Topology, symmetry, and phase transitions in lattice gauge ... - tuprints

5.4 fundamental higgs with fractional electric charge 113
Figure 5.15: (left) The confinement-like and Higgs-like phases are analytically connected in
a SU(2) gauge theory with a Higgs field in the fundamental representation.
(right) Coupling fractional electric charge with respect to a compact U(1) restores
center symmetry and allows for a spontaneous symmetry breaking transition,
which, however, is strongly first order [162].
It is convenient to freeze the length of the Higgs via the unimodular limit λ →
∞, φ † φ → 1, dispensing with the Higgs potential. The complex doublet can then
be expressed as an SU(2) matrix,
Φ =
(
φ 1 −φ ∗ 2
φ 2 φ ∗ 1
such that the gauge-Higgs coupling simplifies to,
− κ 2 ∑ µ,x
)
, (5.44)
1
2 tr (Φ† (x)U µ (x)Φ(x + ˆµ)). (5.45)
The ensuing theory is referred to as the Fradkin-Shenker model. At zero temperature
its (β col , κ) phase diagram contains a confinement-like region for small κ. Here
the potential between fundamental charges rises linearly at intermediate distances,
but flattens asymptotically due to dynamical color charge screening just as in standard
QCD with fundamental quarks.
For large κ, on the other hand, the potential is Yukawa-like. There is no electric
flux tube formation. Instead, color charges are screened by short range gauge interactions.
In both regions the asymptotic states are colorless, but the mechanisms
that explain this are superficially different.
The important point is that these regions are not separated by the spontaneous
breaking of a global gauge symmetry. The Fradkin-Shenker theorem ensures the
existence of a path in parameter space between the confinement-like and Higgslike
regions in which all local observables are analytic [163, 164]. So there is no
thermodynamic phase boundary, and the the ‘flux-tube’ and ‘Yukawa-screening’
pictures must transform smoothly into one another.