Note on loop resummation in de Sitter spacetime with the wave function of the universe approach

We analyze the computation of <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" display="inline"><a:mi>n</a:mi></a:math>-point correlation functions in de Sitter spacetime, including loop corrections, using the wave function of the universe approach. This method consists of two stages employing distinct Feynman rules. First, one must compute the wave-function coefficients using interactions as vertices. Then, in the second stage, one computes correlation functions using wave-function coefficients as vertices. For massless fields, loop corrections in the first stage are free of infrared (IR) divergences, which leads to the question of how this matches the well-known IR behavior of correlators obtained via other methods. By considering a scalar field with an arbitrary potential, we compute <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" display="inline"><c:mi>n</c:mi></c:math>-point correlation functions to first order in the potential but to all orders in loops. We find that although loop integrals in the first stage are indeed IR convergent, the second procedure reintroduces the IR divergence. We discuss how this induces renormalization of the interaction potential such that the final result combining both steps exactly matches the form of <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" display="inline"><e:mi>n</e:mi></e:math>-point functions previously calculated with other methods.