Non-Minimal Dark Sectors: Mediator-Induced Decay Chains and Multi-Jet Collider Signatures

Keith R. Dienes, 2, ∗ Doojin Kim, 3, † Huayang Song, ‡ Shufang Su, § Brooks Thomas, ¶ and David Yaylali ∗∗ Department of Physics, University of Arizona, Tucson, AZ 85721 USA Department of Physics, University of Maryland, College Park, MD 20742 USA Mitchell Institute for Fundamental Physics and Astronomy, Department of Physics and Astronomy, Texas A&M University, College Station, TX 77843 USA Department of Physics, Lafayette College, Easton, PA 18042 USA


I. INTRODUCTION
One of the most exciting implications of the mounting observational evidence [1] for particle dark matter is that particle species beyond those of the Standard Model (SM) likely exist in nature. Nevertheless, despite an impressive array of experiments designed to probe the particle properties of these dark-sector species, the only conclusive evidence we currently have for the existence of dark matter is due to its gravitational influence on visible-sector particles. The fact that no nongravitational signals for dark matter have been definitively observed would suggest that interactions between the dark and visible sectors are highly suppressed. While it is certainly possible that these two sectors communicate with each other only through gravity, it is also possible that they might communicate through some additional field or fields which serve as mediators between the two sectors as well. These mediators play a crucial role in the phenomenology of any scenario in which they appear, providing a portal linking the dark and visible sectors and giving rise to production, scattering, and annihilation processes involving dark-sector particles.
Moreover, while we know very little about how the * dienes@email.arizona.edu † doojin.kim@tamu.edu ‡ huayangs@email.arizona.edu § shufang@email.arizona.edu ¶ thomasbd@lafayette.edu * * yaylali@email.arizona.edu dark and visible sectors interact, we know perhaps even less about the structure of the dark sector itself. While it is possible that the dark sector comprises merely a single particle species, it is also possible that the dark sector is non-minimal either in terms of the number of particle species it contains or the manner in which these species interact with each other. For example, multi-component dark-matter scenarios have recently attracted a great deal of attention -in large part because such scenarios can lead to novel signatures at colliders, direct-detection experiments, and indirect-detection experiments [2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18]. Moreover, the dark sector may also include additional particle species which are not sufficiently long-lived to contribute to the dark-matter abundance at present time, but nevertheless play an important role in the phenomenology of the dark sector.
In this paper, we make a simple but important observation: in scenarios involving non-minimal dark sectors, any mediator which provides a portal linking the dark and visible sectors generically also gives rise to processes through which the particles in the dark sector decay. For example, in scenarios in which the dark-sector particles have similar quantum numbers and interact with the fields of the visible sector via a common mediator, processes generically arise in which heavier dark-sector species decay to final states including both lighter darkmatter components and SM particles. Successive decays of this nature can then lead to extended decay cascades wherein both visible and dark-sector particles are produced at each step. Depending on the masses and couplings of the particles involved, these decay cascades can have a variety of phenomenological consequences.
In this paper, we shall consider the implications of such mediator-induced decay cascades at colliders. In particular, we shall consider a scenario in which the dark sector comprises a large number of matter fields χ n , all of which couple directly to a common mediator particle which also couples to SM quarks. Cascade decays in this scenario give rise to signatures at hadron colliders involving large numbers of hadronic jets in the final state, either with or without significant missing transverse energy / E T . Signatures of this sort can be somewhat challenging to resolve experimentally, since the jet multiplicities associated with such decay cascades can be quite large. Indeed, the energy associated with any new particle produced at a collider is partitioned among the final-state objects that ultimately result from its decays. Thus, as one searches for events with increasing numbers of such objects and adjusts the event-selection criteria accordingly, it becomes more likely that a would-be signal event would be rejected on the grounds that too few of these objects have sufficient transverse momentum p T .
Of particular interest within scenarios of this sort the regime in which the number of particles within the ensemble is relatively large, in which the mass spacings between successively heavier χ n are relatively small, and in which each χ n preferentially decays in such a way that the resulting daughter χ m is only slightly less massive than the parent χ n . Within this regime, the decay of each of the heavier χ n typically proceeds through a long decay chain involving a significant number of steps. Since each step in the decay chain produces one or more quarks or gluons at the parton level, such scenarios give rise to events with large jet multiplicities, distinctive kinematics, and a wealth of jet substructure. The collider signatures which arise from these mediator-induced decay cascades are in many ways qualitatively similar to those which have been shown to arise in scenarios involving large numbers of additional scalar degrees of freedom which couple directly to the SM Higgs field [19,20] and in superymmetric models in which a softly-broken conformal symmetry gives rise to a closely-spaced discretum of squark and gluino states [21]. Furthermore, we note that if the lifetimes of the lighter states in the dark sector are sufficiently long, these events could also involve displaced vertices or substantial missing transverse energy.
A variety of search strategies relevant for the detection of signals involving large jet multiplicities have already been implemented at the LHC. Searches for events involving a large number N jet ≥ 8 of isolated, high-p T jets with or without / E T [22,23] have been performed, motivated in part by the predictions of both R-parityconserving [24][25][26][27][28] and R-parity-violating [29] supersymmetry and in part by the predictions of other scenarios, such as those involving colorons [30] or additional quark generations [31]. Searches have also been performed for events involving significant numbers of highp T final-state objects -regardless of their identityin conjunction with a large scalar sum of p T over all such objects in the event [32,33]. Searches of this sort are motivated largely by the prospect of observing signatures associated with extended objects such as miniature black holes [34,35], string balls [36,37], and sphalerons [38][39][40]. Searches for events involving multiple soft jets originating from a displaced vertex [41,42] have been performed as well, motivated by the predictions of hidden-valley models [43][44][45], scenarios involving strongly-coupled dark sectors [46], and certain realizations of supersymmetry [25,26,47,48].
The bounds obtained from these searches impose nontrivial constraints on scenarios in which multiple darksector states couple to SM quarks via a common mediator as well. Ultimately, however, we shall show that such scenarios can give rise to extended mediator-induced decay cascades while simultaneously remaining consistent with existing constraints from ATLAS and CMS searches in both the monojet and multi-jet channels. Future colliders -or potentially even alternative search strategies at the LHC -could therefore potentially uncover evidence of such extended decay cascades and thereby shed light on the structure of the dark sector.
This paper is organized as follows. In Sect. II, we describe a simple model involving an ensemble of unstable dark-sector particles with similar quantum numbers, along with a mediator through which these particles couple to the fields of the visible sector. We also discuss the processes through which these dark-sector particles can be produced at a hadron collider. In Sect. III, we investigate the decay phenomenology of these dark-sector particles and examine the underlying kinematics and combinatorics of the corresponding mediator-induced decay chains at the parton level. We also discuss several preliminary parton-level constraints on our model. In Sect. IV, we perform a detector-level analysis of the model and identify a number of kinematic collider variables which are particularly suited for resolving multi-jet signatures of these decay chains from the sizable SM background. In Sect. V, we investigate the constraints from existing LHC monojet and multi-jet searches. In Sect. VI, we identify regions of model-parameter space which can potentially be probed by alternative search strategies at the forthcoming LHC run and beyond. Finally, in Sect. VII, we summarize our main results and discuss a number of interesting directions for future work. We also briefly discuss search strategies which could improve the discovery reach for such theories at future colliders and comment on the phenomenological implications of mediator-induced decay cascades at the upcoming LHC run.

II. OUR FRAMEWORK
Many scenarios for physics beyond the SM give rise to large ensembles of decaying states, including theories involving large extra spacetime dimensions, theories involving strongly-coupled hidden sectors, theories involving large spontaneously-broken symmetry groups, and many classes of string theories. Such ensembles also arise in the Dynamical Dark Matter framework [2,3]. In order to incorporate all of these possibilities within our analysis, we shall adopt a fairly model-independent approach towards describing our χ n ensemble. In particular, we shall adopt a set of rather generic parametrizations for the masses and decays of such states.
Toward this end, in this paper we consider an ensemble consisting of N Dirac fermions χ n , with n = 0, 1, . . . , N − 1, where these particles are labeled in order of increasing mass, such that m n+1 > m n for all n. For concreteness, we shall further assume that the masses m n of these ensemble constituents scale across the ensemble according to a general relation of the form with positive m 0 , ∆m, and δ. Thus, the mass spectrum of our ensemble is described by three parameters {m 0 , ∆m, δ}: m 0 is the mass of the lightest ensemble constituent, ∆m controls the overall scale of the mass splittings within the ensemble, and δ is a dimensionless scaling exponent. The general relation in Eq. (2.1) is capable of describing the masses of states χ n in a number of different scenarios for physics beyond the SM. For example, if the χ n are the Kaluza-Klein excitations of a fivedimensional scalar field with four-dimensional mass m compactified on a circle or line segment of radius/length R, we have {m 0 , ∆m, Likewise, if the ensemble constituents are the bound states of a strongly-coupled gauge theory, or even the gauge-neutral bulk (oscillator) states within many classes of string theories, we have δ = 1/2, where ∆m and m 0 are related to the Regge slopes and intercepts of these theories, respectively. Thus δ = 1/2, δ = 1, and δ = 2 serve as particularly compelling "benchmark" values. We shall nevertheless take m 0 , ∆m, and δ to be free parameters in what follows.
Having parametrized the masses of our dark ensemble states χ n , we now turn to consider the manner in which these states interact with the particles of the visiblesector through a mediator. One possibility is that these interactions occur through an s-channel mediator φ. Assuming that the SM fields ψ which couple directly to φ are fermions, the interaction Lagrangian takes the schematic form where c ψ and c mn denote the couplings between the mediator and the fields of the visible and dark sectors, respectively. An alternative possibility is that these interactions take place via a t-channel mediator. The interaction Lagrangian in this case takes the schematic form While both possibilities allow our dark-sector constituents χ n to be produced at colliders -and also potentially allow these states to decay, with the simultaneous emission of visible-sector states [49,50] -the mediator φ in the t-channel case can carry SM charges. If these include color charge, mediator particles can be copiously pair-produced on shell at hadron colliders, and decay cascades precipitated by the subsequent decays of these mediators can therefore contribute significantly to the signal-event rate in the detection channels which are our main interest in this paper. The interaction in Eq. (2.3) is also comparatively minimal, with the production and decay processes occurring through a single common interaction. We shall therefore focus on the case of a t-channel mediator φ in this paper. In particular, we shall assume that each of the χ n couples to an additional heavy scalar mediator particle φ of mass m φ which transforms as a fundamental triplet under the SU (3) c gauge group of the SM. We shall then take the coupling between φ and each of the χ n to be given by the interaction Lagrangian where q ∈ {u, d, s, c, b, t} denotes a SM quark, where P R = 1 2 (1+γ 5 ) is the usual right-handed projection operator, and where c nq is a dimensionless coupling constant which in principle depends both on the identity of the ensemble constituent and on the flavor of the quark. For concreteness, we shall assume that the c nq scale according to the power-law relation where the masses m n are given in Eq. (2.1), where c 0q > 0 is an overall normalization for the couplings and where γ is a scaling exponent. The interaction Lagrangian in Eq. (2.4) simultaneously describes two critical features of our model. First, we see that our mediator field generically allows the heavier ensemble constituents χ n fields to decay to successively lighter constituents, thereby forming a decay chain. Indeed, according to our interaction Lagrangian, each step of the decay chain proceeds through an effective threebody decay process of the form χ k → qq χ involving an off-shell mediator φ particle, where m < m k . Such a decay chain is illustrated in Fig. 1, with each step of the decay resulting in two parton-level jets. Indeed, such a decay chain effectively terminates only when a colliderstable constituent is reached. If the parameters which govern our model are such that each ensemble constituent χ k decays primarily to those daughters χ whose masses m are only slightly less than m k , relatively long decay chains involving multiple successive such decays can develop before a collider-stable constituent is reached, especially if the first constituent χ n that is produced is FIG. 1. A decay chain in which an ensemble constituent χn experiences S successive decays into increasingly lighter constituents. Each individual decay occurs through a three-body process of the form χn k →qqχn k+1 involving an off-shell φ † and resulting in the emission of two quarks (or parton-level "jets"). Each decay chain effectively terminates once a collider-stable constituent is reached.
relatively massive. In such cases, relatively large numbers of parton-level "jets" -i.e., quarks or gluonscan be emitted. We see, then, that any χ n that is produced -unless it happens to be collider-stable -will generate a subsequent decay chain. The only remaining issue therefore concerns the manner in which such χ n particles might be produced at a hadron collider such as the LHC. However, the relevant production processes are also described by our interaction Lagrangian in Eq. (2.4) in conjunction with our assumption that φ is an SU (3) c color triplet. Indeed, given this interaction Lagrangian, there are a number of distinct possibilities for how the production of the χ n might take place: • The χ n may be produced directly via the process pp → χ mχn at leading order. The Feynman diagram for this process is shown in Fig. 2.
• The χ n may be produced via the process pp → φχ m , followed by a decay of the form φ → qχ n . In such cases, one constituent χ n particle is produced directly while the other results from a subsequent φ decay. Two representative Feynman diagrams for such processes are shown in Fig. 3.
• Finally, because the φ particles are SU (3) c triplets, the χ n may also be produced via the process pp → φ † φ followed by decays of the form φ → qχ n and φ † → χ mq . In such cases, both χ m andχ n are produced via the decays of φ particles. A representative Feynman diagram for such a process is shown in Fig. 4.
These different production processes have very different phenomenologies. For example, since the amplitude for each contributing diagram in Fig. 2 is proportional to the product c m c n , the cross-section -and therefore the event rate -for the overall process is proportional to c 4 0 . By contrast, the event rates for the overall processes in Fig. 3 are proportional to c 2 0 when φ is on-shell, since the factor c n from the decay vertex affects the decay width of φ but not the cross-section for pp → φχ m . Finally, the event rate for the process shown in Fig. 4 is essentially independent of c 0 , as φ is an SU (3) color triplet and can therefore be pair-produced through diagrams involving strong-interaction vertices alone.  Another distinction between these processes is the manner in which their overall cross-sections scale with the number of kinematically accessible components χ n within the ensemble. For example, the event rate for the process shown in Fig. 4 is essentially set by the crosssection for the initial process pp → φ † φ and is thus largely insensitive to the multiplicity of states within the ensemble. By contrast, processes such as those shown in Figs. 2 and 3 scale with the multiplicity of the χ n states that are kinematically accessible, as the contributions from the production of each separate constituent χ n must be added together. For large ensembles, this can lead to a significant enhancement of the total cross-sections for such processes.
All of these processes are capable of giving rise to large numbers of parton-level jets, particularly if the χ n that are produced give rise to long subsequent decay chains. Additional parton-level jets may also be produced as initial-state radiation or radiated off any internal lines associated with strongly-interacting particles. However, these different processes differ in the minimum numbers of parton-level jets which may be produced. For example, the direct-production process in Fig. 2 can in principle be entirely jet-free as long as only collider-stable ensemble constituents are produced. Likewise, the processes in Fig. 3 must give rise to at least one jet, and indeed processes of this form involving an on-shell φ particle often turn out to provide the dominant contribution to the pp → χ mχn + j monojet production rate at the LHC within our model. By contrast, the process in Fig. 4 must give rise to at least two jets.
In order to streamline the analysis of our model, we shall make two further assumptions in what follows. First, we shall assume that the χ n couple only to the up quark, taking c 0q = 0 for q = {d, s, c, b, t}. Thus only the c nu coefficients are non-zero, and we shall henceforth adopt the shorthand notation c n ≡ c nu for all n. Second, we shall assume that N , the total number of constituents in our ensemble, is not only finite but also chosen so as to maximize the size of the ensemble while nevertheless ensuring that all of the ensemble constituents {χ 0 , χ 1 , ..., χ N −1 } are kinematically accessible via the decays of φ. In other words, we shall take N to be the largest integer such that where m q is the mass of the final-state (up) quark. While this last assumption is not required for the selfconsistency of our model, we shall see that it simplifies the resulting analysis and leads to an interesting phenomenology. With these simplifications, our framework is characterized by six free parameters: {m 0 , ∆m, δ, m φ , c 0 , γ}. These six parameters determine the masses of the ensemble constituents χ n , the probabilities for producing these different ensemble constituents from the decays of φ, and the branching fractions that govern the possible subsequent decays of these constituents. Indeed, depending on the values of these parameters, many intricate patterns of potential decay chains are possible which collectively contribute to jet production. For example, in some regions of parameter space, the lifetimes of the heavier ensemble states are shorter than those of the lighter states, while in other regions the opposite is true (even though the lightest state is of course stable in all cases). Likewise, in some regions of parameter space, each χ n preferentially decays to daughters χ for which m m n , while in other regions the preferred daughters χ are only slightly lighter than χ n . Finally, in some regions of parameter space, the contributions to jet production coming from the processes illustrated within Figs. 2 and 3 might dominate, while in other regions of parameter space the contributions from the process illustrated within Fig. 4 might dominate. Thus, even though our framework is governed by only the single interaction in Eq. (2.4), this framework is extremely rich and many different resulting phenomenologies are possible.
In our analysis of this framework, we shall be interested primarily in those regions of parameter space which potentially give rise to extended jet cascades at colliders such as the LHC. We shall therefore be interested in those regions of parameter space that give rise to a relatively large number of kinematically accessible ensemble constituents χ n which decay promptly on collider timescales and for which the corresponding decays occur along decay chains involving a relatively large number of steps. Beyond this, however, we will not make any further assumptions concerning the values of these parameters. Of course, within our parameter-space regions of interest, there may exist subregions in which some of the other constituents will have very long lifetimes -lifetimes which potentially exceed the age of the universe. In such cases, these long-lived constituents might serve as potential dark-matter candidates of the sort intrinsic to the Dynamical Dark Matter framework [2,3], with the decay cascades arising from the decays of the shorterlived ensemble constituents potentially serving as a signature of this framework. However, we shall not make any such additional assumption in this paper.

III. DECAY-CHAIN PHENOMENOLOGY AND THE GENERATION OF EXTENDED JET CASCADES
We shall now demonstrate that the model presented in Sect. II is capable of giving rise to extended jet cascades at the LHC. In this section our analysis shall be purely at the parton level, while in Sect. IV we shall pass to the detector level.
In principle, mediator-induced decay cascades can arise from any of the processes illustrated in Figs. 2-4. Of course, our eventual goal in this paper is not merely to demonstrate that cascades of this sort with large jet multiplicities are possible, but that they might emerge while simultaneously satisfying existing LHC monojet and multi-jet constraints. For this, of course, the contributions from all of the processes discussed in Sect. II will ultimately matter. This will be discussed in Sect. VI.
We shall begin by outlining the kinematics and combinatorics of the mediator-induced decay chains precipitated by the production proesses illustrated in Figs. 2-4. We shall then discuss how the emergence of extended decay chains yielding large numbers of jets depends on the parameters which characterize our model, and identify a region of parameter space within which such extended decay chains emerge naturally while satisfying certain in-ternal self-consistency constraints.
A. The structure of the decay chain: Kinematics and combinatorics Each of the processes illustrated in Figs. 2-4 eventually results in decay chains of the sort illustrated in Fig. 1. In cases such as that illustrated in Fig. 2, our ensemble constituents χ m and χ n are produced directly. Each then becomes the heaviest component of a subsequent decay chain. By contrast, in cases such as that illustrated in Fig. 4, the particles that are produced directly are the mediator particles φ and φ † . It is the subsequent decays of these mediators which then trigger the unfolding of our decay chains. Finally, cases such as those illustrated in Fig. 3 exhibit what may be considered a "mixture" between these two production mechanisms.
In this section, rather than analyze each process separately, we shall instead treat them together by focusing on the two primary classes of decays which establish and sustain their decay chains. These are Note that although we have written these decay processes in generality, we shall -as discussed in Sect. II -restrict our attention to the case in which all quarks participating in these processes are up-quarks (i.e., q = q = u) in what follows. For cases involving the initial production of a mediator φ, the first process in Eq. (3.1) in some sense "initializes" the decay chain by producing the heaviest χ n constituent within the chain. This initialization process simultaneously produces one jet. The second process then iteratively generates the subsequent decays -each producing two jets -which collectively give rise to the decay chain through which this heaviest constituent χ n sequentially decays into lighter constituents. By constrast, for cases involving the direct production of an ensemble constituent χ n , only the second process in Eq. (3.1) is relevant for generating the subsequent decay chain. Even with a fixed initial state, each of the decay processes in Eq. (3.1) can result in a variety of different daughter particles. Indeed, starting from a given mediator particle φ, it is possible for any kinematically-allowed constituent χ n to be produced via the first process, each with a different probability. Likewise, a given χ n can generally decay into any lighter constituents via the second process, with each possible daughter state occurring with a different probability as well. The sequential repetitions of this latter process thus lead to a proliferation of independent decay chains, with each decay chain terminating only when the lightest ensemble constituent is ultimately reached. (For practical purposes we may also consider a given decay chain to have effectively terminated if the lifetimes for further decays exceed collider timescales.) Thus, combining these effects, we see that each of the processes sketched in Figs. 2-4 actually spawns a large set of many different possible decay chains, each with its own relative probability for occurring and each potentially producing a different number of jets.
It is not difficult to study these decay chains analytically. Within any particular region of the model parameter space, the first step is to calculate the partial widths Γ φn ≡ Γ(φ † →qχ n ) and Γ n ≡ Γ(χ n →q qχ ) associated with the processes in Eq. (3.1). With q = q = u and with the up-quark treated as having a negligible mass, we find that Γ φn for any n ≤ N − 1 is to a very good approximation given by Likewise, we find that Γ n takes the form Under the assumption that no additional interactions beyond those in Eq. (3.1) contribute non-negligibly to the total width of either φ or the χ n , the total decay width Γ φ of φ is then simply with a corresponding φ lifetime τ φ ≡ 1/Γ φ . Likewise, the total decay width Γ n for each ensemble constituent χ n is simply with a corresponding constituent lifetime τ n ≡ 1/Γ n . Indeed, the lightest ensemble constituent χ 0 is absolutely stable, with Γ 0 = 0. Of course, the results in Eqs. (3.2) and (3.3) assume that the φ and χ n particles have total decay widths which are relatively small compared with their masses. This is a self-consistency constraint which will ultimately be found to hold across our eventual parameter-space regions of interest.
While Γ φ and Γ n determine the overall timescales for particle decays within our model, it is the branching fractions BR φn ≡ Γ φn /Γ φ and BR n ≡ Γ n /Γ n which effectively determine the probabilities associated with the various possible decay chains that can arise. The behavior of these branching fractions is essentially determined by the interplay between two factors. The first of these factors is purely kinematic in origin and arises due to phasespace considerations which suppress the partial widths for decays involving heavier ensemble constituents in the final state. Thus, this factor always decreases as the index which labels this final-state ensemble constituent increases. The second factor arises as a result of the scaling of the individual coupling constants c n in Eq. (2.5) across the ensemble. Depending on the value of the scaling exponent γ, this factor may either increase or decrease with the final-state index.
In the regime in which γ 0, the mediator φ and all of the χ n decay preferentially to χ with relatively small values of . Thus, for these parameters, the corresponding decay chains typically involve only one or a few steps and do not give rise to large multiplicities of jets. By contrast, in the opposite regime in which γ is positive and sufficiently large that the enhancement in c 2 with increasing overcomes the phase-space suppression, decays to χ with intermediate values of are preferred. Within this regime, long decay chains can develop and events involving large numbers of hadronic jets naturally arise.
In Fig. 5, we plot BR n as a function of the daughterparticle mass m for several different choices of γ, holding n fixed. For these plots we have chosen the illustrative values m φ = 1 TeV, m 0 = 100 GeV, ∆m = 10 GeV, δ = 1, and c 0 = 0.1. We have also chosen n = 70 for the parent, implying a parent mass m n = 800 GeV. On the one hand, we observe from Fig. 5 that BR n indeed decreases monotonically with for negative γ -and indeed even for γ = 0 -as expected. On the other hand, we also observe that decays to final states with > 0 are strongly preferred even for γ = 1. Thus, even a moderate positive value of γ is sufficient to ensure that decay cascades with multiple steps will be commonplace. Indeed, the shapes of the curves in Fig. 5 do not depend sensitively on the chosen values of ∆m or δ as long as the number of constituents χ lighter than χ n is sufficiently large. This is because for fixed m φ and m n , the branching fraction BR n can be viewed as a function of the single variable r n . Thus, while changing ∆m and δ changes the values of r n at which this function is evaluated, it has no effect on form of the function itself.
Given our results for the relevant branching fractions, we now have the ingredients with which to calculate the probabilities associated with particular sequences of decays -i.e., particular decay chains -in our model. For simplicity, let us focus on the regime in which all χ n with n > 0 decay promptly within the detector. Under this assumption, each decay chain precipitated by the production of a given ensemble constituent terminates only The results shown here correspond to the parameter values n = 70, m φ = 1 TeV, m0 = 100 GeV, ∆m = 10 GeV, δ = 1, and c0 = 0.1 -a choice of parameters for which the mass of the parent is mn = 800 GeV. We see that when γ is large (even if only moderately so), the couplings c which increase with are able to partially overcome the increasingly severe phase-space suppressions that also arise for larger , allowing the parent χn to decay preferentially to daughters χ with intermediate values of . This phenomenon underpins the existence of decay chains with many intermediate steps, allowing such long decay chains to dominate amongst the set of all possible decay chains that emerge from a given parent χ .
when χ 0 (the lightest element within the ensemble) is produced. Within this regime, then, the probabilityP(S) that such a decay chain will have precisely S steps after the initial production of an ensemble constituent (i.e., the probability that our decay chain proceeds according to a schematic of the form where we of course understand that BR ij = 0 for all j ≥ i and where the initial factor BR (prod) n0 is the relative probability that the specific ensemble constituent χ n0 is originally produced. This last factor depends on the production process, with BR (prod) n0 = BR φn0 in the case of indirect production through the mediator φ and BR (prod) n0 = 1 for direct χ n0 production.
This result then allows us to calculate the probabilities P (N jet ) that each of the processes in Figs. 2-4 yields precisely N jet jets at the parton level. First, we observe that each of these processes directly or indirectly gives rise to two ensemble constituents χ n and χ m . While producing these ensemble constituents, each process also produces a certain number ζ of parton-level jets; indeed ζ = 0, 1, 2 for the processes sketched in Figs. 2, 3, and 4, respectively. Each of these two constituents then spawns a set of decay chains, with each step producing exactly two parton-level jets. Thus, for each process in Figs. 2-4, the corresponding probability P (N jet ) that a single event will yield a specified total number N jet of parton-level jets (from either quarks or anti-quarks) is therefore given by Of course, for each process N jet is restricted to the values In Fig. 6, we plot P (N jet ) as a function of N jet for several different choices of the scaling exponent γ. For this figure we have again taken the illustrative values m φ = 1 TeV, m 0 = 100 GeV, ∆m = 10 GeV, δ = 1, and c 0 = 0.1, which together imply N = 90. For concreteness we have also chosen ζ = 2, corresponding to the process sketched in Fig. 4 for which BR (prod) n0 = Br φ,n0 . For this choice of parameters, we see that the decay cascades initiated by parent-particle decays can indeed give rise to significant numbers of jets at the parton level. Indeed, we observe from this figure that for γ 1, the majority of events in which a pair of mediator particles is produced have N jet 10. Similar results also emerge for the processes in Figs. 2 and 3. We conclude, then, that the example model described in Sect. II is capable of giving rise to extended jet cascades at the parton level. Indeed, the existence of this signature does not require any fine-tuning, and emerges as an intrinsic part of the phenomenology of the model.

B. Constraining the model parameter space
Our analysis in Sect. III A focused on the general kinematic and combinatoric structure of the decay chains that give rise to extended jet cascades in our model. However, there are a number of additional constraints which must also be addressed before we can claim that our model is actually capable of giving rise to signatures involving large jet multiplicities at a collider such as the LHC. Some of these additional constraints are fairly generic, and can be discussed even at the parton level. Indeed, as we shall now demonstrate, satisfactorily addressing these concerns will enable us to place several important additional constraints on the parameter space of our model. However, other constraints are more phenomenological and process-specific, having to do with existing LHC bounds on monojet and multi-jet signatures. Discussion of these latter constraints will therefore be deferred to Sect. V.
As discussed in Sect. II, our model is described by six parameters: {m 0 , ∆m, δ, m φ , c 0 , γ}. The first three of these parameters together describe the entire mass spectrum m n of the ensemble constituents, and the fourth is nothing but the mass m φ of the mediator φ. As we have seen, however, the all-important branching fractions BR φn and BR n depend on only the ratios of these masses. Likewise, the quantity N which sets an upper limit on the number of possible jets that can be produced (and which was defined in Sect. II as the number of ensemble constituents which are kinematically accessible via the decays of φ) also implicitly depends on these ratios. Together, these considerations then govern the choices of mass ratios in our system. However, this still leaves an overall mass scale which we may take to be m φ itself. Likewise, we have not yet constrained the two parameters c 0 and γ which together describe the spectrum of couplings in our model through Eq. (2.5). Of course, we have already seen in Figs. 5 and 6 that only when γ is sufficiently positive and large do our decays preferentially proceed through sufficiently small steps that allow decay chains with sufficiently large numbers of steps to develop. However, this still leaves m φ and c 0 unconstrained. Fortunately, there exist additional phenomenological constraints which will enable us to determine suitable ranges for these two remaining parameters as well.
First, although we have demonstrated how ex-tended mediator-induced decay cascades might potentially emerge from our model, we must also ensure that the overall cross-sections for producing these cascades are sufficiently large that the resulting multi-jet signal could actually be detected over background. While these cross-sections are certainly affected by the cascade probabilities discussed above, their overall magnitudes are set by the simpler cross-sections associated with the subprocesses for the production of the initial states that trigger these cascades. For the diagrams sketched in Figs. 2-4, these production cross-sections are respectively given by Calculating these cross-sections is relatively straightforward, and in Fig. 7 we display our results as functions of m φ for a center-of-mass (CM) energy of √ s = 13 TeV. In particular, the solid curves correspond to the parameter choices m 0 = 500 GeV, ∆m = 50 GeV, c 0 = 0.1, and δ = 1 with γ = 1, while the dashed curves correspond to the same values of m 0 , ∆m, c 0 , and δ, but with γ = 3. We note that since σ φφ has no dependence at leading order on the mass spectrum of the ensemble constituents (and therefore on the values of the parameters m 0 , ∆m, and γ), the corresponding curves for both of these parameter choices are identical. We also note that the wiggles which appear in the curves for σ χχ and σ χφ , especially at small m φ , are the consequence of threshold effects which arise due to the discrete changes in N that occur as m φ changes, in accordance with Eq. (2.6).
We observe from Fig. 7 that the cross-section for φφ pair-production dominates for small m φ , but falls rapidly from σ φφ ∼ 500 fb to σ φφ ∼ 10 −3 fb as the mass of the mediator increases from m φ = 500 GeV to m φ = 2500 GeV. By contrast, the cross-sections for the other two production processes either grow with m φ or fall less sharply over the range of m φ shown. This is primarily a consequence of the corresponding increase in N , which in turn results in more individual production processes involving different χ n . Since increasing γ in turn increases the individual production cross-sections for the heavier χ n , both σ χχ and σ φχ are noticeably larger for γ = 3 than for γ = 1. We also note that across the entire range of m φ shown, σ χχ and σ φχ are both larger than 0.01 fb, indicating that these processes could potentially lead to observable signals at the LHC.
We now turn to examine how general considerations involving the coupling structure of our model serve to constrain the coupling parameter c 0 . Since all of the couplings c n in our model are proportional to c 0 , we see that c 0 serves as an overall proportionality factor for both Γ φ and Γ n . In particular, our results in Eqs. imply that Γ φ ∝ c 2 0 and Γ n ∝ c 4 0 . Fortunately, the value of c 0 is constrained by a number of theoretical consistency conditions and phenomenological constraints. For example, given the perturbative treatment leading to the results in Eqs. (3.2) and (3.3), self-consistency requires that we must impose the perturbativity requirement that c n 4π for all 0 ≤ n ≤ N − 1. Given the general expression in Eq. (2.5), we see that the value of c n generally increases as a function of n for γ > 0 and decreases for γ < 0. For any combination of model parameters we must therefore demand that (3.10) In addition, for cases in which the decay chains are initiated through the direct production of the mediator φ, we are assuming that φ behaves like a physical particle rather than a broad resonance. We must therefore also demand that c 0 be sufficiently small that Γ φ m φ , which in turn requires This latter constraint can occasionally surpass the one in Eq. (3.10). For example, for γ = 0 we learn from Eq. (3.10) that c 0 < ∼ 4π, yet even in such cases Γ φ can occasionally exceed m φ , even with only a few ensemble constituents.
In addition to these criteria for theoretical consistency, there are also a number of further constraints which we shall take into account in defining our region of interest within the full parameter space of our model. We emphasize that these are not necessarily inviolable constraints on the model, but rather conditions which we shall impose either for sake of clarity in simplifying our analysis or in order to restrict our focus within the model parameter space to regions in which long decay chains arise.
For example, in order for a decaying particle ensemble to give rise to observable signatures of mediator-induced decay cascades at the LHC, many of the χ n constituents must of course decay promptly within the detector. In general, the decay length L n of χ n in the detector frame is given by L n ≡ βγcτ n , where τ n = Γ −1 n is the proper lifetime of χ n and where β = v/c and γ = (1 − β 2 ) −1/2 are the usual relativistic factors. Since we shall generally be interested in decay chains with many steps -chains in which the dominant individual decays produce daughters that are not overwhelmingly lighter than their parents -none of the ensemble constituents will be excessively boosted upon production. We can therefore treat the relativistic factor βγ as a mere O(1) numerical coefficient in order to obtain an order-of-magnitude estimate of the bound. This is particularly convenient since these factors generally depend on the detailed structure of the decay chain and therefore differ from one event to the next. We will therefore estimate the characteristic length scale at which a given ensemble constituent decays as cτ n . Broadly speaking, if cτ n 1 cm, a particle of species χ n will typically appear as either a displaced vertex or as / E T at the LHC. By contrast, if cτ n O(1 cm), such a particle will tend to decay promptly within the detector. It is these latter decays which are our focus.
In Fig. 8, we plot the length scales cτ n as functions of n for several different choices of model parameters. The red, green, blue, and black curves correspond to the parameter choices γ = {−1, 0, 1, 2}, respectively. The solid curves correspond to the choice c 0 = 0.02, while the dashed curves correspond to the choice c 0 = 0.1. The values of the remaining model parameters are taken to be m φ = 1 TeV, m 0 = 100 GeV, ∆m = 10 GeV, and δ = 1 for all curves shown. We emphasize that the perturbativity criterion in Eq. (3.10) is satisfied for all curves shown. Note that for the parameters shown, the decay lengths tend to decrease as functions of n. This remains true even if γ = −1, indicating that the total phase space available for the decays of χ n increases with n more rapidly than the associated couplings c n might decrease. For c 0 = 0.02, we see from Fig. 8 that a significant number of the ensemble constituents have cτ n O(1 cm) and therefore do not decay promptly within the detector. Indeed, depending on the amount by which cτ n exceeds O(1 cm), these χ n would either decay a measurable distance away from the primary vertex (thereby giving rise In general, long decay chains can certainly arise even in cases for which the lighter χ n have values of cτ n exceeding O(1 cm). In such cases the decays of relevance for our purposes would simply be the decays of the heavier constituents, with the decays of the lighter constituents subsequently occurring either with displaced vertices or completely outside the detector. Indeed, such situations could potentially give rise to many interesting signatures which will be discussed further in Sect. VII. However, for simplicity in what follows, we shall henceforth restrict our attention to the region of parameter space within which In such cases, all possible decays of our ensemble constituents will occur within the detector, thereby allowing us to regard our decay chains as terminating only when the stable ensemble "ground state" χ 0 is reached. Since τ n ∝ c −4 0 , requiring that our ensemble constituents satisfy the criterion in Eq. (3.12) is tantamount to imposing a lower bound on c 0 for any particular assignment of the remaining parameters which characterize our example model. Indeed, as illustrated in Fig 8, reducing c 0 below this bound only inhibits the decay rates . In general we see that there exists an ample allowed range for c0 within which both constraints can be satisfied simultaneously, but this range becomes increasingly narrow as m φ or γ becomes large or as m0 becomes small. of our ensemble constituents to a point beyond which some of the lighter ensemble constituents will begin to exhibit displaced vertices or decay outside the detector. However, since c 0 is also bounded from above by the perturbativity constraint in Eq. (3.10) and/or by our requirement that Γ φ m φ , we see that there is a tension between these two groups of constraints.
In Fig. 9, we illustrate how the competition between the perturbativity constraint and the prompt-decay constraint play out within the parameter space of our model. While the contours in both panels in Fig. 9 correspond to ∆m = 10 GeV and δ = 1, those in the top panel correspond to the choice m 0 = 10 GeV while those in the bottom panel correspond to the choice m 0 = 100 GeV. As is evident from Fig. 9, there are indeed regions of parameter space within which both the perturbativity constraint and the prompt-decay condition can be simultaneously satisfied. Nevertheless, it is also evident from this figure that as γ increases, a significant tension rapidly develops between these two bounds. As we have already seen, the regions of parameter space within which γ 1 turn out to be the regions in which extended mediator-induced decay cascades develop. As a result, this tension will ultimately have important consequences for our model.
It is also relatively straightforward to understand the differences between the top and bottom panels of Fig. 9. In general, for γ ≥ 0 the perturbativity constraint in Eq. (3.10) depends on the properties of χ N −1 . By contrast, the prompt-decay condition in Eq. (3.12) depends on the properties of χ 1 . Given the functional form for c n in Eq. (2.5), we see that c 1 is essentially insensitive to γ in the ∆m m 0 regime, as indicated in the bottom panel of Fig. 9. Likewise, the perturbativity bound becomes increasingly sensitive to γ as the ratio m N −1 /m 0 increases.
For all of these reasons, we shall limit our attention in this paper to regions of parameter space in which γ > ∼ 1, c 0 = 0.1. Indeed, as we have seen, these are the regions in which the processes illustrated in Figs. 2-4 can give rise to observable signatures involving relatively large numbers of jets at the parton level.

IV. FROM PARTON LEVEL TO DETECTOR LEVEL: WHEN YOU'RE A JET, ARE YOU A JET ALL THE WAY?
While it is certainly instructive to examine the collider phenomenology of our model at the parton level, what ultimately matters, of course, are the signatures that can actually be observed at the detector level. Indeed, not all of the parton-level "jets" produced from mediator-induced decay cascades at the parton level ultimately translate to individual reconstructed jets at the detector level. Moreover effects associated with initialstate radiation (ISR), final-state radiation (FSR), and parton-showering can give rise to additional jets at the detector level. Thus, it is critical that we investigate how the parton-level results we have derived in Sect. III are modified by these considerations at the detector level.
Toward this end, our analysis shall proceed as follows. For any given choice of model parameters, we generate signal events for the initial pair-production processes pp → φχ m , pp → χ mχn , and pp → φ † φ at the √ s = 13 TeV LHC using the MG5@aMC [51] code package. We then evaluate the cross-sections for these processes using this same code package. Due to the complexity of the decay chains which arise in our model, we treat the final-state particles produced during each step of the chain as being strictly on shell and simulate the decay kinematics using our own Monte-Carlo code. We have confirmed that the kinematic distributions obtained using our decay code agree well with those obtained from a full implementation of our model in MG5@aMC in cases in which the decay chains are short and such a comparison is feasible. The resulting set of three-momenta for the final-state particles in each event was then passed to Pythia 8 [52] for parton-showering and hadronization. Detector effects were simulated using Delphes 3 [53]. Jets were reconstructed in FastJet [54] using the antik T clustering algorithm [55] with a jet-radius parameter R = 0.4. This procedure has the practical benefit of allowing us to examine the kinematics of long decay chains. However, it is important to note that this procedure neglects neglects certain considerations which can slightly modify the kinematics of the decay cascades and have O(1) effects on the cross-sections for the relevant final states. First, our procedure neglects the interference between the contribution to the overall amplitude for the process pp → χ mχn + j from pp → φχ m production followed by the decay φ →χ n j of the on-shell φ particle and the contribution from processes similar to pp → χ mχn , but in which an additional quark or gluon is produced as initial-state radiation or radiated off the internal φ line. However, since we find that the former contribution vastly dominates over the latter, the effect of neglecting these interference effects is not expected to be significant. Second, our procedure does not employ any jet-matching scheme in order to correct for double-counting in regions of phase space populated both by matrix-elementgeneration and parton-showering algorithms. This effect is not expected to have a significant impact on our results. Third, our procedure also ignores the possibility that any χ n which appear in decay chains or any of the mediators produced by the processes pp → φχ m or pp → φ † φ could be off shell. Once again, the impact on our results is not expected to be significant.
We begin by examining several experimental observables which are potentially useful for discriminating between signal and SM backgrounds. Clearly, the most distinctive feature of these extended mediator-induced decay cascades is the sheer multiplicity of "jets" at the parton level. Thus, given limited statistics, observables which characterize the overall properties of the event as a whole are likely to provide more distinguishing power than the observables which involve particular combina-tions of the momenta of individual jets in the event, due to the combinatorial issues associated with the latter. We therefore focus primarily on the former class of observables in what follows. These observables include N jet and / E T , the distributions of the magnitude p Tj of the transverse momentum of all jets in the event, and the scalar sum (4.1) In order to assess the extent to which showering, hadronization, and detector effects modify the distributions of p Tj , N jet , / E T , and H T , it is useful to compare the parton-level distributions of these observables to the corresponding detector-level distributions. In constructing the parton-level distributions of all of these collider observables, we consider each quark and anti-quark in the final state to be a "jet", regardless of its proximity in (η j , φ j )-space to any other such "jets" in the event, where η j and φ j respectively denote the pseudorapidity and azimuthal angle of a given jet. Moreover, we impose no cuts on either p Tj or η j . By contrast, in constructing the detector-level distribution of p Tj , we require that every jet in a given event satisfy p Tj > 20 GeV and |η j | < 5. Furthermore, in order to be counted as a jet at the detector level, a would-be jet must be separated from every other, more energetic jet in the event by a distance For purposes of illustration, we identify three representative benchmark points within the parameter space of this model for which these criteria discussed in Sec. III B are satisfied, but for which different classes of production processes dominate the event rate in the multi-jet channel at large N jet . The parameter choices associated with these benchmarks are provided in Table I. Benchmark A is representative of the regime in which both pp → φ † φ and pp → φχ m provide significant contributions to the event rate in the multi-jet channel at large N jet , with these two processes contributing at roughly the same order. Benchmark B is representative of the   regime in which pp → φχ m dominates the event rate, while Benchmark C is representative of the regime in which pp → χ mχn dominates. In Fig. 10, we show the normalized distributions of N jet obtained for Benchmarks A (left panel), B (middle panel), and C (right panel). The distributions shown include the individual contributions from pp → φχ m , pp → χ mχn , and pp → φ † φ, each weighted by the crosssection for the corresponding process. The red histogram in each panel shows the distribution obtained at the parton level (with quarks, anti-quarks, and gluons considered to be "jets"), while the blue histogram shows the corresponding distribution at the detector level.
For Benchmark A, we see from 10 that the partonlevel and detector-level N jet distributions look quite similar and that both of these distributions peak at around N jet = 6. For Benchmark B, by contrast, the partonlevel distribution exhibits local maxima at both N jet = 7 and at N jet = 9. This behavior follows from the fact that processes of the form pp → φχ n , which yield an odd number of parton-level jets, dominate the production crosssection for this benchmark. Moreover, we observe that in going from the parton level to the detector level, the N jet distribution shifts to slightly lower values. Several effects contribute to this reduction in N jet . First, jets associated with soft, isolated quarks or anti-quarks may fall below the p Tj > 20 GeV detector-level threshold for jet identification. Moreover, due to the large multiplicity of jets in these events, the hadrons associated one or more of these jets frequently end up in such close proximity in (η j , φ j ) space that they will be clustered together as a single jet at the detector level. For Benchmark C, the parton-level N jet distribution peaks around N jet = 10, with most of the final states containing even numbers of jets. The distribution is smoothed out at the detector level, but otherwise retains the same overall shape.
One of the primary messages of Fig. 10 is that our benchmarks all give rise to a significant population of events with large jet multiplicities even at the detector level. Indeed, for Benchmarks A, B, and C, we find that the fraction of events for which N jet ≥ 9 at the detector level is 16.3%, 24.3%, and 54.8%, respectively.
In Fig. 11, we show the normalized distributions for the other collider observables we consider in our analysis for our three parameter-space benchmarks. From left to right, the panels in each row of the figure correspond to the observables p Tj , / E T , and H T . The distributions in the top, middle, and bottom rows of the figure correspond to Benchmarks A, B, and C, respectively. The red histogram in each panel once again shows the distribution obtained at the parton level, while the blue histogram shows the corresponding distribution at the detector level.
In interpreting the results displayed in Fig. 11, we begin by noting that the parton-level p Tj distributions for all of our benchmarks are sharply peaked toward small values of p Tj . In other words, as one might expect, given the length of the decay chains in these decay-cascade scenarios, a significant fraction of the quarks and antiquarks produced in these decay chains tend to be extremely soft. However, we also note that the distributions for Benchmarks A and B are more sharply peaked than the distribution for Benchmark C. This is ultimately a result of m φ being larger for this latter benchmark than for the other two. A larger value of m φ implies a larger value of N , and the fact that γ > 0 for Benchmark C implies that production processes involving the heavier χ n present in the ensemble will dominate. The average CM energy associated with any of the production processes in Figs. 2-4 is consequently larger for Benchmark C than it is for Benchmark A or B, which results in a higher average p Tj . We also observe that since a p Tj > 20 GeV threshold is required for jet identification at the detector level, many of the soft "jets" present at the parton level for each of our benchmarks do not translate into jets at the detector level.
(1/σ)(dσ/dp (1/σ)(dσ/dp (1/σ)(dσ/dp  Table I. The distributions in the top, middle, and bottom rows of the figure correspond to Benchmarks A, B, and C, respectively. The red histogram in each panel shows the distribution obtained at the parton level (with quarks, anti-quarks, and gluons considered to be "jets"), while the blue histogram shows the corresponding distribution at the detector level.
In comparison with the p Tj distributions shown in Fig. 11, the corresponding / E T and H T distributions vary more dramatically from one benchmark to the next. Perhaps not unsurprisingly, the parton-level H T distribution for Benchmark C peaks at a higher value H T than do the distributions of this same variable for Benchmarks A and B, again owing to the fact that m φ is larger for this benchmark. More interestingly, however, we also see that the parton-level and detector-level H T distributions for Benchmark C are almost identical, while the detectorlevel H T distributions for Benchmarks A and B differ drastically from the corresponding distributions at parton level. The discrepancy between the parton-level and detector-level H T distributions for these two benchmarks is ultimately a result of the p Tj > 20 GeV threshold for jet-identification at the detector level. As discussed above, the jets produced through mediator-induced decay cascades are have a higher average p Tj for Benchmark C than they do for Benchmarks A or B, and consequently the H T distribution for this benchmark is affected less by the cuts. A similar effect, albeit less pronounced, is also observed in the / E T distributions for our benchmarks. We also note that in general, the detector-level / E T and H T distributions for all three of these benchmarks exhibit slightly longer tails than do the corresponding partonlevel distributions.
The results displayed in Fig. 11 indicate that the shapes of the parton-level p Tj , / E T , and H T distributions resulting from mediator-induced decay cascades vary across the parameter space of our model. Moreover, we see that the extent to which the parton-level and detector-level distributions of the same variable differ also depends non-trivially on the location within that parameter space.

V. DETECTION CHANNELS
A variety of different search strategies sensitive to particular kinds of physics beyond the SM which give rise to large numbers of jets have been implemented by both the ATLAS and CMS collaborations [22,23,32,33,41,42]. Some of these turn out to be more suitable for detecting and constraining the large-jet-multiplicity events produced by the mediator-induced decay cascades in our example model than others.
One such class of search strategies are those primarily tailored to the detection of microscopic black holes and sphalerons. The leading constraints on such exotic objects are currently those from a CMS analysis [33] performed with 35.9 fb −1 of integrated luminosity at √ s = 13 TeV. The constraints obtained from a similar ATLAS study [32] performed with 3.6 fb −1 at the same CM energy are less competitive. These searches turn out to be less effective for our model due to the high H T threshold for signal-event selection: H T > 900 GeV in the CMS search and H T > 800 GeV in the ATLAS search. These cuts are imposed in order to reduce the SM multi-jet background. By contrast, for our signal events, either the H T distribution is peaked below 800 GeV or the signal cross-section is too small to be significant. With only 35.9 fb −1 of integrated luminosity, no meaningful constraints can be derived on our model parameter space from the analysis in Ref. [33].
Another class of search strategies commonly adopted in new-physics searches in channels involving large jet multiplicities are those tailored to the detection of scenarios involving long-lived hidden-sector states [41,42]. In searches of this sort, events are selected on the basis of one or more displaced vertices being present. Such searches can indeed be relevant for the detection of extended decay cascades in our example model, but only within the regime in which one or more of the χ n are sufficiently long-lived that they give rise to such vertices. Since we have focused in this paper on the region of parameter space within which region all of the χ n with n > 0 decay promptly within the ATLAS or CMS detector, such searches also have no bearing on our analysis.
By contrast, it turns out that the search strategies which are particularly relevant for probing the parameter space of our model are those commonly adopted in searches for supersymmetry in the multi-jet + / E T channel. In searches of this sort, signal events are selected primarily on the basis of N jet and / E T . The leading constraints on our model from such searches are currently those from LHC √ s = 13 TeV searches by the ATLAS collaboration [23] with 36.1 fb −1 of integrated luminosity and those by the CMS collaboration [22] with 35.9 fb −1 of integrated luminosity. The ATLAS search turns out to be the more relevant of the two for constraining our example model, primarily because the CMS analysis includes a sizable / E T cut. This leads to a significant reduction in statistics for our signal process.
For this reason, we assess the constraints on our model from the multi-jet channel by modeling our triggering requirements and event-selection criteria after those employed in Ref. [23]. In particular, we adopt the same triggering criteria that we used in constructing the detectorlevel N jet , / E T , and H T distributions in Sect. IV. In addition, primarily in order to reduce the SM multi-jet background, we impose the / E T cut Following Ref. [23], we include only the three-momenta of jets with pseudorapidities in the range |η j | < 4.5 when calculating / E T for a given event; likewise, we include only those jets with p Tj > 40 GeV and |η j | < 2.8 within the scalar sum in Eq. (4.1) when calculating H T . Finally, we impose a cut on the total number of jets in the event which exceed a given p Tj threshold. More specifically, we define N 50 jet to be the number of jets with p Tj > 50 GeV in a given event and N 80 jet to be the number of jets with p Tj > 80 GeV. We then perform an inclusive search involving a number of different signal regions defined by different combinations of the threshold cuts N 50 jet ≥ {8, 9, 10, 11} and N 80 jet ≥ {7, 8, 9}. For each channel, we impose the corresponding constraint on the parameter space of our example model by comparing the number of signal events N s after cuts with the 95% C.L. upper limit on N s in Ref. [23]. We emphasize that these signal regions are equivalent to those adopted Ref. [23] for searches in the "heavy-flavor channel" with N b−tag ≥ 0 -i.e., with no additional b-tagging requirement imposed. By contrast, searches in the "jet-mass channel," which are particularly suited for probing newphysics scenarios involving highly-boosted massive particles which give rise to large-radius jets, are less constraining within our parameter-space region of interest. Highly-boosted φ or χ n particles are not produced at any significant rate within this region, and the requirement that large-radius jets with jet masses above a few hundred GeV be present leads to a significant reduction in signal events.
While the most striking signals to which our example model gives rise would be detected in the multi-jet channel, this model can also give rise to observable signals in other channels relevant for new-physics searches. We must therefore ensure that our model is consistent with the results of existing searches in these channels within our parameter-space region of interest. For example, diagrams of the sort depicted in Fig. 3 contribute to the event rate in the monojet + / E T channel, as do diagrams similar to that shown in Fig. 2 in which an additional quark or gluon is produced as initial-state radiation or radiated off the internal φ line. Such diagrams clearly contribute to the event rate in the monojet + / E T channel whenever the ensemble constituents χ m and χ n in the final state are both stable on collider timescales and therefore appear as / E T within a collider detector. Searches in this channel play an important role in constraining single-particle dark-sector models with a similar FIG. 12. Production cross-sections before and after cuts for the processes pp → φχm and pp → χmχn, calculated for the three benchmarks defined in Table I at the √ s = 13 TeV LHC. The left column shows the cross-sections for these processes before any cuts are applied, while the center and right columns show the corresponding cross-sections after the application of the event-selection criteria associated with the monojet and multi-jet analyses described in the text, respectively. The results displayed in the top, middle, and bottom rows of the figure correspond to Benchmarks A, B, and C, respectively. The bar at the top of each panel shows the individual cross-sections σ(pp → φχm) for different values of the index m, while the density plot below it shows the cross-sections σ(pp → χmχn) for different values of the indices m and n. We emphasize that a different color scheme is used in each column, owing to the significant difference in the overall scale of the cross-sections before and after cuts are applied. mediator coupling structure [56], and thus can be anticipated to play an an important role in constraining the parameter space of our model as well.
Moreover, diagrams of this sort in which χ m and/or χ n decay within the detector can also potentially contribute to the nominal signal-event rate in the monojet + / E T channel. This is because the event-selection criteria adopted in searches in this channel typically permit a small number of additional hadronic jets to be present in the final state. Thus, in assessing the monojet constraints on our example model, we must account for events in which the number of jets collectively produced by the decays of χ m and/or χ n is sufficiently small that these event-selection criteria are satisfied.
The most stringent constraints on our model from searches in the monojet + / E T channel are those obtained by the ATLAS Collaboration with 36.1 fb −1 of integrated luminosity at the √ s = 13 TeV LHC [57]. In assessing the constraints on our example model from searches in the monojet + / E T channel, we model our triggering requirements and event-selection criteria after those employed in Ref. [57]. In particular, we select events in which / E T > 250 GeV and in which the leading jet has p Tj > 250 GeV and |η j | < 2.4. In addition, we require that there exist no more than four jets in the event with p Tj > 30 GeV and |η j | < 2.8. We also impose the criterion ∆φ( / p T , p j ) > 0.4, where ∆φ( / p T , p j ) is the difference in azimuthal angle between the missing-transversemomentum vector / p T and the three-momentum vector p j of any reconstructed jet in the event.
Finally, we note that while the most striking multi-jet signatures which arise in our model are those involving large jet multiplicities, channels involving a more modest number of jets and / E T can also potentially be relevant for constraining the parameter space of our model. Indeed, Fig. 10 indicates that a significant number of events with 5-6 jets can be produced even within regions of parameter space where the peak on the N jet distribution is much higher. The leading constraints of this sort turn out to be those from an ATLAS search [58] for squarks and gluinos in events involving 2-6 hadronic jets and substantial / E T . However, as we shall see, constraints from such moderate-jet-multiplicity searches turn out to be subleading compared to those from the monojet + / E T and multi-jet + / E T searches discussed above. In Fig. 12, we present our results for the individual cross-sections σ(pp → φχ m ) for different values of the index m and the individual cross-sections σ(pp → χ mχn ) for different combinations of the indices m and n for the three benchmarks defined in Table I at the √ s = 13 TeV LHC. The results in the top, middle, and bottom rows of the figure correspond to Benchmarks A, B, and C, respectively. The left panel in each row of the figure shows these cross-sections before any cuts are applied, while the center and right panels in the same row show the corresponding cross-sections after the application of the event-selection criteria associated with searches in the monojet and multi-jet channels, respectively. More specifically, the monojet results shown here correspond the event-selection criteria associated with Signal Region IM1 of Ref. [57] with / E T > 250 GeV, while the multijet results correspond to the Signal Region N 50 jet ≥ 8 of Ref. [23] with N b−tag ≥ 0.
In interpreting the results shown in Fig. 12, we begin by observing that for Benchmark A, the individual crosssections σ(pp → φχ m ) before cuts are larger for heavier χ m , due primarily to the fact that γ is positive. This remains true even after the application of the multi-jet cuts, as shown in the top right panel of the figure. By contrast, after the monojet cuts are applied, σ(pp → φχ 0 ) is by far the largest of the σ(pp → φχ m ) for this benchmark. This is primarily a consequence of the upper limit on N jet included among these cuts. Similar behavior is also apparent for this benchmark within the χχ channel. The results obtained for Benchmark B are qualitatively similar to those obtained for Benchmark A, except that the individual contributions σ(pp → φχ m ) and σ(pp → χ mχn ) involving heavier χ m contribute more significantly even after the monojet cuts. This is primarily a reflection of the fact that γ is larger for Benchmark B than it is for Benchmark A. For Benchmark C, the larger value of m φ implies that the number of states in the ensemble is significantly larger than it is for the other two benchmarks. This larger value of N notwithstanding, the results for this benchmark are also qualitatively similar to those obtained for Benchmark A. The most salient difference between the results obtained for these two benchmarks is the significant decrease in σ(pp → χ mχn ) when both m and n become large. This is simply a reflection of the fact that both of the ensemble constituents are quite heavy in this regime.
The total production cross-sections σ χχ , σ φχ , and σ φφ obtained by summing the contributions from all relevant individual production processes are provided in Table II. The cross-sections before the application of any cuts are provided, as well as the corresponding cross-sections obtained after the application of our monojet and multijet cuts. Once again, the monojet results correspond the event-selection criteria associated with Signal Region IM1 of Ref. [57] with / E T > 250 GeV, while the multijet results correspond to the Signal Region N 50 jet ≥ 8 of Ref. [23] with N b−tag ≥ 0. Current limits on the overall production cross-section from LHC monojet and multijet searches are also included in the bottom row of the figure for purposes of comparison. For Benchmark A, we observe that σ φχ and σ φφ are approximately equal and both much larger than σ χχ before cuts. However,  (3.9) at the √ s = 13 TeV LHC, as well as the corresponding cross-sections after the application of the event-selection associated with the monojet search and multi-jet searches described in the text. Also shown are the corresponding experimental upper limits on the overall production crosssection after cuts for both of these monojet and multi-jet searches.
More importantly, however, we observe that all three of these benchmark points are consistent with LHC limits from both monojet and multi-jet searches, despite the fact that a different production process provides the leading contribution to the overall event rate in the multi-jet channel in each case. Thus, we see that a variety of qualitatively different scenarios which give rise to mediatorinduced decay cascades can be consistent with current constraints and therefore potentially within the discovery reach of future collider searches.

VI. SURVEYING THE PARAMETER SPACE
Having gained from our benchmark studies a sense of the range of phenomenological possibilities which can arise within our model, we now expand our analysis by performing a more systematic survey of the phenomenological possibilities that arise across the full parameter space of this model. The purpose of this survey is not only to assess the impact of current experimental constraints, but also to determine which of the production processes discussed in Sect. II dominates the event rate within different regions. In performing this survey, we shall vary the mediator mass m φ and the scaling exponent γ which determines how the mediator interacts with the fields of the dark sector while holding fixed the parameters m 0 = 500 GeV, ∆m = 50 GeV, and δ = 1 which characterize the internal structure of the dark sector itself. For simplicity, and in order to maintain consistency with the constraints outlined in Sect. III across the (m φ , γ)-plane, we fix c 0 = 0.1. More specifically, we sample m φ and γ at a variety of discrete values within the ranges 0.6 TeV ≤ m φ ≤ 2.5 TeV and 0 ≤ γ ≤ 3.5. For each such combination of m φ and γ, we then evaluate the aggregate cross-sections σ φφ , σ φχ , and σ χχ according to the event-generation and event-selection procedures outlined in Sect. IV. In addition, in order to provide a measure of the fraction of events associated with any particular combination of these parameters have truly large jet multiplicities, we also define the parameter N 10% jet , which represents the maximum value of N jet for which at least 10% of the events in a given data sample have N jet ≥ N 10% jet . The results of this parameter-space survey are shown in Fig. 13. Each individual box within the figure corresponds to a particular combination of m φ and γ. The four numbers displayed within each box indicate the value of N 10% jet at four different stages of our analysis, as indicated in the key at the bottom left of the figure. The number enclosed within a black circle in the upper left of each box indicates the value of N 10% jet at the parton level with no additional cuts, while the number in the upper right indicates the corresponding value obtained at the parton level with the basic trigger cuts p Tj > 20 GeV and |η j | < 2.8 applied. Similarly, the number in the lower left indicates the value of N 10% jet obtained at detector level with the same basic trigger applied, while The number in the lower right indicates the value of N 10% jet obtained after the application of the multi-jet trigger cuts N jet ≥ 5, p Tj > 45 GeV, and |η j | < 2.4. The text at the bottom of each box indicates the relative size of the cross-sections σ φφ , σ φχ , and σ χχ at the parton level, before the application of any cuts. The color of each box indicates which production process dominates the overall crosssection for mediator-induced decay-cascade events after the application of the different sets of event-selection criteria described in the legend at the bottom right of the figure. We note that the event-selection criteria associated with the results shown in the "Multi-Jet" column of the legend include not only the cuts explicitly listed in the heading of that column, but also the cuts associated with the multi-jet trigger.
Comparing the N 10% jet values appearing in the upper left and upper right corners of a given box provides a sense of how rudimentary cuts associated with jet-energy thresholds and detector geometry affect the N jet distribution, while comparing the values shown in the upper left and lower left corners provides information about the effects of ISR, FSR, and parton-showering. We observe that throughout the region of the (m φ , γ)-plane shown in the figure, geometric and jet-energy-threshold effects do not have a significant impact on N 10% jet . We also observe that   The text at the bottom of each box indicates the relative sizes of the cross-sections σ φφ , σ φχ , and σχχ at the parton level, before cuts. The color of each box indicates which production process dominates the overall cross-section for decay-cascade events after the application of the different sets of event-selection criteria described in the legend at the bottom right. As discussed in the text, the thick, black solid contour represents the bound from multi-jet searches, while the thick, black dashed contour represents the corresponding bound from moderate-jet-multiplicity searches. The regions above and to the right of these contours are excluded.
while the effects of ISR, FSR, and parton-showering are less uniform across the (m φ , γ)-plane, leading to an increase in N 10% jet in some regions and a reduction in others, the overall impact on these effects is not particularly dramatic within any region of the plane. The reduction in N 10% jet which results from the application of the multi-jet cuts is typically more pronounced. However, the overall message is that whenever mediator-induced decay chains tend to generate a significant number of "jets" at the parton level, this typically translates into a significant population of events with large jet multiplicities at the detector level as well.
In addition to information about jet multiplicities, Fig. 13 also provides information about how the bounds discussed in Sect. V constrain the parameter space of our model. In particular, the solid black jagged line separates the points within out parameter-space scan which satisfy the bound from the multi-jet search limits derived in Ref. [23] from the points which do not. Similarly, the dashed black jagged line separates the points within out parameter-space scan which satisfy the bound from the moderate-jet-multiplicity search limits derived in Ref. [58] from the points which do not. The regions above and to the right of each contour are excluded by the corresponding constraint. By contrast, we find that the constraints from the monojet search limits derived in Ref. [57] do not exclude any of the parameter space shown.
We see from Fig. 13 that the region of parameter space in which m φ and γ are both large -and in which processes of the form pp → χ mχn dominate the event rateis the region most severely impacted by the constraints from multi-jet searches (which supersede the moderatejet-multiplicity searches throughout the region shown). Nevertheless, we observe that regions of parameter space remain within which such processes dominate the event rate both before and after cuts are applied, while at the same satisfying these constraints. While the values of N 10% jet are largest within this excluded region at all stages of our analysis, we note that there exists a substantial region of the allowed parameter space wherein N 10% jet ≥ 8 even after the application of the multi-jet cuts. This is the region within which m φ is large, γ is small, and processes of the form pp → φχ n dominate the overall event rate. By contrast, within regions of parameter space where m φ is small, N is likewise small and the number of individual processes of the form pp → φχ n or pp → χ mχn which contribute to the overall event rate is comparatively small. As a result, pp → φ † φ tends to dominate the event rate in this region and N 10% jet tends not to be terribly high in comparison with the results obtained for larger values of m φ . That said, we note that reasonably large jet multiplicities can still arise within this region, especially for cases in which γ is large.
The results shown in Fig. 13 demonstrate that while existing LHC searches impose non-trivial constraints on parameter space of our model, there nevertheless exists a substantial region of that parameter space within which extended mediator-induced decay cascades arise without violating these constraints. The prospects for probing these regions of parameter space at future colliders -or through use of alternative search strategies at the LHC -will be discussed in Sect. VII.

VII. CONCLUSIONS AND OUTLOOK
In this paper, we have investigated the collider phenomenology of scenarios in which multiple dark-sector particles with similar quantum numbers couple to the fields of the visible sector via a common massive mediator. In such scenarios, the mediator not only plays an important role in providing a portal through which the dark and visible sectors interact, but also necessarily gives rise to decay processes wherein heavier dark-sector particles decay to final states which include both lighter dark-sector particles and visible-sector fields. In cases in which these visible-sector fields are quarks or gluons, successive decays of this sort give rise to extended decay cascades involving large numbers of hadronic jets at hadron colliders. We have investigated the structure of these mediator-induced decay cascades and examined how existing LHC searches constrain the parameter spaces associated with such scenarios. We have also shown that there exist large regions of parameter space within which all applicable constraints from these searches are satisfied, but within which extended decay cascades of this sort develop and within which jet multiplicities are characteristically large. Thus, striking signatures of this sort could potentially manifest themselves at forthcoming LHC runs or at future colliders. Such signatures could therefore provide a way of probing the properties of the dark sector and the mediator through which it couples to the SM.
Many possible extensions of our analysis can be envisioned. For example, in this study, we have chosen to focus on the region of parameter space in which the number of jets with p T sufficient to satisfy the applicable jet-identification criteria is effectively maximized. Thus, we have chosen our model parameters such that m N −1 < m φ and such that the lifetimes of all χ n with n > 0 are sufficiently short that these particles typically decay promptly within a collider detector. However, it would be interesting to examine the discovery prospects for our model within other regions of parameter space as well -regions within which extended mediator-induced decay cascades still arise, but within which the collider phenomenology nevertheless differs in salient ways.
One such alternative possibility arises in the regime in which m N −1 m φ In such cases, any ensemble constituent χ n initially produced by the decay of an on-shell mediator φ is highly boosted. In this regime, particles produced by the subsequent decay of this χ n will be collimated in the direction of its three-momentum vector. A similar situation can also in principle arise in situations in which significant mass gaps occur within the mass spectrum of χ n . Such possibilities are under investigation [59].
In this connection, we also note that while the results of existing LHC searches are effective in probing and constraining scenarios involving mediator-induced decay cascades, alternative search strategies may be even more efficient in resolving the particular kinds of multi-jet signals which arise in these scenarios from SM backgrounds. For example, a variety of jet-shape variables and other jetsubstructure techniques could potentially provide a way of improving the discovery reach for signatures of these cascades when such highly boosted particles arise. Such techniques can also be advantageous in probing regions in which parton-level "jet" multiplicities N jet are typically so large that multiple jets within the same event inevitably overlap in (η, φ)-space. In such cases, a moderate jet multiplicity at the detector level might therefore belie a much higher value of N jet . The application of jet-substructure techniques will be especially relevant at future hadron colliders with CM energies significantly higher than that of the LHC. However, since the substructure of jets arising from mediator-induced decay cascades differs from the substructure of the jets produced by the decays of the heavy SM particles W ± , Z, and t, alternative jet-shape variables and clustering algorithms may be required [59].
Several additional phenomenological possibilities also arise in the regime in which many of the χ n are sufficiently long-lived that they do not tend to decay promptly within a collider detector. For example, any χ n with a characteristic decay length in the range O(1 cm) L n O(10 m) will give rise to events in which the jet cascades associated with the decays of the more massive, promptly-decaying constituents in the ensemble are accompanied by one or more macroscopically displaced vertices. Moreover, depending on the choice of model parameters, it is possible that the final-state ensemble constituent χ m produced at one of these displaced vertices might itself decay within the detector a macroscopic distance away, and so forth. This could lead to spectacular and completely novel signatures involving multiple displaced vertices arising from the same decay chain. An investigation into the prospects for realizing and detecting such signatures at the LHC and at future colliders is currently underway [60].
Furthermore, any χ n with decay lengths L n O(10 m) will manifest themselves as / E T within a col-lider detector. Whenever additional χ n with n > 0 have decay lengths in this range, the decay chains precipitated by any of the production processes depicted in Figs. 2-4 effectively terminate not merely when a χ 0 particle is produced, but whenever any of these ensemble constituents is produced. This has a salient impact on the resulting multi-jet phenomenology. For example, for the case of pp → φ † φ production discussed in detail in Sect. III, the probability P φ (S) for such a decay chain to involve a particular number of steps S would differ from the result in Eq. (3.7). We also note that long-lived particles within these ensembles could also give rise to observable signals at a dedicated surface detector such as MATH-USLA [17,61] -signals which could then be correlated with large-jet-multiplicity signatures in the main LHC detectors. In all cases, however, our main message is clear. If the dark sector contains multiple components with similar quantum numbers, and if this sector communicates with the visible sector through a mediator, then this mediator has the potential to induce extended decay cascades yielding large multiplicities of SM particles. Moreover, as we have demonstrated in this paper, scenarios of this sort can be consistent with existing constraints. Thus, the detection of the corresponding collider signatures of these scenarios remains a viable future possibility. Such signatures might therefore provide an important route for uncovering and probing not only the dark sector but also the mediator through which it couples to the SM.

ACKNOWLEDGMENTS
The research activities of KRD, DK, HS, SS, and DY are supported in part by the Department of Energy under Grant DE-FG02-13ER41976 (de-sc0009913). The research activities of KRD are also supported in part by the National Science Foundation through its employee IR/D program. The research activities of DK are also supported in part by the Department of Energy under Grant de-sc0010813. The research activities of BT are supported in part by the National Science Foundation under Grant PHY-1720430. Portions of this work were performed at the Aspen Center for Physics, which is supported in part by the National Science Foundation under Grant PHY-1607611. The opinions and conclusions expressed herein are those of the authors, and do not represent any funding agencies.