Exponential and Wolfenstein parametrizations of the CKM matrix

IL NUOVO CIMENTO

VOL. 109A, N. 10

Ottobre 1996

Exponential and Wolfenstein parametrizations of the CKM matrix G. DATTOLI, E. SABIA and A. TORRE ENEA, Dipartimento Innovazione, Settore Fisica Applicata, Centro Ricerche Frascati C.P. 65, 000~4 Frascati, Romo, Italy

(ricevuto il 22 Febbraio 1996; approvato il 22 Aprile 1996)

Summary. -- The derivation of the Wolfenstein form of the CabibboKobayashi-Maskawa matrix ~dfrom exponential parametrization is discussed. It is proved that at order )l3 a one-to-one correspondence exists between the Wolfenstein parameters and the entries of matrix A that generates V. The importance of exponential parametrization i n generating mass matrices is analysed. The usefulness of the entries of A in analysing the experimental data is also discussed. PACS 13.10 - Weak and electromagnetic interactions of leptons.

1. - I n t r o d u c t i o n

An appropriate choice of Cabibbo-Kobayashi-Maskawa (CKM)[1] and mass matrices may help to suggest a dynamical model that underlies the weak mixing mechanism and the quark mass spectrum. It would also be desirable to find simple or, at least, nonartificial criteria, to reduce the number of free parameters in the standard model. An attempt to derive the CKM matrix ~" from a new point of view, which allows one to reduce the free parameters, has been proposed in ref. [2-4]. The key-idea in these papers is that, since V is a unitary operator, it can always be cast in an exponential form by exploiting an anti-Hermitian matrix .4 whose entries are written as powers of the Cab ibbo angle. The foregoing can be reworded more technically by saying that since V is an element of the group U(3), it is in a one-to-one correspondence with an e l e m e n t / 4 of Lie algebra U(3) that is the generator of ~'. Therefore, it is clear that (1)

V = exp [.4] = exp [ - i/4],

/~ = - i/~.

is obviously a Hermitian operator and can be viewed as the Hamiltonian generating the weak rotation. According to ref. [1-4], the couplings to first, second and third quark generation have strengths 2, 22, 2 s, respectively and 2 = 0.22 is the Cabibbo 1425

1426

G. DATTOLI, E. SABIA and A. TORRE

angle. The elements of the CKM matrix have been derived under this assumption and their values shown to be within the limits of the experimental constraints. A clear advantage of the exponential parametrization of ref. [2-4] is that it naturally includes the hierarchical feature inspiring the Wolfenstein parameterization [5] which can be obtained as a particular case of (1). The idea of exponent~l parametrization of ~d, without any assumption on the nature of the elements of A, has also been suggested in ref. [6] where it was exploited as a ,,new tool for analysing quark mixing,. The usefulness of A, referred to as the CKM generating matrix, has also been recognised for the analysis of weak-interaction eigenstates (see ref. [7]) and the construction of mass matrices [6, 7]. In this paper we re-parametrize .4 by following the suggestions of ref. [8] and by preserving the Wolfenstein hierarchy. The entries of A are constrained by comparing them with the available experimental data. We show how classes of mass matrices can be generated by means of the form (1) of the CKM matrix. Finally, we discuss sthe importance of the entries of A in analysing the experimental results.

2. - E x p o n e n t i a l a n d W o l f e n s t e i n f o r m s o f t h e CKM m a t r i x The CKM generating matrix can be written as 0

~

(x;t) 3 exp [i5] /

1 /

(2)

.4 = - (x;t) ~ exp [ - id]

o

(y~.)2

V

where x, y and 5 are left unspec~ed for the moment. The use of the Cayley-Hamilton theorem allows one to obtain V in exact form. However, it is interesting that by exploiting the expansion

(3)

n=o n}

truncated at n = 3 , and by neglecting the o(~ m~5) contributions, we get a Wolfenstein-like form, namely(I), ,~2 24 1 -- - - + - 2 24

(4a) ?--

-;t + -6 -~.a/~o(1 - ~) + ir])

~3 ~ -- - 6 1

--~3/~0( Q + i~])

/~2 _ 2

2

~.2/~o(/~ 2 + iC)

- - , ~ 2 / A 0 ( ~ l -~

~-

iC)

1 2 a 1 - ~/~o,~

(1) The ~J matrix in the form (4a) is not unitary because we have truncated the expansion (3) at the order n = 3 and we have neglected contribution of o(;t m~ 5). This point will be further commented in the concluding remarks.

EXPONENTIAL AND WOLFENSTEIN PARAMETRIZATIONS OF THE CK1VI MATRIX

t~o=Y2, (4b)

1 2

O-

xa cosS, y2

/~1=1---

22 12z x 3 + cos5 6 2 y-~ '

,u,~ = 1

2z 6

1427

xa y=---sinS, yZ

1 22 x 3 2 ~-~ cos 5 ,

C-

)~2 2 0.

It is evident that .4 can be exploited to recover the Wolfenstein parametrization, and the (0, ~],/~0)~parameters can be expressed in terms of x, y and & On the contrary, we can express A in terms of the Wolfenstein parameters, thus finding

(5)

=

0 -2

)~ 0

Ho28W 1 ).2/

-/~o 23W*

/~o22

0o

W=

J'

1 ~-Q-i~.

It is worth stressing that in the (~), ~)-plane the following relation holds: (6)

Q-

+ ~]Z = rZ ,

r=

yZ '

higher-order corrections to the Wolfenstein form will be discussed in the concluding section. The numerical value of the entries of A can be determined by means of the experimental data for CKM. By using, indeed, the relation (7)

1-

1 2

= IVt,

I

and the range of values of IVt, b l reported in ref. [9], we find (8)

y = 0.9 _+ 0.1,

which agrees with the fit of ref. [8, 10]. We can consider (8) as a fairly reliable value, but the restriction is much less clear for x and 5. As is well known the possibility that 5 = 0 (or equivalently y = 0) amounts to excluding the CKM origin of the CP violations. In a recent paper Peccei and Wang[11] have noted that assuming y = 0, the r values compatible with the experimental results are (9)

r = -0.33 _+ 0.08,

or what is the same (10)

5 = 0,

x 3 -= (0.83 _+ 0.08) y z ,

which yield for x, the following range of values: (11)

x - 0.876 _+ 0.093

in agreement with the conclusions of ref. [8], according to which the x parameter

1428

G. DATTOLI, E. SABIAand A. TORRE

should be expected to satisfy the condition x ~< y e (0.8, 1) under the hypothesis of the nonexistence of CP violations of CKM origin. By assuming again 5 = 0 and by neglecting contributions of order o()`3) we can predict I IVc,~l

(12)

_

X 3

__ 1 -

Z',

=-- 0.959 _+ 0.0226,

which is within the experimental limits (0.85 _+ 0.23), see ref. [10]. It is therefore clear that a better determination of this quantity could help to conclude that 5 may be assumed to be zero and then to exclude the CKM origin of the CP violations. Furthermore, again under the hypothesis that 5 = 0, we can show that the parameter x s/y2 is_closely related to the ratio Am~/Amd, where Am~ and Amd are the B ~ - B~ and B d - Bd mass differences. Indeed, according to ref. [11] we find

(13)

Am~

IVt,~_l2l2

Ama - ~

IVt, d

(1-)`2/6-(1/2))`2(xS/Y2))2 )2 (1/2 + x3/y2) 2

_

~

If we neglect the contributions containing it s in the numerator of the last term we get an error of the order of 3%, but obtain the simpler relation Am~

(14)

Am d

----

4~

1 (1 + 2x3/y~) 2 '

)`2

~ is expected to be of the order of unity, however, according to a recent analysis by Forty [12], we assume (15)

~ = 1.3 +_ 0.2.

Furthermore, by means of the approximation leading to eq. (12), we can write xs y2

(16)

-

1 1 Ivu, b I Jr 2 )` Ivc, l

which can be used to get Ams

(17a)

1

_ ~2

Amd

(x § IVu, b I/IVe, b I) 2 '

which can be combined with the previous results to yield +- 0.02 (17b)

Am~ Amd

- -

--

m )`2 '

m

=

0.7 -+ 0 . 1 5

/

\

+-- 0.006

the uncertainty on the values of m is dominated by that on ~ . The errors _+0.02 and _+0.006 come from two different extimations of the ratio IVu. b I/IVe. b I (namely 0.08-+ 0.02 and 0.082-+ 0.006, see ref. [11] for further details). It is now evident that if 5 = 0 the ratio Am~/Amd is predicted to range between 11.3 and 17.5, significantly smaller than the present limits which span

EXPONENTIAL AND WOLFENSTEIN PARAMETRIZATIONS OF THE C ~

150

a)

'\

1429

MATRIX

150

b)

II

100

",. A)

100

-",

-

- - ,5=z/2 ...

I

5O

..B)""

0 -2

....

-1

"

0

'

1

50

2

0

0.2 0.4 0.6 0.8 1.0 1.2 1.4

5

x

c)

60

d)

6 =3.0

10 4

40

~(}6 =2.5

% d-

10 2

5=.

,

"

20

i0~

10 -2

....

I

I

10 -1

i0~

101

x

200

0

.....

,....

.......... I

I

2

4 5

. . . . . . . 1"

6

8

e)

100

0 0.8

1.0

1.2

1.4

1.6

x

Fig. 1. - a) A m s / A m d vs. 5 for three different values of x and y = 0.9. A) x = 0.5, B) x = 0.77, C) x = 0.9. The dashed portion covers the values allowed by the experimental data. b) A m s / A m d vs. x for different values of 5 and y = 0.9. c) A m s / A m d vS. X for different values of 5, y = 0.9. d) A m ~ / A m d vs. 5 for two different values of x, y = 0.9. The region around d = 7~ seems to provide a maximum, e) Am~/Amd vs. X for two different values of 5, y = 0.9. This conf'Lrms the fact that to recover the experimental data large 5 values demand for large x values. within 12.5 and 48. A m o r e a c c u r a t e d e t e r m i n a t i o n of the above-quoted ratio could be helpful to establish the n a t u r e of the C P violations. This conclusion should be t a k e n with s o m e cautions. I n fig. 1 we have r e p o r t e d t h e b e h a v i o u r of A m s / A m d vs. x and 5 (2). T h e region allowed b y the experimental results

(2) To give a better feeling of the dependence of IVt, 81 and IVt, d I on 5 we note that for x = 0.8 and y = 0.9 we have IVt, dl = 9.7"10-Scos(d/2), IVt, 81 - 0.0394 - 1.1.10-~(cos(0.55.5))/ /(1 + 0.2-52).

1430

G. DA'ffI'OLI, E. SABIA

100 80 6O . 0

52

and

A. TORRE

~

1.0 /

0

. 0.4

0 0.6

~ 0.8 r

1.0

1.2

Fig. 2. - (r, 5 ) contour plot of Ams/Amd for y = 0.9.

is dashed and for the chosen 6 interval we can exclude x values below 0.6. It is interesting to note that 6 = 0 seems to provide a minimum (s). In fig. lb) and c) we show Am~/Amd VS. X for different values of 6. When 6 > :r/2 the experimental data are reproduced for large x values. We can however exclude large 6 values by noting that, by keeping 5 ~ 0 and at the order o(2 s) we have (17c)

]Vt,~l _ ,u2 m l ~

IV~,~ I

x3,~ 2c0s6, y2

which becomes larger than 1 for 6 > :~/2, while the experimental data seem to support/~2//zl < 1 as most probable value [10]. More general information may be offered by contour plots diagrams as that of fig. 2 where we have reported the (r, 6) contour plots of Am~/Amd. Figure 2 is not 0.009

ii

0.011

," o

0.4

0.6

0.8 r

1.0

1.2

Fig. 3 . - (r, 5) contour plot of IVu,bl (y =0.9).

(s) The moduli of the entries of ~"do not depend on the sign of 6. We have reported the negative region of 6 to emphasize the symmetry and the role of 6 = 0.

EXPONENTIAL AND W O L F E N S T E I N PARAMETRIZATIONS OF THE

CKM

1431

MATRIX

2.0 5 1.5 1.0 0.5 0.0 0.4

o.~

\ ~

/,Z, 0.6

0.8

1.0

1.2

F

Fig. 4. - (r, 5 ) contour plots of IVu, b I/I Vc, b I, the dashed portion refers to the region allowed by the experimental data (y = 0.9). particularly helpful to draw any conclusion on the region of allowed (r, 5) values. However, by inspecting fig. 3, 4, 7, 8, where we have reported the contour plots of Vu, b and I Vu, b I / I v c, b I, by using the fact that I Vu. b I / I V c, b I = 0.08 +- 0.02, the indication 2.0

~o ~

V a ) / /

/

0.4

1.5

0.6

0.8

1.0

1.2

0.039__~ 0.039~

o~

2 / ,~/~.~Z

o.o,

I /,t/~-( o.o o.~ 1.o1.,.

o.4

1.5

~0 . 0 3 ~

~

/

1.0

o.sTo.doT/ I I

o.orf l(/,

~

~

[o.oo6/ / 1.0

~

~ ~'-~'-0.036

[ ~

o.o3~_____~

,.o

oo0.4

f f f/, ( 7>,(-7~ 0.6

0.8

1.0

1.2

0.4

0.6

0.8

1.0

1.2

Fig. 5.- (r, 5) contour plots (y = 0.9). a) IYt, d[ (Y = 0.9); b) [Vt,~l (Y = 0.9); c) (Vr d) IVc,bl (y=0.87).

(y = 0.9);

1432

G. DATTOLI, E. SABIA and A. TORRE

1/2 (

112 ()~3y2)

1

4

y

~

)~2y2 (1_22/6)

a)

Iv7) we find for the elements V~, d and Vu, f

vud:1

(28) Vu s

4

42

+

1

4

[ 1

+(my

4

-

1

1

So

i x 3 .2sin 5 46

3

)

'

4a (1 1 4 / + / 2 ) 6 + xSy~cosSsin5 4 s 6Y + 120 2 xsy

and it is easily checked that (29)

]Vu, d ]2 + IVy,~ ]2 + )V~,b ]2 = 1 + o(AS).

As to the method exploited to generate the mass matrices, we note that, albeit effective, it is hampered by the ambiguity in the choice of the parameters a. We believe that the method of the weak-interaction eigenstates [15] might be more rigorous, and we will analyse this point in more detail in a forthcoming paper.

REFERENCES [1] CARmBON., Phys. Rev. Lett., 10 (1963) 531; KOVAYASHIM. and MASKAWAT., Prog. Theor. Phys., 79 (1973) 652. [2] DATTOLIG., MAINO G., MARI C. and TORRE A., Nuovo Cimento A, 105 (1992) 1127. [3] DATTOLI G., GIANNESSIL. and MARI C., Nuovo Cimento A, 105 (1992) 1555. [4] DATTOLIG., Nuovo Cimento A, 107 (1994) 1243; see also KIELANOWSRIP., Phys. Rev. Left., 20 (1989) 2184. [5] WOLFENSTEIN L., Phys. Rev. Left., 51 (1983) 1975. [6] KUSENKO A., Phys. Lett. B, 274 (1992) 390. [7] DATTOLIG. and TORRE A., Nuovo Cimento A, 108 (1995) 589. [8] DATTOLIG. and TORRE A., Nuovo Cimento A, 108 (1995) 1001. [9] PARTICLE DATAGROUP, Phys. Rev. D, 50 (1994) 1315. [10] ALI A, CERN-7.7455/94. [11] PECCEI R. D. and WANG K., Phys. Lett. B, 379 (1995) 220. [12] FORTYR., to appear in the Proceedings of the International Conference on High Energy Physics (ICHEP 94), Glasgow, July 1994. [13] LI H. and Yu H. L., Phys. Rev. Lett., 74 (1995) 4388. [14] DIXIT V. V., SANTHANAMT. S. and THACKER W. D., Phys. Rev. D, 43 (1991) 975. [15] FRITSCH A. and PLANCKLJ., Phys. Rev. D, 43 (1991) 3026.