The Universal Extra Dimensional Model with S^2/Z_2

arXiv:0904.1909 (to be published on Nuclear Physics B)
N. Maru, T. N, J. Sato and M. Yamanaka
(and in progress)
Takaaki Nomura(Saitama univ)
collaborators
Nobuhito Maru (Chuo univ)
Joe Sato (Saitama univ)
Masato Yamanaka (ICRR)
２０１０１.２０大阪大学
1. Introduction
1. Introduction
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
One of an attractive candidate in this regard
Universal Extra Dimensional (UED) model
Appelquist, Cheng, Dobrescu PRD67 (2000)
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
1. Introduction
Proposed UED models apply only extra space S 1 / Z 2 T 2 / Z 2
Ex) Minimal UED model
(y)
SM is extended on 5 dim spacetime
R
S1 / Z2
1
Extra space is compactified to S / Z 2
(x)
4 dimensional spacetime
Orbifolding is applied
(identification of (x,y)
(x,-y))
To obtain chiral fermion in 4D, etc
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?
1. Introduction
2
S
/ Z2
We proposed new 6dim UED model with
Why two-sphere orbifold S 2 / Z 2 ?
6 dim UED model is particularly interesting
Suggestion of three generation from anomaly cancellations
B. A. Dobrescu, and E.Poppitz PRL 87 (2001)
Proton stability is guaranteed by a discrete symmetry
of a subgroup of 6D Lorentz group
T. Appelquist, B.A. Dobrescu, E. Ponton and H. U. Yee PRL 87(2001)
Correspondence with other model
This extra space is also used to construct Gauge-Higgs
unification model
(T. N and Joe Sato 2008)
1. Introduction
What is a dark matter candidate in our model?
What is the lightest KK particle?
１ｓｔＫＫ Photon?
1. Introduction
What is a dark matter candidate in our model?
What is the lightest KK particle?
１ｓｔＫＫ Photon?
To confirm dark matter candidate
Calculate Quantum correction of KK mass
Out line
1.Introduction
2.Brief review of UED model with two-sphere
3. Quantum correction to KK mass
4.Summary
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
2. Brief review of UED model with two-sphere
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
M
6
2
S / Z2
M
4
Radius: R
( ,  )
(x )
M 6 Coordinates:
X M  ( x  , ,  )
orbifolding : ( ,  )
(   , )
2. Brief review of UED model with two-sphere
Orbifolding of
S
2


2
S2
S 2 / Z2

 


( ,  )
(   , )
identification
Two fixed points: ( / 2,0)
( / 2,  )
By orbifolding
Each field has a boundary condition
Massless extra component gauge boson is forbidden
2. Brief review of UED model with two-sphere
Gauge group
Set up of the model
SU(3)×SU(2)×U(1)Y×U(1)X
Necessary to obtain massless SM fermions
Fields
Weyl fermions of SO(1,5)
 L ( X ) 

 ( X )  
 R ( X ) 
 R ( X ) 

 ( X )  
 L ( X ) 
6-dim chiral projection op
 PR ( L )
  
 0
 L (R ) :Left(right) handed Weyl fermion of SO(1,3)
0 

PL ( R ) 
Gauge field
AM ( X )  ( A ( X ), A ( X ), A ( X ))
AB  iQ cos
We introduce a background gauge field
: U (1) X generator
Q
(Manton (1979))
It is necessary to obtain massless chiral fermion
2. Brief review of UED model with two-sphere
Field contents and their boundary conditions
under ( ,  ) (   , )
Particle
LL
eR
QL
uR
dR
H
A
A ,
Ex-U(1) cahrge
6-dim Chirality
B.C.
1/2
－
( x,    , )   5  I 2( x, , )
1/2
＋
( x,    , )   5  I 2( x, , )
1/2
－
( x,    , )   5  I 2( x, , )
1/2
＋
( x,    , )   5  I 2( x, , )
1/2
＋
( x,    , )   5  I 2( x, , )
0
H ( x,    , )  H ( x, ,  )
0
A ( x,   , )  A ( x, , )
0
A , ( x,   , )   A , ( x, , )
Corresponding to SM particles
Ex-U(1) charge, 6-dim chirality and boundary condition are
chosen to obtain corresponding SM particles as zero mode
2. Brief review of UED model with two-sphere
Kaluza-Klein mode expansion and KK mass
Gauge field (4-dim components)
A ( x, , )  Y ( , ) A ( x)

lm
lm
Satisfying boundary condition
lm

(i)l m
Y 
(Ylm ( ,  )  (1)l Yl m ( ,  ))
2

lm
A (  , )  A ( , ) Ylm (  , )  Ylm ( , )
Fermion
~  lm ( z,  ) ( x) 


R

(  ) ( x, ,  )    ~ 


(
z
,

)

(
x
)
lm
l ,m 
L

~
~
 ,  are written by Jacobi polynomials
() ( x,   , )   5  I 2() ( x, , )
  R ( X ) 

 
  L ( X ) 
2. Brief review of UED model with two-sphere
KK mass spectrum without quantum correction
KK mass
Alm
M
l
SM
 m

lm
1

l (l  1)

R2
l 1
l
for l  even
for l  odd
(0  m  l )
(0  m  l )
For fields whose zero mode is forbidden
l0
by B.C.
lm
H lm
2
SM
Mass degeneracy
Lightest kk particle is stable by Z2
parity on the orbifold
1( 2)
:linear combination of
A ( )
KK mass spectrum is specified by
angular momentum on two-sphere
3. Quantum correction to KK mass
3. Quantum correction to KK mass
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)
3. Quantum correction to KK mass
Calculation of one loop correction
One loop diagrams for mass correction
Fermion(right-handed lepton)
Gauge boson(U(1)Y)
We calculated these diagrams
3. Quantum correction to KK mass
Compare the structure of loop diagram with mUED case
Ex)
U(1) gauge boson loop for fermion
( p, l , m)
( p, l , m' )
 i( p; l, m)   bulk ( p; l , m; l1 , l2 , m1 )( m,m'  (1)l ' m  m,m' 5 )
l1 ,l2 m1
Bulk contribution
(m conserving)
  bound ( p; l, m; l1, l2 , m1 )( 2m1 ,mm'  (1)l ' m  2m1 ,mm' 5 )
l1 ,l2 m1
Boundary contribution
(m non-conserving)
Similar structure as mUED case
3. Quantum correction to KK mass
Compare the structure of loop diagram with mUED case
Ex)
U(1) gauge boson loop for fermion
( p, l , m)
( p, l , m' )
 i( p; l, m)   bulk ( p; l , m; l1 , l2 , m1 )( m,m'  (1)l ' m  m,m' 5 )
l1 ,l2 m1
Bulk contribution
(m conserving)
  bound ( p; l, m; l1, l2 , m1 )( 2m1 ,mm'  (1)l ' m  2m1 ,mm' 5 )
l1 ,l2 m1
KK mode sum
Sum of (l,m)
Boundary contribution
(m non-conserving)
Bulk: m is conserving
Boundary: m is non-conserving
3. Quantum correction to KK mass
Compare the structure of loop diagram with mUED case
Ex)
U(1) gauge boson loop for fermion
( p, l , m)
( p, l , m' )
 bulk ( p, l , m; l1 , l2 , m1 )  log(2 /  2 )
[  p {I (l1 , m1 ; l2 ,m1  m; l , m) I (l , m; l2 ,m1  m; l1 , m1 ) PR
 I  (l1, m1; l2 ,m1  m; l, m) I  (l, m; l2 ,m1  m; l1, m1 )PL }
 4i 5 Ml1 {I  (l1, m1; l2 ,m1  m; l, m)I (l, m; l2 ,m1  m; l1, m1 )PR
 I (l1, m1; l2 ,m1  m; l, m)I  (l, m; l2 ,m1  m; l1, m1 )PL }]
3. Quantum correction to KK mass
Compare the structure of loop diagram with mUED case
Ex)
U(1) gauge boson loop for fermion
( p, l , m)
( p, l , m' )
 bulk ( p, l , m; l1 , l2 , m1 )  log(2 /  2 )
(log div part)
[  p {I (l1 , m1 ; l2 ,m1  m; l , m) I (l , m; l2 ,m1  m; l1 , m1 ) PR
 I  (l1, m1; l2 ,m1  m; l, m) I  (l, m; l2 ,m1  m; l1, m1 )PL }
 4i 5 Ml1 {I  (l1, m1; l2 ,m1  m; l, m)I (l, m; l2 ,m1  m; l1, m1 )PR
 I (l1, m1; l2 ,m1  m; l, m)I  (l, m; l2 ,m1  m; l1, m1 )PL }]
Vertex factors
I (l1 , m1 ; l2 , m2 ; l3 , m3 )   ~ *l1m1Yl2 m2 ~l3m3 d
m
m (m,m)
eim lm
~
lm ( z,  ) 
C (1  z) 2 (1  z) 2 Pl  m ( z)
2
1
(m,m)
Pl  m
( z ) :Jacobi
Vertices describe angular momentum sum rule
1
polynomial
3. Quantum correction to KK mass
Compare the structure of loop diagram with mUED case
Ex)
U(1) gauge boson loop for fermion
( p, l , m)
( p, l , m' )
 bulk ( p, l , m; l1 , l2 , m1 )  log(2 /  2 )
(log div part)
[  p {I (l1 , m1 ; l2 ,m1  m; l , m) I (l , m; l2 ,m1  m; l1 , m1 ) PR
Other diagrams also have similar feature
 I  (l1, m1; l2 ,m1  m; l, m) I  (l, m; l2 ,m1  m; l1, m1 )PL }
 4i 5 Ml1 {I  (l1, m1; l2 ,m1  m; l, m)I (l, m; l2 ,m1  m; l1, m1 )PR
 I (l1, m1; l2 ,m1  m; l, m)I  (l, m; l2 ,m1  m; l1, m1 )PL }]
Vertex factors
I (l1 , m1 ; l2 , m2 ; l3 , m3 )   ~ *l1m1Yl2 m2 ~l3m3 d
m
m (m,m)
eim lm
~
lm ( z,  ) 
C (1  z) 2 (1  z) 2 Pl  m ( z)
2
1
(m,m)
Pl  m
( z ) :Jacobi
Vertices describe angular momentum sum rule
1
polynomial
3. Quantum correction to KK mass
Qualitative features of the quantum corrections
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
KK mode sum is that of angular momentum numbers
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
Summary
We analyzed one loop quantum correction to KK mass
in two-sphere orbifold UED
One loop diagrams have similar structure as mUED
Bulk contribution + boundary contribution
Difference from mUED case and
mUED case and
T 2 / Z2
UED case
T 2 / Z 2 UED case
Vertex give simple ex-dim momentum conservation
S 2 / Z2
UED case
Vertex give angular momentum summation
We need numerical analysis of the loop diagrams
to confirm dark matter candidate
In progress
2. UED model with two-sphere
Action of the 6D gauge theory
1 MN KL
S   dx sin dd (i DM   2 g g Tr[ FMK FNL ]
4g
 ( DM )* DM   V ()  )
4
M
F MN ( X )   M AN ( X )   N AM ( X )  [ AM ( X ), AN ( X )]
gMN  diag(1,1,1,1,R2 ,R2 sin 2  )
     I2
M :
4   5 1
5   5   2
: M6
metric
(R:radius)
:6-dim gamma
matrix
D     A
DM :
D    A
:covariant
derivative
(3  I 4   3 )
3
D    i cos  A
2
Spin connection term (for
2. UED model with two-sphere
Action of the 6D gauge theory
1 MN KL
S   dx sin dd (i DM   2 g g Tr[ FMK FNL ]
4g
 ( DM )* DM   V ()  )
4
M
F MN ( X )   M AN ( X )   N AM ( X )  [ AM ( X ), AN ( X )]
gMN  diag(1,1,1,1,R2 ,R2 sin 2  )
     I2
M :
4   5 1
5   5   2
D     A
DM :
D    A
: M 6 metric
(R:radius)
:6-dim gamma
matrix
It leads curvature originated
mass of fermion in 4D
:covariant
derivative
(3  I 4   3 )
3
D    i cos  A
2
Spin connection term (for
3. KK mode expansion and KK mass spectrum
Derivation of KK spectrum
Expand each field in terms of KK mode
Specified by angular momentum on twosphre
Each fields are expanded in terms of eigenfunctions of
angular momentum on two-sphere
Integrating extra space and obtain 4-dim
Lagrangian
KK mass spectrum is specified
angular momentum on two-sphere
3. KK mode expansion and KK mass spectrum
Gauge field (ex-dim components)
A ,
Extra space kinetic term for
~
A  A / sin 
2

1
1
1
~
~ 


 1
4

 dx sin dd  sin  ( sin A )  sin   A   2 A    sin  ( sin A )  sin   A 


Substitute gauge field as
1
A ( x, ,  )   1 ( x, ,  ) 
   2 ( x,  ,  )
sin 
1
~
A ( x, ,  )   2 ( x, ,  ) 
1 ( x, ,  )
sin 
2

1
1
1 2 


 1
4
2

 dx sin dd  sin  ( sin 1 )  sin  1   2 A    sin  ( sin 2 )  sin  2 


Written by square of angular momentum operator
3. KK mode expansion and KK mass spectrum
Gauge field (ex-dim components)
Expanding1, 2
as
1,2 ( x, ,  )  
1
lm

Ylm ( ,  )1, 2 ( x)
l (l  1)
lm
Satisfying B. A , (  , )   A , ( , )
C.
These substitution and mode expansion lead KK mass term
for 1, 2 from extra space kinetic term
KK mass
Ml 
M l 0
l (l  1)
R2
For 1
For 2
Massless NG boson
These NG bosons are eatenAlmby
3. KK mode expansion and KK mass spectrum
KK-parity for each field
6-dim Lagrangian has discrete symmetry of
( ,  )
( ,    )
Under the symmetry we can define KK-parity (1)
m
Ex) for gauge field(4-dim components)
A ( x, ,   )  Ylm ( ,   ) Alm, ( x)   (1)m Ylm ( , ) Alm, ( x)
lm
lm
Each mode has KK parity
as
－ for m = odd
＋ for m = even
 m 0
 modd
 'm  0
Not allowed by the parity
Lightest m = odd KK particle is stable
Candidate of the dark matter
3. KK mode expansion and KK mass spectrum
Comparison of mass spectrum with mUED ( S 1 / Z 2 )
Ex) for field with mSM  0
M 2 (1 / R 2 )
M 2 (1 / R 2 )
10
M 2  l (l  1) / R2
M 2  n2 / R2
5
0
5
Model with S
1
/ Z2
0
Model with
2
S / Z2
3. KK mode expansion and KK mass spectrum
Comparison of mass spectrum with mUED (S 1 / Z 2 )
Ex)
for gauge field(4-dim
Discrimination
from othercomponents)
UED models is possible
(Mg=0 for simplicity)
M 2 (1 / R 2 )
M 2 (1 / R 2 )
10
M 2  l (l  1) / R2
M 2  n2 / R2
5
0
5
Model with S
1
/ Z2
0
Model with
2
S / Z2
Different from mUED case and
mUED case and
T 2 / Z2
UED case
T 2 / Z 2 UED case
Vertex give simple ex-dim momentum conservation
S 2 / Z2
UED case
Vertex give angular momentum summation
I (l1 , m1; l2 , m2 ; l3 , m3 )   ~*l1m1Yl2m2 ~l3m3 d
~
~
I  (l1 , m1; l2 , m2 ; l3 , m3 )    *l1m1Yl2m2 l3m3 d
m
m (m,m)
eim lm
~
lm ( z,  ) 
C (1  z) 2 (1  z) 2 Pl  m ( z)
2
1
1
m1
m1 ( m1 , m1 )
eim lm
~
lm ( z,  ) 
C (1  z ) 2 (1  z ) 2 Pl  m
( z)
2
1
(m,m)
Pl  m
( z ) :Jacobi
1
polynomial
2. Brief review of UED model with two-sphere
The condition to obtain massless fermion in 4 dim
Positive curvature
of S 2
Masses of fermions
in four-dim
2. Brief review of UED model with two-sphere
The condition to obtain massless fermion in 4 dim
Positive curvature
of S 2
Masses of fermions The background
gauge field AB
in four-dim

cancel
2. Brief review of UED model with two-sphere
The condition to obtain massless fermion in 4 dim
Positive curvature
of S 2
Masses of fermions The background
gauge field AB
in four-dim

cancel
Spin connection term should be canceled by background
gauge field
3
A  ( X )  i cos  ( X )
2
B

Ex)
1
Q (X )  
2
for
1 1 0 
( X )
Q( X )  
2  0  1
 L ( X ) 

 ( X )  
 R ( X ) 
AB  iQ cos
3  I 4   3
 L does not have mass term from the curvature
3. Quantum correction to KK mass
Propagators on M 4  S 2 / Z2
Fermion

 mm '
 mm '
i
l m
 (1)
 5  I2 
 

2   p  i 5  I 2 M l
 p   i 5  I 2 M l

I2 :２×２ identity
±：corresponding to B.C.
() ( x,   , )   5  I 2() ( x, , )
Gauge field
1  ig 
l
４D：


(

1
)
 mm '
mm '
2
2
2 k  Ml



1 i
l


(

1
)
 mm '
extra：
mm
'
2
2
2 k  Ml
Scalar field

1 i
l


(

1
)
 mm '
mm
'
2
2
2 k  Ml

±：corresponding to B.C.

3. Quantum correction to KK mass
Vertices for U(1) interaction
Fermion-gauge boson(４D)-fermion
gi  A

 ig  I (l1, m1; l2 , m2 ; l3 , m3 )PR( L)  I  (l1, m1; l2 , m2 ; l3 , m3 )PL( R)

I (l1 , m1; l2 , m2 ; l3 , m3 )   ~*l1m1Yl2m2 ~l3m3 d
~
~
I  (l1 , m1; l2 , m2 ; l3 , m3 )    *l1m1Yl2m2 l3m3 d
Fermion-gauge boson(ex)-fermion
A 
 4
5
 
gi  A  
sin  


 ig  C (l1, m1; l2 , m2 ; l3 , m3 )PR( L)  C (l1, m1; l2 , m2 ; l3 , m3 )PL( R)
1( 2)
:linear combination of
A ( )

3. Quantum correction to KK mass
Vertices for U(1) interaction
Scalar-gauge-scalar
gH*  HA  h.c
 ig ( p   p' ) J (l1, m1; l2 , m2 ; l3 , m3 )
J (l1 , m1 ; l2 , m2 ; l3 , m3 )   Y *l1m1Yl2 m2 Yl3m3 d
Fermion-gauge boson(ex)-fermion
g 2 H *HA A
ig 2 g  K (l1, m1; l2 , m2 ; l3 , m3 ; l4 , m4 )

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