& +G_3\left(Q^2 \right) \left(q_{\mu} q_{\nu}-q^2 g_{\mu \nu} \right) \mbox{ ,}\end{aligned}$$ in Eq. and using the spin summation formula $$\begin{aligned}
\label{spinsum2}
\sum_{s} \Delta^{(s)}_{\mu}(p) \overline{\Delta}^{(s)}_{\mu}(p) = &-(\slashed{p}-m_{\Delta}) \left[ g_{\mu \nu} - \frac{1}{3}\gamma_{\mu} \gamma_{\nu}-\frac{ 2 p_{\mu} p_{\nu} }{ 3m_{\Delta}^2 } + \frac{ p_{\mu}\gamma_{\nu}-p_{\nu}\gamma_{\mu} }{ 3 m_{\Delta} } \right]\end{aligned}$$ the contribution $T^{\mu \nu}_{\Delta} e_{\nu}^{(\lambda)}$ can be written as $$\begin{aligned}
\label{GoodContribution}
T^{\mu \nu}_{\Delta} e^{\left( \lambda \right)}_ {\nu}=&-\frac{\lambda_{\Delta} \lambda_{P}}{(m_{\Delta}^2-p^2)(m_{P}^2-(p+q)^2)}(\slashed{p}-m_{\Delta}) \times{\nonumber}\\ \ &\left[ g^{\mu}_{\alpha} - \frac{1}{3}\gamma^{\mu} \gamma_{\alpha}-\frac{ 2 p^{\mu} p_{\alpha} }{ 3m_{\Delta}^2 } + \frac{ p^{\mu} \gamma_{\alpha}-p_{\alpha} \gamma^{\mu} }{ 3 m_{\Delta} } \right]\Gamma^{\alpha \nu}\gamma_5 (\slashed{p}+\slashed{q}-m_{P}) \mbox{ .}\end{aligned}$$ However, it is known that the correlation function is plagued by transitions $p \gamma \to N^*$[@Belyaev:1995ya; @Peters], where ${\left| N^* \right\rangle}$ is a isospin-$\frac{3}{2}$ spin-${\frac{1}{2}}$ state with negative parity. It is advantageous to use the Lorentz structures that do not receive contributions due to transitions to spin-${\frac{1}{2}}^{-}$ states. To find these we consider the overlap of ${\left| N^* \right\rangle}$ with $\eta^{\mu}$, which is defined [@Peters] as $${\left\langle 0 \right|} \eta_{\mu}(0){\left| N^*(p,s) \right\rangle}=\frac{\lambda_{N^*}}{(2\pi)^2}(m_{*}\gamma^{\mu}-4p^{\mu}) N^{*{(s)}}(p) \mbox{ ,}$$ where $\lambda_{N^*}$ is the coupling and $m_{*}$ the mass of the spin-$1/2$ state. The spinor $N^{*{(s)}}(p)$ satisfies the Dirac equation $(\slashed{p}-m_{*})N^{*{(s)}}(p)=0$ and equation . Using the general decomposition of the electromagnetic transition matrix element $$\begin{aligned}
