ϕ4 model on a circle

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 ϕ4 scalar theory in which the mass of 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 ϕ3 term contribution to the vacuum shift at one-loop is of the same order of the contribution from the ϕ4 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 S1×R3 in the 't Hooft ansatz is studied.