Implications of residual CP symmetry for leptogenesis in two right-handed neutrino model

7.2 MB·

by  Cai-Chang Li

USTC-ICTS-17-01
Implications of residual CP symmetry for leptogenesis in two
right-handed neutrino model
Cai-Chang Li ∗, Gui-Jun Ding †
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1 Interdisciplinary Center for Theoretical Study and Department of Modern Physics,
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University of Science and Technology of China, Hefei, Anhui 230026, China
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Abstract

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We analyze the interplay between leptogenesis and residual symmetry in the framework of
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p tworight-handedneutrinomodel. Workingintheflavorbasis, weshowthatalltheleptogenesis
e CPasymmetriesarevanishingforthecaseoftworesidualCPtransformationsoracyclicresidual
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flavorsymmetryintheneutrinosector. IfasingleremnantCPtransformationispreservedinthe
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neutrinosector,theleptonmixingmatrixisdetermineduptoarealorthogonalmatrixmultiplied

1 from the right side. The R-matrix is found to depend on only one real parameter, it can take
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three viable forms, and each entry is either real or pure imaginary. The baryon asymmetry is
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generatedentirelybytheCPviolatingphasesinthemixingmatrixinthisscenario. Weperform
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5 a comprehensive study for the ∆(6n2) flavor group and CP symmetry which are broken to a
8 singleremnantCPtransformationintheneutrinosectorandanabeliansubgroupinthecharged
0 lepton sector. The results for lepton flavor mixing and leptogenesis are presented.
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∗E-mail: [email protected]
†E-mail: [email protected]
1 Introduction
A large amount of experiments with solar, atmospheric, reactor and accelerator neutrinos have
provided compelling evidences for oscillations of neutrinos caused by nonzero neutrino masses and
neutrino mixing [1–3]. Both three flavor neutrino and antineutrino oscillations can be described
by three lepton mixing angles θ , θ and θ , one leptonic Dirac CP violating phase δ, and two
12 13 23
independent mass-squared splittings δm2 ≡ m2 −m2 > 0 and ∆m2 ≡ m2 −(m2 +m2)/2, where
2 1 3 1 2
m are the three neutrino masses, ∆m2 > 0 and ∆m2 < 0 correspond to normal ordering (NO)
1,2,3
and inverted ordering (IO) mass spectrum respectively. All these mixing parameters except δ have
been measured with good accuracy [4–8], the experimentally allowed regions at 3σ confidence level
(taken from Ref. [4]) are:
0.259 ≤sin2θ ≤ 0.359,
12
1.76(1.78)×10−2 ≤sin3θ ≤ 2.95(2.98)×10−2,
13
0.374(0.380) ≤sin2θ ≤ 0.626(0.641),
23
6.99×10−5eV2 ≤ δm2 ≤ 8.18×10−5eV2,
2.23(−2.56)×10−3eV2 ≤ ∆m2 ≤ 2.61(−2.19)×10−3eV2 (1.1)
for NO (IO) neutrino mass spectrum. At present, both T2K [9,10] and NOνA [11] report a weak
evidence for a nearly maximal CP violating phase δ ∼ −π/2, and hits of δ ∼ −π/2 also show up in
the global fit of neutrino oscillation data [4–8]. Moreover, several experiments are being planned
to look for CP violation in neutrino oscillation, including long-baseline facilities, superbeams, and
neutrino factories. The above structure of lepton mixing, so different from the the small mixing
in the quark sector, provides a great theoretical challenge. The idea of flavor symmetry has been
extensively exploited to provide a realistic description of the lepton masses and mixing angles. The
finite discrete non-abelian flavor symmetries have been found to be particularly interesting as they
can naturally lead to certain mixing patterns [12], please see Refs. [13–15] for review.
Although the available data are not yet able to determine the individual neutrino mass m , the
i
neutrino masses are known to be of order eV from tritium endpoint, neutrinoless double beta decay
and cosmological data. The smallness of neutrino masses can be well explained within the see-saw
mechanism [16], in which the Standard Model (SM) is extended by adding new heavy states. The
light neutrino masses are generically suppressed by the large masses of the new states. In type I
seesaw model [16] the extra states are right-handed (RH) neutrinos which have Majorana masses
much larger than the electroweak scale, unlike the standard model fermions which acquire mass
proportional to electroweak symmetry breaking. Apart from elegantly explaining the tiny neutrino
masses,theseesawmechanismprovidesasimpleandattractiveexplanationfortheobservedbaryon
asymmetry of the Universe, one of the most longstanding cosmological puzzles. The CP violating
decays of heavy RH neutrinos can produce a lepton asymmetry in the early universe, which is then
converted into a baryon asymmetry through B+L violating anomalous sphaleron processes at the
electroweak scale. This is the so-called leptogenesis mechanism [17].
It is well-known that in the paradigm of the unflavored thermal leptogenesis the CP phases
in the neutrino Yukawa couplings in general are not related to the the low energy leptonic CP
violating parameters (i.e. Dirac and Majorana phases) in the mixing matrix. However, the low
energy CP phases could play a crucial role in the flavored thermal leptogenesis [18] in which the
flavors of the charged leptons produced in the heavy RH neutrino decays are relevant. In models
withflavorsymmetry,thetotalnumberoffreeparametersisgreatlyreduced,thereforetheobserved
baryon asymmetry could possibly be related to other observable quantities [19]. In general, the
leptogenesis CP asymmetries would vanish if a Klein subgroup of the flavor symmetry group is
preserved in the neutrino sector [20].
RecentstudiesshowthattheextensionofdiscreteflavorsymmetrytoincludeCPsymmetryisa
2
very predictive framework [21–33]. If the given flavor and CP symmetries are broken to an abelian
subgroup and Z ×CP in the charged lepton and neutrino sectors respectively, the resulting lepton
2
mixing matrix would be determined in terms of a free parameter θ whose value can be fixed by the
reactor angle θ . Hence all the lepton mixing angles, Dirac CP violating phase and Majorana CP
13
phases can be predicted [33]. Moreover, other phenomena involving CP phases such as neutrinoless
double decay and leptongenesis are also strongly constrained in this approach [20,31,34]. In fact,
we find that the leptogensis CP asymmetries are exclusively due to the Dirac and Majorana CP
phases in the lepton mixing matrix, and the R-matrix depends on only a single real parameter in
this scenario [20].
In this paper we shall extend upon the work of [20] in which the SM is extended to introduce
three RH neutrinos. Here we shall study the interplay between residual symmetry and leptogenesis
in seesaw model with two RH neutrinos. We find that all the leptogensis CP asymmetries would
be exactly vanishing if two residual CP transformations or a cyclic residual flavor symmetry are
preserved by the seesaw Lagrangian. On the other hand, if only one remnant CP transformation
is preserved in the neutrino sector, all mixing angles and CP phases are then fixed in terms of
three real parameters θ which can take values between 0 and π, and the R-matrix would be
1,2,3
constrained to depend on only one free parameter. The total CP asymmetry (cid:15) ≡ (cid:15) +(cid:15) +(cid:15) in
1 e µ τ
leptogenesis is predicted to be zero. Hence our discussion will be entirely devoted to the flavored
thermal leptogenesis scenario in which the lightest RH neutrino mass is typically in the interval of
109 GeV ≤ M ≤ 1012 GeV. Our approach is quite general and it is independent of the explicit
1
form of the residual symmetries and how the vacuum alignment achieving the residual symmetries
is dynamically realized. In order to show concrete examples, we apply this general formalism to the
flavor group ∆(6n2) combined with CP symmetry which is broken down to an abelian subgroup in
thechargedleptonsectorandaremnantCPtransformationintheneutrinosector. Theexpressions
for lepton mixing matrix as well as mixing parameters in each possible cases are presented. We find
that for small values of the flavor group index n, the experimental data on lepton mixing angles
can be accommodated for certain values of the parameters θ . The corresponding predictions
1,2,3
for the cosmological matter-antimatter asymmetry are discussed.
The rest of the paper is organized as follows. In section 2 we briefly review some generic aspects
ofleptogenesisintwoRHmodelandpresentsomeanalyticapproximationswhichwillbeusedlater.
In section 3 we study the scenario that one residual CP transformation is preserved in the neutrino
sector. Theleptonmixingmatrixisdetermineduptoanarbitraryrealorthogonalmatrixmultiplied
fromtherighthandside. TheR-matrixcontainsonlyonefreeparameter,andeachelementiseither
real or pure imaginary. The total CP asymmetry (cid:15) is vanishing, consequently the unflavored
1
leptogenesis is not feasible unless subleading corrections are taken into account. The scenario of
two remnant CP transformations or a cyclic residual flavor symmetry is discussed in section 4. All
leptogenesis CP asymmetries (cid:15) are found to vanish in both cases. Leptogenesis could become
e,µ,τ
potentially viable only when higher order contributions lift the postulated residual symmetry. In
section 5 we apply our general formalism to the case that the single residual CP transformation
of the neutrino sector arises from the breaking of the most general CP symmetry compatible with
∆(6n2)flavorgroupwhichisbrokendowntoanabeliansubgroupinthechargedleptonsector. The
predictionsforleptonflavormixingandbaryonasymmetryarestudiedanalyticallyandnumerically.
Finally, in section 6 we summarize our main results and draw the conclusions.
2 General set-up of leptogenesis in two right-handed neutrino
model
Seesaw mechanism is a popular extension of the Standard Model (SM) to explain the smallness
of neutrino masses. In the famous type I seesaw mechanism [16], one generally introduces addi-
3
tional three right-handed neutrinos which are singlets under the SM gauge group. Although the
seesaw mechanism describes qualitatively well the observations in neutrino oscillation experiments,
it is quite difficult to make quantitative predictions for neutrino mass and mixing without further
hypothesisforunderlyingdynamics. Thereasonisthattheseesawmechanisminvolvesalargenum-
ber of undetermined parameters at high energies whereas much less parameters could be measured
experimentally.
A intriguing way out of this problem is to simply reduce the number of right-handed neutrinos
from three to two [35–37]. The two right-handed neutrino (2RHN) model can be regarded as a
limiting case of three right-handed neutrinos where one of the RH neutrinos decouples from the
seesaw mechanism either because it is very heavy or because its Yukawa couplings are very weak.
Since the number of free parameters is greatly reduced, the 2RHN model is more predictive than
thestandardscenarioinvolvingthreeRHneutrinos. Namely, thelightestleft-handedneutrinomass
automatically vanishes, while the masses of the other two neutrinos are fixed by δm2 and ∆m2.
Hence only two possible mass spectrums can be obtained
√
(cid:112)
NO : m = 0, m = δm2, m = ∆m2+δm2/2,
1 2 3
(cid:112) (cid:112)
IO : m = −δm2/2−∆m2, m = δm2/2−∆m2, m = 0. (2.1)
1 2 3
Moreover there is only one Majorana CP violating phase corresponding to the phase difference
between these two nonzero mass eigenvalues. The Lagrangian responsible for lepton masses in the
2RHN model takes the following form
1
L = −yαL¯αHlαR−λiαN¯iRH(cid:101)†Lα− 2MiN¯iRNicR+h.c. , (2.2)
where L ≡ (ν ,l )T and l indicate the lepton doublet and singlet fields with flavor α = e,µ,τ
α αL αL αR
respectively, N is the RH neutrino with mass M (i = 1,2), and H ≡ (H+,H0)T is the Higgs
iR i
doublet with H(cid:101) ≡ iσ2H∗. The Yukawa couplings λiα form an arbitrary complex 2×3 matrix, here
we have worked in the basis in which both the Yukawa couplings for the charged leptons and the
Majorana mass matrix for the RH neutrinos are diagonal and real. After electroweak symmetry
breaking, the light neutrino mass matrix is given by the famous seesaw formula
m = v2λTM−1λ = U∗mU†, (2.3)
ν
wherev = 175GeVreferstothevacuumexpectationvalueoftheHiggsfieldH0,M ≡ diag(M ,M )
1 2
and m ≡ diag(m ,m ,m ) with m = 0 for NO and m = 0 for IO, and U is the lepton mixing
1 2 3 1 3
matrix. ItisconvenienttoexpresstheYukawacouplingλintermsoftheneutrinomasseigenvalues,
mixing angles and CP violation phases as1
λ = M1/2Rm1/2U†/v, (2.4)
where R is a 2×3 complex orthogonal matrix having the following structure [38,39]
(cid:18)0 cosθˆ ξsinθˆ(cid:19)
NO : R= , (2.5a)
0 −sinθˆ ξcosθˆ
(cid:18) cosθˆ ξsinθˆ 0(cid:19)
IO : R= , (2.5b)
−sinθˆ ξcosθˆ 0
where θˆis an arbitrary complex number and ξ = ±1. From Eqs. (2.5a, 2.5b) we can check that the
R-matrix satisfies
RRT = diag(1,1), for NO and IO,
RTR = diag(0,1,1), for NO, (2.6)
RTR = diag(1,1,0), for IO.
1For other parameterizations of the neutrino Yukawa coupling, see Ref. [40].
4
Leptogenesis is a natural consequence of the seesaw mechanism, and it provides an elegant expla-
nation for the baryon asymmetry of the Universe [17]. We shall work in the typical N -dominated
1
scenario, and we assume that right-handed neutrinos are hierarchical M (cid:29) M such that the
2 1
asymmetry is dominantly produced from the decays of the lightest RH neutrino N . The phe-
1
nomenology of leptogenesis in 2RHN model has been comprehensively studied [36,37,39,41]. The
flavored CP asymmetries in the decays of N into leptons of different flavors are of the form [42–45]
1
Γ(N → l H)−Γ(N →¯l H¯)
1 α 1 α
(cid:15) ≡
α (cid:80) Γ(N → l H)+Γ(N →¯l H¯)
α 1 α 1 α
(cid:104) (cid:105)
2 (cid:61) (λλ†) λ λ∗
3 (cid:88) M1 1j 1α jα
(cid:39)−
16π M (λλ†)
j 11
j=1
3M1 (cid:61)(cid:0)(cid:80)ij√mimjmjR1iR1jUα∗iUαj(cid:1)
=− , (2.7)
16πv2 (cid:80) m |R |2
j j 1j
where Γ(N → l H) and Γ(N →¯l H¯) with α = e,µ,τ denote the flavored decay rates of N into
1 α 1 α 1
lepton l and anti-lepton ¯l respectively. We notice that (cid:15) is invariant under the transformation
α α α
ξ → −ξ and θˆ → −θˆ. Consequently we shall choose ξ = 1 as an illustration in the following
numerical analysis. Inserting the expression for the Yukawa coupling in Eqs. (2.5a, 2.5b) into
Eq. (2.7), we obtain the CP asymmetry
(cid:26)
3 M
(cid:15) (cid:39) − 1 (m2|U |2−m2|U |2) (cid:61)sin2θˆ
α 16πv2m |cosθˆ|2+m |sinθˆ|2 3 α3 2 α2
2 3
(cid:27)
√ (cid:104) (cid:105)
+ξ m m (m +m )(cid:60)(U∗ U )(cid:61)(sinθˆcosθˆ)+(m −m )(cid:61)(U∗ U )(cid:60)(sinθˆcosθˆ) , (2.8)
2 3 2 3 α2 α3 3 2 α2 α3
for NO and
(cid:26)
3 M
(cid:15) (cid:39) − 1 (m2|U |2−m2|U |2) (cid:61)sin2θˆ (2.9)
α 16πv2m |cosθˆ|2+m |sinθˆ|2 2 α2 1 α1
1 2
(cid:27)
√ (cid:104) (cid:105)
+ξ m m (m +m )(cid:60)(U∗ U )(cid:61)(sinθˆcosθˆ)+(m −m )(cid:61)(U∗ U )(cid:60)(sinθˆcosθˆ) (2.10)
1 2 1 2 α1 α2 2 1 α1 α2
for IO neutrino mass spectrum. If the RH neutrino mass M is large enough (e.g. M > 1012 GeV),
1 1
the interactions mediated by all the three charged lepton Yukawa couplings are out of equilibrium.
As a result, the one flavor approximation rigorously holds, and the total CP asymmetry is
(cid:15) ≡ (cid:88)(cid:15) = − 3M1 (cid:61)(cid:0)(cid:80)im2iR12i(cid:1) , (2.11)
1 α 16πv2 (cid:80) m |R |2
α j j 1j
which is completely independent of the lepton mixing matrix U. For the parametrization of the
R-matrix in Eqs. (2.5a, 2.5b), we have
3M (m2−m2) (cid:61)sin2θˆ
NO : (cid:15) =− 1 3 2 , (2.12a)
1 16πv2m |cosθˆ|2+m |sinθˆ|2
2 3
3M (m2−m2) (cid:61)sin2θˆ
IO : (cid:15) =− 1 2 1 . (2.12b)
1 16πv2m |cosθˆ|2+m |sinθˆ|2
1 2
WeseethatthetotalCPasymmetry(cid:15) wouldvanishwhentheparameterθˆisrealorpureimaginary
1
up to π/2. The total baryon asymmetry is the sum of each individual lepton asymmetry. In the
present paper we will be concerned with temperature window (109 ≤ T ∼ M ≤ 1012) GeV. In
1
5
this range only the τ charged lepton Yukawa interaction is in equilibrium, the e and µ flavors are
indistinguishable, and the final baryon asymmetry is well approximated by [46–49]
(cid:20) (cid:18) (cid:19) (cid:18) (cid:19)(cid:21)
12 417 390
Y (cid:39) − (cid:15) η m + (cid:15) η m , (2.13)
B 37g∗ 2 589(cid:101)2 τ 589(cid:101)τ
where (cid:15) ≡ (cid:15) +(cid:15) , m ≡ m +m and
2 e µ (cid:101)2 (cid:101)e (cid:101)µ
(cid:34)(cid:18) m (cid:19)−1 (cid:18)0.2×10−3eV(cid:19)−1.16 (cid:35)−1
η(m ) (cid:39) (cid:101)α + . (2.14)
(cid:101)α 8.25×10−3eV m
(cid:101)α
Here g is the number of relativistic degrees of freedom, the efficiency factor η(m ) accounts for
∗ (cid:101)α
the washing out of the produced lepton number asymmetries due to the inverse decay and lepton
number violating scattering, and the washout mass m parametrizes the decay rate of N into the
(cid:101)α 1
leptons of flavor α with
m ≡ |λ1α|2v2 = (cid:12)(cid:12)(cid:88)m1/2R U∗ (cid:12)(cid:12)2, α = e,µ,τ. (2.15)
(cid:101)α M (cid:12) i 1i αi(cid:12)
1
i
Plugging Eqs. (2.5a) and (2.5b) into above eqution we find the explicit expressions of the washout
masses are
(cid:12)√ √ (cid:12)2
(cid:12)(cid:12) m2Uα∗2cosθˆ+ξ m3Uα∗3sinθˆ(cid:12)(cid:12) , for NO,
m = (2.16)
(cid:101)α (cid:12)√ √ (cid:12)2
(cid:12) m U∗ cosθˆ+ξ m U∗ sinθˆ(cid:12) , for IO.
(cid:12) 1 α1 2 α2 (cid:12)
3 Leptogenesis with one residual CP transformation
In a series of papers [21–33], it has been shown that the residual CP symmetry of the light neutrino
mass matrix can quite efficiently predict the lepton mixing angles as well as CP violation phases.
If the residual CP symmetry is preserved by the seesaw Lagrangian, leptogenesis would be also
strongly constrained [20,34,50]. We assume that the flavor and CP symmetries are broken at a
scaleabovetheleptogenesisscale. Asaconsequence, leptogenesisoccursinthestandardframework
of the SM plus two heavy RH neutrinos without involving any additional state in its dynamics. In
this section, we shall study the implications of residual CP for leptogenesis in 2RHN model, and
we assume that both the neutrino Yukawa coupling and the Majorana mass term in Eq. (2.2) are
invariant under one generic residual CP transformation defined as
νL (cid:55)−C→P iXνγ0Cν¯LT , NR (cid:55)−C→P iX(cid:98)Nγ0CN¯RT , (3.1)
where ν ≡ (ν ,ν ,ν )T, N ≡ (N ,N )T, C denotes the charge-conjugation matrix, X is a
L eL µL τL R 1R 2R ν
3×3 symmetric unitary matrix to avoid degenerate neutrino masses and X(cid:98)N is a 2×2 symmetric
unitary matrix. For the symmetry to hold, λ and M have to fulfill
X(cid:98)N† λXν = λ∗, X(cid:98)N† MX(cid:98)N∗ = M∗. (3.2)
As we work in the basis in which the RH neutrino mass matrix M is real and diagonal, the residual
CP transformation X(cid:98)R should be diagonal with elements equal to ±1, i.e.,
X(cid:98)N = diag(±1,±1), (3.3)
Notice that conclusion would be not changed even if M is non-diagonal in a concrete flavor sym-
metry model [20]. Thus we can find that the light neutrino mass matrix m given by the seesaw
ν
formula satisfies
XTm X = m∗, (3.4)
ν ν ν ν
6
whichmeans(asexpected)m isinvariantundertheresidualCPtransformationX . Thelightneu-
ν ν
trinomassmatrixcanbediagonalizedbyaunitarytransformationU withm = U∗diag(m ,m ,m )U†.
ν ν ν 1 2 3 ν
Then from Eq. (3.4) we can obtain
(cid:16) (cid:17)T (cid:16) (cid:17)
U†X U∗ diag(m ,m ,m ) U†X U∗ = diag(m ,m ,m ).
ν ν ν 1 2 3 ν ν ν 1 2 3
Note m = 0 for NO and m = 0 for IO in the 2RHN model. Hence U is subject to the following
1 3 ν
constraint from the residual CP transformation X ,
ν
Uν†XνUν∗ = X(cid:98)ν, (3.5)
with
X(cid:98)ν = diag(cid:0)eiα,±1,±1(cid:1) for NO, (3.6)
X(cid:98)ν = diag(cid:0)±1,±1,eiα(cid:1) for IO,
where α is a real parameter in the interval between 0 and 2π. Then it is easy to check that X is a
ν
symmetric and unitary matrix for both NO and IO cases. Moreover, with the definion of R-matrix
in Eq. (2.4), we can derive that the postulated residual symmetry leads to the following constraint
on R as
X(cid:98)NR∗X(cid:98)ν−1 = R, (3.7)
Obvioulsy −X(cid:98)N and −X(cid:98)ν give rise to the same constraint on R as X(cid:98)N and X(cid:98)ν, therefore it is
sufficient to only consider the cases of X(cid:98)N = diag(1,±1), X(cid:98)ν = diag(eiα,±1,±1) for NO and
X(cid:98)ν = diag(±1,±1,eiα) for IO. The explicit forms of the R-matrix for all possible values of X(cid:98)N
and X(cid:98)ν are collected in table 1. We see that there are three admissible forms of the R-matrix
summarized as follows
 (cid:18) (cid:19)
0 cosϑ ξsinϑ
R = for NO,
 0 −sinϑ ξcosϑ
R-1st: (cid:18) (cid:19)
cosϑ ξsinϑ 0
R = for IO,
 −sinϑ ξcosϑ 0
 (cid:18) (cid:19)
0 coshϑ iξsinhϑ
R = ± for NO,
 0 −isinhϑ ξcoshϑ
R-2nd: (cid:18) (cid:19) (3.8)
coshϑ iξsinhϑ 0
R = ± for IO,
 −isinhϑ ξcoshϑ 0
 (cid:18) (cid:19)
0 isinhϑ −ξcoshϑ
R = ± for NO,
 0 coshϑ iξsinhϑ
R-3rd: (cid:18) (cid:19)
isinhϑ −ξcoshϑ 0
R = ± for IO.
 coshϑ iξsinhϑ 0
We would like to point out that the R-matrix is constrained to depend on a single real parameter ϑ
inthissetup. Moreover, fromEq.(2.11)wecanseethatthetotalleptonasymmetry(cid:15) isvanishing,
1
i.e.
(cid:15) = (cid:15) +(cid:15) +(cid:15) = 0. (3.9)
1 e µ τ
As a result, the net baryon asymmetry can not be generated in the one flavor approximation which
is realized when the mass of the lightest right-handed neutrino M is larger than about 1012 GeV,
1
unless the residual CP symmetry is further broken by subleading order corrections. This result is
quite general, it is independent of the explicit form of the residual CP transformation and how the
residual symmetry is dynamically realized.
Next we proceed to determine the lepton mixing matrix from the postulated remnant CP
transformation. SinceX mustbeasymmetricunitarymatrixtoavoiddegenerateneutrinomasses,
ν
by performing the Takagi factorization X can be written as [21,33]
ν
X = Σ ΣT , (3.10)
ν ν ν
7
X(cid:98)N X(cid:98)ν R (NO) R (IO)
(cid:18) (cid:19) (cid:18) (cid:19)
0 cosϑ ξsinϑ cosϑ ξsinϑ 0
diag(1,1) D(1,1)
0 −sinϑ ξcosϑ −sinϑ ξcosϑ 0
diag(1,1) D(1,−1) (cid:55) (cid:55)
diag(1,1) D(−1,1) (cid:55) (cid:55)
diag(1,1) D(−1,−1) (cid:55) (cid:55)
diag(1,−1) D(1,1) (cid:55) (cid:55)
(cid:18) (cid:19) (cid:18) (cid:19)
0 coshϑ iξsinhϑ coshϑ iξsinhϑ 0
diag(1,−1) D(1,−1) ± ±
0 −isinhϑ ξcoshϑ −isinhϑ ξcoshϑ 0
(cid:18) (cid:19) (cid:18) (cid:19)
0 isinhϑ −ξcoshϑ isinhϑ −ξcoshϑ 0
diag(1,−1) D(−1,1) ± ±
0 coshϑ iξsinhϑ coshϑ iξsinhϑ 0
diag(1,−1) D(−1,−1) (cid:55) (cid:55)
Table 1: The explicit form of R-matrix for all possible independent values of X(cid:98)N and X(cid:98)ν, where ϑ is a real free
parameter. The symbol “(cid:55)” denotes that the solution for R−matrix does not exist since it has to fulfill the equality
ofEq.(2.6). ThenotationD(x,y)withx,y=±1referstodiag(eiα,x,y)anddiag(x,y,eiα)forNOandIOrespectively.
whereΣ isaunitarymatrixanditcanbeexpressedintermsoftheeigenvaluesandeigenvectorsof
ν
X [33]. ThustheconstraintontheneutrinodigonalizationmatrixU inEq.(3.5)canbesimplified
ν ν
into
ΣTνUν∗X(cid:98)ν−12 = Σ†νUνX(cid:98)ν21 . (3.11)
The matrices on the two sides of this equation are unitary and complex conjugates of each other.
1
Therefore the combination Σ†νUνX(cid:98)ν2 is a generic real orthogonal matrix, and consequently the
unitary transformation U takes the form [33,50,51]
ν
−1
Uν = ΣνO3×3X(cid:98)ν 2 , (3.12)
where O is a three dimensional real orthogonal matrix, and it can be generally parameterized
3×3
as
   
1 0 0 cosθ 0 sinθ cosθ sinθ 0
2 2 3 3
O3×3(θ1,θ2,θ3) = 0 cosθ1 sinθ1 0 1 0 −sinθ3 cosθ3 0 , (3.13)
0 −sinθ cosθ −sinθ 0 cosθ 0 0 1
1 1 2 2
where θ (i = 1,2,3) are real free parameters in the range of [0,π). In our working basis (usually
i
called leptogenesis basis) where the charged lepton mass matrix is diagonal, lepton flavor mixing
completelyarisesfromtheneutrinosector, andthereforetheleptonmixingmatrixU coincideswith
U . Hence we conclude that the mixing matrix and all mixing angles and CP phases would depend
ν
on three free continuous parameters θ if only one residual CP transformation is preserved in
1,2,3
the neutrino sector. In order to facilitate the discussion of leptogenesis, we separate out the CP
parity matrices X(cid:98)N and X(cid:98)ν and define the following three parameters
U(cid:48) ≡ UX(cid:98)ν12, R(cid:48) ≡ X(cid:98)N−12RX(cid:98)ν12, Ki ≡ (X(cid:98)N)11(X(cid:98)ν−1)ii, i = 1,2,3. (3.14)
We see that R(cid:48) is real and the parameter K is equal to +1, −1 or ±e−iα. As a consequence, the
i
flavored CP asymmetry (cid:15) can be expressed as
α
3M1 (cid:61)(cid:0)(cid:80)ij√mimjmjR1(cid:48)iR1(cid:48)jUα(cid:48)∗iUα(cid:48)jKj(cid:1)
(cid:15) = − , (3.15)
α 16πv2 (cid:80) m R(cid:48)2
j j 1j
8
and the washout mass m˜ is given by
α
(cid:12) (cid:12)2
(cid:12)(cid:88)√ (cid:12)
m = (cid:12) m R(cid:48) U(cid:48) (cid:12) . (3.16)
(cid:101)α (cid:12) i 1i αi(cid:12)
(cid:12) (cid:12)
i
Taking into account that the lightest neutrino is massless in 2RHN model, we find (cid:15) and m˜ can
α α
be written into a rather simple form
NO : (cid:15)α = −136Mπv12WNOINαO, m˜α = (cid:12)(cid:12)√m3R1(cid:48)3Uα(cid:48)3+√m2R1(cid:48)2Uα(cid:48)2(cid:12)(cid:12)2, (3.17a)
IO : (cid:15)α = −136Mπv12WIOIIαO, m˜α = (cid:12)(cid:12)√m2R1(cid:48)2Uα(cid:48)2+√m1R1(cid:48)1Uα(cid:48)1(cid:12)(cid:12)2, (3.17b)
with
√
m m R(cid:48) R(cid:48) (m K −m K )
W = 2 3 12 13 3 3 2 2 , Iα = (cid:61)(U(cid:48) U(cid:48)∗),
NO m R(cid:48)2 +m R(cid:48)2 NO α3 α2
√ 2 12 3 13
m m R(cid:48) R(cid:48) (m K −m K )
W = 1 2 11 12 2 2 1 1 , Iα = (cid:61)(U(cid:48) U(cid:48)∗). (3.18)
IO m R(cid:48)2 +m R(cid:48)2 IO α2 α1
1 11 2 12
The explicit expressions of W and W for the three viable forms of the R-matrix are shown in
NO IO
table 2. Notice that W are fixed by the light neutrino masses m and ϑ which parametrizes
NO,IO 2,3
the R-matrix, and the bilinear invariants Iα depend on the low energy CP phases contained
NO,IO
in the mixing matrix U. As a result, if the signal of CP violation was observed in future neutrino
oscillation experiments or neutrinoless double decay experiments, a nonzero baryon asymmetry is
expectedtobegeneratedthroughleptogenesisinthisframework. Inthefollowing, weshallperform
a general analysis of leptogenesis in the 2RHN model with a generic residual CP transformation,
and the lepton mixing matrix can be parameterized as [52]
 c c s c s e−iδ
12 13 12 13 13
U = −s12c23−c12s13s23eiδ c12c23−s12s13s23eiδ c13s23 diag(1,eiφ2,1), (3.19)
s s −c s c eiδ −c s −s s c eiδ c c
12 23 12 13 23 12 23 12 13 23 13 23
wherec ≡ cosθ ,s ≡ sinθ ,δ andφandtheDiractypeandMajoranatypeCPviolatingphases
ij ij ij ij
respectively. Note that there is only one Majorana CP phase φ in the presence of one massless light
neutrino. Now we discuss the predictions for matter/antimatter asymmetry for each admissible
R-matrix.
• R-1st
In this case, the CP asymmetry parameter (cid:15) for the NO case is given by
α
3M φ
1
(cid:15) = W s c s sin(δ+ ),
e 16πv2 NO 12 13 13 2
(cid:20) (cid:21)
3M φ φ
1
(cid:15) =− W c s s s s sin(δ+ )−c c sin ,
µ 16πv2 NO 13 23 12 13 23 2 12 23 2
(cid:20) (cid:21)
3M φ φ
1
(cid:15) =− W c c s s c sin(δ+ )+c s sin , (3.20)
τ 16πv2 NO 13 23 12 13 23 2 12 23 2
where the expression of W has been listed in table 2. It is easy to check the identity (cid:15) +(cid:15) +
NO e µ
(cid:15) = 0 is fulfilled. Notice that the CP asymmetry (cid:15) is closely related to the lower energy CP
τ α
phases. IfboththeDiracphaseδ andtheMajoranaphaseφaretriviallyzero, alltheasymmetry
9
Mass ordering K (R(cid:48) ,R(cid:48) ,R(cid:48) ) W (W )
i 11 12 13 NO IO
√
NO K = K = 1 (0,cosϑ,ξsinϑ) ξ m2m3(m3−m2)sin2ϑ
R-1st 2 3 2(m2cos2ϑ+m3sin2ϑ)
√
IO K = K = 1 (cosϑ,ξsinϑ,0) ξ m1m2(m2−m1)sin2ϑ
1 2 2(m1cos2ϑ+m2sin2ϑ)
√
NO K = −K = 1 ±(0,coshϑ,−ξsinhϑ) ξ m2m3(m2+m3)sinh2ϑ
R-2nd 2 3 2(m2cosh2ϑ+m3sinh2ϑ)
√
IO K = −K = 1 ±(coshϑ,−ξsinhϑ,0) ξ m1m2(m1+m2)sinh2ϑ
1 2 2(m1cosh2ϑ+m2sinh2ϑ)
√
NO −K = K = 1 ±(0,−sinhϑ,−ξcoshϑ) ξ m2m3(m2+m3)sinh2ϑ
R-3rd 2 3 2(m2sinh2ϑ+m3cosh2ϑ)
√
IO −K = K = 1 ±(−sinhϑ,−ξcoshϑ,0) ξ m1m2(m1+m2)sinh2ϑ
1 2 2(m1sinh2ϑ+m2cosh2ϑ)
Table2: TheparametrizationofthefirstrowofR(cid:48) andthecorrespondingexpressionsofW andW forthethree
NO IO
viable forms of the R−matrix.
parameters (cid:15) , (cid:15) and (cid:15) would be vanishing such that a nonzero baryon asymmetry can not be
e µ τ
generated. The washout mass m for NO takes the form
(cid:101)α
m(cid:101)e = (cid:12)(cid:12)(cid:12)√m2s12c13ei2φ cosϑ+ξ√m3s13e−iδsinϑ(cid:12)(cid:12)(cid:12)2 ,
m(cid:101)µ = (cid:12)(cid:12)(cid:12)√m2(cid:16)c12c23−s12s13s23eiδ(cid:17)ei2φ cosϑ+ξ√m3c13s23sinϑ(cid:12)(cid:12)(cid:12)2 ,
m(cid:101)τ = (cid:12)(cid:12)(cid:12)√m2(cid:16)c12s23+s12s13c23eiδ(cid:17)ei2φ cosϑ−ξ√m3c13c23sinϑ(cid:12)(cid:12)(cid:12)2 . (3.21)
In the same manner, we find (cid:15) for IO spectrum is
α
3M φ
(cid:15) =− 1 W c s c2 sin ,
e 16πv2 IO 12 12 13 2
(cid:20) (cid:21)
−3M φ φ φ
(cid:15) = 1W s c s (c2 sin(δ− )+s2 sin(δ+ ))−c s (c2 −s2 s2 )sin ,
µ 16πv2 IO 13 23 23 12 2 12 2 12 12 23 13 23 2
(cid:20) (cid:21)
3M φ φ φ
(cid:15) = 1 W s c s (c2 sin(δ− )+s2 sin(δ+ ))+c s (s2 −s2 c2 )sin (3.22)
τ 16πv2 IO 13 23 23 12 2 12 2 12 12 23 13 23 2
and for the washout mass m we get
(cid:101)α
m(cid:101)e = c213(cid:12)(cid:12)(cid:12)√m1c12cosϑ+ξ√m2s12ei2φ sinϑ(cid:12)(cid:12)(cid:12)2 ,
m(cid:101)µ = (cid:12)(cid:12)(cid:12)√m1(s12c23+c12s13s23eiδ)cosϑ−ξ√m2(c12c23−s12s13s23eiδ)ei2φ sinϑ(cid:12)(cid:12)(cid:12)2 ,
m(cid:101)τ = (cid:12)(cid:12)(cid:12)√m1(s12s23−c12s13c23eiδ)cosϑ−ξ√m2(c12s23+s12s13c23eiδ)ei2φ sinϑ(cid:12)(cid:12)(cid:12)2 . (3.23)
We see that both (cid:15) and m depend on the CP violating phases δ, φ and the free parameter ϑ.
α (cid:101)α
We display the contour regions for Y /Yobs in the plane φ versus ϑ in figure 1, where the three
B B
mixing angles are taken to their best fit values [4] and the Dirac CP phase δ is either 0 or −π/2.
The neutrino mass spectrum is NO and IO respectively in the first row and second row of this
plot. We choose δ = 0 in the left column and δ = −π/2 in the right column. We find that the
experimentally measured value of the baryon asymmetry can be accommodated in the case of
NO, while Y is too small to account for its observed value for IO.
B
• R-2nd
10