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 –
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