Infrared divergence does not affect the gauge-invariant curvature perturbation

We address the infrared (IR) divergence problem during inflation that appears in the loop corrections to the primordial perturbations. In our previous paper, we claimed that, at least in single field models, the IR divergence is originating from the gauge artifact. Namely, diverging IR corrections should not appear in genuine gauge-invariant observables. We propose here one simple but explicit example of such gauge-invariant quantities. Then, we explicitly calculate such a quantity to find that the IR divergence is absent at the leading order in the slow-roll approximation for the usual scale-invariant vacuum state. At the same time we notice that there is a subtle issue on the gauge invariance in how to specify the initial vacuum state.