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ATDHFB_mess.tex
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ATDHFB_mess.tex
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\documentclass[a4paper,12pt]{book}
\usepackage[utf8]{inputenc}
\usepackage{amsmath,amssymb,latexsym}
\usepackage{graphicx,subfigure}
\usepackage{tcolorbox}
\usepackage{wrapfig}
\usepackage{rotating}
\usepackage[margin=2.5cm]{geometry}
\newtheorem{Equation}{\indent Equation}[section]
\textwidth=6in
\renewcommand{\baselinestretch}{1.5}
\begin{document}
The whole mess written out is:
\begin{align*}
\mathcal{H}_{(1),pq}^{11} &= \left(U_{pi}^\dagger U_{jq} J_{ij\alpha\beta}^{11} + U_{pi}^\dagger V_{jq} \frac{M_{ij\alpha\beta}^{11}}{2} \right)[\chi^{11},f]_{\alpha\beta} -
\left(V_{pi}^\dagger U_{jq} \frac{M_{ij\alpha\beta}^{11*}}{2} + V_{pi}^\dagger V_{jq} J_{ij\alpha\beta}^{11*} \right)[\chi^{11},f]^*_{\alpha\beta} \\
&+ \left(U_{pi}^\dagger U_{jq} J_{ij\alpha\beta}^{22} + U_{pi}^\dagger V_{jq} \frac{M_{ij\alpha\beta}^{22}}{2} \right)[\chi^{22},f]_{\alpha\beta} -
\left(V_{pi}^\dagger U_{jq} \frac{M_{ij\alpha\beta}^{22*}}{2} + V_{pi}^\dagger V_{jq} J_{ij\alpha\beta}^{22*} \right)[\chi^{22},f]^*_{\alpha\beta} \\
&+ \left(U_{pi}^\dagger U_{jq} K_{ij\alpha\beta}^{12} + U_{pi}^\dagger V_{jq} \frac{L_{ij\alpha\beta}^{12}}{2} \right)\left(\chi^{12}-\left\{\chi^{12},f\right\}\right)_{\alpha\beta} \\ &\qquad\qquad\qquad\qquad\qquad\qquad\qquad-
\left(V_{pi}^\dagger U_{jq} \frac{L_{ij\alpha\beta}^{12*}}{2} + V_{pi}^\dagger V_{jq} K_{ij\alpha\beta}^{12*} \right)\left(\chi^{12}-\left\{\chi^{12},f\right\}\right)^*_{\alpha\beta} \\
&+ \left(U_{pi}^\dagger U_{jq} K_{ij\alpha\beta}^{21} + U_{pi}^\dagger V_{jq} \frac{L_{ij\alpha\beta}^{21}}{2} \right)\left(\chi^{21}-\left\{\chi^{21},f\right\}\right)_{\alpha\beta} \\ &\qquad\qquad\qquad\qquad\qquad\qquad\qquad-
\left(V_{pi}^\dagger U_{jq} \frac{L_{ij\alpha\beta}^{21*}}{2} + V_{pi}^\dagger V_{jq} K_{ij\alpha\beta}^{21*} \right)\left(\chi^{21}-\left\{\chi^{21},f\right\}\right)^*_{\alpha\beta}
\end{align*}
\begin{align*}
\mathcal{H}_{(1),pq}^{12} &= \left(U_{pi}^\dagger V^*_{jq} J_{ij\alpha\beta}^{11} + U_{pi}^\dagger U^*_{jq} \frac{M_{ij\alpha\beta}^{11}}{2} \right)[\chi^{11},f]_{\alpha\beta} -
\left(V_{pi}^\dagger V^*_{jq} \frac{M_{ij\alpha\beta}^{11*}}{2} + V_{pi}^\dagger U^*_{jq} J_{ij\alpha\beta}^{11*} \right)[\chi^{11},f]^*_{\alpha\beta} \\
&+ \left(U_{pi}^\dagger V^*_{jq} J_{ij\alpha\beta}^{22} + U_{pi}^\dagger U^*_{jq} \frac{M_{ij\alpha\beta}^{22}}{2} \right)[\chi^{22},f]_{\alpha\beta} -
\left(V_{pi}^\dagger V^*_{jq} \frac{M_{ij\alpha\beta}^{22*}}{2} + V_{pi}^\dagger U^*_{jq} J_{ij\alpha\beta}^{22*} \right)[\chi^{22},f]^*_{\alpha\beta} \\
&+ \left(U_{pi}^\dagger V^*_{jq} K_{ij\alpha\beta}^{12} + U_{pi}^\dagger U^*_{jq} \frac{L_{ij\alpha\beta}^{12}}{2} \right)\left(\chi^{12}-\left\{\chi^{12},f\right\}\right)_{\alpha\beta} \\ &\qquad\qquad\qquad\qquad\qquad\qquad\qquad-
\left(V_{pi}^\dagger V^*_{jq} \frac{L_{ij\alpha\beta}^{12*}}{2} + V_{pi}^\dagger U^*_{jq} K_{ij\alpha\beta}^{12*} \right)\left(\chi^{12}-\left\{\chi^{12},f\right\}\right)^*_{\alpha\beta} \\
&+ \left(U_{pi}^\dagger V^*_{jq} K_{ij\alpha\beta}^{21} + U_{pi}^\dagger U^*_{jq} \frac{L_{ij\alpha\beta}^{21}}{2} \right)\left(\chi^{21}-\left\{\chi^{21},f\right\}\right)_{\alpha\beta} \\ &\qquad\qquad\qquad\qquad\qquad\qquad\qquad-
\left(V_{pi}^\dagger V^*_{jq} \frac{L_{ij\alpha\beta}^{21*}}{2} + V_{pi}^\dagger U^*_{jq} K_{ij\alpha\beta}^{21*} \right)\left(\chi^{21}-\left\{\chi^{21},f\right\}\right)^*_{\alpha\beta}
\end{align*}
\begin{align*}
\mathcal{H}_{(1),pq}^{21} &= \left(V^T_{pi} U_{jq} J_{ij\alpha\beta}^{11} + V_{pi}^T V_{jq} \frac{M_{ij\alpha\beta}^{11}}{2} \right)[\chi^{11},f]_{\alpha\beta} -
\left(U_{pi}^T U_{jq} \frac{M_{ij\alpha\beta}^{11*}}{2} + U_{pi}^T V_{jq} J_{ij\alpha\beta}^{11*} \right)[\chi^{11},f]^*_{\alpha\beta} \\
&+ \left(V_{pi}^T U_{jq} J_{ij\alpha\beta}^{22} + V_{pi}^T V_{jq} \frac{M_{ij\alpha\beta}^{22}}{2} \right)[\chi^{22},f]_{\alpha\beta} -
\left(U_{pi}^T U_{jq} \frac{M_{ij\alpha\beta}^{22*}}{2} + U_{pi}^T V_{jq} J_{ij\alpha\beta}^{22*} \right)[\chi^{22},f]^*_{\alpha\beta} \\
&+ \left(V_{pi}^T U_{jq} K_{ij\alpha\beta}^{12} + V_{pi}^T V_{jq} \frac{L_{ij\alpha\beta}^{12}}{2} \right)\left(\chi^{12}-\left\{\chi^{12},f\right\}\right)_{\alpha\beta} \\ &\qquad\qquad\qquad\qquad\qquad\qquad\qquad-
\left(U_{pi}^T U_{jq} \frac{L_{ij\alpha\beta}^{12*}}{2} + U_{pi}^T V_{jq} K_{ij\alpha\beta}^{12*} \right)\left(\chi^{12}-\left\{\chi^{12},f\right\}\right)^*_{\alpha\beta} \\
&+ \left(V_{pi}^T U_{jq} K_{ij\alpha\beta}^{21} + V_{pi}^T V_{jq} \frac{L_{ij\alpha\beta}^{21}}{2} \right)\left(\chi^{21}-\left\{\chi^{21},f\right\}\right)_{\alpha\beta} \\ &\qquad\qquad\qquad\qquad\qquad\qquad\qquad-
\left(U_{pi}^T U_{jq} \frac{L_{ij\alpha\beta}^{21*}}{2} + U_{pi}^T V_{jq} K_{ij\alpha\beta}^{21*} \right)\left(\chi^{21}-\left\{\chi^{21},f\right\}\right)^*_{\alpha\beta}
\end{align*}
\begin{align*}
\mathcal{H}_{(1),pq}^{22} &= \left(V_{pi}^T V^*_{jq} J_{ij\alpha\beta}^{11} + V_{pi}^T U^*_{jq} \frac{M_{ij\alpha\beta}^{11}}{2} \right)[\chi^{11},f]_{\alpha\beta} -
\left(U_{pi}^T V^*_{jq} \frac{M_{ij\alpha\beta}^{11*}}{2} + U_{pi}^T U^*_{jq} J_{ij\alpha\beta}^{11*} \right)[\chi^{11},f]^*_{\alpha\beta} \\
&+ \left(V_{pi}^T V^*_{jq} J_{ij\alpha\beta}^{22} + V_{pi}^T U^*_{jq} \frac{M_{ij\alpha\beta}^{22}}{2} \right)[\chi^{22},f]_{\alpha\beta} -
\left(U_{pi}^T V^*_{jq} \frac{M_{ij\alpha\beta}^{22*}}{2} + U_{pi}^T U^*_{jq} J_{ij\alpha\beta}^{22*} \right)[\chi^{22},f]^*_{\alpha\beta} \\
&+ \left(V_{pi}^T V^*_{jq} K_{ij\alpha\beta}^{12} + V_{pi}^T U^*_{jq} \frac{L_{ij\alpha\beta}^{12}}{2} \right)\left(\chi^{12}-\left\{\chi^{12},f\right\}\right)_{\alpha\beta} \\ &\qquad\qquad\qquad\qquad\qquad\qquad\qquad-
\left(U_{pi}^T V^*_{jq} \frac{L_{ij\alpha\beta}^{12*}}{2} + U_{pi}^T U^*_{jq} K_{ij\alpha\beta}^{12*} \right)\left(\chi^{12}-\left\{\chi^{12},f\right\}\right)^*_{\alpha\beta} \\
&+ \left(V_{pi}^T V^*_{jq} K_{ij\alpha\beta}^{21} + V_{pi}^T U^*_{jq} \frac{L_{ij\alpha\beta}^{21}}{2} \right)\left(\chi^{21}-\left\{\chi^{21},f\right\}\right)_{\alpha\beta} \\ &\qquad\qquad\qquad\qquad\qquad\qquad\qquad-
\left(U_{pi}^T V^*_{jq} \frac{L_{ij\alpha\beta}^{21*}}{2} + U_{pi}^T U^*_{jq} K_{ij\alpha\beta}^{21*} \right)\left(\chi^{21}-\left\{\chi^{21},f\right\}\right)^*_{\alpha\beta}
\end{align*}
The full tensor mess is:
\begin{sidewaystable}
\begin{equation}
\mathcal{\bar{M}}_{:1} = \frac{1}{4}\left(\begin{array}{c}
(E_b-E_a)(f_a-f_b) + i(f_a-f_b)\left(U_{ai}^\dagger U_{jb} J_{ij\alpha\beta}^{11} + U_{ai}^\dagger V_{jb} \frac{M_{ij\alpha\beta}^{11}}{2} + V_{ai}^\dagger U_{jb} \frac{M_{ij\alpha\beta}^{11*}}{2} + V_{ai}^\dagger V_{jb} J_{ij\alpha\beta}^{11*} \right)(f_\beta-f_\alpha) \\
-i(1-f_a-f_b)\left(U_{ai}^\dagger V^*_{jb} J_{ij\alpha\beta}^{11} + U_{ai}^\dagger U^*_{jb} \frac{M_{ij\alpha\beta}^{11}}{2} + V_{ai}^\dagger V^*_{jb} \frac{M_{ij\alpha\beta}^{11*}}{2} + V_{ai}^\dagger U^*_{jb} J_{ij\alpha\beta}^{11*} \right)(f_\beta-f_\alpha) \\
i(1-f_a-f_b)\left(V^T_{ai} U_{jb} J_{ij\alpha\beta}^{11} + V_{ai}^T V_{jb} \frac{M_{ij\alpha\beta}^{11}}{2} + U_{ai}^T U_{jb} \frac{M_{ij\alpha\beta}^{11*}}{2} + U_{ai}^T V_{jb} J_{ij\alpha\beta}^{11*} \right)(f_\beta-f_\alpha) \\
-i(f_a-f_b)\left(V_{ai}^T V^*_{jb} J_{ij\alpha\beta}^{11} + V_{ai}^T U^*_{jb} \frac{M_{ij\alpha\beta}^{11}}{2} + U_{ai}^T V^*_{jb} \frac{M_{ij\alpha\beta}^{11*}}{2} + U_{ai}^T U^*_{jb} J_{ij\alpha\beta}^{11*} \right)(f_\beta-f_\alpha)
\end{array}\right)
\end{equation}
\begin{equation}
\mathcal{\bar{M}}_{:2} = \frac{1}{4}\left(\begin{array}{c}
i(f_a-f_b)\left(U_{ai}^\dagger U_{jb} K_{ij\alpha\beta}^{12} + U_{ai}^\dagger V_{jb} \frac{L_{ij\alpha\beta}^{12}}{2} - V_{ai}^\dagger U_{jb} \frac{L_{ij\alpha\beta}^{21*}}{2} + V_{ai}^\dagger V_{jb} K_{ij\alpha\beta}^{21*} \right)\left(1-f_\alpha-f_\beta\right) \\
(E_a+E_b)(1-f_a-f_b) - i(1-f_a-f_b)\left(U_{ai}^\dagger V^*_{jb} K_{ij\alpha\beta}^{12} + U_{ai}^\dagger U^*_{jb} \frac{L_{ij\alpha\beta}^{12}}{2} - V_{ai}^\dagger V^*_{jb} \frac{L_{ij\alpha\beta}^{21*}}{2} + V_{ai}^\dagger U^*_{jb} K_{ij\alpha\beta}^{21*} \right)\left(1-f_\alpha-f_\beta\right) \\
i(1-f_a-f_b)\left(V_{ai}^T U_{jb} K_{ij\alpha\beta}^{12} + V_{ai}^T V_{jb} \frac{L_{ij\alpha\beta}^{12}}{2} - U_{ai}^T U_{jb} \frac{L_{ij\alpha\beta}^{21*}}{2} + U_{ai}^T V_{jb} K_{ij\alpha\beta}^{21*} \right)\left(1-f_\alpha-f_\beta\right) \\
-i(f_a-f_b)\left(V_{ai}^T V^*_{jb} K_{ij\alpha\beta}^{12} + V_{ai}^T U^*_{jb} \frac{L_{ij\alpha\beta}^{12}}{2} - U_{ai}^T V^*_{jb} \frac{L_{ij\alpha\beta}^{21*}}{2} + U_{ai}^T U^*_{jb} K_{ij\alpha\beta}^{21*} \right)\left(1-f_\alpha-f_\beta\right)
\end{array}\right)
\end{equation}
\begin{equation}
\mathcal{\bar{M}}_{:3} = \frac{1}{4}\left(\begin{array}{c}
i(f_a-f_b)\left(U_{ai}^\dagger U_{jb} K_{ij\alpha\beta}^{21} + U_{ai}^\dagger V_{jb} \frac{L_{ij\alpha\beta}^{21}}{2} - V_{ai}^\dagger U_{jb} \frac{L_{ij\alpha\beta}^{12*}}{2} + V_{ai}^\dagger V_{jb} K_{ij\alpha\beta}^{12*} \right)\left(1-f_\alpha-f_\beta\right) \\
-i(1-f_a-f_b)\left(U_{ai}^\dagger V^*_{jb} K_{ij\alpha\beta}^{21} + U_{ai}^\dagger U^*_{jb} \frac{L_{ij\alpha\beta}^{21}}{2} - V_{ai}^\dagger V^*_{jb} \frac{L_{ij\alpha\beta}^{12*}}{2} + V_{ai}^\dagger U^*_{jb} K_{ij\alpha\beta}^{12*} \right)\left(1-f_\alpha-f_\beta\right) \\
(E_a+E_b)(1-f_a-f_b) + i(1-f_a-f_b)\left(V_{ai}^T U_{jb} K_{ij\alpha\beta}^{21} + V_{ai}^T V_{jb} \frac{L_{ij\alpha\beta}^{21}}{2} - U_{ai}^T U_{jb} \frac{L_{ij\alpha\beta}^{12*}}{2} + U_{ai}^T V_{jb} K_{ij\alpha\beta}^{12*} \right)\left(1-f_\alpha-f_\beta\right) \\
-i(f_a-f_b)\left(V_{ai}^T V^*_{jb} K_{ij\alpha\beta}^{21} + V_{ai}^T U^*_{jb} \frac{L_{ij\alpha\beta}^{21}}{2} - U_{ai}^T V^*_{jb} \frac{L_{ij\alpha\beta}^{12*}}{2} + U_{ai}^T U^*_{jb} K_{ij\alpha\beta}^{12*} \right)\left(1-f_\alpha-f_\beta\right)
\end{array}\right)
\end{equation}
\begin{equation}
\mathcal{\bar{M}}_{:4} = \frac{1}{4}\left(\begin{array}{c}
i(f_a-f_b)\left(U_{ai}^\dagger U_{jb} J_{ij\alpha\beta}^{22} + U_{ai}^\dagger V_{jb} \frac{M_{ij\alpha\beta}^{22}}{2} + V_{ai}^\dagger U_{jb} \frac{M_{ij\alpha\beta}^{22*}}{2} + V_{ai}^\dagger V_{jb} J_{ij\alpha\beta}^{22*} \right)(f_\beta-f_\alpha) \\
-i(1-f_a-f_b)\left(U_{ai}^\dagger V^*_{jb} J_{ij\alpha\beta}^{22} + U_{ai}^\dagger U^*_{jb} \frac{M_{ij\alpha\beta}^{22}}{2} + V_{ai}^\dagger V^*_{jb} \frac{M_{ij\alpha\beta}^{22*}}{2} + V_{ai}^\dagger U^*_{jb} J_{ij\alpha\beta}^{22*} \right)(f_\beta-f_\alpha) \\
i(1-f_a-f_b)\left(V_{ai}^T U_{jb} J_{ij\alpha\beta}^{22} + V_{ai}^T V_{jb} \frac{M_{ij\alpha\beta}^{22}}{2} + U_{ai}^T U_{jb} \frac{M_{ij\alpha\beta}^{22*}}{2} + U_{ai}^T V_{jb} J_{ij\alpha\beta}^{22*} \right)(f_\beta-f_\alpha)\\
(E_b-E_a)(f_a-f_b) - i(f_a-f_b)\left(V_{ai}^T V^*_{jb} J_{ij\alpha\beta}^{22} + V_{ai}^T U^*_{jb} \frac{M_{ij\alpha\beta}^{22}}{2} + U_{ai}^T V^*_{jb} \frac{M_{ij\alpha\beta}^{22*}}{2} + U_{ai}^T U^*_{jb} J_{ij\alpha\beta}^{22*} \right)(f_\beta-f_\alpha)
\end{array}\right)
\end{equation}
\end{sidewaystable}
\begin{tcolorbox}
\begin{align*}
J^{11}_{ij\alpha\beta} = i\bar{v}_{ikjl}U_{l\alpha}U^\dagger_{\beta k} &\qquad J^{22}_{ij\alpha\beta} = -i\bar{v}_{ikjl}V^*_{l\alpha}V^T_{\beta k} \\
K^{12}_{ij\alpha\beta} = i\bar{v}_{ikjl}U_{l\alpha}V^T_{\beta k} &\qquad K^{21}_{ij\alpha\beta} = -i\bar{v}_{ikjl}V^*_{l\alpha}U^\dagger_{\beta k} \\
L^{12}_{ij\alpha\beta} = i\bar{v}_{ijkl}U_{l\alpha}U^T_{\beta k} &\qquad L^{21}_{ij\alpha\beta} = -i\bar{v}_{ijkl}V^*_{l\alpha}V^\dagger_{\beta k} \\
M^{11}_{ij\alpha\beta} = i\bar{v}_{ijkl}U_{l\alpha}V^\dagger_{\beta k} &\qquad M^{22}_{ij\alpha\beta} = -i\bar{v}_{ijkl}V^*_{l\alpha}U^T_{\beta k} \\
J^{11*}_{ij\alpha\beta} = -i\bar{v}_{ikjl}U^*_{l\alpha}U^T_{\beta k} &\qquad J^{22*}_{ij\alpha\beta} = i\bar{v}_{ikjl}V_{l\alpha}V^\dagger_{\beta k} \\
K^{12*}_{ij\alpha\beta} = -i\bar{v}_{ikjl}U^*_{l\alpha}V^\dagger_{\beta k} &\qquad K^{21*}_{ij\alpha\beta} = i\bar{v}_{ikjl}V_{l\alpha}U^T_{\beta k} \\
L^{12*}_{ij\alpha\beta} = -i\bar{v}_{ijkl}U^*_{l\alpha}U^\dagger_{\beta k} &\qquad L^{21*}_{ij\alpha\beta} = i\bar{v}_{ijkl}V_{l\alpha}V^T_{\beta k} \\
M^{11*}_{ij\alpha\beta} = -i\bar{v}_{ijkl}U^*_{l\alpha}V^T_{\beta k} &\qquad M^{22*}_{ij\alpha\beta} = i\bar{v}_{ijkl}V_{l\alpha}U^\dagger_{\beta k}
\end{align*}
\end{tcolorbox}
\begin{sidewaystable}
\begin{equation}
\mathcal{\bar{M}}_{:1} = \frac{1}{4}\left(\begin{array}{c}
(E_b-E_a)(f_a-f_b) + i(f_a-f_b)\left(U_{ai}^\dagger U_{jb} i\bar{v}_{ikjl}U_{l\alpha}U^\dagger_{\beta k} + U_{ai}^\dagger V_{jb} \frac{i\bar{v}_{ijkl}U_{l\alpha}V^\dagger_{\beta k}}{2} + V_{ai}^\dagger U_{jb} \frac{-i\bar{v}_{ijkl}U^*_{l\alpha}V^T_{\beta k}}{2} + V_{ai}^\dagger V_{jb} -i\bar{v}_{ikjl}U^*_{l\alpha}U^T_{\beta k} \right)(f_\beta-f_\alpha) \\
-i(1-f_a-f_b)\left(U_{ai}^\dagger V^*_{jb} i\bar{v}_{ikjl}U_{l\alpha}U^\dagger_{\beta k} + U_{ai}^\dagger U^*_{jb} \frac{i\bar{v}_{ijkl}U_{l\alpha}V^\dagger_{\beta k}}{2} + V_{ai}^\dagger V^*_{jb} \frac{-i\bar{v}_{ijkl}U^*_{l\alpha}V^T_{\beta k}}{2} + V_{ai}^\dagger U^*_{jb} -i\bar{v}_{ikjl}U^*_{l\alpha}U^T_{\beta k} \right)(f_\beta-f_\alpha) \\
i(1-f_a-f_b)\left(V^T_{ai} U_{jb} i\bar{v}_{ikjl}U_{l\alpha}U^\dagger_{\beta k} + V_{ai}^T V_{jb} \frac{i\bar{v}_{ijkl}U_{l\alpha}V^\dagger_{\beta k}}{2} + U_{ai}^T U_{jb} \frac{-i\bar{v}_{ijkl}U^*_{l\alpha}V^T_{\beta k}}{2} + U_{ai}^T V_{jb} -i\bar{v}_{ikjl}U^*_{l\alpha}U^T_{\beta k} \right)(f_\beta-f_\alpha) \\
-i(f_a-f_b)\left(V_{ai}^T V^*_{jb} i\bar{v}_{ikjl}U_{l\alpha}U^\dagger_{\beta k} + V_{ai}^T U^*_{jb} \frac{i\bar{v}_{ijkl}U_{l\alpha}V^\dagger_{\beta k}}{2} + U_{ai}^T V^*_{jb} \frac{-i\bar{v}_{ijkl}U^*_{l\alpha}V^T_{\beta k}}{2} + U_{ai}^T U^*_{jb} -i\bar{v}_{ikjl}U^*_{l\alpha}U^T_{\beta k} \right)(f_\beta-f_\alpha)
\end{array}\right)
\end{equation}
\begin{equation}
\mathcal{\bar{M}}_{:2} = \frac{1}{4}\left(\begin{array}{c}
i(f_a-f_b)\left(U_{ai}^\dagger U_{jb} i\bar{v}_{ikjl}U_{l\alpha}V^T_{\beta k} + U_{ai}^\dagger V_{jb} \frac{i\bar{v}_{ijkl}U_{l\alpha}U^T_{\beta k}}{2} - V_{ai}^\dagger U_{jb} \frac{i\bar{v}_{ijkl}V_{l\alpha}V^T_{\beta k}}{2} + V_{ai}^\dagger V_{jb} i\bar{v}_{ikjl}V_{l\alpha}U^T_{\beta k} \right)\left(1-f_\alpha-f_\beta\right) \\
(E_a+E_b)(1-f_a-f_b) - i(1-f_a-f_b)\left(U_{ai}^\dagger V^*_{jb} i\bar{v}_{ikjl}U_{l\alpha}V^T_{\beta k} + U_{ai}^\dagger U^*_{jb} \frac{i\bar{v}_{ijkl}U_{l\alpha}U^T_{\beta k}}{2} - V_{ai}^\dagger V^*_{jb} \frac{i\bar{v}_{ijkl}V_{l\alpha}V^T_{\beta k}}{2} + V_{ai}^\dagger U^*_{jb} i\bar{v}_{ikjl}V_{l\alpha}U^T_{\beta k} \right)\left(1-f_\alpha-f_\beta\right) \\
i(1-f_a-f_b)\left(V_{ai}^T U_{jb} i\bar{v}_{ikjl}U_{l\alpha}V^T_{\beta k} + V_{ai}^T V_{jb} \frac{i\bar{v}_{ijkl}U_{l\alpha}U^T_{\beta k}}{2} - U_{ai}^T U_{jb} \frac{i\bar{v}_{ijkl}V_{l\alpha}V^T_{\beta k}}{2} + U_{ai}^T V_{jb} i\bar{v}_{ikjl}V_{l\alpha}U^T_{\beta k} \right)\left(1-f_\alpha-f_\beta\right) \\
-i(f_a-f_b)\left(V_{ai}^T V^*_{jb} i\bar{v}_{ikjl}U_{l\alpha}V^T_{\beta k} + V_{ai}^T U^*_{jb} \frac{i\bar{v}_{ijkl}U_{l\alpha}U^T_{\beta k}}{2} - U_{ai}^T V^*_{jb} \frac{i\bar{v}_{ijkl}V_{l\alpha}V^T_{\beta k}}{2} + U_{ai}^T U^*_{jb} i\bar{v}_{ikjl}V_{l\alpha}U^T_{\beta k} \right)\left(1-f_\alpha-f_\beta\right)
\end{array}\right)
\end{equation}
\begin{equation}
\mathcal{\bar{M}}_{:3} = \frac{1}{4}\left(\begin{array}{c}
i(f_a-f_b)\left(U_{ai}^\dagger U_{jb} -i\bar{v}_{ikjl}V^*_{l\alpha}U^\dagger_{\beta k} + U_{ai}^\dagger V_{jb} \frac{-i\bar{v}_{ijkl}V^*_{l\alpha}V^\dagger_{\beta k}}{2} - V_{ai}^\dagger U_{jb} \frac{-i\bar{v}_{ijkl}U^*_{l\alpha}U^\dagger_{\beta k}}{2} + V_{ai}^\dagger V_{jb} -i\bar{v}_{ikjl}U^*_{l\alpha}V^\dagger_{\beta k} \right)\left(1-f_\alpha-f_\beta\right) \\
-i(1-f_a-f_b)\left(U_{ai}^\dagger V^*_{jb} -i\bar{v}_{ikjl}V^*_{l\alpha}U^\dagger_{\beta k} + U_{ai}^\dagger U^*_{jb} \frac{-i\bar{v}_{ijkl}V^*_{l\alpha}V^\dagger_{\beta k}}{2} - V_{ai}^\dagger V^*_{jb} \frac{-i\bar{v}_{ijkl}U^*_{l\alpha}U^\dagger_{\beta k}}{2} + V_{ai}^\dagger U^*_{jb} -i\bar{v}_{ikjl}U^*_{l\alpha}V^\dagger_{\beta k} \right)\left(1-f_\alpha-f_\beta\right) \\
(E_a+E_b)(1-f_a-f_b) + i(1-f_a-f_b)\left(V_{ai}^T U_{jb} -i\bar{v}_{ikjl}V^*_{l\alpha}U^\dagger_{\beta k} + V_{ai}^T V_{jb} \frac{-i\bar{v}_{ijkl}V^*_{l\alpha}V^\dagger_{\beta k}}{2} - U_{ai}^T U_{jb} \frac{-i\bar{v}_{ijkl}U^*_{l\alpha}U^\dagger_{\beta k}}{2} + U_{ai}^T V_{jb} -i\bar{v}_{ikjl}U^*_{l\alpha}V^\dagger_{\beta k} \right)\left(1-f_\alpha-f_\beta\right) \\
-i(f_a-f_b)\left(V_{ai}^T V^*_{jb} -i\bar{v}_{ikjl}V^*_{l\alpha}U^\dagger_{\beta k} + V_{ai}^T U^*_{jb} \frac{-i\bar{v}_{ijkl}V^*_{l\alpha}V^\dagger_{\beta k}}{2} - U_{ai}^T V^*_{jb} \frac{-i\bar{v}_{ijkl}U^*_{l\alpha}U^\dagger_{\beta k}}{2} + U_{ai}^T U^*_{jb} -i\bar{v}_{ikjl}U^*_{l\alpha}V^\dagger_{\beta k} \right)\left(1-f_\alpha-f_\beta\right)
\end{array}\right)
\end{equation}
\begin{equation}
\mathcal{\bar{M}}_{:4} = \frac{1}{4}\left(\begin{array}{c}
i(f_a-f_b)\left(U_{ai}^\dagger U_{jb} -i\bar{v}_{ikjl}V^*_{l\alpha}V^T_{\beta k} + U_{ai}^\dagger V_{jb} \frac{-i\bar{v}_{ijkl}V^*_{l\alpha}U^T_{\beta k}}{2} + V_{ai}^\dagger U_{jb} \frac{i\bar{v}_{ijkl}V_{l\alpha}U^\dagger_{\beta k}}{2} + V_{ai}^\dagger V_{jb} i\bar{v}_{ikjl}V_{l\alpha}V^\dagger_{\beta k} \right)(f_\beta-f_\alpha) \\
-i(1-f_a-f_b)\left(U_{ai}^\dagger V^*_{jb} -i\bar{v}_{ikjl}V^*_{l\alpha}V^T_{\beta k} + U_{ai}^\dagger U^*_{jb} \frac{-i\bar{v}_{ijkl}V^*_{l\alpha}U^T_{\beta k}}{2} + V_{ai}^\dagger V^*_{jb} \frac{i\bar{v}_{ijkl}V_{l\alpha}U^\dagger_{\beta k}}{2} + V_{ai}^\dagger U^*_{jb} i\bar{v}_{ikjl}V_{l\alpha}V^\dagger_{\beta k} \right)(f_\beta-f_\alpha) \\
i(1-f_a-f_b)\left(V_{ai}^T U_{jb} -i\bar{v}_{ikjl}V^*_{l\alpha}V^T_{\beta k} + V_{ai}^T V_{jb} \frac{-i\bar{v}_{ijkl}V^*_{l\alpha}U^T_{\beta k}}{2} + U_{ai}^T U_{jb} \frac{i\bar{v}_{ijkl}V_{l\alpha}U^\dagger_{\beta k}}{2} + U_{ai}^T V_{jb} i\bar{v}_{ikjl}V_{l\alpha}V^\dagger_{\beta k} \right)(f_\beta-f_\alpha)\\
(E_b-E_a)(f_a-f_b) - i(f_a-f_b)\left(V_{ai}^T V^*_{jb} -i\bar{v}_{ikjl}V^*_{l\alpha}V^T_{\beta k} + V_{ai}^T U^*_{jb} \frac{-i\bar{v}_{ijkl}V^*_{l\alpha}U^T_{\beta k}}{2} + U_{ai}^T V^*_{jb} \frac{i\bar{v}_{ijkl}V_{l\alpha}U^\dagger_{\beta k}}{2} + U_{ai}^T U^*_{jb} i\bar{v}_{ikjl}V_{l\alpha}V^\dagger_{\beta k} \right)(f_\beta-f_\alpha)
\end{array}\right)
\end{equation}
\end{sidewaystable}
\end{document}