

 1 Jun 95

CP-Violation and the Quark Mass Matrices

B. Margolis 1 and S. Punch Department of Physics, McGill University, 3600 University St.,

Montr'eal, Qu'ebec, Canada H3A 2T8

C. Hamzaoui D'epartement de Physique, Universit'e du Qu'ebec `a Montr'eal,

Case Postale 8888, Succ. Centre-Ville,

Montr'eal, Qu'ebec, Canada H3C 3P8

Abstract We study a class of quark mass matrix models with the CP-violating phase determined through making the CP-violating parameter J an extremum. These models assume that mu o/ mc o/ mt and that md o/ ms o/ mb. They have fififi VubV

cb fififi ss q

mu

mc , fififi

Vtd

Vts fififi ss q

md ms

and jVusj ss jVcdj ss q mdms + mumc . The Wolfenstein parameters ae and j are found to be

related by ae ss j2. Finally, we examine a special class of such models where the masses are constrained to be roughly in geometric progression. Further application of the extremal condition to J then leads to q mdm

s ss 3q

mu

mc and hence j ss

1 3 , for maximal J .

The Models

We study here the models of the quark mass matrices of the type

MU = \Lambda U 0B@

0 A 0 A\Lambda B C

0 C 1

1CA

; MD = \Lambda D 0B@

0 D 0 D\Lambda E F

0 F 1

1CA

(1)

which incorporate in a characteristic manner the hierarchy mu o/ mc o/ mt and md o/ ms o/ mb. We can take all matrix elements as real except either A or D, and we choose to make D complex. Then for jDj2 o/ jE \Gamma F 2j o/ 1

md \Lambda D ss j

Dj2j E\Gamma F 2j

ms \Lambda D ss jE \Gamma F

2j mb

\Lambda D ss 1 (2a)

and for A2 o/ jB \Gamma C2j o/ 1

mu

\Lambda U ss

A2j B\Gamma C2j

mc \Lambda U ss jB \Gamma C

2j mt

\Lambda U ss 1 : (2b)

1Presented by B.M. at the MRST Meeting, Rochester, New York, May 8, 1995

We have then

~a2 j i AB\Gamma C2 j

2 ss mu

mc and ~b

2 j i jDj

E\Gamma F 2 j

2 ss md

ms (3)

and taking A to be positive

A ss pmumcmt and jDj ss pmdmsm

b : (4)

Our mass matrices then have the approximate form

MU = 0B@

0 pmumc 0p mumc Bmt Cmt

0 Cmt mt

1CA

; MD = 0B@

0 pmdms eifi 0p mdms e\Gamma ifi Emb F mb

0 F mb mb

1CA

(5a)

with j

B \Gamma C2j ss mcmt and jE \Gamma F 2j ss msm

b : (5b)

The quark mixing matrix is given by the approximate expressions

Vud ss 1p1+~

a2p1+~b2 Vcd ss

~a\Gamma ~be

\Gamma ifip

1+~a2+C2p1+~b2 (6a)

Vus ss ~be

ifi\Gamma ~ap

1+~a2p1+~b2+F 2 Vcs ss

1p 1+~a2+C2p1+~b2+F 2 (6b)

Vub ss ~a(C\Gamma F )p1+~a2p1+F 2 Vcb ss \Gamma (C\Gamma F )p1+~a2+C2p1+F 2 (6c)

Vtd ss \Gamma ~be

\Gamma ifi(C\Gamma F )p

1+C2p1+~b2 Vts ss

(C\Gamma F )p 1+C2p1+~b2+F 2 (6d)

and

Vtb = 1 \Gamma O(~b4) : (6e)

CP-Violation in the Mixing Matrix

CP-violation in the Standard Model involves four quark vertices, and all quark processes exhibiting possible violations depend on the representation-invariant quantity J(a02; b02; c02), where1

\Gamma 4J 2(a02; b02; c02) = *(a02; b02; c02) = a04 + b04 + c04 \Gamma 2a02b02 \Gamma 2b02c02 \Gamma 2c02a02 : (7)

The quantity *(a02; b02; c02) is \Gamma 16A2, where A is the area of the triangle with sides of length a0, b0 and c0. The row formulation has1

a0 = jVffj Vffkj; b0 = jVfij Vfikj and c0 = jVfljVflkj (8)

where ff, fi and fl are any permutation of u, c and t, and where j and k are any two of d, s and b. Stationary points for J obtained from (@J =@*2) = 0, * = a0; b0; c0 lead to right-angled triangles with

a02 = b02 + c02; b02 = c02 + a02 or c02 = a02 + b02 : (9)

Given any two sides of the triangle, these conditions correspond to having maximal areaA and hence maximal J 2. Taking ff; fi; fl = u; c; t respectively and j = d, k = b in Equations (8)

a0 = jVudVubj ss ~a(C\Gamma F )(p1+~

a2)2p1+~b2p1+F 2

b0 = jVcdVcbj ss j~a\Gamma ~be

\Gamma ifij(C\Gamma F )

(p1+~a2+C2)2p1+~b2p1+F 2

c0 = jVtdVtbj ss ~b(C\Gamma F )p1+

C2p1+~b2 :

(10)

We have then that b02 ss c02 + a02 if fi = \Sigma ss2 - i.e., if D is pure imaginary in Equation (1). This extremal condition for J is consistent with experimental results for the Cabibbo angle2 when one compares these with our expressions (6) for Vffi with ff = u; c and i = d; s used in conjunction with Equations (3).

Maximal CP-Violation and the Wolfenstein Parameters

We consider now the Wolfenstein approximation3 to the quark mixing matrix

VW = 0B@

1 \Gamma 12 *2 * *3A(ae \Gamma ij)\Gamma

* 1 \Gamma 12 *2 *2A *3A(1 \Gamma ae \Gamma ij) \Gamma *2A 1 \Gamma O(*4)

1CA

: (11)

We have then a0 = jV

udVubj ss *3Apae2 + j2

b0 = jVcdVcbj ss *3A

c0 = jVtdVtbj ss *3Aq(1 \Gamma ae)2 + j2

(12)

and the conditions (9) lead to

(i) a02 = b02 + c02 yielding ae = 1 : (ii) b02 = c02 + a02 yielding j2 = ae(1 \Gamma ae) : (iii) c02 = a02 + b02 yielding ae = 0 :

(13)

The condition (i) implies that the usual unitary triangle of Figure 1 reduces to a straight line with j = 0.

Figure 1: the unitary triangle. Since in this approximation

J = (*3A)2j (14)

this leads to no appreciable CP-violation. Conditions (ii) and (iii) both lead to right-angled triangles in Figure 1. The model described by Equations (1) and (2) yieldsfifi

fifi VubV

td fifififi

2 ss ` ~a

~b '

2

= ae

2 + j2

(1 \Gamma ae)2 + j2 : (15)

Defining r j (~a=~b) leads then to

j2 = r

2

1 \Gamma r2 (1 \Gamma 2ae) \Gamma ae

2 (16)

which together with result (ii) in Equation (13)

j2 = ae(1 \Gamma ae) yields

j = \Sigma r1+r2 and ae = r

2

1+r2 (17a)

so that

ae = \Sigma rj : (17b)

One can also get the results (17a), (17b) by transforming the Kobayashi-Maskawa (KM) mixing matrix given by Equations (6) with fi = \Gamma ss2 using V 0 = SV S\Gamma 1 with

S = 0B@ \Gamma

e\Gamma i 0 0

0 1 0 0 0 1

1CA

(18a)

where

cos = ~ap~

a2+~b2 and sin =

~bp ~a2+~b2 (18b)

in this way getting the positive sign for j in (17a). The accuracy of our approximations and of the Wolfenstein matrix in Equation (11) is not justified to order r3 for j, and therefore

j ss r = s mum

c /

md

ms (19a)

and

ae ss r2 : (19b)

It follows that since2 mumc , 110 mdms , the value of j , 13 . Maximal CP-Violation in a Special Class of Models

We consider now a sub-class of mass matrices of the form of Equation (1), with

MU = \Lambda U 0B@

0 gUa3eiff 0 gUa3e\Gamma iff a2 C

0 C 1

1CA

; MD = \Lambda D 0B@

0 gDb3eifi 0 gDb3e\Gamma ifi b2 F

0 F 1

1CA

(20)

where gU , gD are of order unity.

These matrices satisfy the inequality conditions of Equations (2) if a, b, C and F , which are taken as real, are considerably less than unity in magnitude, with C2 o/ a2 and F 2 o/ b2. We take a, b, gU and gD as positive. Maximal CP-violation within this framework, as discussed above, requires fi \Gamma ff = \Sigma ss2 . Models of this form have been discussed in various contexts in References (4), (5) and (6). They yield an approximate geometric progression for the quark masses.

It follows from Equations (7) and (10) with the angle fi = \Sigma ss2

jJ j ss ~a~b(F \Gamma C)2 (21a)

where

~a = (a3gU)=(a2 \Gamma C2) and ~b = (b3gD)=(b2 \Gamma F 2) (21b)

using Equations (3) so that

jJj = F C(F \Gamma C)2 " y

3gU

y2 \Gamma 1 # "

z3gD z2 \Gamma 1 # (22a)

where

y = aC and z = bF : (22b)

We assume - and we discuss this further below - that gU = gU (y), gD = gD(z) and that gU = gD = g. We have then that

jJj = F 4x(1 \Gamma x)2 " y

3g(y)

y2 \Gamma 1 # "

z3g(z)

z2 \Gamma 1 # (23)

where x = CF . The maximum in jJj, if one exists, as a function of y and z will yield y = z. In Reference (5), the result y = z follows from symmetry conditions in the gauge structure of the theory presented there. The nature of mass spectra for down versus up quarks requires a2 o/ b2 and therefore C ! F if z = y. The maximum in jJ j as a function of x, for x ! 1, comes at x = 13 . In the above we have fixed F , the largest parameter in our mass matrices (1).

We have from the above, then, that for y = z

a

b =

C F = x =

1 3 : (24)

To keep the approximate geometric progression property of the quark masses, we should have g(y) , 1 at the physical value of y. Taking g(y) = 1, we get the following mass matrices from Equations (20) and (22b) for y = z

MU = \Lambda U 0B@

0 a3 0 a3 a2 a=y

0 a=y 1

1CA

; MD = \Lambda D 0B@

0 \Gamma ib3 0 ib3 b2 b=y

0 b=y 1

1CA

: (25)

Taking b = 3a = 0:21 and y = 4 yields the following fit to the KM matrix absolute values

jVKM j = 0B@

0:975291 0:220908 0:00280085 0:220805 0:974615 0:0370178 0:0072755 0:0364037 0:999311

1CA

: (26)

Using the above values for b and y (or F ) and suitable values for \Lambda U , mt and \Lambda D , mb yields excellent quark mass spectra, approximately given by Equations 2. The value for theCP

-violating parameter

J ss " yy2 \Gamma 1 #

2

x(1 \Gamma x)2b4 ss 2 \Theta 10\Gamma 5 : (27)

Summary

We have presented here a framework and a class of models with texture zeroes constrained by:

(1) an approximate geometric progression of masses for the quarks, (2) a certain symmetry in structure for the up and down mass matrices, and (3) maximal CP-violation consistent with constraints (1) and (2). Consequences of making jJj extremal are that the CP-violating phase fi = ss2 , that the spacing of masses for up quarks relative to down quarks is , (b=a)2 = 9 times as great, and that the CP-violating parameter J , 2 \Theta 10\Gamma 5. This is, of course, much smaller than the maximum of J without constraints7;8, which is (6p3)\Gamma 1.

The fit to the KM matrix above is considered to be at a scale of , 1 GeV. These results can be scaled up to the mass of the top quark (for instance) using renormalization group considerations for the Standard Model. From there on the evolution would depend upon the particular theory. Of course, if one had a particular theory in mind, the symmetry breaking would be at some scale above that of the Standard Model and one would then want to evolve down.

References

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D50 (1994), p 1315.

3. L. Wolfenstein, Phys. Rev. Lett. 51 (1983), p 1945. 4. B. Margolis and S. Punch, Proceedings of the Sixteenth Annual Montr'eal-RochesterSyracuse-Toronto Meeting, McGill University, May 11-13, 1994, published by World Scientific, Singapore (1994), editors J.R. Cudell, K.R. Dienes and B. Margolis, pp 3-11. B. Margolis and S. Punch, Can. J. Phys. 71 (1993), p 172.

5. L. Ibanez and G.G. Ross, "Fermion masses and mixing angles from gauge symmetries",

OUTP-9403; FTUAM94/7 (preprint).

6. P. Bin'etray and P. Ramond, "Yukawa textures and anomalies", LPTHE-ORSAY

94/115; UFIFT-HEP-94-19, final 12-29-94.

7. I. Dunietz, O.W. Greenberg and D. Wu, Phys. Rev. Lett. 55 (1985), p 2935. 8. C. Jarlskog, Phys. Rev. Lett. 55 (1985), p 1039.

