There are actually two things I wanted to address. The first shows up on page 16 of the pdf. I think I recall seeing somewhere that there is a way to code to print across a natural page break? I think this is why the equations are blocking up like this. If not, I can break it up manually, if I have to. Just checking.
The other thing is essentially an opinion. I had to break the lines of the equations up; it's QFT, the equations sometimes just get long. (For example, again please see page 16.) There are a zillion ways to break up a line, and this one doesn't look half bad. Aesthetically speaking, does anyone have suggestions how I might be able to make this look a bit more professional? And, before anyone asks, yes I would like to keep all those steps!
(Sorry, I can't figure out how to get the packages to load properly. I've really got to learn how to do that! I've provided some of the code merely for completeness and if anyone wants to try any edits to show me something. I don't know how it would look printed out: it's just a collection of the coding without any of the text to go with it. I split the lines of the equations using arrays. I couldn't get aligned to look right because the first line just covered too much of the line.)
Thanks for any suggestions!
-Dan
Code: Select all
$\begin{equation}
\begin{array}{l}
{\displaystyle \int\dfrac{d^{\,4}q}{(2\pi^{4})\sqrt{2q_{0}}}\,\bigg(A_{4}^{\nu*}(\textbf{k}_{4})(2\pi)^{4}G\gamma^{\nu}\delta^{4}(q+k_{2}-k_{4})A_{2}^{\nu}(\textbf{k}_{2})\bigg)\dfrac{-i}{(2\pi)^{4}}\dfrac{1}{\sqrt{\textbf{q}^{2}-m^{2}}}}\\
\hphantom{XXXX}\times\bigg(A_{3}^{\mu*}(\textbf{k}_{3})(2\pi)^{4}G\gamma^{\mu}\delta^{4}(k_{1}-q-k_{3})A_{1}^{\mu}(\textbf{k}_{1})\bigg)
\end{array}
\end{equation}$
$\begin{equation}
\begin{array}{l}
{\displaystyle \int\dfrac{d^{\,4}q}{(2\pi^{4})\sqrt{2q_{0}}}\,\bigg(A_{4}^{\nu*}(\textbf{k}_{3})(2\pi)^{4}G\gamma^{\nu}\delta^{4}(q+k_{2}-k_{3})A_{2}^{\nu}(\textbf{k}_{2})\bigg)\dfrac{-i}{(2\pi)^{4}}\dfrac{1}{\sqrt{\textbf{q}^{2}-m^{2}}}}\\
\hphantom{XXXX}\times\bigg(A_{4}^{\mu*}(\textbf{k}_{4})(2\pi)^{4}G\gamma^{\mu}\delta^{4}(k_{1}-q-k_{3})A_{1}^{\mu}(\textbf{k}_{1})\bigg)
\end{array}
\end{equation}$
$\begin{equation}
\begin{array}{l}
{\displaystyle \int\dfrac{d^{\,4}q}{(2\pi)^{4}\sqrt{2q_{0}}}[\overline{u}_{4}(2\pi)^{4}ie\gamma^{\nu}\delta^{4}(q+p_{2}-p_{4})u_{2}]\dfrac{-i}{(2\pi)^{4}}\dfrac{\eta_{\mu\nu}}{q^{2}-i\epsilon}}\\
\hphantom{XXXX}\times[\overline{u}_{3}(2\pi)^{4}ie\gamma^{\mu}\delta^{4}(p_{1}-q-p_{3})u_{1}]
\end{array}
\end{equation}$
$\begin{equation}
{\displaystyle \begin{array}{l}
\int d^{\,4}q\left[\dfrac{\overline{u}_{4}}{(2\pi)^{3/2}}i(2\pi)^{4}e\gamma^{\nu}\delta^{4}(q+k_{2}-p_{4})\dfrac{\epsilon_{2\nu}}{(2\pi)^{3/2}\sqrt{2k_{2}^{0}}}\right]\dfrac{-i}{(2\pi)^{4}}\dfrac{-i\not q+m}{q^{2}+m^{2}-i\epsilon}\\
\hphantom{XXXX}\times\left[\dfrac{\epsilon_{3\mu}^{*}}{(2\pi)^{3/2}\sqrt{2k_{3}^{0}}}i(2\pi)^{4}e\gamma^{\mu}\delta^{4}(p_{1}-q-k_{4})\dfrac{u_{1}}{(2\pi)^{3/2}}\right]
\end{array}}
\end{equation}$
$\begin{equation}
{\displaystyle \begin{aligned}\sum_{s_{\alpha},s_{\beta}}[\overline{u}_{\alpha}\Gamma u_{\beta}]^{*}[\overline{u}_{\alpha}\Gamma u_{\beta}] & =\sum_{s_{\beta}}\left[(\overline{u}_{\beta})^{i}\bigg(\left(\gamma^{0}\Gamma^{\dagger}\gamma^{0}\right)(\not\!p_{\alpha}+m)\Gamma\bigg)_{i}^{k}(u_{\beta})_{k}\right]\\
& =\bigg(\left(\gamma^{0}\Gamma^{\dagger}\gamma^{0}\right)(\not\!p_{\alpha}+m)\Gamma\bigg)_{i}^{k}\left\{ \sum_{s_{\beta}}\left[(\overline{u}_{\beta})^{i}(u_{\beta})_{k}\right]\right\} \\
& =\bigg(\left(\gamma^{0}\Gamma^{\dagger}\gamma^{0}\right)(\not\!p_{\alpha}+m)\Gamma\bigg)_{i}^{k}\left\{ \sum_{s_{\beta}}\left[(u_{\beta})_{k}(\overline{u}_{\beta})^{i}\right]\right\} \\
& =\bigg(\left(\gamma^{0}\Gamma^{\dagger}\gamma^{0}\right)(\not\!p_{\alpha}+m)\Gamma\bigg)_{i}^{k}\left\{ (\not\!p_{\beta}+m)_{k}^{i}\right\} \\
& =Tr\left[\left(\gamma^{0}\Gamma^{\dagger}\gamma^{0}\right)(\not\!p_{\alpha}+m)\Gamma(\not\!p_{\beta}+m)\right]
\end{aligned}
}
\end{equation}$
$\begin{equation}
\begin{aligned}Tr\left[\gamma^{\alpha}\gamma^{\beta}\gamma^{\delta}\gamma^{\epsilon}\right] & =Tr\left[\left(2\eta^{\alpha\beta}-\gamma^{\beta}\gamma^{\alpha}\right)\gamma^{\delta}\gamma^{\epsilon}\right]\\
& =2\eta^{\alpha\beta}Tr\left[\gamma^{\delta}\gamma^{\epsilon}\right]-Tr\left[\gamma^{\beta}\gamma^{\alpha}\gamma^{\delta}\gamma^{\epsilon}\right]\\
& =2\eta^{\alpha\beta}Tr\left[\gamma^{\delta}\gamma^{\epsilon}\right]-Tr\left[\gamma^{\beta}\left(2eta^{\alpha\delta}-\gamma^{\delta}\gamma^{\alpha}\right)\gamma^{\epsilon}\right]\\
& =2\eta^{\alpha\beta}Tr\left[\gamma^{\delta}\gamma^{\epsilon}\right]-2\eta^{\alpha\delta}Tr\left[\gamma^{\beta}\gamma^{\epsilon}\right]+Tr\left[\gamma^{\beta}\gamma^{\delta}\gamma^{\alpha}\gamma^{\epsilon}\right]\\
& =2\eta^{\alpha\beta}Tr\left[\gamma^{\delta}\gamma^{\epsilon}\right]-2\eta^{\alpha\delta}Tr\left[\gamma^{\beta}\gamma^{\epsilon}\right]+Tr\left[\gamma^{\beta}\gamma^{\delta}\left(2\eta^{\alpha\epsilon}-\gamma^{\epsilon}\gamma^{\alpha}\right)\right]\\
& =2\eta^{\alpha\beta}Tr\left[\gamma^{\delta}\gamma^{\epsilon}\right]-2\eta^{\alpha\delta}Tr\left[\gamma^{\beta}\gamma^{\epsilon}\right]+2\eta^{\alpha\epsilon}Tr\left[\gamma^{\beta}\gamma^{\delta}\right]-Tr\left[\gamma^{\beta}\gamma^{\delta}\gamma^{\epsilon}\gamma^{\alpha}\right]
\end{aligned}
\end{equation}$
$\begin{equation}
\mathcal{M}=-\dfrac{e^{2}}{(p_{1}-p_{3})^{2}}[\overline{u}_{4}\gamma_{\mu}u_{2}][\overline{u}_{3}\gamma^{\mu}u_{1}]+\dfrac{e^{2}}{(p_{1}-p_{4})^{2}}[\overline{u}_{3}\gamma_{\mu}u_{2}][\overline{u}_{4}\gamma^{\mu}u_{1}]
\end{equation}$
$\begin{equation}
{\displaystyle \begin{aligned}\left\langle \left|\mathcal{M}\right|^{2}\right\rangle & =\dfrac{1}{4}\sum_{s_{1},s_{2},s_{3},s_{4}}\dfrac{e^{4}}{(p_{1}-p_{3})^{4}}[\overline{u}_{4}\gamma_{\mu}u_{2}]^{*}[\overline{u}_{3}\gamma^{\mu}u_{1}]^{*}[\overline{u}_{4}\gamma_{\nu}u_{2}][\overline{u}_{3}\gamma^{\nu}u_{1}]\\
& \hphantom{XXXX}-\dfrac{e^{4}}{(p_{1}-p_{3})^{2}(p_{1}-p_{4})^{2}}[\overline{u}_{4}\gamma_{\mu}u_{2}]^{*}[\overline{u}_{3}\gamma^{\mu}u_{1}]^{*}[\overline{u}_{3}\gamma_{\nu}u_{2}][\overline{u}_{4}\gamma^{\nu}u_{1}]\\
& \hphantom{XXXX}-\dfrac{e^{4}}{(p_{1}-p_{4})^{2}(p_{1}-p_{3})^{2}}[\overline{u}_{3}\gamma_{\mu}u_{2}]^{*}[\overline{u}_{4}\gamma^{\mu}u_{1}]^{*}[\overline{u}_{4}\gamma_{\nu}u_{2}][\overline{u}_{3}\gamma^{\nu}u_{1}]\\
& \hphantom{XXXX}+\dfrac{e^{4}}{(p_{1}-p_{4})^{4}}[\overline{u}_{3}\gamma_{\mu}u_{2}]^{*}[\overline{u}_{4}\gamma^{\mu}u_{1}]^{*}[\overline{u}_{3}\gamma_{\mu}u_{2}][\overline{u}_{4}\gamma^{\mu}u_{1}]
\end{aligned}
}
\end{equation}$
$\begin{equation}
{\displaystyle \begin{array}{l}
\sum_{s_{1},s_{2},s_{3},s_{4}}[\overline{u}_{4}\gamma_{\mu}u_{2}]^{*}[\overline{u}_{3}\gamma^{\mu}u_{1}]^{*}[\overline{u}_{4}\gamma_{\nu}u_{2}][\overline{u}_{3}\gamma^{\nu}u_{1}]\\
\hphantom{XXXX}=\sum_{s_{1},s_{2},s_{3},s_{4}}[\overline{u}_{2}\left(\gamma^{0}\gamma_{\mu}^{\dagger}\gamma^{0}\right)u_{4}][\overline{u}_{1}\left(\gamma^{0}\gamma^{\mu\dagger}\gamma^{0}\right)u_{3}][\overline{u}_{4}\gamma_{\nu}u_{2}][\overline{u}_{3}\gamma^{\nu}u_{1}]
\end{array}}
\end{equation}$
$\begin{equation}
{\displaystyle \sum_{s_{1},s_{2},s_{3},s_{4}}[\overline{u}_{4}\gamma_{\mu}u_{2}]^{*}[\overline{u}_{3}\gamma^{\mu}u_{1}]^{*}[\overline{u}_{4}\gamma_{\nu}u_{2}][\overline{u}_{3}\gamma^{\nu}u_{1}]=\sum_{s_{1},s_{2},s_{3},s_{4}}[\overline{u}_{2}\gamma_{\mu}u_{4}][\overline{u}_{1}\gamma^{\mu}u_{3}][\overline{u}_{4}\gamma_{\nu}u_{2}][\overline{u}_{3}\gamma^{\nu}u_{1}]}
\end{equation}$
$\begin{equation}
\begin{array}{l}
{\displaystyle \sum_{s_{1},s_{2},s_{3},s_{4}}[\overline{u}_{4}\gamma_{\mu}u_{2}]^{*}[\overline{u}_{3}\gamma^{\mu}u_{1}]^{*}[\overline{u}_{4}\gamma_{\nu}u_{2}][\overline{u}_{3}\gamma^{\nu}u_{1}]}\\
\hphantom{XXXXXXXX}={\displaystyle \sum_{s_{1},s_{2},s_{3},s_{4}}[\overline{u}_{1}\gamma^{\mu}u_{3}][\overline{u}_{3}\gamma^{\nu}u_{1}][\overline{u}_{2}\gamma_{\mu}u_{4}][\overline{u}_{4}\gamma_{\nu}u_{2}]}\\
\hphantom{XXXXXXXX}={\displaystyle \sum_{s_{1},s_{2}}[\overline{u}_{1}\gamma^{\mu}\left(\sum_{s_{3}}u_{3}\overline{u}_{3}\right)\gamma^{\nu}u_{1}][\overline{u}_{2}\gamma_{\mu}\left(\sum_{s_{4}}u_{4}\overline{u}_{4}\right)\gamma_{\nu}u_{2}]}\\
\hphantom{XXXXXXXX}={\displaystyle \sum_{s_{1},s_{2}}[\overline{u}_{1}\gamma^{\mu}(\not\!p_{3}+m)\gamma^{\nu}u_{1}][\overline{u}_{2}\gamma_{\mu}(\not\!p_{4}+m)\gamma_{\nu}u_{2}]}\\
\hphantom{XXXXXXXX}={\displaystyle \left(\sum_{s_{1}}[\overline{u}_{1}\gamma^{\mu}(\not\!p_{3}+m)\gamma^{\nu}u_{1}]\right)\left(\sum_{s_{2}}[\overline{u}_{2}\gamma_{\mu}(\not\!p_{4}+m)\gamma_{\nu}u_{2}]\right)}\\
\hphantom{XXXXXXXX}=Tr[\gamma^{\mu}(\not\!p_{3}+m)\gamma^{\nu}(\not\!p_{1}+m)]\,Tr[\gamma_{\mu}(\not\!p_{4}+m)\gamma_{\nu}(\not\!p_{2}+m)]
\end{array}
\end{equation}$
$\begin{equation}
\begin{array}{l}
{\displaystyle \sum_{s_{1},s_{2},s_{3},s_{4}}[\overline{u}_{4}\gamma_{\mu}u_{2}]^{*}[\overline{u}_{3}\gamma^{\mu}u_{1}]^{*}[\overline{u}_{4}\gamma_{\nu}u_{2}][\overline{u}_{3}\gamma^{\nu}u_{1}]}\\
\hphantom{XXXX}=p_{1\alpha}p_{3\delta}(4\eta^{\mu\delta}\eta^{\nu\alpha}-4\eta^{\mu\nu}\eta^{\delta\alpha}+4\eta^{\mu\alpha}\eta^{\delta\nu})p_{2\beta}p_{4\epsilon}(4\eta_{\mu}^{\epsilon}\eta_{\nu}^{\beta}-4\eta_{\mu\nu}\eta^{\epsilon\beta}+4\eta_{\mu}^{\beta}\eta_{\nu}^{\epsilon})\\
\hphantom{XXXX}=16(p_{1}^{\nu}p_{3}^{\mu}-\eta^{\mu\nu}(p_{1}\cdot p_{3})+p_{1}^{\mu}p_{3}^{\nu})(p_{2\nu}p_{4\mu}-\eta_{\mu\nu}(p_{2}\cdot p_{4})+p_{2\mu}p_{4\nu})\\
\hphantom{XXXX}=32(p_{1}\cdot p_{2})(p_{3}\cdot p_{4})+32(p_{1}\cdot p_{4})(p_{2}\cdot p_{3})
\end{array}
\end{equation}$
$\begin{array}{l}
{\displaystyle \sum_{s_{1},s_{2},s_{3},s_{4}}[\overline{u}_{4}\gamma_{\mu}u_{2}]^{*}[\overline{u}_{3}\gamma^{\mu}u_{1}]^{*}[\overline{u}_{4}\gamma_{\nu}u_{2}][\overline{u}_{3}\gamma^{\nu}u_{1}]}\\
\hphantom{XXXX}=p_{1\alpha}p_{3\delta}Tr[\gamma^{\mu}\gamma^{\delta}\gamma^{\nu}\gamma^{\alpha}]\,p_{2\beta}p_{4\epsilon}Tr[\gamma_{\mu}\gamma^{\epsilon}\gamma_{\nu}\gamma^{\beta})]\\
\hphantom{XXXX}=p_{1\alpha}p_{3\delta}(4\eta^{\mu\delta}\eta^{\nu\alpha}-4\eta^{\mu\nu}\eta^{\delta\alpha}+4\eta^{\mu\alpha}\eta^{\delta\nu})p_{2\beta}p_{4\epsilon}(4\eta_{\mu}^{\epsilon}\eta_{\nu}^{\beta}-4\eta_{\mu\nu}\eta^{\epsilon\beta}+4\eta_{\mu}^{\beta}\eta_{\nu}^{\epsilon})\\
\hphantom{XXXX}=16(p_{1}^{\nu}p_{3}^{\mu}-\eta^{\mu\nu}(p_{1}\cdot p_{3})+p_{1}^{\mu}p_{3}^{\nu})(p_{2\nu}p_{4\mu}-\eta_{\mu\nu}(p_{2}\cdot p_{4})+p_{2\mu}p_{4\nu})\\
\hphantom{XXXX}=32(p_{1}\cdot p_{2})(p_{3}\cdot p_{4})+32(p_{1}\cdot p_{4})(p_{2}\cdot p_{3})
\end{array}
\end{equation}$
$\begin{equation}
\begin{array}{l}
{\displaystyle \sum_{s_{1},s_{2},s_{3},s_{4}}[\overline{u}_{4}\gamma_{\mu}u_{2}]^{*}[\overline{u}_{3}\gamma^{\mu}u_{1}]^{*}[\overline{u}_{3}\gamma_{\nu}u_{2}][\overline{u}_{4}\gamma^{\nu}u_{1}]}\\
\hphantom{XXXX}={\displaystyle \sum_{s_{1},s_{2},s_{3},s_{4}}[\overline{u}_{2}\left(\gamma^{0}\gamma_{\mu}^{\dagger}\gamma^{0}\right)u_{4}][\overline{u}_{1}\left(\gamma^{0}\gamma^{\mu\dagger}\gamma^{0}\right)u_{3}][\overline{u}_{3}\gamma_{\nu}u_{2}][\overline{u}_{4}\gamma^{\nu}u_{1}]}
\end{array}
\end{equation}$
$\begin{equation}
\begin{array}{l}
{\displaystyle \sum_{s_{1},s_{2},s_{3},s_{4}}[\overline{u}_{4}\gamma_{\mu}u_{2}]^{*}[\overline{u}_{3}\gamma^{\mu}u_{1}]^{*}[\overline{u}_{3}\gamma_{\nu}u_{2}][\overline{u}_{4}\gamma^{\nu}u_{1}]}\\
\hphantom{XXXX}={\displaystyle \sum_{s_{1},s_{2},s_{3},s_{4}}[\overline{u}_{2}\left(\gamma^{0}\gamma_{\mu}^{\dagger}\gamma^{0}\right)u_{4}][\overline{u}_{4}\gamma^{\nu}u_{1}][\overline{u}_{1}\left(\gamma^{0}\gamma^{\mu\dagger}\gamma^{0}\right)u_{3}][\overline{u}_{3}\gamma_{\nu}u_{2}]}\\
\hphantom{XXXX}={\displaystyle \sum_{s_{2}}[\overline{u}_{2}\gamma_{\mu}\left(\sum_{s_{4}}u_{4}\overline{u}_{4}\right)\gamma^{\nu}\left(\sum_{s_{1}}u_{1}\overline{u}_{1}\right)\gamma^{\mu}\left(\sum_{s_{3}}u_{3}\overline{u}_{3}\right)\gamma_{\nu}u_{2}]}\\
\hphantom{XXXX}={\displaystyle \sum_{s_{2}}[\overline{u}_{2}\gamma_{\mu}(\not\!p_{4}+m)\gamma^{\nu}(\not\!p_{1}+m)\gamma^{\mu}(\not\!p_{3}+m)\gamma_{\nu}u_{2}]}\\
\hphantom{XXXX}=Tr[\gamma_{\mu}(\not\!p_{4}+m)\gamma^{\nu}(\not\!p_{1}+m)\gamma^{\mu}(\not\!p_{3}+m)\gamma_{\nu}(\not\!p_{2}+m)]
\end{array}
\end{equation}$
$\begin{equation}
\begin{array}{l}
{\displaystyle \sum_{s_{1},s_{2},s_{3},s_{4}}[\overline{u}_{4}\gamma_{\mu}u_{2}]^{*}[\overline{u}_{3}\gamma^{\mu}u_{1}]^{*}[\overline{u}_{3}\gamma_{\nu}u_{2}][\overline{u}_{4}\gamma^{\nu}u_{1}]}\\
\hphantom{XXXX}=p_{1\alpha}p_{2\beta}p_{3\delta}p_{4\epsilon}Tr[\gamma_{\mu}\gamma^{\epsilon}\gamma^{\nu}\gamma^{\alpha}\gamma^{\mu}\gamma^{\delta}\gamma_{\nu}\gamma^{\beta}]\\
\hphantom{XXXX}=p_{1\alpha}p_{2\beta}p_{3\delta}p_{4\epsilon}\bigg(2Tr[\gamma^{\nu}\gamma^{\alpha}\gamma^{\epsilon}\gamma^{\delta}\gamma_{\nu}\gamma^{\beta}]-Tr[\gamma^{\epsilon}\gamma_{\mu}\gamma^{\nu}\gamma^{\alpha}\gamma^{\mu}\gamma^{\delta}\gamma_{\nu}\gamma^{\beta}]\bigg)\\
\hphantom{XXXX}=p_{1\alpha}p_{2\beta}p_{3\delta}p_{4\epsilon}\bigg(2Tr[\gamma^{\nu}\gamma^{\alpha}\gamma^{\epsilon}\gamma^{\delta}\gamma_{\nu}\gamma^{\beta}]-2Tr[\gamma^{\epsilon}\gamma^{\alpha}\gamma^{\nu}\gamma^{\delta}\gamma_{\nu}\gamma^{\beta}]+Tr[\gamma^{\epsilon}\gamma^{\nu}\gamma_{\mu}\gamma^{\alpha}\gamma^{\mu}\gamma^{\delta}\gamma_{\nu}\gamma^{\beta}]\bigg)\\
\hphantom{XXXX}=p_{1\alpha}p_{2\beta}p_{3\delta}p_{4\epsilon}\bigg(2Tr[\gamma^{\nu}\gamma^{\alpha}\gamma^{\epsilon}\gamma^{\delta}\gamma_{\nu}\gamma^{\beta}]-2Tr[\gamma^{\epsilon}\gamma^{\alpha}\gamma^{\nu}\gamma^{\delta}\gamma_{\nu}\gamma^{\beta}]+2Tr[\gamma^{\epsilon}\gamma^{\nu}\gamma^{\alpha}\gamma^{\delta}\gamma_{\nu}\gamma^{\beta}]\\
\hphantom{XXXXXXXXXXXXXXXX}-Tr[\gamma^{\epsilon}\gamma^{\nu}\gamma^{\alpha}\gamma_{\mu}\gamma^{\mu}\gamma^{\delta}\gamma_{\nu}\gamma^{\beta}]\bigg)\\
\hphantom{XXXX}=p_{1\alpha}p_{2\beta}p_{3\delta}p_{4\epsilon}\bigg(2Tr[\gamma^{\nu}\gamma^{\alpha}\gamma^{\epsilon}\gamma^{\delta}\gamma_{\nu}\gamma^{\beta}]-2Tr[\gamma^{\epsilon}\gamma^{\alpha}\gamma^{\nu}\gamma^{\delta}\gamma_{\nu}\gamma^{\beta}]+2Tr[\gamma^{\epsilon}\gamma^{\nu}\gamma^{\alpha}\gamma^{\delta}\gamma_{\nu}\gamma^{\beta}]\\
\hphantom{XXXXXXXXXXXXXXXX}-4Tr[\gamma^{\epsilon}\gamma^{\nu}\gamma^{\alpha}\gamma^{\delta}\gamma_{\nu}\gamma^{\beta}]\bigg)\\
\hphantom{XXXX}=p_{1\alpha}p_{2\beta}p_{3\delta}p_{4\epsilon}\bigg(2Tr[\gamma^{\nu}\gamma^{\alpha}\gamma^{\epsilon}\gamma^{\delta}\gamma_{\nu}\gamma^{\beta}]-2Tr[\gamma^{\epsilon}\gamma^{\alpha}\gamma^{\nu}\gamma^{\delta}\gamma_{\nu}\gamma^{\beta}]-2Tr[\gamma^{\epsilon}\gamma^{\nu}\gamma^{\alpha}\gamma^{\delta}\gamma_{\nu}\gamma^{\beta}]\bigg)
\end{array}
\end{equation}$
$\begin{equation}
\begin{array}{l}
{\displaystyle \sum_{s_{1},s_{2},s_{3},s_{4}}[\overline{u}_{4}\gamma_{\mu}u_{2}]^{*}[\overline{u}_{3}\gamma^{\mu}u_{1}]^{*}[\overline{u}_{3}\gamma_{\nu}u_{2}][\overline{u}_{4}\gamma^{\nu}u_{1}]}\\
\hphantom{XXXX}=p_{1\alpha}p_{2\beta}p_{3\delta}p_{4\epsilon}\bigg(\left\{ 4Tr[\gamma^{\beta}\gamma^{\alpha}\gamma^{\epsilon}\gamma^{\delta}]-2Tr[\gamma^{\nu}\gamma^{\alpha}\gamma^{\epsilon}\gamma^{\delta}\gamma^{\beta}\gamma_{\nu}]\right\} -2Tr[\gamma^{\epsilon}\gamma^{\alpha}\gamma^{\nu}\gamma^{\delta}\gamma_{\nu}\gamma^{\beta}]\\
\hphantom{XXXXXXXXXXXXXXXX}-2Tr[\gamma^{\epsilon}\gamma^{\nu}\gamma^{\alpha}\gamma^{\delta}\gamma_{\nu}\gamma^{\beta}]\bigg)\\
\hphantom{XXXX}=p_{1\alpha}p_{2\beta}p_{3\delta}p_{4\epsilon}\bigg(\left\{ 4Tr[\gamma^{\beta}\gamma^{\alpha}\gamma^{\epsilon}\gamma^{\delta}]-2Tr[\gamma_{\nu}\gamma^{\nu}\gamma^{\alpha}\gamma^{\epsilon}\gamma^{\delta}\gamma^{\beta}]\right\} -2Tr[\gamma^{\epsilon}\gamma^{\alpha}\gamma^{\nu}\gamma^{\delta}\gamma_{\nu}\gamma^{\beta}]\\
\hphantom{XXXXXXXXXXXXXXXX}-2Tr[\gamma^{\epsilon}\gamma^{\nu}\gamma^{\alpha}\gamma^{\delta}\gamma_{\nu}\gamma^{\beta}]\bigg)\\
\hphantom{XXXX}=p_{1\alpha}p_{2\beta}p_{3\delta}p_{4\epsilon}\bigg(\left\{ 4Tr[\gamma^{\beta}\gamma^{\alpha}\gamma^{\epsilon}\gamma^{\delta}]-8Tr[\gamma^{\alpha}\gamma^{\epsilon}\gamma^{\delta}\gamma^{\beta}]\right\} -2Tr[\gamma^{\epsilon}\gamma^{\alpha}\gamma^{\nu}\gamma^{\delta}\gamma_{\nu}\gamma^{\beta}]\\
\hphantom{XXXXXXXXXXXXXXXX}-2Tr[\gamma^{\epsilon}\gamma^{\nu}\gamma^{\alpha}\gamma^{\delta}\gamma_{\nu}\gamma^{\beta}]\bigg)\\
\hphantom{XXXX}=p_{1\alpha}p_{2\beta}p_{3\delta}p_{4\epsilon}\bigg(-4Tr[\gamma^{\beta}\gamma^{\alpha}\gamma^{\epsilon}\gamma^{\delta}]-2Tr[\gamma^{\epsilon}\gamma^{\alpha}\gamma^{\nu}\gamma^{\delta}\gamma_{\nu}\gamma^{\beta}]-2Tr[\gamma^{\epsilon}\gamma^{\nu}\gamma^{\alpha}\gamma^{\delta}\gamma_{\nu}\gamma^{\beta}]\bigg)\\
\hphantom{XXXX}=p_{1\alpha}p_{2\beta}p_{3\delta}p_{4\epsilon}\bigg(-4Tr[\gamma^{\beta}\gamma^{\alpha}\gamma^{\epsilon}\gamma^{\delta}]+\left\{ -4Tr[\gamma^{\epsilon}\gamma^{\alpha}\gamma^{\delta}\gamma^{\beta}]+2Tr[\gamma^{\epsilon}\gamma^{\alpha}\gamma^{\nu}\gamma_{\nu}\gamma^{\delta}\gamma^{\beta}]\right\} \\
\hphantom{XXXXXXXXXXXXXXXX}-2Tr[\gamma^{\epsilon}\gamma^{\nu}\gamma^{\alpha}\gamma^{\delta}\gamma_{\nu}\gamma^{\beta}]\bigg)\\
\hphantom{XXXX}=p_{1\alpha}p_{2\beta}p_{3\delta}p_{4\epsilon}\bigg(-4Tr[\gamma^{\beta}\gamma^{\alpha}\gamma^{\epsilon}\gamma^{\delta}]+\left\{ -4Tr[\gamma^{\epsilon}\gamma^{\alpha}\gamma^{\delta}\gamma^{\beta}]+8Tr[\gamma^{\epsilon}\gamma^{\alpha}\gamma^{\delta}\gamma^{\beta}]\right\} \\
\hphantom{XXXXXXXXXXXXXXXX}-2Tr[\gamma^{\epsilon}\gamma^{\nu}\gamma^{\alpha}\gamma^{\delta}\gamma_{\nu}\gamma^{\beta}]\bigg)\\
\hphantom{XXXX}=p_{1\alpha}p_{2\beta}p_{3\delta}p_{4\epsilon}\bigg(-4Tr[\gamma^{\beta}\gamma^{\alpha}\gamma^{\epsilon}\gamma^{\delta}]+4Tr[\gamma^{\epsilon}\gamma^{\alpha}\gamma^{\delta}\gamma^{\beta}]-2Tr[\gamma^{\epsilon}\gamma^{\nu}\gamma^{\alpha}\gamma^{\delta}\gamma_{\nu}\gamma^{\beta}]\bigg)\\
\hphantom{XXXX}=p_{1\alpha}p_{2\beta}p_{3\delta}p_{4\epsilon}\bigg(-4Tr[\gamma^{\beta}\gamma^{\alpha}\gamma^{\epsilon}\gamma^{\delta}]+4Tr[\gamma^{\epsilon}\gamma^{\alpha}\gamma^{\delta}\gamma^{\beta}]+\left\{ -4Tr[\gamma^{\epsilon}\gamma^{\delta}\gamma^{\alpha}\gamma^{\beta}]\right\} \bigg)\\
\hphantom{XXXX}=p_{1\alpha}p_{2\beta}p_{3\delta}p_{4\epsilon}\bigg(-4Tr[\gamma^{\beta}\gamma^{\alpha}\gamma^{\epsilon}\gamma^{\delta}]+4Tr[\gamma^{\epsilon}\gamma^{\alpha}\gamma^{\delta}\gamma^{\beta}]\\
\hphantom{XXXXXXXX}+\left\{ -4Tr[\gamma^{\epsilon}\gamma^{\delta}\gamma^{\alpha}\gamma^{\beta}]+4Tr[\gamma^{\epsilon}\gamma^{\alpha}\gamma^{\delta}\gamma^{\beta}]-2Tr[\gamma^{\epsilon}\gamma^{\nu}\gamma_{\nu}\gamma^{\alpha}\gamma^{\delta}\gamma^{\beta}]\right\} \bigg)\\
\hphantom{XXXX}=p_{1\alpha}p_{2\beta}p_{3\delta}p_{4\epsilon}\bigg(-4Tr[\gamma^{\beta}\gamma^{\alpha}\gamma^{\epsilon}\gamma^{\delta}]+4Tr[\gamma^{\epsilon}\gamma^{\alpha}\gamma^{\delta}\gamma^{\beta}]\\
\hphantom{XXXXXXXX}+\left\{ -4Tr[\gamma^{\epsilon}\gamma^{\delta}\gamma^{\alpha}\gamma^{\beta}]+4Tr[\gamma^{\epsilon}\gamma^{\alpha}\gamma^{\delta}\gamma^{\beta}]-8Tr[\gamma^{\epsilon}\gamma^{\alpha}\gamma^{\delta}\gamma^{\beta}]\right\} \bigg)\\
\hphantom{XXXX}=p_{1\alpha}p_{2\beta}p_{3\delta}p_{4\epsilon}\bigg(-4Tr[\gamma^{\beta}\gamma^{\alpha}\gamma^{\epsilon}\gamma^{\delta}]+4Tr[\gamma^{\epsilon}\gamma^{\alpha}\gamma^{\delta}\gamma^{\beta}]-4Tr[\gamma^{\epsilon}\gamma^{\delta}\gamma^{\alpha}\gamma^{\beta}]-4Tr[\gamma^{\epsilon}\gamma^{\alpha}\gamma^{\delta}\gamma^{\beta}]\bigg)
\end{array}
\end{equation}$
$\begin{equation}
\begin{array}{l}
{\displaystyle \sum_{s_{1},s_{2},s_{3},s_{4}}[\overline{u}_{4}\gamma_{\mu}u_{2}]^{*}[\overline{u}_{3}\gamma^{\mu}u_{1}]^{*}[\overline{u}_{3}\gamma_{\nu}u_{2}][\overline{u}_{4}\gamma^{\nu}u_{1}]}\\
\hphantom{XXXX}=p_{1\alpha}p_{2\beta}p_{3\delta}p_{4\epsilon}\bigg(\left\{ 4Tr[\gamma^{\beta}\gamma^{\alpha}\gamma^{\epsilon}\gamma^{\delta}]-2Tr[\gamma^{\nu}\gamma^{\alpha}\gamma^{\epsilon}\gamma^{\delta}\gamma^{\beta}\gamma_{\nu}]\right\} -2Tr[\gamma^{\epsilon}\gamma^{\alpha}\gamma^{\nu}\gamma^{\delta}\gamma_{\nu}\gamma^{\beta}]\\
\hphantom{XXXXXXXXXXXXXXXX}-2Tr[\gamma^{\epsilon}\gamma^{\nu}\gamma^{\alpha}\gamma^{\delta}\gamma_{\nu}\gamma^{\beta}]\bigg)\\
\hphantom{XXXX}=p_{1\alpha}p_{2\beta}p_{3\delta}p_{4\epsilon}\bigg(\left\{ 4Tr[\gamma^{\beta}\gamma^{\alpha}\gamma^{\epsilon}\gamma^{\delta}]-2Tr[\gamma_{\nu}\gamma^{\nu}\gamma^{\alpha}\gamma^{\epsilon}\gamma^{\delta}\gamma^{\beta}]\right\} -2Tr[\gamma^{\epsilon}\gamma^{\alpha}\gamma^{\nu}\gamma^{\delta}\gamma_{\nu}\gamma^{\beta}]\\
\hphantom{XXXXXXXXXXXXXXXX}-2Tr[\gamma^{\epsilon}\gamma^{\nu}\gamma^{\alpha}\gamma^{\delta}\gamma_{\nu}\gamma^{\beta}]\bigg)\\
\hphantom{XXXX}=p_{1\alpha}p_{2\beta}p_{3\delta}p_{4\epsilon}\bigg(\left\{ 4Tr[\gamma^{\beta}\gamma^{\alpha}\gamma^{\epsilon}\gamma^{\delta}]-8Tr[\gamma^{\alpha}\gamma^{\epsilon}\gamma^{\delta}\gamma^{\beta}]\right\} -2Tr[\gamma^{\epsilon}\gamma^{\alpha}\gamma^{\nu}\gamma^{\delta}\gamma_{\nu}\gamma^{\beta}]\\
\hphantom{XXXXXXXXXXXXXXXX}-2Tr[\gamma^{\epsilon}\gamma^{\nu}\gamma^{\alpha}\gamma^{\delta}\gamma_{\nu}\gamma^{\beta}]\bigg)\\
\hphantom{XXXX}=p_{1\alpha}p_{2\beta}p_{3\delta}p_{4\epsilon}\bigg(-4Tr[\gamma^{\beta}\gamma^{\alpha}\gamma^{\epsilon}\gamma^{\delta}]-2Tr[\gamma^{\epsilon}\gamma^{\alpha}\gamma^{\nu}\gamma^{\delta}\gamma_{\nu}\gamma^{\beta}]-2Tr[\gamma^{\epsilon}\gamma^{\nu}\gamma^{\alpha}\gamma^{\delta}\gamma_{\nu}\gamma^{\beta}]\bigg)\\
\hphantom{XXXX}=p_{1\alpha}p_{2\beta}p_{3\delta}p_{4\epsilon}\bigg(-4Tr[\gamma^{\beta}\gamma^{\alpha}\gamma^{\epsilon}\gamma^{\delta}]+\left\{ -4Tr[\gamma^{\epsilon}\gamma^{\alpha}\gamma^{\delta}\gamma^{\beta}]+2Tr[\gamma^{\epsilon}\gamma^{\alpha}\gamma^{\nu}\gamma_{\nu}\gamma^{\delta}\gamma^{\beta}]\right\} \\
\hphantom{XXXXXXXXXXXXXXXX}-2Tr[\gamma^{\epsilon}\gamma^{\nu}\gamma^{\alpha}\gamma^{\delta}\gamma_{\nu}\gamma^{\beta}]\bigg)\\
\hphantom{XXXX}=p_{1\alpha}p_{2\beta}p_{3\delta}p_{4\epsilon}\bigg(-4Tr[\gamma^{\beta}\gamma^{\alpha}\gamma^{\epsilon}\gamma^{\delta}]+\left\{ -4Tr[\gamma^{\epsilon}\gamma^{\alpha}\gamma^{\delta}\gamma^{\beta}]+8Tr[\gamma^{\epsilon}\gamma^{\alpha}\gamma^{\delta}\gamma^{\beta}]\right\} \\
\hphantom{XXXXXXXXXXXXXXXX}-2Tr[\gamma^{\epsilon}\gamma^{\nu}\gamma^{\alpha}\gamma^{\delta}\gamma_{\nu}\gamma^{\beta}]\bigg)\\
\hphantom{XXXX}=p_{1\alpha}p_{2\beta}p_{3\delta}p_{4\epsilon}\bigg(-4Tr[\gamma^{\beta}\gamma^{\alpha}\gamma^{\epsilon}\gamma^{\delta}]+4Tr[\gamma^{\epsilon}\gamma^{\alpha}\gamma^{\delta}\gamma^{\beta}]-2Tr[\gamma^{\epsilon}\gamma^{\nu}\gamma^{\alpha}\gamma^{\delta}\gamma_{\nu}\gamma^{\beta}]\bigg)\\
\hphantom{XXXX}=p_{1\alpha}p_{2\beta}p_{3\delta}p_{4\epsilon}\bigg(-4Tr[\gamma^{\beta}\gamma^{\alpha}\gamma^{\epsilon}\gamma^{\delta}]+4Tr[\gamma^{\epsilon}\gamma^{\alpha}\gamma^{\delta}\gamma^{\beta}]\\
\hphantom{XXXXXXXXXXXXXXXX}+\left\{ -4Tr[\gamma^{\epsilon}\gamma^{\delta}\gamma^{\alpha}\gamma^{\beta}]+2Tr[\gamma^{\epsilon}\gamma^{\nu}\gamma^{\alpha}\gamma_{\nu}\gamma^{\delta}\gamma^{\beta}]\right\} \bigg)\\
\hphantom{XXXX}=p_{1\alpha}p_{2\beta}p_{3\delta}p_{4\epsilon}\bigg(-4Tr[\gamma^{\beta}\gamma^{\alpha}\gamma^{\epsilon}\gamma^{\delta}]+4Tr[\gamma^{\epsilon}\gamma^{\alpha}\gamma^{\delta}\gamma^{\beta}]\\
\hphantom{XXXXXXXXXXXXXXXX}+\left\{ -4Tr[\gamma^{\epsilon}\gamma^{\delta}\gamma^{\alpha}\gamma^{\beta}]+4Tr[\gamma^{\epsilon}\gamma^{\alpha}\gamma^{\delta}\gamma^{\beta}]-2Tr[\gamma^{\epsilon}\gamma^{\nu}\gamma_{\nu}\gamma^{\alpha}\gamma^{\delta}\gamma^{\beta}]\right\} \bigg)\\
\hphantom{XXXX}=p_{1\alpha}p_{2\beta}p_{3\delta}p_{4\epsilon}\bigg(-4Tr[\gamma^{\beta}\gamma^{\alpha}\gamma^{\epsilon}\gamma^{\delta}]+4Tr[\gamma^{\epsilon}\gamma^{\alpha}\gamma^{\delta}\gamma^{\beta}]\\
\hphantom{XXXXXXXXXXXXXXXX}+\left\{ -4Tr[\gamma^{\epsilon}\gamma^{\delta}\gamma^{\alpha}\gamma^{\beta}]+4Tr[\gamma^{\epsilon}\gamma^{\alpha}\gamma^{\delta}\gamma^{\beta}]-8Tr[\gamma^{\epsilon}\gamma^{\alpha}\gamma^{\delta}\gamma^{\beta}]\right\} \bigg)\\
\hphantom{XXXX}=p_{1\alpha}p_{2\beta}p_{3\delta}p_{4\epsilon}\bigg(-4Tr[\gamma^{\beta}\gamma^{\alpha}\gamma^{\epsilon}\gamma^{\delta}]+4Tr[\gamma^{\epsilon}\gamma^{\alpha}\gamma^{\delta}\gamma^{\beta}]-4Tr[\gamma^{\epsilon}\gamma^{\delta}\gamma^{\alpha}\gamma^{\beta}]\\
\hphantom{XXXXXXXXXXXXXXXX}-4Tr[\gamma^{\epsilon}\gamma^{\alpha}\gamma^{\delta}\gamma^{\beta}]\bigg)\\
\hphantom{XXXX}=p_{1\alpha}p_{2\beta}p_{3\delta}p_{4\epsilon}\bigg(-4Tr[\gamma^{\beta}\gamma^{\alpha}\gamma^{\epsilon}\gamma^{\delta}]-4Tr[\gamma^{\epsilon}\gamma^{\delta}\gamma^{\alpha}\gamma^{\beta}]\bigg)
\end{array}
\end{equation}$
$\begin{equation}
\begin{array}{l}
{\displaystyle \sum_{s_{1},s_{2},s_{3},s_{4}}[\overline{u}_{4}\gamma_{\mu}u_{2}]^{*}[\overline{u}_{3}\gamma^{\mu}u_{1}]^{*}[\overline{u}_{3}\gamma_{\nu}u_{2}][\overline{u}_{4}\gamma^{\nu}u_{1}]}\\
\hphantom{XXXX}=-4p_{1\alpha}p_{2\beta}p_{3\delta}p_{4\epsilon}\bigg((\eta^{\beta\alpha}\eta^{\epsilon\delta}-\eta^{\beta\epsilon}\eta^{\alpha\delta}+\eta^{\beta\delta}\eta^{\alpha\epsilon})+(\eta^{\epsilon\delta}\eta^{\alpha\beta}-\eta^{\epsilon\alpha}\eta^{\delta\beta}+\eta^{\epsilon\beta}\eta^{\delta\alpha})\bigg)\\
\hphantom{XXXX}=-8p_{1\alpha}p_{2\beta}p_{3\delta}p_{4\epsilon}\eta^{\beta\alpha}\eta^{\epsilon\delta}\\
\hphantom{XXXX}=-8(p_{1}\cdot p_{2})(p_{3}\cdot p_{4})
\end{array}
\end{equation}$
