Probing the Neutrino Mass through the Cross Correlation between the
  Rees-Sciama Effect and Weak Lensing
  Cosmology plays a fundamental role to determine the neutrino mass, therefore
also to determine its mass hierarchy, since the massive neutrino contributes to
the total matter density in the Universe at the background and perturbation
levels, once it becomes non-relativistic. After the non-relativistic transition
the fluctuations are smashed out at the scales $k\gg k_{fs}$. Therefore, the
missing fluctuation in the total matter is imprinted on the large scale
structure, say the suppression of the matter power spectrum $\Delta P/P\approx
-8f_{\nu}$ at the scales $k\gg k_{fs}$. In this paper, instead of considering
the linear perturbation theory, which is well understood in the presence of
neutrino, we propose to use the cross correlation between the Rees-Sciama
effect and weak lensing to probe the neutrino mass. At the small scales, the
density contrast grows faster than the background scale factor $\delta\sim a$,
that makes a sign flipping on $\Phi' \propto \mathcal{H}\delta d\ln
(\delta/a)/d\ln a$, which happens only in the non-linear regime. We show that
the flipping scale in the cross power spectrum between the Rees-Sciama effect
and weak lensing depends on the neutrino mass by assuming the shallow and deep
weak lensing surveys. Our analysis shows that the Deep survey has larger
signal-to-noise ratio $S/N\sim 160$. Finally, we use the Fisher information
matrix to forecast constraint on the neutrino mass.
