$\phi^4$ model on a circle

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.