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Metastable de Sitter vacua from critical scalar theory

Studying the critical scalar theory in four dimensional Euclidean space with the potential term $-gϕ^4$ we show that the theory can not be analytically continued through g=0 from g<0 region to g>0 region. For g>0 although energy is not bounded from below but there exist a classical trajectory with an AdS5 moduli space, corresponding to a metastable local minima of the action. The fluctuation around this solution is governed by a minimally coupled scalar theory on four dimensional de Sitter background with a reversed Mexican hat potential. Since in the weak coupling limit, the partition function picks up contribution only around classical solutions, one can assume that our de Sitter universe corresponds to that local minima which lifetime increases exponentially as the coupling constant tends to zero. Similar results is obtained in the case of critical scalar theory coupled to U(1) gauge field which is essential for people living on flat Euclidean space to observe a de Sitter background by optical instruments.