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How far can the SO(10) two higgs model describe the observed neutrino masses and mixings ?
ELSEVIER
SUPPLEMENTS
Nuclear Physics B (Proc. Suppl.) 111 (2002) 306308
How Far Can the SO(l0) Two Higgs Model Describe the Observed Neutrino Masses and Mixings ? K. Matsudaa, Y. Koideb, T. Fukuyamaa, and H. NishiuraC aDepartment of Physics, Ritsumeikan University, Kusatsu, Shiga, 525-8577 Japan bDepartment of Physics, University of Shizuoka, Shizuoka 422-8526 Japan ‘Department of General Education, Junior College of Osaka Institute of Technology, Asahi-ku, Osaka 535-8585, Japan Can the SO(10) model with one 10 and one 126 Higgs scalars give the observed masses and mixings of quarks and leptons without any other additional Higgs scalars? What is of the greatest interest to us is to know the minimum number of the H&s scalars which can give the observed fermion mass spectra. This problem is systematically investigated, and it is concluded that the present data cannot exclude the SO( 10) model with two Higgs scalars although this model cannot give the best fit values of the data.
1. Introduction The SO(10) grand unified theory (GUT) model seems to us the most attractive model when we take the unification of the quarks and leptons into consideration. However, in order to reproduce the observed quark and lepton masses and mixings, usually, many Higgs scalars are brought into the model. So it is the very crucial problem to know the minimum number of Higgs scalars that can give the observed fermion mass spectra and mixings. A model with one Higgs scalar is obviously ruled out for the description of realistic quark and lepton mass spectra. Two Higgs models were initially discussed by Mohapatra and co-worker [l]. One of the merits of the SO(10) model is that it includes right-handed Majorana neutrinos in the fundamental representation and naturally leads to the seesaw mechanism. Also some papers claim that the two Higgs model ((10 and 126+126}) does not reproduce the large mixing angle of the atmospheric neutrino deficit [2]. Our model has the two Higgs scalars (10 and 126) both of which are symmetric with respect to the family index. Therefore those mass matrices are symmetric whose entries are complex valued. We begin with a short review of our previous work [3]. In the case where two Higgs scalars, 410 and 4126, are incorporated in the SO(10) model,
the mass matrices following forms:
of quarks
Mu = @MO +clMl,
and leptons
have the
Md = MO + MI,
Me = MO-3MI.
(1)
Here MO and Ml are the mass matrices generated by the Higgs scalars $10 and &zs, respectively. Also COand cl are the ratios of VEV’s, co = $/v,”
= (~zb)/(&L
cl = v?/v?
= (&)l(4$;s)r
(2)
and @‘ and tid denote the Higgs scalar components that couple with up and down quarks, respectively. Eliminating MO and Ml from Eq.( l), we obtain
Me =
CdMd
+
cuM, =
Cd(Md
+
KM,),
(3)
where cd=--,
3co+c1 ccl - Cl
t;=-.
4 co - Cl
(4)
By the way, the SO(10) GUT asserts that the Dirac neutrino mass matrix MD is given by the form MD = QMO - 3clM1,
(5)
and the Majorana neutrino mass matrices of the left-handed and right-handed neutrinos, ML and MR, are proportional to the matrix Ml: ML = CLMI,
0920-5632/02/%- see front matter 0 2002 Elsevier Science B.V. All rights reserved. PII SO920-5632(02)01730-9
MR = CRMI,
(6)
K. Matsuda et al. /Nuclear
Physics B (Pnx.
Then the neutrino mass matrix derived from the seesaw mechanism becomes M” = CLMl +
(7)
c&z&e
- ciMi)M~l(c&c
- c&Q?
In the present paper we adopt CL = 0. Since Mu, Md, Me, and M, are complex symmetric matrices, they are diagonal&d by the unitary matrices u,, Ud, and u,, respectively, as
U,TMJJ,, = D, , UzMdUd = Dd , U,TMJJ, = D, , U,TM,,U, = D, ,
(8)
where D,, Dd, D,, and D, are diagonal matrices given by
D, = diag(m,,
m,,
mt),
De
= diag(m,,
m,,
mr),
D,
=
Dd c diag(md, m,,
diag(m,, , m,,, mv7).
v,=lJ~ud*, the relation (3) is rewritten as follows:
=
Cdv,Ddv,j
+
cuDu.
(11)
number of parameters in the SO( 10) model with two Higgs scalars
Since there are only two basic matrices MO and Ml in this model, the number of parameters in Eqs.(3), (7) and (8) is
cd,
Dd, D,, D” h%(,
f~
Vq, A,, Av sum.
3x4 = 12 2+1+2 = 5 4+9+9 = 22
(12)
39
and the number of equations is N(eqs) = 12 x 2 = 24. Therefore the number of free parameters is N(pmt) - N(eqs) = 39 - 24 = 15. On the other hand, the number of the physical parameters that can be determined by experiments is mu, mc,
CKM:
mt,
md,
msy
mbr
‘912, 023, 013, 6,
me, mu,, mTr m,,, mvl,, m,,, MNS: e,,, e,,, e,,, 6, P, P sum.
6 4 6 6 22
3. Numerical
Results
In this calculation, we have selected 02s and 6 in the CKM matrix as input parameters and m,, cd and rc as output parameters because these parameters have much effect on the calculation and are not severely restricted from experiments. We have found that only for the signs of the masses.
=(+,-,+;+,-,-;+,*,*)
(a),
(14)
= (+, -1-i
(b),
(15)
and
(13)
+, -, -_; +, f, k)
there are the solutions of Eq.(ll) that are shown in Fig.1, and the corresponding parameter values (ICdk K) a%
2. The
D,,
where fi and p are Majorana phases in the MakiNakagawa-Sakata (MNS) matrix because there is no rephasing in the neutrino fields YL. To sum up the matter, we discuss the consistency test for 22 physical parameters by using only 15 free parameters.
(CKM)
(10)
(U,tUu)TD,(U$lJu)
301
mb),
(9)
Here the Cabibbc-Kobayashi-Maskawa matrix V, is given by
Suppl.) I1 I (2002) 306308
([cdl,
or
K)
=
for
(a),
(3.15698, -0.019296e
2 64172%),
(3.03577, -0.019398e2~gg570’i),
(16)
(17)
and, for (b), = (3.13307, -0.019314e2.71464’i), or
(3.00558, -0.019420e3~10014Di).
(18) (19)
Here m, = 76.3 MeV for input 02s = 0.0420 rad and 6 = 60” at p = mz (mz is the neutral weak boson mass). For the relation between the values at p = mz and those at p = AX (12, is a unification scale), see Ref.[3]. The remaining purpose is to investigate whether or not these solutions can give reasonable values for observed neutrino masses and mixings. From here we discuss the numerical results of the neutrino msss spectrum and neutrino mass For our example, we use the set in matrix. Eq.(18). Even if other sets are used, our results are scarcely changed. The allowed values of the neutrino mass square differences and lepton flavor mixing angles depict complicated tracks with moving D = argcd (Fig. 2). This figure shows a
K. Matsuda et al. /Nuclear Physics B (Pmt. Suppl.) 111 (2002) 306308
308
general tendency for the lepton flavor mixing angles 13~~and OpT in the MNS matrix to get larger as u approaches 3~12. For illustration we take 0 = 149rr/lOO; then these values become Am2 12 Am&
= 0.15,
sin2(28,,)
= 0.76,
sin2(28,,)
= 0.16.
amX, A%
.“‘b
= 0.85,
sin2(28,,)
m,
.-mc l-41
= 0.75, (20)
There still remain some discrepancies between our results and experiments. However, the purpose of the present paper is to study the general tendency of the fitting and not to pursue a precise data fitting, for the data themselves are not definitive and there are theoretical ambiguities not incorporated in the present data fitting such as the renormalization group effect. In this paper we have discussed to what extent the SO(10) two Higgs scalar model describes the quark-lepton masses and mixing parameters. The consistency test in the quark sector is good. In the lepton sector, the test is not so bad when we adopt the MSW large mixing angle solution of solar neutrino deficit, and this model favors the normal hierarchy of neutrino mass spectrum. We conclude that this model cannot be rejected within the existing data. It should be remarked that all the parameters can be determined from the existing data in principle. acknowledgements The work of K.M. was supported Research Grant No. 10421.
*mu ,
. 6,
Figure 1. The solutions of the quark and lepton masses and the CKM matrix. The and right dots depict the experimental and one combination of the solutions, tively.
E charged left bars bounds respec-
10-1
by the JSPS
lo-'
IO-’
REFERENCES
~ IO-‘0
K.S. Babu and R.N. Mohapatra, Phys. Rev. Lett. 70 2845 (1993); L. Lavoura, Phys. Rev. D48 5440 (1993); D-G Lee and R.N. Mohapatra, Phys. Rev. D51 1353 (1995). B. Brahmachari and R.N. Mohapatra, Phys. Rev. D58 015001 (1998); L. Lavoura, Phys. Rev. D48 5440 (1993). K. Matsuda, Y. Koide, and T. Fukuyama, Phys. Rev. D64 053015 (2001). Particle Data Group, D. E. Groom et al., The European Physical Journal C15, 1 (2000).
L
1O-‘:0_.
d
10 -3
0.01
0.1
1
sill2 28
The relation between our results Figure 2. and the twoflavor oscillation analysis [4] when The circles, triangles, and 0 is moved. stars indicate the values of (Am&, sin2 20,,), and (Amf2, sin2 28,,) at ev(Am& 1sin2 28,,), ery ~12 of u. Here we have set Am& = 1.5 x 10-3[eV2] in every case.
SUPPLEMENTS
Nuclear Physics B (Proc. Suppl.) 111 (2002) 306308
How Far Can the SO(l0) Two Higgs Model Describe the Observed Neutrino Masses and Mixings ? K. Matsudaa, Y. Koideb, T. Fukuyamaa, and H. NishiuraC aDepartment of Physics, Ritsumeikan University, Kusatsu, Shiga, 525-8577 Japan bDepartment of Physics, University of Shizuoka, Shizuoka 422-8526 Japan ‘Department of General Education, Junior College of Osaka Institute of Technology, Asahi-ku, Osaka 535-8585, Japan Can the SO(10) model with one 10 and one 126 Higgs scalars give the observed masses and mixings of quarks and leptons without any other additional Higgs scalars? What is of the greatest interest to us is to know the minimum number of the H&s scalars which can give the observed fermion mass spectra. This problem is systematically investigated, and it is concluded that the present data cannot exclude the SO( 10) model with two Higgs scalars although this model cannot give the best fit values of the data.
1. Introduction The SO(10) grand unified theory (GUT) model seems to us the most attractive model when we take the unification of the quarks and leptons into consideration. However, in order to reproduce the observed quark and lepton masses and mixings, usually, many Higgs scalars are brought into the model. So it is the very crucial problem to know the minimum number of Higgs scalars that can give the observed fermion mass spectra and mixings. A model with one Higgs scalar is obviously ruled out for the description of realistic quark and lepton mass spectra. Two Higgs models were initially discussed by Mohapatra and co-worker [l]. One of the merits of the SO(10) model is that it includes right-handed Majorana neutrinos in the fundamental representation and naturally leads to the seesaw mechanism. Also some papers claim that the two Higgs model ((10 and 126+126}) does not reproduce the large mixing angle of the atmospheric neutrino deficit [2]. Our model has the two Higgs scalars (10 and 126) both of which are symmetric with respect to the family index. Therefore those mass matrices are symmetric whose entries are complex valued. We begin with a short review of our previous work [3]. In the case where two Higgs scalars, 410 and 4126, are incorporated in the SO(10) model,
the mass matrices following forms:
of quarks
Mu = @MO +clMl,
and leptons
have the
Md = MO + MI,
Me = MO-3MI.
(1)
Here MO and Ml are the mass matrices generated by the Higgs scalars $10 and &zs, respectively. Also COand cl are the ratios of VEV’s, co = $/v,”
= (~zb)/(&L
cl = v?/v?
= (&)l(4$;s)r
(2)
and @‘ and tid denote the Higgs scalar components that couple with up and down quarks, respectively. Eliminating MO and Ml from Eq.( l), we obtain
Me =
CdMd
+
cuM, =
Cd(Md
+
KM,),
(3)
where cd=--,
3co+c1 ccl - Cl
t;=-.
4 co - Cl
(4)
By the way, the SO(10) GUT asserts that the Dirac neutrino mass matrix MD is given by the form MD = QMO - 3clM1,
(5)
and the Majorana neutrino mass matrices of the left-handed and right-handed neutrinos, ML and MR, are proportional to the matrix Ml: ML = CLMI,
0920-5632/02/%- see front matter 0 2002 Elsevier Science B.V. All rights reserved. PII SO920-5632(02)01730-9
MR = CRMI,
(6)
K. Matsuda et al. /Nuclear
Physics B (Pnx.
Then the neutrino mass matrix derived from the seesaw mechanism becomes M” = CLMl +
(7)
c&z&e
- ciMi)M~l(c&c
- c&Q?
In the present paper we adopt CL = 0. Since Mu, Md, Me, and M, are complex symmetric matrices, they are diagonal&d by the unitary matrices u,, Ud, and u,, respectively, as
U,TMJJ,, = D, , UzMdUd = Dd , U,TMJJ, = D, , U,TM,,U, = D, ,
(8)
where D,, Dd, D,, and D, are diagonal matrices given by
D, = diag(m,,
m,,
mt),
De
= diag(m,,
m,,
mr),
D,
=
Dd c diag(md, m,,
diag(m,, , m,,, mv7).
v,=lJ~ud*, the relation (3) is rewritten as follows:
=
Cdv,Ddv,j
+
cuDu.
(11)
number of parameters in the SO( 10) model with two Higgs scalars
Since there are only two basic matrices MO and Ml in this model, the number of parameters in Eqs.(3), (7) and (8) is
cd,
Dd, D,, D” h%(,
f~
Vq, A,, Av sum.
3x4 = 12 2+1+2 = 5 4+9+9 = 22
(12)
39
and the number of equations is N(eqs) = 12 x 2 = 24. Therefore the number of free parameters is N(pmt) - N(eqs) = 39 - 24 = 15. On the other hand, the number of the physical parameters that can be determined by experiments is mu, mc,
CKM:
mt,
md,
msy
mbr
‘912, 023, 013, 6,
me, mu,, mTr m,,, mvl,, m,,, MNS: e,,, e,,, e,,, 6, P, P sum.
6 4 6 6 22
3. Numerical
Results
In this calculation, we have selected 02s and 6 in the CKM matrix as input parameters and m,, cd and rc as output parameters because these parameters have much effect on the calculation and are not severely restricted from experiments. We have found that only for the signs of the masses.
=(+,-,+;+,-,-;+,*,*)
(a),
(14)
= (+, -1-i
(b),
(15)
and
(13)
+, -, -_; +, f, k)
there are the solutions of Eq.(ll) that are shown in Fig.1, and the corresponding parameter values (ICdk K) a%
2. The
D,,
where fi and p are Majorana phases in the MakiNakagawa-Sakata (MNS) matrix because there is no rephasing in the neutrino fields YL. To sum up the matter, we discuss the consistency test for 22 physical parameters by using only 15 free parameters.
(CKM)
(10)
(U,tUu)TD,(U$lJu)
301
mb),
(9)
Here the Cabibbc-Kobayashi-Maskawa matrix V, is given by
Suppl.) I1 I (2002) 306308
([cdl,
or
K)
=
for
(a),
(3.15698, -0.019296e
2 64172%),
(3.03577, -0.019398e2~gg570’i),
(16)
(17)
and, for (b), = (3.13307, -0.019314e2.71464’i), or
(3.00558, -0.019420e3~10014Di).
(18) (19)
Here m, = 76.3 MeV for input 02s = 0.0420 rad and 6 = 60” at p = mz (mz is the neutral weak boson mass). For the relation between the values at p = mz and those at p = AX (12, is a unification scale), see Ref.[3]. The remaining purpose is to investigate whether or not these solutions can give reasonable values for observed neutrino masses and mixings. From here we discuss the numerical results of the neutrino msss spectrum and neutrino mass For our example, we use the set in matrix. Eq.(18). Even if other sets are used, our results are scarcely changed. The allowed values of the neutrino mass square differences and lepton flavor mixing angles depict complicated tracks with moving D = argcd (Fig. 2). This figure shows a
K. Matsuda et al. /Nuclear Physics B (Pmt. Suppl.) 111 (2002) 306308
308
general tendency for the lepton flavor mixing angles 13~~and OpT in the MNS matrix to get larger as u approaches 3~12. For illustration we take 0 = 149rr/lOO; then these values become Am2 12 Am&
= 0.15,
sin2(28,,)
= 0.76,
sin2(28,,)
= 0.16.
amX, A%
.“‘b
= 0.85,
sin2(28,,)
m,
.-mc l-41
= 0.75, (20)
There still remain some discrepancies between our results and experiments. However, the purpose of the present paper is to study the general tendency of the fitting and not to pursue a precise data fitting, for the data themselves are not definitive and there are theoretical ambiguities not incorporated in the present data fitting such as the renormalization group effect. In this paper we have discussed to what extent the SO(10) two Higgs scalar model describes the quark-lepton masses and mixing parameters. The consistency test in the quark sector is good. In the lepton sector, the test is not so bad when we adopt the MSW large mixing angle solution of solar neutrino deficit, and this model favors the normal hierarchy of neutrino mass spectrum. We conclude that this model cannot be rejected within the existing data. It should be remarked that all the parameters can be determined from the existing data in principle. acknowledgements The work of K.M. was supported Research Grant No. 10421.
*mu ,
. 6,
Figure 1. The solutions of the quark and lepton masses and the CKM matrix. The and right dots depict the experimental and one combination of the solutions, tively.
E charged left bars bounds respec-
10-1
by the JSPS
lo-'
IO-’
REFERENCES
~ IO-‘0
K.S. Babu and R.N. Mohapatra, Phys. Rev. Lett. 70 2845 (1993); L. Lavoura, Phys. Rev. D48 5440 (1993); D-G Lee and R.N. Mohapatra, Phys. Rev. D51 1353 (1995). B. Brahmachari and R.N. Mohapatra, Phys. Rev. D58 015001 (1998); L. Lavoura, Phys. Rev. D48 5440 (1993). K. Matsuda, Y. Koide, and T. Fukuyama, Phys. Rev. D64 053015 (2001). Particle Data Group, D. E. Groom et al., The European Physical Journal C15, 1 (2000).
L
1O-‘:0_.
d
10 -3
0.01
0.1
1
sill2 28
The relation between our results Figure 2. and the twoflavor oscillation analysis [4] when The circles, triangles, and 0 is moved. stars indicate the values of (Am&, sin2 20,,), and (Amf2, sin2 28,,) at ev(Am& 1sin2 28,,), ery ~12 of u. Here we have set Am& = 1.5 x 10-3[eV2] in every case.