Calculability of quark mixing parameters from general nearest neighbor interaction texture quark mass matrices

27 May 1999

Ž .Physics Letters B 455 1999 264–272

Calculability of quark mixing parameters fromgeneral nearest neighbor interaction texture quark mass matrices

David Dooling, Kyungsik KangDepartment of Physics Brown UniÕersity, ProÕidence, RI 02912, USA

Received 8 January 1999; received in revised form 10 March 1999Editor: M. Cvetic

Abstract

We perform an analysis of general quark mass matrices in the general nearest neighbor interaction form. Excellentagreement with experiment is realized with this general texture, which is neither hermitian nor real-symmetric. We thenpropose a new class of quark mass matrices that contain no additional parameters other than the quark masses themselves,and thus possess calculability, i.e., ensure a relationship between the six quark masses and four flavor-mixing parameters ofthe Standard Model. q 1999 Published by Elsevier Science B.V. All rights reserved.

PACS: 11.30.Er; 12.15.Ff; 12.15.Hh; 14.65.yqKeywords: Quark mass matrices; Calculability; Texture

1. Introduction

A fundamental explanation of the flavor-mixing matrix, the fermion masses and their hierarchical structurepersists to be one of the most challenging and outstanding problems of particle physics today. Within the

Ž .standard model SM the fermion masses, the three flavor-mixing angles and the CP violating phase are freeparameters and no relation exists among them. However, the expectation that ‘‘low-energy’’ quantities whichcan be computed in the SM should remain finite as the masses of intermediate particles go to infinity leads us to

Ž 0 0.suspect that such a relation does hold. For example, DM K yK diverges as m ™`. Since the contributions L t

of the intermediate quark is always multiplied by a flavor-mixing matrix element, a relation between quarkmasses and flavor-mixing parameters could conspire to guarantee that the contribution to the low-energyquantity always remains finite.

As an attempt to derive a relationship between the quark masses and flavor-mixing hierarchies, mass matrixansatze based on flavor democracy with a suitable breaking so as to allow mixing between the quarks of nearest¨

Ž . w xkinship via nearest neighbor interactions NNI was suggested about two decades ago 1 . These early attemptsare the first examples of ‘‘strict calculability’’; i.e., mass matrices such that all flavor-mixing parameters dependsolely on, and are determined by, the quark masses. But the simple symmetric NNI texture leads to theexperimentally violated inequality M -110 GeV, prompting consideration of a less restricted form for thetop

w xmass matrices so as to still achieve calculability, yet be consistent with experiment 2 .

0370-2693r99r$ - see front matter q 1999 Published by Elsevier Science B.V. All rights reserved.Ž .PII: S0370-2693 99 00401-3

( )D. Dooling, K. KangrPhysics Letters B 455 1999 264–272 265

It was later shown that the texture structure of these early ansatze for quark mass matrices, the texture¨structure of the NNI mass matrices, holds in general. Branco et al. have demonstrated that if one does not

w ximpose the assumption of hermiticity, then the NNI texture structure contains no physical assumptions 3 . Forthree fermion generations, one may consider without loss of generality quark mass matrices of the NNI form.This texture structure serves as the general starting point from which additional constraints on the mass matrixelements can be imposed in order to achieve calculability.

In this paper we analyze general quark mass matrices in a modified NNI form to be described in the nextsection. This form is free from physical content, as it corresponds to a particular choice of weak basis in theright-handed chiral quark sector. We perform a numerical fit to the most recent measurement of the

Ž .Cabibbo-Kobayashi-Maskawa CKM matrix, V . The results are in excellent agreement with the observ-CKM

ables. Finally, we propose a new mass matrix ansatze based on our numerical results, to ensure some relation¨among the six quark mass masses and the four observable flavor-mixing parameters, thus reducing the numberof free parameters in the SM.

Our paper is organized as follows. In Section 2 we review the NNI form of mass matrices, pointing out asubtlety that has been overlooked in previous works. In Section 3, we present the quark mass matrices to beanalyzed as well as the resulting flavor-mixing matrix. We present the observables used in our fitting procedure

Ž .as well as the resultant unitary triangle UT parameters. Guided by the numerics of the general case, wepostulate quark mass matrices that ensure calculability.

2. General NNI mass matrices

In a given generic gauge theory, flavor eigenstates do not necessarily coincide with the mass eigenstateswhich can be written in terms of combinations of flavor eigenstates. The mass matrix in a given electric chargecharge sector is not necessarily diagonal in flavor space and involves the coupling between the left-handed andright-handed chiral states of different flavor. It can be shown that in the SM with three generations of fermions,one may perform a biunitary transformation of the quark mass matrices that leaves both the quark massspectrum and the flavor-mixing parameters unchanged, where the new mass matrices in both the up-type and

w xdown-type quark sectors are of the NNI form and are in general neither hermitian nor symmetric 3 . We stressthat this form is merely the consequence of choosing a particularly convenient weak basis that allows us toeliminate some of the redundant free parameters of the SM and has no physical consequences.

Starting from initial completely general complex matrices M X , a necessary condition for this biunitaryŽu,d .transformation to yield the NNI form is the existence of a solution to the eigenvalue equation

M X M X† qkM X M X† U slU 1Ž .Ž . jiu u d d i2 j2

where k is initially assumed to be an arbitrary complex constant. The NNI mass matrices are then given by

0 A eiu 1 012

iu iu2 3A e 0 A eM s 2Ž .21 23u � 0iu iu4 50 A e A e32 33

0 B eiu 1X

012X X

iu iu2 3B e 0 B eM s 3Ž .21 23dX X� 0iu iu4 50 B e B e32 33

M sU †M X V 4Ž .Žu ,d . Žu ,d . Žu ,d .

where explicit forms for U and V , corresponding to the left-handed and right-handed chiral quark rotations,Žu,d .w xrespectively, are given in 3 . Note that U, the left-handed chiral quark rotation matrix, is common to both the

( )D. Dooling, K. KangrPhysics Letters B 455 1999 264–272266

up and down quark sectors. It is the freedom of performing an unitary transformation on the triplet ofright-handed quark fields in both the up and down quark sectors, i.e. the freedom associated with the V ,Žu,d .without affecting the quark masses or the flavor-mixing parameters, that enables this texture structure to be

Ž .possible while still remaining completely general. We can rewrite Eq. 1 for arbitrary complex k as

A2 qA2 qk B2 qB2 slŽ .21 23 21 23

k cos b sin a ysin b cos a yk sin b sin a qcos b cos a s0Ž . Ž . Ž . Ž . Ž . Ž . Ž . Ž .real imag

where b'uX yu

X and a'u yu . The first equation exhibits the functional dependence of l on k and5 3 5 3

provides no restriction on k. It may appear that when choosing a weak-basis that gives NNI forms for the quarkmass matrices, one has two arbitrary degrees of freedom corresponding to k and k . But in fact thisreal imag

two-fold degree of freedom does not exist; k must be either purely real or purely imaginary.Ž . Ž . Ž .For both k and k nonzero, we arrive at the inconsistency tan b sycot b . Inspection of Eq. 1real imag

Ž .shows that if k is purely real, we are guaranteed a solution to 1 because the operator on the left hand side isHermitian and has three linearly independent eigenvectors. However, this guarantee does not hold if k is purelyimaginary. Therefore, we arrive at the condition k is purely real, which results in asb. Thus the number ofparameters in our flavor-mixing matrix is reduced to eleven from twelve.

The above mass matrices with their texture structure and the relationship asb are the most general onesthat exhibit the NNI texture and have the fewest number of independent parameters. At this stage no physicalinputs have been introduced, and the resulting flavor-mixing matrix has five more parameters than are necessaryto achieve strict calculability.

3. Flavor-mixing matrix and numerical fit to observables

We use the rephasing freedom of the quark-fields to minimize the number of parameters entering into theŽ .quark flavor-mixing matrix. We can re-write Eq. 2

i u yu quŽ .i u yuŽ . 0 A 0 5 3 21 4 e 0 012e 0 0 &iŽu .A 0 AiŽu yu . 4M s 'P M P3 5 21 23 0 e 0u Žu.L u Žu.R0 e 0� 0 � 0� 0 iŽu .50 A A0 0 1 0 0 e32 33

5Ž .with an analogous expression for M , and where u

X yuX su yu . The flavor-mixing matrix is written ind 3 5 3 5

terms of the unitary matrices U and U that diagonalize M M † and M M †:Žu.L Žd .L u u d d

U M M †U † 'diag m2 ,m2 ,m2 , U M M †U † 'diag m2 ,m2 ,m2Ž . Ž .Žu.L u u Žu.L u c t Žd .L d d Žd .L d s b

From the above expressions for M , we see thatu,d

&&T† †M M sP M M P 6Ž .u u L u u L

&&TBecause M M is a real symmetric matrix, it can be diagonalized by a real orthogonal matrix R :u u u

&&T T 2 2 2R M M R sdiag m ,m ,m 7Ž .Ž .u u u u u c t

The flavor-mixing matrix can then be expressed as

V sU U † sR P † P RT 8Ž .fym Žu.L Žd .L u Žu.L Žd .L d

≠ ≠ ≠§ z ™ iuu e 0 0z z zV s 9Ž .§ z ™ 0 1 0 d s bfym c � 0� 0 � 0§ z ™ 0 0 1 x x xt


&&† 2 2 2where the z are the normalized eigenvectors of M M with eigenvalues m ,m and m and similarly for theu,c, t u u u c tw x X X

z 4–6 . Note that only this single phase u'u yu yu qu enters the flavor-mixing matrix and isd, s,b 1 4 1 4

responsible for CP violation. Thus, it is sufficient for only one of the four non-vanishing elements coming fromthe second columns of M and M to be complex in order to provide a general parametrization of flavor-mixingu d

and CP violation. For example, one may choose u to be non-zero while all other phases vanish. At this stage,1

the mass matrices contain eleven independent parameters.We must specify the energy scale at which we are evaluating the mass matrices. We use the light quark

w xmasses 7,8 m s4.9"0.53 MeV, m s9.76"0.63 MeV, and m s187"16 MeV, and the heavy quarku d s

masses m s1.467"0.028 GeV, m s6.356"0.08 GeV and m s339"24 GeV all of which correspond toc b tŽ .the masses at a modified minimal subtraction MS renormalization point of 1 GeV. Choosing the energy scale

˜ ˜ Tto be 1 GeV we use the invariance of the trace, minor determinants and determinant of M M under auŽd . uŽd .˜ 2similarity transformation to relate the M to the m in each quark sector. These six relations comprise theuŽd . i j i

first six terms of our x 2 .totw x < < < <In addition, we fit to the following V observables 9 : V s0.9740"0.001, V s0.2196"0.0023,CKM ud u s

< <Vu b< < < < < <V s0.224"0.016, V s1.04"0.16, V s0.0395"0.0017, and s0.08"0.02.cd cs cb< <Vcb

Our method is to fit to the experimental data and compute the corresponding x 2 'Ý12 x 2 with one degreetot is1 i

of freedom arising from twelve experimental values minus eleven independent input parameters.The parameters listed in Table 1 yield a x 2 of only 0.180.tot

The single phase u that minimizes x 2 is y0.716"0.0989 radians. The above parameters also maketot

predictions concerning the mixing of the top quark as follows:

< < 2Vtb)< <s0.9984, V V s0.009422t b td2 2 2< < < < < <V q V q Vtd t s tb

< < < <and V r V s0.2479. The first two predictions above are in excellent agreement with the latest Particle Datat d t s< <Vt dw x w xGroup values 9 , while our value of is slightly higher than that predicted in 10,11 .< <Vt s & &

The numerically determined mass matrices M ,M from above display some interesting properties worthu d< <noting at this point. One can immediately see that they are far from being symmetric, as A differs markedly12

< <in magnitude from A , etc. Of particular interest is the difference between the up and down quark sectors21

vis-a-vis the 3-3 element. Historically, mass matrix ansatze previously proposed in the spirit of calculability¨< < < <have written the 3-3 element as a small perturbation about m ; i.e. as m ye , where epsilon is some smallt,b 3

parameter. Inspection of the above parameters shows that this description works fairly well for the down sector,but is not at all applicable to the up quark sector. In the up quark sector, it is the magnitude of the 3-2 elementthat differs only slightly from m sm . This numerical result and observed structure is a consequence of the top3 t

quark being so heavy.Using just the central values of the quark masses at the 1 GeV scale, the absolute values of the elements of

the flavor-mixing matrix elements are:

0.974007 0.219610 0.00316079< <V s 0.223856 0.975401 0.0395023CKM ž /0.00942966 0.0380247 0.999214

To impose unitarity we need to go through one additional step after varying our input parameters and fittingto the above experimental values. In our flavor-mixing matrix, we have in the matrices

≠ ≠ ≠§ z ™u

z z zand§ z ™ d s bc� 0 � 0§ z ™ x x xt


Table 1

w x w xParameter Value MeV Parameter Value MeV

A 976.99"0.9818 B 44.149"0.195912 12

A 101.07"0.975 B 50.532"1.40621 21

A y1465.5"0.5671 B 304.19"3.285123 235A y3.3811=10 "0.9675 B y3644.1"0.972132 32

A y24678"0.9679 B y5203.4"0.969333 33

explicit expressions involving the m2, the six central values of the quark masses squared at the 1 GeV scale. Thei&& &&T Treal symmetric matrices M M and M M will not have these central values as their exact eigenvalues unlessu u d d

the fit is perfect; i.e. unless Ý6 x 2 s0.is1 i

To ensure unitarity, after having performed the fit to the experimental observations, we evaluate the&& &&† † 2eigenvalues of M M and M M to find the correct m we should use in the z and then re-evaluate ouru u d d i i

expression for V , thus ensuring an unitary flavor-mixing matrix.fym

Although unitary, our derived flavor-mixing matrix contains unphysical phases that must be removed in orderto perform an unitary triangle analysis. After finding the mass matrices parameters that yield the minimum x 2,we put the flavor-mixing matrix into the standard CKM representation advocated by the Particle Data Groupand then into an improved Wolfenstein parametrization from which we find the unitary triangle.

The improved Wolfenstein parametrization can be obtained from the standard CKM representation with thew xfollowing identifications 9,12 :

l's ssinu , Al2 's ssinu , Al3 ry ih 's eyi d ssinu eyi dŽ .12 12 23 23 13 13

with modified Wolfenstein parameters

l2 l2

r'r 1y , h'h 1yž / ž /2 2

˜ ˜ TWith the above parameter values and using the true numerical eigenvalues of the resulting M M , weŽu,d . Žu,d .arrive at the values for the flavor-mixing parameters listed in Table 2.

w xJ is the Jarlskog invariant 13,5 , a rephasing invariant of the mixing matrix, and is given in the standardrepresentation of the quark flavor-mixing matrix as Jss s s c c2 c s s2.748573468=10y5.12 13 23 12 13 23 d

After imposing unitarity as described above, the absolute values of V become:CKM

0.974058 0.226275 0.00316006< <V s 0.226101 0.973303 0.0394979CKM ž /0.00941607 0.0384891 0.999215

As alluded to in the introduction, one of the most appealing features of the NNI mass matrices is the largenumber of texture zeroes. One may take advantage of this texture structure to eliminate three of the five realparameters in each sector, expressing them solely in terms of the m2, up to a sign ambiguity associated with thei

Table 2

w xParameter Value rad Parameter Valuey3u 3.160065=10 h 0.344495913

u 0.22825323 r 0.0042277512y2u 3.95070522=10 a 71.61961823

d 1.5585246 b 19.0835028

l 0.22627623 g 89.2968848y5A 0.771433972 J 2.748573468=10


square root branches. That is, we may use the invariance of the characteristic equation under a similaritytransformation to express A , A , and A in terms of m2 ,m2, and m2, and similarly for the down quark23 32 33 u c t

sector. Choosing A and A to be the as yet undetermined parameters in the up-quark mass matrix, and using12 12

the branch information revealed by Table 1, one easily verifies the following relations:

2'm m m bq b q4cu c tA sy , A sy ,(33 32A A 212 21

2 2 2m m mu c t2 2 2 2 2 2A sy m qm qm yA yA yA y ,)23 u c t 12 21 32 2 2A A12 21

where

m2 m2 m2u c t2 2 2 2b'm qm qm y2 A yu c t 12 2 2A A12 21

and

m2 m2 m2u c t2 2 2 2 2 2 2 2 2 2 2c'A m qm qm yA q ym m ym m ym m .Ž .12 u c t 12 u c u t c t2A12

Similarly, for the down-quark sector, choosing B and B to be the parameters as yet undetermined by the12 21

invariance relations, one finds the following equalities:

2'm m m dq d q4ed s bB sy , B sy ,(33 32B B 212 21

2 2 2m m md s b2 2 2 2 2 2B s m qm qm yB yB yB y ,)23 d s b 12 21 32 2 2B B12 21

where

m2 m2 m2d s b2 2 2 2d'm qm qm y2 B yd s b 12 2 2B B12 21

and

m2 m2 m2d s b2 2 2 2 2 2 2 2 2 2 2e'B m qm qm yB q ym m ym m ym m .Ž .12 d s b 12 d s d b s b2B12

Using the above mass matrices with their associated five total degrees of freedom corresponding to< <A , A , B , B and u , and the six experimental values of V observables as before, we obtain Table 3,12 21 12 21 CKM

Table 3

Parameter Value

A 1229.3"32.3812

A 139.82".09121

B 45.872"3.457312

B 48.797".043321

u y0.67806".009657


Table 4

Parameter Value Parameter Valuey3u 3.146249=10 h 0.351524613

u 0.2229535 r 0.00139412y2u 3.9499098=10 a 70.83438823

d 1.5668302 b 19.392858

l 0.2211108 g 89.772768y5A 0.80770906 J 2.67698466=10

with a x 2 of 2.182. This x 2 is not nearly as good as in the previous case, but the simple explanation for thistot tot

apparent decline in numerical confidence is just a reflection of the slight discrepancy between unitarity and thepresent experimental values. In the previous case, the flavor-mixing matrix could deviate from unitarity to

< <satisfy the experimental constraints; i.e., V having a central value that in itself violates unitarity did not docs

violence to the minimization of x 2 because the requirements of unitarity were not automatically satisfied. Intot

the current situation, unitarity is automatically imposed, and so the increase in x 2 is to be expected. Thetot< <predicted V all lie within the ranges preferred by the Particle Data Group analysis, and as such, any set ofCKM

parameters that fulfill this requirement of agreement with the ranges listed therein is deemed ‘‘acceptable’’,regardless of x 2 values.tot

0.975244 0.22111 0.00314624< <V s 0.220939 0.974488 0.0394887CKM ž /0.00924305 0.0385204 0.999215

Implicit in this prejudicial ‘‘acceptance’’ is the belief that only three generations of quarks exist and that futureexperiments will bring the experimental values into closer agreement with the constraints imposed by unitarity.

In addition, the standard representation parameters as well as the Wolfenstein parameters and Jarlskoginvariant J are predicted to be as listed in Table 4.

Because the mass matrices squared have the correct eigenvalues, the free-parameters A , A , B , B and12 21 12 21

u represent the general parametrisation at the fundamental mass matrices in general. To realize calculability, itis not enough to simply postulate some relation among this set of parameters independent of the quark masses,as such a relation would only serve to relate the flavor-mixing parameters among themselves, and not providethe sought after relation between quark masses and flavor-mixing parameters.

4. Making new mass matrix ansatze¨

The above mass matrices are completely general; to achieve calculability we must find expressions forA , A , B , B and u in terms of the m . Guided by the parameter values in Tables 1 and 3, we postulate the12 21 12 21 i

following mass matrices in terms of the m :i

0 A 0° ¶12

2 2 2m m mu c t2 2 2 2 2 2A 0 y m qm qm yA yA yA y& )21 u c t 12 21 32 2 2A A12 21M su

2'bq b q4c m m mu c t0 y y(¢ ß2 A A12 21


2m mu cwhere A s q , A s m m and(( (ž /12 21 u c2 2

0 B 0° ¶12

2 2 2m m md s b2 2 2 2 2 2B 0 m qm qm yB yB yB y& )21 d s b 12 21 32 2 2B B12 21M sd

2' m m mdq d q4e (b d s0 y y(¢ ß2 B21

where B s m m , B sm q m m and b,c,d and e are defined as before.( (12 d s 21 d d s

Lastly, the single phase u entering the flavor-mixing matrix is postulated to be yB rB . We know that for32 33

only two quark generations there is no CP violating phase in the mixing matrix, so it is natural to expect, withinthe framework of calculability, that u will involve ratios of elements in the mass matrices that only involve the

& &mixings of the third generation quarks. With these postulates for M , M and u , V is found from Eqs.u d fymŽ . Ž .6 – 9 . This constitutes a new mass matrix ansatz that provides a calculable model of flavor-mixing and is inexcellent agreement with the latest experimental results.

< <This new ansatz predicts the following absolute values for V :CKM

0.974427 0.224682 0.00307663< <V s ,0.224513 0.973672 0.0394506CKM ž /0.00923256 0.0384783 0.999217

< < < < 2V Vu b t ds0.0779868 and s0.239942. The x found from the six experimental V measurements is found totot CKM< < < <V Vcb t s

be 5.2489. Because we have no free parameters, there are six degrees of freedom. Such a value for x 2tot

corresponds to ;70% confidence level.The standard representation parameters, Jarlskog invariant J, and the modified Wolfenstein parameters are

predicted to be as listed in Table 5.The flavor-mixing observables predicted from this calculability ansatz in Table 5 are in excellent agreement

with the general result of Table 2.The above mass matrices with their calculability property as well as the low x 2 are very compellingtot

arguments in favor of calculability in the quark sector, but naturally one would like to uncover at least a glimpseof the more fundamental theory beyond the SM that is the source of this calculability. When one considershermitian mass matrices, considerations of family permutation symmetry and its breaking in successive stagesare suggestive explanations of the source of calculability, whereas with general NNI texture mass matrices, eventhough calculability still holds, it is not readily apparent what family symmetry, if any, is responsible. Inconclusion, we have elucidated an important point in the construction of NNI weak-eigenstate quark basis thathas eluded some previous authors. We have then performed an analysis of the experimental data to determinethe mass matrix parameters, and have obtained values for the flavor-mixing and UT parameters that are in

w xexcellent agreement with previous analysis 12 . Finally, we have presented a new class of calculable mass

Table 5

Parameter Value Parameter Valuey3u 3.0766324=10 A 0.78147394713

u 0.226617779 h 0.338014612y2u 3.94622992=10 r 0.0146947023

d 1.52735011 a 73.5544600048y5J 2.652858=10 b 18.934824658

l 0.2246832 g 87.510715288


matrices that also can explain all the experimental data and predict nearly identical values for the flavor-mixingand UT parameters as in the previous case.

Acknowledgements

We thank the hospitality of the School of Physics, Korea Institute of Advanced Study, where much of thiswork was completed. D.D. would like to thank the NSFrKOSEF 1998 Summer Institute in Korea Program forfinancial support. Support for this work was provided in part by US Dept. of Energy Contract DE-FG-02-91ER40688-Task A.

References

w x Ž .1 S. Weinberg, in: L. Motz Ed. , a Festscrift for I.I. Rabi, NY Academy of Sciences, New York, 1977; A.C. Rothman, K. Kang, Phys.Ž . Ž . Ž . Ž .Rev. Lett. 43 1979 1548; Phys. Rev. D 24 1981 167; H. Fritzsch, Phys. Lett. B 73 1978 317; Nucl. Phys. B 155 1979 189; A.

Ž . Ž . Ž .De Rujula, H. Georgi, S.L. Glashow, Ann. Phys. NY 109 1977 258; F. Wilczek, A. Zee, Phys. Lett. B 79 1977 418; H. Georgi,Ž .D.V. Nanpoulos, Nucl. Phys. B 155 1979 52.

w x Ž .2 K. Kang, S.K. Kang, Phys. Rev. D 56 1997 1511, and references therein.w x Ž .3 G.C. Branco, L. Lavoura, F. Mota, Phys. Rev. D 39 1989 3443.w x Ž .4 M. Kobayashi, T. Maskawa, Prog. Theor. Phys. 49 1973 652.w x5 A. Rasin, hep-phr9802356, see also hep-phr9708216.w x6 A. Ali, B. Kayser, hep-phr9806230 to be published in The Particle Century, Institute of Physics Publishing Inc., Bristol and

Ž .Philadelphia, 1998, G. Fraser Ed. .w x7 H. Fusaoka, Y. Koide, hep-phr9706211.w x Ž .8 J. Gasser, H. Leutwyler, Phys. Rep. 87 1982 77.w x Ž .9 Particle Data Group, Caso et al., The European Physical Journal C 3 1998 1.

w x10 H. Fritzsch, hep-phr9807551.w x11 V.V. Kiselev, hep-phr9806523.w x12 R.F. Parodi, P. Roudeau, A. Stocchi, hep-phr9802289.w x Ž .13 C. Jarlskog, Phys. Rev. Lett. 55 1985 1039.

Documents

Calculability of quark mixing parameters from general nearest neighbor interaction texture quark mass matrices