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 $${\mathbb {Q}}_{p^n}$$ . 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, $$p \rightarrow 1$$ and $$p \rightarrow \infty $$ . We show that the $$p\rightarrow 1$$ 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 $$p\rightarrow \infty $$ limit, the theory exhibits very peculiar behavior with emerging logarithmic terms in the effective potential, which has no analogue in real theories.