The CP violating phase δ_{13} and the quark mixing angles θ_{13}, θ_{23} and θ_{12} from flavour permutational symmetry breaking

25 Pages·0.26 MB·English

by  A. Mondragon

The CP violating phase δ and the quark mixing angles θ , θ
13 13 23
and θ from flavor permutational symmetry breaking
12
A. Mondrag´on and E. Rodr´ıguez-J´auregui
Instituto de F´ısica, UNAM, Apdo. Postal 20-364, 01000 M´exico, D.F. M´exico.
(February 1, 2008)
0 Abstract
0
0
2
The phase equivalence of the theoretical mixing matrix Vth derived from
n
a the breaking of the flavour permutational symmetry and the standard
J parametrization VPDG advocated by the Particle Data Group is explicitly

2
exhibited. From here, we derive exact explicit expressions for the three mix-
2
ing angles θ , θ , θ , and the CP violating phase δ in terms of the quark
12 13 23 13
2 mass ratios (m /m ,m /m ,m /m ,m /m ) and the parameters Z 1/2 and
v u t c t d b s b ∗
9 Φ∗ characterizing the preferred symmetry breaking pattern. The computed
2
values for the CP violating phase and the mixing angles are: δ = 75 ,
4 1∗3 ◦
6 sinθ1∗2 = 0.221, sinθ1∗3 = 0.0034, and sinθ2∗3 = 0.040, which coincide almost
0 exactly with the central values of the experimentally determined quantities.
9
9 12.15.Ff, 11.30.Er, 11.30.Hv, 12.15.Hh
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Typeset using REVTEX
1
I. INTRODUCTION
In this paper we are concerned with the functional relations between flavor mixing angles
θ , θ , θ , the CP violating phase δ and the quark masses resulting from the breaking
12 13 23 13
of the flavor permutational symmetry.
In a previous paper [1] different Hermitian mass matrices M of the same modified Fritzsch
q
type were derived from the breaking of the flavor permutational symmetry according to
the symmetry breaking scheme S (3) S (3) S (2) S (2) S (2). In a symmetry
L R L R diag
⊗ ⊃ ⊗ ⊃
adapted basis, different patterns for the breaking of the permutational symmetry give rise to
different mass matrices which differ in the ratio Z1/2 = M /M , and are labeled in terms
23 22
of the irreducible representations of an auxiliary S˜(2) group. Then, diagonalizing the mass
matrices, weobtainexact, explicit expressions fortheelements ofthemixing matrixVth, the
Jarlskog invariant J and the three inner angles α, β and γ of the unitarity triangle in terms
of the quark mass ratios, the symmetry breaking parameter Z1/2 and one CP violating phase
Φ. The numerical values of Z1/2 and Φ which characterize the preferred symmetry breaking
patternwereextracted fromaχ2 fit ofthetheoreticalexpressions Vth totheexperimentally
| |
determined values of the moduli of the elements of the mixing matrix Vexp . In this way,
| |
we obtained an explicit parametrization of the quark mixing matrix in terms of four quark
mass ratios m /m , m /m , m /m , m /m , and the parameters Z1/2 and Φ in excellent
u t c t d b s b
agreement with the experimental information about quark mixings and CP violation in the
K K¯ system and the most recent data on oscillations in the B B¯ system. These same
◦ ◦ ◦ ◦
− −
experimental data are usually represented by means of the standard parametrization of the
mixing matrix [2], VPDG, recommended by the Particle Data Group [3], which is written
in terms of three mixing angles θ ,θ ,θ and one CP violating phase δ . The standard
12 13 23 13
parametrization VPDG, was introduced without taking the functional relations between the
quark masses and the flavour mixing parameters into account. In contrast, these functional
relations are exactly and explicitly exhibited in the theoretical expressions for Vth derived
in our previous work [1]. When the best set of parameters of each parametrization is used,
the moduli of corresponding entries of the two parametrizations are numerically equal and
give an equally good representation of the experimentally determined values of the moduli
of the mixing matrix Vexp . Hence, we are justified in writing
| ij |
Vth. = VPDG , (1.1)
| ij | | ij |
even though Vth has only two free, real linearly independent parameters while the number
of adjustable parameters in VPDG is four.
The invariant measurables of the quark mixing matrix are the moduli of its elements,
i.e., the quantities V , and the Jarlskog invariant J. But even J, up to a sign, is a function
ij
| |
of the moduli [4]. Hence, two different parametrizations, such as Vth and VPDG, are equiv-
ij ij
alent if the moduli of corresponding entries are equal even if the arguments of corresponding
entries are different. This difference is of no physical consequence, it reflects the freedom in
choosing the unobservable phases of the quark fields.
In this paper, it is shown that a suitable rephasing of the quark fields changes Vth into a
new, phase transformed V˜th such that all the matrix elements V˜th are numerically equal to
ij
2
the corresponding VPDG, both in modulus and phase. Once this equality is established, we
ij
solve the equations of transformation for sinθ , sinθ and sinθ in terms of the moduli
12 23 13
V˜th . We also derive exact explicit expressions for the phases of the matrix elements VPDG
| ij | ij
in terms of the phases of the matrix elements of Vth. In this way, we derive exact explicit
ij
analytical expressions for the mixing parameters sinθ , sinθ , sinθ and the CP violating
12 23 13
phase δ of the standard parametrization of the mixing matrix [2] in terms of the quark
13
mass ratios m /m , m /m , m /m , m /m , the flavour symmetry breaking parameter Z 1/2
u t c t d b s b ∗
and the CP violating phase Φ .
∗
The plan of this paper is as follows: In Sec. II, we introduce some basic concepts and
fix the notation by way of a very brief sketch of the group theoretical derivation of mass
matrices with a modified Fritzsch texture. Sect. III is devoted to the derivation of exact,
explicit expressions for the elements of the mixing matrix Vth in terms of the quark mass
ij
ratios and the parameters Z1/2 and Φ characterizing the symmetry breaking pattern. In
Sec.IV, thephaseequivalence ofVth andVPDG isexplicitly exhibited, andasetofequations
expressing the non-vanishing arguments wPDG of VPDG in terms of the arguments wth of
ij ij ij
Vth is derived. Explicit expressions for the mixing parameters sinθ , sinθ , sinθ and the
ij 12 23 13
CP violating phase δ as functions of the quark mass ratios and the parameters Z 1/2 and
13 ∗
Φ characterizing the preferred symmetry breaking scheme are obtained in Sec. V and VI.
∗
Our paper ends in Sec. VII with a summary of results and some conclusions.
II. MASS MATRICES FROM THE BREAKING OF S (3) S (3)
L R
⊗
In the Standard Model, analogous fermions in different generations, say u,c and t or
d,s and b, have completely identical couplings to all gauge bosons of the strong, weak and
electromagnetic interactions. Prior to the introduction of the Higgs boson and mass terms,
the Lagrangian is chiral and invariant with respect to any permutation of the left and right
quark fields. The introduction of a Higgs boson and the Yukawa couplings give mass to the
quarks and leptons when the gauge symmetry is spontaneously broken. The quark mass
term in the Lagrangian, obtained by taking the vacuum expectation value of the Higgs field
in the quark Higgs coupling, gives rise to quark mass matrices M and M ,
d u
= q¯ M q +q¯ M q +h.c. (2.1)
Y d,L d d,R u,L u u,R
L
In this expression, q (x) and q (x) denote the left and right quark d- and u-fields
d,L,R u,L,R
in the current or weak basis, q (x) is a column matrix, its components q (x) are the quark
q q,k
Dirac fields, k is the flavour index. In this basis, the charged hadronic currents,
J q¯ γ q , (2.2)
µ u,L µ d,L
∼
are not changed if both, the d-type and the u-type fields are transformed with the same
unitary matrix.
3
A. Modified Fritzsch texture
A number of authors [1], [ [5]- [23]] have pointed out that realistic quark mass matrices
resultfromtheflavourpermutationalsymmetryS (3) S (3)anditsspontaneousorexplicit
L R
⊗
breaking. The group S(3) treats three objects symmetrically, while the hierarchical nature
of the mass matrices is a consequence of the representation structure 1 2 of S(3), which
⊕
treats the generations differently. Under exact S (3) S (3) symmetry, the mass spectrum
L R
⊗
for either up or down quark sectors consists of one massive particle in a singlet irreducible
representation and a pair of massless particles in a doublet irreducible representation, the
corresponding quark mass matrix with the exact S (3) S (3) symmetry will be denoted
L R
⊗
by M . In order to generate masses for the first and second families, we add the terms
3q
M and M to M . The term M breaks the permutational symmetry S (3) S (3)
2q 1q 3q 2q L R
⊗
down to S (2) S (2) and mixes the singlet and doublet representation of S(3). M
L R 1q
⊗
transforms as the mixed symmetry term in the doublet complex tensorial representation of
S (3) S (3) S (3). Putting the first family in a complex representation will allow us
diag L R
⊂ ⊗
to have a CP violating phase in the mixing matrix. Then, in a symmetry adapted basis ,
M takes the form
q
0 A e iφq 0 0 0 0
q −
M = m A eiφq 0 0 + 0 +δ B
q 3q q   q q q 
−△
0 0 0 0 B δ
   q △q − q (2.3)
0 0 0  0 Aqe−iφq 0 
+ m 0 0 0 = m A eiφq +δ B .
3q  3q q q q q 
−△
0 0 1 0 B 1 δ
q q q
 −△   − 
   
From the strong hierarchy in the masses of the quark families, m >> m > m , we
3q 2q 1q
expect 1 δ to be very close to unity. The entries in the mass matrix may be readily
q
−
expressed in terms of the mass eigenvalues (m , m ,m ) and the small parameter δ .
1q 2q 3q q
−
Computing the invariants of M , trM , trM 2 and detM , we get
q q q q
A2 = m˜ m˜ (1 δ ) 1 , = m˜ m˜ , (2.4)
q 1q 2q − q − △q 2q − 1q
B2 = δ ((1 m˜ +m˜ δ ) m˜ m˜ (1 δ ) 1), (2.5)
q q − 1q 2q − q − 1q 2q − q −
where m˜ = m /m and m˜ = m /m .
1q 1q 3q 2q 2q 3q
If each possible symmetry breaking pattern is now characterized by the ratio
Z 1/2 = B /( +δ ), (2.6)
q q q q
−△
the small parameter δ is obtained as the solution of the cubic equation
q
δ [(1+m˜ m˜ δ )(1 δ ) m˜ m˜ ] Z ( m˜ +m˜ +δ )2 = 0, (2.7)
q 2q 1q q q 1q 2q q 2q 1q q
− − − − − −
which vanishes when Z vanishes. An exact explicit expression for δ as function of the
q q
quark mass ratios and Z is given in [1]. An approximate solution to Eq. (2.7) for δ (Z ),
q q q
valid for small values of Z (Z 10), is
q q
≤
Z (m˜ m˜ )2
q 2q 1q
δ (Z ) − . (2.8)
q q ≈ (1 m˜ )(1+m˜ )+2Z (m˜ m˜ )(1+ 1(m˜ m˜ ))
− 1q 2q q 2q − 1q 2 2q − 1q
4
B. Symmetry breaking pattern
In the symmetry adapted basis, the matrix M , written in term of Z1/2, takes the form
2q q
0 0 0
M = m ( m˜ +m˜ +δ ) 0 1 Z1/2 , (2.9)
2q 3q 2q 1q q  q 
−
0 Z1/2 1
 q − 
 
when Z1/2 vanishes, M is diagonal and there is no mixing of singlet and doublet rep-
q 2q
resentations of S(3). Therefore, in the symmetry adapted basis, the parameter Z1/2 is
q
a measure of the amount of mixing of singlet and doublet irreducible representations of
S (3) S (3) S (3).
diag L R
⊂ ⊗
We may easily give a meaning to Z1/2 in terms of permutations. From Eqs. (2.1) and
q
(2.9), we notice that the symmetry breaking term in the Yukawa Lagrangian, q¯ M q is a
L 2q R
functional of only two fields: 1 q (X)+√2q (X) and 1 √2q (X)+q (X) . Under
√3 2 3 √3 − 2 3
the permutation of these fields, (cid:16)q¯LM2qqR splits in(cid:17)to the su(cid:16)m of an antisymme(cid:17)tric term
q¯ MAq which changes sign, and a symmetric term q¯ MS q , which remains invariant,
L 2q R L 2q R
0 0 0 0 0 0
2
M = m a 0 1 √8 +2b 0 1 1 , (2.10)
2q −9 3q  −   √8 
(cid:26) 0 √8 1 0 1 1 (cid:27)
 − −   √8 − 
   
where a = (δ )(√2Z1/2 1) and b = (δ )(√2Z1/2+2). It is evident that there is
q −△q q − 2 q −△q 2 q
a corresponding decomposition of the mixing parameter Z1/2,
q
Z 1/2 = N Z1/2 +N Z1/2 (2.11)
q Aq A Sq S
with
1 = N +N , (2.12)
Aq Sq
where Z1/2 = √8 is the mixing parameter of the matrix MA, and Z1/2 = 1 is the mixing
A − 2q S √8
parameter of MS . In this way, a unique linear combination of Z1/2 and Z1/2 is associated to
2q A S
thesimmetrybreaking patterncharacterizedbyZ1/2. Thus, thedifferent symmetry breaking
q
patterns defined by M for different values of the mixing parameter Z1/2 are labeled in
2q q
terms of the irreducible representations of the group S˜(2) of permutations of the two fields
in q¯ M q . The pair of numbers (N ,N ) enters as a convenient mathematical label of the
L 2q R A S
symmetry breaking pattern without introducing any assumption about the actual pattern
of S (3) S (3) symmetry breaking realized in nature.
L R
⊗
C. The Jarlskog invariant
The Jarlskog invariant, J, may be computed directly from the commutator of the mass
matrices [4]
5
det i[M ,M ]
u d
J = {− } (2.13)
− F
where
F = (1+m˜ )(1 m˜ )(m˜ +m˜ )(1+m˜ )(1 m˜ )(m˜ +m˜ ). (2.14)
c u c u s d s d
− −
Substitution of the expression (2.3) for M and M , in Eq. (2.13), with Z1/2 = Z1/2 = Z1/2
u d u d
gives
Z m˜u/m˜c m˜d/m˜ssinΦ
J = q 1−δu r 1−δd
(1+m˜ )(1 m˜ )(1+m˜ /m˜ )(1+m˜ )(1 m˜ )(1+m˜ /m˜ )
c u u c s d d s
− −
m˜ m˜
[( +δ )(1 δ ) ( +δ )(1 δ )]2 u c ( +δ )2
u u d d d u d d
× −△ − − −△ − − 1 δ −△
(cid:26) (cid:18) − u(cid:19)
m˜ m˜ m˜ m˜ m˜ m˜
d s ( +δ )2 +2 u c d s( +δ )( +δ )cosΦ . (2.15)
u u u u d d
− 1 δ −△ s1 δ s1 δ −△ −△
(cid:18) − d(cid:19) − u − d (cid:27)
where and δ are defined in Eqs. (2.4) and (2.7). In this, way, an exact closed expression
q q
△
for J in terms of the quark mass ratios, the CP violating phase Φ, and the parameter Z
that characterizes the symmetry breaking pattern is derived.
III. THE MIXING MATRIX
The Hermitian mass matrix M may be written in terms of a real symmetric matrix M¯
q q
and a diagonal matrix of phases P as follows
q
M = P M¯ P , (3.1)
q q q q†
Therealsymmetric matrixM¯ maybebroughttoadiagonalformbymeansofanorthogonal
q
transformation
M¯ = O M OT, (3.2)
q q q,diag q
where
M = m diag[ m˜ , m˜ , 1], (3.3)
q,diag 3q 1q 2q
−
with subscripts 1,2,3 refering to u,c,t in the u-type sector and d,s,b in the d-type sector.
After diagonalization of the mass matrices M , one obtains the mixing matrix Vth as
q
Vth = O TPu dO , (3.4)
u − d
where Pu d is the diagonal matrix of relative phases
−
Pu d = diag[1,eiΦ,eiΦ], (3.5)
−
and
6
Φ = (φ φ ). (3.6)
u d
−
The orthogonal matrix O is given by
q
(m˜ f /D )1/2 (m˜ f /D )1/2 (m˜ m˜ f /D )1/2
2q 1 1 1q 2 2 1q 2q 3 3
−
O = ((1 δ )m˜ f /D )1/2 ((1 δ )m˜ f /D )1/2 ((1 δ )f /D )1/2 , (3.7)
q  q 1q 1 1 q 2q 2 2 q 3 3 
− − −
(m˜ f f /D )1/2 (m˜ f f /D )1/2 (f f /D )1/2
1q 2 3 1 2q 1 3 2 1 2 3
 − − 
 
where
f = 1 m˜ δ , f = 1+m˜ δ , f = δ , (3.8)
1 1q q 2 2q q 3 q
− − −
D = (1 δ )(1 m˜ )(m˜ +m˜ ), (3.9)
1 q 1q 2q 1q
− −
D = (1 δ )(1+m˜ )(m˜ +m˜ ), (3.10)
2 q 2q 2q 1q
−
D = (1 δ )(1+m˜ )(1 m˜ ). (3.11)
3 q 2q 1q
− −
In these expressions, δ and δ are, in principle, functions of the quark mass ratios and
u d
the parameters Z1/2 and Z1/2 respectively. However, in [1] we found that keeping Z1/2 and
u d u
1/2
Z as free, independent parameters gives rise to a continuous ambiguity in the fitting of
d
Vth to the experimental data. To avoid this ambiguity we further assumed that the up and
| ij |
down mass matrices are generated following the same symmetry breaking pattern, that is,
Z1/2 = Z1/2 = Z1/2. (3.12)
u d
Then, from Eqs. (3.4) - (3.12) all matrix elements in Vth may be written in terms of four
quarkmassratiosandonlytwofree, realparameters: theparameterZ1/2 whichcharacterizes
the symmetry breaking pattern in the u- and d-sectors and the CP violating phase Φ. The
computation of Vth is quite straightforward. Here, we will give, in explicit form, only those
ij
elements of Vth which will be of use later. From Eqs. (3.4)-(3.12) we obtain,
1/2
m˜ (1 m˜ δ )m˜ (1+m˜ δ )
Vth = c − u − u d s − d
us − (1 δu)(1 m˜u)(m˜c +m˜u)(1 δd)(1+m˜s)(m˜s +m˜d)!
− − −
1/2 1/2
m˜ m˜ (1 m˜ δ )(1+m˜ δ )
u s u u s d
+ − − −
(1−m˜u)(m˜c +m˜u)(m˜d +m˜s)! (cid:26) (1+m˜s) !
1/2
(1+m˜ δ )δ (1 m˜ δ )δ
+ c − u u − d − d d eiΦ (3.13)
(1−δu)(1−δd)(1+m˜s) ! (cid:27)
1/2
m˜ (1 m˜ δ ) m˜ m˜ δ
Vth = c − u − u d s d
ub (1 δu)(1 m˜u)(m˜c +m˜u)(1 δd)(1+m˜s)(1 m˜d)!
− − − −
1/2
m˜ (1+m˜ δ )δ (1 m˜ δ )(1+m˜ δ )
u c u u d d s d
+ − − − −
(cid:26)− (1−δu)(1−m˜u)(m˜c +m˜u)(1−δd)(1+m˜s)(1−m˜d)!
7
1/2
m˜ (1 m˜ δ )δ
+ u − u − u d eiΦ (3.14)
(1−m˜u)(m˜c +m˜u)(1+m˜s)(1−m˜d)! (cid:27)
1/2
m˜ (1+m˜ δ )m˜ (1+m˜ δ )
Vth = u c − u d s − d
cs (1 δu)(1+m˜c)(m˜c +m˜u)(1 δd)(1+m˜s)(m˜s +m˜d)!
− −
1/2
m˜ δ (1 m˜ δ )m˜ δ (1 m˜ δ )
c u u u s d d d
+ − − − −
(cid:26) (1−δu)(1+m˜c)(m˜c +m˜u)(1−δd)(1+m˜s)(m˜s +m˜d)!
1/2
m˜ (1+m˜ δ )m˜ (1+m˜ δ )
+ c c − u s s − d eiΦ (3.15)
(1+m˜c)(1−m˜u)(1+m˜s)(1−m˜d)! (cid:27)
and
1/2
m˜ (1+m˜ δ ) m˜ m˜ δ
Vth = u c − u d s d
cb − (1 δu)(1+m˜c)(m˜c +m˜u)(1 δd)(1+m˜s)(1 m˜d)!
− − −
1/2
m˜ (1 m˜ δ )δ (1 m˜ δ )(1+m˜ δ )
c u u u d d s d
+ − − − − −
(cid:26)− (1−δu)(1+m˜c)(m˜c +m˜u)(1−δd)(1+m˜s)(1−m˜d)!
1/2
m˜ (1+m˜ δ ) δ
+ c c − u d eiΦ. (3.16)
(m˜c +m˜u)(1+m˜c)(1+m˜s)(1−m˜d)! (cid:27)
A. The “best” symmetry breaking pattern
In order to find the actual pattern of S (3) S (3) symmetry breaking realized in
L R
⊗
nature,we made a χ2 fit of the exact expressions for the moduli of the entries in the mixing
matrix, Vth , the Jarlskog invariant, Jth, and the three inner angles of the unitarity triangle,
| ij |
αth, βth and γth, to the experimentally determined values of Vexp , Jexp, αexp, βexp and γexp.
| ij |
A detailed account of the fitting procedure is given in [1]. Here, we will give only a brief
relation of the main points in the fitting procedure.
For the purpose of calculating quark mass ratios and computing the mixing matrix, it is
convenient to give all quark masses as running masses at some common energy scale [24],
[25]. In the present calculation, following Peccei [24], Fritzsch [26] and the Ba-Bar book [27],
we used the values of the running quark masses evaluated at µ = m .
t
m = 3.25 0.9 MeV m = 760 29.5 MeV m = 171.0 12 GeV
u c t
± ± ±
m = 4.4 0.64 MeV m = 100 6 MeV m = 2.92 0.11 GeV (3.17)
d s b
± ± ±
These values, with theexception of m , m andm , were taken fromthework of Fusaokaand
s c b
Koide [25] see also Fritzsch [26] and Leutwyler [28]. The values of m (m ) and m (m ) were
c t b t
obtained by rescaling to µ = m the recent calculations of m (m ) and m (m ) by Pineda
t c c b b
and Yndur´ain [29] and Yndur´ain [30]. The value of m agrees with the latest determination
s
made by the ALEPH collaboration from a study of τ decays involving kaons [31].
We kept the mass ratios m˜ = m /m , m˜ = m /m and m˜ = m /m fixed at their central
c c t s s b d d b
values
8
m˜ = 0.0044, m˜ = 0.034 and m˜ = 0.0015, (3.18)
c s d
but we took the value
m˜ = 0.000032, (3.19)
u
which is close to its upper bound. We found the following best values for Φ and Z1/2,
1 81
Φ = 90 , Z 1/2 = Z1/2 Z1/2 = . (3.20)
∗ ◦ ∗ 2 S − A s32
h i
corresponding to a value of χ2 0.32. The values of the parameters δ (Z) and δ (Z)
u d
≤
obtained from (3.18), (3.19) and (3.20) are
δ (Z 1/2) = 0.000048, δ (Z 1/2) = 0.00228. (3.21)
u ∗ d ∗
Before proceeding to give the numerical results for the mixing matrix Vth, it will be conve-
nient to stress the following points:
1. The masses of the lighter quarks are the less well determined, while the moduli of
the entries in Vexp with the largest error bars, namely V and V , are the most
| ij | | ub| | td|
sensitive to changes in the ratios m /m and m /m respectively. Hence, the quality
u c d s
of the fit of Vth to Vexp is good (χ2 0.5) even if relatively large changes in the
| ij | | ij | ≤
masses of the lighter quarks are made. The sensitivity of V and V to changes
ub td
| | | |
in m /m and m /m respectively, is reflected in the shape of the unitarity triangle
u c d s
which changes appreciably when the masses of the ligther quarks change within their
uncertainty ranges. The best simultaneous χ2 fit of Vth , Jth, and αth, βth and γth, to
| ij |
the experimentally determined quantities was obtained when the ratio m˜ = m /m is
u u t
taken close to its upper bound, as given in (3.19). Furthermore, the chosen high value
of m˜ gives for the ratio V /V the value
u ub cb
| |
V m
ub u
| | = 0.085 0.009 (3.22)
Vcb ≈ smc ±
| |
in very good agreement with its latest world average [ [32], [33], [34]].
2. As the energy scale changes, say from µ = m to µ = 1 GeV, the running quark masses
t
change appreciably, but since the masses of light and heavy quarks increase almost in
the same proportion, the resulting dependence of the quark mass ratios on the energy
scale is very weak. When the energy scale changes from µ = m to µ = 1 GeV, m˜
t u
and m˜ decrease by about 25% and m˜ and m˜ also decrease but by less than 16%.
d c s
3. In view of the previous considerations, a reasonable range of values for the running
quark mass ratios, evaluated at µ = m = 171 GeV, would be as follows
t
0.000022 m˜ 0.000037
u
≤ ≤
0.0043 m˜ 0.0046
c
≤ ≤
0.0013 m˜ 0.0017
d
≤ ≤
0.032 m˜ 0.036 (3.23)
s
≤ ≤
9
The results of the χ2 fit of the theoretical expressions for Vth , Jth, αth, βth and γth to
| ij |
the experimentally determined quantities is as follows:
The quark mixing matrix computed from the theoretical expresion Vth with the numerical
values of quark mass ratios given in (3.18)and (3.19) and the corresponding best values of
the symmetry breaking parameter, Z 1/2 = 81/32, and the CP-violating phase, Φ = 90 ,
∗ ∗ ◦
is q
0.9753 ei1◦ 0.221 ei158◦ 0.0034 ei84◦
Vth = 0.220 ei112◦ 0.9745 ei89◦ 0.040 ei90◦ (3.24)
 
0.0085 ei270◦ 0.039 ei270◦ 0.9992 ei90◦
 
 
In order to have an estimation of the sensivity of our numerical results to the uncertainty
in the values of the quark mass ratios, we computed the range of values of the matrix of
moduli Vth , corresponding to the range of values of the mass ratios given in (3.23), but
| ij |
keeping Φ and Z1/2 fixed at the values Φ = 90 and Z 1/2 = 81/32. The result is
∗ ◦ ∗
q
0.9735 0.9771 0.2151 0.2263 0.0028 0.0040
− − −
Vth = 0.2151 0.2263 0.9726 0.9764 0.037 0.043 , (3.25)
 
| | − − −
0.0078 0.0093 0.036 0.042 0.9991 0.9993
 − − − 
 
which is to be compared with the experimentally determined values of the matrix of moduli
[3],
0.9745 0.9760 0.217 0.224 0.0018 0.0045
− − −
Vexp = 0.217 0.224 0.9737 0.9753 0.036 0.042 . (3.26)
 
| | − − −
0.004 0.013 0.035 0.042 0.9991 0.9994
 − − − 
 
As is apparent from (3.24), (3.25) and (3.26), the agreement between computed and
experimental values of all entries in the mixing matrix is very good. The estimated range of
variation in the computed values of the moduli of the four entries in the upper left corner
of the matrix Vth is larger than the error band in the corresponding entries of the matrix
| |
of the experimentally determined values of the moduli Vexp . The estimated range of vari-
| |
ation in the computed values of the entries in the third column and the third row of Vth is
| ij |
comparable with the error band of the corresponding entries in the matrix of experimentally
determined values of the moduli, with the exception of the elements Vth and Vth in which
| ub| | td |
case the estimated range of variation due to the uncertainty in the values of the quark mass
ratios is significantly smaller than the error band in the experimentally determined value of
Vexp and Vexp .
| ub | | td |
The value obtained for the Jarlskog invariant is
Jth = 2.8 10 5 (3.27)
−
×
in good agreement with the value Jexp = (3.0 1.3) 10 5sinδ obtained from current data
−
| | ± ×
on CP violation in the K K¯ mixing system [3] and the B B¯ mixing system [27].
◦ ◦ ◦ ◦
− −
For the inner angles of the unitarity triangle, we found the following central values:
10