Section \[sec:general\] $$\begin{split}
 {S^{(1)}}{}_{{M}{N}}^{{P}{Q}}(\theta)=\frac{1}{8 \sinh \theta}\,\Big[&{\tilde S^{(0)}}{}_{{M}{N}}^{{R}{S}}(\theta){\tilde S^{(0)}}{}_{{R}{S}}^{{P}{Q}}(\theta)I(i\pi-\theta)\\  \vphantom{\frac{1}{4 (\e_2\,\rmp_1-\e_1\,\rmp_2)}}
 +&  (-1)^{[{P}][{S}]+[{R}][{S}]}\,{\tilde S^{(0)}}{}_{{M}{R}}^{{S}{P}}(0){\tilde S^{(0)}}{}_{{S}{N}}^{{R}{Q}}(\theta)I(0) \\ 
 \vphantom{\frac{1}{4 (\e_2\,\rmp_1-\e_1\,\rmp_2)}}
 +&(-1)^{[{P}][{R}]+[{Q}][{S}]+[{R}][{S}]+[{P}][{Q}]}{\tilde S^{(0)}}{}_{{M}{R}}^{{S}{Q}}(\theta){\tilde S^{(0)}}{}_{{S}{N}}^{{P}{R}}(\theta)I(\theta)\,\Big]~,\label{eqn:final_ferm_rel}
\end{split}$$ where $\tilde S{}^{(0)} (\theta)= 4 \sinh \theta \ S^{(0)}(\theta)$ and the one-loop integrals are given in . Again this can be done with both the full ($M,N,\ldots = (A,\dot A),\,(B,\dot B)\ldots$) and factor ($M,N,\ldots = A,B,\ldots$) tree-level S-matrices. As expected they give the same one-loop S-matrix parametrized by the following functions $$\begin{aligned}
\no
&  M^{(1)}_1 (\theta) = M^{(1)}_2 (\theta) = \frac{P(\theta)}2
&& M^{(1)}_3 (\theta) = M^{(1)}_4 (\theta) = 0
\\
&  M^{(1)}_5 (\theta) = M^{(1)}_6 (\theta) = \frac{P(\theta)}2
&& M^{(1)}_7 (\theta) = M^{(1)}_8 (\theta) = 0
\label{s222funcol}
\\\no
\\
\no
&  L^{(1)}_1 (\theta) = L^{(1)}_3 (\theta) = - \frac{\pi^2}{2k^2} + P(\theta)
&& L^{(1)}_2 (\theta) = L^{(1)}_4 (\theta) = \tilde P(\theta)
\\
\no
&  L^{(1)}_5 (\theta) = L^{(1)}_7 (\theta) = P(\theta)
&&  L^{(1)}_6 (\theta) = L^{(1)}_8 (\theta) = \tilde P(\theta)
\\
&  L^{(1)}_9 (\theta) = L^{(1)}_{10} (\theta) = 0
&&  L^{(1)}_{11} (\theta) = L^{(1)}_{12} (\theta) = 0
\label{s233funcol}
\\\no
\\\no
&  K^{(1)}_1 (\theta) = - K^{(1)}_3 (\theta) = -\frac{5\pi^2}{8k^2} -\frac{i\pi \theta}{2k^2}  + \frac{P(\theta)}2
&& K^{(1)}_2 (\theta) = - K^{(1)}_4 (\theta) = \frac{\pi^2}{2k^2} + \frac{i\pi\theta}{k^2}
\\
\no
&  K^{(1)}_5 (\theta) = 