Cross-correlation of the thermal Sunyaev--Zel'dovich and CMB lensing signals in Planck PR4 data with robust CIB decontamination

We use the full-mission Planck PR4 data to measure the CMB lensing convergence ($\kappa$)--thermal Sunyaev-Zel'dovich (tSZ, $y$) cross-correlation, $C_\ell^{y\kappa}$. This is only the second measurement to date of this signal, following Hill \& Spergel (2014). We perform the measurement using foreground-cleaned tSZ maps built from the PR4 frequency maps via a tailored needlet internal linear combination (NILC) code in our companion paper, in combination with the Planck PR4 $\kappa$ maps and various systematic-mitigated PR3 $\kappa$ maps. A serious systematic is the residual cosmic infrared background (CIB) in the tSZ map, as the high CIB--$\kappa$ correlation can significantly bias the inferred tSZ--$\kappa$ cross-correlation. We mitigate this by deprojecting the CIB in our NILC, using a moment-deprojection approach to avoid leakage due to incorrect modelling of the CIB frequency dependence. We validate our method on mm-sky simulations. We fit a theoretical halo model to our measurement, finding a best-fit amplitude of $A=0.82\pm0.21$ (for the highest signal-to-noise PR4 $\kappa$ map) or $A=0.56\pm0.24$ (for a PR3 $\kappa$ map built from a tSZ-deprojected CMB map), indicating that the data are consistent with our model within $\sim 1$-$2\sigma$. Although our error bars are similar to those of the 2014 measurement, our method is significantly more robust to CIB contamination. Our moment-deprojection approach lays the foundation for future measurements of this signal with higher signal-to-noise maps from ground-based telescopes, which will precisely probe the astrophysics of the intracluster medium of galaxy groups and clusters in the intermediate-mass ($M\sim 10^{13} -10^{14} h^{-1} M_\odot$), high-$z$ ($z<\sim1.5$, c.f. $z<\sim0.8$ for the tSZ auto-power signal) regime, as well as CIB-decontaminated measurements of tSZ cross-correlations with other large-scale structure probes.