Table Of ContentA fuller flavour treatment of N -dominated
2
leptogenesis
2
Stefan Antuscha, Pasquale Di Barib,c, David A. Jonesb, Steve F. Kingb
1
0
2 a
Max-Planck-Institut fu¨r Physik (Werner-Heisenberg-Institut)
n
a F¨ohringer Ring 6, 80805 Mu¨nchen, Germany
J
1 b School of Physics and Astronomy,University of Southampton, Southampton, SO17 1BJ, U.K.
3
c Department of Physics and Astronomy,University of Sussex, Brighton, BN1 9QH, U.K.
]
h
p February 1, 2012
-
p
e
Abstract
h
[
We discuss N -dominated leptogenesis in the presence of flavour dependent ef-
2 2
v fects that have hitherto been neglected, in particular the off-diagonal entries of the
2
3 flavour coupling matrix that connects the total flavour asymmetries, distributed in
1
different particle species, to the lepton and Higgs doublet asymmetries. We derive
5
. analytical formulae for the final asymmetry including the flavour coupling at the
3
0 N -decay stage as well as at the stage of washout by the lightest right-handed neu-
2
0
1 trino N1. Moreover, we point out that in general part of the electron and muon
:
v asymmetries (phantom terms), can completely escape the wash-out at the produc-
i
X tion and a total B L asymmetry can be generated by the lightest RH neutrino
−
r wash-out yielding so called phantom leptogenesis. However, the phantom terms are
a
proportionaltotheinitialN abundanceandinparticulartheyvanishforinitialzero
2
N -abundance. Takinganyoftheseneweffectsintoaccountcansignificantlymodify
2
the final asymmetry produced by the decays of the next-to-lightest RH neutrinos,
opening up new interesting possibilities for N -dominated thermal leptogenesis.
2
1 Introduction
Leptogenesis [1] is based on a popular extension of the Standard Model, where three
right-handed (RH) neutrinos N , with a Majorana mass term M and Yukawa couplings
Ri
h, are added to the SM Lagrangian,
1
= +iN γ ∂µN h ℓ N Φ˜ M Nc N +h.c. (i = 1,2,3, α = e,µ,τ).
L LSM Ri µ Ri− αi Lα Ri − 2 i Ri Ri
(1)
After spontaneous symmetry breaking, a Dirac mass term m = vh, is generated by the
D
vev v = 174 GeV of the Higgs boson. In the see-saw limit, M m , the spectrum
D
≫
of neutrino mass eigenstates splits in two sets: 3 very heavy neutrinos N ,N and N ,
1 2 3
respectively with masses M M M , almost coinciding with the eigenvalues of M,
1 2 3
≤ ≤
and 3 light neutrinos with masses m m m , the eigenvalues of the light neutrino
1 2 3
≤ ≤
mass matrix given by the see-saw formula [2]
1
m = m mT . (2)
ν − D M D
Neutrino oscillation experiments measure two neutrino mass-squared differences. For
normal schemes one has m2 m2 = ∆m2 and m2 m2 = ∆m2 , whereas for in-
3 − 2 atm 2 − 1 sol
verted schemes one has m2 m2 = ∆m2 and m2 m2 = ∆m2 . For m m
3 − 2 sol 2 − 1 atm 1 ≫ atm ≡
∆m2 +∆m2 = (0.050 0.001)eV [3] the spectrum is quasi-degenerate, while for
atm sol ±
m m ∆m2 = (0.0088 0.0001)eV [3] it is fully hierarchical (normal or in-
p1 ≪ sol ≡ sol ±
verted). The most stringent upper bound on the absolute neutrino mass scale comes from
p
cosmological observations. Recently, quite a conservative upper bound,
m < 0.2eV (95%CL), (3)
1
has been obtained by the WMAP collaboration combining CMB, baryon acoustic oscilla-
tions and supernovae type Ia observations [4].
The CP violating decays of the RH neutrinos into lepton doublets and Higgs bosons
at temperatures T & 100GeV generate a B L asymmetry one third of which, thanks
−
to sphaleron processes, ends up into a baryon asymmetry that can explain the observed
baryon asymmetry of the Universe. This can be expressed in terms of the baryon-to-
photon number ratio and a precise measurement comes from the CMBR anisotropies
observations of WMAP [4],
ηCMB = (6.2 0.15) 10−10. (4)
B ± ×
The predicted baryon-to-photon ratio η is related to the final value of the (B L)
B
−
asymmetry Nf by the relation
B−L
η 0.96 10−2Nf , (5)
B ≃ × B−L
2
where weindicatewithN anyparticlenumber orasymmetryX calculatedinaportionof
X
co-moving volume containing one heavy neutrino in ultra-relativistic thermal equilibrium,
so that e.g. Neq(T M ) = 1.
N2 ≫ 2
If one imposes that the RH neutrino mass spectrum is strongly hierarchical, then there
are two options for successful leptogenesis. A first one is given by the N -dominated sce-
1
nario, where the final asymmetry is dominated by the decays of the lightest RH neutrinos.
The main limitation of this scenario is that successful leptogenesis implies quite a restric-
tive lower bound on the mass of the lightest RH neutrino. Imposing independence of the
final asymmetry of the initial RH neutrino abundance and barring phase cancelations in
the see-saw orthogonal matrix entries the lower bound is given by [5, 6, 7]
M & 3 109GeV. (6)
1
×
This implies in turn a lower bound T & 1.5 109GeV on the reheating temperature as
reh
×
well [8] 1. The lower bound Eq. (6) is typically not respected in models emerging from
grand unified theories. It has therefore been long thought that, within a minimal type I
see-saw mechanism, leptogenesis is not viable within these models [11].
There is however a second option[12], namely the N -dominatedleptogenesis scenario,
2
where the asymmetry is dominantly produced from the decays of the next-to-lightest RH
neutrinos. In this case there is no lower bound on the lightest RH neutrino mass M .
1
Instead this is replaced by a lower bound on the next-to-lightest RH neutrino mass M
2
that still implies a lower bound on the reheating temperature.
TherearetwonecessaryconditionsforasuccessfulN -dominatedleptogenesisscenario.
2
The first one is the presence of (at least) a third heavier RH neutrino N that couples
3
to N in order for the CP asymmetries of N not to be suppressed as (M /M )2. The
2 2 1 2
∝
second necessary condition is to be able to circumvent the wash-out from the lightest
RH neutrinos. There is a particular choice of the see-saw parameters where these two
conditions are maximally satisfied. This corresponds to the limit where the lightest RH
neutrino gets decoupled, as in heavy sequential dominance, an example which we shall
discusslater. Inthiscasethebound,M & 1010GeVwhenestimatedwithouttheinclusion
2
of flavour effects, is saturated. In this limit the wash-out from the lightest RH neutrinos
is totally absent and the CP asymmetries of the N ’s are maximal.
2
In order to have successful N -dominated leptogenesis for choices of the parameters
2
not necessarily close to this maximal case a crucial role is played by lepton flavour effects
[13]. If M 109GeV M , as we will assume, then before the lightest RH neutrino
1 2
≪ ≪
1For a discussion of flavour-dependent leptogenesis in the supersymmetric seesaw scenario and the
corresponding bounds on M and T , see [9, 10].
1 reh
3
wash-out is active, the quantum states of the lepton doublets produced by N -decays get
2
fully incoherent in flavour space [14, 15, 16, 17, 18]. In this way the lightest RH neutrino
wash-out acts separately on each flavour asymmetry and is then much less efficient [13] 2.
It has then been shown recently that within this scenario it is possible to have successful
leptogenesis within models emerging from SO(10) grand-unified theories with interesting
potential predictions on the low energy parameters [21]. Therefore, the relevance of the
N -dominated scenario has been gradually increasing in the last years.
2
In this paper we discuss N -dominated leptogenesis in the presence of flavour depen-
2
dent effects that have hitherto been neglected, in particular the off-diagonal entries of the
flavour coupling matrix that connects the total flavour asymmetries, distributed in differ-
ent particle species, to the lepton and Higgs doublet asymmetries. We derive analytical
formulae for the final asymmetry including the flavour coupling at the N -decay stage as
2
well as at the stage of washout by the lightest RH neutrino N . We point out that in
1
general part of the electron and muon asymmetries will completely escape the wash-out
at the production and a total B L asymmetry can be generated by the lightest RH
−
neutrino wash-out yielding so called phantom leptogenesis. These contributions, that we
call phantom terms, introduce however a strong dependence on the initial conditions as
we explain in detail. Taking of all these new effects into account can enhance the final
asymmetry produced by the decays of the next-to-lightest RH neutrinos by orders of mag-
nitude, opening up new interesting possibilities for N -dominated thermal leptogenesis.
2
We illustrate these effects for two models which describe realistic neutrino masses and
mixing based on sequential dominance.
The layout of the remainder of the paper is as follows. In section 2 we discuss the
production of the asymmetry from N -decays and its subsequent thermal washout at
2
similar temperatures. In section 3 we discuss three flavour projection and the wash-out
stage at lower temperatures relevant to the lightest RH neutrino mass. This is where
the asymmetry which survives from N -decays and washout would typically be expected
2
to be washed out by the lightest RH neutrinos in a flavour independent treatment, but
which typically survives in a flavour-dependent treatment. This conclusion is reinforced
in the fuller flavour treatment here making N dominated leptogenesis even more relevant.
2
The fuller flavour effects of the N -dominated scenario are encoded in a compact master
2
formula presented at the end of this section and partly unpacked in an Appendix. Section
4 applies this master formula to examples where the new effects arising from the flavour
2Notice that if M 109GeV and K 1 the wash-out from the lightest RH neutrino can be still
1 1
≫ ≫
avoided thanks to heavy flavour effects [19, 20]. However,throughout this paper we will always consider
the case M 109GeV which is more interesting with respect to leptogenesis in grand-unified theories.
1
≪
4
couplings and phantom leptogenesis play a prominent role. We focus on examples where,
due to the considered effects, the flavour asymmetry produced dominantly in one flavour
can emerge as an asymmetry in a different flavour, a scenario we refer to as the flavour
swap scenario.
2 Production of the asymmetry from N -decays and
2
washout
In the N -dominated scenario, with M respecting the lower bound of M & 1010GeV and
2 2 2
M 109GeV, one has to distinguish two stages in the calculation of the asymmetry. In
1
≪
a first production stage, at T T M , a B L asymmetry is generated from the N
L 2 2
≃ ∼ −
decays. In a second wash-out stage, at T M , inverse processes involving the lightest
1
∼
RH neutrinos, the N ’s, become effective and wash-out the asymmetry to some level.
1
In the production stage, since we assume 1012GeV M 109GeV, the B L
2
≫ ≫ −
asymmetry isgeneratedfromtheN -decays inthesocalledtwo-flavour regime[14,15,16].
2
In this regime the τ-Yukawa interactions are fast enough to break the coherent evolution
of the tauon component of the lepton quantum states between a decay and the subsequent
inverse decay and light flavour effects have to be taken into account in the calculation of
the final asymmetry. On the other hand the evolution of the muon and of the electron
components superposition is still coherent.
If we indicate with ℓ the quantum state describing the leptons produced by N -
2 2
| i
decays, we can define the flavour branching ratios giving the probability P that ℓ is
2α 2
| i
measured in a flavour eigenstate ℓ as P ℓ ℓ 2. Analogously, indicating with
α 2α α 2
| i ≡ |h | i|
ℓ¯′ the quantum state describing the anti-leptons produced by N -decays, we can define
| 2i 2
the anti-flavour branching ratios as P¯ ℓ¯ ℓ¯′ 2. The tree level contribution is simply
2α ≡ |h α| 2i|
given by the average P0 = (P + P¯ )/2. The total decay width of the N ’s can be
2α 2α 2α 2
expressed in terms of the Dirac mass matrix as
M
Γ = 2 (m† m ) (7)
2 8πv2 D D 22
and is given by the sum Γ = Γ +eΓ¯ of the total decay rate into leptons and of the total
2 2 2
decay rate into anti-leptons respectively. The flavoured decay widths are given by
e
M
Γ = 2 m 2, (8)
2α 8πv2 | Dα2|
and can be also expressed as a suem, Γ = Γ + Γ¯ , of the flavoured decay rate into
2α 2α 2α
leptons and of the flavoured total decay rate into anti-leptons respectively.
e
5
Notice that the branching ratios can then be expressed in terms of the rates as P =
2α
Γ /Γ andP¯ = Γ¯ /Γ¯ . TheflavouredCP asymmetriesfortheN -decaysintoα-leptons
2α 2 2α 2α 2 2
(α = e,µ,τ) are then defined as
Γ Γ
2α 2α
ε − , (9)
2α
≡ − Γ +Γ
2 2
while the total CP asymmetries as 3
¯
Γ Γ
2 2
ε − = ε . (10)
2 ≡ − Γ +Γ¯ 2α
2 2
α
X
The three flavoured CP asymmetries can be calculated using [22]
3 ξ(x ) 2
ε = Im h⋆ h (h†h) j + Im h⋆ h (h†h) , (11)
2α 16π(h†h) α2 αj 2j √x 3(x 1) α2 αj j2
22 j6=2 (cid:26) j j − (cid:27)
X (cid:2) (cid:3) (cid:2) (cid:3)
where x (M /M )2 and
j j 2
≡
2 1+x 2 x
ξ(x) = x (1+x) ln − . (12)
3 x − 1 x
(cid:20) (cid:18) (cid:19) − (cid:21)
The tree-level branching ratios can then be expressed as
Γ m 2
P0 = 2α + (ε2) | Dα2| . (13)
2α Γ O ≃ (m† m )
2 D D 22
e
¯
Defining ∆P P P , it will prove useful to notice that the flavoured asymmetries
2α 2α 2α e
≡ −
can be decomposed as the sum of two terms 4 [15],
∆P
ε = P0 ε 2α , (14)
2α 2α 2 − 2
where the first term is due to an imbalance between the total number of produced leptons
andanti-leptonsandisthereforeproportionaltothetotalCP asymmetry, whilethesecond
3Notice that we define the total and flavoured CP asymmetries with a sign convention in such a way
that they have the same sign respectivelyof the produced B L and ∆ asymmetries rather then of the
α
−
L and L asymmetries.
α
4Thederivationis simple andcanbe helpfulto understandlateronphantomleptogenesis. Ifwewrite
P =P0 +∆P /2 and P =P0 ∆P /2, one has
2α 2α 2α 2α 2α− 2α
P Γ P¯ Γ ∆P
ε = 2α 2− 2α 2 =P0 ε 2α .
2α − Γ +Γ 2α 2α− 2
2 2
Notice that we are correcting a wrong sign in Ref. [7].
6
originates from a different flavour composition of the lepton quantum states with respect
to the CP conjugated anti-leptons quantum states.
Sphaleron processes conserve the flavoured asymmetries ∆ B/3 L (α = e,µ,τ).
α α
≡ −
Therefore, the Boltzmann equations are particularly simple in terms of these quantities
[19]. In the two-flavour regime the electron and the muon components of ℓ evolve
2
| i
coherently and the wash-out from inverse processes producing the N ’s acts then on the
2
sum N N + N . Therefore, it is convenient to define correspondingly P0
∆γ ≡ ∆e ∆µ 2γ ≡
P0 + P0 and ε ε + ε . More generally, any quantity with a subscript ‘γ’ has to
2e 2µ 2γ ≡ 2e 2µ
be meant as the sum of the same quantity calculated for the electron and for the muon
flavour component.
The asymmetry produced by the lightest and by the heaviest RH neutrino decays
is negligible since their CP asymmetries are highly suppressed with the assumed mass
pattern. The set of classic kinetic equations reduces then to a very simple one describing
the asymmetry generated by the N -decays,
2
dN
N2 = D (N Neq), (15)
dz − 2 N2 − N2
2
dN
∆γ = ε D (N Neq) P0 W C(2)N , (16)
dz 2γ 2 N2 − N2 − 2γ 2 γα ∆α
2
α=γ,τ
X
dN
∆τ = ε D (N Neq) P0 W C(2)N . (17)
dz 2τ 2 N2 − N2 − 2τ 2 τα ∆α
2
α=γ,τ
X
where z M /T. The total B L asymmetry can then be calculated as N = N +
2 ≡ 2 − B−L ∆τ
N . The equilibrium abundances are given by Neq = z2 (z )/2, where we indicated
∆γ N2 2 K2 2
with (z ) the modified Bessel functions. Introducing the total decay parameter K
i 2 2
K ≡
Γ (T = 0)/H(T = M ), the decay term D can be expressed as
2 2 2
e Γ 1
2
D (z ) = K z , (18)
2 2 2 2
≡ Hz γ
(cid:28) (cid:29)
e
where 1/γ (z ) is the thermally averaged dilation factor and is given by the ratios
2
h i
(z )/ (z ). Finally, the inverse decays wash-out term is given by
1 2 2 2
K K
1
W (z ) = K (z )z3. (19)
2 2 4 2K1 2 2
The total decay parameter K is related to the Dirac mass matrix by
2
m (m† m )
K = 2 , where m D D 22 (20)
2 2
m ≡ M
⋆ 2
e
e
7
is the effective neutrino mass [23] and m is equilibrium neutrino mass defined by [8, 24]
⋆
16π5/2√g v2
m ∗ 1.08 10−3eV. (21)
⋆
≡ 3√5 M ≃ ×
Pl
It will also prove convenient to introduce the flavoured effective neutrino masses m
2α
≡
P0 m and correspondingly the flavoured decay parameters K P0 K = m /m , so
2α 2 2α ≡ 2α 2 2α ⋆
that m = m and K = K . e
α 2α 2 α 2α 2
Tehe flavour coupling matrix C [7, 9, 15, 19, 25, 26] relates the asymmetriees stored in
P P
the leptonedoubleets and in the Higgs bosons to the ∆ ’s. It is therefore the sum of two
α
contributions,
C = Cℓ +CH , (22)
αβ αβ αβ
the first one connecting the asymmetry in the lepton doublets and the second connecting
the asymmetry in the Higgs bosons. Flavour dynamics couple because the generation
of a leptonic asymmetry into lepton doublets from N decays is necessarily accompanied
i
by a generation of a hypercharge asymmetry into the Higgs bosons and of a baryonic
asymmetry into quarks via sphaleron processes. The asymmetry generated into thelepton
doublets is moreover also redistributed to right handed charged particles. The wash-out
of a specific flavour asymmetry is then influenced by the dynamics of the asymmetries
stored in the other flavours because they are linked primarily through the asymmetry into
the Higgs doublets and secondarily through the asymmetry into quarks.
The condition of chemical equilibrium gives a constraint on the chemical potential
(hence number density asymmetry) of each such species. Solving for all constraints one
obtains the C explicitly. If we indicate with C(2) the coupling matrix in the two-flavour
αβ
regime, the two contributions to the flavour coupling matrix are given by
417/589 120/589 164/589 224/589
Cl(2) = − and Ch(2) = , (23)
30/589 390/589 ! 164/589 224/589 !
−
and summing one obtains
(2) (2)
C C 581/589 104/589
C(2) γγ γτ = . (24)
≡ Cτ(2γ) Cτ(2τ) ! 194/589 614/589 !
A traditional calculation, where flavour coupling is neglected, corresponds to approximat-
ing the C-matrix by the identity matrix. In this case the evolution of the two flavour
asymmetries proceeds uncoupled and they can be easily worked out in an integral form
[7, 8, 27],
N (z ) = Nin e−P20α Rzzi2n dz2′ W2(z2′) +ε κ(z ;K ), (25)
∆α 2 ∆α 2 2α 2 2α
8
where the efficiency factors are given by
κ(z ;K ) = z2 dz′ dNNi e−P20α Rzz′ dz2′′W2(z2′′). (26)
2 2α − 2 dz′ 2
Zz2in 2
We will neglect the first term due the presence of possible initial flavour asymmetries and
assume zin 1. The efficiency factors and therefore the asymmetries get frozen to a
2 ≪
particular value of the temperature given by T = M /z (K ), where [28]
Lα 2 B 2α
zB(K2α) ≃ 2+4K20α.13e−K22.5α = O(1÷10). (27)
Defining T min(T ,T ), the total final B L asymmetry at T is then given by
L Lτ Lγ L
≡ −
NT∼TL ε κ(K )+ε κ(K ). (28)
B−L ≃ 2γ 2γ 2τ 2τ
Assuming an initial thermal N -abundance, the final efficiency factors κ(K ) κ(z =
2 2α 2
≡
,K ) are given approximately by [7]
2α
∞
2 1
κ(K ) 1 exp K z (K ) . (29)
2α 2α B 2α
≃ K z (K ) − −2
2α B 2α (cid:20) (cid:18) (cid:19)(cid:21)
On the other hand, in the case of vanishing initial abundances 5 , the efficiency factors
are the sum of two different contributions, a negative and a positive one,
κf = κf (K ,P0 )+κf (K ,P0 ). (30)
2α − 2 2α + 2 2α
The negative contribution arises from a first stage where N Neq, for z zeq, and is
N2 ≤ N2 2 ≤ 2
given approximately by
κf−(K2,P20α) ≃ −P20 e−3πK82α eP220α NN2(zeq) −1 . (31)
2α (cid:18) (cid:19)
The N -abundance at zeq is well approximated by the expression
2 2
N(K )
N (zeq) N(K ) 2 , (32)
N2 2 ≃ 2 ≡ 2
1+ N(K )
2
that interpolates between the limit K 1, w(cid:16)herepzeq 1(cid:17)and N (zeq) = 1, and the
2 ≫ 2 ≪ N2 2
limitK 1, wherezeq 1andN (zeq) = N(K ) 3πK /4. Thepositivecontribution
2 ≪ 2 ≫ N2 2 2 ≡ 2
arises from a second stage when N Neq, for z zeq, and is approximately given by
N2 ≥ N2 2 ≥ 2
κf+(K2,P20α) ≃ z (K2)K 1−e−K2αzB(K2α2)NN2(zeq) . (33)
B 2α 2α (cid:18) (cid:19)
5These analyticalexpressionsreproduce very well the numericalresults found in [9]. The difference is
at most 30% around K 1 and much smaller than 10% elsewhere.
2α
≃
9
Ifflavour couplingistakenintoaccount, wecanstill solveanalyticallyeqs. (15)performing
the following change of variables
N N U U
∆γ′ = U ∆γ , where U γ′γ γ′τ (34)
N∆τ′ ! N∆τ ! ≡ Uτ′γ Uτ′τ !
is the matrix that diagonalizes
P0 C(2) P0 C(2)
P0 2γ γγ 2γ γτ , (35)
2 ≡ P20τ Cτ(2γ) P20τ Cτ(2τ) !
i.e. U P0U−1 = diag(P0 ,P0 ). In these new variables the two kinetic equations for the
2 2γ′ 2τ′
flavoured asymmetries decouple,
dN
∆γ′ = ε D (N Neq) P0 W N (36)
dz 2γ′ 2 N2 − N2 − 2γ′ 2 ∆γ′
2
dN
∆τ′ = ε D (N Neq) P0 W N , (37)
dz 2τ′ 2 N2 − N2 − 2τ′ 2 ∆τ′
2
where we defined
ε ε
2γ′ 2γ
U . (38)
ε2τ′ ! ≡ ε2τ !
The solutions for the two N are then still given by eq. (25) where, however, now
∆α′
the ‘unprimed’ quantities have to be replaced with the ‘primed’ quantities and therefore
explicitly one has
NT∼TL ε κ(K ), (39)
∆γ′ ≃ 2γ′ 2γ′
NT∼TL ε κ(K ).
∆τ′ ≃ 2τ′ 2τ′
Notice that the B L asymmetry at T T is still given by NT∼TL = NT∼TL +NT∼TL.
− ∼ L B−L ∆γ ∆τ
The two N ’s can be calculated from the two N ’s using the inverse transformation
∆α ∆α′
NT∼TL NT∼TL U−1 U−1
∆γ = U−1 ∆γ′ , where U−1 γγ′ γτ′ . (40)
N∆Tτ∼TL ! N∆Tτ∼′TL ! ≡ Uτ−γ1′ Uτ−τ1′ !
To study the impact of flavour coupling on the final asymmetry, we can calculate the ratio
N
B−L
R (41)
≡ N
(cid:12) B−L|C=I(cid:12)
(cid:12) (cid:12)
between the asymmetry calculated takin(cid:12)g into acco(cid:12)unt flavour coupling, and the asym-
(cid:12) (cid:12)
metry calculated neglecting flavour coupling, corresponding to the assumption C = I.
10
