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arXiv:1308.0612v2 [hepph] 15 Aug 2013
SLACPUB15704
Fermion Portal Dark Matter
Yang Bai
a
and Joshua Berger
b
a
Department of Physics, Unive rsity of Wisconsin, Madison, WI 53706, USA
b
SLAC National Accelerator Laboratory, 2575 Sand Hill Road, Menlo Park, CA 94025, USA
Abstract
We study a class of simpliﬁed dark matter models in which one dark matter particle couples w ith
a mediator and a Standard Mode l fermion. In such models, collider and direct detection searches
probe complimentary regions of parameter space. For Majorana dark matter, direct detectio n
covers the region near mediatordark ma tter degeneracy, while colliders probe regions w ith a la rge
dark matter and media tor mass splitting. For Dirac and complex dark matter, direct detection is
eﬀective for the entire region above the mass threshold, but colliders provide a strong b ound for
dark matter lighter than a few GeV. We also point out that dedicated searches for signatures with
two jets or a monojet not coming from initial state radiation, along missing transverse energy can
cover the remaining pa rameter space for thermal relic dark matter.
1 Introduction
There is now a large body of evidence for the existence of particulate dark matter that interacts
gravitationally and makes up nearly a quarter of the energy density of th e universe [1–4]. On the
other h an d, such dark matter does not exist in the Standard Model (SM), has not been conclusively
observed to interact nongravitationally, and has never been detected, directly or indirectly, in a
laboratory experiment. Determining the physical properties of dark matter therefore constitutes
one of the most pressing q uestions in highenergy physics. Ideally, it would be possible to measure
the nongravitational interactions in several ways: by directly detecting rare ambient d ark matter
scattering events oﬀ a target in underground laboratories, by detecting the products of dark matter
selfannihilation or decay, and by measuring a momentum imbalance in collider events.
Given that so little is known about the properties of d ark matter, the most informative framework
for studying dark matter properties in a s im pliﬁed framework that is as modelindependent as possib le.
One powerful approach is to consider eﬀective operators of d ark matter coupling to quarks and other
SM particles [5–25]. In the absense a light mediator below around one GeV, the eﬀective operator
approach provides an excellent description for interpreting the direct detection experiment results
because of the small exchange momentum in the scattering processes. A simpliﬁed description in terms
of such operators neatly encapsu lates all possible interactions of dark matter with their detectors. On
the other hand, su ch operators may be unsuitable or at least not precise for the purposes of collider
studies [26–28]. T he latest searches for dark matter at ATLAS [21, 29] and CMS [25, 30] have set an
lower bound on the cutoﬀs of the eﬀective operators to be around one TeV. The current constraints
are stringent when compared to the limits derived fr om direct detection experiments, but are still
much below the centerofmass ener gy, 8 TeV, of the latest run at the L HC. When the partonlevel
collision energy is comparable to the mediator mass, the eﬀective theory does not apply and a more
complete description is required.
To go beyond an eﬀective operator approach, one could directly use a concrete underlying model
such as the Minimal Supersymmetric Standard Model (MSSM) to study the complementarity among
diﬀerent experimental probes of dark matter [31]. However, to maintain the modelindependent
paradigm of the eﬀective operator approach, a “Sim pliﬁed Dark Matter Model” could be a better
ch oice. In this approach, one can introduce one or two new particles w ith one or two interactions to
probe the potentially complicated dark matter sector. For dark matter interacting with SM particles,
one of the most important things is the properties of the mediator. The majority of existing stu dies in
the literature h ave the mediator particle couple to two dark matter particles at the same time: Higgs
1
portal [32–34], 2HDM portal [35], axionportal [36], gravitymediated [37], dilatonassisted [38] and
Z
′
mediated [39, 40]
1
[43]. I n this paper, we study s im pliﬁed dark matter models with SM fermions
as the portal particle, which we call Fermion Portal (FP) dark matter. T his class of models is well
motivated and can easily be a part of some un derlying models such as SUSY [44] or extradimension
models [45].
In FP dark matter models, an SM singlet dark matter particle interacts with quarks via a new Q CD
color triplet state. There are several classes of such models depending on the Lorentz properties of the
dark matter: it may be a Dirac fermion, Majorana fermion, complex scalar, real scalar, or vector. The
vector case requires an additional scalar ﬁeld or additional dynamics to provide the vector boson mass
and we do not consider it here. For the real scalar dark matter case, the nonrelativistic interaction
crosssections including selfannihilation and direct detection are highly suppressed and we do not
consider th is case either. Therefore, in this paper we perform detailed studies of the Dirac fermion,
Majorana fermion, and complex scalar cases.
A further speciﬁcation can be made based on species of quarks to which the dark matter couples.
At r enormalizable level, the simplest interactions are to have dark matter particles coup le to only
righthanded fermions. In the nonrelativistic limit, the interactions can be transformed as vector
or axivector couplings between the dark matter particle and th e SM fermion. For either type of
couplings, a dark matter particle that couples exclusively to heavy quarks would have a suppressed
direct detection crosssection. We concentrate on the couplings to up and dow n quarks in this paper
and leave the heavy quark case for future exploration.
The six cases that are considered here constitute a set of models for benchmarking the progress of
dark matter experiments, as well as for studying the complementarity of diﬀerent types of experiments.
Unlike th e eﬀective operator approach, they can be extrapolated to arbitrarily high ener gies while
giving sensible results. Furthermore, they are wellmotivated and simple enough to allow for deep
experimental scr utiny. In th is paper, we demons trate the power of current experiments to probe these
models and determine the allowed parameter space to probe in the future. For each case, we also show
the parameter space to satisfy the dark matter thermal relic abundance. We pay attention to the
potential for current experiments to probe the thermal relic hypothesis. We ﬁnd that the parameter
space for a thermal relic is highly constrained for most of the scenarios considered, bu t that there is
some potential at the moment in a model with Majorana dark matter.
The remainder of this paper is structured as follows. In Section 2, we introduce the Fermion Portal
1
The millicharged dark matter with only one massless gauge boson in this model contradicts quantum gravity, as
shown in Ref. [41] based on arguments in Ref. [42].
2
class of simpliﬁed models. We determine the allowed parameter space for dark matter to be a thermal
relic in Section 3. Current direct detection and collider constraints are determined in Sections 4 an d
5 respectively, with summary plots presented in Section 5. We discuss potential improvement for the
LHC collider searches and conclude in Section 6.
2 Simpliﬁed dark matter model: fermion portal
If the dark matter sector interacts directly with a single fermion in the SM, two particles with diﬀerent
spins are required in the dark matter sector. In this paper, we will concentrate on the quark portal dark
matter an d leave the lepton portal dark matter for future exploration. Restricting to particles with a
spin less than one, there are two general s itu ations: fermionic dark matter with a colortriplet scalar
partner or scalar dark m atter with a colortriplet fermion partner. In the former case, we consider
both Dirac and Majorana dark matter, wh ile for the latter case we only consider a complex scalar dark
matter and skip the real scalar dark matter case [6], which h as a quark mass su ppressed swave or
a dwave or threebody suppressed annihilation rate and a velocity sup pressed direct detection cross
section if the quark masses are neglected.
We begin by considering fermionic dark matter coupled to righthan ded quarks as the portal to
the dark matter sector. T he dark matter candidate may be a Dirac or Majorana fermion, χ, that is
an S M gauge singlet. The mediator is an SU(3)
c
triplet with an app ropriately chosen hypercharge.
The renormalizable operators are
L
fermion
⊃ λ
u
i
φ
u
i
χ
L
u
i
R
+ λ
d
i
φ
d
i
χ
L
d
i
R
+ h.c. , (1)
where u
i
= u, c, t (d
i
= d, s, b) are diﬀerent SM quarks. Since χ is the dark matter candidate, the
partner masses m
φ
i
must be larger th an the dark matter m ass m
χ
. In our analysis, we assume the
branching ratio of th e decay φ
u
i
→ χ¯u
i
and φ
d
i
→ χ
¯
d
i
is 100%. We also require the Yukawa couplings
λ
i
to be less than
√
4π to preserve perturbativity. Since we will concentrate on the ﬁrst generation
quarks, we neglect the ﬂavor index from now on to simplify th e n otation. Using the up quark operator,
the width of φ
u
particle is calculated to be
Γ(φ → χ +
u) =
λ
2
u
16π
(m
2
φ
− m
2
χ
)
2
m
3
φ
, (2)
for both Dirac and Majorana cases.
Similarly, for a complex scalar dark matter, X, and its partner, ψ, a colortriplet Dirac fermion,
we have the interactions
L
scalar
⊃ λ
u
i
X
ψ
u
i
L
u
i
R
+ λ
d
i
Xψ
d
i
L
d
i
R
+ h.c. . (3)
3
For the up quark operator, th e decay width of ψ
u
ﬁeld is
Γ(ψ → X
†
+ u) =
λ
2
u
32π
(m
2
ψ
−m
2
X
)
2
m
3
ψ
. (4)
If the operators in Eqs. (1), (3) are deﬁned in the ﬂavor basis, the quark r ighthanded currents be
come physical and additional (weak) ﬂavor constraints apply to the model parameter space. However,
if they are deﬁned in the quark mass basis, there are no additional ﬂ avor changing processes beyond
the SM. We simply take the mass basis assumption an d ignore the ﬂavor physics constraints. We next
explore the dark matter phenomenology of this class of models, including its thermal relic ab undance,
direct detection an d collider searches. Some other studies for the spindependent direct detection and
indirect detection signatures can be found in Refs. [46, 47].
3 Relic abundance
The complimentarity between dark m atter collider and direct detection searches is independent of the
dark matter thermal history. Since the weakly interacting massive particle (WIMP) is still the best
motivated scenario that generates the observed dark matter relic abu ndance for a weakscale mass, we
calculate the thermal relic abundance for the simpliﬁed fermionportal dark matter. We then compare
the allowed thermal relic parameter space to direct detection and collider bounds.
In the fermionic dark matter case, the main annihilation channel is
χχ → uu for Dirac dark matter.
The dominant contribution to the annihilation crosssection is
1
2
(σv)
χ¯χ
Dirac
=
1
2
"
3 λ
4
u
m
2
χ
32 π (m
2
χ
+ m
2
φ
)
2
+ v
2
λ
4
u
m
2
χ
(−5m
4
χ
− 18m
2
χ
m
2
φ
+ 11m
4
φ
)
256 π (m
2
χ
+ m
2
φ
)
4
#
≡ s + p v
2
, (5)
where v is the relative velocity of two dark matter particles and is typically 0.3 c at the fr eezeout
temperature and 10
−3
c at present. T he factor of 1/2 in Eq. (5) accounts for the fact that Dirac dark
matter is composed of both a particle and an antiparticle. For Majorana dark matter, the annihilation
rate only contains a pwave contribution at leading order in the limit of zero quark masses
(σv)
χχ
Majorana
= v
2
λ
4
u
m
2
χ
(m
4
χ
+ m
4
φ
)
16π (m
2
χ
+ m
2
φ
)
4
≡ p v
2
. (6)
In the nondegenerate parameter space, we only need to care about the dark matter annihilation
rate. The dark matter relic abundance is ap proximately related to the “s” and “p” variables by
Ω
χ
h
2
≈
1.07 × 10
9
GeV M
Pl
√
g
∗
x
F
s + 3 (p − s/4)/x
F
, (7)
4
where the Planck scale is M
Pl
= 1.22×10
19
GeV and g
∗
is the number of relativistic degrees of freedom
at the freezeout temperature and is taken to be 86.25 here. The freezeout temperature x
F
is given
by
x
F
= ln
"
5
4
r
45
8
g
2π
3
M
Pl
m
χ
(s + 6 p/x
F
)
√
g
∗
√
x
F
#
, (8)
where g = 2(4) is th e number of degrees of freedom for the Majorana (Dirac) fermion dark matter.
In the degenerate p arameter space with ∆ ≡ (m
φ
−m
χ
)/m
χ
≪ 1 and comparable to or below the
freezeout temperature 1/x
F
∼ 5%, coannihilation eﬀects become important [48, 49]. Neglecting th e
subleading electroweak interaction, the ann ihilation crosssection for χ + φ
†
→ u + g is given by
(σv)
χ φ
†
=
g
2
s
λ
2
u
24π m
φ
(m
χ
+ m
φ
)
+ v
2
g
2
s
λ
2
u
(29m
2
χ
− 50m
χ
m
φ
+ 9m
2
φ
)
576π m
φ
(m
χ
+ m
φ
)
3
, (9)
for both Dirac and Majorana dark matter. Add itionally, the φ ﬁeld selfannihilation crosssection is
given by
(σv)
φ φ
†
[gg] =
7 g
4
s
216π m
2
φ
− v
2
59 g
4
s
5184π m
2
φ
, (10)
(σv)
φ φ
†
[f
¯
f] = v
2
g
4
s
432π m
2
φ
, for f 6= u , (11)
(σv)
φ φ
†
[u¯u] = v
2
"
g
4
s
432π m
2
φ
−
g
2
s
λ
2
u
108π (m
2
χ
+ m
2
φ
)
+
λ
4
u
m
2
φ
48π (m
2
χ
+ m
2
φ
)
2
#
, (12)
for both Dirac and Majorana dark matter. Here, f represents the SM quarks and we have neglected
all quark masses in our calculation f or a heavy m
φ
with m
φ
≫ m
f
. For Majorana dark matter, there
is an additional annihilation channel with cross section
(σv)
φ φ
[uu] =
λ
4
u
m
2
χ
6π (m
2
φ
+ m
2
χ
)
2
+ v
2
λ
4
u
m
2
χ
(3m
4
χ
−18m
2
χ
m
2
φ
− m
4
φ
)
144π (m
2
φ
+ m
2
χ
)
4
. (13)
Following Refs. [48, 49], we have the eﬀective degrees of freedom as a f unction of th e temperature
parameter x
g
eﬀ
= g
χ
+ g
φ
(1 + ∆)
3/2
e
−x ∆
, (14)
with g
φ
= 6 (we count φ and φ
†
together) and g
χ
= 2(4) for Majorana (Dirac) fermion. The eﬀective
annihilation cross section for the Dirac case is
(σv)
eﬀ
=
1
2
(σv)
χ¯χ
g
2
χ
g
2
eﬀ
+ (σv)
χφ
†
g
χ
g
φ
g
2
eﬀ
(1 + ∆)
3/2
e
−x ∆
+
1
2
(σv)
φφ
†
g
2
φ
g
2
eﬀ
(1 + ∆)
3
e
−2 x ∆
, (15)
5
and for the Majoran a case is
(σv)
eﬀ
= (σv)
χχ
g
2
χ
g
2
eﬀ
+ (σv)
χφ
†
g
χ
g
φ
g
2
eﬀ
(1 + ∆)
3/2
e
−x ∆
+
1
2
[(σv)
φφ
†
+ (σv)
φφ
]
g
2
φ
g
2
eﬀ
(1 + ∆)
3
e
−2 x ∆
, (16)
Variables s
eﬀ
and p
eﬀ
can be constructed by form ing a similar combination to (σv)
eﬀ
. They replace s
and p in Eqs. (7) and (8) for the purposes of calculating the thermal relic abundance.
Fitting to the observed value of Ω
χ
h
2
= 0.1199 ± 0.0027 from Planck [4] and WMAP [3], we show
the allowed values of m
χ
and m
φ
in Fig. 1 for diﬀerent values of couplings. For Dirac dark matter,
0
500
1000
1500
2000
0
500
1000
1500
2000
m
Φ
HGeVL
m
Χ
HGeVL
Dirac fermion dark matter
Λ
u
= 1
0.5
0.75
1.2
100
1000
500
200
2000
300
3000
150
1500
700
100
1000
500
200
2000
300
3000
150
1500
700
m
Φ
HGeVL
m
Χ
HGeVL
Majorana fermion dark matter
from left to right:
Λ
u
=0.5, 0.75, 1.0, 1.2
Figure 1: Left panel: the masses of Dirac fermion dark matter and its partner for diﬀerent choices
of coupling, after ﬁtting the observed dark matter energy fraction, Ω
χ
h
2
= 0.1199 ± 0.0027, from
Planck [4] and WMAP [3]. The blue dotted lines neglect coannihilation eﬀects, while the blue solid
lines include them. The black dotted line is boundary of the region for which m
φ
> m
χ
. Right panel:
the same, but for a Majorana dark matter.
the coannihilation eﬀects have a signiﬁcant eﬀect for small values of λ
u
, but only have a small eﬀect
for lager values of λ
u
. Due to pwave suppression of χχ annihilation, Majorana dark matter mass is
preferred to have either a light mass, below around 600 GeV, or a heavy mass nearly degenerate with
its partner.
For complex scalar dark matter, the annihilation rate of XX
†
→ u
u is also pwave suppressed and
given by
1
2
(σv)
XX
†
complex scalar
=
1
2
"
v
2
λ
4
m
2
X
16 π (m
2
X
+ m
2
ψ
)
2
#
≡ p v
2
. (17)
6
The allowed parameter space f or a thermal relic in the complex scalar case has sim ilar features to the
Majorana case, includin g the coannihilation eﬀects.
4 Dark matter direct detection
For calculation of dark matter direct detection crosssections, one could integrate out the dark matter
partner and calculate the scattering cross sections using the eﬀective oper ators. However, for the
degenerate region, the dark matter partner in the schannel can dramatically increase the scattering
cross section. To capture the resonance eﬀects, we keep the dark matter partner pr op agator in our
calculation.
χ
q
φ
χ
q
χ
q
φ
χ
q
(a) (b)
Figure 2: Feynman diagrams for scattering of a fermion dark matter oﬀ nucleus. Only the left panel in
(a) contributes to the Dirac fermion case, while both (a) and (b) contribute to the Majorana fermion
case.
For the Dirac dark matter case, only the left pan el in Fig. 2 contributes. Both spinindependent
(SI) and spindependent (SD) scattering exist. The leading SI interaction crosssection per nucleon is
given by
σ
Nq
SI
(Dirac) =
λ
u

4
f
2
Nq
µ
2
64 π[(m
2
χ
− m
2
φ
)
2
+ Γ
2
φ
m
2
φ
]
, (18)
where N = p, n; µ is the reduced mass of the dark matternucleon system; f
Nq
is the coeﬃcient related
to the quark operator matrix element inside a nucleon. For the up quark operator at hand, one has
f
p u
= 2 and f
n u
= 1 [44, 50]. The subleading SD interaction cross section is given by
σ
Nq
SD
(Dirac, Majorana) =
3 λ
u

4
∆
2
Nq
µ
2
64 π[(m
2
χ
− m
2
φ
)
2
+ Γ
2
φ
m
2
φ
]
, (19)
with ∆
p
u
= ∆
n
d
= 0.842 ±0.012 and ∆
p
d
= ∆
n
u
= −0.427 ±0.013 [51]. For Majorana dark matter, there
is only an SD scattering cross s ection with the same formula as the SD scattering of the Dirac ferm ion
case.
7
For the complex scalar case, the SI scattering cross section is given by
σ
Nq
SI
(complex scalar) =
λ
u

4
f
2
Nq
m
2
p
32 π[(m
2
X
− m
2
ψ
)
2
+ Γ
2
ψ
m
2
ψ
]
, (20)
while the SD scattering cross section is suppressed by the dark matter velocity and is neglected here.
Searches for SI dark matter interactions with nuclei are particularly constraining when they are
predicted by a given mo del. We include the most stringent SI direct detection constraints fr om
Xenon100 [52] for heavier dark matter masses and Xenon10 [53] for lighter dark matter masses.
Xenon100 [52] is sensitive to crosssections nearly down to 10
−45
cm
2
at a dark matter mass of around
100 GeV. The Xenon10 [53] experiment has some additional sensitivity for low dark m atter masses.
For the S D scattering cross section, we mainly use the limits from SIMPLE [54], COUPP [55], and
PICASSO [56] experiments for coupling to protons and from Xenon100 [57] and CDMS [58, 59] for
coupling to neutrons.
In addition to placing strong constraints, four experiments (DAMA [60], CoGeNT [61], CRESST
II [62], CDMS [63]) have now seen excesses in regions of parameter space already probed by Xenon100
under some assumptions [64]. For the purposes of this study, we ignore the debatable excesses and only
consider the constraints from experiments. Because we study the up quark and down quark operators
separately, isospin symmetry is generically broken. Therefore, we will consider the constraints on dark
matter–proton and dark matter–neutron scattering separately for both SI and SD.
5 Collider constraints
Since the dark matter couples to quarks, it can be produced at colliders. In addition, the colored
mediator may be produced, yielding strong constraints both from associated and pair produ ction.
Except in the regime of an extremely heavy mediator, these channels provide the dominant constraints.
Associated production of the med iator and the dark matter particle, along with radiative contributions
from dark matter pair production, yield a monojet s ignature, while pair production of mediators can
be seen in searches for jets plus missing transverse energy (MET). Example d iagrams for the three
production mechanisms in the case of coupling to up quarks are illustrated in Fig. 3. For simplicity, we
consider the constraints f rom the CMS experiment in the monojet and jets plus MET channels [25,65].
There are comparable constraints from the ATLAS experiment [21,66], though the reach of the CMS
searches is slightly better at present.
8
¯u
χ
χ
u
φ
u
φ
†
u
g
g
u
χ
χ
φ
†
u
g
u
u
χ
χ
φ
u
u
u
g
(a) (b) (c)
Figure 3: The thr ee dark matter particle produ ction mechanisms at hadron colliders. Diagram (a)
has two jets in ﬁnal state, while (b) and (c) provide monojet signatures.
5.1 Estimated limits from monojet on tchange φ exchange
For the fermionic dark matter case and in the heavy m
φ
limit, the Fierztransformed eﬀective operator
λ
u

2
8 m
2
φ
χγ
µ
(1 + γ
5
) χ uγ
µ
(1 − γ
5
) u (21)
is generated. The existing search at the 8 TeV LHC with around 20 fb
−1
constrains the combination of
up q uark and down quark operators. For light dark matter masses below analysis cuts on mon ojet p
T
or /E
T
, the collider production cross section is insensitive to the parity structure of the operators [25].
One can approximately translate the constraints on Λ ∼
√
2 m
φ
/λ
u
 obtained in Ref. [25] to our model
parameter space. For light dark matter masses, the 90% conﬁdence level (CL) constraints on Λ in
Ref. [25] is around 900 GeV, leading to an estimated constraint of m
φ
/λ
u
 & 640 GeV.
5.2 Limits from 2j + E
miss
T
on φ pair production
In the limit of a small dark mattermediator coupling, λ
u
≈ 0, th e only signiﬁ cant diagram yielding
this ﬁnal state is (a) in Fig. 3. T he p roduction crosssection is identical to that of a single squark in
the MSSM. The present bounds on this process from CMS constrain the colored particle m ass to be
above arou nd 500 GeV [67] for a massless neutralino. For λ
u
6= 0, there are additional contributions
from tchannel dark matter exchange and the crosssection for the parton level process u + ¯u → φ + φ
∗
is given by:
σ = −
1
1728πs
3
n
2
q
s(s − 4m
2
φ
)
4g
4
s
(4m
2
φ
− s) + 12g
2
s
λ
2
u
(s + 2m
2
χ
− 2m
2
φ
) + 27λ
4
u
s
+3λ
2
u
16g
2
s
m
2
χ
s + (m
2
φ
− m
2
χ
)
2
+ 9λ
2
u
s(s + 2m
2
χ
− 2m
2
φ
)
log
s −
q
s(s − 4m
2
φ
) + 2m
2
χ
− 2m
2
φ
s +
q
s(s − 4m
2
φ
) + 2m
2
χ
− 2m
2
φ
.
(22)
9
This extra contribution is signiﬁcant for λ
u
= 1 and leads to a much higher sensitivity. We also
note that there is destructive interference for a small value of λ
u
, as shown in Fig. 4 for diﬀerent values
of m
φ
. We therefore anticipate that the experimental limits from jets plus E
miss
T
could become weaker
at some intermediate values of λ
u
.
0.0
0.2
0.4
0.6
0.8
1.0
0
20
40
60
80
100
Λ
u
ΣHpp ® ΦΦ
+
L HfbL
LHC8 TeV
m
Χ
=10 GeV
m
Φ
= 400 GeV
m
Φ
= 500 GeV
m
Φ
= 600 GeV
Figure 4: The pairproduction cross sections of the φ ﬁeld as a function of λ
u
.
To estimate the current bounds on this model, as well as the case of scalar dark matter, we calculate
LO crosssections for the full process using MadGraph [68] with a model constructed by FeynRules [69].
NLO Kfactors calculated u sing Prospino [70] are applied to the pure QCD contr ibution to the cross
section for the cases of fermionic dark matter. The limits provided in [65] are then applied to the
calculated crosssection to obtain an estimate of the current 95% CL exclusion limit. The results of
this analysis are presented below, in Section 5.3.
5.3 Limits from monojet on single φ productions
The dominant prod uction channel for monojets is process (b) in Fig. 3 at a s mall value of λ
u
. The
resulting crosssection at LO for u + g → φ + χ is given by
σ(u + g → φ + χ) =
λ
2
u
g
2
s
768 π s
3
(3s + 2m
2
χ
− 2m
2
φ
)
q
(s + m
2
χ
− m
2
φ
)
2
− 4m
2
χ
s , (23)
where
√
s is the centerofmass energy. In order to estimate the current reach of monojet searches,
we generate events for all tr eelevel diagrams with one quark plus dark matter particles in the ﬁnal
state using MadGraph [68] with the models deﬁned in FeynRules [69]. The events are showered and
hadronized using Pythia [71], then the hadrons are clustered into jets using FastJet [72]. The cuts
described in R ef. [25] are then applied to the events in order to estimate the acceptance times eﬃciency
10
of th at search . The resulting LO signal cross section times estimated eﬃciency and acceptance for each
signal region are compared to the limits s et in Ref. [25]. We present our results for several diﬀerent
scenarios in two ways: ﬁrst in the m
φ
–m
χ
plane and second in the m
χ
–σ
SI(SD)
plane with all limits at
95% CL.
We begin by considering the model with Majorana dark matter and only λ
u
6= 0. For λ
u
= 1, the
exclusion curves are shown in Fig. 5. The dominant constraints come from collider searches in the
monojet and jets + MET channels, as well as dark matter spindependent direct detection searches.
In addition, we show the lines at which the observed dark matter relic abundance is attained assuming
that χ is a thermal relic. The exclusion extends up to scalar masses of around 700 GeV prov ided that
100
200
300
400
500
600
700
0
100
200
300
400
500
m
Φ
HGeVL
m
Χ
HGeVL
Λ = 1
SD, p
Jets + MET
Monojet
COUPP
Thermal relic
10
1
10
0
10
1
10
2
10
3
10
43
10
42
10
41
10
40
10
39
10
38
10
37
10
36
10
35
m
Χ
HGeVL
Σ Hcm
2
L
Λ = 1, SD, p
m
Φ
> 100 GeV
Jets + MET
Monojet
COUPP
SIMPLE
PICASSO
Thermal relic
Figure 5: 95% exclusion limits (except the black solid line from the thermal relic abundance) from
the most sensitive searches for Majorana dark matter with the only coupling to th e up quark with
λ
u
= 1. The left panel is in the m
φ
− m
χ
plane, while the right panel is in the σ −m
χ
plane.
the dark matter is lighter than about 300 GeV. In Fig. 5, we have included the coannihilation eﬀects
for the degenerate spectrum. We show the thermal relic required p arameter space in the black and
solid line in both panels of Fig. 5. In the σ − m
χ
plane, we stop plotting the thermal relic line when
the dark matter mass is close to the mediator mass. There is some parameter space at the moment
where a thermal relic is allowed, for a mediator mass of around 400 GeV, though we s tress that the
thermal relic abundance may be set in other ways. It is important to note that in this model, the
monojet search has a wider reach than the jets + MET search for heavy mediator m asses. This is due
11
to the fact that some of the d iagrams for φφ production are proportional to the Majorana dark matter
mass. In ad dition, up to dark matter masses of around 300 GeV, the domin ant constraint on these
models comes fr om colliders. I n particular, this means that the possibility of light dark matter below
a few GeV is highly constrained. The SD direct detection, jets+MET and monojet are complimentary
as they cover diﬀerent parts of parameter space.
For comparison, in Fig. 6 we show the same exclusions in the mass plane for λ
u
= 0.5. In this case,
the current constr aints are far weaker. Even for the mediator masses below a few hundred GeV, there
is a signiﬁcant allowed fraction of parameter space, which it is important to cover in future search es,
especially at colliders. On the other hand, for su ch a small coupling, it is diﬃcult to obtain the correct
relic abundance via therm al production except in the coannihilation region; an alternate nonther mal
mechan ism could be considered such that dark matter is not overpro duced.
100
200
300
400
500
600
700
0
100
200
300
400
500
m
Φ
HGeVL
m
Χ
HGeVL
Λ = 0.5
SD, p
Jets + MET
Monojet
COUPP
Thermal relic
10
1
10
0
10
1
10
2
10
3
10
43
10
42
10
41
10
40
10
39
10
38
10
37
10
36
10
35
m
Χ
HGeVL
Σ Hcm
2
L
Λ = 0.5, SD, p
m
Φ
> 100 GeV
Jets + MET
Monojet
COUPP
SIMPLE
PICASSO
Thermal relic
Figure 6: The same as Fig. 5 for the up quark case with λ
u
= 0.5.
We also study the same model, but for the down quark case with only λ
d
6= 0. For λ
d
= 1, the
exclusion curves are sh own in Figs. 7. The dominant constraints are the same as in the uptype case.
The constraints are slightly weaker in this case and the jets + MET search dominates f or at high
mediator masses as it is less sensitive to the down quark parton distribution function suppression. In
this case, there is a similar parameter space allowed for a thermal relic.
Next, we consider models with Dirac dark matter and complex scalar dark matter. For these
models, the SI dir ect detection constraints dominate up to very low dark matter masses, independent
12
100
200
300
400
500
600
700
0
100
200
300
400
500
m
Φ
HGeVL
m
Χ
HGeVL
Λ = 1
SD, n
Jets + MET
Monojet
Xenon100
Thermal relic
10
1
10
0
10
1
10
2
10
3
10
43
10
42
10
41
10
40
10
39
10
38
10
37
10
36
10
35
m
Χ
HGeVL
Σ Hcm
2
L
Λ = 1, SD, n
m
Φ
> 100 GeV
Jets + MET
Monojet
X100
CDMS
Thermal relic
Figure 7: 95% exclusion limits from the most sensitive searches for Majorana dark matter with coupling
to the down q uark.
of m
φ
. For λ
u
= 1, the exclusion curves are shown in Figs. 8 and 9. These cases are highly constrained
200
400
600
800
1000
0
100
200
300
400
500
m
Φ
HGeVL
m
Χ
HGeVL
Λ = 1
SI
Jets + MET
Monojet
Xenon100
Thermal relic
10
1
10
0
10
1
10
2
10
3
10
46
10
45
10
44
10
43
10
42
10
41
10
40
10
39
10
38
m
Χ
HGeVL
Σ Hcm
2
L
Λ = 1, SI
Jets + MET
Monojet
Xenon100
Xenon10
Thermal relic
Figure 8: 95% exclusion limits from the most sensitive searches for Dirac dark matter with coupling
to the up quark.
13
200
400
600
800
1000
0
100
200
300
400
500
m
Ψ
HGeVL
m
X
HGeVL
Λ = 1
SI
Jets + MET
Monojet
Xenon100
Thermal relic
10
1
10
0
10
1
10
2
10
3
10
46
10
45
10
44
10
43
10
42
10
41
10
40
10
39
10
38
m
X
HGeVL
Σ Hcm
2
L
Λ = 1, SI
Jets + MET
Monojet
Xenon100
Xenon10
Thermal relic
Figure 9: 95% exclusion limits from the most sensitive searches for complex scalar dark matter with
coupling to the up quark.
by searches for spinindependent scattering, which is u nsupp ressed. Since dark matter interactions
generally violate isospin in our models, the diﬀerent couplings to p rotons and neutrons should be
taken into account in calculating the bou nds. The SI crosssection bounds per nucleon are generally
calculated under th e assumption of isospin, su ch that the proton and neutron crosssections are the
same. In order to take into account isospin violation, we calculate the crosssection for interaction
with a proton and rescale by
σ
DM,nucleon
=
[f
p
Z + f
n
(A − Z)]
2
f
2
p
A
2
σ
DM,p
, (24)
where A and Z are th e mass number and atomic number of the target nucleus respectively. The
dominant SI bounds come from Xe targets, so that A = 131, neglecting small eﬀects from other
comparable or subdominant isotopes, and Z = 54. All scattering cross sections presented in Figs. 8
and 9 are the averaged one, σ
DM,nucleon
.
It is interesting to note that collider bounds take over for light dark matter, below the threshold
of direct detection experiments. In the case of a complex scalar, the low mass bound ﬂattens out in
the crosssection plane since it is not sensitive to the r educed mass of the dark matternucleon system,
but rather the nucleon mass itself, as can be seen from Eq. (20).
14
6 Discussion and conclusions
The signal spectrum from the associated production of dark matter and its partner could be dra
matically diﬀerent from backgrounds. Particularly when the Yukawa coupling is small, associated
production is the dominant part of the signal. Additional kinematic variables can be used to enhance
the dark matter signal in the fermionportal scenario. We use MadGraph5 [68] to generate the dark
matter signal events and shower them in PYTHIA [73]. We then use PGS [74] to perform the fast detector
simulation. After utilizing the basic cuts in Ref. [25], where E
miss
T
> 200 GeV has been imposed, we
calculate the normalized E
miss
T
distributions for several diﬀerent spectra. In the left panel of Fig. 10,
we show the E
miss
T
from the χ + φ associate productions. Because the jet from the decay of φ → χ + j
is energetic, the E
miss
T
distributions have a peakstructure with the peak at around m
φ
/2 for a small
m
χ
. As a comparison, the right panel of the Fig. 10 shows the E
miss
T
distribution without onshell
production of φ. The spectrum is monotonically decreasing in this case, which follows the s hape of
the background although w ith a diﬀerent slope. For a larger m
φ
, the signal spectrum becomes slightly
harder at higher masses. In principle, the peak structure in the left panel can be used to discover dark
200
300
400
500
600
700
0.00
0.05
0.10
0.15
0.20
0.25
0.30
E
T
miss
HGeVL
Fraction of Events
m
Χ
=10 GeV, m
Φ
=400 GeV
m
Χ
=10 GeV, m
Φ
=700 GeV
m
Χ
=10 GeV, m
Φ
=1000 GeV
150
200
250
300
350
400
450
500
0.00
0.05
0.10
0.15
0.20
0.25
E
T
miss
HGeVL
Fraction of Events
m
Χ
=10 GeV, m
Φ
=400 GeV
m
Χ
=10 GeV, m
Φ
=700 GeV
m
Χ
=10 GeV, m
Φ
=1000 GeV
Figure 10: Left panel: the fraction of events after basic cuts as a function of E
miss
T
for the associated
production of χ + φ with φ → χ + j. Right panel: the same as the left one but for the productions of
2χ + j with the jet from ISR.
matter, f or instance performing a “bu mp” search in the E
miss
T
distribution. In practice, the peaks are
too wide to make it feasible. Improvin g the jet energy resolution and E
miss
T
measurement can yield
signiﬁcant boosts in sensitivity.
To explore more fermion portal dark matter parameter space, we emphasize the importance of a
dedicated search of the two jets plus MET signature. As can be seen from the left panel in Fig. 6, for
15
small values of the Yu kawa coupling, the current limit on the colored m ediator mass is weak, around
350 GeV for a dark matter mass at 100 GeV. Additional kinematic variables like m
T
2
can increase th e
search sensitivity [75].
Note added: we note here that during the completion of our paper, another paper [76] appeared
with diﬀerent emphasis: our paper concentrates m ore on the complimentarity of direct detection and
collider searches f or dark matter, while their paper has more focus on the thermal relic parameter
space.
Acknowledgments
Y. Bai is supported by startup funds from the UWMadison. SLAC is operated by Stanford University
for the US Department of Energy under contract DEAC0276SF00515. YB also would like to thank
the Kvali Institute for Theoretical Physics, U. C. Santa Barbara, where part of this work was done.
This research was also supported in part by the National Science Foundation under Grant No. NSF
PHY1125915.
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