A neutrino mass-mixing sum rule from SO(10) and neutrinoless double beta decay

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by  F. Buccella

A neutrino mass-mixing sum rule from SO(10) and neutrinoless double beta decay
F. Buccella,1 M. Chianese,1,2 G. Mangano,1 G. Miele,1,2 S. Morisi,1,2 and P. Santorelli1,2
1INFN, Sezione di Napoli, Complesso Univ. Monte S. Angelo, I-80126 Napoli, Italy
2Dipartimento di Fisica Ettore Pancini, Universita` di Napoli Federico II,
Complesso Univ. Monte S. Angelo, I-80126 Napoli, Italy
(Dated: January 12, 2017)
Minimal SO(10) grand unified models provide phenomenological predictions for neutrino mass
patterns and mixing. These are the outcome of the interplay of several features, namely: i) the
seesaw mechanism; ii) the presence of an intermediate scale where B-L gauge symmetry is broken
and the right-handed neutrinos acquire a Majorana mass; iii) a symmetric Dirac neutrino mass
matrixwhosepatternisclosetotheup-typequarkone. Inthisframeworktwonaturalcharacteristics
emerge. Normal neutrino mass hierarchy is the only allowed, and there is an approximate relation
involving both light-neutrino masses and mixing parameters. This differs from what occurring
7 when horizontal flavour symmetries are invoked. In this case, in fact, neutrino mixing or mass
1 relations have been separately obtained in literature. In this paper we discuss an example of such
0 comprehensive mixing-mass relation in a specific realization of SO(10) and, in particular, analyse
2
its impact on the expected neutrinoless double beta decay effective mass parameter (cid:104)m (cid:105), and
ee

n on the neutrino mass scale. Remarkably a lower limit for the lightest neutrino mass is obtained
a (mlightest (cid:38)7.5×10−4 eV, at 3 σ level).
J

1 PACSnumbers: 12.60.-i;14.60.Pq;14.60.St.
1

] Grand Unified Theories (GUT) embed the Standard (m3 (cid:28) m1 < m2) are still allowed by present data [9–
h
Model (SM) gauge group into (semi-) simple groups of 11],wherebydefinitionthemasseigenstatem istheone
p 3
higher dimension, and provide remarkable insights on is- thatmaximallymixeswithflavoureigenstatesν andν .
- µ τ
p sues which are left unsolved by the - yet extremely suc- Indeed, it is the whole paradigm of fermion mass pat-
e
h cessful - SU(3)c×SU(2)L×U(1)Y theory. Examples of tern and mixing parameters hierarchies that remains a
[ these phenomenological features are the explanation of mystery, a deep question in particle physics known as
electricchargequantization,unificationofthegaugecou- the flavour problem. In the last decades, many ideas
2
plings at some large mass scale and a prediction for the have been put forward as attempts to address this prob-
v
value of the Weinberg angle. Furthermore, GUTs have lem, within GUTs or in different schemes. One inter-
1
9 a smaller set of free parameters with respect to SM, and esting possibility is based on the idea of extending the
4 providenicerelationsamongfermionmassesandmixing. SM gauge group to include a symmetry acting between
0 Finally, they share the property that all matter fields, thethreefamilies,knownashorizontalorflavoursymme-
0
for each generation, can be allocated in just a few of ir- tries. Such a symmetry could be abelian continuous [12]
.
1 reducible group representations (IRR): only two in case or discrete [14], and non-abelian continuous [13] or dis-
0 of SU(5) [1] and Pati-Salam [2, 3] groups, and a single crete [15, 16]. Some years ago, it became very popu-
7
16-dimensional spinorial representation for SO(10) [4, 5] lar to exploit non-abelian discrete symmetry after pre-
1
: (for a review see ref. [6]). liminary experimental indications supporting an almost
v In this paper we focus on the SO(10) GUTs, point- maximal atmospheric neutrino mixing angle θ and a
i 23
X ing out that, adopting minimal and reasonable assump- smallreactorangleθ atthesametime(seeforinstance
13
r tions that will be discussed in the following, two in- ref. [17] and references therein). However, recent experi-
a teresting phenomenological implications about neutrino mental data show a clear deviation from the maximality
masses and mixing emerge: for the atmospheric angle, and indicate a not vanishing
• only normal ν−mass ordering is allowed [7, 8]; sinθ13 ∼ λC with λC ≈ 0.22−0.23 denoting the sinus
of the Cabibbo angle [18]. In view of this, non-abelian
• there is a mixing dependent ν−mass sum rule that flavour symmetries at the present seem to be quite dis-
constraints the allowed region in the plane light- favoured [19].
est neutrino mass eigenstate (m ) vs effective
lightest Typically, flavour symmetries lead to simple relations
mass parameter ((cid:104)m (cid:105)). This eventually affects
ee amongneutrinoparameters,knownasthemassandmix-
the neutrinoless double beta decay rates.
ing sum rules. The presence of neutrino mass sum rules
We know from neutrino oscillations experiments, that at wasfirstobservedinref.[20]andthenstudiedinref.[21].
leasttwoofthethreeactiveneutrinosaremassive. How- A phenomenological classification was given in ref. [22]
ever,theabsoluteneutrinomassscaleisstillunknown,as while a more extensive analysis based on the possible
well as the mass ordering. In fact, both Normal Hierar- neutrino mass mechanism can be found in ref. [23] (for a
chy (NH) (m <m (cid:28)m ) and Inverted Hierarchy (IH) reviewonthisissueseealsoref.[24]). Ontheotherhand,
1 2 3
2
mixing sum rules have been introduced in ref.s [25–28] tains the 10 and 126 IRRs as well.1
(seeref.s[29–34]forarecentdiscussion),andhavestrong This implies that the Dirac neutrino mass matrix
implication for the connection between model building is symmetric.
andexperiments. Itisworthstressingthatinbothcases,
• The Dirac neutrino mass matrix m has approx-
these sum rules may strongly impact the expected rate D
imatively the same structure of the up-type quark
for neutrinoless double beta decays, since they give non
mass matrices M .
trivialrelationsamongthethreeneutrinomassesormix- u
Thisisratheragoodapproximation. Infact,dueto
ing parameters.
the bottom-tau mass unification at the M scale,
The mass and mixing sum rules take respectively the X
the vev of the 10 must be dominant over the 126
general form
one. Thefactthatm ≈M impliesthattheDirac
D u
κ mh+κ (m e−2iα)h+κ (m e−2iβ)h =0, (1) neutrino mass eigenvalues are strongly hierarchical
1 1 2 2 3 3 liketheupquarkcase,andthecorrespondingdiag-
onalizingmatrixhasaCabibbo-likestructure,with
√ √
(1− 2sinθ ) = ρ +λ(1− 2sinθ )cosδ, only the angle in the 1-2 plane is large and of the
23 atm 13
order of λ .
θ = ρ +θ cosδ, (2) C
12 sol 13
• There is an upper limit on the mass of the heaviest
where the neutrino masses mi are generally complex right-handed neutrino MR3 (cid:46)1011 GeV.
quantities, θ13, θ23, θ12 are respectively, the reactor, at- This is related to the intermediate B-L symmetry
mospheric and solar angles, δ is the Dirac phase, and α breaking.
and β are the Majorana phases. Finally, κ , ρ , ρ , λ
i atm sol
From type-I seesaw mechanism we have
and h=−1,−1/2,1/2,1 are parameters that depend on
theparticularflavourscheme,ascanbeseenforinstance 1
m =−m mT , (4)
in ref.s [25–27]. ν DM D
R
Though relations like eq.s (1) or (2) are typically ob-
or
tained within extensions of the SM based on flavor sym-
1
metries approaches it is worth observing that similar re- M =−mT m , (5)
R Dm D
sults can be also obtained in GUTs, where the approach ν
is, so to say, vertical i.e. based on gauge symmetry where m and M are the light neutrino and the right-
ν R
paradigm, rather than horizontal. As an example, in handed mass matrices, respectively. Eq. (4) can be also
ref.s [20, 35, 36] the following constraint has been ob- rewritten as
tained within a class of SO(10) models 1
M =−V†mdiagV∗U UTV†mdiagV∗, (6)
R L D L mdiag L D L
sin2θ cos2θ 1 ν
12 + 12 + =0. (3)
m m e−2iα m e−2iβ where we have assumed that the Dirac neutrino mass
1 2 3 matrix is symmetric, namely m = V†mdiagV , and
D L D L
This is a mass-mixing sum rule, since the coefficients k denoted by U the neutrino mixing matrix. From the
i
of eq. (1) are functions of the mixing oscillation parame- assumption m ≈ M we have that V is similar to
D u L
ters. Thisrelationhasbeenobtainedunderthefollowing the mixing matrix that diagonalises on the left the up-
assumptions – more details can be found in ref. [36]: type quark mass matrix. In first approximation it re-
sults in a rotation in the 1-2 plane with an angle of
• type-I seesaw mechanism is dominant over type-II.
the order of Cabibbo one. Moreover, the mass matrix
In SO(10) there are two contributions to the light mdiag is strongly hierarchical (mdiag) (cid:28)(mdiag) ∼
neutrino mass given by the type-I [37–41] and D D 11,22 D 33
O(m ) and so we get
top
type-II [40–43] seesaw. While right-handed neu-
(cid:12)(cid:18) (cid:19) (cid:12)
tMriXno∼m1a0ss11isGgeeVn,erwahteedreattheanPaintit-eSramlaemdiagtreouspcalies MR3 ≡(MR)33 ≈(cid:12)(cid:12)(cid:12) Umdν1iagUT 33(cid:12)(cid:12)(cid:12) (mdDiag)233. (7)
broken, the type-II contribution is suppressed by
This means that, in order to have M (cid:46) 1011 GeV, a
the mass of the scalar electroweak triplet [44–47]. R3
strong cancellation is required, which reads
Such a mass is proportional to the vev of the 210
sactatlahreIGRURT, wschaicleh. dFriovresthtihserebarseoankitnygpeo-fISsOee(s1a0w) (cid:12)(cid:12)(cid:12)(cid:12)(cid:18)Umd1iagUT(cid:19) (cid:12)(cid:12)(cid:12)(cid:12)(cid:46)10−2eV−1 ≡ε. (8)
ν 33
is more natural in SO(10) with respect to type-II.
This result has been also numerically checked and
confirmed in ref. [8].
1 For studies of SO(10) GUTs with two Higgs doublets see for
• In addition to the 210, the Higgs scalar sector con- instanceref.s[8,48,49].
3
Finally taking the standard Particle Data Group theleptonnumberbytwounits,ismediatedbyMajorana
parametrisation for the lepton mixing matrix U [18], we neutrinomasstermsanditwouldeventuallydemonstrate
have the Majorana nature of neutrinos [50]. The 0νββ decay
rates are proportional to the “effective mass”
(cid:12)(cid:12)(cid:12)(cid:12)mA21 + m2Be−22iα + m3Ce−22iβ(cid:12)(cid:12)(cid:12)(cid:12)(cid:46)ε, (9) (cid:104)mee(cid:105)=(cid:12)(cid:12)Ue21m1+Ue22m2e2iα+Ue23m3e2iβ(cid:12)(cid:12) , (15)
which is a function of the Majorana phases, α and β,
with
and the lightest neutrino mass m , given by m or
lightest 1
A = cosθ12cosθ23sinθ13eiδ−sinθ12sinθ23, (10) m3 in case of Normal and Inverted Hierarchies, respec-
B = sinθ cosθ sinθ eiδ+cosθ sinθ , (11) tively. The Dirac phase δ is included in the matrix ele-
12 23 13 12 23
ment U . The other neutrino masses are given in terms
e3
C = cosθ cosθ , (12)
13 23 of the measured squared mass differences, known from
oscillation experiments. Due to eq. (9) that provides a
which reproduces the relation in eq. (3) assuming ε = 0
√ relation between the three neutrino masses and the Ma-
and sinθ =0, sinθ =1/ 2. Notice that the relation
13 23 jorana phases one gets a bound for the allowed region in
ineq.(9)isageneralizationoftheonereportedineq.(3),
the m –(cid:104)m (cid:105) plane, see fig. 1. In particular, the
andwewilldiscussinthefollowingitsphenomenological lightest ee
solid (dashed) lines bound the region obtained by span-
implications.
ning the 3 σ ranges for the neutrino mixing parameters
In general, there are no theoretical predictions about
given in ref. [9] in case of NH (IH).
themasshierarchyevenforagivenneutrinomassmecha-
In fig. 1 the dotted (dot-dashed) line represents the
nismlikethetype-Iseesaw,butaswehavealreadystated
values for m and (cid:104)m (cid:105) satisfying the relation (9),
inSO(10)GrandUnifiedmodelsonlynormalν−massor- lightest ee
once the NH (IH) best-fit values of the neutrino mixing
dering is allowed [8]. This can be easily understood. In
parameters have been taken into account. Once we al-
SO(10)withjusta10and126inthescalarsector,three
low the neutrino mixing parameters to vary in their 99%
fermionmassmatrices(M , m andm )canbewritten
u D ν C.L. ranges one gets the shaded area reported in fig. 1.
in terms of the remaining two (Md and Ml) as2 In the plot the cosmological bound (cid:80)m <0.17eV, ob-
ν
tained by the Planck Collaboration [51] (vertical line),
M = f [(3+r)M +(1−r)M ],
u u d l
and the constraint (cid:104)m (cid:105)<0.2eV coming from the non-
m = f [3(1−r)M +(1+3r)M ], (13) ee
D u d l observation of the neutrinoless double beta decay in the
mν = fνmD(Md−Ml)−1mD, phase 1 of the GERDA experiment [52] (horizontal line)
are also shown. It is worth observing that the SO(10)
where f , f , r are free parameters that are functions of
u ν relation (9) is in agreement with an inverted hierarchy
thevevof10and126andofYukawamatrices(seeref.[8]
scenario only in case of a quasi-degenerate pattern. As
formoredetails). IfM andM arestronglyhierarchical
d l one can appreciate from fig. 1 an interesting lower limit
this will imply the same for Mu and mD. On the other m (cid:38) 7.5×10−4 eV, at 3 σ is obtained. A similar
lightest
hand, M −M can be whatever, since M and M are
d l l d limit has been derived in left-right models with type-II
quite similar. Yet, the resulting m is also hierarchical
ν seesaw [53].
and therefore, an inverted ordering is very unnatural.
The largest uncertainty in the shaded region of fig. 1
This can be also seen in a different way, starting from
is related to the Dirac phase δ, which still can range at
the relation in eq. (3). As pointed out in ref. [35] one
3 σ in the whole range [0,2π] (see ref.s [9–11]). This
gets
also affects the predicted lower limit on m = m .
lightest 1
m (cid:0)m e−2iα+m e−2iβ(cid:1) We show in fig. 2 the best-fit lines from eq. (9) for
tan2θ =− 1 2 3 , (14) different choices of the parameter δ in the interval [0,π].
12 m e−2iα(m +m e−2iβ)
2 1 3 In particular, for small values of the Dirac phase, the
relation (9) provides two different best-fit lines that
that gives in the IH-limit (m (cid:28) m < m ) a solar
3 1 2
gradually merge into a single best-fit region as the
mixing angle such that |tan2θ | ≈ 1, inconsistent with
12
parameter δ increases. When δ ranges in the interval
the experimental value 0.42±0.07 at 95% C.L. [9].
[π,2π] we get the same allowed lines shown in fig. 2, but
from the bottom-right to the up-left plots. Notice that
Remarkably, the mass relation in eq. (9) leads to a
the smallest values for m are obtained for δ =0.
prediction for the neutrinoless double beta decay 0νββ lightest
rates. Itiswellknownthatthisdecay,withaviolationof
In summary, we have revisited and generalized a neu-
trino mass-mixing relation that naturally emerges in a
SO(10) GUT framework. This kind of relations has
2 Hereweneglectthetype-IIneutrinomasscontributionaccording been pointed out and originally studied in the context
totheconsiderationsgivenabove. of flavour horizontal symmetry extensions of the SM. In
4
and mixing angles are involved in simple sum rules, like
���
the one in eq. (9), and strongly suggest a normal hierar-
excludedbyGERDAI chicalpatternforneutrinomasses. Wehaveanalyzedthe
��-� impact of this constraint on neutrinoless double beta de-
)� cay mass parameter (cid:104)mee(cid:105), and found that a lower limit
IH
� on the absolute neutrino mass scale emerges.
( ��-�
y
〉 d g So far, experiments have not been able to distinguish
��� ure olo between the two neutrino hierarchy schemes, but there
〈 ��-� NH sfavo cosm are good chances that this will be possible in the near
di y future in several experiments, like for instance, Hyper-
b
Kamiokande [54], T2K [55], ORCA [56], PINGU [57].
��-�
��-� ��-� ��-� ��-� ��� In this framework, a possible evidence in favour of an
IH scheme will rule out the class of SO(10) models here
� (��)
��������
presented.
FIG. 1: The solid (dashed) lines bound the allowed region in
them –(cid:104)m (cid:105)planeobtainedbyspanningthe3σranges
lightest ee The authors acknowledge support by the Istituto
for the neutrino mixing parameters [9] in case of NH (IH).
Nazionale di Fisica Nucleare, I.S. QFT-HEP and TAsP,
The dotted (dot-dashed) line is the prediction of eq. (9) on
theeffectivemass,oncetheNH(IH)best-fitvaluesoftheneu- and the PRIN 2012 Theoretical Astroparticle Physics
trino mixing parameters are adopted [9]. The shaded region of the Italian Ministero dell’Istruzione, Universit`a e
represents the 3 σ area obtained according to the neutrino Ricerca.
mass-mixing dependent sum rule of eq. (9).
100 100
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