Is the large-scale structure traced by the BOSS LOWZ galaxies consistent with $\textit{Planck}$?

Recently, several studies reported a significant discrepancy between the clustering and lensing of the Baryon Oscillation Spectroscopic Survey (BOSS) galaxies in the $\textit{Planck}$ cosmology. We construct a simple yet powerful model based on the linear theory to assess whether this discrepancy points toward deviations from $\textit{Planck}$. Focusing on scales $10<R<30$ $h^{-1}\mathrm{Mpc}$, we model the amplitudes of clustering and lensing of BOSS LOWZ galaxies using three parameters: galaxy bias $b_\mathrm{g}$, galaxy-matter cross-correlation coefficient $r_\mathrm{gm}$, and $A$, defined as the ratio between the true and $\textit{Planck}$ values of $\sigma_8$. Using the cross-correlation matrix as a diagnostic, we detect systematic uncertainties that drive spurious correlations among the low-mass galaxies. After building a clean LOWZ sample with $r_\mathrm{gm}\sim1$, we derive a joint constraint of $b_\mathrm{g}$ and $A$ from clustering+lensing, yielding $b_\mathrm{g}=2.47_{-0.30}^{+0.36}$ and $A=0.81_{-0.09}^{+0.10}$, i.e., a $2\sigma$ tension with $\textit{Planck}$. However, due to the strong degeneracy between $b_\mathrm{g}$ and $A$, systematic uncertainties in $b_\mathrm{g}$ could masquerade as a tension with $A=1$. To ascertain this possibility, we develop a new method to measure $b_\mathrm{g}$ from the cluster-galaxy cross-correlation and cluster weak lensing using an overlapping cluster sample. By applying the independent bias measurement ($b_\mathrm{g}=1.76\pm0.22$) as a prior, we successfully break the degeneracy and derive stringent constraints of $b_\mathrm{g}=2.02_{-0.15}^{+0.16}$ and $A=0.96\pm0.07$. Therefore, our result suggests that the large-scale clustering and lensing of LOWZ galaxies are consistent with $\textit{Planck}$, while the different bias estimates may be related to some observational systematics in the target selection.