A simple extension of the standard model is proposed in which all the three
generations of neutrinos are Dirac particles and are naturally light. We then
assume that the neutrino mass matrix is diagonal and degenerate, with a few eV
mass to solve the dark matter problem. The self energy radiative corrections,
however, remove this degeneracy and allow mixing of these neutrinos. The
electroweak radiative corrections then predict a lower bound on the $\nu_\mu -
\nu_e$ mass difference which solves the solar neutrino problem through MSW
mechanism and also predict a lower bound on the $\nu_\tau - \nu_\mu$ mass
difference which is just enough to explain the atmospheric neutrino problem as
reported by super Kamiokande.Comment: 11 pages latex fil

A. Athanassopoulos

A. Joshipura

A. N. Taylor

B. Armbruster

D. G. Lee

D. O. Caldwell

E. Ma

F. Cuypers

H. V. Klapdor-Kleingrothaus

J. Engel

L. Baudis

L. Wolfenstein

M. Davis

M. Günther

P. Anselmann

R. Davis

S. P. Mikheyev

S. T. Petcov

U. Sarkar

Utpal Sarkar

Y. Fukuda

English

arXiv

Crossref

Physical Review D

Degenerate Dirac neutrinos

A simple extension of the standard model is proposed in which all the three generations of neutrinos are Dirac particles and are naturally light. We then assume that the neutrino mass matrix is diagonal and degenerate, with a few eV mass to solve the dark matter problem. The self-energy radiative corrections, however, remove this degeneracy and allow mixing of these neutrinos. The electroweak radiative corrections then predict a lower bound on the νμ-νe mass difference which solves the solar neutrino problem through the MSW mechanism and also predict a lower bound on the ντ-νμ mass difference which is just enough to explain the atmospheric neutrino problem as reported by Super-Kamiokande

Sarkar, Utpal

Publications of the IAS Fellows

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98
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99
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DESY 98-104
August 1998
Degenerate Dirac Neutrinos
Utpal Sarkar(a,b)1
(a)Theory Group, DESY, Notkestraße 85, 22607 Hamburg, Germany
(b)Theory Group, Physical Research Laboratory, Ahmedabad, 380 009, India 2
Abstract
A simple extension of the standard model is proposed in which all
the three generations of neutrinos are Dirac particles and are naturally
light. We then assume that the neutrino mass matrix is diagonal and
degenerate, with a few eV mass to solve the dark matter problem. The
self energy radiative corrections, however, remove this degeneracy and
allow mixing of these neutrinos. The electroweak radiative corrections
then predict a lower bound on the νµ−νe mass difference which solves
the solar neutrino problem through MSW mechanism and also predict
a lower bound on the ντ − νµ mass difference which is just enough
to explain the atmospheric neutrino problem as reported by super
Kamiokande.
1E-mail: utpal@prl.ernet.in
2permanent address
The neutrinoless double beta decay [1] puts severe constraints on the
Majorana mass of the νe. With this constraint it is not possible to explain
simultaneously the dark matter problem [2], solar neutrino problem [3], at-
mospheric neutrino problem [4] and the laboratory bounds on the mixing
angles [5]. Since the preliminary results from KARMEN [6] contradicts the
LSND result [7], we shall not include that in our analysis. If the neutrinos
are Dirac particle there will not be any lepton number violation and hence
there will not be any constraint from the neutrinoless double beta decay. In
that case one can postulate an almost degenerate neutrino scenario [8] to
explain the other problems.
In this article, we consider exactly degenerate Dirac particles with a few
eV mass to explain the dark matter problem. However, this will not allow
any flavour mixing. So we introduce explicit lepton flavour violation, which
will break the mass degeneracy radiatively, which in turn will allow flavour
mixing. The electroweak self-energy corrections will then predict a lower
bound on the mass squared differences between νµ and νe which can solve
the solar neutrino problem through matter enhanced neutrino oscillation [9]
and simultaneously predict a lower bound on the mass squared difference
between ντ and νµ which is just enough to solve the atmospheric neutrino
problem. There is also a similar contribution to mass splitting from flavour
violating radiative corrections, which also gives the maximal neutrino flavour
mixing.
Consider a two generation Majorana neutrino scenario. The neutrino
mass matrix is given by,
Mν =
(
mee meµ
mµe mµµ
)
. (1)
The neutrinoless double beta decay [1] will imply mee < 0.46 eV. If we
consider an almost degenerate neutrino scenario [8], to solve the dark matter
problem [2] we require mµµ ≃ mee. Then the small mixing [5] of νe with νµ
will imply that sin2θeµ ∼
meµ
mµµ
< 0.6 and the masses of the νe and νµ are
2
less than 1 eV. This will not solve the dark matter problem. This problem
is solved if one assumes that the neutrinos are Dirac particles. However, the
main problem of making the neutrinos a Dirac particle is that, in a simple
extension of the standard model the Dirac mass of the neutrinos are related
to the charged lepton masses and hence cannot be small (of the order of a
few eV) naturally.
We now propose a scenario where the neutrinos are Dirac particles and are
naturally light. The left handed neutrinos combine with the right handed
neutrinos through their interactions with a different higgs doublet, which
does not couple to the quarks and charged leptons because of the presence
of an additional U(1) symmetry. This new higgs doublet acquires a small
vacuum expectation value (vev), when the extra U(1) symmetry and the
electroweak symmetry are broken [10], and hence its coupling gives a small
Dirac mass to the neutrinos naturally. There is no lepton number violation
in this scenario and hence there are no Majorana mass of the neutrinos.
We extend the standard model gauge group to include a new U(1) sym-
metry,
Gext ≡ SU(3)c × SU(2)L × U(1)Y × U(1)X .
We also extend the model to include the three right handed neutrinos νiR
(i = 1, 2, 3); four additional singlet fermions Yi (i = 1, 2, 3) and Z, which
are required for purpose of anomaly cancellation; one new higgs doublet η,
a scalar singlet χ and a charged scalar singlet ζ . Transformation properties
of the new particles are presented in table 1. The scalar χ acquires a vev
at a very high scale M, breaks the U(1)X symmetry and give masses to the
extra singlet fields. The mass of the doublet η is also of the order of M, but
it does not acquire any vev to start with. However, after the electroweak
symmetry breaking it acquires a small vev and gives small Dirac masses to
the neutrinos naturally.
The lagrangian contains the quadratic and the quartic terms and the
3
Table 1: transformations of the new particles
Fermions
νiR (1,1,0,2)
Yi (1,1,0,-1)
Z (1,1,0,-3)
Scalars
η (1,2,-1/2,-2)
χ (1,1,0,2)
ζ (1,1,1,-2)
trilinear mixing terms which are given by,
Lscalar = M
2
η η
†η +M2χχ
†χ +m2φφ
†φ
+
1
2
λ1(φ
†φ)2 +
1
2
λ2(η
†η)2 +
1
2
λ3(χ
†χ)2
+
1
2
λ4(φ
†φ)(η†η) +
1
2
λ5(χ
†χ)(η†η) +
1
2
λ6(φ
†φ)(χ†χ)
+ µφ†ηχ. (2)
The scalar χ acquires a vev at some large scale M , which is the only other
mass scale in the model other than the electroweak symmetry breaking scale.
We consider
Mχ ∼Mη ∼< χ >∼M ∼ 10
11GeV.
Then to prevent the usual higgs doublet from acquiring a large mass we
require
µ ∼ mφ ∼ m,
where m is the electroweak syummetry breaking scale.
We may now minimize the potential to determine the vevs of the different
scalar fields. They are given by,
< χ >2∼ −
M2χ
λ3
; < φ >2∼
−λ1m
2
φ + λ4M
2
η
λ1λ2 − λ24
4
and < η >∼
µ < φ >< χ >
M2η
∼ 100eV. (3)
The scalar field η acquires a very small vev naturally in this scenario. As
a result, if it gives a Dirac mass to the neutrinos, then we have a natural
explanation of the smallness of the Dirac mass of the neutrinos. To obtain
the mass of the neutrinos we now write down the Yukawa couplings of the
leptons and the singlet fermions,
L = fνiαliLναRη + gabY ca Ybχ+ g
′
aYaZχ+ feiαliLeαRφ+ hαβ(ναR)
ceβRζ. (4)
The extra singlets Ya and Z get very large masses from the vev of the scalar
χ and they donot couple with the light neutrinos. Since they are decoupled
from the low energy sector, we shall not discuss them from now on. Although
we have to introduce these singlet fields for purpose of anomaly cancellation,
if one can implement this mechanism in a larger theory like grand unified
theory or string theory, one may not require to introduce these fields. The
scalar ζ give radiative mass splitting to neutrinos of different generations,
which in turn gives neutrino flavour mixing.
The first term in this equation gives small Dirac mass to the neutrinos
Mν = fνiα < η > . (5)
We now assume that the Dirac mass matrix is diagonal and degenerate and
to explain the hot component of the dark matter the diagonal elements are
given by fνiαδiα = m. The neutrino Dirac mass matrix then becomes
Mν =

m 0 00 m 0
0 0 m

 .
We further assume that in this basis the Yukawa interactions of the field
ζ is also diagonal, but the charge lepton mass matrix is not diagonal. In
general, it may be possible to diagonalise the charge lepton mass matrix and
5
the neutrino mass matrix simultaneously, since the neutrino mass matrix is
degenerate. However, the coupling of the charged scalar ζ will give radiative
correction, which will break the mass degeneracy and will not allow to make
the charged current interaction diagonal. As a result all the observed mixing
may come from the charge lepton mass matrix and we can only determine
them from experiments. The mixing angle will depend on the amount of
radiative mass splitting.
In general, the charged lepton mass matrix can be diagonalised by a bi-
unitary transformation
V †ikMeiαUαβ = M
diag
ekβ δkβ.
Then the matrix Vik, which diagonalises the matrix MeiαM
†
ejα, will enter
in the charged current interactions. In the basis [eiL, eαR], in which the
charged lepton mass matrix is diagonal, the charge current interaction of the
neutrinos and the charged leptons will be given by,
Lcc = νiLγµV
†
kiekLW
µ. (6)
The charged lepton mixing matrix Vij will introduce neutrino flavour mixing
after the radiative mass splitting.
We now define the basis for the neutrinos [νeiL, ν
e
αR], which has diagonal
charged current interaction and are given by,
νeiL = VikνkL and ν
e
αR = U
ν
αβνβR.
In this basis, the mass matrix is not diagonal. But when the mass matrix is
diagonal and degenerate, we can always make transformations to the right
handed fields and make them diagonal. In the basis [νeiL, ν
e
αR], the neutrino
mass matrix is given by,
Meνij = VijMν .
6
But Vij commutes with Mν and hence we can make a transformation ν
e
αR =
V †βανβR; Vβα = Vij , and diagonalise the mass matrix. However, in the
presence of the radiative corrections due to ζ , this is not possible.
We assume that the couplings of ζ to the right handed leptons, hαβ , to
be diagonal. Although the phenomenology of such dilepton have not been
studied, one can extend the analysis of ref. [11] to constrain the parameters.
If, in addition, we assume that h11 < 10
−5, then there is only one constraint
from the (g − 2)µ, which is h22 > 0.3 for mζ ∼ 100 GeV. For h33 there is no
bound in this scenario and we can consider this to be of the order of 1. With
this choice we get a self energy radiative correction (with the internal loop
containing charged leptons and ζ) to the neutrino mass matrix in the basis
[νiL, ναR], in which the charged current interaction is not diagonal but the
ζ couplings are diagonal, given by
Mνiα =

m+m
ζ
1 0 0
0 m+mζ2 0
0 0 m+mζ3

 (7)
where, mζi = m
h2ii
4pi
e2i
m2
ζ
; ei = e, µ, τ . In this case, it will not be possible
to diagonalise the neutrino mass matrix in the basis in which the charged
current interaction is diagonal. Including the standard model self energy
radiative corrections [12], we can now write down the neutrino mass matrix
in the basis [νeiL, ν
e
αR] as follows,
Meνiα = VijMνiα + diag[m
ew
1 m
ew
2 m
ew
3 ] VijMνjα (8)
where, mewi = αw(m
2
ei
/m2w), ei ≡ e, µ, τ . This mass matrix can now be
diagonalised to get the flavour mixing matrix and the mass ssquared differ-
ence between the different flavours of neutrinos. For any arbitrary choice
of the mixing matrix it is difficult to solve this analytically. So, for pur-
pose of illustration we demonstrate with a two generation example, and then
present some realistic numbers for a three generation scenario which we check
numerically.
7
We consider the µ and τ family in the two generation example. For
Vij =
(
cos θ − sin θ
sin θ cos θ
)
, the mass matrix is given by,
Meνiα =
(
(m+mζ2)(1 +m
ew
2 ) cos θ −(m+m
ζ
3)(1 +m
ew
2 ) sin θ
(m+mζ2)(1 +m
ew
3 ) sin θ (m+m
ζ
3)(1 +m
ew
3 ) cos θ
)
(9)
which can be diagonalised with a bi-unitary transformation. The unitary
matrix, which diagonalises Meνiα(M
e
νjα)
†, gives the neutrino flavour mixing
and is given by,
V eij =
(
cos φ sin φ
sin φ cos φ
)
where,
φ ∼
1
2
tan−1
[
2 cos θ sin θ
(mζ3 −m
ζ
2)
(mζ3 −m
ζ
2) + (m
ew
3 −m
ew
2 )
]
and the mass squared difference is given by,
(m23 −m
2
2)
2 = [2m cos θ sin θ(mζ3 −m
ζ
2)]
2 + [m(mζ3 −m
ζ
2) +m(m
ew
3 −m
ew
2 )]
2.
(10)
It is clear from the expressions that both the electroweak radiative correction
as well as the radiative corrections due to ζ will contribute to the mass
squared difference. If ζ does not contribute to the mass difference, the mixing
angle vanishes as pointed out earlier. For the mixing angle to be maximal
we require the contribution of ζ to be of the same order or more than the
contribution from the electroweak radiative corrections.
The mass difference generated by the standard model self energy radiative
corrections (considering the Dirac mass of the neutrinos to be around 7 eV)
are given by,
∆msol = (m
ew
2 )
2 − (mew1 )
2 = αwm
2m
2
µ −m
2
e
m2w
= 0.6× 10−5
∆matm = (m
ew
3 )
2 − (mew2 )
2 = αwm
2m
2
τ −m
2
µ
m2w
= 1.4× 10−3 (11)
8
The νe − νµ mass difference ∆m
ew
sol is just enough to solve the solar neutrino
problem, while the νµ − ντ mass difference ∆m
ew
atm falls within the solution
suggested by the recent super kamiokande result. If the contribution due
to ζ are of the same order of magnitude (which is the case for mζ ∼ 100
GeV and h11 = 0, h22 ∼ 0.1 and h33 ∼ 1), then we get the maximal mixing
angle for the νµ and ντ oscillations and the relevant mass squared difference
as required by the super kamiokande experiment. For the solar neutrino
problem, the mass squared difference between νe and νµ is just right for
the MSW solution [9], but we only get the small mixing solution. In this
scenario it will be difficult to explain the vacuum oscillation solution of the
solar neutrino problem, since although one can adjust the parameters of the
couplings of ζ to get a small radiative corrections, one cannot change the
electroweak radiative correction. As a result, the mass squared difference
between νe and any other neutrinos will be larger than required, unless there
are new sterile neutrinos, to which the νe oscillates. Since the neutrino masses
arise from a different higgs doublet, the neutrino mass mixing is not related
to the CKM quark mixing matrix.
In summary, we pointed out that if neutrinos are Dirac particles, then we
may start with a degenerate diagonal mass matrix with the diagonal elements
to be around a few eV, so that neutrinos could be the hot dark matter of the
universe. We also presented a model in which neutrinos could be light Dirac
particles naturally. The mass degeneracy is broken by self energy radiative
corrections, which can then allow enough mixing of these neutrinos to solve
the atmospheric neutrino and solar neutrino problems. The standard model
radiative corrections give a lower bound on the mass squared difference which
is just enough to solve both the atmospheric neutrino and solar neutrino
problems.
9
ACKNOWLEDGEMENT
I would like to thank Profs. David Cline, Prof. D.P. Roy and M. Raidal
for discussions. I would also like to thank Prof W. Buchmuller and the
Theory Division, DESY, Hamburg for hospitality. Financial support from
the Alexander von Humboldt Foundation is acknowledged.
10
References
[1] H.V. Klapdor-Kleingrothaus, in Proc Lepton and Baryon number viola-
toin, Trento, April 1998; M. Gu¨nther et al, Phys. Rev. D 55 (1997) 54;
L. Baudis et al, Phys. Lett. B 407 (1997) 219.
[2] J. Engel, S. Pittel and P. Vogel, Int. J. Mod. Phys. E 1 (1992) 1; A.N.
Taylor, M. Rowan-Robinson, Nature 359 (1992) 396; M. Davis, F.J.
Summers and D. Schlegel, Nature 359 (1992) 393.
[3] R. Davis, Prog. Part. Nucl. Phys. 32 (1994) 13; Y. Fukuda et al, Phys.
Rev. Lett. 77 (1996) 1683; P. Anselmann et al, Phys. Lett. B 357 (1995)
237; B 361 (1996) 235.
[4] T. Kajita, Talk presented at Neutrino 1998 Conference; Y. Fukuda et
al, hep-ex/9805006, hep-ex/9805021 and hep-ph/9807003.
[5] Particle Data Group, Eur. Phys. J. C 3 (1998) 1.
[6] R. Armbruster et al, Phys. Rev. C 57 (1998) 3414.
[7] A. Athanassopoulos et al (LSND Collaboration), Phys. Rev. Lett. 77
(1996) 3082, nucl-ex/9706006 and nucl-ex/9709006.
[8] S.T. Petcov and A. Yu. Smirnov, Phys. Lett. B322 (1994) 109; A. Jo-
shipura, Z. Phys. C 64 (1994) 31; D.O. Caldwell and R.N. Mohapatra,
Phys. Rev. D 50 (1994) 3477; D.G. Lee and R.N. Mohapatra, Phys.
Lett. B 229 (1994) 463; A. Joshipura, Phys. Rev. D 51 (1995) 1321.
[9] L. Wolfenstein, Phys. Rev. D 17 (1978) 2369; S.P. Mikheyev and A. Yu.
Smirnov, Yad. Fiz. 42 (1995) 1441 [Sov. J. Nucl. Phys. 42 (1985) 913].
[10] E. Ma and U. Sarkar, Phys. Rev. Lett. 80 (1998) 5716.
[11] F. Cuypers and S. Davidson, Eur. Phys. J. C 2 (1998) 503.
[12] L. Wolfenstein, Nucl.Phys. B 186 (1981) 147; U. Sarkar, Phys. Rev. D
35 (1987) 1528.
11


Indian Academy of Sciences

ar
X
iv
:h
ep
-p
h/
98
08
27
7v
1 
 1
0 
A
ug
 1
99
8
DESY 98-104
August 1998
Degenerate Dirac Neutrinos
Utpal Sarkar(a,b)1
(a)Theory Group, DESY, Notkestraße 85, 22607 Hamburg, Germany
(b)Theory Group, Physical Research Laboratory, Ahmedabad, 380 009, India 2
Abstract
A simple extension of the standard model is proposed in which all
the three generations of neutrinos are Dirac particles and are naturally
light. We then assume that the neutrino mass matrix is diagonal and
degenerate, with a few eV mass to solve the dark matter problem. The
self energy radiative corrections, however, remove this degeneracy and
allow mixing of these neutrinos. The electroweak radiative corrections
then predict a lower bound on the νµ−νe mass difference which solves
the solar neutrino problem through MSW mechanism and also predict
a lower bound on the ντ − νµ mass difference which is just enough
to explain the atmospheric neutrino problem as reported by super
Kamiokande.
1E-mail: utpal@prl.ernet.in
2permanent address
The neutrinoless double beta decay [1] puts severe constraints on the
Majorana mass of the νe. With this constraint it is not possible to explain
simultaneously the dark matter problem [2], solar neutrino problem [3], at-
mospheric neutrino problem [4] and the laboratory bounds on the mixing
angles [5]. Since the preliminary results from KARMEN [6] contradicts the
LSND result [7], we shall not include that in our analysis. If the neutrinos
are Dirac particle there will not be any lepton number violation and hence
there will not be any constraint from the neutrinoless double beta decay. In
that case one can postulate an almost degenerate neutrino scenario [8] to
explain the other problems.
In this article, we consider exactly degenerate Dirac particles with a few
eV mass to explain the dark matter problem. However, this will not allow
any flavour mixing. So we introduce explicit lepton flavour violation, which
will break the mass degeneracy radiatively, which in turn will allow flavour
mixing. The electroweak self-energy corrections will then predict a lower
bound on the mass squared differences between νµ and νe which can solve
the solar neutrino problem through matter enhanced neutrino oscillation [9]
and simultaneously predict a lower bound on the mass squared difference
between ντ and νµ which is just enough to solve the atmospheric neutrino
problem. There is also a similar contribution to mass splitting from flavour
violating radiative corrections, which also gives the maximal neutrino flavour
mixing.
Consider a two generation Majorana neutrino scenario. The neutrino
mass matrix is given by,
Mν =
(
mee meµ
mµe mµµ
)
. (1)
The neutrinoless double beta decay [1] will imply mee < 0.46 eV. If we
consider an almost degenerate neutrino scenario [8], to solve the dark matter
problem [2] we require mµµ ≃ mee. Then the small mixing [5] of νe with νµ
will imply that sin2θeµ ∼
meµ
mµµ
< 0.6 and the masses of the νe and νµ are
2
less than 1 eV. This will not solve the dark matter problem. This problem
is solved if one assumes that the neutrinos are Dirac particles. However, the
main problem of making the neutrinos a Dirac particle is that, in a simple
extension of the standard model the Dirac mass of the neutrinos are related
to the charged lepton masses and hence cannot be small (of the order of a
few eV) naturally.
We now propose a scenario where the neutrinos are Dirac particles and are
naturally light. The left handed neutrinos combine with the right handed
neutrinos through their interactions with a different higgs doublet, which
does not couple to the quarks and charged leptons because of the presence
of an additional U(1) symmetry. This new higgs doublet acquires a small
vacuum expectation value (vev), when the extra U(1) symmetry and the
electroweak symmetry are broken [10], and hence its coupling gives a small
Dirac mass to the neutrinos naturally. There is no lepton number violation
in this scenario and hence there are no Majorana mass of the neutrinos.
We extend the standard model gauge group to include a new U(1) sym-
metry,
Gext ≡ SU(3)c × SU(2)L × U(1)Y × U(1)X .
We also extend the model to include the three right handed neutrinos νiR
(i = 1, 2, 3); four additional singlet fermions Yi (i = 1, 2, 3) and Z, which
are required for purpose of anomaly cancellation; one new higgs doublet η,
a scalar singlet χ and a charged scalar singlet ζ . Transformation properties
of the new particles are presented in table 1. The scalar χ acquires a vev
at a very high scale M, breaks the U(1)X symmetry and give masses to the
extra singlet fields. The mass of the doublet η is also of the order of M, but
it does not acquire any vev to start with. However, after the electroweak
symmetry breaking it acquires a small vev and gives small Dirac masses to
the neutrinos naturally.
The lagrangian contains the quadratic and the quartic terms and the
3
Table 1: transformations of the new particles
Fermions
νiR (1,1,0,2)
Yi (1,1,0,-1)
Z (1,1,0,-3)
Scalars
η (1,2,-1/2,-2)
χ (1,1,0,2)
ζ (1,1,1,-2)
trilinear mixing terms which are given by,
Lscalar = M
2
η η
†η + M2χχ
†χ + m2φφ
†φ
+
1
2
λ1(φ
†φ)2 +
1
2
λ2(η
†η)2 +
1
2
λ3(χ
†χ)2
+
1
2
λ4(φ
†φ)(η†η) +
1
2
λ5(χ
†χ)(η†η) +
1
2
λ6(φ
†φ)(χ†χ)
+ µφ†ηχ. (2)
The scalar χ acquires a vev at some large scale M , which is the only other
mass scale in the model other than the electroweak symmetry breaking scale.
We consider
Mχ ∼ Mη ∼< χ >∼ M ∼ 10
11GeV.
Then to prevent the usual higgs doublet from acquiring a large mass we
require
µ ∼ mφ ∼ m,
where m is the electroweak syummetry breaking scale.
We may now minimize the potential to determine the vevs of the different
scalar fields. They are given by,
< χ >2∼ −
M2χ
λ3
; < φ >2∼
−λ1m
2
φ + λ4M
2
η
λ1λ2 − λ24
4
and < η >∼
µ < φ >< χ >
M2η
∼ 100eV. (3)
The scalar field η acquires a very small vev naturally in this scenario. As
a result, if it gives a Dirac mass to the neutrinos, then we have a natural
explanation of the smallness of the Dirac mass of the neutrinos. To obtain
the mass of the neutrinos we now write down the Yukawa couplings of the
leptons and the singlet fermions,
L = fνiαliLναRη + gabY ca Ybχ + g
′
aYaZχ + feiαliLeαRφ + hαβ(ναR)
ceβRζ. (4)
The extra singlets Ya and Z get very large masses from the vev of the scalar
χ and they donot couple with the light neutrinos. Since they are decoupled
from the low energy sector, we shall not discuss them from now on. Although
we have to introduce these singlet fields for purpose of anomaly cancellation,
if one can implement this mechanism in a larger theory like grand unified
theory or string theory, one may not require to introduce these fields. The
scalar ζ give radiative mass splitting to neutrinos of different generations,
which in turn gives neutrino flavour mixing.
The first term in this equation gives small Dirac mass to the neutrinos
Mν = fνiα < η > . (5)
We now assume that the Dirac mass matrix is diagonal and degenerate and
to explain the hot component of the dark matter the diagonal elements are
given by fνiαδiα = m. The neutrino Dirac mass matrix then becomes
Mν =



m 0 0
0 m 0
0 0 m


 .
We further assume that in this basis the Yukawa interactions of the field
ζ is also diagonal, but the charge lepton mass matrix is not diagonal. In
general, it may be possible to diagonalise the charge lepton mass matrix and
5
the neutrino mass matrix simultaneously, since the neutrino mass matrix is
degenerate. However, the coupling of the charged scalar ζ will give radiative
correction, which will break the mass degeneracy and will not allow to make
the charged current interaction diagonal. As a result all the observed mixing
may come from the charge lepton mass matrix and we can only determine
them from experiments. The mixing angle will depend on the amount of
radiative mass splitting.
In general, the charged lepton mass matrix can be diagonalised by a bi-
unitary transformation
V †ikMeiαUαβ = M
diag
ekβ δkβ.
Then the matrix Vik, which diagonalises the matrix MeiαM
†
ejα, will enter
in the charged current interactions. In the basis [eiL, eαR], in which the
charged lepton mass matrix is diagonal, the charge current interaction of the
neutrinos and the charged leptons will be given by,
Lcc = νiLγµV
†
kiekLW
µ. (6)
The charged lepton mixing matrix Vij will introduce neutrino flavour mixing
after the radiative mass splitting.
We now define the basis for the neutrinos [νeiL, ν
e
αR], which has diagonal
charged current interaction and are given by,
νeiL = VikνkL and ν
e
αR = U
ν
αβνβR.
In this basis, the mass matrix is not diagonal. But when the mass matrix is
diagonal and degenerate, we can always make transformations to the right
handed fields and make them diagonal. In the basis [νeiL, ν
e
αR], the neutrino
mass matrix is given by,
Meνij = VijMν .
6
But Vij commutes with Mν and hence we can make a transformation ν
e
αR =
V †βανβR; Vβα = Vij , and diagonalise the mass matrix. However, in the
presence of the radiative corrections due to ζ , this is not possible.
We assume that the couplings of ζ to the right handed leptons, hαβ , to
be diagonal. Although the phenomenology of such dilepton have not been
studied, one can extend the analysis of ref. [11] to constrain the parameters.
If, in addition, we assume that h11 < 10
−5, then there is only one constraint
from the (g − 2)µ, which is h22 > 0.3 for mζ ∼ 100 GeV. For h33 there is no
bound in this scenario and we can consider this to be of the order of 1. With
this choice we get a self energy radiative correction (with the internal loop
containing charged leptons and ζ) to the neutrino mass matrix in the basis
[νiL, ναR], in which the charged current interaction is not diagonal but the
ζ couplings are diagonal, given by
Mνiα =



m + mζ1 0 0
0 m + mζ2 0
0 0 m + mζ3


 (7)
where, mζi = m
h2ii
4π
e2i
m2
ζ
; ei = e, µ, τ . In this case, it will not be possible
to diagonalise the neutrino mass matrix in the basis in which the charged
current interaction is diagonal. Including the standard model self energy
radiative corrections [12], we can now write down the neutrino mass matrix
in the basis [νeiL, ν
e
αR] as follows,
Meνiα = VijMνiα + diag[m
ew
1 m
ew
2 m
ew
3 ] VijMνjα (8)
where, mewi = αw(m
2
ei
/m2w), ei ≡ e, µ, τ . This mass matrix can now be
diagonalised to get the flavour mixing matrix and the mass ssquared differ-
ence between the different flavours of neutrinos. For any arbitrary choice
of the mixing matrix it is difficult to solve this analytically. So, for pur-
pose of illustration we demonstrate with a two generation example, and then
present some realistic numbers for a three generation scenario which we check
numerically.
7
We consider the µ and τ family in the two generation example. For
Vij =
(
cos θ − sin θ
sin θ cos θ
)
, the mass matrix is given by,
Meνiα =
(
(m + mζ2)(1 + m
ew
2 ) cos θ −(m + m
ζ
3)(1 + m
ew
2 ) sin θ
(m + mζ2)(1 + m
ew
3 ) sin θ (m + m
ζ
3)(1 + m
ew
3 ) cos θ
)
(9)
which can be diagonalised with a bi-unitary transformation. The unitary
matrix, which diagonalises Meνiα(M
e
νjα)
†, gives the neutrino flavour mixing
and is given by,
V eij =
(
cos φ sin φ
sin φ cos φ
)
where,
φ ∼
1
2
tan−1
[
2 cos θ sin θ
(mζ3 − m
ζ
2)
(mζ3 − m
ζ
2) + (m
ew
3 − m
ew
2 )
]
and the mass squared difference is given by,
(m23 − m
2
2)
2 = [2m cos θ sin θ(mζ3 − m
ζ
2)]
2 + [m(mζ3 − m
ζ
2) + m(m
ew
3 − m
ew
2 )]
2.
(10)
It is clear from the expressions that both the electroweak radiative correction
as well as the radiative corrections due to ζ will contribute to the mass
squared difference. If ζ does not contribute to the mass difference, the mixing
angle vanishes as pointed out earlier. For the mixing angle to be maximal
we require the contribution of ζ to be of the same order or more than the
contribution from the electroweak radiative corrections.
The mass difference generated by the standard model self energy radiative
corrections (considering the Dirac mass of the neutrinos to be around 7 eV)
are given by,
∆msol = (m
ew
2 )
2 − (mew1 )
2 = αwm
2 m
2
µ − m
2
e
m2w
= 0.6 × 10−5
∆matm = (m
ew
3 )
2 − (mew2 )
2 = αwm
2 m
2
τ − m
2
µ
m2w
= 1.4 × 10−3 (11)
8
The νe − νµ mass difference ∆m
ew
sol is just enough to solve the solar neutrino
problem, while the νµ − ντ mass difference ∆m
ew
atm falls within the solution
suggested by the recent super kamiokande result. If the contribution due
to ζ are of the same order of magnitude (which is the case for mζ ∼ 100
GeV and h11 = 0, h22 ∼ 0.1 and h33 ∼ 1), then we get the maximal mixing
angle for the νµ and ντ oscillations and the relevant mass squared difference
as required by the super kamiokande experiment. For the solar neutrino
problem, the mass squared difference between νe and νµ is just right for
the MSW solution [9], but we only get the small mixing solution. In this
scenario it will be difficult to explain the vacuum oscillation solution of the
solar neutrino problem, since although one can adjust the parameters of the
couplings of ζ to get a small radiative corrections, one cannot change the
electroweak radiative correction. As a result, the mass squared difference
between νe and any other neutrinos will be larger than required, unless there
are new sterile neutrinos, to which the νe oscillates. Since the neutrino masses
arise from a different higgs doublet, the neutrino mass mixing is not related
to the CKM quark mixing matrix.
In summary, we pointed out that if neutrinos are Dirac particles, then we
may start with a degenerate diagonal mass matrix with the diagonal elements
to be around a few eV, so that neutrinos could be the hot dark matter of the
universe. We also presented a model in which neutrinos could be light Dirac
particles naturally. The mass degeneracy is broken by self energy radiative
corrections, which can then allow enough mixing of these neutrinos to solve
the atmospheric neutrino and solar neutrino problems. The standard model
radiative corrections give a lower bound on the mass squared difference which
is just enough to solve both the atmospheric neutrino and solar neutrino
problems.
9
ACKNOWLEDGEMENT
I would like to thank Profs. David Cline, Prof. D.P. Roy and M. Raidal
for discussions. I would also like to thank Prof W. Buchmuller and the
Theory Division, DESY, Hamburg for hospitality. Financial support from
the Alexander von Humboldt Foundation is acknowledged.
10
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11


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Degenerate Dirac Neutrinos

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