  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, $\chi_0^2\approx 34.1$ and $\chi_T^2\approx 24.0$
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
$15000\deg^2$. 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.
