Can the SO(10) Model with Two Higgs Doublets Reproduce the Observed Fermion Masses?

Abstract

It is usually considered that the SO(10) model with one 10 and one 126 Higgs scalars cannot reproduce the observed quark and charged lepton masses. Against this conventional conjecture, we find solutions of the parameters which can give the observed fermion mass spectra. The SO(10) model with one 10 and one 120 Higgs scalars is also discussed.

Full text

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arXiv:hep-ph/0010026v2 12 Apr 2001
Can the SO(10) Model with Two Higgs Doublets
Reproduce the Observed Fermion Masses?
K. Matsuda, Y. Koide(a), and T. Fukuyama
Department of Physics, Ritsumeikan University, Kusatsu, Shiga, 525-8577 Japan
(a) Department of Physics, University of Shizuoka, Shizuoka 422-8526 Japan
(February 1, 2008)
It is usually considered that the SO(10) model with one 10 and one 126 Higgs scalars cannot
reproduce the observed quark and charged lepton masses. Against this conventional conjecture, we
find solutions of the parameters which can give the observed fermion mass spectra. The SO(10)
model with one 10 and one 120 Higgs scalars is also discussed.
PACS number(s): 12.15.Ff, 12.10.-g, 12.60.-i
I. INTRODUCTION
The grand unification theory (GUT) is very attractive
as a unified description of the fundamental forces in the
nature. Especially, the SO(10) model is the most attrac-
tive to us when we take the unification of the quarks and
leptons into consideration. However, in order to repro-
duce the observed quark and lepton masses and mixings,
usually, a lot of Higgs scalars are brought into the model.
We think that the nature is simple. What is of the great-
est interest to us is to know the minimum number of the
Higgs scalars which can give the observed fermion mass
spectra. A model with one Higgs scalar is obviously ruled
out for the description of the realistic quark and lepton
mass spectra. Then, how is a model with two different
types of Higgs scalars (e.g., 10 and 126 scalars)?
In the SO(10) GUT scenario, a model with one 10 and
one 126 Higgs scalars leads to the relation [1]
Me= cuMu+ cdMd,(1.1)
where Me, Muand Mdare charged lepton, up-quark and
down-quark mass matrices, respectively. It is widely ac-
cepted that there will be almost no solution of cu and
cd which give the observed fermion mass spectra. The
reason is as follows: We take a basis on which the up-
quark mass matrix Mu is diagonal (Mu = Du). Then,
the relation (1.1) is expressed as
?
Me= cuDu+ cd?
Mdis almost diagonal and the mass hi-
erarchy of up-quark sector is much severe than that of
down-quark sector, we observe that the contribution to
the first and the second generation part of?
up-quark part Duis negligible so that it is proportional
to that of?
me/mµ≃md/ms does not reproduce the observed hier-
archical structure of the down-quark and charged lepton
masses [2] such as predicted by Georgi-Jarlskog mass re-
lations mb = mτ, ms = mµ/3 and md = 3me at the
Md.(1.2)
Considering that?
Mefrom the
Md. Thus, the relation (1.1) which predicts
GUT scale [3]. However, the above conclusion is some-
what impatient one. (i) It is too simplified to regard?
as almost diagonal. (ii) We must check a possibility that
the mass relations are satisfied with the opposite signs,
i.e., mb = ±mτ, ms = ±mµ/3 and md = ±3me. (iii)
The mass values at the GUT scale which are evaluated
from the observed values by using the renormalization
group equations show sizable deviations from the Georgi-
Jarskog relations. The purpose of the present paper is to
investigate systematically whether there are solutions of
cuand cdwhich give the realistic quark and lepton masses
or not.
Md
II. OUTLINE OF THE INVESTIGATION
In the SO(10) GUT model with one 10 and one 126
Higgs scalars, the down-quark and down-lepton mass ma-
trices Mdand Meare given by
Md= M0+ M1, Me= M0− 3M1,(2.1)
where M0and M1are mass matrices which are generated
by the 10 and 126 Higgs scalars φ10 and φ126, respec-
tively. Inversely, we obtain
M0=1
4(3Md+ Me), M1=1
4(Md− Me).(2.2)
On the other hand, the up-quark mass matrix Muis given
by
Mu= c0M0+ c1M1,(2.3)
where
c0= vu
c1= vu
0/vd
1/vd
0= ?φu0
1= ?φu0
10?/?φd0
126?/?φd0
10?,
126?,(2.4)
and φuand φddenote Higgs scalar components which
couple with up- and down-quark sectors, respectively.
Therefore, by using the relations Eq.(2.2), we obtain the
relation
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Me= cdMd+ cuMu,(2.5)
where
cd= −3c0+ c1
c0− c1
, cu=
4
c0− c1
.(2.6)
For convenience, first, we investigate the case that the
matrices Mu, Mdand Meare symmetrical matrices at the
unification scale because we assume that they are gener-
ated by the 10 and 126 Higgs. Then, we can diagonalize
those by unitary matrices Uu, Udand Ue, respectively, as
UT
uMuUu= Du, UT
dMdUd= Dd, UT
eMeUe= De,
(2.7)
where Du, Ddand Deare diagonal matrices. Since the
Cabibbo-Kobayashi-Maskawa (CKM) matrix V is given
by
V = UT
uU∗
d,(2.8)
the relation (2.5) is re-written as follows:
(U†
eUu)TDe(U†
eUu) = cdV DdVT+ cuDu.(2.9)
At present, we have almost known the experimental val-
ues of De, Du and V DdV†. Therefore, we obtain the
independent three equations:
TrDeD†
e= |cd|2Tr
?
?
(V DdVT+ κDu)(V DdVT+ κDu)†?
((V DdVT+ κDu)(V DdVT+ κDu)†)2?
?
,(2.10)
Tr(DeD†
e)2= |cd|4Tr
,(2.11)
detDeD†
e= |cd|6det(V DdVT+ κDu)(V DdVT+ κDu)†?
,(2.12)
where κ = cu/cd. By eliminating the parameter cd, we
have two equations for the parameter κ:
(m2
e+ m2
m2
µ+ m2
em2
µ+ m2
µm2
τ)3
µm2
τ)2
τ
=(2.10)3
(2.12),(2.13)
(m2
e+ m2
µ+ m2
2(m2
em2
τ+ m2
τm2
e)=
(2.10)2
(2.10)2− (2.11),(2.14)
where (2.10)3, for instance, means the right-hand side of
(2.10) to the third power. Let us denote the parameter
values of κ evaluated from (2.13) and (2.14) as κA and
κB, respectively. If κAand κBcoincide with each other,
then we have a possibility that the SO(10) GUT model
can reproduce the observed quark and lepton mass spec-
tra. If κA and κB do not so, the SO(10) model with
one 10 and one 126 Higgs scalars is ruled out, and we
must bring more Higgs scalars into the model. Of course,
in the numerical evaluation, the values κA and κB will
have sizable errors, because the observed values De, Du,
Dd and V have experimental errors, and the values at
the GUT scale also have errors. The values κAand κB
are not so sensitive to the renormalization group equa-
tion effect (evolution effect), because those are almost
determined only by the mass ratios. (More details will
be discussed in the Sec. III.) Therefore, we will evaluate
κAand κBby using the center values at µ = mZ in the
Sec. IV. If we find κA≃ κB, we will give further detailed
numerical study only for the case.
III. EVOLUTION EFFECT
The relations (2.13) and (2.14) hold only at the unifica-
tion scale µ = ΛXOn the other hand, we know only the
experimental values of the fermion masses mf and CKM
matrix parameters Vij at the electroweak scale µ = mZ.
For a model which does not have any intermediate en-
ergy scales, we can straightforwardly estimate the values
of mf and Vij at µ = ΛX from those at µ = mZ by the
one-loop renormalization equation
dYf
dt
=
1
16π2(Tf− Gf+ Hf)Yf
(3.1)
where Tf, Gfand Hfdenote contributions from fermion-
loop corrections, vertex corrections due to the gauge
bosons and vertex corrections due to the Higgs boson(s),
respectively.Therefore, we can directly check the re-
lations (2.13) and (2.14) by substituting the observable
quantities mf and Vij at µ = ΛX. However, for a model
which has an intermediate energy scale such as a non-
SUSY model, the values of mf and Vij at µ = ΛX are
highly model-dependent, so that the check of Eqs. (2.13)
and (2.14) cannot be done so straightforwardly.
In this section, we will show that we can approximately
check Eq. (2.13) and (2.14) by using the values of mfand
Vijat µ = mZ, without knowing the explicit values of mf
and Vij at µ = ΛX, as far as the evolutions of mf and
Vijare not singular.
It is well known that in such a conventional model the
evolution effects are approximately described as [4]
m0
mu/mt
u/m0
t
≃m0
c/m0
mc/mt
t
≃ 1 + εu,
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m0
md/mb
|V0
ub|
|Vub|≃|V0
m0
mu/mc
d/m0
b
≃m0
ms/mb
|Vcb|≃|V0
≃m0
md/ms
s/m0
b
≃ 1 + εd,
cb|
td|
|Vtd|≃|V0
d/m0
s
≃|V0
ts|
|Vts|≃ 1 + εd,
|Vus|≃|V0
u/m0
c
us|
cd|
|Vcd|≃ 1 ,(3.2)
where m0
µ = ΛX (µ = mZ). The relations (3.2) hold only for
a model where the Yukawa coupling constant of top
quark, yt≡ (Yu)33, satisfies yt≫ (Yd)ij (i,j = 1,2,3).
The relations (3.2) also hold even in a model which
has an intermediate energy scale ΛI, because, for exam-
ple, when we denote (mu/mt)µ=ΛX/(mu/mt)µ=ΛIand
(mu/mt)µ=ΛI/(mu/mt)µ=mZas 1+εu1 and 1+εu2, re-
spectively, we can obtain (mu/mt)µ=ΛX/(mu/mt)µ=mZ≃
1 + εuwith εu= εu1+ εu2.
By using the approximate relations (3.2) the diagonal-
ized up-quark mass matrix D0
as


00


00
=m0
mt(1 + εuS)Du,
qand V0
ij(mq and Vij) denote the values at
uat µ = ΛX is presented
D0
u= m0
t
m0
u/m0
0
t
00
m0
c/m0
t 0
1




≃ m0
t
mu/mt
0
00
mc/mt 0
1


1 + εu
0
0
00
1 + εu 0
01


t
(3.3)
where
S =


1 0 0
0 1 0
0 0 0

. (3.4)
Similarly, the matrix D0
dis given by
D0
d≃m0
b
mb(1 + εdS)Dd.(3.5)
The CKM matrix V0at µ = ΛXis given by


Vtd(1 + εd) Vts(1 + εd)
≃ (1 + εdS3)V (1 + εdS3) − 2εdS3,
V0≃
1Vus
1
Vub(1 + εd)
Vcb(1 + εd)
1
Vcd


(3.6)
where S3= 1−S and 1 is a 3×3 unit matrix. By using
the relations (3.4) - (3.6), we can obtain the approximate
expression
V0D0
dV0T≃m0
b
mb
?(1 + εd)V DdVT− εdmbS3
?,(3.7)
where we have used the observed hierarchical relations
among the quark mass ratios and CKM matrix parame-
ters. Therefore, the matrix V DdVT+κDuin Eqs. (2.10)-
(2.12) is given by
K0≡ V0D0
≃ (1 + εd)m0
dV0T+ κ0D0
u
b
mb
?V DdVT+ κDu
−εdmbS3+ εuκDuS) ,(3.8)
where
κ =m0
t/mt
m0
b/mb
κ0
1 + εd
.(3.9)
Since the solutions κ are of the order of 10−2as we
show in the next section, we can neglect the term κDuS
compared with V DdVT(note that in order to neglect
the component (DuS)11 it is essential that the sign of
md/ms is positive, because (V DdVT)11 ≃ md+ V2
and V2
us≃ |md/ms|). On the other hand, for such a small
value of κ, the term mbS3cannot be neglected compared
with the term κDu. However, for a small value of εd,
we can find that the solutions κ are substantially not af-
fected by the term εdmbS3. As a result, we obtain the
approximate expression
usms
K0≃ (1 + εd)m0
b
mb
?V DdVT+ κDu
?.(3.10)
Therefore, Eq. (2.13) and (2.14) at µ = ΛX, i.e.,
[(m0
e)2+ (m0
(m0
µ)2+ (m0
µ)2(m2
τ)2]3
τ)2
e)2(m0
=[Tr(K0K0†])]3
det(K0K0†)
,(3.11)
[(m0
e)2+ (m0
µ)2+ (m0
[Tr(K0K0†)]2
µ)2+ (m0
µ)2(m0
τ)2]2
2[(m0
e)2(m0
τ)2+ (m0
τ)2(m0
e)2]
(3.12)
=
[Tr(K0K0†)]2− Tr(K0K0†)2,
are approximately replaced by the relations at µ = mZ:
(m2
e+ m2
m2
µ+ m2
em2
τ)3
µm2
τ
=[Tr(KK†)]3
det(KK†)
,(3.13)
(m2
e+ m2
µ+ m2
[Tr(KK†)]2
Tr[(KK†)]2− Tr(KK†)2,
µ+ m2
µm2
τ)2
2(m2
em2
τ+ m2
τm2
e)
(3.14)
=
where
K = V DdV†+ κDu,(3.15)
and κ is given by Eq. (3.9). This means that when we
find the solution κ at µ = mZ, the solution at µ = ΛX
also exists, no matter whether the model is a SUSY one
or a non-SUSY one. Then, we can obtain the value κ0
at µ = ΛXfrom the relation (3.9) with the solution κ at
µ = mZ.
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IV. NUMERICAL STUDY AT µ = MZ
As mentioned in the preceding section, if the solution
κ exists at the energy scale µ = mZ, the one at µ = ΛX
also exists. Therefore, we investigate the relations (2.13)
and (2.14) at µ = mZ. Note that Eqs.(2.13) and (2.14)
are realized by GUT scale because Eq.(2.7) is broken
at µ = mZ. In the present section, tentatively, we as-
sume that the Yukawa coupling constant Y10and Y126at
µ = mZ keep their forms symmetrical, so that we can
put the observed values Du, Ddand V at µ = mZ into
the relations (2.13) and (2.14). For the fermion masses
at µ = mZ, we use the following values: [5]
mt= 181 ± 13 GeV,mb= 3.00 ± 0.11 GeV,
mc= 677+56
mu= 2.33+0.42
mτ= 1746.7± 0.3 MeV,
mµ= 102.75138± 0.00033 MeV,
me= 0.48684727± 0.00000014 MeV.
−61MeV,ms= 93.4+11.8
md= 4.69+0.60
−13.0MeV,
−0.45MeV,
−0.66MeV, (4.1)
The input values for the CKM matrix parameters have
been taken as [6]
θ12= 0.219 − 0.226,
θ13= 0.002 − 0.005,
θ23= 0.037 − 0.043,
(4.2)
where
V =


c13c12
c13s12
s13e−iδ
s23c13
c23c13
−c23s12− s23c12s13eiδ
s23s12− c23c12s13eiδ
c23c12− s23s12s13eiδ
−s23c12− c23s12s12s13eiδ

,(4.3)
with cij ≡ cosθij and sij ≡ sinθij.
has been performed allowing all the combinations of the
quark mass signatures. Here it should be noted that,
since muis much smaller than mcand mt, the difference
of the sign of muscarcely makes a change of allowed re-
gions. In this calculation, we have selected θ23and δ as
input parameters and ms, cdand κ as output parameters
because the calculation is sensitive to these parameters.
We give the numerical results in Fig 1.
for ms, θ23and δ, we have adopted the center values of
Eq.(4.1) as input values. Moving θ23at intervals of 0.0005
rad and fixing δ = 60◦, we search the solutions where κA
and κBbecome coincident. Our numerical analysis shows
that the solutions exist in the combinations of Table I.
In a table II, we show the nearest solution of ms, θ23and
The calculation
Here, except
δ to the center values of Eq.(4.1).
In the following we perform data fitting for the case
of top line of Table II. Eqs. (2.10)-(2.12) can constrain
only the absolute value of cd. The argument of the pa-
rameter cdmay be decided by taking neutrino sector into
consideration in the future. For the time being, we set
cd≡ |cd|eiσ= e0.107iso that c0becomes a real number:
c0=1 − cd
cu
= 34.7,(4.4)
c1= −3 + cd
cu
= 101.8 − 10.8i.(4.5)
In this case, the mass matrices in MeV are
M0=3V DdVT+ cd(κDu+ V DdVT)
4
=


−12.4 − 0.7i −23.0− 1.8i
−23.0 − 1.8i −91.5− 3.9i
9.6 − 13.2i
9.6 − 13.2i
194.0+ 10.5i
194.0+ 10.5i 1874.9− 180.0i

,(4.6)
M1=V DdVT− cd(κDu+ V DdVT)
4
=


4.19 + 0.69i
7.68 + 1.43i
−3.72 + 4.09i −65.05− 10.48i 1119.67+ 179.98i
7.68 + 1.43i
24.14+ 3.88i
−3.72+ 4.09i
−65.05− 10.48i

.(4.7)
Here, using the condition
246GeV, we can get VEV’s as
?|vu
0|2+ |vd
0|2+ |vu
1|2+ |vd
1|2=
vd
0=
246[GeV]
?(|c0|2+ 1) + (|c1|2+ 1)|ρ|2
(4.8)
with ρ ≡ vd
and 126 become
1/vd
0. Then, the Yukawa couplings about 10
Y10=M0
vd
0
,Y126=M1
vd
1
.(4.9)
We consider that the model should be calculable pertur-
bativly. We can see that every element of the Yukawa
coupling constants (4.9) is smaller than one if we take a
suitable value of |ρ|.
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V. 10 AND 120
In the SO(10) GUT scenario, we can also discuss the
model with one 10 and one 120 by the same method.
The Yukawa couplings of 10 and 120 are symmetric
and antisymmetric, respectively. If we consider a case
that the Yukawa coupling constants of 10 are real and
120 pure imaginary, we can make them Hermitian, i.e.,
Y†
120= Y120. Therefore, by considering
the real vacuum expectation values v10and v120, we can
obtain the Hermitian mass matrices Mu, Mdand Me:
10= Y10 and Y†
Md= M0+ M2, Me= M0− 3M2,
Mu= c0M0+ c2M2.(5.1)
Then, we can diagonalize those by unitary matrices Uu,
Udand Ueas
U†
uMuUu= Du,U†
dMdUd= Dd,U†
eMeUe= De.
(5.2)
Since the CKM matrix V is given by
V = U†
uUd,(5.3)
the relation (5.1) is re-written as follows:
(U†
uUe)De(U†
uUe)†= cdV DdV†+ cuDu.(5.4)
As stated previously, we have almost known the exper-
imental values of De, Du and V DdV†. Therefore, we
obtain the independent three equations:
TrDe= cd[TrDd+ κTrDu],(5.5)
TrD2
e= c2
d[TrD2
d+ 2κTr(DuV DdV†) + κ2TrD2
u], (5.6)
detDe= c3
ddet(V DdV†+ κDu),(5.7)
where κ = cu/cd. For the parameter κ, we have two
equations:
m2
(me+ mµ+ mτ)2
=TrD2
e+ m2
µ+ m2
τ
d+ 2κTr(DuV DdV†) + κ2TrD2
(TrDd+ κTrDu)2
memµmτ
(me+ mµ+ mτ)3=det(V DdV†+ κDu)
u
,(5.8)
(TrDd+ κTrDu)3.(5.9)
Eqs. (5.8) and (5.9) are more simple than Eqs. (2.13)
and (2.14). cdand κ are real since we have assumed the
Mu, Md and Me to be Hermitian. So the calculation
is easier than the case for 10 and 126. The numerical
results are listed in Table III-IV.
VI. SUMMARY AND DISCUSSION
In conclusion, we have investigated whether an SO(10)
model with two Higgs scalars can reproduce the observed
mass spectra of the up- and down-quark sectors and
charged lepton sector or not.
est is to see whether we can find reasonable values of
the parameters cuand cdwhich satisfy the SO(10) rela-
tion (2.5) or not. For the case with one 10 and one 126
scalars, in a parameter κ = cu/cd, we have obtained two
equations (2.13) and (2.14) which hold at the unification
scale µ = ΛX and which are described in terms of the
observable quantities (the fermion masses and CKM ma-
trix parameters). We have sought for the solution of κ
approximately by using the observed fermion masses and
CKM matrix parameters at µ = mZ instead of the ob-
servable quantities at µ = ΛX. Although we have found
no solution for real κ, we have found four solutions for
complex κ which satisfy Eqs. (2.13) and (2.14) within
the experimental errors. Similarly, we have found four
solutions for a model with one 10 and one 120 scalars.
It should be worth while noting that the solutions in the
latter model are real. The latter model is very attractive
because the origin of the CP violation attributes only to
the 120 scalar. In the both models, we can make the
magnitudes of all the Yukawa coupling constants smaller
than one, so that the models are safely calculable under
the perturbation theory.
By the way, note that the numerical results are very
sensitive to the values of msand θ23. For numerical fit-
tings, it is favor that the strange quark mass msis some-
what smaller than the center value ms= 93.4 MeV which
is quoted in Ref. [5].
Also note that the relative sign of md to ms in each
solution is positive, i.e, md/ms> 0 as seen in Tables I
and III. It is well known that a model with a texture
(Md)11= 0 on the nearly diagonal basis of the up-quark
mass matrix Muleads to the relation |Vus| =
[7], where the relative sign is negative, i.e., md/ms< 0.
On the contrary, we can conclude that in the SO(10)
model with two Higgs scalars, we cannot adopt a model
with the texture (Md)11= 0.
In the present paper, we have demonstrated that
the unified description of the quark and charged lep-
ton masses in the SO(10) model with two Higgs scalars
is possible. However, we have not referred to the neu-
trino masses.Concerning this problem, Brahmachari
and Mohapatra have recently showed that one 10 and
one 126 model is incompatible with large νµ-ντ mixing
angle [8]. Since there are many possibilities for neutrino
mass generation mechanism, we are optimistic about this
problem, too. Investigating for a question whether an
SO(10) model with two Higgs scalars can give a unified
description of quark and lepton masses including neu-
trino masses and mixings or not is our next big task.
What is of great inter-
?−md/ms
5