Calibrating Cluster Number Counts with CMB lensing
  CMB Stage-4 experiments will reduce the uncertainties on the gravitational
lensing potential by an order of magnitude compared to current measurements,
and will also produce a Sunyaev-Zel'dovich (SZ) cluster catalog containing
$\sim10^{5}$ objects, two orders of magnitudes higher than what is currently
available. In this paper we propose to combine these two observables and show
that it is possible to calibrate the masses of the full Stage-4 cluster catalog
internally owing to the high signal to noise measurement of the CMB lensing
convergence field. We find that a CMB Stage-4 experiment will constrain the
hydrostatic bias parameter to sub-percent accuracy. We also show constraints on
a non parametric $Y-M$ relationship which could be used to study its evolution
with mass and redshift. Finally we present a joint likelihood for thermal SZ
(tSZ) flux and mass measurements, and show that it could lead to a
$\sim5\sigma$ detection of the lower limit on the sum of the neutrino masses in
the normal hierarchy ($\sum m_{\nu}=60 \textrm{meV}$) once combined with
measurements of the primordial CMB and CMB lensing power spectra.
