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Physics Letters B 560 (2003) 204-213
www. elsevier. com/locate/npe
Gauge-Higgs unification in the left-right model
Ilia Gogoladze^ Yukihiro Mimura, S. Nandi
Physics Department, Oklahoma State University, Stillwater, OK 74078, USA Received 22 January 2003; accepted 17 March 2003 Editor: H. Georgi
Abstract
We construct a supersymmetric left-right model in four dimension with gauge-Higgs unification starting from a SU(3)c x SU (4) w x U(1) B—L gauge symmetry in five dimension. The model has several interesting features, such as, the CKM mixings in the quark sector are naturally small while for the neutrino sector it is not, light neutrino masses can be generated via the seesaw mechanism in the usual way, and the model has a U(1)R symmetry which naturally forbid dimension five proton decay operators. We also discuss the grand unification of our model in SO(12) in five dimensions. © 2003 Published by Elsevier Science B.V.
1. Introduction
Recent topics of the theories in higher dimensions give us a lot of interesting phenomenological pictures. One of the most attractive motivations of extension of dimensions is that the variety of particles in Nature can be understood by means of a geometrical language. For example, gauge fields with the coordinate for the extra dimensions behave as scalar fields in 4 dimension. Since masses of the gauge bosons are prohibited by gauge invariance, the scalar field originated from gauge bosons can be a good candidate of the low energy Higgs fields, which breaks electroweak symmetry. That leads to the idea of the gauge-Higgs unifi-
E-mail addresses: ilia@hep.phy.okstate.edu (I. Gogoladze), mimura@hep.phy.okstate.edu (Y. Mimura), shaown@okstate.edu (S. Nandi).
1 On a leave of absence from: Andronikashvili Institute of Physics, GAS, 380077, Tbilish, Georgia.
cation in the higher-dimensional theories [1-3]. Recent realization of the phenomenological models in higher dimensions makes us encourage to revisit the idea [4-7].
We consider that the extra dimensions are com-pactified in an orbifold in order to make chiral theories in 4D, since 5D fermions include both chiral-ity in 4D language. In an orbifold space, such as S1 /Z2, we can impose boundary conditions at the folding places, and the gauge symmetry can be broken through these boundary conditions [2,8]. Recently, a great deal of works has been done on the gauge symmetry breaking using the orbifold boundary conditions, and these lead to many attractive features of the unified gauge theories in higher dimensions [9,10]. Using the orbifold boundary condition, we can project out unwanted fields such as colored Higgs triplets in the grand unified theories [9]. In such a progress of the higher-dimensional unified theories, interesting ideas of gauge-Higgs unification are suggested. In
0370-2693/03/$ - see front matter © 2003 Published by Elsevier Science B.V. doi:10.1016/S0370-2693(03)00400-3
the higher-dimensional supersymmetric theories, the gauge multiplet contains both vector multiplet and chi-ral supermultiplet in 4D. Assigning the different Z2 parity between vector multiplet and chiral supermultiplets, we can make vector multiplet massless but chi-ral supermultiplets heavy, which means the supersym-metry is broken. If we break gauge symmetry through boundary condition simultaneously, a part of the chiral supermultiplets can have a zero mode which remains massless in the low energy. Then, we can identify such a supermultiplet with the low energy Higgs field. This is the main idea of the gauge-Higgs unification which we consider in this Letter. This idea was realized in the 6D N = 2 supersymmetric theories [5], and more recently in the 5D N = 1 supersymmetric theories [6,7]. The latter scenario gives us an interesting possibility that the gauge and Yukawa coupling constants have the same origin. Since the Yukawa interactions arise from the gauge interaction in the 5D Lagrangian, those two coupling constants are "unified" in the 5D theory. This is a very interesting feature of the 5D gauge-Higgs unified scenario. InRef. [6], the authors considered the theory of SU(3)w and SU(6) as an example of the scenario, but in their models, there are many unwanted fields in the matter hypermultiplets. In order to make the unwanted fields heavy, they need many brane-localized fields for each generation. The brane fields are actually needed to cancel the gauge anomaly in the 4D theory which arises from the zero modes of bulk hypermultiplets, and that means it is not easy to understand the anomaly free structure in their models. In Ref. [7], the authors consider the gauge-Higgs unification in larger gauge group such as E6, E7 and E8.
In this Letter, we construct a supersymmetric left-right model, SU(3)c x SU(2)l x SU(2)r x U(1)b-l in 4D [11,12] with gauge-Higgs unification. We point out that we are unifying the gauge fields with only the Higgs multiplets which give masses to the standard model fermions. We start with a supersymmetric model with the gauge group SU(3)c x SU(4)w x U(1)B-L in 5D. The SU(4)w gauge symmetry is broken down to SU(2)l x SU(2)r x U(1)x by orbifold boundary condition, and then the gauge symmetry in 4D become left-right symmetric gauge group with extra U(1)X symmetry (U(1)X is broken to nothing using suitable brane interactions). In this model, no unwanted zero modes arise from the matter hypermul-tiplets. It is easy to see the anomaly free structure,
and this structure naturally give rise to even number of families of the matter hypermultiplets. The left-right symmetric construction gives us a good picture to the scenario of the 5D gauge-Higgs unification. Our model naturally leads to small CKM mixings in the quark sector, while in the neutrino sector, mixing can be large. The model is nicely grand unified in SO(12) in 5D.
Our Letter is organized as follows: in Section 2, we construct our supersymmetric SU(4)w model in 5D with gauge-Higgs unification and show how orbifold compactification leads to left-right symmetric model in 4D. In Section 3, we discuss the quark and lepton mass matrices and mixings in our model and various other features. Grand unification of our model in SO(12) is contained in Section 4. Section 5 has our conclusions and discussions.
2. Gauge-Higgs unification in SU(4)w gauge theory
In this section, we discuss the construction of the gauge-Higgs unification in 5D N = 1 supersymmetric theory based on the gauge group SU(3)c x SU(4)w x U(1)B-L. We will consider S'/Z2 orbifold, which is constructed by identifying the coordinate of the fifth dimension, y, under two parity transformations: Z2: y ^—y and Z2: y' ^ —y', where y' = y + nR. The orbifold space is regarded as a interval [0,nR] and 4-dimensional walls (we call 4D wall brane) are placed at the folding point y = 0 and y = nR. The 5D N = 1 supersymmetric theory corresponds to 4D N = 2 supersymmetric theory. In 4D language, the N = 2 gauge multiplet contains one N = 1 vector multiplet V(A^,X) and one N = 1 chiral multiplet V(a + iA5,k'). The boundary conditions at 4D walls are given as
(V )<*■->>=(-PP. )("-)• (V) {x"-—>'> = ( -PV-—') (x'-y'>- (')
where P and P' acts on gauge space. Then 4D N = 2 supersymmetry is broken down to N = 1 supersymmetry and non-trivial P and P' breaks gauge
symmetry G down to H. In the case that P = P', the vector multiplets has Z2 x Z2 parity as (+, +) for unbroken gauge symmetry, and (-, -) for broken one G/H. On the other hand, for the chiral multiplet, the signature of the parity is opposite. Since only (+, +) components have massless modes, the chiral multiplet S for the broken generator remains massless. We identify the massless chiral multiplet as Higgs field to break electro-weak symmetry SU(2)L x U(1)Y.
We will apply this 5D gauge-Higgs unification scenario to left-right model. We consider SU(3)c x SU(4)w x U(1)B-L symmetry as the bulk gauge symmetry. We need this U(1)B-L. The reason is that after the orbifold breaking of SU(4)w, although we get an U(1)x, it does not have the right B - L quantum numbers as required by the bi-doublet Higgs in the left-right theory. The boundary condition breaks SU(4)W down to SU(2)l x SU(2)r x U(1)x on the 4D walls, if we use P = P' = diag(1,1, -1, -1). The SU(4) adjoint is decomposed as
15 = (3,1)0 + (1, 3)0 + (1,1)0 + (2, 2)2 + (2, 2)-2,
where the numbers in the subscripts represent U(1)x charges. The two bi-doublets correspond to the broken generator, and their S components of the gauge multiplet remain massless after the compactification. These can be identified to be the bi-doublet Higgs required to gives masses to the fermions in the left-right symmetric theory, and in our model, they originate from the gauge supermultiplet. We define the hypercharge as Y = TR + (B - L)/2, where TR is a generator for SU(2)r . The B - L charges of the bi-doublets are zero since those ones come from SU(4)W gauge multiplet. Thus those bi-doublet quantum numbers are same as the ones needed in the left-right model. Since we do not need the U(1)x symmetry at low energy, we will break U(1)x without mixing withSU(2)R and U(1)B-L. To do that, we add the brane fields z and Z, which have +1 and -1 U(1)x charge and are singlets for the other symmetry.
Next we consider the matter fields. The matter fields should be bulk fields in the scenario of 5D gauge-Higgs unification since the chiral superfield S transforms non-linearly under 5D gauge transformation and we cannot make Yukawa coupling with the chiral field S at 4D walls. So the matter fields are N = 2 hypermultiplets (&, &c), where the & and
&c are the N = 1 chiral multiplets. If we take this non-conjugated field & as fundamental representation under SU(4)W (conjugated field &c is the antifundamental), then & and &c are decomposed as
& = (2,1)1 + (1,2)-1, (3)
&c = (2,1)-1 + (1, 2)1. (4)
The boundary condition is given as follows:
&(x^, -y) = sP&(x^,y),
&c(x'\ -y) = -sP&c(x'\y), (5)
&(x^, -yr) = sP'&(x^,yr),
&c(x-y') = -sP'&c(x^,y'), (6)
where s = ±1. In the case where s = 1, (2,1)1 and (1, 2)1 have (+, +) parity and others have (-, -), and in the case where s = -1, the parity charges are opposite. With this boundary condition, we obtain the left- and right-handed quarks and leptons assigning appropriate color and B - L charges. Choosing s = 1, we have one family of quarks and lep-tons, which are the massless, in the representations (SU(3)c, SU(2)l, SU(2)r)u(1)b-l,u(1)x :
Ql: (3, 2,1)1/3,1, Qcr: (3*, 1, 2)-W,
Il: (1, 2,1)-1,1, IR: (1,1, 2)1,1. (7)
It is important to notice that gauge anomaly with respect to U(\)x arises such as SU(3)2C x U(1)x, SU(2)2l r x U(1)x, and U(1)3x. To cancel the anomaly, we need one more family with s = -1. In other words, fixing s = 1, we need both (3,4)1/3 and (3, 4*)1/3 for quark non-conjugated fields, and need both (1, 4)-1 and (1, 4*)-1 for lepton non-conjugated fields in the representation (SU(3)c, SU(4)w)U(1)b-l . This gives an interesting feature of this model: the number of the generation is even. We note that it is possible that the U(1)x anomaly can be canceled by introducing appropriate brane fields. However, since we have to make those brane fields heavy by folding Dirac mass with another multiplets, this possibility is not economical. This U(1)x gauge anomaly gives us a strong constraint to construct a model.
We now discuss the coupling among the matter fields & , & c and the chiral multiplet S . The action
is written as [13]
S5D = I d x dy
I d40 (Qe-VV + VceVQc )
dz0 Ve {d5 - jls + h.c.
and the "superpotential" term gives us the Yukawa coupling in the 4D theory. Naming the (2, 2)—2 and (2,2)2 in the S multiplet as and respectively, we obtain the following Yukawa coupling for quark:
S4D = I d x
x MiK^ + y2Q-0>20)Q-(R))
+ h.c.,
where we denote that Q±L and Q±R having ±1 charge for U(1)X symmetry, and the subscript (0) as the zero modes. Yukawa couplings for leptons are also written in the same way. We can see that the Yukawa coupling constants y1 and y2 originate from gauge coupling, and thus, the Yukawa and gauge couplings have the same origin. This is the most interesting feature of the 5D gauge-Higgs unification scenario. However, then we find that the Yukawa couplings for all the families have the same values and equal to the gauge coupling. Observation excludes such a situation for the first and second family, and we need to solve the problem. The problem can be solved by introducing bulk masses for the hypermultiplets № and №c such as
S = j d4xdy j
d 2eM(y)VcV + h.c.
where M(y) = M(e(y) + e(nR - y))/2 and e(y) is a step function. With this bulk mass, the zero-mode wave functions of the fermions localize at 4D walls. If we choose M < 0, the zero-mode wave function of № localizes at y = 0 with a profile e-IMy, and for №c, the zero-mode localizes at y = nR with profile
,|M |(y-*R)
. Then we find that the Yukawa couplings
are different from 4D gauge coupling g such as
ttR\M\
sinh(7rtf|M|)^
Therefore, if |M |R is larger than 1, the Yukawa couplings are exponentially suppressed, and thus we can obtain the hierarchical fermion masses.
In this section, we have constructed the 5D gauge-Higgs unification model in the context of our SU(4)w gauge theory. In this 5D scenario, not only the low energy Higgs fields come from gauge multiplet, but also the Yukawa couplings originate from gauge interaction. The fermion mass hierarchy is generated by the bulk masses of the matter fields. But still, we do not have generation mixing for the quarks since the Yukawa couplings come from gauge interactions which do not produce such mixings. We will see how the family mixings are generated in the next section.
3. Quark and lepton mass matrices
In this section, we will investigate the quark and lepton mass matrices of our SU(4)w model. We recall that number of generation of this model should be even number because of U(1)X gauge anomaly. We will consider 4-generation model here. The two of the 4 generations have +1 charge for U(1)X and the others have -1 charge.
We introduce chiral superfields QL, QcR, lL and lcR on the 4D walls. The quantum numbers (SU(3)c, SU(2)L, SU(2)r)U(1)b-l,u{1)x of those fields are the following:
ql{ ^ ^ 1 -1/3^ ha, 2,1)1,0, 7,c
ßR(3,1, 2)1/3,0,
cr(1, 1,2)-1,0.
Then we will have the following superpotential terms for quark:
S = j d4xdy (S(y) + S(y - nR)>
f d 20 ((zQL +zQl + zQ\ + ZQ4l)q + (zQR1 +zQcR2 + zQcR + ~zQcR4)QcR )
+ h.c.
where z and z are the chiral superfields which have ±1 U(1)X charge and singlet for the other symmetry. The superscripts 1, 2, 3, 4 denote the family indices. We also have similar superpotential terms for
leptons. The vacuum expectation values for z and Z break U(\)x symmetry, generate flavor mixings and make the 4th family heavy. From Eqs. (9) and (13), the 5 x 5 quark mass matrices for left-handed quarks (QL,Q2l,Q3l,Q4l,Qr) and right-handed quarks (QL ,QcR-,Qr,Qr,Ql) are obtained to be
Mu,d — diag(y1K^ ,y2Ku
«3 a4 0 /
u,d u,d u,d) y3K{ ,y4K2' )
where Ku,d (a — 1, 2) are the vacuum expectation values of such as
and ai and a' are the vacuum expectation values of z and Z multiplied by order 1 factors. There is an ambiguity how to select which generation gets mass by ^ or 02, but two of the generations should get masses by and the other two by The parameters ai (ai) for the up-type and down-type quarks are same because up- and down-type quarks are in the same multiplets QL (QcR). If we assume the left-right symmetry Q'L ^ QcR* and QL ^ QcR in the same way as ordinary 4D left-right symmetric model, we have a' — a* and the mass matrices are hermitian.
We now calculate the mass eigenvalues of the mass matrices given in Eq. (14). We assume that ai's are much larger scale than y^^. Then, one of the 4 generations decouples around U(\)x breaking scale and the other 3 generations remain at weak scale. We find that weak scale quark masses, assuming y1K1, y2K2 < y3K1 < y4K2,
mtb = -
|a1|2 +|a2|2 + |a3|2
i' |2.
|a1|2 +|a2|2
|a1|2 + |a2|2 + |a3|^ |a1|2 + |a2|2 + |a3|2
x y3K-i,
mu,d =
Vl«l|2+ l«2p
|a 2' | 2
Vl«il2+I«2l2
| a 1' |
|a 2' | 2
Since the mass discrepancy of the up- and down-type quarks comes from the difference of k«U and , the masses of third and second family should come from different ; otherwise, unacceptable relation appears such as mc/mt — ms /mb . The similar relation between first and second generation (or first and third generation) can appear, so we have assumed y1K1 ~ y2K2.
Next we investigate the Cabibbo-Kobayashi-Mas-kawa (CKM) angles. We define the 4 x 4 unitary matrix U and U' such that
(a1,a2,a3,a4)U — (0, 0, 0,a), (a1 ,a2,a'3,a'4)U' — (0, 0, 0,a)
where a — y lai I2, and a' — y |a'|2. The unitary matrices mix the families largely. However, since these mixing matrices are the same for the up- and down-type quarks, the large mixings do not contribute to the CKM matrix. After removing the non-physical large mixing, we have the quark mass matrices:
MU,d —
U tMu,dU'
V 0 0 0 a'
Having 4th generation decoupled, we obtain the 3-ge-neration quark mass matrices:
(mu,d)j — [(U tMu,du ')j —1,2,3. (21)
We can easily calculate the expression of the matrices explicitly, but here we show the approximate structure of the matrices:
mu,d mu,d mu,d mu,d ~ I mu,d mc,s mc,s mc
. (22)
mu,d mc,s mt,b
From Eq. (22), we obtain the CKM angles approximately:
Vus — 0{\)—-,
Vcb~0( 1) —, mb
Vub — O(1)
The 0(1) factors are the functions of the parameters ai and ai. We spoil the good relation for Cabibbo angle Vus ~ jmJTm s, but still we can fit all the CKM angles and quark masses to the observed values by choosing the parameters such as ai and ai. It is worth noting here that smallness of the CKM angle is automatically derived if we have hierarchical Yukawa coupling in the diagonal elements. We do not need to assume the hierarchical coupling in the brane interaction in Eq. (13), thanks to unification of up-and down-type quarks via the gauge-Higgs unification. Furthermore, since we use the exponential profile of the fermion localization to derive the Yukawa hierarchy, we do not need any small numbers in the fundamental theory to derive the quark mass structure.
Now we briefly discuss the lepton sector. We can construct the charged lepton mass matrix and Dirac neutrino mass in a way similar to the quark sector. To have the Majorana mass for the right-handed neutrino, we introduce SU(2)R triplet Higgses A (1,1,3)2,0 and Q (1,1,3)-2 0 whose vacuum expectation values break SU(2)r x U(1)b-l down to U(1)y. Giving appropriate coupling permitted by U(1)X symmetry, we introduce the brane superpotential term at y = 0,
d20 (fijAtpR + AlRlR) + h.c.
where some elements of the coefficient fij should be zero because of U(1)X symmetry. The brane superpotential term generates Majorana mass term for the right-handed neutrino in 4D, and gives small neutrino masses for the light neutrinos through the seesaw mechanism. If the right-handed leptons are localized at y = 0 by appropriate bulk masses, the Majorana mass matrix does not have hierarchical structure. If the eigenvalues of the Majorana mass matrix do not have hierarchy, the neutrino mass after seesaw will be the squared-hierarchy, for example, mV2 : mV3 — mC : m2. Thus, considering recent experiment of atmospheric and solar neutrino oscillation, we need a hierarchical structure in the Majorana mass matrix. If the right-handed leptons are localized at y = nR and the brane interaction is introduced only at y = 0, the Majorana mass matrix can have a hierarchical structure. The large mixing angles for neutrino oscillation can be originated from this Majorana mass structure. We note
that the 4th generation lepton mass (which we call a in the quark sector) should be large enough, so the 4th neutrino mass after seesaw should be higher than weak scale.
Finally, we will make a comment about unification of gauge and Yukawa couplings.2 Noting that the effective top Yukawa coupling in 4D is always smaller than 4D gauge coupling at 1 /R scale, we have to consider additional brane-localized gauge kinetic terms or the RG evolution of the SU(2)L gauge coupling and top Yukawa coupling. We mention about the latter case. Since left-right models has one more set of Higgs doublets rather than MSSM and we have (heavy) 4th generation in our model building, the absolute value of beta function for SU(2) L is larger than MSSM. Thus, we can easily make the gauge coupling at 1/R large enough to produce the observed value of the top quark mass. We make one more comment about bottom-tau ratio. Since down-type quark and charged lepton are not unified at this stage, there is no reason Yukawa couplings for bottom and tau are unified at 1 /R scale. But still, it is interesting that we have a possibility to realize the bottom-tau ratio, since all the 5D Yukawa couplings are unified.
4. Possible unified models
In this section, we will consider possible unified model of the SU(4)w theory which we have discussed. Our model fits nicely in a SO(12) unification. It is well known that ordinary 4D left-right symmetric model can be embedded in the S0(10). However, our left-right model cannot embedded in S0(10) since the gauge symmetry of our model is SU(3)c x SU(4)w x U(1)b-l, which rank is 6. Furthermore, the S0(10) adjoint 45 does not include uncoloredbi-doublet Higgs, and the 45 does not include MSSM doublet Higgs neither. Thus, we cannot construct the gauge-Higgs unification in S0(10) scheme. We also comment that E6 adjoint includes the MSSM Higgs doublets, but does not include bi-doublet for left-right models.
2 The numerical calculation of gauge and Yukawa unification in a 4D model is demonstrated in the Ref.['4].
Since the symmetry SU(3)c x U(1)b-l can be easily unified to the SU(4)c as in the Pati-Salam model [11], we can consider gauge-Higgs unification scenario in SU(4)c x SU(4)W model.3 The bi-doublet Higgs fields are included in the 4D N = 2 gauge multiplet of SU(4)W again, and the boundary conditions are given in the same way as in Eq. (1). All quarks and leptons are embedded in the hypermul-tiplet of (4, 4) representation. To cancel the U(1)X anomaly, we need both (4, 4) and (4, 4*) in the same way as we have discussed before, and the number of generation of the bulk fields should be again even. Those bulk matter multiplets do not include any unwanted fields. After orbifolding, the theory become Pati-Salam model (with U(1)X symmetry), and the quarks and leptons are unified into one multiplet in each chirality. We can also break SU(4)c down to SU(3)c x U(1)B-L through boundary condition, if we choose P = (1, 1,1, 1) and P' = (1, 1,1, -1) with respect to SU(4)c gauge space. Here we take P = P' since we do not want to make colored Higgs massless.
The SU(4)c x SU(4)W gauge theory can be unified into a SO(12) gauge theory. In order to realize the gauge-Higgs unification in the 5D SO(12) theory, we use the boundary conditions for 4D N = 2 gauge multiplet (V, S)
{x'\ -y) =
1 , (x^
(x'\y + 2nR) =
TVT TST
-1) (x^y)
and for N = 2 hypermultiplets (&, &c),
y (x^, -y) = PV(x^,y), yc(x'\ -y) = -Pyc(x'\y),
decomposed as
66 = 1o + 45o + 102 + 10-2, S0(10) x U(1)x,
66 = (15,1) + (1,15) + (6, 6), SU(4)c x SU(4)W.
For the N = 1 vector multiplet V, the P and T should be assigned as P,T = + for unbroken generators and P,T = - for the broken ones. Since the gauge fields for the broken generator become massive for each decomposition, the S0(12) is broken down to SU(4)c x SU(2)l x SU(2)r x U(1)x as a result. The vector multiplet V and chiral scalar field S are decomposed into the Pati-Salam gauge symmetry (with U(1)X) with (P, T) signature in the following:
V = (1,1,1)0+,+) + (1, 3,1)0+,+) + (1,1, 3)0+,+) + (6, 2, 2)0+,-) + (15,1,1)0+,+)
+ (1, 2, 2)2 ,+) + (1, 2, 2)-2,+) + (6,1,1)2-,-) + (6,1,1)--2,-),
S = (1,1,1)0 ,+) + (1, 3,1)0 ,+) + (1,1, 3)0 ,+) + (6, 2, 2)0-,-) + (15,1, DO-^
+ (1, 2, 2)2+,+) + (1, 2, 2)-+2,+) + (6,1,1)2+,-) + (6,1,1)-+2,-).
Then we find that, for the chiral superfield S, only SU(4)c-singlet bi-doublets have zero modes and those bi-doublets can be identified as low energy Higgs fields and their Yukawa couplings with matter fields come from gauge interaction in the same way as Section 2. The matter hypermultiplets are given as 32 and 32' representation of SO(12). Those representations are decomposed into
V(x'\y + 2nR) = TV(x'\y), yc (x^,y + 2nR) = Tyc (x^,y).
We take that P breaks SO(12) down to SO(10) x U(1)x, and T breaks SO(12) down to SU(4)c x SU(4)w. The adjoint representation 66 of SO(12) is
32 = 161 + 16*-1, S0(10) x U(1)X, 32 = (4, 4) + (4*, 4*), SU(4)c x SU(4)V
32 = 16-1 + 16*
S0(10) x U(1)X,
32' = (4, 4*) + (4, 4*), SU(4)c x SU(4)V
3 The supersymmetric SU(4)c x SU(4)W in 5D is also discussed intheRef. [15].
Taking the P and T appropriately, the hypermulti-plets (№, &c) for 32 are decomposed into SU(4)c x
SU(2)l x SU(2)r x U(1)x as №32 — (4, 2,1)1+,+) + (4*, 1, 2)1+,-)
+ (4*, 2,1)--,-) + (4,1, 2)-7+\ (33)
№3c2 — (4*, 2,1)(-;+] + (4,1, 2)--,-)
+ (4, 2,1)1+,-) + (4*, 1, 2)1+,+), (34)
and for the hypermultiplet 32'
^32' — (4, 2, t)-^ + (4*, 1, 2)(;->
+ (4*, 2,1)1-,-) + (4,1, 2)1-,+), (35)
^3c2' — (4*, 2,1)i1-,+) + (4,1, 2)1-,->
+ (4, 2,1)-+1,-> + (4*, 1, 2)-+1,+). (36)
Then we find that only (4,2,1)±1 and (4*, 1, 2)±1 in which all the left- and right-handed quarks and leptons are included have massless zero modes. Here again, we encounter the U(1)x anomaly if we have only 32 but not 32', vice versa. Since each 32 and 32' gives one generation of fermions, the number of generation which comes from bulk hypermultiplets is even number as a result. The generation mixings for quark and lepton are introduced by adding brane fields in the same way as Section 3. We note that this SO(12) unification gives one version of realization of gauge-Higgs unification in the 4D S0(10) gauge theory. The S0(10) x U(1)x decomposition of S0(12) adjoint 66 includes two vector representation 10, which can be identified to the Higgs fields in the 4D S0(10) unified models. The 10's include bi-doublets Higgses for left-right model (or we should say Pati-Salam model), and the colored partners are projected out by using another boundary condition T. Since extra U(1)x symmetry does not mix with electroweak gauge sector, the weak mixing angle prediction is same as in ordinary S0(10) theory that the gauge symmetry is broken down to Pati-Salam symmetry, if we neglect the brane-localized gauge kinetic terms with large cutoff scale. Though all the fermions are unified in spinor representations, the left- and right-handed fermions are separated to the non-conjugated and conjugated chiral superfields № and №c in our construction.
We make a comment about another possible unified gauge group of our SU(3)c x SU(4)w x U(1)B-L. One can consider the unified gauge group such as SU(7)
or SU(8). However, those extensions of our model are not straightforward, since we have to care about U(1)x anomaly. We have to have both (3, 4)1/3 and (3, 4*)1/3 forthe quark representation to cancel the 4D U(1)x anomaly, but it seems difficult to make both with correct B - L charges in the decomposition of SU(7) representation. In SU(8) case, we do not have to care about B - L charge, but still it is difficult to make anomaly free set of zero modes. For all that, we can make the U(1)x anomaly free by introducing appropriate brane fields, but that is not a simple extension of our SU(4)w models.
5. Conclusion and discussion
We have considered the 5D supersymmetric SU(4)w gauge theories. The gauge group SU(4)w is broken down to left-right symmetric gauge symmetry SU(2)l x SU(2)r x U(1)x by orbifold boundary conditions, and the model is reduced to supersymmetric left-right model (or Pati-Salam model) with extra U(1)x in 4D. In building the model, we employ a scenario of gauge-Higgs unification suggested in Ref. [6]. In this scenario, the Higgs fields which break electroweak gauge symmetry are unified to the gauge sector in the 4D N = 2 supermultiplet. Furthermore, the 4D Yukawa interactions, which we need in order to give masses to the fermions by Higgs mechanism, arise from gauge interaction in 5D Lagrangian. This is the most interesting feature of this type of gauge-Higgs unification. The smallness of the 4D Yukawa coupling constants for 1st and 2nd generations can be understood by the fermion localization along the 5D coordinate: the left- and right-chiral fermions are separated to the different 4D walls. Since the localization of the fermion gives us a exponential profile with respect to the 5D coordinate, it is easy to realize the smallness of the coupling for the 1st family. The model-building, in which we employ that the Yukawa coupling originates from gauge coupling in 5D, leads us naturally to the world in which the standard gauge group in 4D is unified in larger gauge group in higher dimension.
The usual left-right symmetry, which is embedded in S0(10) gauge symmetry for example, helps us to understand why CKM matrix is close to the identity matrix. Actually, the CKM matrix is exactly same
as identity matrix with only one Higgs bi-doublet in supersymmetric left-right model (only one 10 Higgs in SO(10) model). Furthermore, the quark mass ratios are same between up- and down-type quark, for example, mc/mt = ms/mb, if we have only one Higgs multiplet in the models. Since those things do not match to observation, we have to add at least one more Higgs into the models. It is interesting that two Higgs multiplets are contained in the gauge-Higgs unification scenario. However, in ordinary 4D construction of the left-right models with two Higgs multiplets, we spoil the reason that the CKM matrix is close to identity matrix if the two Yukawa couplings with two different Higgs are general. Thus, many people consider texture assumptions of the Yukawa couplings or flavor symmetries in order to justify the small CKM angles. In our model in this Letter, we do not need such a texture or flavor symmetry to explain the smallness of the CKM angles as we have seen in Section 3. The reason is in the following: the 4D Yukawa interaction which arise from gauge interaction in 5D is flavor diagonal. Though the generation mixing is caused from brane interaction, the brane interaction does not make large mixing angle to the CKM matrix because of the left-right symmetry even if the generation mixing of the bulk hypermultiplets is large for each up- and down-type quarks. Eventually, the CKM matrix is close to identity matrix, as long as the quark masses are hierarchical. The quark mass hierarchy can be obtained from fermion localization. This is one of the interesting feature of our left-right model with gauge-Higgs unification.
One more interesting feature of our SU(4)w model is that particle contents of the zero modes in the matter hypermultiplets are enough and sufficient to construct 4D left-right symmetric model (or Pati-Salam model). We do not have to add brane fields to cancel 4D gauge anomaly. Instead, each fermion family has to have a pair with opposite U(1)x charge to cancel 4D U(1)x gauge anomaly. This leads to the number of families of bulk hypermultiplet to be even number. In order to make family mixing, we have to have at least one family of brane fields for quarks and leptons. The brane fields fold Dirac masses with bulk hypermultiplets and the number of families at low energy is equal to the difference between the number of bulk hypermultiplets and the number of brane superfields. Thus, in the minimal choice of
particle contents, the model with family mixings has 3 families in the weak scale (we obtain the 3-family as the minimal model with family mixing).
Finally, we comment about U(1)R symmetry. The 5D N = 1 supersymmetry corresponds to 4D N = 2 supersymmetry, and the boundary condition breaks the N = 2 supersymmetry down to N = 1 supersymmetry. The U(1)R symmetry can arise as a subgroup of the SU(2)r in N = 2 supersymmetry algebra. We can assign the U(1)R charge to the 4D fields appropriately. The U(1)R symmetry can prohibit the dimension five proton decay operators, and of course, there is no dimension four operator in left-right model.
Acknowledgements
We thank K.S. Babu, N. Haba, J. Lykken and S. Raby for useful discussions. Y.M. acknowledges the warm hospitality and support of the KEK Theory Group during his visit there. This work was supported in part by US DOE Grants # DE-FG030-98ER-41076 and DE-FG-02-01ER-45684.
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