Effective Potential for Quantum Scalar Fields in a de Sitter Geometry

We study the quantum theory of an O(N) scalar field on de Sitter geometry at leading order in a nonperturbative 1/N expansion. This resums the infinite series of so-called superdaisy loop diagrams. We obtain the de Sitter symmetric solutions of the corresponding, properly renormalized, dynamical field equations and compute the complete effective potential. Because of its self-interactions, the field acquires a strictly positive square mass which screens potential infrared divergences. Moreover, strongly enhanced ultralong-wavelength fluctuations prevent the existence of a spontaneously broken symmetry state in any dimension.