$O(N)$ model in Euclidean de Sitter space: beyond the leading infrared approximation

We consider an $O(N)$ scalar field model with quartic interaction in $d$-dimensional Euclidean de Sitter space. In order to avoid the problems of the standard perturbative calculations for light and massless fields, we generalize to the $O(N)$ theory a systematic method introduced previously for a single field, which treats the zero modes exactly and the nonzero modes perturbatively. We compute the two-point functions taking into account not only the leading infrared contribution, coming from the self-interaction of the zero modes, but also corrections due to the interaction of the ultraviolet modes. For the model defined in the corresponding Lorentzian de Sitter spacetime, we obtain the two-point functions by analytical continuation. We point out that a partial resummation of the leading secular terms (which necessarily involves nonzero modes) is required to obtain a decay at large distances for massless fields. We implement this resummation along with a systematic double expansion in an effective coupling constant $\sqrt\lambda$ and in $1/N$. We explicitly perform the calculation up to the next-to-next-to-leading order in $\sqrt\lambda$ and up to next-to-leading to order $1/N$. The results reduce to those known in the leading infrared approximation. We also show that they coincide with the ones obtained directly in Lorentzian de Sitter spacetime in the large $N$ limit, provided the same renormalization scheme is used.