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Using the Matsubara formalism, we consider the massive $(\lambda\phi^{4})_{D}$ vector $N$-component model in the large $N$ limit, the systembeing confined between two infinite paralell planes. We investigate thebehavior of the coupling constant as a function of the separation $L$ betweenthe planes. For the Wick-ordered model in $D = 3$ we are able to give an exactformula to the $L$-dependence of the coupling constant. For thenon-Wick-ordered model we indicate how expressions for the coupling constantand the mass can be obtained for arbitrary dimension $D$ in the small-$L$regime. Closed exact formulas for the $L$-dependent renormalized couplingconstant and mass are obtained in $D = 3$ and their behaviors as functions of$L$ are displayed. We are also able to obtainn in generic dimension $D$, anequation for the critical value of $L$ corresponding to a second order phasetransition in terms of the Riemann $zeta$-function. In $D = 3$ arenormalization is done and an explicit formula for the critical $L$ is given.Comment: 16 pages, 4 figure

Boundary dependence of the coupling constant and the mass in the vectorN-component (λφ4)Dtheory

Boundary dependence of the coupling constant and the mass in the vectorN-component (λφ⁴)_Dtheory