Takaaki Nomura(Saitama univ) collaborators Joe Sato (Saitama univ) Nobuhito Maru (Chuo univ) Masato...
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Transcript of 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
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
2. Brief review of UED model with two-sphere
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
2. Brief review of UED model with two-sphere
(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
2. Brief review of UED model with two-sphere
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
2. Brief review of UED model with two-sphere
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. Brief review of UED model with two-sphere
)2(1 :linear combination of )(A
Lightest kk particle is stable by Z2 parity on the orbifold
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)
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
3. Quantum correction to KK mass
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
3. Quantum correction to KK mass
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)KK mode sum Sum of (l,m)
Bulk: m is conservingBoundary: m is non-conserving
3. Quantum correction to KK mass
Compare the structure of loop diagram with mUED case Ex)
),,( mlp )',,( mlpU(1) gauge boson loop for fermion
R
bulk
PmlmmlmlImlmmlmlIp
mllmlp
),;,;,(),;,;,({[
)/log(),,;,,(
11121211
22121
}),;,;,(),;,;,( 11121211 LPmlmmlmlImlmmlmlI
Rl PmlmmlmlImlmmlmlIMi ),;,;,(),;,;,({4 111212115 1
}]),;,;,(),;,;,( 11121211 LPmlmmlmlImlmmlmlI
3. Quantum correction to KK mass
Compare the structure of loop diagram with mUED case Ex)
),,( mlp )',,( mlp
R
bulk
PmlmmlmlImlmmlmlIp
mllmlp
),;,;,(),;,;,({[
)/log(),,;,,(
11121211
22121
}),;,;,(),;,;,( 11121211 LPmlmmlmlImlmmlmlI
Rl PmlmmlmlImlmmlmlIMi ),;,;,(),;,;,({4 111212115 1
}]),;,;,(),;,;,( 11121211 LPmlmmlmlImlmmlmlI
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)
3. Quantum correction to KK mass
Compare the structure of loop diagram with mUED case Ex)
),,( mlp )',,( mlp
R
bulk
PmlmmlmlImlmmlmlIp
mllmlp
),;,;,(),;,;,({[
)/log(),,;,,(
11121211
22121
}),;,;,(),;,;,( 11121211 LPmlmmlmlImlmmlmlI
Rl PmlmmlmlImlmmlmlIMi ),;,;,(),;,;,({4 111212115 1
}]),;,;,(),;,;,( 11121211 LPmlmmlmlImlmmlmlI
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)
Other diagrams also have similar feature
3. Quantum correction to KK mass
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)
3. KK mode expansion and KK mass spectrum
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
3. KK mode expansion and KK mass spectrum
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
3. KK mode expansion and KK mass spectrum
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
3. KK mode expansion and KK mass spectrum
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
3. KK mode expansion and KK mass spectrum
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
)(),( zP mmml :Jacobi
polynomial
The condition to obtain massless fermion in 4 dim
Positive curvature of
2SMasses 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
2SMasses of fermions in four-dim
The background gauge field
BA
cancel
2. Brief review of UED model with two-sphere
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
2. Brief review of UED model with two-sphere
The condition to obtain massless fermion in 4 dim
Positive curvature of
2SMasses of fermions in four-dim
The background gauge field
BA
cancel
3. Quantum correction to KK mass
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
3. Quantum correction to KK mass