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CKM matrix exponential parametrization and Euler angles

G. DATTOLI, E. SABIAand A. TORRE

ENEA, Dipartimento Innovazione, Divisione Fisica Applicata, Centro Ricerche Frascati C.P. 65, 00044 Frascati, Roma, Italy

(ricevuto l’11 Marzo 1997; approvato il 5 Maggio 1997)

Summary. — We show that the exponential parametrization of the CKM matrix allows to establish exact relations between the Euler weak rotation angles and the entries of the CKM generating matrix, which has already been shown to include the hierarchy features of the Wolfenstein parametrization. The analysis includes

CP-violating effects and its usefulness to treat the experimental data is also

proved.

PACS 12.15 – Electroweak interactions.

1. – Introduction

In a number of previous papers we have shown that the exponential parametrization of the CKM matrix offers a useful tool to deal with the weak quark mixing [1-3] and provides an effective analytical method to derive mass matrices and weak interaction eigenstates [4]. The usefulness of this parametrization was also stressed in ref. [5] and in ref. [6] its flexibility to analyze the experimental data has been proved.

One of the characterizing elements of the exponential parametrization of the CKM is that it naturally incorporates the Wolfenstein hierarchy [7]. In particular a one-to-one correspondence between the entries of the CKM-generating matrix and the Wolfenstein parameters has been pointed out in ref. [6].

This paper is addressed to the solution of a further problem, namely that of deriving the explicit relation between the mixing angles and the entries of the generating CKM matrix. We will indeed show that a Euler-type rotation matrix naturally follows from the exponential parametrization. It will also be shown that the weak Euler angles can be written in terms of the entries of the generating CKM matrix, which includes the hierarchical dependence in terms of power of the parameter l, linked to the Cabibbo angle [8] and specifying the mixing of the first quark generation.

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The paper is organized as follows in sect. 2 we derive the Euler rotation without the inclusion of the CP-violating phase, which is included in sect. 3. Final comments and the comparison with previous analysis will be presented in sect. 4.

2. – CKM matrix and Euler angles: the d 40 case

The CKM matrix V× and its generating partner A× are linked by the relations [1, 6]

V×_{4 e}A×, (1)

where A× is an antisymmetric 333 matrix specified by

A×₄

u

0 2V3 2V2 V3 0 V1 V2 2V1 0

ˆ

`

˜

. (2)

All the entries of A× are assumed real because we are neglecting the CP violation effects, furthermore we assume that the Vj are proportional to powers of the coupling

strength between u and s quarks, namely

V34 l , V14 (yl)2, V24 (xl)3. (3)

As is well known l is fixed by the successful relation

l 4

o

mu ms

`0.22 , (4)

while, in the absence of CP violations, x and y are close to unity and range between the limits [9, 10]

x Gy (0.960.1) .

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As also noted in ref. [9] the matrix (2) can be written as a linear combination of the

SO(3) generators, namely

A×_{4 2 iV}1J×12 iV2J×21 iV3J×3, (6)

where the J×iare specified by the following matrix realization:

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and by the rule of commutation

[J×l, J×m] 4iel , m , kJ×k

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with el , m , k being the Ricci tensor. According to eqs. (1), (6)-(8) the matrix V× can be

viewed as a rotation matrix in the 3-dimensional Euclidean space, induced by a torque vector V with components

V f(V1, V2, V3) . (9)

As is well known rotations can be described in terms of Euler angles (a , b , g) and the relevant matrix takes the form (Ch4 cos h , Sh4 sin h)

V×₄

u

CgCbCa2 SgSa CgCbSa1 SgCa 2CgSb 2SgCbCa2 CgSa 2SgCbSa1 CgCa SgSb SbCa SbSa Cb

v

. (10)

It is now important to specify the link between the components of V and the Euler angles. By using the Pauli matrices, i.e. the minimal representation of the angular momentum operators J×l, we obtain the following spinor image of A× :

V2 11 V221 V234

k

l21 (yl)41 (xl)6. (12c)

The 2 32 matrix image of the Euler matrix can be written as [11] R×(a, b, g) 4

u

e

2(iO2 )(a 1 g)_{cos (bO2)}

e(iO2)(a2g)_{sin (bO2)}

2e2(iO2 )(a 2 g)sin (bO2)

e(iO2)(a1g)_{cos (bO2)}

v

.

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Therefore, by comparing eqs. (12) and (13) we get

.

`

/

`

(iO2)(a2g)_{sin (bO2) .}

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The above relations provide the following identifications:

.

`

/

`

´

cos b 4122 NVN 2 2 V23 NV×N2

k

sin

g

NVN 2

h

h

. (15)

A power series expansion yields (1₎

.

`

By using the values

l 40.22, x Gy (0.8, 1)

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obtained from the analysis of the numerical data [12, 13], we can obtain

.

/

´

0 .031887 GbG0.0494574 , 0 .064207 Ga1pO2 G0.106528 , 20 .326573 G g 2 pO2 G 2 0 .284225 . (18) (1) Note that g 2a 2 4 p 2 2n 40

!

Q ₍₂₁₎n ( 2 n 11)

g

x3 y2l

h

2 n 11 .

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Furthermore by using eq. (16) up to O(l5_{) we find for V}× V×₄

.

`

´

1 2 l 2 2 1 l4 24 2l 1 l 3 6 2l3m0( 1 2r) l 2 l 3 6 1 2 l 2 2 2 l4 2

g

m 2 02 1 12

h

l2m0m2 2l3m0r 2l2m0m1 1 2 1 2m 2 0l4

ˆ

`

˜

, (19a) where (19b) m04y2, r4 1 2 2 x3 y2 , m141 2 l2 6 1 1 2l 2 x 3 y2 , m241 2 l2 6 2 1 2l 2 x 3 y2 , Equation (19a) provides a Wolfenstein-like form of the CKM matrix in the limit in which the CP violation, are assumed to be not of CKM origin. For further comments the reader is addressed to refs. [6].

3. – The case d c 0

In the previous section we have assumed that A× is a real matrix, the CP violating effects are included by modifying the generating CKM matrix as follows [6]:

A×₄

u

0 2l 2(xl)3e2id l 0 (yl)2 (xl)3eid 2(yl)2 0

v

. (20)

The presence of the complex contribution creates some difficulties in the sense that we cannot exploit the SO(3) symmetry to obtain the CKM matrix in terms of Euler angles. The problem can be solved in this case too, but instead of SO(3) we should use SU(3) or the methods of ref. [7] where we have discussed the theory of weak-interaction eigenstates [14].

The exact solution, albeit possible, yields a too complicated result to be useful. We will see that an approximate solution provides a simpler mathematical tool and a deeper physical insight.

We can decompose the CKM generating matrix as follows:

A × 3 . (21)

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We can approximate V× by using a naive disentanglement, namely V×_{4 e}A× C eA×1_eA×21 A×3 1 E×1C eA × 1_eA×2_eA×3 1 E×11 E×2. (22)

The error of the first disentanglement is

E×1C 1 2[A × 1, A×21 A×3] Co(l3) , (23a)

while for the second

E×2C 1 2[A × 2, A×3] Co(l5) . (23b) In conclusion we find V×1 , 2 , 3C eA × 1_eA×2_eA×3_{1 o(l}3_{) .} (24)

It is worth noting that

VA×1 , 2 , 34 eA ×

1_eA×2_eA×3

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_l4 1 2 (yl) 4 2

v

,

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v

, which provides an alternative Wolfenstein-like parametrization.

We must underline that the procedure we have adopted may be considered doubtful, we have indeed used the naive disentanglement with validity o(l3_{) and then} we have expanded eqs. (26) and (27) up to o(l5_{). Notwithstanding the result we have} obtained are close to those derived in ref. [6] with a more rigorous expansion procedure. We can overcome the above problem by using a more accurate disentanglement procedure. By exploiting the symmetric split approximation, namely [15] eA× 1 B× C eA×O2eB× eA× O2 1 1 24

[

A × 1 2 B× , [A× , B×]

]

, (29) we can write V×_{C (e}A× 3O2_eA×2O2_eA×1O2_{) Q (e}A×1O2_eA×2O2_eA×3O2_{) 1o(l}4_{) ,} (30)

which can be considered a fairly accurate approximation which also preserves, at any order in l, the unitarity of V×. By using the corresponding matrix forms and expanding up to o(l4) we obtain from eq. (30) a form which almost coincides with that obtained in

(2_{) The six possible combinations can be linked to the various “standard” parametrization of the}

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ref. [6] and reported below (3₎ V×_C

u

1 2 l 2 2 1 l4 24 2l 1 l 3 6 2l3y2( 1 2r1ih) l 2 l 3 6 1 2 l 2 2 2 l4 2

g

y 4 2 1 12

h

l2y2(m21 iC) 2l3y2 (r 1ih) 2l2y2(m11 iC) 1 2 1 2y 4 l4

v

, (31) where

.

`

/

`

´

r 4 1 2 2 x3 y2cos d , h 42 x3 y2sin d , m14 1 2 l2 6 1 1 2l 2 x 3 y2cos d , m24 1 2 l2 6 2 1 2l 2 x 3 y2cos d , C 42 l2 2 h , (32)

it is worth noting that (31) has been derived by a direct expansion of eq. (1).

4. – Concluding remarks

The results of the previous section are particularly useful to generate unitary approximation of the CKM matrix in Wolfenstein-like forms. However higher-order perturbative approximations may be exploited to deal with the experimental data, in appendix A we provide the entries of V× in which unitarity is kept up to the accuracy

O(l8_{). It is worth noting that}

.

`

/

`

´

V×1 , 24 l 2 l3 6 1

g

1 2x 3_y2 cos d 2 1 6y 4 1 1 120 1 i 2x 3_y2_{sin d}

h

_l5_, V×2 , 14 2 l 1 l3 6 1

g

1 2x 3_y2 cos d 1 1 6 y 4 2 1 120 2 i 2x 3_y2_{sin d}

h

_l5_. (33)

This result is compatible with the experimental data according to which [12, 13] NV×1 , 2N 4 0 .2205 6 0.0018 and NV×2 , 1N 4 0 .204 6 0.017.

(3_{) V}×_{differs from V}_{× only for minor details in the coefficients m} 1 , 2.

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We want also to emphasize that, within the present context, the Jorlskog rephasing invariant [16, 17] is linked to the determinant of the A× matrix which is given by (4)

Ndet A×N 4 2 x3y2l6sin d . (34)

Furthermore the off-axis asymmetries of V× are [18] (5)

NV×i , jN22 NV×j , iN2a A1, NVA × i , jN22 NVA × j , iN2a A2 (35)

are easily computed as (6₎

A14 2 x3y2l6cos d , A24 l2

k

1 2y4

l2 3

l

. (36)

Before closing the paper we want to comment on the modified form of the Wolfenstein matrix proposed in ref. [18], which in our notation writes

V×x4

.

`

´

1 2 1 2 l 2 2 1 8 l 4 2l( 1 2 l4x3_eid₎ y2l3

g

`

˜

. (37)

Albeit this form may have some similarity with the generalizations proposed in this paper and in ref. [6], its main difference with respect to our forms stems from the fact that when we switch off the interaction with (b , t) family by setting x 4y40 we do not recover the lowest-order expansion of a rotation matrix having l as mixing angle.

(4_{) By recalling that the Wolfenstein parameters A and h are linked to (x , y , d) by y}2

4 A ,

x3

Oy2sin d 4h, we obtain Ndet A×N 4 2 A2l6_{h .}

(5) By VA×we denote the matrix VA×4 V×R× , R×4

u

0 0 1 0 1 0 1 0 0

v

.

(6_{) Since r 41/22 (x}3_/y2_{) cos d we can write A}

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AP P E N D I X A

In the following we give the entries of V× in a form which preserves unitarity up to O(l8₎

.

`

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