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

88 fractional electric charge and quark confinement
unified theories, and the new topological possibilities. Prototypical unified theories
provide instructive, motivating examples of how a topological link between the
Standard Model gauge groups may emerge in practice. Their details are not, however,
essential to our investigation of whether the inclusion of fractional electric
charge allows for the resurrection of a center symmetry breaking deconfinement
transition in the presence of dynamical quarks. For this, the vital assumption is
that the Standard Model gauge groups which are inherited from higher energy
physics are compact, irrespective of their origin. For a preliminary study, we implement
2-color lattice models with fractional electric charge with respect to a compact
U(1) gauge group as conceptual playgrounds.
5.1 center symmetry breaking by quarks
The explicit breaking of the center symmetry by quark fields in an SU(N) gauge
theory is best understood in analogy with the ordering of spins by an external
magnetic field. Consider the lattice action for Wilson quarks in the fundamental
representation of SU(N), Eq. (2.31). Integrating over the quark fields gives their
contribution to the weight of a given gauge configuration in the partition function,
∫
∏ dψ(x)d ¯ψ(x) exp(−a 4 ∑
x
x,y
¯ψ(x)(1 − κD(x|y))ψ(y)) = det(1 − κD), (5.1)
where the hopping matrix D(x|y) contains the parallel transporters connecting
neighboring sites. This fermion determinant may be re-expressed as an effective
action in terms of a power series. The formal result is [31],
det(1 − κD) = exp(tr [ln(1 − κD)]) = exp(−
which converges when the hopping parameter is small,
κ =
∞
∑
j=1
1
j κj tr[D j ]), (5.2)
1
2(am + 4) < 1 8 , (5.3)
i.e., for large bare quark masses m.
Each term in the exponential in Eq. (5.2) is the product of parallel transporters
U µ along a closed quark loop. Large loops are exponentially suppressed in powers
of the hopping parameter κ, which plays the role of the inverse quark mass in
lattice units. The loops which close within the lattice volume merely renormalize
the lattice gauge coupling β. They are plaquettes and higher dimensional analogs
that are likewise invariant under global center transformations that multiply every
link in a timeslice by a Z N phase (cf. Eq. (3.8)).
Loops that wind around the lattice are another story. In the thermodynamic limit,
L → ∞, the relevant loops are those wrapping around the compact temporal 1/T
direction. The shortest possible winding gives a Polyakov loop corresponding to
a static quark worldline. Since loops in opposite temporal directions are treated
equally, the hopping expansion in Eq. (5.2) contributes terms of the form,
∝ −κ N t
(P(⃗x) + P(⃗x) † ) = −2κ N t
Re P(⃗x), (5.4)