aainstitutetext: PRISMA Cluster of Excellence & Mainz Institute for
Theoretical Physics, Johannes Gutenberg University, 55099 Mainz,
Germanybbinstitutetext: Institute for Theoretical Physics (ITP), Karlsruhe
Institute of Technology, Engesserstraße 7, D-76128 Karlsruhe,
Germanyccinstitutetext: Institute for Nuclear Physics (IKP), Karlsruhe
Institute of Technology, Hermann-von-Helmholtz-Platz 1, D-76344 Eggenstein-
Leopoldshafen, Germany

# Hunting for Dark Matter Coannihilation by Mixing Dijet Resonances and
Missing Transverse Energy

Malte Buschmann a Sonia El Hedri a Anna Kaminska a Jia Liu a Maikel de Vries a
Xiao-Ping Wang a Felix Yu b,c José Zurita<EMAIL_ADDRESS>elhed001@uni-
mainz.de<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

Simplified models of the dark matter (co)annihilation mechanism predict
striking new collider signatures untested by current searches. These models,
which were codified in the coannihilation codex, provide the basis for a dark
matter (DM) discovery program at the Large Hadron Collider (LHC) driven by the
measured DM relic density. In this work, we study an exemplary model featuring
$s$-channel DM coannihilation through a scalar diquark mediator as a
representative case study of scenarios with strongly interacting
coannihilation partners. We discuss the full phenomenology of the model,
ranging from low energy flavor constraints, vacuum stability requirements, and
precision Higgs effects to direct detection and indirect detection prospects.
Moreover, motivated by the relic density calculation, we find significant
portions of parameter space are compatible with current collider constraints
and can be probed by future searches, including a proposed analysis for the
novel signature of a dijet resonance accompanied by missing transverse energy
(MET). Our results show that the $13$ TeV LHC with $100~{}\mathrm{fb}^{-1}$
luminosity should be sensitive to mediators as heavy as $1$ TeV and dark
matter in the 400–500 GeV range. The combination of searches for single and
paired dijet peaks, non-resonant jets + MET excesses, and our novel resonant
dijet + MET signature have strong coverage of the motivated relic density
region, reflecting the tight connections between particles determining the
dark matter abundance and their experimental signatures at the LHC.

††preprint: MITP/16-039
TTP/16-019

##  1 Introduction

With the resumption of the Large Hadron Collider (LHC) operating at
$\sqrt{s}=13$ TeV, the hunt for collider signals of dark matter (DM)
production has again begun in earnest. A positive signal of dark matter at the
LHC would spur a revolution in particle physics and astrophysics, although the
huge breadth of dark matter models and their concomitant collider signatures
makes designing a search strategy a daunting task. In particular, most of our
current knowledge voids possible interactions of the DM, rendering it
colorless and electrically neutral, while the only concrete DM measurement is
its relic density, $\Omega h^{2}=0.1198\pm 0.0026$ Ade:2015xua ;
Agashe:2014kda .

Nevertheless, the relic density requirement has not featured prominently in
collider searches thus far. In particular, the dark matter relic density can
be driven by all interactions of dark sector particles, while collider
searches based on effective operators Beltran:2010ww ; Goodman:2010yf ;
Bai:2010hh ; Goodman:2010ku ; Bartels:2012ui  or dark matter pair annihilation
simplified models Alves:2011wf ; An:2013xka ; DiFranzo:2013vra ;
deSimone:2014pda ; Abdallah:2014hon ; Buckley:2014fba ; Harris:2014hga ;
Garny:2015wea ; Abdallah:2015ter  eschew such complications. As a result, the
collider searches for dark matter are mainly variations on a theme of missing
transverse energy Khachatryan:2014rra ; Khachatryan:2014rwa ; Aad:2014tda ;
Aad:2015zva ; CMS:2015jdt ; Abercrombie:2015wmb  instead of being driven by
the known dark matter relic density.

In reference Baker:2015qna , we established a bottom-up framework for dark
matter discovery at the LHC based on a simplified model treatment of the
(co)annihilation mechanism of thermal relic dark matter. This approach
uncovered several new signatures ripe for analysis, and more importantly,
ensures a direct connection between the relic density calculation and the
collider signatures associated with the model. One novelty with this
construction was the inclusion of DM coannihilation Griest:1990kh  in addition
to pair annihilation, as the mere presence of the coannihilation partner
together with the mediator radically adds to the complexity and variety of DM
collider signatures. We stress that, unlike the dark matter field, the
coannihilating partner and the mediator of the coannihilation diagram can have
color and electromagnetic charges. In this way, the collider signatures of the
dark matter, coannihilation partner, and the mediator are driven by production
and decay modes dictated by the coannihilation codex Baker:2015qna .

Because the relic density measurement motivates the simplified model
construction, we use the $\Omega h^{2}$ calculation to inform the most
promising parameter space for a dark matter discovery at the LHC. In this way,
naive expectations about DM phenomenology from the weakly-interacting massive
particle (WIMP) miracle (see reference Bertone:2004pz  for a review) are
sharpened into concrete predictions for dark sector particle masses and
couplings with real discovery prospects at the LHC. Our approach also
complements the broader community efforts at exploring non-WIMP phenomenology
at colliders (see, e.g., references Feng:2008ya ; Feng:2009mn ; Bell:2013wua ;
Hochberg:2014dra ; Izaguirre:2015zva ).

In this work, we continue our exploration of the models presented in the
codex, again focusing on the case when the coannihilation partner and the
$s$-channel mediator are colored particles. In contrast with the $s$-channel
leptoquark case study presented in reference Baker:2015qna , though, we study
an $s$-channel diquark mediator with Yukawa couplings to first generation
quarks. Since the dark matter pair annihilation rate is suppressed, the DM
relic density will mainly be set by processes involving only strong
interactions. As a result, the dark sector mass scale can be readily estimated
for a wide range of strongly interacting models which depend only on the
$SU(3)_{C}$ representation of the coannihilation partner and its relative
splitting with the DM particle. For splittings of about ${\cal O}(1-10)\%$ of
the dark matter mass, this scale ranges from several TeV down to a few
hundreds of GeV, which is the prime target space for colored particle searches
at LHC Eichten:1984eu .

This model exemplifies the approach underpinning the coannihilation codex
Baker:2015qna , where the relic density calculation points to the region of
parameter space of interest for collider searches. Moreover, this simplified
model exhibits a novel dijet resonance + missing transverse energy (MET)
signature, which thus far remains an unexplored search channel at the LHC. In
contrast to ad hoc models exhibiting this collider signature Gupta:2015lfa ;
Autran:2015mfa ; Bai:2015nfa , our model readily generates the resonant dijet
and MET final state at colliders by recycling vertices used in the
coannihilation diagram. Other signals in direct detection and indirect
detection experiments are also predicted by reordering the topology of the
coannihilation diagram. In this way, we highlight the versatility and power of
the coannihilation approach where all features of the DM annihilation
mechanism connect to phenomenological signatures at dark matter experiments.

In section 2, we present the field content, Lagrangian, and general
phenomenology of our $s$-channel diquark mediated DM coannihilation model. We
show the relic density results for this model in section 3, demonstrating that
favorable regions in parameter space are within reach of the LHC. We also
briefly comment on the direct detection and indirect detection prospects. In
section 4, we review the existing collider bounds for our $s$-channel diquark
mediator and our color triplet coannihilation partner, and also extrapolate
these bounds to $100$ fb-1 of 13 TeV LHC luminosity. We study the sensitivity
of our proposed dijet resonance + MET collider signature in section 5,
highlighting the fact that this novel signature both probes new parameter
space outside the reach of current searches as well as makes the DM
coannihilation connection manifest. We conclude in section 6.

##  2 Diquark mediated coannihilation

As emphasized in the coannihilation codex Baker:2015qna , simplified models
that explicitly model the dark matter annihilation mechanism offer unique
phenomenology previously neglected in dark matter studies. The codex
classifies dark matter simplified models by the Standard Model (SM) gauge
charges of the dark sector and mediator fields and the $s$-channel,
$t$-channel, four point, or hybrid (both $s$-channel and $t$-channel) topology
of the coannihilation diagram. Having such a classification enables a clear
and thorough exploration of the signature space of dark matter at colliders,
aiding in the prioritization of searches and helping to identify unexplored
final states. These experimental signatures are in fact guaranteed in this
framework, since they result from stitching together production modes and
decay vertices dictated by SM gauge interactions and the coannihilation
diagram.

###  2.1 An $s$-channel example: the ST6 model

---
Field | $(SU(3)_{C},SU(2)_{L},U(1)_{Y})$ |  Spin assignment
DM | (1, 1, 0) |  Majorana fermion
X | (3, 1, -2/3) |  Dirac fermion
M | (3, 1, -2/3) |  Scalar
Table 1: Field content, Standard Model gauge quantum numbers, and spin
assignments for the scalar diquark case study ST6. Electric charge is defined
as $Q\equiv T_{3}+\dfrac{1}{2}Y$, where $T_{3}$ is the third component of weak
isospin.

One exemplary model to consider is ST6 (“$s$-channel mediator, color triplet
model 6” from table 4 of reference Baker:2015qna ), which features an
$s$-channel coannihilation topology with a fermionic color triplet
coannihilation partner X, scalar color triplet mediator M, and a fermionic SM
gauge singlet dark matter DM. The field definitions are shown in table 1. This
model is especially attractive to consider because both the coannihilation
partner X and the mediator M can be pair-produced at the LHC via strong
interactions. The corresponding large pair production rates for these
particles could lead to immediate LHC discovery prospects. Moreover, this
model juxtaposes signatures reminiscent of supersymmetry with unique collider
signatures characteristic of $s$-channel coannihilation, which highlights the
importance of our bottom-up approach to DM model building. Other $s$-channel
color triplet, sextet, and octet models in the coannihilation codex
Baker:2015qna  also share much of the same phenomenology as ST6, and thus the
analysis we present is representative of many models detailed in the codex.

Given the field content in table 1, the Lagrangian for this model is

$\displaystyle\mathcal{L}$
$\displaystyle=\mathcal{L}_{\text{DM}}+\mathcal{L}_{\text{X}}+\mathcal{L}_{\text{M}}+\mathcal{L}_{\text{vis}}+\mathcal{L}_{\text{dark}}\
,$ $\displaystyle\mathcal{L}_{\mathrm{DM}}$
$\displaystyle=\frac{i}{2}\overline{\mathrm{DM}}\not{\partial}\mathrm{DM}-\frac{m_{\mathrm{DM}}}{2}\overline{\mathrm{DM}}\,\mathrm{DM}\
,$ $\displaystyle\mathcal{L}_{\mathrm{X}}$
$\displaystyle=i\overline{\mathrm{X}}\not{D}\mathrm{X}-m_{\mathrm{X}}\overline{\mathrm{X}}\,\mathrm{X}\
,$ $\displaystyle\mathcal{L}_{\mathrm{M}}$
$\displaystyle=\left(D_{\mu}\mathrm{M}\right)^{\dagger}\left(D^{\mu}\mathrm{M}\right)-m_{\mathrm{M}}^{2}\mathrm{M}^{*}\,\mathrm{M}-\lambda_{\mathrm{HM}}\left(H^{\dagger}H\right)\mathrm{M}^{*}\mathrm{M}-\lambda_{\mathrm{M}}\left(\mathrm{M}^{*}\mathrm{M}\right)^{2}\
,$ $\displaystyle\mathcal{L}_{\mathrm{vis}}$
$\displaystyle=-\varepsilon_{abc}\mathrm{M}_{a}\overline{u_{b}^{C}}\left({Y_{ud}^{L}P_{L}+Y_{ud}^{R}P_{R}}\right)d_{c}-Y_{u\ell}\mathrm{M}^{*}_{a}\overline{\ell_{R}^{C}}u_{R}^{a}-Y_{QL}\mathrm{M}^{*}_{b}\left({\overline{L_{L}^{C}}_{\alpha}\varepsilon^{\alpha\beta}(Q_{L})_{\beta}^{b}}\right)+\mathrm{h.c.}\
,$ $\displaystyle\mathcal{L}_{\mathrm{dark}}$
$\displaystyle=-y_{D}\overline{\mathrm{X}}\,\mathrm{DM}\,\mathrm{M}+\mathrm{h.c.}\
,$ (1)

where $H$ denotes the SM Higgs field, $a$, $b$, and $c$ denote color indices,
$\alpha$ and $\beta$ denote $SU(2)_{L}$ weak isospin indices, $C$ is the
charge conjugation operator, and we have suppressed flavor indices on
$Y_{ud}^{L}$, $Y_{ud}^{R}$, $Y_{u\ell}$ and $Y_{QL}$. The Yukawa interactions
in $\mathcal{L}_{\mathrm{vis}}$ are written in the mass basis for all quarks
and leptons. We need a Cabibbo-Kobayashi-Maskawa (CKM) rotation to write
$d_{L}$ in the mass basis, therefore $Q_{L}=\left({u_{L},V_{CKM}d_{L}}\right)$
for mass eigenstates.111Note that $SU(2)_{L}$ invariance can be made manifest
in $\mathcal{L}_{\text{vis}}$ by rewriting the first Lagrangian term as
$-Y_{QQ}^{ij}\varepsilon_{abc}M_{a}\left({\overline{Q_{L}^{C}}_{\alpha}^{b,i}\varepsilon^{\alpha\beta}(Q_{L})_{\beta}^{c,j}}\right)-Y_{ud}^{R}\varepsilon_{abc}M_{a}\left({\overline{u_{R}^{C}}_{b}d_{Rc}}\right)$,
and identifying $Y_{ud}^{L}$ with $2Y_{QQ}V_{\text{CKM}}$.

The gauge quantum numbers allow for additional couplings between X or DM and
the SM fields. For example, X shares the same quantum numbers as the right-
handed down-type quarks, and hence a Yukawa interaction between X, the SM
Higgs, and $Q_{L}$ would lead to mixing between X and the SM down-type quarks
after the Higgs acquires a vev. Then, the DM particle would decay to three
quarks via a mixing angle insertion, which is phenomenologically ruled out.
Separately, DM could have a Yukawa interaction with the Higgs and the left-
handed lepton doublet, which would immediately lead to mixing with active
neutrinos or other sterile neutrino phenomenology, depending on the mass
generation mechanism for SM neutrinos. In order to ensure our DM particle is
an appropriate dark matter candidate, we impose a global $\mathbb{Z}_{2}$
symmetry under which both the DM and X are odd while the mediator and all SM
particles are even. Once this $\mathbb{Z}_{2}$ symmetry is introduced, only
the interactions shown in equation 2.1 remain.

The field content in table 1 and Lagrangian in equation 2.1 lead to the
$s$-channel coannihilation diagram depicted in figure 1. We reiterate that the
prescription from reference Baker:2015qna  for constructing simplified models
is based solely on the coannihilation topology and SM gauge representation
assignments. As a result, the parameter space of a given model may have
significant constraints coming from flavor physics, proton decay bounds, Higgs
physics, and vacuum stability, among others. We discuss these questions in the
remainder of this section and will find that interesting and viable parameter
space regions are still allowed for model ST6, from which we motivate our LHC
study focused on dijet resonances and MET.

$y_{D}$$y_{ud}$XDM$\bar{u}$$\bar{d}$M Figure 1: Coannihilation diagram for the
model ST6, with a diquark mediator M, coannihilation partner X, and dark
matter DM. The Yukawa couplings $y_{D}$ and $y_{ud}$ denote the mediator
couplings to dark sector particles and SM particles, respectively.

###  2.2 Low energy constraints

In this section, we investigate the constraints on the flavor structure of the
Yukawa matrices shown in $\mathcal{L}_{\text{vis}}$ in equation 2.1. First, we
note that the simultaneous presence of all Yukawa matrices in
$\mathcal{L}_{\text{vis}}$ generates diquark-mediated diagrams that induce
proton decay, which is very strongly constrained. For example, if
$(Y_{QL})_{11}$ and $(Y_{ud}^{L})_{11}=(Y_{ud}^{R})_{11}=y_{ud}$ are each
nonzero, our mediator generates the decay process $p\to e^{+}\pi^{0}$, which
has a minimum lifetime of at least $8.2\times 10^{33}$ years Agashe:2014kda .
Correspondingly, the product of these couplings is limited to be
$\sqrt{y_{ud}(Y_{QL})_{11}}<4\times 10^{-17}(m_{\text{M}}/\text{GeV})$. As
usual, however, these proton decay bounds can be avoided by extending the
accidental baryon and lepton global symmetries of the SM to the new physics
particles. If we charge M with baryon number $B=2/3$ and $L=0$, for instance,
the $Y_{QL}$ and $Y_{u\ell}$ matrices identically vanish. We will hence turn
off these couplings, enabling the dijet resonance signature at the LHC instead
of the leptoquark resonance, given that the leptoquark case study was
presented in reference Baker:2015qna .

Next, we focus on possible flavor-violating entries in $Y_{ud}^{L}$ and
$Y_{ud}^{R}$, which are highly constrained by low energy flavor violation
probes. Because our mediator has electric charge $-1/3$, it does not induce
any tree-level $\Delta F=2$ meson mixing, which instead occurs via box
diagrams. For instance, simultaneous $(Y_{ud}^{L})_{11}$ and
$(Y_{ud}^{L})_{12}$ diquark couplings induce large $K^{0}$–$\bar{K}^{0}$
mixing, leading to the constraint $m_{M}\gtrsim
10^{3}(Y_{ud}^{L})_{11}(Y_{ud}^{L})_{12}$ TeV Bona:2007vi ; Isidori:2010kg ;
Giudice:2011ak . If the $K^{0}$–$\bar{K}^{0}$ box diagram involves both
chiralities, the experimental constraint is more stringent, requiring
$m_{M}\gtrsim 10^{4}(Y_{ud}^{ij})^{2}$ TeV where $i$, $j$ are light quark
flavor indices. Multiple diagonal entries in $Y_{ud}^{L}$ are also
constrained, since nonzero $(Y_{ud}^{L})_{11}$ and $(Y_{ud}^{L})_{22}$
couplings also lead to $K^{0}$–$\bar{K}^{0}$ mixing via $M$ and $W^{\pm}$
mixed box diagrams. The resulting bound requires $m_{M}\gtrsim
100\sqrt{(Y_{ud}^{L})_{11}(Y_{ud}^{L})_{22}}$ TeV. Diagonal entries in
$Y_{ud}^{R}$ are much less constrained by this diagram because of a chiral
quark mass suppression.

We note that while arbitrary flavor structures in $\mathcal{L}_{\text{vis}}$
are incompatible with proton decay and flavor violation constraints, these
flavor aspects are orthogonal to the primary relic density motivation of the
model. Thus, for the remainder of the section, in addition to the $B=2/3$ and
$L=0$ global charges of M, we will set
$(Y_{ud}^{L})_{11}=(Y_{ud}^{R})_{11}=y_{ud}$ and set all other entries of
$Y_{ud}^{L}$ and $Y_{ud}^{R}$ to zero to satisfy these flavor constraints.
This flavor structure, while sufficient for our purposes in discussing the
relic density connection to LHC signatures, is overly restrictive given the
constraints above, however, and fuller discussions of possible flavor aspects
of scalar diquarks can be found in references Isidori:2010kg ; Giudice:2011ak
. Finally, electroweak oblique corrections can also constrain the X and M
masses in our model, but these corrections decouple quickly for X and M masses
even moderately above the electroweak scale Arnold:2009ay .

###  2.3 Vacuum stability and precision Higgs physics

We now consider the impact of the Higgs portal quartic coupling
$\lambda_{\text{HM}}$ in $\mathcal{L}_{\text{M}}$ for both the stability of
the electroweak vacuum and precision Higgs physics. While this term is again
irrelevant for the suitability of this coannihilation model regarding relic
density, the magnitude of this coupling cannot be constrained by global or
gauge symmetries and can have significant phenomenological effects.

First, we note that a large, negative value for $\lambda_{\text{HM}}$ will
destabilize the electroweak vacuum and cause the diquark mediator to obtain a
vacuum expectation value (vev), which is phenomenologically unacceptable.
Hence, we require $m_{\text{M}}^{2}+\lambda_{\text{HM}}v^{2}/2>0$, where $v$
is the Higgs vev, $H=\frac{1}{\sqrt{2}}(v+h)$. In addition, we impose
$\lambda_{M}>0$ and
$\lambda_{\text{HM}}+\sqrt{\lambda_{\text{H}}\lambda_{\text{M}}}>0$, with
$\lambda_{\text{H}}$ the SM Higgs quartic coupling, to ensure a global vacuum
stability Gunion:1989we .

The quartic coupling $\lambda_{\text{HM}}$ is also constrained from Higgs
physics, since the mediator M propagates in the loop-induced gluon fusion
Higgs production process. In the limit where the top mass and $m_{\text{M}}$
are heavy compared to the Higgs mass, we can calculate the gluon signal
strength
$\mu_{ggF}\approx\left|{1+c_{M}\lambda_{\text{MH}}v^{2}/(8m_{\text{M}}^{2})}\right|^{2}$
Dobrescu:2011aa ; Kumar:2012ww , where $c_{M}=2\times 4/3$ is proportional to
the quadratic Casimir for a color triplet complex scalar. The ATLAS and CMS
combination of Higgs measurements constrain $\mu_{ggF}=1.03^{+0.17}_{-0.15}$
CMS:2015kwa , which implies $-0.8<\lambda_{\text{HM}}<1.2$ for
$m_{\text{M}}=500$ GeV. The Higgs decay rate to photons is affected by the
charged mediator running in the loop. However, since contributions to this
decay mode from intermediate scalars are suppressed by
$v^{2}/m_{\text{M}}^{2}$ Djouadi:2005gj  and the electric charge, the effect
of the mediator on the Higgs diphoton decay width will be small even for large
$\lambda_{\text{HM}}$.

As the quartic couplings are not relevant for the collider analysis or relic
density calculation, we will simplify the remaining discussion by neglecting
the terms $\lambda_{\text{HM}}$ and $\lambda_{\text{M}}$. If, however, Higgs
data begins to show significant deviations from the SM expectation and a dijet
signal emerges from our proposed analysis in section 5, we can revisit the
impact of $\lambda_{\text{HM}}$ to both fit the deviations to our model and
predict additional signatures given by the coannihilation topology.

###  2.4 Phenomenological features

As described in the previous sections, the various Yukawa structures and
quartic couplings appearing in $\mathcal{L}_{\text{vis}}$ and
$\mathcal{L}_{\text{M}}$ can lead to unacceptable proton decay, flavor
violation, vacuum instability, and Higgs physics effects. We readily satisfy
all of these bounds, however, by giving baryon number $2/3$ and lepton number
$0$ to M, restricting $(Y_{ud}^{L})_{11}=(Y_{ud}^{R})_{11}=y_{ud}$ as the only
nonzero entries in $Y_{ud}^{L}$ and $Y_{ud}^{R}$, and neglecting the quartic
scalar couplings in $\mathcal{L}_{\text{M}}$.

These simplifications lead to a small number of physical parameters in our
simplified model: the masses of the mediator, X, and DM, the visible coupling
$y_{ud}$, and the dark coupling $y_{D}$. We define
$\Delta\equiv(m_{\text{X}}-m_{\text{DM}})/m_{\text{DM}}$ to be the fractional
mass splitting between the coannihilation partner X and the dark matter, and
we always assume X is heavier than DM, which ensures the stability of the DM
particle. Since DM is a SM gauge singlet, it is only produced at the LHC in
the decays of X or M, where both X and M can be pair-produced from SM gauge
interactions. We also can singly-produce M via the coupling $y_{ud}$, giving a
dijet resonance or the very difficult M $\to(jj)_{\text{soft}}+\not{E}_{T}$
channel.

The coannihilation partner X has only a single decay mode, X $\to$ DM
$\bar{u}\bar{d}$, which is prompt as long as $\Delta$ is not extremely small.
Pair production of X then leads to the $2(jj)_{\text{soft}}+\not{E}_{T}$ final
state, where the softness of the outgoing jets is correlated with $\Delta$. If
$\Delta$ is small, these soft jets will escape detection and the primary
signature will be $\not{E}_{T}$ accompanied with one or two jets from
initial/final state radiation. This signature is already probed by the ATLAS
and CMS monojet searches Khachatryan:2014rra ; Aad:2015zva  and by the lowest
multiplicity signal regions of the multijet + MET searches Aad:2014wea ;
Khachatryan:2016kdk ; Atlas:2016rxq ; Chatrchyan:2014lfa . For large $\Delta$
and large DM mass, the jets from the X decay can be hard enough to pass
detector thresholds, leading to signatures akin to gluino pair production in
the minimal supersymmetric standard model (MSSM) Alves:2011wf ; Aad:2014wea .
Moreover, because DM is a SM gauge singlet, $\Delta$ plays a dominant role in
determining the DM relic density via the X–X and X–DM annihilation modes in
the effective DM annihilation cross section Griest:1990kh .

The diquark mediator can either decay via the visible coupling $y_{ud}$ or the
dark sector coupling $y_{D}$. The corresponding partial widths, assuming
massless quarks, are

$\displaystyle\Gamma\left(\mathrm{M}\to\bar{u}\bar{d}\right)=$
$\displaystyle\frac{y_{ud}^{2}}{4\pi}m_{\mathrm{M}}\ ,$
$\displaystyle\Gamma\left(\mathrm{M}\to\mathrm{X}\,\mathrm{DM}\right)=$
$\displaystyle\frac{y_{D}^{2}}{8\pi}m_{\mathrm{M}}K(\Delta,\tau_{\mathrm{DM}})\
,$ (2)

where, for on-shell M $\to$ X DM decays, $\tau_{\mathrm{DM}}\equiv
m_{\mathrm{DM}}^{2}/m_{\mathrm{M}}^{2}$ and

$\displaystyle K(\Delta,\tau_{\mathrm{DM}})\equiv$
$\displaystyle\left(1-\Delta^{2}\tau_{\mathrm{DM}}\right)^{\frac{1}{2}}\left(1-(2+\Delta)^{2}\tau_{\mathrm{DM}}\right)^{\frac{3}{2}}\
.$ (3)

The possible final states for M pair production are $2(jj)_{\text{res}}$,
$(jj)_{\text{res}}+(jj)_{\text{soft}}+\not{E}_{T}$, and
$2(jj)_{\text{soft}}+\not{E}_{T}$, where we denote the decay products from X
as soft because of the fractional mass splitting $\Delta$. While the pair
production of dijet resonances has been a traditional new physics signature,
the mixed decay $(jj)_{\text{res}}+(jj)_{\text{soft}}+\not{E}_{T}$ gives a
striking dijet resonance + MET final state which remains unexplored. Section 5
will present a proposed analysis and the experimental prospects for this
interesting mixed decay signature.

The relative rates between the three M M∗ final states are dictated by the
visible branching fraction $B$,

$\displaystyle B\equiv$
$\displaystyle\mathrm{Br}\left(\mathrm{M}\to\bar{u}\bar{d}\right)=\frac{y_{ud}^{2}}{y_{ud}^{2}+\frac{1}{2}y_{D}^{2}K(\Delta,\tau_{\mathrm{DM}})}=\frac{B_{0}}{B_{0}+(1-B_{0})K(\Delta,\tau_{\mathrm{DM}})}\
,$ $\displaystyle B_{0}\equiv$
$\displaystyle\left.\mathrm{Br}\left(\mathrm{M}\to\bar{u}\bar{d}\right)\right|_{m_{\mathrm{DM,X}}=0}=\frac{y_{ud}^{2}}{y_{ud}^{2}+\frac{1}{2}y_{D}^{2}}\
,$ (4)

where we introduce $B_{0}$ to represent the mediator M branching fraction into
SM particles at zero dark matter mass and coannihilation partner mass. We note
that the couplings $y_{ud}$ and $y_{D}$ can be traded for $B_{0}$ or $B$ and
the total width of M, which are natural variables to discuss the collider
constraints and prospects on M. Hence, we adopt the language of
$m_{\text{DM}}$, $\Delta$, $m_{\text{M}}$, $B_{0}$ or $B$, and
$\Gamma_{\text{M}}$ to tighten the connection between the relic density
calculation and collider probes.

We implement the diquark triplet model in FeynRules v2.3.18 Christensen:2008py
; Alloul:2013bka  with the Lagrangian in equation 2.1 using the
simplifications summarized in the beginning of this subsection. This
implementation is publicly available in the FeynRules model database
website:fr:database , where an interface with Monte Carlo generators is also
available using the UFO Degrande:2011ua  format. Our relic density
calculations in section 3 are performed using MicrOMEGAs v4.1.8
Belanger:2014vza  through CalcHEP v3.6.25 Belyaev:2012qa  output of FeynRules.
We also performed cross-checks of our relic density calculation using MadDM
v2.0 Backovic:2013dpa ; Backovic:2015tpt . The collider recasting discussion
in section 4 and the mixed decay study shown in section 5 use simulated Monte
Carlo events generated from MadGraph5 v1.5.14 Alwall:2011uj ; Alwall:2014hca .
We shower and hadronize the events using Pythia v8.2 Sjostrand:2007gs  and
perform detector simulation with Delphes v3.2 deFavereau:2013fsa  using the
default CMS parameter card. When indicated, matching between the hard matrix
element and the parton shower is performed using the MLM matching scheme
Mangano:2002ea .

##  3 Relic density

The central concern for the simplified models developed in the coannihilation
codex Baker:2015qna  is the dark matter relic density. By modeling the dark
matter annihilation mechanism, we can directly connect the parameter space of
a given simplified model to phenomenological signals in direct detection
experiments, indirect detection experiments, and colliders. Hence, the relic
density calculation features strongly in motivating our discussion of
experimental signatures, and we can demonstrate the particular interplay
between different terrestrial and astrophysical probes manifest in our diquark
mediator case study.

In this vein, we calculate the parameter space that gives the correct dark
matter relic density, $\Omega h^{2}=0.1198\pm 0.0026$ Ade:2015xua ;
Agashe:2014kda  for the ST6 model, subject to the Lagrangian flavor
restrictions discussed in section 2. The three fields, DM, X, and M, and two
couplings, $y_{ud}$ and $y_{D}$, lead to several channels that determine the
final DM relic density. Since the dark matter candidate is a pure SM gauge
singlet fermion, the usual pair annihilation channel $\overline{\text{DM}}$ DM
$\to$ $\overline{\text{SM}}$ SM vanishes at tree level and can be neglected.
Obtaining the appropriate relic density for DM therefore depends crucially on
X and M. In particular, DM and X readily stay in thermal equilibrium with SM
particles until thermal freeze out from Hubble expansion in the early
universe, where the process that freezes out last determines the relic density
Griest:1990kh .

There are four categories of two-to-two (co)annihilation channels relevant for
the diquark model:

DM X $\to$ SM1 SM2:

the standard coannihilation channel where DM and X coannihilate to SM
particles $\bar{u}\bar{d}$ through the $s$-channel diquark mediator. This
channel is dominant only when the diagram is resonant, that is when
$m_{\mathrm{DM}}+m_{\mathrm{X}}\sim m_{\mathrm{M}}$.

$\overline{\text{X}}$ X $\to$ $\overline{\text{SM}}$ SM:

pair annihilation of X into SM pairs. Since X is a color triplet, pair
annihilation of X into gluons and quarks is typically dominant, especially for
low X masses when other channels are kinematically closed. As long as DM and X
are in chemical equilibrium, this channel sets a lower bound on
$m_{\mathrm{DM}}$ for a given value of $\Delta$: ever lighter DM and X masses
cause $\Omega h^{2}$ to fall below the measured DM relic density and hence
underclose the universe.

DM X $\to$ M $V_{\mathrm{SM}}$:

real radiation of a SM gauge boson ($V_{\mathrm{SM}}=g$, $\gamma$, $Z$) from
the on-shell production of M. This category of channels opens up whenever
$m_{\mathrm{DM}}+m_{\mathrm{X}}\gtrsim m_{\mathrm{M}}$ and is generated
whenever X or M radiates off a SM gauge boson. Since X and M are colored, this
category is dominated by the gluon radiation diagram.

$\overline{\text{DM}}$ DM $\to$ M∗ M:

DM pair annihilation to a pair of mediators. The last channel only opens up
when $m_{\mathrm{DM}}\gtrsim m_{\mathrm{M}}$. This pair annihilation mode of
DM is generated through a $t$-channel X exchange and gives the only
contribution to the effective coannihilation cross section which is not
exponentially suppressed by $\Delta$. The $\overline{\text{X}}$ X $\to$ M∗ M
process also exists but is subdominant to X pair annihilation to SM particles.

We note that the Sommerfeld effect modifies the cross section for the process
$\overline{\text{X}}$ X $\to\bar{q}q$, $gg$, where the $t$-channel exchange of
soft gluons can potentially lead to significant non-perturbative effects. We
use the calculation of the Sommerfeld enhancement factors for colored
particles in reference deSimone:2014pda  and implement this in MicrOMEGAs
Ibarra:2015nca . In contrast to the results of reference deSimone:2014pda ,
however, we find that the Sommerfeld enhancement effects are less than a few
percent on the relic density for the parameters we consider because of a
cancellation between the enhancement of the cross section in the channel
$\overline{\text{X}}$ X $\to gg$ and the suppression in the channel
$\overline{\text{X}}$ X $\to\bar{q}q$. We therefore neglect this effect in our
relic density calculations.

Figure 2: Upper panel: the relic density in the triplet diquark model ST6 as a
function of the dark matter mass for given values of $m_{\text{M}}$,
fractional mass splitting $\Delta$, and $y_{ud}=y_{D}=y$, where the gray
horizontal band shows the Planck measurement with $3\sigma$ uncertainty
Ade:2015xua . Lower panel: the relative contributions from the four different
channels described in the text are shown for a single benchmark point.

In figure 2, we plot the dark matter relic density as a function of the DM
mass for various choices of mediator mass, $\Delta$, and Yukawa couplings
$y_{ud}=y_{D}=y$. We also overlay the measured relic density with $3\sigma$
uncertainties from Planck Ade:2015xua  as a gray horizontal band. In the
bottom panel, the relative contributions from the four different annihilation
categories are displayed for a single benchmark point. This breakdown of the
overall relic density calculation into constituent annihilation and
coannihilation channels reinforces the importance of explicitly accounting for
all fields and diagrams that contribute to the dark matter relic density.

###  3.1 Relic density favored regions

We now use the relic density calculation to motivate masses and couplings to
probe with collider experiments, while direct detection and indirect detection
prospects are discussed in section 3.2. From figure 2, we see that
perturbative Yukawa couplings for TeV-scale mediators and electroweak scale
dark sector particles generally give the correct DM relic density. As a
result, because M and X can be pair produced via strong interactions at the
LHC, we expect a dynamic interplay between the parameter space region favored
by the relic density calculation, the current exclusion bounds from ATLAS and
CMS, and the discovery prospects from future searches. Following section 2, we
adopt $m_{\text{DM}}$, $\Delta$, $m_{\text{M}}$, $B_{0}$ and
$\Gamma_{\text{M}}$ as the five parameters of our model. Recall $B_{0}$ from
equation 4 is the particular coupling relation defining the zero mass visible
branching fraction of the diquark mediator, and
$\Gamma_{\text{M}}=\Gamma\left(\text{M}\to\bar{u}\bar{d}\right)+\Gamma\left(\text{M}\to\text{DM}\text{X}\right)$
from equation 2 is the total mediator width.

Figure 3: Parameter space consistent with the measured relic density within
$3\sigma$ from Planck Ade:2015xua , for fixed $B_{0}=0.1$ and $\Delta=0.125$
in the $m_{\text{DM}}$ versus $m_{\text{M}}$ plane, with different overlays of
maximal $\Gamma_{\text{M}}/m_{\text{M}}$.

In figure 3, we scan the $m_{\text{DM}}$ vs. $m_{\text{M}}$ plane for
$B_{0}=0.1$ and $\Delta=0.125$, where $B_{0}=0.1$ is equivalent to requiring
$y_{D}^{2}=18y_{ud}^{2}$. We overlay regions with
$\Gamma_{\text{M}}/m_{\text{M}}\leq 10^{-1}$, $10^{-2}$, $10^{-3}$, and
$10^{-4}$, which effectively scan over $y_{ud}$ and $y_{D}$ given the
proportionality above. We see that heavy DM masses, which generally overclose
the universe as depicted from the asymptotic behavior in figure 2, still
recover the correct $\Omega h^{2}$ if the DM X $\to$ SM1 SM2, DM X $\to$ M
$V_{\text{SM}}$, and $\overline{\text{DM}}$ DM $\to$ M∗ M channels are large
enough. If the width of the mediator is small, however, the DM X $\to$ M
$V_{\text{SM}}$ and $\overline{\text{DM}}$ DM $\to$ M∗ M channels become
negligible. Then, the correct relic density is only found in the resonant
coannihilation region, or when the everpresent $\overline{\text{X}}$ X $\to$
$\overline{\text{SM}}$ SM channel together with the given $\Delta$ produce the
appropriate effective DM annihilation cross section to fall into the measured
band of $\Omega h^{2}$. The upper and lower lobes for
$\Gamma_{\text{M}}/m_{\text{M}}\leq 10^{-3}$ show the separate impact of the
$\overline{\text{DM}}$ DM $\to$ M∗ M and DM X $\to$ M $V_{\text{SM}}$
channels, respectively, avoiding overclosure of the universe by supplementing
the $\overline{\text{X}}$ X $\to$ $\overline{\text{SM}}$ SM channel. Because
our model always includes the $\overline{\text{X}}$ X $\to$
$\overline{\text{SM}}$ SM channel, and because we always assume a sufficient
$y_{D}$ coupling to keep DM and X in chemical equilibrium Baker:2015qna , we
underclose the universe for $\mathcal{O}(100\text{ GeV})$ DM masses for
$\Delta=0.125$, regardless of $B_{0}$. For these light DM masses, the SM gauge
interactions ensure that the $\overline{\text{X}}$ X $\to$
$\overline{\text{SM}}$ SM process rapidly depletes the DM relic density. To
have a viable DM model in this underclosure region, we would either have to
increase $\Delta$, which expressly suppresses the $\overline{\text{X}}$ X
$\to$ $\overline{\text{SM}}$ SM contribution to the effective DM annihilation
cross section, or we would require additional field content outside of our
simplified model.

Figure 4: Parameter space consistent with the measured relic density within
$3\sigma$ from Planck Ade:2015xua , for fixed $B_{0}=0.1$ and scanning over
$\Gamma_{\text{M}}/m_{\text{M}}\leq 10^{-4}$, with different choices of
$\Delta$ overlaid.

This interesting dependence on $\Delta$ is shown in figure 4, where we have
chosen $B_{0}=0.1$ and scanned the Yukawa couplings such that
$\Gamma_{\text{M}}/m_{\text{M}}\leq 10^{-4}$. Most strikingly, we see that the
choice of $\Delta$ sensitively changes the boundary between underclosure and
overclosure delineated by the $\overline{\text{X}}$ X $\to$
$\overline{\text{SM}}$ SM process, shown in the horizontal shaded bands. The
diagonal bands aligned with $m_{\text{DM}}+m_{\text{X}}\sim m_{\text{M}}$
correspond to the resonant coannihilation process. We see that the horizontal
bands become wider for larger $m_{\text{DM}}$, and this growth is larger on
the left side of the coannihilation funnel compared to the right side. The
overall width of the bands is determined by the relative flattening of the
relic density curve as $m_{\text{DM}}$ increases, as evident in figure 2. On
the left side of the coannihilation funnel, however, the opening of the DM X
$\to$ M $V_{\mathrm{SM}}$ and $\overline{\text{DM}}$ DM $\to$ M∗ M channels
enhance the effective annihilation cross section and avoid DM overclosure of
the universe.

We see from figure 4 that the choice of $\Delta$ effectively sets the dark
matter mass scale at which the correct relic abundance is achieved. On the
other hand, $m_{\text{DM}}$ and $\Delta$ directly set the energy scale of the
X decay products in the X $\to\bar{u}\bar{d}$ DM decay and hence control the
complementarity between searches for X pair production and probes for the
diquark mediator. Intriguingly, the combination of $m_{\text{DM}}$ from $300$
GeV to $500$ GeV and $\Delta\sim 0.1$ results in a spectrum of X decay
products that straddles the jet thresholds used in current multijets and MET
searches Aad:2014wea . Therefore, we present a detailed discussion of the
current collider bounds for signatures of both X and M in section 4.

Finally, we briefly comment on generalizing these conclusions to the entire
set of codex models with a strongly interacting coannihilation partner X and a
SM gauge singlet DM. Given X and DM are in chemical equilibrium, the
$\overline{\text{X}}$ X $\to$ $\overline{\text{SM}}$ SM annihilation channel
will always dominate for low DM masses and hence connect the choice of
$\Delta$ to a particular relic density motivated energy scale for
$m_{\mathrm{DM}}$, leading to the behavior seen in figure 4. In general,
larger QCD representations for X will increase the efficacy of this
coannihilation channel, driving the relic density motivated scale of
$m_{\mathrm{DM}}$ even lower for a given $\Delta$. Moreover, the different X
and DM representations will change the relative importance of the non-
perturbative Sommerfeld effect deSimone:2014pda ; Ibarra:2015nca . A full
quantitative description of relic density constraints for dark matter models
with a strongly interacting coannihilation partner will be the subject of a
future work.

###  3.2 Direct and indirect detection prospects

We now address the direct detection prospects for the ST6 $s$-channel diquark
mediator DM model. For $\Delta\sim 0.1$ and X and DM masses in the few hundred
GeV range, the coannihilation partner X decays promptly and will have
vanishing relic density today. Since DM has no SM gauge quantum numbers, all
of the DM direct detection interactions are loop-induced and scale at least
with the new physics coupling $y_{D}$. These loop-induced diagrams, though,
are not inherently responsible for determining the dark matter relic density
and thus the relic density requirement will not have a meaningful bearing on
the direct detection prospects.

Since the momentum transfer in DM–nucleon scattering is
$\mathcal{O}(100~{}\mathrm{MeV})$, and our DM mass scale is at least 100 GeV,
we can use an effective field theory description based on the Lagrangian in
equation 2.1 and integrating out M, X, and the high momentum modes of DM to
analyze the leading DM direct detection prospects. We find three loop-induced
operators that can drive DM direct detection:

$\overline{\text{DM}}$ DM $H^{\dagger}H$:

This is the leading dimension-five operator, which is loop-induced by X and M
internal legs and requires a nonzero Higgs portal coupling
$\lambda_{\text{HM}}$. This operator scales as
$(y_{D}^{2}\lambda_{\text{HM}})/(16\pi^{2}m_{\text{M}})$ and effects DM direct
detection once the Higgs acquires a vev. Since the quartic coupling
$\lambda_{\text{HM}}$ is arbitrary, however, the impact of this operator for
direct detection can be completely negligible.

$\overline{\text{DM}}$ DM $\bar{u}u$, $\overline{\text{DM}}$ DM $\bar{d}d$:

These dimension-six operators arise from insertions of X, M and quark fields
and can be written schematically as
$\left[\overline{\text{DM}}\,\Gamma^{\text{DM}}\,\text{DM}\right]\times\left[\bar{q}\,\Gamma^{q}\,q\right]$,
where $\Gamma^{\text{DM}}$ and $\Gamma^{q}$ enumerate the possible Lorentz-
invariant contractions (see, e.g. references Goodman:2010yf ; Goodman:2010ku
). Because the DM is a Majorana particle, however, the pure vector
$\Gamma^{\text{DM}}=\gamma^{\mu}$ and tensor
$\Gamma^{\text{DM}}=\sigma^{\mu\nu}$ contractions vanish. The axial vector
current $\Gamma^{\text{DM}}=\gamma^{\mu}\gamma^{5}$ is also absent, because
the hypercharge couplings of X and M induce purely vector $Z$ and photon
currents after electroweak symmetry breaking. The remaining possibilities are
scalar bilinears, which require a chiral flip in the quark sector and are thus
suppressed by $m_{q}/m_{\text{M}}^{3}$, which is too small to be constrained
by direct detection.

$\overline{\text{DM}}$ DM $G_{\mu\nu}^{a}G^{a,\mu\nu}$:

This dimension-seven operator is induced by X and M internal legs. After
integrating out the internal legs, this operator scales with
$(\alpha_{s}y_{D}^{2})/(4\pi m_{\text{M}}^{3})$, and the strong suppression by
the loop factor and three powers of $m_{M}$ implies we can safely ignore its
contribution to the DM direct detection cross section. Similarly, the
dimension-seven operator with $B_{\mu\nu}B^{\mu\nu}$ instead of
$G_{\mu\nu}^{a}G^{a,\mu\nu}$ is further suppressed by
$\alpha^{\prime}/\alpha_{s}$.

In summary, the direct detection interactions in our model are all driven by
higher dimensional operators, and if $\lambda_{\text{HM}}$ is small, none of
these operators are expected to give a meaningful direct detection DM–nucleon
cross section.

Regarding indirect detection signals, we can have loop-induced pair
annihilation of DM using the operators listed above. The first operator
results in final state Higgs bosons or any pair of SM gauge bosons or matter
particles, while the second and third operators give final state quarks and
gluons, and the aforementioned $\overline{\text{DM}}$ DM
$B_{\mu\nu}B^{\mu\nu}$ dimension-seven operator gives photons and $Z$ bosons.
If $m_{\mathrm{DM}}<m_{\mathrm{M}}$, then DM can annihilate to
$ud\bar{u}\bar{d}$ via one or two off-shell diquark mediators, but this
process is phase-space suppressed compared to the dimension-six scalar
bilinear operator. The operator scaling behavior from the direct detection
discussion also applies for the indirect detection signals, given the lack of
resonant enhancement in these processes. If $m_{\mathrm{DM}}>m_{\mathrm{M}}$,
we have the pair annihilation of DM into M∗ M, which then decay to quarks.
Because DM is a Majorana fermion, however, this annihilation channel is
$p$-wave suppressed. Therefore, the indirect detection signals from this model
are dim.

##  4 Existing searches

In this section, we discuss the existing collider constraints on our triplet
diquark model. The visible mediator decays are constrained by searches for
single and pair-produced dijet resonances, while the coannihilation partner X
and the DM are probed via monojet searches as well as multijets + MET
searches. These collider signatures arise from stitching together gauge
interactions of M and X with the interaction vertices inherent in the
coannihilation diagram in figure 1.

Following the discussion in section 2.4, we recast the searches for single
dijet resonances, pair-produced dijet resonances, monojets, and MSSM gluinos
to constrain the masses and couplings of our diquark mediator, coannihilation
partner, and the DM. A striking signature absent in current searches by ATLAS
and CMS, however, is the promising mixed decay channel from pair-produced
mediators, which we present in section 5.

###  4.1 Single and paired dijet resonance searches

Dijet resonance searches set bounds on the mediator mass and couplings in this
model. Our relic density results in figures 3 and 4 point to mediator masses
of 200 GeV to 1200 GeV as especially attractive to target via dijet resonance
searches. Furthermore, since our diquark mediator is a color triplet, single
dijet searches and paired dijet searches probe complementary sets of new
physics couplings.

Many dijet resonance searches constrain the $pp\to\mathrm{M}\to\bar{u}\bar{d}$
channel (for an overview, see reference Dobrescu:2013coa , and see reference
Chala:2015ama  regarding the DM context). The latest $8$ TeV search from CMS
Khachatryan:2016ecr  uses data scouting to constrain dijet resonances as light
as $500$ GeV, and the $8$ TeV ATLAS search Aad:2014aqa  uses pre-scaled jet
triggers to obtain limits on resonances as light as $250$ GeV. The $ud$
diquark nature of our mediator dictates stronger single resonance constraints
from $pp$ colliders instead of $p\bar{p}$ colliders like
Sp$\overline{\mathrm{p}}$S and Tevatron, hence we do not present the UA2
Alitti:1993pn , CDF Run 1 Abe:1997hm , or CDF Run 2 Aaltonen:2008dn
constraints.

In the CMS scouting analysis, events are required to have $H_{T}>250$ GeV,
where $H_{T}$ is the scalar $p_{T}$ sum of jets with $p_{T}>30$ GeV and
$|\eta|<2.5$ and jets are clustered using the anti-$k_{T}$ algorithm with
distance parameter $R=0.5$. The leading and subleading jets in $p_{T}$ are
used as seeds to form wide jets, which add together jets closer than $R=1.1$
to the nearer seed jet. The wide jets must have $\Delta\eta<1.3$ to reduce QCD
background and an invariant mass larger than 390 GeV to ensure a smoothly
falling spectrum. The ATLAS search clusters jets with the anti-$k_{T}$
algorithm and $R=0.6$, and events must have two jets with $p_{T}>50$ GeV and
rapidity $|y|<2.8$. These jets must have a rapidity separation $\Delta y<1.2$
and an invariant mass $m_{jj}>250$ GeV.

Given a narrow mediator M, with $\Gamma_{\text{M}}/m_{\text{M}}\lesssim
10^{-1}$, the single dijet production rate scales as
$\sigma(y_{0},m_{\text{M}})(y_{ud}^{2}/y_{0}^{2})B$, where
$\sigma(y_{0},m_{\text{M}})$ is the cross section calculated using a reference
visible coupling $y_{0}$ and mediator mass $m_{\text{M}}$ and $B$ is the
visible branching fraction as defined in equation 4. For diquarks decaying
purely to dijets, $B=100\%$ and the bound on the coupling is extracted by
equating
$\sigma(y_{0},m_{\text{M}})(y_{ud,\text{excl}}^{2}/y_{0}^{2})=\sigma_{\text{excl}}$
for an excluded cross section $\sigma_{\text{excl}}$ from experiment. When the
X DM decay is kinematically accessible, though, it is more convenient to
constrain the visible branching fraction $B$ for fixed choices of the total
width $\Gamma_{\text{M}}$, since these parameters encode the entire dependence
of the signal dijet rate. The excluded visible branching fraction is
determined by equating
$\sigma(y_{0},m_{\text{M}})(y_{ud}^{2}/y_{0}^{2})B_{\text{excl}}=\sigma(y_{0},m_{\text{M}})B_{\text{excl}}^{2}\Gamma_{\text{M}}/\Gamma_{0}=\sigma_{\text{excl}}$,
where $\Gamma_{0}=y_{0}^{2}\text{M}/(4\pi)$ is the visible width determined by
the reference coupling $y_{0}$. These exclusions from CMS Khachatryan:2016ecr
and ATLAS Aad:2014aqa  are shown in figure 5 for
$\Gamma_{\text{M}}/m_{\text{M}}=3\times 10^{-4}$, $5\times 10^{-4}$, and
$10^{-3}$. We see that larger total mediator widths lead to stronger
exclusions, since the overall dijet resonance rate increases with the total
mediator width. Nevertheless, single dijet constraints are not sensitive to
total widths smaller than $\Gamma_{\text{M}}/m_{\text{M}}=10^{-4}$, hence we
use $\Gamma_{\text{M}}/m_{\text{M}}=10^{-4}$ as an upper limit on the total
mediator width in our calculations in section 5.

Pair-produced diquarks are dominantly produced via color interactions, and the
corresponding paired dijet signal scales with $B^{2}$ if the dark decay M
$\to$ X DM is kinematically open. For
$m_{\text{M}}<m_{\text{DM}}+m_{\text{X}}$, though, the dark decay is closed
and paired dijet searches probe mediator masses independently of the visible
coupling $y_{ud}$. The single and paired dijet searches are otherwise
complementary, given that the single dijet mass reach is typically higher than
the paired dijet mass reach.

There are paired dijet searches by CMS using $7$ TeV Chatrchyan:2013izb  and
$8$ TeV Khachatryan:2014lpa  data. Each dijet pair must fulfill requirements
on $\Delta R_{jj}$ and then the combination of the two pairs that minimizes
$\Delta m/m_{\mathrm{avg}}$ is selected. To suppress the QCD continuum
background, a further cut on the $p_{T}$ imbalance versus the average mass of
the dijet pairs is imposed, ensuring a smoothly falling distribution in the
average mass. This distribution is used to obtain cross section limits on a
pair-produced dijet resonance with masses between $200$ GeV and $1000$ GeV for
the $8$ TeV search, which supersedes the old result. There is also an ATLAS
$7$ TeV analysis ATLAS:2012ds , but this covers the mass window ranging from
$150$ GeV to $350$ GeV and is less constraining than the $8$ TeV search by
CMS.

Figure 5: The existing limits from searches for paired dijet resonances are
shown, where the CMS search (blue line) Khachatryan:2014lpa  is most
constraining. The orange line shows the projected exclusion by the same search
at $13$ TeV and $100$ fb-1 of integrated luminosity. We also show the envelope
of dijet resonance searches by CMS (red) Khachatryan:2016ecr  and ATLAS
(purple) Aad:2014aqa  as exclusion contours for different choices of
$\Gamma_{\mathrm{M}}/m_{\mathrm{M}}=3\times 10^{-4}$ (dotted), $5\times
10^{-4}$ (dashed), and $10^{-3}$ (solid). Only the most constraining dijet
resonance search is displayed for a given mediator mass.

Hence, we use the most constraining $8$ TeV CMS analysis Khachatryan:2014lpa ,
which gives the cross section limit as $\sigma\times B^{2}$. We recast their
limit using NLL-fast v2.1 Borschensky:2016xy ; Beenakker:2009ha ;
Kulesza:2009kq ; Kulesza:2008jb  in combination with CTEQ v6.6 Nadolsky:2008zw
parton distribution functions (PDFs) to calculate $m_{\mathrm{M}}$ dependent
next-to-leading order (NLO) cross sections at $8$ TeV LHC. The resulting
limits are shown in figure 5 as the shaded blue region, where we have
delineated the disjoint low mass and high mass search regions. We see that
even low mass mediators are unconstrained by the data if the visible branching
fraction is less than 40%.

Using Collider Reach Salam:2014xy , we estimate the 13 TeV sensitivity for
pair produced dijet resonance searches, assuming an integrated luminosity of
$100$ fb-1. A similar projection for the dijet resonance limits is not given
because these would only alter the maximum value for
$\Gamma_{\mathrm{M}}/m_{\mathrm{M}}$, and because the mass sensitivity using
data scouting techniques by CMS or prescaled jet triggers by ATLAS has not
been demonstrated at 13 TeV. Both the low mass and high mass paired dijet
search regions are scaled upward, although this is an oversimplification of
the potential search strategy at 13 TeV LHC. We expect these rough limits are
nonetheless indicative of the possible sensitivity, given that Collider Reach
simply uses PDF scalings to mimic the production of on-shell resonances when
the background and signal are generated by the same partonic scattering
channel. The projection is shown as an orange contour in figure 5, where
again, visible branching fractions below 40% are not expected to be
constrained.

###  4.2 Jets + MET searches

In contrast to the resonance signatures from our diquark mediator, the
coannihilation partner X decays to soft jets and missing transverse energy
with a 100% branching fraction. Pair production of X can then be constrained
by monojet searches as well as multijet + MET searches. The resolved nature of
the soft jets depends on the fractional mass splitting $\Delta$ and the X
mass, which influences the tradeoff in signal sensitivity between the monojet
and multijet + MET searches. Jets from initial state radiation (ISR), however,
skew the search sensitivity to the multijet + MET searches, especially for
light X masses.

Since X is a color triplet fermion, we can calculate the pair production cross
section for X using Top++ v2.0 Czakon:2011xx  interfaced with LHAPDF v6.1
Buckley:2014ana  and the NNPDF v3.0 Ball:2014uwa  set to obtain NLO signal
cross sections for arbitrary X masses. Using CheckMate v1.2 Drees:2013wra , we
find the 8 TeV jets + $\not{E}_{T}$ search by ATLAS Aad:2014wea  imposes the
strongest constraints on the X mass for various choices of $\Delta$. This
search considers a wide range of signal regions characterized by the number of
hard jets observed in the event and the value of the effective mass
$m_{\mathrm{eff}}$, defined as the scalar $p_{T}$ sum of all jets with
$p_{T}>40$ GeV plus the $\not{E}_{T}$. The most powerful region for our signal
is the “2-jet medium” region, that, in particular, requires
$m_{\mathrm{eff}}>1.2$ TeV in addition of two hard jets. Since $\Delta$ is
typically small, the two hard jets observed for our signature come from ISR.
The relatively relaxed event selection implies this search region is
systematics limited, where ATLAS quotes a $6.6\%$ systematic uncertainty for a
background expectation of $760$ events. The related “2-jet tight” and “4-jet
loose” regions, with $m_{\text{eff}}$ cuts of $1.6$ and $1.0$ TeV,
respectively, are also sensitive to our final state but are less powerful than
the “2-jet medium” region and have larger systematic uncertainties of about
8%. We remark that although the strongest bounds for this search are obtained
by tagging on ISR jets, the traditional ATLAS monojet search Aad:2015zva
leads to much weaker bounds for our choices of $\Delta$. On the other hand, we
expect that dedicated analyses beyond the monojet and multijet + MET searches
can improve the sensitivity to compressed spectra Nath:2016kfp ;
Delgado:2016gqn .

We show the 95% confidence level (C.L.) exclusion limits on the X mass for
$\Delta=0.1$, $0.125$, and $0.15$ in table 2. Note we neglect the impact of
the mediator pair production here, which can give the same final state via the
M∗ M $\to 2(jj)_{\mathrm{soft}}\not{E}_{T}$ decay chain. Since this decay is
only kinematically open once $m_{\mathrm{M}}\geq
m_{\mathrm{X}}+m_{\mathrm{DM}}$, this process pays the suppression factors of
a much smaller pair production cross section of M and the dark decay branching
fraction $(1-B)^{2}$ compared to the X pair production rate. Our results rule
out X masses below about 390 GeV for the $\Delta\sim 0.1$ values motivated by
figure 4.

---
Search  | $\Delta=0.1$ | $\Delta=0.125$ |  $\Delta=0.15$
$8$ TeV  |  $384$ GeV |  $396$ GeV |  $392$ GeV
$13$ TeV  |  $398$ GeV |  $399$ GeV |  $396$ GeV
$13$ TeV, $100$ fb-1 projection  |  $464$ GeV |  $468$ GeV |  $477$ GeV
Table 2: The 95% C.L. exclusion limits on $m_{\mathrm{X}}$ from $8$ TeV and
$13$ TeV jets + MET searches by ATLAS Aad:2014wea ; Atlas:2016rxq  at the LHC,
as well the projected exclusion reach for $100$ fb-1 of $13$ TeV luminosity.

The ATLAS and CMS collaborations have both updated their jets + MET searches
at $13$ TeV Atlas:2016rxq ; Khachatryan:2016kdk , which are not included in
the CheckMate catalogue yet. Given the slightly larger luminosity of ATLAS, we
recast the ATLAS search only and expect the CMS bounds to be similar. Using
the Monte Carlo pipeline described in the end of section 2.4, we cross check
our MLM matched sample of signal events with the cut flow efficiencies of the
signal regions described in reference Atlas:2016rxq . We use our derived
signal acceptance fractions to compute 95% C.L. exclusions on
$m_{\mathrm{X}}$. Our model is most constrained by the “2-jet loose” and
“2-jet medium” signal regions of reference Atlas:2016rxq , which require at
least two hard jets + MET as well as a loose or medium cut on the effective
mass. Our limits on $m_{\mathrm{X}}$ are shown in the middle row of table 2,
showing a minor improvement over the 8 TeV results.

In order to compare with the projected sensitivity of our mixed mediator
search described in section 5, we extrapolate the jets + MET search to $100$
fb-1 of integrated luminosity. Assuming the current systematic uncertainties
are unchanged, we rescale the background and signal events by the luminosity
ratio for the 2-jet loose signal region to obtain a projected 95% C.L.
exclusion on the X mass. Since the 2-jet loose and 2-jet medium signal regions
are already dominated by systematic uncertainties, the increase in integrated
luminosity does not significantly improve the bounds on the visible cross
section and thereby on $m_{\mathrm{X}}$. Our results are in the last row of
table 2, which show that X masses of about 470 GeV are expected to be probed
in future multijet + MET searches.

###  4.3 Summary of conventional searches

To summarize, we have studied the existing constraints and future sensitivity
from the conventional LHC searches. We first discussed the dijet resonance
searches targeting single and pair production of the mediator, where the
bounds are collected in figure 5. The single production bounds constrain
$B\lesssim 35\%$ for $\Gamma_{\text{M}}/m_{\text{M}}=10^{-3}$, although this
bound is completely absent once $\Gamma_{\text{M}}/m_{\text{M}}=10^{-4}$.
Current pair production limits constrain mediators below 360 GeV for $B$
varying between about 45% and 100%, though this bound is strongest only for
mediators near 200 GeV. The projected reach of the paired dijet search with
100 fb-1 of 13 TeV data covers mediator masses between 360 GeV to 720 GeV, but
this relies on a simple extrapolation of the low mass and high mass reach from
8 TeV. We conclude that mediators heavier than 360 GeV with
$\Gamma_{\text{M}}/m_{\text{M}}\leq 10^{-4}$ are unconstrained by the current
single and paired dijet searches, which leaves a large, open parameter space
to explore with future dijet searches and our proposed mixed decay search
described in section 5.

We then discussed the multijet + MET searches, which target X pair production
and the X $\to(jj)_{\text{soft}}\not{E}_{T}$ decay. Our resulting limits on
the X mass, shown in table 2, have a mild dependence on $\Delta$, since
$\Delta$ sets the hardness of the jets from the X decay. The jets + MET
searches, however, are already dominated by systematic uncertainties, and so
they are not expected to improve dramatically with 100 fb-1 of 13 TeV data.
Our recasting and projection of the ATLAS 13 TeV search Atlas:2016rxq  shows
an improvement on the $m_{\text{X}}$ limit from 400 GeV to 460 GeV. The
results should be juxtaposed with figure 3, where the $\Delta=0.125$ to
$\Delta=0.10$ relic density regions straddle the current and expected
exclusion limits from the jets + MET search. This coincidence in the relic
density motivated region indicates that the LHC has excellent prospects for
discovering a colored coannihilation mechanism akin to our $s$-channel diquark
model, and it also emphasizes the importance of the overlooked dijet resonance
+ MET channel, which we discuss next.

##  5 Mixed decay phenomenology

As highlighted in section 2.4, pair production of our diquark mediator can
lead to the unique signature of a dijet resonance accompanied by missing
transverse energy. The resonance + MET final state was also previously
emphasized as a general prediction of coannihilation models with an
$s$-channel mediator Baker:2015qna . The resonance + MET signature is also
particularly motivated because this channel directly tests the coannihilation
diagram from figure 1. Since the rate for this signature is sensitive to the
ratio of the dark and visible new physics couplings via the mediator branching
fractions, combining results from the mixed and the dijet resonance searches
would be central to determining the underlying dark matter Lagrangian.
Moreover, the dijet resonance + MET signature together with the relic density
calculations presented in section 3 and the single dijet, paired dijets, and
jets + MET signatures discussed in section 4 unify into a compelling picture
of a bottom-up discovery of dark matter at the LHC. This synthesis will be
presented in section 5.4.

The mixed decay of pair-produced mediators leads to a final state of
$(jj)_{\text{res}}+(jj)_{\text{soft}}+\not{E}_{T}$, as shown in figure 6. The
typical momenta of the soft jets is determined by the mass splitting between X
and DM. As described in section 3, and in most of the parameter space favored
by relic density requirements, these momenta are close to the various jet
$p_{T}$ thresholds used by ATLAS and CMS. The primary goal of our analysis is
to identify the dijet resonance and large MET signature, while the possible
discrimination of additional soft signal jets from X decay will be a minor
secondary consideration.

$g$$g$M$\text{M}^{\ast}$$u$$d$DMXDM$\bar{u}$$\bar{d}$$\left.\begin{array}[]{c}{}\hfil\\\
{}\hfil\\\ {}\hfil\\\ {}\hfil\\\ \end{array}\right\\}$soft jets +
MET$\Bigg{\\}}$dijet resonance Figure 6: An example process for the mixed
decay signature from pair produced mediators. One mediator decays visibly to a
dijet resonance, and the other decays to DM and X, giving soft jets and
missing transverse energy.

###  5.1 Event generation

Signal events are generated using the procedure and tools described in section
2. Following the relic density favored regions in figure 4, we focus on
$\Delta=0.1$, $0.125$, and $0.15$. Given the jets + MET recasting performed in
section 4.2, with results summarized in table 2, we know that X is already
excluded to about 400 GeV for these choices of $\Delta$, and hence we focus
our study on mediators in the several hundred GeV to TeV mass range.

Our new physics signal is characterized by a strong dijet resonance recoiling
against large MET. While the intrinsic width of our mediator is constrained to
be small by dijet resonance searches, as discussed in section 4.1, we expect
the resolution on our narrow resonance to be dominated by jet energy scale
uncertainty as well as jet clustering contamination from ISR, final state
radiation (FSR), and the X decay products. For our mediator mass range and
choices of $\Delta$, we find these effects are not significant enough to
warrant a complex jet substructure analysis to sharpen the dijet peak
structure, and we instead ameliorate these effects by optimizing the jet
clustering algorithm and clustering radius. Our events are clustered using the
anti-$k_{T}$ algorithm Cacciari:2008gp  with $R=0.5$.

Given the final state of a dijet resonance and MET, we study the following
backgrounds:

  * •

$(Z\rightarrow\nu\nu)+1,2~{}\mathrm{jets}$: this background is similar to the
final state that we are considering but will typically have a lower
$\not{E}_{T}$ and no resonant structure.

  * •

$(W^{\pm}\rightarrow\ell^{\pm}\nu)+1,2~{}\mathrm{jets}$: this background can
be significantly reduced by a lepton veto. If the lepton is not reconstructed,
though, the associated signature is similar to the
$(Z\rightarrow\nu\nu)+\mathrm{jets}$ background.

  * •

QCD multijets: while this background has no intrinsic MET, QCD can easily
produce fake MET if a jet is badly reconstructed or mismeasured. This
$\not{E}_{T}$ distribution follows a different slope than the vector boson +
jets backgrounds above and should lead to a smooth continuum in the dijet
invariant mass spectrum.

We do not simulate other backgrounds, which are expected to be subdominant to
those listed above. In particular, $t\bar{t}+\text{jets}$ will be readily
reduced by vetoing heavy flavor jets, while the remaining semi-leptonic and
fully leptonic $t\bar{t}$ events will only contribute if the leptons are lost,
which renders it subdominant to the $W^{\pm}+\text{jets}$ background.
Similarly, diboson production where one boson decays invisibly or leptonically
and the other hadronically will be a small correction to the $Z+\text{jets}$
or $W^{\pm}+\text{jets}$ backgrounds, respectively. To properly model the
associated jet distributions for each background, we use MLM matching
Mangano:2002ea  to avoid double-counting jets from the hard matrix element and
jets from the parton shower. The $Z$ and $W^{\pm}$ backgrounds are matched up
to two jets and the QCD multijet background is matched up to three jets, each
using a matching scale of $20$ GeV.

In order for the Monte Carlo generation to efficiently populate the tails of
the background distributions, we sample the phase space following the
procedure described in reference Avetisyan:2013onh . We modify MadGraph to
implement an $S_{T}^{*}$ cut at generator level, where $S_{T}^{*}$ is the
scalar $p_{T}$ sum of all generator level particles. We partition the MC
generation in $S_{T}^{*}$ bins, which follow

$\sigma_{i}=\sigma(\mathtt{stmin}_{i}<S_{T}^{*}<\mathtt{stmax}_{i})\gtrsim
0.9\times\sigma(\mathtt{stmin}_{i}<S_{T}^{*})\ ,$ (5)

where $\sigma_{i}$ is the cross section in the $i$-th bin and
$\mathtt{stmax}_{i}$, $\mathtt{stmin}_{i}$ are the bin edges. The final
overflow bin must satisfy $N>10\times\sigma_{\mathrm{overflow}}\times L$,
where $N$ is the total number of events to be generated in the bin and $L$ is
the desired integrated luminosity.

We use NLL-fast v3.1 Borschensky:2016xy ; Beenakker:2009ha ; Kulesza:2009kq ;
Kulesza:2008jb ; Beenakker:2015rna  in combination with NNPDF v3.0
Ball:2014uwa  to calculate $m_{\mathrm{M}}$ dependent NLO-cross sections for
pair-production of the mediator at $13$ TeV. For our mediator mass range, the
$K$-factor is relatively flat, varying between $1.58$ and $1.65$ for
$m_{\text{M}}$ from $500$ to $1500$ GeV. The background $K$-factors are
obtained from MCFM v6.8 Campbell:2010ff  for processes involving electroweak
gauge bosons and from references Ellis:1992en ; Giele:1994gf  for QCD
multijets. The background $K$-factors are found to be $1.15$ for the vector
boson backgrounds and $1.3$ for QCD multijets.

We apply mild preselection cuts on both the signal and backgrounds, requiring
events to have at least two jets, $\not{E}_{T}>100$ GeV and the leading jet
$p_{T}$ greater than $80$ GeV. These cuts only increase the efficiency of our
Monte Carlo event generation and will be superseded by the analysis level
cuts. Signal samples are generated with $m_{\text{M}}$ starting at $500$ GeV,
which is the lowest mass our simulation can confidently probe a dijet
resonance, and $m_{\text{DM}}$ starting at $25$ GeV.222Pythia 8 is not able to
correctly process the X decay for the tiny mass splittings associated with
lighter DM masses, but in principle, our search strategy will still be
sensitive to this region of parameter space.

###  5.2 The mixed decay search strategy

As outlined in 5.1, our search strategy targets a signal exhibiting a large
amount of $\not{E}_{T}$, at least two hard jets with a resonant structure, and
possible additional resolved jets from X decays, ISR, and FSR. Our main
discriminants between signal and background will be MET and the resonant dijet
peak over a continuum dijet background. We will also employ angular cuts to
improve the overall signal to background ratio in the dijet invariant mass
spectrum.

Our cut flow table is shown in table 3. In the first row, we show the 13 TeV
cross sections for each process after applying preselection cuts and
$K$-factors. For comparison, we also show a signal benchmark point with
$(m_{\mathrm{DM}},m_{\mathrm{X}},m_{\mathrm{M}})=(379,417,900)$ GeV.
Sequential cuts and cumulative acceptance efficiencies are shown in subsequent
rows.

We first apply a lepton veto, which mainly reduces the leptonic $W^{\pm}+$
jets background by about $30$%. This low suppression rate is due to the tight
lepton isolation requirements of the CMS Delphes card (summed $p_{T}$ of
tracks within a cone of $R=0.5$ must be less than $10$% of the lepton
$p_{T}$). We expect a veto on leptons using medium or loose identification
criteria would improve the $W^{\pm}+$ jets rejection and negligibly impact our
signal acceptance. Next, since our diquark mediator is coupled only to first
generation quarks, we apply a $b$-jet veto, which is expected to significantly
suppress the subdominant $t\bar{t}$ background. We use the $b$-tagging
efficiencies quoted in reference ATL-PHYS-PUB-2015-022 , with a maximum
tagging efficiency of $85$% and a mistag rate of $0.2$%.

We require at least two hard jets, where hard jets have $p_{T}>100$ GeV and
$|\eta|<2.5$. This hard jet requirement suppresses the remaining vector boson
plus jets backgrounds by about $85$% and the QCD background by about $68$%
while retaining more than $95$% of the extant signal. We also cut on events
with $\not{E}_{T}$ arising primarily from jet mismeasurement by requiring
$\Delta\phi(j_{i},\not{E}_{T})>0.2$ for all hard jets $j_{i}$. As expected,
this cut significantly reduces QCD, by about a factor of four. We model jet
mismeasurements using the default parameters from the CMS card of
Delphes-3.2.0. Note that, as mentioned in Baker:2015qna , this modeling leads
to an overestimate of the tail of the QCD $\not{E}_{T}$ distribution. Our
results can therefore be considered conservative.

Finally, we require the rapidity difference between the two leading hard jets
to be less than $1.3$, which mimics the cut used in the dijet resonance
searches and reflects the kinematic configuration expected from figure 6.
After these general cuts, the cumulative acceptances for the different
backgrounds are of the order of $5\%$. We obtain our final sensitivity
estimates after simultaneously optimizing the $\not{E}_{T}$ and
$m_{j_{1}j_{2}}$ mass window cuts for each signal benchmark.

---
| signal |  $Z_{\nu\nu}$ \+ jets |  $W^{+}_{\ell\nu}$ \+ jets |  $W^{-}_{\ell\nu}$ \+ jets |  QCD
$\sigma_{\text{NLO}}$ (pb)  | $7.60\\!\times\\!10^{-3}$ | $93.21$ | $157.33$ | $75.48$ |  $6948.76$
$\ell$ veto  | $100$ | $100$ | $71.4$ | $69.0$ |  $100$
$b$-jet veto  | $86.9$ | $89.3$ | $66.8$ | $64.5$ |  $87.3$
$p_{T}(j_{1}),\ \\!p_{T}(j_{2})\\!>\\!100$ GeV  | $83.0$ | $11.8$ | $9.76$ | $9.68$ |  $27.6$
$\Delta\phi(j_{i},\not{E}_{T})\\!>\\!0.2$ | $75.0$ | $10.9$ | $8.50$ | $7.99$ |  $6.20$
$\left|\Delta\eta(j_{1},j_{2})\right|\\!<\\!1.3$ | $51.5$ | $6.36$ | $5.01$ | $4.82$ |  $3.61$
$\not{E}_{T}\\!>\\!900~{}\mathrm{GeV}$ | $6.29~{}\\!(48)$ | $3.51\\!\times\\!10^{-3}~{}\\!(327)$ | $7.74\\!\times\\!10^{-4}~{}\\!(122)$ | $2.83\\!\times\\!10^{-4}~{}\\!(21)$ |  $1.71\\!\times\\!10^{-5}~{}\\!(119)$
Mass window  | $3.56~{}\\!(27)$ | $3.35\\!\times\\!10^{-4}~{}\\!(31)$ | $8.70\\!\times\\!10^{-5}~{}\\!(14)$ | $1.93\\!\times\\!10^{-5}~{}\\!(2)$ |  $1.28\\!\times\\!10^{-6}~{}\\!(9)$
Table 3: Detailed cut flow for a signal benchmark with
$(m_{\mathrm{DM}},m_{\mathrm{X}},m_{\mathrm{M}})$ = (379, 417, 900) GeV and
the $Z+\text{jets}$, $W^{\pm}+\text{jets}$, and QCD multijet backgrounds. The
cross sections quoted include the preliminary cuts described in section 5.1
and $K$-factors for 13 TeV LHC. The $\not{E}_{T}$ and $m_{j_{1}j_{2}}$ cuts
are optimized for each signal point individually, and we quote both the
efficiency (in %) and the number of events at $100$ fb-1 luminosity for each
process. The dijet mass window considered for this benchmark point is
$\left[812,987\right]$ GeV.

In the left panel of figure 7, we show the stacked $\not{E}_{T}$ distribution
after cuts for the different backgrounds, and overlay our signal benchmark as
a black outline. Cutting on $\not{E}_{T}$ significantly removes the
backgrounds, where the rejection factors range from $10^{3}$ for $Z+$ jets to
$10^{5}$ for QCD, while the benchmark signal is reduced by less than an order
of magnitude, giving a signal to background ratio of $\mathcal{O}(10\%)$. The
optimal $\not{E}_{T}$ cut is correlated with the mediator mass. Furthermore,
for low DM masses, jets from the X decay might be boosted enough to pass the
detection threshold, thus the amount of $\not{E}_{T}$ in signal events is also
expected to mildly decrease with the DM mass. Note that, at this point of the
search, our analysis strategy is similar to the one used in the existing
multijet + MET searches, but the resonant structure of the mediator provides a
significant handle to dramatically improve the mediator mass reach.

Figure 7: Signal and background $\not{E}_{T}$ (left) and $m_{j_{1}j_{2}}$
(right) distributions for a signal benchmark model with
$(m_{\mathrm{DM}},m_{\mathrm{X}},m_{\mathrm{M}})$ = (379, 417, 900) GeV. The
signal distribution is shown in black outline. The stacked background
distributions are from top to bottom: $Z+$ jets (red), $W^{+}+$ jets (orange),
$W^{-}+$ jets (green) and QCD (blue). The vertical dashed line at $900$ GeV on
the left panel represents the optimal $\not{E}_{T}$ cut for this benchmark
point. The $m_{j_{1}j_{2}}$ distribution is shown for events that passed this
$\not{E}_{T}$ cut.

In the right panel of figure 7, we show the $m_{j_{1}j_{2}}$ distributions for
the signal and the backgrounds after a $\not{E}_{T}>900$ GeV cut. Since the
background jets do not come from resonances, the background dijet mass
spectrum falls smoothly. The signal, however, exhibits a wide peak around the
mass of the mediator particle and can be easily identified on top of the
backgrounds. To impose an appropriate mass window for the dijet mass spectrum,
we fit the signal $m_{j_{1}j_{2}}$ distribution after cuts by a Crystal Ball
function Gaiser:1982yw ,

$s(x)=N\times\begin{cases}\exp\left[-\frac{(x-\bar{x})^{2}}{2\sigma^{2}}\right]\text{
if }\frac{x-\bar{x}}{\sigma}>-\alpha\ ,\\\
\left(\frac{n}{|\alpha|}\right)^{n}\exp\left(-\frac{|\alpha|^{2}}{2}\right)\left(\frac{n}{|\alpha|}-|\alpha|-\frac{x-\bar{x}}{\sigma}\right)^{-n}\text{
if }\frac{x-\bar{x}}{\sigma}\leq-\alpha\ .\end{cases}$ (6)

Here $N$ is the normalization, $\bar{x}$ and $\sigma$ are Gaussian parameters,
and $\alpha$ and $n$ are power law parameters, all determined by the fit. We
integrate the number of events within $2\sigma$ of the Gaussian mean,
$\bar{x}$, for the signal and the background samples to calculate the
significance,

$\mathcal{S}=\frac{S}{\sqrt{\sum B_{i}+(\sum\mathrm{sys}_{i}\times
B_{i})^{2}}}\ ,$ (7)

where $\mathrm{sys}_{i}$ is the systematic uncertainty for a given background
$B_{i}$, and $S$ and $B_{i}$ are the number of signal and background events in
the mass window, respectively. Following reference Atlas:2016rxq , we take
$\mathrm{sys}_{i}=5\%$ for the $W$ and $Z$ backgrounds, while we use $20\%$
for the QCD multijet background, because of the lack of simulated events in
the tail of the $\not{E}_{T}$ distribution.

We optimize the $\not{E}_{T}$ cut by maximizing the dijet resonance
significance $\mathcal{S}$. The last two rows of table 3 show the efficiencies
as well as expected number of events for 100 fb-1 of 13 TeV LHC data given the
benchmark-specific $\not{E}_{T}>900$ GeV cut and the dijet mass window cut,
$812\text{ GeV }\leq m_{j_{1}j_{2}}\leq 987\text{ GeV}$.

In order to estimate robustly the number of background events in the mass
window after the $\not{E}_{T}$ cut without being limited by Monte Carlo
statistics, we fit each background $m_{j_{1}j_{2}}$ distribution by a falling
function of the form Aad:2014aqa

$b(x)=p_{0}(1-x)^{p_{1}}x^{p_{2}+p_{3}\ln x}\ ,$ (8)

where $x=m_{j_{1}j_{2}}/\sqrt{s}$. For each background, we then integrate
$b(x)$ with the optimal $p_{i}$ parameters over the chosen mass window to
obtain the number of background events after the $m_{j_{1}j_{2}}$ cuts.

###  5.3 Sensitivity of the mixed decay search

The results of our collider study for $\Delta=0.125$ are shown in figure 8 for
$100$ fb-1 of 13 TeV luminosity. We illustrate the mixed sensitivity with
three contours corresponding to $B_{0}=0.1$, $B_{0}=0.5$, and $B=0.5$. Recall
that fixing $B_{0}$ is equivalent to fixing a particular coupling combination
of $y_{ud}$ and $y_{D}$ via equation 4, while the physical branching fraction
of the mediator to dijets depends on $B_{0}$ and the phase space of the M
$\to$ X DM decay. Alternately, fixing $B$ requires the couplings to change as
a function of the M and DM masses. In particular, as we move along the $B=0.5$
contour in figure 8, $B_{0}$ is monotonically decreasing to zero up to the
phase space boundary of $(2+\Delta)m_{\text{DM}}=m_{\text{M}}$.

Figure 8: Comparison in LHC sensitivity between the existing jets + MET
searches and the newly proposed mixed topology in the DM mass vs. mediator
mass plane with $\Delta=0.125$. The mixed search sensitivity is shown for
$B_{0}=0.1$, $B_{0}=0.5$, and
$B=\text{Br}\left(\text{M}\to\bar{u}\bar{d}\right)=0.5$ as an ideal scenario.
For given values of $m_{\mathrm{DM}}$ and $m_{\mathrm{M}}$, this ideal rate
can always be ensured by choosing $B_{0}$ accordingly.

As expected, by changing from $B_{0}=0.5$ to $B_{0}=0.1$, the expected reach
of the dijet resonance + MET search moves closer to the phase space boundary
of $(2+\Delta)m_{\text{DM}}=m_{\text{M}}$. The $B_{0}=0.5$ reach is coincident
with the $B=0.5$ reach for low DM masses since the phase space suppression of
$K(\Delta,\tau_{\text{DM}})$ is negligible. By construction, the intersection
point between the $B_{0}=0.1$ and $B=0.5$ contours indicates when the phase
space suppression factor maximizes the overall rate for the mixed decay
signature from M pair production, $\sigma(pp\to\text{M}^{*}\text{M})\times
2B(1-B)\xrightarrow[B=0.5]{}0.5\times\sigma(pp\to\text{M}^{*}\text{M})$. The
$B=0.5$ choice also fixes the paired dijet resonance rate to
$0.25\times\sigma(pp\to\text{M}^{*}\text{M})$ when the M $\to$ X DM decay is
on-shell. The $B_{0}=0.1$ and $B=0.5$ contours indicate that a significant
portion of parameter space in the $m_{\text{DM}},m_{\text{M}}$ plane will be
tested by the mixed decay search from section 5.2. This search is also
complementary to the current and future reach from the jets + MET searches for
X pair production, indicated by the shaded and horizontal lines at
$m_{\text{DM}}\sim 360$ GeV and $m_{\text{DM}}\sim 420$ GeV in figure 8,
respectively. These two searches test integral processes predicted by the
coannihilation mechanism, and the simultaneous observation of both channels
would be a striking sign of $s$-channel coannihilation at the LHC.

###  5.4 Connecting the relic density to collider searches

As emphasized in reference Baker:2015qna  and throughout this paper, our
simplified models based on modeling the DM annihilation mechanism allow us to
develop collider searches that target the parameter space favored by the
measured $\Omega h^{2}$. Moreover, for our model, direct detection and indrect
detection signals are suppressed compared to the collider probes, and thus the
relic density parameter space will mainly be confronted by the existing LHC
searches discussed in section 4 and our proposed mixed decay analysis from
section 5.2. This bottom-up approach directly connects the relic density
requirement to the LHC signature space, enforcing the role of LHC as a DM
producing machine.

As discussed in section 3, DM (co)annihilation to SM particles occurs through
numerous subprocesses with different dependencies in the new physics couplings
$y_{ud}$ and $y_{D}$. These couplings, though, are significantly constrained
by the dijet resonance searches described in section 4.1. As a result, in the
parameter space region where DM has the correct relic density and satisfies
the dijet constraints, the relic density will be driven by
$\overline{\text{X}}$ X annihilation via strong interactions or DM X resonant
coannihilation. This was illustrated in figure 4, where the chosen $B_{0}=0.1$
parameter and $\Gamma_{\text{M}}/m_{\text{M}}\leq 10^{-4}$ scan led to $\Omega
h^{2}$ favored regions for each value of $\Delta$. The relic density analysis
points to $\Delta\sim 10\%$ for DM and mediator masses within reach of the
LHC, although the exact preferred region depends sensitively on $\Delta$, as
shown in figure 4.

The prospects of different LHC searches as well as the region allowed by relic
density constraints are shown in figure 9 for $0.10\leq\Delta\leq 0.125$. For
this $\Delta$ range, the correct relic density is obtained for dark matter
masses between $300$ to $500$ GeV or in the funnel region. Following the
discussion in section 3.1, the relic density region constricts to a horizontal
band driven by $\overline{\text{X}}$ X $\to$ $\overline{\text{SM}}$ SM and a
coannihilation funnel parallel to the $(2+\Delta)m_{\text{DM}}=m_{\text{M}}$
line. Furthermore, $\Delta=0.125$ corresponds to the lower edge of the
horizontal band, while $\Delta=0.1$ corresponds to the upper edge.

While $m_{\text{DM}}\lesssim 360$ GeV is excluded by the recasted 13 TeV jets
+ MET search by ATLAS discussed in section 4.2, the projected limit of this
search with 100 fb-1 luminosity is only expected to improve to
$m_{\text{DM}}\sim 425$ GeV and will only cover part of the favored relic
density region. Similarly, the relic density region is excluded for
$m_{\text{M}}\lesssim 360$ GeV by the 8 TeV search for paired dijet
resonances, but this search weakens dramatically below the kinematic threshold
$(2+\Delta)m_{\text{DM}}=m_{\text{M}}$ because of the rapid turn-on of the M
$\to$ X DM decay for our choice of $B_{0}=0.1$. Nevertheless, the 100 fb-1
projection for paired dijet searches should reach mediator masses of about 720
GeV, giving an independent test of the relic density region above the
$(2+\Delta)m_{\text{DM}}=m_{\text{M}}$ line compared to the 100 fb-1 jets +
MET projection.

Moreover, for on-shell decays of the mediator to X DM, the new mediator mixed
decay search introduced in sections 5.2 and 5.3 probes an orthogonal decay
mode to the paired dijet search and also complements and enhances the jets +
MET reach. We remark that fixing $B=0.5$ gave a uniform reach for the mediator
mixed search at about $m_{\text{M}}=1$ TeV, as shown in figure 8, and so the
mixed decay search clearly has parameter space sensitivity beyond that
expected from the jets + MET projection. In the fortunate case of excesses in
both the mixed decay search and the jets + MET search, an immediate goal would
be to test whether the mediator is consistent with an $s$-channel
coannihilation mechanism. Multiple, well-motivated searches covering
complementary signatures of the same relic density region is a hallmark
feature of simplified models in our coannihilation framework, since these
signatures result from recycling vertices and gauge number assignments
inherent in the DM coannihilation diagram.

Figure 9: Summary plot for $0.1\leq\Delta\leq 0.125$ and $B_{0}=0.1$ in the
dark matter mass versus mediator mass plane. The relic density region is
determined by scanning over mediator couplings requiring
$\Gamma_{\text{M}}/m_{\text{M}}\leq 10^{-4}$ and matching the Planck
measurement of $\Omega h^{2}$, allowing a $3\sigma$ deviation from the central
value. The current 13 TeV, 3.2 fb-1 limit from recasting the ATLAS jets + MET
search Atlas:2016rxq  and the projection with 100 fb-1 luminosity are shown in
solid and dashed blue lines, adopting the strongest limit according to the
variation in $\Delta$. The current 8 TeV, 20 fb-1 and future 13 TeV, 100 fb-1
paired dijet limits are shown as mostly vertical orange contours. Our proposed
search for the novel signature of a dijet resonance + MET is shown for 100
fb-1 of 13 TeV data as a red contour with shading to indicate the dependence
on $\Delta$. The black diagonal lines span the kinematic threshold
$(2+\Delta)m_{\text{DM}}=m_{\text{M}}$ for on-shell decays of the mediator to
X DM.

##  6 Conclusions

The enigmatic dark matter question remains at the forefront of particle
physics research today. Discovering the particle properties of dark matter
would represent a pinnacle achievement in the fields of particle physics and
astrophysics. Hence it is crucially important to identify all viable dark
matter search strategies for present and future experiments.

Building on the general classification of simplified coannihilating dark
matter models Baker:2015qna , we studied the relic density predictions,
indirect and direct detection constraints, and collider phenomenology of a
dark matter particle coannihilating with a strongly interacting partner. We
adopted model ST6 from reference Baker:2015qna  as a case study, where the DM
is an electroweak singlet Majorana fermion, the coannihilation partner X is a
color triplet fermion, and the $s$-channel mediator M is a scalar color
triplet. Besides the three masses, the only other relevant parameters in this
simplified model are the $y_{D}$ coupling between the mediator, X, and DM, and
the $y_{ud}$ coupling between the mediator and the SM up and down quarks,
ensuring consistency with flavor constraints and precision Higgs physics.

Dark matter models with a strongly interacting coannihilation partner have two
striking features regarding relic density and collider signatures. First, the
coannihilation paradigm provides an explicit counterpart to the WIMP miracle,
giving weak scale masses and the correct relic density unencumbered by weak
couplings or weak scale cross sections. In particular, the relic density is
determined by four types of (co)annihilation channels. The DM X $\to$ SM1 SM2
channel dominates in the funnel region, where
$m_{\mathrm{DM}}+m_{\mathrm{X}}\approx m_{\mathrm{M}}$. The DM X $\to$ M
$V_{\mathrm{SM}}$ and $\overline{\text{DM}}$ DM $\to$ M∗ M processes can
become relevant for large DM masses, above the resonant coannihilation funnel.
The $\overline{\text{X}}$ X $\to$ $\overline{\text{SM}}$ SM process, since X
has a color charge, efficiently annihilates X particles into quark and gluon
pairs, and is most important in the low $m_{\mathrm{DM}}$ region. This process
determines the underclosure bound on $m_{\mathrm{DM}}$ for a given fractional
X–DM mass splitting $\Delta$ and can push the preferred dark matter mass into
the multi-TeV range for $\Delta\lesssim 0.01$. For $0.1\leq\Delta\leq 0.125$,
though, the relic density motivates dark matter masses $m_{\mathrm{DM}}\sim
300-500$ GeV, consistent with direct and indirect detection constraints and
within the current and near future reach of LHC searches.

The existence of a colored coannihilation partner and a colored mediator M
ensures strong pair production of these particles at hadron colliders, and
since M couples to first generation quarks, single mediator production is also
possible. These production modes, when considered in tandem with the decay
processes dictated by the $s$-channel coannihilation diagram, present very
promising and interesting targets for LHC searches.

The conventional searches at ATLAS and CMS already provide interesting
constraints on the parameter space of the model. We have seen that single
dijet searches constrain a combination of the mediator width and its dark and
visible couplings, which in turn reduces the importance of the DM X $\to$ M
$V_{\text{SM}}$ and $\overline{\text{DM}}$ DM $\to$ M∗ M subprocesses in the
relic density calculation and narrows the width of the coannihilation funnel.
We have also seen that X pair production is probed in jets + MET searches,
where, for $0.1\leq\Delta\leq 0.125$, X masses below about 400 GeV are
currently ruled out, and X masses around 470 GeV should be probed with 100
fb-1 of 13 TeV data. In addition, M pair production gives rise to a paired
dijet resonance signature, where the current searches constrain M masses below
360 GeV and can potentially reach 720 GeV, as long as the visible branching
fraction is 100%.

Once the M $\to$ X DM decay is kinematically open, however, our proposed
mediator mixed decay search pushes the mediator mass sensitivity to roughly 1
TeV. This can surpass the projected reach of the jets + MET search in covering
the parameter space favored by the relic density calculation. This dijet
resonance plus MET signal is characteristic of many $s$-channel coannihilation
models featuring a colored mediator. Moreover, this signature provides the
only direct link between visible matter and the dark matter, and measurements
of the rates and kinematics in this final state would be crucial for a
determination of the dark matter Lagrangian.

Finally, we stress the intricate interplay between the relic density analysis,
the fractional mass splitting parameter $\Delta$, and the resulting reach by
LHC searches, which are entirely driven the coannihilation simplified model.
The minimal assumptions of the coannihilation framework Baker:2015qna  suffice
to characterize completely the dark matter annihilation process, ensuring that
the resulting collider signatures are directly motivated and driven by the
potential discovery of thermal dark matter.

###### Acknowledgements.

We would like to thank Mihailo Backovic, Alexander Pukhov, Peter Skands and
Stefan Vogl for useful correspondence. This research is supported by the
Cluster of Excellence Precision Physics, Fundamental Interactions and
Structure of Matter (PRISMA-EXC 1098), by the ERC Advanced Grant EFT4LHC of
the European Research Council, and by the Mainz Institute for Theoretical
Physics. The work of MB and JL is supported by the German Research Foundation
(DFG) in the framework of the Research Unit “New Physics at the Large Hadron
Collider” (FOR 2239) and of Grant No. KO 4820/1–1. MB would furthermore like
to thank the Fermilab Theory Group for their hospitality. SEH would also like
to acknowledge the hospitality of the Berkeley Center for Theoretical Physics
and the Theoretical Physics Department at LBNL.

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