Leptogenesis and $CP$ violation in $SU(5)$ models with lepton flavor mixing originating from the right-handed sector

1.2 MB·

by  Heinrich Päs

DO-TH 14/01
Leptogenesis and CP violation in SU(5) models with lepton flavor mixing originating
from the right-handed sector
H. P¨as∗ and E. Schumacher†
Fakult¨at fu¨r Physik, Technische Universit¨at Dortmund, 44221 Dortmund, Germany
We discuss neutrino masses and mixing in the context of seesaw type I models with three right-
handed Majorana neutrinos and an approximately diagonal Dirac sector. This ansatz is motivated
by the idea that the flavor structure in the right-handed Majorana masses is responsible for the
largemixingangles,whereasthesmallmixingangleθ originatesfromtheDiracYukawacouplings
13
in analogy to the quark sector. To obtain θ ≈ 0.15 we study a possible SU(5) grand unified
13
theoryrealizationwithaU(1)×Z(cid:48) ×Z(cid:48)(cid:48)×Z(cid:48)(cid:48)(cid:48) flavorsymmetryandincludeacomplexperturbation
2 2 2
parameter in the Dirac mass matrix. The consequences for CP violating phases and effects on
leptogenesis are investigated.
4 PACSnumbers: 11.30.Hv,12.10.-g,14.60.St
1
0
2

n
I. INTRODUCTION
u
J
4 Neutrinos are special in several respects: First, they are much lighter than the charged leptons and quarks and as
2 they do not carry any unbroken quantum number they can be Majorana particles. These properties are exploited in
the seesaw mechanism where the heavy Majorana masses of right-handed neutrinos drive the effective masses of left-

] handedneutrinosdowntoorbelowtheeVscale. Moreover,themismatchbetweenneutrinoandchargedleptonmixing
h
parametrizedinthePontecorvo-Maki-Nakagawa-Sakata(PMNS)matrixexhibitslargeorevenmaximalmixingangles,
p
- instarkcontrasttothesmallCabibbo-Kobayashi-Maskawa(CKM)mixinginthequarksector. Afterrecentdatafrom
p reactor experiments have revealed a nonvanishing neutrino mixing angle θ and thus rule out exact tribimaximal
13
e
(TBM) mixing [1–3], many attempts have been made to explain the finite θ . While the idea of anarchy is becoming
h 13
[ more attractive, the most popular approaches are still based on discrete symmetries, as summarized, e.g., in [4] or
[5]. Models that intend to explain the leptonic mixing pattern can be constructed using large symmetry groups such

2
as ∆(96) or D . At the expense of predictivity, these models contain several free parameters to account for all the
v 14
physical observables. Alternatively, models can be based on smaller symmetry groups, e.g., A or S , which yield
8 4 4
specificpatternssuchastheTBMorgoldenratiomixingandincludeperturbationsinordertogeneratethenecessary
2
3 deviations from these structures.
2 In this paper we analyze the possibility that also the large lepton mixing arises from the right-handed Majorana
. sector. To this end we study a generic type I seesaw model with three heavy right-handed neutrinos:
1
0
m =mT M−1m , (1)
4 ν D R D
1
: wheremD denotestheDiracmassmatrixandMR thematrixoftheright-handedneutrinos. Inapreviouspublication
v
[6] the Dirac matrix has been assumed to be diagonal, and it has been shown that this can give rise to TBM. We
i
X adjust this pattern by perturbing the Dirac mass matrix with a complex parameter in order to accommodate a finite
r θ13. The philosophy behind this ansatz is to have the small mixing originate from the Dirac sector, in analogy to the
a small CKM mixing in the quark sector, while the Majorana property of the right-handed neutrinos is responsible for
thelargemixinganglesθ andθ . ThestructureoftheDiracsectorismotivatedbyanSU(5)GrandUnifiedTheory
23 12
(GUT) embedding, which accounts not only for leptonic mixing but also for observables of the quark sector. Related
works exist, where the Majorana mass terms of the neutrino sector account for the large leptonic mixing angles, e.g.,
[7, 8]
The paper is organized as follows: We start by describing the outline of the model in Sec. II proposing a possible
realization in an SU(5) GUT. In Sec. III we briefly summarize the methods used to analyze CP violation in our
model and then focus on leptogenesis in Sec. IV. Sec. V deals with the numerical analysis and with the implications
for neutrinoless double-beta decay (0νββ). Conclusions are drawn in Sec. VI.
∗Electronicaddress: [email protected]
†Electronicaddress: [email protected]
2
II. OUTLINE OF THE MODEL
If we consider m and the mass matrix of the charged leptons to be diagonal, all mixing in the neutrino sector
D
originatesfromM . AnexplicitexamplecanbeconstructedbyusinganSU(5)GUTwithU(1)×Z(cid:48) ×Z(cid:48)(cid:48)×Z(cid:48)(cid:48)(cid:48) flavor
R 2 2 2
symmetry in close analogy to the model published in [9]. All standard model particles, including the right-handed
neutrinos, areaccommodatedinthe10⊕5⊕1multipletsofSU(5). Assigningappropriatequantumnumbersunder
these symmetry groups yields the quark mixing matrix in the first approximation, where the off-diagonal elements
are suppressed by a Froggatt-Nielsen (FN) mechanism [10]. In this framework the fermion masses are generated
through couplings to additional scalar fields η with universal vacuum expectation values (VEVs) u. These VEVs
i
are suppressed by a large messenger scale Λ such that u ≈ λ = 0.22, consequently leading to suppression factors in
Λ
the fermion mass term ∝ λn, where n denotes the sum of the fermion field charges under the corresponding flavor
symmetries. It has been shown, e.g., in [11, 12] that the combination of an SU(5) with a U(1) symmetry can give
FN
rise to maximal mixing in the lepton sector due to specific U(1) charge assignments.
FN
The matrix structures generated by the FN mechanism must be consistent with the following approximate mass
relations:
m :m :m ≈ λ8 :λ4 :1, m :m :m ≈ λ4 :λ2 :1, m :m :m ≈ λ4 :λ2 :1. (2)
u c t d s b e µ τ
Following Ref. [9], assigning the following U(1) charges to the SU(5) multiplets, and (-1) for the flavon fields, leads
to the desired structures of the Yukawa matrices in the quark sector,
10 : I : 4 II : 2 III : 0, (3)
5 : I : 3 II : 3 III : 3, (4)
where I,II and III specify the family number. The powers of the suppression factors λ entering the Lagrangian
depend on the Z charges ρ ,ρ and ρ of the respective flavons η(cid:48),η(cid:48)(cid:48), and η(cid:48)(cid:48)(cid:48). The sum rules for cyclic groups Z
2 1 2 3 2
are
ρ +ρ =(ρ +ρ ) mod 2, (5)
i j i j
where ρ again are the Z charges of the interacting fermions ψ . This results in
i,j n i,j
 
λ8 λ6+ρ1+ρ2 λ4+ρ1+ρ3
10i⊗10j : Yu ∝λ6+ρ1+ρ2 λ4 λ2+ρ3+ρ2 , (6)
λ4+ρ1+ρ3 λ2+ρ3+ρ2 1
 
λ7 λ7+ρ1+ρ2 λ7+ρ1+ρ3
10i⊗5j : Yd ∝λ5+ρ1+ρ2 λ5 λ5+ρ3+ρ2 . (7)
λ3+ρ1+ρ3 λ3+ρ3+ρ2 λ3
Since the Yukawa matrix of the charged leptons Y ∼5 ⊗10 ∼YT is hierarchical as well, large lepton mixing must
l i j d
beaconsequenceofthespecificstructureoftheneutrinosector. Toestablishthestructureofthelatter, letusdenote
the U(1) charges of the right-handed neutrinos NR in the SU(5) singlet 1 as e ,e , and e , respectively. According
1,2,3 1 2 3
to Ref. [9] the right-handed neutrinos carry no Z charges as the seesaw scale is lower than the messenger scale Λ
2
at which the flavons η receive their VEVs u. The relevant matrix structures in the neutrino sector arise from the
i
following products:
 
λ3+e1 λ3+e2+ρ1 λ3+e3+ρ1
5i⊗1j : YD ∝λ3+e1+ρ2 λ3+e2 λ3+e3+ρ2 , (8)
λ3+e1+ρ3 λ3+e2+ρ3 λ3+e3
 
λ2e1 λe1+e2 λe1+e3
1i⊗1j : YR ∝λe2+e1 λ2e2 λe2+e3 . (9)
λe3+e1 λe3+e2 λ2e3
By choosing
e =0; e =e =1; ρ =ρ =ρ =1, (10)
3 1 2 1 2 3
we can motivate a hierarchical structure of the Dirac matrix YD with two perturbation parameters assigned to the
elements YD and YD:
12 23
   
λ λ2 λ λ2 λ2 λ
YD ∝λ3·λ2 λ λ, YR ∝λ2 λ2 λ. (11)
λ2 λ2 1 λ λ 1
3
Following our charge assignments from Eqs. (10) and (11) we choose a specific Dirac mass pattern for our further
analysis,
 
λ 0 (cid:15)
mD = 0 λ γ ·v, (12)
0 0 1
with λ ≈ 0.22 and v = 2√46GeV, where (cid:15) generates a nonzero θ13 value and γ cancels the effects of (cid:15) on the large
2
mixing angles θ and θ . In analogy to the quark sector we assume that all complex phases except for one can be
12 23
absorbed into the interacting lepton fields. A single complex phase φ in (cid:15)=|(cid:15)|eiφ remains, whereas γ ∈R.
The procedure to accommodate the experimentally determined mixing angles can be summarized in three steps. It
has been demonstrated in Ref. [6] that a diagonal Dirac sector can give rise to TBM mixing. Therefore by adopting
m =diag(λ,λ,1)·v (13)
D
we first determine the right-handed mass matrix M to yield exact TBM mixing with θ and θ in the allowed 3σ
R 12 23
ranges [13],
θ ∈[0.543,0.626], θ ∈[0.625,0.956], θ ∈[0.125,0.173]. (14)
12 23 13
We can derive an analytical expression for M−1 from the condition that m must be diagonalizable with U :
R ν TBM
M−1 =(mT)−1U m(cid:48) UT m−1 (15)
R D TBM ν TBM D
 2m1+m2 −m1+m2 −m1+m2 
1 λ2 λ2 λ
=  −m1+m2 m1+2m2+3m3 m1+2m2−3m3  and (16)
3v2 λ2 2λ2 2λ
−m1+m2 m1+2m2−3m3 m1+2m2+3m3
λ 2λ 2
 λ2(m1+2m2) λ2(m1−m2) λ(m1−m2) 
MR = v32  λ2(mmm111mm−22m2) 12λ2(cid:16)(cid:16)m11m+1mm222 + m33(cid:17)(cid:17) 21λ(cid:16)(cid:16)m11m+1mm222 − m33(cid:17)(cid:17) (17)
λ(m1−m2) 1λ 1 + 2 − 3 1 1 + 2 + 3
m1m2 2 m1 m2 m3 2 m1 m2 m3
where m(cid:48) =diag(m ,m ,m ) serves as an input parameter. The matrix M given in Eq. (17) is in good agreement
ν 1 2 3 R
withY ofEq. (11). Theinitialvaluesforthelightneutrinomassesm inm(cid:48) areselectedaccordingtotheexperimental
R i ν
bounds on ∆m2 and ∆m2 [14]
12 23
(cid:40)
+2.46×10−3(eV)2 (NH)
∆m2 =7.59×10−5(eV)2, ∆m2 = . (18)
12 23 −2.37×10−3(eV)2 (IH)
The model exhibits different behavior in the various mass regimes, where the following cases are considered:
1. Degenerate masses (deg): m ≈m ≈m >0,
1 2 3
2. Normal hierarchy (NH): m (cid:29)m ≈m ≈0,
3 2 1
3. Inverse hierarchy (IH): m ≈m (cid:29)m ≈0.
1 2 3
In the second step we include the perturbation parameters (cid:15) and γ in the Dirac mass matrix m according to Eq.
D
(12) to produce θ ≈0.15=θExp, while M remains unchanged. This guarantees that θ is solely affected by Dirac
13 13 R 13
couplings.
The parameters |(cid:15)|,φ and γ then are fitted to the current experimental 3σ bounds on the mixing angles θ given
ij
in Eq. (14) for neutrino mass eigenstates m ∈ [10−3,10−1]eV. This mass region includes the NH and IH scenarios
i
as well as the degenerate mass regime being consistent with cosmological bounds on the neutrino masses [15] and a
successful leptogenesis scenario [16]. The bounds on the mixing angles constrain the parameter space of (cid:15) to small
regions, enabling predictions on the observable CP phases. Assuming that leptogenesis successfully generates the
correct baryon asymmetry of the Universe, additional constraints arise that confine these intervals even further.
4
III. CP VIOLATION
The PMNS matrix depends on four parameters: three mixing angles θ and one CP violating phase δ. In the case
ij
of three additional right-handed neutrinos the mixing matrix includes two supplementary Majorana phases, leading
to the conventional parametrization of the PMNS matrix
 c c s c s e−iδ 
12 13 12 13 13
UPMNS =−s12c23−c12s23s13eiδ c12c23−s12s23s13eiδ s23c13 ·P , (19)
s s −c c s eiδ −c s −s c s eiδ c c
12 23 12 23 13 12 23 12 23 13 23 13
where P = diag(eiα,eiβ,1) contains the Majorana phases α and β, and s and c are abbreviations for sinθ and
ij ij ij
cosθ , respectively.
ij
Inourmodelthecomplexperturbationparameter(cid:15)providesthesourceofCP violationintheneutrinosectorandleads
toCP violatingphasesinU. Themixingmatrixcanbeextractedfromm(cid:48) =UT m U usingaTakagidecomposition,
ν ν
which is applicable for complex symmetric matrices [17]. The real diagonal matrix m(cid:48) contains the non-negative
ν
square roots of the eigenvalues of m m†. However, the mixing matrix we receive from our numerical analysis does
ν ν
not resemble the usual convention of the PMNS matrix given in Eq. (19). We can extract the mixing angles θ and
ij
CP phases δ, α and β using rephasing invariants and properties of the unitarity triangles of U, previously explored
in [18, 19]. By defining ”Majorana-type“ phases ζ and ξ , i∈(1,2,3),
i i
ζ ≡Arg(U U∗ ), ζ ≡Arg(U U∗ ), ζ ≡Arg(U U∗ ), (20)
1 e1 e2 2 µ1 µ2 3 τ1 τ2
ξ ≡Arg(U U∗ ), ξ ≡Arg(U U∗ ), ξ ≡Arg(U U∗ ), (21)
1 e1 e3 2 µ1 µ3 3 τ1 τ3
we can express the mixing angles and δ according to [20] as a function of ζ and ξ :
i i
|sin(ξ −ξ )||sin(−ζ +ξ +ζ −ξ )||sin(ξ −ξ )|
tan2θ = 1 2 2 2 3 3 1 3 , (22)
12 |sin(−ζ +ξ +ζ −ξ )||sin(ξ −ξ )||sin(−ζ +ξ +ζ −ξ )|
1 1 2 2 2 3 1 1 3 3
|sin(ξ −ξ )||sin(−ζ +ξ +ζ −ξ )||sin(ζ −ζ )|
tan2θ = 1 3 1 1 3 3 1 2 , (23)
23 |sin(−ζ +ξ +ζ −ξ )||sin(ξ −ξ )||sin(ζ −ζ )|
1 1 2 2 1 2 1 3
|sin(ξ −ξ )||sin(ζ −ζ )||sin(ζ −ζ )|
tan2θ = 2 3 1 3 1 2 ·sin2θ , (24)
13 |sin(ξ −ξ )||sin(ξ −ξ )||sin(ζ −ζ )| 12
1 3 1 2 2 3
1 |cosθ cosθ sinθ |2|sin(ξ −ξ )||sin(ξ −ξ )|
|sinδ|= 12 13 13 1 3 1 2 . (25)
8 |sin(2θ )sin(2θ )sin(2θ )cosθ ||sin(ξ −ξ )|
12 13 23 13 2 3
These formulas are valid regardless of the parametrization of the matrix U. To extract the Majorana phases α and
β we recall that they rotate the Majorana unitarity triangles in the complex plane. Hence, one can conclude that
expressions for the phases α and β result from (with U :=U )
PMNS,ij Pij
(cid:18)U U∗ (cid:19) (cid:18)U U∗ (cid:19) (cid:18)U U∗ (cid:19)
Arg P11 P13 =α, Arg P12 P13 =β, Arg P11 P12 =α−β. (26)
U U∗ U U∗ U U∗
11 13 12 13 11 12
IV. LEPTOGENESIS
Leptogenesis [21] explains the present matter asymmetry by assuming that a lepton asymmetry in the early Universe
is converted into a baryon asymmetry through B+L violating sphaleron processes. Leptogenesis is closely connected
to the seesaw mechanism as it relies on the existence of heavy right-handed Majorana neutrinos N decaying into
i
leptonsl andscalarfieldsφviaN →l +φ. Observationsofthecosmicmicrowavebackgroundallowforanestimate
α i α
of the baryon asymmetry [22]
n −n
YCMB = B B =(8.79±0.44)·10−11, (27)
∆B s
wheresistheentropydensityoftheuniverseandn denotetheabundancesofbaryonsandantibaryons,respectively.
B,B
The baryon asymmetry can be calculated in terms of the CP asymmetry σ and the efficiency factor κ [23],
iα i
Y =σ κ 10−3. (28)
∆B iα i
5
The efficiency factor κ measures the effect of washout processes depending on the ”washout regime”. For a general
i
system it is given by the solution of the Boltzmann equations for leptogenesis, which characterize the competition of
production and washout of the N ’s. In this paper the following approximations are used, as described in Ref. [23]:
i
m m
m<m ∩ m <m : κ ≈ i(cid:101) weak washout (29)
(cid:101) ∗ i ∗ i m2
∗
m
m>m ∩ m <m : κ ≈ i intermediate washout (30)
(cid:101) ∗ i ∗ i m
∗
m
m>m ∩ m >m : κ ≈ ∗ strong washout (31)
(cid:101) ∗ i ∗ i m
i
with m ≈1.1×10−3eV and the sum of the light neutrino masses m=(cid:80) m .
∗ (cid:101) i i
The CP asymmetry σ generated by a heavy right-handed neutrino N that decays into a lepton with flavor α reads
iα i
explicitly [23]
(cid:40) (cid:32) (cid:33) (cid:32) (cid:33)(cid:41)
σ ≡ 1 1 (cid:88) Im(cid:2)(YD†YD) Y YD∗(cid:3)g Mj2 +Im(cid:2)(YD†YD) YDYD∗(cid:3) Mi2 . (32)
iα 8π(YD†YD) ji αi αj M2 ij αi αj M2−M2
ii j(cid:54)=i i i j
The Yukawa couplings matrix YD is related to the Dirac matrix by YD = mD. Because of the Majorana nature
v
of the N , contributions to the CP asymmetry arise only from higher-order interferences of their decays. The loop
i
corrections are included in the function g(x)
√ (cid:20) 1 (cid:18) 1(cid:19)(cid:21)
g(x)= x +1−(1+x)ln 1+ . (33)
1−x x
Inamodelwiththreeadditionalright-handedneutrinos,theneutralizingeffectoftheN ontheasymmetrygenerated
2,3
inthedecayofN canbeneglectedifM >T ,thereheatingtemperature,andM (cid:28)M . Inourapproximation
1 2,3 reh 1 2,3
the second term in Eq. (32) vanishes due to zeros in the pattern of YD.
Model independent limits on neutrino masses can be inferred from the upper bound of the CP asymmetry σ . As
iα
stated in Sec. II, in the single lepton flavor approximation with hierarchical right-handed neutrino masses M a
i
successfulleptogenesismechanismisconfinedtothemassregion10−3eV< m < 0.1eV[16]. Wewillthereforefocus
i
our attention on this interval.
Because of the specific structure of m given in Eq. (12), we obtain a small number of contributions to the CP
D
asymmetry. With
     
λ 0 |(cid:15)|eiφ λ 0 |(cid:15)|e−iφ λ2 0 |(cid:15)|eiφλ
YD = 0 λ γ , YD∗ = 0 λ γ , and Y†Y = 0 λ2 γλ , (34)
0 0 1 0 0 1 |(cid:15)|e−iφλ γλ 1
we receive nonvanishing contributions to σ only if α=1:
iα
(cid:88) |(cid:15)|2sin(2φ)(cid:18) (cid:18)M2(cid:19) (cid:18)M2(cid:19)(cid:19)
σ = σ = λ2·g 1 −g 3 , (35)
1 i1 8π M2 M2
i 3 1
resultingina|(cid:15)|2sin(2φ)dependenceofY . Notethattheparameterγ doesnotaffectthegeneratedCP asymmetry.
∆B
V. NUMERICAL ANALYSIS
For the numerical analysis we use the following ordering schemes:
(cid:113) (cid:113)
NH: m =m , m = m2+∆m2 , m = m2+∆m2 +∆m2 , (36)
1 0 2 0 12 3 0 12 23
(cid:113) (cid:113)
IH: m =m , m = m2+∆m2 +∆m2 , m = m2+∆m2 , (37)
3 0 2 0 12 23 1 0 23
(cid:112) (cid:112)
where m ∈[10−3,10−1]eV. In the regions of larger masses, where m (cid:29) ∆m2 , ∆m2 the mass eigenstates are
0 i 12 23
degenerate. To explain the small neutrino masses through heavy right-handed neutrinos, which are compatible with
6
the leptogenesis mechanism described in Sec. IV, the Dirac masses have to be at the GeV scale. As explained in Sec.
I, we assume TBM mixing with a diagonal Dirac sector in order to determine M from Eq. (15).
R
In the cases of m =10−3eV, where the NH and IH scenarios are relevant, and m =0.1eV (degenerate masses) the
0 0
right-handed mass eigenstates are given by
NH: M =4.934·1013GeV, M =3.942·1014GeV, M =7.311·1015GeV, (38)
R,1 R,2 R,3
IH: M =2.890·1013GeV, M =5.452·1013GeV, M =1.595·1016GeV, (39)
R,1 R,2 R,3
deg: M =1.365·1013GeV, M =1.447·1013GeV, M =2.830·1014GeV, (40)
R,1 R,2 R,3
revealing a hierarchy among the the heavy right-handed neutrino masses, fulfilling the conditions for our approxima-
tions in leptogenesis.
The parameter spaces of the perturbation parameters |(cid:15)|, φ, and γ are scanned for combinations that reproduce the
neutrino mixing angles θ within the current 3σ bounds, see Eq. (14). Assuming that leptogenesis successfully
ij
generates the correct baryon asymmetry of the Universe, given in Eq. (27), we can confine these regions even further
to make precise predictions on physical observables.
Thecomparisonoflow-scaleexperimentaldataonneutrinoparameterswithcalculationscarriedoutattheGUTscale
requires taking into account the renormalization group (RG) evolution of the leptonic mixing parameters. The RG
running effects on the mixing parameters have been considered in various publications [24–26] and can be significant
especially in the case of a degenerate mass spectrum. The leptonic mixing angles are expected to run faster than
their quark equivalents for they are larger and the neutrino mass differences are particularly tiny. The generic
enhancement for the evolution of the angles can be estimated analytically, which has been done, for instance, in
[25]. For our numerical analysis of the degenerate mass regime we calculate the light neutrino masses at the seesaw
scale using Eq. (1) and run the matrix m down to the electroweak scale before determining the mixing parameters
ν
according to Sec. III. This allows for a better comparison with experimental data and a more robust analysis. The
effects of RG running are implemented using REAP 1.8.4, generously provided by [25].
Regarding leptogenesis, Eq. (31) is used to compute the efficiency factor κ since m > 10−3 > m∗. The resulting
i 0
parameter spaces complying with Eq. (27) are listed in Table I, where δ,α,β denote the Dirac and Majorana CP
phases,respectively,correspondingtotheparameterregionsof|(cid:15)|,φ,andγ. Theallowedcombinationsof|(cid:15)|andφas
wellasthegeneratedbaryonasymmetryaredepictedinFig. 1. Theblueareasdenoteparametervaluesthataccount
for neutrino mixing, while the red color indicates combinations that also lead to successful leptogenesis.
3333....0000 YY YY
BB 3333....0000 BB
1100(cid:45)(cid:45)99 1100(cid:45)(cid:45)1100
2222....5555
2222....5555
2222....0000
2222....0000
1100(cid:45)(cid:45)1122 1100(cid:45)(cid:45)1122
ΦΦΦΦ
1111....5555 ΦΦΦΦ
1111....5555
YY (cid:64)(cid:64)1100(cid:45)(cid:45)1111(cid:68)(cid:68) YY (cid:64)(cid:64)1100(cid:45)(cid:45)1111(cid:68)(cid:68)
BB BB
1111....0000 99..33 1111....0000 99..33
0000....5555 0000....5555
88..44 88..44
0000....0000 0000....0000
0000....00002222 0000....00004444 0000....00006666 0000....00008888 0000....00002222 0000....00003333 0000....00004444 0000....00005555 0000....00006666 0000....00007777 0000....00008888
(cid:200)(cid:200)(cid:200)(cid:200)ΕΕΕΕ(cid:200)(cid:200)(cid:200)(cid:200) (cid:200)(cid:200)(cid:200)(cid:200)ΕΕΕΕ(cid:200)(cid:200)(cid:200)(cid:200)
(a)Y∆B(|(cid:15)|,φ),NH (b)Y∆B(|(cid:15)|,φ),IH
Figure1: (a): Parameterspaceof|(cid:15)|andφcompatiblewiththe3σrangesoftheθ inthecaseofNHwithm ∈[0.00,0.05] eV.
ij 0
Thewhiteareasdidnotyieldmodelscompatiblewiththeexperimentalbounds. The(red)lineoverlyingthecontourplotmarks
parametervaluesthatsuccessfullygeneratethebaryonasymmetryY =(8.79±0.44)×10−11 inleptogenesis. (b)Parameter
∆B
space in the IH case.
7
m |(cid:15)| φ γ
0
NH [0.018,0.050] [0.018,0.060] [0.063,0.565]∪[2.639,3.079] [0.02,0.20]
IH [0.002,0.044] [0.034,0.076] [0.251,1.445]∪[1.696,2.890] [−0.38,0.00]
δ α β
NH [0.016,0.083] [0.039,0.192]∪[3.118,3.133] [0.003,0.157]
IH [0.205,1.293] [0.448,3.112] [0.009,0.067]∪[0.402,2.404]
Table I: Parameter spaces of the fit parameters |(cid:15)|, φ and γ for successful leptogenesis and the CP phases δ,α, and β corre-
sponding to these intervals.
In a previous publication [6] it was found that models with a diagonal Dirac sector and TBM mixing favor very light
neutrino masses. Although in principle all considered hierarchies are compatible with the experimental bounds on
the neutrino mixing angles in our model, the statistics summarized in the following support the previous statement:
NH: m ∈[0.00,0.05] eV, n=46250, n =19, (41)
0 L
IH: m ∈[0.00,0.05] eV, n=3768, n =229, (42)
0 L
deg: m ∈[0.05,0.10] eV, n=189, n =0, (43)
0 L
where n and n count the number of events with and without leptogenesis, respectively, for each considered mass
L
regime. TheneutrinomixingisbestaccountedforbytheNHscenario,whereasIHisfavoredifleptogenesissuccessfully
explains the baryon asymmetry of the Universe. The RG analysis reveals that the model cannot accommodate the
leptonic mixing angles for m >0.056 eV, practically ruling out degenerate neutrino masses as a suitable scenario.
0
Because of the number of combinations accommodating correct neutrino mixing as well as the baryon asymmetry of
the early universe, predictions are the most reliable in the IH scenario. The predictions on the CP phases listed in
Table I can be used for further studies; e.g., the Majorana phases α and β affect the observables measured in 0νββ
experiments. The amplitude of these processes is proportional to the effective Majorana mass
(cid:88)
m = U2 m . (44)
ββ ei i
i
Cancellation can occur in NH depending on the size of the Majorana phases; however, in this particular case we
predictsmalllowenergyCP violationascanbeseenfromTableI.Accordingtoourresultsabovewegiveanestimate
of the effective Majorana mass in the considered hierarchies,
mNH ∈[0.048,0.063] eV, mIH ∈[0.026,0.050] eV. (45)
ββ ββ
Note again that due to Eqs. (41) - (43) the predictions are most reliable in the IH scenario. The bounds on m are
ββ
well below the current upper limit (cid:104)m (cid:105)(cid:46)0.19−0.45 eV given by the EXO-200 experiment at 90% confidence level
ββ
[27], but partly accessible in next-generation 0νββ decay experiments.
VI. CONCLUSIONS
SU(5) inspired seesaw type I models with an almost diagonal Dirac sector and three heavy right-handed neutrinos
successfully accommodate θExp. The large neutrino mixing angles are ascribed to the structure of the right-handed
13
sector, while the small mixing angle is generated by a complex perturbation parameter (cid:15) and a real parameter γ
assigned to the off-diagonal elements of m . The structure of the perturbed Dirac mass matrix is obtained from an
D
embeddinginanSU(5)×U(1)×Z(cid:48) ×Z(cid:48)(cid:48)×Z(cid:48)(cid:48)(cid:48) symmetry,wherethehierarchyamongthefermionfamiliesisgenerated
2 2 2
by a Froggatt-Nielsen mechanism. The parameter (cid:15) also provides a possible source of CP violation in the leptonic
sector, enabling predictions on the Dirac CP phase δ and the Majorana phases α and β.
The numerical analysis shows that the neutrino mixing parameters can in principle be reproduced in all considered
hierarchiesformassesuptom ≈0.056eV. However, theNHscenarioisstronglyfavoredoverIHandthedegenerate
0
massregimeaccommodatingthecorrectneutrinomixinganglesθ . Furtherconstraintscanarisefromtherequirement
ij
that leptogenesis generates the baryon asymmetry of the Universe, which eventually results in a preference of IH and
rules out degenerate neutrino masses entirely. Regarding the phases δ, α, and β the CP violation in the IH case can
be large, whereas for NH we predict small CP violation in all phases. Since the degenerate mass regime is excluded,
the resulting effective Majorana mass m is partly accessible by next-generation 0νββ experiments in IH scenarios.
ββ
The leptogenesis scenario discussed in this paper is in good agreement with experimental and cosmological bounds,
although a more realistic scenario with higher precision may improve the final predictions.
8
VII. ACKNOWLEDGMENTS
H.P. was supported by DFG Grant No. PA 803/6-1. We thank Gautam Bhattacharyya for many helpful comments
both on the original idea as well as on the actual work.
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