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.