solutions to Eqs. (\[ireeqed\], \[ireem\]) are known. With the notations that we have used they can be taken directly from Ref. [@egt]. Hence, we are left with the only unknown amplitude ${A'}_j$ in Eq. (\[iree\]). Using Eqs. (\[ireeqed\], \[ireem\]), we arrive at the IREE for ${A'}_j$, namely: $$\label{ireeaprime}
{A'}_j(s/M^2, \eta) = \tilde{A}_j(s/M^2) - \tilde{A}_j^{(QED)}(s/M^2) + 
{G'}_j(s/M^2, \eta) + {Q'}_j(s/M^2, \eta) ~.$$

In order to solve Eq. (\[ireeaprime\]), it is more convenient to use the Sommerfeld-Watson transform. As long as one considers the positive signature amplitudes, this transform formally coincides with the Mellin transform. It is convenient to use different forms of this transform for the invariant amplitudes we consider: $$\begin{aligned}
\label{mellinaqed}
A_j^{(QED)}(s/ \mu^2) &=& 
\int_{-\imath \infty}^{\imath \infty} \frac{d \omega}{2\pi\imath}\,
\Big( \frac {s}{\mu^2}\Big)^{\omega} f_j^{(0)} (\omega) ~, 
\\ \nonumber \\
\label{mellinaqedtilde}
\tilde{A}_j^{(QED)}(s/ M^2) &=& 
\int_{-\imath \infty}^{\imath \infty} \frac{d \omega}{2\pi\imath}\,
\Big( \frac {s}{M^2}\Big)^{\omega} f_j^{(0)}(\omega) ~,
\\ \nonumber \\
\label{mellinatilde}
 \tilde{A}_j(s/ M^2) &=& 
\int_{-\imath \infty}^{\imath \infty} \frac{d \omega}{2\pi\imath}\,
\Big( \frac {s}{M^2}\Big)^{\omega} f_j(\omega) ~,
\\ \nonumber \\
\label{mellinaprime}
{A'}_j(s/M^2, \eta) &=& 
\int_{-\imath \infty}^{\imath \infty} \frac{d \omega}{2\pi\imath}\,
\Big( \frac {s}{M^2}\Big)^{\omega} F_j (\omega, \eta) ~. \end{aligned}$$

Combining Eqs. (\[mellinaqed\]) to (\[mellinaprime\]) with Eq. (\[ireeaprime\]) we arrive at the following equation for the Mellin amplitude $F_j (\omega, \eta)$: 