Theory, phenomenology, and experimental avenues for dark showers: a Snowmass 2021 report

In this work, we consider the case of a strongly coupled dark/hidden sector, which extends the Standard Model (SM) by adding an additional non-Abelian gauge group. These extensions generally contain matter fields, much like the SM quarks, and gauge fields similar to the SM gluons. We focus on the exploration of such sectors where the dark particles are produced at the LHC through a portal and undergo rapid hadronization within the dark sector before decaying back, at least in part and potentially with sizeable lifetimes, to SM particles, giving a range of possibly spectacular signatures such as emerging or semi-visible jets. Other, non-QCD-like scenarios leading to soft unclustered energy patterns or glueballs are also discussed. After a review of the theory,  existing benchmarks and constraints, this work addresses how to build consistent benchmarks from the underlying physical parameters and present new developments for the Hidden Valley module, along with jet substructure studies. Finally, a series of improved search strategies is presented in order to pave the way for a better exploration of the dark showers at the LHC.

§ INTRODUCTION

As the experimental program of the LHC searches for physics beyond the Standard Model (SM) is maturing, the community has started devoting significant effort to investigating alternative models and their associated phenomenology, especially those providing exotic signatures which would not have been directly addressed yet in the existing searches. Of interest here is the case of a strongly coupled dark sector or hidden sector, which extends the SM with an additional non-Abelian gauge group. Considering the non-trivial structure of the SM QCD, we should be open to the idea of a potentially complicated dark sector via non-Abelian gauge groups. These extensions generally contain matter fields, much like the SM quarks and gauge fields similar to the SM gluons. There are no a priory expectations on the gauge group dimension (number of colors), or that of matter fields (number of flavors) that the theory may have.

When the dark sector confines below some confinement scale () dark hadrons are formed, and depending on the symmetries of the theory, some of them could be stable leading to dark matter candidates. To allow for the production of dark states at the LHC, which could be either dark quarks or hadrons, the dark sector is coupled to the SM via a portal.
The realisation of associated LHC phenomenology of such dark/hidden sector is however very much dependent on the details of the model. Nevertheless, some generic expectations can be set. For example, at the LHC, cases where dark quark masses () and corresponding confinement scale are much smaller than the collider centre-of-mass energy ($\mqdark \lesssim \lamdark \ll \sqrt{s}$) lead to spectacular signatures in terms of emerging or semi-visible jets <cit.>. Increasing implies heavier bound states, which in turn decreases the final state multiplicities for a given $\sqrt{s}$, as the allowed phase space decreases. This means that as the limit $\lamdark \sim \sqrt{s}$ is approached, depending on the relevant production mechanisms, $2\to 2$ SM initial state to dark meson final state processes become prevalent and resonance-like searches for dark bound states may prove useful <cit.>. Finally, cases where $\mqdark \gg \lamdark,  \mqdark \lesssim \sqrt{s}$ lead to unusual signals known as quirks <cit.>. If the strongly-interacting sector is non-QCD like, other signatures such as Soft Unclustered Energy Patterns are also possible <cit.>. For a review pertaining to this discussion see <cit.>.

In this work, we focus on LHC exploration of such dark/hidden sectors where $\lamdark \ll \sqrt{s}$. In such cases, dark quarks could be produced at the LHC through a portal and undergo rapid hadronization within the dark sector before decaying back, at least in part and potentially with sizeable lifetimes, to SM particles. Models with such hidden dark sectors have been discussed e.g. in the context of twin Higgs models <cit.>, composite and/or asymmetric dark matter scenarios <cit.>, and string theory <cit.>. Some example studies focusing on hidden valley phenomenology can be found in <cit.> with additional examples referred to elsewhere in this write-up. The above references show a rather large activity throughout the last four decades.

These hidden valley models differ from most other Beyond the Standard Model scenarios because the infrared (IR) parameters of the theory can not be computed from ultraviolet (UV) definitions using perturbative techniques. This is in contrast to many other models e.g. MSSM, which have a well-defined relationships between UV and IR, even if their parameter space is high dimensional. The other well known approach to characterise new physics is the use of simplified models which do not rely on such top-down priors but are defined by their minimality; these may not be effective for hidden valley scenarios either,  due to the inherent dependence on UV parameters in strongly interacting theories. Therefore, neither principles used otherwise to analyse new physics scenarios apply to hidden valley models.

While on the one hand these considerations motivate avenues for model-building with applications to shortcomings of the SM such as e.g. dark matter, LHC searches for hidden valleys are primarily motivated by the exotic phenomenology as stated above. Concretely, this means that we do not have a strong theory prior on e.g. the number of colors and flavors, the mass hierarchies amongst the matter fields, or the possible patterns of flavor breaking. Out of the multiple choices at hand, however, it is nevertheless possible to try and build internally coherent models and develop tools to predict their phenomenology and guide the searches.

Despite the complexity and the challenges in analysing such non-Abelian new sectors, there has been an increased activity in the recent years that has focused on  understanding the signature parameter space of such models, both on the theory and the experimental sides. This report presents some of these developments, particularly concentrating on jet-like signatures, and puts in perspective the efforts necessary to make systematic progress in understanding, classifying and searching for such non-Abelian scenarios. Throughout this report we will consider dark sector scenarios where the dark quarks are uncharged under any SM group and the dark sector communicates to the SM via an additional mediator.

The report is organised as follows. QCD-like dark-sector scenarios are first reviewed in Section 2, addressing the theories in the s- and t-channels, and the existing benchmarks and limits for such models. Section 3 will address two possibilities of dark sector beyond the QCD-like scenarios: soft unclustered energy patterns (SUEP) and glueballs. Section 4 will be devoted to simulation tool limitations and how to build consistent benchmarks from the underlying physical parameters for semi-visible jets. After a discussion of consistent parameter setting, some improvements to the Hidden Valley module and their validation will be presented, followed by some phenomenological studies on the jet substructure effects of varying the physical parameters. Finally, a series of improved search strategies will be discussed in Section 5, based on event-level variables, deep neural networks, autoencoder-based anomaly detection or better triggering algorithms.

§ QCD-LIKE SCENARIOS OF DARK SECTOR

In SM QCD, the strong coupling constant $\alpha_s$ becomes weaker as the energy increases. This is known as asymptotic freedom. New non-Abelian sectors which display such asymptotic freedom fall into the category of QCD-like scenarios.

This section outlines possible exotic signatures that QCD-like dark sector scenarios may exhibit, including a discussion of benchmark models that have been or are currently being employed by the community. We also discuss existing limits in the context of these benchmarks. These signatures and results in turn provide a strong motivation behind detailed studies of such scenarios, motivating further theory effort, as will be discussed in sec. <ref>.

§.§ Existing constraints

Contributors: Giuliano Gustavino, Steven Lowette,  Kevin Pedro, Pedro Schwaller,  Andrii Usachov, Carlos Vázquez Sierra

The discussion so far has not only established that QCD-like dark sectors are theoretically interesting, but that they can also lead to exotic signatures for the experiments. In light of this observation, several searches at the LHC have already been carried out, with many other studies also underway. The search results are currently reported using one or more of the benchmarks discussed in  <ref>. In this Section, we illustrate some of the public search results, and existing constraints on the theory landscape.
Given that the searches rely on generic signal characteristics, it may be possible to reinterpret the results in terms of other, UV/IR coherent models.

§.§.§ ATLAS search program

While no direct constraints have been published so far by the ATLAS Collaboration on these models, a broad set of semi-visible jet scenarios in both $t$- and $s$-channel production processes are being studied by the collaboration as mentioned in the previous section. Besides possible dedicated searches, the recasting of previously published results in other channels might prove useful in constraining the parameter space. Indeed, existing exclusion limits obtained in analyses looking at di-jet <cit.> and +jet <cit.> final states should already be able to constrain a phase-space predicted by dark QCD models in different ranges. Furthermore, some studies also focus on scenarios with emerging jets, where dark hadrons have a non-negligible lifetime.

§.§.§ CMS search for emerging jets

The CMS emerging jet search <cit.> follows the class of models introduced in Ref. <cit.>, as described in Section <ref>.
The search considers the following parameter variations: $\mbifun = 400\text{--}2000\,\GeV$, $\mpidark = 1\text{--}10\,\GeV$, $\ctpidark = 1\text{--}1000\unit{mm}$.

The search requires four high-jets and triggers on such events using  the scalar sum of their momenta, .
Per-jet quantities indicating displaced tracks—including the 2D impact parameter, the 3D impact parameter significance, and the fraction of the from prompt tracks associated with the primary vertex—are used to identify or “tag” jets as emerging.
The signal region definition requires either two jets tagged as emerging, or one jet tagged as emerging along with substantial missing transverse momentum.
The latter option increases sensitivity to models with larger values where many dark hadrons decay outside of the detector.
The misidentification rate for this tagging procedure is measured and used to estimate the QCD multijet background.
Heavy flavor jets from B hadrons are found to be misidentified more frequently, as expected because of their non-negligible decay lengths.
Multiple signal regions with different selection requirements are defined, and for each model, the signal region with the highest expected sensitivity is used.

95% CL limits on the cross section for pair production of leading to emerging jets, in the plane of versus (here called $\text{c}\tau_{\pi_\text{DK}}$ and $\mathrm{m_{X_{DK}}}$) where (here called $m_{\pi_\text{DK}}$) is set to 5 GeV. Reproduced from Ref. <cit.>.

The search results are shown in Fig. <ref>.
Using a $13\,\TeV$ dataset with $16.1\,\ifb$ of integrated luminosity, no significant excess above the SM prediction is observed.
This search excludes models with $400 < \mbifun < 1250\,\GeV$ for $5 < \ctpidark < 255\unit{mm}$ at 95% confidence level (CL).
It is found that the limits are similar over the range of values explored.
Work is ongoing to incorporate the remainder of the LHC Run 2 dataset, up to $138\,\ifb$, and to improve the sensitivity to other models, such as Ref. <cit.>, which is summarized in Section <ref>.

§.§.§ CMS search for semi-visible jets

The CMS search for semi-visible jets <cit.> is based on the class of models introduced in Refs. <cit.>, as described in Section <ref>.

The search requires two high-wide jets and triggers on the jet and the .
This dijet system is combined with the missing transverse momentum to compute the transverse mass ,
which has a falling spectrum for the SM backgrounds, while the signal has a kinematic edge at the mediator mass.
The QCD multijet background is rejected by requiring high values of the transverse ratio $\RT = \met/\MT$,
while the electroweak backgrounds (, , ) are rejected by vetoing identified and isolated leptons (, ) and requiring a small minimum angle between the jets and the .
Various sources of instrumental background and misreconstruction, which introduce artificial , are also rejected.
Two signal regions are defined in terms of the variable to provide additional sensitivity.

95% CL limits on the cross section for $\PZprime\rightarrow\text{semi-visible jets}$ in a two-dimensional plane with variations of and from the model-independent (left) and model-dependent (right) searches, where $\mpidark=\mrhodark=20\,\GeV$ (here called $\mathrm{m_{dark}}$) and $\adark=\adarkPeak$ (here called $\alpha_\text{dark}$). Reproduced from Ref. <cit.>.

The full $13\,\TeV$ dataset of $138\,\ifb$ is analyzed and no indication of a resonance in the spectrum is observed.
Both model-independent and model-dependent results are obtained, as shown in Fig. <ref>.
The former excludes models with $1.5 < \mZprime < 4.0\,\TeV$ and $0.07 < \rinv < 0.53$ at 95% CL, depending on the other model parameters.
The latter uses a boosted decision tree (BDT) that combines jet substructure variables to tag jets as semi-visible.
It extends the exclusions to $1.5 < \mZprime < 5.1\,\TeV$ and $0.01 < \rinv < 0.77$ for the specific models described in Section <ref> that were used to train the BDT.

§.§.§ CMS search for SIMPs as a link to signatures of trackless jets

Dark sector models could give rise to experimental signatures where jets are formed with visible regular hadrons arising from decays of hidden-sector particles that are long-lived, and thus make these hadrons appear displaced within the jet. With sufficient displacement, jets can arise in which none of the tracks of the constituent charged hadrons can be reconstructed, thus making the jets appear neutral. Such neutral jets are extremely rare among high-momentum jets from regular standard-model quarks or gluons, and can thus be a very sensitive probe of physics beyond the standard model from dark sectors.

The CMS collaboration has performed a search for a pair of such trackless jets <cit.> using $16.1\,\ifb$ of integrated luminosity recorded in 2016. The signature probed consisted of a pair of back-to-back high-momentum trackless jets, where experimentally the trackless nature was sought by looking for the ratio of the jet energy carried by charged particles to the energy carried by neutral particles to be less than 5%. To illustrate the effectiveness of this requirement in suppressing standard model QCD background jets, a background rejection of over $10^5$ in data was reported for this 5% requirement.

This CMS search for trackless jets was inspired by and interpreted in a model proposing a new interaction through a low-mass mediator with a new dark matter fermion <cit.>. The interaction leads to very high interaction cross sections, which are not necessarily excluded, though many model assumptions need to be made to avoid the many cosmology, particle physics and astrophysical constraints. The model considered is not a dark sector model per se, as indeed there is no decay back from a dark sector leading to missing charged hadrons, but rather the jets constitute solely of SIMP particles interacting in the calorimeters, thus generating neutral jets. The aspect of a displaced decay is thus missing, though the similar experimental signature does potentially impact dark sector searches.

The interpretation of the CMS trackless jets search in the SIMPs model has been found to be difficult at large SIMP masses, above ${\sim}100\,\GeV$. At such high masses, the modeling of the SIMP-nucleon interaction is complicated by the SIMP mass, and the approximation used in the CMS analysis, which involved treating the SIMP in the Geant simulation as a massive neutron-like object, becomes exploratory. As such, the strong exclusion limits on the SIMP pair production cross section as a function of the SIMP mass, obtained by CMS in absence of an excess of trackless jets in the analyzes data, shown in Fig. <ref>, are reported with this caveat above $100\,\GeV$.

Cross section upper limits for the production of a pair of SIMPs (represented by $\chi$) in the context of the CMS search for trackless jets. See text for details. Figure taken from <cit.>.

§.§.§ Collider constraints on $t$-channel models

A plethora of new physics searches are ongoing at the LHC and have pushed the limits on the masses of new particles above the TeV scale in many cases. While dark showers are a spectacular and unique signature, existing new physics searches still retain some sensitivity in regions of parameter space where the signal looks SM-like.

For $t$-channel dark sectors <cit.>, often the leading production mode is pair production of the heavy mediator, which gives rise to a signature with two ordinary QCD jets and two dark showers, as shown in Fig.<ref>. It is clear that missing energy searches should become efficient in the limit where the dark sector particles become very long lived, while searches for prompt multijet signals can probe the regime of very short lifetimes. Due to the stochastic nature of particle decays, these searches also retain some sensitivity in the intermediate lifetime regime.

Constraints on the mass of a $t$-channel mediator (here called $X$) as a function of the dark sector particle lifetime. See text for details. Figure taken from <cit.>.

A recast of a di-jet plus MET search <cit.>, a search for paired prompt di-jet resonances <cit.> and of the dedicated emerging jets search <cit.> was performed in Ref. <cit.>. As can be seen in Fig. <ref>, the dedicated emerging jets search performs best in the intermediate lifetime regime, while the recast searches can probe all the remaining range of lifetimes. As expected, mediator masses below the TeV scale are already strongly constrained. In the short lifetime regime, the constraints are weaker. This is partially because the published search uses only a limited amount of data, but also because fighting the QCD multi-jet background is hard. Another option to constrain the $t$-channel mediators in the regime of short dark sector lifetimes is through their contribution to angular correlations in dijet events <cit.>, when the mediator is exchanged in the $t$-channel instead. This constraint however will depend on the magnitude of the Yukawa coupling of the mediator.

The $t$-channel mediators naturally allow couplings that break SM flavor symmetries. While most flavor violating couplings are constrained by low-energy flavor observables <cit.>, the LHC has the potential to constrain flavor changing couplings of the top quark <cit.>. Such scenarios could also be of interest for searches for dark showers produced in association with tops or even stemming from exotic top decays.

§.§.§ Existing constraints and projections from LHCb

The LHCb experiment, originally designed for heavy-flavor physics, has shown its potential as a general purpose detector in the recent years. An excellent secondary vertex resolution, low momentum thresholds and particle identification capabilities make LHCb a natural candidate to search for dark QCD signatures in the low-mass region. These searches are now becoming part of the LHCb physics program, where world-leading constraints have already been set for hidden valley scenarios, with reinterpretations in other dark sector models as well as a large number of very encouraging sensitivity projections, described in the following paragraphs.

LHCb has published a search for low-mass resonances decaying into pairs of muons, using 5.1 fb$^{-1}$ of data collected at 13 TeV <cit.>. In this article, a model-independent search of both prompt and displaced resonances $X$ is performed. For the displaced case, the secondary $X~{\to}~\mu^+\mu^-$ decay vertex is required to be transversely displaced from the primary vertex in the range $12<\rho_T<30$ mm, allowing for the resonance to become a long-lived signature in the detector. Then, limits on the cross-section $\sigma(X~{\to}~\mu^+\mu^-)$ are placed, and interpreted in various production models. One of these models in the regard of dark QCD is that of a hidden valley scenario, where constraints are set on the kinetic mixing strength $\gamma-Z_{HV}$ between a heavy hidden valley boson $Z_{HV}$ with photon-like couplings, and a photon, fixing the average multiplicity of hidden valley hadrons to a value of 10.
These are the most stringent constraints placed up to date, for masses of a composite hidden valley vector boson, $X$, up to 3 GeV, as presented in figure <ref>.

Upper bounds at 90$\%$ C.L. on the kinetic mixing strength $\gamma-Z_{HV}$. The grey box shows a vetoed region due to the large doubly misidentified $K_S^0$ background. Figure taken from ref. <cit.>.

These results have been also re-interpreted in the context of Z-initiated dark showers, assuming various benchmark scenarios <cit.>. In Figure <ref>, projections for one of the scenarios are shown, showing the capabilities of LHCb to probe Z branching fractions down to 10$^{-7}$ during the high-luminosity phase.

Projections at 90% C.L. of LHCb sensitivity to a muon rich dark showers initiated by the decay of a Z into dark quarks (here called $\psi'$). In this benchmark scenario, a dark pion mass (here called $\hat{\pi}$) of 650 MeV, an average multiplicity of 7 and a branching fraction of the dark pion into two muons of 96%, are assumed. More details can be found in ref. <cit.>.

In a more general sense, the capabilities of LHCb to probe other dark QCD models are summarized in Ref. <cit.>, in the context of benchmark scenarios featuring a range of dark hadron and mediator masses, for different assumptions on the average dark hadron multiplicity in the dark sector.
The projections described in this major report show the outstanding potential of the LHCb experiment to place very stringent constraints in the low-mass range, in complementarity with ATLAS and CMS.

§ DARK SECTOR BEYOND QCD-LIKE SCENARIOS

§ SIMULATION TOOL LIMITATIONS AND HOW TO BUILD CONSISTENT BENCHMARKS FROM THE UNDERLYING PHYSICAL PARAMETERS FOR SEMI-VISIBLE JETS

§ IMPROVED SEARCH STRATEGIES

The wide variety of signatures coming from the dark/hidden sector scenarios considered throughout this work also motivates advanced techniques which may enable us to distinguish between signal and background at the LHC. These techniques may involve new kinematic variables, jet substructure information (as briefly discussed in sec. <ref>), machine learning, or advanced triggering strategies. In this Section, we illustrate some the  avenues which have been explored in the literature, using the dark/hidden sector parametrizations presented in sec. <ref>. It would be of great interest to also perform such analyses in the light of the new developments presented in  sec. <ref>.

§.§ Event-level variables

Contributors: Hugues Beauchesne, Giovanni Grilli di Cortona

Semi-visible jets are a characteristic signature of many confining dark sectors and consist of jets of visible hadrons intermixed with invisible stable particles. Up to now, two main search strategies have been pursued: tagging semi-visible jets (see e.g. Refs. <cit.>) and exploiting the special relation between the azimuthal direction of the semi-visible jets and the missing transverse momentum (see e.g. Refs. <cit.>). In Ref. <cit.>, it was shown that these two approaches can be combined to define new event-level variables that considerably increase the sensitivity of semi-visible jet searches. The central idea is that semi-visible jets are responsible for most of in signals and that tagging specifies which jets are semi-visible. The tagging information then predicts the direction and magnitude of , which can be compared to its measurement. In this section, we present a summary of Ref. <cit.> and refer to it for technical details.

For illustration purposes, consider the following benchmark model. Assume a new confining group $\mathcal{G}$. Introduce a dark quark $\Pqdark$ that is a fundamental of $\mathcal{G}$ and neutral under the Standard Model gauge groups. Introduce a scalar mediator $S$ that is an antifundamental of $\mathcal{G}$ and has an hypercharge of $-1$. These fields allow the Lagrangian

\begin{equation}\label{eq:LagrangianDecay}
\mathcal{L} = \lambda_i S^\dagger \Paqdark P_R E_i + \text{h.c.},
\end{equation}

where $E_i$ are the Standard Model leptons. Assume for simplicity that the only non-negligible $\lambda_i$ is the one corresponding to the electron. If the mediators are pair-produced, they will each decay to an electron and a dark quark. The experimental signature will then be two electrons and two semi-visible jets. This is similar to the signature of leptoquark pair-production and as such preselection cuts are applied based on typical leptoquark cuts. The event is also required to contain two jets tagged as semi-visible. We focus on the background. Events are generated using <cit.>,  8 <cit.> and  3 <cit.>. The Hidden Valley module of is used with the following parameters:

Setting             Value      Setting            Value

NGauge              3          Dark pion mass     10 Gev

nFlav               1          Dark rho mass      21 GeV

FSR                 On         pTminFSR           11 GeV

alphaOrder          1          fragment           On

Lambda              10 GeV     probVec            0.75

Dark quark mass     10 GeV

All other parameters are left to their default value. Finally, we define $1 - \rinv$ as the average fraction of the dark pions that decay back to Standard Model particles.

Consider a signal event. Label the transverse momenta of the two dark quarks produced from the decay of the two $S$ as $\mathbf{p}_\mathrm{T}^{\PqdarkI}$, with $i \in \{1, 2\}$. Each $\Pqdark$ leads to a visible jet of transverse momentum $\mathbf{p}_\mathrm{T}^{D_i}\sim(1 - \rinv) \mathbf{p}_\mathrm{T}^{\PqdarkI}$ and a contribution to $\mathbf{\met}$ of $\sim \rinv \mathbf{p}_\mathrm{T}^{\PqdarkI}$. This gives

\begin{equation}\label{eq:vMETexp}
\met \sim \frac{\rinv}{1 - \rinv}\mathbf{p}_\mathrm{T}^{D_1} + \frac{\rinv}{1 - \rinv}\mathbf{p}_\mathrm{T}^{D_2}.
\end{equation}

Consider the decomposition $\met = a_1 \mathbf{p}_\mathrm{T}^{D_1} + a_2 \mathbf{p}_\mathrm{T}^{D_2}$. The coefficients $a_1$ and $a_2$ should then peak at $\sim \rinv/(1 - \rinv)$ and can be combined in a single test statistics. This could be done in multiple ways, but a simple and powerful one is to train a fully supervised neural network on the $a_1$ and $a_2$ of both the signal and the background. Alternatively, one can encode much of the same reasoning in a single variable. Define

\begin{equation}\label{eq:DeltaPhi}
\Delta \phi = \left|\phi_{\mathbf{p}_\mathrm{T}^{D}} - \phi_{\met}\right|,
\end{equation}

where $\phi_{\mathbf{p}_\mathrm{T}^{D}}$ ($\phi_{\met}$) is the azimuthal angle of $\mathbf{p}_\mathrm{T}^{D_1}  + \mathbf{p}_\mathrm{T}^{D_2}$ ($\met$). This quantity should peak at 0 for the signal, but unfortunately contains no information on the norm of $\met$.

We introduce two comparisons. First, the standard procedure up to now has been to compute the minimal difference in azimuthal angle between $\met$ and the leading jets <cit.>

\begin{equation}\label{eq:CLLM}
\Delta \phi_{\text{CLLM}} = \min_{i \leq 4}\left\{\left|\phi_{\mathbf{p}_\mathrm{T}^{j_i}}  - \phi_{\met}\right|\right\},
\end{equation}

where in this case four jets are considered. Second, we consider a supervised neural network using $x = \{\phi_{\mathbf{p}^{D_1}}, \phi_{\mathbf{p}^{D_2}}, \eta_{\mathbf{p}^{D_1}}, \eta_{\mathbf{p}^{D_2}}, \phi_{\met}\}$. This is only meant as a comparison, as fully supervised neural network are susceptible to simulation artefacts and sculpting.

ROC curves for $\Delta \phi$ (blue), $\Delta \phi_{CLLM}$ (orange) and for the neural networks using the input variables $a_1$ and $a_2$ (green) or the set $x$ (red) for a mediator mass (here called $L$) of 500 GeV. Taken from Ref. <cit.>.

Receiver Operating Characteristic (ROC) curves are shown in Fig. <ref> for different values of $\rinv$. As can be seen, the coefficients $a_1$ and $a_2$ typically provide the strongest results. They sometimes exceed the fully supervised neural network by exploiting information on the magnitude of the momenta which are not provided to the neural network. The coefficients outperform the standard approach of $\Delta \phi_{\text{CLLM}}$  by an order of magnitude for a signal rejection rate of 0.5. The variable $\Delta \phi$ also generally outperforms $\Delta \phi_{\text{CLLM}}$.

§.§ Casting a graph net to catch dark showers

Contributors: Elias Bernreuther

To increase the sensitivity to dark shower signals consisting of promptly decaying dark hadrons, it is crucial to reduce the large QCD background. While backgrounds from mismeasured QCD jets mimic the signal with regards to event-level observables, such as $\Delta\phi$, differences are expected at the level of jet substructure. These can arise from differences in the shower evolution between QCD and the dark sector, the presence of visibly decaying heavy dark mesons in the jets, or invisible dark hadrons that are interspersed with visible particles. See e.g. Refs. <cit.> for recent studies of dark shower signals in terms of classic jet substructure variables. In contrast, advances in tagging jets with modern machine learning techniques make use of low-level properties of jet constituents. Here, we summarize the results of Ref. <cit.>, which studies the potential of deep neural networks for identifying semi-visible jets from dark showers.

As a benchmark, dark showers of nearly mass-degenerate GeV-scale dark mesons which are produced at the LHC via a heavy $\PZprime$ vector mediator with mass on the TeV scale were considered. The underlying dark sector is the Aachen model summarized in Section <ref> and motivated by cosmological and experimental constraints <cit.>.
The dark quark production process $p p \to \Pqdark\Paqdark$ was simulated with MadGraph5 2.6.4 <cit.> using a UFO file generated with FeynRules <cit.> and performing MLM matching with up to one additional hard jet. Showering and hadronization, both in QCD and in the hidden sector, were carried out using  8.240 <cit.>. The settings used in 's Hidden Valley module for a signal with dark meson mass $\mdark$ are summarized in table <ref>. The parameter probVector was set to 0.5 such that 25 % of dark mesons are unstable, flavor-diagonal vector mesons as predicted by the benchmark model. Jet clustering is performed by FastJet <cit.> using the anti-$k_T$ algorithm with jet radius $R=0.8$.

Settings of the Hidden Valley module used for generating the dark shower signal in Ref. <cit.>.

setting     value     setting     value     setting     value     setting     value

FSR     on     probVector     0.5      Ngauge     3     Lambda     $\mdark$

fragment     on     dark pion mass     $\mdark$     nFlav     2     pTminFSR     $1.1 \mdark$

alphaOrder     1     dark rho mass     $\mdark$     spinFv     0

A priori, it is not clear what the optimal jet representation and neural network architecture are to optimally distinguish dark shower jets from QCD jets. In Ref. <cit.> it was shown that a dynamic graph convolutional neural network (DGCNN) <cit.> operating on particle clouds outperforms convolutional neural networks (CNNs) based on jet images <cit.> and a network operating on ordered lists of Lorentz vectors <cit.>. While a standard CNN carries out convolutions over neighboring pixels in a jet image, a DGCNN performs convolutions over edges of a graph constructed from jet constituents that are neighbors in feature space. While graph networks also represent the state of the art in tagging boosted top jets <cit.>, their advantage over a CNN or a Lorentz Layer network is considerably larger in identifying semi-visible jets. A comparison of ROC curves showing the QCD jet background rejection $1/\epsilon_B$ as a function of the dark shower signal efficiency $\epsilon_S$ for $\mdark = 5$ GeV  is shown in figure <ref>.

Left: ROC curves showing the semi-visible jet tagging efficiency $\epsilon_S$ and QCD background rejection $1/\epsilon_B$ for a DGCNN compared to a CNN and a LoLa network operating on jet images and Lorentz vectors, respectively. Right: ROC curves for a DGCNN trained on a mixed sample containing a number of different dark meson masses $\mdark$ (here called $m_\mathrm{meson}$) and tested on dark showers with $\mdark$ as stated in the legend (dashed lines), compared to a DGCNN trained and tested on dark showers with identical $\mdark$ (solid lines). Figures taken from Ref. <cit.>.

Since the parameters of the dark sector are a priori unknown it is a crucial question how well the classification performance of the DGCNN generalizes to dark showers with different parameter values than were used for training. Varying and $\mdark$, the performance continuously degrades the further the parameters of the dark showers in the test sample are from those in the training sample. While the effect is modest for , it is much more substantial for the dark meson mass. For example, for a network trained with $\mdark=5$ GeV, the background rejection rate for signal efficiencies between 0.1 and 0.3 is reduced by nearly an order of magnitude when tested on samples with $\mdark=20$ GeV. This suggests that the network learns to reconstruct this mass from the jet constituents. Importantly, this behavior can be mitigated by training the network on mixed samples which contain jets with a range of different dark meson masses. This yields a much more general classifier as reflected in the ROC curves in figure <ref>.

Finally, it was investigated how much the sensitivity of an experimental search for dark showers can be improved by applying a DGCNN as a semi-visible jet tagger. As an example, an ATLAS search for mono-jet events with a luminosity of $36.1$ fb$^{-1}$ <cit.> was considered, which is sensitive to signal events where one of the two dark showers remains invisible and, thus, $\Delta\phi \approx \pi$. For an event to be accepted, it had to fulfil the original selection criteria of the search and contain at least one fat jet that is classified as a semi-visible jet by the network. The training sample consisted of jets from a dark shower signal with the benchmark parameters stated in Section <ref> and from the dominant $\PZ$+jets background. The expected number of background events with and without the DGCNN tagger is shown in table <ref> for the signal region EM4, which is the region most sensitive to the signal when $\mZprime = 1$ TeV. In addition, the table compares the resulting expected 95 % CL limit $S^{95}_\mathrm{exp}$ on the number of signal events in the region with and without the tagger and shows the corresponding improvement of the projected limit on the dark quark production cross section. In the benchmark scenario shown in table <ref> a DGCNN for tagging semi-visible jets can improve the sensitivity of the search to dark showers by more than one order of magnitude.

Number of background events $B$ with systematic uncertainty in the signal region EM4 of the search with and without the dark shower tagger and corresponding expected 95 % CL limit $S^{95}_\mathrm{exp}$ on the number of signal events. In addition, the improvement in the limit on the dark quark production cross section for the benchmark scenario described in the main text is shown relative to the search without a tagger. Table adapted from Ref. <cit.>.

$B$     $S^{95}_\mathrm{exp}$     $(\sigma^{95}_\mathrm{exp})^\mathrm{w/o  NN}/\sigma^{95}_\mathrm{exp}$

without DGCNN tagger      $27640 \pm 610$     $1239$     1

with DGCNN tagger     $12.1 \pm 0.3$     $8.2$     $19.7$

§.§ Autoencoders for semi-visible jets

Contributors: Annapaola de Cosa, Jeremi Niedziela, Kevin Pedro

Semi-visible jets arise from Hidden Valley models of dark matter, which include strong interaction in the dark sector. They constitute a challenging experimental signature in which a fraction of jet constituents is invisible to the detector, leading to missing transverse energy being aligned with the jet.

The details of the kinematics are mainly affected by the following theory parameters: (the mass of the mediator), (the mass of the dark hadrons) and (the fraction of stable, invisible dark hadrons). However, a large total number of unknown theory parameters leads to a vast model space with a huge number of possible scenarios that can easily evade any constraints from e.g. cosmological measurements. Since it is impractical to perform dedicated searches for all possible model variations, we propose to use autoencoders (AE) as anomalous jets taggers instead <cit.>.

The autoencoder-based anomaly detection strategy is robust against both detector effects and details of the model implementation. AEs are designed to detect objects significantly different from the training sample, without prior knowledge of signal characteristics. For reference, the AE introduced here is compared to a Boosted Decision Tree (BDT) trained on the QCD background and a mixture of different signals. For completeness, we have also studied alternative anomaly detection techniques, namely Variational Autoencoders (VAE) and Principal Component Analysis (PCA).

All architectures mentioned above were trained on high-level properties of jets: $\eta$ and $\phi$ coordinates and invariant mass , as well as jet substructure variables: jet dispersion , jet ellipse minor and major axes, EFP1, and ECF ratios: C2 and D2. We have also considered including four-momenta of jet constituents in the training, but they were ultimately discarded since no improvement was observed.

Left and middle panels: comparison of AUC values from the autoencoder and a BDT trained on a mixture of all signal models with $\mdark = 20\,\GeV$. Right panel: AUC values for a BDT trained on a signal with parameters different from the signal used for testing, as indicated by the arrows. For example, the AUC value presented in the top left corner of the table comes from a model trained on the sample from the lower right corner. This figure is reproduced from Ref. <cit.>.

The performance of different approaches is quantified by comparing the area under the ROC (receiver operator characteristic) curve (AUC), shown in Fig. <ref>. It was demonstrated that an AE-based jet tagger can provide satisfactory performance, compared with the fully supervised BDT approach. The PCA proved to be less efficient than other approaches. The VAE was found give the best results when trained exclusively on reconstruction loss, leading its variance to collapse to zero and therefore becoming equivalent to a regular AE.

Comparison of AUC values of the autoencoder and BDT with varying values (here called $m_{\text{dark}}$). The BDT was trained on a mixture of all signals with $\mdark = 20\,\GeV$. This figure is reproduced from Ref. <cit.>.

Robustness against unknown model parameters was also assessed. As shown in the rightmost panel of Fig. <ref> and in Fig. <ref>, in certain cases the AE can outperform the BDT when the latter was trained on an incorrect signal hypothesis. Another interesting observation that can be made in the right panel of Fig. <ref> is that a BDT trained on = 0.3 and tested on = 0.7 performs better then the one trained on a mixture of different signals (left panel). This is caused by the fact that the low signal is more similar to the background, and therefore the BDT has to learn how to distinguish between the two more precisely. This results in a performance boost when tested on an easier case of large .

§.§ Autoencoders for SUEP

Contributors: Jared Barron, David Curtin, Gregor Kasieczka, Tilman Plehn, Aris G.B. Spourdalakis

§.§.§ Searching for Hadronic SUEP

The theoretical motivation and experimental phenomenology of SUEPs are described in section <ref>. Strategies to overcome the experimental challenges of searches for SUEP at the LHC are still being developed. For the nightmare scenario of prompt, hadronically decaying SUEP, a search strategy was proposed in <cit.>, employing an autoencoder neural network as an anomaly detector.

An autoencoder is an unsupervised neural network trained on background events, which attempts to minimize the difference between its output and input. Ideally the autoencoder learns to do this efficiently only for inputs that are similar to its training data, so that when evaluated on an event from outside the background distribution, a high reconstruction error flags the event as anomalous. In the case of a search for SUEP, the background events are soft, highly isotropic QCD events. The unsupervised nature of this analysis avoids the model dependence that comes from using signal simulation to develop a classifier.

§.§.§ Signal Generation

While the use of unsupervised machine learning techniques removes the need for signal events in the training dataset, a simulated signal dataset is still necessary to evaluate the autoencoder's performance as an anomaly detector. For this purpose, SUEP events were generated using a statistical toy model of the dark shower.

The highly isotropic hadronic SUEP toy model simulated events were generated beginning with the production of Higgs bosons in association with a or boson, simulated at center-of-mass energy 14 TeV in 8 <cit.>. The vector boson was then required to decay leptonically. The hard lepton(s) from the vector boson decay were used to sidestep the issue of how to trigger on SUEP for this analysis. The decay of the Higgs to a shower of dark mesons was performed with the  plugin in 8.243, which models the dark shower as being a completely isotropic cloud with Boltzmann-distributed momenta as was presented in Eq. <ref>, and for which

the parameter $\Tdark$ controls the energy distribution of the dark mesons, and represents the Hagedorn temperature of the dark sector. Only one flavor of dark meson is assumed, with mass $\mdark$. Each dark meson was then forced to decay hadronically to a $u\bar{u}$ quark pair. From this point the parton showering and hadronization were performed by as normal. Signal simulation was generated for $\mdark$ from 0.4 GeV to 8 GeV, and for $\Tdark/\mdark$ from 0.25 to 4. Detector simulation was performed with 3 with CMS detector settings <cit.>. Due to the difficulty of disentangling the highly diffuse energy depositions of SUEP from pile-up, only charged track information was used for the analysis.

§.§.§ Background Generation

The simulated QCD background events necessary for the training and test datasets were created by generating di-jet plus lepton(s) events with a reduced jet $\pt$ threshold of 15 GeV in  with hadronization by 8 and detector simulation by 3 <cit.>.

§.§.§ Analysis

A trigger-level selection was applied to all simulated events requiring at least one charged lepton with $\pt>40$ GeV, or two opposite-charged leptons with $\pt>30(20)$ GeV, as well as hadronic $\HT>30$ GeV. A further set of pre-selection cuts were then applied. Before feeding events into the autoencoder as training data, 98% of the initial simulated background events were discarded by cutting on three high-level observables that encode the essential features of SUEP.

First, the multiplicity of charged tracks was required to be $N_{charged}\geq 70$. Second, the event ring isotropy variable introduced in <cit.>, measuring the Wasserstein distance between a given event and a uniformly isotropic distribution of energy, was required to be $\mathcal{I}<0.07$. Finally, the inter-particle $\Delta R_{ij}$ distance averaged over all pairs of tracks in the event was required to be $\overline{\Delta R}>3$. Signal efficiency of these cuts varied from $1-30\%$ with $\mdark$ and $\Tdark$.

A fully connected autoencoder with five layers was trained using QCD background events that passed the pre-selection cuts as training data. Each event was represented using a modified inter-particle distance matrix $\Delta \tilde R_{ij}$ of the 70 highest-$\pt$ charged tracks in the event.

\begin{equation}
\label{e.inputfeatures}
\Delta \tilde R_{ij} \equiv \left\{
\begin{array}{lll}
\Delta R_{ij}=\sqrt{(\Delta \eta_{ij})^{2} + (\Delta \phi_{ij})^{2}} & & i > j\\
p_{T,i}/\mathrm{GeV} \ \ \   & \mathrm{for} \ \ \  & i = j\\
0 &  & i < j
\end{array} \right.
\end{equation}

A modified mean-squared-error loss quantified the reconstruction error for each event.

§.§.§ Results

After training on background events, the autoencoder was fed test data including both background events and signal events across the range of simulated $\mdark$ and $\Tdark$ points. Using the reconstruction loss as an anomaly score, ROC curves were constructed for each parameter point. To estimate the physical sensitivity of the model, the minimal excludable branching ratio of Higgs to SUEP for which $S/\sqrt{B + u_{sys}B^{2}}>2$ was computed. Statistical uncertainties due to the limited size of the simulated background sample became dominant as the cut threshold was increased before the classifier's performance began to deteriorate, indicating that the sensitivity of a real search using this method could be even higher than we report here.

Minimum excludable $\mathrm{Br}(h \to \mathrm{SUEP})$ at the HL-LHC, assuming $1\%$ systematic uncertainty on QCD background for fully connected autoencoder.

As Figure <ref> illustrates, this autoencoder-based analysis could exclude Higgs branching ratios to SUEP down to 1% for dark meson mass $\mdark<~ 1$ GeV and $\Tdark/\mdark<~ 1$. If the dark shower temperature $\Tdark$ is $<~0.5 \mdark$, branching ratios down to 5% could be probed for $\mdark$ up to $\approx 8$ GeV.

Using a neural network architecture and event representation tailored to the essential characteristics of the SUEP signature, but without relying on the details of any signal simulation model, this study demonstrates that even the maximally challenging scenario of entirely prompt and hadronic SUEP can be probed at the HL-LHC.

§.§ Triggering on Emerging Jets

Contributors: Daniel Stolarski

The original Emerging Jets (EJ) theory paper <cit.> as well as the CMS search <cit.>
(see also section <ref> of this white paper)
considers a model with a colored mediator $\Pbifun$ that is pair produced at the LHC, which leads to a final state with two QCD jets and two EJs. Those works also consider mediator masses in the regime of $\mbifun \gtrsim 600$ GeV. Given those assumptions, the vast majority of EJ events have substantial $\HT$ and thus the trigger efficiency is very high. In this section we consider relaxing both of the above assumptions and explore how one can still trigger on Emerging Jets.

In <cit.>, an $s$-channel mediator was considered
(see also section <ref> of this report),
focusing on a $\PZprime$ that couples to the quark current in the SM and the dark quark current. Such a mediator produces events that typically do not include additional hard jets. That work also considered the possibility of relatively light $\PZprime$ down to masses of $\sim 50$ GeV. The typical $\HT$ of such events, particularly in the light $\PZprime$ regime, is considerably lower than typical trigger thresholds at the LHC experiments, and other techniques are needed to increase the trigger efficiency.

The events were generated using a modified spin-1 mediator model[https://github.com/smsharma/SemivisibleJets] <cit.> implemented using the  <cit.> package. The hard process is generated with  <cit.> using a centre of mass energy of 13 TeV. This output is interfaced to the Hidden Valley <cit.> module of 8 <cit.>, which simulates showering and hadronization in the dark sector as well as decays of dark hadrons to either other dark hadrons or to SM states. The $\PZprime$ mass is varied and a $\PZprime$ width of $\Gamma_{\PZprime} = \mZprime/100$ is used. The remaining dark sector parameters are varied across a few benchmark models shown in Table I of <cit.>.

Initial state radiation (ISR) in QCD or EW is included at leading order in the hard processes. The resulting hadrons are clustered into jets using the Anti-$k_{t}$ algorithm <cit.> implemented in  <cit.> with a jet angular parameter $R = 0.4$ and a maximum pseudorapidity of $|\eta| < 2.49$ to be compatible with the ATLAS inner tracker. MLM matching and merging procedure <cit.> is employed for extra QCD radiation with  of $\mZprime$/10. A crude detector volume cut is implemented at the 8 stage for which particles that are outside of a cylinder of $(r = 3000$ mm$, z = 3000$ mm) are considered stable.

Two main strategies are explored to increase the trigger efficiency. The first is exploiting the possibility of SM radiation from the initial state. While electroweak ($\PW/\PZ/\gamma$) radiation was explored, the most effective strategy was to use additional QCD radiation. This radiation can increase the trigger efficiency in two complimentary ways. First, the additional hard jet(s) can be used to trigger on directly. Second, the emerging jets tend to be boosted and carry more energy. This in turn will will increase the $\HT$ ($\met$) if the dark pion states are short (long) lived.

(Figure 4 from <cit.>) Cross section times efficiency of various processes (leading order, 1-jet ISR, 2-jet ISR, electro weak ISR) scaled by their respective leading order process. The left plot uses the (here called MET) trigger and Model A which has a dark pion lifetime of 150 mm. The right uses the $\HT$ trigger and Model B which has a dark pion lifetime of 5 mm. The dotted line is the leading order process. On the right plot, the blue region on the left has zero of events that we simulated pass the trigger and thus an efficiency $\epsilon \lesssim \epsilon_{\rm{ min}} = 1/ 400$ $000$.

Using ATLAS trigger thresholds from <cit.>, we estimate the improvement in rate achieved by including radiation and the results are shown in Fig. <ref>. In addition to increasing the trigger efficiency, events with extra radiation have reduced rates, therefore Fig. <ref> shows the ratio of the cross section times trigger efficiency for events with radiation to those without. We see that the largest improvement is additional radiation of two extra jets (green line). The left panel is a benchmark with a dark pion lifetime of 150 mm (Model A) and uses the missing energy trigger. We see that for a light $\PZprime$, more than an order of magnitude improvement in rate is possible. The right side is a model with a dark pion lifetime of 5 mm (Model B) and uses the $\HT$ trigger. In that benchmark, the efficiency of the leading order process is below what was simulated for $\mZprime \lesssim 350$ GeV, and the improvement is potentially even larger.

The first method considered above uses existing triggers, but <cit.> also considers implementing new triggers using modern machine learning techniques.
As ISR is no longer relevant, 8's hidden valley production process $f\bar{f} \rightarrow \PZprime$ processes is employed to generate events. Regardless of the lifetime of the dark pions, the detector subsystem with the largest number of decays is the inner tracker. Therefore, the strategy employed (which is also similar to that of <cit.> proposed for $b$-tagging), is to use the tracker information but not reconstruct tracks. Rather <cit.> proposes to use hit patterns in different layers of the tracker as an input to a support vector machine[Other machine learning techniques gave similar results.] (SVM) from the TMVA toolkit <cit.>.

A proper detector simulation of the inner tracker is outside of the scope, but a crude detector simulation with code used in <cit.> which encompasses the ATLAS tracker from the Inner B-layer (IBL) to the Transition Radiation Tracker (TRT). This detector simulation assumes simple models of energy loss through each thin layer of the detector.

When proposing new triggers, backgrounds must also be considered, and the main background for this strategy is $b\bar{b}$ jets as they have a very large rate and also produce displaced hadrons.
Simulations are performed using $ gg \rightarrow b\bar{b}$ with 8's heavy flavor hard $b\bar{b}$ processes.  The inclusive background cross section is taken from the 8. Pileup is added to both signal and background events with 8's minimum bias events. For each signal or background event, a number of minimum bias events are added randomly sampled from a poisson distribution with mean of $\mu =  50$, mimicking the Run 2 conditions.

(Figure 8 from <cit.>) On the left, discrimination of signal (blue) from $b\bar{b}$ background (green) using a support vector machine. The flat bars (points) correspond to the training (test) set. On the right, Receiver Operation Characteristic ROC for four different lifetimes of signal. At a given background efficiency, the expected signal efficiencies increase as the dark pion lifetimes lower. The required background rejection is estimated to lie between the horizontal dotted lines. Both figures use a mediator mass of 500 GeV.

The left panel of Fig. <ref> shows the ability of the SVM to distinguish signal in blue from the dominant $b\bar{b}$ background. On the right panel we show the ROC curve as a function of lifetime. A background rejection of $\sim 10^{-2}-10^{-3}$ is needed for a novel high level trigger, and we see that efficiencies of $\mathcal{O}(10\%)$ are achievable, with larger efficiencies at lower dark pions lifetimes. It is also found that using an SVM trained on one signal benchmark can also give good acceptance for other signal benchmarks, showing great promise for such a new trigger.

§ SUMMARY AND PERSPECTIVES

In this report, we have summarised the work performed in the context of the Dark Shower Snowmass project: it is the first comprehensive effort to gather the large, pre-existing theoretical, phenomenological and experimental communities working in this field, following initial discussions in the LHC Long-lived Particles Working Group <cit.> and also some presentations in the LHC Dark Matter Working Group. This report also concretely describes pathways for a systematic exploration of strongly interacting theories. In this context, we mainly concentrated on QCD-like scenarios leading to jetty signatures at the LHC, but we also discussed signatures such as SUEPs and glueballs which are typically associated with non-QCD like theories.

QCD-like scenarios, which are the main focus of this report, are inherently non-trivial to analyse due their non-perturbative nature. In such theories, confinement in the IR leads to bound states whose masses and interactions are governed by the UV dynamics. While the SM QCD has been analysed in great detail in terms of UV versus IR parametrizations, little is known for arbitrary gauge groups and flavor contents. Nevertheless, due to the interesting new signatures the strongly interacting scenarios could produce at the LHC, their phenomenology is being actively explored.

In this context, we began this report (see sec. <ref>) with a review of the existing efforts and phenomenological parametrizations of QCD-like scenarios. We qualitatively illustrated the phenomenological differences obtained for various mediator mechanisms, giving rise to exotic LHC signatures such as emerging or  semi-visible jets. We also discussed some existing experimental results constraining these models and ongoing efforts to search for these signatures.

If the dark sector is instead non QCD-like, other classes of spectacular signatures can be obtained in terms of SUEPs or glueballs. These were discussed sec. <ref>, in which original phenomenological SUEP studies were presented, along with recent preliminary simulation tools for these scenarios.

After this overview of existing efforts and of the signature landscape, the report also addressed in sec. <ref> possible pathways for consistent theory frameworks, especially concentrating on semi-visible jets. In that section, lattice calculations, chiral perturbation theory and an analysis of symmetry breaking due to SM-DS portals were combined to exemplify avenues in theoretical model building. Improvements to the 8 Hidden Valley module, made in the context of this Dark Shower Snowmass project, were also presented along with their validation. Combining the theory developments  with the new Hidden Valley module, we then illustrated their impact on the phenomenology of semi-visible jets.

In the final section of the report, sec. <ref>, we discussed some proposed improvements to LHC search strategies. These include efforts using machine learning, trigger considerations and the definition of new event level variables.

Strongly-interacting dark sectors are an exciting class of scenarios in which a vibrant community of theorists, phenomenologists and experimentalists is being invested. They could lead to spectacular signatures which have not yet been systematically explored at the LHC. In view of the large phenomenological interest  of such theories, a more concentrated effort in theoretical work is needed, covering  model building and classification of associated LHC signatures, a deeper understanding of hadronization physics, as well as studies of cross correlation with open problems of the SM such as the nature of dark matter. It is clear from this report that such a work involves communication among experts in SM QCD, lattice, and collider physics as well as in dark matter. We hope that our report lays down the foundations for such a wider exchange, and that this may help devising better strategies that could ultimately lead to a breakthrough in finding signals of strongly-interacting theories.