IX}}_{11}$ $\displaystyle=0\,,$ $\displaystyle
c^{\text{IX}}_{12}$
$\displaystyle=\frac{4m_{1}^{3}m_{2}^{3}\left(2\sigma^{2}-1\right)^{3}(m_{1}+m_{2})}{\left(\sigma^{2}-1\right)\epsilon^{3}q}\,,$
$\displaystyle c^{\text{IX}}_{13}$
$\displaystyle=-\frac{4m_{1}^{3}m_{2}^{3}\left(2\sigma^{2}-1\right)^{3}(m_{1}+m_{2})}{\left(\sigma^{2}-1\right)\epsilon^{3}q}\,,$
$\displaystyle c^{\text{IX}}_{14}$
$\displaystyle=-\frac{4m_{1}^{3}m_{2}^{3}\left(2\sigma^{2}-1\right)(m_{1}+m_{2})}{\left(\sigma^{2}-1\right)^{3/2}\epsilon^{4}q}c^{\rm
IX}_{14,1}\,,$ $\displaystyle c^{\text{IX}}_{15}$
$\displaystyle=\frac{8m_{1}^{3}m_{2}^{3}\left(1-2\sigma^{2}\right)^{2}(m_{1}+m_{2})\left(\left(2\sigma^{2}-1\right)\epsilon-2\sigma(\sigma+1)\right)}{(\sigma+1)\sqrt{\sigma^{2}-1}\epsilon^{4}q}\,.$

where

$\displaystyle c^{\rm IX}_{1,1}$
$\displaystyle=\epsilon\Big{[}6\left(4\sigma^{6}-3\sigma^{2}+1\right)\epsilon^{2}-\left(60\sigma^{6}-80\sigma^{4}+35\sigma^{2}-11\right)\epsilon+36\sigma^{6}-64\sigma^{4}+34\sigma^{2}-7\Big{]}\,,$
(278) $\displaystyle c^{\rm IX}_{1,2}$
$\displaystyle=4\sigma\left(2\sigma^{2}-1\right)(2\epsilon-1)\big{(}\left(12\sigma^{2}-6\right)\epsilon^{2}+\left(23-16\sigma^{2}\right)\epsilon-6\sigma^{2}+7\big{)}\,,$
$\displaystyle c^{\rm IX}_{2,1}$
$\displaystyle=\sigma\big{(}2\left(2\sigma^{2}-1\right)^{3}\epsilon^{3}+\left(14\sigma^{2}-15\right)\left(1-2\sigma^{2}\right)^{2}\epsilon^{2}$
$\displaystyle\qquad\;\;+\left(-96\sigma^{6}+152\sigma^{4}-64\sigma^{2}+8\right)\epsilon+2\left(\sigma^{2}-1\right)\big{)}\,,$
$\displaystyle c^{\rm IX}_{2,2}$
$\displaystyle=4\big{(}2\left(2\sigma^{2}-1\right)^{3}\epsilon^{3}+\left(8\sigma^{4}-10\sigma^{2}+3\right)\sigma^{2}\epsilon^{2}$
$\displaystyle\qquad\;\;+\left(-60\sigma^{6}+94\sigma^{4}-40\sigma^{2}+5\right)\epsilon+26\sigma^{6}-43\sigma^{4}+19\sigma^{2}-2\big{)}\,,$
$\displaystyle c^{\rm IX}_{3,1}$
$\displaystyle=\sigma\left(1-2\sigma^{2}\right)^{2}\left(\left(24\sigma^{2}-12\right)\epsilon^{2}+\left(28-24\sigma^{2}\right)\epsilon-2\sigma^{2}+1\right)\,,$
$\displaystyle c^{\rm IX}_{3,2}$
$\displaystyle=2\left(2\sigma^{2}-1\right)\left[\left(8\sigma^{4}-12\sigma^{2}+6\right)\epsilon-\left(1-2\sigma^{2}\right)^{2}\right]\,,$
$\displaystyle c^{\rm IX}_{4,1}$
$\displaystyle=\epsilon\left(2\sigma^{2}-1\right)\Big{[}12\left(2\sigma^{2}-1\right)^{3}\epsilon^{2}-4\left(4\sigma^{4}-\sigma^{2}-2\right)\epsilon-144\sigma^{6}+296\sigma^{4}-158\sigma^{2}+5\Big{]}\,,$
$\displaystyle c^{\rm IX}_{4,2}$
$\displaystyle=4\sigma\Big{[}12\left(2\sigma^{2}-1\right)^{3}\epsilon^{3}-8\left(1-2\sigma^{2}\right)^{2}\left(4\sigma^{2}-5\right)\epsilon^{2}$
$\displaystyle\qquad\;\;+\left(-8\sigma^{6}-16\sigma^{4}+34\sigma^{2}-15\right)\epsilon+\left(6\sigma^{2}-7\right)\left(1-2\sigma^{2}\right)^{2}\Big{]}\,,$
$\displaystyle c^{\rm IX}_{5,1}$
$\displaystyle=\sigma\Big{[}-2\left(2\sigma^{2}-1\right)^{3}\epsilon^{3}+\left(34\sigma^{2}-33\right)\left(1-2\sigma^{2}\right)^{2}\epsilon^{2}$
$\displaystyle\qquad\;\;-8\left(18\sigma^{6}-30\sigma^{4}+13\sigma^{2}-1\right)\epsilon-2\sigma^{2}+2\Big{]}\,,$
$\displaystyle c^{\rm IX}_{5,2}$
$\displaystyle=4\Big{[}-2\left(2\sigma^{2}-1\right)^{3}\epsilon^{3}+\left(-8\sigma^{6}+14\sigma^{4}-9\sigma^{2}+2\right)\epsilon^{2}$
$\displaystyle\qquad\;\;+\left(64\sigma^{6}-104\sigma^{4}+44\sigma^{2}-3\right)\epsilon-28\sigma^{6}+44\sigma^{4}-17\sigma^{2}+1\Big{]}\,,$
$\displaystyle c^{\rm IX}_{6,1}$
$\displaystyle=\sigma\Big{[}\left(12\sigma^{2}-6\right)\epsilon^{2}+\left(13-10\sigma^{2}\right)\epsilon-2\sigma^{2}+1\Big{]}\,,$
$\displaystyle c^{\rm IX}_{6,2}$
$\displaystyle=8\left(1-2\sigma^{2}\right)^{2}\epsilon^{2}+\left(4-4\sigma^{2}\right)\epsilon-2\left(1-2\sigma^{2}\right)^{2}\,,$
$\displaystyle c^{\rm IX}_{7,1}$
$\displaystyle=\epsilon(2\sigma^{2}-1)\Big{[}-12\left(1-2\sigma^{2}\right)^{2}\left(\sigma^{2}-2\right)\epsilon^{2}$
$\displaystyle\qquad\qquad\qquad\
-2\left(60\sigma^{6}-88\sigma^{4}+37\sigma^{2}-7\right)\epsilon+216\sigma^{6}-424\sigma^{4}+226\sigma^{2}-19\Big{]}\,,$
$\displaystyle c^{\rm IX}_{7,2}$
$\displaystyle=4\sigma\Big{[}12\left(2\sigma^{2}-1\right)^{3}\epsilon^{3}-8\left(1-2\sigma^{2}\right)^{2}\left(7\sigma^{2}-8\right)\epsilon^{2}$
$\displaystyle\qquad\quad\;\;+\left(40\sigma^{6}-88\sigma^{4}+46\sigma^{2}-3\right)\epsilon+\left(6\sigma^{2}-7\right)\left(1-2\sigma^{2}\right)^{2}\Big{]}\,,$
$\displaystyle c^{\rm IX}_{8,1}$
$\displaystyle=\epsilon\Big{[}36\left(\sigma^{2}-2\right)\left(1-2\sigma^{2}\right)^{2}\epsilon^{2}$
$\displaystyle\qquad\;\;-6\left(52\sigma^{6}-80\sigma^{4}+23\sigma^{2}+5\right)\epsilon+168\sigma^{6}-320\sigma^{4}+158\sigma^{2}-5\Big{]}\,,$
$\displaystyle c^{\rm IX}_{8,2}$
$\displaystyle=-4\sigma\left(2\sigma^{2}-1\right)(6\epsilon-1)\Big{[}\left(12\sigma^{2}-6\right)\epsilon^{2}+\left(2\sigma^{2}+5\right)\epsilon-6\sigma^{2}+7\Big{]}\,,$
$\displaystyle c^{\rm IX}_{9,1}$
$\displaystyle=\sigma\left[4\left(1-2\sigma^{2}\right)^{2}\epsilon^{2}-4\left(4\sigma^{4}-8\sigma^{2}+3\right)\epsilon+12\sigma^{4}-20\sigma^{2}+5\right]\,,$
$\displaystyle c^{\rm IX}_{9,2}$
$\displaystyle=-8\left(1-2\sigma^{2}\right)^{2}\epsilon^{2}+8\left(1-2\sigma^{2}\right)^{2}\epsilon+8\sigma^{4}-8\sigma^{2}-2\,,$
$\displaystyle c^{\rm IX}_{10,1}$
$\displaystyle=2\left(2\sigma^{2}-1\right)\epsilon\,,$ $\displaystyle c^{\rm
IX}_{10,2}$
$\displaystyle=\sigma\left[\left(4\sigma^{2}-2\right)\epsilon+6\sigma^{2}-7\right]\,,$
$\displaystyle c^{\rm IX}_{14,1}$
$\displaystyle=2(2\sigma+1)\left(1-2\sigma^{2}\right)^{2}\epsilon-12\sigma^{5}+20\sigma^{3}-\sigma^{2}-7\sigma+1\,.$

####  D.1.4 $\overline{\rm IX}$

The coefficients are

$\displaystyle c^{\overline{\rm IX}}_{1}$
$\displaystyle=-\frac{m_{1}^{2}m_{2}^{2}\left(2\sigma^{2}-1\right)}{6\left(\sigma^{2}-1\right)^{2}\epsilon^{4}(2\epsilon-1)}\Big{[}c^{\overline{\rm
IX}}_{1,1}(m_{1}^{2}+m_{2}^{2})+c^{\overline{\rm
IX}}_{1,2}m_{1}m_{2}\Big{]}\,,$ (279) $\displaystyle c^{\overline{\rm
IX}}_{2}$
$\displaystyle=-\frac{2m_{1}^{2}m_{2}^{2}}{\left(\sigma^{2}-1\right)^{3/2}\epsilon^{4}(2\epsilon-1)}\Big{[}c^{\overline{\rm
IX}}_{2,1}(m_{1}^{2}+m_{2}^{2})+c^{\overline{\rm
IX}}_{2,2}m_{1}m_{2}.\Big{]}\,,$ $\displaystyle c^{\overline{\rm IX}}_{3}$
$\displaystyle=\frac{m_{1}^{2}m_{2}^{2}\left(2\sigma^{2}-1\right)}{2\left(\sigma^{2}-1\right)^{3/2}\epsilon^{3}(2\epsilon-1)}\Big{[}c^{\overline{\rm
IX}}_{3,1}(m_{1}^{2}+m_{2}^{2})+c^{\overline{\rm IX}}_{3,2}m_{1}m_{2}.\Big{]}$
$\displaystyle c^{\overline{\rm IX}}_{4}$
$\displaystyle=\frac{m_{1}^{2}m_{2}^{2}}{6\left(\sigma^{2}-1\right)^{2}\epsilon^{4}(2\epsilon-1)}\Big{[}c^{\overline{\rm
IX}}_{4,1}(m_{1}^{2}+m_{2}^{2})+c^{\overline{\rm
IX}}_{4,2}m_{1}m_{2}\Big{]}\,,$ $\displaystyle c^{\overline{\rm IX}}_{5}$
$\displaystyle=-\frac{2m_{1}^{2}m_{2}^{2}}{\left(\sigma^{2}-1\right)^{3/2}\epsilon^{4}(2\epsilon-1)}\Big{[}c^{\overline{\rm
IX}}_{5,1}(m_{1}^{2}+m_{2}^{2})+c^{\overline{\rm
IX}}_{5,2}m_{1}m_{2}\Big{]}\,,$ $\displaystyle c^{\overline{\rm IX}}_{6}$
$\displaystyle=\frac{m_{1}^{2}m_{2}^{2}\left(2\sigma^{2}-1\right)}{\left(\sigma^{2}-1\right)^{3/2}\epsilon^{3}(2\epsilon-1)}\Big{[}c^{\overline{\rm
IX}}_{6,1}(m_{1}^{2}+m_{2}^{2})+c^{\overline{\rm
IX}}_{6,2}m_{1}m_{2}\Big{]}\,,$ $\displaystyle c^{\overline{\rm IX}}_{7}$
$\displaystyle=\frac{m_{1}^{2}m_{2}^{2}}{6\left(\sigma^{2}-1\right)^{2}\epsilon^{4}(2\epsilon-1)}\Big{[}c^{\overline{\rm
IX}}_{7,1}(m_{1}^{2}+m_{2}^{2})+c^{\overline{\rm
IX}}_{7,2}m_{1}m_{2}\Big{]}\,,$ $\displaystyle c^{\overline{\rm IX}}_{8}$
$\displaystyle=-\frac{m_{1}^{2}m_{2}^{2}\left(2\sigma^{2}-1\right)}{6\left(\sigma^{2}-1\right)^{2}\epsilon^{4}(6\epsilon-1)}\Big{[}c^{\overline{\rm
IX}}_{8,1}(m_{1}^{2}+m_{2}^{2})+c^{\overline{\rm
IX}}_{8,2}m_{1}m_{2}\Big{]}\,,$ $\displaystyle c^{\overline{\rm IX}}_{9}$
$\displaystyle=-\frac{m_{1}^{2}m_{2}^{2}\left(2\sigma^{2}-1\right)}{3\left(\sigma^{2}-1\right)^{3/2}\epsilon^{3}(2\epsilon-1)}\Big{[}c^{\overline{\rm
IX}}_{9,1}(m_{1}^{2}+m_{2}^{2})+c^{\overline{\rm
IX}}_{9,2}m_{1}m_{2}\Big{]}\,,$ $\displaystyle c^{\overline{\rm IX}}_{10}$
$\displaystyle=4m_{1}^{2}m_{2}^{2}\left(2\sigma^{2}-1\right)^{2}\Big{[}\frac{2m_{1}^{2}m_{2}^{2}\left(1-2\sigma^{2}\right)}{\left(\sigma^{2}-1\right)\epsilon^{4}q^{2}}+\frac{c^{\overline{\rm
IX}}_{10,1}\left(m_{1}^{2}+m_{2}^{2}\right)+c^{\overline{\rm
IX}}_{10,2}m_{1}m_{2}}{3\left(\sigma^{2}-1\right)^{2}\epsilon^{4}}\Big{]}\,,$
$\displaystyle c^{\overline{\rm IX}}_{11}$ $\displaystyle=0\,,$ $\displaystyle
c^{\overline{\rm IX}}_{12}$
$\displaystyle=\frac{4m_{1}^{3}m_{2}^{3}\left(2\sigma^{2}-1\right)^{3}(m_{1}+m_{2})}{\left(\sigma^{2}-1\right)\epsilon^{3}q}\,,$
$\displaystyle c^{\overline{\rm IX}}_{13}$
$\displaystyle=-\frac{4m_{1}^{3}m_{2}^{3}\left(2\sigma^{2}-1\right)^{3}(m_{1}+m_{2})}{\left(\sigma^{2}-1\right)\epsilon^{3}q}\,,$
$\displaystyle c^{\overline{\rm IX}}_{14}$
$\displaystyle=-\frac{4m_{1}^{3}m_{2}^{3}\left(2\sigma^{2}-1\right)(m_{1}+m_{2})}{\left(\sigma^{2}-1\right)^{3/2}\epsilon^{4}q}c^{\overline{\rm
IX}}_{14,1}\,,$ $\displaystyle c^{\overline{\rm IX}}_{15}$
$\displaystyle=\frac{8m_{1}^{3}m_{2}^{3}\left(1-2\sigma^{2}\right)^{2}(m_{1}+m_{2})\left(\left(2\sigma^{2}-1\right)\epsilon-2\sigma(\sigma-1)\right)}{(\sigma-1)\sqrt{\sigma^{2}-1}\epsilon^{4}q}\,.$

where

$\displaystyle c^{\overline{\rm IX}}_{1,1}$
$\displaystyle=\epsilon\left(6\left(4\sigma^{6}-3\sigma^{2}+1\right)\epsilon^{2}+\left(-60\sigma^{6}+80\sigma^{4}-35\sigma^{2}+11\right)\epsilon+36\sigma^{6}-64\sigma^{4}+34\sigma^{2}-7\right)\,,$
(280) $\displaystyle c^{\overline{\rm IX}}_{1,2}$
$\displaystyle=-4\sigma\left(2\sigma^{2}-1\right)(2\epsilon-1)\big{(}\left(12\sigma^{2}-6\right)\epsilon^{2}+\left(23-16\sigma^{2}\right)\epsilon-6\sigma^{2}+7\big{)}\,,$
$\displaystyle c^{\overline{\rm IX}}_{2,1}$
$\displaystyle=-\sigma\big{(}2\left(2\sigma^{2}-1\right)^{3}\epsilon^{3}+\left(14\sigma^{2}-15\right)\left(1-2\sigma^{2}\right)^{2}\epsilon^{2}$
$\displaystyle\qquad\;\;+\left(-96\sigma^{6}+152\sigma^{4}-64\sigma^{2}+8\right)\epsilon+2\left(\sigma^{2}-1\right)\big{)}\,,$
$\displaystyle c^{\overline{\rm IX}}_{2,2}$
$\displaystyle=4\big{(}2\left(2\sigma^{2}-1\right)^{3}\epsilon^{3}+\left(8\sigma^{4}-10\sigma^{2}+3\right)\sigma^{2}\epsilon^{2}$
$\displaystyle\qquad\;\;+\left(-60\sigma^{6}+94\sigma^{4}-40\sigma^{2}+5\right)\epsilon+26\sigma^{6}-43\sigma^{4}+19\sigma^{2}-2\big{)}\,,$
$\displaystyle c^{\overline{\rm IX}}_{3,1}$
$\displaystyle=-\sigma\left(1-2\sigma^{2}\right)^{2}\left(\left(24\sigma^{2}-12\right)\epsilon^{2}+\left(28-24\sigma^{2}\right)\epsilon-2\sigma^{2}+1\right)\,,$
$\displaystyle c^{\overline{\rm IX}}_{3,2}$
$\displaystyle=2\left(2\sigma^{2}-1\right)\left(\left(8\sigma^{4}-12\sigma^{2}+6\right)\epsilon-\left(1-2\sigma^{2}\right)^{2}\right)\,,$
$\displaystyle c^{\overline{\rm IX}}_{4,1}$
$\displaystyle=\epsilon\left(2\sigma^{2}-1\right)\big{(}12\left(2\sigma^{2}-1\right)^{3}\epsilon^{2}+4\left(-4\sigma^{4}+\sigma^{2}+2\right)\epsilon-144\sigma^{6}+296\sigma^{4}-158\sigma^{2}+5\big{)}\,,$
$\displaystyle c^{\overline{\rm IX}}_{4,2}$
$\displaystyle=-4\sigma\big{(}12\left(2\sigma^{2}-1\right)^{3}\epsilon^{3}-8\left(1-2\sigma^{2}\right)^{2}\left(4\sigma^{2}-5\right)\epsilon^{2}$
$\displaystyle\qquad\;\;+\left(-8\sigma^{6}-16\sigma^{4}+34\sigma^{2}-15\right)\epsilon+\left(6\sigma^{2}-7\right)\left(1-2\sigma^{2}\right)^{2}\big{)}\,,$
$\displaystyle c^{\overline{\rm IX}}_{5,1}$
$\displaystyle=-\sigma\big{(}-2\left(2\sigma^{2}-1\right)^{3}\epsilon^{3}+\left(34\sigma^{2}-33\right)\left(1-2\sigma^{2}\right)^{2}\epsilon^{2}$
$\displaystyle\qquad\;\;+\left(-144\sigma^{6}+240\sigma^{4}-104\sigma^{2}+8\right)\epsilon-2\sigma^{2}+2\big{)}\,,$
$\displaystyle c^{\overline{\rm IX}}_{5,2}$
$\displaystyle=4\big{(}-2\left(2\sigma^{2}-1\right)^{3}\epsilon^{3}+\left(-8\sigma^{6}+14\sigma^{4}-9\sigma^{2}+2\right)\epsilon^{2}$
$\displaystyle\qquad\;\;+\left(64\sigma^{6}-104\sigma^{4}+44\sigma^{2}-3\right)\epsilon-28\sigma^{6}+44\sigma^{4}-17\sigma^{2}+1\big{)}\,,$
$\displaystyle c^{\overline{\rm IX}}_{6,1}$
$\displaystyle=-\sigma\left(\left(12\sigma^{2}-6\right)\epsilon^{2}+\left(13-10\sigma^{2}\right)\epsilon-2\sigma^{2}+1\right)\,,$
$\displaystyle c^{\overline{\rm IX}}_{6,2}$
$\displaystyle=8\left(1-2\sigma^{2}\right)^{2}\epsilon^{2}+\left(4-4\sigma^{2}\right)\epsilon-2\left(1-2\sigma^{2}\right)^{2}\,,$
$\displaystyle c^{\overline{\rm IX}}_{7,1}$
$\displaystyle=\epsilon\left(2\sigma^{2}-1\right)\big{(}-12\left(1-2\sigma^{2}\right)^{2}\left(\sigma^{2}-2\right)\epsilon^{2}$
$\displaystyle\qquad\qquad\qquad\quad+2\left(-60\sigma^{6}+88\sigma^{4}-37\sigma^{2}+7\right)\epsilon+216\sigma^{6}-424\sigma^{4}+226\sigma^{2}-19\big{)}\,,$
$\displaystyle c^{\overline{\rm IX}}_{7,2}$
$\displaystyle=-4\sigma\big{(}12\left(2\sigma^{2}-1\right)^{3}\epsilon^{3}-8\left(1-2\sigma^{2}\right)^{2}\left(7\sigma^{2}-8\right)\epsilon^{2}$
$\displaystyle\qquad\quad\;\;+\left(40\sigma^{6}-88\sigma^{4}+46\sigma^{2}-3\right)\epsilon+\left(6\sigma^{2}-7\right)\left(1-2\sigma^{2}\right)^{2}\big{)}\,.$
$\displaystyle c^{\overline{\rm IX}}_{8,1}$
$\displaystyle=\epsilon\big{(}36\left(\sigma^{2}-2\right)\left(1-2\sigma^{2}\right)^{2}\epsilon^{2}$
$\displaystyle\qquad\;\;-6\left(52\sigma^{6}-80\sigma^{4}+23\sigma^{2}+5\right)\epsilon+168\sigma^{6}-320\sigma^{4}+158\sigma^{2}-5\big{)}\,,$
$\displaystyle c^{\overline{\rm IX}}_{8,2}$
$\displaystyle=4\sigma\left(2\sigma^{2}-1\right)(6\epsilon-1)\left(\left(12\sigma^{2}-6\right)\epsilon^{2}+\left(2\sigma^{2}+5\right)\epsilon-6\sigma^{2}+7\right)\,,$
$\displaystyle c^{\overline{\rm IX}}_{9,1}$
$\displaystyle=-\sigma\left(4\left(1-2\sigma^{2}\right)^{2}\epsilon^{2}-4\left(4\sigma^{4}-8\sigma^{2}+3\right)\epsilon+12\sigma^{4}-20\sigma^{2}+5\right)\,,$
$\displaystyle c^{\overline{\rm IX}}_{9,2}$
$\displaystyle=-8\left(1-2\sigma^{2}\right)^{2}\epsilon^{2}+8\left(1-2\sigma^{2}\right)^{2}\epsilon+8\sigma^{4}-8\sigma^{2}-2\,,$
$\displaystyle c^{\overline{\rm IX}}_{10,1}$
$\displaystyle=2\left(2\sigma^{2}-1\right)\epsilon\,,$ $\displaystyle
c^{\overline{\rm IX}}_{10,2}$
$\displaystyle=-\sigma\left(\left(4\sigma^{2}-2\right)\epsilon+6\sigma^{2}-7\right)\,,$
$\displaystyle c^{\overline{\rm IX}}_{14,1}$
$\displaystyle=2(2\sigma+1)\left(1-2\sigma^{2}\right)^{2}\epsilon+12\sigma^{5}-20\sigma^{3}-\sigma^{2}+7\sigma+1\,.$

###  D.2 H-class

The H class consists of two sectors, H and $\overline{\rm H}$. We present
their coefficients of the IBP reduction for the loop integrands. The master
integrals are decomposed into the even and the odd terms in $q$, and the odd
terms do not contribute to the amplitudes. Thus we focus on the even terms
only.

####  D.2.1 H sector

The coefficients of the IBP reduction of the amplitude for the H sector
defined in (203) are given by

$\displaystyle c^{\text{H}}_{1}$
$\displaystyle=\frac{2m_{1}^{3}m_{2}^{3}\sigma}{(\epsilon-1)\epsilon^{4}(4\epsilon^{2}-8\epsilon+3)^{2}(9\epsilon^{2}-9\epsilon+2)}c^{\rm
H}_{1,1}\,,$ (281) $\displaystyle c^{\text{H}}_{2}$
$\displaystyle=-\frac{8m_{1}^{3}m_{2}^{3}\sigma}{(\epsilon-1)\epsilon^{3}(2\epsilon-3)(2\epsilon-1)(4\epsilon-1)}c^{\rm
H}_{2,1}\,,$ $\displaystyle c^{\text{H}}_{3}$
$\displaystyle=-\frac{4m_{1}^{3}m_{2}^{3}\sigma}{\epsilon^{4}(4\epsilon^{2}-8\epsilon+3)^{2}}c^{\rm
H}_{3,1}$ $\displaystyle c^{\text{H}}_{4}$
$\displaystyle=\frac{m_{1}^{3}m_{2}^{3}}{8(\epsilon-1)^{2}\epsilon^{4}(4\epsilon^{2}-8\epsilon+3)}c^{\rm
H}_{4,1}\,,$ $\displaystyle c^{\text{H}}_{5}$
$\displaystyle=\frac{32m_{1}^{3}m_{2}^{3}}{\sqrt{\sigma^{2}-1}(\epsilon-1)\epsilon^{4}(2\epsilon-3)(2\epsilon-1)}c^{\rm
H}_{5,1}\,,$ $\displaystyle c^{\text{H}}_{6}$
$\displaystyle=\frac{8m_{1}^{3}m_{2}^{3}}{\sqrt{\sigma^{2}-1}(\epsilon-1)\epsilon^{4}(2\epsilon-3)(2\epsilon-1)}c^{\rm
H}_{6,1}\,,$ $\displaystyle c^{\text{H}}_{7}$
$\displaystyle=\frac{32m_{1}^{3}m_{2}^{3}\sigma\left(\left(4\sigma^{2}-10\right)\epsilon^{2}+\left(15-8\sigma^{2}\right)\epsilon-6\right)}{(\epsilon-1)\epsilon^{3}(2\epsilon-3)(2\epsilon-1)}\,,$
$\displaystyle c^{\text{H}}_{8}$
$\displaystyle=\frac{8m_{1}^{3}m_{2}^{3}}{\sqrt{\sigma^{2}-1}(\epsilon-1)\epsilon^{4}(2\epsilon-3)(2\epsilon-1)(4\epsilon-1)}c^{\rm
H}_{8,1}\,,$ $\displaystyle c^{\text{H}}_{9}$
$\displaystyle=\frac{4m_{1}^{3}m_{2}^{3}\left(-3\left(1-2\sigma^{2}\right)^{2}+4\left(4\sigma^{4}-3\sigma^{2}-2\right)\epsilon^{2}+\left(-12\sigma^{4}+8\sigma^{2}+12\right)\epsilon\right)}{\sqrt{\sigma^{2}-1}\epsilon^{4}(4\epsilon^{2}-8\epsilon+3)}\,,$
$\displaystyle c^{\text{H}}_{10}$
$\displaystyle=-\frac{8m_{1}^{3}m_{2}^{3}\sigma\left(4\left(4\sigma^{2}+3\right)\epsilon^{2}+\left(-22\sigma^{2}-19\right)\epsilon+6\right)}{\epsilon^{4}(4\epsilon^{2}-8\epsilon+3)}\,.$

where

$\displaystyle c^{\rm H}_{1,1}$
$\displaystyle=-96\left(3\sigma^{2}+37\right)\epsilon^{7}+8\left(187\sigma^{2}+2117\right)\epsilon^{6}-60\left(56\sigma^{2}+549\right)\epsilon^{5}+38\left(97\sigma^{2}+897\right)\epsilon^{4}$
(282)
$\displaystyle\quad+\left(-1564\sigma^{2}-20197\right)\epsilon^{3}+\left(6773-18\sigma^{2}\right)\epsilon^{2}+6\left(18\sigma^{2}-193\right)\epsilon+72\,,$
$\displaystyle c^{\rm H}_{2,1}$
$\displaystyle=\left(36\sigma^{4}+60\sigma^{2}-4\right)\epsilon^{3}+\left(-92\sigma^{4}-146\sigma^{2}+2\right)\epsilon^{2}$
$\displaystyle\quad+\left(56\sigma^{4}+142\sigma^{2}-9\right)\epsilon-54\sigma^{2}+9\,,$
$\displaystyle c^{\rm H}_{3,1}$
$\displaystyle=8\left(9\sigma^{2}+4\right)\epsilon^{4}-4\left(68\sigma^{2}+41\right)\epsilon^{3}+\left(258\sigma^{2}+224\right)\epsilon^{2}-3\left(18\sigma^{2}+35\right)\epsilon+18\,,$
$\displaystyle c^{\rm H}_{4,1}$
$\displaystyle=\left(3328\sigma^{2}+368\right)\epsilon^{4}-8\left(1424\sigma^{2}+185\right)\epsilon^{3}$
$\displaystyle\quad+4\left(3377\sigma^{2}+502\right)\epsilon^{2}-2\left(3060\sigma^{2}+539\right)\epsilon+675\sigma^{2}+183\,,$
$\displaystyle c^{\rm H}_{5,1}$
$\displaystyle=\left(4\sigma^{4}-23\sigma^{2}-21\right)\epsilon^{3}+\left(14\sigma^{2}+46\right)\epsilon^{2}+\left(-22\sigma^{4}-\sigma^{2}-31\right)\epsilon+6\sigma^{4}-3\sigma^{2}+6\,,$
$\displaystyle c^{\rm H}_{6,1}$
$\displaystyle=3\left(1-2\sigma^{2}\right)^{2}+4\left(6\sigma^{4}-\sigma^{2}-6\right)\epsilon^{3}+\left(-50\sigma^{4}+12\sigma^{2}+50\right)\epsilon^{2}+\left(22\sigma^{4}-4\sigma^{2}-29\right)\epsilon\,,$
$\displaystyle c^{\rm H}_{8,1}$
$\displaystyle=-4\left(9\sigma^{6}-45\sigma^{4}+27\sigma^{2}+13\right)\epsilon^{4}+4\left(23\sigma^{6}-112\sigma^{4}+68\sigma^{2}+34\right)\epsilon^{3}$
$\displaystyle\quad-2\left(28\sigma^{6}-129\sigma^{4}+68\sigma^{2}+61\right)\epsilon^{2}+\left(26\sigma^{4}-44\sigma^{2}+41\right)\epsilon-3\left(1-2\sigma^{2}\right)^{2}\,.$

####  D.2.2 $\overline{\rm H}$ sector

Similarly, the coefficients of the IBP reduction of the amplitude for the
$\overline{\rm H}$ sector defined in (205) are given by

$\displaystyle c^{\overline{\rm H}}_{1}$
$\displaystyle=-\frac{2m_{1}^{3}m_{2}^{3}\sigma}{(\epsilon-1)\epsilon^{4}(4\epsilon^{2}-8\epsilon+3)^{2}(9\epsilon^{2}-9\epsilon+2)}c^{\overline{\rm
H}}_{1,1}\,,$ (283) $\displaystyle c^{\overline{\rm H}}_{2}$
$\displaystyle=\frac{8m_{1}^{3}m_{2}^{3}\sigma}{(\epsilon-1)\epsilon^{3}(2\epsilon-3)(2\epsilon-1)(4\epsilon-1)}c^{\overline{\rm
H}}_{2,1}\,,$ $\displaystyle c^{\overline{\rm H}}_{3}$
$\displaystyle=\frac{4m_{1}^{3}m_{2}^{3}\sigma}{\epsilon^{4}(4\epsilon^{2}-8\epsilon+3)^{2}}c^{\overline{\rm
H}}_{3,1}$ $\displaystyle c^{\overline{\rm H}}_{4}$
$\displaystyle=\frac{m_{1}^{3}m_{2}^{3}}{8(\epsilon-1)^{2}\epsilon^{4}(4\epsilon^{2}-8\epsilon+3)}c^{\overline{\rm
H}}_{4,1}\,,$ $\displaystyle c^{\overline{\rm H}}_{5}$
$\displaystyle=-\frac{32m_{1}^{3}m_{2}^{3}}{\sqrt{\sigma^{2}-1}(\epsilon-1)\epsilon^{4}(2\epsilon-3)(2\epsilon-1)}c^{\overline{\rm
H}}_{5,1}\,,$ $\displaystyle c^{\overline{\rm H}}_{6}$
$\displaystyle=-\frac{8m_{1}^{3}m_{2}^{3}}{\sqrt{\sigma^{2}-1}(\epsilon-1)\epsilon^{4}(2\epsilon-3)(2\epsilon-1)}c^{\overline{\rm
H}}_{6,1}\,,$ $\displaystyle c^{\overline{\rm H}}_{7}$
$\displaystyle=-\frac{32m_{1}^{3}m_{2}^{3}\sigma\left(4\sigma^{2}\epsilon^{2}-10\epsilon^{2}-8\sigma^{2}\epsilon+15\epsilon-6\right)}{(\epsilon-1)\epsilon^{3}(2\epsilon-3)(2\epsilon-1)}\,,$
$\displaystyle c^{\overline{\rm H}}_{8}$
$\displaystyle=-\frac{8m_{1}^{3}m_{2}^{3}}{\sqrt{\sigma^{2}-1}(\epsilon-1)\epsilon^{4}(2\epsilon-3)(2\epsilon-1)(4\epsilon-1)}c^{\overline{\rm
H}}_{8,1}\,,$ $\displaystyle c^{\overline{\rm H}}_{9}$
$\displaystyle=-\frac{4m_{1}^{3}m_{2}^{3}\left(-12\sigma^{4}+12\sigma^{2}+16\sigma^{4}\epsilon^{2}-12\sigma^{2}\epsilon^{2}-8\epsilon^{2}-12\sigma^{4}\epsilon+8\sigma^{2}\epsilon+12\epsilon-3\right)}{\sqrt{\sigma^{2}-1}\epsilon^{4}(4\epsilon^{2}-8\epsilon+3)}\,,$
$\displaystyle c^{\overline{\rm H}}_{10}$
$\displaystyle=-\frac{8m_{1}^{3}m_{2}^{3}\sigma\left(16\sigma^{2}\epsilon^{2}+12\epsilon^{2}-22\sigma^{2}\epsilon-19\epsilon+6\right)}{\epsilon^{4}(4\epsilon^{2}-8\epsilon+3)}\,.$

where

$\displaystyle c^{\overline{\rm H}}_{1,1}$
$\displaystyle=-96\left(3\sigma^{2}+37\right)\epsilon^{7}+8\left(187\sigma^{2}+2117\right)\epsilon^{6}-60\left(56\sigma^{2}+549\right)\epsilon^{5}+38\left(97\sigma^{2}+897\right)\epsilon^{4}$
(284)
$\displaystyle\quad+\left(-1564\sigma^{2}-20197\right)\epsilon^{3}+\left(6773-18\sigma^{2}\right)\epsilon^{2}+6\left(18\sigma^{2}-193\right)\epsilon+72\,,$
$\displaystyle c^{\overline{\rm H}}_{2,1}$
$\displaystyle=\left(36\sigma^{4}+60\sigma^{2}-4\right)\epsilon^{3}+\left(-92\sigma^{4}-146\sigma^{2}+2\right)\epsilon^{2}$
$\displaystyle\quad+\left(56\sigma^{4}+142\sigma^{2}-9\right)\epsilon-54\sigma^{2}+9\,,$
$\displaystyle c^{\overline{\rm H}}_{3,1}$
$\displaystyle=8\left(9\sigma^{2}+4\right)\epsilon^{4}-4\left(68\sigma^{2}+41\right)\epsilon^{3}+\left(258\sigma^{2}+224\right)\epsilon^{2}-3\left(18\sigma^{2}+35\right)\epsilon+18\,,$
$\displaystyle c^{\overline{\rm H}}_{4,1}$
$\displaystyle=\left(3328\sigma^{2}+368\right)\epsilon^{4}-8\left(1424\sigma^{2}+185\right)\epsilon^{3}$
$\displaystyle\quad+4\left(3377\sigma^{2}+502\right)\epsilon^{2}-2\left(3060\sigma^{2}+539\right)\epsilon+675\sigma^{2}+183\,,$
$\displaystyle c^{\overline{\rm H}}_{5,1}$
$\displaystyle=\left(4\sigma^{4}-23\sigma^{2}-21\right)\epsilon^{3}+\left(14\sigma^{2}+46\right)\epsilon^{2}+\left(-22\sigma^{4}-\sigma^{2}-31\right)\epsilon+6\sigma^{4}-3\sigma^{2}+6\,,$
$\displaystyle c^{\overline{\rm H}}_{6,1}$
$\displaystyle=3\left(1-2\sigma^{2}\right)^{2}+4\left(6\sigma^{4}-\sigma^{2}-6\right)\epsilon^{3}+\left(-50\sigma^{4}+12\sigma^{2}+50\right)\epsilon^{2}+\left(22\sigma^{4}-4\sigma^{2}-29\right)\epsilon\,,$
$\displaystyle c^{\overline{\rm H}}_{8,1}$
$\displaystyle=-4\left(9\sigma^{6}-45\sigma^{4}+27\sigma^{2}+13\right)\epsilon^{4}+4\left(23\sigma^{6}-112\sigma^{4}+68\sigma^{2}+34\right)\epsilon^{3}$
$\displaystyle\quad-2\left(28\sigma^{6}-129\sigma^{4}+68\sigma^{2}+61\right)\epsilon^{2}+\left(26\sigma^{4}-44\sigma^{2}+41\right)\epsilon-3\left(1-2\sigma^{2}\right)^{2}\,.$

###  D.3 IY-class

We now consider the coefficients of the IBP reduction of the amplitude in the
IY-class. There are the horizontal mirror dual pairs, and we focus on the
three independent sectors: IY, $\overline{\rm IY}$ and /Y. Note that the
coefficients for the  IY class can be derived from the coefficients in the IY-
class by the vertical mirror dual relations (108).

####  D.3.1 IY sector

The coefficients of the IBP reduction of the amplitude for the IY sector
defined in (214) are given by

$\displaystyle c^{\text{IY}}_{1}$
$\displaystyle=\frac{m_{1}^{3}m_{2}^{2}}{12\left(\sigma^{2}-1\right)(1-2\epsilon)^{2}(\epsilon-1)\epsilon^{3}(2\epsilon-3)(2\epsilon+1)(9\epsilon^{2}-9\epsilon+2)}\Big{[}c^{\rm
IY}_{1,1}m_{1}+c^{\rm IY}_{1,2}m_{2}\Big{]}\,,$ (285) $\displaystyle
c^{\text{IY}}_{2}$
$\displaystyle=\frac{4m_{1}^{3}m_{2}^{2}}{\sqrt{\sigma^{2}-1}(\epsilon-1)\epsilon^{4}(2\epsilon-1)}\Big{[}c^{\rm
IY}_{2,1}m_{1}+c^{\rm IY}_{2,2}m_{2}\Big{]}\,,$ $\displaystyle
c^{\text{IY}}_{3}$
$\displaystyle=\frac{2m_{1}^{3}m_{2}^{2}}{\sqrt{\sigma^{2}-1}(\epsilon-1)\epsilon^{3}(2\epsilon-1)}\Big{[}c^{\rm
IY}_{3,1}m_{1}+c^{\rm IY}_{3,2}m_{2}\Big{]}\,,$ $\displaystyle
c^{\text{IY}}_{4}$
$\displaystyle=\frac{2m_{1}^{3}m_{2}^{2}}{\left(\sigma^{2}-1\right)(\epsilon-1)\epsilon^{3}(2\epsilon-1)}\Big{[}c^{\rm
IY}_{4,1}m_{1}+c^{\rm IY}_{4,2}m_{2}\Big{]}\,,$ $\displaystyle
c^{\text{IY}}_{5}$
$\displaystyle=\frac{2m_{1}^{3}m_{2}^{3}\left(2\sigma^{2}-1\right)\left(\left(32\sigma^{2}-16\right)\epsilon^{2}+\left(38-84\sigma^{2}\right)\epsilon+54\sigma^{2}-17\right)}{3\sqrt{\sigma^{2}-1}\epsilon^{3}(4\epsilon^{2}-8\epsilon+3)}\,,$
$\displaystyle c^{\text{IY}}_{6}$
$\displaystyle=\frac{2m_{1}^{3}m_{2}^{2}}{3\left(\sigma^{2}-1\right)(\epsilon-1)\epsilon^{3}(2\epsilon+1)(6\epsilon-5)(6\epsilon-1)}\Big{[}c^{\rm
IY}_{6,1}m_{1}+c^{\rm IY}_{6,2}m_{2}\Big{]}\,,$ $\displaystyle
c^{\text{IY}}_{7}$
$\displaystyle=-\frac{2m_{1}^{4}m_{2}^{3}\left(2\sigma^{2}-1\right)\left(\left(4\sigma^{2}-6\sigma-1\right)\epsilon^{2}+(10\sigma^{2}-8\sigma+6)\epsilon+4(\sigma-1)\sigma-2\right)}{(\epsilon-1)\epsilon^{4}(2\epsilon-1)q}\,,$
$\displaystyle c^{\text{IY}}_{8}$
$\displaystyle=-\frac{4m_{1}^{4}m_{2}^{3}\left(2\sigma^{2}-1\right)\left(4\sigma\epsilon^{2}+(3-7\sigma)\epsilon+2\sigma-1\right)}{(\epsilon-1)\epsilon^{4}(2\epsilon-1)q}\,,$
$\displaystyle c^{\text{IY}}_{9}$
$\displaystyle=\frac{2m_{1}^{4}m_{2}^{3}\left(2\sigma^{2}-1\right)\left(\left(16\sigma^{2}-8\right)\epsilon-15\sigma^{2}+7\right)}{\sqrt{\sigma^{2}-1}(\epsilon-1)\epsilon^{4}q}\,,$
$\displaystyle c^{\text{IY}}_{10}$
$\displaystyle=\frac{m_{1}^{3}m_{2}^{2}}{(\epsilon-1)\epsilon^{3}(2\epsilon-1)(2\epsilon+1)(4\epsilon-3)(4\epsilon-1)}\Big{[}c^{\rm
IY}_{10,1}m_{1}+c^{\rm IY}_{10,2}m_{2}\Big{]}\,,$ $\displaystyle
c^{\text{IY}}_{11}$
$\displaystyle=\frac{2m_{1}^{3}m_{2}^{2}}{\sqrt{\sigma^{2}-1}(\epsilon-1)\epsilon^{3}\left(8\epsilon^{2}-6\epsilon+1\right)}\Big{[}c^{\rm
IY}_{11,1}m_{1}+c^{\rm IY}_{11,2}m_{2}\Big{]}\,,$ $\displaystyle
c^{\text{IY}}_{12}$
$\displaystyle=-\frac{4m_{1}^{4}m_{2}^{3}\sigma\left(2\sigma^{2}-1\right)(1-2\epsilon)^{2}(3\epsilon-4)}{(\epsilon-1)\epsilon^{4}(2\epsilon-1)^{2}q}\,,$
$\displaystyle c^{\text{IY}}_{13}$
$\displaystyle=\frac{4m_{1}^{4}m_{2}^{3}\left(2\sigma^{2}-1\right)\left((10\sigma-2)\epsilon^{2}+(4-12\sigma)\epsilon+2\sigma-1\right)}{(\epsilon-1)\epsilon^{4}(2\epsilon-1)q}\,.$

where

$\displaystyle c^{\rm IY}_{1,1}$
$\displaystyle=-432\left(32\sigma^{6}-18\sigma^{4}-13\sigma^{2}+7\right)\epsilon^{7}+24\left(2904\sigma^{6}-2366\sigma^{4}-297\sigma^{2}+335\right)\epsilon^{6}$
(286)
$\displaystyle\quad-4\left(32760\sigma^{6}-34268\sigma^{4}+5187\sigma^{2}+1265\right)\epsilon^{5}$
$\displaystyle\quad+2\left(52920\sigma^{6}-67802\sigma^{4}+21173\sigma^{2}-819\right)\epsilon^{4}$
$\displaystyle\quad+\left(-19656\sigma^{6}+37910\sigma^{4}-20080\sigma^{2}+2282\right)\epsilon^{3}$
$\displaystyle\quad+\left(-22176\sigma^{6}+24662\sigma^{4}-5245\sigma^{2}-265\right)\epsilon^{2}$
$\displaystyle\quad+\left(13320\sigma^{6}-18176\sigma^{4}+6241\sigma^{2}-209\right)\epsilon$
$\displaystyle\quad-6\left(360\sigma^{6}-532\sigma^{4}+203\sigma^{2}-7\right)\,,$
$\displaystyle c^{\rm IY}_{1,2}$
$\displaystyle=-864\left(10\sigma^{5}-7\sigma^{3}+\sigma\right)\epsilon^{7}+864\sigma^{3}\left(41\sigma^{2}-25\right)\epsilon^{6}+24\sigma\left(-1894\sigma^{4}+621\sigma^{2}+449\right)\epsilon^{5}$
$\displaystyle\quad+8\sigma\left(1621\sigma^{4}+2293\sigma^{2}-2546\right)\epsilon^{4}+8\sigma\left(1444\sigma^{4}-2696\sigma^{2}+1309\right)\epsilon^{3}$
$\displaystyle\quad+24\sigma\left(-298\sigma^{4}+85\sigma^{2}+87\right)\epsilon^{2}+24\sigma\left(27\sigma^{4}+144\sigma^{2}-122\right)\epsilon+144\sigma\left(\sigma^{4}-6\sigma^{2}+4\right)\,,$
$\displaystyle c^{\rm IY}_{2,1}$
$\displaystyle=\sigma\left(-8\sigma^{2}+\left(12\sigma^{2}-24\sigma^{4}\right)\epsilon^{3}+\left(60\sigma^{4}-44\sigma^{2}+16\right)\epsilon^{2}+\left(-36\sigma^{4}+40\sigma^{2}-13\right)\epsilon+4\right)\,,$
$\displaystyle c^{\rm IY}_{2,2}$
$\displaystyle=\left(-8\sigma^{4}+20\sigma^{2}-8\right)\epsilon^{3}+2\left(6\sigma^{4}-8\sigma^{2}+5\right)\epsilon^{2}+\left(12\sigma^{4}+6\sigma^{2}-11\right)\epsilon-4\left(\sigma^{4}+\sigma^{2}\right)+3\,,$
$\displaystyle c^{\rm IY}_{3,1}$
$\displaystyle=\left(8\sigma^{4}-4\sigma^{2}\right)\epsilon^{2}-4\left(5\sigma^{4}-5\sigma^{2}+1\right)\epsilon+12\sigma^{4}-16\sigma^{2}+5\,,$
$\displaystyle c^{\rm IY}_{3,2}$
$\displaystyle=\left(8\sigma^{4}-4\sigma^{2}\right)\epsilon^{2}-2\left(8\sigma^{4}-7\sigma^{2}+1\right)\epsilon+4\sigma^{4}-2\sigma^{2}-1\,,$
$\displaystyle c^{\rm IY}_{4,1}$
$\displaystyle=-4\left(2\sigma^{2}-1\right)\left(2\sigma^{4}-1\right)\epsilon^{2}+\left(40\sigma^{6}-40\sigma^{4}+4\sigma^{2}+2\right)\epsilon-2\sigma^{2}\left(12\sigma^{4}-16\sigma^{2}+5\right)\,,$
$\displaystyle c^{\rm IY}_{4,2}$
$\displaystyle=\sigma\left(-4\left(1-2\sigma^{2}\right)^{2}\epsilon^{2}+\left(30\sigma^{4}-29\sigma^{2}+5\right)\epsilon+-6\sigma^{4}+\sigma^{2}+3\right)\,,$
$\displaystyle c^{\rm IY}_{6,1}$
$\displaystyle=-432\left(1-2\sigma^{2}\right)^{2}\epsilon^{4}+24\left(132\sigma^{4}-136\sigma^{2}+31\right)\epsilon^{3}-4\left(260\sigma^{4}-294\sigma^{2}+49\right)\epsilon^{2}$
$\displaystyle\quad+\left(-784\sigma^{4}+792\sigma^{2}-194\right)+6\left(62\sigma^{4}-69\sigma^{2}+12\right)\epsilon\,,$
$\displaystyle c^{\rm IY}_{6,2}$
$\displaystyle=-432\sigma\left(1-2\sigma^{2}\right)^{2}\epsilon^{4}+108\sigma\left(34\sigma^{4}-39\sigma^{2}+11\right)\epsilon^{3}-12\sigma\left(144\sigma^{4}-205\sigma^{2}+66\right)\epsilon^{2}$
$\displaystyle\quad+\left(-770\sigma^{5}+781\sigma^{3}-197\sigma\right)\epsilon+5\sigma\left(110\sigma^{4}-151\sigma^{2}+47\right)\,,$
$\displaystyle c^{\rm IY}_{10,1}$
$\displaystyle=128\left(10\sigma^{4}-7\sigma^{2}+1\right)\epsilon^{5}-8\left(352\sigma^{4}-303\sigma^{2}+46\right)\epsilon^{4}+8\left(194\sigma^{4}-203\sigma^{2}+27\right)\epsilon^{3}$
$\displaystyle\quad+\left(392\sigma^{4}-278\sigma^{2}+66\right)\epsilon^{2}+\left(-480\sigma^{4}+474\sigma^{2}-61\right)\epsilon+72\sigma^{4}-76\sigma^{2}+7\,,$
$\displaystyle c^{\rm IY}_{10,2}$
$\displaystyle=4\sigma(4\epsilon-3)\big{(}\left(40\sigma^{4}-52\sigma^{2}+16\right)\epsilon^{4}+\left(-84\sigma^{4}+142\sigma^{2}-50\right)\epsilon^{3}$
$\displaystyle\qquad\qquad\qquad\;+12\left(\sigma^{4}-4\sigma^{2}+2\right)\epsilon^{2}+\left(32\sigma^{4}-46\sigma^{2}+14\right)\epsilon+8\sigma^{2}-5\big{)}\,,$
$\displaystyle c^{\rm IY}_{11,1}$
$\displaystyle=\sigma\big{(}\left(80\sigma^{4}-56\sigma^{2}+8\right)\epsilon^{3}-4\left(39\sigma^{4}-37\sigma^{2}+8\right)\epsilon^{2}$
$\displaystyle\qquad\;\;+4\left(22\sigma^{4}-26\sigma^{2}+7\right)\epsilon-12\sigma^{4}+16\sigma^{2}-5\big{)}\,,$
$\displaystyle c^{\rm IY}_{11,2}$
$\displaystyle=4\left(10\sigma^{6}+3\sigma^{4}-6\sigma^{2}+1\right)\epsilon^{3}+\left(-104\sigma^{6}+48\sigma^{4}+26\sigma^{2}-10\right)\epsilon^{2}$
$\displaystyle\quad+2\left(32\sigma^{6}-28\sigma^{4}+\sigma^{2}+1\right)\epsilon-4\sigma^{4}+2\sigma^{2}+1\,.$

####  D.3.2 $\overline{\rm IY}$ sector

The coefficients of the IBP reduction of the amplitude for the $\overline{\rm
IY}$ sector defined in (214) are given by

$\displaystyle c^{\overline{\text{IY}}}_{1}$
$\displaystyle=\frac{m_{1}^{3}m_{2}^{2}}{12\left(\sigma^{2}-1\right)(1-2\epsilon)^{2}(\epsilon-1)\epsilon^{3}(2\epsilon-3)(2\epsilon+1)(9\epsilon^{2}-9\epsilon+2)}\Big{[}c^{\overline{\rm
IY}}_{1,1}m_{1}+c^{\overline{\rm IY}}_{1,2}m_{2}\Big{]}\,,$ (287)
$\displaystyle c^{\overline{\text{IY}}}_{2}$
$\displaystyle=-\frac{4m_{1}^{3}m_{2}^{2}}{\sqrt{\sigma^{2}-1}(\epsilon-1)\epsilon^{4}(2\epsilon-1)}\Big{[}c^{\overline{\rm
IY}}_{2,1}m_{1}+c^{\overline{\rm IY}}_{2,2}m_{2}\Big{]}\,,$ $\displaystyle
c^{\overline{\text{IY}}}_{3}$