  We cross-correlate cosmic microwave background (CMB) lensing and galaxy weak
lensing maps using the Planck 2013 and 2015 data and the 154 deg^2
Canada-France-Hawaii Telescope Lensing Survey (CFHTLenS). This measurement
probes large-scale structure at intermediate redshifts ~0.9, between the high-
and low-redshift peaks of the CMB and CFHTLenS lensing kernels, respectively.
Using the noise properties of these data sets and standard Planck 2015 LCDM
cosmological parameters, we forecast a signal-to-noise ratio ~4.6 for the
cross-correlation. We find that the noise level of our actual measurement
agrees well with this estimate, but the amplitude of the signal lies well below
the theoretical prediction. The best-fit amplitudes of our measured
cross-correlations are $A_{2013}=0.48\pm0.26$ and $A_{2015}=0.44\pm0.22$ using
the 2013 and 2015 Planck CMB lensing maps, respectively, where $A=1$
corresponds to the fiducial Planck 2015 LCDM prediction. Due to the low
measured amplitude, the detection significance is moderate (~2$\sigma$) and the
data are in tension with the theoretical prediction (~2-$2.5\sigma$). The
tension is reduced somewhat when compared to predictions using WMAP9
parameters, for which we find $A_{2013}=0.56\pm0.30$ and
$A_{2015}=0.52\pm0.26$. We consider various systematic effects, finding that
photometric redshift uncertainties, contamination by intrinsic alignments, and
effects due to the masking of galaxy clusters in the Planck 2015 CMB lensing
reconstruction are able to help resolve the tension at a significant level
(~10% each). An overall multiplicative bias in the CFHTLenS shear data could
also play a role, which can be tested with existing data. We close with
forecasts for measurements of this cross-correlation using ongoing and future
weak lensing surveys, which will definitively test the significance of the
tension in our results with respect to LCDM.
