PHYSICAL REVIEW D 67, 077301 ~2003!

Hierarchy and up-down parallelism of quark mass matrices

Jian-wei MeiInstitute of High Energy Physics, Chinese Academy of Sciences, P.O. Box 918 (4), Beijing 100039, China

Zhi-zhong Xing*CCAST (World Laboratory), P.O. Box 8730, Beijing 100080, China

and Institute of High Energy Physics, Chinese Academy of Sciences, P.O. Box 918 (4), Beijing 100039, China†

~Received 28 January 2003; published 21 April 2003!

In view of the quark mass hierarchy and the assumption of up-down parallelism, we derive two phenom-enologically favored patterns of Hermitian quark mass matrices from the quark flavor mixing matrix. Wecompare one of them with two existingAnsatzeproposed by Rosner and Worah and by Robertset al., and findthat only the latter is consistent with the present experimental data.

DOI: 10.1103/PhysRevD.67.077301 PACS number~s!: 12.15.Ff, 12.10.Kt

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The masses and flavor mixing angles of six quarksfree parameters in the standard electroweak model, butvalues can directly or indirectly be determined from a nuber of experiments@1#. The fact that three quark masseseach~up or down! sector perform a strong hierarchy is a bpuzzle to particle physicists; so is the hierarchy of thquark mixing angles. Theoretical attempts to solvepuzzle, e.g., those starting from supersymmetric grand ucation theories or from superstring theories@2#, are encour-aging but have not proven to be very successful. Phenenologically, the common approach is to find out simptextures of quark mass matrices, from which a bridgetween the hierarchy of quark mixing angles and that of qumasses can naturally be established@3#. The flavor symme-tries hidden in such textures might finally provide us wuseful hints towards the underlying dynamics responsiblethe generation of quark masses and the origin ofCP viola-tion.

The Cabibbo-Kobayashi-Maskawa~CKM! quark mixingmatrix V arises from the mismatch between the diagonalition of the up-type quark mass matrixMU and that of thedown-type quark mass matrixMD @4#. Without loss of gen-erality, one may arrangeMU andMD to be Hermitian in theframework of the standard model or its extensions whhave no flavor-changing right-handed currents@5#. There ex-ist two reciprocal ways to explore possible relations betwthe hierarchy ofV and that ofMU andMD :

~a! Taking account of a specific texture of the Hermitiquark mass matrixMU or MD , which might result from anew flavor symmetry and its explicit breaking or rely osome plausible theoretical arguments, one may do the untransformation

OU† MUOU5S du 0 0

0 dc 0

0 0 d t

D , OD† MDOD5S dd 0 0

0 ds 0

0 0 db

D , ~1!

whered i stands for the mass eigenvalue of thei quark. Thenthe CKM matrix V is given asV5OU

† OD , from which the

*Electronic address: xingzz@mail.ihep.ac.cn†Mailing address.

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quark mixing angles can be expressed in terms of the raof quark masses~and other unfixed parameters ofMU andMD). This normal approach has extensively been discussin the literature@3#.

~b! Taking account of the experimentally allowed patteof V and making some proper assumptions to decomposVinto V5OU

† OD ~e.g., the principle of ‘‘naturalness’’ disfavorany severe fine-tuning of or delicate cancellations betwrelevant parameters in the unitary transformation matriOU and OD), one may construct the Hermitian quark mamatricesMU andMD through

MU5OUS du 0 0

0 dc 0

0 0 d t

D OU† , MD5ODS dd 0 0

0 ds 0

0 0 db

D OD† . ~2!

This reverseapproach has also attracted some attention inliterature@6#.

In this Brief Report, we present a further study of threlationship between the CKM matrix and the quark mamatrices by following approach~b!. Our main goal is to de-rive the phenomenologically favored textures of Hermitiquark mass matrices from the CKM matrix. We are guidby the hierarchy of quark masses, the hierarchy of the CKmatrix elements and the aforementioned principle of natuness. The only assumption that we need to make is the stural parallelism between up- and down-quark mass maces. Such an empirical assumption should essentially beif two quark sectors are governed by the same dynamics.arrive at two explicit textures of quark mass matrices, whwere not obtained in Ref.@6#. One of them is particularlyinteresting, as its form is somehow similar to theAnsatzeproposed by Rosner and Worah@7# and by Robertset al. @8#.A careful comparison shows that our pattern of quark mmatrices is compatible with theAnsatzadvocated in Ref.@8#,while the Rosner-Worah pattern is not favored by the presexperimental data.

The hierarchy of quark flavor mixing can clearly be sefrom the Wolfenstein parametrization of the CKM matrV @9#:

©2003 The American Physical Society01-1

V'

12 12 l22 1

8 l4 l pl3

2l 12 1 l22 1 ~114A2!l4 Al2 ~3!

BRIEF REPORTS PHYSICAL REVIEW D67, 077301 ~2003!

S 2 8

ql3 2Al21 12 ~2q2A!l4 12 1

2 A2l4D

ta

e

ca

ofs

ct

ca

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t

wherep[A(r2 ih), q[A(12r2 ih), and the terms of orbelow O(l5) have been neglected. Current experimendata yieldl'0.22, A'0.83, r'0.17, andh'0.36 @10#.Thus we have

upu5AAr21h2'0.33, uqu5AA~12r!21h2'0.75.~4!

We find thatupu/uqu'2l holds anduVubu5upul3 is actuallyof O(l4).

To construct the Hermitian quark mass matricesMU andMD from V through Eq.~2!, the key point is to determine thunitary transformation matricesOU andOD . As the physicalquark masses (mu ,mc ,mt) or (md ,ms ,mb) approximatelyperform a geometrical hierarchy at a common energy s~e.g., the electroweak scalem5MZ @3#!,

mu/mc;mc/mt;z2, md/ms;ms/mb;l2, ~5!

wherez is of O(l2) and can be defined asz[kl2 with k;O(1), we areallowed to expand the matrix elementsMU or MD in powers ofz or l. Provided two quark sectorare governed by the same dynamics,MU and MD are thenexpected to have approximately parallel structures charaized respectively by the perturbative parametersz and l.This up-down parallelism would be exact, if the geometrirelations in Eq.~5! held exactly~i.e., mu /mc5mc /mt5z2

and md /ms5ms /mb5l2). Note that the parallelism betweenMU andMD implies the parallelism betweenOU andOD , whose textures can respectively be expanded in teof z andl. Since the CKM matrixV5OU

† OD deviates fromthe unity matrix only at theO(l) level, bothOU and ODmust be close to the unity matrix in the spirit of naturalneIt is therefore instructive to expandOU andOD as follows:

OU5I 1 (n51

`

~ZnUzn!, OD5I 1 (

n51

`

~ZnDln!, ~6!

where I stands for the unity matrix, andZnU or Zn

D ~for n51,2,3, . . . ) denotes the coefficient matrix ofzn or ln.While Zn

U;ZnD should in general hold, the simplest pos

bility is ZnU5Zn

D([Zn), equivalent to the exact parallelismbetweenOU andOD . In this especially interesting case, thexpressions ofZ1 , Z2 , Z3, andZ4 can concretely be determined from Eqs.~3! and ~6!:

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Z15S 0 1 0

21 0 0

0 0 0D , Z25S 2 1

2 k 0

2k 2 12 A

0 2A 0D ,

Z35S 2k 0 p

0 2k 0

q 0 0D ,

Z45S 2 18 ~114k2! 2 1

2 k~112k2! Ak

12 k~122k2! 2 1

8 ~114k214A2! Ak2

0 2Ak21 12 ~A22p* ! 2 1

2 A2D .

~7!

We see that the complexCP-violating phase entersOU orOD at the level ofO(z3) or O(l3). As a straightfowardconsequence ofz;O(l2), the contribution ofOD to V isdominant over that ofOU to V.

With the help of Eqs.~6! and ~7!, we are now able toderive the hierarchical textures ofMU andMD from Eq.~2!.Note that the quark mass eigenvalues (du ,dc ,d t) and(dd ,ds ,db) in Eq. ~2! may be either positive or negativeWithout loss of generality, we taked t5mt anddb5mb . Theother four mass eigenvalues can be expressed, in view of~5!, as follows:

du5r uz4mt , dc5r cz2mt , ~8!

dd5r dl4mb , ds5r sl2mb , ~9!

whereur uu;ur cu;ur du;ur su;O(1) holds. Then we arrive athe results ofMU andMD :

MU'mtS ~r u1r c!z4 r cz

3 pzz3

r cz3 r cz

2 Az2

pz* z3 Az2 1D ,

MD'mbS ~r d1r s!l4 r sl

3 pll3

r sl3 r sl

2 Al2

pl* l3 Al2 1D , ~10!

where

r c[r c~11kz!, r s[r s~11kl!, ~11!

pz[p1kAz, pl[p1kAl. ~12!

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BRIEF REPORTS PHYSICAL REVIEW D67, 077301 ~2003!

We see that the parallelism betweenMU andMD in Eq. ~10!is not exact~e.g.,pz'p5” pl), even though the exact paralelism betweenOU andOD has been assumed in our calclations.

Note that the sign uncertainties of quark mass eigenvamay lead to two distinct textures ofMU and MD . If r u ~orr d) and r c ~or r s) have the same sign, the~1,1! element ofMU ~or MD) amounts approximately to 2r cz

4 ~or 2r sl4). In

this case, we obtain

MU'mtS 2r cz4 r cz

3 pzz3

r cz3 r cz

2 Az2

pz* z3 Az2 1D ,

MD'mbS 2r sl4 r sl

3 pll3

r sl3 r sl

2 Al2

pl* l3 Al2 1D . ~13!

If the signs ofr u ~or r d) and r c ~or r s) are opposite to eachother, however, a very significant cancellation must appbetween them.1 In this case, the~1,1! elements ofMU andMD are expected to be of or belowO(z5) and O(l5), re-spectively. Then we are led to a somehow simpler texturequark mass matrices

MU'mtS 0 r cz3 pzz

3

r cz3 r cz

2 Az2

pz* z3 Az2 1D ,

MD'mbS 0 r sl3 pll3

r sl3 r sl

2 Al2

pl* l3 Al2 1D . ~14!

The textures of Hermitian quark mass matrices in Eqs.~13!and ~14! result naturally from the quark mass hierarchy athe up-down parallelism. Therefore they can be regardetwo promising candidates for the ‘‘true’’ quark mass matricin an underlying effective theory of quark mass generationlow energies.

Now let us compare the texture of quark mass matriobtained in Eq.~14! with two existingAnsatzeproposed byRosner and Worah~RW! in Ref. @7# and by Roberts, Ro-manino, Ross, and Velasco-Sevilla~RRRV! in Ref. @8#. Notethat quark mass matrices of both RW and RRRV formssymmetric instead of Hermitian. Their consequencesquark flavor mixing andCP violation are essentially unchanged, however, if the Hermiticity is imposed on them@8#.For this reason, we consider the Hermitian versions ofRW and RRRVAnsatzeand compare them with Eq.~14!.

1This cancellation does not involve two different quark sectothus it has no conflict with the principle of naturalness mentionbefore.

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The RW ansatz of quark mass matrices is based ocomposite model of spin-1/2 particles@7# and its Hermitianform can be written as

MU5S 0 A2aU aU

A2aU* bU A2bU

aU* A2bU gU

D ,

MD5S 0 A2aD aD

A2aD* bD A2bD

aD* A2bD gD

D , ~15!

whereaU and aD are complex parameters so as to accomodate the observed effects ofCP violation in the quarksector@1#. Comparing between Eqs.~14! and ~15!, we findthat the latter could basically be reproduced from the formif the following conditions were satisfied:

u r cuupzu

'A

ur cu'A2,

u r suuplu

'A

ur su'A2. ~16!

In view of Eqs.~8! and~9! as well as Eqs.~11! and~12!, weimmediately realize that the relations in Eq.~16! are impos-sible to hold. Hence the RW texture is actuallyincompatiblewith ours given in Eq.~14!. Does this incompatibility implythe disagreement between the RWAnsatzand current experi-mental data? We find that the answer is affirmative. Since~1,2! and~1,3! elements ofMU or MD in Eq. ~15! are com-parable in magnitude, we are led to the prediction

uVtdu'uVubu;~1/A2!l3 ~17!

in the leading-order approximation. Such a result is obously inconsistent with the present experimental data, whrequireuVubu;O(l4) anduVub /Vtdu'upu/uqu'2l. Thus theRW texture of quark mass matrices, no matter whether iHermitian or symmetric, is no longer favored in phenomenology.

The Hermitian RRRV ansatz of quark mass matrices@8#takes the form2

MU5mtS 0 bUeU3 cUeU

4

bU* eU3 eU

2 aUeU2

cU* eU4 aU* eU

2 1D ,

MD5mbS 0 bDeD3 cDeD

4

bD* eD3 eD

2 aDeD2

cD* eD4 aD* eD

2 1D , ~18!

whereeU'Amc /mt andeD'Ams /mb stand respectively forthe expansion parameters of up- and down-quark sectorsthe remaining parameters (aU ,bU ,cU) and (aD ,bD ,cD) are,d

2This texture is similar to the one discussed in Ref.@11#.

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BRIEF REPORTS PHYSICAL REVIEW D67, 077301 ~2003!

all of O(1) in magnitude. From Eqs.~8! and ~9!, we obtainthe relation betweeneU ~or eD) andz ~or l) as follows:

eU'zAur cu, eD'lAur su. ~19!

In addition, we find

uaUu'A

ur cu, ubUu'

11kz

Aur cu, ucUu'

upzu

zur cu2, ~20!

uaDu'A

ur su, ubDu'

11kl

Aur su, ucDu'

uplu

lur su2. ~21!

It is obvious that the moduli of (aU ,bU ,cU), and(aD ,bD ,cD) are all ofO(1), consistent with the assumptiomade in Ref.@8#.3 Hence the RRRV ansatz iscompatiblewith our texture of quark mass matrices given in Eq.~14!and favored by current experimental data on quark flamixing.

It is worth remarking that a simplification of the RRRAnsatzto the following four-zero texture:

MU5mtS 0 bUeU3 0

bU* eU3 eU

2 aUeU2

0 aU* eU2 1

D ,

MD5mbS 0 bDeD3 0

bD* eD3 eD

2 aDeD2

0 aD* eD2 1

D ~22!

would give rise to the predictionuVubu/uVcbu'Amu /mc

3ucUu;l/z;1/l may be ;4.5, but it has little impact on theglobal fit of the RRRVAnsatzto current experimental data@8#.

e..,

a,

einsc

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<0.06 @12# for reasonable values ofmu andmc @13#, whichis difficult to agree with the present experimental resuVub /Vcbuex'0.09 @1#. Such a discrepancy implies that thinteresting four-zero pattern of quark mass matrices in~22! might no longer be favored.4

Comparing Eq.~18! with Eqs.~15! and ~22!, we see thatthe ~1,3! elements ofMU andMD should be neither largethan or comparable with their neighboring~1,2! elements norvanishing or negligibly small. This observation is particlarly true forMD , which contributes dominantly to the CKMmatrix V.

We have derived two phenomenologically-favored tetures of Hermitian quark mass matrices from the CKM mtrix. Our starting points of view include the hierarchy oquark masses, the hierarchy of the CKM matrix elementsthe principle of naturalness. The main assumption thathave made is the structural parallelism between up-down-quark mass matrices, which is expected to be trutwo quark sectors are governed by the same dynamics.

We have compared one of the obtained textures withexistingAnsatzeof quark mass matrices, proposed by Rosnand Worah and by Robertset al. It turns out that the RosnerWorahAnsatzis no more favored in phenomenology, whitheAnsatzof Robertset al. is in good agreement with currenexperimental data. We hope that our results for the structuof quark mass matrices may serve as useful guides to mbuilding, from which some deeper understanding of qumasses, flavor mixing, andCP violation can finally beachieved.

This work was supported in part by the National NatuScience Foundation of China.

4Note, however, that the general four-zero texture of Hermitquark mass matrices can still be in good agreement with curexperimental data. See Ref.@14# for detailed discussions.

ys.

ipe,

@1# Particle Data Group, K. Hagiwaraet al., Phys. Rev. D66,010001~2002!.

@2# See, e.g., S. Dimopoulos, L. J. Hall, and S. Raby, Phys. RLett. 68, 1984~1992!; M. Leurer, Y. Nir, and N. Seiberg, NuclPhys.B398, 319 ~1993!; L. E. Ibanez and G. G. Ross, PhysLett. B 332, 100 ~1994!; K. S. Babu and R. N. MohapatraPhys. Rev. Lett.74, 2418~1995!; T. Kobayashi and Z. Z. Xing,Mod. Phys. Lett. A12, 561 ~1997!; Int. J. Mod. Phys. A13,2201 ~1998!; K. Matsuda, T. Fukuyama, and H. NishiurPhys. Rev. D61, 053001~2000!; D. Cremades, L. E. Iba´nez,and F. Marchesano, hep-ph/0212064, and references ther

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@5# H. Fritzsch and Z. Z. Xing, Nucl. Phys.B556, 49 ~1999!, andreferences therein.

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v.

.h

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@7# J. L. Rosner and M. Worah, Phys. Rev. D46, 1131~1992!.@8# R. G. Robertset al., Nucl. Phys.B615, 358 ~2001!.@9# L. Wolfenstein, Phys. Rev. Lett.51, 1945~1983!; A. J. Buras,

M. E. Lautenbacher, and G. Ostermaier, Phys. Rev. D50, 3433~1994!; Z. Z. Xing, ibid. 51, 3958~1995!.

@10# A. J. Buras, hep-ph/0210291, and references therein.@11# G. C. Branco, D. Emmanuel-Costa, and R. Gonzalez Fel

Phys. Lett. B483, 87 ~2000!.@12# See, e.g., D. Du and Z. Z. Xing, Phys. Rev. D48, 2349~1993!;

L. J. Hall and A. Rasin, Phys. Lett. B315, 164 ~1993!; H.Fritzsch and Z. Z. Xing,ibid. 353, 114 ~1995!; 413, 396~1997!.

@13# J. Gasser and H. Leutwyler, Phys. Rep., Phys. Lett.87C, 77~1982!; H. Leutwyler, Phys. Lett. B378, 313 ~1996!.

@14# H. Fritzsch and Z. Z. Xing, Phys. Lett. B555, 63 ~2003!.

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