- #36

- 1,131

- 158

All is left is to work out the term (let me drop the 2 factor for the explicit computation) ##\varepsilon_{\mu\nu\rho\sigma}g_s f_{ilm}F_i^{\mu \nu}A_l^{\rho} A_m^{\sigma}##

\begin{align*}

\varepsilon_{\mu\nu\rho\sigma}g_s f_{ilm}F_i^{\mu \nu}A_l^{\rho} A_m^{\sigma} &= \varepsilon_{\mu\nu\rho\sigma}g_s f_{ilm}(\partial^\mu A^\nu_i - \partial^\nu A^\mu_i)A_l^{\rho} A_m^{\sigma} \\

&= 2 \varepsilon_{\mu\nu\rho\sigma} g_s f_{ilm} \partial^\mu A^\nu_i A_l^{\rho} A_m^{\sigma}\\

&= \frac 2 3 \partial^\mu (\varepsilon_{\mu\nu\rho\sigma} g_s f_{ilm} A^\nu_i A_l^{\rho} A_m^{\sigma})\\

&= \partial^\mu (\varepsilon_{\mu\nu\rho\sigma} A^\nu_i A_l^{\rho} A_m^{\sigma}) -\frac 1 3 \partial^\mu (\varepsilon_{\mu\nu\rho\sigma} A^\nu_i A_l^{\rho} A_m^{\sigma})

\end{align*}

This smells really good! (at least to me!)

We finally have

\begin{align*}

G \cdot \tilde G &= \varepsilon_{\mu \nu \rho \sigma} G_i^{\mu \nu}G_i^{\rho \sigma}\\

&=\varepsilon_{\mu \nu \rho \sigma} (F_i^{\mu \nu} + g_s f_{ijk} A_j^{\mu} A_k^{\nu})(F_i^{\rho \sigma} + g_s f_{ilm} A_l^{\rho} A_m^{\sigma})\\

&=\varepsilon_{\mu \nu \rho \sigma} (F_i^{\mu \nu}F_i^{\rho \sigma} + g_s f_{ilm}F_i^{\mu \nu}A_l^{\rho} A_m^{\sigma} + g_s f_{ijk} A_j^{\mu} A_k^{\nu} F_i^{\rho \sigma} + g^2_sf_{ijk}f_{ilm}A_j^{\mu} A_k^{\nu}A_l^{\rho} A_m^{\sigma})\\

&=\varepsilon_{\mu \nu \rho \sigma} (F_i^{\mu \nu}F_i^{\rho \sigma} + g_s f_{ilm}F_i^{\mu \nu}A_l^{\rho} A_m^{\sigma} + g_s f_{ijk} A_j^{\mu} A_k^{\nu} F_i^{\rho \sigma})\\

&= \varepsilon_{\mu \nu \rho \sigma} (F_i^{\mu \nu}F_i^{\rho \sigma} + 2g_s f_{ilm}F_i^{\mu \nu}A_l^{\rho} A_m^{\sigma})\\

&= 2\partial_{\mu}(\varepsilon^{\mu \nu \rho \sigma} A_{\nu}^i [G_{\rho \sigma}^i - \frac{g_s}{3}f_{ijk} A_{j\rho} A_{k\sigma}])

\end{align*}

Mmm so my result has a 2 extra factor...

This issue is simply solved by redefining ##G \cdot \tilde G := \frac 1 2 \varepsilon_{\mu \nu \rho \sigma} G_i^{\mu \nu}G_i^{\rho \sigma}## but I might have missed some detail in the computation and that is why I get the 2 extra factor

What do you think?