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The cross-correlation between cosmic microwave background (CMB) gravitationallensing and large-scale structure tracers will be an important cosmologicalprobe in the coming years. Quadratic estimators provide a simple and powerful(if suboptimal) way to reconstruct the CMB lensing potential and are widelyused. For Gaussian fields, the cross-correlation of a quadratic-estimator CMBlensing reconstruction with a tracer is exactly unbiased if the power spectraare known and consistent analytic lensing mode response functions are used.However, the bispectrum induced by non-linear large-scale structure growth andpost-Born lensing can introduce an additional bias term ($N_L^{(3/2)}$) in thecross-correlation spectrum, similar to the $N_L^{(3/2)}$ bias in theauto-spectrum demonstrated in recent works. We give analytic flat-sky resultsfor the cross-correlation bias using approximate models for the post-Born andlarge-scale structure cross-bispectra, and compare with N-body simulationresults using ray-tracing techniques. We show that the bias can be at the5-15\% level in all large-scale structure cross-correlations using small-scaleCMB temperature lensing reconstruction, but is substantially reduced usingpolarization-based lensing estimators or simple foreground-projectedtemperature estimators. The relative magnitude of these effects is almost threetimes higher than in the CMB lensing auto-correlation, but is small enough thatit can be modelled to sufficient precision using simple analytic models. Weshow that $N_L^{(3/2)}$ effects in cross-correlation will be detected with highsignificance when using data of future surveys and could affect systematiceffects marginalization in cosmic shear measurements mimicking galaxy intrinsicalignment.

CMB lensing reconstruction biases in cross-correlation with large-scale structure probes