Waterfall instability in F-term hybrid inflation ($m_{\pm }^2\simeq |g{|}^2(|S{|}^2\pm |\zeta |)$)

In the small-field global approximation, the waterfall fluctuations on $u=v=0$ have the displayed masses. One becomes negative when $|S{|}^2<|\zeta |$, driving the fields towards the supersymmetric vacuum branch. In the canonical supergravity model, the exact masses follow from expanding the exponential and Kähler-covariant derivatives; their Planck-suppressed corrections should not be discarded unless the corresponding field and symmetry-breaking scales are small.