1. No category

Outline
The left-right symmetric seesaw mechanism
Thomas Konstandin, KTH, Stockholm
Akhmedov, Blennow, H¨
allgren, T.K., Ohlsson,
hep-ph/0612194,
JHEP, 04:022, 2007.
Outline
Outline
1
Introduction and Motivation
Seesaw mechanism
Leptogenesis
2
Left-right Symmetric Models
Left-right symmetric seesaw mechanism
Leptogenesis
Stability
3
Conclusions
Conclusions
Introduction
Left-right Symmetric Models
Conclusions
Seesaw mechanism
Mass generation by the seesaw mechanism
Neutrino masses are limited by cosmological bounds to be
X
mνi . 1 eV.
i
If this mass is only due to the Higgs mechanism, the corresponding
Yukawa coupling would be
y=
mν
∼ 10−12 .
v
Introduction
Left-right Symmetric Models
Conclusions
Seesaw mechanism
Seesaw mechanism
This issue can be elegantly resolved by adding a Majorana mass
term for the right-handed neutrino to the Lagrangian
1
L 3 νR mN (νR )c +
2
h.c.
what leads to the following mass matrix
0 mD
νL
(νL )c νR
T m
mD
(νR )c
N
and after block diagonalization in leading order (mN mD )
T
−mD m1 mD
0
ν
c
N
.
ν N
Nc
0
mN
Introduction
Left-right Symmetric Models
Conclusions
Seesaw mechanism
Seesaw mechanism
The smallness of the neutrino masses in the type I seesaw
mν = −mD
1 T
m
mN D
can be explained by a large (GUT) scale
mN ∼ 1014 GeV,
mν ∼ 1 eV,
mD ∼ 100 GeV.
Additionally, since the Majorana mass term is lepton number
violating and CP-violating, this mechanism provides the possibility
to explain the baryon asymmetry of the Universe (BAU) via
leptogenesis.
Introduction
Left-right Symmetric Models
Conclusions
Leptogenesis
Sakharov conditions
The seesaw mechanism provides all prerequisites to produce a
baryon asymmetry
When the temperature of the Universe drops below the mass
of the right-handed neutrinos, the neutrinos become
over-abundant and decay out-of-equilibrium
This decay is lepton number violating (mN 6= 0)
∗)
This decay is CP-violating (mN 6= mN
The sphaleron process converts the lepton asymmetry into a
baryon asymmetry
Introduction
Left-right Symmetric Models
Conclusions
Leptogenesis
Leptogenesis
The baryon-to-photon ratio, ηB = (6.1 ± 0.2) × 10−10 , can in
leading order be parametrized as
ηB = 3 × 10−2 ηN
where η denotes the efficiency factor of the decays of the lightest
right-handed neutrino and N the CP asymmetry in its decays into
leptons and Higgs particles
N =
Γ(N1 → l H) − Γ(N1 → ¯l H ∗ )
.
Γ(N1 → l H) + Γ(N1 → ¯l H ∗ )
Introduction
Left-right Symmetric Models
Conclusions
Leptogenesis
Decay asymmetry
Fukugita, Yanagida, ’86
The L and CP violating part of the decay rate results from a cross
term between the tree level decay amplitude and the following loop
diagrams
Lh
Lh
Hu
j
νR
νR1
(a)
j
νR
νR1
Lf
Hu
Hu
Hu
Lf
(b)
and can in the limit mN1 mN2 be written
N =
†
∗) ]
3mN1 Im[(mD mν mD
11
.
†
2
16πv
(mD mD )11
Introduction
Left-right Symmetric Models
Leptogenesis
Decay asymmetry
Davidson, Ibarra, ’02
In the type I seesaw model, the decay asymmetry fulfills the
Davidson-Ibarra-bound
p
q
2 |
3mN1 |∆matm
2 | ≈ 0.05 eV.
N <
,
|∆matm
16πv 2
This leads to the fact that for viable leptogenesis, N ∼ 10−7 , a
lower bound on the mass of the lightest right-handed is given by
mN1 & 108 GeV
for typical efficiency factors of η . 1. Besides, this bound is not
easily saturated.
Conclusions
Introduction
Left-right Symmetric Models
Conclusions
Leptogenesis
Gravitino bound in SUSY models
Moroi, Murayama, Yamaguchi, ’93
Kawasaki, Kohri, Moroi, ’05
In supersymmetric models, the decay
of the gravitino into the LSP poses
constraints on the reheating
temperature (dark matter
over-abundance, BBN constraints).
Depending on the parameters of the
mSUGRA model this bound lies in
the range
TR . 107 GeV toTR . 1010 GeV ↔ mN1 > 108 GeV.
Introduction
Left-right Symmetric Models
Conclusions
Leptogenesis
Hierarchies in Yukawa couplings and tuning
Motivated by GUT models, one expects a similar hierarchical
structure for the Yukawa coupling of the neutrino as found for the
other fermions
yup ∼ yν , ydown ∼ yl
Since the neutrino mass matrix has only a mild hierarchy (at least
between the two largest elements) and
mν = −mD
1 T
m
mN D
the Majorana mass mN needs to have the doubled hierarchy of mD .
Hence, hierarchical neutrino Yukawa couplings seem to be
unnatural in the seesaw framework.
Introduction
Left-right Symmetric Models
Conclusions
Leptogenesis
Open questions in the seesaw mechanism
Hierarchy in the neutrino Yukawa coupling
A hierarchy in the Yukawa couplings, as expected from GUT
models requires the doubled hierarchy in the Majorana mass term,
what is an unnatural situation.
Gravitino bound
In supersymmetric models, thermal leptogenesis requires rather
high reheating temperatures, that can lead to over-closure of the
Universe with gravitinos.
Introduction
Left-right Symmetric Models
Conclusions
Left-right symmetric seesaw mechanism
Mass generation by the seesaw mechanism
The left-right symmetric framework is based on the gauge group
SU(2)L × SU(2)R × SU(3)color × U(1)B−L
and contains the following (color singlet) Higgs fields
0
Φ =v
Φ(2, 2, 0)
0
∆ = vL
∆L (3, 1, −2)
L0 ∆ R = vR
∆R (1, 3, 2)
By spontaneous symmetry breaking, the neutral components of the
Higgs fields obtain vacuum expectation values
¯α ΦLβ + f αβ LT
L ⊃ f αβ RαT C iτ2 ∆R Rβ + y αβ R
α C iτ2 ∆L Lβ + h.c..
Introduction
Left-right Symmetric Models
Conclusions
Left-right symmetric seesaw mechanism
Mass generation by the seesaw mechanism
T and the
The left-right symmetry imposes in this case mD = mD
seesaw relation reads
1
vL
− mD
mD = mνII + mνI .
mν = mN
vR
mN
Strategy: Use mD = mup motivated by GUTs/prejudice. The
remaining parameters of the model are then:
The lightest neutrinos mass m0
The hierarchy (normal/inverted) of the light neutrinos
Five Majorana phases (three more than in pure type I)
The ratio vR /vL
This seesaw relation is invertible for given mν and mD and leads to
2n solutions for mN in the case of n flavors (with same low energy
phenomenology).
Akhemdov, Frigerio, ’05
Introduction
Left-right Symmetric Models
Conclusions
Left-right symmetric seesaw mechanism
Small vR /vL
In the one-flavor case the solution has a two-fold ambiguity
s
mν vR
mν2 vR 2
2 vR
mN =
±
+ mD
2 vL
4
vL
vL
cancellation: |mI | ≈ |mII | |mν |,
domination: mI ≈ mν or mII ≈ mν
mν vvRL
14
10
mN [GeV]
12
10
2
mD
mν
10
10
8
mD
10
q
vR
vL
vR
vL
6
10
14
10
16
10
18
10
≈
20
2
4mD
mν2
22
10
10
vR/vL
24
10
26
10
Introduction
Left-right Symmetric Models
Conclusions
Left-right symmetric seesaw mechanism
Inversion formula
In the three-flavor case, the solutions can be given in closed form
what requires to find the roots of a quartic equation.
Analogously to the one-flavor case, the solution in the three-flavor
case have a eight-fold ambiguity (y = yup ).
18
0.75
10
mN
3
mN
2
mN
16
10
0.5
1
12
ui
10
i
mN [GeV]
14
10
10
10
0.25
8
10
6
10
4
10
14
10
16
10
18
10
20
22
10 10
vR/vL
24
10
26
10
28
10
0
14
10
16
10
18
10
20
22
10 10
vR/vL
24
10
26
10
Pure type II ’+ + +’: Gravitino problem for vR /vL > 1021 .
28
10
Introduction
Left-right Symmetric Models
Conclusions
Left-right symmetric seesaw mechanism
Pure type I
The pure type I solution (’− − −’) shows a large spread in the
right-handed masses and a strong suppression of mixing angles.
16
10
mN
3
mN
2
mN
14
12
10
-3
4×10
ui
1
10
8
2×10
10
i
mN [GeV]
10
-3
10
6
10
4
10
14
10
16
10
18
10
20
22
10
10
vR/vL
24
10
26
10
0
14
10
16
10
Leptogenesis possible? Fine-tuning needed?
18
10
20
22
10
10
vR/vL
24
10
26
10
Introduction
Left-right Symmetric Models
Conclusions
Left-right symmetric seesaw mechanism
Mixed solution
In addition there are six mixed cases.
16
0.6
10
mN
3
mN
2
mN
14
12
10
0.4
1
ui
10
10
i
mN [GeV]
10
8
0.2
10
6
10
4
10
14
10
16
10
18
10
20
22
10
10
vR/vL
24
10
26
10
0
14
10
16
10
18
10
20
22
10
10
vR/vL
24
10
’− − +’: Gravitino bound eventually fulfilled. Leptogenesis
possible? Stability for vR /vL > 1018 ?
26
10
Introduction
Left-right Symmetric Models
Conclusions
Leptogenesis
Decay asymmetry
Hambye, Senjanovic, ’03
Antusch, King, ’04
The L and CP violating part of the decay rate results from a cross
term between the tree level decay amplitude and the following loop
diagrams
Lh
Lh
Hu
j
νR
νR1
(a)
j
νR
νR1
Lf
Hu
Hu
Hu
(b)
∆
νR1
Lf
Hu
Hu
Lh
(c)
and the Higgs triplet leads to an additional contribution.
Lf
Introduction
Left-right Symmetric Models
Conclusions
Leptogenesis
Leptogenesis
Antusch, King, ’04
In the limit mN1 mN2 , m∆ the asymmetry can be written
N 1 =
†
∗) ]
3mN1 Im[(mD (mνI + mνII )mD
11
.
†
2
16πv
(m mD )11
D
Due to the modified seesaw formula, the DI-bound is avoided, but
leads to the slightly weaker bound
N 1 .
3mN1 mν,max
16πv 2
and hence
mN1 & 3 × 107 GeV.
This bound is only slightly better than in the pure type I case, but
easier to saturate due to additional Majorana phases.
Introduction
Left-right Symmetric Models
Conclusions
Leptogenesis
Additional Majorana phases
¨ llgren, T.K., Ohlsson, ’06
Akhmedov, Blennow, Ha
For example, in the one-flavor case, the relative phase κ between
mD and mν cannot be removed and can even lead to leptogenesis
for the type II dominated solution.
N =
3 mν mN
sin(4κ) ,
16π v 2
ηB = 1.7 × 10−6 eV
2
mν mN
sin(4κ).
|mD |2 v 2
Thus, it is possible to saturate the Antusch/King bound and to
reproduce the observed baryon asymmetry e.g. with the values
(κ = π/8)
|y | = 10−4 ,
m0 = 0.1 eV ,
vR
= 1.7 × 1020 .
vL
Introduction
Left-right Symmetric Models
Conclusions
Leptogenesis
Leptogenesis
The same mechanism is operative in the two-flavor case in the limit
2
4 mD,1
m02
2
4 mD,2
vR
,
vL
m02
for the two solutions of type ’±+’ as long as the second eigenvalue
of the Yukawa coupling y2 > 5 × 10−4 .
16
0.6
10
mN
3
mN
2
mN
14
12
0.4
1
ui
10
10
i
mN [GeV]
10
10
8
0.2
10
6
10
4
10
14
10
16
10
18
10
20
22
10
10
vR/vL
24
10
26
10
0
14
10
16
10
18
10
20
22
10
10
vR/vL
24
10
26
10
Introduction
Left-right Symmetric Models
Leptogenesis
Leptogenesis
10
10
ηB
-7
10
10
-8
-9
-10
10
10
κ = π/4
κ = π/8
-6
10
10
-5
-11
-12
10
18
10
19
vR/vL
10
20
10
21
The numerical evaluation for the three-flavor case gives for the
solution ’− − +’ ( m0 = 0.1 eV, y = yu ). Leptogenesis is
generally viable for the four solutions of type ’± ± +’.
Conclusions
Introduction
Left-right Symmetric Models
Conclusions
Stability
Stability measure
¨ llgren, T.K., Ohlsson, ’06
Akhmedov, Blennow, Ha
To quantify tuning we use the following stability measure
v
1/3 u X
det mN u 12
∂ml 2
t
.
Q=
det mν ∂Mk
k,l=1
where ml and Mk determine the light and heavy neutrino mass
matrices according to
X
X
mN =
(Mk + iMk+N )Tk , mν =
(mk + imk+N )Tk ,
k
k
and Tk , k ∈ [1, 6], form a normalized basis of the complex
symmetric 3 × 3 matrices.
Introduction
Left-right Symmetric Models
Conclusions
Conclusions
Summary
The qualitative analysis mostly depends on the fact the y = yup
has a large hierarchy.
7
10
(++-)
(-+-)
(+--)
(---)
(+-+)
(--+)
(-++)
(+++)
6
10
5
10
4
Q
10
3
10
2
10
1
10
0
10
14
10
16
10
18
10
20
10 10
vR/vL
22
24
10
26
10
28
10
Introduction
Left-right Symmetric Models
Conclusions
Conclusions
Summary
The qualitative analysis mostly depends on the fact the y = yup
has a large hierarchy.
7
10
(++-)
(-+-)
(+--)
(---)
(+-+)
(--+)
(-++)
(+++)
6
10
5
10
4
Q
10
stability, leptogenesis
3
10
2
10
1
10
0
10
14
10
16
10
18
10
20
10 10
vR/vL
’± ± −’: Unstable, no leptogenesis
22
24
10
26
10
28
10
Introduction
Left-right Symmetric Models
Conclusions
Conclusions
Summary
The qualitative analysis mostly depends on the fact the y = yup
has a large hierarchy.
7
10
(++-)
(-+-)
(+--)
(---)
(+-+)
(--+)
(-++)
(+++)
6
10
5
10
4
stability
Q
10
3
10
2
10
1
10
0
10
14
10
16
10
18
10
20
10 10
vR/vL
’± − +’, vR /vL 1023 : Unstable
22
24
10
26
10
28
10
Introduction
Left-right Symmetric Models
Conclusions
Conclusions
Summary
The qualitative analysis mostly depends on the fact the y = yup
has a large hierarchy.
7
10
(++-)
(-+-)
(+--)
(---)
(+-+)
(--+)
(-++)
(+++)
6
10
5
10
4
Q
10
3
10
2
10
gravitinos
1
10
0
10
14
10
16
10
18
10
20
10 10
vR/vL
22
24
10
26
10
28
10
’± + +’, vR /vL 1023 : Gravitino bound violated
Introduction
Left-right Symmetric Models
Conclusions
Conclusions
Summary
The qualitative analysis mostly depends on the fact the y = yup
has a large hierarchy.
7
10
(++-)
(-+-)
(+--)
(---)
(+-+)
(--+)
(-++)
(+++)
6
10
5
10
4
Q
10
3
10
2
10
1
10
0
10
14
10
16
10
18
10
20
10 10
vR/vL
22
24
10
26
10
’± ± +’, 1016 vR /vL 1023 : Sweet spot
28
10
Introduction
Left-right Symmetric Models
Conclusions
Conclusions
Conclusion
In case of hierarchical Yukawa coupling for the neutrinos, the
left-right symmetric type I+II framework constitutes a model that
compared to the pure type I framework has the following
properties:
GUT embedding
A large hierarchy in the neutrino Yukawa coupling is naturally
implemented. The required tuning is reduced from Q ∼ 105 to
Q ∼ 102 .
SUSY embedding - Gravitino bound
The lower bound on the lightest right-handed neutrino mass is
slightly relaxed from mN1 > 5 × 108 GeV to mN1 > 1 × 108 and
the Antusch/King bound can be easily saturated using the
additional Majorana phases.