質量行列と世代間対称性－

tri-bimaximal 混合の世代構造と理論の現状 －Tri-bimaximal Neutrino Mixingand Flavor Symmetry

2007.11.2 @ICRR
Koichi Yoshioka (Kyoto univ)

0. Introduction
Current exp results

model
model
Top-Down approachhigh-energy models aremodel
undetectable and indistinguishable.....
(only remnants are found)
model
model
H.Murayama (2006)

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more "reasonable" approach (this talk) ：Extract and Identify theparticle (neutrino)
property (generation structure)
in Nature (in fundamental theory)

Flavor Theory
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Plan of talk1. Lepton masses and mixing
2. Tri-bi (TB) maximal generation mixing
3. Flavor symmetry
4. TB mixing and Neutrino flavor structure
5. Summary

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1. Lepton masses and mixing: charged lepton mass matrix: neutrino mass matrix (Majorana or Dirac)(similar exp for Dirac)

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Current experimental data
best fitlarge
maximalsmall(3 sigma)

Maltoni et al [ hep-ph/0405172 ver.6 (Sep 2007) ]
0.40
0.38
0.26
0.23
ver.1 2

KamLanddata..... 5 6

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2. Tri-bi maximal generation mixing
Harrison-Perkins-Scott(2002)

uniquezclose to the best fit values !
z

no Dirac CP phase

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Comparison to the democratic mixing: democratic mixing
Harari-Haut-Weyers
Koide
Fukugita-Tanimoto-Yanagida:

2 mass eigen leptons are
tri-maximal mixture of
bi-maximal mixture ofandis a TB mixing matrix in the charged-lepton sector[ but experimentally disfavored (by the solar angle) ]7/19

Theoretical perspective"parameterfitting"

TB mixing

good theoretical motivationz a key to looking for "hidden" flavor structurez flavor symmetry plays important rolez(cf.)

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3. Flavor symmetryStandard Model ＝ gauge sector ＋ Yukawa sector
gauge symz abelianflavor sym

or non-abelian ?

abelian: discriminate between generations
non-abelian : connect different generationsz continuousor discrete ?

continuous
discrete: free rotation between generations: definite meaning of generations

Non-Abelian Discrete flavor symmetry isappropriate for (lepton) flavor physics

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The quark sector
Log(m)For example :updown

1st

2nd

3rdz large mass hierarchyz small mixingi.e. "separate" generations
non-abelian flavor sym ....
U(1) Froggatt-Nielsen mechanism ○?

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4. TB mixing and Neutrino flavor structure
What is the form offor TB mixing?

z integer (inter-family related) matrix elements

non-abelian discrete flavor sym
z the exp data (Vtri-bi) implies the neutrino mixing

is determined independently of mass eigenvalues
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TB mixing indicates that Flavor structure

S3orA4is hidden in the lepton sector

S3 : permutation of 3 objects(3 generations)A4 : even permutation of 4 objects
3-dim representation x 1
1-dim representation x 3
suitable for 3 generations
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TB mixing and S3z The 1st and 2nd terms are S3 symmetric(the most general S3-invariant forms)
z The 3rd is not(S3 flavor sym breaking)
KEY for Vtri-bi(cf.) gauge sym breaking
in the SM gauge sector

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But...
z fine tuning required experimentallythe 3rd termz any subgroup of S3 does not work .....

Extra property of neutrinos :zChen-Wolfenstein* the 3rd term negligible* degenerate neutrino masses
z Magic matrixLamz Twisted flavors

Haba-Watanabe-KY

z Extra higgs contributions

Mohapatra-Nasri-Yu:

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TB mixing and A4A4 structure is hidden :z The 3rd term is A4 symmetricz A 3-dim higgs gives the general A4-symmetric

Majorana mass term:

A4 flavor sym breaking→ Z2 :

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Other approaches* different flavor structure( tri-bi maximal ) = ( tri maximal ) - ( maximal )

from

from

cubic roots of unity

Vtri-bi is divided by 2 pieces
both of which may be theoretically justified(e.g. almost all A4 work adopt this flavor structure)

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* different flavor symmetry(discrete, non-abelian)# of group elements= many people= some people= curious people
models become complex
z little advantagezNature does not likethese complex structures...?
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Deviations from exact TB mixing
z TB mixing is a (good) approximationz TB mixing is exact only in flavor-symmetric theory(flavor sym breaking is important)z Uncertainty in exp data
Deviations from TB mixing
z decreasing

a clue todeeperunderstanding(really needed...?)z nonzeroz (Dirac phase) CP violation

Future neutrino experiments
will give further information

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5. Summary
The tri-bimaximal generation mixing
z good approximation of(close to the best-fit values)z theoretically well motivatedz indicates definite symmetric structure of Yukawa sectorz non-abelian discrete flavor symmetry
z S3

or A4
z independent of mass eigenvalues
z incompatible with exact flavor symmetryz important to understand flavor symmetry breaking
z deviation from exact
z nonzeroz CP violation in future neutrino exp.
will provide deeper understanding of flavor physics

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