  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 order in 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.
