$\phi^4$ model on a circle

TL;DR

This paper studies the effective three-dimensional $^4$ scalar theory derived from a four-dimensional critical scalar theory at finite temperature, analyzing vacuum structures, parity violation effects, and connections to SU(2) instantons.

Contribution

It introduces a novel analysis of the effective $^4$ theory at finite temperature, including vacuum shifts, parity violation effects, and links to instanton solutions.

Findings

Effective mass proportional to $n/T$ in the scalar theory.

Parity violation leads to a cubic term in the potential.

Vacuum shifts from one-loop corrections are comparable to quartic contributions, but diminish for large $n$.

Abstract

The four dimensional critical scalar theory at equilibrium with a thermal bath at temperature $T$ is considered. The thermal equilibrium state is labeled by $n$ the winding number of the vacua around the compact imaginary-time direction which compactification radius is 1/T. The effective action for zero modes is a three dimensional $\phi^4$ scalar theory in which the mass of the the scalar field is proportional to $n/T$ resembling the Kaluza-Klein dimensional reduction. Similar results are obtained for the theory at zero temperature but in a one-dimensional potential well. Since parity is violated by the vacua with odd vacuum number $n$, in such cases there is also a cubic term in the effective potential. The $\phi^3$-term contribution to the vacuum shift at one-loop is of the same order of the contribution from the $\phi^4$-term in terms of the coupling constant of the four dimensional theory but becomes negligible as $n$ tends to infinity. Finally, the relation between the scalar classical vacua and the corresponding SU(2) instantons on $S^1\times{\mathbb R}^3$ in the 't Hooft ansatz is studied.