Gauge-Higgs Grand Unification

$SO(11)$ gauge-Higgs grand unification in the Randall-Sundrum warped space is proposed. Orbifold boundary conditions and one brane scalar field reduce $SO(11)$ to the standard model symmetry, which is further broken to $SU(3)_C \times U(1)_{EM}$ by the Hosotani mechanism. In a minimal model quarks and leptons are contained in a multiplet in ${\bf 32}$ of $SO(11)$ in each generation. Proton decay is naturally suppressed by a conserved fermion number.

There remain uneasy features in the Higgs boson sector in the SM. Unlike such gauge bosons as the photon, W boson, Z boson, and gluons, whose dynamics is governed by the gauge principle, the Higgs boson is an elementary scalar field for which there lacks an underlying fundamental principle. The Higgs couplings of quarks and leptons as well as the Higgs self-couplings are not regulated by any principle. At the quantum level, there arise huge corrections to the Higgs boson mass, which have to be canceled and tuned by hand to obtain the observed 125 GeV mass. One way to achieve natural stabilization of the Higgs boson mass against quantum corrections is to invoke supersymmetry, and many investigations have been made along this line. In this paper, we focus on an alternative approach, the gauge-Higgs unification [1][2][3][4][5].
The Higgs boson is unified with gauge bosons in the gauge-Higgs unification, which is formulated as a gauge theory in five or more dimensions. When the extra-dimensional space is not simply connected, an Aharonov-Bohm (AB) phase in the extra-dimensional space plays the role of the Higgs boson, breaking part of the non-Abelian gauge symmetry. The 4D fluctuation mode of the AB phase appears as a Higgs boson in four dimensions at low energies. In other words, the Higgs boson is part of the extra-dimensional component of gauge potentials, whose dynamics is controlled by the gauge principle. The gauge invariance guarantees the periodic nature of physics associated with the AB phase in the extra dimension, which we denote as θ H . The value of θ H is determined dynamically, from the location of the global minimum of the effective potential V eff (θ H ). At the classical (tree) level, V eff (θ H ) is completely flat, as θ H is an AB phase yielding vanishing field strengths. At the quantum level, V eff (θ H ) becomes nontrivial as the particle spectra and their interactions depend on θ H . It has been shown that the θ H -dependent part of V eff (θ H ) is finite at the one-loop level, free from ultraviolet divergence even in five or more dimensions as a consequence of the gauge invariance. Nontrivial minimum θ min H induces gauge symmetry breaking in general. The mass of the corresponding 4D Higgs boson, proportional to the second derivative of V eff (θ H ) at the minimum, becomes finite irrespective of the cutoff scale in a theory, giving a way to solve the gauge hierarchy problem. This mechanism of dynamical gauge symmetry breaking is called the Hosotani mechanism.
Gauge-Higgs unification models of electroweak interactions have been constructed [6][7][8][9][10][11][12][13]. The orbifold structure of the extra-dimensional space is vital to have chiral fermions, and natural realization of dynamical EW symmetry breaking is achieved in the 5D Randall-Sundrum (RS) warped spacetime. The most promising is the SO(5) × U (1) X gauge-Higgs unification in RS, which is consistent with the observation at low energies provided its AB phase θ H 0.1. The model accommodates the custodial symmetry, and gives almost the same couplings in the gauge sector as the SM. It has been shown that one-loop corrections to the Higgs-boson decay to γ γ due to the running of an infinite number of Kaluza-Klein (KK) excitation modes of the W boson and top quark turn out to be finite and very small, being consistent with the present LHC data [10]. The model predicts Kaluza-Klein excitations of the Z boson and photon as Z events with broad widths in the mass range 5-8 TeV; a dark-matter candidate (dark fermion) of a mass 2-3 TeV and other signals such as anomalous Higgs couplings are predicted as well [12][13][14][15][16][17].
With the gauge-Higgs EW unification model at hand, the next step is to incorporate strong interactions to achieve gauge-Higgs grand unification [18][19][20][21][22][23][24][25][26]. There are models of gauge-Higgs grand unification in five dimensions with gauge group SU (6), which breaks down to SU (3) C × SU (2) L × U (1) Y × U (1) X by the orbifold boundary condition on S 1 /Z 2 . Burdman and Nomura [20] showed that the EW Higgs doublet emerges. Haba et al. [21,22] and Lim and Maru [23] showed that dynamical EW symmetry is achieved with extra matter fields, though they yield exotic particles at low energies. Kojima et al. [24] have proposed an alternative model with SU (5) × SU (5) symmetry. Grand unification in the composite Higgs scenario has been discussed by Frigerio et al. [25]. Yamamoto [26] has attempted to dynamically derive orbifold boundary conditions in gauge-Higgs unification models.
In this paper, we propose a new model of gauge-Higgs grand unification in RS with gauge symmetry SO(11) that carries over the good features of SO(5) × U (1) X gauge-Higgs EW unification. We show that the EW symmetry breaking is induced even in the pure gauge theory by the Hosotani mechanism, in sharp contrast to other models. Quarks and leptons are implemented in a minimal set of fermion multiplets. Proton decay is naturally suppressed by the conservation of a new fermion number.

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Downloaded from https://academic.oup.com/ptep/article-abstract/2015/11/111B01/2606880 by guest on 29 July 2018 The SO(11) gauge potential, A M , expressed as an 11 × 11 antisymmetric Hermitian matrix, satisfies the orbifold boundary condition (BC) given by P 0 and P 1 break SO (11) to SO (10) and SO(4) × SO (7), respectively. In all, the symmetry is bro- Fermions are introduced in the bulk in 32 and 11 of SO(11), 32 and 11 . We introduce a scalar field in 16 of SO(10), 16 , on the Planck brane. To make the matter content in 32 and 16 transparent, let us adopt the following representation of SO (11) Here σ 0 = I 2 and σ 1,2,3 are Pauli matrices. Note that 11 = −i 1 · · · 10 . The SO(11) generators in the spinorial representation are given by T sp jk = − 1 2 i j k . In this representation, the upper and lower half-components of 32 correspond to 16 and 16 of SO (10). 32 and 11 satisfy 32 (x, y j − y) = −P sp j γ 5 32 (x, y j + y), 11 (x, y j − y) = η 11 j P j γ 5 11 (x, y j + y), Here γ 5 = ±1 correspond to right-and left-handed Lorentz spinors, and η 11 j = ±1. The action in the bulk is given by where L g.f. and L gh are gauge fixing and ghost terms.
The action for 16 is given by where   (10) and SU (5) content of each field is also indicated. abc T bc ± T 4a . In the spinorial representation The content of 32 is easily determined by examining Q EM in the representation (2) with BC (3). The result is summarized in Table 1. BC at y = 0 with P sp 0 admits parity-even left-handed (righthanded) modes only for 16 16 of SO (10), whereas BC at y = L with P sp 1 admits parity-even left-handed (right-handed) modes only for SU (2) L (SU (2) R ) doublets. In Table 1, a field with a hat has an opposite charge to the corresponding one without a hat. For instance, u j andû j have Q EM = + 2 3 and − 2 3 , respectively. Notice that all leptons and quarks in SM, but nothing additional, appear as zero modes in 32 . In the SU (5) grand unified theory (GUT) in 4D, the 5 (10) multiplet contains L and d c L q L , u c L , e c L so that gauge interactions alter the quark/lepton number and u L → u c L , d c L → e L , etc transitions are induced. In the present case, these processes do not occur and proton decay is suppressed. Indeed, proton decay is forbidden to all orders, provided that the 32 , 11 fermion number N is conserved. All of u, d, e − have N = +1 in the current model. Although the fermion number current is anomalous, its effect on proton decay is expected to be negligible, as in the case of baryon number nonconservation in SM.
In the gauge-field sector, BC (1) alone leads to zero modes (4D massless gauge fields) in SO(4) × SO (6), some of which become massive due to 16 2 on the Planck brane generates mass terms of the form L gauge mass = −δ(y) 1 4 g 2 w 2 (A α μ ) 2 . We assume that gw/ √ L is much larger than the KK mass scale m KK = π kz −1 L . In this case, even if A α μ is parity even at y = 0, the boundary condition effectively becomes a Dirichlet condition and the lowest KK mode acquires a mass of O(m KK ) [7,9]. It is straightforward to check that all gauge fields in SO(4) × SO (6) C μ becomes massive due to 16 = 0. B Y μ is a gauge field of U (1) Y . After EW symmetry breaking by the Hosotani mechanism, A 34 μ mixes with A 4,11 μ . The photon is given by In terms of the SU (2) L coupling g w = g/ √ L in 4D, the U (1) EM and U (1) Y couplings are e = (3/8) 1/2 g w and g Y = (3/5) 1/2 g w . The Weinberg angle is given by sin 2 θ W = 3/8. 16 = 0 breaks SO (10) to SU (5) on the Planck brane. We add a comment that there appear 21 would-be Nambu-Goldstone (NG) bosons associated with this symmetry breaking, 9 of which are eaten by gauge fields in SO(4) × SO (6) There remain 12 uneaten NG modes corresponding to a complex scalar field with the same SM quantum numbers (3, 2) 1/6 as a quark doublet. They are massless at the tree level, but would acquire masses at the quantum level. Further, they are color-confined. It is expected that these colored scalars and quarks form colorsinglet bound states, whose dynamics can be explored by collider experiments. Evaluation of the masses of these new bound states, as well as deriving their experimental consequences, is reserved for future investigation. We note that 16 = 0 also gives large brane mass terms for gauge fields in SO (10) = 1, . . . , 4), admits a zero mode, and yields a nonvanishing Aharonov-Bohm (AB) phase playing the role of 4D Higgs fields. AB phases are defined as phases of eigenvalues ofŴ = Pexp ig L −L dy A y · P 1 P 0 , which are invariant under gauge transformations preserving the orbifold BC [2,21]. We expand A 4,11 y (x, y) as where is identified with the neutral Higgs boson in four dimensions. Insertion of (10) intoŴ shows that θ H is the AB phase. A gauge transformation generated by shifts θ H to θ H + β, and changes BC matrices to P 0 = e −2iβT 4,11 P 0 and P 1 = P 1 . Note that T 4,11 = σ 2 in the 4-11 subspace in the vectorial representation, and T sp 4,11 = − 1 2 σ 0 ⊗ σ 2 ⊗ σ 1 ⊗ σ 1 ⊗ σ 2 in the spinorial representation. The boundary conditions in (1) and (3) are preserved provided β = 2π n (n: an integer). The gauge invariance guarantees the periodicity in θ H for physical quantities.
With the mass spectrum at hand, one can evaluate V eff (θ H ) at 1-loop in the standard method [7,10]. There is a distinct feature in the spectrum in the gauge-field sector. In the gauge-Higgs grand unification, there are six Y towers with the spectrum (16) where the lowest modes have the smallest mass for cos θ H = 0. This leads to an important consequence that even in pure gauge theory the EW symmetry is spontaneously broken by the Hosotani mechanism. V eff (θ H ) evaluated with (14)-(17) has the global minimum at θ H = ± 1 2 π ; see Fig. 1. This has never happened in the gauge-Higgs EW unification models. 32 does not affect this behavior very much in the absence of brane interactions. Contributions from particles with the upper spectrum in (18) and those with the lower spectrum almost cancel numerically in V eff (θ H ) for z L 1. 11 with η 11 0 η 11 1 = 1 (−1) in (19) strengthens (weakens) the EW symmetry breaking.
At this stage, however, quarks and leptons have degenerate masses. The degeneracy is lifted by interactions on the Planck brane (at y = 0) that must respect SO (10) invariance. Let us decompose 32 into 16 and 16 of SO(10): , which, in combination with mixing of neutral components in 32 , may induce Majorana masses for neutrinos. However, it has to be kept in mind that such terms may lead to proton decay at higher loops. As mentioned above, V eff (θ H ) is minimized 7 at θ H = ± 1 2 π in pure gauge theory. θ H = ± 1 2 π , however, leads to a stable Higgs boson due to the H parity [27,28], which is excluded phenomenologically. A desirable value of θ H can be achieved by an appropriate choice of η 11 0 η 11 1 and inclusion of brane interactions for 32 and 11 . Alternatively, one may introduce fermions ( 55 , 11 , 32 ) such that quarks and leptons are dominantly contained in ( 55 , 11 ).
In this paper, we have presented the SO(11) gauge-Higgs grand unification model that generalizes the SO(5) × U (1) X gauge-Higgs EW unification. The orbifold boundary condition and brane scalar 16 reduce the SO(11) symmetry directly to the SM symmetry. The 4D Higgs doublet appears as the extra-dimensional component of the gauge potentials with custodial symmetry. The EW symmetry is spontaneously broken by the Hosotani mechanism, even in the pure gauge theory. We have presented a model with 32 and 11 for quarks and leptons. Proton decay is suppressed by the fermion number N conservation in the absence of Majorana masses. The effect of the fermion number current anomaly for proton decay is expected to be small. Although neutrino Majorana masses lead to proton decay at higher loops, the contribution will be suppressed by large Majorana masses and the loop effect. There remains a task to pin down the parameters of the model to reproduce the observed Higgs boson mass and quark-lepton spectrum, and derive phenomenological predictions. Further, the masses of the colored would-be NG bosons from 16 and color-singlet bound states need to be clarified and the consistency with experimental results at LHC needs to be examined. We will come back to these issues in forthcoming papers.