We discuss an extended SU(2)&#215;U(1) model which naturally leads to mass scales and mixing angles relevant for understanding both the solar and atmospheric neutrino anomalies in terms of the vacuum oscillations of the three known neutrinos. The model uses a softly broken Le-L&#956;-L&#964; symmetry and contains a heavy scale MH ~1015 GeV. The Le-L&#956;-L&#964; symmetric neutrino masses solve the atmospheric neutrino anomaly while breaking of Le-L&#956;-L&#964; generates highly suppressed radiative mass scale &#916;S ~10&#8722;10 eV2 needed for the vacuum solution of the solar neutrino problem. All the neutrino masses in the model are inversely related to MH, thus providing seesaw-type of masses without invoking any heavy right-handed neutrinos. Possible embedding of the model into an SU(5) grand unified theory is discussed

Joshipura, A. S.

Rindani, S. D.

English

Publications of the IAS Fellows

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PRL-TH-98/011
Vacuum Solutions of Neutrino Anomalies
Through a Softly Broken U(1) symmetry
Anjan S. Joshipura and Saurabh D. Rindani
Theoretical Physics Group, Physical Research Laboratory,
Navarangpura, Ahmedabad, 380 009, India.
Abstract
We discuss an extended SU(2)×U(1) model which naturally leads
to mass scales and mixing angles relevant for understanding both the
solar and atmospheric neutrino anomalies in terms of the vacuum os-
cillations of the three known neutrinos. The model uses a softly broken
Le − Lµ − Lτ symmetry and contains a heavy scale MH ∼ 1015GeV.
The Le − Lµ − Lτ symmetric neutrino masses solve the atmospheric
neutrino anomaly while breaking of Le − Lµ − Lτ generates highly
suppressed radiative mass scale ∆S ∼ 10−10 eV2 needed for the vac-
uum solution of the solar neutrino problem. All the neutrino masses
in the model are inversely related to MH , thus providing seesaw-type
of masses without invoking any heavy right-handed neutrinos. Pos-
sible embedding of the model into an SU(5) grand unified theory is
discussed.
1
Recent results on the oscillations of the muon neutrino seen at the Su-
perkamioka [1] may be taken as the first experimental evidence for physics
beyond the standard electroweak model. It is attractive to suppose that
these are indirect hints into grand unification. The neutrino mass in the
SU(3)×SU(2)×U(1) theory can be characterized by a five dimensional op-
erator which leads tomν ∼ 〈φ〉2M , 〈φ〉 ∼ 250GeV being the electroweak and the
M some heavy scale. The identification of M with a scale MH ∼ 1015GeV
in grand unified theory nicely fits in [2, 3] with the neutrino mass scale√
∆A ∼ 0.07 eV seen at the Superkamioka.
The seesaw model based on grand unified SO(10) theory leads to the
above dimension five term in which M is determined by the right-handed
neutrino masses. Apart from providing an overall scale, this model also
relates [4] hierarchy among neutrino masses to that in the masses of the
other (up quarks in the minimal case) fermions. This feature of SO(10) can
indeed provide another scale ∆S needed to solve the solar neutrino problem.
In the simplest SO(10) model one expects ∆S
∆A
∼
(
mc
mt
)4
. ∆A ∼ 10−3 eV2 then
automatically leads to a ∆S required for the vacuum solution [5] to the solar
neutrino problem. The two large mixing angles needed in this case are not
generic features of the seesaw model but could come out under reasonable
assumptions. [6, 7].
The above attractive features of SO(10) related to neutrino masses are not
shared by generic SU(5)-based grand unified models. It is possible in these
models to obtain neutrino masses and also to understand their overall scale
in terms of the grand unified scale simply by adding a heavy 15 -dimensional
Higgs field [2, 4, 8] . But one cannot easily relate hierarchy in ∆S and ∆A to
the known fermion masses as in the SO(10) case. Our aim here is to present
a simple SU(5) scheme which does this. While the mechanism we discuss is
more general, we give a specific example in which (a) ∆S
∆A
gets related to the
charged lepton masses and (b) two large mixing angles come out naturally.
The natural value for the ∆S
∆A
is close to 10−7 resulting in vacuum oscillations
as the cause for both the solar and atmospheric neutrino deficits.
To simplify the matter we shall first discuss a scheme based on the stan-
dard SU(2)×U(1) model and discuss its SU(5) generalization later on. We
need to extend SU(2) × U(1) model in two ways. We enlarge it with two
extra multiplets of scalar fields namely a triplet ∆ and an additional doublet
field φ2. We also impose a global Le − Lµ − Lτ symmetry. This symmetry
2
has been recognized [2, 6] to provide under reasonable assumptions two large
mixing angles needed for the vacuum solutions of the neutrino anomalies.
It leads to a pair of degenerate neutrinos with a common mass m0 which
determine the atmospheric neutrino mass scale. m0 is inversely related to
the grand unified scale MH in the manner discussed below.
Keeping SU(5) unification in mind, we assume the triplet to be very
heavy, with mass ∼ MH . But such a heavy triplet can influence the low
energy theory crucially by generating Le−Lµ−Lτ symmetric neutrino mass
matrix at tree level and departure from it at one-loop level.
The leptonic Yukawa couplings in the model are given by
−LY = 1
2
fij l¯
c′
iL ∆ l
′
jL + Γ
a
ij l¯
′
iLe
′
jRφ
a + H.c. , (1)
where a = 1, 2 label the Higgs doublets and ∆ is 2× 2 matrix in the SU(2)
space. The Le − Lµ − Lτ symmetry allows the following Yukawa textures:
f ≡ M
ν
0
〈∆0〉 =
m0
〈∆0〉


0 c s
c 0 0
s 0 0

 ;
Γ1 ≡ M
l
1
〈φ01〉
=
1
〈φ01〉


m1 0 0
0 m2 m23
0 m32 m3

 ;
Γ2 ≡ M
l
2
〈φ02〉
=
1
〈φ02〉


0 m12 m13
0 0 0
0 0 0

 , (2)
where we have chosen the Le−Lµ−Lτ charge 2 for the field φ2 and zero for
φ1 and ∆.
The tree level neutrino mass matrix is Le − Lµ − Lτ symmetric and can
be diagonalized by
Uν =


1/
√
2 −1/√2 0
c/
√
2 c/
√
2 −s
s/
√
2 s/
√
2 c

 . (3)
3
If mixing among charged leptons is small then Uν provides the bimaximal
mixing [9] for c ∼ s ∼ 1√
2
and can therefore simultaneously solve the solar
and atmospheric neutrino anomaly through the vacuum oscillations.
The atmospheric scale m0 is determined in the model by the vacuum
expectation value (vev) of ∆0. This is driven by the following scalar potential
V = M2aφ
†
aφa +M
2
HTr.∆
†∆
+ λa(φ
†
aφa)
2 + λ∆Tr.(∆
†∆)2 + ....
−
[
µabφ
T
a∆φb + c.c.
]
. (4)
The terms not explicitly written in the above equations correspond to some
of the quartic terms involving ∆ and quartic cross terms for the doublet
fields. The trilinear terms in eq.(4) are of crucial importance. Firstly, they
induce a small vev for the neutral Higgs ∆0 leading to a degenerate pair of
neutrinos. In addition, they softly break the lepton number and Le−Lµ−Lτ
symmetry. This breaking makes the model phenomenologically acceptable
which otherwise would have contained a doublet plus triplet majoron already
ruled out at LEP. In addition, the Le − Lµ − Lτ breaking by trilinear terms
also generate radiative corrections to the neutrino mass matrix which result
in the splitting of the degenerate pairs and solves the solar neutrino problem.
The triplet vev following from eq.(4) after minimization is of the order
〈∆0〉 ∼ 〈φ1〉〈φ2〉
MH
, (5)
where µab are assumed to be of the same order as the (large) triplet mass
MH . The neutrino mass generated at tree level thus displays the seesaw
type dependence on the heavy scale. Specifically, one gets through eq.(2)
m0 ∼ 3(10−1−10−2) eV for mH ∼ 1014−1015GeV and 〈φ1〉 ∼ 〈φ2〉 providing
the atmospheric neutrino scale.
The tree level neutrino mass matrix following from eq.(2) is Le−Lµ−Lτ
symmetric but the presence of a vev for φ2 breaks this symmetry in the
charged lepton mass matrix. This breaking ultimately gets communicated to
the neutrino mass matrix at the one-loop level. This occurs through the one
loop diagrams shown in Fig.(1).
Let us define the charge lepton mass eigenstates as eiL,R ≡ U †L,Riα e′αL,R
where
UL†(M l1 +M
l
2)U
R ≡ UL†M lUR = M l0 , (6)
4
M l0 being the diagonal charged lepton mass matrix. The soft breaking of
Le − Lµ − Lτ through µ22,23 and vev for φ2 results in finite and calculable
corrections to Le − Lµ − Lτ breaking entries of the neutrino mass matrix
Mν . In order to evaluate these, it is convenient to work with the original
(massless) neutrino flavor basis and treat the mass term Mν0 as an additional
interaction. The Ha in Fig.1 refer to the mass eigenstates of the charged
Higgs fields H ′a ≡ (φ+1 , φ+2 ,∆+) = OabHb.
We have evaluated diagrams of Fig. 1 in the Rξ gauge. Each of the
diagrams gives a finite correction to the Le − Lµ − Lτ breaking elements in
Mν0 and their sum is gauge independent. One finds,
(Mν)11 =
g2
16π2M2W
(Mν0U
LM l0M
l†
0 U
†L)11 ×(
1− 3lnMW
M3
− 4M
2
WO22
g2〈φ02〉
(√
2O32
〈∆0〉 −
O12
2〈φ01〉
)
ln
M2
M3
)
(Mν)ij =
g2
32π2M2W
(
(Mν0U
LM l0M
l†
0 U
†L)ij + (Mν0U
LM l0M
l†
0 U
†L)ji
)
×
(
1− 3lnMW
M3
− 4M
2
WO12
g2〈φ01〉
(√
2O32
〈∆0〉 −
O22
2〈φ02〉
)
ln
M2
M3
)
. (7)
i, j in the above equation take the value 2 and 3 only. M l0 and U
L are defined
in eq.(6). We have repeatedly used the orthogonality of the matrices UL,R
and O in arriving at finite result. M2,3 refer to the masses of the two physical
charged Higgs fields one of which is very heavy, i.e. M3 ∼MH . Terms cubic
in neutrino masses are neglected in writing the above results.
Although the heavy field decouples in the limitMH very large, its residual
mixing of order MW
MH
∼ m0
MW
with the doublet fields influences the radiative
masses. This is explicit in the above equations through the presence of the
tree level neutrino mass matrix. This has the consequence that the radiatively
generated mass terms also display the basic seesaw structure present at the
tree level.
The contributions in eq.(7) depend on all three charged lepton masses
but the contribution due to tau lepton dominates over the rest unless UL31 is
enormously suppressed . We shall assume dominance of this contribution.
The (logarithmic) contribution of the W diagram is similar in magnitude to
the Higgs contributions containing elements of O if the mixing among doublet
5
fields φ1,2 is O(1). Hence for the numerical estimate we shall concentrate on
the lnMW
M3
term. The radiatively corrected neutrino mass matrix then has
the structure
Mν ≈ m0


2ǫs c s
c 0 ǫc
s ǫc 2ǫs

 . (8)
We have implicitly assumed a real UL and U
L
33 ≫ UL23 in writing the above
structure. The parameter ǫ is defined as
ǫ ≡ − 3g
2m2τ
32M2Wπ
2
ln
MW
M3
UL13U
L
33 ∼ (7× 10−5)UL13UL33 (9)
when M3 ∼ 1015GeV.
Let us now look at the phenomenological consequences. As already men-
tioned, ∆A ≡ m20 ∼ 10−2 − 10−3 eV2 follow when Higgs mass MH is in the
range 1014 − 1015GeV. The radiatively corrected mass matrix also implies:
∆S
∆A
∼ 8ǫs ∼ (4× 10−4)UL31UL33 ≤ 2× 10−4. (10)
The mixing among neutrinos is governed by
Kl ≡ U †LUν . (11)
The ratio ∆S
∆A
depends upon unknown values of the mixing among charged
leptons. The scale required for the vacuum solution follows if the mixing
element U31 is small. Indeed, Uij ∼ Uji ∼ O(mimj ) , for i < j leads to
∆S ∼ 10−7∆A ∼ 10−9 − 10−10 eV2 .
The leptonic Kobayashi-Maskawa matrix K is also approximately given in
this case by Uν which provides the required bimaximal mixing. Thus model
under consideration leads to vacuum solution to the solar neutrino problem
for natural value of the relevant parameters.
Unlike the vacuum case, the MSW [10] solution does not follow naturally
in the model. To see this, let us concentrate on the approximate result
eq.(10). If UL33U
L
31 is less than O(1) then one does not get a ∆S in the range
required for the MSW to work inside the Sun even when ∆A is close to its
6
upper limit of 10−2eV2. Moreover, the charged-lepton mixings being small,
the relevant [11] effective mixing angle sin2 2θS ≡ 4K
2
e1K
2
e2
(1−K2
e3
)2
is close to 1 in this
case, and one gets energy independent suppression already ruled out [12, 13]
at the 99% CL.
On the other hand, if mixing in the charged-lepton sector, specifically
UL31,33, is large, there is a possibility that the large mixing among the neutrinos
can be compensated by the large mixing among the charged leptons. The
effective mixing angle in that case can be appreciably less than 45◦. Recent
global fit to new experimental results does allow large mixing angle solution if
one does not include the Superkamioka results on the day-night asymmetry
in the fit. However, in that case, the allowed value of ∆A is even smaller
than in the small-angle case. Specifically, the allowed range for large mixing
solution is given by [12]
0.6 < sin2 2θS < 0.8 ; 8× 10−5 eV2 < ∆S < 2× 10−4 eV2.
It follows that even though proper choice of UL31 can lead to the correct
sin2 2θS, eq.(10) cannot lead to the ∆S in the required range. There is the
possibility that the Higgs contribution we have neglected might, for some
choice of Higgs and charged-lepton mixing, give rise to ∆S in the allowed
region. However, this would be a marginal case.
The generalization of above results to SU(5) model is straightforward.
As an illustration, consider a model with a 15-plet ∆ and two Higgs 5¯-plets
φ1,2. The Le − Lµ − Lτ symmetry can be replaced by a U(1)H symmetry
under which three generations of 5¯-plet of fermions carry charges (1,-1,-1)
respectively while the corresponding 10-plets have opposite U(1)H charges.
φ2 carries charge 2 and rest of the fields are taken neutral. In this case down
quarks together with the charged lepton have the mass structure given by
M l while up-quark masses are given by the following Yukawa couplings:
−Lu = Γua ij10i10jφ∗a , (12)
where a = 1, 2 label the two 5¯-plets of Higgs. The Le − Lµ − Lτ symmetry
allows the following Yukawa textures:
Γu1 ≡


0 β1 β2
β1 0 0
β2 0 0

 ;
7
Γu2 ≡


0 0 0
0 β22 β23
0 β23 β33

 . (13)
It follows that the additional U(1)H symmetry does not lead to any prediction
in the quark sector but allows a general structure for the quark masses and
mixing.
The trilinear terms in (4) are allowed by SU(5) but break the U(1)H
softly. All the previous considerations on the tree level as well radiative
neutrino masses go through. However there are additional diagrams similar
to Fig. 1 contributing to the neutrino masses. These are obtained from
above by replacing W boson, charged leptons and colour singlet Higgs by
the heavy charge-1/3 X-bosons, d-quarks and the colour triplet Higgs bosons
respectively. The contribution of these is suppressed due to heavy X mass
and due to the fact that colour-triplet Higgs have comparable masses. This
is to be contrasted with Fig. 1 which contributes large logarithmic factor
due to vastly different Higgs masses in the loop, see eq. (7). Thus previous
considerations based on the SU(2)× U(1) model remain valid in this case.
We have discussed here a specific case of the Le−Lµ−Lτ symmetry in view
of its phenomenological interest. But the suggested basic mechanism provides
a nice scheme to generate pseudo-Dirac structure for neutrino masses in which
some symmetry (e.g. Le−Lµ) leads to Dirac structure and its violation in the
charged lepton sector leads to splitting among the degenerate pair radiatively.
Since there have been numerous schemes [14] for radiative neutrino masses,
it is appropriate to contrast the present one from the rest. Large class of ra-
diative models [15] use the original mechanisms proposed by Zee [16] and by
Babu [17]. The violation of lepton number at tree level gets communicated
radiatively to neutrinos in these schemes. Here, neutrinos have lepton num-
ber violating but Le − Lµ − Lτ symmetric masses at tree level and breaking
of Le − Lµ − Lτ symmetry gets communicated radiatively.
The most noteworthy feature of the present scheme is the dependence of
the radiative corrections on the tree-level neutrino and the charged lepton
masses. The former is absent in Zee type of models and radiatively generated
contribution is controlled only by the charged lepton masses. This feature
makes the radiative contribution here quite small and allows one to obtain
extremely suppressed solar neutrino scale relevant for the vacuum solution
of the solar neutrino problem.
8
The conventional radiative models need introduction of additional singly
and doubly charged Higgs fields with masses near electroweak scale. Here the
role of the charged singlet is played by corresponding field in the triplet which
is very heavy. Thus the present scheme does not predict a light exotic charged
Higgs. Theoretically, the conventional models are not easily amenable to
grand unification in contrast to the present case. The present scheme is
similar in spirit to the seesaw model based on SO(10) . In spite of the
absence of the right-handed neutrino , the model presented here contains
seesaw structure for all the neutrino masses and these masses are closely
linked to the mass of the charged leptons. This makes the model fairly
predictive and leads to a simultaneous solution for the solar and atmospheric
anomalies which to date provide the strongest hints to believe that neutrinos
are massive.
References
[1] Y. Fukuda et al., Phys. Lett. B436, 33 (1998) (hep-ex/9805006); Phys.
Rev. Lett. 81 (1998) 1158 (hep-ex/9805021), Phys. Rev. Lett. 81 (1998)
1562 (hep-ph/9807003) and S. Hatakeyama et al. Phys. Rev. Lett. 81
(1998) 2016 (hep-ex/9806038).
[2] A. S. Joshipura, hep-ph/9808261
[3] F. Wilczek, hep-ph/9809509.
[4] See e.g. “Massive Neutrinos in Physics and Astrophysics”, R. N. Moha-
patra and Palash B. Pal, World Scientific (1991).
[5] V. Barger, R. J. N. Philips and K. Whisnant, Phys. Rev. D24, 538
(1981) ; S. L. Glashow and L. M. Krauss, Phys. Lett. B190, 199 (1987);
A.Acker, S. Pakvasa and J. Pantaleone, Phys. Rev. D43, 1754 (1991) ;
P. I. Krastev and S. Petcov, Phys. Lett. B286, 85 (1992).
[6] R. Barbieri et al. hep-ph/9807235.
[7] Y. Nomura and T. Yanagida, hep-ph/9807325.
[8] Ernest Ma and Utpal Sarkar, Phys. Rev. Lett. 80 (1998) 5716.
9
[9] V. Barger, S. Pakvasa, T. J. Weiler and K. Whisnant, hep-ph/9806387.
[10] L. Wolfenstein, Phys. Rev. D17, 2369 (1978) ; S. Mikheyev and A. Yu.
Smirnov, Sov. J. Nucl. Physics, 42, 913 (1985).
[11] Q. Y. Liu and A. Yu. Smirnov, hep-ph/9712493.
[12] J. N. Bahcall, P. I. Krastev and A. Yu. Smirnov, hep-ph/9807216.
[13] C. Giunti, hep-ph/9810272.
[14] A review and reference for models with radiatively generated neutrino
masses can be found in K. S. Babu and E. Ma, Mod. Phys. Lett, A4 ,
1975 (1989).
[15] J. T. Peltoniemi and J. W. F. Valle, Nucl. Phys. B406, 409 (1993); J.
T. Peltoniemi, D. Tommasini and J. W. F. Valle, Phys. Lett. B298,
383 (1993); J. T. Peltonemi, A. Yu. Smirnov and J. W. F. Valle, Phys.
Lett. B286, 321 (1992); D.O. Caldwell and R.N. Mohapatra, Phys.
Rev. D48, 3259 (1993) ; N. Gaur et al, hep-ph/9806272; Ernest Ma,
hep-ph/9807386.
[16] A. Zee, Phys. Lett. 93B, 389 (1980).
[17] K. S. Babu, Phys. Lett. B203, 132 (1988).
10
Ljα
H
a
+
ν Ljνk L α
_
eRiRi
H _
a
ν Ljνk L α
_
e
Riν
c
Rk ν LjRi
WW
X X
+
ν cν c
_
ν
e
c
α
+
e
α
+ ν LjRi
H _
a
ν Ljα
_
eRi
H
a
+
ν c
e +
X X
ν cν Rk
cν c
ν
Figure 1: 1-loop diagrams contributing to the Le −Lµ −Lτ breaking entries
of the neutrino mass matrix.
11


Vacuum solutions of neutrino anomalies through a softly broken U (1) symmetry

Indian Academy of Sciences

ar
X
iv
:h
ep
-p
h/
98
11
25
2v
1 
 5
 N
ov
 1
99
8
hep-ph/9811252
PRL-TH-98/011
Vacuum Solutions of Neutrino Anomalies
Through a Softly Broken U(1) symmetry
Anjan S. Joshipura and Saurabh D. Rindani
Theoretical Physics Group, Physical Research Laboratory,
Navarangpura, Ahmedabad, 380 009, India.
Abstract
We discuss an extended SU(2)×U(1) model which naturally leads
to mass scales and mixing angles relevant for understanding both the
solar and atmospheric neutrino anomalies in terms of the vacuum os-
cillations of the three known neutrinos. The model uses a softly broken
Le − Lµ − Lτ symmetry and contains a heavy scale MH ∼ 1015 GeV.
The Le − Lµ − Lτ symmetric neutrino masses solve the atmospheric
neutrino anomaly while breaking of Le − Lµ − Lτ generates highly
suppressed radiative mass scale ∆S ∼ 10−10 eV2 needed for the vac-
uum solution of the solar neutrino problem. All the neutrino masses
in the model are inversely related to MH , thus providing seesaw-type
of masses without invoking any heavy right-handed neutrinos. Pos-
sible embedding of the model into an SU(5) grand unified theory is
discussed.
1
Recent results on the oscillations of the muon neutrino seen at the Su-
perkamioka [1] may be taken as the first experimental evidence for physics
beyond the standard electroweak model. It is attractive to suppose that
these are indirect hints into grand unification. The neutrino mass in the
SU(3)×SU(2)×U(1) theory can be characterized by a five dimensional op-
erator which leads to mν ∼ 〈φ〉
2
M
, 〈φ〉 ∼ 250 GeV being the electroweak and the
M some heavy scale. The identification of M with a scale MH ∼ 1015 GeV
in grand unified theory nicely fits in [2, 3] with the neutrino mass scale√
∆A ∼ 0.07 eV seen at the Superkamioka.
The seesaw model based on grand unified SO(10) theory leads to the
above dimension five term in which M is determined by the right-handed
neutrino masses. Apart from providing an overall scale, this model also
relates [4] hierarchy among neutrino masses to that in the masses of the
other (up quarks in the minimal case) fermions. This feature of SO(10) can
indeed provide another scale ∆S needed to solve the solar neutrino problem.
In the simplest SO(10) model one expects ∆S
∆A
∼
(
mc
mt
)4
. ∆A ∼ 10−3 eV2 then
automatically leads to a ∆S required for the vacuum solution [5] to the solar
neutrino problem. The two large mixing angles needed in this case are not
generic features of the seesaw model but could come out under reasonable
assumptions. [6, 7].
The above attractive features of SO(10) related to neutrino masses are not
shared by generic SU(5)-based grand unified models. It is possible in these
models to obtain neutrino masses and also to understand their overall scale
in terms of the grand unified scale simply by adding a heavy 15 -dimensional
Higgs field [2, 4, 8] . But one cannot easily relate hierarchy in ∆S and ∆A to
the known fermion masses as in the SO(10) case. Our aim here is to present
a simple SU(5) scheme which does this. While the mechanism we discuss is
more general, we give a specific example in which (a) ∆S
∆A
gets related to the
charged lepton masses and (b) two large mixing angles come out naturally.
The natural value for the ∆S
∆A
is close to 10−7 resulting in vacuum oscillations
as the cause for both the solar and atmospheric neutrino deficits.
To simplify the matter we shall first discuss a scheme based on the stan-
dard SU(2) ×U(1) model and discuss its SU(5) generalization later on. We
need to extend SU(2) × U(1) model in two ways. We enlarge it with two
extra multiplets of scalar fields namely a triplet ∆ and an additional doublet
field φ2. We also impose a global Le − Lµ − Lτ symmetry. This symmetry
2
has been recognized [2, 6] to provide under reasonable assumptions two large
mixing angles needed for the vacuum solutions of the neutrino anomalies.
It leads to a pair of degenerate neutrinos with a common mass m0 which
determine the atmospheric neutrino mass scale. m0 is inversely related to
the grand unified scale MH in the manner discussed below.
Keeping SU(5) unification in mind, we assume the triplet to be very
heavy, with mass ∼ MH . But such a heavy triplet can influence the low
energy theory crucially by generating Le −Lµ −Lτ symmetric neutrino mass
matrix at tree level and departure from it at one-loop level.
The leptonic Yukawa couplings in the model are given by
−LY =
1
2
fij l̄
c′
iL ∆ l
′
jL + Γ
a
ij l̄
′
iLe
′
jRφ
a + H.c. , (1)
where a = 1, 2 label the Higgs doublets and ∆ is 2 × 2 matrix in the SU(2)
space. The Le − Lµ − Lτ symmetry allows the following Yukawa textures:
f ≡ M
ν
0
〈∆0〉 =
m0
〈∆0〉




0 c s
c 0 0
s 0 0




;
Γ1 ≡
M l1
〈φ01〉
=
1
〈φ01〉




m1 0 0
0 m2 m23
0 m32 m3




;
Γ2 ≡
M l2
〈φ02〉
=
1
〈φ02〉




0 m12 m13
0 0 0
0 0 0




, (2)
where we have chosen the Le −Lµ −Lτ charge 2 for the field φ2 and zero for
φ1 and ∆.
The tree level neutrino mass matrix is Le − Lµ − Lτ symmetric and can
be diagonalized by
Uν =




1/
√
2 −1/
√
2 0
c/
√
2 c/
√
2 −s
s/
√
2 s/
√
2 c




. (3)
3
If mixing among charged leptons is small then Uν provides the bimaximal
mixing [9] for c ∼ s ∼ 1√
2
and can therefore simultaneously solve the solar
and atmospheric neutrino anomaly through the vacuum oscillations.
The atmospheric scale m0 is determined in the model by the vacuum
expectation value (vev) of ∆0. This is driven by the following scalar potential
V = M2aφ
†
aφa + M
2
HTr.∆
†∆
+ λa(φ
†
aφa)
2 + λ∆Tr.(∆
†∆)2 + ....
−
[
µabφ
T
a ∆φb + c.c.
]
. (4)
The terms not explicitly written in the above equations correspond to some
of the quartic terms involving ∆ and quartic cross terms for the doublet
fields. The trilinear terms in eq.(4) are of crucial importance. Firstly, they
induce a small vev for the neutral Higgs ∆0 leading to a degenerate pair of
neutrinos. In addition, they softly break the lepton number and Le−Lµ−Lτ
symmetry. This breaking makes the model phenomenologically acceptable
which otherwise would have contained a doublet plus triplet majoron already
ruled out at LEP. In addition, the Le − Lµ − Lτ breaking by trilinear terms
also generate radiative corrections to the neutrino mass matrix which result
in the splitting of the degenerate pairs and solves the solar neutrino problem.
The triplet vev following from eq.(4) after minimization is of the order
〈∆0〉 ∼ 〈φ1〉〈φ2〉
MH
, (5)
where µab are assumed to be of the same order as the (large) triplet mass
MH . The neutrino mass generated at tree level thus displays the seesaw
type dependence on the heavy scale. Specifically, one gets through eq.(2)
m0 ∼ 3(10−1−10−2) eV for mH ∼ 1014−1015 GeV and 〈φ1〉 ∼ 〈φ2〉 providing
the atmospheric neutrino scale.
The tree level neutrino mass matrix following from eq.(2) is Le −Lµ −Lτ
symmetric but the presence of a vev for φ2 breaks this symmetry in the
charged lepton mass matrix. This breaking ultimately gets communicated to
the neutrino mass matrix at the one-loop level. This occurs through the one
loop diagrams shown in Fig.(1).
Let us define the charge lepton mass eigenstates as eiL,R ≡ U †L,Riα e′αL,R
where
UL†(M l1 + M
l
2)U
R ≡ UL†M lUR = M l0 , (6)
4
M l0 being the diagonal charged lepton mass matrix. The soft breaking of
Le − Lµ − Lτ through µ22,23 and vev for φ2 results in finite and calculable
corrections to Le − Lµ − Lτ breaking entries of the neutrino mass matrix
Mν . In order to evaluate these, it is convenient to work with the original
(massless) neutrino flavor basis and treat the mass term Mν0 as an additional
interaction. The Ha in Fig.1 refer to the mass eigenstates of the charged
Higgs fields H ′a ≡ (φ+1 , φ+2 , ∆+) = OabHb.
We have evaluated diagrams of Fig. 1 in the Rξ gauge. Each of the
diagrams gives a finite correction to the Le − Lµ − Lτ breaking elements in
Mν0 and their sum is gauge independent. One finds,
(Mν)11 =
g2
16π2M2W
(Mν0 U
LM l0M
l†
0 U
†L)11 ×
(
1 − 3lnMW
M3
− 4M
2
WO22
g2〈φ02〉
(√
2O32
〈∆0〉 −
O12
2〈φ01〉
)
ln
M2
M3
)
(Mν)ij =
g2
32π2M2W
(
(Mν0 U
LM l0M
l†
0 U
†L)ij + (M
ν
0 U
LM l0M
l†
0 U
†L)ji
)
×
(
1 − 3lnMW
M3
− 4M
2
WO12
g2〈φ01〉
(√
2O32
〈∆0〉 −
O22
2〈φ02〉
)
ln
M2
M3
)
. (7)
i, j in the above equation take the value 2 and 3 only. M l0 and U
L are defined
in eq.(6). We have repeatedly used the orthogonality of the matrices UL,R
and O in arriving at finite result. M2,3 refer to the masses of the two physical
charged Higgs fields one of which is very heavy, i.e. M3 ∼ MH . Terms cubic
in neutrino masses are neglected in writing the above results.
Although the heavy field decouples in the limit MH very large, its residual
mixing of order MW
MH
∼ m0
MW
with the doublet fields influences the radiative
masses. This is explicit in the above equations through the presence of the
tree level neutrino mass matrix. This has the consequence that the radiatively
generated mass terms also display the basic seesaw structure present at the
tree level.
The contributions in eq.(7) depend on all three charged lepton masses
but the contribution due to tau lepton dominates over the rest unless UL31 is
enormously suppressed . We shall assume dominance of this contribution.
The (logarithmic) contribution of the W diagram is similar in magnitude to
the Higgs contributions containing elements of O if the mixing among doublet
5
fields φ1,2 is O(1). Hence for the numerical estimate we shall concentrate on
the lnMW
M3
term. The radiatively corrected neutrino mass matrix then has
the structure
Mν ≈ m0




2ǫs c s
c 0 ǫc
s ǫc 2ǫs




. (8)
We have implicitly assumed a real UL and U
L
33 ≫ UL23 in writing the above
structure. The parameter ǫ is defined as
ǫ ≡ − 3g
2m2τ
32M2W π
2
ln
MW
M3
UL13U
L
33 ∼ (7 × 10−5)UL13UL33 (9)
when M3 ∼ 1015 GeV.
Let us now look at the phenomenological consequences. As already men-
tioned, ∆A ≡ m20 ∼ 10−2 − 10−3 eV2 follow when Higgs mass MH is in the
range 1014 − 1015 GeV. The radiatively corrected mass matrix also implies:
∆S
∆A
∼ 8ǫs ∼ (4 × 10−4)UL31UL33 ≤ 2 × 10−4. (10)
The mixing among neutrinos is governed by
Kl ≡ U †LUν . (11)
The ratio ∆S
∆A
depends upon unknown values of the mixing among charged
leptons. The scale required for the vacuum solution follows if the mixing
element U31 is small. Indeed, Uij ∼ Uji ∼ O(mimj ) , for i < j leads to
∆S ∼ 10−7∆A ∼ 10−9 − 10−10 eV2 .
The leptonic Kobayashi-Maskawa matrix K is also approximately given in
this case by Uν which provides the required bimaximal mixing. Thus model
under consideration leads to vacuum solution to the solar neutrino problem
for natural value of the relevant parameters.
Unlike the vacuum case, the MSW [10] solution does not follow naturally
in the model. To see this, let us concentrate on the approximate result
eq.(10). If UL33U
L
31 is less than O(1) then one does not get a ∆S in the range
required for the MSW to work inside the Sun even when ∆A is close to its
6
upper limit of 10−2eV2. Moreover, the charged-lepton mixings being small,
the relevant [11] effective mixing angle sin2 2θS ≡ 4K
2
e1K
2
e2
(1−K2
e3
)2
is close to 1 in this
case, and one gets energy independent suppression already ruled out [12, 13]
at the 99% CL.
On the other hand, if mixing in the charged-lepton sector, specifically
UL31,33, is large, there is a possibility that the large mixing among the neutrinos
can be compensated by the large mixing among the charged leptons. The
effective mixing angle in that case can be appreciably less than 45◦. Recent
global fit to new experimental results does allow large mixing angle solution if
one does not include the Superkamioka results on the day-night asymmetry
in the fit. However, in that case, the allowed value of ∆A is even smaller
than in the small-angle case. Specifically, the allowed range for large mixing
solution is given by [12]
0.6 < sin2 2θS < 0.8 ; 8 × 10−5 eV2 < ∆S < 2 × 10−4 eV2.
It follows that even though proper choice of UL31 can lead to the correct
sin2 2θS, eq.(10) cannot lead to the ∆S in the required range. There is the
possibility that the Higgs contribution we have neglected might, for some
choice of Higgs and charged-lepton mixing, give rise to ∆S in the allowed
region. However, this would be a marginal case.
The generalization of above results to SU(5) model is straightforward.
As an illustration, consider a model with a 15-plet ∆ and two Higgs 5̄-plets
φ1,2. The Le − Lµ − Lτ symmetry can be replaced by a U(1)H symmetry
under which three generations of 5̄-plet of fermions carry charges (1,-1,-1)
respectively while the corresponding 10-plets have opposite U(1)H charges.
φ2 carries charge 2 and rest of the fields are taken neutral. In this case down
quarks together with the charged lepton have the mass structure given by
M l while up-quark masses are given by the following Yukawa couplings:
−Lu = Γua ij10i10jφ∗a , (12)
where a = 1, 2 label the two 5̄-plets of Higgs. The Le − Lµ − Lτ symmetry
allows the following Yukawa textures:
Γu1 ≡




0 β1 β2
β1 0 0
β2 0 0




;
7
Γu2 ≡




0 0 0
0 β22 β23
0 β23 β33




. (13)
It follows that the additional U(1)H symmetry does not lead to any prediction
in the quark sector but allows a general structure for the quark masses and
mixing.
The trilinear terms in (4) are allowed by SU(5) but break the U(1)H
softly. All the previous considerations on the tree level as well radiative
neutrino masses go through. However there are additional diagrams similar
to Fig. 1 contributing to the neutrino masses. These are obtained from
above by replacing W boson, charged leptons and colour singlet Higgs by
the heavy charge-1/3 X-bosons, d-quarks and the colour triplet Higgs bosons
respectively. The contribution of these is suppressed due to heavy X mass
and due to the fact that colour-triplet Higgs have comparable masses. This
is to be contrasted with Fig. 1 which contributes large logarithmic factor
due to vastly different Higgs masses in the loop, see eq. (7). Thus previous
considerations based on the SU(2) × U(1) model remain valid in this case.
We have discussed here a specific case of the Le−Lµ−Lτ symmetry in view
of its phenomenological interest. But the suggested basic mechanism provides
a nice scheme to generate pseudo-Dirac structure for neutrino masses in which
some symmetry (e.g. Le−Lµ) leads to Dirac structure and its violation in the
charged lepton sector leads to splitting among the degenerate pair radiatively.
Since there have been numerous schemes [14] for radiative neutrino masses,
it is appropriate to contrast the present one from the rest. Large class of ra-
diative models [15] use the original mechanisms proposed by Zee [16] and by
Babu [17]. The violation of lepton number at tree level gets communicated
radiatively to neutrinos in these schemes. Here, neutrinos have lepton num-
ber violating but Le − Lµ − Lτ symmetric masses at tree level and breaking
of Le − Lµ − Lτ symmetry gets communicated radiatively.
The most noteworthy feature of the present scheme is the dependence of
the radiative corrections on the tree-level neutrino and the charged lepton
masses. The former is absent in Zee type of models and radiatively generated
contribution is controlled only by the charged lepton masses. This feature
makes the radiative contribution here quite small and allows one to obtain
extremely suppressed solar neutrino scale relevant for the vacuum solution
of the solar neutrino problem.
8
The conventional radiative models need introduction of additional singly
and doubly charged Higgs fields with masses near electroweak scale. Here the
role of the charged singlet is played by corresponding field in the triplet which
is very heavy. Thus the present scheme does not predict a light exotic charged
Higgs. Theoretically, the conventional models are not easily amenable to
grand unification in contrast to the present case. The present scheme is
similar in spirit to the seesaw model based on SO(10) . In spite of the
absence of the right-handed neutrino , the model presented here contains
seesaw structure for all the neutrino masses and these masses are closely
linked to the mass of the charged leptons. This makes the model fairly
predictive and leads to a simultaneous solution for the solar and atmospheric
anomalies which to date provide the strongest hints to believe that neutrinos
are massive.
References
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(1998) 2016 (hep-ex/9806038).
[2] A. S. Joshipura, hep-ph/9808261
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9
[9] V. Barger, S. Pakvasa, T. J. Weiler and K. Whisnant, hep-ph/9806387.
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[13] C. Giunti, hep-ph/9810272.
[14] A review and reference for models with radiatively generated neutrino
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1975 (1989).
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T. Peltoniemi, D. Tommasini and J. W. F. Valle, Phys. Lett. B298,
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hep-ph/9807386.
[16] A. Zee, Phys. Lett. 93B, 389 (1980).
[17] K. S. Babu, Phys. Lett. B203, 132 (1988).
10
Ljα
Ha
+
ν Ljνk L α
_
eRiRi
H
_
a
ν Ljνk L α
_
e
Riν cRk ν LjRi
WW
X X
+
ν cν c
_
ν
e
c
α
+
eα
+ ν LjRi
H
_
a
ν Ljα
_
eRi
Ha
+
ν c
e +
X X
ν cν Rk
cν c
ν
Figure 1: 1-loop diagrams contributing to the Le −Lµ −Lτ breaking entries
of the neutrino mass matrix.
11


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Vacuum solutions of neutrino anomalies through a softly broken U (1) symmetry

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