Scalar-field source Legendre transform ($\Gamma [m]=F[h]+\int hm,m=-\delta F/\delta h$)

For a real statistical source convention $Z[h]=\int \mathcal{D}\varphi e^{-H_0[\varphi ]+\int h\varphi }$, define $F=-\log Z$ and the displayed Legendre transform. On a differentiable stable branch, $\delta \Gamma /\delta m=h$. In the Landau approximation, the branch expression is ${\Gamma }_{\mathrm{L}}[m]=H_0[m]$. The exact Legendre effective free energy is convex; a bare nonconvex Landau free energy is a local approximation and requires branch or coexistence interpretation.