  The one-point probability distribution function (PDF) is a powerful summary
statistic for non-Gaussian cosmological fields, such as the weak lensing (WL)
convergence reconstructed from galaxy shapes or cosmic microwave background
(CMB) maps. Thus far, no analytic model has been developed that successfully
describes the high-convergence tail of the WL convergence PDF for small
smoothing scales from first principles. Here, we present a halo-model formalism
to compute the WL convergence PDF, building upon our previous results for the
thermal Sunyaev-Zel'dovich field. Furthermore, we extend our formalism to
analytically compute the covariance matrix of the convergence PDF. Comparisons
to numerical simulations generally confirm the validity of our formalism in the
non-Gaussian, positive tail of the WL convergence PDF, but also reveal the
convergence PDF's strong sensitivity to small-scale systematic effects in the
simulations (e.g., due to finite resolution). Finally, we present a simple
Fisher forecast for a Rubin-Observatory-like survey, based on our new analytic
model. Considering the $\{A_s, \Omega_m, \Sigma m_\nu\}$ parameter space and
assuming a Planck CMB prior on $A_s$ only, we forecast a marginalized
constraint $\sigma(\Sigma m_\nu) \approx 0.08$ eV from the WL convergence PDF
alone, even after marginalizing over parameters describing the halo
concentration-mass relation. This error bar on the neutrino mass sum is
comparable to the minimum value allowed in the normal hierarchy, illustrating
the strong constraining power of the WL convergence PDF. We make our code
publicly available at https://github.com/leanderthiele/hmpdf.
