theory can be computed from (\[eq:S2\])[^10],

$$\begin{aligned}
\langle A^{\alpha}_{ij}(t^{\prime})A^{\beta}_{kl}(t)\rangle &= \Delta^{il,kj}_{\alpha}(t^{\prime}-t)\delta^{\alpha\bar{\beta}},\\
\langle (\phi^{\alpha}_a(t^{\prime}))_{ij}(\phi^{\beta}_b(t))_{kl}\rangle &= \Delta^{il,kj}_{\alpha}(t^{\prime}-t)\delta^{\alpha\bar{\beta}}\delta_{ab},\\
\langle (\psi^{\alpha}_I(t^{\prime}))_{ij}(\psi^{\beta}_J(t))^{\dag}_{kl}\rangle &=
\Theta^{il,kj}_{\alpha}(t^{\prime}-t)\delta^{\alpha\beta}\delta_{I\!J}.\end{aligned}$$

We also used the following correlators,

$$\begin{aligned}
\label{eq:PropDerivative}
\begin{array}{ll}
\langle D_tA^{\alpha}_{ij}(t^{\prime})A^{\beta}_{kl}(t)\rangle &= -\langle A^{\alpha}_{ij}(t^{\prime})D_tA^{\beta}_{kl}(t)\rangle = D_t\Delta^{il,kj}_{\alpha}(t^{\prime}-t)\delta^{\alpha\bar{\beta}},\\
\langle D_tA^{\alpha}_{ij}(t^{\prime})D_tA^{\beta}_{kl}(t)\rangle &= \delta(t^{\prime}-t)\delta^{\alpha\bar{\beta}}\delta_{il}\delta_{kj}-{(j_{\alpha}+1)}^2\Delta^{il,kj}_{\alpha}(t^{\prime}-t)\delta^{\alpha\bar{\beta}},\\
\langle D_t\phi^{a\alpha}_{ij}(t^{\prime})\phi^{b\beta}_{kl}(t))\rangle &=
-\langle \phi^{a\alpha}_{ij}(t^{\prime})D_t\phi^{b\beta}_{kl}(t))\rangle
=D_t\Delta^{il,kj}_{\alpha}(t^{\prime}-t)\delta^{\alpha\bar{\beta}}\delta^{ab},\\
\langle D_t\phi^{a\alpha}_{ij}(t^{\prime})D_t\phi^{b\beta}_{kl}(t)\rangle &=
\delta(t^{\prime}-t)\delta^{\alpha\bar{\beta}}\delta_{il}\delta_{kj}\delta^{ab}-\\
&~~-{(j_{\alpha}+1)}^2\Delta^{il,kj}_{\alpha}(t^{\prime}-t)\delta^{\alpha\bar{\beta}}\delta^{ab}.
\end{array}\end{aligned}$$

The bosonic propagator $\Delta(t)$ is a periodic function with period $\beta$. For $0\leq t\leq\beta$ it is defined as, $$\begin{aligned}
\label{eq:BosPropOferConvention}
\begin{array}{c}
\Delta^{il,kj}_{\lambda}(t)\equiv\left(\frac{e^{i\bar{\alpha}t}}{2{(j_{\lambda}+1)}}\left(\frac{e^{-{(j_{\lambda}+1)} t}}{1-e^{i\beta\bar{\alpha}}e^{-\beta{(j_{\lambda}+1)}}}\right.\right.
\left.\left.-\frac{e^{{(j_{\lambda}+1)} t}}{1-e^{i\beta\bar{\alpha}}e^{\beta{(j_{\lambda}+1)}}}\right)\right)^{il,kj},
\end{array}\end{aligned}$$ where $\bar{\alpha}\equiv\alpha\otimes\mathbbm{1}-\mathbbm{1}\otimes\alpha$ and a term in the expansion in powers of $e^{i\beta\bar{\alpha}}=e^{i\beta\alpha}\otimes e^{-i\beta\alpha}$ is understood to carry indices by putting them directly on the tensor product ($(A\otimes B)_{il,kj}\equiv A_{il}B_{kj}$).

The fermionic propagator $\Theta(t)$ is anti-periodic with (anti)period $\beta$. We can define it over $t\in(-\beta,\beta]$ (and continue periodically) , $$\begin{aligned}
\Theta_{\lambda}(t)=e^{i\bar{\alpha} t}\times\begin{cases} \frac{-ie^{-{\epsilon_{\lambda}(j_{\lambda}+\frac{1}{2})} t}}{1+e^{i\beta\bar{\alpha}}e^{-\beta{\epsilon_{\lambda}(j_{\lambda}+\frac{1}{2})}}} & t\in(0,\beta]\\
\frac{ie^{-{\epsilon_{\lambda}(j_{\lambda}+\frac{1}{2})} t}}{1+e^{-i\beta\bar{\alpha}}e^{\beta{\epsilon_{\lambda}(j_{\lambda}+\frac{1}{2})}}} & t\in(-\beta,0]\end{cases}\end{aligned}$$ where indices should be put on it by the same prescription as in the bosonic case.

Note that the fermionic propagator, and the derivative of the bosonic propagator, are ambiguous at $t=0$. We found that this ambiguity only leads to an overall normalization of the partition function, which we fix by hand anyway[^11].

Counterterms action {#ssec:CountertermsAction}
-------------------

As usual in perturbation theory, our 2-loop 