Page 1
arXiv:hep-ph/0610155v2 16 Oct 2006
Preprint typeset in JHEP style - HYPER VERSION
UFIFT-HEP-06-16
UMD-HEP-06-055
February 2, 2008
Mixed Dark Matter in Universal Extra Dimension Models
with TeV Scale WRand Z′
Ken Hsieh1, R.N. Mohapatra1and Salah Nasri2
1Department of Physics, University of Maryland, College Park, MD 20742, USA
2Department of Physics, University of Florida, Gainesville, FL, 32611
E-mail: kenhsieh@physics.umd.edu, rmohapat@physics.umd.edu,
snasri@phy.ufl.edu
Abstract: We show that in a class of universal extra dimension (UED) models that solves
both the neutrino mass and proton decay problems using low scale left-right symmetry, the
dark matter of the Universe consists of an admixture of KK photon and KK right-handed
neutrinos. We present a full calculation of the dark matter density in these models taking
into account the co-annihilation effects due to near by states such as the scalar partner of
the KK photon as well as fermion states near the right-handed KK neutrino. Using the
value of the relic CDM density, we obtain upper limits on R−1of about 400 − 650 GeV
and MZ′ ≤ 1.5 TeV, both being accessible to LHC. For a region in this parameter space
where the KK right-handed neutrino contributes significantly to the total relic density of
dark matter, we obtain a lower bound on the dark matter-nucleon scattering cross section
of 10−44cm2, which can be probed by the next round of dark matter search experiments.
Keywords: Dark Matter, Cosmology of Theories beyond the SM, Field Theories in
Higher Dimensions, Beyond Standard Model, Compactification and String Models.
Page 2
Contents
1. Introduction2
2. Set up of the Model3
3.Spectrum of Particles
3.1Gauge and Higgs Particles at the Zeroth KK Level
3.2 Gauge and Higgs Particles at the First KK Level
3.3 Spectrum of Matter Fields
3.3.1Possible Dark Matter Candidates
6
7
8
10
10
4. Dark Matter Candidate I: ν2L,2R
4.1Annihilation Channels of ν2L,2R
4.2 Co-annihilation Contributions to the Relic Density of ν2L,2R
4.3 Important Differences in Comparison to Standard Analysis
11
11
13
14
5.Dark Matter Candidate II: B(11)
5.1 (Co)-Annihilation Channels of B(11)
5.2 (Co)-Annihilation Channels of B(11)
(-)
or B(11)
µ
14
15
16
µ
(-)
6. Numerical Results of Relic Density
6.1B(11)
6.2B(11)
17
17
18
µ -ν(01)
(-)-ν(01)
2L,2RDark Matter
2L,2RDark Matter
7. Direct Detection of Two-Component Dark Matter18
8. Some Phenomenological Implications20
9.Conclusions 21
10. Acknowledgement 21
Appendices21
A. Fields on T2/Z2× Z′
2
22
B. Normalization of Fields and Couplings
B.1 Matter Fields
B.2 Gauge Bosons
B.3 Normalization of Couplings
23
23
24
25
– 1 –
Page 3
1. Introduction
Understanding the dark constituent of the Universe is one of the major problems of physics
beyond the Standard Model (SM). While in the supersymmetric extensions of the Standard
Model, the lightest supersymmetric partner (LSP) of the standard model fields is one of the
most well motivated candidates for the cold dark matter (CDM), it is by no means unique
and other viable CDM candidates have been proposed in the literature [1–3]. It is hoped
that the Large Hadron Collider (LHC) will provide evidence for supersymmetry making
the case for this particle stronger. Nonetheless, at this point, different candidates must be
studied in order to isolate their possibly different signatures in other experiments in order
to make a proper identification of the true candidate. With this goal in mind, in this paper,
we continue our study [4] of a class of dark matter candidates [1,5], which arises in models
with extra dimensions [6–8], the so-called universal extra dimensional (UED) models [9].
The UED models lead to a very different kind of TeV scale physics and will also be
explored at LHC. These models have hidden extra spatial dimensions with sizes of order
of an inverse TeV with all SM fields residing in all the dimensions. There could be one
or two such extra dimensions and they are compactified with radius R−1≤ TeV [9]. It
has recently been pointed out [1] that the lightest Kaluza-Klein (KK) particles of these
models being stable can serve as viable dark matter candidates. This result is nontrivial
due to the fact that the dark matter relic abundance is determined by the interactions
in the theory which are predetermined by the Standard Model. It turns out that in the
minimal, 5D extra dimension UED models based on the standard model gauge group, the
first KK mode of the hypercharge boson is the dark matter candidate provided the inverse
size of the extra dimension is less than a TeV [1].
A generic phenomenological problem with 5D UED models based on the Standard
Model gauge group is that they can lead to rapid proton decay as well as unsuppressed
neutrino masses. One way to cure the rapid proton decay problem is to consider six
dimensions [10] where the two extra spatial dimensions lead to a new U(1) global symmetry
that suppresses the strength of all baryon number nonconserving operators. On the other
hand both the neutrino mass and the proton decay problem can be solved simultaneously if
we extend the gauge group of the six dimensional model to SU(2)L×SU(2)R×U(1)B−L[11].
This avoids having to invoke a seventh warped extra dimension solely for the purpose of
solving the neutrino mass problem [12]. With appropriate orbifolding, a neutrino mass
comes out to be of the desired order due to a combination two factors: the existence of
B − L gauge symmetry and the orbifolding that keeps the left-handed singlet neutrino as
a zero-mode which forbids the lower dimensional operators that could give unsuppressed
neutrino mass. Another advantage of the 6D models over the 5D ones is that cancellation
of gravitational anomaly automatically leads to the existence of the right-handed neutrinos
[13] needed for generating neutrino masses.
In a recent paper [4], we pointed out that the 6D UED models with an extended
gauge group [11] provide a two-component picture of dark matter consisting of a KK right-
handed neutrino and a KK hypercharge boson. We presented a detailed calculation of the
– 2 –
Page 4
relic abundance of both the νKK
the dark matter in the cryogenic detectors in these models. The two main results of this
calculation [4] are that: (i) present experimental limits on the value of the relic density [14]
imply very stringent limits on the the two fundamental parameters of the theory i.e. R−1
and the second Z′-boson associated with the extended gauge group i.e. R−1≤ 550 GeV
and MZ′ ≤ 1.2 TeV and (ii) for one particular region in this parameter range where the
relic density of the KK right-handed neutrino contributes significantly to the total relic
density of the dark matter, the DM-nucleon cross-section is greater than 10−44cm2, and is
accessible to the next round of dark matter searches. Thus combined with LHC results for
an extra Z′search, the direct dark matter search experiments could rule out this model.
This result is to be contrasted with that of minimal 5-D UED models, where the above
experiments will only rule out a part of the parameter space. Discovery of two components
to dark matter should also have implications for cosmology of structure formation.
In this paper, we extend the work of ref. [4] in several ways: (i) we update our calcu-
lations taking into account the co-annihilation effect of nearby states; (ii) a feature unique
to six and higher dimensional models is the presence of physical scalar KK states of gauge
bosons degenerate at the tree level with γKKstate and will therefore impact the discussion
of KK dark matter. Its couplings to matter have different Lorentz structure and therefore
contribute in different ways to the relic density. We discuss the relative significance of the
scalar state and its effect on the relic density calculation of the previous paper [4] for both
the cases when it is lighter and heavier than the γKKstate. (iii) We also comment on the
extra W and Z′boson phenomenology in the model.
This paper is organized as follows: in Section 2, we review the basic set up of the
model [11]. In Section 3, we present the spectrum of states at tree level. In Sections 4 and
5, we discuss the relic density of νKKstates and the hypercharge vector and pseudoscalar,
respectively. In Section 6, we give the overall picture of dark matter in these models in
terms of relic abundance and rates of direct detection. Section 7 discusses the signals such
two-component dark matter would give in direct detection experiments. In Section 8, we
give the phenomenology of the model for colliders, especially the Z′and WRproduction
and decays. Finally, in Section 9 we present our conclusions.
R
and the BKK
Y
as well as the cross section for scattering of
2. Set up of the Model
We choose the gauge group of the model to be SU(3)c× SU(2)L×SU(2)R×U(1)B-Lwith
matter content per generation as follows:
Q1,−,Q′
ψ1,−,ψ′
1,−= (3,2,1,1
1,−= (1,2,1,−1); ψ2,+,ψ′
3); Q2,+,Q′
2,+= (3,1,2,1
3);
2,+= (1,1,2,−1);(2.1)
where, within parenthesis, we have written the quantum numbers that correspond to each
group factor, respectively and the subscript gives the six dimensional chirality to cancel
gravitational anomaly in six dimensions. We denote the gauge bosons as GM, W±
and BM, for SU(3)c, SU(2)L, SU(2)Rand U(1)B-Lrespectively, where M = 0,1,2,3,4,5
1,M, W±
2,M,
– 3 –
Page 5
denotes the six space-time indices. We will also use the following short hand notations:
Greek letters µ,ν,··· = 0,1,2,3 to denote usual four dimensions indices, as usual, and
lower case Latin letters a,b,··· = 4,5 for those of the extra space dimensions. We will also
use ? y to denote the (x4,x5) coordinates of a point in the extra space.
First, we compactify the extra x4, x5dimensions into a torus, T2, with equal radii, R,
by imposing periodicity conditions, ϕ(x4,x5) = ϕ(x4+2πR,x5) = ϕ(x4,x5+2πR) for any
field ϕ. This has the effect of breaking the original SO(1,5) Lorentz symmetry group of the
six dimensional space into the subgroup SO(1,3) × Z4, where the last factor corresponds
to the group of discrete rotations in the x4-x5plane, by angles of kπ/2 for k = 0,1,2,3.
This is a subgroup of the continuous U(1)45rotational symmetry contained in SO(1,5).
The remaining SO(1,3) symmetry gives the usual 4D Lorentz invariance. The presence of
the surviving Z4symmetry leads to suppression of proton decay [10] as well as neutrino
mass [11].
Employing the further orbifolding conditions :
Z2: y → −y
Z′
2:
(2.2)
?
(x4,x5)′→ −(x4,x5)′
y′= y − (πR/2,πR/2)
We can project out the zero modes and obtain the KK modes by assigning appropriate
Z2× Z′
In the effective 4D theory the mass of each mode has the form: m2
N = ? n2= n2
2and m0is the Higgs vacuum expectation value (vev) contribution to
mass, and the physical mass of the zero mode.
We assign the following Z2× Z′
Gµ(+,+); Bµ(+,+); W3,±
Ga(−,−); Ba(−,−); W3,±
2quantum numbers to the fields.
N= m2
0+N
R2; with
1+ n2
2charges to the various fields:
1,µ(+,+); W3
1,a(−,−); W3
2,µ(+,+); W±
2,µ(+,−);
2,a(−,+).
2,a(−,−); W±
(2.3)
For quarks we choose,
Q1L≡
?
?
u1L(+,+)
d1L(+,+)
?
?
; Q′
1L≡
?
?
u′
d′
1L(+,−)
1L(+,−)
u′
d′
?
?
; Q1R≡
?
?
u1R(−,−)
d1R(−,−)
u2R(+,+)
d2R(+,−)
?
?
; Q′
1R≡
?
?
u′
d′
1R(−,+)
1R(−,+)
u′
d′
?
?
;
Q2L≡
u2L(−,−)
d2L(−,+)
; Q′
2L≡
2L(−,+)
2L(−,−)
; Q2R≡; Q′
2R≡
2R(+,−)
2R(+,+)
; (2.4)
and for leptons:
ψ1L≡
?
?
ν1L(+,+)
e1L(+,+)
?
?
; ψ′
1L≡
?
?
ν′
e′
1L(−,+)
1L(−,+)
ν′
e′
?
?
; ψ1R≡
?
?
ν1R(−,−)
e1R(−,−)
ν2R(+,−)
e2R(+,+)
?
?
; ψ′
1R≡
?
?
ν′
e′
1R(+,−)
1R(+,−)
ν′
e′
?
?
;
ψ2L≡
ν2L(−,+)
e2L(−,−)
; ψ′
2L≡
2L(+,+)
2L(+,−)
; ψ2R≡; ψ′
2R≡
2R(−,−)
2R(−,+)
.(2.5)
– 4 –
Page 6
The zero modes i.e. (+,+) fields corresponds to the standard model fields along with
an extra singlet neutrino which is left-handed. They will have zero mass prior to gauge
symmetry breaking. The singlet neutrino state being a left-handed (instead of right-handed
as in the usual case) has important implications for neutrino mass. For example, the
conventional Dirac mass term¯LHνRis not present due to the selection rules of the model
and Lorentz invariance. Similarly, L˜Hν2Lis forbidden by gauge invariance as is the operator
(LH)2. Thus neutrino mass comes only from much higher dimensional terms.
For the Higgs bosons, we choose a bidoublet, which will be needed to give masses to
fermions and break the standard model symmetry and and a pair of doublets χL,Rwith
the following Z2× Z′
?
φ−
χ−
2quantum numbers:
φ ≡
φ0
u(+,+) φ+
u(+,+) φ0
d(+,−)
d(+,−)
?
;χL≡
?
χ0
L(−,+)
L(−,+)
?
;χR≡
?
χ0
χ−
R(+,+)
R(+,−)
?
, (2.6)
and the following charge assignment under the gauge group,
φ = (1,2,2,0),
χL= (1,2,1,−1),χR= (1,1,2,−1).(2.7)
At the zero mode level, only the SM doublet (φ0
expectation values (vev) of these fields, namely ?φ0
symmetry and the extra U(1)′
lowest KK modes of all the particles and their masses is shown in Fig. 1 with the following
identification of modes in Table 1.
u,φ−
u) and a singlet χ0
u? = vwkand ?χ0
Rappear. The vacuum
R? = vR, break the SM
Ygauge group, respectively. A diagram that illustrates the
R− 1
(++)
(+−) (−+)
(−−)
R− 1
2
Φ(11)(−−)
Φ(00)(++)
Φ(11)(++)
Φ(10)(+−)
,
Φ(01)(+−)
Φ(01)(−+) Φ(10)(−+)
,
0
MASS
Figure 1: The masses of lowest KK-modes of 6D model.
The most general Yukawa couplings in the model are
hu¯Q1φQ2+ hd¯Q1˜φQ′
2+ he¯ψ1˜φψ2+ h′
u¯Q′
1φQ′
2+ h′
d¯Q′
1˜φQ2+ h′
e¯ψ′
1˜φψ′
2+ h.c.; (2.8)
– 5 –
Page 7
where˜φ ≡ τ2φ∗τ2is the charge conjugate field of φ. A six dimensional realization of the
left-right symmetry, which interchanges the subscripts: 1 ↔ 2, is obtained provided the
3 × 3 Yukawa coupling matrices satisfy the constraints: hu = h†
h′e= h
u; h′u= h
′†
u; he = h†
e;
′†
e; hd = h
′†
d. At the zero mode level one obtains the SM Yukawa couplings
L = hu¯QφuuR+ hd¯Q˜φudR+ he¯L˜φueR+ h.c.
It is important to notice that in the above equation hu,eare hermitian matrices, while hdis
not. The vev of φugives mass to the charged fermions of the model. As far as the neutrino
mass is concerned, the lowest dimensional gauge invariant operator in six-D that gives rise
to neutrino mass after compactificaion has the form ψT
vwkv2
R
M2
∗(M2
get neutrino masses of order ∼ eV without fine tuning. Furthermore, it predicts that the
neutrino mass is Dirac (predominantly) rather than Majorana type.
As there are a large number of KK
modes, one may worry whether or not
electroweak precision constraints in terms
of S and T parameters are satisfied. It
has been shown that in the minimal uni-
versal extra dimension (MUED) the KK
contributions to the T parameter almost
cancel for heavier standard model Higgs
[9,15]. However, it was found that in the
MUED for Higgs mass heavier than 300
GeV the lightest Kaluza-Klein particle is
the charged KK Higgs [16]. The abun-
dance of such charged massive particles
are inconsistent with big bang nucleosyn-
thesis as well as other cosmological obser-
vations for masses less than a TeV [17].
This lead to the conclusion that the com-
pactification scale 1/R > 400 GeV for
mH > 300 GeV. To our knowledge, there has been no such analysis for the 6−D mod-
els similar to ours, and it is outside the scope of the current paper to perform a complete
analysis regarding the electroweak constraints. Therefore, we leave the investigation of this
open issue for future work.
(2.9)
1,Lφψ2,Lχ2
Rand leads to neutrino
mass mν≃ λ
∗R)3. For M∗R ∼ 100 and M∗∼ 10 TeV, vR∼ 2 TeV and λ ∼ 10−3, we
(Z2,Z′
2) Particle Content
Q1L; u2R; d′
Gµ; Bµ; W3,±
φ0
u; χ0
Q′
W±
2,µ;
φ+
d; φ0
Q′
W±
2,a;
χ0
L
Q1R; u2L; d′
Ga; Ba; W3,±
(++)
2R; ψ1L; e2R; ν′
2L;
1,µ; W3
2,µ;
u; φ−
R
(+−)
1L; u′
2R; d2R; ψ′
1R; ν2R; e′
2R;
d; χ-
R
(−+)
1R; u′
2L; d2L; ψ′
1L; ν2L; e′
2R;
L; χ-
(−−)
2L; ψ1R; ν1R; e2L;
1,a; W3
2,a
Table 1: Particle content of 6D model separated
by Z2× Z′
2parities.
3. Spectrum of Particles
Once the extra dimensions are compactified, the KK modes are labelled by the quanta of
momenta in the extra dimensions. As we have two such extra spatial dimensions, the KK
modes are labelled by two integers, and we will denote a KK mode as φ(mn), where m
(n) is the momentum in the quantized unit of R−1along the fifth (sixth) dimension. A
– 6 –
Page 8
detailed expansion of a field in the 6D theory into KK mode is presented in the Appendix.
Generally, φ(mn)would receive a (mass)2of the order (m2+ n2)R−2.
3.1 Gauge and Higgs Particles at the Zeroth KK Level
In the gauge basis, we have the zero-mode gauge bosons: B(00)
After symmetry-breaking, we will have the usual SM gauge bosons: one exactly massless
gauge boson, A(00)
µ
, one pair of massive, charged vector boson W±,(00)
neutral guage boson Z(00)
µ
. In addition, we will have another neutral gauge boson Z′(00)
as well as mixing between Z(00)
µ
and Z′(00)
µ
.
In this subsection we calculate the zeroth-mode gauge boson masses and mixings from
Higgs mechanism (and drop the (00) superscript throughout this subsection). The relevant
terms are
(B-L)µ,W±,3(00)
L,µ
, and W3(00)
R,µ.
L,µ
, and one massive
µ
,
Lh= Tr[(Dµφ)†Dµφ] + (DµχR)∗DµχR+ (DµχL)∗DµχL
(3.1)
where
Dµφ = ∂µφ − igL(− →τ ·− →
?
φ−
WLµ)φ + igRφ(− →τ ·− →WRµ),
φ =
φ0
uφ+
uφ0
d
d
?
,
DµχL=
?
?
∂µ− igL(− →τ ·− − →WL,µ) + i(1
2)gBLB(B-L),µ
??
??
χ0
χ−
L
L
?
?
,
DµχR=∂µ− igR(− →τ ·− − →WR,µ) + i(1
2)gBLB(B-L),µ
χ0
χ−
R
R
,
− →τ ·− →Wµ=1
2
?
W3
µ
√2W+
−W3
µ
√2W−
µµ
?
. (3.2)
With vev of the fields ?φ0
the gauge bosons:
u? = vwand ?χ0
R? = vR, we obtain the following mass terms for
L =1
2v2
w(W+
L,µW−
L,µ) +1
2(v2
w+ v2
R)(W+
R,µW−
R,µ)
+1
2
?
W3
L,µW3
R,µB(B-L)µ
?
1
2g2
2gLgRv2
0
Lv2
w
−1
2g2
−1
2gLgRv2
R(v2
2(gRgBL)v2
w
0
−1
w
1
w+ v2
R) −1
R
2(gRgBL)v2
1
2g2
R
BLv2
R
W3,µ
L
W3,µ
R
Bµ
(B-L)
. (3.3)
The exact expressions of the mass eigenvalues and the compositions of the eigenstates
(Aµ,Zµ,Z′
R,µ) are rather complicated, and we make the
approximation of vR≫ vw. In this approximation, we find the relations,
µ) in terms of (B(B-L)µ,W3
L,µ,W3
Aµ
Zµ
Z′
µ
= U†
G
W0
W0
Bµ
1,µ
2,µ
,(3.4)
– 7 –
Page 9
where
U†
G=
sinθw
cosθw−sinθw0
0
cosθw 0
01
1
0 sinθR cosθR
0 cosθR−sinθR
00
, (3.5)
and
tanθR≡gBL
gR
,g2
Y≡
g2
BL+ g2
BLg2
R
g2
R
, and tanθw≡gY
gL.(3.6)
It is easy to understand UG intuitively. In the limit vw ≪ vR, the symmetry-breaking
occurs in two stages, corresponding to the two matrices in UG. First, we have SU(2)L×
SU(2)R× U(1)B-L→ SU(2)L× U(1)Y, where a linear combination of B(B-L),µand W3
acquire a mass to become Z′
µ, while the orthogonal combination, BY,µ, remains massless
and serves as the gauge boson of the residual group U(1)Y. Then we have the standard
electroweak breaking of SU(2)L×U(1)Y → U(1)em, giving us massive Zµand the massless
photon Aµ. Using UG, we can simplify the mass matrix enormously
R,µ
U†
GM2UG=
000
0M2
Z
−
g2
R
√
(g2
L+g2
Y)(g2
M2
R+g2
BL)M2
Z
0 −
g2
R
√
(g2
L+g2
Y)(g2
R+g2
BL)M2
Z
Z′
, (3.7)
where we have defined the mass eigenvalues (up to O(vw/vR)2)
M2
Z=?g2
M2
L+ g2
Y
?v2
w
2
Z′ = (g2
BL+ g2
R)v2
R
2
+
g4
R
(g2
BL+ g2
R)
v2
2.
w
(3.8)
Here we see that we have explicitly decoupled Aµ, and it remains massless exactly. Although
we have defined MZto be same as the tree-level mass of Z-boson of the Standard Model,
here Zµis strictly speaking not an eigenstate because of the Z − Z′mixing. Such mixing
would be important, as we will see, for the calculation of relic density and the direct
detection rates of the dark matter of the model. However, in the limit v2
will be working with, we can treat the defined masses and states in Eq. 3.8 as eigenvalues
and eigenstates, and treat the mixing terms perturbatively in powers of (v2
The only zero-mode Higgs bosons in the model are φ0(00)
the six degrees of freedoms are eaten and the remaining physical Higgs particles are the
real parts of φ0(00)
u
and χ0,(00)
R
. The masses are these particles are determined from the
potential, and are free parameters, whose values, however, do not affect the calculations of
the relic density and direct detection rates of the dark matter.
R≫ v2
wthat we
w/v2
R).
u
,φ−(00)
u
, and χ0,(00)
R
. Four of
3.2 Gauge and Higgs Particles at the First KK Level
We first consider the question of whether KK modes of Higgs bosons acquire vevs. The
zero modes Higgs bosons acquire vevs due to negative mass-squared terms in the potential.
– 8 –
Page 10
The higher KK modes of the Higgs bosons φ(mn), however, have an additional mass-squared
contribution of the form (m2+ n2)R−2. Therefore, if the negative mass-squared term in
the potential is smaller in magnitude than R−2, then none of the higher Higgs KK modes
would acquire vevs. We will assume this is the case in our calculations, and the only fields
that acquire vevs are φ0(00)
u
and χ0,(00)
R
, the zero-modes of neutral Higgs fields.
Here we will only consider the details of those gauge bosons in the (11) KK modes,
and in this subsection it is understood that we have the superscript (11). That is, we do
not consider the (01) and (10) modes of W±
R,µ,5,6. For a compact notation that will be
convenient later on, for the scalar partners (G5and G6) of a generic vector gauge boson
(Gµ), we form the combinations
G(±)≡
1
√2(G5± G6). (3.9)
In the absence of Higgs mechanism, G(+)will be eaten by Gµat the corresponding KK-
level, while G(−)will be left as a physical degree of freedom. Qualitatively, W±
linear combination of W±
while the two remaining orthogonal directions are left as physical degrees of freedom.
At the (11)-level, before symmetry breaking, we have the modes
R,µeats a
R,(+), W±
R,(−), and χ−
R(all fields with the superscript (01) and (10)),
Neutral Gauge Bosons:W3
W3
L,µ,W3
L,(+),W3
W±
Lµ
W±
R,µ,Bµ
L,(−),W3
Neutral Scalars:
R,(+),W3
R,(−),B(+),B(−),φ0
u,χ0
R
Charged Gauge Bosons:
Charged Scalars:
L,(+),W±
L,(−),φ−
u. (3.10)
Three (two) linear combinations of the neutral (charged) scalars would be eaten, leaving
seven (four) degrees of freedom (note that the Higgs fields are complex). Since only the
zero-mode Higgs acquire vevs, the Higgs mechanism contribution to the mass matrix of the
neutral gauge bosons is same as that in Eq. 3.3, and we have an additional contribution of
2R−213×3. We can diagonalize the (mass)2matrix up to O(v2
matrix UGand obtain the eigenvalues to be those in Eq. 3.8 with the additional 2R−2.
Of the neutral scalars, we have several sets of particles that do not mix with members
of other sets at tree level:
w/v2
R) using the same unitary
Set 1:
1
√2Re[φ0
W3
u],
1
√2Re[χ0
R]
Set 2:
L,(−),W3
W3
R,(−),B(−)
R,(+),B(+), Set 3:
L,(+),W3
1
√2Im[φ0
u],
1
√2Im[χ0
R]. (3.11)
The squared-mass of particles in Set 1 are simply 2R−2in addition to the squared-masses
of corresponding particles at (00)-modes. The mass matrix of particles in Set 2 are exactly
that of the neutral gauge bosons, with a lightest mode of A(−)with mass mA(−)= mAµ=
√2R−1. Three linear combinations of particles in Set 3 are eaten, and the two remaining
particles have masses that will depend on the Higgs potential. As is the case with the
– 9 –
Page 11
zeroth-modes, as long as these Higgs are heavier than the lightest gauge bosons, the values
of their masses will not affect our results about the dark matter of the model.
3.3 Spectrum of Matter Fields
Because there is no yukawa coupling between the Higgs doublet χRand matter, at tree level
all mass terms arise from the momentum in the extra dimensions and vw. The structure
of the yukawa couplings, with the Z2× Z′
fields have the SM spectrum. As for the higher modes, the mass terms arising from the
extra dimension connect the left- and right-handed components of a 6D chiral fermion
Ψ±, where ± denotes 6D chirality. The mass terms arising from electroweak symmetry-
breaking, however, connects left- and right-handed components of two different 6D chiral
fields. Taking the electron as an example, the mass matrix of the electron KK modes in
the basis {e1Le1Re2Le2R} (with e1Land e2Rhaving zero modes) is
?
Generalizing this, we see that the (mn) modes have masses
2orbifolding ensures that the zero-mode matter
Me(1) =
?
e1Le1Re2Le2R
?
√2
0R−1
0
yev2
0
0yev2
0
−R−1
0
√2
R−1
0
yev2
yev2
0
−R−1
√2
√2
√2
e1L
e1R
e2L
e2R
=
e1e2
??
R−1yev2
yev2
√2−R−1
??
e1
e2
?
. (3.12)
mf(mn) =
?N2
R2± m2
f(00)
?1/2
, (3.13)
where N2= m2+ n2and m2
f(00)= y2
f
v2
2is the zero-mode mass of the fermion.
w
3.3.1 Possible Dark Matter Candidates
In order to see the dark matter candidates in our model, we look at the spectrum of the
KK modes (see Fig. 1). There are two classes of KK modes of interest whose stability is
guaranteed by KK parity: the ones with (−,−) and (±,∓) Z2× Z′
The former have mass√2R−1and the latter R−1. We see from Fig. 1 that the first class
of particles are the first KK mode of the hypercharge gauge boson BY and the second are
the right handed neutrinos ν2L, 2R. The presence of the RH neutrino dark matter makes
the model predictive and testable as we will see quantitatively in what follows. The basic
idea is that νRannihilation proceeds primarily via the exchange of the Z′boson. So as
the Z′boson mass gets larger, the annihilation rate goes down very fast (like M−4
the ν2’s overclose the Universe. Also since there are lower limits on the Z′mass from
collider searches [25], the ν2’s contribute a minimum amount to the ΩDM. This leads to
a two-component picture of dark matter and also adds to direct scattering cross section
making the dark matter detectable. Below we make these comments more quantitative
and present our detailed results.
2quantum numbers.
Z′ ) and
– 10 –
Page 12
4. Dark Matter Candidate I: ν2L,2R
4.1 Annihilation Channels of ν2L,2R
Since the yukawa couplings are small, except for the top-quark coupling, we only consider
annihilations through gauge-mediated processes. For completeness, we first list the cou-
plings between matter fields and the neutral vector gauge bosons. For matter fields charged
under SU(2)1, we have
LSU(2)1
ffB
= (qγµPLq)
??
T3
L+YBL
2
??
R+ g2
gLgRgBL
?g2
Lg2
R+ g2
BLg2
L+ g2
BLg2
?
R
?
Aµ
+
?
T3
Lg2
Lg2
L(g2
BL) −YBL
L+ g2
?
2g2
?g2
Rg2
BL
?g2
2
R+ g2
BLg2
BLg2
?
R BL+ g2
R
Zµ
+
?
YBL
g2
BL
?g2
BL+ g2
R
Z′
µ
. (4.1)
And for matter fields charged under SU(2)2,
LSU(2)2
ffB
= (qγµPRq)
??
T3
R+YBL
2
??
gLgRgBL
?g2
??
2g2
Lg2
R+ g2
BLg2
L+ g2
BLg2
R
?
Aµ
+
?
?
−T3
R−YBL
2
g2
Rg2
BL
?g2
BL
?
Lg2
R+ g2
?
BLg2
L+ g2
BLg2
R
?g2
BL+ g2
R
?
Zµ
+
−T3
Rg2
?g2
R+YBL
BL+ g2
R
Z′
µ
, (4.2)
where T3
respectively. We choose this notation because SU(2)1is to be identified with SU(2)Lof
the Standard Model, even though there are right-handed particles that are charged under
the SU(2)1group. Also, YBL= +1/3 for quarks and YBL= −1 for leptons.
Using these formulas, the gauge interaction of the dark matter candidates ν2L,2R is
given by the six-dimensional Lagrangian
L= ±1
2and T3
R= ±1
2are the quantum number for the SU(2)1and SU(2)2groups
Lν= −1
2(νγµν)
g2
R+ g2
?g2
BL
BL+ g2
R
Z′
µ. (4.3)
We first notice that ν2L,2Rcouple as a Lorentz vector. Second, we see that ν2L,2Rdo not
couple to Aµnor Zµas expected because ν2L,2Rare singlets under the SM gauge group.
There is a small coupling between ν2L,2Rand Zµdue to Z′
of evaluating annihilation cross sections, we can safely ignore this mixing, as we will show.
However, this mixing will be important when we consider the direct detection of ν2L,2R. In
addition, we have the charged-current interaction, similar to the SM case
µ− Zµmixing. For the purpose
LCC=gR
√2
?
ν2γµW+
2,µPRe2+ e2γµW−
2,µPRν2
?
.(4.4)
– 11 –
Page 13
f
νR
KK
f
νR
KK
__
_
Z’
νR
KK
νR
KK
__
eR
eR
WR
KK
_
Figure 2: Diagrams of annihilation channels of νKK
2L,2Rto SM fermion-antifermion pairs.
Even though e2is a Dirac spinor, its left-handed component has Z2×Z′
and the annihilation is kinematically forbidden.
Although we have two independent Dirac fermions for dark matter, ν(10)
couple the same way to Z′
µand have the same annihilation channels. The only difference
is that, for charged current processes, ν(01)(ν(10)) couples to W±,(01)
dominant contribution to the total annihilation cross section of ν2L,Ris s-channel process
mediated by Z′
µ, as shown in Fig. 2. The thermal-averaged cross section for ?σ(ν2ν2→
ff)vrel?, where f is any chiral SM fermion except the right-handed electron eR, is
g2
2charge of e2L(−−),
2
and ν(01)
2
, they
2,µ
(W±,(10))
2,µ
). The
σ(ν2ν2→ ff)vrel=
(ννZ′µ)g2
(ffZ′µ)
12π
s + 2M2
(s − M2
ν
Z′)2, (4.5)
and with s = 4M2
ν+ M2
νv2
rel, we expand in v2
rel,
σ(ν2ν2→ ff)vrel=
g2
(ννZ′µ)g2
(ffZ′µ)
2π
M2
ν− M2
ν
(4M2
Z′)2
?
1 + v2
rel
?1
6−
2M2
ν− M2
ν
4M2
Z′
??
.(4.6)
For the final state eReR, we have a t-channel process through charged-current in ad-
dition to the s-channel neutral-current process (see Fig. 2). The cross-section therefore
involves three pieces: two due to the s and t channels and another from the interference,
denoted by σss, σttand σstrespectively. Of these, σsshas the same form as Eq. 4.6, and
we have
σ(ν2ν2→ eRer)ttvrel=
g4
32π
R
M2
(M2+ M2
WR)2
?
1 + v2
rel
?3M4+ M2M2
3(M2+ M2
WR+ M4
WR)2
WR− 7M2M2
WR)
(4.7)
WR
??
,
σstvrel=
g(ννZ′µ)g(eReRZ′µ)g2
4π(4M2− M2
R
Z′)(M2+ M2
WR)
?
M2− v2
rel
M2(40M4+ M2
WRM2
Z′+ 8M2M2
Z′)(M2+ M2
Z′)
12(4M2− M2
?
Due to Z − Z′, there can also be annihilation of KK neutrino into SM Higgs, charged
bosons, as well as fermion-antifermion pairs. The diagrams for these processes are shown
in Fig. 3. In the limit that vw≪ vR, we can work to the leading-order in the expansion
of O(v2
mass-insertion. In terms of Feynman diagrams, these annihilation channels are s-channel
processes, where a pair KK neutrino annihilates into a Z′-boson, which propagates to the
w/v2
R), where we can estimate these processes by treating the Z − Z′mixing as a
– 12 –
Page 14
νR
νR
KK
νR
KK
__
Z’
f
_
f
Z
νR
__
KK
νR
KK
__
Z’
Z
h
h
νR
KK
KK
__
Z’
Z
h
h
νR
KK
νR
KK
Z’
Z
W+
W−
Figure 3: Diagrams of annihilation channels of νKK
2L,2Rto SM particles through Z − Z′mixing.
mixing vertex, converting Z′to Z, which then decays into h∗h (both neutral and charged),
massless W+W−or ff. Compared to the amplitude of annihilation of KK neutrino into
SM fermions without Z − Z′mixing, the annihilation through mixing have effectively a
replaced propagator
1
(s − M2
Z′)→
1
(s − M2
Z′)δM2
1
(s − M2
Z)
(4.8)
where
δM2≡
g2
R
?(g2
L+ g2
Y)(g2
R+ g2
BL)M2
Z, (4.9)
is the off-diagonal element in the Z − Z′(mass)2matrix. Since s ∼ 4M2
annihilation cross section into transverse gauge bosons and the Higgs bosons are suppressed
by a factor of M4
The same is true for the annihilation to fermion-antifermion pairs of the SM; we can ignore
the effects of Z − Z′mixing in these channels. As for the longitudinal modes, the ratio of
annihilation cross-sections of the longitudinal modes of the gauge bosons to the one single
mode of SM fermion-antifermion pair is roughly
ν= 4R−2, the
Z/s2∼ (100GeV )4/16(500GeV )4∼ 10−4, and can therefore be neglected.
σ(νKKνKK→ W+W−)
σ(νKKνKK→ ff)
2for gR= 0.7gL. As there is only one annihilation mode into the longi-
tudinal modes of the charged gauge bosons, whereas there are many annihilation channels
to the SM fermion-antifermion pairs, the total annihilation cross section is dominated by
the SM fermion-antifermion contributions.
∼
?δM2
m2
W
?2
.(4.10)
This ratio is about1
4.2 Co-annihilation Contributions to the Relic Density of ν2L,2R
In the MUED model, the KK mode of the left-handed electron, eKK
degenerate with the KK mode of the left-handed neutrino. The self- and co-annihilation
L, is expected to be nearly
– 13 –
Page 15
contribution of eKK
such effects do not significantly alter the qualitative results, and that νKK
different mass can still account for the observed relic density. (However, νKK
by the direct detection experiments. This will be discussed in detail in Section 7.)
For our current model, the story is different. As can be seen in Eq. 2.5, e2L,2R, the part-
ners of ν2L,2Runder SU(2)2, carry different quantum numbers under the Z2× Z′
and thus do not have (10) nor (01) modes. There are states that are nearly degenerate
with ν2L,2R, such as the e′states. However, these states interact with ν2L,2Ronly through
Yukawa interactions, which can be ignored. Therefore, we expect effects of self- and co-
annihilation with ν2L,2Rnearby states to be even smaller than the MUED case, and ignore
all such effects in our analysis.
Lhas been studied in the literature [1] [5], where it is shown that including
L
with a slightly
is ruled out
L
2orbifold,
4.3 Important Differences in Comparison to Standard Analysis
We note here that our our analysis of the annihilation channels for νKK
of [1] and [20] for νKK
L
in two important ways.
process is mediated by Z-boson of the SM, whose mass can be ignored, whereas we have
s-channel processes mediated by Z′, whose mass is significantly larger than the mass of
our dark matter candidate in the region of interest. Second, to a good approximation we
can discard t,u-channel processes mediated by charged gauge bosons W±
has contributions both from R−1and vR. To see this, let us make the approximation
m2
diagram with a s-channel diagram σstwith that coming from the square of an s-channel
diagram σss,
2L,2Rdiffer from those
First, in their analysis, the s-channel
2, because m2
W±
2
W±= m2
Z′ + R−2, then we compare the cross section involving the product of a t or u
σss
σst
≈m2
4m2
ν+ m2
ν− m2
W±
Z′
=2(R−1)2+ m2
4(R−1)2− m2
Z′
Z′. (4.11)
Then σss≫ σstwould require that
2(R−1)2+ m2
4(R−1)2− m2
Z′
Z′
≫ 1→m2
Z′ ≫ 2(R−1)2, (4.12)
which is satisfied in the region of interest in the parameter space. Similarly, the cross
section involving two t− or u-channel diagrams, σtt,σuuor σtuis small compared to σss.
5. Dark Matter Candidate II: B
(11)
(-)
or B(11)
µ
The lightest (11) mode is either B(11)
in Reference [24] found that B(11)
orbifold in that particular case, and may not apply to Z2× Z′
Instead of performing the radiative corrections to determine which of the two particles is
lighter, we will do a phenomenological study exploring both of these cases. To simplify the
notation, we will often discard the (11) superscript in the fields.
(-) or B(11)
is heavier than B(11)
µ , depending on radiative corrections. Although
µ
(-), this result is specific the choice of
2orbifold that we have here.
– 14 –
Page 16
5.1 (Co)-Annihilation Channels of B(11)
µ
When vw≪ R−1, B(11)
effects. The annihilation channels and cross sections of B(11)
in [1] and [5], and in this subsection we summarize their results.
µ
is same as A(11)
Y,µ, the KK mode of the photon up to small mixing
have been studied in detail
µ
f
fKK
Bµ
Bµ
KK
KK
f
_
Bµ
KK
Bµ
KK
f
f
_
fKK
Figure 4: Annihilation channels of a pair of B(11)
µ
into SM fermion-antifermion pair.
B(11)
µ
can annihilate itself into a fermion-antifermion pair through t- and u-channel
processes mediated by the (11) mode of the fermion (Fig. 4). It is important to note that
the left- and right-handed fermions of SM have separate massive KK modes with vector-
like couplings to the zero-mode fermion and B(11)
written as
µ . The annihilation cross section can be
σ(B(11)
µ B(11)
µ
→ ff) = g4
1(Y2
L+ Y4
R)Nc
10(2M2
f+ s)ArcTanh(β) − 7sβ
72πs2β2
, (5.1)
where Mf =
the final state (3 for quarks and 1 for leptons), and YL,Ris the hypercharge of the left- and
right-handed fermion. Summing over all SM fermions gives
√2R−1is the mass of the KK-fermion exchanged, Ncis the color factor in
?
There are also annihilation channels to Higgs through t- and u-channel processes me-
diated by a (11) mode of the Higgs boson as well as a quartic interaction. The annihilation
cross section is given by
f∈SM
Nc(Y4
L+ Y4
R) = 3(Y4
eL+ Y4
eR+ Y4
νL+ 3(Y4
uL+ Y4
uR+ Y4
dL+ Y4
dR)) =95
18. (5.2)
σ(B(11)
µ B(11)
µ
→ h∗h) =
g4
6πβs,
1Y4
φ
(5.3)
where Yφ= 1/2 is the hypercharge of the Higgs doublet. By summing over two complex
Higgs doublets, we have taken into account the annihilation into the longitudinal zero
modes of the W and Z gauge bosons.
In the MUED, the KK mode with mass closest to B(1)
handed electron e(1)
R
when radiative corrections are included [22]. However, compared to
the case without co-annihilation, the qualitative results of the relic density due to B(1)
remains the same when one includes the co-annihilation e(1)
out by [1], this is because there are only two channel of such co-annihilation, leading to a
small co-annihilation cross section, and thus small change in the relic density for a fixed
R−1.
µ is the KK mode of the right-
µ
RB(1)
µ → eRAµ[1]. As pointed
– 15 –
Page 17
In our case, we expect B(11)
B(11)
µ
in addition to e(11)
B(11)
(-)
→ XX is significant as B(11)
and u-channel processes mediated by a KK fermion. The co-annihilation cross section to
fermion-antifermion pair is
(-)
(which has no MUED analog) to be close in mass to
1R(the analog of e(1)
R
in MUED). Furthermore, the co-annihilation
µ B(11)
(-)
can annihilate to all SM fermions through t-
µ B(11)
?
f∈SM
σ(B(11)
µ B(11)
(-) → ff) = g4
1
95
18
ArcTan(β)
12πsβ2
. (5.4)
Although the co-annihilation effect was overlooked in [4], the most important conclu-
sions of our previous work remain the same, as we will show later.
5.2 (Co)-Annihilation Channels of B(11)
(-)
The coupling of B(11)
component notation)
(-)
to matter fields in the full 6D-Lagrangian is given by (in four-
L6D=g1Y
=g1Y
√2BY,-
?Ψ−(iγ5− 1)Ψ−+ Ψ+(iγ5+ 1)Ψ+
√2BY,−
?
?(−i − 1)Ψ-LΨ-R+ (i − 1)Ψ-RΨ-L+ (−i + 1)Ψ+LΨ+R+ (i + 1)Ψ+RΨ+L
?.
(5.5)
In terms of KK-modes, B(11)
and its annihilation channels to fermions will proceed through t− and u−processes medi-
ated by a KK-fermion. The annihilation cross section is
(-) will couple to fermion fields in (00-fermion)(11-fermion) pairs,
?
f∈SM
σ(B(11)
(-)B(11)
(-) → ff) = g4
1
95
18
2(2M2
f+ s)ArcTan(β) − 3sβ
2πsβ2
. (5.6)
In the non-relativistic limit, this cross-section is p-wave suppressed. There is also annihi-
lation to a pair of Higgs bosons through the quartic coupling
L4D= g2
1Y2
HB(11)
(-)B(11)
(-)H†(00)H(00), (5.7)
and this gives a cross section of
?σvrel? =g4
1Y4
2πs.
H
(5.8)
Because the annihilation of B(11)
density resulting B(11)
we must rely on co-annihilation channels such as B(11)
density, as we will see in the next section.
(-)
to fermion modes is p-wave suppressed, the relic
self-annihilation channels would in general be too high. Therefore,
µ B(11)
(-)
(-) → XX to obtain observed relic
– 16 –
Page 18
6. Numerical Results of Relic Density
The main free parameters of our theory are R−1and MZ′, and the mass-splitting ∆ ≡
(MB(11)
µ
−MB(11)
(-) (-)
can satisfy the constraint
)/MB(11)
. In addition, we have gRor gBLas a free parameter as long as we
g2
1=
g2
BLg2
BL+ g2
R
g2
R
.
6.1 B(11)
µ
-ν(01)
2L,2RDark Matter
600 8001000
MZ? ?GeV?
1200 1400 1600
300
400
500
600
700
R?1?GeV?
s?res
20%20%
30%
40%
50%
60%
70%
gR?0.7 gL
??0.05
0 0.020.04 0.06 0.080.1
?
0.02
0.04
0.06
0.08
0.10
0.12
?BYh2
0.02
0.04
0.06
0.08
0.10
0.12
R?1?400 GeV
R?1?500 GeV
R?1?600 GeV
Figure 5: The plot on the left shows the contour in the R−1− MZ′ plane that corresponds to
ΩνL,Rh2+ΩBYh2being the observed dark matter. The intersection of the red lines with the contour
indicate the the fraction of KK neutrinos in the dark matter. The plot on the right shows ΩBYh2
as a function of ∆ for various values of R−1.
For ∆ = 0.05, we present the allowed region in R−1− MZ′ space that gives the
observed dark matter relic density in the first plot of Fig. 5. Since both B(11)
can independently give the correct relic density without co-annihilation from other modes
with almost degenerate mass, varying ∆ does not affect the qualitative results of what
we present below. The independence of ΩB(11)
µ
of Fig. 5. For small values of MZ′, the annihilation of ν(01)
dark matter is B(11)
µ
having a mass of roughly
2Mν(01) = 2R−1= MZ′, the annihilation of ν(01)
contribution to dark matter relic density is minimal. Away from the line of s-channel
resonance, the contribution of ν(01)
to decrease the relic density due to B(11)
µ , keeping the total relic density within the allowed
range.
µ
and ν(01)
2L,2R
h2on ∆ as can be seen in the second plot
2L,2Ris efficient and most of the
√2R−1∼ 700 GeV. In fact, along the line
2L,2Rhas an s-channel resonance, and its
2L,2Rto the relic density increases, and R−1decreases so as
– 17 –
Page 19
The current experimental bound on the massive, neutral, vector boson is MZ′ > 800
GeV. If we further impose the bound that R−1> 400 GeV, the allowed region in the
parameter space is very limited.
6.2 B(11)
(-)-ν(01)
2L,2RDark Matter
8001000
MZ? ?GeV?
12001400
350
400
450
500
550
600
R?1?GeV?
0.5%
1%
2%
5%
0 0.020.04 0.060.080.1
?
0.05
0.1
0.15
0.2
?BYh2
300 GeV400 GeV
500 GeV
Figure 6: The plot on the left shows the allowed region in the parameter space that gives rise to
the observed dark matter relic density for gR= 0.7gLand different values of ∆. On the right, we
plot the relic density due to B(11)
(-)
as a function of the mass-splitting ∆ for various values of R−1.
As stated earlier, B(11)
observed relic abundance. However, there is significant co-annihilation process B(11)
ff. In Fig. 6, we show contours that give the observed relic density for various values of
∆. We see that when B(11)
(-)
and B(11)
µ
are nearly degenerate to less than 5%, then the
distribution of dark matter among ν(01)
the mass splitting between B(11)
(-)
and B(11)
µ
is larger than 5%, however, the model is ruled
out as we can not obtain the observed relic density without violating R > 400 GeV bound.
When B(11)
(-)
and B(11)
µ
are nearly degenerate, ν(KK)
observed relic density when MZ′ is about 1.2 TeV and R−1∼ 400 GeV.
(-) by itself can not annihilate efficiently enough to account for the
(-)B(11)
µ
→
2L,2Rand B(11)
(-)
is similar to the previous case. When
2L,2Rcan still contribute significantly to the
7. Direct Detection of Two-Component Dark Matter
As we have a two-component dark matter, the total dark matter-nucleon cross section is
given by
σn= κνRσνR+ κBσB, (7.1)
– 18 –
Page 20
where σνR(B)is the spin-independent KK neutrino (hypercharge vector or pseudoscalar)-
nucleon scattering cross section, and
κνR≡
ΩνRh2
ΩνRh2+ ΩBh2, (7.2)
is the fractional contribution of the KK neutrino relic density to the total relic density of
the dark matter. κB is similarly defined. As pointed out in Ref. [1], σB is of the order
σB∼ 10−10pb, and we will find that σνR≫ σB. Therefore, it is a good approximation to
take σnas
σn≈ κνRσνR. (7.3)
The elastic cross section between ν2L,2Rand a nucleon inside a nucleus N(A,Z) is given
by
σ0=b2
Nm2
πA2,
n
(7.4)
where bN= Zbp+ (A − Z)bnand bp,nis the effective four-fermion coupling between ν2L,2R
and nucleon. They are given by bp= 2bu+ bdand bn= bu+ 2bd. In our case, although
ν2L,2Ronly couples to Z′
We can including the effects of mixing up to order of O(M2
as perturbations and include one vertex mixing. In this case, we have
µat leading order, we have to taken into account the Z−Z′mixing.
Z/M2
Z′) by treating the mixing
bq=
1
2M2
Z′g(ν2ν2Z′)
?
(g(qLqLZ′)+ g(qRqRZ′)) − (g(qLqLZ)+ g(qRqRZ))δM2
M2
Z
+ O
?M2
Z
M2
Z′
??
,
(7.5)
where
δM2≡
g2
R
?(g2
L+ g2
Y)(g2
R+ g2
BL)M2
Z
(7.6)
is the mixing between Z and Z′(see Eq. 3.7).
The prospects of direct detection of B(11)
lated detection rates are beyond the reach of current experiments. As for B(11)
there is no s-wave for elastic scattering B(11)
by a factor of v2
will dominate that of both B(11)
µ
and B(11)
lightest KK-mode of sterile neutrino as dark matter candidate could be detected directly
in the current and the next rounds of direct-detection experiments if its relic density is
significant compared to the observed total relic density, in contrast to other dark matter
candidates in the literature, such as the neutralino of MSSM or the lightest KK-mode of
the photon.
In Fig. 7, we show the direct detection cross section as a function of MZ′ for both
cases where B(11)
µ
and B(11)
µ
has been studied extensively, and the calcu-
(-), because
(-)N → B(11)
(-)N, the cross-section is suppressed
rel∼ 10−5. Therefore, we expect that the direct detection rates of ν(10)
(-). This is one of the main points of our work: the
2L,2R
(-) is the lighter of the two. The horizontal lines correspond to the
– 19 –
Page 21
8009001000 1100
MZ'?GeV?
12001300 1400
10?44
10?43
10?42
ΚΝRΣΝR?cm2?
10?44
10?43
10?42
gR?0.7 gL
??0.05
800 9001000 1100
MZ'?GeV?
12001300 1400
10?44
10?43
10?42
ΚΝΣΝR?n
S
?cm2?
Direct Detection of Ν2
?01?– BY,???
?11?Dark Matter
10?44
10?43
10?42
Figure 7: The plot on the left (right) shows the dark matter-nucleon cross-section as a function of
MZ′ for the case where B(11)
µ
(B(11)
R−1and −0.05 < ∆ < 0 that gives the observed relic density. The horizontal lines correspond to
the upper bounds on σnfrom CDMS II for dark matter candidates with masses 300 and 500 GeV
(-)) is lightest (11) mode. The plots scan over different values of
upper bounds on σnfrom CDMS II for dark matter candidates with masses 300 and 500
GeV, which are about 4×10−43cm2and 7×10−43cm2, respectively.
A particularly interesting region in the parameter space is R ∼ 400 GeV and MZ′ ∼
1200 GeV. Here, the KK-sterile neutrino contributes to roughly half of the relic density.
This admixture of dark matter is just below the current experimental bound from direct
detection, as shown in Fig. 5, when we use the CDMS II bound that dark matter-nucleon
spin-independent cross-section must not exceed 4 × 10−43cm2for a 400 GeV dark matter.
8. Some Phenomenological Implications
In this section, we give a qualitative comparision of the phenomenological implications
of this model with those of the conventional left-right symmetric models [26].
long been recognized that two important characteristic predictions of the left-right models
are the presence of TeV scale WRand Z′gauge bosons which can be detectable in high
energy colliders [28]. In addition to the collider signatures of generic UED models [27],
two predictions characteristic of the model discussed here which differ from those of the
earlier models are : (i) The mass of Z′has an upper bound of about 1.5 TeV and a more
spectacular one where (ii) the WRin this model, being a KK excitation, does not couple
to a pair of the known standard model fermions which are zero modes. This property of
the WRhas a major phenomenological impact and will require a completely new analysis
of constraints on this e.g. the well known KL−KSmass difference constraint on MWR[29]
does not apply here since the mixed WL− WR exchange box graph responsible for the
new contribution to KL− KSmass difference does not exist. The box graph where both
exchange particles are WR’s exists but its contribution to the ∆S = 2 Hamiltonian is
It has
– 20 –
Page 22
suppressed compared to the left-handed one by a factor
weak bound on MWR.
Also, the bounds from muon and beta decay [30] are nonexistent for the same reason
because there is no tree level WR contribution to these processes. Furthermore, in this
model, there is no WL− WRmixing unlike the conventional left-right models.
Because of this property, the decay modes and production mechanism of the WRare
also very different from the case of the conventional left-right model, while the decay modes
and production mechanism of the Z′remain the same. We do not discuss the Z′case which
has been very widely discussed in literature.
?MWL
MWR
?4
and gives only a very
The WRwill have a mass given by As far as the WRis concerned, it is given by the
formula M2
WR∼
to u′
decay modes¯e′2Lν′
and the R−1. The leptonic decay mode will look very similar to the supersymmetric case
where pair-produced sleptons will decay to a lepton and the neutralino. The hadronic
channel will however look different from the squark case. The further details of the collider
signature of our model is currently under investigation and will be presented separately.
cos2θW
cos2θWM2
2R¯d′2R, u2L¯d2L, ¯ e2L,Rν2L,R, and¯e′2L,Rν′
2Land u2R¯d2Rwill dominate depending on the precise value of WRmass
Z′. Furthermore, it can only be pair produced and will decay
2L,R. For sub-TeV WR, only the
2L¯d′2L, u2R¯d2R, u′
9. Conclusions
In summary, we have studied the profile of cold dark matter candidates in a Universal Extra
Dimension model with a low-scale extra WRand Z′. There are two possible candidates:
νKK
R
and either the B(11)
µ
depending on which one receives less radiative corrections.
We have done detailed calculation of the relic density of these particles as a function
of the parameters of the model which are gR, R−1and MZ′. We find upper limits on
these parameters where the above KK modes can be cold dark matter of the universe.
In discussing the relic abundance, we have considered the co-annihilation effect of nearby
states. We also calculate the direct detection cross-section in current underground detectors
for the entire allowed parameter range in the model and we find that, for the case where
KK neutrino contributes significantly to the total relic density, the lowest possible value of
the cross-section predicted by our model is accessible to the current and/or planned direct
search experiments. Therefore, the most interesting region of our model can not only be
tested in the colliders but also these dark matter experiments. Combined with LHC search
for the Z′of left-right model, dark matter experiments could rule out this model.
(-) or B(11)
10. Acknowledgement
The work of K.H and R.N.M is supported by the National Science Foundation grant no.
Phy-0354401. S.N is supported by the DOE grant no. Phy. DE-FG02-97ER41029.
– 21 –
Page 23
A. Fields on T2/Z2× Z′
2
For convenience of type-setting, we define the functions
c(i,j) ≡ cosix5+ jx6
c′(i,j) ≡ cosix′5+ jx′6
R
,s(i,j) ≡ sinix5+ jx6
s′(i,j) ≡ sinix′5+ jx′6
R
,
R
,
R
,(A.1)
And for reference we will make use of this integral
?2πR
0
dx5
?2πR
0
dx6c(i,j)c(m,n) =
?2πR
0
dx5
?2πR
0
dx6s(i,j)s(m,n) = 2π2R2δimδjn(A.2)
for positive integers i,j,m and n, extensively. We have the compactified space 2πR×2πR
by imposing the periodic boundary conditions on the fields
φ(xµ,x4,x5) = φ(xµ,x4+ 2πR,x5) = φ(xµ,x4,x5+ 2πR). (A.3)
The periodic boundary conditions mean that we can write the fields in the form of
φ(xµ,x4,x5) =
?
n,m
?
c(n,m)ϕ(nm)(xµ) + s(n,m)˜ ϕ(nm)(xµ)
?
.(A.4)
On top of the periodic boundary conditions, we impose two orbifolding symmetries on our
theory
Z2: y → −y,Z′
2: y′→ −y′.(A.5)
with y′= y − (πR/2,πR/2). Demanding that the Lagrangian be invariant under the orb-
ifolding symmetries, we can assign parities to the fields under the discrete transformations
and remove roughly half of the KK modes in Eq. A.4. The choices of signs are motivated
by the desired phenomenology. In our case, we have two orbifolding symmetries, so we can
assign two signs to a given field. There are four possibilities: (+,±) and (−,±), and we
examine each case separately.
For (+,±) case, we have the general expansion
φ(+,±)(xµ,x4,x5) =
1
2πR
∞
?
∞
?
n,m=0
?
??
c(n,m)ϕ(nm)?
c′(n,m)cos(m + n)π
=
1
2πR
n,m=0
2
− s′(n,m)sin(m + n)π
2
?
ϕ(nm)
?
.
(A.6)
So we see that for (+,+) fields, we need n + m and n − m to be even.
– 22 –
Page 24
For (+,−) fields, we need n + m and n − m to be odd. For (−,±) case, we have the
general expansion
φ(−,±)(xµ,x4,x5) =
1
2πR
∞
?
∞
?
n+m≥1
?
??
s(n,m)ϕ(nm)?
s′(n,m)cos(m + n)π
=
1
2πR
n+m≥1
2
+ c′(n,m)sin(m + n)π
2
?
ϕ(nm)
?
.
(A.7)
So we see that for (−,+) fields, we need n + m and n − m to be odd. For (−,−) fields,
we need n + m and n − m to be even. Of course, for the (−,±) cases, we can not have
(m,n) = (0,0) mode.
B. Normalization of Fields and Couplings
B.1 Matter Fields
The dark matter candidates of the theory are the first KK modes of the neutrinos charged
under SU(2)2. They have the Z2×Z′
we see each of ν2L,2Rhas two independent modes:− →
two independent Dirac particles in the sense that there is no mixing at tree level in the
effective 4D theory.
We expand the kinetic energy term
2charges: ν2L(−,+),ν2R(+,−). If we let− →
n = (1,0) and− →
n = (n,m),
n = (0,1). These are
L6D-KE= iΨΓM∂MΨ
= i
?
Ψ−Ψ+
??
0γµ∂µ+ iγ5∂5+ ∂6
0γµ∂µ+ iγ5∂5− ∂6
??
Ψ+
Ψ−
?
= iΨ−(γµ∂µ+ iγ5∂5+ ∂6)Ψ−+ iΨ+(γµ∂µ+ iγ5∂5− ∂6)Ψ+.(B.1)
Note that Ψ is an eight-component object, while Ψ±are four-component, six-dimensional
chiral spinors. We denote six-dimensional chirality by ± and four-dimensional chirality by
L,R. Each six-dimensional chiral spinor is vector-like in the four-dimensional sense, and
each is a Dirac spinor. Since our dark matter candidate is of (-1) 6D-chirality, we only deal
with the first part of the kinetic energy term, and drop the subscript.
Since we are after the coefficients, we expand in detail the first KK mode of the dark
matter candidate.
L6D-KE⊃ iΨ−(γµ∂µ+ iγ5∂5+ ∂6)Ψ−
?
= i(Ψ†
= i
Ψ†
LΨ†
R
??
i∂5+ ∂6
σµ∂µ
σµ∂µ
−i∂5+ ∂6
Rσµ∂µΨR+ Ψ†
??
ΨR
ΨL
?
Lσµ∂µΨL+ Ψ†
L(i∂5+ ∂6)ΨR+ ΨR(−i∂5+ ∂6)ΨL) (B.2)
– 23 –
Page 25
At this point, we use the KK-expansions. Noting the charge assignments ν2L(−,+),ν2R(+,−),
we expand the fields as
ν2R=
1
√2πR
1
√2πR
?c(1,0)ν(10)
?s(1,0)ν(10)
2R(xµ) + c(0,1)ν(01)
2R(xµ)?,
2L (xµ)?.
ν2L=
2L (xµ) + is(0,1)ν(01)
(B.3)
The four-dimensional effective Lagrangian is obtained by inserting the expansion of Eq. B.3
into Eq. B.2, and integrate x5and x6from 0 to 2πR. Following this procedure, we obtain
L4D-eff=
?2πR
0
dx5
?2πR
0
dx6L6D-KE= i
?
ν†(10)
2L
σµ∂µν(10)
2L + ν†(01)
2L
σµ∂µν(01)
2L + ν†(10)
2R σµ∂µν(10)
2R + ν†(01)
2R σµ∂µν(01)
2R
?
−1
R
?
(ν†(10)
2L
ν(10)
2R − ν†(10)
2R ν(10)
2L) − (ν†(01)
2L
ν(01)
2R − ν†(01)
2R ν(01)
2L)
?
(B.4)
.
¿From this calculation, we see that we have two independent Dirac neutrinos that do not
mix with each other: ν(01)and ν(10).
Following the same procedure, we can find the normalization of scalars.
Φ(+,+) =
1
2πRφ(00)+
1
√2πR
?
m,n
cosmx5+ nx6
R
φ(mn),
Φ(+,−) =
1
√2πR
?
?
?
m,n
cosmx5+ nx6
R
φ(mn),
Φ(−,+) =
1
√2πR
m,n
sinmx5+ nx6
R
φ(mn),
Φ(−,−) =
1
√2πR
m,n
cosmx5+ nx6
R
φ(mn). (B.5)
Again, the rules of m,n in the previous section for fermions apply to the scalars.
B.2 Gauge Bosons
As we are only interested in the normalization of the gauge fields, we consider a generic
gauge boson, AM, associated with an U(1) symmetry. We then have the expansion
Lgauge= −1
= −1
= −1
+1
2(∂µA5∂µA5+ ∂5Aµ∂5Aµ− ∂5Aµ∂µA5− ∂µA5∂5Aµ)
+1
2(∂µA6∂µA6+ ∂6Aµ∂6Aµ− ∂6Aµ∂µA6− ∂µA6∂6Aµ)
−1
4FMNFMN
4FµνFµν−1
4FµνFµν
2F5µF5µ−1
2F6µF6µ−1
2F56F56
2(∂5A6∂5A6+ ∂6A5∂6A5− 2∂5A6∂6A5) (B.6)
– 24 –
Page 26
Notice that we have made the changes A5,6= −A5,6,∂5,6= −∂5,6, so that A5,6should
be treated as real, scalar fields. We will work with this equation for the various gauge
particles.
As with the case of the neutral gauge bosons in our theory, we assign the Z2× Z′
parities to be Aµ(++) and A5,6(−−), and obtain the following lowest KK modes:
2
A(00)
µ , A(11)
µ , A(20)
A(11)
5
, A(20)
A(11)
6
, A(20)
µ , A(02)
, A(02)
, A(02)
µ ,
55
,
66
,(B.7)
and the KK-mode expansions for these states
Aµ=
1
2πRA(00)
1
√2πR
µ
+
1
√2πR
?
cos2x5
RA(20)
µ
+ cos2x6
RA(02)
µ
+ cosx5+ x6
R
A(11)
µ
?
,
A5,6=
?
sin2x5
RA(20)
5,6+ sin2x6
RA(02)
5,6+ sinx5+ x6
R
A(11)
5,6
?
. (B.8)
One may check that the normalization gives canonical fields for the scalars AKK
For the four-dimensional Lagrangian we insert the expansion of Eq. B.8 into Eq. B.6
and integrate over x5and x6. In additional to canonical kinetic energy terms, we obtain
the masses of these modes.
5,6.
L =1
−1
2
?4
?4
R2A(20)
µ Aµ,(20)+
4
R2A(02)
µ Aµ,(02)+
2
R2A(11)
µ Aµ,(11)
?
)2− 21
2R2(A(20)
5
)2+
4
R2(A(02)
6
)2+
1
R2(A(11)
5
)2+
1
R2(A(11)
6
R2A(11)
5
A(11)
6
?
(B.9)
We note first that we have some massless modes in A(02)
the fact that we do not have terms ∂5A5and ∂6A6in FMNFMN. As in the case of 5D UED
models, these modes are eaten by the corresponding KK modes of A(mn)
massive. Here we note that the linear combination of
is eaten by Bµ(11). Generally, at each KK level, one linear combination of AKK
(and any corresponding KK modes of Higgs particle, if there is Higgs mechanism) is eaten
by AKK
µ, while the orthogonal KK combination remains a physical mode, and is a potential
DM candidate if it is indeed the lightest KK mode.
6
and A(20)
5
. This can be traced to
µ
so they can become
) is also massless, and
1
√2(A(11)
5
+A(11)
6
5
and AKK
6
B.3 Normalization of Couplings
In six-dimensional Lagrangian, both the yukawa and gauge couplings are dimensionful.
We find the correct normalization by equating the 4D couplings to the effective 4D cou-
pling resulting from integrating over x5and x6. For example, consider a generic yukawa
interaction in the 6D theory
L6D-Yukawa= y6DΨ1ΦΨ2
– 25 –
Page 27
where y6Dhas dimension [M]−1. The coupling involving the zero-modes in the effective
theory is then
L4D-Yukawa=
?2πR
0
dx5
?2πR
0
dx6
y6D
(2πR)3ψ
(00)
1 φ(00)ψ(00)
2
=
y6D
2πRψ
(00)
1 φ(00)ψ(00)
2
.(B.10)
So effectively we have y4D= y6D(2πR)−1. Note that this is general: for the SM couplings
in the 4D effective theory, all fields are (00) and have a normalization (2πR)−1, so the
effective 4D couplings obtained after integrating over x5and x6are simply the 6D couplings
multiplied by (2πR). By the same reasoning, we also have λ6D= (2πR)2λ4Dfor the quartic
coupling in the potential.
In general, the coupling between higher modes will come with extra factors resulting
from integrating over x5and x6. However, the most important case for our purpose of
calculating annihilation diagrams involve couplings between two fermions and a boson
where exactly one of field is a (00) mode, and the two other fields are both (mn) mode with
m,n nonzero. Suppose we have a coupling in the 6D Lagrangian of the form L6D= g6DΨΦΨ,
where g6Dhas dimension of [M]−1, and we impose that g6D= g4D(2πR). In the 4D effective
theory we have L4D= g4Dψ
of the corresponding 6D fields in capital letters. The effective coupling between the KK
modes g4Dψ
(00)φ(mn)ψ(mn), where the lower-case fields are the KK-modes
(00)φ(mn)ψ(mn)in the effective 4D theory is then
L4D-eff=
?2πR
0
dx5
?2πR
0
dx6
g4D(2πR)
(2πR)(√2πR)2c2(m,n)ψ
(00)
1 φ(mn)ψ(mn)=g4Dψ
(00)
1 φ(mn)ψ(mn).
(B.11)
So we see that there is no additional factors compared to the case with all (00)-modes.
References
[1] G. Servant and T. M. P. Tait, Nucl. Phys. B 650, 391 (2003); H. C. Cheng, J. L. Feng and
K. T. Matchev, Phys. Rev. Lett. 89, 211301 (2002);
[2] K. Agashe and G. Servant, Phys. Rev. Lett. 93, 231805 (2004).
[3] R. N. Mohapatra and V. L. Teplitz, Phys. Rev. D 62, 063506 (2000); R. Foot, Int. J. Mod.
Phys. D 13, 2161 (2004); Z. Berezhiani, P. Ciarcelluti, D. Comelli and F. L. Villante, Int. J.
Mod. Phys. D 14, 107 (2005); A. Y. Ignatiev and R. R. Volkas, Phys. Rev. D 68, 023518
(2003).
[4] K. Hsieh, R. N. Mohapatra and S. Nasri, arXiv:hep-ph/0604154; Phys.Rev. D74, 066004
(2006).
[5] K. Kong and K. T. Matchev, JHEP 0601, 038 (2006).
[6] I. Antoniadis, Phys. Lett. B 246, 377 (1990).
[7] I. Antoniadis, N. Arkani-Hamed, S. Dimopoulos and G. R. Dvali, Phys. Lett. B 436, 257
(1998) [arXiv:hep-ph/9804398].
– 26 –
Page 28
[8] N. Arkani-Hamed, S. Dimopoulos and G. R. Dvali, Phys. Lett. B 429, 263 (1998)
[arXiv:hep-ph/9803315].
[9] T. Appelquist, H. C. Cheng and B. A. Dobrescu, Phys. Rev. D 64, 035002 (2001).
[10] T. Appelquist, B. A. Dobrescu, E. Ponton and H. U. Yee, Phys. Rev. Lett. 87, 181802
(2001).
[11] R. N. Mohapatra and A. Perez-Lorenzana, Phys. Rev. D 67, 075015 (2003).
[12] T. Appelquist, B. A. Dobrescu, E. Ponton and H. U. Yee, Phys. Rev. D 65, 105019 (2002).
[13] B. A. Dobrescu and E. Poppitz, Phys. Rev. Lett. 87, 031801 (2001).
[14] D. N. Spergel et al., arXiv:astro-ph/0603449.
[15] I. Gogoladze and C. Macesanu, arXiv:hep-ph/0605207.
[16] M. Kakizaki, S. Matsumoto and M. Senami, Phys. Rev. D 74, 023504 (2006).
[17] A. Gould, B. T. Draine, R. W. Romani and S. Nussinov, Phys. Lett. B 238, 337 (1990);
S. Dimopoulos, D. Eichler, R. Esmailzadeh and G. D. Starkman, Phys. Rev. D 41, 2388
(1990); A. Kudo and M. Yamaguchi, Phys. Lett. B 516, 151 (2001).
[18] D. S. Akerib et al. [CDMS Collaboration], Phys. Rev. Lett. 96, 011302 (2006)
[19] T. Appelquist and H. U. Yee, Phys. Rev. D 67, 055002 (2003).
[20] H. C. Cheng, J. L. Feng and K. T. Matchev, [1].
[21] F. Burnell and G. D. Kribs, Phys. Rev. D 73, 015001 (2006).
[22] H. C. Cheng, K. T. Matchev and M. Schmaltz, Phys. Rev. D 66, 036005 (2002).
[23] Particle Data Group, Phys. Lett. 592 B, 1 (2004).
[24] E. Ponton and L. Wang, arXiv:hep-ph/0512304.
[25] For a review, see J. Erler and P. langacker, Particle Data Group, Zeit. fur Phys. C 15, 1
(2000).
[26] J. C. Pati and A. Salam, Phys. Rev. D 10, 275 (1974); R. N. Mohapatra and J. C. Pati,
Phys. Rev. D 11, 566, 2558 (1975); G. Senjanovi´ c and R. N. Mohapatra, Phys. Rev. D 12,
1502 (1975).
[27] C. Macesanu, C. D. McMullen and S. Nandi, Phys. Lett. B 546, 253 (2002)
[arXiv:hep-ph/0207269].
[28] For discussion of WRand Z′properties of the left-right models, see B. kayser and J.
Gunion, Proceedings of the Snowmass workshop, 1984, ed. R. Donaldson et al.; M. Cvetic
and P. Langacker, Phys. Rev. D 46, 4943 (1992); [Erratum-ibid. D 48, 4484 (1993)];
T. G. Rizzo, arXiv:hep-ph/0610104.
[29] G. Beall, M. Bander and A. Soni, Phys. Rev. Lett. 48, 848 (1982).
[30] M. A. B. Beg, R. V. Budny, R. N. Mohapatra and A. Sirlin, Phys. Rev. Lett. 38, 1252
(1977) [Erratum-ibid. 39, 54 (1977)].
– 27 –
Download full-text