Euclidean effective action (${\Gamma }_E[\phi ]$)

For $Z_E[J]=\int \mathcal{D}\varphi e^{-S_E[\varphi ]+\int J\varphi }$, define $W_E=\log Z_E$, $\phi =\delta W_E/\delta J$ and the Legendre transform ${\Gamma }_E[\phi ]=\int J\phi -W_E[J]$. Its functional derivatives generate Euclidean one-particle irreducible vertices. A bosonic Gaussian integral about a background gives the one-loop term $\frac{1}{2}\mathrm{Tr}\log S_E^{(2)}[\phi ]$. This is the Euclidean version of the quantum effective action, with signs fixed by $e^{-S_E}$ rather than $e^{iS}$.