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Cosmology plays a fundamental role to determine the neutrino mass, thereforealso to determine its mass hierarchy, since the massive neutrino contributes tothe total matter density in the Universe at the background and perturbationlevels, once it becomes non-relativistic. After the non-relativistic transitionthe fluctuations are smashed out at the scales $k\gg k_{fs}$. Therefore, themissing fluctuation in the total matter is imprinted on the large scalestructure, 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 consideringthe linear perturbation theory, which is well understood in the presence ofneutrino, we propose to use the cross correlation between the Rees-Sciamaeffect and weak lensing to probe the neutrino mass. At the small scales, thedensity 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 thatthe flipping scale in the cross power spectrum between the Rees-Sciama effectand weak lensing depends on the neutrino mass by assuming the shallow and deepweak lensing surveys. Our analysis shows that the Deep survey has largersignal-to-noise ratio $S/N\sim 160$. Finally, we use the Fisher informationmatrix to forecast constraint on the neutrino mass.

Probing the Neutrino Mass through the Cross Correlation between the Rees-Sciama Effect and Weak Lensing