Coleman-Weinberg potential in $p$-adic field theory

In this paper, we study $\lambda {\varphi }^4$ 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 a structure very similar to that of its real cousin. We also study two formal limits of the effective potential, $p\to 1$ and $p \rightarrow \infty$\mathrm{.}Weshowthatthe$p\rightarrow 1$ 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 $p\to \mathrm{\infty }$ limit, the theory exhibits very peculiar behavior with emerging logarithmic terms in the effective potential, which has no analog in real theories.