The Lyman-$\alpha$ power spectrum - CMB lensing convergence
  cross-correlation
  We investigate the three-point correlation between the Lyman-$\alpha$ forest
and the CMB weak lensing ($\delta_F \delta_F \kappa$) expressed as the
cross-correlation between the CMB weak lensing field and local variations in
the forest power spectrum. In addition to the standard gravitational bispectrum
term, we note the existence of a non-standard systematic term coming from
mis-estimation of the mean flux over the finite length of Lyman-$\alpha$
skewers. We numerically calculate the angular cross-power spectrum and discuss
its features. We integrate it into zero-lag correlation function and compare
our predictions with recent results by Doux et al.. We find that our
predictions are statistically consistent with the measurement, and including
the systematic term improves the agreement with the measurement. We comment on
the implication of the response of the Lyman-$\alpha$ forest power spectrum to
the long-wavelength density perturbations.
