$\delta$-functions $\delta(\text{p}_1 - \text{p}_3)\delta(\text{p}_2 - \text{p}_4)$ (see comment above ), it then follows that a candidate expression for the one-loop S-matrix elements is given by the following simple sum of products of two tree-level amplitudes weighted by scalar bubble integrals. $$\begin{aligned}
 {S^{(1)}}{}_{{M}{N}}^{{P}{Q}}(p_1,p_2)=\frac{1}{8 (\e_2\,\rmp_1-\e_1\,\rmp_2)}\,\Big[&{\tilde S^{(0)}}{}_{{M}{N}}^{{R}{S}}(p_1,p_2){\tilde S^{(0)}}{}_{{R}{S}}^{{P}{Q}}(p_1,p_2)I(p_1+p_2)\nonumber\\ \vphantom{\frac{1}{4 (\e_2\,\rmp_1-\e_1\,\rmp_2)}}  +& {\tilde S^{(0)}}{}_{{M}{R}}^{{S}{P}}(p_1,p_1){\tilde S^{(0)}}{}_{{S}{N}}^{{R}{Q}}(p_1,p_2)I(0) \nonumber\\
 \vphantom{\frac{1}{4 (\e_2\,\rmp_1-\e_1\,\rmp_2)}}
 +&{\tilde S^{(0)}}{}_{{M}{R}}^{{S}{Q}}(p_1,p_2){\tilde S^{(0)}}{}_{{S}{N}}^{{P}{R}}(p_1,p_2)I(p_1-p_2)\,\Big]
 \label{eqn:final}~,\end{aligned}$$ where $\tilde S^{(0)}(p_1,p_2)=4 (\e_2\,\rmp_1-\e_1\,\rmp_2) S^{(0)}(p_1,p_2)$. The denominator on the right-hand side comes from the Jacobian $J(p_1,p_2)$ assuming a standard relativistic dispersion relation (for the theories we consider, at one-loop this is indeed the case), and the one-loop integrals read explicitly $$\begin{aligned}
 I(p_1+p_2)&=\frac{i\pi-{\operatorname{arsinh}}(\e_2\,\rmp_1-\e_1\,\rmp_2)}{4\pi i\,(\e_2\,\rmp_1-\e_1\,\rmp_2)}\,,\label{bubblep}\\
 I(0)&=\frac{1}{4\pi i}\,,\\
 I(p_1-p_2)&=\frac{{\operatorname{arsinh}}(\e_2\,\rmp_1-\e_1\,\rmp_2)}{4\pi i\,(\e_2\,\rmp_1-\e_1\,\rmp_2)}~.\label{bubblem}\end{aligned}$$ For theories including fermionic fields, the above derivation holds up to signs. To be precise the sign prescription is given as follows: $$\begin{aligned}
\no
 {S^{(1)}}{}_{{M}{N}}^{{P}{Q}}(p_1,p_2)=&\frac{1}{8 (\e_2\,\rmp_1-\e_1\,\rmp_2)}\,\Big[{\tilde S^{(0)}}{}_{{M}{N}}^{{R}{S}}(p_1,p_2){\tilde S^{(0)}}{}_{{R}{S}}^{{P}{Q}}(p_1,p_2)I(p_1+p_2)\nonumber\\\label{eqn:final_ferm}  \vphantom{\frac{1}{4 (\e_2\,\rmp_1-\e_1\,\rmp_2)}}
 +&  (-1)^{[{P}][{S}]+[{R}][{S}]}\,{\tilde S^{(0)}}{}_{{M}{R}}^{{S}{P}}(p_1,p_1){\tilde S^{(0)}}{}_{{S}{N}}^{{R}{Q}}(p_1,p_2)I(0) \\ \no
 \vphantom{\frac{1}{4 (\e_2\,\rmp_1-\e_1\,\rmp_2)}}
 +&(-1)^{[{P}][{R}]+[{Q}][{S}]+[{R}][{S}]+[{P}][{Q}]}{\tilde S^{(0)}}{}_{{M}{R}}^{{S}{Q}}(p_1,p_2){\tilde S^{(0)}}{}_{{S}{N}}^{{P}{R}}(p_1,p_2)I(p_1-p_2)\,\Big]~,\end{aligned}$$ where $[{M}] = 0$ for a boson and $1$ for a fermion.

The expressions and above, as is clear from the second term, are not invariant under the interchange of $p_1$ and $p_2$ along with the corresponding flavor indices. Furthermore, if we choose the alternative solution of the conservation $\delta$-function in (\[eqn:tch1\]), namely $\ell_2=\ell_1+p_4-p_2$, 