LHC Searches for Dark Sector Showers

This paper proposes a new search program for dark sector parton showers at the Large Hadron Collider (LHC). These signatures arise in theories characterized by strong dynamics in a hidden sector, such as Hidden Valley models. A dark parton shower can be composed of both invisible dark matter particles as well as dark sector states that decay to Standard Model particles via a portal. The focus here is on the specific case of 'semi-visible jets,' jet-like collider objects where the visible states in the shower are Standard Model hadrons. We present a Simplified Model-like parametrization for the LHC observables and propose targeted search strategies for regions of parameter space that are not covered by existing analyses. Following the 'mono-$X$' literature, the portal is modeled using either an effective field theoretic contact operator approach or with one of two ultraviolet completions; sensitivity projections are provided for all three cases. We additionally highlight that the LHC has a unique advantage over direct detection experiments in the search for this class of dark matter theories.


I. Introduction
The Large Hadron Collider (LHC) provides a unique opportunity to discover dark matter (DM) and study its properties. To date, LHC DM searches have largely been focused on Weakly Interacting Massive Particles (WIMPs), neutral particles with weak-scale mass and interactions. The signature of WIMPs at the LHC is relatively clean: they simply leave the detector and their presence is inferred by enforcing transverse momentum conservation in each collision. In contrast, non-WIMP scenarios can lead to very different collider signatures that require their own dedicated analyses. An additional challenge lies in organizing the enormous variety of self-consistent theories into a finite number of inclusive searches. To this end, we focus on a broad class of models characterized by strong dynamics in a hidden dark sector. We present a proposal for a new analysis framework that builds upon the existing DM program at the LHC and targets the distinctive phenomenology of these models.
We will assume that the strongly coupled hidden sector includes some families of dark quarks that bind into dark hadrons at energies below a dark confinement scale Λ d . While the dark hadrons interact strongly with each other, they interact only weakly with visible states through the portal. If a dark quark is produced with transverse momentum p T Λ d in an LHC collision, it will shower and ultimately hadronize, producing collimated sprays of dark hadrons. These states are invisible at colliders unless they can decay to the Standard Model. Depending on the symmetries of the theory, some fraction of these states are likely to be stable, providing good DM candidates. However, many of the hadrons should decay back to the visible sector through the portal coupling, which is required to produce the dark quarks in the first place. Their decays may lead to a hadronic shower with DM interspersed amongst the visible states.
Characterizing the individual shower constituents is difficult because of the large number of nearly collinear, low-p T states. Greater success can be achieved by clustering the final states into jets and focusing on generic properties of the shower as a whole. Figure 1  illustrates a range of allowed final states that can result, depending on the detailed particle content and parameter choices of the dark sector. In the left-most diagram, all the hadrons are unstable and decay to light quarks. The result looks very much like an ordinary QCD jet, although differences exist at the substructure level. In the right-most diagram, all the dark hadrons are collider stable 1 and do not result in any direct visible signatures-in fact, these would be nearly indistinguishable from WIMP signatures, as we emphasize below. The central diagram illustrates what happens when some fraction of the dark hadrons decay to quarks. The result is a cluster of visible hadronic states that would be constructed as a jet, albeit an unusual one. Because this jet has dark hadrons interspersed throughout, we refer to it as a 'semi-visible' jet [50]. Figure 1 illustrates the case for hadronic decay modes, but the same holds for any decay scenario. One can, for example, consider dark hadron decays to heavy quarks, leptons, or photons.
In the following, we present a search program for strongly interacting dark sectors that yield semi-visible jets. We will see that semi-visible jets generally lead to a new collider signal topology where the total momentum of the DM is correlated with the momentum of the visible states. In Sec. II, we introduce a simplified parametrization that covers the phase space realized by these theories. Motivated by the standard LHC WIMP searches (referred to as 'mono-X' searches, where X can be a jet(s), a weak gauge boson, etc.), we focus on several different production channels. To begin, we remain agnostic about the new states that connect the dark sector to the Standard Model and rely on an effective 1 The DM candidate proposed here is not necessarily assumed to constitute all of the observed relic density. theory framework where the interaction is modeled by a contact operator; this is discussed in Sec. III. In Sec. IV, we consider dedicated searches for two ultraviolet (UV) completions of   this contact operator. Throughout, we emphasize the complementarity with standard LHC   searches, indicating regions of parameter space where current analyses already have coverage, and other regions where new dedicated analyses are required. In Sec. V, we show that direct detection experiments have limited sensitivity to these DM models, thereby emphasizing the critical role played by a dedicated LHC program. We conclude in Sec. VI with a discussion of additional final states, as well as control regions. Two Appendices are included. The first demonstrates the convergence of the separate UV models in the contact-operator limit, and the second shows the insensitivity of our search to variations in the dark sector parameters.
For the reader that would like to simulate the signal Monte Carlo used here, we provide all generation files at https://github.com/smsharma/SemivisibleJets.

II. Signatures of Dark Sector Parton Showers
Building an experimental program that systematically searches for all strongly coupled dark sectors is not feasible due to the large number of possible models. This motivates inclusive searches with non-trivial signal efficiency to a wide range of scenarios. The key is to realize that not all elements of a spectrum of new particle states and their ensuing interactions affect observable signatures at a collider. This is why Simplified Models are now broadly used for supersymmetry [51] and WIMP searches [52][53][54][55]. The complicated dynamics of a dark sector have a limited number of effects on collider observables, primarily impacting the multiplicity of the final state, the fraction of invisible final-state particles, and the average p T of these states. A search that targets these variables yields inclusive bounds in parameter space that can later be recast for any particular theoretical model.
The remainder of this section provides concrete details on how to map an example dark sector Lagrangian onto a simplified parametrization, 2 and then translate it into Monte Carlo events. The discussion is naturally divided into three parts. Sec. II A describes the hidden-sector dynamics, Sec. II B focuses on the portal, and Sec. II C details the signal and background generation, and describes the limit-setting procedure.
2 By definition, a Simplified Model is written in terms of physical observables that are directly related to Lagrangian parameters. It is not possible to do so for the dark sectors we consider here, as some of the observables depend on non-perturbative physics. For this reason, we refer to our proposal as a 'simplified parametrization,' even though it shares the same guiding principles as a Simplified Model.

A. Dark Sector Dynamics
This section elucidates the dark sector physics. For illustration, we consider a toy scenario where the dark sector is an SU (2) d gauge theory with coupling α d = g 2 d /(4π), containing two fermionic states χ a = χ 1,2 in the fundamental representation: where G d µν is the dark gluon field strength, and M d,a is the mass for the χ a ; we assume that the dark quarks have a common mass M d . Similar to QCD, the fermions act as dark quarks that interact strongly with coupling strength α d .  [23,30,31,56]. Fortunately, most of the details concerning the spectrum are irrelevant for collider observables; we focus on three aspects that do have an effect. 3 It is important to distinguish between bound states that do or do not decay back into Standard Model particles. Clearly, a stable state (or one that decays only within the hidden sector) leaves the detector without a trace and results in missing energy. If it does decay to the Standard Model, the decay products can be observed directly. Basic symmetry arguments can be used to determine the stability of the hadrons formed from pairs of the χ a . For generic masses, the dark-isospin number U (1) 1−2 and dark-baryon number U (1) 1+2 (where "1" and "2" refer to the flavor indices) are accidental symmetries of the theory. For instance, the mesons χ 1 χ † 2 and χ † 1 χ 2 are charged under dark-isospin, and can be stabilized if these symmetries are preserved. Similarly, the baryons χ 1 χ 2 and χ † 1 χ † 2 can be stable because they are charged under dark-baryon number. By contrast, the mesons χ † 1 χ 1 and χ † 2 χ 2 are not charged under either symmetry and are thus expected to decay.
Additionally, different spin and CP configurations of the bound states are possible.
For example, the dark hadrons can form scalar, pseudoscalar, vector and/or higher spin combinations. In what follows, we assume that the DM is the lightest stable scalar dark hadron, η d . The spin quantum numbers can determine aspects of the decay parametrics. For example, vector mesons could decay promptly if coupled to the Standard Model through a vector portal, while the decay of an unstable (pseudo)scalar would be suppressed by additional mass insertions. The decay of higher spin states may also be suppressed by loop factors if they cannot decay within the hidden sector. This implies that generically some displaced vertices are expected, which could provide additional handles for improving signal discrimination. However, we choose to design searches that are insensitive to the presence of displaced vertices, which can be strongly model-dependent. In practice, we treat all decays as prompt throughout the rest of the paper.
The relative number of stable and unstable states in the dark sector can vary significantly depending on the details of the theory. For example, one can generalize this toy model to an arbitrary confining sector with any number of colors, N c , or flavors, N f . Extending the flavor symmetry to U (1) N f would naively result in N f "uncharged" mesons and N f (N f −1)/2 "charged" mesons. This impacts the fraction of possible stable to unstable states in the hidden sector, thereby changing the amount of observed missing energy. In addition, there should also be baryons, although their production in the shower will tend to be suppressed.
Introducing a mass splitting between the various mesons can also alter the multiplicity of the final state and the relative fraction of stable and unstable states. Following the Lund string model for fragmentation, the yield of a meson is exponentially sensitive to the meson mass; heavier mesons are exponentially less likely to be produced during hadronization [58]. This is captured by the suppression factor for estimating the ratio of χ 2 to χ 1 production: ( When the mass splitting between χ 1 and χ 2 is large compared to the dark confinement scale, the production of stable dark mesons (e.g., χ 1 χ † 2 and χ † 1 χ 2 ) is suppressed. This in turn reduces the number of invisible states in the dark parton shower.
To capture the variation in the number of stable to unstable states in dark sector models, we introduce the following parameter: If the dark hadrons decay entirely to visible states, then r inv → 0. The opposite limit arises when none of the dark hadrons decay back to the Standard Model (on collider timescales).
In this limit, r inv → 1, and this scenario would be indistinguishable from WIMPs.
The two important parameters that remain are (1) the characteristic mass scale for the dark hadrons, M d , and (2) the dark strong coupling, α d . 4 Both affect the number 4 While these are both physical parameters, we prefer to think of them in the same spirit as r inv . This is justified since the mapping between the real Lagrangian parameters and what is actually computed by the simulation is an unsolved problem and is certainly not captured using current state-of-the-art tools. of dark hadrons that are produced during the dark shower, which subsequently impacts the multiplicity of the dark jet. These parameters also alter the relative p T of the states produced in the shower, which manifests in the detector as the amount of missing and visible energy of the final states. For simplicity, we assume that none of the dark hadron resonance structure is relevant, such that the collider observables are insensitive to any mass splittings in the dark sector spectrum. This assumption only applies if the hard interaction scale of the new-physics event is much larger than the confining scale Λ d . Furthermore, we assume that √ŝ Λ d for the events that populate the signal region such that the perturbative shower is a good approximation. This is true for the searches described below due to the strong kinematic cuts.   Armed with this parametrization of the dark sector physics, we next turn to the details of the portal that connects it to the Standard Model.

B. Portal to the Dark Sector
The portal describes how the hidden sector communicates with the visible Standard Model states. This determines the production channels at the LHC and implies a particular set of decay modes. Following the mono-X literature, we study the three portals illustrated in Fig. 3. Specifically, we consider the contact operator limit [59][60][61] where the mediator is integrated out, as well as two UV completions of this operator [62][63][64][65][66][67][68][69][70][71][72][73][74][75][76][77][78]. To summarize, a strongly interacting hidden sector can be described by three dark sector parameters (α d , M d , and r inv ) and a portal parameter (Λ). While we simulate an SU (2) d sector to derive the results that follow for concreteness, this same approach can be applied to any strongly-interacting hidden sector that decays back to Standard Model quarks. This provides a powerful framework in which the collider results can be presented in terms of generic parameters that can be mapped onto a range of strongly interacting dark sector theories.

C. Event Generation and Sensitivity Estimation
Signal and background events are generated using MadGraph5 aMC@NLO [79] with parton distribution functions NN23LO1 [80] and are showered using Pythia8 [81]. To simulate the dark sector shower and hadronization, we use the Hidden Valley module [82,83] in Pythia8, where we have implemented the running of the dark coupling α d as in [47]. All events are then passed through the DELPHES3 [84] detector simulator with CMS settings. Jets are initially clustered using the anti-k T [85] algorithm with R = 0.5 [86].
To perform the searches described in this paper, we must implement r inv within our simulation framework. First, we shower and hadronize in the dark sector, producing dark mesons. Next, we decay all the dark mesons either to a quark pair or to invisible DM particles. The invisible branching ratio is equal to r inv .
We generate 20,000 signal events, unless otherwise specified, for each parameter point Both of these choices improve the background statistics in the high missing energy ( E T ) tail.
We validate our electroweak and t t background samples by comparing against Monte Carlo in [88]. We use the E T > 250 GeV signal region in that study to calibrate the K-factors (accounting for NLO corrections to the overall cross section) for our backgrounds, finding values of 1.0, 1.1 and 1.7 for the W ± + jets, Z + jets and t t + jets samples, respectively. We obtain a K-factor ∼1.0 for QCD by matching to the di-jet distributions in [89], and make the conservative choice to not implement a K-factor for the signal.
There are two kinds of searches described in the following sections. For the cut-and-count approaches, we treat the background as an Asimov dataset to obtain the expected exclusion reach, following [90]. Given the number of expected signal(background) events, s(b), we then compute the Poisson log-likelihood ratio, L(s + b, b), of the signal hypothesis to the background-only hypothesis. A 95% confidence limit is set by varying the number of signal events such that L(s + b, b) = 2 2 . In the large background limit, and a standard 2σ Gaussian limit is recovered. To compute the expected exclusion reach for the shape analysis in Sec. IV A, we treat the background as an Asimov dataset and the final Poisson log-likelihood ratio is computed by summing over the contribution from each bin.
Because we are primarily interested in comparing different search strategies, as opposed to the precise numbers provided by the projections themselves, this simple treatment of the statistics suffices. For simplicity, no systematic errors are included in the searches proposed here. A detailed study of the relevant systematic uncertainties is beyond the scope of this paper and will require careful study in any experimental implementation of this proposal.

III. Dark Sector Showers from Contact Operators
In this section, we consider the case where the portal is modeled as a contact operator, and show that it leads to semi-visible jets. We focus on the following dimension-six operator: where Λ is the characteristic dimensionful scale for the operator, and the c ijab are O(1) couplings that encode the possible flavor structures. As discussed in Sec. II A above, the DM η d is a scalar bound state comprised of the χ's. Of course, a variety of operators can be written that span a range of effective interactions and spin states of χ a . While the following analysis can be repeated for these different scenarios, we focus on the vector contact operator as an illustrative example. We also restrict ourselves to the production mode u u, d d → χ χ, to quarks and thus there is no parton-level missing energy (neglecting neutrinos that are produced from heavy-flavor quark decays). When r inv = 1, all the dark hadrons are collider stable. Initial-state-radiation (ISR) is required to observe such events, as in the standard WIMP case. The ISR jet boosts the dark hadrons in the antipodal direction, leading to non-vanishing missing energy that is oriented opposite the jet. 5 In the intermediate r inv scenario, two back-to-back semi-visible jets are produced and the missing energy points in the direction of the jet that contains the most stable mesons.
To study this behavior quantitatively, we generate events for the vector contact operator by taking the large-mass limit for an s-channel mediator (see Appendix A for further details), with 60,000 signal events produced over the range of r inv values. The mapping from cross section to Λ is then evaluated for c = 1. The left panel of Fig. 5 shows the ∆φ distributions for the signal, where and ∆φ j i , E T is the angle in the rapidity-azimuthal angle plane between the p T of the i th jet and the missing transverse momentum vector. When r inv = 1, the missing energy is typically oriented opposite to the hardest jet in the event, as expected for the ISR regime.
As r inv decreases, the distribution in ∆φ becomes peaked towards zero, demonstrating that 5 The ISR spectrum for r inv = 1 is not identical to that for a WIMP. While the number of WIMPs produced in each event is constant, the number of dark hadrons produced in a shower varies from event to event, which can affect the E T spectrum [91]. To study the projected sensitivity for the vector contact operator, we perform an optimized cut analysis on two separate signal regions-one with ∆φ > 0.4 and the other with ∆φ < 0.4. The former is the standard requirement for most current searches at the LHC, and is implemented to minimize contamination from jet-energy mis-measurement. This is exemplified by the left panel of Fig. 6, where the QCD background falls off steeply with ∆φ. Requiring ∆φ > 0.4 removes a significant fraction of the high-E T QCD events, as demonstrated in the right panel of Fig. 6. Even when ∆φ < 0.4, however, there is a negligible contribution from QCD above E T ∼ 800 GeV. In contrast, the top background is less steep and the electroweak background is nearly isotropic such that cutting on ∆φ has a less significant effect. Note that the signal populates the control region currently utilized by standard searches when ∆φ < 0.4, which can significantly complicate the background determination in a data analysis. We comment on this further in Sec. VI.
Considering two separate regions with ∆φ greater/less than 0.4 allows us to study the complementarity between the two approaches. At the trigger level, we require E T > 200 GeV and a jet with p T > 250 GeV and |η| < 2.8. Additionally, events containing isolated electrons(muons) with |η| < 2.5 and p T > 20(10) GeV are vetoed. We optimize the missing energy cut to maximize the signal sensitivity for a given r inv . The cut is chosen from the list E T > [400, 600, 800, 1000, 1200] GeV; however, in cases where ∆φ < 0.4, the minimum    how the searches change when the contact operator is resolved into the s-channel and tchannel UV completions. As we will see, the s-channel model motivates a significantly different strategy, while the t-channel model is covered by the same simple E T -driven approach that we used for the contact operator limit.

IV. Dark Sector Showers from Resolved Contact Operators
Next, we resolve the contact operator at tree-level with two simple UV completions.
We characterize these two cases as s-channel and t-channel, which refer to the Feynman diagrams that dominate the production of p p → χ χ at the LHC for the two models.

A. s-channel
A pair of dark quarks can be produced through a new heavy resonance, Z , that couples to the Standard Model baryon-number current and the DM flavor-number current via where g q,χ are coupling constants and i, a are flavor indices. The Z can potentially couple to other visible states, but we focus on the quark current here as we are interested in purely hadronic events. It is worth emphasizing that Eq. (6) is a simple phenomenological parametrization. Specifically, we remain agnostic about the new particle content that is needed to appropriately cancel anomalies-see [92] for a recent discussion-and do not model-build the mixing structure that is required to give g q = g χ . We assume that the Higgs sector which gives the Z its mass does not impact the collider signatures, and thus do not specify it. In this subsection, we revisit the analysis first proposed in [50] for this s-channel production mode to explore its complementarity with existing LHC searches, as well as the contact operator case.
We generate events for the s-channel production in MadGraph using the DMsimp [93][94][95] model file implemented through FeynRules [96], taking as fixed g q = 0.1 and g χ = 1; note that the Z width is calculated self-consistently in the generation. When the Z decays predominantly to visible quarks, di-jet searches provide the best sensitivity regardless of the details of the dark sector. In this case, r inv → 0 and the final state resembles two QCD jets whose invariant mass (M jj ) reconstructs the Z mass. Following the ATLAS di-jet analysis [89], we require that the p T of the leading and sub-leading jets be at least 440 and 60 GeV, respectively, at the trigger and preselection level. We further require that |∆y| < 1.2 between the two leading jets. The left panel of Fig. 8 shows the invariant mass distribution for M Z = 1.5 TeV, taking r inv = 0 and 0.3. As the invisible fraction increases, the width of the signal's invariant mass distribution broadens, reducing the sensitivity of a bump hunt.
where the p i are free parameters. The best-fit distributions are shown in Fig. 8.
The left panel of Fig. 9 shows the limits on Λ (or, correspondingly, m Z ). In order to compute this limit, we fix the couplings to be g q = 0.1 and g χ = 1, and scan over the Z mass. The bounds from the M T analysis (solid red) are strongest for r inv ∼ 0.3. The M T search loses sensitivity as r inv → 0 because no stable hadrons are produced in the dark shower and the E T requirement is consequently too strong. In this regime, however, the M jj analysis proves to be useful, with sensitivity peaking at r inv = 0 (solid yellow). For comparison, we also show the limits from the contact operator analysis discussed in Sec. III.
For most values of r inv , either the M jj or M T analysis does considerably better. However, the contact operator search provides the strongest bounds near r inv ∼ 1. In this limit, the M T analysis loses sensitivity as events tend to fail the jet number and p T cut.
The right panel of Fig. 9 shows the bounds on the production cross section, as a function of Z mass for r inv = 0.3 using the M T search (yellow) and for r inv = 0.1 using the M jj search (red). When computing these limits, we fix the mass and vary the production cross section.
We assume a fixed signal shape and branching ratio derived with g q = 0.1 and g χ = 1; this is a good approximation for the range of cross sections excluded by the two search strategies.
The production cross section for the mediator is shown in dashed black, for the benchmark case with g q = 0.1. When the Z becomes sufficiently heavy, the vector mesons in the shower can manifest displaced decays [50] for 6 where B ∼ 10 is the average boost factor as computed by the simulation, 7 m ρ d is the mass of the vector meson, and the inequality is saturated for a lab-frame displacement of a millimeter.
This parameter range is indicated by the blue shaded region in Fig. 9. If the cross section limit reaches this level of sensitivity, a search that relies on displaced signatures should be implemented, perhaps along the lines of the proposed strategies for emerging jets [47]. We stress that Eq. (8) is a rough estimate and that the value of g q depends quite sensitively on the vector meson mass, which we simply take to be m ρ d = 20 GeV in the figure.
Now that we understand how the search strategy and sensitivity changes for a scenario described by an s-channel UV completion, we move on to the example of a model where the dark quark pair production occurs via a t-channel diagram.

B. t-channel
The collider physics for the t-channel UV completion is governed by the coupling where a, b are DM-flavor indices, i, j are Standard Model-flavor indices, and q Rj represents both up-and down-type quarks. The dark and visible sectors communicate via the scalar bi-fundamental Φ bi , which is in the fundamental representation under both visible QCD and the dark non-Abelian gauge group. For simplicity, we have only introduced a coupling to the right-handed quarks, which requires the Φ bi to carry hypercharge. There is no obstruction to coupling with left-handed quarks q Lj ; this would require the Φ bi to form electroweak doublets, which is not considered here. Additionally, we take all the flavor structure to be proportional to the identity λ ijab = λ δ ij δ ab and assume a common mass M Φ for the scalar bi-fundamentals.
A variety of production modes are possible for this scenario. In addition to direct pair-production of the dark quarks, the bi-fundamentals may also be directly produced if they are light enough. For example, the Φ can be pair-produced via its coupling to visible gluons/quarks (g g, qq → Φ Φ * ) or associatively (q g → Φ χ). The large number of Ratio of the t-channel direct production cross section to the total cross section as a function of the bi-fundamental mass. The total cross section includes processes with additional quarks in the final states through Φ ai → χ a q i . The t-channel production mode accounts for a larger proportion of the total cross section as the mediator mass is increased towards the contact operator limit. (Right) The parton-level invariant mass distribution for the χ χ particles, m χχ , for M Φ = 500 GeV and 100 TeV (the contact operator limit) and λ = 1. One can clearly identify the threshold as each production channel turns on. This shows that the high χ χ tail falls off more rapidly when M Φ is light, which results in weaker limits.
production modes results in a complicated dependence of the production cross section on M Φ . This behavior is demonstrated in the left panel of Fig. 10, which plots the fractional contribution of the t-channel direct production process as a function of M Φ for two choices of λ. In the Appendix, we show how large M Φ must become such that the t-channel and schannel distributions are identical, demonstrating that the contact operator limit is reached for masses of O(10 TeV).
In practice, when generating events for this model, we produce matched samples of p p → χ χ + jets events with 0, 1, and 2 jets. This implies that production modes involving one    To assess the reach for this model, we optimize a search with cuts that are motivated by standard jets + E T analyses, e.g. [98]. After applying a trigger-level cut of p T,1 > 250 GeV and E T > 200 GeV, we optimize the signal reach by scanning in E T > [600, 800, 1000, 1200] GeV. We repeat this procedure for the case where ∆φ < 0.4 and > 0.4.
As in the contact operator case, when ∆φ < 0.4 we restrict ourselves to E T ≥ 800 GeV.
This is identical to the search strategy for the contact operator limit, presented in Sec. III above. We also investigated the impact of additional cuts on H T , as well as the p T of the jets. We find improved performance for smaller values of r inv when cuts on the p T of the third and fourth jets are imposed because they target the additional hard jets produced by the intermediate Φ states. For example, at r inv = 0.2, the ∆φ < 0.4 limit on M Φ improves from ∼ 1000 to ∼ 1500 GeV with these additional cuts. We only show the results for the optimized E T cuts (and not the additional jet p T cuts) so that the comparison with the contact operator search is transparent. The cut-flow for a few benchmarks is provided in Table III.
The left panel of Fig. 11 shows the projected sensitivity bounds on the bi-fundamental mass, as a function of r inv . For r inv 0.8, the search with ∆φ < 0.4 is more powerful, but ∆φ > 0.4 does better at higher invisible fractions, as expected. We also compare the results to the expected reach for the contact operator limit. At first glance, it would appear that the contact operator approach yields additional sensitivity, even though new channels are Note that for r inv → 0, a search strategy that does not have a minimum E T requirement should be investigated. The mapping onto the contact operator limit is Λ = 2 M Φ /λ. (Right) The 95% exclusion limits on the production cross section as a function of M Φ for r inv = 0.9 (red) or 0.5 (yellow) corresponding to ∆φ > 0.4 (solid) and ∆φ < 0.4 (dashed). The total production cross section is shown as the dot-dashed black line. present for the full UV complete model. However, this is spurious as the contact operator is not a good approximation for the mass scales relevant at the LHC. In particular, the apparent improvement in the contact operator limit is an artifact of the larger tail in the m χχ distribution illustrated in Fig. 10. The right panel of Fig. 11 shows the corresponding 95% exclusion limit on the production cross section, as a function of M Φ . We see explicitly that the ∆φ < 0.4 cut gives improved sensitivity when r inv = 0.5, but that the reverse is true when r inv = 0.9. For this UV completion, the ρ d generally will not be displaced until M Φ is larger than O(10 TeV) for λ = 1 (as can be inferred from Eq. (8) which is relevant in this model as well), which is well outside our expected sensitivity. Additionally, the QCD pair-production of Φ is present for arbitrarily small values of λ. As a result, we do not include a displaced region in Fig. 11.
There is potential room for improvement beyond the search presented here. For example, a more sophisticated strategy could be devised to target small r inv . There is the additional complication that the dark shower tends to wash out the anticipated gains in sensitivity resulting from the additional production modes. It may be that less inclusive variables, such as M T 2 [99] or its variants, could yield improved reach in certain regions of parameter space. We leave these investigations to future work. This completes our discussion of the collider projections for semi-visible jets. The next section demonstrates that the direct detection of the η d is highly suppressed.

V. Complementarity with Direct Detection Experiments
Collider searches for DM in the contact operator limit (q q → χ χ) are interesting in large part due to their complementarity with direct detection searches (q χ → q χ). A comparison of the limits derived using both experimental approaches has been explored in detail for the case of mono-X signatures [52][53][54]. When the DM is composite, the comparison is complicated by the fact that q 2 is the squared momentum transfer at the LHC(direct detection experiment). In other words, the DM degrees of freedom are dark quarks at LHC energies, but become dark mesons at the scales probed by direct detection experiments. The rest of this section provides some non-perturbative arguments to estimate the size of the direct detection rates for the strongly interacting models of interest here. We will show that the direct detection rates are highly suppressed and fall below the neutrino background. This section implicitly assumes that the η d comprises all of the DM.
Our goal is to compute the scattering of the composite DM particle, η d , off a Standard Model nucleus for the vector contact operator given in Eq. (4). It is worth noting that this was among the portals suggested in the first paper on direct detection, and was excluded long ago for non-composite DM interacting via the Standard Model Z boson [100]. For the composite DM candidate studied here, additional factors of momentum suppress the rate and make the model safe from direct detection.
From Eq. (4), the direct detection scattering rate depends on the matrix element of a vector current involving the η d . Let the initial(final) momentum of the η d be k(k ) such that the total momentum is P µ = (k + k) µ and the momentum transfer to the nucleus is q µ = (k − k ) µ . By Lorentz invariance, the matrix element of interest requires the presence of an object that carries a vector index; P µ and q µ are the only vectors that are available.
Hence, the composite matrix element must take the form 8 where F d 1 (q 2 ) and G d (q 2 ) are DM form factors. Note that the form factors only depend on q 2 , which can be related to P 2 using m 2 η d . We use the standard notation that F 1 is the electric form factor; if η d had carried spin, there would be the possibility of an F 2 (q 2 ) magnetic form factor and its contribution to the current would be proportional to the spin vector. The requirement that Eq. (10) vanish by current conservation ∂ µ J µ = 0 is directly related to the stability of the DM and imposes that G d (q 2 ) = 0.
In the limit of small momentum transfer (q 2 Λ 2 d ), the remaining form factor can be expanded to first order as The first term in Eq. (11) since the integral over the wavefunctions ψ η d yields q 2 = 0 by orthogonality. As we argued in Sec. II A, the DM is neutral with respect to this current, which immediately implies that Importantly, the higher-order contributions to F d 1 (q 2 ) are non-zero. Physically, as q 2 increases from zero, the structure of the meson begins to reveal itself, as in deep inelastic scattering. Because the partons are charged under the symmetry of interest, this leads to a non-zero contribution to F d 1 (q 2 ). In Standard Model physics, this is usually couched in terms of a non-zero "charge radius," so we will use the same language here. For example, in the case of the Standard Model neutron, F n 1 (q 2 ) = q 2 × (−R 2 n /6 + κ/(4m 2 n )) + O(q 4 ) [101,102], where R n is the charge radius of the neutron, κ is the dimensionless magnetic moment of the neutron, and m n is the mass of the neutron. Because the dark meson η d is a scalar, the magnetic moment is zero and the only contribution to F η d 1 (q 2 ) at O(q 2 ) is proportional to the square of the charge radius. For concreteness, we will assume that R d ∼ 1/Λ d , which is reasonable up to order-one numbers since this is the only scale of relevance for the dark meson (under our assumption that M d ∼ Λ d ).
The net result of these arguments is that the cross section is suppressed by four powers of the momentum exchange. Using these parametrics, we can make a rough estimate of the spin-independent direct detection cross section per nucleon: Noting that the neutrino background begins to dominate at cross sections of 10 −45 to 10 −49 for a DM mass of 10 and 100 GeV respectively [103], this is a very challenging signal to observe at a direct detection experiment.
The result in Eq. (13) clearly applies for the s-channel UV completion. Unsurprisingly, the situation for the t-channel case is very similar. In the heavy-mediator limit, the DMquark effective interaction can be written in a useful form by applying the Fierz identities: The DM matrix element for the vector current is the same as in Eq. (10); there are no axialvector contributions because there are no combinations of η d that yield the correct Lorentz and parity structure as the quark-level operator χ γ µ γ 5 χ. Because the vector operator is the only one that contributes, this means that the direct detection estimate in Eq. (13) applies in this case as well.
As we have seen, the direct detection signals for these composite DM models are highly suppressed for the operators considered in this work, which suggests that the LHC provides a unique opportunity for discovery. It is worth emphasizing, however, that the arguments in this section rely on the assumption that the DM is a scalar and ignore the possibility of inelastic transitions between the DM to a nearby state in the dark hadronic spectrum.
While the latter can provide a potential detection window, the detection rate depends on the mass splittings of the lightest states [104]. Because we remain agnostic to the details of the dark spectrum, we do not consider this possibility here.

VI. Conclusions
This paper proposes a comprehensive discovery program for dark sector parton showers at the LHC. Such signatures arise in a broad range of theories, but an inclusive search program can be designed by using a simplified parametrization of the dark sector and portal physics.
The LHC observables depend primarily on four parameters that divide into: • Dark Sector Parameters: The dark sector strong coupling constant (α d ), the dark hadron mass scale (M d ), and the ratio of invisible to visible hadrons that are produced in the parton shower (r inv ).
• Portal Parameter : The operator scale (Λ) associated with the portal interaction.
In the spirit of mono-X searches, we consider the contact operator limit, and then UV complete this portal with either an s-or t-channel mediator. Targeted search strategies can improve the sensitivity reach to the resolved operators, at the expense of being less model independent.
We focused specifically on the scenario where the visible states produced in the dark parton shower are light quarks, and the visible hadronic shower is aligned with a collimated spray of DM particles, forming "semi-visible jets." In this case, the missing energy typically points in the same direction as one of the jets in the event, resulting in low signal efficiency under standard preselection cuts for jets + E T searches, which require ∆φ > 0.4. We show that reversing this requirement to ∆φ < 0.4 significantly improves the signal reach for a wide range of r inv for both the contact operator and its UV completions. We demonstrate these gains by optimizing search strategies over simple cuts in jet number, E T , and H T .
While it has been demonstrated that these cuts are sufficient to cover the variety of phase space that can be realized by Simplified Models with weakly coupled DM [105], it is entirely possible that more detailed searches would improve the sensitivity to semi-visible signals. For example, variables such as razor [106] or α T [107, 108] might provide additional handles. Furthermore, developing a search that directly targets the small r inv region would be interesting to investigate. Strategies that use the substructure of the semi-visible jets could lead to further improvements, although one must be careful to not rely on detailed features of the dark hadronization given the large uncertainty implicit in modeling these non-perturbative effects.
Semi-visible jets populate the control region typically utilized by ATLAS and CMS in standard jets + E T studies. Therefore, care needs to be taken to establish a datadriven background strategy for these new types of searches. To avoid complications in the projections made in this paper, we cut aggressively on the missing energy to eliminate the QCD background in the ∆φ < 0.4 region for the contact operator and t-channel searches, and relied on a bump-hunt for the s-channel search. A more sophisticated determination of the background uncertainties could potentially relax the missing energy cuts used here and improve the signal reach. One possibility 9 is to use a high-statistics photon+jets sample to determine the missing energy contribution from QCD-specifically, the photon energy could be measured and used to constrain the energy of the jets in the event. This sample could then be used to characterize the QCD background in the signal region with ∆φ < 0.4, where an isolated photon veto would ensure orthogonality with the control region. A detailed experimental study is needed to establish the viability of this method.
This paper focused on the spectacular under-explored collider signatures that result from a strongly interacting hidden sector. We studied the vector contact operator and two of its UV completions, but a variety of other operators are possible [109][110][111][112][113][114]  operators. In addition, astrophysical probes, which we have not discussed here, may also shed light on these non-minimal sectors, either through cascades produced in annihilation events [44,115] or self-interactions [116].
As we have demonstrated, the LHC can play a unique and critical role in the discovery of hidden dark sectors. The framework laid out in this paper provides an exciting opportunity to extend the current DM program at the LHC to these new model frontiers.

Acknowledgments
We are especially grateful to O. Mattelaer, whose prompt and helpful feedback regarding MadGraph was critical for the completion of this work; the patches he provided enabled us to use the bias and gridpack modules together and to successfully generate the t-channel events. We also thank A.

A. Approaching the Contact Operator Limit
In this appendix, we study how the contact operator limit is approached for the s/tchannel UV completions. In the large M Z (M Φ ) limit, the χ χ production can be described by the effective contact operator by integrating out the Z (Φ). In the Z case, this gives where we have taken c = 1 as defined in Eq. (4). For the t-channel case, the spin structure is different since only q R couples to the dark sector. The contact operator in this case is where P L,R are the projection operators for the corresponding helicity component. The extra factor of 2 is present so that when Λ Z = Λ Φ , the total production cross sections for the two cases are equal. Because the protons are not polarized at the LHC, the helicity structures do not lead to differences in the distributions of interest here.
To illustrate how quickly the contact operator limit is approached, Fig. 12 and 13 show the normalized E T and ∆φ distributions for different values of the mediator mass and r inv .
For the s-channel model, we take g q = 0.1 and g χ = 1, and for the t-channel model, we take λ = 1. For low masses, there are significant differences between the two cases. This is due to different production channels along with modifications to the m χχ distributions. As the masses increase to O(10 TeV), the distributions converge to the universal contact operator limit. We take M Z = 100 TeV for the contact operator event generation.

B. Search Insensitivity to Dark Shower Parameters
In this Appendix, we provide a concrete illustration that the searches studied here are insensitive to the detailed choices made for the dark sector parameters, and thus are inclusive. In particular, we vary the following parameters: the dark confinement scale