- Email: [email protected]
A model of flavor mixing
Volume 177, n u m b e r 2
PHYSICS LETTERS B
11 September 1986
A M O D E L OF FLAVOR M I X I N G Eduard MASSO Departament de Fisica Tebrica, Universitat Autbnoma de Barcelona, Bellaterra (Barcelona), Spain Received 26 May 1986
A model of quark mass matrices is presented where flavor mixings can be expanded m powers of small parameters,. These parameters characterize the observed quark mass pattern. The weak mixing matrix has a very simple form. The correlation of the weak mixing angles and CP-violating phases with the main features of the quark spectrum may be easily studied. An upper bound is predicted on the top-quark mass, ml'4 45 GeV, and a rate for the b--*u transition close to its experimental limit.
The origin of the observed fermion spectrum and the flavor mixing phenomena constitutes one of the major problems of particle physics [1]. In the S U ( 2 ) x U ( 1 ) standard model all these observables have to be determined by experiment. A more basic theory containing the standard model should shed more light on the mass problem. Since such an extended theory is still unknown, it is useful to consider constraints on the low-energy mass matrices. Much effort has been addressed to restrict the quark sector. These restrictions generate relations among quark masses and weak mixings. Some of the regularities observed in the quark families are incorporated into the Fritzsch mass matrix [2]. The general structure of this quark mass matrix is 0 M=
A2
AI 0
0 ] Bl •
0
B2
C
(1)
where IA~I~.IBiI~.ICI. It has been shown that (1)is compatible with the experimental information we have on quark masses and mixings [3] :~. Motivated by properties of mass matrices in some unified the++~These phenomenological studies were performed with [A~I = IA21and IB, I = IB_,I in (1). In fact, it is this restricted form the one originallyproposed in ref. [2].
0370-2693/86/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
ories, Stech suggested specific relations involving mass matrices which are also phenomenologically consistent [4]. These relations are Mu=M*u=h~u,
MD=M~=aMu+A,
(2)
where o~is a constant and A an antisymmetric matrix. There is a wide class of theoretical scenarios that lead to the constraints (1) [5-7] and (2) [4,8]. It is certainly interesting to ask whether the restrictions (1) and (2) can be imposed simultaneously and, if so, what are the consequences of this combined model. A partial answer to these questions has been given recently by Gronau, Johnson and Schechter [9]. These authors have shown that (i) for any number of families, (1) and (2) are compatible and the weak mixing matrix is a function of the quark masses and (ii) for three generations, the CP-nonconserving phase defined in ref. [10] is near the "maximal" value n/2. In this note we would like to show that the combined Fritzsch-Stech model has some remarkable features. Flavor mixings, i.e., Cabibbo angles and CPviolating phases, can be expanded in terms of small parameters. These parameters characterize the observed quark mass pattern. As a consequence, the weak mixing matrix adopts a very simple form and its correlation with the main features of the quark spectrum may be easily studied. We will show that the model is phenomenologically consistent, although it is quite constrained by experiment.
183
Volume 177, number 2
PHYSICS LETTERS B
The simultaneous imposition of (1) and (2) leads to very specific and restricted mass matrices. We have
Mt,=t
{o o 1 W 0
0 x
x 1
11 September 1986
tg20 = y2/X~ = ( s / b - c/t)/(c/t).
(7)
We have
,
P*Mz)P=M{~,
Mt~=Mt~T=M() * .
(8)
We now rotate Mu and M~ to diagonal forms 0 W-iZ 0
Mo=b
W+iZ 0 X-iY
0 ] X+iY , 1
(3)
VTM~,V= Diag(0, - c, t), V' ~M(~V' = Diag(d,-s, b).
(9)
The orthogonal matrices V and V' can be found in ref. [5]. Finally, the weak mixing matrix is
where W 2 = uc]t 2 ,
W 2 + Z 2 = ds/b 2 , K = VxPV ' .
X2=c/t,
X2 + y 2 = s / b .
(4)
Here, u, d, c, s, t and b are the quark masses normalized a t # = 1 GeV. The fact that the quark weak basis, in which ME and MD are expressed in (4), differs from the quark mass basis gives rise to flavour mixing. This is described by the weak mixing matrix
K=
Kud Kcd [K,d
Kus Kcs K,s
Kub Kcb K~b
] •
(5)
We would like now to determine the magnitudes of the elements of K. We will first present our results when u = 0 (which implies W=0). In this limiting case, the important features of the Fritzsch-Stech model are most easily seen. The precise effects of the non-vanishing up-quark mass will be considered below. To find the weak mixing matrix we start transforming MD tO a real and symmetric matrix. When u = O, we use the phase-diagonal matrix
P = Diag(e i°', e i0-~, eiO~) 0,=~(0+n),
02=~(0-n/2),
(6)
q~3---- ~(20+ n/2),
(10)
One obtains K=
Ri 1 iR2sin 0 e-io R~R2sin ¢~ R2sin 0 - i e -'~
.
(ll)
This is the weak mixing matrix in the limiting case u=0. When u ~ 0 it has to be modified (see eq. (16) below). In (11) we have defined the parameters R~=X2+ y2=s/b,
R2=gR24~d/s,
(12)
and ~ is defined in (7). In (11) and in the rest of the paper the first row and column of K are real. These are our phase conventions for the quark fields. In the weak mixing matrix (11) we kept only the leading terms in R~ and R 2. This can be justified because these two parameters are small. Indeed, from (12) we see that R1 and R2 are mass ratios of quarks belonging to different families. These mass ratios are small because the quark masses cluster in generations. The smallness of R ~and R2 describes the threegeneration hierarchy of the quark spectrum. We would like now to find other features of the quark spectrum which may be important when discussing the weak mixing matrix of our model. We start quoting the quark masses, normalized at # = 1GeV, from ref. [ 11 ]. They are u=5.1+l.5MeV,
d=8.9+_2.6MeV,
where :2 c=l.35_+0.05GeV,
s=0.175_+0.055GeV,
b=5.3_+0.1 GeV. :" The PETRAlower bound on the top-quark mass ensures that the RHS of(7) is positive. 184
To these we should add the "tentative" values
(13a)
Volume 177, n u m b e r 2
m,° = 30-50 G e V .
PHYSICS LETTERS B
(13b)
Here, m~° is the "pole" mass of the top quark. (To a very good approximation, rn,p is half of the toponium mass.) We can see now the smallness of R~ and R2. Taking the central values of the quark masses we have displayed, we get R, - 1/4 and R2-~ 1/5. Let us now examine another type of mass ratios. For each family, we consider the quotient among the masses of the up- and down-type quark. We have three such ratios. In our combined model, it can be shown that, in the limiting case of vanishing weak mixings, those ratios would be identical :3. In fact, the parameter ~ in (7) is a measure of how close this limit is realized in the case of the second and third generations. Eq. (7) is equivalent to sin2~= (s/b-c/t)(s/b).
(14)
Taking the central values of the quark masses in (13) and evolving the top quark mass with A4 = 100 MeV in the four-flavor regime ~4 we get sin2q~=l/3. Expanding the matrix K in (11 ) in powers of sin 0 amounts to setting Real(e io)=cos ~ 1 . This approximation is 20% offwhen sin20= 1/3. The parameter ~ in (14) is relatively small. This can be considered as the consequence of a regularity in the spectrum (13). Indeed, we have that the relation s/b ~-c/t is approximately fulfilled. This regularity cannot be extrapolated to the first family. Using the quark masses in (3) we find d/s,>u/c. It is the anomalous lightness of the up-quark that breaks the regularity. We will treat the up-quark mass as a perturbation in the matrix K in (11 ), which was evaluated with u = 0. The appropriate parameter for these purposes is the angle sin2z = (u/d)/(c/s) .
(15 )
The central values of the quark masses in ( 13 ) imply sin Z -~ 1/4. Expanding in powers of ~ and introducing a non~ The relationship among vanishing mixings and identical up-down mass ratios has been discussed in a variety of models (see ref. [4] and references therein, see also ref. [12]). :4 Here and in the rest of the paper we evolve the top-quark mass m~(~t) in the following way. Between/1= 1 GeV and 5 GeV we use A~= 100 MeV and four effective flavours. Between/2= 5 GeV and l~= m,(m0 we use five flavors and the corresponding As.
11 September 1986
vanishing up-quark mass through the parameter x in (15) we get
1
K=
- R j R~R2(R2+sinOsinz) 1 iR2sin R~R2sin ~ R2sin ~ -i R1
(16) The matrix K is the weak mixing matrix of the combined Fritzsch-Stech model, with mass matrices exhibited in ( 3 ). For each element of K, we only display the leading term in a power expansion of the small parameters R~, R2, sin 0 and sin Z. We stress that our expansion and the precise parameters we use in the expansion are suggested by the structure of the mass matrices :5. The family hierarchy is described by R~ and R2. The pattern of the up-down mass ratios is described by two angles. The parameter sin ¢, is associated with a regularity of that pattern and sin Z describes the lightness of the up-quark. These four combinations of mass ratios describe the general features of the quark spectrum. The four parameters determine the full spectrum up to a global rescaling of the d-, s- and b-quark masses and an independent rescaling of the u-, c-, and t-quark masses. The weak mixing matrix in (16) has the nice properties of simplicity and of being determined by the general structure of the quark masses. We should now check its phenomenological consistency. We have, using (16) and (13 ), that Iguol = R l = (d/s)1/2=0.23+0.01 =0,.,
(17)
where 0e is the usual Cabibbo angle. The successful relation in (17 ) was in fact the starting motivation to consider the Fritzsch-type ansatz [2]. The experimental status of the mixing Igcbl is by now very clear. The relatively long B-lifetime [ 14 ] Za/10-12
s=0.6-2.5
(18)
implies [ 15 ] 0.04~< JKcblexo~<0.08.
(19)
The significant difference in the magnitude of (17) and (19) can be easily understood in the framework of our model. Indeed, from (16) we predict :5 This should be distinguished from expansions of the weak mixing matrix [ 13].
185
Volume 177, number 2
Ig, bl/Igudl =(RJR,) sin 0 .
PHYSICS LETTERS B
(20)
Since R~ and R2 are roughly o f the same order o f magnitude, the suppression o f the b ~ c transitions is due to the smallness o f ¢ (or to the relation s/b~-c/t). The agreement between the F r i t z s c h - S t e c h model and experiment can be m a d e quantitative. The experimental range in (19) implies an u p p e r and lower b o u n d on 0. Using (14), this translates into a range o f top-quark masses in which the m o d e l reproduces (18) and (19). W i t h A4= 100 MeV we o b t a i n the prediction 24 GeV~
(21)
which is consistent with (13b). The selected range in (21 ) and the masses in (13a) allow us to d e t e r m i n e K,u. O u r p r e d i c t i o n is 0.002~< [K, bl ~<0.013,
(22)
to be c o m p a r e d with the e x p e r i m e n t a l value [ 16] IK, blexo~<0.013.
(23)
Eqs. (21) and (22) are the m a i n predictions o f our model. The former is, in practical terms, an upper b o u n d on the top-quark mass o f 45 GeV. O u r prediction for the rate o f the b--,u transition is close to its experimental limit. Eq. (22) will be tested by future experiments seeking for b--,u transitions. We finally address our attention to the CP-nonconservation originated in the mass matrices (3). It is the phase-diagonal matrix P which carries the information on CP-nonconservation. This can be seen in (10). In the limit u = 0 (sin x = 0 ) a n d sin 0 = 0 we get P=e
in/6Diag(i, 1, 1).
(24)
The n/2-phase difference in the elements o f P makes the invariant phase defined in ref. [10] to be z~/2. Gronau, Johnson and Schechter, which were the first to suggest the c o m b i n e d F r i t z s c h - S t e c h model, have studied the relationship between this m a x i m a l value and the quark spectrum [9]. Corrections to (24) are small and calculable in our model. Some pieces o f these corrections control the non-conservation o f CP in d e t e r m i n e d systems. F o r instance, to describe CP-violation in the K°-I~ ° complex we need to go b e y o n d the leading terms displayed in the weak mixing matrix (16). We need to 186
11 September 1986
know the small phases which will a p p e a r in the c - s and t - s flavor mixings. These are easily evaluated. We find Kcs= 1 + i 6 ,
Kts=RzsinO+i~',
(25)
with 3=R~sin 0(R~+sin 0 sinz) , ~'= -R2(R~+sin ~ sinz) .
(26)
These equations reflect the fact that the CP-violating phases can also be expanded in terms o f our four basic parameters. We have checked that (25) and (26) are consistent with the experimental CP-nonconserving parameters ~. This will be'presented elsewhere [ 17 ]. After this work was c o m p l e t e d we received two reports where the phenomenology o f a fourth generation in the F r i t z s c h - S t e c h model is studied [ 18 ]. I a m indebted to Jon Bagger for helpful remarks. This work was partially supported by the research projects CAICYT.
References [ 1] See, e.g.H. Harari, in: Proc. twelfth SLAC Summer Institute, ed. P.M. McDonough (1984). [2] H. Fritzsch, Phys. Lett. B 73 (1978) 317; Nucl. Phys. B 155 (1979) 189; see also L.-F. Li, Phys. Lett. B 84 (1979) 461. [3] M. Shin, Phys. Lett. B 145 (1984) 285; H. Fritzsch, Phys. Lett. B 166 (1986) 423. [4] B. Stech, Phys. Lett. B 130 (1983) 189. [ 5] H. Georgi and D.V. Nanopoulos, Nucl. Phys. B 155 (1979) 52. [6] S. Dimopoulos, Phys. Lett. B 129 (1983) 417; J. Bagger, S. Dimopoulos, H. Georgi and S. Raby, in: Proc. Fifth Workshop on Grand unification, eds. K. Kang, H. Fried and P. Frampton (World Scientific, Singapore, 1984); H. Georgi, A. Nelson and M. Shin,, Phys. Lett. B 150 (1985) 306. [7] A. Davidson, V. Nair and K. Wali, Phys. Rev. D 29 (1984) 1504, 1513; V. Nair, L. Michel and K. Wali, Phys. Lett. B 138 (1984) 128. [ 8 ] B. Stech, in: Flavour mixing in weak interactions, ed. L. L. Chau (Plenum, New York, 1984). [ 9] M. Gronau, R. Johnson and J. Schechter, Phys. Rev. Lett. 54 (1985) 2176. [ 10] M. Gronau and J. Schechter, Phys. Rev. Len. 54 (1985) 385, 1209 (E). [ 11 ] J. Gasser and H. Leutwyler, Phys. Rep. 87 (1982) 27.
Volume 177, number 2
PHYSICS LETTERS B
[ 12 ] T.P. Cheng and L.-F. Li, Phys. Rev. Lett. 55 ( 1985 ) 2249. [ 13] L. Wolfenstein, Phys. Rev. Lett. 51 (1983) 1945. [ 14 ] E. Fern~indez et al., Phys. Rev. Left. 51 ( 1983 ) 1022; N.S. Lockyer et al., Phys. Rev. Lett. 51 ( 1983 ) 1316; M. Althoffet al., Phys. Lett. B 149 (1984) 524; D.E. Klem et al., Phys. Rev. Lett. 53 (1984) 1873; W. Barrel et al., preprint DESY 86-001 (1986).
11 September 1986
[15] See, e.g., E. de Rafael, in: Lectures on quark flavor mixing in the standard model, preprint MPI-PAE/PTh 72/84 (1984). [16] C. Klopfenstein et al., Phys. Lett. B 130 (1983) 444; A. Chen et al., Phys. Rev. Lett. 52 (1984) 1084. [ 17 ] E. Mass6, in preparation. [ 18] K. Kang and M. Shin, Phys. Lett. B 165 (1985) 383; R. Johnson, J. Schechter and M. Gronau, Phys. Rev. D33 (1986) 2641.
187
PHYSICS LETTERS B
11 September 1986
A M O D E L OF FLAVOR M I X I N G Eduard MASSO Departament de Fisica Tebrica, Universitat Autbnoma de Barcelona, Bellaterra (Barcelona), Spain Received 26 May 1986
A model of quark mass matrices is presented where flavor mixings can be expanded m powers of small parameters,. These parameters characterize the observed quark mass pattern. The weak mixing matrix has a very simple form. The correlation of the weak mixing angles and CP-violating phases with the main features of the quark spectrum may be easily studied. An upper bound is predicted on the top-quark mass, ml'4 45 GeV, and a rate for the b--*u transition close to its experimental limit.
The origin of the observed fermion spectrum and the flavor mixing phenomena constitutes one of the major problems of particle physics [1]. In the S U ( 2 ) x U ( 1 ) standard model all these observables have to be determined by experiment. A more basic theory containing the standard model should shed more light on the mass problem. Since such an extended theory is still unknown, it is useful to consider constraints on the low-energy mass matrices. Much effort has been addressed to restrict the quark sector. These restrictions generate relations among quark masses and weak mixings. Some of the regularities observed in the quark families are incorporated into the Fritzsch mass matrix [2]. The general structure of this quark mass matrix is 0 M=
A2
AI 0
0 ] Bl •
0
B2
C
(1)
where IA~I~.IBiI~.ICI. It has been shown that (1)is compatible with the experimental information we have on quark masses and mixings [3] :~. Motivated by properties of mass matrices in some unified the++~These phenomenological studies were performed with [A~I = IA21and IB, I = IB_,I in (1). In fact, it is this restricted form the one originallyproposed in ref. [2].
0370-2693/86/$ 03.50 © Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
ories, Stech suggested specific relations involving mass matrices which are also phenomenologically consistent [4]. These relations are Mu=M*u=h~u,
MD=M~=aMu+A,
(2)
where o~is a constant and A an antisymmetric matrix. There is a wide class of theoretical scenarios that lead to the constraints (1) [5-7] and (2) [4,8]. It is certainly interesting to ask whether the restrictions (1) and (2) can be imposed simultaneously and, if so, what are the consequences of this combined model. A partial answer to these questions has been given recently by Gronau, Johnson and Schechter [9]. These authors have shown that (i) for any number of families, (1) and (2) are compatible and the weak mixing matrix is a function of the quark masses and (ii) for three generations, the CP-nonconserving phase defined in ref. [10] is near the "maximal" value n/2. In this note we would like to show that the combined Fritzsch-Stech model has some remarkable features. Flavor mixings, i.e., Cabibbo angles and CPviolating phases, can be expanded in terms of small parameters. These parameters characterize the observed quark mass pattern. As a consequence, the weak mixing matrix adopts a very simple form and its correlation with the main features of the quark spectrum may be easily studied. We will show that the model is phenomenologically consistent, although it is quite constrained by experiment.
183
Volume 177, number 2
PHYSICS LETTERS B
The simultaneous imposition of (1) and (2) leads to very specific and restricted mass matrices. We have
Mt,=t
{o o 1 W 0
0 x
x 1
11 September 1986
tg20 = y2/X~ = ( s / b - c/t)/(c/t).
(7)
We have
,
P*Mz)P=M{~,
Mt~=Mt~T=M() * .
(8)
We now rotate Mu and M~ to diagonal forms 0 W-iZ 0
Mo=b
W+iZ 0 X-iY
0 ] X+iY , 1
(3)
VTM~,V= Diag(0, - c, t), V' ~M(~V' = Diag(d,-s, b).
(9)
The orthogonal matrices V and V' can be found in ref. [5]. Finally, the weak mixing matrix is
where W 2 = uc]t 2 ,
W 2 + Z 2 = ds/b 2 , K = VxPV ' .
X2=c/t,
X2 + y 2 = s / b .
(4)
Here, u, d, c, s, t and b are the quark masses normalized a t # = 1 GeV. The fact that the quark weak basis, in which ME and MD are expressed in (4), differs from the quark mass basis gives rise to flavour mixing. This is described by the weak mixing matrix
K=
Kud Kcd [K,d
Kus Kcs K,s
Kub Kcb K~b
] •
(5)
We would like now to determine the magnitudes of the elements of K. We will first present our results when u = 0 (which implies W=0). In this limiting case, the important features of the Fritzsch-Stech model are most easily seen. The precise effects of the non-vanishing up-quark mass will be considered below. To find the weak mixing matrix we start transforming MD tO a real and symmetric matrix. When u = O, we use the phase-diagonal matrix
P = Diag(e i°', e i0-~, eiO~) 0,=~(0+n),
02=~(0-n/2),
(6)
q~3---- ~(20+ n/2),
(10)
One obtains K=
Ri 1 iR2sin 0 e-io R~R2sin ¢~ R2sin 0 - i e -'~
.
(ll)
This is the weak mixing matrix in the limiting case u=0. When u ~ 0 it has to be modified (see eq. (16) below). In (11) we have defined the parameters R~=X2+ y2=s/b,
R2=gR24~d/s,
(12)
and ~ is defined in (7). In (11) and in the rest of the paper the first row and column of K are real. These are our phase conventions for the quark fields. In the weak mixing matrix (11) we kept only the leading terms in R~ and R 2. This can be justified because these two parameters are small. Indeed, from (12) we see that R1 and R2 are mass ratios of quarks belonging to different families. These mass ratios are small because the quark masses cluster in generations. The smallness of R ~and R2 describes the threegeneration hierarchy of the quark spectrum. We would like now to find other features of the quark spectrum which may be important when discussing the weak mixing matrix of our model. We start quoting the quark masses, normalized at # = 1GeV, from ref. [ 11 ]. They are u=5.1+l.5MeV,
d=8.9+_2.6MeV,
where :2 c=l.35_+0.05GeV,
s=0.175_+0.055GeV,
b=5.3_+0.1 GeV. :" The PETRAlower bound on the top-quark mass ensures that the RHS of(7) is positive. 184
To these we should add the "tentative" values
(13a)
Volume 177, n u m b e r 2
m,° = 30-50 G e V .
PHYSICS LETTERS B
(13b)
Here, m~° is the "pole" mass of the top quark. (To a very good approximation, rn,p is half of the toponium mass.) We can see now the smallness of R~ and R2. Taking the central values of the quark masses we have displayed, we get R, - 1/4 and R2-~ 1/5. Let us now examine another type of mass ratios. For each family, we consider the quotient among the masses of the up- and down-type quark. We have three such ratios. In our combined model, it can be shown that, in the limiting case of vanishing weak mixings, those ratios would be identical :3. In fact, the parameter ~ in (7) is a measure of how close this limit is realized in the case of the second and third generations. Eq. (7) is equivalent to sin2~= (s/b-c/t)(s/b).
(14)
Taking the central values of the quark masses in (13) and evolving the top quark mass with A4 = 100 MeV in the four-flavor regime ~4 we get sin2q~=l/3. Expanding the matrix K in (11 ) in powers of sin 0 amounts to setting Real(e io)=cos ~ 1 . This approximation is 20% offwhen sin20= 1/3. The parameter ~ in (14) is relatively small. This can be considered as the consequence of a regularity in the spectrum (13). Indeed, we have that the relation s/b ~-c/t is approximately fulfilled. This regularity cannot be extrapolated to the first family. Using the quark masses in (3) we find d/s,>u/c. It is the anomalous lightness of the up-quark that breaks the regularity. We will treat the up-quark mass as a perturbation in the matrix K in (11 ), which was evaluated with u = 0. The appropriate parameter for these purposes is the angle sin2z = (u/d)/(c/s) .
(15 )
The central values of the quark masses in ( 13 ) imply sin Z -~ 1/4. Expanding in powers of ~ and introducing a non~ The relationship among vanishing mixings and identical up-down mass ratios has been discussed in a variety of models (see ref. [4] and references therein, see also ref. [12]). :4 Here and in the rest of the paper we evolve the top-quark mass m~(~t) in the following way. Between/1= 1 GeV and 5 GeV we use A~= 100 MeV and four effective flavours. Between/2= 5 GeV and l~= m,(m0 we use five flavors and the corresponding As.
11 September 1986
vanishing up-quark mass through the parameter x in (15) we get
1
K=
- R j R~R2(R2+sinOsinz) 1 iR2sin R~R2sin ~ R2sin ~ -i R1
(16) The matrix K is the weak mixing matrix of the combined Fritzsch-Stech model, with mass matrices exhibited in ( 3 ). For each element of K, we only display the leading term in a power expansion of the small parameters R~, R2, sin 0 and sin Z. We stress that our expansion and the precise parameters we use in the expansion are suggested by the structure of the mass matrices :5. The family hierarchy is described by R~ and R2. The pattern of the up-down mass ratios is described by two angles. The parameter sin ¢, is associated with a regularity of that pattern and sin Z describes the lightness of the up-quark. These four combinations of mass ratios describe the general features of the quark spectrum. The four parameters determine the full spectrum up to a global rescaling of the d-, s- and b-quark masses and an independent rescaling of the u-, c-, and t-quark masses. The weak mixing matrix in (16) has the nice properties of simplicity and of being determined by the general structure of the quark masses. We should now check its phenomenological consistency. We have, using (16) and (13 ), that Iguol = R l = (d/s)1/2=0.23+0.01 =0,.,
(17)
where 0e is the usual Cabibbo angle. The successful relation in (17 ) was in fact the starting motivation to consider the Fritzsch-type ansatz [2]. The experimental status of the mixing Igcbl is by now very clear. The relatively long B-lifetime [ 14 ] Za/10-12
s=0.6-2.5
(18)
implies [ 15 ] 0.04~< JKcblexo~<0.08.
(19)
The significant difference in the magnitude of (17) and (19) can be easily understood in the framework of our model. Indeed, from (16) we predict :5 This should be distinguished from expansions of the weak mixing matrix [ 13].
185
Volume 177, number 2
Ig, bl/Igudl =(RJR,) sin 0 .
PHYSICS LETTERS B
(20)
Since R~ and R2 are roughly o f the same order o f magnitude, the suppression o f the b ~ c transitions is due to the smallness o f ¢ (or to the relation s/b~-c/t). The agreement between the F r i t z s c h - S t e c h model and experiment can be m a d e quantitative. The experimental range in (19) implies an u p p e r and lower b o u n d on 0. Using (14), this translates into a range o f top-quark masses in which the m o d e l reproduces (18) and (19). W i t h A4= 100 MeV we o b t a i n the prediction 24 GeV~
(21)
which is consistent with (13b). The selected range in (21 ) and the masses in (13a) allow us to d e t e r m i n e K,u. O u r p r e d i c t i o n is 0.002~< [K, bl ~<0.013,
(22)
to be c o m p a r e d with the e x p e r i m e n t a l value [ 16] IK, blexo~<0.013.
(23)
Eqs. (21) and (22) are the m a i n predictions o f our model. The former is, in practical terms, an upper b o u n d on the top-quark mass o f 45 GeV. O u r prediction for the rate o f the b--,u transition is close to its experimental limit. Eq. (22) will be tested by future experiments seeking for b--,u transitions. We finally address our attention to the CP-nonconservation originated in the mass matrices (3). It is the phase-diagonal matrix P which carries the information on CP-nonconservation. This can be seen in (10). In the limit u = 0 (sin x = 0 ) a n d sin 0 = 0 we get P=e
in/6Diag(i, 1, 1).
(24)
The n/2-phase difference in the elements o f P makes the invariant phase defined in ref. [10] to be z~/2. Gronau, Johnson and Schechter, which were the first to suggest the c o m b i n e d F r i t z s c h - S t e c h model, have studied the relationship between this m a x i m a l value and the quark spectrum [9]. Corrections to (24) are small and calculable in our model. Some pieces o f these corrections control the non-conservation o f CP in d e t e r m i n e d systems. F o r instance, to describe CP-violation in the K°-I~ ° complex we need to go b e y o n d the leading terms displayed in the weak mixing matrix (16). We need to 186
11 September 1986
know the small phases which will a p p e a r in the c - s and t - s flavor mixings. These are easily evaluated. We find Kcs= 1 + i 6 ,
Kts=RzsinO+i~',
(25)
with 3=R~sin 0(R~+sin 0 sinz) , ~'= -R2(R~+sin ~ sinz) .
(26)
These equations reflect the fact that the CP-violating phases can also be expanded in terms o f our four basic parameters. We have checked that (25) and (26) are consistent with the experimental CP-nonconserving parameters ~. This will be'presented elsewhere [ 17 ]. After this work was c o m p l e t e d we received two reports where the phenomenology o f a fourth generation in the F r i t z s c h - S t e c h model is studied [ 18 ]. I a m indebted to Jon Bagger for helpful remarks. This work was partially supported by the research projects CAICYT.
References [ 1] See, e.g.H. Harari, in: Proc. twelfth SLAC Summer Institute, ed. P.M. McDonough (1984). [2] H. Fritzsch, Phys. Lett. B 73 (1978) 317; Nucl. Phys. B 155 (1979) 189; see also L.-F. Li, Phys. Lett. B 84 (1979) 461. [3] M. Shin, Phys. Lett. B 145 (1984) 285; H. Fritzsch, Phys. Lett. B 166 (1986) 423. [4] B. Stech, Phys. Lett. B 130 (1983) 189. [ 5] H. Georgi and D.V. Nanopoulos, Nucl. Phys. B 155 (1979) 52. [6] S. Dimopoulos, Phys. Lett. B 129 (1983) 417; J. Bagger, S. Dimopoulos, H. Georgi and S. Raby, in: Proc. Fifth Workshop on Grand unification, eds. K. Kang, H. Fried and P. Frampton (World Scientific, Singapore, 1984); H. Georgi, A. Nelson and M. Shin,, Phys. Lett. B 150 (1985) 306. [7] A. Davidson, V. Nair and K. Wali, Phys. Rev. D 29 (1984) 1504, 1513; V. Nair, L. Michel and K. Wali, Phys. Lett. B 138 (1984) 128. [ 8 ] B. Stech, in: Flavour mixing in weak interactions, ed. L. L. Chau (Plenum, New York, 1984). [ 9] M. Gronau, R. Johnson and J. Schechter, Phys. Rev. Lett. 54 (1985) 2176. [ 10] M. Gronau and J. Schechter, Phys. Rev. Len. 54 (1985) 385, 1209 (E). [ 11 ] J. Gasser and H. Leutwyler, Phys. Rep. 87 (1982) 27.
Volume 177, number 2
PHYSICS LETTERS B
[ 12 ] T.P. Cheng and L.-F. Li, Phys. Rev. Lett. 55 ( 1985 ) 2249. [ 13] L. Wolfenstein, Phys. Rev. Lett. 51 (1983) 1945. [ 14 ] E. Fern~indez et al., Phys. Rev. Left. 51 ( 1983 ) 1022; N.S. Lockyer et al., Phys. Rev. Lett. 51 ( 1983 ) 1316; M. Althoffet al., Phys. Lett. B 149 (1984) 524; D.E. Klem et al., Phys. Rev. Lett. 53 (1984) 1873; W. Barrel et al., preprint DESY 86-001 (1986).
11 September 1986
[15] See, e.g., E. de Rafael, in: Lectures on quark flavor mixing in the standard model, preprint MPI-PAE/PTh 72/84 (1984). [16] C. Klopfenstein et al., Phys. Lett. B 130 (1983) 444; A. Chen et al., Phys. Rev. Lett. 52 (1984) 1084. [ 17 ] E. Mass6, in preparation. [ 18] K. Kang and M. Shin, Phys. Lett. B 165 (1985) 383; R. Johnson, J. Schechter and M. Gronau, Phys. Rev. D33 (1986) 2641.
187