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Aspects of QED and non-Abelian gauge theories in S1 and S1 spacetimes

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JournalPhysical Review D
DatePublished - 1988
Issue number10
Volume37
Number of pages19
Pages (from-to)2853-2871
Original languageEnglish

Abstract

We study various aspects of Abelian and non-Abelian gauge field theories in flat spacetime with the topology of S1×Rn (n=3,4). We first discuss the effective potential for the electromagnetic field in S1×R3 spacetime, with the objective of finding the dependence of coupling constants on the size L of the compactified dimension. It is first necessary to determine the stable vacuum. In particular, a charged scalar field satisfies an effective periodic boundary condition, while a charged spinor field satisfies an effective antiperiodic boundary condition at the stable vacuum. Then we study the effective coupling constants which depend on the size L of the compactified dimension at one-loop level for QED in S1×R3 spacetime. In the presence of a charged scalar field, the one-loop correction of one of the effective coupling constants for the lowest Fourier component of the electromagnetic field behaves for small L like 1/L, rather than lnL. This is because the lowest Fourier component of the scalar field is constant in the S1 direction. The effective coupling constants for higher Fourier components have logarithmic L dependence, but due to collinear singularities (i.e., singularities in self-energy occurring in the limit of zero mass when the virtual particles are on shell and have parallel momenta) the coefficient of the lnL term differs from component to component. We also point out some ambiguities in defining the effective coupling constants. Finally we discuss the effective potential of non-Abelian gauge theories with a charged spinor field in S1×R4 spacetime. We find that if periodic boundary conditions are imposed on the spinor field in the beginning, the local gauge symmetry will be spontaneously broken in some cases, and we give an explicit model in which this occurs. This spontaneous symmetry breaking depends on the gauge group and its representations, and does not occur for an Abelian gauge field.

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