Boundary dependence of the coupling constant and the mass in the vector N-component $(\lambda {\varphi }^4{)}_D$ theory

Abstract

Using the Matsubara formalism, we consider the massive $(\lambda {\varphi }^4{)}_D$ vector $N$-component model in the large $N$ limit, the system being confined between two infinite paralell planes. We investigate the behavior of the coupling constant as a function of the separation $L$ between the planes. For the Wick-ordered model in $D=3$ we are able to give an exact formula to the $L$-dependence of the coupling constant. For the non-Wick-ordered model we indicate how expressions for the coupling constant and the mass can be obtained for arbitrary dimension $D$ in the small-$L$ regime. Closed exact formulas for the $L$-dependent renormalized coupling constant 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 $D$, an equation for the critical value of $L$ corresponding to a second order phase transition in terms of the Riemann $zeta$-function. In $D=3$ a renormalization is done and an explicit formula for the critical $L$ is given.

16 pages, 4 figures

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• Fermionic determinant with a linear domain wall in 2+1 dimensions
• Vacuum Stress Tensor of a Scalar Field in a Rectangular Waveguide
• Vacuum Polarization of a Scalar Field in a Rectangular Waveguide

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