The mass spectra of the standard model particles are reproduced in the gauge–Higgs grand unification in six-dimensional warped space without introducing exotic light fermions. Light neutrino masses are explained by the gauge–Higgs seesaw mechanism. We evaluate the effective potential of the four-dimensional Higgs boson appearing as a fluctuation mode of the Aharonov–Bohm phase in the extra-dimensional space, and show that the dynamical electroweak symmetry breaking takes place with the Higgs boson mass GeV and . The Kaluza–Klein mass scale in the fifth dimension is approximately given by .
B40B41B42B431. Introduction
The discovery of the last of the standard model (SM) particles, the Higgs boson, seems to imply that the non-vanishing vacuum expectation value (VEV) of the Higgs field spontaneously breaks the electroweak (EW) gauge symmetry to the electromagnetic gauge symmetry . Almost all observational data from low-energy experiments are consistent with the SM. The SM is a good low-energy effective theory. However, it is not clear whether or not the Higgs boson is a genuine fundamental scalar field, which usually suffer from the so-called gauge hierarchy problem.
There are several proposals to overcome the problem by making use of symmetries. One of them is the gauge–Higgs unification. In this theory, the Higgs boson is identified with part of the extra-dimensional component of gauge fields in higher-dimensional spacetime [1–5]. It is described as a four-dimensional (4D) fluctuation mode of the Aharonov–Bohm (AB) phase along the extra-dimensional space.
The EW unification of the SM is formulated as the gauge–Higgs unification in the five-dimensional (5D) Randall–Sundrum (RS) warped space [6–16]. According to Refs. [11–14], its phenomenology at low energies under the mass scale of the first Kaluza–Klein (KK) modes is almost the same as in the SM for the AB phase . bosons, which are the first KK modes of , , and , are predicted around the TeV range for . bosons can be produced at the 14 TeV LHC [15]. At electron–positron linear colliders at energies of GeV TeV, the interference effects among , , and bosons give distinct signals of the gauge–Higgs unification [16].
To incorporate the gauge symmetry in higher-dimensional gauge theories and gauge–Higgs unification scenario, grand unified theories (GUTs) based on a GUT gauge group have been discussed in Refs. [17–37]. The gauge–Higgs grand unified theory (GHGUT) is proposed in the 5D RS warped space in Ref. [32]. The EW Higgs boson is identified with part of the fifth-dimensional component of the gauge bosons. The gauge symmetry is reduced first by two different orbifold boundary conditions (BCs) on the ultraviolet (UV) and infrared (IR) branes in the RS space. It is reduced to by the BC on the UV brane, and to by the BC on the IR brane, the resultant symmetry being . Secondly, the brane scalar on the UV brane, which is an spinor , spontaneously breaks to by the Higgs mechanism on the UV brane. As a net result the gauge symmetry is reduced to the SM gauge symmetry , which is dynamically broken to by the Hosotani mechanism.
Zero modes of 5D fermions in the bulk are identified with each generation of quarks and leptons. In Ref. [35] it is found that the observed mass spectra of the quarks and leptons can be reproduced, while additional exotic particles appear having unacceptably small masses.
Recently, to avoid these light exotic particles, GHGUT in six-dimensional (6D) hybrid warped space has been proposed in Ref. [38]. For neutrinos a new seesaw mechanism in 6D hybrid warped space has been formulated by using a 5D symplectic Majorana fermion [39], which generalizes the well-known 4D seesaw mechanism [40].
This paper is organized as follows. In Sect. 2, 6D GHGUT is introduced. The matter content is specified and the action is given that contains both 6D bulk and 5D brane terms. In Sect. 3, a summary is given for the mass spectrum of the 4D gauge and scalar bosons originating from the 6D gauge bosons. In Sect. 4, we derive the mass spectrum of 4D SM fermions. By using these mass spectra, we evaluate the effective potential in Sect. 5 to show that the dynamical EW symmetry breaking takes place and the 4D Higgs boson mass GeV is obtained. Section 6 is devoted to a summary and discussions. In the Appendix, the basics of the KK expansion in 6D warped space are explained.
2. Six-dimensional gauge–Higgs grand unification theory
We construct an gauge-Higgs grand unified model on the six-dimensional hybrid warped space introduced in Ref. [38]. The metric of generalized RS space [38,41] is given by
where is a warped factor and . satisfies and . The fifth dimension with the coordinate behaves as the EW dimension, whereas the sixth dimension with the coordinate is and behaves as the GUT dimension. Two spacetime points and are identified by the transformation. As a result, the spacetime has the same topology as the orbifold . The spacetime in Eq. (2.1) solves the Einstein equations with brane tensions at and . The bulk region is the anti-de-Sitter space with a negative cosmological constant . The five-dimensional branes at and have the same topology as .
The extra-dimensional space has four fixed points under : , , , and . In terms of the conformal coordinate in the region , the metric becomes
The fifth- and sixth-dimensional KK mass scales are given by and . The warp factor is supposed to be large; . is expected to be a GUT scale while is TeV so that .
Parity transformations around the four fixed points are defined as . Only three of the four parity transformations are independent. They satisfy the relation .
We adopt orbifold BCs such that and , which enables us to avoid the problem of unwanted light exotic fermions mentioned in the Introduction. More specifically, we choose BCs such that and break to and , respectively. The orbifold BCs reduce symmetry to the Pati–Salam symmetry . We note that the fifth- and sixth-dimensional loop translations and are related to the s by and .
2.1. 6D bulk and 5D brane fields and orbifold boundary conditions
The matter content in the GHGUT consists of 6D gauge bosons , 6D Weyl fermions , 6D Dirac fermions and in the six-dimensional bulk space, and a 5D brane scalar boson and 5D brane symplectic Majorana fermions on the UV brane [38]. Their orbifold BCs are listed below:
For the 6D gauge boson , the orbifold BCs are given by
where in the vector representation we take the orbifold BCs and as
By using , the parity assignment of , , and are summarized in Table 1.
For the four 6D spinor 32 bulk Weyl fermions , the orbifold BCs are given by
where . Six-dimensional Dirac matrices satisfy , and , , . The s are
To ensure the 6D chiral anomaly cancellation, we assign for ; for . We take , , and . The parity assignment
of the 4D left- and right-handed components of are summarized in Table 2. We find that has zero modes, corresponding to one generation of quarks and leptons for each . The corresponding names adopted in Ref. [35] are also listed in Table 2 for .
For the three 6D vector 11 bulk Dirac fermions and , the orbifold BCs are given by
where for ; for . The parity assignments of the 4D left- and right-handed components of are summarized in Table 3.
For the 5D spinor brane scalar field on the UV brane at , , the orbifold BCs are given by
where , . The components of have zero modes, one of which corresponds to the of . It is responsible for reducing to at the UV brane . The BCs of are summarized in Table 4.
For the 5D singlet fermions () at , their orbifold BCs are given by
These brane fields also satisfy the 5D symplectic Majorana condition , , where :
Here the generation index has been suppressed.
Parity assignment of , , and in .
(3,1,1)
(1,3,1)
(1,1,15)
(2,2,6)
(2,2,1)
(1,1,6)
Parity assignment in Eq. (2.7) of in .
Parity assignment for in .
Left
Right
Left
Right
name
Left
Right
Left
Right
name
Parity assignment for
in .
BCs
2.2. Action
The action consists of the 6D bulk and 5D brane terms.
2.2.1. Bulk terms
The bulk part of the action is given by
where and are bulk actions of gauge and fermion fields, respectively. The action of gauge field 55 is
where , , , and the field strength is defined by
We take the following gauge-fixing and ghost terms:
where , , and . and , where and .
The action of fermions in spinor and vector representations 32 and 11, , , and , is given by
where . , , and are bulk mass parameters, and is a covariant derivative given by
where and for .
Here, we introduce defined by
which turns out to be convenient for discussing mass spectra for fermions in Sect. 4. The bulk part of the bulk fermion action becomes
2.2.2. Action for the brane scalar <inline-formula><tex-math notation="LaTeX" id="ImEquation311"><![CDATA[$\Phi_{\bf 32}$]]></tex-math></inline-formula> and the Higgs mechanism
We consider the 5D brane terms for a brane scalar :
where and are constants, and
We denote the 5D brane scalar field as
and assume that the component of for develops the non-vanishing VEV:
where one component of of corresponds to of . The VEV breaks to . Note that branching rules of and representations for are given by and , respectively. (For further information see, e.g., Refs. [42–44].)
We also use the 5D brane scalar field related to by
For ,
The combination of the non-vanishing VEV on the UV brane (at ) and the orbifold BCs reduces to the SM gauge group .
2.2.3. Action for the singlet brane fermion <inline-formula><tex-math notation="LaTeX" id="ImEquation339"><![CDATA[$\chi_{\bf 1}$]]></tex-math></inline-formula>
The action for , which satisfies the symplectic Majorana condition (2.11), is
where is a constant matrix.
2.2.4. Brane masses and interactions
On the UV brane there can be -invariant brane interactions among the bulk fermions, the singlet brane fermion, and the brane scalar. We consider
where the s and s are coupling constants. Note that and are forbidden by the parity assignment of and shown in Table 3.
2.2.5. Mass terms for fermions on the UV brane
From the action in Eqs. (2.26) and (2.27), the quadratic mass terms on the UV brane for the bulk and brane fermions with are given by
where
Here, , , , , , and . For the sake of simplicity has been denoted as . All the parameters are dimensionless quantities, whereas , , and have the dimension of mass.
2.2.6. Mass terms for gauge bosons on the UV brane
By replacing with its VEV in Eq. (2.20), the mass terms for the 4D components of the gauge fields can be read off as
where the notation is the same as in Ref. [35]. We omit them here. All the 31 components of obtain large brane masses via . Each mass term changes the fifth-dimensional BCs on the UV brane for the corresponding fields. Similarly, from the action in Eq. (2.20), we find the mass terms for the sixth-dimensional component of the :
These mass terms give masses for the corresponding fields by changing the BCs. We shall see more details in Section 3.
2.3. EW Higgs boson and twisted gauge
The orbifold BCs break to , and, further, the non-vanishing VEV of the 5D brane scalar reduces to . The bilinear terms of the action of gauge fields in Eq. (2.13) are written down as
Here, run over so that . The fourth- and sixth-dimensional components and have additional bilinear terms that come from the 5D brane scalar term in Eq. (2.20) with . The parity assignments of , , and are summarized in Table 1. The components of and have parity . The components of obtain masses through the brane mass terms in Eq. (2.31), so only the components of , which is identified with the SM Higgs scalar fields, have zero modes. Three of them are absorbed by and bosons. The remaining neutral component, the observed Higgs boson, also becomes massive by radiative corrections through the Hosotani mechanism as shown below. We note that the components of get additional masses of order by radiative corrections at one loop. The components , , and of have parity . is spontaneously broken to by the non-vanishing VEV of the component of .
The zero modes of are physical degrees of freedom which cannot be gauged away. By using the KK mode expansion discussed in Appendix A.1, the sixth-dimensional modes (-independent modes) of are expanded, in terms of mode functions for the parity boundary condition, as
where the zero mode function in the fifth dimension is given by . The four-component real field is identified with the EW Higgs doublet field in the SM. We will show later that the dynamically obtain the non-vanishing VEV . The VEV breaks to . are absorbed by and bosons.
Under a general gauge transformation , new gauge potentials satisfy the following new BCs:
It is convenient to work in the twisted gauge discussed, e.g., in Ref. [35]. With the following large gauge transformation , is transformed to :
The new BC matrices are given by
For fields in the vector and spinor representations, the BC matrices are given by
where
3. Mass spectrum of bosons
In this section we examine the spectrum of gauge fields, particularly for the sixth-dimensional KK modes (-independent modes), because we are interested in the mass spectrum of the SM particles and the effective potential . In the following we omit the modes which are odd under the sixth-dimensional loop translation . Those modes have masses .
The BCs for gauge fields in the absence of the brane terms are given by
at and . In the presence of the brane mass terms on the UV brane, the Neumann BC is modified to an effective Dirichlet BC for the fifth-dimensional zero mode of the components of and [35]. The equation of motion for in the -coordinate is given in the form
where the right-hand side involves interaction terms. Suppose that is parity even, namely . Note that is parity odd so that is parity even. By integrating the equation and taking the limit , one finds . In the -coordinate the Neumann condition is changed to the effective Dirichlet condition
Note that in the current six-dimensional model the mass dimensions of and are , , so that is dimensionless. When , develops a cusp at . The lowest mode of becomes massive, with mass . The value of depends on the brane mass terms. The BCs for the gauge fields are summarized in Table 5.
Venn diagram for the symmetry-breaking pattern and summary of the BCs of the gauge field. and stand for Neumann and Dirichlet conditions, respectively. represents the effective Dirichlet condition given by Eq. (3.3).
In the twisted gauge , where . is defined in Eq. (2.35). The , , and components of the gauge fields are changed as
while the other components remain unchanged. As in Ref. [35], the BCs at determine wave functions for , , and :
where and are defined by Eqs. (A.2) and (A.3) in Appendix A.1.
From the BCs for the gauge fields summarized in Table 5, the components (2) , (4) , and (6) are parity odd under the sixth-dimensional loop translation , while the components (1) , (3) , and (5) are parity even under . Thus, we find that the low-lying modes of the components (2), (4), and (6) of have masses of , while the low-lying modes of the components (1), (3), and (5) of have masses less than or around .
We summarize the formulas determining the mass spectrum of the sixth-dimensional modes of , for which the arguments in Ref. [35] remain intact. We use the same notation as Ref. [35]. and () stand for and components of , respectively.
components.
(i) : For and towers, there are sixth-dimensional modes, and there are also fifth-dimensional zero modes in the absence of brane terms. In the presence of the brane terms in Eq. (2.30), their mass spectra of the fifth-dimensional modes are given by
where . Let us denote the average magnitude of and by and , respectively. When , the spectrum is determined by
Indeed, in the typical cases examined below, so that the condition is well satisfied for . The mass of the boson is given by
(ii) : For , , and towers, there are sixth-dimensional modes, and there are also fifth-dimensional zero modes in the absence of brane terms. Here, and , where . In the presence of the brane terms in Eq. (2.30), the mass spectra of the fifth-dimensional modes are given by
For ,
The mass of the boson is given by
The relation for the tower in Eq. (3.10) can be written, by using , as
The value is valid at the GUT scale. The gauge coupling constants evolve as the energy scale, following the renormalization group equation (RGE). It is not clear whether evolves to the observed value at low energies as there are KK modes which do not respect below the GUT scale. We leave solving the RGE for future investigation. When we evaluate the effective potential at the EW scale in Sect. 5, we adopt the formula in Eq. (3.12), where the observed value, , at the EW scale is inserted.
(iii) : For the tower, there are sixth-dimensional modes, but there are no fifth-dimensional zero modes. The mass spectra of the fifth-dimensional modes are given by
(iv) For gluons, there are sixth-dimensional modes, and there are also fifth-dimensional zero modes which correspond to 4D gluons. The mass spectrum of the fifth-dimensional massive modes is given by
(v) For -gluons, there are sixth-dimensional modes, and there are also fifth-dimensional zero modes in the absence of brane terms. In the presence of the brane terms in Eq. (2.30), the mass spectrum of the fifth-dimensional modes is given by
For ,
(vi) For -bosons, there are no sixth-dimensional modes.
(vii) For -bosons, there are no sixth-dimensional modes.
(viii) For -bosons, there are no sixth-dimensional modes.
components.
(i) and : There are sixth-dimensional modes, but there are no fifth-dimensional zero modes. The mass spectrum of the fifth-dimensional modes is given by
(ii) : There are sixth-dimensional modes for . There are no fifth-dimensional zero modes for , while there are fifth-dimensional zero modes for . The mass spectra of the fifth-dimensional modes are given by
(iii) : Higgs tower. There are sixth-dimensional modes, and there is also a fifth-dimensional zero mode. The mass spectrum of the fifth-dimensional modes is given by
(iv) : There are no sixth-dimensional modes.
(v) , : There are no sixth-dimensional modes.
components.
(i) and : There are sixth-dimensional modes, but there are no fifth-dimensional zero modes. The mass spectra of the fifth-dimensional modes are given by
(ii) : There are sixth-dimensional modes for and . There are no fifth-dimensional zero modes for , while there are fifth-dimensional zero modes for in the absence of brane terms. In the presence of the brane terms in Eq. (2.31), one finds that at ,
Thus, the mass spectra of the fifth-dimensional modes are given by
The average magnitude of () is much larger than that of (). In typical cases . Hence the spectrum is determined by
(iii) : -Higgs tower. There are sixth-dimensional modes, and there are also fifth-dimensional zero modes in the absence of brane terms. In the presence of the brane terms in Eq. (2.31), the mass spectra of the fifth-dimensional modes are given by
which is well approximated by
The components () acquire large one-loop corrections of order to their masses so that their contributions to the -dependent part of become negligible and may be dropped.
(iv) : There are no sixth-dimensional modes.
(v) , : There are no sixth-dimensional modes.
4. Mass spectrum of fermions
In this section we determine the mass spectrum of quarks and leptons. For up-type quarks, there are no brane interactions on the UV brane as in the 5D GHGUT. For down-type quarks, charged leptons, and neutrinos, both 6D bulk mass terms and brane mass terms on the UV brane are important to reproduce the observed mass spectra.
In the following, we will calculate mass spectra for one set of a 6D Weyl fermion , 6D Dirac fermions and , and 5D singlet , which is identified with one generation of the SM quarks and leptons. We will also discuss mass spectra for the 6D Weyl fermion denoted by . We will omit the superscripts and . We are interested in EW-scale physics and we keep only the sixth-dimensional KK modes because the masses of sixth-dimensional modes are much larger than as . That is, we will discuss mass spectra for the and components of , the components of , the and components of , and the and components of . We will denote , , , and as , , , and , respectively. In this paper we assume that while can be negative. The calculation method employed in the following is the same as employed in the 5D GHGUT discussed in Ref. [35].
4.1. Up-type quark
(i) :
From the action in Eq. (2.19), we find the equations of motion for up-type quarks with :
Here, , , and
where , are the pairs defined in Eq. (2.38). For an up-type quark with , namely for the top quark, the equations become
In the twisted gauge, the equations of motion become
where
and the relation between the twisted gauge and the original one is given by
Let us check KK mode expansions for and . Since has the parity assignment , we can expand it by using parity even combinations:
where
0) \cr
u_{0L}^C(x,y)&(n=0) \cr
\displaystyle \frac{1}{\sqrt{2}} \big\{ u_{-n L}^C(x,y) + i u_{-n L}^S(x,y) \big\} &(n< 0) ,
\end{cases} \cr
&u_{nL}(x,-y)=u_{-nL}(x,y) , u_{nL}(x,L-y)=u_{-nL}(x,L+y) .
\end{align}]]>
Note that and satisfy the same parity property as and in Eq. (A.13), and , are defined in Eq. (A.14). On the other hand, has the parity assignment so that one can expand it by using parity odd combinations:
where
0) \cr
u_{0R}^S(x,y)&(n=0) \cr
\displaystyle \frac{1}{\sqrt{2}} \big\{ u_{-n R}^S(x,y) + i u_{-n R}^C(x,y) \big\} &(n< 0) ,
\end{cases} \cr
&u_{nR}(x,-y)=-u_{-nR}(x,y) ,
u_{nR}(x,L-y)=-u_{-nR}(x,L+y) .
\end{align}]]>
To investigate the physics at energies , we keep only sixth-dimensional KK modes, and replace by :
They have the following BCs at :
where . The equations of motion in the twisted gauge of Eq. (4.6) for the fields become
It follows that , etc. The BCs at in the twisted gauge are the same as those in the original gauge:
In terms of the basis functions , given in Appendix B of Ref. [35], which satisfy
one can write the mode functions for as
The equations of motion in Eq. (4.13) in the bulk yield
so that one finds
The BCs at in the twisted gauge follow from those in the original gauge:
We write the above equations in matrix form:
where we have used shorthand notation such as standing for . In the following we will use the same notation. From , we find the mass formula of the up-type quarks:
The same formula holds for with . The formula in Eq. (4.22) is the same as in the 5D GHGUT in Ref. [35]. For , the up-type quark mass spectrum is given by
\frac{1}{2} . \\
\end{array}
\right.
\end{align}]]>
4.2. Down-type quark
(ii) :
Consider with . Parity even modes at with are , , , and . From the action in Eq. (2.19) and the and terms in Eq. (2.28), one finds the equations of motion for down-type quarks:
For sixth-dimensional KK modes, the equations of motion reduce to
To obtain BCs at , we integrate the above (a), (d), (f), and (g) in the vicinity of for parity-odd fields:
For parity-even fields, we evaluate the equations of motion at by using the above conditions:
where we used the equations of motion (d) and (g) at .
We recall that in the twisted gauge all fields obey free-field equations in the bulk. Their eigenmodes are determined by the BCs on the IR brane. The mass spectra can be fixed by the BCs at the UV brane. In the twisted gauge the mode functions are given by
where satisfy the relations in Eq. (4.16) and . The BCs at in the twisted gauge are converted to
From , we find the mass spectrum formula for the down-type quarks:
The same formula is obtained for with . For , the down-type quark mass spectrum is given by
\frac{1}{2} . \end{cases}
\end{align}]]>
Combining Eqs. (4.23) and (4.31), one finds
4.3. Charged lepton
(iii) :
Parity even modes at with are , , , and . From the action in Eq. (2.19) and the and terms in Eq. (2.28), one finds the equations of motion for charged leptons:
For sixth-dimensional KK modes, the equations of motion for charged leptons reduce to
To obtain boundary conditions at , we integrate the above (a), (d), (e), (h) in the vicinity of for parity-odd fields:
For parity-even fields, we calculate the equations of motion at by using the above conditions:
where we used the equations of motion (d) and (h) at .
By using the BCs on the IR brane, the mode functions of charged leptons in the twisted gauge are given by
As in the case of down-type quarks, the BCs at in the twisted gauge are converted to
From , we find the mass spectrum formula for the charged leptons:
The mass spectrum of charged leptons, for which , is given by
\frac{1}{2}.
\end{cases}
\end{align}]]>
Here we have made use of , which assures , and have assumed that for \frac{1}{2}$]]> and for so that . Combining Eqs. (4.23) and (4.40), one finds
\frac{1}{2} , \cr
z_L^{c_2 - \frac{1}{2}}\, \displaystyle \frac{\mu_{\bf 11}'}{\widetilde{\mu}_2}\,
\sqrt{\frac{1 - 2 c_2}{2 c_0 - 1}}
&\mbox{for}\ c_0 > \frac{1}{2},~ c_2 < \frac{1}{2} , \cr
\displaystyle \frac{\mu_{\bf 11}'}{\widetilde{\mu}_2}\,
\sqrt{\frac{2 c_2 - 1}{2 c_0 - 1}}
&\mbox{for}\ c_0 >\frac{1}{2},~ c_2 > \frac{1}{2}.
\end{cases}
\end{align}]]>
It will be seen below that in a typical example for the third generation and .
One comment is in order about the masses of exotic charged leptons with charge . No zero modes exist for and , and all modes have masses larger than . On the other hand have zero modes. They acquire masses through the Hosotani mechanism and brane interactions. We suppose that and/or is sufficiently large that their lightest masses are .
4.4. Neutrino
(iv) :
The 6D spinor and vector bulk fermion and and a 5D singlet symplectic Majorana brane fermion appear in this sector. From the action in Eqs. (2.19) and in Eq. (2.28), the equations of motion for the bulk fermions become
The equations for the 5D brane symplectic Majorana fermion are
Note that has a zero mode but has no zero mode in the sixth-dimensional direction.
For sixth-dimensional KK modes, the equations of motion become
Here, is given by Eq. (4.2). To find the form of for in , let us recall that the original and twisted gauges are related by where . Hence,
As
one finds
It follows that
and so on.
To obtain boundary conditions at , we integrate the above (a), (d), (e), (f), (j), (m), (n), and (o) () in the vicinity of for parity-odd (in ) fields :
For parity-even (in ) fields , we evaluate the equations of motion at by making use of the above conditions to find:
at .
There are nine fields in the neutral fermion sector which intertwine with each other. To simplify the discussions, we consider the case in which . In the limit, the equations of motion and boundary conditions in the neutral sector, Eqs. (4.44), (4.49), and (4.50), split into two parts. The first sector, the neutrino sector-1, contains , in and in , while the second sector, the neutrino sector-2, contains other components in and . mixes the neutrino sector-1 and sector-2.
In the twisted gauge, the mode functions are determined by the BCs on the IR brane. The mass spectra can be fixed by the BCs on the UV brane. For the neutrino sector-1, by using the boundary conditions at , their mode functions can be written as
Here, are chosen in the neutrino sector-1 such that is satisfied where is defined in Eq. (2.11). One can take to be real. In this case, is satisfied so that equation (q) in Eq. (4.44) implies that
With this identity the first relation in Eq. (4.50) can be rewritten as
Setting , one finds that the BCs at in the twisted gauge can be written as
From , we find the mass spectrum formula for the neutrino sector-1:
This gives the gauge–Higgs seesaw mechanism for the small neutrino masses. For and , the neutrino mass is given by
where is given by Eq. (4.23). As shown in Ref. [38], the gauge–Higgs seesaw mechanism is characterized by a mass matrix
in the 4D effective theory in which and . The Majorana mass may take a moderate value.
Next, we consider the neutrino sector-2. The BCs at the IR brane determine the mode functions in the twisted gauge to be
By making use of Eqs. (4.47) and (4.48), the BCs at the UV brane are expressed as
where
Note that
From the mass spectra for the neutrino sector-2 are found to be
In the limit, or in the absence of the brane interactions, Eq. (4.62) becomes
For large , it becomes
For the neutrino sector-1 and sector-2 mix through the boundary conditions. One needs to solve
is evaluated to be
where . In evaluating the effective potential , we shall use the approximate formula for small .
4.5. Dark fermion
(v)
The 6D Weyl fermion , which may be called a dark fermion multiplet, has sixth-dimensional KK modes. For one need not introduce brane interactions. We consider the action in Eq. (2.19) for the dark fermion sector . From the parity assignment shown in Table 2, the mass formula for the dark fermions is found to be
For , the dark fermion mass is given by
1/2 .
\end{cases}
\end{align}]]>
As in the case of the 5D GHGUT discussed in Ref. [35], the bulk mass parameter of the dark fermions, , must be relatively small not to become light exotic particles.
5. Effective potential
We evaluate the Higgs effective potential by using the mass spectrum formulas of the sixth-dimensional KK modes of the gauge bosons and fermions. The evaluation is done in the same manner as in Ref. [35].
One-loop effective potential from each KK tower is given by [9,45,46]:
where is the mass spectrum of the KK tower and we take the and sign for bosons and fermions, respectively. When the mass spectrum is determined by , Eq. (5.1) can be written as
where . We utilize the mass spectra of the bulk gauge and fermions obtained in Sects. 3 and 4. Note that only -dependent mass spectra contribute to the -dependent part of . It contains the bulk gauge field, spinor, and vector fermion fields.
We summarize those mass spectra. The gauge boson contribution is almost the same as in the 5D GHGUT. The difference from the 5D case is that the boson contributions can be neglected in the current scheme as they have masses of . The relevant mass spectra of gauge fields and are given, from Eqs. (3.7), (3.12), and (3.18), by
The mass spectra of up- and down-type quarks, charged leptons, neutrinos, and dark fermions are given, from Eqs. (4.22), (4.30), (4.39), (4.55), (4.62), and (4.67), by
where has been assumed for (iv-1) and (iv-2). (v) is the dark fermion mass spectrum. stands for the bulk mass of the 6D spinor bulk Weyl fermion .
We evaluate the effective potential . In the gauge is decomposed into
Here,
where and are modified Bessel functions.
The fermion part is evaluated in a similar manner. Following the decomposition in Eq. (5.4), and taking into account four degrees of freedom for each Dirac fermion and the color factor 3 for quarks, one can write:
where
As is not a real function, has been converted to the integral involving .
The Higgs mass is determined, at the minimum of , by
Note that the 4D neutral Higgs field is related to in Eq. (2.33) by
We give results for the effective potential and the Higgs mass for typical sample values of the parameters. In the following calculation, we take into account the sixth-dimensional modes of 6D bulk gauge bosons , the third-generation spinor and vector fermions , , , and dark fermions . Contributions coming from the first and second generations of quarks and leptons are numerically negligible. Some of the parameters in the fermion sector are fixed by the observed masses of quarks and leptons in their approximate forms given in Eqs. (4.23), (4.31), (4.40), and (4.56).
The calculation algorithm is almost the same as in 5D GHGUT given in Sect. 5 of Ref. [35]. We are interested in the effective potential at the electroweak scale. Some of the relations derived in this paper are those at the GUT scale. The value of evolves from at the GUT scale to the observed value at the electroweak scale by the renormalization group equation (RGE). The RGE analysis of the coupling constants is beyond the scope of the current paper. In this paper we content ourselves with inserting the observed value of into the formulas for and . As input parameters we take , , , , , , and . The evaluation is carried out in the following steps:
(1) We first pick the values for and . So far consistent sets of parameters have been found only for .
(2) From , Eq. (3.12), and are determined.
(3) is fixed from by Eq. (4.23).
(4) Some of the parameters in the brane interactions on the UV brane remain free. We take , , and GeV. These three parameters are kept fixed in the evaluation below. We temporarily assign a value for . Then, , , and are determined by Eqs. (4.32), (4.41), and (4.56), or by
(5) Given the value of the bulk mass parameter of the dark fermion multiplet, the effective potential can be evaluated. We adjust the value of such that the minimum of is located at the initial value of . In this manner a consistent parameter set has been obtained.
(6) The Higgs boson mass is determined from (5.9), which can be viewed as a function of , and , namely .
(7) By demanding that the resulting is the observed value, namely GeV, is determined with the given and .
(8) Take a different . By repeating the above procedure one finds a new value for .
The physical value of needs to be determined, for instance, from observation of the bosons (the first KK bosons ) and their masses. In the gauge–Higgs electroweak unification many of the physical quantities depend on the value of , but not on the details of the parameters in the theory. It has been shown that TeV (TeV) to be consistent with the current LHC data [15,16].
The results for the effective potential in the gauge are depicted in Fig. 1, for which , , and GeV. The bulk and brane mass parameters are chosen as
The effective potential in the gauge in which and the minimum of is located at . For the top two figures, the blue solid line shows the effective potential with both gauge and fermion effects, and the red and green dashed lines represent the contributions of fermions and gauge fields, respectively. For the bottom left figure, the green solid line represents the total contributions of gauge fields, and the red and purple dashed lines represent and boson contributions. In the bottom right figure the red solid line represents the total contribution of all fermions, the green dashed lines represent contributions of the top quark, the blue dashed line represents that of the neutrino sector-2, and the black dashed line represents the contribution of dark fermions. Contributions coming from other fermions are negligible.
The resultant , , and are given by
In Table 6, we have tabulated various sets of the parameters which yield GeV. As increases, decreases. One sees in order for 8\,$]]>TeV to be consistent with the current LHC data. We have observed that cannot be too small in order to reproduce GeV.
Parameter sets which give dynamical EW symmetry breaking with GeV. Here , , and GeV. can be varied without affecting and . For , for instance, a set of and yields GeV, , and .
[TeV]
[GeV]
0.05
0.1730
24.63
0.3289
0.5960
125.17
1.138
0.06
0.1590
20.53
0.3292
0.5828
125.07
1.046
0.07
0.1480
17.60
0.3294
0.5714
125.13
0.9736
0.08
0.1400
15.40
0.3297
0.5625
125.03
0.9208
0.09
0.1330
13.70
0.3300
0.5544
125.08
0.8746
0.10
0.1270
12.33
0.3303
0.5471
125.17
0.8350
0.11
0.1225
11.21
0.3307
0.5414
125.05
0.8052
0.12
0.1180
10.28
0.3311
0.5356
125.14
0.7755
0.13
0.1145
9.495
0.3315
0.5311
125.06
0.7523
0.14
0.1110
8.821
0.3320
0.5264
125.12
0.7291
0.15
0.1080
8.236
0.3325
0.5224
125.12
0.7091
0.16
0.1055
7.725
0.3331
0.5192
125.03
0.6925
0.17
0.1030
7.275
0.3337
0.5158
125.02
0.6759
0.18
0.1005
6.875
0.3343
0.5125
125.11
0.6593
0.19
0.0985
6.517
0.3349
0.5099
125.05
0.6459
0.20
0.0965
6.195
0.3356
0.5073
125.05
0.6326
0.21
0.0945
5.903
0.3363
0.5047
125.21
0.6192
0.22
0.0930
5.639
0.3371
0.5030
125.00
0.6092
0.23
0.0910
5.398
0.3379
0.5003
125.17
0.5959
0.24
0.0895
5.177
0.3387
0.4986
125.14
0.5858
0.25
0.0880
4.974
0.3395
0.4969
125.16
0.5758
0.26
0.0867
4.786
0.3404
0.4956
125.11
0.5671
0.27
0.0855
4.613
0.3413
0.4945
125.04
0.5590
0.28
0.0840
4.452
0.3423
0.4928
125.16
0.5490
0.29
0.0830
4.302
0.3432
0.4921
125.05
0.5423
0.30
0.0818
4.163
0.3442
0.4911
125.07
0.5342
In Figure 2, is plotted as a function of . The data points in Table 6 are fitted by
in the range . A similar relation has been found in the gauge–Higgs EW unification where TeV and [11,12]. The difference between the two is expected to originate from the different gauge group structure of the models and the different matter content. So far we have found consistent parameter sets only for and .
The Kaluza–Klein mass scale as a function of . The red dots represent points in Table 6 which reproduce GeV. The blue thick dashed line represents the fitting curve given in Eq. (5.14). The black thin dashed line indicates found in the gauge–Higgs EW unification [11,12].
6. Summary and discussions
In this paper, we discussed 6D GHGUT in the 6D hybrid warped space. The GUT gauge symmetry is reduced to the Pati–Salam symmetry by the orbifold BCs, and the symmetry is spontaneously broken to the SM gauge symmetry by the non-vanishing VEV of the 5D brane scalar field on the UV brane. Finally, the EW gauge symmetry is broken to by the Hosotani mechanism. The SM Higgs boson appears as a zero mode of the fifth-dimensional components of the gauge field.
The SM fermions are contained in , , , . The SM fermions as well as the and bosons acquire masses via the Hosotani mechanism. In Sect. 4, we showed that the quark and lepton mass spectra can be reproduced by choosing the parameters of the bulk masses and the brane interactions. In the 5D GHGUT in Ref. [35], there necessarily have arisen light exotic vector-like fermions accompanied by SM fermions, which is experimentally unacceptable. In contrast to the case of the 5D GHGUT, the non-SM fermions have masses of either or in the current 6D model. Thus the problem of light exotic fermions has been solved.
From the analysis of the effective potential in Sect. 5, we have seen that the EW symmetry is dynamically broken by the AB phase . The Higgs boson mass GeV can be obtained for , namely for TeV.
In the present paper we have treated each generation of quarks and leptons independently. It is necessary to incorporate the flavor mixing in the gauge–Higgs grand unification. The mixing arises from the brane interactions. For quarks and charged leptons the mixing would result from the matrix structure of various s and s in the brane interactions in Eq. (2.27) of , , and . On the other hand, the dominant mixing in the neutrino sector would arise from the matrix structure in the mass associated with the symplectic Majorana fields in Eq. (2.26). The large mixing angles observed in the neutrino sector might be related to this special circumstance.
We would like to add a comment on the effective potential of the AB phase along the sixth dimension. From Table 5, not only the components of the fifth-dimensional gauge field but also those of the sixth-dimensional gauge field have zero modes in the absence of brane interactions. As in the effective potential for the AB phase along the fifth dimension, the bulk gauge bosons and the bulk fermions , , and , contribute to . If there were only gauge fields, it can be shown that the minimum of would be located at . In other words, the EW symmetry would be broken by the sixth-dimensional AB phase and the and bosons would acquire masses of . The situation is similar to that in the effective potential in the 5D GHGUT [32,35]. Further, the contribution from the bulk spinor fermions to has little dependence, as is inferred from Table 2. The -dependent contributions coming from a set of bulk vector fermions and almost cancel each other, as can be deduced from Table 3. Hence, with the minimal fermion content presented in this paper the minimum of would be located at around . This undesirable result can be avoided by introducing additional bulk vector fermions with appropriate boundary conditions.
The phase plays a crucial role in the gauge–Higgs grand unification. It is found that the 5D KK mass scale is determined by in a very good approximation by the formula in Eq. (5.14). Similar relations are expected for the 4D Higgs cubic and quartic couplings as in the gauge–Higgs EW unification. For we have recognized the difference between the gauge–Higgs EW and grand unification. We need experimental data to find which one is preferred.
We will come back to these issues in the near future.
Acknowledgments
We would like to thank Hisaki Hatanaka for critical reading of the manuscript and valuable remarks. This work was supported in part by Japan Society for the Promotion of Science Grant-in-Aid for Scientific Research No. 15K05052.
Funding
Open Access funding: SCOAP.
Appendix A. Basics for 6D Kaluza–Klein expansion
We present KK mode expansions in 6D hybrid warped space with the metric given in Eq. (2.1).
A.1. 6D gauge fields
The equations of motion for free gauge fields in the 6D hybrid warped space are given by
where and are defined in Eq. (2.32). These and differ from and in the 5d case. Basis mode functions in the -coordinate in the 6d case are given, except for zero modes, by
where . They can be expressed as
and satisfy
These basis functions have been employed in the text. We note that the basis functions for fermions, and , remain the same as in the 5d case defined in Ref.[35].
A.2. 6D scalar fields
The action of a massless 6D scalar field is given by
where . The equation of motion for the scalar field is given by
It follows that () satisfies
The Bessel equation is given by
whose solutions are given by the Bessel functions and . Hence, a mode function can be written as , where satisfies
The orbifold boundary condition for the scalar field is given by
In the -coordinate, satisfies the same boundary condition as Eq.(A.9), and Eq.(A.6) becomes
Due caution must be taken as . If is parity even in the -coordinate, the Neumann () condition becomes, in the -coordinate (),
as can be confirmed by integrating Eq. (A.10) over from to (or from to ). If is parity odd in the -coordinate, it satisfies the Dirichlet () condition at or .
(i)
The 6D scalar field is expanded as
where and satisfy
and Fourier modes and are given by
and satisfy the Neumann and Dirichlet conditions at , respectively. There is a zero mode for :
which corresponds to a mode constant in the fifth dimension. Other modes can be written as
Eigenvalues 0$]]> and the relative coefficient are determined by the boundary conditions. Noting a recursion relation
one finds eigenvalues () from
Thus is expanded as
where satisfies
There is one massless mode with .
(ii)
The expansion is given by
There is no massless mode.
(iii)
where
Note that the mass spectrum is given by
(iv)
A.3. 6D fermion field
Let us consider a free 6D Weyl fermion with :
Its action is given by
where is defined in Eq. (4.5). As an example we consider the case in which the BCs are given by
The equations of motion are
To find the eigenmodes, we separate 4D and the fifth- and sixth-dimensional parts as
It follows that
In the -coordinate, and satisfy the BCs
(i)
In this case,
The mode functions can be written as
where the are defined in Eq. (A.14). and satisfy the BCs
By inserting Eq. (A.33) into Eq. (A.28), one finds equations for in the -coordinate ():
It follows that
The equations for the zero () modes in the sixth dimension, and , are reduced to the Bessel equation. In terms of the basis functions , given in Appendix B of Ref. [35], solutions satisfying the BCs are given by
0 ,
\end{align}]]>
where is a normalization constant. There is one massless mode for the right-handed component. For the modes the solutions are more involved.
(ii)
In this case the mode functions of and can be written as
The equations for the zero () modes in the sixth dimension, and , are
and solutions satisfying the BCs are given by
0 ,
\end{align}]]>
where is a normalization constant. There is one massless mode for the left-handed component.
(iii)
In this case the mode functions can be written as
where are defined in Eq. (A.22). There are no zero modes in the sixth dimension.
(iv)
In this case the mode functions can be written as
There are no zero modes in the sixth dimension.
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