UCRHEP-T204
December 1997





"Pattern of Texture Zeroes in Quark Mass Matrices


With a Divergent Top-Yukawa Coupling at the GUT Scale"




Bipin R. Desai1 and D. P. Roy1,2



1 Department of Physics, University of California

Riverside, California 92521, USA

2 Tata Institute of Fundamental Research, Mumbai 400 005, India





Abstract


In a SUSY GUT model responsible for generating symmetric quark mass matrices at the

GUT scale ( = ) we assume that the top-Yukawa coupling, t, becomes infinite at that scale.

As a consequence, the MSSM renormalization group equations for quark Yukawa couplings

exhibit hierarchical solutions which lead to a pattern of texture zeroes in quark mass matrices at 

= similar to one of the solutions of Ramond, Roberts and Ross. The evolution in energy scale

to low energies shows excellent agreement between the measured quantities involving the scale-

independent ratios of CKM matrix elements and their predicted values in terms of quark mass

ratios. It is, noted that the t condensate model of Bardeen, et al., predicts an infinite t at  =

implying, for our model, that at  = , both the symmetry of Yukawa matrices and condensate

dynamics may have a common origin.


I. INTRODUCTION


In the quark-sector of the Standard Model, to which we will confine our attention, there are
ten parameters that are arbitrary but experimentally measurable: six quark masses, along with three
mixing angles and one phase angle of the CKM (Cabbibo-Kobayashi-Masakawa) matrix. An
enormous body of literature exists today dealing with attempts to reduce this arbitrariness by
invoking simplified structures for the up (U) and down (D) - sector Yukawa matrices at a higher
scale (e.g., the Grand Unified Theory (GUT) scale). [1][2]


Typically, the 3 x 3 Yukawa matrices are assumed to be symmetric at the GUT scale and to
have zeroes at specific locations ("texture zeroes")[3]. While the symmetry of the Yukawa matrices
is a natural assumption in Grand Unified Theories at the level of SO(10) and beyond, there is as yet
no obvious mechanism that can accommodate the texture zeroes.


The above two assumptions, however, reduce the number of independent parameters
allowing relations to develop at the GUT scale between quark masses, which are the eigenvalues of
the Yukawa matrices, and the mixing angles, which are related to the rotation matrices
diagonalizing these Yukawa matrices. These relations are then evolved, down to low-energy scales,
using the renormalization group (RG) equations, and compared with experiments.


One such model, which we will frequently refer to, is that of Ramond, Roberts, and Ross
(RRR) [2] in which the structures of U and D are systematically analyzed with maximum number
of allowed texture zeroes. They find that there are five possible solutions at the GUT scale each
with five texture zeroes in the combined U-D system that are consistent with the observed
experimental values for the quark masses and CKM angles.


We show that if the Yukawa coupling, t, of the top quark goes to infinity at the GUT scale
then, in order to be symmetric at that scale, the Yukawa matrices must have texture zeroes. The U-
D- system then resembles solution 4 of RRR. Specifically, we find


o o o a o
U
=

;

=

D a b o (I.1)


o o o d


where the matrix elements above are functions of t = ln (/1 Gev). All the texture zeroes, except
for that of D11, can be attributed to the infinite t boundary condition in the solution of the RG


equation at the GUT scale.


With regard to our assumption about an infinite t at the GUT scale we note that there are
physical circumstances under which such a behavior is predicted. For example, Bardeen et al [4][5]
have observed that the existence of a Higgs boson as a t condensate within a SUSY-GUT model
implies



t

as






where the mechanism responsible for condensate formation is assumed to arise from GUT scale
dynamics. Our assumption about Yukawa matrices then implies that the dynamics that gives rise to
the condensate also gives rise to symmetric Yukawa matrices at the GUT scale.


Before proceeding to discuss our model in detail we summarize below the role of texture
zeroes in a 2 x 2 system and then discuss the main results of RRR.


For a 2 x 2 matrix there exists the so-called sea-saw mechanism [6] which generates the
mass-hierarchy and relates the mixing angle to these masses. Here the mass matrix is symmetric
and has a zero in (11)-position,

o b
Y
=
d

<

<

b

;


b d


If we write the diagonalized version as


diag m1 o
Y
=


o m2

then


m = (I.2)
1 b
2 /d
, m
=
d

2


which exhibits the desired mass hierarchy, m1<<m2. For the diagonalizing matrix

c - s
R =

;
c
=
cos

, s =
sin


s c

one finds


b
s (I.3)
d



Expressing it in terms of m1 and m2 we find


m
s =
1 (I.4)
m2



which relates the mixing angle to the ratio of masses.


The above example of a single texture zero in a symmetric 2 x 2 matrix illustrates the
crucial role these zeroes play in relating different experimental quantities. By reducing the total
number of independent parameters in Y to just two it establishes a relationship between the three
parameters m1, m2, and s given by (I.4)


For a more general symmetric 2 x 2 matrix

a b
Y =
(I.5)
b d


with a << b << d, the relation (I.2) remains correct but to satisfy (I.3) and (I.4) one must have


b

<

<

a 2 /d (I.6)



in which case the situation is equivalent to having a texture zero (a = o)


For the more relevant case of a 3 x 3 matrix RRR, in the above-mentioned investigation,
find five separate solutions consistent with experiments, each with a total of five texture zeroes for
the combined U - D - system.


Our prediction for U and D given by (I.1) corresponds to solution 4 of RRR whose general
structure, for both U and D, is with x << y, z << w


o x o
Y
=
x y z (I.7)

o z w


Reviewing the discussion in RRR we note that the eigenvalues m1, m2, and m3, in


m1 o o
diag
Y
=
o m2 o


o o m3

are given by


z2 x2
m (I.8)
3 =
,
w m
2 =
y - , m
1 =

w m2

and the diagonalizing matrix, written in the same manner as in the 2 x 2 sea-saw case, is given by


c -
s o
1 o o

=

R s c o

o c -
s (I.9)

o o 1 o s c

As with the 2 x 2 case one can show that


x
s =

z2
y -
w (I.10)
z
s =
w

In the following, for the D-sector, the rotation matrix will be designated as Rd and the
corresponding angles as = 1, = 4. For the U-sector the corresponding quantities will be taken
as Ru with = 2, = 3. The CKM matrix can now be constructed as

Vckm =
R (I.11)
u P R+
d





where P is the (diagonal) matrix involving the phases i.e.,


i
o o
e 1 o o
i

=

P o o
o -i o
e e (I.12)
-i
o o 1 o o e


The resulting CKM matrix (I.11) is


-i i i
c1 c

-

2 s
1 s2 e
s1 e
c

+

1 s2 s (

2 s
-

3 s4 e
)
-i i -i i


Vckm - c (I.13)
1 s2
-
s1 e
- s 1s2 e
c

+
1 c
2 c
3 c
4
+
s
3 s4 e
s3
-
s4 e

-i -i i
s (
1 s -

3 s4 e
) - c (

1 s
-

3 s4 e
) c3 c

+

4 s
3 s4 e




II Renormalization Group Equations for U, D and Vckm


The RG equations for U and D in the MSSM scheme are well known and are given to one-loop
[2][7] by


dU
16 2 [-

=
2
c
(II.1)
i g 3

+

i UU +
+
DD+ Tr(3

+
UU + U
)]
dt i



dD
16 2 [-

=
2
c
' (II.2)
i g 3

+

i DD+ UU

+
+ Tr(3

+
DD+ )] D
dt i





where we have ignored the leptonic sector. The gi's are the gauge couplings and the constants ci, '
ci

are given by

13 16
c
=
, ,
3

i
15 3
(II.3)
7 16
c
=
' , ,
3

i
15 3

Olechewski and Pokorski [8] have derived the RG equations for Vckm. Expressing the matrix
elements in the form Vij (i,j = 1,2,3) they obtain, for MSSM

d
16 2 V (

-

=
2

+
2
) (II.4)
ij t b V 32)

23,

31,

13,

=
(ij

ij
dt


Of particular interest to us as we shall see later, are the ratios V
31 /V23 and
V
13 /V23 . From the above RG equations it can easily be shown that the t-derivatives of these

ratios identically vanish. By expressing the matrix elements in terms of quark indices this implies
that


V td const.

=

V cb
(II.5)
V ub const.

=

V cb

The RG equation for V12 gives

d V ~ (
0
(II.6)
12 V 4 )
12
dt

The term on the right hand side above is extremely small (V12 0.22 at low energies). If we
neglect it then we obtain

V (II.7)
us const.


We turn to the quark mass matrices in the next two sections.



III The Quark Masses at the GUT Scale


We now solve for the eigenvalues (the quark masses or the corresponding Yukawa
couplings i) of U and D from (II.1) and (II.2). If we take


>

>
b)

s,
d,

c,

u,

=

(i

t i


then, ignoring the gauge and D- contributions in (II.1) near the GUT scale, one can solve the RG
equations for the eigenvalues of U given by


o o
u
diag
U
=
o o
c

o o t

For t one obtains


d
2
16 t
=
3
3 +
(

3
+
+

) (III.1)
t 2u 2c 2t 2t
dt



Keeping only the most dominant term, we have


d 3
t 3t

=
(III.2)
dt 8 2

If, as discussed in the Introduction, we impose the boundary condition at the GUT scale

ln (III.3)
t

as
t





then


2 1
(III.4)
t =

3 ( ln t

-
)1/2

This is the same result as obtained by Bardeen et al [4][5]. We note that there are no
arbitrary constants due to the nature of the non-linear equation and the boundary condition.




d
2
16 c =
3
3 +
3(
+
+

) (III.5)
c 2u 2c 2t c
dt

As for the other two couplings we have for c, near  =
and similarly for u. To obtain the behavior of c vis-a-vis t one notes that the gauge contributions
cancel out from the RG equations for the ratio

c
R
=

ct t

Consequently, keeping, once again, only the most dominant term one obtains from (III.1)
and (III.5)

dR
ct - 3 2
=
t R (III.6)
ct
dt 16 2


which gives


1/4
R . ln
ct const

=
(
t

-
)



Thus, unlike t, c is given only up to an arbitrary multiplicative constant. It is interesting to


and therefore,
const.
(III.7)
c
=
(
ln t
-
)1/4

note from (III.7) that


ln
c
as
t




but with a smaller power than t. As noted above the gauge couplings are canceled out exactly in
(III.6) so that, as one evolves from the GUT scale to lower energies, there are no gauge terms
present in Rct. The gauge contributions to c is, therefore, inherited entirely through t.


Similarly for u,



1/4
R ln
ut =
const.
(
t

-
)

and


const.
(III.8)
u =
(ln - t )1/4

The arbitrary constants can be determined from the experimentally known values for the
masses mc and mu.


For the D-sector we write


o o
d
diag
D
=
o o
s

o o b

and use (II.2). Even though U and D cannot be diagonlized simultaneously we can, to a very good
approximation, take UU+ to be given entirely by t so that the only non-zero contribution from U in
(II.2) comes from (UU+)33 = .


Solving (II.2) at the GUT scale keeping only the UU+ contribution, one obtains


const.
(III.9)
b
=

( ln -
t
)1/12

and, through the ratios Rcb and Rdb,


,
const.

=
=
const.
(III.10)
s d


One can also consider the ratios Ruc (=u/c) and Rds (= d/s). In the RG equations for Ruc
(Rds) the gauge couplings as well as the dominant t and b contributions cancel out so that


dRuc dR

=

0

=
ds
dt dt

Thus these ratios are constant consistent with the results obtained earlier. Converting them
to masses we find that since the -dependent terms divide out in each of the two ratios that


mu const.

=

mc (III.11)
md const.

=

ms



IV The Full U- and D- Matrices at the GUT Scale and the Texture Zeroes


Consider the RG equation for the full D-matrix near the GUT scale in which only the t -
contribution is kept[2][7].


We then have

o o o

dD
16 2 UU

=
+
=

D o o o
D (IV.1)
dt
o o 2
t

For D33 this gives the same results as b

1
D ~ (IV.2)
33 (ln - t )1/12


For D31 and D13 we find


2 dD31 2 2 dD
16 =
13

t D31 ;
16
0

=

dt dt

Substituting the expression for t from (III.4) the solutions are given by


c31
D (IV.3)
31
=

, D13 c

=
13
( ln t
-
)1/12

where c31 and c13 are constants.


If, as we have assumed, the Yukawa matrices are symmetric at the GUT scale then, at t = ln, one
must have

D (IV.4)
31
=
D13

A comparison of the equations (IV.3) and (IV.4) at t = ln clearly shows that the two can be
reconciled only if c13 = c31 = 0 [10]. Therefore, at the GUT Scale

D31
=
D13 0

=


Similarly, one can show that


D32
=
D23 0

=




The remaining matrix elements of D are constants [11] which, however, can be made to
satisfy the symmetry requirement without being zero.


The structure of the D-matrix is then


D11 D12 o


=

D D21 D22 o


o o D33

It has two texture zeroes which, as we demonstrated, arise naturally as a combined
consequence of the t - boundary condition and the symmetry requirement at the GUT Scale.


Since the sub-matrix Dij (i,j = 1,2) is in a (disjointed) block form and its matrix elements are
constants, we can choose it to be of the sea-saw form, discussed in the Introduction, by


assuming D11 = 0.


We can then write the D-matrix at the GUT Scale in the form



o a o


=

D a b o (IV.5)

o o d

where,



b

,
a ~
const.


(IV.6)

1
d
~ (ln - t )1/12


We write the U-matrix in the symmetric form at the GUT Scale



U

=

(IV.7)



and consider the RG equation (II.1) at the GUT Scale, ignoring the gauge and the D-contributions,



dU
16
2
UU

[3

=
+ Tr(3

+
UU + U
)]
dt

It is easy to check that the right hand side is symmetric and, as a result, so is the left hand
side.


In the approximation that is dominant one finds




2 1

=
(IV.8)
3 ( ln -
t
)1/2



which, not surprisingly, is the same expression as that of t in (III.4).


One can also determine to be


a

=
1 (IV.9)
( ln t
-
)1/2



where a1 is a constant which, in order to maintain the correct hierarchy, must satisfy <<, i.e.,

2
a
<

<
(IV.10)
1 3

In Appendix A it is shown that


a
2 3 1
1 a

=

+
2 (IV.11)
2 ( ln t
-
)1/2 ( ln t
-
)1/4

where a2 is a constant and a1 is the same constant as in (IV.9). Both the terms above must be kept
in order to give the correct behavior for the eigenvalue c because, when one diagonalizes U, one
obtains


2


=
-
c
(IV.12)

1

~ (ln - t )1/4

which reproduces the behavior derived in (III.7).


In appendix A it is also shown that


1
, ~ (ln - t )1/4
(IV.13)
~ const.

Thus in comparison to the other matrix elements as well as to u (see III.8) is small at the GUT
Scale and can, therefore, be neglected.


In Appendix B it is shown, furthermore, that because is of the same order as its
contribution to both the mass eigenvalues and to Vckm is negligible compared to that of .


Thus, to an excellent approximation, one can take


, ~ 0
(IV.14)



In summary then, the structure of the U-matrix at the GUT Scale is given by


o o
U

=
(IV.15)


o

where


1 1 1
~ , ~ , ~ (IV.16)
( ln t
-
)1/4 ( ln -
t
)1/2 ( ln -
t
)1/2

and


2 1
=

3 ( ln -
t
)1/2




V. Determination of Vckm and the Predictions of the Model


As demonstrated in the previous sections our model predicts the U-D matrices to have the
same structure as that of solution 4 of RRR.


One can take over the results already derived in the Introduction for this case, noting that,
for Vckm, s1 and s4 are involved with the D-sector, and s2 and s3 with the U-sector.


Comparing (IV.5) with (I.7) we find s1 and s4 from (I.10)


md
s
=

1
ms

(V.1)

s 0

=

4





Since the ratio md/ms is a finite constant at the GUT scale and remains so as we evolve
down to the lowest energy,  ~ 1 Gev, consistent with perturbation theory, s1 is also a constant
which can be determined from the experimentally observed mass values.


Similarly, comparing (IV.15) with (I.7) we find from (I.10)


mu
s
=
(V.2)
2
mc


which once again remains a constant down to  ~ 1 Gev. However, one cannot predict s3(=/)
from the quark masses.


With s4 = 0 the CKM matrix of (I.13) reduces to

-i i
c1 c

-

2 s
1 s2 e
s1 e
c

+

1 s2 s
2 s3
-i i


Vckm - c (V.3)
1 s2
-
s1 e
- s 1s2 e
c

+
1 c
2 c
3 c
4 s3

s
1 s3 - c 1s3 c3 c
4


While two of the four parameters, s1 and s2 are predicted in terms of the quark mass ratios,
one has to determine s3 and the phase angle, , from the experimental data.


The model predictions for s1 and s2 can be translated into two predictions for the ratios of
the CKM matrix elements i.e.,


V ub mu
s

=

=

2
V cd mc (V.4)
V td md
s

=

=

1
V cd ms



Since the ratios on the left and right sides above are scale-independent (see (II.5) and
(III.11)) the above relations derived at the GUT scale remain valid at low energies and can be
compared with the measured values.


The current estimates of the running quark masses at an energy scale of  ~ 1 Gev are [12]

m 5.1

=
 MeV
0.9

, 
u m
9.3

=
MeV
1.4

d (V.5)
m 175

=
 MeV
25
, 
s m
1230

=
MeV
50

c


where the u, d masses are estimated from chiral perturbation theory [13] and the s, c masses via
QCD sum rules [14]. The resulting ratios give


s ,
0.23

=
(V.6)
1 s 0.064

=

2


with an uncertainty of  10% in each case. These are in good agreement with the CKM matrix
elements [15].


V td V
0.22

=

,
0.08
ub 0.08

=
 0.02
(V.7)
V cb V cb

particularly for the 2nd quantity. Here the 1st ratio has been obtained from

V   (V.8)
td 0.009

=
,
0.003
V cb 0.041

=
0.003


where the estimate of V o o
td is from mix
B - B ing assuming the dominant contribution to come
d d
from Standard Model physics (t-exchange). The error bar comes mainly from the theoretical
uncertainty in the hadronic matrix elements. Note that due to the small values of s2 relative to s1,
one can predict the magnitude of Vus to be in the range s1  s2 irrespective of the phase angle ,
which is again in agreement with the experimental value

V  (V.9)
us
0.2205

=
0.0018


Note that the ratios of the quark masses as well as those of the CKM matrix elements of
(V.4) are scale independent quantities. The same is true of V
us . Therefore, one can make the

above comparison at any desired scale of energy. A recent compilation of the quark masses at  =
Mz [12] gives


m 2.

=

u 33 MeV
+.42
,
-.45 m
4.

=

d 69 MeV
+.60 ,
-.66

(V.10)

ms 93.

=
4- MeV
11.8
+
13.0 , m
c 677

=
+56
61
- MeV

one can easily check that the resulting s1 and s2 are in close agreement with (V.6) and hence with
the CKM matrix element of (V.7) and (V.9).


Finally, the unknown parameters s
3 is fixed by the measured value of Vcb = 0.041  .003,

while the phase angle can be determined from

2 2 1/2
V cos (V.10)
us (

=
s1 +
s
2 +
2
s1 s2 )

In particular the precise value of the mass ratio ms/md = 18.9 0.8 as suggested by
Leutwyler [13] implies s1 to be given by (V.1) to  2%. This would imply



cos ~ -
0.3 (V.11)



One can re-express the above quantities in terms of the Wolfenstein parameters for the
CKM matrix.


2 3

A ( i
-
)

-

1
2

2 2
Vckm =

-
A

-

1
2


A 3
(1
- - i ) -
A 2 1



Thus


V td 2 2
s
1
=

=

-

(1 )
+

V cb

V ub 2 2
s
2
=

=

+

V (V.12)
cb

s
3
=
V
=
A 2
cb

2
-
s21 - s2
cos
=
2

2 s 1s2

The measurements for the CP-violating phase are often expressed in terms of and . By
taking the values of s
1 and s2 from (V.6) and of = (Vus ) from (V.9) we obtain and as

follows


-
<

.08
;

.08

<
-

<

.25 .25

<
(V.11)

This compares favorable with the measured values [16]

10
.

=
,
+.13 .

=
(V.12)
-.39 34+.06
-.09


It is clear from the RG equations of the CKM parameters that only s1, s2 and (and,
therefore, and ) are scale independent quantities.


For the sake of completeness, we shall conclude with a brief discussion of the quark masses
and their scale dependence in our model.


VI. QUARK MASSES


The top-Yukawa coupling evolution down from an infinite GUT-scale value to low energy
has already been discussed by Bardeen et al [4][5]. A rough estimate of t at  = 102 Gev can be
obtained based on quasi-IR fixed point in the RG equations for t.


In the Standard Model this is given by t = 4/3 g3, keeping only the SU(3) gauge coupling,

g3. This yields t = 1.6 or mt = t v/ 2 = 278 Gev where v(=246 Gev) is the Higgs vacuum
expectation value.


In the Minimal Supersymmetric Standard Model of present interest, the fixed point value of

t is given in terms of the dominant gauge coupling g3 as t = 8 / 3 g3 = 1.13, the resulting top
quark mass is


v
sin
t
m
=
197

=
sin GeV
(VI.1)
t 2

where tan is the ratio of the vacuum-expectation values of the two Higgs doublets of MSSM.
This is consistent with the experimental value of mt = 175 GeV.


As far as the remaining quark-Yukawa couplings are concerned, each has an arbitrary
multiplicative constant which can be determined by fitting them to the experimentally measured
values. However, md/ms and mu/mc remain a constant as they evolve down from the GUT Scale. It
is interesting to note that c, u and b (though not s or d) are divergent at  = leading to the
question whether cc, uu and bb - condensates exist or, since they are coupled through unitarity to t_,
whether they contribute to the same (t_) condensate.


VII. CONCLUSION


We have shown, using the one loop MSSM renormalization group equations, that the
combined effect of a symmetric Yukawa matrix and an infinite t at the GUT Scale produces
texture zeroes in U and D through a set of hierarchical solutions. The U-D system then resembles
one of the solutions (solution 4) analyzed by RRR. The experimentally measured scale-
independent ratios V
td /Vcb and Vub /Vcb show excellent agreement with their predicted

values m / m and m / m respectively. Other predictions of the model compare favorably
d s u c

with experiments.


VIII. ACKNOWLEDGEMENT


One of us (BRD) thanks Dr. E. Keith and Dr. A. Khachatourian for discussions. We also
thank Dr. E. Ma for his comments. This work was supported in part by the U.S. Department of
Energy under grant number DE-FG0394ER40837.


Appendix A



We first note that the solution of the equation


(A.1)

d
16 2
= 3 2
N ,
dt
where N is a constant (=1 or 2 in our case) and the (33) element of U,


2 1
= (A.2)
3 ( ln - t )1/2



is given by


const.

=
(A.3)
( ln t

-
)N/4


The equations (II.1) satisfied by the individual matrix elements of U in (IV.7) from (IV.12),
keeping only the and 2 terms, (neglecting gauge and D-contributions) are

d
16 3

=
2
2 3

+
2
dt


d
2
16
3

=

3

+
2
dt


d
16 2
3

=
2
3

+
2
dt (A.4)


d
2
16
3

=
6

+
2
dt


d
2
16 3(

=

+

) 6

+
2
dt



A priori if we keep only the 2 -term above then from (A.1) and (A.3) we obtain


1 1
~ ~ ~ ;
~ ~ (A.5)
(
ln t

-
)1/4 ( ln t
-
)1/2



However, a closer look at the driving terms in the equations above (e.g. the first term
proportional to ) shows that they can alter the behavior from what is given by (A.5)


We show in what follows that there exists a set of self-consistent hierarchical solutions in
which >> imply >> and vice versa, and that it is these solutions that are relevant to our
problem.


In particular we will show that


1 1
~ const
. ;
, ~
;
~
( ln - t )1/4 ( ln - t )1/2


as well as


1
~ (ln - t )1/2


(i) and


In equation (A.4) if we neglected the driving term in equation for , we obtain from (A.1)
and (A.3)


a

=
1 (A.6)
( ln t
-
)1/2

we will come back later to the question of the driving term.


For , if we keep the driving term then the equation for it is of the form



d
=

+

A B
(A.7)
dt



whose solution is


t t
[

=
t
d
t A
e- dt" B
+
A] e d t B (A.8)



where A is a constant, and A and B are functions of t.


The solution for , after substituting the expression for and and using (A.8) is,


a2 3 1
1 a

=
+
2 (A.9)
2 ( ln - t )1/2 ( ln t
-
)1/4



where a2 is a constant and a1 is the same constant which appears in (A.6)


(ii) , ,


In the equation for in (A.4) if we neglect the driving term, we obtain

1
~ (A.10)
( ln t

-
)1/4


we will come back later to the assumption about the driving term.


To consider the behavior of at the GUT scale we first note that if we ignored the driving
term then we would get the behavior given by (A.5)


From (A.4) we combine the equations for and (keeping their driving terms) to write the
equation for their ratio,



=

x (A.11)


which is given by


dx
16 2 =

3

-

(1
x2 ) -

3
x (A.12)
dt

If were of the type given by (A.5) and given by (A.6) then keeping the
1 1
leading as well as next to the leading( ) terms in both we would
1/2
( ln - t ) ( ln - t )1/4

expect the ratio x to behave as


1
b(

+

a

=

x ln t

-
)4 (A.13)



However, then the left hand side (LHS) of (A.12) would behave as


1
LHS
(A.14)
( ln t

-
)3/4


and the right hand side (RHS) from the knowledge of , and , as


const. a

-

(1 2 ) const. a

RHS
-
(A.15)
( ln t
-
)3/4 ( ln t)

-


To satisfy equation (A.12) (LHS=RHS),one must have,a=0, and, therefore,


b(

=

x ln t

-
)1/4
(A.16)



Given the behavior of already determined in (A.6), this implies that

1

~ (A.17)
( ln t

-
)1/4


so that does not have the behavior of the type given by (A.5) but rather a much milder behavior
given by (A.17).


To determine , we combine the equations for and in (A.4) and write the equation for
their ratio (keeping the driving terms)



y
=

which is of the form




dy
3 2 3
16 2 =
-
y (A.18)
dt


If the behavior for given by (A.5) were correct then since is determined by (A.10) the
ratio is given by


a

=
y b

+
(
ln t

-
)1/4
(A.19)

1
where we kept the leading as well as next-to-leading (~constant) terms in and .
1/4
( ln - t )

The left-hand-side of (A.18) now behaves as


1
LHS
(ln t - )3/4


on the other hand


1 1
RHS
-
a

( ln t
-
)3/4 ( ln t)

-




therefore, to have


LHS = RHS


one must have a'=0 and therefore


b

=
y ( ln t

-
)1/4
(A.20)



so that does not have the behavior given by (A.5) but rather


const




at the GUT scale.


Looking a posteriori at assumptions made in the and equations of neglecting the driving
terms we note, after solving the equations exactly (using (A.7) and (A.8)), that the only
modifications occurring in the solutions are of the type ln(ln-t) which is extremely mild and can
be neglected.


Finally, looking at the entire set of equations contained in (A.4) as an ensemble our
conclusion about the hierarchy of the matrix elements can be made highly plausible from the nature


of the hierarchy of the driving terms: >> in the driving terms in the first three equations of (A.4)
implies >> >> and, in turn, >> in the driving term in the last two equations implies >>
.


Appendix B



We show below that, for the U-matrix, even though we have


1
~ ~ (ln -t )1/4


the contribution of compared to is negligible in obtaining both the mass-eigenvalues of U and
the relevant CKM elements(e.g.V
ub ).



As we have already demonstrated, we can take = 0,in the U-matrix. To facilitate
comparison between the contribution of and consider two separate possibilities


o o o o
U

(a)
=
U

(b)

,
=
o


o

The rotation parameters s1 and s4 in CKM are involved only with the D-sector and we have
already shown that
s4 = 0


The rotation parameter s2 and s3, on the other hand, involve only the U-sector and the
dependence of CKM parameters on those angles is different depending on whether one considers
case (a) or case (b).


We will compare case (a) and case (b) in terms of contribution of and to the two
physical parameters, m
u and Vub .



Case (a). We have already considered this case in the Introduction where we also derived
the CKM matrix,(I.13). With s4=0, we have from the results of the Introduction


m

(i) | u
V | =
(B.1)
ub s
2 s =
const.
3
m
c



(ii) The contribution to mu is


2
1
m
u =
1/4
m ln
c ( t

-
)



Case (b) The CKM matrix (with s4=0) is

i
c1 c

2 s1 c

e
3 c

4 s
2
-i


Vckm - -
s (B.2)
1 e
c1 c
3 c
4 s3
-i
- c
1 s
+

2 s
1 s -
3 c
1 s3 c2 c
3 c
4


where one can show that


s =
,
2 s =

3


To compare the above quantities we find



(i) |
V | =

ub s =
0

2

(ii) contribution to mu is

2 2

m (B.3)
u =
=
(
(

) )
2
m
- c


so that even if we find that for this case there is an extra multiplicative factor (see IV.10)

3 a2
1 <

<
1 (B.4)
(2 )2

Therefore for both the parameters, V
ub and mu, the contribution coming from (b) is

negligible compared to case (a)[17].


We can, therefore, for all practical purposes, take

0
(B.5)


REFERENCES


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[9] Based on their RG equations it is well known that the gauge couplings are very small at
large t.


[10] Because dD13/dt vanishes, the other possibility, that of an infinite c13, is not physically
acceptable as it would imply D13 to remain infinite below .


[11] This is true even when the contributions of other Yukawa couplings (e.g., c, u) are
included.


[12] H. Fusaoka and Y. Koide, AMU-97-02, .


[13] H. Leutwyler, CERN-TH/96-25, .


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[17] One can confirm this result by comparing RRR's solution 5, which corresponds to our case
(b) and requires = 4 ( 0.22) with solution 4 which corresponds to case (a) and requires
= 2 6 (both normalized to = 1 and both with the same D-matrix structure as ours).
Thus the contribution of (b) will be comparable to (a) only when is more than an order of
magnitude larger than , but not when it is of the same order.



