Coleman–Weinberg potential in p-adic field theory

In this paper, we study λ ϕ 4 scalar field theory defined on the unramified extension of p-adic numbers Q p n . For different “space-time” dimensions n , we compute one-loop quantum corrections to the effective potential. Surprisingly, despite the unusual properties of non-Archimedean geometry, the Coleman–Weinberg potential of p-adic field theory has structure very similar to that of its real cousin. We also study two formal limits of the effective potential, p → 1 and p → ∞ . We show that the p → 1 limit allows to reconstruct the canonical result for real field theory from the p-adic effective potential and provide an explanation of this fact. On the other hand, in the p → ∞ limit, the theory exhibits very peculiar behavior with emerging logarithmic terms in the effective potential, which has no analogue in real theories.