Flavor & CP Violations in Gauge-Higgs Unification

Nobuhito Maru (Chuo University)

1/7/2011 Workshop “Physics beyond the Standard Model & Predictable Observables”@Kobe

Flavor & CP Violations in Gauge-Higgs Unification

Nobuhito Maru (Chuo University)

1/7/2011 Workshop “Physics beyond the Standard Model & Predictable Observables”@Kobe

ReferencesReferences

“Flavor Mixing in Gauge-Higgs Unification”

Y.Adachi, N.Kurahashi, C.S.Lim & N.M.JHEP1011 150 (2010)

“CP Violation due to Compactification”

C.S.Lim, N.M. & K.NishiwakiPRD81 076006 (2010)

“Neutron Electric Dipole Moment in the Gauge-Higgs Unification”

Y.Adachi, C.S.Lim & N.M.PRD80 055024 (2010)

“Flavor Mixing in Gauge-Higgs Unification”

Y.Adachi, N.Kurahashi, C.S.Lim & N.M.JHEP1011 150 (2010)

“CP Violation due to Compactification”

C.S.Lim, N.M. & K.NishiwakiPRD81 076006 (2010)

“Neutron Electric Dipole Moment in the Gauge-Higgs Unification”

Y.Adachi, C.S.Lim & N.M.PRD80 055024 (2010)

�Introduction�Flavor Violation�Mechanism, mixing

�CP Violation�by Higgs VEV�by Compactification

�Summary

�Introduction�Flavor Violation�Mechanism, mixing

�CP Violation�by Higgs VEV�by Compactification

�Summary

PLAN

0 0K K−

IntroductionIntroduction

☆ Hierarchy problem

☆ Origin of fermion mass hierarchy & flavor mixings

☆ Origin of CP violation

☆ Hierarchy problem

☆ Origin of fermion mass hierarchy & flavor mixings

☆ Origin of CP violation

Unsettled problems in Higgs sector of the SM

We now discuss “Gauge-Higgs unification” scenario as an attractive candidate of New Physics, which is expected to shed some lights on these problems

by higher dimensional gauge symmetry

We now discuss “Gauge-Higgs unification” scenario as an attractive candidate of New Physics, which is expected to shed some lights on these problems

by higher dimensional gauge symmetry

Gauge-Higgs UnificationGauge-Higgs Unification

Zero mode of the extra component of higher dimensional gauge field = SM Higgs

Zero mode of the extra component of higher dimensional gauge field = SM Higgs

� Higgs mass is forbidden at tree level

� Radiatively generated Higgs mass is finite& cutoff independent

due to higher dim. gauge symmetry

� Higgs mass is forbidden at tree level

� Radiatively generated Higgs mass is finite& cutoff independent

due to higher dim. gauge symmetry

Manton (79), Fairlie (79),Hosotani(83)

This fact opens up an avenue to solve the hierarchy problem w/o SUSY

Hatanaka, Inami & Lim (98)

�D-dim QED on S^1@1-loop Hatanaka, Inami & Lim (1998)

�5D Non-Abelian gauge theory on S^1/Z2@1-loop　 Gersdorff, Irges & Quiros (2002)

�6D Non-Abelian gauge theory on T^2@1-loopAntoniadis, Benakli & Quiros (2001)

�6D Scalar QED on S^2@1-loop Lim, N.M. & Hasegawa (2006)

�5D QED on S^1@2-loopN.M. & Yamashita (2006); Hosotani, N.M., Takenaga & Yamashita (2007)

�5D Gravity on S^1 (GGH) Hasegawa, Lim & N.M. (2004)

…

�D-dim QED on S^1@1-loop Hatanaka, Inami & Lim (1998)

�5D Non-Abelian gauge theory on S^1/Z2@1-loop　 Gersdorff, Irges & Quiros (2002)

�6D Non-Abelian gauge theory on T^2@1-loopAntoniadis, Benakli & Quiros (2001)

�6D Scalar QED on S^2@1-loop Lim, N.M. & Hasegawa (2006)

�5D QED on S^1@2-loopN.M. & Yamashita (2006); Hosotani, N.M., Takenaga & Yamashita (2007)

�5D Gravity on S^1 (GGH) Hasegawa, Lim & N.M. (2004)

…

Explicit calculations of Higgs massExplicit calculations of Higgs mass

In gauge-Higgs unification, the following issues are nontrivial

1: Realizing Yukawa hierarchy

2: Accommodating the flavor mixing

3: CP violation

1: Realizing Yukawa hierarchy

2: Accommodating the flavor mixing

3: CP violation

∵ Yukawa coupling is originated from the gauge coupling

which is real and universal for all flavors

∵ Yukawa coupling is originated from the gauge coupling

which is real and universal for all flavors

In this talk, we discuss how these issues are incorporated

in the gauge-Higgs unification scenario

Flavor ViolationFlavor Violation

Adachi, Kurahashi, Lim & Maru, JHEP1011 015 (2010)

As a new feature of higher dimensional models with Z2 orbifold, Z2-odd bulk masses are allowed

( ) ( )( ):i i iM y yε ψ ψ ε sign function

with Mi being different depending on each flavor

New source of flavor violation specific to higher dimensional models

New source of flavor violation specific to higher dimensional models

The bulk mass controls the location of 0 mode fermions localization

( ) ( ) ( ) ( )0 0

2 2,1 1

i i

i i

M y M yi iL RM R M R

M Mf y e f y e

e eπ π−

−= =− − y

0 πR

RL

Small overlap

4D effective Yukawa coupling4D effective Yukawa coupling

( ) ( ) ( ) ( )0 044 42 MR

R

L R

R

Y g dyf y f y MRg e gπ

π

ππ−

−= ≈ ≤∫

No need of unnatural fine-tuning for 5D parameters due to exponential suppression

Getting a viable top mass is not trivial, but possible Cacciapaglia, Csaki & Park (2005)

No need of unnatural fine-tuning for 5D parameters due to exponential suppression

Getting a viable top mass is not trivial, but possible Cacciapaglia, Csaki & Park (2005)

At first glance, the bulk masses can be off-diagonal in flavor space,

which seems to generate flavor mixing

Unfortunately, it is not the case:For each representation R of the gauge group,

a general form of bulk mass terms

can be always diagonalized by a suitable unitary transformation,leaving the kinetic term invariant

Unfortunately, it is not the case:For each representation R of the gauge group,

a general form of bulk mass terms

can be always diagonalized by a suitable unitary transformation,leaving the kinetic term invariant

( ) ( ) ( ) ( )ij i jM R y R Rε ψ ψ

We are led to introduce brane localized mass terms,

which are the sources of flavor mixing &are also necessary to make exotics heavy

as will be seen below

We are led to introduce brane localized mass terms,

which are the sources of flavor mixing &are also necessary to make exotics heavy

as will be seen below

ModelModel

5D SU(3) model compactified on1

2S Z

( )( )

3

6

3

6

i i i

i i i i

Q d

Q u

ψ

ψ

= ⊕

= Σ ⊕ ⊕

N-generations of bulk fermion are introduced

(i = 1, …, N)

Need to eliminate the redundant quark doublets (Q) and exotics (∑)

Brane localized mass termsBrane localized mass termsBurdman & Nomura (2003)

3i i Dψ=L ( )( ) 3 6

i iiM y i Dε ψ ψ− + ( )( )( ) ( ) ( ) ( )3 6

6

2 , ,i j jR ij

ii

L ij L

y

y RQ x Q x y y

M

Q xδ π η λ

ε ψ−

+ + + ⋯

Brane localized fields

Brane mass matrices(off-diagonal elementsare generically allowed)

“Flavor mixing”

( ) [ ] ( )3

6

3

6

1 3

2 4

0

,

HR R diag

SM LL

H QR R

SM LL

Q Qy Q y Q m

QQ

Q QU Q Q

Q

U U

U UQ

δ η λ δ ′ =

′= =

∼

QBM L

“2Nx2N unitary matrix”

“generation dependent” bulk masses

( ) ( ) ( ) ( )( )6 6

5 3 5 6

0 0 0 065 3 4

i i i iYukawa y y

i j i jy R SM R

ii ij ii ijd SMu

g A d Q g

Y U Y

A u Q

g A d Q u QU

= +

→ +

L

( ) ( )0 0 2iR i iii i RM

L RRY dyf f RM e

π ππ

π −

−= ≈∫

( )†

3 † † †3 3 4 4†

4

ˆ1

ˆd dR d dL

CKM uL dL N N

u uR u uL

Y V Y U VV V V U U U U

Y V Y U V×

= = + ==

Yukawa couplingYukawa coupling

4D effective Yukawas YuU3, YdU4 are diagonalized in a usual way

Yukawa coupling with flavor mixing

M3,6 ∝ 1 (Yu,d ∝ 1) case (flavor symmetry restored)M3,6 ∝ 1 (Yu,d ∝ 1) case (flavor symmetry restored)

( )

† †3 43 4

† † † †3 3 3 1

† † †

†

4 4 4

†

†

ˆ ˆ ˆ

ˆ ˆ ˆ

1

d dR dL d d dL dL U U U UuL dL

u

CKM uL dL dL dL

uR uL u u uL uL

Y V U V Y Y V U U VV V

Y V U V Y Y V

V V

U V

V V

U

V

+ = → → ∝

→

⇒ = ∝ =

∼ ∼

∼ ∼

No mixing

To get flavor mixing,

we need non-degenerate bulk massesas well as the off-diagonal brane masses(characteristic to gauge-Higgs unification)

To get flavor mixing,

we need non-degenerate bulk massesas well as the off-diagonal brane masses(characteristic to gauge-Higgs unification)

LessonLesson

Natural flavor conservationNatural flavor conservation

FCNC has played crucial roles in the discussion of the viability of New Physics

We ask if “natural flavor conservation” is satisfied, i.e. if FCNC processes at tree level are forbidden

Glashow-Weinberg conditionGlashow-Weinberg condition

“Fermions with the same electric charges, chirality should have the same isospin (I3)”

“Fermions with the same electric charges, chirality should have the same isospin (I3)”

Glashow & Weinberg (1977)

3 = 2L1/6 (Q) (+) + 1L-1/3 (-)

2R1/6 (-) + 1R-1/3 (dR) (+)

6* = 3L-1/3 (-) + 2L1/6 (Q) (+) + 1L2/3 (-)

3R-1/3 (+) + 2R1/6 (-) + 1R2/3 (uR) (+)

3 = 2L1/6 (Q) (+) + 1L-1/3 (-)

2R1/6 (-) + 1R-1/3 (dR) (+)

6* = 3L-1/3 (-) + 2L1/6 (Q) (+) + 1L2/3 (-)

3R-1/3 (+) + 2R1/6 (-) + 1R2/3 (uR) (+)

Parity assignmentParity assignment

The condition is satisfied in the down sectorrelevant to K-Kbar mixing @tree level

( ) ( ) ( )03 2 3 3 4 33 1 3RR R↑ ↓−⊕ ⊕ −Same quantum number as down quark

“Not” the end of the story

“exotic”

0 0K K− mixing

FCNC at tree level even in QCD sector

( ) ( ) ( ) ( ) ( )( )( ) ( ) ( ) ( )( )( ) ( ) ( ) ( ) ( )

0 0†

0 0 0 0† † † †3 3 4 4

0 0 0 0 0

0 0

0 0

2n

a i i i ia asstro

dR RR dRij

ng R R L L

a n i jas R R

a n ndL LL dL dL L

n i jas L L d

ijL R

gG d d d d

V I V

V U I U

R

g G d d

g G d d V V U I U V

µ µµ

µµ

µµ

γ λ γ λπ

γ λ

γ λ +

+

+

⊃ +L

0 mode sector: No mixing O.K.

Nonzero KK gluon couplings induce nontrivial flavor mixing

⇒ mixing@tree level0 0K K−

( )0Rs ( )0

Rd

( )0Ls( )0

Ld

( )a nGµ

( ) ( ) ( )

( ) ( )( )20 0

22

2 3 1 0 0 2 0 022

13cos

1sin 2

15.8 10

Ri n i

RR RR

n nKRR RR

d s

n

K S K K dRn

I dy f y

mI Im CR f m

n

R

m

ny

R

m

π

π

π α θ

π −

− =

− +

−∆ −

<

∫

∑∼

x (exp)

MeV

Dominant process of mixing0 0K K−

( )1

R>O 10 TeV

Chiral enhancementfactor ～ 25

“GIM-like” mechanism in GHU“GIM-like” mechanism in GHU

The lower bound for the compactification scale is smaller than that from naïve order estimate

( )( )

2

22 5

sin cos 1

10300c c

ccM Te

M TV

eVθ θ

≤ ⇒ ≥

This apparent discrepancy can be understood

since the “GIM-like” mechanism works in GHU

i.e. FCNC process is automatically suppressed for light generation of quarks

( ) ( ) ( )( )

( )( ) ( )

( ) ( ) ( )

1 2

1 2

2 2

21 0 0 2 0 0

21

2

2 21 1 2

2 2

2

1 2 1 2

1

2

12

RM RM

n

n nKK RR RR

n

iRM RM

S R I In

M M M MRM

R M M M M

e e

e e

π π

π π

π

π

π π

∞

− −

− −

=

−≡ −

−

− +−

−

+

−

∑

≃

≫

Light quark masses are obtained from the large bulk massesthrough the factor exp[-πRM]

In the large bulk mass limit (πRM >> 1), the KK mode sum can be approximated as follows

2

22

ii qRM

W

me

mπ− ⇔ similar to

GIM suppression

2 2

2c u

W

m m

m

−

exponentialsuppression!!

More intuitive understanding of “GIM-like” suppression

More intuitive understanding of “GIM-like” suppression

FCNC is controlled by the factor

( ) ( )( )21 0 0 2 0 0n nRR RRI I− ( )0 0 2

2

1cos

1

i

i

R ii n M yRR RM

R

M nI dy e y

RR e

π

πππ −

= −∫

In πMR >> 1 limit & consider only for small mode index n

Width “1/M” of 0 mode function

Period 2πR/n of KK gluon mode function≪

Almost flat KK gluon mode function for fast exponential dumping 0 mode fermions

Almost flavor universal(similar to 0 mode sector, RS-GIM)

CP ViolationCP Violation

Adachi, Lim & Maru, PRD80 055024 (2010)

Lim, Maru & Nishiwaki, PRD81 076006 (2010)

In gauge-Higgs unification, Yukawa coupling is provided by real gauge coupling

↓

The theory is CP invariant, no explicit breaking

The two mechanisms of “spontaneous” CP violation are now discussed

The two mechanisms of “spontaneous” CP violation are now discussed

1: CP violation by <Ay>

2: CP violation due to Compactification

CP violation by Higgs VEVCP violation by Higgs VEV

Adachi, Lim & Maru (2010)

CP can be broken due to the VEV of CP odd HiggsCP can be broken due to the VEV of CP odd Higgs

Consider 5D theory

0 2 *: iψ γ γ ψ→CP

( ) ( ) ( ) ( ): , , , , ,Tyyx y x y A A A Aµ µ

µ µ→ → − −CP

yA is CP odd and leads to CP violation

5D CP trf = 4D CP trf

Accordingly, CP trf for the space-time coordinate & the gauge field are found as

To get physical CP violation,

we need both bulk masses & Higgs VEVTo get physical CP violation,

we need both bulk masses & Higgs VEV

LessonLesson

CommentComment

In the case of vanishing bulk mass, the coupling of Ay can be the scalar coupling

by a chiral rotation of matter fermion↓

Ay is CP even

To check this expectation, neutron EDM (P & CP violating quantity) is calculated

in 5D SU(3) gauge-Higgs modelwith a fermion in the 3-dim. representation

All of internal lines are KK modes

( )3

6 2 2654.6 10 2.9 10yN

Me Rd g R e cA m

π− − × < × ⋅

≃

Mc > 2.6TeVMc > 2.6TeV

Already generate at 1-loop (e.g. at least 3-loop in SM) & even with one generation (e.g. at least 3 gen. in SM)

Already generate at 1-loop (e.g. at least 3-loop in SM) & even with one generation (e.g. at least 3 gen. in SM)!!

CP violation due to CompactificationCP violation due to Compactification

Lim, Maru & Nishiwaki (2010)

If the extra space has a complex structure, CP can be broken due to the geometry of compactified space

If the extra space has a complex structure, CP can be broken due to the geometry of compactified space

Consider the simplest case: 6D

5 55 5

2 1 25 5

0 0 0, ,

0 0 0y zi

I i ii

µµ µ

µ

γ γ γγ γ σ γ σγ γ γ

Γ = ⊗ = Γ = ⊗ = Γ = ⊗ = −

( )† TM MC CΓ = − Γ

�

4

0 22

C

C iγ γ σ= ⊗

Gamma matrices:

Charge conjugation matrix satisfying

is4D case is not reproduced

( )6 ,TψΨ = ΨMixing of ψ & Ψ

( )6 ,TψΨ = Ψ

( ) ( )06 6 6 43 3 6: , : TCσγ σΨ → ⊗ Ψ Ψ → ⊗ ΨP C

Modified C, P transformation for 6D Dirac fermion

( ) ( ) ( ) ( ), ,: , , , , :y z y z y z y z→ −→P C CP

C, P transformation for extra coordinates

Introducing a complex coordinate ω = y + iz

CP: ω→ω*CP: ω→ω*CP transformation becomes

complex conjugation

Lim (1991)

Consider an orbifold compactification2

4T Z

CP symmetry is incompatiblewith Z4 orbifold condition

(If Z2 instead of Z4 is considered,CP is not broken ∵ π = -π)

CP symmetry is incompatiblewith Z4 orbifold condition

(If Z2 instead of Z4 is considered,CP is not broken ∵ π = -π)

To check this expectation, we have shown in a model of 6D U(1) gauge-Higgs on T^2/Z4

� CP phase remained in the interaction vertex after the re-phasing of fields

� Nonzero Jarskog-type parameter� CP is broken even with one generation

→ New mechanism different from that of KM

� CP phase remained in the interaction vertex after the re-phasing of fields

� Nonzero Jarskog-type parameter� CP is broken even with one generation

→ New mechanism different from that of KM

SummarySummary

● Gauge-Higgs unification predicts finite Higgs mass andis an attractive scenario for the physics beyond the SM

● As Yukawa coupling is universal, Yukawa hierarchy and flavor mixing are challenging issues

● Yukawa hierarchy is realized through exp[-MR]by the fermion localization at different points

● Non-degenerate bulk masses are new sources of flavor violation beyond the Glashow-Weinberg argument & lead to FCNC at tree level

● In the case of mixing, nonzero KK gluon exchange at tree level yields the amplitude suppressed by the compactification scale and the data put its lower bound like O(10TeV)

● Gauge-Higgs unification predicts finite Higgs mass andis an attractive scenario for the physics beyond the SM

● As Yukawa coupling is universal, Yukawa hierarchy and flavor mixing are challenging issues

● Yukawa hierarchy is realized through exp[-MR]by the fermion localization at different points

● Non-degenerate bulk masses are new sources of flavor violation beyond the Glashow-Weinberg argument & lead to FCNC at tree level

● In the case of mixing, nonzero KK gluon exchange at tree level yields the amplitude suppressed by the compactification scale and the data put its lower bound like O(10TeV)

0 0K K−

� “GIM-like” mechanism works in GHU as well

� As Yukawa coupling is real, CP violation is also a challenging issue

� Two new mechanisms of spontaneous CP violationby the VEV of Higgs & by the compactification

were proposed

� These mechanisms of CP violation are beyond that of KM, & might be relevant for Baryon asymmetry of the universe

� “GIM-like” mechanism works in GHU as well

� As Yukawa coupling is real, CP violation is also a challenging issue

� Two new mechanisms of spontaneous CP violationby the VEV of Higgs & by the compactification

were proposed

� These mechanisms of CP violation are beyond that of KM, & might be relevant for Baryon asymmetry of the universe

� “GIM-like” mechanism works in GHU as well

� As Yukawa coupling is real, CP violation is also a challenging issue

� Two new mechanisms of spontaneous CP violationby the VEV of Higgs & by the compactification

were proposed

� These mechanisms of CP violation are beyond that of KM, & might be relevant for Baryon asymmetry of the universe

� “GIM-like” mechanism works in GHU as well

� As Yukawa coupling is real, CP violation is also a challenging issue

� Two new mechanisms of spontaneous CP violationby the VEV of Higgs & by the compactification

were proposed

� These mechanisms of CP violation are beyond that of KM, & might be relevant for Baryon asymmetry of the universe

Thank you very muchThank you very muchfor your attention!!for your attention!!

Backup SlidesBackup Slides

No reason to choose the same bulk masses for different representations, 3 & 6*

Natural choice if we have some GUT where 3 & 6* are embedded into

a single representation of the GUT group

Sp(8) → Sp(6) x SU(2) → SU(3) x U(1) x SU(2)Sp(8) → Sp(6) x SU(2) → SU(3) x U(1) x SU(2)

36 → (1, 3) + (21, 1) + (6, 2) → (1, 3) + (1 + 6 + 6* + 8, 1) + (3 + 3*, 2)

36 → (1, 3) + (21, 1) + (6, 2) → (1, 3) + (1 + 6 + 6* + 8, 1) + (3 + 3*, 2)

3 & 6* of SU(3) can be embedded into an adjoint representation 36 of Sp(8)

( )1

2MN M

MN MF F i D= − + ΨΓ ΨL Tr ( )5,M iµγ γΓ =

[ ]( )( )

( )

1

5

1 2 3

, , 0,1,2,3,5

2 : :

, ,

MN M N N M D M N

a a aM M M M M

T

F F F ig A A M N

D ig A A A λ λ

ψ ψ ψ

+= ∂ − ∂ − =

= ∂ − =

Ψ =

Gell-Mann matrices

LagrangianLagrangian

Boundary conditions:Boundary conditions:

Chiral fermionsChiral fermions

( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )

( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )

( ) ( )( ) ( )( ) ( )

1 1

1 2 2

3 3

, ,, , , , , ,

, , , , , , , , , ,

, , , , , , , ,

L R

D L R

L R

A Aµ

ψ ψψ ψψ ψ

+

+ + + − − + + + + − − − − − − + + = + + + + − − = − − − − + + Ψ = + + + − − − − − − + + + + + + − − − − + + +

SU(3) → SU(2) x U(1)SU(3) → SU(2) x U(1) HiggsHiggs

( ) ( ) ( ) ( )( ) ( )( )

( )

( )

( )

( ) ( ) ( )( )

2 3 2

3

3

3

1

1

3

2 3

3 3

1

1

0

0

3

2 22 3 2 3

0 0

, , 0

0

,

2 3 3

2

2 2 2

,

n

n

nn

n

n

n nn

nn

n n

m m

m m

BW W W

W B

i m

i

i

W BW

W B W B

g

W

µµ µ µ

µ µ µ µµ

µ µ µ

µµ

µµ

µµ

µµ

ψ ψ

ψγψ γ

ψ ψ

γ ψ

ψ

ψ∞

+ +

−

−

=

+−

+

− − −

∂ − = ∂ − ∂ −

+

+

− + − −

∑ ɶ ɶ ɶ

ɶ

ɶ ɶ

L4D

fermion

( )

( )

( )

( ) ( ) ( )

( )

( ) ( ) 55

3

3

2

1

5

22

32

6

1 11 , 1 , ,

2 2

n

L

n

n

L

n

g git t b i m b t LbW b LtW t Lt b Lb W

gt Lt b Lb b Rb B

gnL R m g

R

µ

µ µ µ µ µ µµ µ µ µ µ

µ µ µµ

ψ

γ

γ γ γ γ γ γ

γ γ γ

γ γ

ψ

ψ

+ −

+ ∂ + ∂ − + + + +

+ + −

≡ − ≡ + = =

ɶ

ɶ

( )1,

22Wm gv M

Rπ= =

4D fermion effective Lagrangian in terms of mass eigenbasis

( )

( )

( )

( )

( )

( )

1 1

2 2

3 3

2 0 01

0 1 12

0 1 1

n n

n n

n n

ψ ψ

ψ ψ

ψ ψ

= −

ɶ

ɶ

ɶ

Derivation of chiral suppressionDerivation of chiral suppression

( ) ( ), , ,

1 1

6 2a a

a

s Ld s Rd s Ld s Rd s Ld s Rdµ µ µα β α µ β α α β µ β α β β µ ααβ α β

α β α βλ γ λ γ γ γ γ γ′ ′′ ′

′ ′= − +∑ ∑ i i

1 1

6 2a aij kl ij kl il jkλ λ δ δ δ δ= − +

( ) ( )( ) ( ) ( ) ( )

( ) ( ) ( ) ( )( )5 5

2 2

2 0 0 0 0

40 0

2

1

2

a b a b

a bab

a b a b a b a b

a b a bab abn

a b a b a b a b

a b a bab ab

a

ab

K s L d s R d K

K s L d n n s R d K K s L d n n s R d K

K s L d s R d K K s L d s R d K

K s d s d K K s R d

µα α β µ α

µ µα α β µ α α β β µ α

µ µα α β µ α α β β µ α

µα α β µ β α β

γ γ

γ γ γ γ

γ γ γ γ

γ γ γ γ

′ ′

′ ′

′ ′ ′ ′

′ ′ ′ ′

′ ′ ′ ′

′ ′ ′ ′

′′

= −

≈ −

= − −

∑

i

( )

( )2

2 5 5

2 2

2 2 2

1

2 2 3 3

0 0

1

0

1

4

0

1

6 6

b a b

a b

ab abKK

K

K KK K K K K K

d s d s

s L d K

fp K s d s d K

m

m mf m f m f m

m m m m

β α

µα α β β

δ δ γ γ

′′

= − − +

= + ≈ + +

4-Fermi operator

Hadronic matrix element

“vacuumsaturation”

Fierz transformation

5 5 55

25 5 5

10 0 , 0 0

32

10 , 0 0

32

KK

K

K K

d sK

fj K s d K p s d K s d K

m

f ms d K s d K s d K

m mm

µ µ µ µ µα α α β αβ α α

α α α β αβ α α

γ γ γ γ δ γ γ

γ γ δ γ

= = =

= − =+

2

4 32

cos sin 0 cos sin 1 0,

sin cos 0 sin cos 0 1

a aU U

b b

θ θ θ θθ θ θ θ

′ ′− − − = = ′ ′ −

† †3 3 4 4 1U U U U+ =

( )

( )

200

32

200

4

†

†

2

†

†

0 cos sin 1 0

0 sin cos 0 1

0 cos sin 0

0 sin cos 0

ˆ

ˆ

d dR dL dR dLRL

RLu uR uL uR uL

Y V V V V

Y V

c a

V

I Ud b

c aV VI U

d b

θ θθ θ

θ θθ θ

− − −

′ ′ − ′ ′

= =

= =

Yukawa couplings

Parameterize rotation angles as (CP invariance is assumed)

with satisfying a unitarity condition

( ) ( ) ( )00 0 0 2iR i i i RM

RL L RRI dyf f RM e

π ππ

π −

−= ≈∫

Now, we focus on 2 generation case

2 2 † 2 2 †

ˆ ˆˆ ˆ ˆ ˆ ˆdet , det ,

ˆ ˆ ˆ ˆˆ ˆ ˆ ˆ, ,

u c u d s d i i W

u c u u d s d d c dL uL

m m Y m m Y m m m

m m TrY Y m m TrY Y θ θ θ

= = ≡

+ = + = = −

( )( )( )( )

( ) ( ) ( )( )

( )( ) ( )( )

2 2 2 2 2 2 2 2 2 2 2 2

2 2 2 2 2 2 2 2 2 2 2

2 2 2 2 2 2 2 2 2 2 2

2 2 2 2

2 2 2 2

ˆ ˆ ˆ ˆ, 1 1

ˆ ˆ sin

ˆ ˆ 1 1 sin

tan 2 tan 2tan 2

1 tan 2 tan 2

2 1 1 sin costan 2

1 cos

u c d s

u c

d s

dL uLc

dL uL

dL

m m a b c d m m a b c d

m m a c b d a b c d

m m a c b d a b c d

a b d c

a c d

θ

θ

θ θθθ θ

θ θθ

θ

= = − −

′+ = + − − −

+ = − + − − − −

−=+

− − −=

− +

( ) ( )( )

( )( ) ( )

2 2 2 2 2 2

2 2

2 2 2 2 2 2 2 2 2 2

sin 1 sin cos

2 sin costan 2

cos sin sin cosuL

b c d

ab d c

a c d b c d

θ θ θ

θ θθ

θ θ θ θ

− − +

′ ′−=

′ ′ ′ ′+ − +

Parameter fitting (6 parameters - 5 observables = 1 parameter)

( )

( )( ) ( )( )( ) ( )( )

( )( ) ( )

2 2

2 2 2 2 2 2 2 2 2 2

2 2 2 2 2 2 2 2 2 2

2 2

2 2

2 2

tan 2 tan 2tan 2

1 tan 2 tan 2

2 1 1 sin costan 2

1 cos sin 1 sin cos

2 sin costan 2

cos sin sin cos

dL uLc

dL uL

dL

uL

a b

a c d b

d c

d c

d c

c d

ab

a c d b c d

θ θθθ θ

θ θθ

θ θ θ θ

θ θθ

θ θ θ θ

−=+

− − =

∝

− + − − +

′ ′=

′ ′ ′ ′+

−

−

−

+

−

Mixing again

Universal bulk mass limit (c=d)Universal bulk mass limit (c=d)

tan2θc = 0 i.e. θc = 0 tan2θc = 0 i.e. θc = 0

Check

( )0Ls ( )0

Rs( )0Rd ( )0

Rd

( )0Ls( )0

Ls( )0Ld

( )0Rs

( )0Ld

( )0Ld

0 5K d sγ∼

Mass insertion is necessary

2 2 2106 1

497 5d s

K

m m MeV

m MeV

+ ≈ ≈

“Chiral suppression”

0K 0K

Left-Left type Left-Right type

“Dominant”

(Right-Right type is also suppressed similarly)

3 types of amplitudes should be considered

( )a nGµ

( )a nGµ

( )( ) ( ) ( )

( ) ( ) ( ) ( )

2

221 0 0 2 0 02 3 25

4 2

22 1 0 0 2 0 042

2

12 sin 2

9

17.5 10 sin 2

SK eff

n

n nKS K K dR RR RR

nd s

n

n ndR RR RR

n

m K K

B mCR B f m I I

m m n

Rf C R I Inπ

π α θ

π θ

∆ =∆ =

− − − − +

− × −

∑

∑

∼

∼

KK modes

L

( ) ( )( )

( ) ( ) ( )( )2 2 2 2 2 2

2 2 2 2

20 0

1 1 11 sin 2 cos 1 sin 2 sin 1 1 cos 2 sin 2

2 2 2

1 1 1sin 2 cos sin 2 sin

2 2 2

cos1

dL dL dL

dL dL

Ri n i

RR RR

C a b a b

a b a

I dy f yn

yRR

π

π

θ θ θ θ θ θ

θ θ θ θ

π −

= − − + − − − −

′ ′− + −

=

∫

cos 2 sin 2dL

b θ θ ′

KL-KS mass differenceWe have calculated KL-KS mass difference from the dominant left-right type amplitude

B4,5: Bag parameters (B4=0.81, B5=0.56)

( )1min0 60.2R TeV−≤ ≤

Solution 1

( )1min0 60.2R TeV−≤ ≤

Solution 1

θ=0

2 20

32 2

cos sin 1 0 1 0

sin cos 0 1 0 1

a aU

b b

θθ θθ θ

= − − − = → − −

( )( )

2†

2

2 2

† †

2 2

0 1 0ˆ ,0 0 1

1 0ˆ ˆ 1, . .

0 10

d dR dL

d d dR dR dR dR

c aY V V

d b

a cY Y V V V i e

b dθ

− ′ = −

− ∴ = ⇒ = −

=

( ) ( ) ( ) ( ) 21 0 0 2 0 02 2

2sin 2 0

1n

n nK S K K RR RR

ndRm R f m I I

nθα

− ∆ − = ∑∼KK modes

Comment on a point 1/R=0Comment on a point 1/R=0

No constraint of 1/R from the left-right type…

Vanishing mixingin down sector

But, we have to notice that the L-L type or R-R type dominates over the L-R one at some value of sinθ’

2 2 2106 1

497 5d s

K

m m MeV

m MeV

+ ≈ ≈

Chiral suppression

(Right-Right type is also suppressed similarly)

Most stringent lower bound of 1/R from LL

～ 60TeV x 1/5～ O(10TeV)

Most stringent lower bound of 1/R from LL

～ 60TeV x 1/5～ O(10TeV)

( )0Ls ( )0

Rs( )0Rd ( )0

Rd

( )0Ls( )0

Ls( )0Ld

( )0Rs

( )0Ld

( )0Ld

0 5K s dγ∼

0K 0K

Left-Left type Left-Right type

( )a nGµ( )a nGµ

( )1min19.2 171R TeV−≤ ≤

Solution 2

In the extreme case of |sinθ’|～1the bulk mass of 2nd generationhappens to be relatively small

(see plot, d～1)↓

“GIM-like” mechanism does not work↓

Severe lower bound for the compactification scale

In the extreme case of |sinθ’|～1the bulk mass of 2nd generationhappens to be relatively small

(see plot, d～1)↓

“GIM-like” mechanism does not work(No exponential suppression)

↓

Severe lower bound for the compactification scale