An analysis of quark mass matrices and B-B mixing

Nuclear Physics B306 (1988) 14-50 North-Holland, Amsterdam

AN ANALYSIS OF QUARK MASS MATRICES AND B-B MIXING Yosef NIR of Science, and Stanford Accelerator Center, Stanford University, Stanford, Cal. 94305, USA

Received 7 August 1987 (Revised 15 February 1988)

We study the implications of the recent B,-%, mixing data on the quark sector parameters - mt, sis and 8 - within the three-generation standard model. We derive limits on several experimental quantities: r,, al/e, branching ratios of rare K and B decays. The Fritzsch scheme for quark mass matrices is consistent with the experimental data only if several experimental and theoretical quantities assume values at the limit of their allowed ranges. This allows a unique solution: m, - 85 GeV, sts - 0.0035, 6 - 100°, with many definite predictions. The Stech scheme and its extension by Gronau, Johnson and Schechter are inconsistent with the experimental constraints. We study how a direct measurement of m, or si3 will improve our constraints and predictions.

1. Introdnctlon

The standard model with three generations of fermions has undergone numerous experimental tests. All these tests have confirmed again and again the validity of the standard model predictions of low-energy phenomena. The predictions of the minimal standard model can all be given in terms of eighteen independent parameters. Of these, ten are related to the quark sector of the theory. The ten parameters describing the quark sector of the three-generation standard model consist of six quark masses, three mixing angles and one Kobayashi-Maskawa (KM) phase. The quark masses are best determined from meson spectroscopy and chiral perturbation theory. The mixing angles are best determined from flavorchanging tree-level decays of mesons. We call these “direct ” measurements. Further information on the value of the parameters comes from processes which only occur at the loop level of the standard model. We call such information “indirect” measurements.

Seven of the quark sector parameters are relatively well known. These are five of the quark masses (given here in GeV) md - 0.009,

m,-0.15,

mu-0.005,

m,-

1.5,

0550-3213/88/$03.50OElsevier Science Publishers B.V. (North-Holland Physics Publishing Division)

mb-5,

0.1)

Y. Nit / B-B mixing

15

and two of the mixing angles s12 - 0.22,

s:3 - 0.05.

(1.2)

We gave here only the order of magnitude of these parameters. Later we comment on the accuracy of our knowledge of some of them. All the above values result from direct measurements. Three parameters are rather poorly determined. The top quark has not yet been found. There is a lower limit on its mass, coming from its non-observation in e+e annihilations: m t >I 23 GeV. If the difference between the squared masses of the t-quark and the b-quark were too large, the predicted value of sin20w would deviate too much from its measured value. This gives [1] m t ~<180 GeV. We will refer to the bounds 23 GeV ~
(1.3)

as "direct" bounds, although the upper limit comes from an indirect measurement. The third mixing angle, s13, has only a direct upper limit on its value s13 ~<0.011.

(1.4)

We will later explain how this limit is determined. The phase 8 has only very loose bounds on its value, coming from the measurement of the CP-violating parameter e. Recently, the ARGUS collaboration has found a positive evidence for Bd-Bd mixing [2]. This result significantly contributes to our knowledge of the quark sector parameters. The Combination of the B-B mixing data and of the e value gives constraints on the allowed values of the three poorly determined parameters. In the near future, a large amount of additional information is expected to be found from further studies of B physics. Another possible source of information is the continuous search for the top quark. The first purpose of this paper is to analyze the implications of the existing ARGUS data on the quark sector parameters and to understand how near future experiments will further restrict the allowed domain of these parameters. The implications of the B-B mixing data on the standard model parameters were previously discussed in several papers [3-6]. We, here, give a much more detailed analysis and a broader scope of results. In the standard model, mixing angles and quark masses are unrelated. Several models beyond the standard model suggest special forms of mass matrices. These mass matrices have less than ten independent parameters, thus giving relations among quark masses, mixing angles and phases. Consequently, they are more constrained than the standard model, and one could check if their predictions are consistent with the experimental values of the quark sector parameters. In this paper, we consider four such schemes: the Fritzsch scheme [7] with eight indepen-

16

Y. Nir / B-B mixing

dent parameters, the Stech scheme [8] with seven independent parameters, the Fritzsch-Shin scheme [9] and the Gronau-Johnson-Schechter (GJS) scheme [10], each with six independent parameters, We check whether the predictions of these mass matrices are consistent with the new ARGUS results and with all earlier data. For the schemes which do not fail to account for the observed B-B mixing, we study their predictions for the results of future experiments. The general lines of our calculations and our main results were given in a previous paper [5]. Here, we give the details of these calculations, a much more detailed explanation of the different considerations and additional results and predictions. The paper is organized as follows. In sect. 2 we describe the constraints on the quark sector parameters. We emphasize the ambiguities involved in inferring values for s23 and s13 from direct measurements. We review the constraints on mr, $13 and 8 coming from th~ e measurement, and carefully analyze those resulting from the B - B mixing measurement. In sect. 3 we give predictions for future experiments (Bs-Bs mixing, direct CP violation and rare K and B decays), based on the allowed range of parameters derived in sect. 2. In sect. 4 we explain the idea of schemes for quark mass matrices. We explain the relations between the "physical" quark masses (involved in the physical processes which serve as measurements for mixing angles) and "running" quark masses (which appear in the mass matrices). In sect. 5 we study the consistency of the Fritzsch scheme (and its extension by Shin) with the new experimental data. In sect. 6 we study the consistency of the Stech scheme (and its extension by GJS) with these data. In sect. 7 we give our conclusions, and describe further experimental tests expected in the near future.

2. The quark mixing matrix in the standard model 2.1. PARAMETRIZATION

We use an approximate form [11,12] for the CKM matrix

V =

1

S12

S13e - i S

--S12--S23S13 ei8

1

S23

$12S23--$13 ei8

--$23

1

,

(2.1)

where sij = sin 0,7. The second term in Vcd is only relevant in the calculation of the CP-violating parameter e. 2.2. "DIRECT" MEASUREMENTS

The main advantage of the parametrization (2.1) is that direct measurements of the matrix elements Vus, Vcb and Vub are trivially translated into the values of st2, SEa and sx3, respectively.

17

Y. Nir / B-B mixing An analysis [13] of the K¢3 and hyperon decays gives for s12 = [V~I: S12

=

0.220 -4- 0.002.

(2.2)

The value of s23 = [V~b[ can be extracted from the semileptonic B-meson partial width [14-19]. We follow the calculation by The Particle Data Group [19] using the free quark model. Model dependence and QCD corrections are ignored (for the calculation of QCD corrections see, e.g. ref. [20]). One assumes that the partial width is given by the b-quark W-mediated decay. Then (s23) 2 is given by

(S23)2 [ ~ where

F(m2/m 2) is

J[

'l'b

mbF(me/mb)

(2.3)

a phase space factor

F(x)

= 1 - 8x + 8x 3 - x 4 - 12x21n(x).

(2.4)

We use the experimental data [17] BR(b ~ cE~e) = 0.121 + 0.008,

(2.5)

% = (1.16 _+ 0.16) × 10 -12 s,

(2.6)

and [18]

The quark masses m b and m e that should be used in our different calculations are subject to m a n y theoretical uncertainties [15,17]. We use mc = 1.5 -4- 0.2 GeV,

m b = 5.0 ± 0.3 GeV.

(2.7)

However, it is unlikely that in any single calculation we should use m c and m b at their opposite limits. The meson masses are close to the respective upper limits in eq. (2.7), while the physical quark masses are close to the lower limits. Thus, for the ratio mJmb, relevant to eq. (2.3), we take a smaller range

mc/m b -----0.30 -4-0.03.

(2.8)

Using the full range of the ratio (2.8) we get

F(m2/m 2) -- 0.52 -4- 0.07.

(2.9)

Adding the errors in quadrature we get from eq. (2.3) (s2~) 2= (1.85,4,0.65) × 1 0 -3,

(2.10)

Y. Nir / B-B mixing

18

which gives $ 23

= 0 . t~43 +0"°°7 v - 0.009 •

(2.11)

We note that our analysis will be rather sensitive to the upper bound on Sz3. The non-observation of charmless B-decays [17, 21]

r(b -~ ue~,) r(b -~ c¢~t) gives an upper bound on sx3 =

0.08

(2.12)

I Vubl

s13 ~ [0.08F(mE/m E)] 1/2s23.

(2.13)

In order to avoid the large uncertainties involved in the determination of s23, we will use the ratio between s13 and s23 rather than s13 itself

IVubl q - Vcb---T I -where we used

sx--2~ [O.08F(m2/m2)]l/2~0.22,

(2.14)

$23

F(m2/m 2) ~ 0.59. As

s23 ~ 0.050

s13 -<.<0.011.

(2.15)

This bound strengthens the validity of the approximation we made while replacing {Vus[, [ Veb[ and I Vub{ with s12, s23 and s13; this approximation is actually good to order 10 -4 . To summarize, direct measurements give sx2 = 0.220 + 0.002,

s23 = n"-"n43 +0.0o7 -o.009,

s13 ~<0.011,

(2.16)

We will often use (s23) 2 = (1.85 d: 0.65) × 10 -3 ,

$13

q = - - ~< 0.22.

(2;17)

S23

2.3. " I N D I R E C T "

MEASUREMENTS

The experimental results that we use to further limit the allowed range of the quark sector parameters are the measurements of the CP-violating parameter e, and the B - B mixing parameter x d. The standard model contributions to these parameters come from box diagrams and are GIM-suppressed. Relating these experimental results to the CKM parameters depends on the assumption that there are no significant contributions from diagrams beyond the standard model.

Y. N i r / B-B mixing

19

2.3.1. CP-violation: e. The expression for e [22] with the parametrization (2.1) is

lel

-- C . B K • [$2312q sin8 { [ 7/3f3(Yt) - 7/1] ycsl2 + r12Ytf2(Yt)(s23)2[s12 - qcos 8] }, (2.18)

where yi = M E ,

i = c,t,

]

3 yt(1 +y,) 1 + f2(Y,) = 1

4 (1 - y t ) 2

f3(Yt)--ln

~

4 1 -yt~ 1 +

'

ln(y t

,

G~f2MK M2 C -= 6~r 2¢~- A M K

(2.19)

Eq. (2.18) gives a relation among the three undetermined parameters of the quark sector, (rot, q, 8). Other parameters in eq. (2.18) may be divided into two groups (a) Parameters which are known to a high accuracy (uncertainties of a few percent or less). These are s12 (eq. (2.2)) and

lel

- 2.3 × 10 -3,

C=4x 171

=

0.7,

10 4 , 712 = 0.6,

~/3 = 0.4,

(2.20)

T h e three parameters 7/i are Q C D corrections [23]. The dimensionless constant C was calculated using G F = 1.166 x 10 -5 GeV -2 , f ~ = (0.16 GeV)2;

M w = 82 G e V ,

M K = 0.498 G e V ,

A M K = 3.52 X 10 -15 G e V .

(2.21)

(b) Parameters with large uncertainties. These are m c (eq. (2.7)), (s23) 2 (eq. (2.10)) and the B K parameter, which is usually estimated to be in the range ~< B K ~< 1.

(2.22)

Y. Nit / B-B mixing

20

(o)

,\,)

m t =120GeV,

m t = 55 GeV, BK= I

0.4

BK=I

I

-

'~ 0 . 2

'

I

"

I

~

c;r

I

0

(b)

I

=0.20, BK=I

I

q =0.06,BK=I

2O0

~- ioo

0

0

50

I00

150 0

50

I00

150

8

Fig. 1. Examples of the e bounds. (a) Allowed range of q=s13/s23 and 8 for m t = 55,120 GeV and B K = 1. Heavy and light solid lines give, respectively, the central value and the edges of the band allowed by e. The dotted line is the direct upper limit on q. The shaded area is the allowed region. (b) Allowed range of m t and 8 for q=0.20,0.06 and B K = I . Heavy and light solid lines give, respectively, the central value and the edges of the band allowed by ~. The dotted lines are the direct limits on m t. The shaded area is the allowed region.

Either or b o t h of m t and q may be soon directly measured. Thus, we present the "e-relation" (eq. (2.18)) in two ways. (i) We fix m t and get a curve in the q - 8 plane. We show two examples in fig. la; in order to demonstrate the e-relation for both low and high values of m t we select m t = 55 G e V and m t = 120 GeV. We take here B K = 1. Using the central values, s23 = 0.043 and m c = 1.5 GeV, we get the central curve. Taking the maximal values, s23 = 0.050 and mc = 1.7 GeV, gives a lower bound on q, while taking minimal values, s23 = 0.034 and m~ = 1.3 GeV, gives an upper bound. Values of (8, q) within the b a n d (and below q = 0.22) are allowed. Comparing the two m t values, one can see that the higher the m t, the lower the curves. (ii) We fix q and get a curve in the m t-8 plane. We show two examples in fig. l b ; we choose q = 0.20 and 0.06 to show the e-relation for high and low q-values. Again, we take B K = 1. Taking for s23 and m¢ the central, maximal or minimal values, gives the central curve, a lower bound and an upper bound on mr, respectively. The region within the bounds and with 23 GeV~< mt~< 180 GeV is allowed. Lower q-values give higher curves.

Y. Nir / B-B mixing

21

We studied the implications of the e-relation for the whole range of m t and q in a similar manner to the one given in the above examples. Additional examples are given in figs. 4 - 9 (together with information from the xa data). The results by themselves are known. We present them here because we will later be interested in combining these results with those derived from the new B - B mixing data. 2.3.2. B-B mixing: x d. The A R G U S collaboration has recently observed Bd-Bd mixing with [2] rd = 0.21 ± 0.08.

(2.23)

We will use the parameter x d ----A M / F which is related to rd by

rd = 2 + x 2 '

(2.24)

6 2 Xd

_

- ~'b~

F

2

2

.

71MB( B nfn ) MwYt f2 ( Yt )l Vtd VtbI

2 Q

(2.25)

We neglect here the contributions of lighter intermediate quarks (u, c ) a n d corrections of order mb//mt 2 z due to external momenta. Note that while K-K. mixing depends on Re[Mlz(K°)], the B - B mixing depends on [Mx2(B°)l. In our parametrization [ Vtd]2 = (S23)2[S~2 + q 2 _ 2s12qcos 6] '

(2.26)

while I Vtbl2 = 1 to a high accuracy. Eqs. (2.25) and (2.26) give a relation among the three parameters rot, q, 8. Again, the other parameters in these equations may be divided into two groups. (a) Parameters which are well known. In addition to the previous parameters, GF, n w and s12, we now use M B = 5.28 GeV,

71= 0.85,

(2.27)

where , / i s a QCD correction [14]. (b) Parameters with large experimental errors (xa, %s223) or theoretical ambiguities (BBf~). From eqs. (2.23) and (2.24) we get

x a = 0.73 ± 0.18.

(2.28)

One should note that s23 and 'rb appear only in the combination [Tb($23)2], which does not depend on %, as can be readily seen from eq. (2.3). Therefore, the error on this combination is somewhat smaller than on (s2~) 2 alone %(s23) 2 = (3.25 ± 1.10) X l 0 9 GeV -x,

(2.29)

Y. Nir / B-B mixing

22

where errors on the quantifies in eq. (2.3) were added in quadrature. The hadronic parameter B B (analogous to B r of the kaon system) is believed to be close to 1. However, there is much uncertainty involved in the calculation of the B decay constant fB [24-28]. We will use

BBf~ =

(0.15 _+ 0.05 GeV) 2.

(2.30)

We note that our constraints on the quark sector parameters strongly depend on the upper limit in eq. (2.30). We can write

xa/C,=Ytf2(Yt)[s22 + q2_

2sl2qcos 8] '

(2.31)

2

(2.32)

where 2

C'-

[%(s23)] GF

2

bl 2

= 5 • 1 +7"° -3,-

We used here the full range of ['l'b(S23) 2] and Bnf 2. We will use eq. (2.31) with the numerical values (2.28) and (2.32) as our "xa-relation" among (mr, q, 8). As the product in eq. (2.29) does not depend on %, improved measurements of the b-hfetime will not affect the xd-relation, but may - rather paradoxically - affect the e-relation through a possible shift in s23 values. We present the xa-relation in two ways, demonstrated in fig. 2. (i) We select a value for m t (again, we take as examples m t = 55,120 GeV). This is shown in fig. 2a. Using x a -- 0.73 and C' -- 5.1 we get the central curve in the q-8 plane. The edges of the band are derived using (x d = 0.55, C' -- 12.1) and (x a = 0.91, C ' = 1.5). For the low m t the upper bound lies far above the direct upper limit q ~< 0.22 and is not shown in the figure. For the high m t there is no lower bound on q, namely q = 0 is allowed. The allowed (8, q) values are those within the band and below q = 0.22. (ii) We select a value for q. Again, we take as our examples q = 0.20, 0.06. This is shown in fig. 2b. Using x d = 0.73 and C' = 5.1 we get the central curve in the m t-8 plane. A lower bound on m t is derived from (Xd----0.55, C ' = 12.1). An upper b o u n d on m t is derived from (x d --0.91, C ' = 1.5). For low q values the upper bound lies far above the upper limit m t ~< 180 GeV, and therefore it is not shown in the figure. The allowed (8, mr) values are those within the band and with 23 GeV -N
Y, Nir / B - B mixing

23

(a) m t =55 GeV

m I = 120 GeV

I ~'~.

\1 0.4--

I

xNI

\

-

\

\

I

~ 4

\ \

\

ro

,,T 0.2 "'" .................... " : ' " ~ U ~ i ~ J ~ t~r

o

I

(b)

I

I

= 0.20

q =0.06

\

~

~-

200 .oo°,

fi- I00

..........

0

f ...........

I ...........

50

I00 8

........... i ...........

i ....... 150

0

50

i Y ......... I00

rT". 150

8

Fig. 2. Examples of the xa bounds. (a) Allowed range of q=st3/s23 and 8 for mt=55,120 GeV. Heavy and fight dashed lines give, respectively, the central value and the edges of the band allowed by x d. For m t = 55 GeV the upper bound is outside the graph: The dotted line is the direct upper limit on q. The shaded area is the allowed region. (b) Allowed range of m t and 8 for q = 0.20, 0.06. Heavy and light dashed lines give, respectively, the central value and the edges of the band allowed by x a. For q = 0.06 the upper bound is outside of the graph. The dotted lines are the direct limits on mt. The shaded area is the allowed region.

(b) No range of either q or 8 is excluded for all m cvalues. In particular, q = 0, = 0 is an allowed solution for large enough mr. The xd-relation by itself cannot give us direct information on CP violation. (c) The upper bounds on m t and on q are always less stringent than the direct bounds. 2.4. RESULTS

In the last section, we analyzed the relations among the three undetermined parameters of the quark sector, mr, q and 8. Now we combine the "e-relation" (eq. (2.18)) with the "xa-relation" (eq. (2.31)), and require that values of the three parameters would be consistent with both simultaneously. In addition they should, of course, obey the direct limits, 23 GeV ~
Y. Nit / B-B mixing

24

(a)

m t =55 GeV, BK= I

0.4

"\ ,',

-

m t = 120 GeV, BK= I

/'/F-

.........

J

", X

X

......... :"

J

....

............

~-~ 0.2 ii

I

0

b)

I

F--

=0.20, BK=I

T''='='I='~

I

q =0.06, BK=I

200

- 100 E

0

0

50

I00

8

150

0

50

I00

150

B

Fig. 3. (a) Allowed range of q=st3/s23 and 8 for m t = 55,120 GeV and B K = 1. The meaning of the lines are as in figs. l(a) and 2(a). The central solution (the intersection point of the two central curves) is denoted by a solid circle. For m t = 55 GeV the central solution lies outside of the direct bounds. The shaded area is the final allowed region. (b) Allowed range of m t and 8 for q = 0.20,0.06 and B K = 1. The meaning of the lines are as in figs l(b) and 2(b). The central solution is denoted by a solid circle. The shaded area is the final allowed region.

(b) The most likely values of (rot, q, 8). These are the intersection points of the central curves (provided they are within the direct bounds), corresponding to central values of all parameters in eqs. (2.18) and (2.31). As an example, we take the combined information of figs. 1 and 2, and show the results in fig. 3. With m t fixed (fig. 3a) at 55 GeV, the central solution lies outside the direct bounds (as is the case for all m t~< 82 GeV). The value of q should be above 0.13 and the phase 8 is in the range 105°-165 °. With m t = 120 GeV the central solution is q = 0.11, 8 = 155 °. Values of q above 0.03 are allowed. This limit is determined by the e-bound only, as is the case for m t >I 100 GeV. The phase $ should be in the range 40 ° ~< $ ~< 175 °. With q fixed (fig. 3b) at 0.20, the central solution is m t = 87 GeV, 8 = 162 °. Values of m t above 45 GeV and 35 ° <~ $ ~< 178 ° are allowed. With q fixed at 0.06, the central solution is m t = 151 GeV, $ = 138 °. Values of m t above 76 G e V and 25 ° ~/~ ~< 165 ° are allowed. As can b e seen from these examples, the x d bounds are significant for low m t values and for large q values. At large m t and small q values, most of the excluded region of parameters is a consequence of the e bounds.

Y. Nir / B - B mixing

25

mt =65GeV, BK=I

m t--45c~v, BK= I

,\,

,

y

I

I

I

",,I

r

I/

0.4

,WO.2

I

I

m =I80GeV, BK=I

mt=85GeV , BK= I

I

x

x

I

0.4 m

"D~ 0 . 2

]

0 0

I ~-I-I00 8

5O

150

0

50

I00 8

150

Fig. 4. Allowed range and central solutions of q=sl3/sz3 and 8 for m t = 4 5 , 6 5 , 8 5 , 1 8 0 GeV and B K = 1. Solid lines give the e bounds. Dashed lines give the x d bounds. The dotted line gives the direct bound. The central solution is denoted by a solid circle. The shaded area is the final allowed region. m r = 6 5 GeV, BK=0.7

m t = 4 5 G e v , BK=0.7

0.4

\

"-. /

• °, •, • o

~'0.2

\, /

~

r

o

.

i

.

~

.

i

I

m =85 GeV, Bg=0.7

0.4

~ n o-

I

I

m t =180 GeV, BK=0.7

i

0.2

0

50

I00 8

150

0

50

I00 8

150

Fig. 5. Allowed range of q = s13/s23 and 8 for m t = 45, 65, 85,180 GeV and BK = 0.7. The meaning of the lines, the solid circle and the shaded area are as in fig. 4.

Y. Nir / B-B mixing

26

ml = 45 GeV, BK=0.4

I

I\

mr=65 GeV, BK= 0.4

I/

"\l

/

0.4

I~

I/

I

I

u)

& 0.2

._" • . . . . . . . . . •~ . . . L "

"".2"

i

I

I

I

I

m t =85GeV, BK=0.4

m t =I80GeV, BK=0.4

0.4

17

",,

/

"-.

/

. . . . . .

o.z

\

o

I 0

50

t" ~ - I IO 0

0

150

1

I

I

50

I O0

150

8

Fig. 6. Allowed range of q = s13/s23 and 8 for m t 45,65, 85,180 GeV and BK = 0.4. The meaning of the lines, the solid circle and the shaded area are as in fig 4. There is no allowed region for m t = 45 GeV. =

W e use the same m e t h o d to give results for several values of m t and q. In figs. 4 - 6 we show the allowed regions in the q - 8 plane for m t = 45, 65, 85,180 G e V with B K --- 1, 0.7, 0.4. W e also give the central solutions. I n figs. 7 - 9 we show the allowed regions and the central solutions in the m t - 8 p l a n e for q = 0.18, 0.14, 0.10, 0.04 with B K = 1, 0.7, 0.4. F o r each selected value of q we derive a lower b o u n d on m t" T h e collection of these points f r o m all q values gives a curve in the m t - q plane. This curve, shown in fig. 10, c a n b e used in two ways. (i) O n c e we have a value for q (or an u p p e r bound), the curve gives the lower b o u n d on m t. (ii) O n c e we have a value for m t (or all u p p e r bound), the curve gives a lower b o u n d on q. If the actual q and m t lie b o t h on this curve, it is required that all parameters, s23, m c, x d a n d B n f ~ , should simultaneously have values which correspond to their e x p e r i m e n t a l or theoretical limits. T h e collection of all central solutions gives another curve. It is m o r e likely that we w o u l d find q and m t on this curve, as it corresponds to central values of all p a r a m e t e r s involved. This curve is also shown in fig. 10.

Y. Nir / B-B mixing q =0.18, BK=I

27

q =0.14, BK=I

200

,oof \ or"

.... q-'O.lO, BK=I

=0.04, BK=I

200

-I00 E

o

........... 0

r .......... I.......... 50 I O0

"l............... 150 0

i .......... i .......... r 50 I O0 150 8

....

Fig. 7. Allowed range of m t and 8 for q=0.18,0.14,0.10,0.04 and B K = I . Solid lines give the e bounds. D a s h e d lines give the x d bounds. Dotted lines give the direct bounds on m t. The central solution is denoted by a solid circle. The shaded area is the final allowed region. =0.14, B =0.7

q =0.18, BK = 0.7

200

I00

" "° f ""° ~........""°'*"°"

£ ,..o°°o°°..,°°°,...°.°°°°.°o°..°.°.o°

I q=O. lO, BK=0.7 200

I

I

I

I

q= 0 . 0 4 , BK =0.7

_-:°

. . . . . .

I00 g i . . . . . . . . . . . f ........... i . . . . . . . . . . . i ....... 0 50 I00 150 0 B

. . . . . . . . . .

f . . . . . . . . . . . r . . . . . . . . . . . I ....... ,50 I00 150

Fig. 8. Allowed range of m t and 8 for q = 0.18,0.14,0.10,0.04 and B K = 0.7. The meanlng of the lines, the solid circle and the shaded area are as in fig. 7.

Y. Nir / B-B mixing

28 =0.18,

BK=

q : 0.14. BK = 0.4

0.4

250

100

**°.°°~**°°°o°.°°o°°ooo°o°.°°o°°°...°oo°°,

= 0.10,

200

q =0.04.8K= 0.4

BK = 0.4

:..,~, . . . . . . . . .

.... • °..o ...... .oo..o.. ...... °,....°°.°,

I00 °°.°°°..°°°°.°o°.°°°°°°°°°°°°o°°.o°°

0

.....

I

I

I

50

I00

150

o...o°~°°°...°°°oo°oo°°

0

......

°°o°..°°.°°°,

I

I

I

50

I00

150

Fig. 9. Allowed range of m t and ~ for q = 0.18,0.14,0.10,0.04 and BK = 0.4. The meaning of the fines, the solid circle and the shaded area are as in fig. 7. For q = 0.04 there is no allowed region.

T o summarize, we get the following bounds on m t and q 43 GeV ~< m t ~< 180 G e V , 0.02 ~< q < 0.22.

(2.33)

However, the m t and q values are more likely to be found in the range 83 GeV ~< m t ~< 180 G e V , 0.04 ~< q < 0.22. 8

~00 - -

=1.0 i

(2.34)

BK=0.7 I

i

I =

i

B I

'

I;

=0.4 i

I'

=

I:

•~ ~%~!~!~.~..~,:~.,.~.,..~..,:?,~. . . . . . ,~.~,.,,.o.,,.,,..~.~.~.,~.~.~,." . . . . . . . . ~:~,.~..,.~~ . . , . ~ : ~i~!i!::i~i!i~H~#~:i::i::~!ili~i~ - ....:::::::::::::::::::::::::::::::::::" ................. ~.......................... ~:::~':':"!i~:~. ~ i '............. : ~ : ~ ~:':::~*::::::~ ...........

150 100

-

50 ,..oo....o.ooo.......o,..oo~.o~..°....ooo..ooo..oooo....o°

,

0

I

0.1

i

I:

0.2

t

0

I

:.oo

I

0.1

,°,...°°°.°°°°..°°.°°°o°°%,°°

I-:

0.2

i

0

I

0.1

i

l: 0.2

q = $13/S~.3

Fig. 10. M o w e d range (shaded) of q and m t for BK = 1, 0.7, 0.4 and the curve (dot-dash) representing the central values. Dotted fines represent direct bounds while the solid curve is the indirect bound from the combination of the e and the Xd bounds.

29

Y. Nit / B-B mixing

The phase 8 is probably in the range 108°-175 °, but values as small as 20 ° and as large as 178 ° cannot be excluded.

3. Predictions There are several experimental quantities that depend on the values of the unknown parameters (rot, q, 8). However, at present they cannot constrain these parameters for two possible reasons. Either the theoretical calculations of these quantifies involve additional uncertainties, or the experiments have not yet reached the needed level of sensitivity. Thus, instead of getting constraints we give here predictions for these processes, based on the bounds that we have derived. 3.1. Bs-~ MIXING: Xs Although the calculation of both x a and x~ is subject to large uncertainties, the ratio between these quantities is expected to be well-approximated by Xs Iv l 2 x-~ = -[Vrd '-~"

(3.1)

One assumes here that the lifetime %, the mass rob, the B B parameter and the decay constant fB, are all equal for Bd and Bs. In our parametrization Xs

($23) 2

1

X"-d= ]St2S23-- st3eiSI2 = 1s12-- qeia[ 2"

(3.2)

The ratio is a function of q = $13/$23 and 8 only, and is independent of m t and s23. This ratio is minimized when 8 is taken as close to 180 ° as possible (the phase 8 cannot be exactly 180 ° because it would lead to e = 0) x s

Xd

>1

1

>/4.7,

(3.3)

(S12 + q)2

where the second inequality results from q .<<0.22. This gives xs >1 2.6 ~ rs >/0.77.

(3.4)

Thus, a large Bs-Bs mixing is expected, independently of m r Actually, for most of the range of the parameters we expect the mixing to be near-maximal. With O - 90 °

Y. Nir / B - B mixing

30

1.0 0.8 0.6 r$

0.4 0.2 o

I

o

I 0.1

I

I 0.2

q=s13/S23 Fig. 11. Lower limits on rs for 8 - 180 ° (solid line) and 8 - 90 ° (dotted line) as a function of q = st3/s23.

we get rs >1 0.94. In fig. 11 we give lower limits on rs for 0.02 ~
3.2. D I R E C T CP-VIOLATION: ~ ' / ,

The contribution from the "penguin" diagram to e' gives [16] e --

-

6.0C"

,

~

(3.5)

S12

where C"

[ Im~6][ = [ ----0~.1J[ 1.0 GeV 3

(3.6)

The "penguin" operator is Q6, while the Wilson coefficient is C6, where the CKM factor was explicitly taken out. The calculation of both C6 and 0r~rlQ61K °) is subject to major theoretical ambiguities, on which we have nothing to add here. Our analysis concerns only the product of the mixing angles and the phase. The CKM factor is maximal for s23 ~ 0.05, q ~ 0.22 and 8 dose to 90 °. This would give g / e - 1.5 × 10-2C ''. A lower limit on the CKM factor is derived when q i s at its lower limit - 0.02 (in which case sin 8 - O(1) from the e bound). We get e'/e - 8 x 10-4C ''. This lower limit is determined by the e bound, so at this stage the B - B mixing data do not improve the predictions. We will, however, return to the subject when we study the Fritzsch scheme in subsect. 5.4.

Y. Nir / B-B mixing

31

The three recent experimental results are [29] ~f

-- = ( - 0 . 4 6 ± 0.53 ± 0.24) × 10 -2, E

~J

-- = (+0.17 + 0.82) × 10 -2,

-- - (+0.35 + 0.30 + 0.20) × 10 -2.

(3.7)

£

3.3. K+"-' or+ p~

The branching ratio for the decay K + ~ ~r+p~ is [22]

(3.8)

3 Iz3.2v;v~z~(ej, z,)l 2

BR( K÷-~ ~r+~'~) = 2x2BR( K+--> rr°e+~'e) E i-1

IV.~l ~

'

where z i = [m( gi)]Z/M2w and

D(y,z)=

8 z-ykl-z] 1

+~

y y__( 4-y z- y ~ l - y

ln(z)+¼Y+~s 1

)2 3] + 1+

1-z

1-y

yln(y).

(3.9)

-~(1

We use [19] a

X= 4~r sin20w = 0.0025, B R ( K + ~ r ° e + p e ) =0.048,

(3.10)

and get [22] B R ( K + ~ ¢r+p~) -- 6.0 ×

lO-7Ebi,

(3.11)

i

where

]Vus]2

[

----D(Yc, zi)+ s2....~ 3s12 VtdD(Yt'gi)

(3.12)

The z i can be put to zero, except for z, in D(yc, z,), where the non-negligible mass

32

Y. Nit

/ B-B

mixing

of the ~" gives a smaller D

D(yc, ze)=D(y¢,z,)=3.9XlO -3,

D(yc,z,)---3.1XlO -3.

(3.13)

The second term in/3,, depends on all three parameters (mr, q, 8). The two terms in /3~ give comparable contributions. To get an upper limit on the branching ratio we replace (in eq. (3.12)) Vtd with I~dl and put for s23 and I Vml their upper limits. An upper limit on I Vtdl is derived from the direct limits on s23 and s13, I Vml ~< 0.022. An indirect upper limit on I Vml is derived from the xd-relation (eq. (2.25)) with x d = 0.91, % = 1 ps and Bar 2 = (0.1 GeV) 2. It is more stringent than the first limit only for m t >1 140 GeV. The upper limit on the branching ratio is shown in fig. 12a. It varies from 9 x 10 -zl for m t = 45 GeV to 5.3 × 10 - l ° for m t = 180 GeV. A lower limit is derived by taking the lower limits of both (s23/s12) IVtd] and (s2Js12)Re(Vtd). We note that the lower limit on the real part can be considerably smaller than that on the absolute value. The lower limit on I~dl is derived from eq. (2.25) with x d = 0.55, % = 1.32 ps and BBf 2 = (0.2 GeV) 2. As for Re(Vtd), we numerically searched for the minimum in all the allowed range of the parameters (s23, q, 8) and checked for consistency with the lower bound on I Vtd]. The lower bound on the branching ratio is also shown in fig 12a and is in the range

(6-7) × 10 -zz. T~6

I

I

I

I

I

i

~8-I-

÷t 4

}

re.

............. 7

~ 4

m 2

f

Ib~

x

9_

° o

I

i

I

i

I

_o

i

~

i

I

i

/

"~10 J:l

1

_ 1I

J~

(dl

0

J

0 50

I I00 mt

I

I 150

(GeV)

I

200

0

i

50

I00 mt

150

!00

(GeV)

Fig. 12. (a) Upper and lower limits on BR(K +---, ~+pF) as a function of mt. Co) The upper limit on the short-distance contribution to BR(K°L--,/~ +/~-) as a function of mt (solid line). The dotted line gives the experimental value for this branchlnE ratio. The dot-dashed line gives the calculated upper limit on the short-distunce contribution. (c) The branching ratio BR(b-~ sv~) as a function of m t. (d) The branching ratio BRCo-* s'r) as a function of mr. The solid line gives the results without QCD corrections. The dashed line includes QCD corrections, with the assumption m t "~ Mw.

Y. Nir / B-B mixing

33

The limits on B R ( K + ~ ¢r+t,~) were previously studied in ref. [3]. The range of parameters used here and in ref. [3] is slightly different. When this difference is taken into account the upper limits agree, but our lower limit is lower by approximately a factor of 2. For the central values of the parameters we find BR(K + ~ ¢r+~,~) = (1.4-1.8) × 10 -1° . The present experimental upper limit is [19] B R ( K + ~ ¢r+~,~) ~< 1.4 × 10 -7.

(3.14)

3.4 K ° ---,/~+~tThe short-distance contribution to the decay K ° ---,/t+/~- is given by [22]

BR( K °

#+/~-)SD =

-"

4 2~'(K°)BR(K+~/~+p~,)tReE3-2Vj*VjdC(yj)j2[ ] (3.15) x ,(it+) i v~l~ ,

where

~_y ln(Y)+¼Y+n(l_y ~ .-

C(y)=

, 16,

In addition to the previous parameters we now use [19] BR(K + ~ #+p~,) = 0.64, 'r(KL) = 5.2 X 10 -8 S, ~'(K +) = 1.2 X 10 -s s.

(3.17)

We get BR(K°~#

-F

--

/~ ) S D = 6 . 7 × 1 0 - 5 C ,

(3.18)

where

6=

, Vus,2

s Re(Vui) C(y,)] -- [C(yo) + 1232

(3.19)

C(y¢)

We find = 3.3 x 10 -4 and C(yt) = O(1) so, as in the case of K ~ ¢rJ,~, the two terms in C are of the same order of magnitude. We find the limits on the short-distance contribution to the branching ratio in methods similar to those described in subsect. 3.3. The upper limit is shown in fig. 12b. It varies from 2 x 10 -x° for m r = 4 5 GeV to 5.6 × 10 -9 for m r = 180 GeV. The experimental result is [19] BR(K ° ~/~ +#-) = (9.1 + 1.8) × 10 "9.

(3.20)

Y. Nit / B-B mixing

34

However, one usually assumes [30] that the important contribution to this mode comes not from the short-distance diagrams but from the two-photon intermediate state. Subtracting the calculated two-photon contribution implies [30] that the short-distance contribution should be less than 5.6 × 10 -9. Even our upper limit is consistent with this bound. The central solutions of (mr, q, 8 ) give B R ( K ° ~ P+/~-)SD = (4.8-13.5)× 10 -1°, and a lower limit can be put of order 1 × 10 -x°. These results indeed support the importance of the two-photon intermediate state to this decay. 3.5. b ~ sp~ The branching ratio for the decay b --+ sv~ is given by [31] 3 *I~jbD(Yy, zi)[ 2 Zj.EVjs B R ( b - ' sv~) -- ×2BR(b-' (u'c)¢~+) i=t ]~ IV.hi 2 + t(mE/m2)tV~bl 2" (3.21)

a

Using BR(b ~ (u,c)e~e) = 0 . 1 2 ,

(3.22)

BR(b ~ sv~) = 7.5 × 10 -7 X 3/)B,

(3.23)

we get

where ^

D B-

~ . 2 Vj*V}bD(Yj, z = 0)12 = i Vubl2 +

F(m2/m2)lVcbl2

[D(Yt,O)-D(yc,O)] 2 [D(Yt,0)] 2 -- F( m2/m 2)" q2 + F(m2/m 2) (3.24)

As the contribution of the c-quark is very small, the effect of putting m.+.--,+0 is negligible. The branching ratio is, to a good approximation, a function of m t only. This is shown in fig. 12c. The branching ratio varies from 2 × 10 -6 for m t = 45 GeV to 4.6 × 10 -5 for m t = 180 GeV. The quark level decay discussed here may be realized in the hadronic level by B ~ Kv~, or by more complicated final hadronlc states such as K~r, K~tr etc. 3.6. b ~ s y The branching ratio for the decay b -~ sy is given b y [32]

3a

IE~_2 ~**~bFz(yj) l~ IV.hi 2 + F(me/mb)l V~bl '

(3.25)

Y. Nir / B-B mixing

35

where

F2(Y)=

2 3(y-1)

y 2 ( 1 - 2~y) 7 1 ] In(y). + 4 ( y - 1 ) 2 + 2 ( y - 1) 3 y + ( y - l ) 4

(3.26)

Using the known parameters we get BR(b ~ sy) = 4.5 x 10-4F

(3.27)

where

:=

E~=2Vy*VjbF2(Y/) I2 ivobl2 + F(m /m ,)lV bl

[F2(Yt)-F2(Yc)]2 =

,: +

[F2(yt)]2.

(3.28)

=

In the last approximation we used F2(yc)= 1.9 x 10 - 4 << F2(Yt) = O(10-1). For m t < M w one cannot ignore QCD corrections. The main effect is to replace [33]

F2(Yt) --* [F2(Yt) +

4 ?In(yt/y.)],

(3.29)

where YB = 342/M2. The QCD correction always dominates the F2(Yt) term. Again, we are in a situation where the branching ratio depends on m t o n l y . This is shown in fig. 12d. The dashed line shows the results with the QCD correction. This calculation becomes unreliable when m t approaches M w. The branching ratio varies from 1 x 10 -5 (8 x 10 -5 with QCD corrections) for m t = 45 GeV to 1.3 x 1 0 - 4 for m t ---- 180 GeV. The quark level process b ~ s't is realized in the hadronic decays B ~ KU)y, where K u) are excited states of the kaon. The decay into the ground state K ° is forbidden by angular momentum considerations. The calculation of the exclusive modes involves further hadronic uncertainties [34]. In conclusion, we analyzed the implications of the parameters ( m r , q, 8 ) o n several experimental measurements. The rs and e'/e values are determined by q and 8. The rs parameter is independent of m t, while e'/e has only a weak (logarithmic) dependence on mt. The rare B decays are determined by m t and insensitive to q values. The rare K decays are functions of all thre~ parameters. 4. Relations among masses and angles 4.1. SCHEMES FOR MASS MATRICES

Within the standard model (SM), the quark sector is described by ten free parameters. In the physical (mass) basis, these are the six quark masses, threa mixing angles and one phase. These parameters can all be experimentally determined. Whatever their experimental values are, the SM remains self-consistent.

36

Y. Nir / B-B mixing

In the interaction basis, our parameters are entries of the yet undiagonalized mass matrices. If we had some theoretical principle from which we could determine the mass matrices, we would predict the values of the physical parameters. In several schemes of mass matrices, the number of independent entries of the mass matrices is less than ten; either some entries vanish or there are relations among the nonvanishing entries. These schemes provide us with relations among quark masses, angles and phases (the motivation for such relations is discussed, for example, in ref. [351). One should check that these relations are consistent with the experimental data. If the relations suggested by a certain scheme are not compatible with the experimental constraints, then either the scheme is incorrect, or the use of its predictions should await the finding of additional new physics. We note that the existence of new physics beyond the SM is inherent in the suggestion of schemes for quark mass matrices. The validity of our discussion lies in the assumption that this new physics itself does not significantly contribute to CP-violation and B-B mixing. This is the case, for example, if the new physics takes place at a sufficiently large energy scale. We discuss four schemes for mass matrices; the Fritzsch scheme [7] with two predictions, the Fritzsch scheme with the Shin phases [9] with four predictions, the Stech scheme [8] with three predictions, and the Gronau-Johnson-Schechter (GJS) scheme [10] with four predictions. These schemes were consistent with all previous data [7-10, 36, 37]. We check in sects. 5 and 6 their consistency with the new B-B mixing measurement (combined with earlier data). 4.2. QUARK MASSES

The masses of the quarks, which are eigenvalues of the mass matrices to be discussed, are not the physical masses but parameters in the lagrangian. This means that they are running masses which should all be taken at a single energy scale. In the different schemes, the three mixing angles and the phase depend on mass ratios rather than on the masses themselves. As mass ratios are, to a good approximation, independent of the energy scale, the scale itself can be arbitrarily chosen. Following ref. [39] we use md/m s = 0.051 + 0.004, mu/m c = 0.0038 + 0.0012, ms/m

b=

0.033 + 0.011.

(4.1)

In our calculations we ignore the relatively small error on md/m~, but we consider the full range of the other ratios. The relations we get involve the undetermined mass ratio rnc/m t. In order to confront these relations with the x d and e bounds, involving the physical mass of

37

Y. Nir / B-B mixing

the t quark m vhys, t we must (a) Specify me at a certain energy scale #. We take # = 1 GeV. The value of m ¢ ( # = 1 GeV) is given in ref. [39] mc(ja = 1 GeV) = 1.35 + 0.05 OeV.

(4.2)

(b) Translate the relations involving m c / m t into relations which depend on the running top mass rot( g --- 1 GeV). (c) Write these relations in terms of the physical mass of the t quark. The relation between the physical mass and the running mass, including a first order Q C D correction, is [39] m tplays = ~ I t ( # = rot) 1 Jr

.

(4.3)

In order to relate mt(/x -- mr) to mr( # = 1 GeV) we use the usual equation for the running mass [39]

m(/~)=~

2 ,Y0hiL+a 8r,/fLl-2 'o I---~o3 ~+~---~-'~1~"]

,

(4.4)

where /~o = 11 - }N/,

T0=2,

/~1 = 102 - :3~ N/,

L= ln( Va2),

A = 0.1 GeV,

(4.5)

200

150 (.9

I00 E 5O

J

0 0

I mt[/~

Fig. 13. The physical mass of the t quark,

I

I00

.I

I

. I

500

200 = I GeV" I

(GeV)

a function of the running mass at I GeV, mt(p = 1 GeV).

m phys, as

38

Y. Nir / B-B mixing

and ~ is the renormalization group invariant mass. The physical mass of the top quark, as a function of its running mass at 1 GeV, is given in fig. 13. A good approximation for rntPhY~in the interesting range between 40 GeV and 180 GeV is mtphys - 0.6m t (# = 1 GeV) (we use the full expression in our calculations). In what follows we, again, denote the physical mass of the t quark by m r 5. The Fritzsch scheme for quark masses

5.1. THE EIGHT PARAMETERS We study mass matrices of the form [38, 37]

a

M u=

0

,

M d=

bU

i0

a-

aae -i~

0

0)

0

bae;*2 .

bde -ie~2

(5.1)

cd

The six real parameters (a u, b u, c", a d, b d, Cd) Can be expressed in terms of the six quark masses rot,

,

(5.2) The approximation is good to O ( m J m s ) - (~), The two phases (q~l, alP2) c a n be expressed in terms of the quarks masses and the two known mixing angles, st2 and 823 S12 ~" ] m ~ s

S23

e -iot rn~m~ ,

(5.3)

--I m ~m~ - e - i ¢ 2 ~ / ~ c I"

(5.4)

V mt I

Thus, all eight parameters' of the Fritzsch scheme are expressible in terms of seven known parameters (five quark masses and two mixing angles), and the yet unknown mass of the top quark. Consequently, for every selected value of rot, we get predictions for the unknown mixing angle, s13

s13

mb

+

V mc

mb

e -i.2

V mt ]

,

(5.5)

and for the phase ~ [37] sin 8 sin 4'1 s12s23/st3 - cos ~ --" cos q~t - ~/rndmc/msmu "

(5.6)

Y. Nir / B-B mixing

39

In this section we find whether the predictions (5.5) and (5.6) are consistent with the allowed ranges obtained in sect. 2. 5.2. SIMPLE LIMITS ON THE RANGE OF PARAMETERS Eq. (5.4) gives a lower limit on the unknown mass ratio me.l/rot

m(;m

--

mt

>t

- -

mb

-

s23

(5.7)

Using m s / m b >10.022 and s23 ~< 0.050 we get m c

- - >t 0.01 ~ mr( # = 1 GeV) ~< 145 GeV. m t,

(5.8)

For other values of m J m b and $23 we get more stringent bounds. For example, the central values m J m b = 0.033 and s23 = 0.043 give mt(/x --- 1 GeV) ~< 73 GeV. We find that the bound (5.8) is translated into a bound on the physical mass m t ~< 88 GeV, so we study the predictions of the Fritzsch scheme only for

(5.9)

43 GeV ~< m t ~< 88 GeV.

The lower limit is the one derived from the e and x d bounds in sect. 2. As mentioned above, other values of m J m b and s23 lead to more stringent bounds. The upper bound corresponding to central values, m t(# = 1 GeV) ~< 73 GeV, leads to a bound on the physical mass m t ~< 47 GeV. Eq. (5.9) considerably reduces the range of parameters that we need to study. It is possible to further reduce this range without exact calculations by considering eq. (5.5). First, we rewrite eq. (5.5) in the following way

q~

(5.10)

~

$23

$23 \ m b

V mc

The phase ~3 is defined through s23e-i.3 =--~ upper limit on q

q <~ ~

(

m---b]

1] ~

- e-ie~2~-Jmt. This gives an

--~¢ "

(5.11)

Using r n J m b <~0.044, m J m c <~0.005 and ,f:23~ 0.034 we get q ~<0.13, so we study the predictions of the Fritzsch scheme only for 0.02 < q ~< 0.13.

(5.12)

40

Y. Nit / B-B mixing

The lower bound is the one derived from the e and x a bounds. For other values of m J m b and s23 we get more stringent bounds. For example, the central values m J m b = 0.033 and s23 = 0.043 give q ~<0.10. In fig. 6 we gave the allowed region in the m t-q plane. The two bounds (5.9) and (5.12) overlap within this allowed region only for 56 GeV ~
(5.13)

Thus, in order to check consistency of the Fritzsch scheme with the allowed (rot, q, 8) values, we need to make explicit calculations only within the bounds (5.13). 5.3. RESULTS We calculated the values of q and 8 predicted by cqs. (5.10) and (5.6). This was done as follows. (a) We select a value for m J m ¢ and find ~t from eq. (5.3). We note that the sign of ~1 is undetermined. (b) We select values for m J m b, m J m t and s23 and find ~2 from eq. (5.4). Again, the sign of ~2 is undetermined. (c) We find q from cq. (5.10). The two possible relative signs between ~t and ~2 give two possible solutions. (d) We find 8 by solving cq. (5.6). For each of the two solutions of q, there are two possible solutions for 8 corresponding to the two possible signs of ~t. Thus, for each selected set of parameters s23, m J m t , m J m b and m u / m c we have several possible solutions for q and & On the other hand, for each set of values of m t and s23 we have an allowed region for q and 8 from the experimental measurements (as given, for example, in fig. 4). We then check whether the Fritzsch predictions are within the allowed range. We find that for almost all values of the parameters the Fritzsch solutions are below the experimental lower bounds. In order to get a consistent solution the following are needed. (a) The mass ratio m J m b should be close to its lower limit, m J m b - 0.022. (b) The mixing angle s23 should be close to its upper limit, s23 ~ 0.050. (c) The B - B mixing parameter x d should be close to its lower limit, x d ~ 0.55. (d) The B decay constant fn should be close to the upper limit of its theoretical evaluation, B B f 2 - (0.20 GeV) 2. (e) The B K constant should be dose to the upper limit of its theoretical evaluation, B K - 1. Only if all of these conditions are simultaneously fulfilledis there a very narrow region of (m t,q, 8) which is consistent with both the Fritzsch relations and the

Y. Nir / B-B mixing rnt=85GeV,

0.5

I

'

q =O.OL 8K~-I

BK~I i

I

u

250

I

\

0.4

41

|,

i

,

I

'

I

/

(a)

................................ I'"* (b)

20C

\ \ \

,,,, 0.3

\ ==°= ==,, = . . ° ~ H , .

*.**°.°,.,o°=,..,e,

g ~oo

" 0.2

o"

0.I

-

2',J I

0 0

I

50

J

I "~'a~

I00

"r ~

50-

T

0

150

] 0

I 50

l

l J I I00 150

I

8 Fig. 14. (a) Allowed range of q and 8 for m t = 85 CreV, B K ~<1. The small shaded area is the overlap of the region allowed by the Fritzsch scheme and the region allowed by the data. Higher m t values are excluded by the Fritzsch predictions. For lower m t values there is no overlap between the two allowed regions. Solid, dashed and dotted fines are defined as in fig. 4. (1o) Allowed range of m t and 8 for q = 0.07, B K ~ 1. The small shaded area is the overlap of the region allowed by the Fritzseh scheme and the region allowed by the data. For other values of q there is no overlap between the two allowed regions. Solid, dashed and dotted lines are defined as in fig. 7.

experimental data. This region is within the following bounds 82 G e V ~< m t ~< 88 G e V ,

0.06 ~< q ~< 0.08, 96 ° ~< 8 ~< 110 ° .

(5.14)

In fig. 14 we show the Fritzseh solutions consistent with the experimental data for m t = 85 GeV and for q = 0.07. We remind the reader that when we gave the e and the x d bounds we did not add errors in quadrature but allowed for the full range of all parameters. The consistency of the Fritzsch scheme and the three-generation standard model requires many experimental and theoretical quantities to simultaneously assume values at the limits of their allowed ranges. This is an unlikely situation. However, the Fritzsch scheme cannot be completely excluded. In conclusion, if the Fritzsch scheme is to be consistent with the standard model, then the three unknown parameters of the quark sector should assume the values m t - 85 G e V ,

q - 0.07,

8 - 100 ° .

(5.15)

The bounds on other known parameters of the quark sector become more stringent s23 - 0.05,

mJm

b-

0.22.

(5.16)

Y. Nir / B-B mixing

42

The values of 823 and q give s13 - 0.0035. In addition, there are no ambiguities in the e-relation (B K - 1) and in the xd-relation (x d - 0.55, B B f 2 - (0.2 GeV)2). 5.4. P R E D I C T I O N S

The Fritzsch scheme provided us with a full determination of all the parameters of the quark sector in the three-generation standard model. Thus, it predicts the results of future experiments which depend on these parameters. We now give the predictions for all the experiments described in sect. 3. We use the values given in eqs. (5.15) and (5.16). (1) The x s parameter assumes the value (eq. (3.2)) Xd

xs

(s12)2

-- 11 ~ r s = 0.98.

(5.17)

Thus, B,-B s mixing is near maximal. (2) For the CP-violating parameter e ' / e we get (eq. (3.5)) e t

-- - 0.47 × 10- 2C".

(5.18)

e

If C " - 1, this is close to the level studied by present experiments. (3) .The calculation of the rare kaon decays is simplified by taking Yt "= m 2t/nw2 --1. In the expression (3.8) for B R ( K + ~ ~r+v~) we have D ( y t = 1, z = 0 ) = ~ . ^ Consequently/)e = Dr = 5.7 × 10 -5 and / ) = 4.6 × 10 -5. This gives BR(K + ~ ~r+ r~) = 6.0 × 10 -7 •/3, = 9.6 × 10-11.

(5.19)

i

(4) For the K ° --,/~+/~- decay (eq. (3.15)), we have C(yt -~ 1) = ~ --* C = 4.0 × 10 -6 , and consequently BR( K i ~ / ~ +/~- )SD = 6.7X 10-s(~-- 2.7 X 10 -x°,

(5.20)

which is consistent with the assumption that the two-photon intermediate state dominates this mode. We note that, as m t -- Mw, QCD corrections to the above calculations are small and may be neglected. (5) In the calculation of the rare B decays we again put Yt ~ 1. As discussed before, the other parameters do not affect the results, because we normaliTe our results to the semi-leptonic branching ratio. For the b --, spy mode (eq. (3.23)), we have b B = 2 . 0 / F ( m ~ / m ~ ) and consequently BR(b ~ sv~) = 7.5 × 10 -7 ×

3bB=

8.7 × 10 -6.

(5.21)

Y. Nir / B-B mixing

43

(6) For the radiative decay (eq. (3.25)) we have F2(y t = 1) = ~ . As rn t - M w we ignore the QCD corrections. We get BR(b ---, sv) = 4.5 × 10-4ff = 4 × 10 -5 .

(5.22)

QCD corrections are likely to give a somewhat higher value for this branching ratio. (7) We note that the low value of s13 - 0.0035 predicted by the Fritzsch scheme indicates that non-charmed B decays will not be observed in the near future. In particular r ( b ~ ue~t) 0.006 ~< r ( b ~ cg~e) ~<0.015,

(5.23)

where we have used the full range of q (eq. (5.14)) and F(m~/m~) (eq. (2.9)). If there is a direct observation of such a mode in the near future, it is likely to give a lower bound on q = sx3/s23 which is higher than the upper bound of the Fritzsch scheme, q ~< 0,08. Thus, such an observation may exclude the Fritzsch scheme. As for non-leptonic non-charmed decays, the calculation of their branching ratios is subject to large uncertainties. Using the calculations and the parameters presented in ref. [40], we find BRCB° --* 7r+rr -) - 1 × 10 -5, BR('-B° ~ rr+# -) - 3 × 10 - s , BR(-B° -+ vt+A{) - 3 × 10 -5 , B R ( ~ ° - , o + t , - ) - 2 × lO - 5 ,

BR(B ° --, o+Ai-)

- 3 X 1 0 -5.

(5.24)

5.5. THE FRITZSCH SCHEME WITH THE SHIN PHASES

In our calculations of the Fritzsch scheme parameters, we find 14,11 - 7 8 ° and

14,21 - 6 °. The proximity of these phases to the Shin ansatz for the phases [9] 14,11 =90°,

14,21 =0°

(5.25)

makes it interesting to check whether this ansatz can be accommodated with the experimental data. The values of the six real parameters are the same as in the general Fritzsch scheme (eq. (5.2)). The mixing angle sa2 is given by

md mu] 1/z.

--+ $12 ~

ms

F£/c ]

(5.26)

44

Y. Nit / B-B mixing

As all three parameters in this equation are known, we should check this relation for consistency. We find that it suggests that both mass ratios ~hould be close to their lower limits, m d / m , - 0 . 0 4 7 and m J m c -0.0026, while the mixing angle is at its upper limit, s12 - 0.222. The relation for s23 $23 m

--

(5.27)

,

becomes a prediction for m c / m t ; actually, the lower limit in eq. (5.7) now becomes an equality. As we know already that the Fritzseh scheme is consistent with the experimental data only for Sz3 - 0.05, ms/mr, - 0.022, we find mr( ~ = 1

GeV)

- 145

GeV

--* m t - 8 8 G e V .

A s for the u n k n o w n mixing angle, $13 w e get

sl3

1

$23

ms 3 md +

m],J2

mb ]

me

ms

= 0.053.

(5.28)

(5.29)

F o r the phase 8 w e have

sin8

~ ,/ ~msmu

SlZ/q -- COS 8

V mdmc

mr=88 GeV, B K ~_1

0,5 I,~. ~1

I

~

I

---, 8 = 92 ° .

(5.30)

I I

O.4

o.3 ~o " o.2 O.I 0 0

50

I00

150

8 Fig. 15. Allowed range of q and 8 for m t = 88 GeV, B K ¢ 1. The solid circle gives the Shin prediction. Higher m t values are excluded by the Shin predictions. For lower m t values the Shin solution lies even further out of the allowed region. Solid, dashed and dotted lines and the shaded area are defined as

m fig. 4.

45

Y. Nit / B-B mixing

The allowed region for q - 8 with m t ---88 GeV is shown in fig. 15. The Slain solution lies outside of the allowed region. It is consistent with the e bound but not with the x a bound. However, it lies very close to the allowed region, and as we ignored corrections of O ( m J m d ) to the Fritzsch predictions we conclude that the Shin ansatz is still not completely ruled out. If we have an experimental lower bound on the ratio q in the near future, it may invalidate the Fritzsch matrices with the Shin phases.

6. The Stech scheme for quark masses

6.1. RELATIONS AMONG ANGLES AND MASSES The Stech scheme suggests that in the interaction basis the quark mass matrices have the following properties M u = MUt = M uT, M d=M

atffiaM

(6.1)

u+A,

where A is an antisymmetric matrix and a is a constant. Without loss of generality, this implies mass matrices of the form [8]

M u=

(o00 / -m

0

e

0

,

M d=

/.m a

m t

-ia

-am

- ib

- ic

c

ic

(6.2)

am t

The relations are most easily obtained [8] from (6.3)

V M a(di~) v * = a M ~ diag) + . d .

"r,~, matrix V is the C K M mixing matrix (eq. (2.1)). The matrix .4 is hermitian and z~ithout diagonal elements, but in general it is not antisymmetric. Equating the (3, 3) element of both sides in eq. (6.3) gives a ~ m b / / m t . Comparing the (1,1) element gives m d - (s12)2ms + (Sl3)2mb =

a m u ---, s12 ..---

which is consistent with the known values of s12 and Comparing the (2, 2) element gives

md/m

ctm c ~

s23 ~-

(6.4)

me mt

(6.5)

s.

~m~ (SlZ)2md -- rn s + (S23)2mb ---- _

,

46

Y. Nit / B-B mixing

The third relation is derived by noting [8] that the matrix A, being hermitian and with no diagonal elements, fulfills Re[det A] = 0

Re[(-s12m s + s23sxaeinmb)(s12s23ms + s13e-iSmb)(s23mb)] = 0 --* COS8 ---

ms s12

+

mb (S23)2q

mb q

ms

(6.6)

$12

6.2. R E S U L T S

For any given values of $23 and ms//m b w e have a prediction for m J m t (eq. (6.5)). This is translated into a value for m t ( / x = 1 GeV) and then into the physical mt, in the manner described in subsect. 4.2. An upper limit on m t is obtained when s23 = 0.050, ms/m b = 0.022 m t(/x ---- 1

GeV) ~<72 GeV ~ m t ~<46 GeV.

(6.7)

As the experimental data gave m t t> 43 GeV, there is only a very small "window" open for the Stech scheme (6.8)

43 GeV ~
Lower values of $23 and higher values of ms/m b give smaller m t and are excluded by the experimental data. For m t as low as 45 GeV, the CKM-matrix element Vtd should be near its upper limit in order to give large enough B-B mixing (see eqs. (2.25) and (2.26)). This

m t =

4.5 GeV, B K 5-1

0.5

I~\

!

I

~

I

\t 0.4

'

0.3 ee)

"

i'"",,

J

..... l...........: ....... 0.2 0.1 \

I

0 0

I 50

I

i ~''-~---L-I00 150

8 Fig. 16. Allowed range (shaded) of q and 8 for m t ~ 45 GeV, B K g I and the curve (dot-dash) representing the Stech scheme. The two do not overlap. Higher m t values are excluded by the Stech scheme. Lower m t values are excluded by the data. Solid, dashed and dotted lines are defined as in fig. 4.

Y. Nit / B-B mixing

47

requires q c o s S - - - 0 . 2 0 which, given the known values of s12 and m J m b , is clearly inconsistent with the prediction (6.6). In fig 16 we show the region allowed for q-8 with m t - 45 GeV. We also show the curve representing the Stech scheme (eq. (6.5)). There is no overlap between the Stech prediction and the region allowed by experimental data. Thus we conclude: within a three generation standard model, the Stech scheme is excluded. 6.3. THE GRONAU-JOHNSON-SCHECHTERSCHEME Gronau, Johnson and Schechter [10] suggested to have mass matrices of the Fritzsch form with the Stech constraints (6.1). The parameters in the GJS scheme obey all the constraints of both the Fritzsch scheme and the Stech scheme. Consequently, if any of these schemes is excluded, so is the GJS scheme. Thus, we conclude that the GJS form of mass matrices is ruled out together with the Stech form. 7. Conclusions

We studied the quark sector parameters within the three-generation standard model. We analyzed the constraints coming from two types of measurements: direct constraints from tree level decay widths, and indirect constraints from e and x a measurements. We find that the new B-~I mixing data, when combined with the CP-violation data, give the following new bound m t >/43 GeV.

(7.1)

In addition, for any given value of m t, we found the allowed region for $13/S23 and 8. Alternatively, for any given value of s13/s23, we give the allowed region of m t and 8. The new constraints allow us to improve the predictions of future experimental results. The Bs-Bs mixing is expected to be near maximal. The ratio e'/e can be at the level of present experiments, but much smaller values are stillpossible. We get the following bounds on rare-decays branching ratios 5 × 10 - n ~
1 x 10 -5 ~ sy) ~< 1.3 × 10 -4 .

(7.2)

Different schemes for quark mass matrices give additional relations among quark masses, mixing angles and phases. Thus, the range of parameters is further con-

Y. Nir / B-B mixing

48

strained. We find that the Fritzsch scheme is inconsistent with the experimental data for most of the range of the different parameters. Only if many parameters simultaneously assume values near their experimental or theoretical bounds, BK ~ 1, Bar ~ -(0.2 GeV) 2, x d -0.55, s23- 0.05 and m J m b -0.022, do we get a very small allowed region of parameters near the values mt

~

85 GeV,

$13 ~ 0.0035,

8 - 100 ° .

(7.3)

As all parameters are practically determined by this unique solution, it gives very definite predictions for all the experiments mentioned above r~ - 0.98, e'/e - 0.47 X 10-2C '' , B R ( K + ~ ¢r+v~) - 9.6 X 10 - n ,

BR( K° --*/~+/~-)SD - 2.7 × 10 -1°, BR(b ~ sv~) - 8.7 × 10 -6,

BR(b -* sT) - 4 × 10 -5,

r(b -* ue t) r ( b -* c¢~t)

0.01.

(7.4)

The Fritzsch solution with the Shin phases is more unlikely to be consistent with the experimental data but it is not completely ruled out. The Stech form is excluded by the new data. Even when all parameters are taken to their extreme values, the Stech relations cannot be fulfilled within the allowed range of parameters. The GJS extension of the Stech scheme is also ruled out. All our bounds and conclusions are valid within the three-generation standard model. They remain valid in theories beyond the standard model provided that there are no significant contributions to CP violation and to B-B mixing from processes other than those present in the standard model (see ref. [6] for recent studies of B-B mixing in theories beyond the standard model). Our results cannot be applied if a fourth generation exists. We gave here improved constraints on the three parameters m t, q and 8. Two of these, m t and q = st3/s23, may soon be directly measured. If m t is found to be less than 40 GeV, new physics beyond the three-generation standard model is implied. The prediction that rs is near maximal is valid in the standard model as well as in m a n y extensions of it. If we find rs < 0.77, the most likely possibility is the existence of a fourth generation.

Y. Nir / B-B mixing

49

Many near future experiments may rule out the Fritzsch scheme. A direct observation of non charmed b decays (implying q > 0.08) is the most reasonable candidate to do that. If we find m t < 82 GeV or rs < 0.98 the Fritzsch scheme will be excluded. If we find better ways to calculate s23 and get s23 < 0.05, again the Fritzsch scheme will not be valid. However, if this better calculation gives an upper bound which is weaker than the one we use (or if experiments find a shorter b lifetime than is quoted now), the constraints on the Fritzsch scheme may be loosened up a little. We expect the accumulation of new data from B physics to provide us with a much better knowledge of the standard model parameters. We should be able to decide whether a three-generation standard model still gives a satisfactory picture of the experimental data and whether the Fritzsch scheme is consistent with this picture. I thank Ikaros Bigi and Fred Gilman for valuable discussions. I am especially grateful to Haim Harari for his advice and help. I acknowledge the support of a Fulbright fellowship.

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