"Extra Dimensions and Compositeness as a Basis for Hierarchy in Quark
Mass Matrices"

Bipin R. Desai and Alexander R. Vaucher

Department of Physics, University of California
Riverside, California 92521, USA
(September, 2000)


Abstract

Strong gauge and top-Yukawa couplings predicted in the
presence of extra dimensions lead to a trickle-down effect of
the top-coupling through the renormalization group equations
for the quark Yukawa matrices. The matrix elements for the
u and d-quarks get progressively smaller as one moves away
from the dominant (33)-element revealing a hierarchical
pattern for the mass matrices.





1


The possible existence of extra dimensions has raised an intriguing
question whether the gauge coupling constants a c g2 4 h become very
i i p
large at some scale [1]. If, indeed, that happens then it makes sense that the
quark-Yukawa couplings, particularly the largest of them, the top-coupling
should behave the same way [1], [2].

A simple way to motivate this and to obtain g in the process is to first
t
consider the behavior of a , the largest of the three gauge couplings, and
3
relate it to the top coupling. Beyond m m , ( 1
R , R being the radius of
0 m0
the extra dimension), a is given by a power law [1]
3


~ ~ d
F b b X L O
3
1 1 m 3 m 3 d m
a ( ) ( M ) ln ln (1)
z M 1P
3 m a 3 p
2 HG I
Mz KJ F
p
2 HG I
m KJ F
2 HG I
M KJ P
0 pd m0
N Q

where d represents additional dimensions, X is a known function of d , and
d
~
b 6 4
3 h , where h represents the generation of chiral fermions of the
MSSM. For h 2, a 1( ) decreases rapidly for . Specifically, for
3 m m m0
h 2, and d 1 ( X 2 ) one can write (1) in this region as
d


2
a 1 c et (2)
3 3 p

where c is a known quantity and t lnb g. Expanding near the zero at
3 m m0
t t , we can write
0


2
a 1 t0
e bt tg (3)
3 p 0


If g is the top-Yukawa coupling, and g2 4 , then in MSSM
t a t t p

1
d 3 8
1 a t
a (4)
dt t p p
3 a 1
3


Clearly then, because of (3), the right hand side would develop a pole and
become infinite at t t . This is untenable as would have the rather
0 a t
unphysical behavior of diving down to zero at that point (that is a 1 becomes
t
infinite). The sensible alternative is to have a 1 vanish at t t as well (i.e.
t 0
both couplings share the same singular point).


2


Writing

a 1 c bt tg (5)
t t 0


we find from (3) and (4) that

d 3 4
1 ct e t
a 0 (6)
dt t p 3

Because t is very large one can ignore the second term on the right so that
0


d 3
a 1 (7)
dt t p

therefore

3
a 1 bt tg
t p 0

which implies 3
c in (5) justifying a posteriori neglecting the second right
t p
hand side term in (6). The top coupling is now given by

2
g p
t 3bt tg (8)
0


This result, not surprisingly, is identical to that of Bardeen et al for the
compositeness condition obtained through the vanishing of the Higgs
renormalization parameter, Z , at the composite scale (Z 0; g ) [4].
H H t
And it is independent of the manner in which the gauge couplings become
large.

What we have said above simply re-iterates the predictions of strong
top-dynamics in the presence of extra dimensions [1], [2], [5].

Since m does not enter explicitly in expression (8) we re-define t and
0
t in what follows
0





3


t t lnbm 1Gevg U
V|
t t lnb 1Gevg (9)
0 0 W|

where following [4] we call, , the composite scale.

Since we know the explicit form of g near the boundary, and know
t
that it is singular there, we can use Frobenius-type method to solve the
renormalization group equations (RGE) for the quark-Yukawa couplings.
Eventhough it has often been stated that the RGE equations can not shed
important light on the hierarchy question [6], the knowledge of g turns out
t
to provide very interesting results [7].

We find a definite trickle-down effect of the top-coupling as one
moves away from the (33)-element of the u-quark matrix (U) into the other
matrix elements revealing a hierarchical pattern. The same thing happens to
the d-quark matrix (D).

We will first consider the U-Matrix where, because of the dominance
of the top-coupling, the (33) element, to an excellent approximation, is given
by g itself.
t

Fu u u
11 12 13 I
U Gu u u J (10)
HG 21 22 23J
u u u
31 32 33 K


We will choose U to be symmetric, so that

u u
ij ji


and

2
u g p
33 t 3bt tg (11)
0





The RGE for U , for MSSM as an example, is given by




4


L
dU
16 2 2
p c g 3UU DD Tr 3UU +
c h U (12)
dt i i
NM OQP
i


In order to demonstrate the hierarchy, we write the RGE in terms of the
ratios

u
r ij , dr r i (13)
ij u ij ji
33


This also has the effect of eliminating the contribution of the gauge
couplings. If we define

t
x 1 (14)
t0

then we obtain the following RGEs

dr 1
11 cr r2h (15a)
dx 4x 11 13


dr 1
12 br r r g (15b)
dx 4x 12 23 13


dr 1
13 r r (15c)
dx 4x 23 12

dr 1
22 cr r2h (15d)
dx 4x 22 23


dr 1
23 br r r r g (15e)
dx 4x 13 12 23 22


We expect r 1 (except r 1). If we write
ij 33


r c xnij (16)
ij ij


near x=0 then, since

1
u x 2
~ (17)
33





5


from (11), the exponents n will be positive, and because of the square-root
ij
singularity of 1 1
u , they will be 0, , ,... etc.
33 4 2

Inserting (16) in equations (15a) through (15e) we note that each
equation will provide two relations. One will relate the n 's through the
ij
equality of the powers of x on both sides and the other will involve mostly
the c 's.
ij


We find the following relations between the coefficients, keeping c23
as the input, and keeping only the leading terms [8]

c c2 U
22 23
c c c V|
13 12 23

c c2 c2 W| (18)
11 12 23


We note immediately that, since we expect

c 1,
ij


there is a definite hierarchical pattern. This is even more evident when we
examine the n 's. The U-matrix reads, after we put in all the results, as
ij
(normalized to u 1)
33

F 1 1 1
2 2 I
c c x 2 c x 4 c c x
G 4
G 12 23 12 12 23
1 J
4 2 J
U c x c c
G (19)
G 12 23 23 J
1
4 J
c c x c
HG 1
12 23 23 KJ

near x=0.


We notice that as we move away from the (33)-element the matrix
elements get progressively smaller until we reach the (11)-element which is
the smallest. The quark masses can be estimated from the diagonal
elements,



6


1
m :m :m c2 c2 x 2 : c2 :1 (20)
u c t 12 23 23


which shows a definite hierarchical pattern.

One often expresses the mass hierarchy in terms of the CKM
parameters. In order to connect to the CKM elements, therefore, we write
the RGE for V in MSSM which reads
CKM

d U
16 2 3
p ln V
b g O Vc h
12 12
dt V|| (21)
d
16 2 2
p ln V
d i g , i(j 1323
, 31
, 32
, )
dt ij t W||

where we keep only the most dominant term on the right.

We write the CKM matrix in the Wolfenstein representation

F 2
l 3 I
1 l Al (r i
G )
G h
2 J
2 J
V G l
l 1 A 2
l J (22)
CKM GG 2 J
A 3
l 1
( r ih) A 2
l 1 J
HG KJ

Because V is very small, (21) implies that it remains a constant
12 l . The
constants c in (16) may, therefore, be related to
ij l ,

c f (l) (23)
ij ij


Considering specifically the (23)-term in (22), to avoid complex elements
involving r and h, the relations (11) and (21) state that

dV 1
23 V (24)
dx 12x 23

therefore,

1
V x12
~ (25)
23





7


From the Wolfenstein parameterization this implies, since l is a constant,
that

1
A ~ x12 (26)

The matrix (19) can be written as

F c2 c2 A6 c A3 c c A3
12 23 12 12 23 I
G 3 2 J
U ~ c A c c
G (27)
12 23 23 J
HG 3 1 J
c c A c
12 13 23 K

which shows the hierarchical behavior without the x dependence.

From (23) if we take c 2
~ 2
~
12 l , and c23 l then the mass ratio (20)
would be

m :m :m A6 8
l : 4
l :1
u c t


which is consistent with the standard hierarchical representation of the U-
quark masses.


Turning now to the D-matrix

Fd d d
11 12 13 I
D Gd d d J (28)
HG 21 22 23J
d d d
31 32 33 K


the corresponding RGE is given by

L
dD
16 2 ' 2
p c g 3DD UU Tr 3DD+
c h D (29)
dt i i
NM OQP
i


It can be shown from the above equation that

d d (30)
ij ji


i.e. D is not symmetric.


8


Once again, as in the case of g and u , we conclude that, to an
t 33
excellent approximation,

d g (31)
33 b


where g is the b-quark coupling.
b


From equation (29), keeping the dominant g contribution on the right it is
t
easy to verify that

1
d x 12
~ (32)
33


which has a singularity at x=0, like u , but a much milder one.
33


If we write

d
r ij (33)
ij d33

then from (29) we obtain the following eight equations, where on the right
we have kept only the most dominant terms

dr 1
11 br r g (34a)
11 13r
dx 12x 31
dr 1
12 br r g (34b)
12 13r
dx 12x 32
dr 1
13 br r g (34c)
dx 12x 13 13
dr 1
21 br r g (34d)
21 23r
dx 12x 31
dr 1
22 br r g (34e)
22 23r
dx 12x 32
dr 1
23 br r g (34f)
dx 12x 23 23
dr 1
31 br r g (34g)
31 23r r
23 23r
dx 12x 21
dr 1
32 br r g (34h)
32 23r r
23 23r
dx 12x 22





9


Once again expressing

r c xnij (35)
ij ij


we obtain from the earlier results for r (normalizing d 1)
ij 33

F 1 1 1 1 1
2 1
3 3 4 I
c c c x c c x c c x
GG 3 12 23 21 3 12 23 2 12 23 J
1 1 J
D G c x12 c x12 c
G (36)
21 22 23 J
G 1 1 J
c c x12 c c x12 1
23 21 23 22 J
HG KJ
and F 1 4 1 2 4 1 3I
c c c A c c A c c A
G 3 12 23 21 3 12 23 2 12 23 J
D ~ G c A c A c
G (37)
21 22 23 JJ
c c A c c A
HG 1
23 21 23 22 KJ

Again, a definite hierarchical pattern is exhibited in the above solution
because the matrix elements get progressively smaller through the
contributions of c , c and the powers of x or A. Furthermore, unlike the U-
ij ij
matrix, the dependence on x or A is much milder in D.

The above results are obtained near the composite scale x=0.
However, because c and c 's are expected to be constants, matrices (27) and
ij ij
(37) containing the Wolfenstein parameter A maybe continuable to lower
energies. This remains to be explored.

One could try to get more quantitative in discussing our results e.g. by
assuming certain specific dependence for the constants c and c so that our
ij ij
matrices reproduce the representations of U and D that are normally given in
terms of l . And one could invoke texture zeroes and introduce special
models. We will take these up in a later work.

What is quite remarkable, however, is that once we know the specific
singular form of g (and g ) at the composite scale, an inherent hierarchical
t b
form for the Yukawa matrices emerges simply based on the RGEs.


10


We thank Dr. Utpal Sarkar for many helpful discussions.

This work was supported in part by the U.S. Department of Energy
under contract No: DE-FG03-94ER40837.





11


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ph/000638.

[3] N. Arkani-Hamed, S. Dimopoulos, and G. Dvali, Phys. Lett. B429,
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[4] W. Bardeen, C. Hill, and M. Lindner, Phys. Rev. D41, 1642(1990).

[5] H-C. Cheng, B. Doberscu, and C. Hill .

[6] C. Frogatt and H. Nielsen, Nucl. Phys. B147, 277(1979).

[7] B.R. Desai and D.P. Roy Phys Rev. D58, 113007(1998).

[8] For example, 2
e j
r has a term c x c 4
23 which is simply replaced by c ,
23 23 23
because the power is extremely small.





12



