takaaki nomura(saitama univ) collaborators joe sato (saitama univ) nobuhito maru (chuo univ) masato...

球面オービフォールドを用いた６次元 UED模型における KK 質量への量子補正の計算

Takaaki Nomura(Saitama univ)collaborators

Joe Sato (Saitama univ)

Nobuhito Maru (Chuo univ)

Masato Yamanaka (ICRR)

arXiv:0904.1909 (to be published on Nuclear Physics B)N. Maru, T. N, J. Sato and M. Yamanaka(and in progress)

２０１０　１ . ２０　大阪大学

1. Introduction

Universal Extra Dimensional (UED) model

1. Introduction

Simple extension of SM to higher dimensional spacetime

Introducing compact extra space

All the SM particles can propagate on extra space

Providing a candidate of the dark matter

as a stable lightest Kaluza-Klein (KK) particle

One of an attractive candidate in this regard

Appelquist, Cheng, Dobrescu PRD67 (2000)

Existence of Dark Matter requires new physics It can not be explained by the Standard Model

We need new physics beyond the SM to describe DM

physics

Ex)Minimal UED modelSM is extended on 5 dim spacetime

Extra space is compactified to

Orbifolding is applied

R

4 dimensional spacetime

21 / ZS

(identification of (x,y) (x,-y))

(x)

(y)

To obtain chiral fermion in 4D, etc

1. Introduction

Proposed UED models apply only extra space 21 / ZS 2

2 / ZT

Application of other extra spaces is interesting for asking

Which extra space is more plausible

to describe dark matter physics?Which space is consistent with other

experimental results?

21 / ZS

1. Introduction

6 dim UED model is particularly interesting Suggestion of three generation from anomaly

cancellationsProton stability is guaranteed by a discrete symmetry

of a subgroup of 6D Lorentz group

B. A. Dobrescu, and E.Poppitz PRL 87 (2001)

T. Appelquist, B.A. Dobrescu, E. Ponton and H. U. Yee PRL 87(2001)

(T. N and Joe Sato 2008)

We proposed new 6dim UED model with

22 / ZS

Why two-sphere orbifold ?

22 / ZS

Correspondence with other model

This extra space is also used to construct Gauge-

Higgs

unification model

1. Introduction

What is a dark matter candidate in our model?

１ｓｔ　ＫＫ　 Photon?

What is the lightest KK particle?

1. Introduction

What is a dark matter candidate in our model?

１ｓｔ　ＫＫ　 Photon?

To confirm dark matter candidate

Calculate Quantum correction of KK mass

What is the lightest KK particle?

1.Introduction

2.Brief review of UED model with two-

sphere

3. Quantum correction to KK mass

4.Summary

Out line

2. Brief review of The UED model with two-sphere

arXiv:0904.1909 (to be published on Nuclear Physics B)N. Maru, T. N, J. Sato and M. Yamanaka

Universal Extra Dimensional(UED) Model with two-sphere(S2) orbifold

Extension of SM to 6-dimensional spacetime

Extra-space is compactified to S2/Z2

All the SM particles propagate extra-space

4M

22 / ZS

6M

Coordinates:6M ),,( xX M

)( x

),(

orbifolding : ),( ),(

Radius: R

2. Brief review of UED model with two-sphere

Orbifolding of 2S

2S

),( ),( identification

22 / ZS

Two fixed points: )0,2/( ),2/(

By orbifolding

Each field has a boundary condition Massless extra component gauge boson is

forbidden

2


Set up of the modelGauge

group SU(3)×SU(2)×U(1)Y×U(1)X

Necessary to obtain massless SM fermions

Weyl fermions of SO(1,5)

)(

)()(

X

XX

R

L

)(RL :Left(right) handed Weyl fermion of SO(1,3)

Fields

)(

)()(

X

XX

L

R

Gauge field

))(),(),(()( XAXAXAXAM

We introduce a background gauge field cosiQAB

It is necessary to obtain massless chiral fermion

: generator Q XU )1(

)(

)(

0

0

RL

LR

P

P

6-dim chiral projection op


(Manton (1979))

,AA

RdRuLQReLL

Field contents and their boundary conditions under

),( ),(

Particle Ex-U(1) cahrge 6-dim Chirality B.C.

1/2 － 1/2 ＋ 1/2 － 1/2 ＋ 1/2 ＋

0

0

0

),,(),,( 25 xIx

),,(),,( 25 xIx

),,(),,( 25 xIx

),,(),,( 25 xIx

),,(),,( 25 xIx

Corresponding to SM particles

),,(),,( xHxH

),,(),,( xAxA

),,(),,( ,, xAxA

H

Ex-U(1) charge, 6-dim chirality and boundary condition are chosen to obtain corresponding SM particles as zero mode


Kaluza-Klein mode expansion and KK mass

Gauge field (4-dim components) )(),(),,( xAYxA

lm

lmlm

Satisfying boundary condition ),(),( lmlm YY),(),( AA

)),()1(),((2

)( mll

lm

ml

lm YYi

Y

Fermion

ml Llm

Rlm

xz

xzx

,

)(

)(),(~

)(),(~),,(

),,(),,( )(25

)( xIx

)(

)(

X

X

L

R

~,~ are written by Jacobi polynomials


KK mass spectrum without quantum correction

lmAKK mass Mass degeneracy

lm1

lm

lmH

1l for evenl )0( lml for oddl )0( lm2

2 )1(

R

llmM SM

lSM

For fields whose zero mode is forbiddenby B.C. 0l

KK mass spectrum is specified by angular momentum on two-sphere


)2(1 :linear combination of )(A

Lightest kk particle is stable by Z2 parity on the orbifold


We calculate quantum correction to KK mass We focus on U(1)Y interection

To confirm 1st KK photon (U(1)Y gauge boson) is the lightest one

1st KK gluon would be heavy because of non-abelian gauge interection

We must confirm 1st KK photon can be lighter than right handed lepton

As a first step

We compare the structure of one loop diagram with that of mUED case (H.Cheng, K.T.Matchev and

M.Schmaltz 2002)

Calculation of one loop correction One loop diagrams for mass

correction Fermion(right-handed lepton)

Gauge boson(U(1)Y)

We calculated these diagrams



Compare the structure of loop diagram with mUED case Ex) Ex)

),,( mlp )',,( mlpU(1) gauge boson loop for fermion

21 1,5',

'',121 ))1((),,;,;(),;(

llmm

mlmm

mbulk mllmlpmlpi

21

11

1,5',2

'',2121 ))1((),,;,;(

llmmm

mlmmm

mbound mllmlp

Bulk contribution(m conserving)

Boundary contribution(m non-conserving)

Similar structure as mUED case


Compare the structure of loop diagram with mUED case Ex) Ex)


21 1,5',

'',121 ))1((),,;,;(),;(

llmm

mlmm

mbulk mllmlpmlpi

21

11

1,5',2

'',2121 ))1((),,;,;(

llmmm

mlmmm

mbound mllmlp

Bulk contribution(m conserving)

Boundary contribution(m non-conserving)KK mode sum Sum of (l,m)

Bulk: m is conservingBoundary: m is non-conserving


Compare the structure of loop diagram with mUED case Ex)


R

bulk

PmlmmlmlImlmmlmlIp

mllmlp

),;,;,(),;,;,({[

)/log(),,;,,(

11121211

22121

}),;,;,(),;,;,( 11121211 LPmlmmlmlImlmmlmlI

Rl PmlmmlmlImlmmlmlIMi ),;,;,(),;,;,({4 111212115 1

}]),;,;,(),;,;,( 11121211 LPmlmmlmlImlmmlmlI



),,( mlp )',,( mlp

R

bulk

PmlmmlmlImlmmlmlIp

mllmlp

),;,;,(),;,;,({[

)/log(),,;,,(

11121211

22121




U(1) gauge boson loop for fermion

Vertex factors dYmlmlmlI mlmlml

332211

~~),;,;,( *332211

)()1()1(2

),(~ ),(2

1

2

1

zPzzCe

z mm

ml

mmlmim

lm

)(),( zP mmml :Jacobi

polynomialVertices describe angular momentum sum rule

(log div part)



),,( mlp )',,( mlp

R

bulk

PmlmmlmlImlmmlmlIp

mllmlp

),;,;,(),;,;,({[

)/log(),,;,,(

11121211

22121




U(1) gauge boson loop for fermion

Vertex factors dYmlmlmlI mlmlml

332211

~~),;,;,( *332211

)()1()1(2

),(~ ),(2

1

2

1

zPzzCe

z mm

ml

mmlmim

lm


polynomialVertices describe angular momentum sum rule

(log div part)

Other diagrams also have similar feature


Qualitative features of the quantum corrections

KK mode sum is that of angular momentum numbers

Overall structure is similar to mUED There are bulk contribution and boundary contribution

KK photon receive negative mass correction

First KK photon would be the Dark matter candidate

Vertices factor express angular momentum sum rule

# of KK mode in loop is increased compared to mUED

We need numerical analysis of the loop diagrams to estimate KK mass spectrum

SummaryWe analyzed one loop quantum correction to KK mass

in two-sphere orbifold UED

Difference from mUED case and UED case 22 / ZT

mUED case and UED case 22 / ZT

Vertex give simple ex-dim momentum conservation

UED case 22 / ZS

Vertex give angular momentum summation

One loop diagrams have similar structure as mUED

Bulk contribution + boundary contribution

We need numerical analysis of the loop diagramsto confirm dark matter candidate

In progress

6M

][4

1(sin

24

NLMKKLMN

MM FFTrgg

gDidddxS

:6-dim gamma matrix

:covariant derivative 　

)](),([)()()( XAXAXAXAXF NMMNNMMN

)sin,,1,1,1,1( 222 RRdiaggMN

:M

:MD

: metric

255

154

2

I

AiD

AD

AD

cos23

Spin connection term (for fermion)

)( 343 I

(R:radius)

Action of the 6D gauge theory

))()( * VDD MM

2. UED model with two-sphere

][4

1(sin

24

NLMKKLMN

MM FFTrgg

gDidddxS

:6-dim gamma matrix

:covariant derivative 　

)](),([)()()( XAXAXAXAXF NMMNNMMN

)sin,,1,1,1,1( 222 RRdiaggMN

:M

:MD

: metric6M

255

154

2

I

AiD

AD

AD

cos23

Spin connection term (for fermion)

)( 343 I

(R:radius)

Action of the 6D gauge theory

))()( * VDD MM

2. UED model with two-sphere

It leads curvature originated mass of fermion in 4D

Derivation of KK spectrum Expand each field in terms of KK mode

Specified by angular momentum on two-sphre

Integrating extra space and obtain 4-dim Lagrangian

KK mass spectrum is specified

Each fields are expanded in terms of eigenfunctions of angular momentum on two-sphere

angular momentum on two-sphere

3. KK mode expansion and KK mass spectrum

Gauge field (ex-dim components)


AAAAAdddx~

sin

1)sin(

sin

12

sin

1)

~sin(

sin

1sin

24

Extra space kinetic term for ,A sin/~

AA

Substitute gauge field as

),,(sin

1),,(),,(

~12

xxxA

),,(sin

1),,(),,( 21

xxxA

2

22

2

12

14

sin

1)sin(

sin

12

sin

1)sin(

sin

1sin

Adddx

Written by square of angular momentum operator

Gauge field (ex-dim components)

These substitution and mode expansion lead KK mass termfor from extra space kinetic term

)(),()1(

1),,( 2,12,1 xY

llx

lm

lmlm

2

)1(

R

llM l

KK mass


Expanding as2,1

Satisfying B. C.

),(),( ,, AA

2,1

For 1

0lM For 2 Massless NG boson

These NG bosons are eaten by

lmA

KK-parity for each field6-dim Lagrangian has discrete symmetry of

),( ),(

Under the symmetry we can define KK-paritym)1(

Ex) for gauge field(4-dim components)

)(),()1()(),(),,( ,, xAYxAYxAlm

lmlm

m

lm

lmlm

Each mode has KK parity as－ for m = odd＋ for m = even

Lightest m = odd KK particle is stable

oddm

0m

0' m

Candidate of the dark matter

Not allowed by the parity


Comparison of mass spectrum with mUED ( ) Ex) for field with

)/1( 22 RM

0

5

10

Model with 21 / ZS Model with 2

2 / ZS

)/1( 22 RM

21 / ZS

0

5

222 / RnM

22 /)1( RllM


0SMm

Comparison of mass spectrum with mUED ( ) Ex) for gauge field(4-dim components) (Mg=0 for simplicity)

)/1( 22 RM

0

5

10

Model with 21 / ZS Model with 2

2 / ZS

)/1( 22 RM

21 / ZS

0

5

222 / RnM

22 /)1( RllM


Ex) for gauge field(4-dim components) (Mg=0 for simplicity) Ex) for gauge field(4-dim components) (Mg=0 for simplicity)

Discrimination from other UED models is possible

Different from mUED case and UED case 22 / ZT

mUED case and UED case 22 / ZT

Vertex give simple ex-dim momentum conservation

UED case 22 / ZS

Vertex give angular momentum summation

dYmlmlmlI

dYmlmlmlI

mlmlml

mlmlml

332211

332211

~~),;,;,(

~~),;,;,(

*332211

*332211

)()1()1(2

),(~ ),(2

1

2

1

zPzzCe

z mm

ml

mmlmim

lm

)()1()1(2

),(~ )1,1(1

2

11

2

1

zPzzCe

z mm

ml

mmlmim

lm


polynomial

The condition to obtain massless fermion in 4 dim

Positive curvature of

2SMasses of fermions in four-dim





The background gauge field

BA

cancel


Spin connection term should be canceled by background gauge field

Ex)2

1)(XQ for

)(

)()(

X

XX

R

L

L does not have mass term from the curvature

)(10

01

2

1)( XXQ

)(cos2

)( 3 XiXAB

cosiQAB

343 I





The background gauge field

BA

cancel


Propagators on 224 / ZSM

2525

'

25

' )1(2

IMIipMIip

i

l

mmml

l

mm

Fermion

I2 : ２ × ２ identity ± ： corresponding to B.C. ),,(),,( )(25

)( xIx

Gauge field

''22)1(

2

1mm

lmm

lMk

ig

''22)1(

2

1mm

lmm

lMk

i

４ D ： extra ：

Scalar field

''22)1(

2

1mm

lmm

lMk

i

± ： corresponding to B.C.

Vertices for U(1) interaction Fermion-gauge boson( ４ D)-fermion

Aig

)(332211)(332211 ),;,;,(),;,;,( RLLR PmlmlmlIPmlmlmlIig

dYmlmlmlI

dYmlmlmlI

mlmlml

mlmlml

332211

332211

~~),;,;,(

~~),;,;,(

*332211

*332211

Fermion-gauge boson(ex)-fermion

sin

54 AAig

)(332211)(332211 ),;,;,(),;,;,( RLLR PmlmlmlCPmlmlmlCig

)2(1 :linear combination of )(A


Vertices for U(1) interaction Scalar-gauge-scalar

chHAgH .*

),;,;,()'( 332211 mlmlmlJppig

dYYYmlmlmlJ mlmlml332211

*332211 ),;,;,(

Fermion-gauge boson(ex)-fermion

AHAHg *2

),;,;,;,( 443322112 mlmlmlmlKgig


takaaki nomura(saitama univ) collaborators joe sato (saitama univ) nobuhito maru (chuo univ) masato...

Documents