CP violation and leptogenesis in three
generation seesaw model
広島大学
由宇 朗大
01/09/2016
Contents
• Seesaw model
• Asymmetries
CP violation in neutrino oscillation
Leptogenesis
• Jarlskog parameter and Δ as indicator of CP asymmetry
Derivation of Δ
• Specific models
Four zero texture model
Single massless neutrino model
• Summary and further task
Introduction for Neutrino
It is known that neutrino has tiny but non-zero mass.
・Neutrino mass is established by neutrino oscillation.
However, Standard Model can not explain neutrino mass
The physics beyond Standard Model is required.
more concretely,
• In the framework of Standard Model, only left-handed neutrino
is introduced.
• There is no term which describes neutrino mass.
Seesaw mechanism has been suggested to solve the problem.
Seesaw mechanism
Minkowski, ’77; Gell-Mann, Ramond, Slansky, Yanagida; Glashow; Mohapatra, Senjanovic ‘79
• heavy right-handed neutrino ν𝑅
• Majorana particle
Lagrangian
𝑚𝑅 ,𝑚𝐿 : Majorana masses
𝑚𝐷 : Dirac mass
Only the neutrino with tiny masse remains in
effective low energy theory
If one of the mass eigenvalues gets heavier, the other one becomes
lighter, and vice versa.
This is the reason of the name “seesaw” for the mechanism.
in three generation model (𝑒, 𝜇, 𝜏)
6×6 mass matrix
We define the effective mass
𝑛𝑖 : mass eigenvalues
This unitary matrix which diagonalize 𝑚𝑒𝑓𝑓 , is named Maki-NakagawaSakata matrix (MNS matrix).
・MNS matrix describes the flavor mixing between
weak-basis and mass-basis.
A conventional parametrization for MNS matrix 𝑉 𝑀𝑁𝑆 is
where 𝑆𝑖𝑗 = sin θ𝑖𝑗 , 𝑐𝑖𝑗 = cos θ𝑖𝑗
• θ12 , θ23 , θ13 are mixing angles.
• δ is a CP phase.
CP violation related to neutrino sector
CP transformation is the simultaneous operation of Charge
conjugation (C) and Parity (P).
CP asymmetry is generally written as,
𝑃 𝐴 → 𝐵 − 𝑃(𝐴 → 𝐵)
(𝐴 and 𝐵 are CP conjugates of 𝐴 and 𝐵.)
(1) CP asymmetry in neutrino oscillation, which is an observable
in experiments.
Fukugita, Yanagida ‘86
(2)CP asymmetry in leptogenesis, which is one of the scenarios
explaining Baryon Asymmetry of the Universe (BAU).
As a characteristic indicator for CP violation, we introduce
Jarlskog parameter 𝐽. C.Jarlskog ‘85
Transition probabilities via
neutrino oscillation
CP asymmetry
Transition probabilities via
anti-neutrino oscillation
CP asymmetry is proportional to 𝐽 written with mixing angles and CP phase.
Jarlskog parameter
𝐽
2
2
2
For 𝐴𝐶𝑃 ≠ 0, ∆≡ 𝐽∆𝑛12
∆𝑛23
∆𝑛31
≠ 0.
𝐽 can be expressed in terms of 𝑚𝑒𝑓𝑓 and mass eigenvalues.
We define its numerator, which consists of the
components of 𝑚𝑒𝑓𝑓 .
We derive the general form of Δ in terms of three generation
seesaw parameters.
By new variables
Dirac mass matrix
We have to notice that 𝑈 is not necessarily
to be unitary nor hermite matrix.
values, which have mass dimension
a diagonal matrix
hermite matrix 𝐴 by 𝑈
Its elements and properties
Δ has been calculated in previous work in minimal seesaw model with two
S.Kaneko, S.Kang, D.Kimura, T.Morozumi, M.Tanimoto Cosmological
right handed Majorana neutrinos. T.Fujihara,
family asymmetry and CP violation. Physical review D72,016006(2005)
Dirac mass matrix has
the form of 3×2 matrix
Introducing two heavy right handed neutrino masses 𝑀1 and 𝑀2 .
Hence,
I extend the calculation of Δ to the 3×3 case.
The result of Δ in terms of seesaw parameters
𝑝, 𝑞, 𝑟 denote 1, 2, or 3.
means the sum of all 6 patterns of
substitution among 𝑝, 𝑞, 𝑟 .
{𝑝,𝑞,𝑟}
𝑝 ≠ 𝑞 ≠ 𝑟.
𝑒, μ,τ are each flavor.
means the sum of 3 patterns of
periodic substitution for (𝑒, μ,τ) .
{𝑒,μ,τ}
The difference between 3×3 model and 3×2 model
First two terms are already
involved in previous model.
Due to the third right handed
neutrino mass, new mass scale
𝑋3 appears.
𝑋1𝑚 𝑋2𝑛 𝑋3𝑙 (𝑚 + 𝑛 + 𝑙 = 6, 𝑚, 𝑛, 𝑙 ≠ 0)
From third to the last terms do not
appear in the 3×2 model.
Leptogenesis (generation of lepton number asymmetry)
The lepton number asymmetry 𝜀𝑖𝑘 (𝑘 = 1,2,3 𝑖 = 𝑒, 𝜇, 𝜏) is related to the
difference of the partial decay width of heavy right handed neutrino.
𝜈Rｋ
𝑙𝑖− 2
𝑙𝑖+ 2
ー 𝜈ｋ
R
𝜙+
Electron number asymmetry (𝑖 = 𝑒),
𝜙−
non-zero.
T.Endoh, T.Morozumi, Z.Xiong ‘2004
In order to 𝜀𝑒𝑘 ≠ 0, at least two of the matrix elements of 𝑚𝐷𝑒𝑘 must be non-zero.
We impose reasonable conditions on the model to make it more
simplify, while it can explain the CP violation and also
leptogenesis at the same time.
We consider specific cases, which do not contradict to any
experimental input.
(1) Four zero texture model
(2) Single massless neutrino model
(1) Four zero texture model
For example, we take as many matrix elements of 𝑈 zero as possible, while electron
number asymmetry is dominant contribution (e-leptogenesis).
All three elements on the first row component must not be zero for e-leptogenesis.
The possible textures are,
6 patterns
We can substitute maximum 4 matrix elements for 0.
As four zero texture, we can mention such patterns as below,
However, those three patterns can not explain CP violation, since for any pattern Δ＝0.
They can not explain the CP violation.
For instance, we pick one pattern up,
10 parameters in general
The remaining elements are generally complex,
but by the property of 𝑈, we can reduce parameters.
4 parameters
θ1 , θ2 , δ1 , δ2
accompanying ∆ with this case
∆
(2) Single massless neutrino model
Even if only one of mass eigenvalues equals zero, it dose not contradict to the
experimental facts.
The advantage of this model is that
• Two mass squared differences measured by experiments correspond to the mass
squared itself.
taking 𝑛1 = 0, 𝑛2 ≠ 0, 𝑛3 ≠ 0,
2
2
Δ𝑛12
=-𝑛22 , Δ𝑛13
=-𝑛32
• We can reduce the diagonalization of 3×3 complex matrix to that of 2×2 matrix.
The necessary and sufficient condition for that at least one of neutrinos is
massless is
If 𝑈 is not regular, it can be rewritten as
an unitary matrix
a matrix whose first row
components are zero
We reach 2×2 symmetric matrix.
the matrix which real-diagonalize the 2×2 matrix
the MNS matrix
We obtain the two non-zero mass eigenvalues 𝑛2 and 𝑛3 .
Summary
• The most general form of Δ, which is a characteristic
indicator of CP violation was derived.
• I considered two simplified models, which guarantee CP
violation and leptogenesis, by means of three generation
seesaw model.
(1) four zero texture model and (2) single massless
neutrino model
• I expressed the MNS matrix and neutrino mass eigenvalues
in terms of seesaw model.
Further task
• I will perform the concrete numerical analysis as for above
two models.