Flavor Physics from Warped Extra Dimensionsweb.ics.purdue.edu/~markru/chang.pdf · A short tour into the Extra dimensions. Chang, Sanghyeon Flavor Physics from Warped Extra Dimensions

IntroductionExtra Dimensions

Bulk Standard ModelFermion mass and mixing in Bulk SM

Flavor physics in warped extra-dimension

Flavor Physics from Warped Extra Dimensions

Talk at Purdue University

Chang, Sanghyeon

Yonsei University

November 14, 2007

Chang, Sanghyeon Flavor Physics from Warped Extra Dimensions

Table of Contents

Introduction

Extra Dimensions

Bulk Standard Model

Fermion mass and mixing in Bulk SM

What is Extra Dimension?Hierarchy ProblemSupersymmetry

What is Extra Dimension?

For us, a rope is 1-D, but for ant, it is 2-D space.

Hierarchy Problem

I Why Gravity is so weak?

Weak Scale (MW ) = 1000 GeV

Planck Scale (Mpl) = 1019 GeV

I Weak scale : Standard Model

I Above Planck Scale : Quantum Gravity

I Between two scales : GUT, Supersymmetry, Superstring etc.

There is no experimental clue on the intermediate scale.

Hierarchy Problem

That’s not what they wanted to prove.

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The Problem of Higgs Mass

In Standard Model(SM),

Higgs mass correction is quadratic divergent to cutoff scale.

M2Higgs −→ M2Higgs + cΛ2cut

We need an upper bound of Λcut to be around 1 TeV.

In SM, Mpl is a natural cut-off.

The Problem of Higgs Mass

In Standard Model(SM),

Higgs mass correction is quadratic divergent to cutoff scale.

M2Higgs −→ M2Higgs + cΛ2cut

We need an upper bound of Λcut to be around 1 TeV.

In SM, Mpl is a natural cut-off.

Supersymmetry(SUSY)

To solve Hierarchy problem, SUSY was introduced.

SUSY is a symmetry between boson and fermion.

Each particle in SM has a superpartner

with the same quantum number but different spin.

Each loop-correction by scalar loop is cancelled by higgsino loop.

(soft) SUSY breaking generates natural cut-off at ∼TeV.

Supersymmetry(SUSY)

Not exactly replacing SUSY, but a simple idea was brought by

Arkani-Hamed, Dimopoulos and Dvali PLB 429 (1998).

ADD suggest that the hierarchy problem can be solved by

existence of extra spatial dimensions.

Not exactly replacing SUSY, but a simple idea was brought by

Arkani-Hamed, Dimopoulos and Dvali PLB 429 (1998).

ADD suggest that the hierarchy problem can be solved by

existence of extra spatial dimensions.

AAD ModelStructure of flat extra-dimensionRandall-Sundrum modelRS1 modelWhere do we live in RS1 space?

The FLATLAND

Flat Extra dimensions

If our Universe has a thickness R,

we will find extra dimensions in a small scale r < R.

Let the number of extra dimensions n.

Then the gravity become stronger in small range.

G1

r, (r > R)

G(4+n)1

rn+1, (r < R)

G1

r, (r > R)

G(4+n)1

rn+1, (r < R)

G1

r, (r > R)

G(4+n)1

rn+1, (r < R)

V

Gravity

ElectroMagnetic

R r

The 4D gravity can be weakened by extra-dimensions.

A short tour into the Extra dimensions

How to compactify extra-dimensions is not yet understood.

It is hard to comprehend due to its “large” size.

How to compactify extra-dimensions is not yet understood.

It is hard to comprehend due to its “large” size.

We are here

Gravity is confined here

Our (3 + 1)D universe

We are here

Gravity

Flat extra dimension S1 (Arkani-Hamed et.al.)

We are hereGravity is localized

RS1 Warped extra dimension S1/Z2

All SM filed should be confined on the 3+1D brane except gravity

This can be related with the string theory with brane solutions

As you know, string need to be compactified.

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The object of this model is to solve the hierarchy problem.

To explicit, to keep the Higgs mass 1-loop contribution TeV

which should be the QG scale.

In short,QG scale = TeV

Existence of massive Kaluza-Kelin(KK) graviton is cosmologically

dangerous.

Randall-Sundrum model (RS1)

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Randall-Sundrum model (RS1)Randall and Sundrum PRL (1999)

RS1 is model with one warped extra-dimension.

ds2 = e−2σ(y)(dt2 − dx2)− dy2,

σ(y) = k |y | is the warp factork ∼ Mpl is the curvature of warped space.

The 5th direction y is bounded and the bulk is AdS5

All SM fields are confined on TeV brane (IR boundary)

Gravity resides in the bulk.

TeV scale KK gravitons.

ds2 = e−2σ(y)(dt2 − dx2)− dy2,

TeV scale KK gravitons.

ds2 = e−2σ(y)(dt2 − dx2)− dy2,

TeV scale KK gravitons.Chang, Sanghyeon Flavor Physics from Warped Extra Dimensions

Where do we live in RS1?

UV(IR) brane is actually a UV(IR) cut-off (boundary).

There is no reason for SM to be confined.

If the energy level is close to TeV,

SM fields behave more tamed in the bulk.

But, how can we develop bulk SM?

The conformal coordinate of z ≡ eσ/k is useful.

ds25 =1

(kz)2

(dt2 − d~x 2 − dz2) .

y is confined in 0 ≤ y ≤ L

z is also bounded in 1/k ≤ z ≤ 1/T .T ≡ e−kLk ≡ � k.

With kL ≈ 35, the warp factor � ≡ e−kL reduces T at TeV scalefrom k at Planck scale:

With this scaling the gauge hierarchy problem is answered.

ds25 =1

(kz)2

(dt2 − d~x 2 − dz2) .

ds25 =1

(kz)2

(dt2 − d~x 2 − dz2) .

Fields in RS1 BulkExperimental signalsBulk SM with Custodial isospinBulk SM KK modeHiggsless model

Bulk fields in RS1

I Bulk scalar field was introduced at early stage of RS model.

Goldberger and Wise, (1999).

I Then Bulk gauge field was introduced by

Davoudiasl,Hewett and Rizzo, (1999).

I And bulk fermion was followed soon after.

SC, Hisano, Nakano, Okada and Yamaguchi, (1999),

Grossman and Neubert (1999).

Bulk fields in RS1

Bulk gauge bosons

The action for a 5D U(1) gauge field

Sgauge =∫

d4xdz√

G[−14gMPgNQFMNFPQ + 12M2gMNAMAN] ,

where FMN = ∂MAN − ∂NAM .

The general mass term M2(z),

M2(z) = aUV2k δ(z − zUV) + aIR2k δ(z − zIR) + b2k2,

the dimensionless b and aUV(aIR) are bulk mass and localized mass

on the UV (IR) brane.

Bulk gauge bosons

The action for a 5D U(1) gauge field

Sgauge =∫

d4xdz√

G[−14gMPgNQFMNFPQ + 12M2gMNAMAN] ,

where FMN = ∂MAN − ∂NAM .

The general mass term M2(z),

M2(z) = aUV2k δ(z − zUV) + aIR2k δ(z − zIR) + b2k2,

the dimensionless b and aUV(aIR) are bulk mass and localized mass

on the UV (IR) brane.

The KK expansion of the dimension 3/2 field AM(x , z) is

Aν(x , z) =√

k∑

n A(n)ν (x)f

(n)A (z)

With the equation of motion for mode function

−z∂z(

1z ∂z f

(n)A (z)

)+ M

2(z)k2z2

f(n)A (z) = m

(n)A

2f

(n)A (z),

The mode function can be obtained

f(n)A (z) =

z

N(n)A

[Jν(m

(n)A z) + β

(n)A Yν(m

(n)A z)

],

where ν =√

1 + b2, b = 0 if the gauge symmetry is conserved in

the bulk.

Aν(x , z) =√

k∑

n A(n)ν (x)f

(n)A (z)

−z∂z(

1z ∂z f

(n)A (z)

)+ M

2(z)k2z2

f(n)A (z) = m

(n)A

2f

(n)A (z),

f(n)A (z) =

z

N(n)A

[Jν(m

(n)A z) + β

(n)A Yν(m

(n)A z)

],

where ν =√

the bulk.

Aν(x , z) =√

k∑

n A(n)ν (x)f

(n)A (z)

−z∂z(

1z ∂z f

(n)A (z)

)+ M

2(z)k2z2

f(n)A (z) = m

(n)A

2f

(n)A (z),

f(n)A (z) =

z

N(n)A

[Jν(m

(n)A z) + β

(n)A Yν(m

(n)A z)

],

where ν =√

the bulk.

Fermions in RS1 Bulk

Sfermion =∫

d4xdy[Ψ̂eσ iγµ∂µΨ̂− 12 Ψ̂γ5∂y Ψ̂ + 12 (∂y Ψ̂)γ5Ψ̂ + mDΨ̂Ψ̂

]

Bulk Lagrangian should respect the Z2 parity

γ5Ψ(x ,−y) = ±Ψ(x , y)

The bulk fermion can be divided into two chiral components,

Ψ̂ = Ψ̂L + Ψ̂R

which can be expanded to KK modes

Ψ̂(x , y)L(R) =√

k∑n

ψ(n)L(R)(x)f

(n)L(R)(y).

Sfermion =∫

]Bulk Lagrangian should respect the Z2 parity

γ5Ψ(x ,−y) = ±Ψ(x , y)

Ψ̂ = Ψ̂L + Ψ̂R

Ψ̂(x , y)L(R) =√

k∑n

ψ(n)L(R)(x)f

(n)L(R)(y).

Sfermion =∫

γ5Ψ(x ,−y) = ±Ψ(x , y)

Ψ̂ = Ψ̂L + Ψ̂R

Ψ̂(x , y)L(R) =√

k∑n

ψ(n)L(R)(x)f

(n)L(R)(y).

Sfermion =∫

γ5Ψ(x ,−y) = ±Ψ(x , y)

Ψ̂ = Ψ̂L + Ψ̂R

Ψ̂(x , y)L(R) =√

k∑n

ψ(n)L(R)(x)f

(n)L(R)(y).

SM fields

Parity of RS1 Warped extra dimension S1/Z2Chang, Sanghyeon Flavor Physics from Warped Extra Dimensions

Parity of RS1 Warped extra dimension S1/Z2

I The 5-th D parity should be conserved.

I Bulk fermion with odd parity cannot have a zero mode.

I The zero mode is chiral (thus massless).

I Chiral fermion is automatically localized on IR boundary.

I All bulk gauge boson have (massless) zero mode.

I (Massless) SM fields can be induced from bulk fields (except

the Higgs).

Will KK fields survive experimental bounds?

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Experimental signals

Energy scale is warped down at IR boundary (y = y0).

Mple−ky0 ∼ TeV

The QG scale is about TeV.

Bulk field generates KK modes with about TeV scale mass gap.

Quark KK mode contribution to MW /MZ conflicts with

experimental data

Mple−ky0 ∼ TeV

experimental data

Mple−ky0 ∼ TeV

experimental data

Custodial isospin

In 2003, Agashe et.al. suggested that if there is

SU(2)L×SU(2)R ×U(1)B−L gauge symmetry on S1/Z2×Z′2.

There exist Ads5/CFT conformal dual global SU(2) custodial

isospin.

Which protect MW /MZ ratio from KK mode contributions.

Agashe, Delgado, May and Sundrum, (2003)

Custodial isospin

In 2003, Agashe et.al. suggested that if there is

SU(2)L×SU(2)R ×U(1)B−L gauge symmetry on S1/Z2×Z′2.

There exist Ads5/CFT conformal dual global SU(2) custodial

isospin.

Which protect MW /MZ ratio from KK mode contributions.

Agashe, Delgado, May and Sundrum, (2003)

Other than custodial isospin,

this model has another important feature in it.

Two independent parities on each boundary (UV,IR).

(±,±) and (±,∓)

For bulk fermions, this gives symmetry

γ5Ψ(x ,−y) = ±Ψ(x , y) , γ5Ψ(x , L− y) = ±Ψ(x , L + y)

(±,±) and (±,∓)

We are here

Gravity is localized

Parities of Agashe et.al. Warped extra dimension S1/Z2 × Z′2Chang, Sanghyeon Flavor Physics from Warped Extra Dimensions

We are here

Gravity is localized

Parities of Agashe et.al. Warped extra dimension S1/Z2 × Z′2Chang, Sanghyeon Flavor Physics from Warped Extra Dimensions

Gauge boson KK mode mass on S1/Z2 × Z′2

246810m(n)A =T

(++)(+�) (�+) (��)aIR!1aUV!1

aIR;UV!1Chang, Sanghyeon Flavor Physics from Warped Extra Dimensions

Only (++) parity bulk field can have zero mode.

The gauge symmetry in 4D is broken for all other parities.

Thus WR fields are massive without bulk mass term.

The (+−) field contains a very light KK mode.

If the boundary mass term turns on and start to increase,

each mode mass transform to the other parity

Only (++) parity bulk field can have zero mode.

The gauge symmetry in 4D is broken for all other parities.

Thus WR fields are massive without bulk mass term.

The (+−) field contains a very light KK mode.

If the boundary mass term turns on and start to increase,

each mode mass transform to the other parity

The boundary mass modifies the KK modes masses

(++)

(−+)

(−−)

(+−)

aUV →∞

aIR →∞

aUV →∞

KK mass of bulk gauge boson

m(1)A(++) ≈ 2.45 T , m

(2)A(++) ≈ 5.57 T , m

(3)A(++) ≈ 8.70 T ,

m(1)A(+−) ≈ 0.24 T , m

(2)A(+−) ≈ 3.88 T , m

(3)A(+−) ≈ 7.06 T ,

m(1)A(−+) ≈ 2.40 T , m

(2)A(−+) ≈ 5.52 T , m

(3)A(−+) ≈ 8.65 T ,

m(1)A(−−) ≈ 3.83 T , m

(2)A(−−) ≈ 7.02 T , m

(3)A(−−) ≈ 10.17 T .

If T ∼ TeV, m(1)A(+−) can be about 100 GeV. (Without Higgs)

m(1)A(++) ≈ 2.45 T , m

(2)A(++) ≈ 5.57 T , m

(3)A(++) ≈ 8.70 T ,

m(1)A(+−) ≈ 0.24 T , m

(2)A(+−) ≈ 3.88 T , m

(3)A(+−) ≈ 7.06 T ,

m(1)A(−+) ≈ 2.40 T , m

(2)A(−+) ≈ 5.52 T , m

(3)A(−+) ≈ 8.65 T ,

m(1)A(−−) ≈ 3.83 T , m

(2)A(−−) ≈ 7.02 T , m

(3)A(−−) ≈ 10.17 T .

If T ∼ TeV, m(1)A(+−) can be about 100 GeV.

(Without Higgs)

m(1)A(++) ≈ 2.45 T , m

(2)A(++) ≈ 5.57 T , m

(3)A(++) ≈ 8.70 T ,

m(1)A(+−) ≈ 0.24 T , m

(2)A(+−) ≈ 3.88 T , m

(3)A(+−) ≈ 7.06 T ,

m(1)A(−+) ≈ 2.40 T , m

(2)A(−+) ≈ 5.52 T , m

(3)A(−+) ≈ 8.65 T ,

m(1)A(−−) ≈ 3.83 T , m

(2)A(−−) ≈ 7.02 T , m

(3)A(−−) ≈ 10.17 T .

If T ∼ TeV, m(1)A(+−) can be about 100 GeV. (Without Higgs)

Bulk fermion can have four different Z2 × Z′2 parity sets,Ψ̂i (x , y) =

√k∑

n[ψ(n)iL (x)f

(n)iL (y) + ψ

(n)iR (x)f

(n)iR (y)]

i = 1, 2 represent the parallel conditions (±±)fiL has (±±) parity and fiR has (∓∓),

i = 3, 4 represent the crossed conditions, where (±∓)fiL has (±∓) parity and fiR has (∓±).

f(n)iL (z) =

√z

N(n)i

[Jci + 12

(m(n)i z) + β

(n)i Yci + 12

(m(n)i z)

],

f(n)iR (z) =

√z

N(n)i

[Jci− 12

(m(n)i z) + β

(n)i Yci− 12

(m(n)i z)

].

ci is the bulk fermion mass.

√k∑

n[ψ(n)iL (x)f

(n)iL (y) + ψ

(n)iR (x)f

(n)iR (y)]

f(n)iL (z) =

√z

N(n)i

[Jci + 12

(m(n)i z) + β

(n)i Yci + 12

(m(n)i z)

],

f(n)iR (z) =

√z

N(n)i

[Jci− 12

(m(n)i z) + β

(n)i Yci− 12

(m(n)i z)

].

ci is the bulk fermion mass.

√k∑

n[ψ(n)iL (x)f

(n)iL (y) + ψ

(n)iR (x)f

(n)iR (y)]

f(n)iL (z) =

√z

N(n)i

[Jci + 12

(m(n)i z) + β

(n)i Yci + 12

(m(n)i z)

],

f(n)iR (z) =

√z

N(n)i

[Jci− 12

(m(n)i z) + β

(n)i Yci− 12

(m(n)i z)

].

ci is the bulk fermion mass.Chang, Sanghyeon Flavor Physics from Warped Extra Dimensions

m(0)1;2m(1)1m(2)1m(3)1

m(1)2m(2)2m(3)2

massinunitofT

0.80.60.40.20-0.2-0.4-0.6-0.8

1086420 m(1)3m(2)3m

(3)3m(1)4m(2)4m

(3)4KKmassinunitofT

0.80.60.40.20-0.2-0.4-0.6-0.8

1086420Figure: KK mass spectra of bulk fermion without Higgs in unit of T

Only (++) parity bulk mode function can be a zero mode.

The gauge field with other parity is massive in 4D.

(+−) parity mode has a light gauge boson without Higgs.

The fermions with other than (++) parity become massive in 4D.

(+−) and (−+) mode can have very light KK modes.

Higgs should be confined on IR boundary

Bulk Higgs scalar brings back hierarchy problem again.

Only (++) parity bulk mode function can be a zero mode.

The gauge field with other parity is massive in 4D.

(+−) parity mode has a light gauge boson without Higgs.

The fermions with other than (++) parity become massive in 4D.

(+−) and (−+) mode can have very light KK modes.

Higgs should be confined on IR boundary

Bulk Higgs scalar brings back hierarchy problem again.

Higgsless Standard Model

If the VEV of Higgs become very large, SM fields

((++) mode) simulate the (+−) KK modes without Higgs field.

Since (+−) has light gauge boson, and can have light fermionsThere is a possible Higgsless SM (Higgsless gauge symmetry

breaking).

Unfortunately, due to the parity conservation,

one of the fermion doublet component should have zero mass.

breaking).

Basic assumptionsQuark mass and mixingLepton mass and mixingBound from Lepton flavor violation

Basic assumptions

1. All mass hierarchy are generated from bulk mass structure:

All 5D SM parameters are considered O(1).

2. There should not be a order changing cancellation during the

matrix algebra between mixing and mass matrices.

With this two simple assumptions, we construct a bulk SM in

warped RS1.

Basic assumptions

warped RS1.

Basic assumptions

warped RS1.

Universal Yukawa coupling

The SM fermion mass and mixing is decided by

both bulk fermion masses and Yukawa coupling with Higgs.

We assume “(almost) universal Yukawa coupling”.

There is no hierarchy within Yukawa coupling matrix.

yf '

1 1 1

.

Universal Yukawa coupling

The SM fermion mass and mixing is decided by

both bulk fermion masses and Yukawa coupling with Higgs.

We assume “(almost) universal Yukawa coupling”.

There is no hierarchy within Yukawa coupling matrix.

yf '

1 1 1

.

The fermions mass matrix is generated from,

Mij/vW ' FL(ci )× FR(cj),

where

FL(ci ) = �ci−1/2

√2ci−1

1−�2ci−1 , FR(ci ) = �−ci−1/2

√2ci +1

�−2ci−1−1 ,

ci is mass of bulk fermion fi .

Left-handed and right handed SM fermion comes from different

bulk fermions.

where

FL(ci ) = �ci−1/2

√2ci−1

1−�2ci−1 , FR(ci ) = �−ci−1/2

√2ci +1

�−2ci−1−1 ,

bulk fermions.

where

FL(ci ) = �ci−1/2

√2ci−1

1−�2ci−1 , FR(ci ) = �−ci−1/2

√2ci +1

�−2ci−1−1 ,

bulk fermions.

FL(c) as a function of bulk mass c .

0

2

4

6

8

–0.4 –0.2 0 0.2 0.4 0.6 0.8

FL(c)

FL(0.5)

c

If we increase (−)ci , FL(R)(ci ) decrease slowly until (−)ci = 1/2.

For (−)ci > 1/2, it decrease fast in power of �(−)ci .

Without a large hierarchy in bulk fermion mass ci ,

All SM fermion masses can be generated.

If (−)ci < 0, the 1st KK mode can be very light. m(1) � TeV.

Model with larger symmetry might contains light neutral

and stable KK mode, i.e. WIMPS.

If we increase (−)ci , FL(R)(ci ) decrease slowly until (−)ci = 1/2.

For (−)ci > 1/2, it decrease fast in power of �(−)ci .

Without a large hierarchy in bulk fermion mass ci ,

All SM fermion masses can be generated.

If (−)ci < 0, the 1st KK mode can be very light. m(1) � TeV.

Model with larger symmetry might contains light neutral

and stable KK mode, i.e. WIMPS.

WIMPS are ...

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Bulk fermions

Qi contains left handed quark

Ui ,Di contains right handed quark

Li contains lepton doublet

Ei contains lepton singlet

Ni contains right handed neutrino

Quark mass and mixing

(Ma)ij ' vW FL(cQi )FR(cAj),

where a = u, d and A = U,D.

UTqLMqUqR = Mdiagq for q = u, d .

The CKM matrix is defined as K = UTuLUdL.

K '

1 λ λ3

λ 1 λ2

λ3 λ2 1

,For simplified Wolfenstein parameterization and λ ' 0.22.

Quark mass and mixing

(Ma)ij ' vW FL(cQi )FR(cAj),

where a = u, d and A = U,D.

UTqLMqUqR = Mdiagq for q = u, d .

The CKM matrix is defined as K = UTuLUdL.

K '

1 λ λ3

λ 1 λ2

λ3 λ2 1

,For simplified Wolfenstein parameterization and λ ' 0.22.

We also assume that there is no order changing cancellation

in matrix diagonalization or other algebras.

i.e. 1 is a approximation of a number between√λ and 1/

√λ.

Thus the order λ in matrix will not change after the algebra.

This simple assumption leads to a unique bulk mass structure for

SM fermions.

We also assume that there is no order changing cancellation

in matrix diagonalization or other algebras.

i.e. 1 is a approximation of a number between√λ and 1/

√λ.

Thus the order λ in matrix will not change after the algebra.

This simple assumption leads to a unique bulk mass structure for

SM fermions.

Since uL and dL has almost the same matrix form

UuL ' UdL '

1 λ λ3

λ 1 λ2

λ3 λ2 1

.

Write the quark masses with λ

Mdiagu = diag(mu,mc ,mt) ' vW diag(λ8, λ3.5, 1),Mdiagd = diag(md ,ms ,mb) ' vW diag(λ7, λ5, λ2.5).

With known masses of SM fermions,

UuL ' UdL '

1 λ λ3

λ 1 λ2

λ3 λ2 1

.

UuL ' UdL '

1 λ λ3

λ 1 λ2

λ3 λ2 1

.

Following hierarchical relations are obtained:

FL(cQ1) : FL(cQ2) : FL(cQ3) ' λ3 : λ2 : 1,FR(cA1) : FR(cA2) : FR(cA3) ' λ3 : λ2 : 1,FR(cD1)

2 + FR(cD2)2 + FR(cD2)

2 'λ5[FR(cU1)

2 + FR(cU2)2 + FR(cu2)

2].

The SM quark mixing matrices can be approximated as

(UqL)ij(i≤j) ≈FL(cQi )

FL(cQj ), (UqR)ij(i≤j) ≈

FR(cAi )

FR(cAj ),

where A = U,D.

From various experimental constraints, (e.g. Z → bb̄)

The bulk quark masses which satisfy all SM parameters can be

determined for T ∼ TeV.(SC, Kim, Yamaguchi)

cQ1 ' 0.61, cQ2 ' 0.56, cQ3 ' 0.3± 0.03.

cU1 ' −0.70, cU2 ' −0.52, 0

From various experimental constraints, (e.g. Z → bb̄)

The bulk quark masses which satisfy all SM parameters can be

determined for T ∼ TeV.(SC, Kim, Yamaguchi)

cQ1 ' 0.61, cQ2 ' 0.56, cQ3 ' 0.3± 0.03.

cU1 ' −0.70, cU2 ' −0.52, 0

Lepton mass matrix

UMNS = U†eUν ,

where

M†νMν = Uν(Mdiagν )2U

†ν , M

†eMe = Ue(M

diage )2U

†e .

MNS matrix can be approximated as

|UMNS| ∼

1 1 λm

1 1 1

where the experimental constraint on Ue3 gives m > 1.3.

Universal lepton mass matrices are

(Mν)ij ' vW FL(cLi )FR(cNj),(Me)ij ' vW FL(cLi )FR(cEj).

Individual neutrino masses are not yet measured.

But from the neutrino oscillation

∆m2sol = m22 −m21 ' 7.5× 10−5 eV2,

∆m2atm = |m23 −m22| ' 2.5× 10−3 eV2.

Neutrino mass can be either Normal Hierarchy(NH) ( m1 ' 0 )Or Inverse Hierarchy(IH) (m3 = 0)

∆m2sol = m22 −m21 ' 7.5× 10−5 eV2,

∆m2atm = |m23 −m22| ' 2.5× 10−3 eV2.

∆m2sol = m22 −m21 ' 7.5× 10−5 eV2,

∆m2atm = |m23 −m22| ' 2.5× 10−3 eV2.

Assumption of no order changing cancellation in matrix product

allows only NH for neutrino mass.

MTν Mν ∝

λ2n λn λn

λn 1 1

(NH).Also we parameterize lepton masses

Mdiagν = diag(m1,m2,m3) = vW diag(< λ20.5, λ20.5, λ19),

Mdiage = diag(me ,mµ,mτ ) = vW diag(λ8.5, λ5, λ3).

Assumption of no order changing cancellation in matrix product

allows only NH for neutrino mass.

MTν Mν ∝

λ2n λn λn

λn 1 1

(NH).Also we parameterize lepton masses

Mdiagν = diag(m1,m2,m3) = vW diag(< λ20.5, λ20.5, λ19),

Mdiage = diag(me ,mµ,mτ ) = vW diag(λ8.5, λ5, λ3).

Maximal mixing between 2 and 3 flavors suggests the mixing

Uf '

1 λaf λn

λbf 1 1

λcf 1 1

,with f = e, ν.

To have a desired mass hierarchy,

1.5 + aν & n, 2 + ae & n.

Unlike quark mass, lepton mass has some ambiguity.

Maximal mixing between 2 and 3 flavors suggests the mixing

Uf '

1 λaf λn

λbf 1 1

λcf 1 1

,with f = e, ν.

To have a desired mass hierarchy,

1.5 + aν & n, 2 + ae & n.

Unlike quark mass, lepton mass has some ambiguity.

With MNS matrix UMNS = UTe Uν

λaν ∼ λbν + λcν ∼ 1, λbe + λce . λm, λn . λm.

The allowed values of n and m in a narrow range

1.3 . m . n . 1.5.

Thus, as a representative value, we expect

Ue3 ' λm ' 0.10− 0.14,

which is almost marginal to experimental bound.

1.3 . m . n . 1.5.

Ue3 ' λm ' 0.10− 0.14,

1.3 . m . n . 1.5.

Ue3 ' λm ' 0.10− 0.14,

Lepton flavor violation

If we focus the KK mass within detectable range 2− 3 TeV,The Lepton flavor violation(LFV) processes gives a strong bound.

Γ(τ → 3e)Γ(µ→ eνµν̄e) ' (K

(1)L11K

(1)L12)

2

(mZ

M(1)A

)4

The K(1)Lij factor represents li -lj -A

(1) vertex,

K(n)Lij =

∑3k=1

(UeL)ik

ĝ (n)(cLk )(U†eL

)kj,

where effective coupling of KK gauge mode is,

ĝ(n)L (cfi ) =

√kL∫

dzk[f

(0)L (z , cfi )

]2f

(n)A (z) ≡ ĝ (n)(cfi ).

ĝ (n)(c) saturates around -0.2 if c > 0.55.

–0.5

0

0.5

1

1.5

2

2.5

0.2 0.3 0.4 0.5 0.6

ĝ(c)

c

if KK gauge boson is experimentally testable, i.e. M(1)A ' 2-3 TeV,

we can specify its mixing matrix

UeL '

1 δ δ

< δ2 1 1

δ 1 1

,where δ = λm ' 0.1,

and bulk lepton masses (SC, Kim, Song)

cL1 ' 0.59, cL2 ' 0.5, cL3 ' 0.5,

cE1 ' −0.74, cE2 ' −0.65, cE3 ' −0.55.

cN2 ' −1.2, cN3 ' −1.1.

if KK gauge boson is experimentally testable, i.e. M(1)A ' 2-3 TeV,

we can specify its mixing matrix

UeL '

1 δ δ

< δ2 1 1

δ 1 1

,where δ = λm ' 0.1, and bulk lepton masses (SC, Kim, Song)

cL1 ' 0.59, cL2 ' 0.5, cL3 ' 0.5,

cE1 ' −0.74, cE2 ' −0.65, cE3 ' −0.55.

cN2 ' −1.2, cN3 ' −1.1.

Experimental signal of flavor violation?B mixingresults

Is there experimental application of this model?

Theoretical or Experimental?

Loading..

llaw1.aviMedia File (video/avi)

B0q − B̄0q mixing data

The current experimental results are well measured

∆Mexpd = (0.507± 0.004) ps−1 ,∆Mexps =

[17.33+0.42−0.21(stat)± 0.07(syst)

]ps−1 .

There is no tree level contribution to B0q − B̄0q in SM.

Can be a good test for the RS bulk SM.

B0q − B̄0q mixing at tree level by bulk gluon

q̄

b

b̄

q

G(n)

The KK gluon contribution to B0q − B̄0q is tree level while SMcontribution is from box diagram.

For the B0q − B̄0q transition amplitude Mqbb̄ defined by

〈B0q |H∆B=2eff |B̄0q〉 = 2MBqMqbb̄,

where

∆Mq = 2 |Mqbb̄|.

“mixing-induced” CP violation by phase

φq = arg(Mq

bb̄

).

B0q − B̄0q mixing from the SM box diagrams and the RS KK gluons:

Mqbb̄

= Mq,SMbb̄

(1 +

Mq,RSbb̄

Mq,SMbb̄

).

We parameterize the new physics effect by

rqeiσq =

Mq,RSbb̄

Mq,SMbb̄

,

where rq ≥ 0 and σq is real.

The rq and σq are constrained by the experimental result for ∆Mq

and the theoretical calculation of the ∆MSMq ,

of which the ratio is defined by ρq:

ρq ≡∣∣∣∣ ∆Mq∆MSMq

∣∣∣∣ =∣∣∣∣∣M

q,SM

bb̄+ Mq,RS

bb̄

Mq,SMbb̄

∣∣∣∣∣ = √1 + 2rq cosσq + r2q .

φd can be divided into the SM phase and that from New Physics

(NP) contributions.

φq = φSMq + φ

NPq = φ

SMq + arg(1 + rqe

iσq ).

φNPq is determined by rq and σq.

sinφNPq =rq sinσq√

1 + 2rq cosσq + r2q

,

cosφNPq =1 + rq cosσq√

1 + 2rq cosσq + r2q

.

SM amplitude is

Mq,SMbb̄

=G2Fm

2W

12π2MBq η̂

B B̂Bq f2Bq

(V ∗tqVtb)2S0(xt) ,

where xt = m2top/m

2W , and S0 is an “Inami–Lim” function .

CKM components are

|V ∗tdVtb| = (8.6±1.3)×10−3 , |V ∗tsVtb| = (41.3±0.7)×10−3.

Short-distance QCD correction η̂B = 0.552

B̂Bd,s f2Bd,s

is determined from lattice simulation by JLQCD

collaboration.

New physics effect becomes

rqeiσq ≡ M

q,RS

bb̄

Mq,SMbb̄

=16π2

NC

8g2sg4S0(xt)

m2Wκ233κ

2q3

∑n=1

(ĝ (n)(cQ3)

m(n)A

)2.

O(1) ambiguity and phases in mixings are absorbed in√λ < κij <

1√λ

.

(UqL)ij = κijVCKMij ,

φd associated with B0d − B̄0d is well measured,

(sinφd)ccs̄ = sin(2β + φNPd ) = 0.687± 0.032,

where φSMd = 2β.

φNPd is estimated by (Ball and Fleischer, 2006)

φNPd∣∣incl

= −(10.1± 4.6)◦, φNPd∣∣excl

= −(2.5± 8.0)◦.

New Physics parameters are constrained by the CP phases.

1

2

3

4

5

6

7

8

9

10

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2

MKK

[TeV]

c=0.32, κ =1/ 22

ρd

(JLQCD)

φd

incl

π π π π π π π π π π

σd

Figure: Allowed parameter space of (σd , MKK ). Red lines satisfy the

observed ρd and blue lines for φNPd |incl, with 1σ uncertainty.

1

2

3

4

5

6

7

8

9

10

0.2 0.4 0.6 0.8 1.2 1.4 1.6 1.8 2

MKK

[TeV]

σs

c=0.32, κ =1/ 22

π π π π π π π π π π

ρs

(JLQCD)

0

Figure: Allowed parameter space of (σs , MKK ) from the observed ∆Ms .

Results

For the choice of optimistic parameters for bulk SM,

cQ3 = 0.32, κ = 1/√

2

B0d − B̄0d bounds allow KK mode mass as low as 3 TeV.

Bs mixing gives a weaker bound than Bd mixng.

Summary

1 SM field can be resides on S1/Z2 × Z′2 bulk space.

2 Fermion bulk masses Can be estimate from SM parameters.

3 In universal Yukawa coupling model, Ue3 ∼ 0.1 in the neutrinomixing matrix.

4 B0d − B̄0d constrains 1st KK mode M(1)A >∼ 3 TeV.

The last one is rather disappointing, since it will be impossible to

detect for LC and will be hard for LHC.

Summary

1 SM field can be resides on S1/Z2 × Z′2 bulk space.2 Fermion bulk masses Can be estimate from SM parameters.

Summary

Credit for the movie files

I NBC TV series Law & Order

Episode : Big Bang

I WB TV series Angel

Episode : Supersymmetry

Introduction

Extra Dimensions

Bulk Standard Model

Fermion mass and mixing in Bulk SM

Documents

Flavor Physics from Warped Extra Dimensionsweb.ics.purdue.edu/~markru/chang.pdf · A short tour into the Extra dimensions. Chang, Sanghyeon Flavor Physics from Warped Extra Dimensions