Coleman–Weinberg potential in p-adic field theory

• Main Authors: Dmitry S. Ageev, Andrey A. Bagrov, Askar A. Iliasov
• Format: Article
• Language: English
• Published: SpringerOpen 2020-09-01
• Series: European Physical Journal C: Particles and Fields
• Online Access: http://link.springer.com/article/10.1140/epjc/s10052-020-08442-5

Abstract In this paper, we study $$\lambda \phi ^4$$ λ ϕ 4 scalar field theory defined on the unramified extension of p-adic numbers $${\mathbb {Q}}_{p^n}$$ 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, $$p \rightarrow 1$$ p → 1 and $$p \rightarrow \infty $$ p → ∞ . We show that the $$p\rightarrow 1$$ 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 $$p\rightarrow \infty $$ p → ∞ limit, the theory exhibits very peculiar behavior with emerging logarithmic terms in the effective potential, which has no analogue in real theories.