We investigate how the Fritzsch ansatz for the quark mass matrices can be
modified in the least possible way to accommodate the observed large top quark
mass and the measured values of the CKM elements. As one of the solutions, we
find that the \{23\} and the \{32\} elements of the up quark mass matrix are
unequal. The rest of the assumptions are same as in Fritzsch ansatz. In this
formalism we have an extra parameter i.e. the ratio of the \{{23\}} and the
\{{32\}} element, which gets fixed by the large top quark mass. The predicted
values for $\frac{V_{ub}}{V_{cb}}$ , $\frac{V_{td}}{V_{ts}}$ from this new
ansatz are in the correct experimental range even for the smaller values of
$\tan \beta $. In the end, we write down the $SO(10)$ motivated superpotential
for these new mass matrices.Comment: 10 pages, Latex, Figure available on reques

Dutta, B.

Nandi, S.

English

arXiv

We investigate how the Fritzsch ansatz for the quark mass matrices can be modified in the least possible way to accommodate the observed large top quark mass and the measured values of the CKM elements. As one of the solutions, we find that the \{23\} and the \{32\} elements of the up quark mass matrix are unequal. The rest of the assumptions are same as in Fritzsch ansatz. In this formalism we have an extra parameter i.e. the ratio of the \{{23\}} and the \{{32\}} element, which gets fixed by the large top quark mass. The predicted values for \frac{V_{ub}}{V_{cb}} , \frac{V_{td}}{V_{ts}} from this new ansatz are in the correct experimental range even for the smaller values of \tan \beta . In the end, we write down the SO(10) motivated superpotential for these new mass matrices

Dutta, B

Nandi, S

CERN Document Server

July 1995 OSU Preprint 304
A new ansatz: Fritzsch Mass Matrices
with least modication
B. Dutta and S. Nandi
Department of Physics
Oklahoma State University
Stillwater, OK 74078
ABSTRACT
We investigate how the Fritzsch ansatz for the quark mass matrices can be
modied in the least possible way to accommodate the observed large top quark
mass and the measured values of the CKM elements. As one of the solutions,
we nd that the f23g and the f32g elements of the up quark mass matrix are
unequal. The rest of the assumptions are same as in Fritzsch ansatz. In this
formalism we have an extra parameter i.e. the ratio of the f23g and the f32g
element, which gets xed by the large top quark mass. The predicted values for
Vub
Vcb
, VtdVts from this new ansatz are in the correct experimental range even for
the smaller values of tan. In the end, we write down the SO(10) motivated
superpotential for these new mass matrices
A simple approach for including the observed hierarchy of quark masses and mix-
ing angles was suggested by Fritzsch[1]. It prescribes a form for the mass matrices
which has certain amount of predictive power. It can predict the top mass based on
the Vcb element of the CKM matrix as input. The top mass according to this ansatz
can not be more than 90 GeV. The recent CDF data[2] however shows that the top
mass (pole mass) is above 160 GeV (mt = 176  8 (stat)  10 (syst)GeV). So, the
simplest version is clearly excluded.
This situation improves if one realizes these mass matrices at the GUT scale[3].
The eect of the running from the weak scale upto GUT scale is utilized to incor-
porate a somewhat heavier top in the theory. But the improvement is not enough
to incorporate the recently discovered heavy top quark. If we try to use this top
mass as input in this GUT scenario, the predicted value of Vcb is far o from the
experimentally predicted range which is Vcb = 0:0400  0:0025  0:0020[4]. In Fig.1
we show the range of Vcb values for the range of the top mass mt from 155 to 185 GeV
, where mt stands for the running mass. The relation between the running mass and
the pole mass is given by
mt = mt (mt)

1 +
4
3
3


(1)
Increasing the value of tan improves the situation a little better as shown in
Fig.1. However, to get the correct prediction for Vcb, one has to go beyond tan  65,
where the theory loses the perturbative nature [5].
In this letter we propose a modication of this Fritzsch ansatz in the least possible
sense. We make the f23g and the f32g element asymmetrical in the up quark mass
matrix. It then looks like:
Mu = PuUQu (2)
where
U =
0BB@
0 au 0
au 0 bu
0 b
0
u cu
1CCA
The zeros are described later in the superpotential. The expressions a; b; b
0
; c are
real numbers while P and Q are diagonal phase matrices. Among the quark phases
contained in P and Q, only two are relevant for quark mixing, we denote them by  
and  .The down matrix is kept the same as in ref.[1], i.e.
Md = PdDQd (3)
where
D =
0BB@
0 ad 0
ad 0 bd
0 bd cd
1CCA
2
We also observe the same hierarchical form as has been exercised in the Fritzsch
ansatz, i.e. c b  b
0
 a: We realize these matrices at the GUT scale.
The real matrix U is diagonalized by the bi-orthogonal transformationRuUR
0−1
u =
Udiag producing the eigenvalues fmu;−mc;mtg.Using the hierarchy of masses mu 
mc  mt we obtain
Ru =
0BB@
1 su1 − 
u
1s
u
2 s
u
1s
u
2 + 
u
1
−su1 1 s
u
2
−u1 −s
u
2 1
1CCA
R
0
u =
0BB@
1 su1 − 
u
2s
0u
2 s1s
0u
2 + 
u
2
−su1 1 s
0u
2
−u2 −s
0u
2 1
1CCA
where su1  sin’
u
1 =
q
mu
mc
and su2  sin’
u
2 = −
q
mc
mt
and we have set cos’ui  1:
Similarly u1 
mcsu1 s
u
2
rmt
; u2 = 
u
1r , s
0u
2 =
su2
r
; and r  b
b
0 .
The down quark mass matrix D is diagonalized by the orthogonal transformation
RdDR
−1
d = D
diag producing the eigenvalues fmd;−ms;mbg. Using the heirarchy of
masses md ms  mb , we obtain
Rd =
0BB@
1 sd1 − 
d
1s
d
2 s
d
1s
d
2 + 
d
1
−sd1 1 s
d
2
−d1 −s
d
2 1
1CCA
sd1  sin’
d
1 =
q
md
ms
and sd2  sin’
d
2 = −
q
ms
mb
and d1 =
mssd1s
d
2
mb
. VCKM at the
unication scale in terms of the mass ratios is given by:
Ru
0BB@
1
ei
ei
1CCAR−1d (4)
We nd the expressions for V 0cb; V
0
ub; V
0
td ; V
0
ts as :V 0cb =

s
ms
mb
− ei
s
mcr
mt
 (5)
where  =  − ;V 0ub =
msmb
s
md
mb
+ ei 
 s
ms
mb
s
mu
mc
− ei
 s
mur
mt
−
mc
mt
s
mu
mtr
!! (6)
where  = ;
V 0td =
−mcmt
s
mu
mtr
− ei 
s
md
ms
 s
ms
mb
− ei
s
mcr
mt
+

ms
mb
 3
2
! (7)
3
V 0ts =

s
mcr
mt
− ei
s
ms
mb
 (8)
The zeros in the superscript indicates the GUT value. All the masses are at the GUT
scale ( 1016). We run them to the top scale ( 170GeV ), and also assume that the
MSUSY = mt.
To accomplish the running, we write down the Yukawa sector which has the generic
form
LY = qLHuuuR + qLHdddR + ‘LH‘deR + h:c: (9)
where Hu; Hd; H‘ denote the 3  3 Yukawa coupling matrices for the up quarks,
down quarks and charged leptons. The one loop evolution equations for the Yukawa
matrices take the form (t  ‘n(=MG))[6]:
162
dHu
dt
=
h
Tr(3HuH
y
u + 3aHdH
y
d + aH‘H
y
‘ )
+
3
2
(bHuH
y
u + cHdH
y
d)−GU

Hu
162
dHd
dt
=
h
Tr(3aHuH
y
u + 3HdH
y
d +H‘H
y
‘ )
+
3
2
(bHdH
y
d + cHuH
y
u)−GD

Hd (10)
162
dH‘
dt
=
h
Tr(3aHuH
y
u + 3HdH
y
d +H‘H
y
‘ )
+
3
2
bH‘H
y
‘ −GE

H‘ :
For the minimal supersymmetric standard model (MSSM) under consideration the
coecients a; b; c are given by
(a; b; c) = (0; 2; 2
3
) (11)
and the quantities GU ; GD and GE are given by:
GU =
13
15
g21 + 3g
2
2 +
16
3
g23
GD =
7
15
g21 + 3g
2
2 +
16
3
g23
GE =
9
5
g21 + 3g
2
2 ;
(12)
The gauge couplings gi (above) obey the standard one loop renormalization group
equations:
4
82
dg2i
dt
= big
4
i ; i = 1; 2; 3 (13)
where
(b1; b2; b3) = (
33
5
; 1;−3) forMSSM (14)
From eqn.(9), one can compute the evolution equations for the eigenvalues of the
Yukawa coupling matrices[7][8]:
162
dfi
dt
= fi
243 X
j=u;c;t
f2j + 3a
X
=d;s;b
f2 + a
X
b=e;;
f2b −GU
+
3
2
bg2i +
3
2
c
X
=d;s;b
f2 j Vi j
2
35
162
df
dt
= f
243a X
j=u;c;t
f2j + 3
X
=d;s;b
f2 +
X
b=e;;
f2b −GD
+
3
2
bf2 +
3
2
c
X
j=u;c;t
f2j j Vj j
2
35
162
dfa
dt
= fa
243a X
j=u;c;t
f2j + 3
X
=d;s;b
f2 +
X
b=e;;
f2b −GE
+
3
2
bf2a

(15)
where i = (u; c; t);  = (d; s; b); a = (e; ; ).
We will also need the evolution equations for the elements of the CKM matrix[7][8]:
162
d
dt
j Vi j
2 = 3c
24X
j 6=i
X
=d;s;b
f2i + f
2
j
f2i − f
2
j
f2Re

ViV

jVjV

i

+
X
 6=
X
j=u;c;t
f2 + f
2

f2 − f
2

f2jRe

V jVjViV

i
35 : (16)
The above expressions would simplify considerably if we exploit the hierarchy in
the Yukawa couplings (fb  fs  fd, etc) and in the CKM matrix elements. If only
the leading terms are kept, one obtains the following approximate expressions for the
evolution of the various mass ratios and the mixing angles:
162
d
dt

m
mb

= −
3
2

m
mb
 
bf2b + cf
2
t );  = d; s
162
d
dt

mi
mt

= −
3
2

mi
mt

(bf2t + cf
2
b ); i = u; c
5
162
d
dt

md
ms

= −
3
2

md
ms
 
bf2s + cf
2
c + cf
2
t j Vts j
2

162
d
dt

mu
mc

= −
3
2

mu
mc
 
bf2c + cf
2
s + cf
2
b j Vcb j
2

162
d
dt
j Vi j = −
3
2
c j Vi j

f2t + f
2
b

(i) = (ub); (cb); (td); (ts)
162
d
dt
j Vus j = −
3
2
c j Vus j
 
f2c + f
2
s + f
2
t
j Vtd j2 − j Vub j2
j Vus j2
!
162
d
dt
j Vcd j = −
3
2
c j Vcd j
 
f2c + f
2
s + f
2
b
j Vub j2 − j Vtd j2
j Vcd j2
!
: (17)
One comment is necessary here:
jVubj
jVcbj
and jVtdjjVtsj do not run in one loop.
The low energy i.e. the mt scale values of the Yukawa couplings are :
b(mt) =
p
2mb(mb)
bv cos
;  (mt) =
p
2m (m)
v cos
; t(mt) =
p
2mt(mt)
v sin
(18)
where f =mf(mf)=mf (mt) gives the running of the masses below  = mt; obtained
from 3-loop QCD and 1 loop QED evolution, for heavy flavors f = t; b; c; : For light
flavors f = s; d; e;  we stop at  =1 GeV and dene f =mf(1GeV )=mf (mt) . For
3 = 0:118; b ’ 1:5; c ’ 2:1; s = d = u ’ 2:4: The running mass values are
mb(mb) = 4:25  0:15GeV;m (m) = 1:7777GeV;mc(mc) = 1:2GeV;ms(1GeV ) ’
0:175GeV;mu(1GeV ) ’ 0:006GeV;md(1GeV ) ’ 0:008GeV [9].
We solve for r

= b
b
0

from V 0cb using eqn.(5) for a range of values of ’ as shown
in Table 1. The values of ’ are chosen so that r is real. We use these values of r
to predict V 0ub(eqn.(6)),V
0
td(eqn.(7)) and V
0
ts(eqn.(8)). We then calculate the values of
Vub, Vcb, Vtd and Vts at the low scale using eqn.(17). In Table 1 , we present the values
of Vub
Vcb
and Vtd
Vts
from our ansatz and compare them with the experimental values. The
dependence of any prediction on  is negligible,  is kept xed at =2. As shown in
the Table 1, the prediction of the model is in excellent agreement with the current
experimental ranges for an wide range of tan. The model will be tested further as
the experimental ranges are narrowed down in the future.
Motivated by the supersymmetric SO(10) Grand Unifying group, we can write
down the superpotential for the mass matrices
W = f (10
00
)
12  1 2’1000 + f
(10
0
)
23  2 3’100 + f
(120)
23  2 3’120 + f
10
33 3 3’10 + h:c:
(19)
Here, ’120 has the vev only in the up direction. The zeros are produced by the discrete
symmetry. From eqn.(19), we obtain the following form of the mass matrices.
6
U =
0BB@
0 10
00
0
10
00
0 10
0
; 120
0 10
0
; 120 10
1CCA ; D =
0BB@
0 10
00
0
10
00
0 10
0
0 10
0
10
1CCA (20)
Here, the entries in an element correspond to the Higgs elds contributing to that
element. Since f120g is an antisymmetric representation , the f23g and f32g elements
are asymmetric.
In conclusion we summarize that if f23g and f32g elements of the up quark sector
of the Fritzsch mass matrices are unequal, one can predict Vub
Vcb
and Vtd
Vts
in the correct
experimental range even with a heavy top in the theory. Moreover, for this ansatz,
tan  need not be very high, it can lie anywhere between 0.6 and 65. Also, it is
possible to write a superpotential for this ansatz.
This research was supported in part by the US Department of Energy, Grant Number
DE-FG02-94ER40852.
7
References
[1] H. Fritzsch, Phys. Lett. B 70, 436 (1977) and Nucl. Phys. B 155, 189 (1979).
[2] F. Abe and et.al. Phys. Rev. D 74, 2627 (1995).
[3] K. S. Babu and Q. Sha, Phys. Rev. D 47, 5004 (1993) and Nucl. Phys. B 311,
172 (1993).
[4] J. R. Patterson, Proceedings of XXV II International conference on High Energy
Physics, Glasgow, UK 20-27 July, 1994 p.149.
[5] V. Barger, Talk presented at the Workshop on Physics at Current Accelerator
and the Supercollider, Argonne, June, 1993; V. Barger, M. S. Berger and P.
Ohmann, Phys. Rev. D 47, 1094 (1993).
[6] T. P. Cheng, E. Eichten, and L-F. Li, Phys. Rev. D 9, 225 (1974); M. Mahachacek
and M. Vaughn, Nucl. Phys. B 236, 221 (1984).
[7] E. Ma and S. Pakvasa, Phys. Lett. B 86, 43 (1979) and Phys. Rev. D 20, 2899
(1979).
[8] K. S. Babu, Z. Phys. C 35, 69 (1987);
K. Sakai, Z. Phys. C 32, 149 91986);
B. Grzadkowski, M. Lindner and S. Theisen, Phys. Lett. B 198, 64 (1987);
M. Olechowski and S. Pokorski, Phys. Lett. B 257, 388 (1991).
[9] J. Gasser and H. Leutwyler, Phys. Rep 87, 77 (1982).
8
TABLE CAPTION
Table 1
The predicted values of Vub
Vcb
Vtd
Vts
from the new ansatz are compared with
their experimental values for dierent values of tan and tan
FIGURE CAPTION
Figure 1
The predicted values of Vcb from Fritzsch ansatz is plotted as functions of
mt, for values of tan=3 and 60. The experimental range of Vcb is also
shown. To be conservative, we have taken the deviation from the central
value of Vcb to be 0.006 instead of 0.003.
9
Table 1:
tan  tan  predicted Vub
Vcb
predicted Vtd
Vts
Exptal.range of Vub
Vcb
Exptal.range of Vtd
Vts
0 0.069 0.23
3 0.18 0.082 0.23 0.03-0.137 0.11-0.36
0.23 0.084 0.22
0 0.069 0.23
20 0.18 0.080 0.22
0.24 0.084 0.22
0 0.068 0.23
40 0.18 0.078 0.23
0.24 0.082 0.22
0 0.067 0.22
60 0.18 0.072 0.22
0.26 0.075 0.22
10


A new ansatz: Fritzsch Mass Matrices with least modification

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