Neutrinoless double beta decay in SO(10) inspired seesaw models

0.11 MB·English

by  F. Buccella

DSF-44/2003
CERN-TH-2004-015
Bounds for neutrinoless double beta decay in SO(10) inspired
5
0
see-saw models
0
2

n F. Buccella and D. Falcone
a
J
Dipartimento di Scienze Fisiche, Universita` di Napoli,
2

Via Cintia, Napoli, Italy, and INFN, Sezione di Napoli
3
v
9
5
1
4
By requiring the lower limit for the lightest right-handed neutrino mass, obtained
0
4
0 in the baryogenesis from leptogenesis scenario, and a Dirac neutrino mass matrix
/
h similar to the up-quark mass matrix we predict small values for the ν mass and
p e
-
p for the matrix element mee responsible of the neutrinoless double beta decay, mνe
e
h around 5 10−3 eV and m smaller than 10−3 eV, respectively. The allowed range
ee
: ·
v
i for the mass of the heaviest right-handed neutrino is centered around the value of
X
r the scale of B - L breaking in the SO(10) gauge theory with Pati-Salam intermediate
a
symmetry.
2
The robust experimental evidence [1, 2] for neutrino oscillations [3], with increasing
precision on the square mass differences and mixing angles
∆m2 = 7 10−5eV2 (1)
s ·
2
(tanθ ) = 0.4 (2)
s
∆m2 = 2.6 10−3eV2 (3)
a ·
2
(tanθ ) = 1 (4)
a
should be a milestone for the extension of the standard model. Indeed the smallness of
neutrino masses points in the direction of theories like the unified SO(10) model [4], where
the see-saw formula [5]
m = mTM−1m (5)
L − D R D
arises naturally with the elements of the Dirac mass matrix of the order of the charged
fermion masses and the elements of the Majorana mass matrix at the scale of the spon-
taneous symmetry breaking of B L. Right-handed neutrinos of high mass have been
−
advocated in the baryogenesis from leptogenesis scenario [6] and a lower limit for the mass
8
of the lightest neutrino around 5 10 GeV has been found [7].
·
In a recent paper Akhmedov, Frigerio and Smirnov [8] studied the compatibility of
the seesaw mechanism with what is known on neutrino oscillations and the leptogenesis
scenario, and reached the conclusion that the two lightest states should be almost de-
generate. Here we study the consistency of the see-saw formula with the leptogenesis
framework within the hypothesis that m has strong similarities with m , as expected in
D u
SO(10) theories. We do not require strict equality (which would follow from the assump-
tion that the electroweak Higgs transforms as 10 representations), since it does not hold
for m and m , but assume, as in [8], the eigenvalues of m to be 1 MeV, 400 MeV and
d e D
100 GeV, respectively, with the same hierarchical pattern as the up quarks, and small
mixing angles for the matrices V and V which diagonalize m in the basis where m is
R L D e
diagonal:
m = V diag(m ) V†. (6)
D R D L
We define
M = (diag(m )) (7)
h D hh
and m (i = 1,2,3) the eigenvalues of m . One has for the mass of the lightest neutrino
i L
the upper limit
2
3 (M1M2M3) 3
m1 < m1m2m3 = < (0.23 eV) (8)
| | | | MRMRMR
1 2 3
3
comparable with the limit coming from astrophysics [9]. From Eq.(6) and the inverse
see-saw formula
M = m m−1mT (9)
R − D L D
it is easy to derive
MR = VRVRM M AL (10)
ab − af bg f g fg
where
AL = (m−1) V∗LV∗L (11)
fg L cd cf dg
which shows the intriguing property that the part proportional to M M of all the matrix
f g
elements of M isalso proportionalto thesame factorAL . This part may bediagonalized
R fg
by the transformation induced by any unitary matrix with (VR,VR,VR) in the last line.
13 23 33
If we take θatm = 45o, θ13 = 0o and define c = cosθs, s = sinθs, we get the following
elements of the effective neutrino mass matrix:
2 2
mee = c m1 +s m2 (12)
meµ = cs(m1 m2)/√2 (13)
−
meτ = cs(m1 m2)/√2 (14)
−
2 2
mµµ = (s m1 +c m2 +m3)/2 (15)
2 2
mµτ = (s m1 +c m2 m3)/2 (16)
−
2 2
mττ = (s m1 +c m2 +m3)/2, (17)
and a similar expression for m−1 with the exchange m 1/m . From the expression for
L i → i
m−1 we get
L
s 1 c 1 1 1
AL = [cVL∗+ (VL∗+VL∗)]2 +[sVL∗ (VL∗+VL∗)]2 + [VL∗ VL∗]2 =
33 13 23 33 13 23 33 23 33
√2 m1 −√2 m2 2 − m3
2 2
1 s c 1
[VL∗]2( + + )+... (18)
33
2 m1 m2 m3
where we have selected the term proportional to (VL∗)2, since we assume that, as it
33
happens in the CKM matrix [10], the largest matrix elements are the diagonal ones.
2
From now on we shall take tan θ = 0.4 [1].
s
If m1 (tan2θs) m2 , the lightest eigenvalue of MR would be M3R M32/7m1 and
| | ≪ | | ≃
then the product
2
MRMR 7(M2M1) 2.6 1015 GeV2, (19)
2 1
≃ m2m3 ∼ ·
| |
4
which implies MR < 5.1 107 GeV, an order of magnitude smaller than the lower limit
1
·
required in the leptogenesis scenario.
To get some cancellation in the r.h.s. of Eq.(18), m1 should be at least (2/5) m2 .
| | ≃ | |
This would imply for the product of the eigenvalues of M
R
2
MRMRMR 21(M3M2M1) 1030GeV3, (20)
3 2 1
≤ 10∆m2 ∆m2 ≃
sp a
and therefore, in order to be MR 5 108 GeV, one should have
1
≥ ·
MR 4 1012GeV (21)
3
≤ ·
which requires, according to Eq.(18),
2 2
s c 1
+ + < 0.8eV−1 (22)
|m1 m2 m3|
and a cancellation between the three contributions, in fact 1 > 4 eV−1. We should also
|m3|
have
MRMR 2 1021GeV2 (23)
3 2
≤ ·
which implies that the coefficient of (M3M2)2 in the expression of M3RM2R be less than
1.25 eV−2.
For hermitean matrices, if the lightest eigenvalue is smaller than the other two, the
product of them is approximately the sum of the minors corresponding to the three
diagonal matrix elements. Let us consider then
2 2 2
ǫ ǫ ǫ M M
MRMR (MR)2 = abz rsp tnq M M M M VRVRA˜L = ǫ2 (VR)2 2 3 A˜L +...
aa bb − ab 4(m1m2m3) r s t n pz qz pq abz 1z m1m2m3 11
(24)
where A˜Lpq = ViLpVjLqmLij, which shows that the terms proportional to (M3M2)2 for all the
minors considered are multiplied by the factor
1 s
[cVL + (VL +VL)]2+
11 21 31
|m2m3 √2
1 c
[sVL (VL +VL)]2+
11 21 31
m1m3 − √2
1
[(VL VL)]2 < 1.25 eV−2. (25)
21 31
2m1m2 − |
Since we have
1
25 eV−2, (26)
m2m3 ≥
| |
5
to get the coefficient of (M3M2)2 smaller than 1.25 eV−2, we need a cancellation between
the three terms in Eq.(25). Let us see if it is possible to obey at the same time the limits
given by Eqs.(22) and (25) with m given by Eqs. (13-18) and the form
L
cosρ sinρ 0
 
VL = sinρ cosρ 0 (27)
 
− 
 
0 0 1
 
which looks like the CKM matrix up the order λ. Eq.(25) takes the form
1
(c2(cosρ)2 +s2(sinρ)2 √2cscosρsinρ)+
|m2m3 −
2
1 c
(s2(cosρ)2 + (sinρ)2 +√2cscosρsinρ)+
m1m2 2
2
1 (sinρ)
< 1.25 (eV)−2 (28)
m1m2 2 |
In [11] one found the solutions of Eq.(22) with a vanishing r.h.s. consistent with
m2 m2 = ∆m2 = 7 10−5 eV2 (29)
2 − 1 s ·
m2 c2m2 s2m2 = 2.6 10−3 eV2, (30)
3 2 1
− − ·
one with the same sign for m2 and m3 and [12]
2
m1 2 m1 s
= (tanθ ) , (31)
m2 − s − m3c2 ≃ −c2 +0.16
the other with the same sign for m1 and m3, m3 larger than the other two, and
m1 m2, (32)
≃ −
3
m1 cos2θsm3 = m3. (33)
≃ 7
These two solutions areslightly modified by the small upper bound in ther.h.s. of Eq.(22)
into
m1 (tanθ )2c2 ±0.8·eV−1p∆m2s = −2±0.02 (34)
s
m2 ≃ − c2 + ∆m2s 6.12
q∆m2a
m1 cos2θ (1 0.8 eV−1 ∆m2) = 3(1 0.04). (35)
m3 ≃ s ± · p a 7 ±
There are also solutions, with m1 and m2 with the same sign and m1 > 0.11 eV or > 2.5
| |
eV (which is excluded by astrophysics [9]). However only the solution of Eq.(22) with the
6
same sign for m2 and m3 satisfies Eq.(28) with a small value of ρ. By considering only
the term proportional to (VL∗)2 in Eq.(19), one neglects terms at the order
33
4√2cVL∗ VL∗
13 13
9 (36)
sV∗L ≃ VL∗
33 33
1 1
for the relative coefficients of and and
m1 m2
4VL∗
23
(37)
VL∗
33
1 1
for the relative coefficients of and . Should one take for the ratios of the matrix
m1 m3
elements of VL the corresponding values of the CKM matrix, these uncertainties would
be 10% and 100% and would effect the first and the second term in the denominator of
the r.h.s. of Eq.(31), respectively. By taking into account all the uncertainties one has
m1
0.45 < < 0.25 (38)
− m2 −
which implies
4.23 10−3 < m1 < 2.16 10−3 eV (39)
· − ·
8.64 10−3 < m2 < 9.4 10−3 eV (40)
· ·
5.15 10−2 < m3 < 5.18 10−2 eV. (41)
· ·
We predict a small value for the neutrinoless double beta decay parameter
Mee = sin2θsm2 +cos2θsm1 = m1m2/m3 0.8eV−1m1m2 (42)
− ±
which implies M to be in the range (3.5 8) 10−4eV and a slightly larger value for the
ee
− ·
effective electron neutrino mass
2 2
mνe = sin θs|m2|+cos θs|m1| (43)
in the range (4 5.6) 10−3eV. For simplicity we have considered real m ’s, but our results
i
− ·
would not change by considering complex m ’s.
i
By taking AL = 0 and A˜L = 0 one gets
33 11
√2sc(m2 m1)
tanρ = − (44)
−m3 +c2m2 +s2m1
in the range ( 0.15, 0.12) about twice smaller than tanθ .
c
− −
¿From the identity
AL33AL22 (AL23)2 = A˜L11/m1m2m3 (45)
−
7
2
it follows that, if we ask that no term proportional to M is present in M and no term
3 R
2
proportional to (M3M2) is contained in the diagonal minors of MR, there is no term
2
proportional to (M3M2) in MR. In conclusion, the largest contribution to MR would be
the ones proportional to M3M1 and M22 which are multiplied by the factors AL31 and AL22,
respectively:
MaRb = −M3M1VaR1VbR3AL31 −M22VaR2VbR2AL22 +... (46)
2
where we have not written the terms proportional to M2M1 and M1. In general
DetMR = (M3M2M1)2DetAL (47)
−
and therefore
DetAL = (m1m2m3)−1. (48)
For real M , which is not what we want to get the CP violation needed to produce
R
baryogenesis, but is a property of the CKM matrix at the first order in sinθ , we have
c
the eigenvalues of MR as M22AL22 and (M3M1)/ AL22m3m2m1. Since
− ± p
2 2
c s
AL = sin2ρ( + ), (49)
22
m1 m2
8 10
the eigenvalues are in the ranges (5 6.8) 10 GeV and (4 5) 10 GeV, respectively.
− · ± − ·
8
The lightest right-handed neutrino is just above the lower limit 5 10 GeV from leptoge-
·
nesis. We have deduced the consequences of AL = 0 and A˜L = 0 to display an example
33 11
8
with the lightest right-handed neutrino mass larger than 5 10 GeV.
·
10
In general the mass of the heaviest right-handed neutrino will be in the range 10
12
- 5 10 GeV with the geometrical mean of the two extrema very near to the scale of
·
B L symmetry breaking in the SO(10) unified theory with Pati-Salam [13] intermediate
−
11
symmetry, 3 10 GeV [14].
·
Our assumption of small mixing values for VL and on the value of the product of the
two smallest eigenvalues of m ( 4 102MeV2 ) excludes the high values of MR proposed
D 3
·
in the literature [15].
The upper limit given by Eq.(21) implies a small coefficient for the contribution pro-
portional to M2 to the matrix elements of MR, which is welcome to get the lepton
3
asymmetry needed for leptogenesis, which would vanish, should one neglect the other
contributions.
We want to stress that the small values found for AL and A˜L are not unnatural
33 11
tunings; just the opposite: since Eq.(5) gives m in terms of MR (and of mD), the fact
L
that the matrix elements of MR ( as well as its eigenvalues ) are not so different, as it
happens for m , makes the choice described here a natural one.
L
8
Aknowledgements.
One of us (FB) aknowledges a clarifying discussion with Prof. Ferruccio Feruglio and
the hospitality at the theory division of CERN, where part of this work has been done.
9
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