\mathbf{L_4} &= -\frac{1}{2}\beta g^2 \mathrm{tr}(\rho^{a\dag}\rho^a) F^{\bar{\alpha}\beta\gamma}F^{\alpha\bar{\beta}\bar{\gamma}}\int_0^{\beta} dt \left\{\Theta_{\alpha}^{ij}(t)\Delta_{\gamma}^{jk}(t)\Theta_{\beta}^{ki}(t)+\Theta_{\alpha}^{ij}(t)\Delta_{\gamma}^{ki}(t)\Theta_{\beta}^{jk}(t)\right\},\label{eq:L4}\\
\mathbf{L_{6a}} &= -\frac{1}{4}\beta g^2 \mathbf{Q}^{aabb}\left(B^{\alpha\bar{\alpha}\gamma}B^{\beta\bar{\beta}\bar{\gamma}}+B^{\alpha\beta\gamma}B^{\bar{\alpha}\bar{\beta}\bar{\gamma}}\right)\Delta^{ij}_{\alpha}(0)\Delta^{jk}_{\beta}(0),\label{eq:L6a}\\
\mathbf{L_{6b}} &= \beta g^2 N_s \frac{B^{\alpha\beta\gamma}B^{\bar{\alpha}\bar{\beta}\bar{\gamma}}}{j_\gamma(j_\gamma+2)}\left(D_{\tau}\Delta^{ij}_{\alpha}(0)D_{\tau}\Delta^{ik}_{\beta}(0)+(j_{\beta}+1)^2\Delta^{ij}_{\alpha}(0)\Delta^{jk}_{\beta}(0)\right),\label{eq:L6b}\\
\mathbf{L_{7}} &= \beta g^2 N_s D^{\alpha\bar{\alpha}\gamma}B^{\beta\bar{\beta}\bar{\gamma}}\Delta^{ij}_{\alpha}(0)\Delta^{jk}_{\beta}(0),\label{eq:L7}\\
\mathbf{L_8} &=\beta g^2 N_f\frac{F^{\alpha\beta\gamma}F^{\beta\alpha\bar{\gamma}}}{2j_{\gamma}(j_{\gamma}+2)} \left( \Theta^{ij}_{\alpha}(0)\Theta^{ik}_{\beta}(0)+\Theta^{ij}_{\alpha}(0)\Theta^{kj}_{\beta}(0)\right).\label{eq:L8}\end{aligned}$$

Each diagram has three index loops, and the short-hand indexing on the propagators indicates how a term in their expansion in powers of $e^{i\beta\alpha}\otimes e^{-i\beta\alpha}$ should be grouped into traces. The notation $\Delta^{ij}$ means that the left side of the tensor product in the expansion belongs to the index loop $i$ and the right side to the index loop $j$. Also, the propagators above should be understood to be multiplied by the appropriate regulators.

We can sum analytically over all angular momentum quantum numbers, except the total angular momentum, by using the identities derived in appendix \[sapp:AMSums\]. We then have to expand the propagators in powers of $e^{i\beta\alpha}\otimes e^{-i\beta\alpha}$ and extract the coefficients $f_n(x)$ and $f_{nm}(x)$ defined in (\[eq:2LoopEffAction\]). The results are expressed as sums over the total angular momentum numbers constrained by some selection rules from the integrals over 3 spherical harmonics.

In all diagrams we find that we can write $f_n(x)=f_{1,+}(x^n)+(-1)^{n+1}f_{1,-}(x^n)$. The expressions for those are long and not very illuminating. They are summarized in appendix \[app:2LoopDiagrams\].

