  We introduce two kurt-spectra to probe fourth-order statistics of weak
lensing convergence maps. Using state-of-the-art numerical simulations, we
study the shapes of these kurt-spectra as a function of source redshifts and
smoothing angular scales. We employ a pseudo-$C_{\ell}$ approach to estimate
the spectra from realistic convergence maps in the presence of an observational
mask and noise for stage-IV large-scale structure surveys. We compare these
results against theoretical predictions calculated using the FFTLog formalism,
and find that a simple nonlinear clustering model-the hierarchical ansatz-can
reproduce the numerical trends for the kurt-spectra in the nonlinear regime. In
addition, we provide estimators for beyond fourth-order spectra where no
definitive analytical results are available, and present corresponding results
from numerical simulations.
