Coleman-Weinberg potential in p-adic field theory

In this paper, we study lambda phi(4) scalar field theory defined on the unramified extension of p-adic numbers Q(pn). 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 -> infinity. 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 -> infinity limit, the theory exhibits very peculiar behavior with emerging logarithmic terms in the effective potential, which has no analogue in real theories.