Gauge-Higgs Seesaw Mechanism in Six-Dimensional Grand Unification

TL;DR

This paper presents a six-dimensional $SO(11)$ gauge-Higgs grand unification model that naturally generates small neutrino masses through a novel seesaw mechanism within the gauge-Higgs framework.

Contribution

It introduces a new six-dimensional gauge-Higgs grand unification model with a unique seesaw mechanism for neutrino mass generation.

Findings

Neutrino masses are naturally small due to the new seesaw mechanism.

The model unifies electroweak and grand-unification dimensions in a six-dimensional space.

Fermions are incorporated in specific representations of $SO(11)$.

Abstract

$SO(11)$ gauge-Higgs grand unification is formulated in the six-dimensional hybrid warped space in which the fifth and sixth dimensions play as the electroweak and grand-unification dimensions. Fermions are introduced in ${\bf 32}$, ${\bf 11}$ and ${\bf 1}$ of $SO(11)$. Small neutrino masses naturally emerge as a result of a new seesaw mechanism in the gauge-Higgs unification which is characterized by a $3 \times 3$ mass matrix.

Equations

$$d{s}^2$$

$$\left(\begin{array}{c}{A}_{\mu }\\ A_y\\ A_v\end{array}\right)(x,y_j-y,v_j-v)=P_j\left(\begin{array}{c}{A}_{\mu }\\ -A_y\\ -A_v\end{array}\right)(x,y_j+y,v_j+v)P_j^{-1}$$

$${\Psi }_{32}^{\alpha }(x,y_j-y,v_j-v)={\eta }_j^{\alpha }\bar{\gamma }{P}_j^{sp}{\Psi }_{32}^{\alpha }(x,y_j+y,v_j+v)$$

$$\int d^{4}x{\int }_0^{2\pi R_6}dv{\int }_1^{z_L}\frac{dz}{k}i(-{\eta }^{†},{\xi }^{†})$$

$$\times \left(\begin{array}{cc}-k{\hat{D}}_{-}(c)+i{D}_v& {\sigma }^{\mu }{D}_{\mu }\\ {\bar{\sigma }}^{\mu }{D}_{\mu }& -k{\hat{D}}_{+}(c)+i{D}_v\end{array}\right)\left(\begin{array}{c}\xi \\ \eta \end{array}\right)$$

$${\chi }^C=\tilde{\chi }\equiv i\bar{\Gamma }\chi ,$$

$${\mathcal{L}}_m=-\frac{m_B}{\sqrt{k}}(\overline{\chi }{\nu }^{\prime }+{\overline{\nu }}^{\prime }\chi ){|}_{y=0}-\frac{1}{2}M\overline{\chi }\chi .$$

$$\frac{i}{2}({\nu }_{0L}^{c†},{\nu }_{0R}^{\prime †},{\eta }_{0L}^{c†})\left(\begin{array}{ccc}& m_D& \\ m_D& & {\tilde{m}}_B\\ & {\tilde{m}}_B& M\end{array}\right)\left(\begin{array}{c}{\nu }_{0L}\\ {\nu }_{0R}^{\prime c}\\ {\eta }_{0L}\end{array}\right)+h.c.$$

$${\kappa }_{\mathrm{small}}\sim \frac{m_D^{2}M}{{\tilde{m}}_B^2},$$

$$-k{\hat{D}}_{-}(c_{32})\left(\begin{array}{c}{\nu }_R\\ {\nu }_R^{\prime }\end{array}\right)+\sigma \partial \left(\begin{array}{c}{\nu }_L\\ {\nu }_L^{\prime }\end{array}\right)=0,$$

$$\bar{\sigma }\partial \left(\begin{array}{c}{\nu }_R\\ {\nu }_R^{\prime }\end{array}\right)-k{\hat{D}}_{+}(c_{32})\left(\begin{array}{c}{\nu }_L\\ {\nu }_L^{\prime }\end{array}\right)=\delta (y)\left(\begin{array}{c}0\\ m_B{\eta }_L/\sqrt{k}\end{array}\right),$$

$$\displaystyle\Big{\{}\sigma\partial\,\eta_{L}-\frac{m_{B}}{\sqrt{k}}\nu_{R}^{\prime}-M\eta_{L}^{c}\Big{\}}\delta(y)=0~{}.$$

$${\nu }_R(x,1^{+})=0,$$

$${\hat{D}}_{+}{\nu }_L(x,1^{+})=D_{+}{\nu }_L(x,1)+\frac{i}{2}{\theta }^{\prime }(1^{+}){\nu }_L^{\prime }(x,1^{+})=0,$$

$$-2{\nu }_L^{\prime }(x,1^{+})=\frac{m_B}{\sqrt{k}}{\eta }_L(x),$$

$$\displaystyle kD_{-}\nu_{R}^{\prime}(x,1^{+})=-\frac{m_{B}}{2\sqrt{k}}\Big{\{}\frac{m_{B}}{\sqrt{k}}\nu_{R}^{\prime}(x,1)+M\eta_{L}^{c}(x)\Big{\}}$$

$$\left(\begin{array}{c}{\tilde{\nu }}_R\\ {\tilde{\nu }}_R^{\prime }\end{array}\right)$$

$$\left(\begin{array}{c}{\tilde{\nu }}_L\\ {\tilde{\nu }}_L^{\prime }\end{array}\right)$$

$${\eta }_L$$

$$K=\left(\begin{array}{ccc}\cos \frac{1}{2}{\theta }_H{S}_R& \sin \frac{1}{2}{\theta }_H{C}_R& 0\\ -\sin \frac{1}{2}{\theta }_H{C}_L& \cos \frac{1}{2}{\theta }_H{S}_L& \frac{1}{2}{m}_B/k\\ -\sin \frac{1}{2}{\theta }_H{S}_R& \cos \frac{1}{2}{\theta }_H{C}_R& -(k\lambda +M)/m_B\end{array}\right).$$

$$\displaystyle(k\lambda+M)\Big{\{}S_{L}S_{R}+\sin^{2}\hbox{$\frac{1}{2}$}\theta_{H}\Big{\}}+\frac{m_{B}^{2}}{2k}S_{R}C_{R}=0~{}.$$

$$\lambda =-\frac{2kM}{m_B^2}{\sin }^2\frac{1}{2}{\theta }_H\{\begin{array}{ll}(1-2c)z_L^{2c-1}& \mathrm{for}c<\frac{1}{2},\\ 2c-1& \mathrm{for}c>\frac{1}{2}.\end{array}$$

$$m_{\nu }=-\frac{m_D^{2}M{z}_L^{2c+1}}{(c+\frac{1}{2})m_B^2}.$$

Full text

††preprint: OU-HET 936, MISC-2017-06

The discovery of the Higgs boson at LHC supports the scenario of the unification of electromagnetic and weak forces. In the standard model (SM) the electroweak (EW) gauge symmetry, $SU(2)_{L}\times U(1)_{Y}$, is spontaneously broken to $U(1)_{\rm EM}$ by the vacuum expectation value (VEV) of the Higgs scalar field. Although almost all observational data at low energies, including the data from 13 TeV LHC, are consistent with the SM, it is not clear whether the discovered Higgs boson is precisely what is introduced in the SM. The Higgs boson sector of the SM lacks a principle which governs and regulates the Higgs interactions with itself and other fields, in quite contrast to the gauge field sector which is controlled by the gauge principle. At the quantum level the mass of the Higgs boson $m_{H}$ acquires large corrections which have to be canceled by fine-tuning a bare mass in the theory. It is called the gauge-hierarchy problem.

There are many proposals to overcome these problems. One possible scenario is the gauge-Higgs unification in which the Higgs boson is identified with a part of the extra-dimensional component of gauge fields defined in higher dimensional spacetime.YH1 ; Davies1 ; Hatanaka1998 The Higgs boson appears as a four-dimensional fluctuation mode of the Aharonov-Bohm (AB) phase $\theta_{H}$ along the extra-dimensional space. It acquires a finite mass at the quantum level, independent of the cutoff scale.

In the EW interactions the $SO(5)\times U(1)$ gauge-Higgs unification in the five-dimensional Randall-Sundrum (RS) warped space has been formulated.ACP2005 ; Medina2007 ; HS2007 ; FHHOS2013 It has been shown to give almost the same phenomenology at low energies as the SM for \theta_{H}\hskip 3.00003pt\raisebox{1.72218pt}{<}\raisebox{-3.01385pt}{\sim}\hskip 3.00003pt0.1. It gives many predictions to be explored and confirmed in the coming experiments at LHC and in future experiments at $e^{+}e^{-}$ colliders. For instance, it predicts the $Z^{\prime}$ bosons (the first KK modes of $\gamma$, $Z$ and $Z_{R}$) and $W^{\prime}$ boson (the first KK modes of $W$) around 7 to 8 TeV range for $\theta_{H}=0.1$ to $0.07$, and larger forward-backward asymmetry in $e^{+}e^{-}$ collisions at 250 GeV$\sim 1\,$TeV than in the SM.

As a next step it is natural to incorporate strong interaction to achieve gauge-Higgs grand unification (GHGU). The mere fact of charge quantization in the quark-lepton spectrum strongly indicates grand unification. Such attempts have been already made.Burdman2003 ; Lim2007 ; Yamashita2011 ; Serra2011 ; HY2015a ; Yamatsu2016a ; Furui2016 Recently the $SO(11)$ gauge-Higgs grand unification model in the five-dimensional Randall-Sundrum warped space has been proposed, in which the EW Higgs boson emerges from the fifth dimensional component of the gauge potentials and many good features of the $SO(5)\times U(1)$ gauge-Higgs EW unification are carried over.HY2015a ; Furui2016 The breaking of the symmetry of grand unification is achieved there by imposing different orbifold boundary conditions at the UV and IR branes in the RS space. On the UV brane $SO(11)$ is broken to $SO(10)$, whereas on the IR brane to $SO(4)\times SO(7)$. As a result the remaining gauge symmetry becomes $SO(4)\times SO(6)$. The brane scalar in 16 of $SO(10)$ is introduced on the UV brane, which spontaneously breaks $SO(10)$ to $SU(5)$, leaving the SM gauge symmetry as a whole. It is found that proton decay is strictly forbidden in the minimal model. The mass spectrum of quarks and leptons is realized in the combination of the Hosotani mechanism and $SO(10)$-invariant interactions on the UV brane.

However, as a consequence of the two distinct orbifold boundary conditions imposed on the UV and IR branes, there necessarily emerge light exotic fermions ($\hat{u}$) of charge -\hbox{\frac{2}{3}}e. $\hat{u}$ has parity either $(+,-)$ or $(-,+)$. It does not cause a problem in flat space, but in the RS space its mass $m_{\hat{u}}$ turns out to be about m_{u}\cot\hbox{\frac{1}{2}}\theta_{H}, which contradicts with the observation for $\theta_{H}\sim 0.1$. The problem is unavoidable in the five-dimensional RS space.Furui2016 Furthermore it is difficult to naturally explain small masses of neutrinos.

To overcome these difficulties we propose gauge-Higgs grand unification in the six-dimensional hybrid warped space. Consider six-dimensional spacetime with a metric

[TABLE]

where $\eta_{\mu\nu}={\rm diag~{}}(-1,1,1,1)$, $\sigma(y)=\sigma(y+2L_{5})=\sigma(-y)$, and $\sigma(y)=k|y|$ for$|y|\leq L_{5}$. We identify spacetime points $(x^{\mu},y,v)$, $(x^{\mu},y+2L_{5},v)$, $(x^{\mu},y,v+2\pi R_{6})$, and $(x^{\mu},-y,-v)$. The spacetime has the same topology as $M^{4}\times(T^{2}/Z_{2})$. It naturally allows to have chiral fermions in four dimensions. There appear four fixed points in the extra-dimensional space under $Z_{2}$ parity; $(y_{0},v_{0})=(0,0)$, $(y_{1},v_{1})=(L_{5},0)$, $(y_{2},v_{2})=(0,\pi R_{6})$ and $(y_{3},v_{3})=(L_{5},\pi R_{6})$. Parity $P_{j}$ around each fixed point is defined by $(x^{\mu},y_{j}+y,v_{j}+v)\rightarrow(x^{\mu},y_{j}-y,v_{j}-v)$. The metric (1) solves the Einstein equations with the brane tension at $y=0$ and $L_{5}$ and a negative cosmological constant $\Lambda=-10k^{2}$. There are five-dimensional branes at $y=0$ and $L_{5}$, each of which has topology of $M^{4}\times S^{1}$. The spacetime (1) generalizes the RS space, and is called as the hybrid warped space hereafter.

There appear two Kaluza-Klein mass scales $m_{{\rm KK}_{5}}=\pi k/(e^{kL_{5}}-1)$ and $m_{{\rm KK}_{6}}=R_{6}^{-1}$ in the fifth and sixth dimensions. We suppose that the warp factor is large; $z_{L}=e^{kL_{5}}\gg 1$. $m_{{\rm KK}_{5}}$ turns out to be $6\sim 10\,$TeV as in the $SO(5)\times U(1)$ gauge-Higgs EW unification. $m_{{\rm KK}_{6}}$ is expected to be a GUT scale, and therefore $m_{{\rm KK}_{6}}\gg m_{{\rm KK}_{5}}$. Not all $P_{j}$’s are independent. It is easy to see that $P_{3}=P_{2}P_{0}P_{1}=P_{1}P_{0}P_{2}$. Further loop translations along the fifth and sixth dimensions, $U_{5}$: $(x^{\mu},y,v)\rightarrow(x^{\mu},y+2L_{5},v)$ and $U_{6}$: $(x^{\mu},y,v)\rightarrow(x^{\mu},y,v+2\pi R_{6})$, are related by $U_{5}=P_{1}P_{0}=P_{3}P_{2}$ and $U_{6}=P_{2}P_{0}=P_{3}P_{1}$.

We consider $SO(11)$ gauge theory in the hybrid warped space. Gauge potentials satisfy

[TABLE]

where $P_{j}{\rm~{}or~{}}-P_{j}\in SO(11)$ and $(P_{j})^{2}=1$. We take, in the vectorial representation, $P_{0}^{vec}=P_{1}^{vec}=I_{4}\oplus(-I_{7})$ and $P_{2}^{vec}=P_{3}^{vec}=I_{10}\oplus(-I_{1})$. Note $U_{5}=P_{1}P_{0}=1$. The choice $P_{0}=P_{1}$ and $P_{2}=P_{3}$ enables us to avoid light exotic particles in the warped space. $SO(11)$ is reduced to $SO(4)\times SO(7)$ by $(P_{0},P_{1})$, and to $SO(10)$ by $(P_{2},P_{3})$. As in the 5d model, the orbifold boundary conditions reduce $SO(11)$ to $SO(4)\times SO(6)$. In the representation of the $SO(11)$ Clifford algebra in Ref. Furui2016 , the corresponding $P_{j}$’s in the spinorial representation are given by $P_{0}^{sp}=P_{1}^{sp}=I_{2}\otimes\sigma^{3}\otimes I_{8}$ and $P_{2}^{sp}=P_{3}^{sp}=I_{16}\otimes\sigma^{3}$.

Four fermion multiplets in the spinor representation, $\Psi_{\bf 32}^{\alpha}(x,y,v)$ ($\alpha=1\sim 4$), are introduced. Three of them ($\alpha=1,2,3$) contain three generations of quarks and leptons. Dirac matrices in 6 dimensions satisfy $\{\Gamma^{a},\Gamma^{b}\}=2\eta^{ab}$ ($\eta^{ab}={\rm diag~{}}(-1,1,1,1,1,1)$). 6d chirality matrix is given by $\gamma^{7}_{6d}=I_{4}\otimes\sigma^{3}$, which is related to 4d chirality matrix $\gamma^{5}_{4d}=I_{2}\otimes\sigma^{3}\otimes I_{2}$ by $\gamma^{5}_{4d}\gamma^{7}_{6d}=\gamma^{7}_{6d}\gamma^{5}_{4d}=-i\Gamma^{5}\Gamma^{6}\equiv\bar{\gamma}$. The orbifold boundary conditions are

[TABLE]

where $\eta_{j}^{\alpha}=\pm 1$. We impose the 6d Weyl condition such that $\gamma^{7}_{6d}=+1\,(-1)$ for $\alpha=1,2\,(3,4)$, to ensure the cancellation of 6d chiral anomaly. In the current representation the upper (lower) half of $\Psi_{\bf 32}$ corresponds to ${\bf 16}$ ($\overline{\bf 16}$) of $SO(10)$. We choose $\eta_{j}^{1,2}=-1$, $\eta_{j}^{3}=1$, and $\eta_{0,2}^{4}=-\eta_{1,3}^{4}=1$. One finds that the three generations ($\alpha=1,2,3$) have zero modes corresponding to quarks and leptons such that all left-handed $SU(2)_{L}$ doublets are in ${\bf 16}$ and all right-handed $SU(2)_{L}$ singlets are in $\overline{\bf 16}$. The zero mode structure is the same as in the 5d model of Refs. HY2015a ; Furui2016 . The exotic particle components encountered in the 5d model, denoted by the $\hat{~{}}$ symbol there, have all $P_{0}=P_{1}=-P_{2}=-P_{3}$. Fields with $P_{0}=P_{1}=-P_{2}=-P_{3}$ are expanded in Fourier series of either \cos(n+\hbox{\frac{1}{2}})v/R_{6} or \sin(n+\hbox{\frac{1}{2}})v/R_{6} so that the mass of each mode is equal to or greater than 1/2R_{6}=\hbox{\frac{1}{2}}m_{{\rm KK}_{6}}\gg m_{{\rm KK}_{5}}. The problem of the appearance of light exotic fermions in the 5d GHGU is solved. The fourth generation $\Psi_{\bf 32}^{4}$ does not have any zero mode. Its lightest mode has a mass of order of m_{t}\cot\hbox{\frac{1}{2}}\theta_{H}. $\Psi_{\bf 32}^{4}$ plays the role of dark fermions in the $SO(5)\times U(1)$ gauge-Higgs EW model, and is necessary to have 6d anomaly cancellation as well.

Dirac fermions in the vector representation, $\Psi_{\bf 11}^{\beta}$ and $\Psi_{\bf 11}^{\prime\beta}$ ($\beta=1,2,3$), are also introduced in the 6d bulk. The boundary conditions are given by formulas similar to (3) with $\eta_{j}^{\alpha}\bar{\gamma}P_{j}^{sp}$ replaced by $\eta\bar{\gamma}P_{j}^{vec}$ where $\eta=+(-)$ for $\Psi_{\bf 11}^{\beta}$ ($\Psi_{\bf 11}^{\prime\beta}$). The fermion spectrum at low energies is the quark-lepton spectrum in the SM.

There are brane fields defined on the UV brane at $y=0$. A single brane scalar field in the spinor representaion of $SO(11)$, $\Phi_{\bf 32}(x,v)$, is introduced. Its VEV, $w/\sqrt{\pi R_{6}}$, spontaneously breaks $SO(11)$ to $SU(5)$. As a result $SO(11)$ is reduced to the SM symmetry, ${\cal G}_{\rm SM}=SU(2)_{L}\times SU(3)_{C}\times U(1)_{Y}$. One needs to assume only $w\gg m_{{\rm KK}_{5}}$. We note that the $SO(11)$ gauge invariance on the UV brane is demanded as the brane covers the bulk region $0<v<\pi R_{6}$. $\Phi_{\bf 32}$ satisfies the boundary condition $\Phi_{\bf 32}(x,v_{j}-v)=\eta_{j}P_{0}^{sp}\Phi_{\bf 32}(x,v_{j}+v)$ ($j=0,2$) where $(\eta_{0},\eta_{2})=(-,+)$. The $SU(5)$ singlet component of 16 in $SO(10)$ has a zero mode and develops nonvanishing VEV.

In addition to the SM gauge fields, $A_{y}^{a\,11}$ and $A_{v}^{a\,11}$ ($a=1\sim 4$) have zero modes. The zero modes of $A_{y}^{a\,11}$ ($a=1\sim 3$) are absorbed by the $W$ and $Z$ bosons. The zero mode of $A_{y}^{4\,11}$ becomes the 4d Higgs field. Its mass, $m_{H}=125\,$GeV, is generated at one loop. The zero modes of $A_{v}^{a\,11}$ ($a=1\sim 4$) are 4d scalars with masses of $O(g_{4}m_{{\rm KK}_{6}})$ generated at the one loop level. Eight components of complex $\Phi_{\bf 32}$ have zero modes under the boundary conditions. Among them the $SU(5)$ singlet component acquires nonvanishing VEV. Nine components out of the 16 real components are absorbed by $SO(4)\times SO(6)/{\cal G}_{\rm SM}$ gauge fields. The remaining 6 ($=16-1-9$) pseudo-Nambu-Goldstone bosons acquire masses of $O(g_{4}w)$ at the one loop level.

The action of a fermion field $\Psi$ in 6d bulk is given by \int d^{6}x\sqrt{-\det G}\,\overline{\Psi}\big{\{}\Gamma^{a}{E_{a}}^{M}\big{(}D_{M}+\frac{1}{8}\Omega_{bcM}[\Gamma^{b},\Gamma^{c}]\big{)}+ic\sigma^{\prime}(y)\Gamma^{6}\big{\}}\Psi where $\overline{\Psi}=i\Psi^{\dagger}\Gamma^{0}$, $D_{M}=\partial_{M}-igA_{M}$, and $\Omega_{bcM}$’s are spin-connections. The last term with the coefficient $c$ represents a bulk vector mass in the hybrid warped space, which generalizes a bulk scalar mass in the RS space. For $\Psi_{\bf 11}$ an additional mass term $m_{\bf 11}\overline{\Psi}_{\bf 11}\Psi_{\bf 11}$ is allowed. It is most convenient to work in the conformal coordinate $z=e^{ky}$ in the fundamental region $1\leq z\leq z_{L},0\leq v<2\pi R_{6}$. For a 6d Weyl fermion with $\gamma_{6d}^{7}=+$, we write $\Psi=z^{5/2}(\xi,\eta,0,0)$ where $\xi,\eta$ are two-component right- and left-handed spinors. The action becomes

[TABLE]

where $\hat{D}_{\pm}(c)=\pm D_{z}+(c/z)$. We stress that the bulk vector mass term $\overline{\Psi}ic\sigma^{\prime}(y)\Gamma^{6}\Psi$ precisely plays the role of the bulk scalar mass in the 5d GHGU in the RS space.

The parity $P_{0}=P_{2}=+1$ components of the 6d bulk fermion fields, $\Psi_{\bf 32}^{\alpha}$, $\Psi_{\bf 11}^{\beta}$ and $\Psi_{\bf 11}^{\prime\beta}$, have brane interactions with the brane scalar $\Phi_{\bf 32}$ on the UV brane at $y=0$. They take the $SO(11)$ invariant form such as $\delta(y)\overline{\Psi}_{\bf 32}\Gamma^{a}\Phi_{\bf 32}(\Psi_{\bf 11})_{a}$. With \raise 0.68889pt\hbox{\langle}\lower 0.68889pt\hbox{}\Phi_{\bf 32}\raise 0.68889pt\hbox{\rangle}\lower 0.68889pt\hbox{}\not=0 they generate mass mixing among $\Psi_{\bf 32}$ and $\Psi_{\bf 11}$. It can be shown that the observed mass spectrum of quarks and charged leptons is reproduced in the combination of the Hosotani mechanism, brane interactions and $\overline{\Psi}_{\bf 11}\Psi_{\bf 11}$ terms, which will be reported separately. In the present paper we focus on the neutral fermion sector, and show how small neutrino masses are generated by a new seesaw mechanism.

$SO(11)$ singlet, eight-component brane fermions $\chi^{\beta}(x,v)$ ($\beta=1\sim 3$) are introduced on the UV brane for three generations. $\chi^{\beta}$ satisfies $\chi^{\beta}(x,v_{j}-v)=\bar{\gamma}\,\chi^{\beta}(x,v_{j}+v)$ ($j=0,2$). For the sake of simplicity in notation we suppress the generation index $\alpha,\beta$ hereafter. With $\chi$ written in terms of two-component spinors as $\chi=(\xi_{+},\eta_{+},\xi_{-},\eta_{-})$, the orbifold boundary condition implies that only $\xi_{+}$ and $\eta_{-}$ have zero modes ($v$-independent modes). Recall that charge conjugation $C$ in six dimensions is given by $\chi^{C}=e^{i\delta_{C}}(-\sigma^{2}\eta_{+}^{*},-\sigma^{2}\xi_{+}^{*},\sigma^{2}\eta_{-}^{*},\sigma^{2}\xi_{-}^{*})$. $\chi$ lives on the five-dimensional brane so that one can impose the simplectic Majorana condition Peskin1998 on $\chi$,

[TABLE]

where $\bar{\Gamma}=\gamma_{4d}^{5}\Gamma^{6}$. As $\bar{\Gamma}$ commutes with $\Gamma^{a}$ ($a=0,1,2,3,6$) and anti-commutes with $\Gamma^{5}$, $\tilde{\chi}$ transforms as a 5d spinor. One finds that $\xi_{+}=\eta_{-}^{c}=-e^{i\delta_{C}}\sigma^{2}\eta_{-}^{*}$ and $\eta_{+}=\xi_{-}^{c}=e^{i\delta_{C}}\sigma^{2}\xi_{-}^{*}$. Combined with the boundary conditions, $\chi$ has only one independent zero mode in $\eta_{-}$.

For $\chi$ field a mass term \hbox{\frac{1}{2}}M\overline{\chi}\chi\delta(y) is allowed. Further there is a brane interaction $-\delta(y)\kappa_{B}\{\overline{\chi}\Phi_{32}^{\dagger}\Psi_{\bf 32}+h.c.\}$, which generates, with \raise 0.68889pt\hbox{\langle}\lower 0.68889pt\hbox{}\Phi_{32}\raise 0.68889pt\hbox{\rangle}\lower 0.68889pt\hbox{}\not=0, mass mixing among $\chi$ and $\nu^{\prime}$ where $(\nu^{\prime},e^{\prime})$ is an $SU(2)_{L}$-singlet and $SU(2)_{R}$-doublet in $\Psi_{\bf 32}$.Furui2016 As a result the action for $\chi$ is given by $\int d^{4}x\int_{0}^{2\pi R_{6}}dv({\cal L}_{k}+{\cal L}_{m})$ where {\cal L}_{k}=\hbox{\frac{1}{2}}\overline{\chi}(\Gamma^{\mu}\partial_{\mu}+\Gamma^{6}\partial_{v})\chi and

[TABLE]

The relevant fields determining the observed neutrino are $\chi$, $\nu^{\prime}$ and $\nu$ of $(\nu,e)$ which is an $SU(2)_{L}$-doublet and $SU(2)_{R}$-singlet in $\Psi_{\bf 32}$. Zero modes are contained in $\nu_{L}$, $\nu_{R}^{\prime}$ and $\eta_{-}$. We would like to stress that $M\overline{\chi}\chi$ gives rise to $M(-i\eta_{-}^{\dagger}\eta_{-}^{c}+i\eta_{-}^{c\dagger}\eta_{-})$, namely a Majorana mass in four dimensions.

Due care must be taken in deriving mass eigenvalues, as $\nu$ and $\nu^{\prime}$ are 6d fields whereas $\chi$ is a 5d field. Before giving rigorous treatment, it is instructive to present an effective theory in four dimensions. Let $(\nu_{0L},\nu_{0R}^{\prime},\eta_{0L})$ be canonically normalized 4d fields associated with the zero modes of $(\nu,\nu^{\prime},\chi)$. In perturbative expansion $\eta_{-}(x,v)=(2\pi R_{6})^{-1/2}\eta_{0L}(x)+\cdots$ and $\nu_{R}^{\prime}(x,z,v)=(k/2\pi R_{6})^{1/2}z^{5/2}u_{R}^{(0)}(z;c_{\bf 32})\nu_{0R}^{\prime}(x)+\cdots$ where $c_{\bf 32}$ is the bulk vector mass parameter of $\Psi_{\bf 32}$ and the mode function $u_{R}^{(0)}(z;c)$ is normalized by $\int_{1}^{z_{L}}dzu_{R}^{(0)}(z;c)^{2}=1$. With $\theta_{H}\not=0$, $\nu$ and $\nu^{\prime}$ mix by the Hosotani mechanism, generating an effective Dirac mass $m_{D}(i\nu_{0L}^{\dagger}\nu_{0R}^{\prime}-i\nu_{0R}^{\prime\dagger}\nu_{0L})$ just as other quark and lepton pairs do. Then the mass terms are written as

[TABLE]

where $\tilde{m}_{B}=m_{B}u_{R}^{(0)}(1;c_{\bf 32})$. $u_{R}^{(0)}(z;c)$ is given by $(2c+1)^{1/2}z_{L}^{-1/2}(z/z_{L})^{c}$ in free theory. Mass eigenvalues $\kappa$ are determined by $\kappa^{3}-M\kappa^{2}-(\tilde{m}_{B}^{2}+m_{D}^{2})\kappa+m_{D}^{2}M=0$. In particular, for $m_{D}\ll M,\tilde{m}_{B}$, one finds \kappa_{\rm large}\sim\hbox{\frac{1}{2}}\big{\{}M\pm\sqrt{M^{2}+4\tilde{m}_{B}^{2}}\big{\}} and

[TABLE]

which is identified as the small neutrino mass. This should be compared with the seesaw mechanism in four dimensions Minkowski1977 which is typically characterized by a $2\times 2$ matrix in each generation. Note that the Majorana mass $M$ appears in the numerator of (9). In the gauge-Higgs grand unification, an effective Majorana mass $\tilde{m}_{B}^{2}/M$ for $\nu_{R}^{\prime}$ is induced through $\nu_{R}^{\prime}\rightarrow\eta_{L}\rightarrow\eta_{L}^{c}\rightarrow\nu_{R}^{\prime c}$.

To confirm the above picture and derive a precise formula, we return to the equations in the six-dimensional hybrid warped space. Each field is expanded in a Fourier series in the sixth coordinate $v$. Only fields with $P_{0}P_{2}=P_{1}P_{3}=+1$ have zero modes. All other modes have large masses \geq\hbox{\frac{1}{2}}m_{{\rm KK}_{6}}. Supposing that $\tilde{m}_{B},M\ll m_{{\rm KK}_{6}}$, we safely retain only $v$-independent modes. Further we assume, for the sake of simplicity, that the mixing of $\nu$ and $\nu^{\prime}$ with other neutral component of $\Psi_{\bf 32}$ and $\Psi_{\bf 11}$, induced by brane interactions, may be neglected. After factoring out $z^{5/2}$ for $\nu$ and $\nu^{\prime}$ as done in (5), one finds

[TABLE]

Here $-gA_{z}=\frac{1}{2}\theta^{\prime}(z)\tau_{1}$ in $\hat{D}_{\pm}(c_{\bf 32})$ where $\theta(z)=\theta_{H}(z_{L}^{2}-z^{2})/(z_{L}^{2}-1)$. $\nu_{L}$ and $\nu_{R}^{\prime}$ are parity even at $y=0$ and $L_{5}$, whereas $\nu_{R}$ and $\nu_{L}^{\prime}$ are parity odd. It follows from (12) that at $y=\epsilon$ ($z=1^{+}$)

[TABLE]

where $D_{\pm}=\pm(\partial/\partial_{z})+(c_{\bf 32}/z)$. $\nu_{L}^{\prime}$ develops a discontinuity at $y=0$ due to the brane interaction with $\eta_{L}$.

At this stage it is convenient to move to the twisted gauge in which (\tilde{\nu},\tilde{\nu}^{\prime})^{t}=\exp\big{\{}\frac{i}{2}\theta(z)\tau_{1}\big{\}}(\nu,\nu^{\prime})^{t} and $\tilde{A}_{z}=0$. $\tilde{\nu}$ and $\tilde{\nu}^{\prime}$ satisfy free equations in the bulk $1<z\leq z_{L}$. The boundary conditions at $z=z_{L}$ remain unchanged, and are given by $D_{+}\tilde{\nu}_{L}=\tilde{\nu}_{R}=\tilde{\nu}_{L}^{\prime}=D_{-}\tilde{\nu}_{R}^{\prime}=0$. As a consequence solutions to (12) satisfying the boundary conditions at $z_{L}$ can be expressed, in terms of basis functions defined in Appendix B of Ref. Furui2016 , as

[TABLE]

where $c=c_{\bf 32}$, $D_{+}(C_{L},S_{L})=\lambda(S_{R},C_{R})$, $D_{-}(C_{R},S_{R})=\lambda(S_{L},C_{L})$, and $C_{L}C_{R}-S_{L}S_{R}=1$. $C_{L}=C_{R}=1$ and $S_{L}=S_{R}=0$ at $z=z_{L}$. $f_{R,L}$ satisfy $\sigma\partial f_{L}=k\lambda f_{R}$, $\bar{\sigma}\partial f_{R}=k\lambda f_{L}$ and $f_{L}^{c}=f_{R}$. $\alpha_{\eta}$ is taken to be real.

It follows from (16) that $K(\alpha_{\nu},\alpha_{\nu^{\prime}},\alpha_{\eta})^{t}=0$ where

[TABLE]

Here $S_{R}=S_{R}(1;\lambda,c)$ etc. Mass eigenvalues $k\lambda$ are determined by $\det K=0$;

[TABLE]

For $\lambda z_{L}\ll 1$ and $k\lambda\ll|M|$ one finds that

[TABLE]

The Dirac mass $m_{D}$ acquired through the Hosotani mechanism in the neutral sector is the same as the mass $k\lambda_{D}$ of the up-type quark determined by S_{L}S_{R}+\sin^{2}\hbox{\frac{1}{2}}\theta_{H}=0.Furui2016 One finds that \lambda_{D}z_{L}=\sqrt{1-4c^{2}}\sin\hbox{\frac{1}{2}}\theta_{H} for c<\hbox{\frac{1}{2}} and \sqrt{4c^{2}-1}z_{L}^{-c+(1/2)}\sin\hbox{\frac{1}{2}}\theta_{H} for c>\hbox{\frac{1}{2}}. We recall that c>\hbox{\frac{1}{2}} (<\hbox{\frac{1}{2}}) for $u,c$ ($t$) quarks. Thus the small neutrino mass is given by the gauge-Higgs seesaw formula

[TABLE]

If the estimate $u_{R}^{(0)}(1;c)\sim(2c+1)^{1/2}z_{L}^{-1/2-c}$ in free theory were inserted into $\tilde{m}_{B}$ in the formula (9) in the 4d effective theory, the extra factor $\frac{1}{2}$ would appear, which may be understood as a result of the mixing among $\nu_{R}^{\prime}$ and $\eta_{L}^{c}$.

In this paper the $SO(11)$ gauge-Higgs grand unification has been formulated in the six-dimensional hybrid warped space. The seesaw mechanism for neutrinos naturally emerges, whose structure is characterized by a $3\times 3$ mass matrix. Details, concerning the spectrum of quarks and leptons, evaluation of $V_{\rm eff}(\theta_{H})$, and dynamical EW symmetry breaking, will be reported separately.