to the energy-momentum tensor, whereby the parameter $T$ in our equations is not the true temperature (that would be measured by an observed moving with the local rest frame). To obtain the physical correction to $T^{\mu\nu}$, which we call $\Pi_{2,phys}^{\mu\nu}$, we must perform a temperature shift. Putting $T=T_{phys}-\delta T$ with $\delta T d\rho/dT = \Pi_2^{00}$, the true correction reads \_[2,phys]{}\^[ij]{}= \_2\^[ij]{}-\^[ij]{}\_2\^[00]{} which is traceless. Taking into account the above we find

\_[2,phys]{}\^[ij]{} &= L \[\^[(1)i]{}\_k \^[(1)kj]{}-\^[ij]{}\^[(1)lm]{}\^[(1)]{}\_[lm]{}\]\
& +T\^[ij]{}\^[a(1)]{} \^[a(1)]{} \[piphys\]

with L = To obtain the last term in Eq. (\[piphys\]) we have used the identity $\left\langle \omega^2 \right\rangle = 2T\left\langle \omega \right\rangle $. Note also that $\Pi_{2,phys}^{ij}$ is transverse. For simplicity, in what follows we will drop the subindex “phys” in $\Pi_{2,phys}^{ij}$.

The quadratic contribution to the color current can be computed directly from Eqs. (\[jdfull\]), (\[zeta20\]) and (\[zeta2b\]). We get J\_[2c]{}\^ = u\^d\^[ab]{}\_c \^[a(1)]{} \^[b(1)]{} \[j2htot\] where we have used that $\lambda^{(1)j}_j = 0$ so that the last term in Eq. (\[jdfull\]) drops out.

Using the closure picked out by the EPP, we have completed the task of expressing the currents $\rm{tr}(\mathbf{T}^{\mu\nu})$ and $\mathbf{J}^\mu$ in terms of the variables of the effective theory.

### Summary of the EPP 