ON THE INSTABILITY OF THE VACUUM IN MULTIDIMENSIONAL SCALAR THEORIES

The one-loop effective potential in a scalar theory with quartic inter action on the space M(4) X T-n for n = 2 is calculated and is shown to be unbounded from below. This is an indication of a possible instabil ity of the vacuum of the lambda phi(4) model on M(4), when it is regar ded as a low-energy sector of the theory obtained by dimensional reduc tion of the original six-dimensional one. It is argued that, in some r ange of the values of the parameters of the theory, higher-loop correc tions do not change this result qualitatively. The issue of stability for other values of the number n of extra dimensions is also discussed .