Predicting the CIB-$phi$ contamination in the cross-correlation of the tSZ effect and $phi$

The recent release of {it Planck} data gives access to a full sky coverage of the thermal Sunyaev-Zeldovich (tSZ) effect and of the cosmic microwave background (CMB) lensing potential ($phi$). The cross-correlation of these two probes of the large-scale structures in the Universe is a powerful tool for testing cosmological models, especially in the context of the difference between galaxy clusters and CMB for the best-fitting cosmological parameters. However, the tSZ effect maps are highly contaminated by cosmic infra-red background (CIB) fluctuations. Unlike other astrophysical components, the spatial distribution of CIB varies with frequency. Thus it cannot be completely removed from a tSZ Compton parameter map, which is constructed from a linear combination of multiple frequency maps. We have estimated the contamination of the CIB-$phi$ correlation in the tSZ-$phi$ power-spectrum. We considered linear combinations that reconstruct the tSZ Compton parameter from {it Planck} frequency maps. We conclude that even in an optimistic case, the CIB-$phi$ contamination is significant with respect to the tSZ-$phi$ signal itself. Consequently, We stress that tSZ-$phi$ analyses that are based on Compton parameter maps are highly limited by the bias produced by CIB-$phi$ contamination.