AMUP-94-09 US-94-08 December 1994 5 9 Quark Mass Matrix with a Structure of a 9 1 Rank One Matrix Plus a Unit Matrix ∗ n a J 4 1 1 Hideo Fusaoka† v 9 9 Department of Physics, Aichi Medical University 2 1 Nagakute, Aichi 480-11, Japan 0 5 9 and / h p Yoshio Koide‡ - p e Department of Physics, University of Shizuoka h : 52-1 Yada, Shizuoka 422, Japan v i X r a Abstract A quark mass matrix model M = M1/2O M1/2 is proposed where q e q e M1/2 = diag(√m ,√m ,√m ) and O is a unit matrix plus a rank one e e µ τ q matrix. Up- and down-quark mass matrices M and M are described u d in terms of charged lepton masses and additional three parameters (one in M and two in M ). The model can predict reasonable quark mass u d ratios (not only m /m , m /m , m /m and m /m , but also m /m ) u c c t d s s b u d and Kobayashi-Maskawa matrix elements. ∗ To be published in Mod. Phys. Lett. (1995). † E-mail: [email protected] ‡ E-mail: [email protected] 1 Recent observation of top quark mass by the CDF Collaboration [1] has brought a realistic study of quark mass matrix model within our reach more and more. Now, of the ten independent observable quantities in three-quark-family scheme, we have already possessed experimental knowledge of nine quantities, i.e. except for one parameter (a CP-violation phase parameter). One of the criteria of model-building is how we can describe those observable quantities with a few parameters as possible. From the phenomenological point of view, for example, the Rosner-Worah model [2] (and its democratic-type version [3]), which has six adjustableparameters,canprovidesatisfactorypredictionsforsixquarkmassesand four independent observable quantities of Kobayashi-Maskawa (KM) [4] matrix. Recently, one of the authors (Y.K.) has proposed a lepton and quark mass matrix model [5]: M = mf GO G , (1) f 0 f G = diag(g ,g ,g ) , (2) 1 2 3 O = 1+3a X(φ ) , (3) f f f 1 0 0 1 eiφ 1 1 1 =  0 1 0  , X(φ) =  e−iφ 1 1  , (4) 3  0 0 1   1 1 1          where f = ν,e,u, and d are indices for neutrinos, charged leptons, up- and down- quarks, respectively. In the charged lepton mass matrix M , the parameter a is e e chosen as a = 0, so that the parameters meg2 are fixed by charged lepton masses e 0 i me (me = m , me = m , me = m ) as meg2 = me. Since the phase parameters i 1 e 2 µ 3 τ 0 i i φ are fixed at φ = 0 and φ = π/2 and the parameters mu and md are fixed q u d 0 0 as mu = md, the model includes only two adjustable parameters (a and a ) and 0 0 u d provides reasonable values of quark mass ratios (not absolute values) and KM matrix parameters. In spite of such phenomenological success, the following points in Ref. [5] are still unsatisfactory to us: (i) X(φ ) is not a rank one matrix so that we must d propose a complicated mechanism to explain the origin of this term. (ii) There are no reasonable explanation for nonzero phase terms which exist only in (1, 2) and (2, 1) matrix elements of X(φ ). This ansatz is contrary to the philosophy of d “democratic”. (iii) Even though we accept that O = 1+3a X(π/2) is simple, the d d 2 inverse matrix O−1 is not simple and it is difficult to account for M = mf GO G d f 0 f in a seesaw type model M mM−1m with m G and M O−1. f ≃ F ∝ F ∝ f In this paper we search for a possible form of the matrices O in (1) with f the following conditions: (a) The matrix form of each term in O is as simple as f possible. (b) The number of hierarchically different terms is as few as possible. In a Higgs mechanism model, the latter condition means Higgs fields are as few as possible. The simplest form of O is a unit matrix, but it leads to M = M = M and f u d e fails to give the good predictions for quark mass spectrum and KM matrix. The next simple form of O is a unit matrix plus a rank one matrix, which agrees with f the condition (b). Therefore it is meaningful to study quark masses and mixing in the case that O (q = u,d) is given by a unit matrix plus a rank one matrix with a q complex coefficient. In the present paper, we propose the following quark mass matrix: M = (mf/me)M1/2O M1/2 , (5) f 0 0 e f e M1/2 = diag(√m ,√m ,√m ) , (6) e e µ τ O = 1+3a eiαfX , (7) f f where X X(0) is a rank one matrix. The inverse matrices of O are also simple f ≡ O−1 = 1+3b eiβfX , (8) f f where b eiβf = a eiαf/(1+3a eiαf) . (9) f f f − Ina seesaw type modelwith heavy fermions, theinverse matrices O−1 become more f fundamental quantity than O and the parameters b and β are more important f f f than a and α . f f In the model of Ref. [5], there are only two adjustable parameters (a , a ) u d because of the ansatz φ = π/2, but it is difficult to give a reasonable explanation d for the mass matrices in model building. In the present model, three parameters (a , a , α ) are necessary but a matrix form of O−1 as well as O are simple. This u d d f f makes model building easy. If one feel the “democratic” type matrix form X in the present model (7) somewhat mysterious, one may alternatively consider a diagonal matrix form 3 diag(0,0,1) by taking a suitable transformation of family basis, because the unit matrix term 1 in (7) is unchanged under this transformation. The reason that we consider a democratic matrix from in O is motivated f by only a phenomenological reason suggested in Ref. [6], i.e., by the fact that for up-quark mass matrix with α = 0, we can obtain the successful mass relation [6] u m /m 3m /4m , (10) u c e µ ≃ for a small value of ε 1/a . Substituting the quark mass values which is given u u ≡ in eq. (12), the left hand side of eq. (10) is 4.0 10−3, while the right hand side of × eq. (10) is 3.6 10−3. Note that the ratio m /m is insensitive to the parameter u c × a . The parameter ε 1/a is determined by the mass ratio u u u ≡ m /m 2(m /m )ε . (11) c t µ τ u ≃ The quark mass values [7] at an electroweak symmetry breaking energy scale µ = Λ φ0 = (√2G )−1/2/√2 = 174 GeV are W F ≡ h i m = 0.0024 0.0005 GeV , m = 0.605 0.009 GeV , m = 174 10+13 GeV , u ± c ± t ± −12 m = 0.0042 0.0005 GeV , m = 0.0851 0.014 GeV , m = 2.87 0.03 GeV , d s b ± ± ± (12) where φ0 is a vacuum expectation value of a Higgs scalar field φ0 in the standard h i (4) model and we have used Λ = 0.26 GeV. MS Differently from the model given in Ref. [5], down-quark mass matrix M d with α = 0 in the present model is not Hermitian. We will demonstrate that the d 6 present model with the form (7) also can provide reasonable predictions of quark mass ratios and KM matrix by adjusting our parameters a , a and α (i.e., b , b u d d u d and β ). d In the present model, a case a 1/2 can provide phenomenologically d ≃ − interesting predictions as seen below. For small values of α and ε (2+a−1), | d| d ≡ − d we obtain the down-quark mass ratios m 1 2m m s e µ κ 1 48 , (13) mb ≃ 2 − s 3m2τ ! m 16m m 2m m d e µ e µ 1+96 , (14) ms ≃ κ2 m2τ s 3m2τ ! 4 where α ε 2 κ = sin2 d + d . (15) s 2 4 (cid:18) (cid:19) We also obtain m m m m d s e µ 4 , (16) m2 ≃ m2 b τ as a relation which is insensitive to the small parameters α and ε . The left hand d d | | side of eq. (16) is 4.3 10−5, while the right hand side of eq. (16) is 6.8 10−5 × × with the quark mass values (12). Furthermore, we canobtainratiosofup-quarks todown-quarks, forexample, m /m 6κ 12m /m . (17) u d s b ≃ ∼ Suitable choice of small values of ε and α ensures m /m O(1) in spite of d d u d ∼ m m . From (9), a small value ε = 1/ a 0 means b 1/3, while a t b u u u ≫ | | | | ≃ ≃ − small value ε = 2 + a−1 0 means b 1. It is noted that, in spite of the | d| | d | ≃ d ≃ − large ratio of m /m , the ratio of b /b is not so large, i.e., b /b 3. t b d u d u ≃ Then, let us discuss the KM matrix elements V . The KM matrix V is given ij by V = UuPUd† , (18) L L where Uu and Ud are defined by L L UuM M†Uu† = diag(m2,m2,m2) , UdM M†Ud† = diag(m2,m2,m2) , (19) L u u L u c t L d d L d s b respectively, and P is a phase matrix. Here, we have considered that the quark basis for the mass matrix (5) can, in general, deviate from the quark basis of weak interactions by some phase rotations, The simplest case P =diag(1,1,1) cannot provide reasonable predictions of V . When we take ij | | P = diag(1,1, 1) , (20) − we can obtain reasonable predictions for both quark mass ratios and KM matrix elements, although it is an open question why such a phase inversion is caused on the third family quark. The predictions of V are sensitive to every value of ij | | ε , ε and α , so that it is not adequate to express V as simple approximate u d d ij | | relations such as those in (10)–(11) and (13)–(17). Therefore, we will show only 5 numerical results for V . For example, by taking a = 28.65, a = 0.4682, ij u d | | − α = 0 and α = 7.96◦ (b = 0.3295, b = 1.072, β = 0 and β = 18.5◦), which u d u d u d − − are chosen by fitting the quark mass ratios, we obtain the following predictions of quark masses, KM matrix elements V and the rephasing-invariant quantity J ij | | [8]: m = 0.00228 GeV , m = 0.591 GeV , m = 170 GeV , u c t (21) m = 0.00429 GeV , m = 0.0875 GeV , m = 3.02 GeV , d s b V = 0.223 , V = 0.0542 , V = 0.00309 , us cb ub | | | | | | (22) V = 0.0146 , V /V = 0.0570 , J = 2.30 10−5 . td ub cb | | | | × The prediction V = 0.0542 in (22) is somewhat large in comparison with the cb | | experimental value V = 0.040 0.005 [9]. If we use P = (1,1, eiδ) with a small cb | | ± − phase value δ instead of P = (1,1, 1), we can obtain more excellent predictions − without changing predictions of quark masses in (21): for example, when we take δ = 4.4◦, we obtain − V = 0.223 , V = 0.0400 , V = 0.00274 , (23) us cb ub | | | | | | V = 0.0111 , V /V = 0.0686 , J = 1.55 10−5 . td ub cb | | | | × In the numerical predictions of quark masses (21), we have used a common enhancement factor of quark masses to lepton masses, mu/me = md/me = 3, in 0 0 0 0 order to compare with quark mass values at µ = Λ (12). It is an open question W why we can set the factor mq/me as just three. Although we are happy if we 0 0 can explain such the factor mq/me = 3 by evolving quark and lepton masses from 0 0 µ = Λ to µ = Λ , unfortunately, it is not likely to derive such a large factor 3 X W ∼ from the conventional renormalization calculation. At present, we have no theory to determine the parameters a and α . For f f charged leptons, we must take a = 0. For quarks, we have chosen a from the e q phenomenological parameter fitting. However, in the present stage, we do not provide any unified understanding for a and α , i.e., they are nothing more than f f phenomenological parameters. In conclusion, quark mass ratios and KM matrix elements can be fitted only by three parameters a , a and α (b , b and β ) fairly well. If we take a seesaw- u d d u d d type model, we must consider that the parameters b and β in O−1 are more f f f 6 fundamental ones rather than a and α in O . Then it is worth while that we f f f can obtain a large ratio of m /m together with a reasonable ratio m /m without t b u d taking so hierarchically different values of b and b , i.e., with taking b 1/3 u d u ≃ − and b 1, in contrast to a 30 and a 1/2 in the case of GO G picture. d u d f ≃ − ≃ ≃ − Acknowledgments The authors are grateful to Professor M. Tanimoto for the simulating com- ments. This work was supported by the Grant-in-Aid for Scientific Research, Min- istry of Education, Science and Culture, Japan (No.06640407). References and Footnote [ 1 ]CDF Collaboration, F. Abe et al., Phys. Rev. Lett. 73, 225 (1994); Phys. Rev. D50, 2966 (1994). [ 2 ]J. L. Rosner and M. Worah, Phys. Rev. D46, 1131 (1992). [ 3 ]Y. Koide, in International Symposium on Extended Objects and Bound Sys- tems, Proceedings, Karuizawa, Japan, 1992, edited by O. Hara, S. Ishida, and S.Naka(WorldScientific, Singapore, 1992); K.Matumoto, Prog.Theor.Phys. 89, 269 (1993); Y. Koide and H. Fusaoka, Phys. Rev. D48, 432 (1993). [ 4 ]M. Kobayashi and T. Maskawa, Prog. Theor. Phys. 49, 652 (1973). [ 5 ]Y. Koide, Phys. Rev. D49, 2638 (1994). [ 6 ]Y. Koide, Mod. Phys. Lett. A8, 2071 (1993). [ 7 ]For light quark masses m (µ), we have used the values at µ = 1 GeV, m = q u 5.6 1.1 MeV, m = 9.9 1.1 MeV and m = 199 33 MeV: C. A.Dominquez d s ± ± ± and E. de Rafael, Annals of Physics 174, 372 (1987). 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