The left-right symmetric seesaw mechanism · Seesaw mechanism Mass generation by the seesaw mechanism Neutrino masses are limited by cosmological bounds to be X i mν i. 1 eV. If

Outline

The left-right symmetric seesaw mechanism

Thomas Konstandin, KTH, Stockholm

Akhmedov, Blennow, Hallgren, T.K., Ohlsson,hep-ph/0612194,

JHEP, 04:022, 2007.

Outline

Outline

1 Introduction and MotivationSeesaw mechanismLeptogenesis

2 Left-right Symmetric ModelsLeft-right symmetric seesaw mechanismLeptogenesisStability

3 ConclusionsConclusions

Introduction Left-right Symmetric Models Conclusions

Seesaw mechanism

Mass generation by the seesaw mechanism

Neutrino masses are limited by cosmological bounds to be

∑

i

mνi. 1 eV.

If this mass is only due to the Higgs mechanism, the correspondingYukawa coupling would be

y =mν

v∼ 10−12.


Seesaw mechanism

Seesaw mechanism

This issue can be elegantly resolved by adding a Majorana massterm for the right-handed neutrino to the Lagrangian

L 31

2νR mN (νR)c + h.c.

what leads to the following mass matrix

(

(νL)c νR

)

(

0 mD

mTD mN

)(

νL

(νR)c

)

and after block diagonalization in leading order (mN � mD)

(

νc N)

(

−mD1

mNmT

D 0

0 mN

)(

νNc

)

.


Seesaw mechanism

Seesaw mechanism

The smallness of the neutrino masses in the type I seesaw

mν = −mD

1

mN

mTD

can be explained by a large (GUT) scale

mN ∼ 1014GeV, mν ∼ 1 eV, mD ∼ 100 GeV.

Additionally, since the Majorana mass term is lepton numberviolating and CP-violating, this mechanism provides the possibilityto explain the baryon asymmetry of the Universe (BAU) vialeptogenesis.


Leptogenesis

Sakharov conditions

The seesaw mechanism provides all prerequisites to produce abaryon asymmetry

When the temperature of the Universe drops below the massof the right-handed neutrinos, the neutrinos becomeover-abundant and decay out-of-equilibrium

This decay is lepton number violating (mN 6= 0)

This decay is CP-violating (mN 6= m∗N)

The sphaleron process converts the lepton asymmetry into abaryon asymmetry


Leptogenesis

Leptogenesis

The baryon-to-photon ratio, ηB = (6.1 ± 0.2) × 10−10, can inleading order be parametrized as

ηB = 3 × 10−2ηεN

where η denotes the efficiency factor of the decays of the lightestright-handed neutrino and εN the CP asymmetry in its decays intoleptons and Higgs particles

εN =Γ(N1 → l H) − Γ(N1 → l H∗)

Γ(N1 → l H) + Γ(N1 → l H∗).


Leptogenesis

Decay asymmetry

Fukugita, Yanagida, ’86

The L and CP violating part of the decay rate results from a crossterm between the tree level decay amplitude and the following loopdiagrams

νR

Hu

Lh

νjR

Hu

Lf

1

(a)

νR

Hu

Lh

νjR

Hu

Lf

1

(b)

and can in the limit mN1� mN2

be written

εN =3mN1

16πv2

Im[(m†D

mνm∗D)11]

(m†DmD)11

.


Leptogenesis

Decay asymmetry

Davidson, Ibarra, ’02

In the type I seesaw model, the decay asymmetry fulfills theDavidson-Ibarra-bound

εN <3mN1

√

|∆m2atm

|

16πv2,

√

|∆m2atm

| ≈ 0.05 eV.

This leads to the fact that for viable leptogenesis, εN ∼ 10−7, alower 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 noteasily saturated.


Leptogenesis

Gravitino bound in SUSY models

Moroi, Murayama, Yamaguchi, ’93

Kawasaki, Kohri, Moroi, ’05

In supersymmetric models, the decayof the gravitino into the LSP posesconstraints on the reheatingtemperature (dark matterover-abundance, BBN constraints).Depending on the parameters of the

mSUGRA model this bound lies inthe range

TR . 107GeV toTR . 1010

GeV ↔ mN1> 108

GeV.


Leptogenesis

Hierarchies in Yukawa couplings and tuning

Motivated by GUT models, one expects a similar hierarchicalstructure for the Yukawa coupling of the neutrino as found for theother fermions

yup ∼ yν , ydown ∼ yl

Since the neutrino mass matrix has only a mild hierarchy (at leastbetween the two largest elements) and

mν = −mD

1

mN

mTD

the Majorana mass mN needs to have the doubled hierarchy of mD .

Hence, hierarchical neutrino Yukawa couplings seem to beunnatural in the seesaw framework.


Leptogenesis

Open questions in the seesaw mechanism

Hierarchy in the neutrino Yukawa coupling

A hierarchy in the Yukawa couplings, as expected from GUTmodels requires the doubled hierarchy in the Majorana mass term,what is an unnatural situation.

Gravitino bound

In supersymmetric models, thermal leptogenesis requires ratherhigh reheating temperatures, that can lead to over-closure of theUniverse with gravitinos.


Left-right symmetric 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

Φ(2, 2, 0)

∆L(3, 1,−2)

∆R(1, 3, 2)

⟨

Φ0⟩

= v⟨

∆0L

⟩

= vL⟨

∆0R

⟩

= vR

By spontaneous symmetry breaking, the neutral components of theHiggs fields obtain vacuum expectation values

L ⊃ f αβRTα C iτ2∆RRβ + yαβRαΦLβ + f αβLT

α C iτ2∆LLβ + h.c..




The left-right symmetry imposes in this case mD = mTD and the

seesaw relation reads

mν = mN

vL

vR

− mD

1

mN

mD = mII

ν + mI

ν .

Strategy: Use mD = mup motivated by GUTs/prejudice. Theremaining 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 to2n solutions for mN in the case of n flavors (with same low energyphenomenology).Akhemdov, Frigerio, ’05



Small vR/vL

In the one-flavor case the solution has a two-fold ambiguity

mN =mν

2

vR

vL

±

√

m2ν

4

(

vR

vL

)2

+ m2D

vR

vL

cancellation: |mI| ≈ |mII| � |mν |, domination: mI ≈ mν or mII ≈ mν

1014

1016

1018

1020

1022

1024

1026

vR/v

L

106

108

1010

1012

1014

mN

[G

eV]

mD

√

vR

vL

mν

vR

vL

m2D

mν

vR

vL≈

4m2D

m2ν



Inversion formula

In the three-flavor case, the solutions can be given in closed formwhat requires to find the roots of a quartic equation.

Analogously to the one-flavor case, the solution in the three-flavorcase have a eight-fold ambiguity (y = yup).

1014

1016

1018

1020

1022

1024

1026

1028

vR/v

L

104

106

108

1010

1012

1014

1016

1018

mN

i [G

eV]

mN

3m

N2

mN

1

1014

1016

1018

1020

1022

1024

1026

1028

vR/v

L

0

0.25

0.5

0.75

�ui

Pure type II ’+ + +’: Gravitino problem for vR/vL > 1021.



Pure type I

The pure type I solution (’− −−’) shows a large spread in theright-handed masses and a strong suppression of mixing angles.

1014

1016

1018

1020

1022

1024

1026

vR/v

L

104

106

108

1010

1012

1014

1016

mN

i [G

eV]

mN

3m

N2

mN

1

1014

1016

1018

1020

1022

1024

1026

vR/v

L

0

2×10-3

4×10-3

�ui

Leptogenesis possible? Fine-tuning needed?



Mixed solution

In addition there are six mixed cases.

1014

1016

1018

1020

1022

1024

1026

vR/v

L

104

106

108

1010

1012

1014

1016

mN

i [G

eV]

mN

3m

N2

mN

1

1014

1016

1018

1020

1022

1024

1026

vR/v

L

0

0.2

0.4

0.6

�ui

’−− +’: Gravitino bound eventually fulfilled. Leptogenesispossible? Stability for vR/vL > 1018?


Leptogenesis

Decay asymmetry

Hambye, Senjanovic, ’03

Antusch, King, ’04

The L and CP violating part of the decay rate results from a crossterm between the tree level decay amplitude and the following loopdiagrams

νR

Hu

Lh

νjR

Hu

Lf

1

(a)

νR

Hu

Lh

νjR

Hu

Lf

1

(b)

νR

Lh

Hu

∆

Hu

Lf

1

(c)

and the Higgs triplet leads to an additional contribution.


Leptogenesis

Leptogenesis

Antusch, King, ’04

In the limit mN1� mN2

,m∆ the asymmetry can be written

εN1=

3mN1

16πv2

Im[(m†D (mI

ν + mIIν )m∗

D)11]

(m†DmD)11

.

Due to the modified seesaw formula, the DI-bound is avoided, butleads to the slightly weaker bound

εN1.

3mN1mν,max

16πv2

and hencemN1

& 3 × 107GeV.

This bound is only slightly better than in the pure type I case, buteasier to saturate due to additional Majorana phases.


Leptogenesis

Additional Majorana phases

Akhmedov, Blennow, Hallgren, T.K., Ohlsson, ’06

For example, in the one-flavor case, the relative phase κ betweenmD and mν cannot be removed and can even lead to leptogenesisfor the type II dominated solution.

εN =3

16π

mνmN

v2sin(4κ) , ηB = 1.7 × 10−6

eVmνm2

N

|mD |2 v2sin(4κ).

Thus, it is possible to saturate the Antusch/King bound and toreproduce the observed baryon asymmetry e.g. with the values(κ = π/8)

|y | = 10−4 , m0 = 0.1 eV ,vR

vL

= 1.7 × 1020.


Leptogenesis

Leptogenesis

The same mechanism is operative in the two-flavor case in the limit

4m2D,1

m20

�vR

vL

�4m2

D,2

m20

,

for the two solutions of type ’±+’ as long as the second eigenvalueof the Yukawa coupling y2 > 5 × 10−4.

1014

1016

1018

1020

1022

1024

1026

vR/v

L

104

106

108

1010

1012

1014

1016

mN

i [G

eV]

mN

3m

N2

mN

1

1014

1016

1018

1020

1022

1024

1026

vR/v

L

0

0.2

0.4

0.6

�ui


Leptogenesis

Leptogenesis

1018

1019

1020

1021

vR/v

L

10-12

10-11

10-10

10-9

10-8

10-7

10-6

10-5

η B

κ = π/4κ = π/8

The numerical evaluation for the three-flavor case gives for thesolution ’− − +’ ( m0 = 0.1 eV, y = yu ). Leptogenesis isgenerally viable for the four solutions of type ’±± +’.


Stability

Stability measure

Akhmedov, Blennow, Hallgren, T.K., Ohlsson, ’06

To quantify tuning we use the following stability measure

Q =

∣

∣

∣

∣

det mN

det mν

∣

∣

∣

∣

1/3

√

√

√

√

12∑

k,l=1

(

∂ml

∂Mk

)2

.

where ml and Mk determine the light and heavy neutrino massmatrices according to

mN =∑

k

(Mk + iMk+N)Tk , mν =∑

k

(mk + imk+N)Tk ,

and Tk , k ∈ [1, 6], form a normalized basis of the complexsymmetric 3 × 3 matrices.


Conclusions

Summary

The qualitative analysis mostly depends on the fact the y = yup

has a large hierarchy.

1014

1016

1018

1020

1022

1024

1026

1028

vR/v

L

100

101

102

103

104

105

106

107

Q

(++-)(-+-)(+--)(---)(+-+)(--+)(-++)(+++)


Conclusions

Summary



1014

1016

1018

1020

1022

1024

1026

1028

vR/v

L

100

101

102

103

104

105

106

107

Q

(++-)(-+-)(+--)(---)(+-+)(--+)(-++)(+++)

stability, leptogenesis

’±±−’: Unstable, no leptogenesis


Conclusions

Summary



1014

1016

1018

1020

1022

1024

1026

1028

vR/v

L

100

101

102

103

104

105

106

107

Q

(++-)(-+-)(+--)(---)(+-+)(--+)(-++)(+++) stability

’±− +’, vR/vL � 1023: Unstable


Conclusions

Summary



1014

1016

1018

1020

1022

1024

1026

1028

vR/v

L

100

101

102

103

104

105

106

107

Q

(++-)(-+-)(+--)(---)(+-+)(--+)(-++)(+++)

gravitinos

’± + +’, vR/vL � 1023: Gravitino bound violated


Conclusions

Summary



1014

1016

1018

1020

1022

1024

1026

1028

vR/v

L

100

101

102

103

104

105

106

107

Q

(++-)(-+-)(+--)(---)(+-+)(--+)(-++)(+++)

’±± +’, 1016 � vR/vL � 1023: Sweet spot


Conclusions

Conclusion

In case of hierarchical Yukawa coupling for the neutrinos, theleft-right symmetric type I+II framework constitutes a model thatcompared to the pure type I framework has the followingproperties:

GUT embedding

A large hierarchy in the neutrino Yukawa coupling is naturallyimplemented. The required tuning is reduced from Q ∼ 105 toQ ∼ 102.

SUSY embedding - Gravitino bound

The lower bound on the lightest right-handed neutrino mass isslightly relaxed from mN1

> 5 × 108 GeV to mN1> 1 × 108 and

the Antusch/King bound can be easily saturated using theadditional Majorana phases.

Documents

The left-right symmetric seesaw mechanism · Seesaw mechanism Mass generation by the seesaw mechanism Neutrino masses are limited by cosmological bounds to be X i mν i. 1 eV. If