Splitting of the connection in gauge theories with broken symmetry

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Abstract
We obtain a fully geometric analog of the Higgs mechanism whereby a symmetry-breaking Higgs field is used to impart mass to gauge fields. We do this by showing that under fairly general hypotheses a symmetry-breaking Higgs field φ on a ‘‘principal bundle with connection’’ (P,ω) allows the decomposition of the connection ω into a pair (ω′,τ) where ω′ is a connection on P that reduces to a φ-subbundle of P and where τ is a tensorial field on P. The gauge fields that remain massless are identified with the components of ω′ while the gauge fields that acquire mass are identified with the components of τ. This decomposition of the connection is exploited in the case where the group of the bundle is the conformal group which scales some fixed metric of arbitrary signature. The geometry of such a bundle with connection generalizes Weyl geometry and provides a bundle setting for conformal gauge theories. We then show that the Weinberg–Salam electroweak theory can be recast as a conformal gauge theory. A primary feature of our conformal version of the Weinberg–Salam theory is that it provides a geometrical interpretation of the surviving component of the Higgs scalar field as an infinitesimal conformal factor.

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