  Several recent cosmological analyses have found tension between constraints
derived from the Canada-France-Hawaii Telescope Lensing Survey (CFHTLenS) data
and those derived from other data sets, such as the Planck cosmic microwave
background (CMB) temperature anisotropies. Similarly, a direct
cross-correlation of the CFHTLenS data with Planck CMB lensing data yielded an
anomalously low amplitude compared to expectations based on Planck or
WMAP-derived cosmological parameters (Liu & Hill 2015). One potential
explanation for these results is a multiplicative bias afflicting the CFHTLenS
galaxy shape measurements, from which shears are inferred. Simulations are used
in the CFHTLenS pipeline to calibrate such biases, but no data-driven
constraints have been presented to date. In this paper, we cross-correlate
CFHTLenS galaxy density maps with CFHTLenS shear maps and Planck CMB lensing
maps to calibrate an additional multiplicative shear bias (m) in CFHTLenS
(beyond the multiplicative correction that has already been applied to the
CFHTLenS galaxy shears), following methods suggested by Vallinotto (2012) and
Das et al. (2013). We analyze three magnitude-limited galaxy samples, finding
2-4sigma evidence for m<1 using the deepest sample (i<24), while the others are
consistent with m=1 (no bias). This matches the expectation that the shapes of
faint galaxies are the most prone to measurement biases. Our results for m are
essentially independent of the assumed cosmology, and only weakly sensitive to
assumptions about the galaxy bias. We consider three galaxy bias models,
finding in all cases that the best-fit multiplicative shear bias is less than
unity. A value of m~0.9 would suffice to reconcile the amplitude of density
fluctuations inferred from the CFHTLenS shear two-point statistics with that
inferred from Planck CMB temperature data. This scenario is consistent with our
results.
