Nonabelian Dark Matter with Resonant Annihilation

17 Pages·2014·0.25 MB·English

Nonabelian Dark Matter with Resonant Annihilation
Cheng-Wei Chiang,1,2,3 Takaaki Nomura,1,4 and Jusak Tandean5
1Department of Physics and Center for Mathematics and Theoretical Physics,
National Central University, Chungli 320, Taiwan
4
1 2Institute of Physics, Academia Sinica, Taipei 115, Taiwan
0
2 3Physics Division, National Center for Theoretical Sciences, Hsinchu 300, Taiwan

b
e 4Department of Physics, National Cheng-Kung University, Tainan 701, Taiwan
F
5Department of Physics and Center for Theoretical Sciences,
2
2 National Taiwan University, Taipei 106, Taiwan

]
h Abstract
p
We construct a model based on an extra gauge symmetry, SU(2) U(1) , which can provide gauge
- X B−L
p ×
bosons to serve as weakly-interacting massive particle dark matter. The stability of the dark matter
e
h is naturally guaranteed by a discrete Z symmetry that is a subgroup of SU(2) . The dark matter
2 X
[
interacts with standard model fermions by exchanging gauge bosons which are linear combinations of

2 SU(2) U(1) gaugebosons. Withtheappropriatechoiceofrepresentationforthenewscalarmultiplet
X B−L
v ×
whosevacuumexpectation valuespontaneouslybreakstheSU(2) symmetry,therelation betweenthenew
2 X
8 gauge boson masses can naturally lead to resonant pair annihilation of the dark matter. After exploring
8
the parameter space of the new gauge couplings subject to constraints from collider data and the observed
0
. relic density, we use the results to evaluate the cross section of the dark matter scattering off nucleons and
6
0 compare it with data from the latest direct detection experiments. We find allowed parameter regions that
3
can be probed by future direct searches for dark matter and LHC searches for new particles.
1
:
v
i
X
r
a
1
I. INTRODUCTION
The standard model (SM) of particle physics has been very successful in describing an enormous
amountofexperimentaldataatenergiesupto (100)GeV.Thereare,however, questionsremaining
O
that require physics beyond the minimal SM to address. Among the outstanding issues are the
explanations for the astronomical evidence of dark matter (DM) and for the numerous experimental
indications of neutrino mass [1]. It is then of great interest to explore a new physics scenario in
which the DM and neutrino sectors are intimately connected.
Previously, wehaveconsideredasimplemodelwhichprovidesnotonlyDMofthepopularweakly-
interacting massive particle (WIMP) type, but also a means to endow neutrinos with mass [2]. The
DM candidate belongs to a complex scalar singlet stabilized by a Z symmetry that is not imposed
2
in an ad hoc way, but instead emerges from an extra Abelian gauge group related to baryon number
minus lepton number, U(1) , that is spontaneously broken by the nonzero vacuum expectation
B−L
value (VEV) of a new scalar field, in the Krauss-Wilczek manner [3]. Light neutrino masses are
produced via the well-known seesaw mechanism [4], which is triggered with the involvement of the
same new scalar field after the addition of right-handed neutrinos. The DM relic density receives
contributions mainly from diagrams mediated by the Higgs boson and also those mediated by the
U(1) gauge boson, Z′. It turns out that constraints from collider data and the observed relic
B−L
density together imply that the Z′ mass has to be in the resonance region of the Z′-mediated DM
annihilation, namely about twice the DM mass. Furthermore, results from DM direct detection
experiments and Higgs data from the LHC favor the dominance of the Z′-exchange contributions
to the relic density. All this motivates us to look for a different possible scenario in which the
resonance condition can be fulfilled naturally.
In this paper, we demonstrate that such a possibility can be realized in a model where the
role of WIMP DM is played by massive gauge bosons associated with a nonabelian symmetry.
Although most of the WIMP DM candidates proposed in the literature are either fermionic or
spinless, those with spin one have also been considered before [5]. Here we construct a model based
on the gauge group G SU(2) U(1) , where G refers to the SM group and the extra
SM X B−L SM
× ×
gauge symmetries offer gauge bosons which can act as WIMP candidates. The stability of the
DM is naturally maintained by a discrete Z symmetry which is a subgroup of the new nonabelian
2
gauge group, SU(2) . This Z symmetry appears after the spontaneous breaking of SU(2) by the
X 2 X
nonzero VEV of a new scalar multiplet, following the Krauss-Wilczek mechanism [3]. Then the
Z -odd gauge bosons associated with SU(2) can serve as DM if they are lighter than other Z -odd
2 X 2
particles in themodel. Since SM fermions are charged under U(1) , the DMcan interact with SM
B−L
fermions at tree level by exchanging gauge bosons which are obtained from the linear combinations
of SU(2) and U(1) gauge fields. Thus, the new gauge interactions are responsible for both
X B−L
the relic abundance and the DM interactions with nucleons. Another interesting feature of the
model is that, the DM being made up of SU(2) gauge bosons, its mass is related to the masses of
X
the mediating gauge bosons, implying that resonant pair annihilation can be naturally achieved by
choosingsuitablerepresentationsofthescalarfieldsinvolvedinthebreakingoftheSU(2) U(1)
X B−L
×
gauge symmetry and ensuring that their VEVs are sufficiently well separated. What’s more, the
presence of the U(1) gauge symmetry requires the introduction of right-handed neutrinos for
B−L
gauge-anomaly cancellation, which in turn participate in the type-I seesaw mechanism to generate
2
light neutrino masses [4], with the right-handed neutrino masses being connected to the U(1)
B−L
breaking scale. This model turns out to have sufficient parameter space that is consistent with
current collider, relic density, and DM direct search data. Therefore, it can be probed further by
ongoingorfutureDMdirect detectionexperiments, andsomeofthenewparticlesmaybeobservable
at the LHC with sufficient luminosities.
This paper is organized as follows. The next section contains the details of our model which
possesses WIMP DM composed of the gauge bosons of an extra nonabelian gauge symmetry. We
explain how the choices of the new particles and their quantum numbers can naturally translate into
resonant annihilation of the DM. In Section III, we examine constraints on the new gauge couplings
from collider data. In Section IV, we deal with the relic density of our DM candidates and extract
the parameter values allowed by its observed value. In Section V, we use the results to predict
the DM-nucleon scattering cross-section and compare it with current data from direct detection
experiments. In Section VI, we comment on the collider phenomenology of the new particles in our
model. We conclude in Section VII with the summary of our study and some more discussion.
II. A MODEL OF DARK MASSIVE GAUGE BOSON
Compared to the SM, the new model contains the additional gauge group SU(2) U(1) ,
X B−L
×
where X refers to themassive gaugebosonthatserves asthe DM, whereas B andLstandfor baryon
and lepton numbers, respectively. We denote the gauge fields associated with SU(2) and U(1)
X B−L
by Cµ and Eµ, respectively, k = 1,2,3, and their coupling constants g and g . The model also
k X B−L
has new complex scalar fields S and Φ as well as three extra fermions ν , all of which are singlets
5 kR
under the SM gauge group, but carry nonzero U(1) charges. Under SU(2) transformations,
B−L X
S is a singlet, while Φ is a five-plet represented by the column matrix Φ = φ ,φ ,φ ,φ ,φ T,
5 5 2 1 0 −1 −2
where φ corresponds to the eigenvalue T = a of the third generator of SU(2) . In Table I we
a 3X (cid:0) X (cid:1)
collect the SU(2) U(1) quantum number assignments for the fermions, scalars, and new gauge
X B−L
×
bosons in the model, with H being the usual scalar doublet.
The renormalizable Lagrangian for S and Φ , with H included in the potential , is
5
V
= µS † S + µΦ † Φ , (1)
L D Dµ D 5 Dµ 5 − V
where (cid:0) (cid:1) (cid:0) (cid:1)
µS = ∂µS +2ig EµS , µΦ = ∂µΦ +ig Cµ (5)Φ +ig Eµ (5) Φ , (2)
D B−L D 5 5 X k Tk 5 B−L QB−L 5
= µ2Φ†Φ + λ S 2 µ2 S 2 + λ H†H µ2 H†H + (other quartic terms) . (3)
V − Φ 5 5 S| | − S | | H − H
(cid:0) (cid:1) (cid:0) (cid:1)
f ν H S φ φ φ φ φ X X† C E
SM R 2 1 0 −1 −2 3
SU(2) U(1) 1[B L] 1[ 1] 1[0] 1[2] 5[2] 5[2] 5[2] 5[2] 5[2] 3[0] 3[0] 3[0] 1[0]
X B−L
− −
T 0 0 0 0 2 1 0 1 2 1 1 0 0
(cid:2)3X (cid:3)
− − −
ZX + + + + + + + + +
2 − − − −
TABLE I: The charge assignments under SU(2) U(1) and ZX parity of the fermions, scalars and
X× B−L 2
newgauge bosonsinthemodel, with f referringto SMfermions, X = (C iC )/√2, andT denoting
SM 1− 2 3X
the eigenvalue of the third generator of SU(2) .
X
3
In µΦ above, summation over k = 1,2,3 is implicit, and (5) and (5) are matrices for the
D 5 Tk QB−L
generators of SU(2) and U(1) , respectively, acting on Φ , where
X B−L 5
0 2 0 0 0 0 2 0 0 0
−
2 0 √6 0 0 2 0 √6 0 0
1   i  − 
(5) = 0 √6 0 √6 0 , (5) = 0 √6 0 √6 0 ,
T1 2   T2 2  − 
0 0 √6 0 2 0 0 √6 0 2
   − 
0 0 0 2 0 0 0 0 2 0 
   
(5)  (5)  
= diag(2,1,0, 1, 2) , = diag(2,2,2,2,2) . (4)
T3 − − QB−L
In this paper, we consider the scenario in which the SU(2) U(1) gauge symmetry is spon-
X B−L
×
taneously broken according to
SU(2) U(1) hSi SU(2) ZB−L hΦ5i ZX ZB−L , (5)
X × B−L −−→ X × 2 −−→ 2 × 2
where S = v /√2 and Φ = v ,0,0,0,0 T/√2 are the VEVs of S and Φ , with v v > 0.
h i S h 5i Φ 5 S ≫ Φ
Since Φ = 0 occurs via its T = 2 component, φ = 0, the ZX symmetry emerges naturally
h 5i 6 3X(cid:0) (cid:1) h 2i 6 2
as a subgroup of SU(2) and the particles with even (odd) T values will be ZX even (odd), as
X 3X 2
Table I shows. On the other hand, ZB−L is the remnant of U(1) after S = 0, as discussed
2 B−L h i 6
in Ref.[2], but does not play a role in the stabilization of X. Thus, in this scenario the remaining
ZX guarantees the stability of the lightest ZX-odd particle(s), which can therefore act as DM. Here
2 2
we choose the gauge boson X = (C iC )/√2 and its conjugate X† to be the DM, hence tacitly
1 2
−
taking the ZX-odd scalar bosons to be more massive than X. It is worth mentioning that we would
2
arriveat thesame results below if Φ = 0 throughits T = 2 component instead. As forH, its
5 3X
h i 6 −
VEV is also nonvanishing and breaks the electroweak symmetry just as in the SM. We assume that
the other parameters in the potential are such that the vacuum has the above desired properties,
V
leaving a detailed analysis of for future work.
V
After SU(2) U(1) spontaneously breaks into ZX ZB−L, the new gauge bosons acquire
X× B−L 2 × 2
in the mass terms
L
= Φ† g Cµ (5) +g Eµ (5) g C (5) +g E (5) Φ + 4g2 E2 S 2
Lm 5 X k Tk B−L QB−L X k′µTk′ B−L µQB−L 5 B−L h i
h ih i
(cid:10) (cid:11) 1 4g2 v2 4g g v2 C(cid:10) (cid:11)
= g2 v2X†Xµ + Cµ Eµ X Φ X B−L Φ 3µ . (6)
X Φ µ 2 3 4g g v2 4g2 v2 +v2 E
X B−L Φ B−L Φ S ! µ!
(cid:0) (cid:1)
From the last line, upon diagonalizing the 2 2 matrix in the(cid:0)second t(cid:1)erm, we obtain the eigenvalues
×
m2 = g2 v2 , (7)
X X Φ
m2 = 2g2 v2 +2g2 v2 +v2 2 g2 v2 g2 v2 +v2 2 +4g2 g2 v4 , (8)
ZL,ZH X Φ B−L Φ S ∓ X Φ − B−L Φ S X B−L Φ
q
assuming that m < m for th(cid:0)e mass ei(cid:1)genstat(cid:2)es Z and Z (cid:0)which a(cid:1)re(cid:3) given by
ZL ZH L H
Z cosθ sinθ C
L = 3 , (9)
Z sinθ cosθ E
(cid:18) H(cid:19) (cid:18)− (cid:19)(cid:18) (cid:19)
2g g R v2
tan(2θ) = X B−L v , R = Φ . (10)
g2 R g2 (1+R ) v v2
X v − B−L v S
4
In this study, we focus on the case in which v2 v2 and g g , implying that
S ≫ Φ X ∼ B−L
g
θ X R , (11)
| | ≃ g v
B−L
m2 4m2 (1 R ) , (12)
ZL ≃ X − v
g2
m2 4m2 B−L (1+R ) . (13)
ZH ≃ X g2 R v
X v
Accordingly, with R 1, we obtain the mass relation
v
≪
m 2m , (14)
ZL ≃ X
which naturally leads to resonant annihilation of the DM pair via the Z -mediated contribution.
L
It is worth noting that the five-plet Φ is the minimal choice of SU(2) representation that can
5 X
result in the resonant relation in Eq.(14). In general, for an SU(2) isospin value T and its third
X X
component T , one would get m2 /m2 T (T + 1) T2 / 2T2 assuming small mixing
3X X ZL ≃ X X − 3X 3X
angle θ, in analogy to the ρ parameter in the electroweak sector [1].
(cid:2) (cid:3) (cid:0) (cid:1)
The neutrino mass-generating sector is the same as that given in Ref.[2], the relevant Lagrangian
having the form
= iλ ν¯ HTτ L 1λ′ ν¯ (ν )cS† + H.c., (15)
Lmν kl kR 2 lL − 2 kl kR lR
(′)
wheresummationover k,l = 1,2,3 isimplicit, λ arefreeparameters,τ isthesecondPaulimatrix,
kl 2
L represents a lepton doublet, and the superscript c indicates charge conjugation. The Dirac and
lL
Majorana mass matrices from these terms are = λv /√2 and = λ′v /√2, respectively,
MD H MνR S
where v is the VEV of H. Hence v sets the mass scale of the right-handed neutrinos, ν . In our
H S kR
examples later on, we will see what values of v are compatible with the observed relic density and
S
collider data.
Since X is our chosen candidate for DM and interacts with SM fermions by exchanging the Z
L,H
bosons at tree level, in the following two sections we evaluate the new gauge couplings subject to
collider and relic density data. Subsequently, we use the allowed values of the couplings to predict
the cross section of the DM-nucleon scattering and compare it with the existing results of DM direct
detection experiments.
III. CONSTRAINTS FROM COLLIDER EXPERIMENTS
The gauge bosons Z and Z interact with SM fermions at tree level with coupling constants
L H
g sinθ and g cosθ, respectively, according to the Feynman rules listed in Appendix A. It
B−L B−L
follows that measurements on processes mediated by Z and Z can offer constraints on these
L H
couplings. Significant restrictions may be available from the data on e+e− and hadron collisions
into fermion pairs, which we treat in this section.
We first look at the constraints from e+e− ff¯ scattering. In this work we assume that mixing
→
between the Z boson and Z is negligible, but we will comment on the impact of kinetic mixing
L,H
between them later on and discuss it further in Appendix B. In the absence of the mixing, the new
gauge couplings have no effects on the Z-pole observables at leading order. On the other hand,
5
the measurements of e+e− ff¯ at LEPII with center-of-mass energies from 130 to 207 GeV are
→
relevant [6]. Weemploy thedata onthe cross section andforward-backward asymmetry for f = µ,τ
and on the cross section for f = quark. To evaluate the limits on the new couplings, we include
boththe Z andZ contributions to the scattering amplitude, their couplings and masses satisfying
L H
the relations in Eqs. (11)-(13). Although Z is much heavier than Z , the fermionic couplings of
H L
the latter can be much smaller than those of the former to compensate for the suppression of the
Z contribution to the amplitude due to its bigger mass. In the examples presented below, the Z
H H
contributions to e+e− ff¯ turn out to dominate the Z ones.
L
→
For definiteness and simplicity, hereafter we set g = g 0. Adopting the 90% confidence-
X B−L ≥
level (CL) ranges of the LEPII measurements [6] and using the formulas given in Ref.[7], but with
s-dependent Z and Z widths [6], we then scan the m and R space. To illustrate the results,
L,H X v
we display in Figure 1 the upper limits on g versus m for R = 10−2 (red dashed curve) and
X X v
10−3 (blue dashed curve) on the left and right sides, respectively. The horizontal, straight portions
of the curves correspond to the perturbativity requirement, g < √4π.
X
The most recent data from the LHC on the cross-section of the Drell-Yan (DY) process in
proton-proton collisions at √s = 7TeV with 4.5fb−1 of integrated luminosity have revealed no
discrepancy from the SM expectations and therefore no evidence of Z bosons [8]. Consequently,
L,H
we follow the same analysis as in Refs. [2, 9] to derive upper bounds on the coupling constants
using the SM cross-section. In the present case, we can consider the Z and Z contributions
L H
separately because we focus on events with dilepton invariant mass around m or m where their
ZL ZH
effects are of different orders for small R . Thus respective constraints are obtained for the pairs
v
m ,g sinθ and m ,g cosθ . To estimate the DY cross-section numerically, we utilize
ZL B−L ZH B−L
the CalcHEP package [10] by incorporating the new particles and Feynman rules of our model.
(cid:0) (cid:1) (cid:0) (cid:1)
Then we apply the one-bin log likelihood LL = 2[N ln(N/ν)+ν N], where N (ν) is the number
−
10 10
Rv=10-2 Rv=10-3
1 1
X 0.1 X 0.1
g g
0.01 0.01
DYconstraint DYconstraint
LEPconstraint LEPconstraint
0.001 0.001
10 20 50 100 200 500 1000 10 20 50 100 200 500 1000
m m
X HGeVL X HGeVL
FIG.1: Upperlimitsong versusm fromLEPIIandLHCdataon e+e− ff¯ andDrell-Yanscattering,
X X →
respectively, for R = 10−2 (left) and 10−3 (right) under the assumption that g = g , compared to
v X B−L
the corresponding values of g (solid curves) consistent with the observed relic density. The horizontal,
X
straight portions of the dashed and dotted curves correspond to the perturbativity condition, g <√4π.
X
6
of events predicted by the SM (SM plus the Z or Z boson) in the ℓ+ℓ− invariant mass window
L H
of 20% around the expected Z or Z mass, with √s = 7TeV and 4.5fb−1 of luminosity. The
L H
±
upper limit on the cross-section is obtained from the solved value of ν for each Z or Z mass, after
L H
adopting LL = 2.7 which corresponds to the 90% CL.
We find that the Z contribution to the DY process yields a stricter bound on g as a function
H B−L
of m , as the Z contribution is strongly suppressed by the small θ . We show the resulting upper-
ZH L | |
limits on g = g in Figure 1, where m is related to m by Eq.(13), for R = 10−2 (red
X B−L X ZH v
dotted curve) and 10−3 (blue dotted curve) on the left and right, respectively. We notice that the
(cid:0) (cid:1)
limit in the R = 10−2 case becomes large at m 5GeV corresponding to m m where
v X ∼ ZH ∼ Z
the SM background is large.
IV. RESONANT DARK MATTER ANNIHILATION AND RELIC DENSITY
Now we estimate the relic density of the DM particle, X, in order to search for the model
parameter space consistent with the observed relic density. The thermal relic abundance is found
by solving the Boltzmann equation which describes the number density of the DM. We employ the
approximate solution to the Boltzmann equation for the present-day relic density Ω, given by [11]1
∞
1.07 109 σv
Ωh2 = × , J = dx h i ,
√g∗ mPlJ GeV Zxf x2
0.038gm m σv
x = ln X Plh i , (16)
f √g x
∗ f
where h denotes the Hubble constant in units of 100km/s/Mpc, g is the number of relativistic
∗
degrees of freedom below the freeze-out temperature T = m /x , m = 1.22 1019 GeV is the
f X f Pl ×
Planck mass, g = 3 to account for X having spin-1, and σv is the thermal average of the DM
h i
annihilation cross-section. More explicitly [12],
x ∞
σv = ds √s s 4m2 K √sx/m σ , (17)
h i 8m5 K2(x) − X 1 X ann
X 2 Z4m2X
(cid:0) (cid:1) (cid:0) (cid:1)
where K is the modified Bessel function of the second kind of order i and σ represents the cross
i ann
section of X†X annihilation into all possible final states.
Under the assumptions made inSection II, we find that the main contributions to σ come from
ann
the s-channel transitions X†X Z∗ f f¯ . Although Z -mediated diagrams also contribute,
→ L → SM SM H
in this case they can be neglected because of the suppression due to m m and their lack of
ZH ≫ ZL
the resonance enhancement of the Z -mediated diagrams in the nonrelativistic region √s 2m
L ∼ X
due to m 2m . Thus, with the Feynman rules in Appendix A, we arrive at
ZL ≃ X
σ = gX2 gB2−Lcos2θsin2θ s−4m2X s−4m2f s2 +20m2X s+12m4X
ann 432π q(cid:0) m4(cid:1)(cid:0)s (cid:1) s m2 2 +Γ2 m2
Xf X − ZL ZL ZL
s+2m2 VˆZL 2 + (cid:0)s 4m2 (cid:1)AˆZL 2 Nf , (18)
× f f − f f c
h i
(cid:0) (cid:1)(cid:12) (cid:12) (cid:0) (cid:1)(cid:12) (cid:12)
(cid:12) (cid:12) (cid:12) (cid:12)
1
For a more accurate approximation, see [27].
7
where the sum is over all fermions with masses m < m and color factors Nf, the couplings
f X c
VˆZL and AˆZL are given in Eq. (A2), and Γ is the width of Z . Now, since m2 = 4m2 (1 R )
f f ZL L ZL X − v
and s 4m2 according to Eqs. (12) and (17), respectively, in the denominator of σ above we
≥ X ann
have s m2 2 16m4 R2. From the collider bounds on g = g derived in the previous
− ZL ≥ X v B−L X
section, we find that for the mass range of interest 16m4 R2 Γ2 m2 . Consequently, the Γ
(cid:0) (cid:1) X v ≫ ZL ZL ZL
term can be neglected in the calculation of Eq.(17).
With Eqs. (16)-(18), we can extract the g ,m regions compatible with the observed Ω. Its
X X
most recent value has been determined by the Planck Collaboration from the Planck measurement
(cid:0) (cid:1)
and other data to be Ωh2 = 0.1187 0.0017 [13]. Accordingly, we require the relic density of X
±
to satisfy the 90% CL (confidence level) range of its experimental value, 0.1159 Ωh2 0.1215.
≤ ≤
As mentioned in the preceding section, for simplicity we take g = g , implying that θ R .
X B−L | | ≃ v
The plots in Figure 1 display the resulting g values allowed by the relic data for R = 10−2 (red
X v
solid curve) and 10−3 (blue solid curve) on the left and right panels, respectively. One can see
¯
that, although the ffZ couplings are suppressed by the small mixing angle, θ 1, the observed
L
| | ≪
relic density can be reproduced with moderate-sized couplings g = g = (0.1)- (1) over
B−L X O O
m 1000GeV due to the resonance enhancement. This can be partly understood from the fact
X ≤
that in the resonance region the denominator of σ is dominated by the term 4m2 m2 2 R2
ann X− ZL ∝ v
which approximately cancels the R2 factor in the numerator.
v (cid:0) (cid:1)
In Figure 1, we can also compare the coupling ranges that fulfill the requirements from both
the collider and relic density data. Evidently, the constraints from LEPII data restrict the allowed
masses to m & 400(220) GeV with couplings of (1) for R = 10−2 10−3 . The cases with
X O v
R . 10−4 and m 1000GeV are excluded by the LEPII constraints.
v X ≤ (cid:0) (cid:1)
Since we have the relation m = g v √R from Eqs. (7) and (10), it is interesting to explore
X X S v
the values of v subject to the same experimental requirements. We illustrate this in Figure 2
S
obtained with the allowed g ranges in Figure 1. Hence v should be between about 5 and 10 TeV
X S
in order to satisfy both the collider and relic data. This suggests that our model is compatible with
the TeV-scale type-I seesaw scenario.
V. DIRECT DETECTION OF DARK MATTER
The direct detection of X relies on its scattering off a nucleon N elastically, XN XN, which
→
proceeds from Z exchanges in the t channel. Since m m , the Z contribution can be
L,H ZH ≫ ZL H
neglected. It follows that in the nonrelativistic limit the cross section of XN XN is
→
g2 g2 cos2θsin2θµ2 g4 R2µ2
σN = X B−L XN X v XN , (19)
el πm4 ≃ 16πm4
ZL X
where µ = m m / m +m and we have made use of N u¯γαu+d¯γαd N = 3N¯γαN [14],
XN X N X N h | | i
the other quarks having vanishing contributions. This indicates that σN is not sensitive to g for
(cid:0) (cid:1) el B−L
fixed R 1.
v
≪
In Figure 3 we plot σN as a function of m for the allowed parameter regions in Figure 1, the
el X
red and blue strips belonging to the R = 10−2 and 10−3 cases, respectively. Also shown are the
v
recent data from DM direct searches. Clearly, much of the σN prediction still escapes the existing
el
8
15000
10000
LV R =10-3
v
e
G
H
S
v
R =10-2
v
5000
200 300 500 700 1000
m HGeVL
X
FIG. 2: Values of v versus m satisfying the requirements from both the collider and relic density data
S X
and corresponding to the allowed g regions in Figure 1.
X
constraints, including the strictest ones from XENON100 [17] and LUX [20], but it will be probed
more stringently by future direct searches such as XENON1T [25].
Before moving on, we would like to make a few remarks regarding the potential implications of
mixing between the SM and extra gauge bosons in our model. Since none of the scalar fields in the
theory carries both the electroweak and new quantum numbers, there is no mass mixing between
the SM and new gauge bosons. In contrast, as discussed in Appendix B, kinetic mixing between
the U(1) and U(1) gauge bosons can occur both at tree and loop levels. We find that the
Y B−L
impact of this mixing is not significant on the results above for the allowed values of the new gauge
couplings and Z mass. Especially, the relation m 2m is unaffected. We further find that,
L ZL ≃ X
although the Z mass is sensitive to the kinetic mixing, being enhanced by it, the effect can be
H
minimized if the mixing parameter has a magnitude below 0.5. Our rough estimate of the relevant
loop diagram in Appendix B suggests that mixing size of order 0.5 is not atypical. Lastly, since
the X annihilation and X-nucleon scattering processes are dominated by the Z contributions, the
L
increased m would not be important for them. It follows that it is reasonable to neglect the
ZH
impact of the kinetic mixing.
VI. COMMENTS ON COLLIDER PHENOMENOLOGY
In this section, we briefly discuss how the extra scalar and gauge bosons in our model may be
produced and detected at the LHC. The new scalar bosons coming from Φ and S comprise twelve
5
degrees of freedom in total. Four of them are “eaten” by the new gauge bosons, making them
massive. The remaining extra scalar bosons can be expressed as φeven and φodd, which are linear
i j
combinations of ZX-even and -odd particles, respectively. Since two of the new massive gauge
2
bosons are ZX even and the other two ZX odd, there are six φeven’s and two φodd’s which are
2 2
9
DAMA
10-40
CRESST
CoGeNT
XENON10
CDMSII
10-42 CDMSSi
CRESST
L
2
m
c CDMSGe
H 10-44 Rv=10-3 Rv=10-2
Nel
Σ XENON100
LUX
10-46
XENON1T
10-48
10 20 50 100 200 500 1000
mX HGeVL
FIG. 3: Cross-section σN of XN XN scattering corresponding to the allowed parameter regions
el →
in Figure 1. The predicted cross-sections are compared to 90%CL upper-limits from XENON10 (green
dashed-dotted curve) [15, 16], XENON100 (black short-dashed curve) [17], CDMS Ge (red long-dashed
curves) [18], CDMS Si (blue solid curve) [19], and LUX (purple dashed-double-dotted curve) [20]. The
predictionisalsocomparedtothe90%CL(magenta) signalregionsuggested byCoGeNT [21],agray patch
compatible with the DAMA modulation signal at the 3σ level [22], two 2σ-confidence (light brown) areas
representing CRESST-II data [23], and a cyan area for a possible signal at 90%CL from CDMS II [24].
Also plotted is the XENON1T projected sensitivity (brown dashed-triple-dotted curve) [25].
physical. In this study we do not specify the new scalars’ mass spectrum, but one could obtain it
by doing a detailed analysis of the scalar potential.
Taking into account the ZX parities of the new particles, we find the decay patterns
2
¯
Z f f , (20)
L → SM SM
Z f f¯ , XX†, φevenφeven, φoddφodd , (21)
H → SM SM i j i j
φeven Z Z , XX†, X(†)φodd, Z φeven, φevenφeven, φoddφodd , (22)
i → L,H L,H i L,H i j k j k
φodd Z X, φevenX, φevenφodd , (23)
i → L,H j j k
where the particles on the right-hand sides may be off-shell depending on the masses involved.
Throughout we have assumed that X is lighter than new scalar bosons, and so Z decays mostly
L
to SM fermions. Since the couplings of Z to the fermions are proportional to their B L
L,H
−
numbers, Z tend to decay into leptons rather than quarks, as the decay rates of Z into
L,H L,H
a charged lepton pair and into a quark-antiquark pair, with relatively negligible masses, are related
by Γ : Γ 1 : 3(1/3)2 = 3 : 1. The decay branching fractions of the scalar bosons
ZL,H→ℓ+ℓ− ZL,H→qq¯≃
depend on their mass spectrum and couplings in the potential.
10