#### Predictive leptogenesis from minimal lepton flavour violation

Revised: May
Predictive leptogenesis from minimal lepton avour
L. Merlo 0 1 2
S. Rosauro-Alcaraz 0 1 2
Field Theories
0 Cantoblanco , 28049, Madrid , Spain
1 Universidad Autonoma de Madrid
2 Departamento de F sica Teorica and Instituto de F sica Teorica, IFT-UAM/CSIC
A predictive Leptogenesis scenario is presented based on the Minimal Lepton Flavour Violation symmetry. In the realisation with three right-handed neutrinos transforming under the same avour symmetry of the lepton electroweak doublets, lepton masses and PMNS mixing parameters can be described according to the current data, including a large Dirac CP phase. The observed matter-antimatter asymmetry of the Universe can be achieved through Leptogenesis, with the CP asymmetry parameter " described in terms of only lepton masses, mixings and phases, plus two real parameters of the low-energy e ective description. This is in contrast with the large majority of models present in the literature, where " depends on several high-energy parameters, preventing a direct connection between low-energy observables and the baryon to photon ratio today. Recovering the correct amount of baryon asymmetry in the Universe constrains the Majorana phases of the PMNS matrix within speci c ranges of values: clear predictions for the neutrinoless double beta decay emerge, representing a potential smoking gun for this framework.
Beyond Standard Model; Neutrino Physics; Global Symmetries; E ective
1 Introduction 2 3 4
5
Conclusions
A Lowering Tmax
The minimal lepton
avour violation with vectorial SU(
3
)V
2.1
A suitable basis for leptogenesis
Baryogenesis trough leptogenesis
Numerical analysis
4.1
Low-energy phenomenology
Despite being so tiny, this non-vanishing value poses one of the most relevant unsolved
questions in particle physics and cosmology today: why are there more baryons than
antibaryons in the present Universe?
In 1967, Sakharov rst suggested that the baryon asymmetry of the Universe (BAU)
might not represent some sort of initial condition, but could be understood in terms of
microphysical laws that ful l the following three conditions [3]:
- Baryon number violation
{ 1 {
1
Introduction
The uncertainties on the measurements for the cosmic abundances of the lightest elements
have improved considerably in the last decades, posing stringent constraints on the thermal
history of the very early Universe. The observed abundances of protium, deuterium, 3He,
4He and Lithium, besides well agreeing with the predictions of the standard Big Bang
Nucleosynthesis [1], allow to deduce the value of the baryon to photon ratio today,
where NB;B; are the number densities of baryons, anti-baryons and photons, respectively.
An independent determination of B is provided by the CMB measurements [2] that agrees
with the value extracted from the lightest element abundances:
B
NB
N
NB ;
B = (6:11
0:04)
nesis [6] (see refs. [7{12] for update reviews on the subject), besides being promising in a
large part of the associated parameter space, represents a framework where also other open
problems of the SM of particle physics may
nd a solution: on the one hand the origin of
neutrino masses and on the other hand the Flavour problem.
The small, but non-vanishing, masses of the light active neutrinos represent an
experimental evidence of the incompleteness of the SM. The introduction of right-handed (RH)
neutrinos a la type I Seesaw mechanism [13{17] is an elegant approach that explains the
smallness of the active neutrino masses through the largeness of the masses of their RH
counterparts. This mechanism provides the ingredients to explain the present amount of
BAU: there is a leptonic source of CP violation and a source of Lepton number violation;
RH neutrino decays may occur out-of-equilibrium, when the temperature drops below their
masses. In consequence, out-of-equilibrium decays of the RH neutrinos might produce a
lepton asymmetry that is then partially converted into BAU through non-perturbative
sphaleron e ects [18]: in the SM context,
B =
1
cs
cs
L
with
cs
8NF + 4
22NF + 13
< 1 ;
with NF the number of avour species considered. It follows that the more anti-leptons are
produced, the more baryons are generated, with a rate that is approximately close to 1=3.
The basic quantity in Leptogenesis is the parameter " that measures the amount of
CP asymmetry generated in the decays of the RH neutrinos R [6]: indicating with
and
the decay rates of R into leptons and antileptons respectively,
where `L, and
stand for the SU(
2
)L-doublet left-handed (LH) leptons and the
SU(
2
)Ldoublet Higgs eld, the CP asymmetry parameter is given by
with
(a) the
avour (RH neutrino mass) index. The analytic expression of the CP
parameter " depends on the product y , with
the Dirac neutrino Yukawa coupling in
the mass basis for the RH neutrinos and for the charged leptons. In the convention where
the active neutrino mass term is de ned by
a
a
"(a)
L
( Ra ! `L
Ra ! `L
)
;
P
P
a
a + a
a ;
12 Lc m
{ 2 {
L + h.c. ;
(1.3)
(1.4)
(1.5)
(1.6)
where the mass matrix is diagonalised by the PMNS matrix U according to
m^
= U T m U ;
(1.7)
(1.8)
(the \^" symbol is adopted here and in the following to refer to diagonal matrices),
is a matrix in avour space that can be written in the Casas-Ibarra parametrisation as
follows [19]:
=
p
2
v
U m^ 1=2RM^ 1=2 ;
where v = 246 GeV is the EW vacuum expectation value (VEV), R a complex orthogonal
matrix, and M^ the diagonal mass matrix of the RH neutrinos. This expression depends
on 9 low-energy parameters, i.e. 3 active neutrino masses, 3 mixing angles, 1 CP violating
Dirac phase and 2 CP violating Majorana phases, and on 9 high-energy ones, corresponding
to the 3 RH neutrino masses and the 6 parameters of the matrix R. The latter is typically
independent from the low-energy quantities and its parameters are arbitrary. In general,
this prevents to uniquely determine the parameter " in terms of low-energy observables
and the RH neutrino masses.
The use of avour symmetries helps improving the predictivity in this scenario: as
a avour symmetry rules the interactions among the di erent fermion generations, the R
matrix might be (partially) xed, allowing to predict the value of " (almost) just in function
of neutrino masses, mixings and phases. Some examples can be found in refs. [20{31] (see
also refs. [32, 33] for predictive scenario not involving avour symmetries).
The aim of the present paper is to investigate on a speci c scenario where a
continuous non-Abelian group is implemented in the Lagrangian as a global avour symmetry,
providing an exceptionally predictive framework for both Leptogenesis and low-energy
observables. The symmetry under consideration is the one of the so-called Minimal Flavour
Violation (MFV) in the lepton sector (MLFV), considering the type I Seesaw mechanism
with three RH neutrinos. The MFV ansatz [34] consists in assuming that any source of
avour and CP violation in any theory Beyond the SM (BSM) is the one in the SM, i.e. the
Yukawa couplings. This concept has been technically formulated in terms of the
avour
symmetry of the fermion kinetic terms of a given Lagrangian [35]: the
avour group is
a product of a U(
3
) factor for each fermion in the spectrum, and it is U(
3
)6 [36{40] for
the type I Seesaw mechanism with 3 RH neutrinos. The Yukawa interactions are the only
terms of the renormalisable Lagrangian that are not invariant under the
avour
symmetry, unless the Yukawa couplings are promoted to be elds, dubbed spurions, transforming
non-trivially under U(
3
)6. In the original proposal [35], the Yukawa spurions are
dimensionless, non-dynamical elds that acquire background values (they could be interpreted
as VEVs if the spurions were promoted to be dynamical elds [41{44]), breaking explicitly
the avour symmetry, and reproducing the measured values of fermion masses and PMNS
angles and phases.
In the quark sector, any non-renormalisable operator containing fermion elds is,
eventually, made invariant under the avour symmetry by the insertion of suitable powers of
the Yukawa spurions. Once the latter acquire their background values, the strength of the
e ects induced by such e ective operators is suppressed by speci c combinations of quark
{ 3 {
masses, mixing angles and CP violating phase. In consequence, the cut-o scale that
suppresses any non-renormalisable operator can be of the order of a few TeV [35, 45{58],
instead of hundreds of TeV as in the generic case [59].
In the lepton sector, with the addition of three RH neutrinos, the predictive power of
the MLFV is lost in the most generic case. Indeed, three quantities, and not only two as
in the quark case, need to be promoted to spurions, i.e. the charged lepton Yukawa, the
neutrino Dirac Yukawa and the RH neutrino Majorana mass matrices. A simple parameter
counting reveals that it is not possible to uniquely determine the three spurion backgrounds
in terms of lepton masses and PMNS parameters. This prevents to link the coe cients of
avour changing operators with low-energy quantities, with the consequent loss of
predictivity. A way out is to reduce the symmetry content: in refs. [36, 38] the non-Abelian part
of the U(
3
) symmetry associated to the RH neutrinos was substituted for a simpler SO(
3
)
plus the hypothesis of CP conservation; in ref. [40], instead, it was identi ed with the one
of the lepton SU(
2
)L doublets, thus considering a vectorial SU(
3
)V
avour symmetry. Both
approaches allow to reduce the number of spurions to two, restoring the predictivity of the
models: the e ects of any
avour changing e ective operator can be described in terms
of lepton masses and PMNS parameters [36{40, 60, 61]. An updated phenomenological
analysis of these two di erent MLFV realisations has been recently presented in ref. [62].
A fundamental distinction between them is that the CP conservation hypothesis of the
SO(
3
)
CP version is disfavoured by the recent indication of a CP non-conserving Dirac
phase in the PNMS matrix [63{69].
Leptogenesis in the MLFV context has already been investigated in ref. [70] (see also
refs. [71{74]), considering the SO(
3
)
CP version: in order to guarantee a leptonic source
of CP violation necessary to explain the measured BAU, the CP conservation hypothesis
has been relaxed; in consequence, the precise prediction of avour e ects at low-energy
in terms of lepton masses, mixing and phases has been lost. The aim of this paper is to
investigate Leptogenesis in the SU(
3
)V MLFV version introduced in ref. [40], where no
additional hypothesis on CP is made and the present indication for the Dirac CP phase of
the PMNS can be ful lled.
The rest of the paper is structured as follows. The SU(
3
)V MLFV scenario under
consideration is described in Sect 2. The Leptogenesis CP asymmetry parameter " and
the Boltzmann equations are discussed in section 3. The numerical results are presented
in section 4, showing that a correct value for the BAU is achieved only in a part of the
allowed parameter space, testable with (non-)observation of the neutrinoless double beta
decay. Concluding remarks can be found in section 5.
2
The minimal lepton
avour violation with vectorial SU(
3
)V
The use of avour symmetries to explain the avour puzzle in the SM goes back to 1978,
when Froggatt and Nielsen [75] rst introduced a single U(
1
) factor to describe the quark
mass hierarchies and the CKM mixing matrix. Subsequent analyses also included the lepton
sector [76{81], where however a larger freedom is present due to the lack of knowledge of
some neutrino parameters. At the beginning of this century, the use of avour discrete
{ 4 {
symmetries became very popular due to the high predictive power in the lepton sector of
this kind of models [82{85]. These constructions have been later extended to the quark
sector, attempting to provide a uni ed explanation of the avour puzzle [86{90, 90{99], and
they have been shown to be contexts where
avour violating processes are under control
with new physics at the TeV scale [100{108]. Only in 2011, with the discovery of a
nonvanishing and relatively large leptonic reactor angle [109{113], strong doubts raised on the
goodness of non-Abelian discrete models to describe Nature.
In this panorama, the idea of MFV1 experienced a new revival of interest: this context
is more predictive than models based on the Froggatt-Nielsen U(
1
) and escapes from the
rigidity of the discrete constructions. This section will summarise the main aspects of the
MLFV scenario presented in ref. [40], xing at the same time the notation used throughout
this paper.
Considering the SM spectrum supplemented with three RH neutrinos, the avour
symmetry characterising the SU(
3
)V MLFV scenario is GF
NA
GFA where
GF
NA
A
GF
SU(
3
)V
U(
1
)Y
SU(
3
)eR
U(
1
)L
U(
1
)eR :
(2.1)
A
GF
(2.2)
(2.3)
(2.4)
The distinction between Abelian and non-Abelian terms re ects the fact that the
nonAbelian symmetry factors deal exclusively with the inter-generation hierarchies [41{44],
while the Abelian ones may explain the hierarchies between the third generation fermions,
such as the ratio m =mt. The choice of GFA in eq. (2.1) is the result of using the freedom of
rearranging the U(
1
) factors in order to identify the hypercharge, the Lepton number and
transformations that act globally on the RH charged lepton elds only.
with the lepton eld transformations under GF
NA
U(
1
)L
The part of the Lagrangian containing the kinetic terms is invariant under GF
the lepton masses. The Type I Seesaw Lagrangian, which can be written as [40]
L = e`L YeeR + `L eY
R +
1
2 L RcYM R + h.c. ;
describes the light active neutrino masses at low-energy through the so-called Weinberg
operator [119],
O5 =
1
2
`L ~ Y
YM1 Y T
L
~T `c
L
+ h.c. ;
where e
i 2 , Ye, Y and YM are 3
3 matrices in the avour space, L is the scale
of lepton number violation and e is a constant that will be associated to the breaking of
the U(
1
)eR symmetry. By the rst Shur's lemma, as `L and R transform as triplets under
1Despite being so predictive, the MFV only describes masses and mixings, but does not explain their
respect can be found in refs. [41{44] (see also refs. [114{118]).
the two parameters e and L to be spurion elds, i.e. non-dynamical elds that transform
non-trivially under GF
NA
GFA. Selecting the spurion transformations under GF
NA as
and under U(
1
)L
ues, breaking explicitly the avour symmetry: in the charged lepton mass basis,
Y^e =
L are respectively a dimensionless quantity and a mass. Notice that the
same symbols have been used for the couplings in eq. (2.3), for the spurions in eqs. (2.5)
and (2.6), and for their background values in eq. (2.7): it will be clear which is the meaning
associated to each symbol in the formulae that follow.
An estimate of e and of L follows by assuming that the largest eigenvalues of Ye and
of YM are . 1:2 then
where
m2atm
of order L.
2:5
10 3 eV2 [67, 68] is the atmospheric squared mass di erence of the
light active neutrinos and the \&" symbol re ects the fact that the absolute neutrino mass
scale is still unknown. Within this setup, the expected mass scale of the RH neutrinos is
In the spirit of the MLFV, any non-renormalisable operator can be made invariant
under the
avour symmetry by inserting suitable combinations of the spurions.
Once
the latter acquire background values, the strength of each operator gets suppressed by
a combination of lepton masses and PMNS parameters. These extra suppressions allow
to predict the rates for rare radiative lepton decays and lepton conversion in nuclei in
2Considering values larger than 1 would imply that multiple products of Yukawa spurions would be
more relevant than the single spurions themselves, and then they should be treated in a non-perturbative
approach [47].
{ 6 {
agreement with present data with a new physics scale that suppresses the e ective operators
as low as the TeV (see ref. [62] for a recent update).
Spurion insertions can be introduced not only in e ective operators, but also in the
renormalisable terms of the Lagrangian.3 In particular, the introduction of spurions in the
Dirac Yukawa term will be shown to be necessary in order to achieve successful
Leptogenesis. Considering only the most relevant contributions, the Dirac Yukawa term can be
written as
`L e 1 + c1Y^eY^ey + c2YMy YM
R ;
where c1;2 are dimensionless real parameters that are taken to be smaller than 1 in order
to enforce a perturbative approach.4
eq. (2.7) holds in rst approximation.
2.1
A suitable basis for leptogenesis
The explicit computation of the " parameter that controls the amount of CP asymmetry
generated in the RH neutrino decays is typically performed in the mass basis for charged
leptons and for RH neutrinos. The mass Lagrangian in this basis reads
Within this hypothesis, the expression for YM in
L = e`L Y^eeR + `L e
R +
1
2 L RcY^M R + h.c. ;
where
is the Dirac neutrino Yukawa in this basis. Considering the background values of
the spurions in eq. (2.7),
reads
(2.9)
(2.10)
(2.11)
(2.12)
where Y^e is de ned in eq. (2.7), while
= U 1 + c1U yY^e2U + c2Y^M2
;
^
The two parameters c1 and c2 control the complex contributions coming from the PMNS
matrix and the real contributions coming from the diagonal RH neutrino mass matrix,
respectively. They are expected to be of the same order of magnitude and they will be
taken equal to each other in what follows in order to simplify the study of the parameter
space. It will be shown a posteriori that relaxing this condition has not relevant impact
on the results as far as they are taken of the same order of magnitude.
3Some operators that are non-renormalisable in the description considered here appear in the list of the
renormalisable ones if a non-SM Higgs eld is considered, as described in the so-called Higgs E ective Field
Theory [120{135]. As shown in refs. [124, 126, 127, 133, 136{140], a di erent phenomenology is expected
with a non-SM Higgs in the spectrum. In the present paper, however, the standard formulation with a
SU(
2
)L-doublet Higgs is retained.
4In ref. [71], considering the SO(
3
)
CP version of MLFV, the equivalent of the coe cients c1;2 have
been shown to be generated by radiative corrections during the evolution of the Lagrangian parameters.
{ 7 {
The relevance of the spurion insertions becomes evident computing the value of three
speci c weak-base invariants [20], related to the CP violation responsible for Leptogenesis:
I1 =Im
I2 =Im
I3 =Im
Tr h y Y^M3 T
Tr h y Y^M5 T
Tr h y Y^M5 T
^ i
YM
^ i
YM
Y^M3 i
three invariants together with the parameter " would vanish.
It is straightforward to show that the three invariants depend on the combinations
was taken without the spurions insertions, then
= U and the
Baryogenesis trough leptogenesis
The prediction for the baryon asymmetry in the Universe requires to compute the CP
asymmetry parameter " and to take into consideration its evolution during the expansion
of the Universe, which depends on the interactions that are in thermal equilibrium at
di erent temperatures. With this respect, the value of the RH neutrino mass scale
L is
a fundamental parameter as it identi es di erent avour regimes [141{147]: the lower L
is, the more relevant the avour composition of the charged leptons produced in the RH
neutrino decays is. For the SU(
3
)V MLFV framework, L & 1014 GeV and it corresponds
to the so-called un avoured regime, where the charged lepton avour does not play any
role. Indeed, the only relevant interactions at these energies are the Yukawa ones, which
induce RH neutrino decays, and the gauge ones that are
avour blind: lepton and
antilepton quantum states propagate coherently between the production in decays and the
later absorption from inverse decays.
In addition, the scale
L identi es the reheating temperature necessary for the
thermal production of the RH neutrinos [148, 149]: once the temperature drops below Ma,
the thermal production of the corresponding RH neutrino Na becomes irrelevant. This
allows to identify a lower bound on the reheating temperature at about 1013 14 GeV in
the MLFV scenario under consideration. The usually quoted upper bound of 106 10 GeV
does not apply as it is exclusively connected to the so-called gravitino problem in
supersymmetry [150{152].
Besides
L, the splitting between the RH neutrino masses is also relevant: when the
spectrum is highly hierarchical then the asymmetries produced by the heaviest states
are typically (partially) washed out by the inverse decay of the lightest states (i.e.
`L (`L ) +
(i.e. `L +
( ) !
Ra) and by the 2 $ 2 scattering mediated by the lightest states
$ `L + ); when instead the spectrum is degenerate, a resonance in the
decay rate is present [153{160], which, however, is diluted due to the washout e ects of all the
three RH neutrinos. In the framework under consideration, depending on the mass of the
lightest active neutrino, the spectrum varies from hierarchical to degenerate and therefore
the computation of B is not straightforward. In particular, when the heavier RH neutrinos
also contribute to the nal asymmetry, the avour composition of the three RH neutrinos is
{ 8 {
relevant and need to be taken into consideration [29, 141, 161{163]: part of the asymmetry
generated by a heavier RH neutrino may escape the washout from a lighter one; moreover,
part of the nal asymmetry may not come from the production in the RH neutrino
decays, but from the dilution e ects. The density matrix formalism [141, 143, 163{166] (see
ref. [167] for an alternative avour-covariant formalism) turns out to be extremely e ective
in these cases, and thus for the MLFV framework under discussion: it allows to
calculate the asymmetry not only in the well de nite regimes with a hierarchical or degenerate
RH neutrino spectrum, but also in the intermediate cases, describing together the lepton
quantum states and the thermal bath.
In the rest of this section, the density matrix approach will be adopted following
xing the notation and illustrating the procedure to follow, while in the next
section the results of the numerical simulation will be presented. In the present analysis
several contributions will not be considered, as their impact is not relevant for the results
presented here: they are due to
L = 1 scatterings [168{171], thermal corrections [148,
172], momentum dependence [171, 173], and quantum kinetic e ects [174{177].
The baryon-to-photon number ratio at recombination, whose best experimental
determination is reported in eq. (1.2), can be written in terms of the nal B
L asymmetry
density NB L as
f
f
NB L
B = cs N rec ' 0:0096 NB L
;
with cs = 28=79 de ned in eq. (1.3) for NF = 3, and N rec
at recombination.
The nal B
L asymmetry results from the sum of the asymmetries generated by the
three RH neutrinos, in case partially washed out by the inverse decays [141, 178]. It can
be calculated solving the following system of four di erential equations:
' 37 the photon number density
d(NB L)
dz
dNNa =
dz
= "(a)Da[z] NNa
Da[z] NNa
N Neqa
N Neqa
Wa[z] nP(a)0; NB L
o
(a = 1; 2; 3) :
equilibrium, that is N Neqa [z
xa = Ma2=Ml2ight, is given by [149, 163]
The parameter z is the ratio between the lightest RH neutrino mass Mlight and the
temperature of the bath, i.e. z
Mlight=T . NX is any particle number or asymmetry X calculated
in a portion of co-moving volume containing one RH neutrino in ultra-relativistic thermal
1] = 1. The expression for N Neqa [za] at a za
pxaz, with
f
2
(3.1)
(3.2)
(3.3)
(3.4)
(3.5)
where Kn[za] is a modi ed Bessel function, satisfying to
The Da[z] terms are the RH neutrino decay factors [179]
N Neqa [za] =
21 za2 K2[za] ;
za2 y00 + za y0
za2 + n2 y = 0 :
where K1[za] is also a modi ed Bessel function, and H[za] is the Hubble expansion rate of
the Universe given by
where g? = gSM = 106:75 is the total number of degrees of freedom and MP l = 1:22
1019 GeV the Planck mass. The second expression on the right-hand side of eq. (3.5)
contains the total decay parameters Ka that measure the strength of the washout: they
are de ned as the ratio between the total decay widths of the RH neutrinos calculated at
a temperature much smaller than Ma and the Hubble parameter at T = Ma, when the RH
neutrinos start to become non-relativistic: explicitly,
where the total decay rates aD [za] read [180]
1 the RH neutrinos decay and inverse-decay many times before entering the
nonrelativistic regime: in consequence their abundance is close to the equilibrium distribution
and this case is dubbed strong washout regime. On the other side, for Ka
1, called weak
washout regime, the majority of the RH neutrinos decay completely out-of-equilibrium,
already in the non-relativist regime, and therefore their equilibrium abundance is
exponentially suppressed by the Boltzmann factor. Introducing the notation of the so-called
e ective washout parameter [169] and of the equilibrium neutrino mass [181{183],
mea
m? =
2
v2 ( y )aa ;
Ma
16 5=2pg? v
2
p
the total decay parameter can be written as
The Wa[z] terms are the washout factors due to inverse decays [148, 180, 184] and
L = 2 processes [148, 180, 184], which provide the two relevant e ects for these values of
the RH neutrino masses:
where the two factors are de ned as
Ka = mea :
m?
Wa[z]
WaID[z] +
Wa[z] ;
1
4
WaID[z] =
Ka K1[za] za3
Wa[z] ' z2 Ma me2a
a
for
za
1 ;
with
=
p
3 5MP l
1:202 the Apery constant. The inverse decay processes are relevant when
they are in equilibrium, i.e. Wa[z] & 1, and this occurs only in the strong washout regime
for Ka > 3. Instead, in the weak washout regime, Wa[z] < 1 and the inverse decays are
always irrelevant. On the other side, the
L = 2 processes have a relevant e ect only for
z & z
1, where z is determined by
The P
(a)0 factors are the avour projectors along the `a direction de ned by
WaID[z ] =
me =0:51 MeV ;
m
m
=105:66 MeV ;
=1776:86
0:12 MeV ;
(3.15)
HJEP07(218)36
(3.14)
(3.16)
Ma2
M 2
b
(3.17)
(4.1)
terms are considered.
(a) =
16 ( y )aa b6=a
i
Ma2
h
X
h
Ma2 Ma2
a b
Mb2
y
M 2 2 + (Ma a + Mb b)
b
2
a b
#
1 +
ba
y
where
is the Dirac Yukawa in eq. (2.11). The su x \0" indicates that only the leading
Finally, the avoured CP asymmetry parameters "(a) are given by [153{156, 156{160]
y
i Mb
ab Ma
Mb2
Ma2
1 +
ln 1 +
a b
y
ab
a b
ba
i
Ma2 Ma2
Mb2
Ma2
M 2 2 + (Ma a + Mb b)
b
2
)
;
where the Kadano -Bayn regulator [185], that is the term in the denominator containing
the RH neutrino decay rates
a, plays an important role when the spectrum is almost
degenerate. Di erent regulators can be considered, depending on the formalism chosen:
the one used in the previous expression prevents the arising of any divergence in the
doub, which instead occurs within the classical
bly degenerate limit Ma ! Mb and
a !
Boltzmann approach.
4
Numerical analysis
This section contains the results of the numerical analysis rst focussing on the baryon
asymmetry and then on the neutrinoless double beta decay.
The input data used are the PDG values for the charged lepton masses [186]
columns refer to the NO and IO, respectively. The notation chosen is
ms2ol
m22
m21 and
m2atm
m23
m21 for NO and
m2atm
m22
m23 for IO. The errors reported correspond to
the 1 uncertainties.
where the electron and muon masses are shown without errors as the sensitivities are
completely negligible, and the results of the neutrino oscillation t from ref. [67] (see also
refs. [68, 69]) reported in table 1. In the analysis that follows, all these input parameters
have been taken at their central values.
the PDG parametrisation of the PMNS matrix,
U = R23( 23) R13( 13; C`P) R12( 12) P ;
where Rij is a 3
3 rotation in the avour space in the ij sector of an angle ij and P is
the diagonal matrix containing the Majorana CP phases de ned by
P = diag 1; ei 221 ; ei 231
:
0:02195+00::0000007754
0:02212+00::0000007743
0:3070+:001:0213
0:572+00::002218
281+3303
7:40+00::2210
2:483+00::003345
neutrino mass is presently unknown. Moreover, it is still an open issue the ordering of the
neutrino mass eigenstates: the so-called Normal Ordering (NO) refers to the case when
m 1 < m 2
m 3 while the Inverse Ordering (IO) to the case when m 3
m 1 < m 2 .
The labelling of the three i is determined by the avour content of each mass eigenstate:
1 is the state with the largest contamination of e
; 2 is the one with an almost equally
composition of the three avours; 3 is the one almost exclusively de ned as a equal mixture
of
and
. The diagonal active neutrino mass matrix can thus be written in terms of
(4.2)
(4.3)
0 q
0
m^ IO = BB
the lightest neutrino mass as follows: for the NO and IO respectively,
0
0
0
0
q
ms2ol
0
0
To match with the notation typically adopted in Leptogenesis, a di erent convention
is chosen for the labelling of the RH neutrino mass eigenstates. For both NO and IO,
N1 always refers to the lightest eigenstate, N2 to the next to lightest and the N3 to the
heaviest. In consequence, Y^M in eq. (2.12) takes a di erent de nition in terms of the
three RH neutrino masses depending on the ordering of the spectrum: for the NO and IO
respectively,
LY^MNO
LY^ MIO
diag(M3; M2; M1)
diag(M2; M1; M3) :
(4.4)
(4.5)
The lepton number violation scale L, the spurion background value Y^M and the active
neutrino masses are linked together by eq. (2.12). In consequence it is possible to identify
a range of values for the lightest neutrino mass, given a value for the scale
L and requiring
that the largest entry of Y^M is of order 1, according to the MLFV construction illustrated in
section 2. Figure 1 shows the pro les of the RH neutrino masses as a function of the lightest
active neutrino mass mlight for a NO spectrum. The plot for the IO case is very similar:
the only di erence is that the line corresponding to the next-to-lightest RH neutrino (in
red) almost overlaps with the one of the lightest (in blue). The horizontal lines represent
di erent values for the
L scale, L = 1015; 1016; 1017 GeV, and their crossing with the
line of the heaviest RH neutrino mass (in green) identi es the lowest value that mlight can
take satisfying (Y^M )ii
1.
Figure 1 shows that the lower bound on
L reported in eq. (2.8) corresponds to the
lightest RH neutrino line (in blue) for mlight . 0:03 eV. An upper bound on
L can be
taken, in full generality, to be at the Planck scale. However, such a large L is not consistent
with the hypothesis of thermal production of RH neutrinos, as the temperature of the
Universe should be at least of the same order of magnitude as their masses. In the numerical
analysis that follows, the lepton number violation scale is taken at
corresponding heaviest RH neutrino mass satis es M3 < 1016 GeV and the range of values
for the lightest active neutrino mass is mlight 2 [0:003; 0:2] eV. In consequence, as shown
in
gure 1, all the three RH neutrinos may contribute to the baryon asymmetry. Further
discussion on the maximal temperature of the Universe and on the thermal production of
L = 1016 GeV: the
the RH neutrinos will follow at the end of next section.
101140-4
0.010
mlight (eV)
mlight. The blue (red) [green] continuous line corresponds to the lightest (next-to-lightest) [heaviest]
RH neutrino. The horizontal lines represent di erent values for the lepton number violation scale:
the dashed one refers to
= 1015 GeV, while the dotted to
= 1016 GeV, and the dot-dashed to
= 1017 GeV. The shaded areas are regions where the condition (Y^M )ii
speci c cases are illustrated for L = 1015 GeV, 1016 GeV, 1017 GeV.
1 does not hold: three
4.1
Baryon asymmetry in the Universe
This subsection is devoted to illustrate the results of the numerical analysis on the baryon
asymmetry in the Universe. Under the assumption that the reheating temperature is close
to the maximal temperature Tmax at a given instant, and solving the Boltzmann equations
in eq. (3.2) with the initial condition on z = Mlightest=Tmax & 0:06, the lepton asymmetry
due to the outof- equilibrium decay of the three RH neutrinos is partially washed out by
inverse decays and
L = 2 processes."
Figure 2 shows the pro les of WaID (continuos lines) and
Wa (dashed lines) as a
function of za: the value for za at which continuos and dashed lines cross is z
10 and
it corresponds to the temperature at which the washout due to inverse decays starts to
be less relevant than the dilution e ect due to the
L = 2 processes. The
Wa lines
start from za = 5, satisfying the condition za
1 as discussed below eq. (3.14). The
pro les in
gure 2 correspond to a speci c choice for the lepton number violation scale,
L = 1016 GeV, the lightest active neutrino mass, mlight = 0:003 eV, and the coe cients
c1 = c2 = 0:01, and it refers to the NO spectrum. Considering the IO spectrum, the main
di erence resides in that the lines corresponding to the lightest and the next-to-lightest
neutrinos (blue and red) almost overlap. Lowering
L, taking larger values for mlight or
taking di erent values for c1;2, but still smaller than 0:1, does not change substantially the
plot. Instead, for values c1;2
1, the washout e ects of the heaviest neutrino become more
relevant, although not changing the global picture. It follows from the fact that so large
c1;2 values induce large o -diagonal entries in
in eq. (2.11) and then the RH neutrino
avour directions have larger overlap.
The standard procedure consists in solving the Boltzmann equations with a nal value
za = +1, even if this not e ective from a computational point a view. However, it is
possible to identify a value zmax such that B is practically constant for za > zmax. The
Wa (dashed lines) as a function of za. The colours
refer to the RH neutrino mass eigenstate in the NO case: the blue (red) [green] continuous line
corresponds to the lightest (next-to-lightest) [heaviest] RH neutrino. The lepton number violation
scale is
xed to L = 1016 GeV, the lightest active neutrino mass to mlight = 0:003 eV, which
corresponds to zin = 0:06, and the coe cients c1 = c2 = 0:01.
0.100
a0.001
Δ
,
IDa10-5
10-7
10-9
1.×10-8
5.×10-9
B
η
0
-5.×10-9
-1.×10-8
0.1
1
10
100
z
B as a function of z 2 [0:06; 100] for three benchmark points in the parameter space:
the green line corresponds to 21 =
and 31 =
=4; the blue line corresponds to
21 = 7 =4 and
31 =
=2; the red line to
21 = 3 =4 and
31 = 5 =4. Continuous (dashed) lines correspond to
the NO (IO) case. The mass of the lightest active neutrino is
xed to mlight = 0:02 eV, while the
remaining input parameters have been taken at their central values as reported in table 1.
pro le of B as a function of za is shown in gure 3 for three distinct benchmark points in
the parameter space: in a good approximation zmax = 20 and this value will be adopted in
the rest of the analysis.
Moreover, gure 3 leads to the conclusion that B strongly depends on the speci c
benchmark point chosen and in consequence one may expect that only a small
percentage of points in the whole parameter space accommodates the current determination of
B. This is re ected in the scatter plots in
gure 4 that show
B as a function of the
lightest active neutrino mass, for c1 = c2 = 0:01 (details on the input parameters can
be found in the caption): values for B consistent with data, represented by the black
points in the plots, can be found for mlightest 2 [0:003; 0:04] eV in the NO case and for
a) B vs mlight for the NO case.
b) B vs mlight for the IO case.
bottom. In black the points where
B falls inside its experimental determination at 3
error.
Charged lepton masses and neutrino oscillation parameters have been taken at their central value
as in table 1, 0:01 . z < 20, c1 = c2 = 0:01 and the Majorana CP phases randomly vary in their
dominium.
mlightest 2 [0:004; 0:012] eV in the IO case.
B cannot take values in the white region
above the coloured ones, while any arbitrary smaller value is not excluded, although much
smaller ones would correspond to ne-tuned situations where cancellations between the
nal contributions to B occur.
The cuspids at mlight
0:008 eV in the NO and at mlight
0:012 eV in the IO do not
correspond to any cancellation in the "
parameters, but they arise as a numerical output
during the resolution of the Boltzmann equations.
Figure 5 shows the correlations existing between the Majorana CP phases and the
lightest active neutrino mass for the NO case in 5a and for the IO in 5b and 5c, and between
the two Majorana phases for the only IO case in 5d. The 31 phase does not manifest any
relevant correlation for the NO case. The plots suggest the presence of speci c regions of
the parameter space corresponding to a successful baryogenesis. For the NO case, one may
conclude that 21 and mlight are highly correlated and, for a given value of mlight, 21 varies
0.005
0.010
the only points that satisfy B within its experimental determination at 3 error.
only inside a small interval. This is not the case for the Majorana phases in the IO case,
where the allowed parameter space is much wider; however, the strong correlation between
them in gure 5d identi es speci c regions of values where B agrees with data at 3 .
The scatter plots shown in gures 4 and 5 are obtained with the Dirac CP phase within
its 1 con dence level, that nowadays is a large interval of
60 and
80 for the NO and
IO respectively. These results have a very mild dependence on the value of this phase: by
comparing the speci c predictions for distinct xed values of C`P, no relevant di erences
can be appreciated.
On the other hand, these plots highly depend on the values of c1 = c2: for smaller
values, for example c1 = c2 = 0:001, B is predicted to be smaller than its experimental
determination at 3 in the whole range for mlight and for both NO and IO; for larger
values, for example c1 = c2 = 0:1, points with B = 6
10 10 can be found for any value of
mlight and in both NO and IO, but no correlation between Majorana phases and mlight are
present. In the latter case, a successful description of BAU is the result of an occasional
cancellation between the contributions to B obtained solving the density matrix equations
in eq. (3.2).
The subjacent hypothesis to the numerical result shown above is that the maximal
temperature of the Universe is Tmax =
L = 1016 GeV, implying that the three RH
1.×10-9
5.×10-10
eV
5G
101
=
Tax
m
B(1.×10-10
η
eV
5101G
=
ax
m
B(1.×10-10
T
η
5G
++++
+++++
normal hierarchical case, in the center the inverse hierarchical one, and on the right the degenerate
spectrum case. The red dashed line represents the diagonal to easier drive the eye on the values
when the two computed
B have the same value. The black continuous lines delimit the 3
value
for the experimental determination of B. The two parameters c1 and c2 have been
xed at 0:01,
while two values for mlight have been considered, mlight = 0:006 eV for the
rst two plots and
mlight = 0:2 eV for the one on the right. Each point in the plots corresponds to a given random
choice of the rest of parameters.
neutrinos are thermally produced and contribute to the nal value of B. If a lower value
for Tmax is taken, then the heaviest neutrinos may not be thermally produced and their
contributions would be negligible. Figure 6 shows the e ect on the
nal value of
B of
lowering the value of Tmax, for a normal hierarchical active neutrino spectrum on the left,
for an inverse hierarchical one in the middle, and for a degenerate spectrum on the right.
The axes represent the nal value of B considering Tmax = 1016 GeV and Tmax = 1015 GeV.
The two parameters c1 and c2 have been xed at 0:01, while two values for mlight have been
considered, mlight = 0:006 eV for the rst two plots and mlight = 0:2 eV for the one on
the right. Each point in the plots corresponds to a given random choice of the rest of
parameters: in this way, it is possible to clearly identify on the nal value of B the impact
of the temperature dependence and therefore the impact of the heavier sterile neutrinos.
The diagonal red line drives the eye to tell when
B is larger for Tmax = 1016 GeV or for
Tmax = 1015 GeV: if the points align along the diagonal, then either the heaviest sterile
neutrino would not contribute to the
nal value of B or the three of them are thermally
produced even considering the lowest temperature case; if all the points cover the region
on the right of the diagonal, then the heaviest sterile neutrino does have an impact and
its contribution sums constructively with the ones from the lightest states; in the opposite
case, i.e. all the points on the left of the diagonal, its contribution sums destructively with
the other ones.
Focussing
rst on the normal hierarchical case (plot on the left), the points cover
an area along the diagonal, with a small preference for B at Tmax = 1016 GeV. Any
xed value of B at Tmax = 1016 GeV corresponds to the same values of B at Tmax =
1015 GeV, whiting a factor 2 3. Moreover, there are points where the B matches with the
experimentally allowed regiones (inside the parallel continuous black lines) and many others
where this does not occurs. This lets conclude that the value of B strongly depends on
the speci c set of parameters, especially Majorana phases, considered, as already pointed
out in
gure 4. Moreover, the value for B with Tmax = 1016 GeV, where all the three
sterile neutrinos contribute, are within a factor 2
3 similar to the ones for
Tmax = 1015 GeV, where only the lightest ones are relevant. The small preference for the
region where B with Tmax = 1016 GeV indicates that the impact of the heaviest sterile
neutrino is often not negligible and slightly increases the nal value of B. It follows that
gure 4(a), where the points show that B spans a few order of magnitudes, is a good
representative for this scenario with Tmax = 1016 15 GeV and for a hierarchical spectrum.
For the inverse hierarchical case (plot in the middle), the largest majority of the points
cover the region for B with Tmax = 1016 GeV, indicating that the heaviest sterile neutrino
typically contributes to the nal value of B, increasing its value. Moreover, only for Tmax =
1016 GeV, B reaches the experimentally allowed region, indicating that the heaviest sterile
neutrino contributions are necessary. As a result, gure 4(b) fairly represents only the case
with Tmax = 1016 GeV.
Both the plots in
degenerate spectrum.
Finally, focussing to the degenerate spectrum (plot on the right), all the points strictly
align with the diagonal, indicating that
B does not change for Tmax = 1015 GeV or
1016 GeV. This was expected because for mlight = 0:2 eV all the three sterile neutrinos have
masses below Tmax = 1015 GeV and therefore are the three of them thermally generated.
gure 4 well represent this scenario with Tmax = 1016 15 GeV for the
The plots equivalent to those in gures 4 and 5(a) for Tmax = 1015 GeV can be found
in appendix A. As can be seen, the NO case is essentially una ected by the change of the
temperature, while the IO one presents a di erence for small values of mlight where B does
not reach the experimental band.
4.2
Low-energy phenomenology
The reduction of the allowed parameter space for the Majorana phases in the c1 = c2 = 0:01
case,
gure 5, has an impact on the predictions for the neutrinoless double beta decay
e ective mass mee, de ned by
jmeej = c213 c212 m 1 + c123 s122 m 2 ei 21 + s123 m 3 ei( 31 2 C`P) ;
(4.6)
where cij and sij stand for cos ij and sin ij , respectively. The investigation on this decay
has received a strong impulse in the last decades and numerous experiments are currently
competing to probe the existence of this process, as its observation would automatically
infer that neutrinos have (at least partly) Majorana nature [187]. Table 2 reports the lower
bounds on jmeej sensitivity for near future 0 2 experiments that will be considered in the
following.
Figure 7 shows the pro le of jmeej as a function of the lightest active neutrino mass
mlight in 7a for the NO and in 7b for the IO, while as a function of the Majorana phases
in 7c for the NO and in 7d and 7e for the IO. For both the mass orderings, describing
HJEP07(218)36
Experiment
CUORE [188]
GERDA-II [189]
LUCIFER [190]
MAJORANA D. [191]
NEXT [192]
AMoRE [193]
nEXO [194]
PandaX-III [195]
SNO+ [196]
SuperNEMO [197]
Isotope
130Te
76Ge
82Se
76Ge
136Xe
100Mo
136Xe
136Xe
130Te
82Se
jmeej [ eV]
0:073
0:008
0:11
0:20
0:13
0:12
0:084
0:011
0:082
0:076
0:084
0:01
0:02
0:01
0:01
0:008
0:001
0:009
0:007
0:008
successfully the amount of BAU leaves viable only the hierarchical regime. For the NO,
gure 7a, jmeej can take values only below 0:04 eV, while a lower bound at about 4 10 4 eV
seems plausible, as con rmed in gure 7c, although the point density is poor in this region:
interestingly, it appears a region precluded for 0:0095 eV . mlight . 0:035 eV. For the IO,
gure 7b, the parameter space corresponding to
B inside its experimental determination
at 3
is con ned in a well-de ned region between 0:005 eV . mlight . 0:01 eV and
0:018 eV . jmeej . 0:05 eV.
Complementary information can be extracted in the plots with jmeej as a function of
the Majorana phases. For the NO, gure 7c, only jmeej vs
21 shows a correlation: only
values for 21 in the interval [ =8; 3 =4] leads to larger values of jmeej, while smaller
values may be described for almost any
21. For the IO, gures 7d and 7e, a correlation
between jmeej and both the Majorana phases is present and the allowed parameter space
is limited in relatively small regions.
An observation of the neutrinoless double beta decay in the present experiments, if
fully interpreted in terms of Majorana neutrino exchange, would be crucial to determine
the values of the Majorna phases for which a successful BAU occurs. Once determined the
ordering of the active neutrino mass spectrum, a larger value for jmeej would favour values
of 21 in the interval
31 in the interval
[ =8; 3 =4] for the NO and
[
=2; =2] in the IO, and values of
[ =8; ] in the only IO. The determination of the value for the lightest
active neutrino mass would help reducing these interval: if mlight is found relatively large,
then only the NO scenario would be compatible with a successful explanation of the BAU,
while the IO case would be then excluded.
HJEP07(218)36
a) jmeej vs mlight for the NO case.
b) jmeej vs mlight for the IO case.
c) jmeej vs 21 for the NO case.
d) jmeej vs 21 for the IO case.
e) jmeej vs 31 for the IO case.
function of the Majorana phases for the NO case in c) and for the IO one in d) and e).
5
Conclusions
The MFV ansatz works extraordinary well in the quark sector accommodating a huge
amount of experimental measurements. If an underlying dynamics is the reason behind
this hypothesis, then it is natural to expect a similar mechanism at work also in the lepton
sector. Two distinct versions of the MLFV can be considered when the SM spectrum is
extended by the three RH neutrinos: only if the latter transform under the same symmetry
of the lepton electroweak doublets [43], SU(
3
)`L
SU(
3
)NR ! SU(
3
)V , then violation of
the CP symmetry can be described according to the recent experimental indication.
The presence of non-vanishing CP violating phases in the leptonic mixing may be
the missing ingredient in the SM to successfully describe the baryon asymmetry in the
Universe. In this paper, baryogenesis through Leptogenesis has been considered for the
rst time within the context of the SU(
3
)V MLFV framework, resulting in a very predictive
setup where the " parameter that describes the amount of CP violation in Leptogenesis
only depends on low-energy parameters: charged lepton and active neutrino masses, PMNS
parameters and two parameters of the low-energy e ective description.
Fixing the two e ective parameters at their natural value 0:01, when a baryon to
photon ratio today agrees with its experimental determination at 3
then correlations
between the Majorana phases and the lightest active neutrino mass arise. The latter can
be analysed considering the impact in the neutrinoless double beta decay observable: only
selected regions of the whole jmeej vs mlight parameter space correspond to values that are
consistent with a successful baryogenesis. In the NO case, only upper bounds on jmeej and
mlight can be identi ed: jmeej . 0:04 eV and mlight . 0:04 eV. Instead, in the IO case,
jmeej can take values only inside a much smaller interval [0:02; 0:05] eV corresponding to
a narrow interval for mlight that is [0:004; 0:012] eV. These regions will be tested only in
several years as the sensitivity required is of the order of that one expected by the nEXO
experiment.
Acknowledgments
The authors warmly thank Pasquale di Bari, Mattias Blennow, Enrique Fernandez
Mart nez, Pilar Hernandez, Olga Mena and Nuria Rius for discussions and suggestions.
They also thank the HPC-Hydra cluster at IFT. L.M. thanks the department of Physics
and Astronomy of the Universita degli Studi di Padova and the Fermilab Theory Division
for hospitality during the writing up of the paper. L.M. acknowledges partial
nancial
support by the Spanish MINECO through the \Ramon y Cajal" programme
(RYC-201517173), by the European Union's Horizon 2020 research and innovation programme under
the Marie Sklodowska-Curie grant agreements No 690575 and No 674896, and by the
Spanish \Agencia Estatal de Investigacion" (AEI) and the EU \Fondo Europeo de Desarrollo
Regional" (FEDER) through the project FPA2016-78645-P, and through the Centro de
excelencia Severo Ochoa Program under grant SEV-2016-0597.
A
Lowering Tmax
Lowering Tmax implies that the heaviest sterile neutrinos may not be thermally produced,
preventing in this way their contributions to the
nal value of B. Figure 8 shows the
results for Tmax = 1015 GeV. Comparing these plots with those in
gure 4, the NO case
is essentially una ected by this change, as also con rmed by the correlation plot showing
the behaviour of the Majorana phase
21 vs mlight when compared with the equivalent
plot in gure 5(a). The IO case presents a sustancial di erence, as B does not reach the
experimental band for small values of mlight.
a) B vs mlight for the NO case.
b) B vs mlight for the IO case.
c) 21 vs mlight for the NO case.
Figure 8.
B as a function of the lightest neutrino mass for the NO on the top and IO in the
middle. In black the points where
B falls inside its experimental determination at 3
error. The
correlation between
21 and mlight in the bottom for the NO case only: the points corresponds to
the black ones in the rst plot with
B inside its experimental value. Charged lepton masses and
neutrino oscillation parameters have been taken at their central value as in table 1, 0:01 . z < 20,
c1 = c2 = 0:01 and the Majorana CP phases randomly vary in their dominium.
Open Access.
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