We propose a novel method to detect cosmic magnification signals by cross-correlating foreground convergence fields constructed from galaxy shear measurements with background galaxy positional distributions, namely shear-number density correlation. We apply it to the Hyper Suprime-Cam Subaru Strategic Program (HSC-SSP) survey data. With 27 non-independent data points and their full covariance, $chi_0^2approx 34.1$ and $chi_T^2approx 24.0$ with respect to the null and the cosmological model with the parameters from HSC shear correlation analyses in Hamana et al. 2020 (arXiv:1906.06041), respectively. The Bayes factor of the two is $log_{10}B_{T0}approx 2.2$ assuming equal model probabilities of null and HSC cosmology, showing a clear detection of the magnification signals. Theoretically, the ratio of the shear-number density and shear-shear correlations can provide a constraint on the effective multiplicative shear bias $bar m$ using internal data themselves. We demonstrate the idea with the signals from our HSC-SSP mock simulations and rescaling the statistical uncertainties to a survey of $15000deg^2$. For two-bin analyses with background galaxies brighter than $m_{lim}=23$, the combined analyses lead to a forecasted constraint of $sigma(bar m) sim 0.032$, $2.3$ times tighter than that of using the shear-shear correlation alone. Correspondingly, $sigma(S_8)$ with $S_8=sigma_8(Omega_mathrm{m}/0.3)^{0.5}$ is tightened by $sim 2.1$ times. Importantly, the joint constraint on $bar m$ is nearly independent of cosmological parameters. Our studies therefore point to the importance of including the shear-number density correlation in weak lensing analyses, which can provide valuable consistency tests of observational data, and thus to solidify the derived cosmological constraints.
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