Canonical-Kähler correction to an F-term hybrid valley ($V(S,0,0)=|g\zeta {|}^2{e}^x(1-x+x^2),x=|S{|}^2/M_p^2$)

The supergravity F-term potential on the inflationary branch has the displayed exact form. Its expansion is $|g\zeta {|}^2(1+x^2/2+O(x^3))$, so the inflaton quadratic mass cancels at tree level for this canonical model. For $x>0$ its derivative is $|g\zeta {|}^2{e}^{x}x(1+x)>0$; the branch is therefore a transverse valley, not a stationary supersymmetry-breaking vacuum.