$μ-τ$ Symmetry and Radiatively Generated Leptogenesis

0.26 MB·English

by  Y. H. Ahn

µ τ Symmetry and Radiatively Generated Leptogenesis
−
Y. H. Ahn1,∗, Sin Kyu Kang2,†, C. S. Kim1,‡ and Jake Lee2,§
1 Department of Physics, Yonsei University, Seoul 120-749, Korea
2 Center for Quantum Spacetime, Sogang University, Seoul 121-742, Korea
7 (Dated: February 2, 2008)
0
0
Abstract
2

n
We consider a µ τ symmetry in neutrino sectors realized at the GUT scale in the context
a −
J
of a seesaw model. In our scenario, the exact µ τ symmetry realized in the basis where the
5 −
1
charged lepton and heavy Majorana neutrino mass matrices are diagonal leads to vanishing lepton

3
asymmetries. We find that, in the minimal supersymmetric extension of the seesaw model with
v
7
0 large tanβ, the renormalization group (RG) evolution from the GUT scale to seesaw scale can
0
0 induce a successful leptogenesis even without introducing any symmetry breaking terms by hand,
1
6 whereas such RG effects lead to tiny deviations of θ and θ from π/4 and zero, respectively. It is
23 13
0
/
h shown that the right amount of the baryon asymmetry ηB can be achieved via so-called resonant
p
- leptogenesis, which can be realized at rather low seesaw scale with large tanβ in our scenario so
p
e
that the well-known gravitino problem is safely avoided.
h
:
v
Xi PACS numbers: 14.60.Pq,11.30.Fs, 11.10.Hi, 98.80.Cq, 13.35.Hb
r
a
∗ E-mail: [email protected]
† E-mail: [email protected]
‡ E-mail: [email protected]
§ E-mail: [email protected]
1
I. INTRODUCTION
Recent precise neutrino experiments appear to show robust evidence for the neutrino
oscillation. The present neutrino experimental data [1, 2, 3] exhibit that the atmospheric
neutrino deficit points toward a maximal mixing between the tau and muon neutrinos.
However, the solar neutrino deficit favors a not-so-maximal mixing between the electron
and muon neutrinos. In addition, although we do not have yet any firm evidence for the
neutrino oscillation arisen from the 1st and 3rd generation flavor mixing, there is a bound
on the mixing element U from CHOOZ reactor experiment, U < 0.2 [4]. Although
e3 e3
| |
neutrinos have gradually revealed their properties in various experiments since the historic
Super-Kamiokandeconfirmationofneutrinooscillations[1],propertiesrelatedtotheleptonic
CP violation are completely unknown yet. To understand in detail the neutrino mixings
observed in various oscillation experiments is one of the most interesting issues in particle
physics. The large values of θ and θ may be telling us about some underlying new
sol atm
symmetries of leptons which are not present in the quark sector, and may provide a clue to
understanding the nature of quark-lepton complementarity beyond the standard model.
Recently, there have been some attempts to explain the maximal mixing of the atmo-
spheric neutrinos and very tiny value of the 3rd mixing element U by introducing some
e3
approximate discrete symmetries [5, 6] or the mass splitting among the heavy Majorana
neutrinos in the seesaw framework [7]. In the basis where charged leptons are mass eigen-
states, the µ τ interchange symmetry has become useful in understanding the maximal
−
atmospheric neutrino mixing and the smallness of U [8, 9, 10, 11, 12]. The mass difference
e3
between the muon and the tau leptons, of course, breaks this symmetry in such a basis. So
we expect this symmetry to be an approximate one, and thus it must hold only for the neu-
trino sector. To generate non-vanishing but tiny mixing element U , in the literatures [11]
e3
the authors introduced µ τ symmetry breaking terms in leptonic mass matrices by hand
−
at tree level. We have also proposed a scheme for breaking of µ τ symmetry through an
−
appropriate CP phase in neutrino Dirac-Yukawa matrix so as to achieve both non-vanishing
U and successful leptogenesis [12]. In our scheme, µ τ symmetry breaking factor asso-
e3
−
ciated with the CP phase is essential to achieve both non-vanishing U and leptogenesis.
e3
However, besides the soft breaking terms, the µ τ symmetry is still approximate one in
−
the sense that its breaking effects in the lepton sector can arise via the radiative correc-
2
tions generated by the charged lepton Yukawa couplings which are not subject to the µ τ
−
symmetry.
In this work we propose that the precise µ τ symmetry, imposed in Ref. [12], exists
−
only at high energy scale such as the GUT scale and a renormalization group (RG) evolution
from high scale to low scale gives rise to the breaking of µ τ symmetry in the lepton
−
sector without introducing any ad hoc soft symmetry breaking terms. However, it turns
out that the RG effects in the standard model (SM) and even its minimal supersymmetric
extension are quite meager such that the size of U and the deviation of θ from the
e3 23
maximal mixing are tiny1. In this paper, however, we shall show that such small RG effecs
in supersymmetric seesaw model can lead to successful leptogenesis which is absent in the
exact µ τ symmetry, whereas lepton asymmetry generated in the context of the SM is too
−
small toachieve successful leptogenesis. Wenotethat theleptogenesis realized inour scheme
is, in fact, a kind of radiatively induced leptogenesis which has been discussed in [13, 14].
As will be shown later, in our scheme both real and imaginary parts of the combination of
neutrino Dirac Yukawa matrix (Y Y†) , which are needed for leptogenesis, are zero in the
ν ν jk
limit of the exact µ τ symmetry at tree level. We note that each of them is generated via
−
RG effects proportional to tan2β at low energy. Thus, the lepton asymmetry generated in
our scheme is proportional to tan4β and it can be enhanced by taking large value of tanβ.
This observation is different from the results in [13, 14], in which only real part of (Y Y†)
ν ν jk
is radiatively generated and thus lepton asymmetry is proportional to tan2β.
This paper is organized as follows. In Sec. II, we present a supersymmetric seesaw model
reflecting µ τ symmetry at a high energy scale such as the GUT scale. The discussion for
−
RG evolution from high scale such as the GUT scale to low scale is given in Sec. III. In Sec.
IV, we show how successful leptgenesis can be radiatively induced in our scheme. Numerical
results and conclusion are given in Sec. V.
1
This is so mainly because our scheme reflects normal hierarchical light neutrino mass spectrum.
3
II. SUPERSYMMETRIC SEESAW MODEL WITH µ τ SYMMETRY REALIZED
−
AT THE GUT SCALE
Let us begin by considering a supersymmetric version of the seesaw model, which is given
as the following leptonic superpotential:
1
W = lcY L H +NcY L H NcTM Nc , (1)
lepton L l · 1 L ν · 2 − 2 L R L
b b b b b b b b
where the family indices have been omitted and L , j = e,µ,τ 1,2,3 stand for the chiral
j
≡
super-multiplets of the SU(2) doublet lepton fieblds, H are the Higgs doublet fields with
L 1,2
hypercharge 1/2, Nc and lc are the super-multiplebt of the SU(2) singlet neutrino and
∓ jL jL L
charged lepton field,brespectibvely. In the above superpotential, M is the heavy Majorana
R
neutrinomassmatrix, andY andY arethe3 3chargedleptonandneutrinoDiracYukawa
l ν
×
matrices, respectively. After spontaneous symmetry breaking, the seesaw mechanism leads
to the following effective light neutrino mass term,
m = YTM−1Y υ2 , (2)
eff − ν R ν 2
where υ is the vacuum expectation value of the Higgs field with positive hypercharge and
2
denoted as υ = υsinβ with υ 174 GeV.
2
≈
Let us impose the µ τ symmetry for the neutrino sectors in the basis in which both the
−
charged lepton mass and heavy Majorana mass matrices are diagonal, and then the neutrino
Dirac-Yukawa matrix and the heavy Majorana neutrino mass matrix are given as
y y y M 0 0
11 12 12 1
   
Y = y y y , M = 0 M 0 , (3)
ν 12 22 23 R 2
   
y y y   0 0 M 
 12 23 22   2 
   
where the elements y of the neutrino Dirac-Yukawa matrix are all complex in general. As
ij
is shown in Ref. [12], the µ τ symmetry imposed as above is responsible for the neutrino
−
mixing pattern with θ = 45◦ and θ = 0◦ after seesawing. Here, we assume that the above
23 13
matricesEq. (3)reflectingtheµ τ symmetryarerealizedattheGUTscale, Q = 2 1016
GUT
− ×
GeV. As is also shown in [12], the seesaw model based on Eq. (3) leads to only the normal
hierarchical light neutrino mass spectrum because we take diagonal form of heavy Majorana
neutrino mass matrix [12, 15]. Thus, the RG effects on the neutrino mixing matrix U
PMNS
are expected to be very small even in the supersymmetric case. However, as will be shown
4
later, such small RG effects can trigger leptogenesis which is absent in the case of the exact
µ τ symmetry. With those exact µ τ symmetric structures in the neutrino sectors, we
− −
shall show that a successful leptogenesis could be achieved solely through the RG running
effects between the GUT and the seesaw scales without being in conflict with experimental
low energy constraints.
III. RELEVANT RGE’S IN MSSM
In the minimal supersymmetric standard model (MSSM), the radiative behavior of the
heavyMajorananeutrinosmassmatrixM isdictatedbythefollowingRGequation[16, 17]:
R
d
M = 2[(Y Y†)M +M (Y Y†)T], (4)
dt R ν ν R R ν ν
where t = 1 ln(Q/Q ) with an arbitrary renormalization scale Q. The RG equation
16π2 GUT
for the neutrino Dirac-Yukawa matrix can be written as
dY 3
ν = Y [(T 3g2 g2)+(Y†Y +3Y†Y )], (5)
dt ν − 2 − 5 1 l l ν ν
where T = Tr(3Y†Y + Y†Y ), and g ,g are the SU(2) and U(1) gauge coupling con-
u u ν ν 2 1 L Y
stants, respectively.
For our convenience, let us re-formulate the RG equation, Eq. (4), in the basis where
M is diagonal. Since M is symmetric, it can be diagonalized with a unitary matrix V,
R R
VTM V = diag(M ,M ,M ). (6)
R 1 2 3
Note that as the structure of mass matrix M changes with the evolution of the scale, that
R
of the unitary matrix V depends on the scale, too. The RG evolution of the unitary matrix
V(t) can be written as
d
V = VA, (7)
dt
where matrix A is anti-hermitian, A† = A, due to the unitarity of V. Then, differentiating
−
Eq. (6), we obtain
dM δ
i ij = ATM +M A +2 VT[(Y Y†)M +M (Y Y†)T]V . (8)
dt ij j i ij { ν ν R R ν ν }ij
5
It immediately follows from the anti-hermiticity of A that A = 0 in Eq. (8). Absorbing
ii
the unitary transformation into the neutrino Dirac-Yukawa coupling
Y VTY , (9)
ν ν
≡
the real diagonal part of Eq. (8) becomes
dM
i = 4M (Y Y†) . (10)
dt i ν ν ii
On the other hand, the off-diagonal part of Eq. (8) leads to
M +M M M
A = 2 k jRe[(Y Y†) ]+2i j − kIm[(Y Y†) ], (j = k). (11)
jk M M ν ν jk M +M ν ν jk 6
k j j k
−
Note that the real part of A is singular for the degenerate cases with M = M , and the
jk j k
RG equation for Y in M diagonal basis is written as
ν R
dY 3
ν = Y [(T 3g2 g2)+(Y†Y +3Y†Y )]+ATY . (12)
dt ν − 2 − 5 1 l l ν ν ν
ThesingularityinRe[A ]canbeeliminatedwiththehelpofanappropriaterotationbetween
jk
degenerate heavy Majorana neutrino states. Such a rotation does not change any physics
and it is equivalent to absorbing the rotation matrix R into the neutrino Dirac-Yukawa
matrix Y ,
ν
Y Y = RY , (13)
ν ν ν
→
e
where the matrix R, particularly rotating 2nd and 3rd generations of heavy Majorana neu-
trinos, can be parameterized as
1 0 0
 
R(x) = 0 cosx sinx . (14)
 
 
0 sinx cosx
 
 − 
Then, the singularity in the real part of A is indeed removed when the rotation angle x is
jk
taken to fulfill the condition,
Re[(Y Y†) ] = 0, for any pair j,k corresponding to M = M . (15)
ν ν jk j k
e e
For our purpose, let us parameterize Y at the GUT scale as follows:
ν
ρeiϕ11 ωeiϕ12 ωeiϕ12
 
Y = d ωeiϕ12 κeiϕ22 eiϕ23 , (16)
ν
 
ωeiϕ12 eiϕ23 κeiϕ22 
 
 
6
where ϕ denote CP phases in Y , and define the following useful hermitian parameter
ij ν
H H H
11 12 12
 
H (Y Y†) = d2 H∗ H H , (17)
≡ ν ν  12 22 23 
H∗ H H 
 12 23 22 
 
where
H = ρ2 +2ω2,
11
H = ρωei(ϕ11−ϕ12) +ωκei(ϕ12−ϕ22) +ωei(ϕ12−ϕ23),
12
H = ω2 +κ2 +1,
22
H = ω2 +2κcos(ϕ ϕ ). (18)
23 22 23
−
As shown in [12], the hermitian parameter H in the limit of the exact µ τ symmetry leads
−
to vanishing lepton asymmetry which is disastrous for successful leptogenesis. To generate
non-vanishing lepton asymmetry, we need to break the exact degeneracy of the masses of
2nd and 3rd heavy Majorana neutrinos and the µ τ symmetric texture of Y proposed in
ν
−
Eq. (16). In our scenario, as will be shown later, only the RG evolution, without including
any ad hoc soft breaking terms, is responsible for such a breaking required for successful
leptogenesis.
For our µ τ symmetric Y given in Eq. (16), the angle satisfying the condition Eq. (15)
ν
−
is x = π/4. Without a loss of generality, taking x = π/4, we obtain
±
H √2H 0
11 12
 
H (Y Y†) = RHRT = √2H∗ H +H 0 . (19)
≡ ν ν 12 23 22
 
 
e e e 0 0 H +H
 23 22 
 − 
It is obvious from Eq. (19) that Re[H ] = 0 and thus the singularity in A does not
23(32) 23(32)
appear. We also note that Im[H e] = 0, which is due to the µ τ symmetric structure of
23(32)
−
H 2. Aswillbeshownlater,sinceetheCPasymmetryrequiredforleptogenesisisproportional
teo Im[H2 ] = 2Re[H ]Im[H ], both real and imaginary parts of H should be nonzero for
23 23 23 23
successeful leptogeneesis. Inethis work, we shall show that non-vanieshing values of them can
be generated through the RG evolution.
2 In Ref. [13, 14], the authors considered the radiatively induced leptogenesis based on arbitrary textures
of neutrino Dirac-Yukawa matrix for which Im[H23] needs not to be zero in general.
e
7
Now, let us consider RG effects which may play an important role in successful leptoge-
nesis. First, the parameter δ = 1 M /M reflecting the mass splitting of the degenerate
N 3 2
−
heavy Majorana neutrinos is governed by the following RGE which can be derived from Eq.
(10),
dδ
N
= 4(1 δ )[H H ] 8Re[H ]. (20)
N 22 33 23
dt − − ≃
e e
The solution of the RGE (20) is approximately given by
δ = 8d2 ω2+2κcos(ϕ ϕ ) t, (21)
N 22 23
{ − }·
where we used Eq. (18). Note that the radiative splitting of degenerate heavy Majorana
neutrinos masses depends particularly on the phase difference, ϕ ϕ .
22 23
−
RGE of the parameter H is written as
dH e 3
= 2[(T 3g2 g2)H +Y (Y†Y )Y† +3H2]+ATH +HA∗. (22)
dt − 2 − 5 1 ν l l ν
e
e e e e e e
Considering the structure of H in Eq. (19), up to non-zero leading contributions in the right
side of Eq. (22), RGE of H eis given by
23
dRe[He]
23 y2Re[(Y Y∗ )],
dt ≃ τ ν23 ν33
e
dIm[H ] e e
23 2Im[(Y Y†Y Y†) ] 2y2Im[(Y Y∗ )]. (23)
dt ≃ ν l l ν 23 ≃ τ ν23 ν33
e
e e e e
In terms of the parameters in Eq. (18), the radiatively generated H is given approximately
23
by e
κ2 1
Re[H ] y2d2 − t,
23 ≃ τ 2 ·
Im[He ] 2y2d2κsin(ϕ ϕ ) t. (24)
23 ≃ τ 23 − 22 ·
e
Interestingly enough, the radiatively generated Im[H ] is proportional to sin(ϕ ϕ ).
23 23 22
−
e
IV. RADIATIVELY INDUCED RESONANT LEPTOGENESIS
When two lightest heavy Majorana neutrinos are nearly degenerate, the CP asymmetry
through their decays gets dominant contributions from self-energy diagrams and can be
written as [18, 19, 20, 21]
Γ(N lϕ) Γ(N lϕ†) Im[(Y Y†)2 ] Γ2 −1
ε = i → − i → ν ν ik 1+ k , (25)
i Γ(Ni → lϕ)+Γ(Ni → lϕ†) ≃ Xk6=i 16π(YνYν†)iiδN,ik(cid:16) 4Mi2δN2,ik(cid:17)
8
where Γ is the tree-level decay width of the k-th right-handed neutrino,
k
(Y Y†) M
Γ = ν ν kk k, (26)
k
8π
and δ is a parameter which denotes the degree of the mass splitting between two degen-
N,ik
erate heavy Majorana neutrinos,
M
k
δ 1 . (27)
N,ik
≡ − M
i
As shown in [12], the neutrino Dirac-Yukawa matrix and the heavy Majorana neutrino mass
matrix given in the forms of Eq. (3) are consistent with neutrino oscillation data only when
M M M . Here we note that it is rather difficult to realize naturally such an inverted
1 2 3
≫ ≃
hierarchy of the heavy Majorana neutrino mass spectrum in GUT models. For the mass
hierarchy M M M , the decay of N takes place in thermal equilibrium and thus the
1 2 3 1
≫ ≃
lepton asymmetry required for successful leptogenesis will be accomplished by ε given as
2(3)
follows:
Im[(Y Y†)2 ] Γ2 −1
ε = ν ν 23 1+ 3(2) , (28)
2(3) 16π(YνYν†)22(33)δN(cid:16) 4M22(3)δN2 (cid:17)
where δ = δ . From Eqs. (19,21,24), the lepton asymmetry is given by
N N,23
Im[(H )2]
23
ε
2(3)
≃ 16πH δ
2e2(33) N
y4κ(κ2 1)sin(∆ϕ) t
τe − · , (29)
≃ 64π ω2+2κcos(∆ϕ) h
2(3)
{ }·
where ∆ϕ ϕ ϕ , and two parameters h are defined as
23 22 2(3)
≡ −
h = H /d2 = 1+κ2 +2ω2+2κcos(∆ϕ),
2 22
h = He /d2 = 1+κ2 2κcos(∆ϕ). (30)
3 33
−
e
In Eq. (29) we neglected the term containing the decay width since it turns out to be very
small in our scenario. Note that due to the opposite sign of the term 2κcos(∆ϕ) in h and
2
h , either h orh becomes larger depending on the sign of cos(∆ϕ). This implies that either
3 2 3
of the two degenerate heavy Majorana neutrinos, N or N , would dominantly contribute
2 3
to the leptogenesis over two distinct regions of ∆ϕ. More specifically, for ∆ϕ < 90◦ or
∆ϕ > 270◦, ǫ is dominant over ǫ because of h h . Otherwise, ǫ is dominant over
2 3 2 3 3
≪
ǫ . However, in our scenario as shown in Fig. 1, only the former case (ǫ ǫ ) is allowed,
2 2 3
≫
9
mainly because of the experimental constraint ∆m2 /m2 1, as will be shown later in
sol atm ≪
detail.
We remark that the radiatively induced lepton asymmetry ǫ is proportional to y4 =
i τ
y4 (1 + tan2β)2, and thus for large tanβ it can be highly enhanced and proportional
τ,SM
to tan4β. Furthermore, it has an explicit dependence of the evolution scale t. These two
points are different from what was obtained in [13, 14], where the neutrino Dirac -Yukawa
matrix has been arbitrary chosen so that Im[H ] could be initially non-zero and the lepton
23
asymmetry became proportional to y2 at leaeding order and, at the same time, the scale
τ
dependence was cancelled out.
The resulting baryon-to-photon ratio is estimated in the context of MSSM to be
η 1.67 10−2 ε κ , (31)
B i i
≃ − × ·
Xi
where the efficiency factor κ describes the washout of the produced lepton asymmetry ε .
i i
The efficiency in generating the resultant baryon asymmetry is usually controlled by the
parameter defined as
Γ m˜
i i
K = , (32)
i
≡ H m
∗
where H is the Hubble constant, and m , the so-called effective neutrino mass, is given by
i
e [m m† ]
m = D D ii, (33)
i
M
i
e
and m is defined as
∗
5
16π2 1 υ2
m = g2 1.08 10−3 eV, (34)
∗ ∗
3√5 M ≃ ×
Planck
where we adopted M = 1.22 1019 GeV and the effective number of degrees of freedom
Planck
×
g g = 106.75, g 2g . Although most analyses on baryogenesis via leptoge-
∗ ∗SM ∗MSSM ∗SM
≃ ≃
nesis conservatively consider K < 1, much larger values of K , even larger than 103, can be
i i
tolerated [21].
From the actual numerical calculations, we find that our scenario resides in the so-called
strong washout regime with
K & 1, K & 10. (35)
2 3
10