Consequences of minimal seesaw with complex μτ antisymmetry of neutrinos
HJE
Consequences of minimal seesaw with complex antisymmetry of neutrinos
Rome Samanta 0 2
Probir Roy 1 2
Ambar Ghosal 0 2
0 Astroparticle Physics and Cosmology Division, Saha Institute of Nuclear Physics , HBNI
1 Center for Astroparticle Physics and Space Science, Bose Institute
2 Kolkata 700064 , India
We propose a complex extension of trino Majorana matrix M . The latter can be realized for the Lagrangian by appropriate CP transformations on the neutrino elds. The resultant form of M is shown to be simply related to that with a complex (CP) extension of cal phenomenological consequences, though their group theoretic origins are quite di erent. We investigate those consequences in detail for the minimal seesaw induced by two strongly hierarchical rightchiral neutrinos N1 and N2 with the result that the Dirac phase is maximal while the two Majorana phases are either 0 or . We further provide an uptodate discussion of the 0 process visavis ongoing and forthcoming experiments. Finally, a thorough treatment is given of baryogenesis via leptogenesis in this scenario, primarily with the assumption that the lepton asymmetry produced by the decays of N1 only matters here with the asymmetry produced by N2 being washed out. Tight upper and lower bounds on the mass of N1 are obtained from the constraint of obtaining the correct observed range of the baryon asymmetry parameter and the role played by N2 is elucidated thereafter. The mildly hierarchical rightchiral neutrino case (including the quasidegenerate possibility) is discussed in an appendix.
Beyond Standard Model; CP violation; Neutrino Physics

1 Introduction
2
3
4
5
6
7
8
Complex extension of antisymmetry
Neutrino mixing angles and phases from M CP
A
Origin of neutrino masses from a minimal seesaw
Neutrinoless double beta decay
Baryogenesis via leptogenesis
Numerical analysis: methodology and discussion
Concluding comments and discussion
A Discussion of the case with mildly hierarchical RH neutrinos
masses. The atmospheric mixing angle is now pinned around its maximal value of =4 and
the solar mixing angle around the tribimaximal value while the reactor mixing angle is
known to be signi cantly nonzero and close to 80. The current trend of the data [4] suggests
that the Dirac CP phase could be close to 3 =2 but a de nitive statement is yet to emerge.
A speci c prediction on the value of the latter will be very welcome. It is not known yet
whether the light neutrinos are Dirac or Majorana in nature while relentless searches for the
decisive neutrinoless double
decay signal continue. For the latter case the two Majorana
phases of the neutrinos also need to be predicted. Light Majorana neutrino masses can
be generated by the seesaw mechanism [5{8] and a minimal version [9{21] with just two
heavy rightchiral (RH) neutrinos seems especially attractive. Further, the formulation of
a viable scheme of baryogenesis via leptogenesis within this scenario is a challenging task.
There has also been a substantial amount of work with discrete
avor symmetries of the
light neutrino Majorana mass matrix: speci cally real
permutation symmetry [22{31]
and its complex (CP) extension [32{34] as well as real
permutation antisymmetry [35]
but not the complex (CP) extension of that. This last mentioned topic will be the subject
of our attention in this paper with the aim of predicting the neutrino CP phases.
{ 1 {
with
(M
LCl = C LlT and the subscripts l; m spanning the lepton avor indices e, ,
the subscript L denotes leftchiral neutrino
elds. M
is a complex symmetric matrix
6
= M
= M T ) in lepton
avor space. It can be put into a diagonal form by a
similarity transformation with a unitary matrix U :
U T M U = M d
diag (m1; m2; m3):
Here mi (i = 1; 2; 3) are real and we assume that mi
0. We work in the basis in which
charged leptons are mass diagonal. We are motivated by a avorbased model constructed
by Mohapatra and Nishi [42], which could accommodate a diagonal charged lepton mass
matrix as well as a CPtransformed
interchange symmetry. Now we can relate U to the
P M N S mixing matrix UP MNS:
U = P UP MNS
P B s12c23
0
The neutrino mass terms in the Lagrangian density read
21 LCl(M )lm Lm + h:c:
GT M G = M ;
G = B0 0 1C :
0
A
GT M G = M :
Ll ! iGlm 0 LCm;
{ 2 {
(1.1)
while
(1.2)
where P = diag (ei 1 ; ei 2 ei 3 ) is an unphysical diagonal phase matrix and cij
sij
sin ij with the mixing angles ij = [0; =2]. We work within the PDG convention [36]
but denote our Majorana phases by
and . CPviolation enters through nontrivial values
of the Dirac phase
and of the Majorana phases ;
with ; ;
= [0; 2 ].
Real
symmetry [22{31] for M implies that
where G is a generator of a Z2 symmetry e ecting
interchange. In the neutrino avor
space G has the form
A substantial amount of phenomenological work has been done following the consequences
of (1.4). Additionally, its possible group theoretic origin from a more fundamental
symmetry such as A4 have been investigated [37{40]. However, this avor symmetry leads to the
prediction that 13 = 0 which has now been excluded at more than 5.2 [41]. A way out
was proposed [32{34] in terms of its complex (CP) extension (CP ) with the postulate
The above can be realized as a Lagrangian symmetry by means of a CP transformation on
the neutrino elds as
where l; m are avor indices and
of (1.6) have been investigated in ref. [42].
Lm = C LTm. Detailed phenomenological consequences
C
Let us move on to real permutation antisymmetry [35] which proposes that Note that the antisymmetry condition in (1.8) can written as a symmetry condition
of work has earlier been done using [43, 44] the real
antisymmetry idea  including
its application to the neutrino masses and mixing as well as its possible group theoretic
origin from a more fundamental avor symmetry such as A5. However, the major
phenomenological problem with exact real
antisymmetry is that it leads to a maximal solar
neutrino mixing angle 12 =
=4 as well as two degenerate light neutrinos  in con ict
with experiment [45]. Perturbative modi cations, in attempts to address these problems,
unfortunately lead to a proliferation of extra unknown parameters. It is therefore highly
desirable to propose an extension of this symmetry which is exact and therefore has the
beauty of minimizing the number of input parameters. This is what we aim to do in
this paper by proposing a complex (CP) extension of
avor antisymmetry CP
working out its various phenomenological implications. Complex extensions of
A and
symmetry [32{34] and scaling symmetry [46{48] as well as their consequences have been worked
out earlier. That is the direction of our thrust here for
antisymmetry.
We consider a complex (CP) extension of antisymmetry (CP
A) in the neutrino
mass matrix. We show that this extension leads to a form of M
which is very simply
related to that of M for the CP
case. Moreover, this form allows neutrino mixing angles
that are perfectly compatible with experiment both for a normal and for an inverted mass
ordering. Additionally, speci c statements can be made on CP violation in the neutrino
sector. The Majorana phases
and
have to be 0 or
while Dirac CP violation has to
be maximal with the phase
being either =2 or 3 =2. Further, reasonably nondegenerate
values for the three neutrino masses can be generated by incorporating the minimal seesaw
mechanism [9{21] implemented through two heavy rightchiral neutrinos NR` (` = 1; 2)
with a 2
2 Majorana matrix MR. (In case there is a third heavy Majorana neutrino, that
is assumed to be much heavier and hence totally decoupled). De nitive predictions can be
made on neutrinoless double beta decay for both types of mass ordering. Finally, a realistic
scenario of baryogenesis via leptogenesis can be drawn and an acceptable value of the baryon
asymmetry parameter YB can be derived. Though the phenomenological consequences of
with some new results related to the scenario of baryogenesis via leptogenesis will be useful.
The new features in our work are (i) the demonstration that M CP
A and M CP
have identical phenomenological consequences despite there origin from di erent residual
symmetries (Z4 and Z2 respectively) and (ii) the use of the minimal seesaw with two heavy
righthanded neutrinos (and consequently one massless left handed neutrino) to explore
M CP
and M CP A
are identical, we feel that an uptodate detailed discussion of these along
{ 3 {
those consequences  in particular
decay and baryogenesis via leptogenesis. Let
us highlight here what we propose to do in this paper. We plan to discuss the complex
antisymmetry which has been analyzed so far in literature with
its perturbative modi cations only. Then we shall show how the resultant M CP A is
simply related to M CP
 the neutrino Majorana mass matrix from the complex (CP)
extension of
symmetry  with identical phenomenological consequences despite the
fact that their respective real components have almost entirely di erent predictions. We
further emphasize the fact that CP
and CP
A are implemented with di erent residual
symmetry generators, namely Z2 and Z4 respectively. Thus the corresponding high energy
theory for these residual CP symmetries would likely be di erent. We then work out the
consequences of CP
A in the framework of a minimal seesaw which leads to a vanishing
value of one of the light neutrino masses and a very constraint range of the sum of the light
neutrino masses as well. We also make an uptodate comparison of our conclusions on
0
decay with ongoing and forthcoming searches. We shall do a full parameter scan of the
3
2 Dirac mass matrix mD in the minimal seesaw scenario using the uptodate neutrino
oscillation 3
global t data. This in turn will lead us to perform a detailed computation
related to the process baryogenesis via leptogenesis in our work which will result in new
interesting upper and lower bounds on the mass of N1. We shall also stress that these
bounds could be erased if we consider a mildly hierarchical RH neutrino spectrum. We
shall discuss the e ect of N2 on the
nal baryon asymmetry YB, in particular on the
obtained upper and lower bounds on M1 from the standard N1 decay scenario.
The rest of the paper is organized as follows. In section 2 we explain the above
mentioned complex extension CP
A. section 3 contains a discussion of how the neutrino
mixing angles and CP violating phases originate from CP
A. In section 4 we discuss the
origin of the neutrino masses from the minimal seesaw mechanism. The phenomenon of
neutrinoless double beta decay is treated in section 5. In section 6 we discuss baryogenesis
via leptogenesis. Constraints on our model parameter space from all these phenomena are
derived by numerical analysis in section 7. The nal section 8 contains a discussion of our
conclusions. In an appendix we discuss what happens to our results if the right handed
neutrinos are mildly hierarchical or quasidegenerate in mass.
2
Complex extension of
antisymmetry
We propose a complex extension of (1.8), namely
T M G = M :
The complex invariance condition in (2.1) can be obtained by the means of a CP
transformation [49{65] on the neutrino elds as
As we will see, since the real part of the resultant complex matrix exhibits
antisymmetry,
we call the implemented CP symmetry as a complex extended
antisymmetry or simply
Ll ! iGlm 0 LCm:
{ 4 {
complex
antisymmetry. This complex
antisymmetry CP
to be broken in the charged lepton sector. Given that our charged lepton mass matrix M` is
diagonal, a replacement of M
by M` in (2.1) would immediately lead to the unacceptable
result m
= m . There is an additional desirable reason for breaking CP
A in M`. A
A, generated by G, needs
nonzero Dirac CP violation is equivalent to
Tr [H ; H`]3 6= 0;
where the hermitian combinations are introduced as H
common CP symmetry G in both the sectors would imply
= M yM , H` = M`yM` [66, 67]. A
GT HT G
= H ; GT H`T G
= H`:
From (2.4) it follows that Tr[H ; H`]3 = 0 which leads to sin
= 0 i.e. a vanishing Dirac
CP violation. Though this is still a possibility, it goes against the current trend of the
data [4]. The most general structure of M
that satis es the CP
A condition (2.1) can be
worked out to be
(2.3)
(2.4)
(2.5)
(2.6)
(2.7)
(2.8)
where A; D are real and B; C are complex mass dimensional quantities which are a priori
unknown. The matrix M CP A can also be written as
B1
B1 0
A
iB
iB
iC
D
iB 1
D C
A
iC
M CP :
GT (iM CP A )G = (iM CP A
) :
B iD
B 1
iD C ;
A
C
Therefore the phenomenological consequences of a complex (CP)
symmetric form of M
and a complex antisymmetric form of the same would be identical. Nevertheless, we deem
it worthwhile to give a detailed updated discussion of its phenomenological consequences
and highlight some new e ects such as the role of another heavy RH neutrino N2 on the
process of baryogenesis via leptogenesis in a standard N1leptogenesis scenario.
{ 5 {
GU
= U d~;
~
dlm =
lm:
d~ = diag (d~1; d~2; d~3);
0iUe1 iUe2 iUe3
1
0d~1Ue1 d~2Ue2 d~3Ue3;
1
BiU 1 iU 2 iU 3AC = B@dd~~11UU 11 dd~~22UU 22 dd~~33UU 33 CA
@
iU 1 iU 2 iU 3
iUe1 = d~1Ue1; iUe2 = d~2Ue2; iUe3 = d~3Ue3:
iU 1 = d~1U 1; iU 2 = d~2U 2; iU 3 = d~3U 3:
where each d~i (i = 1; 2; 3) can be +1 or
G equal to i times G as given in (1.5), namely
1. Eq. (3.1) can explicitly be written, by taking
which is equivalent to six independent equations:
Neutrino mixing angles and phases from
M CP
Eqs. (1.2) and (2.1) together imply [32{34] that
where
Let us take
(3.1)
(3.2)
(3.3)
(3.4)
(3.5)
(3.6)
(3.7)
(3.8)
(3.9)
(3.10)
(3.11)
(3.12)
(3.13)
In order to calculate the Majorana phases in a way that avoids the unphysical phases,
it is useful to construct two rephasing invariants [68{71]
By using (3.5), I1;2 can be written as free of the unphysical phases, namely
I1 = Ue1Ue2; I2 = Ue1Ue3:
I1 = d~1d~2Ue1Ue2; I2 = d~1d~3Ue1Ue3:
After equating the two di erent expressions for I1;2 in (3.7) and (3.8), we obtain
Eqs. (3.9) and (3.10) imply
Thus
I1 = c12s12c123e i 2 = d~1d~2c12s12c123ei 2 ;
I2 = c12s13c13ei(
2 ) = d~1d~3c12s13c13e i(
2 ):
e
i = d~1d~2; e2i(
=2) = d~1d~3
d~1d~2 = +1 )
=4, i.e. a maximal atmospheric mixing. Incorporating this last result,
the modulus square of the rst or the second equality in (3.6) leads after some algebra to
Given the experimentally observed nonvanishing values for all the mixing angles, (3.15)
leads to a maximal Dirac CPviolation
cos = 0 i:e:
= =2 or 3 =2:
~
cos
0
0
0
0
Taking the modulus squared of the third equality in (3.6), namely jU 3j = jU 3j, we
Origin of neutrino masses from a minimal seesaw
matrix MR. We work in a basis in which MR is real, positive and diagonal [72, 73], i.e.,
MR = diag (M1; M2), M1;2 > 0. With mD as the Dirac mass matrix, the neutrino mass
;N
Lmass = NRi(mD)i lL +
mass matrix is given by the standard seesaw relation
In this case (2.1) is satis ed through the symmetry transformation on mD as
M
=
mTDMR 1mD:
mDG =
imD;
{ 7 {
so long as MR 1 is real.
parametrized as
below:
The most general form of mD that satis es (4.3) can be
mD =
where the parameters a1;2, b1;2 and 1;2 are real.
The form of the e ective light neutrino mass matrix M
that now emerges is given
(4.4)
A :
(4.5)
(4.6)
(5.1)
In (4.5) we have introduced new real parameters x1;2 and y1;2 which are obtained by scaling
a1;2 and b1;2 with the square roots of the respective RH neutrino masses M1;2, i.e.
a1;2
pM1;2
= x1;2; pM1;2
b1;2
= y1;2:
1
T10=2
= GjMeej2jMj2me 2:
Here G is the twobody phase space factor and Mee is the (
1,1
) element of the e ective light
neutrino mass matrix M , cf. (1.1). Moreover, M is the nuclear matrix element (NME)
and me is the electron mass. Mee can be written within our convention as
Mee = c122c213m1 + s122c213m2e
i + s123m3ei( 2 ):
The lightest neutrino mass, either m1 for a normal mass ordering or m3 for an inverted
mass ordering, has to vanish since det M CP A = 0. Furthermore, one of the phases of
(say 1) can be rotated by the phase matrix P
= diag (1; ei ; e i ) with the choice
1 =
. Thus we are left with only the phase di erence 2
1 in M . We can now
rename 2
and 2 =
1 as . Without loss of generality, this is also equivalent to the choice 1 = 0
in mD. From now on we shall use this rede ned phase
for both M
and mD.
5
Neutrinoless double beta decay
The rare 0 process can arise from the following decay of a nucleus
(A; Z)
! (A; Z + 2) + 2e :
In (5.1) lepton number is violated by two units. Unlike in neutrinoful double
which is a sequence of two single
decays, nal state neutrinos are absent in the
process. The latter can go through via an appropriate neutrino loop only if the light
neutrinos have Majorana masses. Therefore any observation of such a decay will unambiguously
establish the Majorana nature of the light neutrinos. The halflife, corresponding to
decay, can be expressed as
p2ei =4(x1y1ei 1 +x2y2ei 2 ) ip2ei =4(x1y1e i 1 +x2y2e i 2 )1
Signi cant upper limits on jMeej are available from ongoing search experiments for
0 decay. KamLANDZen [74] and EXO [75] had earlier constrained this value to be
< 0:35 eV. But the most impressive upper bound till date is provided by GERDA
phaseII data [76]: Mee < 0:098 eV. As explained in section 3, we have four sets of values
for the three CP violating phases ; ; in the neutrino sector corresponding to the four
independent d~ matrices. Furthermore, we need to consider both kinds of light neutrino
mass ordering: normal and inverted. Thus we shall have eight sets of predictions for jMeej
from our modelled M . These will be detailed in our section on numerical analysis.
At this stage it may be useful to point out how (5.3) simpli es in our model for
the speci c cases of normal and inverted mass ordering subject to the condition given in
eq. (3.16). For a normal mass ordering, we have m1 = 0 and further
= 0; = 0;
= ; =
: jMeej = s412c413m22 + s143m32
2s122s123c123m2m3
= 0; = ;
= ; = 0 : jMeej = s412c413m22 + s143m32 + 2s122s123c123m2m3
1=2 (Normal);
1=2 (Normal):
(5.4)
(5.5)
Note that the value of jMeej becomes somewhat less here since the terms involving m3 are
suppressed by the powers of s13. For an inverted mass ordering, m3=0 and jMeej becomes
independent of
and . Indeed, we have
=
j
= 0 : jMeej = c123 c412m12 + s142m22 + 2c122s122m1m2
: jMeej = c123 c142m12 + s142m22
2c122s122m1m2
1=2 (Inverted);
1=2 (Inverted):
Since
m221
m232j, in this case we can assume m1
m2
two allowed values of , we have
p
j m232j. Thus, for the
= 0 : jMeej '
=
: jMeej '
q
q
j m232jc123 (Inverted);
j m232jc123[f1
2s122g2] (Inverted):
We see that jMeej for
=
is suppressed here relative to its value in the
= 0 case.
6
Baryogenesis via leptogenesis
To start with, we recall the observed range of YB = (nB
nB)=s
the ratio of baryonic
minus antibaryonic number density to the entropy density  namely
8:55
10 11 < YB < 8:77
CP violating decays from heavy Majorana neutrinos that are out of equilibrium generate
a lepton asymmetry [77{79]. The latter is later converted into a baryon asymmetry by
sphaleron transitions [80]. The appropriate part of the Lagrangian for the process can be
written as
L = i NRi ~ylL +
,
being the Higgs doublet. The possible decays of Ni
tree level, one loop self energy and one loop vertex diagrams [77] for the decay of Ni. It
has the general expression [81]
"i =
1
X
i = v=p2 (so that mD = v =p2) and xij = Mj2=Mi2. In
addition, the loop function g(xij ) has the standard expression
pxij
1
xij
g(xij ) =
+ f (xij )
f (xij ) = pxij 1
(1 + xij ) ln
1 + xij
xij
Before proceeding further in the calculation of "i in our scenario, we need to
address some important issues related to leptogenesis. For hierarchical RH neutrino masses
M2
M1 (some discussion of the mildly hierarchical RH neutrino case including
quasidegenerate masses is given later in the appendix), it can be shown that only the decays of
N1 matter for the creation of lepton asymmetry while the latter created from the heavier
neutrinos gets washed out [82] signi cantly. Therefore, in general, only "1 is the pertinent
quantity in a hierarchical leptogenesis scenario. Nevertheless, there are certain
circumstances in which the decays of N2;3 do a ect the nal baryon asymmetry [87{89].
Furthermore, avor plays an important role in the phenomenon of leptogenesis [83{86]. Assuming
the temperature scale of the process to be T
M1, the rates of the charged lepton Yukawa
interaction categorize leptogenesis into the following three categories.
1) Un avored leptogenesis : T
M1 > 1012 GeV, when all interactions with all avors
are out of equilibrium: in this case all the avors are indistinguishable; therefore the
total CP asymmetry is a sum over all avors, i.e. "1 = P "1 and the nal baryon
asymmetry YB is proportional to "1.
2)  avored leptogenesis: 109 GeV < T
M1 < 1012 GeV, when only the
avor is
in equilibrium and hence distinguishable. In this regime there are two pertinent CP
asymmetry parameters; "1 and "(
12
) = "e1 + "1 . The nal baryon asymmetry YB may
be approximated as [83]
where the washout masses m~2; and "(
12
) are de ned as
m~ 2 = M1 1 j(mD)1ej2 + j(mD)1 j2 ; m~
= M1 1j(mD)1 j2; "(
12
) =
"1 = "e1 + "1 :
X
=e;
(6.6)
(6.7)
In order to know the nature of the washout processes, it is convenient to de ne two
washout parameters K2; = m~2; =10 3 relevant to this mass regime. Further, ( m~2)
and ( m~ ) are the e ciency factors that account for the inverse decay and the lepton
number violating scattering processes while g is the number of relativistic degrees
of freedom in the thermal bath having a value g
106:75 in the SM.
3) Fully avored leptogenesis: T
M1 < 109 GeV, when in addition to the
avor, the
avor is also in equilibrium
thus all the three avors are distinguishable. Again
for the evaluation of the nal baryon asymmetry YB in this regime, we make use of the
approximate analytic formula for YB presented in ref. [83]. In the T
M1 < 109 GeV
344
537
m~
+ "1
344
537
m~
;
(6.8)
(6.9)
(6.10)
(6.11)
(6.12)
(6.13)
(6.14)
(6.15)
to be
to be
Now in our minimal seesaw scheme, assuming a speci c hierarchy of the RH neutrino
masses, namely M2=M1 ' 103, the nal YB is calculated from (6.12), (6.6) and (6.8)
where the washout masses m~
are de ned as
m~
= j(mD)1 j2 ;
M1
= e; ; :
We now focus on the calculation of the quantities related to the leptogenesis in
our model. The avor sum over
leads the rst term in the r.h.s. of (6.3) to be
proportional to Im(hij )2 and the second term to vanish. This is since
X Im[hji(mD)i (mD)j ] = Im[hjihij ] = Im[hjihji] = Imjhjij2 = 0:
g0(x12) = g(x12) + (1
x12) 1:
g0(M22=M12) =
In fact, in our model the matrix h = mDmyD is real as given by
h =
2(a21 + b21)
2(a1a2 + b1b2 cos )
2(a1a2 + b1b2 cos )
2(a22 + b22)
!
:
Therefore the avor summed CP asymmetry parameter "1 = P "1 vanishes, i.e.,
un avored leptogenesis does not occur in this complex (CP) extended
antisymmetry scheme. Using (4.4) and (6.3), the avored CP asymmetries can be calculated
"e1 = 0; "1 =
4 v2
g0(x12) (a1a2 + b1b2 cos )b1b2 sin
=
"1;
where g0(x12) is given by
It is useful to simplify (6.13) for a hierarchical RH neutrino scheme to
In our primary analysis, the e ect of the heavy neutrino (N2) on the produced nal
baryon asymmetry has been neglected with the assumption that the asymmetry produced
by the decays of N2 get washed out [82]. We now give a brief discussion on how the heavy
neutrino N2 can a ect the nal baryon asymmetry YB. As elaborated below, there are two
ways in which the e ect of N2 might arise: indirect and direct. We rst discuss the indirect
e ect. Though the neutrino oscillation data are tted with the rescaled parameters of (4.6),
in order to compute the quantities related to leptogenesis such as "1 , we need to evaluate
the parameters of the Dirac mass matrix elements. Given a set of rescaled parameters,
the latter can be generated by varying M1;2 in (4.6). It is thus interesting to see whether
the nal baryon asymmetry is a ected by the chosen mass ratios of the RH neutrinos. We
nd that the nal YB is not particularly sensitive to M2. A relook at (6.14) reminds us
that the second term is suppressed compared to the rst term, since the former is of the
order of x121. Thus, taking only the rst term of (6.14) into consideration, the
avored
CP asymmetry parameters of (6.12) can be simpli ed in terms of the rescaled parameters
of (4.6) as
"1 =
for the fully avored regime.
12
Since all rescaled parameters in (6.17) are xed by the 3
practically insensitive to the value of M2. Nevertheless, for a precise numerical
computation of the
nal baryon asymmetry, we need to take into account the e ect of the second
term in (6.14). The sensitivity of YB to the magnitude of the second term of (6.14) for
di erent mass hierarchical schemes of the RH neutrinos will be discussed in detail in the
numerical section 7.
We now turn to discuss the direct e ect of N2. We have so far focused on the lepton
asymmetry produced by the decay of the lightest of the heavy neutrinos. It is shown in
ref. [89] that, due to a decoherence e ect, the amount of lepton asymmetry, generated
by N2 decays, gets protected against N1washout. The latter therefore survives down to
the electroweak scale and contributes to the nal baryon asymmetry. For this procedure
to work out, two washout parameters
satisfy the condition
1 = h11M1 1m
1 and
2 = h22M2 1m
1
1 and
teractions destroy the coherence among the states produced by N2; hence a part of the
lepton asymmetry produced by N2 becomes blind to the N1washout and survives
orthogonal to N1states. On the other hand, a mild washout of the lepton asymmetry, produced
by N2 due to N2related interactions, is represented by the
1 condition. For such
a mild washout scenario, a sizable lepton asymmetry generated by N2 survives through
blue) color represents the parameter space for normal (inverted) mass ordering.
the N1leptogenesis phase and hence contributes to the nal baryon asymmetry. We shall
elaborate on the validity of these conditions in our model in the following section.
7
Numerical analysis: methodology and discussion
In order to check the viability of our theoretical assumptions and consequent outcomes, we
present a numerical analysis in substantial detail. Our method of analysis and organization
are as follows. First, we utilize the 3
values of the globally tted neutrino oscillation data
presented in table 2 to constrain the parameter space in terms of the rescaled parameters
de ned in (4.6). For numerical computation, we make use of the exact analytical formulae
for the light neutrino masses and mixing angles presented in ref. [90]. It is seen that, in
this complex extended
antisymmetry scheme, an appreciable region of the of parameter
space could be well tted within the 3
range of the global oscillation data (see gure 1)
for each of the mass orderings. We next discuss the predictions of the present model in
the context of the
0 experiments for both mass orderings. In order to estimate the
value of YB, we make use of these constrained rescaled parameters with a subtlety. For
the computation of YB we need to evaluate the parameters of mD (i.e., a1;2, b1;2) and
Parameters
6:93{7:96
6:93{7:96
7:37
7:37
(eV2)
2:411{2:646
2:39{2:624
2:52
2:50
Mi separately. Since we have only constrained the rescaled parameters, for a given set of
rescaled parameters, there remains a freedom to make various sets of independent choices
for the elements of mD along with Mi. Keeping this in mind, we explore two di erent
numerical ways to discuss leptogenesis and its consequent outcomes. First, we choose a
speci c hierarchical mass spectrum for the RH neutrinos: M2=M1 = 103. Then, for a xed
value of M1, we use the entire parameter space for the rescaled parameters to generate
the elements of mD which are explicitly used to compute the
nal YB. This leads to a
lower bound on M1 below which YB in the observed range cannot be generated. In another
approach, instead of taking the entire rescaled parameter space, we focus only on that
set of rescaled parameters which corresponds to a positive value of YB (the sign of YB
depends upon the rescaled parameters) and observables that lie near their best t values.
Then by varying M1, we generate the corresponding parameters of mD using (4.6). Here
we consider the same hierarchical scenario for the RH neutrinos as considered in the rst
approach. Now, for each value of M1 and the corresponding parameters of mD, we obtain
a value for the nal baryon asymmetry YB. Since YB has an observed upper and a lower
bound, we end up with an upper and a lower bound for M1 also. Finally, we provide
a numerical discussion regarding the e ects of the heavy neutrino N2 on the nal YB as
explained analytically in the previous section. We next present the numerical results of
our analysis in much more detailed and a systematic way.
As discussed in section 3, there are four sets of CP violating phases for the four
independent d~ matrices. Thus we get four di erent plots for each mass of the orderings
of the light neutrinos. In
gure 2 we present the plots of jMeej vs. the sum of the light
neutrino masses ( imi) for each mass ordering. Since the lightest neutrino mass is zero
in each case, the other two masses (m2 and m3 for normal ordering and m2 and m1 for
inverted ordering) get xed in a very narrow range by the oscillation constraints on
m221
and j
m223j. It is evident from
gure 2 that jMeej in each plot leads to an upper limit which
is beyond the reach of the GERDA phaseII. However, predictions of our model could be
probed by the combined GERDA + MAJORANA experiments [91]. The sensitivity reach
of other promising experiments such as LEGEND200 (40 meV), LEGEND1K (17 meV)
and nEXO (9 meV) [92] are also shown in gure 2. For each case, the entire parameter space
corresponding to an inverted neutrino mass ordering could be ruled out by the nEXO reach.
We now come to the numerical discussion of baryogenesis via avored leptogenesis. As
mentioned in the beginning of this section, we have performed the numerical computation
pertaining to leptogenesis in two di erent ways. In one way, we have taken a particular
value of M1 and compute the nal YB for the entire rescaled parameter space constrained
by the oscillation data. In the second way, we have used those values of the rescaled
parameters for which the low energy neutrino observables predicted from our model lie
close to their best t values dictated by the oscillation data in table 2. To facilitate this
purpose, we de ne a variable 2 that measures the deviation of the parameters from their
best t values:
2 =
5
X
i=1
Oi(th)
Oi
Oi(bf ) 2
:
(7.1)
In (7.1) Oi denotes the ith neutrino oscillation observable from among
m221;
m232; 12; 23
and 13 and the summation runs over all such observables. The parenthetical th stands for
the numerical value of the observable predicted in our model, whereas bf denotes the best
t value (cf. table 2).
Oi in the denominator represents the measured 1
range of Oi.
Primarily for numerical computation, we choose M2=M1 = 103. However, as indicated in
the previous section, we also present a detailed discussion regarding the sensitivity of YB to
the chosen hierarchy of Mi. Next, we calculate 2 as a function of the primed parameters
for their entire constrained range. Then, for a xed value of M1, we choose that set of
rescaled parameters which corresponds to the minimum value of 2 and a positive value of
YB. For that particular
2 and the corresponding set of rescaled parameters, we are then
able to generate a large set of elements of mD by varying M1 over a wide range and can
calculate YB for each value of M1. An organized discussion is given in what follows.
Computation of YB for a normal mass ordering of light neutrinos.
M1 < 109 GeV. In this regime, all three lepton
avors (e; ; ) are distinguishable.
Since "e1 = 0, we need to individually evaluate "1; only. However, due to the imposed
antisymmetry, two washout parameters m~
would be equal. Thus on account of
the relation in (6.16), the nal baryon asymmetry YB vanishes.
109 GeV < M1 < 1012 GeV. For the evaluation of YB here, we have to look rst at
the washout parameters K and K2 = Ke + K . As shown in the rst plot in the left panel
of gure 3, the entire allowed range of these parameters prefers to lie in K ,K2 > 1 region.
Thus the e ciency factor in (6.6) can be written in a strong washout scenario [83] as
( m~ ) =
"
0:55
10 3 1:16#
m~
;
(7.2)
where
= ; 2. As elaborated in the previous section, the assumed strong hierarchy of RH
neutrinos makes the second r.h.s. term in (6.14) much smaller than the rst term. Hence
the
nal CP asymmetry could be simpli ed to the form as in (6.17) so that the nal YB
in (6.15) is practically proportional to the free parameter M1. Now for a xed value of M1,
we compute YB for the entire rescaled parameter space. In
gure 3, the variation of YB
with the rescaled parameters is shown for a representative value of M1 = 1011 GeV. Any
further lowering of the value of M1 would cause these plots (except the rst plot in the
left panel) to condense along YB axis due to the addressed proportionality of YB with M1.
Thus, below a certain value of M1, one would end up with a value for YB which is below
the lower end 8:55
on M1 to be 6:21
10 11 of the observed range for the latter. We nd this lower bound
1010 GeV for which the peak of a YB vs. ; x1;2; y1;2 curve in gure 3
just touches the red stripe that represents the experimental observed range of YB.
Next, we concentrate on the other way which is a search for a set of rescaled parameters
2
that corresponds to the low energy neutrino observables close to their best t values and
hence the minimum value of
. For this purpose, we take a particular set from the
rescaled parameter space, calculate the corresponding
2 using (7.1) and then compute
YB. We have found that
2min should be 0:397 for YB to be positive. A complete data set
of the rescaled parameters and corresponding values of the observables are tabulated in
table 3 for 2
min = 0:397.
Given the rescaled data set for the
2min, M1 is varied widely to secure YB in the
observed range. For each value of M1, a set of values of the parameters in the elements of
mD is generated. The nal YB is then calculated for each value of M1 and the corresponding
parameters of mD. A careful surveillance of the plot in gure 4 leads to the conclusion that
we can obtain an upper and a lower bound on M1 corresponding to the observed constraint
on YB. In order to realize this fact more clearly, two straight lines have been drawn
the plots represent the variation of YB with the rescaled parameters for a representative value
M1 = 1011 GeV.
x1
0:040
x2
0:014
observables
2min = 0:397
y1
0:01
13
8:420
y2
0:155
12
33:040
1140
m221
105
7:47 (eV)2
2
min
0:397
j
m31j
2
103
2:55 (eV)2
2 = 0:397 for normal mass ordering.
parallel to the abscissa in the mentioned plot: one at YB = 8:55
10 11 and the other at
YB = 8:77
10 11. The values of M1, where the straight lines connect the YB vs. M1 curve,
yield the allowed upper and lower bounds on M1, namely (M1)upper = 7:35
(M1)lower = 7:19
1010 GeV. Again, the near linearity of the YB vs. M1 curve in gure 4
follows from the previously explained approximate proportionality of YB with M1. One
might also ask about the narrow range for M1 as observed in gure 4. Note that in this plot
1010 GeV and
we have presented our result for a particular set of rescaled parameters (with
2min = 0:397).
In principle, one could take the entire rescaled parameter space of our model and compute
the corresponding results on YB and M1 for each set of the mentioned parameters. In that
case the range of M1 would not be as narrow as shown in gure 4.
M1 > 1012 GeV. In this regime YB is zero since the
avored sum CP asymmetry
parameter P "
1 vanishes. Obviously, YB might be generated in this regime also if one
consider small breaking of CP symmetry in the neutrino sector as discussed in ref. [93].
Computation of YB for an inverted mass ordering of light neutrinos. In this case
also the observed range of YB cannot be generated for M1 < 109 GeV and M1 > 1012 GeV
owing to reasons similar to those explained in the case of a normal ordering. However, we
nd that in the case of an inverted ordering, YB cannot be generated in the observed range
even if we consider a
for a value M1 = 9:9
avored regime, i.e., 109 GeV < M1 < 1012 GeV. Numerically,
1011 GeV, YB is computed to be YB = 8:20
10 11. Thus from a
hierarchical leptogenesis perspective, an inverted mass ordering is disfavored in our model
with a complex (CP) extended antisymmetry.
The e ect of N2 on YB.
As mentioned in the previous section, there are two di erent
ways in which the heavy RH neutrino N2 might a ect the nal value of YB. In the rst,
which we name as the indirect e ect, the
nal YB becomes practically insensitive to the
mass of N2 since the second term is suppressed compared to the rst term in (6.14). Now
"1 can be written in a simpler form which is independent of M2. c.f, (6.17); hence it does
CaseI: Normal light neutrino ordering
Hierarchies !
Upper bound (GeV)
Lower bound (GeV)
M2=M1 = 102
M2=M1 = 103
M2=M1 = 104
7:32
7:16
1010
1010
7:35
7:19
1010
1010
7:38
7:20
1010
1010
neutrinos. The plot in the left is for M2=M1 = 102 and the plot in the right is for M2=M1 = 104.
not depend the mass ratio M2=M1. However, for a precise computation of YB, we need to
consider the term neglected in (6.14); that in turn motivates us to perform a quick check of
the RH neutrino mass hierarchy sensitivity of the produced value of YB. For this purpose,
in addition to the standard hierarchical case, i.e. M2=M1 = 103, we calculate YB for two
other di erent mass hierarchical schemes, M2=M1 = 102 and M2=M2 = 104. From
gure 5
we can infer that though the chosen mass ratios of the RH neutrinos are altered, changes
in the lower and upper bounds on M1 are practically insigni cant. For the allowed normal
light neutrino mass ordering, the variation of YB with M1 for di erent mass ratios of the
RH neutrinos has been presented in table 4. It is obvious from the entries of table 4 that a
slight di erence in the upper and lower bounds on M1 in a particular column, as compared
to the other column, arises due the dependence of M2 on the second term in (6.14). For a
xed value of M1, the contribution from the second term in (6.14) is larger for M2=M1 = 102
and smaller for M2=M1 = 104, as compared to the standard M2=M1 = 103 case. Hence
for M2=M1 = 102, the slope of the YB vs. M1 curve is larger than for M2=M1 = 103.
Consequently, for the allowed range of YB, both the upper and the lower bounds get slightly
left shifted on the M1axis (compared to the standard M2=M1 = 103 case). Proceeding in
the same way, we obtain somewhat right shifted bounds for M2=M1 = 104 case.
In contrast, in the direct e ect, any asymmetry produced by N2 survives provided the
conditions
1
1 and
1, cf. (6.18), are satis ed. From
gure 6 we observe that the
allowed parametric region prefers large values of
2 in excess of 10 except at the bottom
(green band). Thus the condition
1 is violated in most of the region. Moreover the
2min = 0:397, for which we calculate nal YB strongly violates
amount of parameter space with
2 < 10 corresponds to values of 2 above 0.9 which is
much higher than
2min for which we compute YB in the observed range. Therefore, in our
nal result, any direct e ect of N2 is not signi cant. Note that there is nothing special
about 2 = 0:9. The issue we are trying to address here, is that there are indeed some data
points in the model parameter space for which the conditions for N2 leptogenesis could be
satis ed. However, the minimum value of 2 for those data sets is 0.9. This means that the
corresponding observables are away from their best t values (though well within 1 ) and
thus the obtained bounds on M1 (e.g. gure 4) will not be a ected by N2 leptogenesis.
However, if one goes beyond 2
0:9, the asymmetry produced by N2 could play a crucial role.
We would like to conclude this section by comparing our results on leptogenesis with
those obtained earlier in previous literature in case of a
avored CP symmetry. Existing
references such as [42, 72, 73, 93] also discuss leptogenesis within the framework of residual
CP symmetry (in particular CP
) and point out the nonoccurrence of un avored
leptogenesis and only the viability of the
avored scenario similar to our proposal of a exact
antisymmetry in the neutrino sector. However, the
nal numerical analysis is di erent
from our case. In particular, all the mentioned references mainly focus on the three
neutrino case where one cannot x the Yukawa couplings only with the oscillation data. Thus
any
nal result on leptogenesis requires other assumptions to constrain all the Yukawas.
We focus on the two RH neutrino case, namely the minimal seesaw mechanism, where the
entire Yukawa parameter (rescaled by RH neutrino masses) space could be constrained
by the neutrino oscillation data. Hence all the results obtained, in particular for the RH
neutrino masses, are exactly dictated by the oscillation data. For a hierarchical RH mass
spectrum, refs. [72, 73] shows a variation of YB with a single model parameter for a xed
value of M1 and best t values of the oscillation parameters. However, here we focus on
the bounds on M1 for the entire parameter space as well as the for the parameter set that
corresponds to the observables which lie near to their best t values. For the rst case,
we obtain a lower bound on M1 while in the other, we obtain an upper as well as a lower
bound on M1. In addition, we have done a thorough study of the RH neutrino hierarchy
sensitivity of the nal YB and showed the possible changes in the bounds on the lightest
RH neutrino M1 for three di erent RH neutrino hierarchical mass spectra. We have also
showed that, for this minimal seesaw with a complex
antisymmetry, the inverted mass
ordering is not a viable option as far as hierarchical leptogenesis is concerned. We are not
within the framework of a Grand Uni ed Theory (GUT) such as SO(10), where the lepton
asymmetry generated by the next to light RH neutrino (N2), is a natural requirement to
produce correct value of YB [94, 95]. Nevertheless, we opt for fast N1 interactions which
are responsible for the survival of the lepton asymmetry generated by N2 [89]. For this CP
symmetric framework we have showed for the rst time that there could be a parameter
space left for which N2 leptogenesis might a ect the
nal value of YB (though a rigorous
study of the N2 leptogenesis is beyond the scope of this paper). Ref. [42] concluded that
for the mass regime M1 < 109 GeV, a resonant leptogenesis is only possible if one considers
breaking in CP
, since in this regime the muon and tauon washout parameters are of
equal strength. In our proposal of CP
A also, this conclusion is true. However, as we
show in the appendix, unlike the hierarchical RH neutrino mass spectrum, RH neutrinos
with a mild hierarchy could also result in a successful leptogenesis for M1
showed that an inverted mass ordering could then be a viable option for a successful
leptogenesis. We also comment on the strength of the mild hierarchy by solving numerically
the formulae for YB within the framework of avor diagonal RH neutrinos. In our
analysis, the sign of YB depends upon the Yukawa parameters. In this context we refer to [93]
which shows how, within the framework of a CP symmetry, the sign of YB depends upon
109 GeV. We
the observables.
8
Concluding comments and discussion
In this paper the complex (CP) extension of
M CP
A which is simply related to M CP
antisymmetry has been shown to yield a
 the Majorana mass matrix from the complex
symmetry with both having identical phenomenological consequences.
These phenomenological consequences of CP
A have been worked out within a minimal
seesaw scheme with two strongly or mildly hierarchical RH neutrinos N1 and N2. We have
further investigated baryogenesis via leptogenesis in this scenario and derived upper and
lower bounds on the mass of N1.
To summarize, we have proposed a new idea, namely a complex extended
antisymmetry, pertaining to the neutrino sector and have worked out its consequences. Unlike
the real
antisymmetry, we envisage there is no need for any breaking of it in the
neutrino sector. Atmospheric neutrino mixing is predicted to be maximal ( 23 =
=4) in this
scheme while the solar and reactor mixing angles ( 12 and 13 respectively) can be t to
their observed values. Neutrino masses get generated via the minimal seesaw mechanism
with two heavy rightchiral neutrinos. The lightest neutrino is predicted to be massless
while the two other neutrino masses can be t to the observed range of values of j
and
m212 both for a normal and an inverted mass ordering. Concrete predictions are made
m232j
for neutrinoless double beta decay: the ongoing experiments are not expected to observe it
though the planned nEXO experiment may have a chance to do so. Finally, we have made a
detailed quantitative examination of baryogenesis via leptogenesis in our scheme including
the indirect and direct e ects of the heavier RH neutrino N2.  avored leptogenesis with
a normal mass ordering turns out to be the only viable possibility that can generate YB in
the observed range in a hierarchical leptogenesis scenario.
Acknowledgments
The work of RS and AG is supported by the Department of Atomic Energy (DAE),
Government of India. The work of PR has been supported by the Indian National Science
Academy.
HJEP06(218)5
A
Discussion of the case with mildly hierarchical RH neutrinos
In the text we have dealt with a strongly hierarchical RH neutrino mass spectrum and
found only the
avored regime to be viable in producing the correct YB for a normal light
neutrino mass ordering. Since in our chosen basis [72, 73], RH neutrinos are nondegenerate,
it would also be interesting to study leptogenesis with a mildly hierarchical including a
quasidegenerate NR mass spectrum. We will see later in this discussion that RH neutrinos
which are not strongly hierarchical might obliterate all the new bounds on M1 that we
obtained earlier.
In general, a quasidegenerate RH neutrino mass spectrum is considered for studying
leptogenesis in a low energy seesaw scenario (resonant leptogenesis [81]); here the RH
neutrinos could have masses O(TeV).
However, in our analysis, we cannot lower the
RH neutrino masses below 109 GeV, since that would correspond to the fully
avored
regime where the two washout parameters m~
are the same due to the imposed
antisymmetry, thereby implying a vanishing YB cf. (6.16). However, depending on the
chosen mild mass splitting of the RH neutrinos, we can lower the lightest RH neutrino mass
down to 109 GeV below which the muon charged lepton
avor equilibriates. In scenarios
where the RH neutrinos are not strongly hierarchical, instead of (6.3), it is useful to use
the general formula for the CP asymmetry parameter [81] "i as
"i =
1
X Imfhij (mD)i (mD)j g f (xij ) +
"
pxij (1
xij )
xij )2 + hj2j (16 2v4) 1
#
+
xij )Imfhji(mD)i (mD)j g
xij )2 + hj2j (16 2v4) 1
(1
:
(A.1)
Note that, unlike (6.3) the above equation is valid for degenerate RH neutrinos also.
Taking into account the contribution from both the RH neutrinos, we have performed
a numerical study to
nd the
nal YB for the lowest allowed value of M1(= 109 GeV). It
turns out that for a normal light neutrino mass ordering, M2 could at most be
17:5M1
to produce the observed lower bound 8:55
10 11 of YB cf. (6.1). One can see that the
obtained mass spectrum is fairly hierarchical though the hierarchy is not very strong. Of
course any number smaller than 17.5 would result in an enhancement of the produced CP
asymmetry. Thus the observed range of YB could be generated with a quasidegenerate
RH mass spectrum too. Interestingly, an inverted light neutrino mass ordering which is
disfavoured for a strongly hierarchical RH neutrino mass spectrum is now a perfectly viable
scenario since we relax the strong hierarchy assumption. Again, as in the previous case,
i.e., for M1 = 109 GeV, it is numerically found that one needs M2 6 1:8M1 in order to
produce the observed lower bound on YB. Note that, unlike in the case of a normal light
neutrino mass ordering, the RH neutrino mass spectrum here favours a mild hierarchical
scenario as we lower the value of M1. We could also point out that here we have considered
the avor diagonal RH neutrinos to calculate the asymmetry. Nevertheless, for a resonant
leptogenesis scenario, a full avorcovariant treatment might play an important role [84{86].
As a concluding remark, we may mention once again that, owing to the imposed
symmetry, a fully avored leptogenesis is not possible for M1 < 109 GeV even if we consider
strongly degenerate RH neutrinos. Nevertheless, a small breaking of the symmetry [93],
or somewhat a more moderate version of the symmetry such as the scaling ansatz [46{48]
will cause a deviation from m~
= m~
cf. (6.16) and will imply a nonvanishing YB. In such
cases leptogenesis with heavily degenerate RH neutrinos (resonant leptogenesis) could be
an interesting topic to study.
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