PhysicsFlavor in Symmetry Broken 3S

Hirotaka Sugawara JSPS Office inHirotaka Sugawara, JSPS Office in Washington DC

W k b i d i h T hi kiWork being done with Toshiaki Kaneko, KEK, Japan

Long time ago I made a proposal together with S Pakvasa toLong time ago、I made a proposal together with S.Pakvasa, to understand flavor physics based on non-Abelian discrete symmetry such as .3Sy y

S.Pakvasa and H.Sugawara, Phys.Lett.B73,61(1978); B82,105(1979)B82,105(1979)

Recently, many attempts are being made to understand, at least in the leptonic part of flavor physics based on all kinds of nonin the leptonic part of flavor physics based on all kinds of non-Abelian discrete symmetries.

We still do not kno :We still do not know:

(1) What is the right symmetry?

(2) How is it broken?

(3) What is the origin of the symmetry?(3) What is the origin of the symmetry?

isbreakingitsofwayspecificithsymmetry wS(1): thatshow we work,continuing In this

d iflsector lepton andsector quark both with cosistent

isbreakingitsofway specificith symmetry wS(1) 3

illsymmetry w theoforigin theingunderstand The (2)dynamics.flavor

broken.is symmetry gauge way on then restrictio strong a give

S of ABC 3

)}2,3,1(),3,2,1(),1,3(),3,2(),2,1(,1{:3S

++ ψψψψψψ ,,:function wave 321

=++ αψψψ

3:singlet 321

⎞⎛⎟⎞

⎜⎛ −ψψ

3

21

⎟⎟⎠

⎞⎜⎜⎝

⎛=

⎟⎟⎟

⎜⎜⎜

+β

ψψψ 22:doublet ⎟

⎠⎜⎝⎟

⎟⎠

⎜⎜⎝

−+ γψψψ6

2 321

⎠⎝

iHihliHi

3 :issinglet SisHiggswhen termmassor coupling Higgs

++R

3

)(ag

γγββαα b LRLRL

⎟⎟⎞

⎜⎜⎛

⎟⎟⎞

⎜⎜⎛

−−−−

+⎟⎟⎞

⎜⎜⎛

=1

)121112

111111

)(( ψψ

ψψψ ba

⎟⎟

⎠⎜⎜

⎝⎟⎟

⎠⎜⎜

⎝ −−−−+

⎟⎟

⎠⎜⎜

⎝

=

3

2321 )211121

3111111

3)(,,(

ψψψψψ

matrixmixingfor thephaseArbitraryKMbroken? be scan How 3

violationCP matrixmixing for thephase-Arbitrary-KM

→matrix massmatrix mixing ←

brokenareSandCPbothmatrixmass thetophasespossible all Allow

→ brokenareSandCPboth 3→

Then

⎞⎛ 131211 δδδ iii bb

Then

⎟⎟⎞

⎜⎜⎛ 131211

δδδ

δδδ

iii

iii bebeae

⎟⎟

⎜⎜= 232221 δδδ iii beaebeM

⎟⎠

⎜⎝

333231 δδδ iii aebebe ⎠⎝

showsalgebrasimple :gfk

showsalgebrasimple

⎞⎛

:

hkfgfk

MM ⎟⎟⎞

⎜⎜⎛

∗

khhkfMM⎟⎟

⎠⎜⎜

⎝

=khg ⎟⎠

⎜⎝

realk :khkgkf ≤≤≤ Re,Re,Re

:massesquarkthetoparameterstheofrelationThe

131

:massesquark the toparameterstheofrelation The

sk =

2222

1

13

Asshgf −≡−=++

3

21

123

Bhff h

Asshgf ≡=++

32131 327

with

Bsssshgffgh −≡+−=+

21

23

23

22

22

212

23

22

211 , mmmmmmsmmms

with++=++=

23

22

213 mmms =

The phase convention and the number of variables

( k )(quark sector)Th l hThere are only two phases one can adjust j

:currentweak ,,

d

diL

uiL ψγψ μ

usly.simultaneochangedbemust andofphase ,,d

iLu

iL ψψ

12214parametersofnumber)dandu (27,,,

==×−−−hgfk

46(masses)sobservableofnumber 122-14parametersofnumber

+===

( )

We can p t either p q ark or do n q ark f h realWe can put either up-quark or down quark f, h real but not both.

Up quark f,h real caseUp quark f,h real case

⎟⎟⎞

⎜⎜⎛ ×− −101305.09999.011 4

*

i

⎟⎟

⎠⎜⎜

⎝ ×+=

− 19999.0101305.09999.09999.011)(

4

*

iMM u

⎟⎟⎟⎞

⎜⎜⎜⎛

+−++

= 07300.09965.0102453.09994.009677.09935.002453.09994.01

)( * iiii

MM d⎟⎠

⎜⎝ −+ 107300.09965.009677.09935.0 ii

⎟⎞

⎜⎛ ×−++ −105025057740032790517301913060120 5 iii

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

−×−−−−×++

= −

5773.08048.01380.0102513.05774.003280.02875.01913.07392.0105025.05774.003279.05173.01913.06012.0

5 iiiiii

Uu

⎟⎟⎞

⎜⎜⎛

+−−−+++

=

⎠⎝

042080576001236016580391806857005607.05746.01932.05864.03792.03763.0

5773.08048.01380.0

iiiiii

U⎟⎟

⎠⎜⎜

⎝ −+=5772.07589.03014.0

04208.05760.01236.01658.03918.06857.0 iiiUd

⎟⎞

⎜⎛ − 003269000129302252097430 i

matrixMixing

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

×−−−×−−

−

−

9991.0107367.00404.0003182.0008012.004117.09735.0101381.02251.0

003269.0001293.02252.09743.0

3

3

iii

i

⎠⎝ × 9991.0107367.00404.0003182.0008012.0 ii

Parameter Experimental CalculateParameter Experimental Calculateλ (Cabibbo) 0.2252 0.225195

A 0.8116 0.811834ρ 0.139 0.136225η 0.341 0.343641

Lepton caseLepton case

(1)neutrino sector

Assume large Majorana masses of right handed neutrino but zero majorana masses of left handed neutrino

)(0

))(,( ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛=

MMM

CM CR

LMTD

DTC

RTL ν

νννν

0

)(

)(

11⎟⎟⎞

⎜⎜⎛

⎟⎟⎞

⎜⎜⎛⎟⎟⎞

⎜⎜⎛

=⎠⎝⎠⎝

VVM

MMD

TMM

R

λ

)()( ,

)(

2112

22

−==

⎟⎟⎠

⎜⎜⎝

=⎟⎟⎠

⎜⎜⎝⎟⎟⎠

⎜⎜⎝

VMVMVVM

VVMMMTDD

MTD

λλ

λ

0)()(

get, we, smallFor

111 =− −

VVVMMM TDMD λ

λ

02 ≈V

⎟⎞

⎜⎛1

D

cc

⎟⎟⎟

⎜⎜⎜

= 1νD ccmM

⎞⎛

⎟⎠

⎜⎝ 1cc

⎟⎟⎞

⎜⎜⎛ 1 21

δδ

δδ ηηii

ii

M

ee

⎟⎟

⎜⎜= 1 31

δδ

δδ ηηii

iiM eeaM⎟⎠

⎜⎝ 132 δδ ηη ii ee

ie ii i )1)(1( δχη δ ⋅⋅⋅++−=

ii i

i

))(1())((

χδχη++≈

i i

i

1))((

δχ

+≡complex

i

i

i

:1

δδ+

complexi :δ

])()(

121112

[21212

⎟⎟⎞

⎜⎜⎛

++−

+⎟⎟⎞

⎜⎜⎛−

= xxxxxxxx

imM χεν ])(

)(211

121[

3232

3131⎟⎟

⎠⎜⎜

⎝ +−+−+

⎟⎟

⎠⎜⎜

⎝ −−−=

xxxxxxxxi

azM χν

“Real” part and “imaginary” part of the matrix commute with each other.

- We can diagonalize this by aｎ orthogonal matrix

111

222

ε

qqqqqqzc

−−−+=

+=

1 321321

η δii eq

qqqqqqzi=

+=

21 δδδη

iiii

i

xq

−+= ++

part.imaginary without definedares' Hereδ

0xxx:casehierarchy Normal (1) 133221 =++ xxx

1,11

1,2(2

13

2

32

32

ν ⎟⎟⎞

⎜⎜⎛

⎟⎟⎞

⎜⎜⎛

⎟⎟⎞

⎜⎜⎛

⎟⎟⎞

⎜⎜⎛

+−+

= xx

xxxx

U

secodthe3eeigen valuantocorresponscolumnfirstThe

)(,

13,(

32

33

32

321

χ

ν⎟⎟

⎠⎜⎜

⎝⎟⎟

⎠⎜⎜

⎝ +−⎟⎟

⎠⎜⎜

⎝⎟⎟

⎠⎜⎜

⎝ − xxN

xxN

bi i lild0hW).2i(x-3 toone third thezeroand eeigen valu column to

secodthe,3eeigen valuan tocorresponscolumn first The

321χχ

++ xx

notation)ordinary in ( xTherefore,case.bimaximal-triexactly tocorrespond0that xseeWe

132

2

θ=

case. bimaximal- tri thefromdeviation theindicates

x:hierarchy Inverted (2) 321

⎞⎛

== xx

31

21

61

⎟⎟⎟⎞

⎜⎜⎜⎛

123

12

16

1ν

⎟⎟⎟⎟

⎜⎜⎜⎜

−=U

3i3i ldldhdfiTh3

1032

⎟⎟

⎠⎜⎜

⎝−

e.eigen valu zero column to third theand 3ix-3eeigen valutocorrespondcolumnssecond theandfirst The iχ

solution.bimaximal-trithetocorrespondnot doesThis

We have to study the charged lepton matrix to get the mixing matrix.

The fact that the neutrino part is by itself gives good experimental fit

means that we must get identity matrix from the charged lepton sector.

Charged lepton family members are all s(3) singlet rather than singlet doublet. All the quarks and neutrinos are singlet-doublet.

scscc 1313121312 ⎟⎞

⎜⎛

ccsssc-scssscssccsccs-U

231323131223122313122312

231323131223122313122312⎟⎟⎟

⎠⎜⎜⎜

⎝ −−−−=

ccss

: smallfor get weccssscscss

13

231323131223122313122312 ⎠⎝s

cc

κθ3

sin 132 =

κθ435.0tan 12

2 +=

κθ 221tan 232 +=

132sin θ

0.025

‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐‐≈0.02

3

122tan θ

132

122 sin

435.0tan θθ +=

0.550.5

Kamland data taken from M.C.Gonzalez-Garcia etal.arXiv:1001.4524v3,page9

The most important question is “Why S(3)?”

There are two kinds of symmetry in nature:

(1)Guage symmetry which is good at high energy(short distance).

(2)Non-gauge symmetry such as baryon number conservation,

CP invariance, parity etc.. These are good only in lower energy region(long distance.distance.

For example,

Baryon number conservation gets vilated when quarks and lepton go into a y g q p gsame multiplet. This makes sense at high energy.

CP is violated when 3rd generation quarks gets into the same multiplet as the other two generationsthe other two generations.

S(3) must belong to this category.

One possibility

)72,3()27,3()1,8()78,1(248638

+++→

×→ ESUE

:phases"for except "unbroken remain may S following Theenergy.high at broken be will This

group.flavor theas thisuse

i

3

3

3

SUSUWe

Tiππ

.

)3(31

e(1,2)

811

22i- 1

−+=

=

etc

IT

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:Questionmodel. standard down tobroken

is group unfied grand how sconstraint strong give willThisSUn rather tha Uusing are fact weIn 33

:matrix mass tolead willmultiplets Higgs ofsort (2)What singlets? belepton chargedCan (1)

n?observatioour with consistent is which model a make topossibleit IsQ

⎞⎛ gfk?

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛=∗

khghkfgfk

MM