Progress in Gauge-Higgs Unification on the Lattice
We study a five-dimensional pure gauge theory formulated on the orbifold and discretized on the lattice by means of Monte Carlo simulations. The gauge symmetry is explicitly broken to at the orbifold boundaries. The action is the Wilson plaquette action with a modified weight for the boundary plaquettes. We study the phase transition and present results for the spectrum and the shape of the static potential on the boundary. The latter is sensitive to the presence of a massive Z-boson, in good agreement with the directly measured Z-boson mass. The results may support an alternative view of the lattice orbifold (stemming from its mean-field study) as a 5d bosonic superconductor.
Progress in Gauge-Higgs Unification on the Lattice
Department of Physics, National Technical University of Athens
Zografou Campus, GR-15780 Athens, Greece
1 The lattice orbifold and its symmetries
We use a five-dimensional (5d) anisotropic Euclidean lattice with points. The lattice spacing is in the four-dimensional (4d) hyperplanes orthogonal to the extra dimension and along the extra dimension. The gauge links connect the lattice points with . The Euclidean index runs over the temporal and spatial () directions. We will also use Greek indices to denote the directions. The lattice is assumed to be periodic except for the extra dimension, which is an interval with boundaries originating from an orbifold projection . The physical length of the extra dimension is . We employ the anisotropic Wilson plaquette action which is defined as
where the weight factor is for the boundary plaquettes and otherwise. Only counterclockwise oriented plaquettes are summed over and is the identity matrix. The anisotropy parameter is in the classical continuum limit . Instead of we will use the equivalent parameter pair . Along the extra dimension with coordinate the orbifold projection specifies Dirichlet boundary conditions
at and with the projection matrix
Thus, at the interval ends, the gauge symmetry is explicitly broken to a subgroup of which is left invariant by group conjugation with . The lattice of the orbifolded gauge theory is schematically represented in Fig. 1.
There are three types of gauge links: 4d links contained in the two boundaries at and , hybrid extra dimensional links (with one end on a boundary transforming under and the other end in the bulk transforming under ) and the remaining bulk links.
In addition to the local gauge symmetry, the orbifold theory possesses the following global symmetries
is a center transformation in the 4d hyperplanes by an element of the center of . is a reflection with respect to the middle of the orbifold interval . The fixed point symmetry is . On the (“left”) boundary at , is defined as
and on the (“right”) boundary at , is defined as
In order that are consistent symmetry transformations, a boundary gauge link should remain in the group after conjugation by . Moreover, the transformations have to commute with the orbifold projection and has to satisfy 
where is an element of the center of , i.e. . If or equivalently
for example , the fixed point transformations are the stick symmetries introduced in  and we denote them by . If or equivalently , then the transformations are global gauge transfomations.
In order to build lattice operators for the scalar (Higgs) and vector (gauge boson) particles, see , we will use the boundary-to-boundary-line
and the orbifolded Polyakov loop111The subscript () indicates that an operator is defined on the () boundary.
Higgs operators, with spin , charge conjugation and spatial parity , are defined by
where . Inspired by , we define a gauge boson operator, with , and , by
where . Note that while is odd under both stick symmetries and , is odd under and even under .
Another quantity which will be used in particular to study dimensional reduction from five to four dimensions is the static potential extracted from Wilson loops. We take the latter to be defined in a 4d hyperplane, so the static potential depends on the coordinate .
In order to build a larger variational basis, the links used to construct the Higgs and gauge boson operators and the Wilson loops are smeared with a HYP smearing  which does not use the temporal links and is adapted for the orbifold. In particular the smearing parameters are set to , and and the spatial links in the 4d hyperplanes are not smeared along the extra dimension.
2 Phase diagram
Fig. 2 shows the phase diagram based on the solutions obtained for the mean-field background . Due to the orbifold boundaries, we distinguish a background , in the 4d hyperplanes and , along the extra dimension. For the details of the mean-field formulation we refer to . The mean-field phase diagram in Fig. 2 has four phases:
, : confined phase (white color);
, : deconfined phase (red color);
, : layered phase, cf.  (blue color);
, : hybrid phase (green color).
The mean-field sees only bulk phase transitions and is not sensitive to compactification effects. Approaching the phase boundary from the deconfined phase, we define the critical exponent of the inverse correlation length, which is given by the Higgs mass (and is measured from the Euclidean time correlator of the operator in Eq. (1)). For , we measure , the Higgs mass and the background do not vanish as the phase transition is approached. This means that the phase transition is of first order (the lattice spacing does not go to zero). For , the layered phase appears at the phase boundary and becomes . Moreover the Higgs mass and the background tend to zero approaching the boundary of the deconfined/layered phases. This is consistent with a second order phase transition. Indeed in , lines of constant physics (LCPs) have been constructed along which the continuum limit was taken. In particular on a LCP with (which is the current experimental value) we find that the Higgs mass is finite without supersymmetry and predict a state of mass . The mean-field calculations show that a LCP with at does not exist.
2.2 Monte Carlo
Fig. 3 shows the phase diagram of the 4d plaquette on the boundary in the plane. We simulated a orbifold in the parameter region and . Along the (or ) line we did a finer scan. For each parameter point we did two runs (hot and cold start) with 500 thermalization and 4000 measurement steps. Two steps are separated by two iterations of one heatbath and 12 overrelaxation update sweeps.
There is a bulk phase transition of first order222 The corresponding transition with periodic boundary conditions along the extra dimension (torus) was studied in , see also .. It is signalled by a hysteresis. At the hysteresis is between and , at slightly smaller values than on the torus, cf. .
Fig. 4 shows the phase diagram of the absolute value of defined in Eq. (1) (left plot) and of defined in Eq. (1) (right plot). The lattices are , the same as in Fig. 3, and the links entering the operators are smeared by 10 iterations of HYP smearing. Vertical lines are plotted to show the statistical errors. Comparing to Fig. 3, we see that both operators show an hysteresis at the first order bulk phase transition. The operators and have finite volume effects when becomes large (which we know from a comparison with lattices) and we therefore restrict the range of parameters plotted. The irregular behavior at is probably due to finite volume effects.
A more detailed investigation of the phase diagram is under way.
On an isotropic () orbifolded lattice we measure the spectrum close to the bulk phase transition () using the operators defined in Eq. (1) and Eq. (1). The statistics is of 2000 measurements separated by two update steps (each update step consists of one heatbath and 16 overrelaxation sweeps). The variational basis is constructed using smeared links with 5, 15 and 30 HYP smearing iterations.
The masses of the Higgs and gauge boson in units of are shown in Fig. 5. Together with the ground states, also the first excited states were resolved. We find a nonzero gauge-boson mass and the value of does not decrease with the lattice size . This implies that the Higgs mechanism is at work. The mass hierarchy is not the one measured by the experiments at CERN, we find . The masses of the excited states of the Higgs and the Z-boson are approximately equal.
At we also measured the static potential on the boundary. The data (blue points) are shown on the left plot of Fig. 6. In order to get a qualitative understanding of the shape of the potential, we perform 4d Yukawa (red dashed line), 4d Coulomb (black continued line) and 5d Coulomb (green dash-dotted line) global fits333 The best 5d Yukawa fit has a zero Yukawa mass and is therefore not considered.. In the Yukawa non-linear fit, the Yukawa mass is a fit parameter. The preferred global fit is the 4d Yukawa. The of the 4d Yukawa fit is shown on the right plot of Fig. 6 as function of the Yukawa mass . The value of which minimizes the is consistent with the directly measured value of , cf. Fig. 5.
After we proved the existence of the Higgs mechanism, by finding a nonzero mass for the gauge boson, the question of its origin arises. In particular Elitzur’s theorem  tells that only global symmetries can be spontaneously broken on the lattice and phase transitions are characterized by gauge invariant order parameters. In Section 1 we have identified global symmetries, the fixed point symmetries and , which contain the stick symmetries and . In the deconfined phase, where we measure the masses, the stick symmetries are spontaneously broken, see Section 2.2. This breaking induces the breaking of the other global symmetries, which are global gauge transformations, this is the origin of the Higgs mechanism. The Polyakov loop in Eq. (1) is the order parameter for confinement/deconfinement. The deconfinement phase can be a Coulomb or a Higgs phase. We conjecture that the operator in Eq. (1), is the order parameter of the Higgs phase, namely the Higgs mechanism happens when .
The Higgs mechanism on the orbifold seems to have a different origin than the Hosotani mechanism , which was formulated in perturbation theory and works only when fermions are included, see also the lattice study of . The orbifold mechanism of spontaneous symmetry breaking could be related to a bosonic superconductor. The mean-field background breaks translation invariance, like a crystal, see Fig. 7. Gauge fluctuations around the mean-field background are like phonons, the Polyakov loop (Higgs) with charge is like a Cooper pair and the Higgs mechanism, due to gauge-Higgs interaction, happens like in a superconductor slab, where the photon becomes massive.
We have presented a Monte Carlo study of the 5d orbifold, extending previous results from . A second order phase transition is not found. Instead there is a line of bulk first order phase transitions and a new transition for anisotropy which is signalled by the boundary (but not by the bulk) 4d plaquette. The boundary transition is also of first order where we did simulations.
The spectrum is measured so far only at for an orbifold with , where we find a massive gauge boson which is heavier than the Higgs scalar. The mass of the gauge boson is consistent with the mass extracted from a 4d Yukawa fit to the boundary static potential, thus supporting dimensional reduction. It remains to be seen how the mass hierarchy is at , where in the mean-field calculation it was possible to reproduce the experimentally measured masses for the Higgs and gauge boson.
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