Mean-Field Gauge Interactions in Five Dimensions II.
Nikos Irges, Francesco Knechtli and Kyoko Yoneyama
1. Department of Physics
National Technical University of Athens
Zografou Campus, GR-15780 Athens Greece
2. Department of Physics, Bergische Universität Wuppertal
Gaussstr. 20, D-42119 Wuppertal, Germany
e-mail: firstname.lastname@example.org, email@example.com,
We study Gauge-Higgs Unification in five dimensions on the lattice by means of the mean-field expansion. We formulate it for the case of an pure gauge theory and orbifold boundary conditions along the extra dimension, which explicitly break the gauge symmetry to on the boundaries. Our main result is that the gauge boson mass computed from the static potential along four-dimensional hyperplanes is nonzero implying spontaneous symmetry breaking. This observation supports earlier data from Monte Carlo simulations .
The phase diagram of five-dimensional gauge theories is surprisingly rich. On an infinite, hypercubic, anisotropic lattice it is parametrized by the two dimensionless parameters and , where is the lattice coupling and the anisotropy parameter. In part I of this work  we explored the phase diagram of a five-dimensional gauge theory with fully periodic boundary conditions, using an expansion around a mean-field background . We concentrated on the regime where and where a line of second order phase transitions was observed. In the vicinity of this phase transition the system reduces dimensionally to four dimensions and in the continuum limit the physics is consistent with what one would expect from the lightest states with four dimensional quantum numbers. For the fully periodic system this would be a four dimensional gauge theory coupled to an adjoint scalar in the confined phase .
In this work we extend the construction of  by changing the boundary conditions in the fifth dimension from periodic to orbifold. The embedding of the orbifold projection in the geometry introduces boundaries at the ”ends” of the fifth dimension (which is now an interval) and its embedding into the gauge group as well known by now , alters the field content surviving on the boundaries. In this respect, for there are at least two possibilities. One is when the orbifold action is such that the adjoint set of scalars (i.e. the extra-dimensional components of the gauge field) is projected out at the boundaries and one is left there with just a pure gauge theory. This construction allows one to carry out an analysis similar to  but in a context that is directly generalizable to QCD once fermions are also added. The other possibility is to use the orbifold action to project out some of the gauge fields and some of the scalars. With such a choice it is possible to realize a field content similar to the one of the bosonic sector of the Standard Model. At a first stage we consider an bulk gauge group. The orbifold then leaves a theory coupled to a complex scalar on the boundaries. This setup could serve as the simplest prototype of the Higgs mechanism.
The idea that the Standard Model Higgs particle may be the remnant of an extra dimensional gauge field is not new . Also, investigations of five-dimensional gauge theories with a lattice regularization have both an analytical and a Monte Carlo past. The analytical work has been concentrated around the question of the existence or not of a layered phase  and around the existence or not of an ultraviolet fixed point . Monte Carlo simulations have been mostly looking for the layered phase , for first order bulk  (the pioneering work in this direction) or second order  (finite temperature and bulk) phase transitions and dimensional reduction via localization . All of these lattice investigations have been carried out on fully periodic lattices. Recently there has been interest also in lattices with orbifold boundary conditions , . Our motivation to look more carefully at the phase diagram of five-dimensional orbifold gauge theories stems from the fact that Coleman-Weinberg computations , continuum perturbation theory at one loop  and exploratory lattice Monte Carlo simulations  indicate that once a Higgs mass is generated by quantum effects, it seems to remain finite, despite the non-renormalizable nature of the higher dimensional theory.
A first attempt to probe analytically the regime away from the perturbative point, in order to see if there is a dynamical mechanism of spontaneous symmetry breaking (SSB) triggered by the gauge field (without the presence of fermions) was made in . The idea there was to use the Symanzik lattice effective action  ( is the lattice spacing and the number of lattice points in the fifth dimension)
which is determined by a finite number of dimensionless coefficient functions on an infinite spatial isotropic lattice, provided that one can consistently truncate the expansion. The ansatz in  was to truncate the expansion after the first two higher dimensional operators: one of order corresponding to the lowest order dimension five boundary counterterm parametrized by the coefficient function and one of order corresponding to the dimension six bulk operator parametrized by the coefficient function . To extract information about SSB in this setup, one assumes a vacuum expectation value (vev) for one of the components of the five-dimensional gauge field and uses the truncated expansion expanded around this vev to compute a Coleman-Weinberg type potential for the dimensionless quantity
with the five-dimensional coupling, the size of the fifth dimension and we have written this formula for an anisotropic lattice with lattice spacings and ( in the classical limit). One then finds the preferred value for by minimizing [14, 18] and calls it . As a consequence, the otherwise massless gauge boson develops a mass due to this vev equal to
This is the Hosotani mechanism, applied to the case of the orbifold. The result of the analysis of , performed at , was that indeed there exist values of and that yield a ”Mexican hat” Higgs potential that triggers SSB with the Higgs particle having a mass of similar order as the gauge boson. In particular, it was shown that a non-zero is able to trigger SSB by itself, by shifting from integer (for which there is no SSB) to half integer. This could be the main phenomenological gain from the complications encountered by entering in the interior of the phase diagram, in view of the fact that in the conventional continuum approaches where one takes , one necessarily needs fermions in order to trigger SSB  (i.e. a non-integer ) and even if SSB is achieved, the Higgs typically turns out to be generically too light . In the absence of a non-perturbative control of the theory, in  the coefficients were treated as free parameters. Non-perturbatively however they are not free parameters and it is not guaranteed that the quantum theory generates values for the coefficients that trigger SSB.
In this work we improve on these approximations by computing the Wilson loop and the mass spectrum of the lightest states, in the mean-field expansion far from the five-dimensional perturbative point, close to the bulk phase transition. The formalism involved is very similar to the one developed in  and therefore will be heavily used. We show that the mean-field expansion predicts the spontaneous breaking of the boundary gauge symmetry already in the pure gauge system and allows for a Higgs like scalar of similar mass as the mass of the broken gauge field. With infinite four-dimensional lattices, the parameters of our model are , and . To the extent that the mean-field expansion is a good description of the non-perturbative system, any result stemming from this approach should be taken seriously. In fact, the first exploratory Monte Carlo studies of the orbifold theory  had earlier reached similar conclusions.
In Section 2 we give a short review of the mean-field expansion formalism. In Section 3 we apply the general formalism to the five-dimensional lattice gauge theory with orbifold boundary conditions along the extra dimension. In Section 4 we present our numerical results and in Section 5 our conclusions. In the Appendices we detail the mean-field calculations of the propagator with orbifold boundary conditions and of the mass spectrum.
2 A short review of the mean-field formalism
The partition function of a gauge theory on the lattice is
where is the Wilson plaquette action and denotes oriented plaquettes (i.e. each plaquette is counted with two orientations). In the mean-field approach  the link variables are traded for the complex quantities and and in terms of these one rewrites the partition function as
where the effective mean-field action is defined via
The mean-field or zeroth order approximation amounts to finding the minimum of the effective action when
The zeroth order saddle point solution or ”mean-field background” can be easily obtained by taking derivatives of Eq. (5) with respect to and and require them to vanish. One then has
The above two equations are the ones that make the action extremal and define the mean-field solution to zeroth order. The free energy per lattice site is
At 0’th order we simply have
Gaussian fluctuations are defined by setting
We impose a covariant gauge fixing on . In  it was shown that this is equivalent to gauge-fix the original links . The integral
introduces the pieces of the propagator
which will be used extensively later. The quadratic part of the effective action is , and denote the dimensionalities of the fluctuation variables and .
We would like to compute the expectation value of observables
in the mean-field expansion. To first order it is given by the formal expression 
and the second derivative of the observable is taken in the mean-field background. is the contribution from the gauge fixing term. actually turns out to be proportional to the unit matrix and drops out from all expressions. The free energy at this order becomes
To extract the mass spectrum, we denote a generic, gauge invariant, time dependent observable as and its connected version as . Defining the correlator
to first order in the fluctuations the expression reduces to with
where the notation means one derivative acting on each of the and . The mass of the lowest lying state is then
It turns out that in order to extract the mass of the vector one needs to go to second order in the mean-field expansion. Physical expectation values are formally given at this order by 
To extract the mass from the connected correlator is straightforward. Again, all time independent contribution (self energies) cancel from connected correlators and at the end the mass is obtained from
where is the next to leading order correction to the -boson correlator.
3 The lattice orbifold in the mean-field expansion
We will now apply the formalism we described in general terms to a specific example: an lattice gauge theory in 5 dimensions with Dirichlet boundary conditions for certain components of the gauge field along the fifth dimension.
The discretized version of the orbifold defined on a five-dimensional Euclidean lattice was constructed in . The points on the lattice are labeled by integer coordinates but we will often use the notation for the time component. The periodic spatial directions () have dimensionless extent and the time-like direction () has extent . The fifth dimension () has extent . The gauge-unfixed anisotropic mean-field Wilson plaquette action reads
where the effective mean-field actions and are computed in Appendix A. The couplings are defined as
In this work we will parametrize the infinite anisotropic lattice by the parameters and , where and . A gauge transformation acts on a bulk link as
on a boundary link as
and on a link whose one end is in the bulk and the other touches the boundary as
One possibility is to derive the orbifold theory from its parent circle theory. In this setup a general link satisfies the orbifold projection condition
where the reflection property about the origin of the fifth dimension is
The transformation property under group conjugation is
The boundary conditions that the above projections imply at their fixed points amount to Dirichlet boundary conditions for some of the links at the orbifold boundary hyperplanes. Consequently the gauge group variables at the boundaries are restricted to the subgroup of invariant under group conjugation by a constant matrix with the property that is an element of the center of . Only gauge transformations that commute with are still a symmetry at the boundaries and thus there, the orbifold breaks explicitly the gauge group. In general, a bulk group breaks by to an equal rank subgroup on the two boundaries .
For , which is the gauge group of our focus, we will take . This means that at the orbifold fixed points the gauge group is broken to the subgroup parametrized by , where are compact phases. In the continuum limit this implies that (the “ gauge boson”) and (the “Higgs”) satisfy Neumann boundary conditions and and Dirichlet ones.
In the mean-field approach, we parametrize the fluctuating fields in the bulk as
The are the Pauli matrices. On the boundaries instead, we use the parameterizations
with . For later convenience we define the line
and introduce the matrices
For the computation of the and Higgs masses we first define the orbifold projected Polyakov loop
satisfying , in terms of which we define the field and then the displaced Polyakov loop 
which assigns a vector and a gauge index to the observable appropriate to a gauge boson. The Higgs observable is derived from the averaged over space and time location connected correlator
and the -boson from the correlator
Out of the above defined objects one can straightforwardly extract the masses using the general results of the previous section.
3.1 The mean-field background
In order to determine the background we need the effective potentials and . The effective potentials in the bulk are the same as on the torus, while on the boundaries they are
For details see Appendix A.
Translation invariance along the dimensions means that we can parametrize the saddle point solution, which minimizes , as follows : for (four-dimensional links)
and for (extra-dimensional links)
The action at zeroth order reads (, )
The minimization equations lead to the following relations: for
A prime on or denotes differentiation with respect to its argument. Similarly, for we have
For (four-dimensional links)
For (extra-dimensional links)
3.2 Observables from fluctuations around the background
In sect. 2 we described the general formalism for computing observables from fluctuations around the mean-field background. Here we apply this formalism to our case and give the results for the free energy, the scalar and vector masses.
The lattice propagator, Fourier transformed along the spatial and time directions is an object that contains the information about the boundary conditions. We denote its components as
The momenta are four dimensional momenta, however we will further split the momenta into their time and spatial components: whenever it is necessary, for clarity. For a more detailed computation of the propagator we divert the reader at this point to Appendix B.
The free energy
The Higgs and -boson masses
For the Higgs, there is a non-trivial contribution already at first order. The result is
where is the Polyakov loop Eq. (38) evaluated on the background and is defined in Eq. (LABEL:Pisymbols). The above correlator does not contain torons since the 00 component of the propagator does not contain any.
For the the result is
where is defined in Eq. (LABEL:Pisymbols). It contains regularizable torons, that is simultaneous zero modes in the propagator and the observable, whose contribution vanish in the infinite lattice volume limit.
Derivations can be found in Appendix C.
The static potential
On the orbifold we have the three types of static potentials. Here we will be interested in the potentials extracted from Wilson loops in the four-dimensional hyperplanes, along either one of the boundaries and in the middle of the orbifold (i.e. at ). We consider the Wilson loops of size along one of the three spatial dimensions and we average over the possible orientations. The exchange contribution (to ) at is
and the self energy contributions
As for the torus , to first order we would like to compute
where , which is to be substituted in the general expression for the first order corrected static potential
Applied to the gauge boson exchange between two static charges on the boundary the general formula reduces to
The formula for the static potential for the static potential along the four dimensional hyperplane in the middle of the orbifold is similar to Eq. (63), the difference being that the background and the propagator are evaluated at and in the last line of Eq. (63) there is a sum over all the gauge components .
4 Spontaneous symmetry breaking
4.1 The phase diagram
Using the equations that determine the mean-field background in Section 3.1, one can extract the leading order approximation to the phase diagram. The equations are solved iteratively and numerically. The confined phase is defined as the phase where for all . When and for all , we define the layered phase. The Coulomb phase is defined where and for all . We do not find a phase where and for all . The background is sensitive to and . On Fig. 1 we plot the phase diagram with color code, red for the confined phase, blue for the layered phase and white for the Coulomb phase. Green is used where for some reason the iterative process does not converge to a solution. For the rest of this paper we will stay in the Coulomb phase.