Vacuum Structure of Two-Dimensional ${\varphi }^4$ Theory on the Orbifold $S^1/Z_2$

Abstract

We consider the vacuum structure of two-dimensional ${\varphi }^4$ theory on $S^1/Z_2$ both in the bosonic and the supersymmetric cases. When the size of the orbifold is varied, a phase transition occurs at $L_c=2\pi /m$, where $m$ is the mass of $\varphi$. For $L<L_c$, there is a unique vacuum, while for $L>L_c$, there are two degenerate vacua. We also obtain the 1-loop quantum corrections around these vacuum solutions, exactly in the case of $L<L_c$ and perturbatively for $L$ greater than but close to $L_c$. Including the fermions we find that the "chiral" zero modes around the fixed points are different for $L<L_c$ and $L>L_c$. As for the quantum corrections, the fermionic contributions cancel the singular part of the bosonic contributions at L=0. Then the total quantum correction has a minimum at the critical length $L_c$.

Revtex, 15 pages, 3 eps figures

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