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SCIENCE ^DIRECT6

Physics Letters B 638 (2006) 492-496

www.elsevier.com/locate/physletb

Universality of strength for Yukawa couplings with extra down-type

quark singlets

Katsuichi Higuchia, Masato Senamib, Katsuji Yamamotoc*

a Department of Literature, Kobe Kaisei College, Kobe 657-0805, Japan b Institute for Cosmic Ray Research, University of Tokyo, Kashiwa, Chiba 277-8582, Japan c Department of Nuclear Engineering, Kyoto University, Kyoto 606-8501, Japan

Received 5 April 2006; accepted 1 June 2006

Available online 13 June 2006

Editor: T. Yanagida

Abstract

We investigate the quark masses and mixings by including vector-like down-type quark singlets in universality of strength for Yukawa couplings (USY). In contrast with the standard model with USY, the sufficient CP violation is obtained for the Cabibbo-Kobayashi-Maskawa matrix through the mixing between the ordinary quarks and quark singlets. The top-bottom mass hierarchy mt > mb also appears naturally in the USY scheme with the down-type quark singlets. © 2006 Elsevier B.V. All rights reserved.

PACS: 12.15.Ff; 14.80.-j; 12.60.-i; 12.15.Hh

Keywords: Quark singlets; Cabibbo-Kobayashi-Maskawa matrix; CP violation; E6

The explanation of the masses and mixings of quarks and leptons is one of the fundamental issues in particle physics. Many notable ideas to address this problem have been investigated, including the universality of strength for Yukawa couplings (USY) [1,2]. In the standard model with USY, the nearly democratic quark mass matrices [3] are provided, and the quark masses and the magnitude of the Cabibbo-Kobayashi-Maskawa (CKM) matrix are really reproduced with the suitable USY phases. However, the USY scheme seems to confront some difficulties within the context of the standard model. Some reasonable explanation should be presented for the topbottom mass hierarchy mt ^ mb; it is simply attributed to the hierarchy of the Yukawa couplings between the up and down sectors with one Higgs doublet, or a large ratio of the vacuum expectation values (VEV's) of two Higgs doublets. More seriously, it is quite difficult to obtain the sufficient CP violation for the CKM matrix in the standard model with USY [1,4], which

* Corresponding author. E-mail address: yamamoto@nucleng.kyoto-u.ac.jp (K. Yamamoto).

is essentially due to the fact that the USY phases are small to provide the quark masses except for the third generation.

In this Letter, we present a new look at the USY scheme by including exotic ingredients. Specifically, we investigate an extension of the standard model with extra down-type quark singlets [5-8]. The standard model contains three generations of the ordinary quarks, left-handed doublets qiL = (uiL, dil)T and right-handed singlets uiR, diR (i = 1, 2, 3), and a Higgs doublet H. In addition, ND vector-like down-type quark singlets DaL and DaR (a = 4,..., 3 + ND) and a Higgs singlet S are included [7,8], which may be accommodated in E6-type models. We will show that the actual quark masses and CKM matrix are indeed obtained in the USY scheme with extra down-type quark singlets. In particular, through the d-D mixing the sufficient CP violation for the CKM matrix is provided from some large USY phases of the Yukawa couplings with the Higgs singlet S. (This mixing mechanism to transmit the CP violation is considered in Refs. [6,7].) The top-bottom hierarchy mt ^ mb also appears naturally in the USY scheme (or more generally flavor democracy) due to the existence of extra down-type quark singlets but no such up-type quark singlets as in the 27 of E6.

0370-2693/$ - see front matter © 2006 Elsevier B.V. All rights reserved. doi:10.1016/j.physletb.2006.06.002

The Yukawa couplings of quarks and Higgs fields with USY are given by

Ly = -qiLAjujRH - qtLAduDjr^H*) - DaLA^jDjrS + H.c.,

Au = xuuJ^U

-e r j,

Ad ei<fifj AD- XRj*.

n ' ■'-'-aJ

XR e'^Dj 3

3 'J 3

where J = j,b with Dj = dj and Db = Db. The respective types of Yukawa couplings are specified with the strengths Xu, Xd, XD and USY phases , $dJ, . The couplings DaLDJRS* are excluded here for definiteness if S is a complex field. This is really the case for the supersymmetric model with a pair of Higgs doublets. The quark mass matrices are given from Eq. (1) as

Mu = (Xv/'i)^), Mv = (Xv/^^J ,

where (H= v, (S) = vs (the possible phase is absorbed by ), and k = vs/v. We investigate the case of Xu = Xd = XD = X for definiteness, while the result is readily extended for different Xu, Xd, XD.

We first consider the up-type quark mass matrix

Mu = Mu{°) + AMu = i 1 1 lj + AMu,

where the perturbation part is given as AMu ~ ii>u with the small USY phase matrix (<&u)ij = $u. Henceforth the quark mass terms are presented to be dimensionless measured in unit of Xv/3 (~ mt/3). The up-type quark mass matrix is relevantly expressed in the hierarchical basis by making a suitable transformation [1,2]:

Mu = UlMuUqIu ~ diag(0, °, 3) + i$uIu,

where <J>u = ul$uUq, Uq = U{1)[12]U(V2)[23], Iu = diag(-i, -i, 1), and "diag" represents a diagonal matrix. The unitary transformation U(a)[IJ] between the Ith and Jth quarks is specified with a 2 x 2 matrix

U(a) =

2 v -a*

supplemented with the right dimension, 3 x 3 for the up sector and {3 + Nd) x {3 + ND) for the down sector. We note here that by suitably choosing the phases of qiL's and ujR's, the USY phases are taken in general as $u3 = -u - and = - ^, giving {<Pu)i3 = {<Pu)3j = 0. In this USY

phase convention, the pre-factor i for <i>u is practically removed with Iu, and the up-type quark mass matrix in Eq. (4) is given as

/<Pu1 <pu2 °N

Mu = VuL diag {mu ,mc,mt)VuR ~ I $c2 °

\ 0 0 3/

with $uij = (@u)ij (u 1 = u, u2 = c, u3 = t).

The quark mass hierarchy mu ^ mc ^ mt for the nearly democratic Mu in Eq. (3) is understood in terms of the sequential

breakings of the permutation symmetry sqL among the left-handed quark doublets [9]:

Sql ^ Sq2l ^ non-

The democratic and S^l invariant Mu (0) provides the top mass. Then, for the USY phases in Eq. (5) the S^l invariant terms $cj and the small S^l breaking ones $uj provide the charm and up masses, respectively, as

'mc » \$uj \ ~ mu

with (VuL) 12 ~ m u/mc « 1 (Vul ^ 1).

We next investigate the down sector including two singlet D's, while the essential features are valid for ND > 2. The USY scheme with only one D is, however, unsatisfactory, still providing the too small CP violation for the CKM matrix. This is because the USY phases in AD with the Higgs singlet S are all eliminated away by rephasing DJR's. Then, the remaining USY phases should be small to provide the ordinary quark masses just as in the standard model with USY.

The down-type quark mass matrix is given as

MD = MD {0) + AMD .

The main part has a quasi-democratic form

Mv {0) =

1 1 1 1 1

1 1 1 1 1

1 1 1 1 1

K K K K K

\K K K K k)

with k = vs/v. The remaining part AMd is provided with the USY phase matrix , (®v)u = $dJ and )aJ = . In accordance with Eq. (4) for the up sector, the mass matrix of down sector is transformed as

Md = U(1)j45] UlMV UqU{V3 )[34]U{2)[45]IV

Md A'dD

AdD Mr

where Id = diag(-i, -i, -i, -e-10, 1) with 0 to be fixed below in Eq. (14). The USY phase matrix is transformed in the same way to &x> Id . This transformation respects the SU(2)w x U(1)y gauge symmetry without dL-DL mixing. The main part is given in this basis as

/0 0 0 0 0 \ 0 0 0 0 0 0 0 0 0 v/T5 0 0 0 0 0 -JWKJ

providing four (3 + Nd - 1) zero eigenvalues. Hence, in contrast with the flavor democracy in the standard model, the bottom quark no longer acquires so a heavy mass as the top quark. This reasonably explains the top-bottom hierarchy mt > mb in the USY scheme. It is also noticed that one (ND - 1) D should obtain a mass from the USY phases as well as the ordinary d's.

The USY phases in Ad with the Higgs doublet H are supposed to be small to provide the ordinary quark masses and

M> {0) =

mixings. On the other hand, those in AD with the Higgs singlet S may be large to provide the significant CP violation for the CKM matrix through the d-D mixing. It is convenient here to make <DJJ = 0 by rephasing VJR's. We may also take for simplicity ^ <D4 ^ 0 under the approximate SpR among V1r-V4r together with the rephasing of V4l (though not essential for the desired CP violation). That is, in this convention

K\, KI «

~ 1. = 0.

The submatrix MD in MAd is given as

md — ■ K '|A| A

10 + A

A = |A|e = exp[i<4D5] - 1 (14)

with 0 = [«4D5| + n)/2] Sign[04D5]. Then, the masses of the heavy quarks, almost the singlets, are given as

mD1 ~ (2/^10)| A|k, mD2 ~V10k. (15)

The submatrix Md for the ordinary quarks is given as

Md = Vd? diag(md°),m(0),mb0))yir)t

1 <Pd\ <d2 <d3 \ <s1 <s2 <s3 I (16)

\<b1 <b2 <b3/

with m<d°) ~ mdi (d1 = d, d2 = s, d3 = b), where is the

3 x 3 submatrix of &d. (The pre-factor i is removed for t^

with Id .) In accordance with the up sector, the hierarchical

quark masses and mixings may be reproduced in terms of Sq3L

and Sq2L in Eq. (6) as

^3) ic

|<bj | ~ mb » |<sj | ~ms » ^dj | ~md.

The left-handed mixing Vd(L0) in Eq. (16) is taken as the pre-CKM matrix (Particle Data Group convention) with the vanishing complex phase S^ ~ 0 and the mixing angles 0(j ) ~ mdi/mdj (i < j) from Eq. (17). Then, by including the d-D mixing effects as seen below, the CKM matrix with sufficient CP violation can be reproduced with reasonable values of 0i(j0), which are adjustable in terms of the USY phases. The right-handed mixing Vd(R0), on the other hand, may be absorbed practically into djR's without physical effects.

The d-D mixing terms in Eq. (10) are given as

AdD — k

i^D1d 0D1S ^D1b\ \&D2d <pD2s $D2b)

(-ie i0<dD1 i(PdD2

-i e-i0$sD1 _ i<sD2

-i e-i0<bD1 i<bD2 +v/15

where <Dkdj = (®v)3+k,j and $diDt = (&v)i,3+k. These d-D mixing terms provide certain corrections to Md, which may be evaluated perturbatively as

(SMd)ij — (A'dD)i4(AdD)4j/mDk.

Then, mainly through D1 , significant imaginary parts are provided to Vub and Vtd for the desired CP violation as

lm[Vub ] — Im[Vd — Im[ (Md) 13 /(Md)33~\ 4>dDi 0Dib cos 9

-0.003.

In total, the left-handed mixing VdL for the ordinary diL's is determined as the 3 x 3 submatrix of the unitary matrix to diagonalize the entire Md in Eq. (10) [5-8]. Then, the weak charged current mixing matrix V (CKM matrix) for the ordinary quarks is given (VuL ~ 1) by

V = vJLVd l.

Here, the case of diagonal Md in Eq. (16) (V(L = vjr = 1) may be specifically interesting, where the CKM mixing emerges entirely from the d-D mixing in the hierarchical basis. In this case, Vus, in particular, is estimated as

|Vus | — ■

l(SMd)121

Vu^/Vc^

|mf ) + (&Md)22r l cos91

„(0)

where the relations |m^) + (SMd)22l > | Im[(SMd)22]|, and fe U^sD! | — K&Md)1jUKSMdhj| — |Vub/Vcb| are considered.

The d-D mixing also induces small corrections to the weak neutral currents, which are related to the unitarity violation of VdL [5-8]. We estimate, in particular, |(VjL VdL)33 -

1| — |(Vdl Vjl)33 - 1| — |(AdD)35(A/5)|2/m2D1 + |(AdD)35 |2/

m2D2 — 3/k2, where the correction to (A'dD)34 through the Dir-D2r mixing — | A/51 is included. Then, in order to suppress the correction to Rb for Z ^ bb to be less than 0.1%,

(17) k = vS/v > 50

is required, implying mD1 > 1 TeV with |A| > 0.5. This hierarchy of the VEV's may be realized naturally in some su-persymmetric model with an extra gauge symmetry (c E6) spontaneously broken by (S) = vS. The quark singlet with mD1 ~ 1 TeV may provide a sizable contribution to the neutron electric dipole moment, while the effect on e'/e will be small enough [6,7].

A numerical result is obtained for the CKM matrix with the CP violation angles as

/ 0.9746 0.2240 0.0037 \ |V |=l 0.2239 0.9738 0.0400) , 0.0078 0.0395 0.9986

a = 96.8°

P = 23.6°

Y = 59.6°,

(19) and the rephasing invariant CP violation measure J = 2.81 x 10-5. The USY phases are taken suitably with k = 50; =

3[Uq diag(mu/mt,mc/mt, 0)U[]ij for Mu (tu) in Eq. (5) with Vul = Vur = 1; <f2, <3, <d4 with ^ = 0 for Md (t^) in Eq. (16) with V{(L = V(R) = 1 and (mf /md,mf /ms, mf]/mb) = (1.026,2.067, 1.008); <d5 = WjjD with (4>dD,4>sD,4>bD) = (-0.01,-0.107,0) for AdD; (<D1 ,<D2,

cific structures in the hierarchical basis as

Fig. 1. The CP violation angles j and y of the CKM matrix versus the USY phase $45 are shown, where $dD is taken typically as -0.003 (right), -0.01 (left), -0.02 (middle).

$D3,$44,$45) = (0, 0, 0.0405, -0.0139, 0.911) with = 0 for AdD and MD. The quark masses are obtained as

mu = 3 MeV, mc = 1.25 GeV, mt = 177 GeV, md = 6 MeV, ms = 100 MeV, mb = 4.25 GeV, mD1 = 1.66 TeV, mD2 = 9.18 TeV

with mR1 /mDl — \A\/5 (\A\ = 0.88). The result of a USY phase space scan is also shown in Fig. 1 for the CP violation angles j (lower) and y (upper) versus the USY phase $45. The USY phase values are taken as in the above example. In particular, $dD is taken typically as -0.003 (right), -0.01 (left), -0.02 (middle) with $bD = 0, by considering \$d1 b\ < 0.1 in Eq. (21) with $b3 — mb/(mt/3), and \$sd1 \ — (\Vcb\/\Vub\)\$dD1 \ < 0.1. Then, $sD, $43, $44 ($41 = $42 = 0) and mS0)/ms are adjusted for \ Vus \= 0.224, \ Vcb \= 0.040 and \Vub\ = 0.0037. (The small $bD may be eliminated by rephas-ing D5R, which is almost compensated with a slight shift of $45 by $bD/V3 ~ some degree.) We have found suitable USY phase values to reproduce the quark masses and CKM matrix with j — 210-240 and y — 40°-90°. This range of j is really reproduced by the Particle Data Group convention with Y — ¿13 and \ Vus \, \ Vcb\, \ Vub\ for 012, 023, 013. (Solutions are not found for y < 40° or > 90° with our computation algorithm.) As seen in Fig. 1, $45 takes the maximal value $45 for given $dD, providing y = Y. (We have evaluated Y — 66°-71° and $R5 — 53°-62° for -0.03 < $dD < -0.002.) This corresponds to the condition mS0) + Re[(^Md)22] = 0 in Eq. (23), specifying Y — n - 00 = (n - $45)/2, as verified by calculating VdL roughly with Eq. (20). The mass of the lighter quark singlet is estimated as mD1 ~ 1.6 TeV (k/50) for $45 ~ 50°. These results are valid for the experimentally determined range of \ Vus \, \Vcb\, \Vub\.

The USY structure may be realized just above the elec-troweak scale as in some large extra dimension models [10, 11]. On the other hand, if it is given at a very high unification scale, the robustness under renormalization group should be considered. We note that the Yukawa couplings have the spe-

'0 0 0

k0 0 3

/0 0 0 0 0 \

0 0 0 0 0

0 0 0 0

0 0 0 * *

0 0 0 * VTÔ + *

Av = ■

Here, "*" denotes the dominant terms with the large USY phases for mD 's and the CP violation, while "0" the perturbation ones with the small USY phases for the ordinary quark masses, except for mt, and mixings. By setting the small USY phases to be zero, chiral symmetries U(2)qL x U(2)uR x U(3)dR really appear. In particular, U(2)qL may break as U(2)qL ^ U(1)q 1L ^ non in accordance with Eq. (6). By virtue of these approximate symmetries the above USY structure is almost maintained under the renormalization group evolution. Then, by including the renormalization group corrections, the suitable USY phase values will be found at the unification scale in some reasonable range to reproduce the quark masses and CKM matrix with sufficient CP violation, as investigated so far.

In summary, we have investigated the quark masses and mixings in the USY scheme by including vector-like down-type quark singlets. In contrast with the standard model with USY, the sufficient CP violation is obtained for the CKM matrix through the mixing between the ordinary down-type quarks and quark singlets. Two or more quark singlets are needed to have the relevant large USY phases for the desired CP violation. These quark singlets may have masses ~ TeV, to be discovered in the future collider experiments [12]. We have shown that with rather flexible choices of the USY phase values the actual quark masses and CKM matrix are really reproduced. Then, it is interesting for further investigations to invoke some textures and flavor symmetries for the USY phases so as to derive some predictive relations among the quark masses and mixings. The top-bottom hierarchy mt ^ mb also appears naturally in the USY scheme in the presence of extra down-type quark singlets but no extra up-type quark singlets. Furthermore, in the USY scheme (or more generally flavor democracy), the fermion mass hierarchy may be extended as mt ^ mb ~ mT if vector-like lepton doublets are also present. In E6-type models, such down-type quark singlets and lepton doublets are indeed accommodated in the 27 representation.

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