Bias to CMB lensing from lensed foregrounds

Extragalactic foregrounds are known to constitute a limiting systematic in temperature-based cosmic microwave background (CMB) lensing with AdvACT, SPT-3G, Simons Observatory, and CMB S4. Furthermore, since these foregrounds are emitted at cosmological distances, they are also themselves lensed. The correlation between this foreground lensing and CMB lensing causes an additional bias in CMB lensing estimators. In this paper, we quantify for the first time this ``lensed foreground bias'' for the standard CMB lensing quadratic estimator, the CMB shear, and the CMB magnification estimators, in the case of Simons Observatory and in the absence of multifrequency component separation. This percent-level bias is highly significant in the cross-correlation of CMB lensing with LSST galaxies and comparable to the statistical uncertainty in the CMB lensing autospectrum. We discuss various mitigation strategies and show that ``lensed foreground bias-hardening'' methods can reduce this bias at some cost in signal to noise. The code used to generate our theory curves is publicly available.https://github.com/EmmanuelSchaan/LensedForegroundBias.