Abstract
In SU($N$) gauge-Higgs theories, with a single Higgs field in the fundamental representation, there exists in addition to the local gauge symmetry a global SU(2) symmetry, at $N=2$, and a global U(1) symmetry, for $N \ne 2$. We construct a gauge-invariant order parameter for the breaking of these global symmetries in the Higgs sector, and calculate numerically the transition lines, in coupling-constant space, for SU(2) and SU(3) gauge theories with unimodular Higgs fields. The order parameter is non-local, and therefore its non-analyticity does not violate the theorem proved by Osterwalder and Seiler. We then show that there exists a transition, in gauge-Higgs theories, between two types of confinement: ordinary color neutrality in the Higgs region, and a stronger condition, which we have called "separation-of-charge confinement," in the confinement region. We conjecture that the symmetry-breaking transition coincides with the transition between these two physically different types of confinement.
Highlights
Contrary to statements found in some textbooks, a local gauge symmetry cannot be broken spontaneously, as shown long ago by Elitzur [1]
There are remnant global symmetries which can break spontaneously, but the locations of the corresponding transition lines are gauge dependent [2], which makes a physical interpretation of such transitions dubious
We will construct a gauge-invariant order parameter which is sensitive to these symmetry breakings, and map out the transition line in coupling constant space for SU(2) and SU(3) gauge-Higgs theories with a single unimodular Higgs field
Summary
Contrary to statements found in some textbooks, a local gauge symmetry cannot be broken spontaneously, as shown long ago by Elitzur [1]. We will construct a gauge-invariant order parameter which is sensitive to these symmetry breakings, and map out the transition line in coupling constant space for SU(2) and SU(3) gauge-Higgs theories with a single unimodular Higgs field This raises the question of the physical distinction between the symmetric and broken phases of a gaugeHiggs theory. The answer is that the global symmetry can be broken in the Higgs sector, without breaking (and giving rise to Goldstone modes) in the full theory To explain this point, let us begin by noting that the partition function Zðβ; γÞ of the gauge-Higgs theory can be regarded as the weighted sum of partition functions Zspinðγ; UÞ of a spin system in a background gauge field, i.e., Z. How can we tell whether the global symmetry is spontaneously broken in these spin systems? If we denote the VEV of the φðxÞ field in the background gauge field as φðx; UÞ, where φðx; UÞ
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