Coleman\u2013Weinberg potential in p-adic field theory

Askar A Iliasov,Andrey A Bagrov,Dmitry S Ageev

doi:10.1140/epjc/s10052-020-08442-5

Abstract

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 prightarrow 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 prightarrow infty limit, the theory exhibits very peculiar behavior with emerging logarithmic terms in the effective potential, which has no analogue in real theories.

Highlights

A possible relevance of non-Archimedean geometry and padic number theory within different contexts of theoretical physics is being discussed for more than 30 years
We have studied one-loop effective potential in the realvalued scalar field theory over unramified extension Qpn of p-adic numbers
By computing the effective potential one can gain information on quantum behavior of field theory, since it provides a transparent representation of such concepts as symmetry breaking and renormalization group flow

Summary

Introduction

A possible relevance of non-Archimedean geometry and padic number theory within different contexts of theoretical physics is being discussed for more than 30 years. Adding counterterms Aφb and Bφb and solving (15), we obtain renormalized one-loop correction to the effective potential of the following form: ΔV = λ2φb log p. The reason is that in two dimensions, we have to impose mass renormalization condition at some scale φ0 = 0 This leads us to appearance of ∼ 1/t2 term in the renormalized potential which comes with different relative coefficients in p-adic and in real field theories: λφb2 4π (27). A peculiar feature of the p → ∞ limit is the logarithmic term in the effective potential for all “space-time” dimensions It does not have an analogue in the conventional real field theory, but does not lead to any unusual or pathological behavior causing neither symmetry breaking nor singularities in the potential if λ > 0

Euler–Maclaurin estimate of the effective potential

Summary and discussion