Signatures of the gauge-Higgs unification at LHC and future colliders are explored. The Kaluza-Klein (KK) mass spectra of and the Higgs self-couplings obey universality relations with the Aharonov-Bohm phase in the fifth dimension. The current data at low energies and at LHC indicate . Couplings of quarks and leptons to KK gauge bosons are determined. Three neutral gauge bosons, the first KK modes , , and , appear as bosons in dilepton events at LHC. For , the mass and decay width of , , and are (5.73, 482), (6.07, 342), and (6.08 TeV, 886 GeV), respectively. For their masses are 8.00 8.61 TeV. An excess of events in the dilepton invariant mass should be observed in the search at the upgraded LHC at 14TeV.
21 May 2014 OU-HET 806, KIAS-P14007
of the SO(5)U(1) gauge-Higgs unification
Shuichiro Funatsu, Hisaki Hatanaka,
Yuta Orikasa and Takuya Shimotani
Department of Physics, Osaka University, Toyonaka, Osaka 560-0043, Japan
School of Physics, KIAS, Seoul 130-722, Republic of Korea
The discovery of the Higgs boson of a mass around 126GeV[1, 2] supports the scenario of unification of forces and symmetry breaking envisioned in the standard model (SM) of electroweak interactions. Experimental data so far are consistent with what the SM describes, but more data are necessary to pin down whether or not the discovered boson is definitively the Higgs boson in the SM. Other scenarios such as supersymmetric models[3, 4], little Higgs models-, composite Higgs models-, warped extra-dimension models-, and UED models- have been proposed in anticipation of physics beyond the SM. It is urgent to derive and predict new phenomena which can be observed and checked in the experiments at the upgraded 14TeV LHC.
The gauge-Higgs unification is formulated in higher-dimensional gauge theory -. The four-dimensional Higgs boson appears as a part of the extra-dimensional component of gauge fields, being unified with four-dimensional gauge fields such as , and . Dynamics of the Higgs boson are governed by the gauge principle. Most viable is the gauge-Higgs unification in the Randall-Sundrum warped space,-. At low energies it yields almost the same physics as the SM, being consistent with LHC data and others. Higgs couplings to gauge bosons, quarks and leptons at the tree level are suppressed by a common factor , where is the Aharonov-Bohm phase in the extra dimension. All of the precision measurements[9, 39], the tree-unitary constraint, and the search- indicate . Branching fractions of various decay modes of the Higgs boson remain nearly the same as in the SM, and the signal strengths of the Higgs decay modes relative to the SM are . We note that though the gauge-Higgs unification model has similarity to the composite Higgs models, it is more restrictive and has more predictive power.
To distinguish the gauge-Higgs unification from the SM we examine the prediction of new particles. It has been pointed out that the first Kaluza-Klein (KK) modes of and , denoted as and , must appear around 6TeV for . In this paper we give detailed analysis of production of , and at the upgraded LHC. Here is the gauge boson associated with , which does not have a zero mode. It will be shown that , and have large widths and can be seen as or signals. Once their masses are determined, the value of is fixed from the universality relations, which leads to further prediction of the Higgs self-couplings, etc.. Many other signals of gauge-Higgs unification have been discussed in the literature-.
In Sec.2 the action of the model is given. In addition to quark-lepton multiplets in the vector representation of , fermion multiplets in the spinor representation of are introduced to realize the observed unstable Higgs boson. In Sec.3 the effective potential is evaluated and relevant parameters of the model are determined. It is shown that there appear universality relations among , the KK mass scale , , , and Higgs cubic and quartic couplings. In Sec.4 dilepton () signals at LHC in the so-called search are examined. In the gauge-Higgs unification , and appear as bosons. Their masses are around 6 TeV (8 TeV) for (0.073), and they have large decay widths. We show that they must be found in the upgraded LHC at 14 TeV. Sec.5 is devoted to conclusions. In the Appendixes we summarize KK mass spectra, wave functions and gauge couplings of gauge fields, quark-leptons, and -spinor fermions.
The model is defined in the Randall-Sundrum warped spacetime  with the metric
where , , and for . The Planck brane and TeV brane are located at and , respectively. In the bulk region, , the cosmological constant is given by . The warp factor is large; . The KK mass scale is given by . In the fundamental region the metric can be written, in terms of the conformal coordinate , as
The gauge symmetry in the bulk region is given by with the corresponding gauge fields , and and gauge couplings and . Quark-lepton multiplets are introduced in the vector representation 5 of , whereas additional fermions are introduced in the spinor representation 4 of [34, 36, 37]. The gauge symmetry is partially broken to by orbifold boundary conditions. On the Planck brane at () there live right-handed brane fermions and brane scalar , which are (2,1) and (1,2) representation of , respectively. The brane interactions are manifestly gauge-invariant under . The brane scalar spontaneously breaks to by , which, in turn, induces mixing among and and makes all exotic fermions acquire masses of . The resultant theory at low energies (TeV) has the SM gauge symmetry with the SM matter content. All anomalies are cancelled. Finally the gauge symmetry is dynamically broken to by the Hosotani mechanism.
The bulk part of the action is given by
The gauge fixing and ghost terms are denoted as functionals with subscripts gf and gh, respectively. , , and . The gauge fixing function is taken as with a background field (), . for quark-multiplets and otherwise. The gauge fields are decomposed as
where and are the generators of and , respectively. The electric charge is given by
In the fermion part and matrices are given by
The term in the action (2.8) gives a bulk kink mass. The dimensionless parameter plays an important role in controlling profiles of fermion wave functions.
The orbifold boundary conditions at and are given by
The symmetry is reduced to by the orbifold boundary conditions. At this stage the four-dimensional components of the five-dimensional gauge fields have zero modes in , whereas the extra-dimensional components have zero modes in , or (). The latter contains the four-dimensional Higgs field, which is a doublet both in and in . Without loss of generality one can set when the EW symmetry is spontaneously broken by the Hosotani mechanism. The zero modes of (a = 1,2,3) are absorbed by and bosons. The four-dimensional neutral Higgs field is a fluctuation mode of the Wilson line phase ,
Here the wave function of the four-dimensional Higgs boson is given by for and . is the dimensionless 4 dimensional coupling.
Quark-lepton multiplets are in the vector representation of . They are decomposed into vectors and singlets. One vector multiplet contains two doublets. In each generation
where the subscripts denote . We choose the bulk mass parameters such that and in each generation. With the boundary condition in (2.17), zero modes appear in
On the Planck brane there exist the brane scalar in (1,2) representation of with and the brane fermions in (2,1) representation of .
where the subscripts denote . ’s are triplets. With these brane fermions all four-dimensional anomalies in are cancelled.
The brane part of the action is given by
breaks to . It also induces mass mixing on the brane
where define brane mass parameters. In the present paper we assume that the brane interactions are diagonal in the generation of quarks and leptons. In this case all of and can be taken to be real and positive without loss of generality. As far as , only and become relevant at low energies.
As shown in Sec.3, the effective potential is minimized at , thereby the electroweak symmetry breaking taking place. The gauge fields are expanded in KK towers. In particular, four-dimensional components of the gauge fields are expanded, in the twisted gauge, as
Here we have introduced such that , and where ’s are generators of in the tensorial representation. The and towers contain and . The other towers do not contain light modes. Each of the towers splits into two KK towers at . In all, (2.37) contains 11 KK towers. Details of wave functions of each KK tower are tabulated in Appendix B.
The fermion are introduced in the spinor representation of unlike other fields in the bulk which are in the vector or adjoint representations. As explained in the next section, the existence of in addition to the other bulk fields leads to nontrivial dependence of the effective potential on and to the instability of the four-dimensional Higgs boson. The boundary condition in (2.17) implies that there is no zero mode for and that the lowest KK modes of dominantly couple to the gauge bosons. If the boundary condition were taken, then the lowest KK modes of would dominantly couple to the gauge bosons. The lowest, neutral component of turns out stable and becomes the dark matter of the Universe, as will be explained in a separate paper. For this reason the -spinor fermion is called a dark fermion.
3 Higgs Boson and the Universality
As explained in (2.20), the extra-dimensional component contains the four-dimensional Higgs field,
The value of is determined by the location of the global minimum of the effective potential . The Higgs boson mass is given by
In this section we explain how the parameters of the model are determined, and show that universality relations appear among , the KK mass , the masses of and , and the Higgs self-couplings.
Let us first consider the case in which all -spinor fermions (dark fermions) are degenerate at the tree level, i.e. (). At the one-loop level only the KK towers whose mass spectra depend on contribute to the effective potential . Those spectra are given by (B.9) for the tower, (B.11) for the tower, (B.26) for the tower, (C.8) for the top quark tower, (C.16) for the bottom quark tower, and (C.20) for the tower or the dark fermions. Contributions of other quarks and leptons turn out exponentially suppressed and negligible.
The relevant parameters of the model are , , , , , , and , from which is determined. Other brane mass parameters are irrelevant so long as . These eight parameters are chosen such that , , , , , and take the observed values. (To be precise, is determined by global fit.) This procedure leaves two parameters, say and , free. The procedure is highly involved as everything must be determined at the global minimum of , which, however, is to be found after all parameters are specified. In other words, all parameters must be determined self-consistently.
First we note that with those given parameters, the one–loop effective potential is given by
where and and are modified Bessel functions. In the following we take the ’t Hooft–Feynman gauge .
We adopt the following algorithm to find consistent solutions. We fix the two parameters and .
Suppose that the minimum of is located at . Equation (B.11) and determine , which fixes by the boson mass .
Now in (3.11) is evaluated with being a parameter. is determined by the condition
which assures that the minimum of is located at .
With these parameters the Higgs boson mass is evaluated from (3.3). This gives , which, in general, differs from the observed value GeV.
We vary the value and repeat the procedure from step 1 until we get GeV.
In this manner the value at the minimum is determined as . All other quantities such as the mass specta of all KK towers, gauge couplings of all particles, and Yukawa couplings of all fermions are determined as functions of , . Determined values for , , , etc. are tabulated in Table 1 in the case of .
Smaller and correspond to heavier masses of the top quark and dark fermions and and give larger contributions to . As gets larger, () becomes larger (smaller) with fixed , as the contribution from each dark fermion to becomes small. Given , only a limited region for is allowed. For one cannot reproduce the Higgs mass 126GeV when becomes too small. When , one cannot reproduce the top quark mass for .
Dark fermions may not be degenerate. Suppose that multiplets have the bulk mass , and multiplets have . Small difference between and can yield a substantial difference in masses, whereas is almost unaffected. For instance, when , a difference leads to (). The dark fermion masses and in the case of and are tabulated in Table 2. It is found that the numerical values of , , , , , and are the same as those in Table 1 to the accuracy of three digits.
3.2 The universality
As described above, various quantities such as , , the mass spectra, Higgs cubic and quartic self-couplings , and Yukawa couplings are determined as functions of and in the case of degenerate dark fermions. In other words they depend not only on , but also on how dark fermions are introduced, which could spoil the predictability of the model. Surprisingly it has been found in Ref.  that universal relations are held among , , , , , , and irrespective of . This property is called the universality. It implies that once one of these quantities is determined from experiments, then other quantities are predicted, irrespective of the details of the dark fermion sector. The mass spectrum of dark fermions, , on the other hand, sensitively depends on .
It is most enlightening to express these universal relations as functions of . The masses , , , are expressed in the form of