\gamma_5-ep\;\gamma^{\mu}\slashed{q}\slashed{p} \gamma_5+4 \left(qp\;p^{\mu}\slashed{e} \gamma_5-ep\;p^{\mu}\slashed{q} \gamma_5 \right)
 {\nonumber}\\
\mathcal{R}_{2}=&\;p^2 \left(qp\;\gamma^{\mu}\slashed{e} \gamma_5-ep\;\gamma^{\mu} \slashed{q}\gamma_5 \right)+4 \left(qp\;p^{\mu}\slashed{e}\slashed{p} \gamma_5-ep\;p^{\mu} \slashed{q}\slashed{p}\gamma_5 \right)
{\nonumber}\\
\mathcal{R}_{3}= &\;qp\;\gamma^{\mu}\slashed{e} \gamma_5-ep\;\gamma^{\mu} \slashed{q}\gamma_5+2 \left(p^{\mu}\slashed{e}\slashed{q}\gamma_5 \right) -\frac{1}{2} \left( \gamma^{\mu}\slashed{e}\slashed{q}\slashed{p}\gamma_5\right)
{\nonumber}\\
\mathcal{R}_{4}=&\;4 \left(p^{\mu}\slashed{e}\slashed{q}\slashed{p}\gamma_5 \right)-p^2 \left( \gamma^{\mu}\slashed{e}\slashed{q} \gamma_5\right) +2 \left( qp\;\gamma^{\mu}\slashed{e}\slashed{p} \gamma_5-ep\;\gamma^{\mu} \slashed{q}\slashed{p}\gamma_5\right)
{\nonumber}\\
\mathcal{R}_{5}=& \;qp \left( \gamma^{\mu}\slashed{e}\slashed{q} \gamma_5 \right) -4\left(qp\;e^{\mu}\slashed{q} \gamma_5-ep\;q^{\mu} \slashed{q}\gamma_5 \right)
{\nonumber}\\
\mathcal{R}_{6}=&\;\gamma^{\mu}\slashed{e}\slashed{q} \gamma_5-2 \left(e^{\mu} \slashed{q}\gamma_5-q^{\mu} \slashed{e}\gamma_5 \right)
{\nonumber}\\
\mathcal{R}_{7}= &\;qp\;\gamma^{\mu}\slashed{e} \gamma_5-ep\;\gamma^{\mu} \slashed{q}\gamma_5-4 \left(  qp\;e^{\mu} \gamma_5-ep\;q^{\mu} \gamma_5\right)
{\nonumber}\\
\mathcal{R}_{8}=&\;4\left(qp\;e^{\mu}\slashed{q}\slashed{p} \gamma_5-ep\;q^{\mu}\slashed{q}\slashed{p}\gamma_5 \right) - qp \left(\gamma^{\mu}\slashed{e}\slashed{q}\slashed{p}\gamma_5 \right) 
{\nonumber}\\
\mathcal{R}_{9}=&\;2 \left(q^{\mu}\slashed{e}\slashed{p} \gamma_5-e^{\mu} \slashed{q}\slashed{p}\gamma_5 \right) +\gamma^{\mu}\slashed{e}\slashed{q}\slashed{p}\gamma_5
{\nonumber}\\
\mathcal{R}_{10}=&\;4 \left( qp\;e^{\mu}\slashed{p} \gamma_5-ep\;q^{\mu} \slashed{p}\gamma_5 \right)- qp\;\gamma^{\mu}\slashed{e}\slashed{q} \gamma_5+ep\;\gamma^{\mu}\slashed{q}\slashed{p} \gamma_5
{\nonumber}\\
\mathcal{R}_{11}=&\;q^{\mu}\slashed{e}\slashed{q}\slashed{p}\gamma_5
{\nonumber}\\
\label{BASIS}
\mathcal{R}_{12}=&\;q^{\mu}\slashed{e}\slashed{q}\gamma_5 \mbox{.}\end{aligned}$$ Here we inserted the explicit expression for $f_{\mu \nu}$, see Eq., which is advantageous for the further calculation.

### The $p \gamma \to \Delta^{+}$ contribution

The contribution of the nucleon-$\Delta$-transition to the correlation function, Eq., can be obtained readily from Eq.. Using the basis , the result has the form: $$\begin{aligned}
\label{GoodTerm}
T^{\mu \nu}_{\Delta}(p,q) e^{\left( \lambda \right)}_{\nu} =& \;- \lambda_{\Delta} \lambda_P  \frac{1}{\left(m_{\Delta}^2-p^2 \right) \left(m_{P}^2-(p+q)^2 \right)} \
\nonumber \\
&\times\left[ \left( \frac{(p+q)^2 G_{1}(0) }{96 m_{\Delta}^2 \pi^4}-\frac{m_{P} G_{2}(0) }{192 \pi^4}+ \frac{(p+q)^2 G_{2}(0) }{192 m_{\Delta} \pi^4} \right)\cdot  \mathcal{R}_{1} \right. 
\nonumber \\
&+ \left(  -\frac{m_{P} G_{1}(0)}{96 m_{\Delta}^2 \pi^4}+\frac{ G_{2}(0)}{192 \pi^4}-\frac{m_{P} G_{2}(0)}{192 m_{\Delta} \pi^4}\right)\cdot  \mathcal{R}_{2}
\nonumber \\
&+\left( -\frac{m_{P}G_{1}(0)}{48 