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PHYSICS LETTERS B

elsevier

Physics Letters B 563 (2003) 209-216

www. elsevier. com/locate/npe

Quarks and leptons between branes and bulk

T. Asakaa, W. Buchmüllerb, L. Covib

a Institute of Theoretical Physics, University of Lausanne, Lausanne, Switzerland b Deutsches Elektronen-Synchrotron DESY, Hamburg, Germany

Received 15 April 2002; accepted 4 May 2003

Editor: P.V. Landshoff

Abstract

We study a supersymmetric S0(10) gauge theory in six dimensions compactified on an orbifold. Three sequential quark-lepton families are localized at the three fixpoints where S0(10) is broken to its three GUT subgroups. Split bulk multiplets yield the Higgs doublets of the standard model and as additional states lepton doublets and down-quark singlets. The physical quarks and leptons are mixtures of brane and bulk states. The model naturally explains small quark mixings together with large lepton mixings in the charged current. A small hierarchy of neutrino masses is obtained due to the different down-quark and up-quark mass hierarchies. None of the usual GUT relations between fermion masses holds exactly. © 2003 Elsevier Science B.V. All rights reserved.

The explanation of the masses and mixings of quarks and leptons remains a challenge for theories which go beyond the standard model [1,2]. In principle, grand unified theories (GUTs) appear as the natural framework to address this question. However, as much work on this topic has demonstrated, all simple GUT relations for fermion mass matrices are badly violated and, within the conventional framework of four-dimensional (4d) unified theories, a complicated Higgs sector is needed to achieve consistency with experiment.

In this Letter we shall address the flavour problem in the context of a supersymmetric S0(10) GUT in six dimensions compactified on an orbifold [3,4]. A new ingredient of orbifold GUTs is the presence of split bulk multiplets whose mixings with complete GUT multiplets can significantly modify ordinary GUT mass relations [5,6]. This extends the well-known mechanism of mixing with vectorlike multiplets [7]. Several analyses of the flavour structure of orbifold GUTs have already been carried out (cf., e.g., [8-12]). In 5d theories large bulk mass terms can lead to a localization of zero modes at one of the two boundary branes, which can explain fermion mass hierarchies [13]. In this way a realistic 'lopsided' structure of Yukawa matrices can be achieved [14].

'Lopsided' fermion mass matrices, mostly based on an abelian generation symmetry [15], have received much attention in recent years (cf. [16-22]). In the context of SU(5) GUTs they introduce a large mixing of left-handed leptons and right-handed down quarks, which leads to small mixings among the left-handed down-quarks. In this way the observed large mixings in the leptonic charged current can be reconciled with the small CKM mixings in the quark current. The mechanism of flavour mixing, which we describe below, is also based on large mixings

E-mail address: takehiko.asaka@ipt.unil.ch (T. Asaka).

0370-2693/03/$ - see front matter © 2003 Elsevier Science B.V. All rights reserved. doi:10.1016/S0370-2693(03)00644-0

OGG[GGG]

'GG I *JGG J

Oi [SO(IO)] Ops [GPS]

Fig. 1. The three S0(10) subgroups at the corresponding fixpoints of the orbifold T2/(Z2 x ZpS x Z?G).

of left-handed leptons and right-handed down quarks. However, these mixings do not respect SU(5) and they are not controlled by a single hierarchy parameter. In this way a different pattern of mixings is achieved with several characteristic predictions for the neutrino sector.

Let us now consider S0(10) gauge theory in 6d with N = 1 supersymmetry compactified on the orbifold T2/(Z\ x ZpS x ZGG) [3,4]. The theory has four fixed points, OI, OGG, Ofl and OPS, located at the four corners of a 'pillow' corresponding to the two compact dimensions (cf. Fig. 1). At OI only supersymmetry is broken whereas S0(10) remains unbroken. At OGG, Ofl and OPS S0(10) is broken to its three GUT subgroups Ggg = SU(5) x U(1)x, flipped SU(5), Gfl = SU(5) x U(1)', and Gps = SU(4) x SU(2) x SU(2), respectively. The intersection of all these GUT groups yields the standard model group with an additional U(1) factor, GSM' = SU(3) x SU(2) x U(1)Y x U(1)X, as unbroken gauge symmetry below the compactification scale. B — L, the difference of baryon and lepton number, is a linear combination of Y and X.

The field content of the theory is strongly constrained by the required cancellation of irreducible bulk and brane anomalies [23]. Motivated by the embedding of all field quantum numbers into the adjoint representation of E8 [24], we have 6 10-plets, H1,...,H6, and 4 16-plets, ,$,$c as bulk hypermultiplets, accompanied by 3 16-plets , i = 1,..., 3, of brane fields. Vacuum expectation values of 0 and 0c break B — L. The electroweak gauge group is broken by expectation values of H1 and H2.

Compared to [24] we have added an additional pair of bulk 16-plets, $ and $c together with two 10-plets, H5 and H6, to cancel bulk anomalies. This is still compatible with the embedding in E8, and it corresponds to the largest number of bulk fields consistent with the cancellation of anomalies. Note that both the irreducible and reducible 6d gauge anomalies vanish.

The parities of H5, H6 and $ are listed in Table 1. $c has the same parities as $. The corresponding zero modes

The zero modes of the fields 0, 0c, H1 ,...,H4 are given in [24]. They are the color triplets and singlets Dc, Nc,

D, N, H^, H2, G3 and G4.

Fermion masses and mixings are determined by brane superpotentials. The allowed terms are restricted by R-invariance and an additional U(1)x symmetry [24]. The corresponding charges of the superfields are given in Table 2. The fields H1, H2, 0 and @c, which acquire a vacuum expectation value, have vanishing R-charge. All matter fields have R-charge one. Since and $ have the same charges we combine them to the quartet (fa) = (fi,$), a = 1,..., 4. The most general brane superpotential up to quartic terms is then given by

Table 1

Parity assignments for the bulk hypermultiplets H5, H6 and $

S0(10)

(1, 2, 2)

5*—2

(1,2,2)

(6, 1, 1)

5*—2

Z2PS Z

(6,1,1) 5+2 G

Z2 Z + — + +

S0(10) GPS

(4,2,1) 10—1

Z2PS Z

(4,2,1) 5*+3 L

(4*, 1,2) 10—1

(4*, 1,2) 5*+3,1—5

Table 2

Charge assignments for the symmetries U(1) r and U(1)y

H\ H2 0c H3 <p H4 ti 4>c 4> H5 H6

R 0 0 0 2 0 2 1 1 1 1 1

X —2a —2a —a 2a a —2a a —a a 2a —2a

W = MdH5H6 + Mlafa<pc + MuIh Ih + M2:JI2Ih + jh^irpHi + + fa<PiraH6

+ fc<Pc<t>cH5 + fD<Pc<PcH3 + f°<P<PH4 + + + ^-HrH2H2

2 M* M* M*

k3 k4 k5 2d 2U

M* 25 M* M* M* M*

gd gu kd kl kl

+ jfWHM + ^H5H2 + + + W, (2)

where we choose M* > 1/R5,6 ~ ^GUT to be the cutoff of the 6d theory, and the bulk fields have been properly normalized. All the volume factors due to the 6d fields are absorbed into the unknown couplings and we will not use them to explain the hierarchies. When the bulk fields are replaced by their zero modes only 9 of the 23 terms appearing in the superpotential remain. Although we have written the superpotential in terms of S0(10) multiplets, on the different branes the Yukawa couplings h(1) and h(2) split into h(d),h(e) and h(u\h(D\ respectively. Some of these couplings are equal due to GUT relations on the corresponding brane.

The main idea to generate fermion mass matrices is now as follows. We consider the case that the three sequential 16-plets are located on the three branes where S0(10) is broken to its three GUT subgroups. As an example, we place at OGG, f2 at Ofl and at OPS. The three 'families' are then separated by distances large compared to the cutoff scale M*. Hence, they can only have diagonal Yukawa couplings with the bulk Higgs fields. Direct mixings are exponentially suppressed. However, the brane fields can mix with the bulk zero modes for which we expect no suppression. These mixings take place only among left-handed leptons and right-handed down-quarks. This leads to a characteristic pattern of mass matrices which we shall now explore.

If B — L is broken, as discussed in [24], (0c) = (<P) = vN, and the bulk zero modes Nc, N, (D, Gc) and (Dc,G) acquire masses O(vN). After electroweak symmetry breaking, with (Hf) = v1, (H2) = v2, the remaining

states have the following mass terms,

W = damdaßdß + e°aKß

+ n>aßVß + uCimtjuj + ~nCiMijnCj-

Here m

are 4 x 4 matrices,

(Kv 0 0

0 hd22V1 0 st ft*

0 0 h33 V1

V /1 VN /2 VN /3 VN Md /

(hdnV1 0 0 h14VA

0 h22V1 0 h24V1

0 0 h33 V1 h34 V1 ,

K M[ m2 M3 m4 /

( hD1V2 0 0 hD4V2^

0 h2u2V2 0 h2D4V2

0 0 hU V2 h3D4V2

V m1 m2 M3 m4

whereas mu and mN are diagonal 3 x 3 matrices, 0

/h"nV2

tu = o hu12v2 \ 0 0

hU3V2'

/hN hL nn Mt

hN 22 Mt

hN % / n33 Mt '

In the matrices md, me and mD we have neglected subleading corrections O(vN/M*). The diagonal elements satisfy four GUT relations which correspond to the unbroken SU(5), flipped SU(5) and Pati-Salam subgroups of S0(10).

The crucial feature of the matrices md, me and mD are the mixings between the six brane states and the two bulk states. The first three rows of the matrices are proportional to the electroweak scale. The corresponding Yukawa couplings have to be hierarchical in order to obtain a realistic spectrum of quark and lepton masses. This corresponds to different strengths of the Yukawa couplings at the different fixpoints of the orbifold. The fourth row, proportional to Md, Ml and vN, is of order the unification scale and, we assume, non-hierarchical.

The mass matrices md, me and mD are of the form

/11 0 0 ¡11 \

0 12 0 1 2

0 0 13 ¡13

where \i , ji = O(v1,2) and Mi = O{AGUT). To diagonalize the matrix m it is convenient to define a set of four-dimensional unit vectors as follows:

(Mi ...M4) = Me4, eaeß = eayeßy = Saß. Using the orthogonal matrices (a, ß = 1,..., 4, i = 1,..., 3),

1 - J v2

Vaß = (eß)a, Uaß = &aß--—&an{eAHM + e44/Xi)Sßi + 01

M M M M 2

we can now perform a change of basis which yields for the mass matrix,

where the 3 x 3 matrix m is given by

(¡1£T + l iej\

+ ¡i-iel I • (12)

M3eJ + l 3 ej /

Here the three-vectors ea, a = 1,---, 4, are determined by the four-vectors ei, i = 1,---, 3, with (ea)i = (ei)a. Note that m is composed of three row vectors of hierarchical length, a structure familiar from lopsided fermion mass models.

lhe hierarchy of the row vectors suggests to perform a further change of basis such that all remaining mixings are small. Three orthogonal three-vectors in, ejej = ei kejk = Sj, can be defined by writing the matrix m in the following form:

' ¡1(ye1 + eT + pel) \

¡2(eT + aeT) I • (13)

¡3eT /

The parameters ¡i are O(ji, jji) and therefore again hierarchical. With respect to this new basis the matrix m has triangular form:

/ ¡1Y j1 ¡10 \

m = I 0 ¡2 ¡2a I • (14)

\ 0 0 ¡3 /

For our discussion of mass eigenvalues and mixing angles we shall need the two matrices mmT and mTm, which in the basis ei are both hierarchical:

(¡1(1 + p2 + Y2) ¡1j2(1 + ap) ¡1j3P\

j1j2(1 + aP) ¡2(1 + a2) ¡2¡зa I , (15)

¡1l3p ¡2 ¡3a ¡3 /

/ Hy 2 ¡2y HPY I

T™ = I ¡2Y ¡2 + ¡2 ¡2a + ¡2p I • (16)

vi2py ¡2a + ¡2p ¡3 + ¡2a2 + ¡2p 2

Consider now the up-quark mass matrix. We concentrate on the case of large tanp = v2/v1 ~ 50, such that hd3 ~ Wi3. The diagonal elements of the mass matrices (4), (5), (6) and (7) are partially connected by the GUT relations which hold on the different branes. For simplicity, we therefore assume universally,

¡1 : ¡2 : ¡3 ~ mu : mc : mt- (17)

It is well known that the hierarchy of down-quark and charged lepton masses is substantially smaller than the up-quark mass hierarchy. Given the scaling (17) of the diagonal elements and the structure of md and me this implies that the down-quark and charged lepton mass matrices must be dominated by the off-diagonal elements. Hence, we assume again universally,

¡1 < ¡1 ~ ji 1, ¡2 < ¡2 ~ ¡2, ¡3 ~ ¡3- (18)

The parameters ¡1,2 of the matrix m are then dominated by the mixing terms jj 1,2, i.e., ¡1,2 ~ jl 1,2.

Since the up-quark matrix mu is diagonal the CKM quark mixing matrix is given by the matrix V which diagonalizes md(md)T. From Eq. (15) one reads off for the two larger masses

mb ~ jj3, ms ~ jj2, (19)

and for the mixing angles

jj 1 12 \\ 1 V«i = &c~izL, (20)

1 2 1 3 13

Using mb, ms and 0c ~ 0.2 as input one obtains for the two remaining mixing angles

Vcb ~ — ~ 2 x 10"2, Vub~0c— ~4x 10"3, (21)

in agreement with analyses of weak decays [25] up to a factor of two, which is beyond the predictivity of our approach.

The smallest eigenvalue vanishes in the limit \1,i2 ^ 0, since in this case two vectors of the matrix m become parallel, with p = a and y = 0. Choosing, for simplicity, \1 /11 < \2/12, one has for non-zero \1 ,\2, 12 mcmb

Y ~ — ~-~ 0.1. (22)

jj 2 mtms

This relation will also be important in our analysis of the neutrino masses. For the down-quark mass one obtains

md 12 \11 mcmb

— — (23)

ms 12 12 mtms

consistent with data [1].

The charged lepton mass matrix me is very similar to the down-quark mass matrix. The main difference is that now there are large mixings between the 'left-handed' states ei. To obtain the contribution of the charged leptons to the leptonic mixing matrix we consider the matrix (me)Tme as given in Eq. (16) in the basis ei. For the two large eigenvalues of me one has mT ~ jj3 ~ mb and

m\ ~ 12 ~ ms. These relations are consistent with data within our accuracy. A potential problem is the smallness of the electron mass, i.e., me/m\ ~ 0.1md/ms. The smallest eigenvalue of me is again given by me/m1 ~ (\211 /12). However, in our model the usual SU(5) relations do not hold for the second row of the mass matrices. Hence, the electron mass is not determined by down quark masses.

Using the diagonal and off-diagonal elements of the mass matrices as determined from up- and down-quark mass matrices, we can now discuss the implications for neutrino masses. The heavy Majorana neutrinos scale like up-quarks (cf. (7)),

M3 : M2 : M1 ~ mt : mc : mu. (24)

The light neutrino masses are given by the seesaw relation

mv = -(mDf-^mD. (25)

The structure of the charged lepton and the Dirac neutrino mass matrices (cf. (5), (6)) is the same. Both matrices lead to large mixings between the 'left-handed' states. In order to determine the leptonic mixing matrix we discuss the Dirac neutrino matrix in the basis ei where the remaining mixings of the left-handed charged leptons is small by construction (cf. (16)).

The Dirac neutrino mass matrix can be written as (cf. (12)),

/ P1e[ + P1«T\

. (26)

P2eT + p2eT

\P3eT + p3eT /

Here the parameters pi, pi are expected to have the same hierarchy as \i , \i. However, in general these parameters will differ by factors 0(1) since there the entries of me and mD arise from different Yukawa couplings in the superpotential. This implies for the matrix m D, with respect to the vectors ei,

/ Pi(AeT + DeT + e T) \

mD = p2(BeT + EeT + e TT) , (27)

Vp3(ceT + FeT + e T)/

where pi ~ pi. Hence, with respect to the basis \ the matrix mD has no longer triangular form,

(Api mD = Bp2

Dpi pi \

Ep2 -2 . (28)

\Cp3 Fp3 p3 J

Generically, the parameters A, ...,F are all 0(1). All we know is that for \12 = p12 = 0 the first two row vectors are parallel, with A = B = C = 0 and D = E. For \1,2,p1,2 = 0 one has analogous to the charged lepton mass matrix (cf. (22)),

P2 \2 A, B,C ~ — ~ — ~ y ~ 0.1.

From Eqs. (25) and (28) one now obtains for the light neutrino mass matrix,

-m v = (m )

71 Mi + " + L

AD^ + BE^ + CF-j^

-2 -2 -2 /iniiL_|_ ft T7 f2- + C 77 fl.

+ F2P2. , F

Mi M2 ^ r

. ¡¡ft. D M2

.rft ^ M3

£)iL.

. f£2. . c M2

-2 -2 -2 Mi +iiM2 +CM3 -2 -2 -2 ^ Ml ' M2. M3

222 PL , £i_ , fl / Mi M2 M3 7

Using Eq. (29) one immediately sees the order of magnitude of the different entries,

/Y2 Y Y\

mv ~ I Y 1 1 I m3, (31)

\ Y 1 1/

where m3 is the largest neutrino mass, i.e., m1 < m2 < m3. It is well known that such a matrix can account for all neutrino data. It has previously been derived based on a U(1) family symmetry [16,17] and also by requiring a compensation between the Dirac and Majorana neutrino mass hierarchies [26,27].

Consider now the parameters in the matrix (30). The mass matrices md, me and mD have the same structure with large off-diagonal entries. For simplicity, we therefore assume for the mass parameters \ and -i have a similar hierarchy, approximately given by the down-quark masses, i.e., : -2 : -3 ~ md : ms: mb. One then obtains

msmt 2

M2 m2bmc M1 -2

This corresponds to the picture of sequential heavy neutrino dominance [28]. It yields large 2-3 mixing,

sin2©231. The largest neutrino mass is m3 ~ m2/M3. Identifying m3 with ^A«^ ~ 0.05 eV one obtains

for the heavy Majorana masses M3 ~ 1015 GeV, M2 ~ 3 x 1012 GeV and M1 ~ 1010 GeV. The second neutrino mass is m2 ~ 0.01 eV, which is consistent with data within our accuracy.

Since the 2-3 determinant is small the matrix (30) can also account for the LMA MSW-solution of the solar neutrino problem [20]. As all neutrino masses are rather close to each other, with unknown coefficients 0(1), a

precise prediction of the mixing angle 012 and the smallest neutrino mass is not possible. Generically, one has sin2012 ~ Ym3/m2 and m1 = 0(y2m2). On the other hand, a definitive prediction of the matrix (30) is a rather large 1-3 mixing angle, 013 ~ y ~ 0.1.

Decays of the lightest right-handed neutrinos may be the origin of the baryon asymmetry of the universe [29]. In addition to the mass M1 ~ 1010 GeV the relevant quantities are the CP-asymmetry e1 and the effective neutrino mass m 1 = (mD^mD)11/M1. One easily obtains e1 ~ 0.1M1/M3 ~ 10-6 and m 1 ~ 0.2m3. These are the typical parameters of thermal leptogenesis [30].

Starting from three sequential families located at three different fixpoints of an orbifold, we have shown that the mixing with split bulk multiplets can lead to a characteristic pattern of quark and lepton mass matrices which can account for small quark mixings together with large lepton mixings in the charged current. Correspondingly, the quark mass hierarchies are large whereas the small neutrino mass hierarchy follows from the difference of down-quark and up-quark mass hierarchies. The dynamical origin of the hierarchy of Yukawa couplings at the different branes remains to be understood.

Acknowledgements

We would like to thank A. Hebecker and D. Wyler for helpful discussions.

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