Dark Mesons at the LHC

A new, strongly-coupled dark sector could be accessible to LHC searches now. These dark sectors consist of composites formed from constituents that are charged under the electroweak group and interact with the Higgs, but are neutral under Standard Model color. In these scenarios, the most promising target is the dark meson sector, consisting of dark vector-mesons as well as dark pions. In this paper we study dark meson production and decay at the LHC in theories that preserve a global SU(2) dark flavor symmetry. Dark pions can be pair-produced through resonant dark vector meson production, $p p\to\rho_D\to\pi_D\pi_D$, and decay in one of two distinct ways: gaugephobic, when $\pi_D\to f\bar{f}'$ generally dominates; or gaugephilic, when $\pi_D\to W+h,Z+h$ dominates once kinematically open. Unlike QCD, the decay $\pi^0_D\to\gamma\gamma$ is virtually absent due to the dark flavor symmetry. We recast a vast set of LHC searches to determine the current constraints on dark meson production and decay. When $m_{\rho_D}$ is slightly heavier than $2 m_{\pi_D}$ and $\rho_D^{\pm,0}$ kinetically mixes with the weak gauge bosons, the 8 TeV same-sign lepton search strategy sets the best bound, $m_{\pi_D}>500$ GeV. Yet, when only the $\rho^0_D$ kinetically mixes with hypercharge, we find the strongest LHC bound is $m_{\pi_D}>130$ GeV, that is only slightly better than what LEP II achieved. We find the relative insensitivity of LHC searches, especially at 13 TeV, can be blamed mainly on their penchant for high mass objects or large MET. Dedicated searches would undoubtedly yield substantially improved sensitivity. We provide a GitHub page to speed the implementation of these searches in future LHC analyses. Our findings provide a strong motivation for model-independent searches of the form $pp\to A\to B+C\to SM\, SM+SM\, SM$ where the theoretical prejudice is for SM to be a t,b,$\tau$ or W,Z,h.

Another use is to simply characterize generic strongly-coupled-like signals as targets for LHC and future colliders. Vector-like confinement [88] pioneered this study in the context of vector-like fermions that transform under part of the SM group as well as under a new, strongly-coupled group with scales near or above the electroweak scale. Further explorations into the phenomenology and especially the meson sector included [20,38,55,73,[89][90][91][92][93][94][95][96][97]. In theories with somewhat lower confinement scales, the dark sector may lead to dark showers and related phenomena [98][99][100][101][102][103][104], displaced signals [105,106] and potentially intriguing spectroscopy [47,107,108]. Spectacular "quirky" signals can arise in theories with a very low confinement scale [109,110]. The latter theories may also lead to a high multiplicity of soft particles that are tricky to observe [111][112][113].
In a companion paper [114], we develop dark sectors whose (ultraviolet) strongly-coupled sector preserves a SU (2) dark flavor symmetry. These theories are mapped into a low energy effective theory that provides the leading interactions of the dark mesons with the Standard Model. A dark sector that is of particular interest to us is Stealth Dark Matter [75]. In this theory, there is a new, strongly-coupled dark sector that consists of vector-like fermions that transform under both the new dark group as well as the electroweak part of the SM, and crucially, also permit Higgs interactions (from Yukawa couplings or higher dimensional operators). Others have also pursued dark sectors with vector-like fermions that permit Higgs interactions for a variety of purposes [20,23,43,85,115].
The dark meson sector of the Stealth Dark Matter theory has several intriguing properties due to the accidental symmetries of the model. Like vector-like confinement [88] the dark sector is free of constraints from precision electroweak observables and Higgs coupling measurements so long as the vector-like mass is dominant. Unlike vector-like confinement, however, the Higgs interactions break the global (species) symmetries of the dark sector, permitting dark pions to decay into SM states. Provided the vector-like masses are smaller than ∼ 4πf , where f is the scale of the new strong interaction, we can organize the states using chiral perturbation theory. In this paper we focus on the most phenomenologically relevant states: the (lightest) triplet of pseudoscalar pions π a D and the heavier triplet of vector mesons ρ a D [116,117]. The scales of the theory, as we will see, are comparable to or somewhat larger than the electroweak scale.
The presence of a SU (2) dark flavor symmetry arises from global symmetries of the ultraviolet strongly-coupled sector. For example, a strongly-coupled sector that contains two flavors of dark fermions with identical (current) masses has a global SU (2) × SU (2) symmetry that is broken by the condensate to a SU (2) dark flavor symmetry [114]. This is just like QCD with its two light flavors of quarks with nearly equal (current) masses. In Ref. [114], we demonstrate strongly-coupled theories where the SU (2) dark flavor symmetry can be identified as an exact custodial symmetry of the dark sector. That is, the Higgs multiplet interacts with the dark flavors such that the SO(3) ∼ SU (2) c is not further broken by the dark sector. Consequently, the dark sector's meson degrees of freedom can be categorized in custodial symmetric representations. Again considering the example of theories with two flavors of dark fermions, the meson sector contains dark pions and one set of dark vector mesons in a triplet representation of the SU (2) dark flavor symmetry. Unlike QCD, however, the vector-like nature of the dark sector permits two possibilities for gauging the global flavor symmetry: the entire SU (2) could be gauged (the SU (2) L weak interaction) or just the U (1) (as in U (1) B hypercharge). 1 This leads to two distinct low energy effective theories of dark mesons: In the latter case, obviously only the U (1) subgroup is gauged, but since we assume the dark sector respects the full global SU (2), we'll refer to this as the SU (2) R model.
In the meson sector the dark pion states can be pair-produced, either via Drell-Yan or resonantly via mixing of the ρ with SM electroweak gauge bosons. The dark pion decays can be categorized into two distinct possibilities: "gaugephobic", when π D → ff dominates; or "gaugephilic", when π → W + h, Z + h dominates once kinematically open. The decay π 0 D → γγ is highly suppressed due to the dark flavor symmetry [114]. For a wide range of parameters, the interaction between single dark pions and the SM is small enough to make single pion production phenomenologically irrelevant, and yet, the interaction can be easily large enough that the dark pions decay promptly back to SM states. We also briefly comment on the possibility that dark pions are sufficiently long-lived so as to modify their phenomenological signature.
Dark mesons are therefore an example of new physics that must be pair produced with ∼ weak strength and decay back to multiple SM particles (only). The combination of a relatively low production cross section and complex final states with no BSM sources of missing energy leads to weak LHC constraints. We perform a detailed breakdown of which LHC searches could potentially set bounds on dark mesons. For the searches with potential, we recast the searches and estimate the bounds for some benchmark dark meson scenarios. For the searches that fail, we identify why. This latter step is useful as we find many 13 TeV analysis are insensitive to dark mesons because their cut thresholds are too high.
The layout of the rest of this paper is as follows. In Sec. 2 we introduce our phenomenological dark meson model and its relevant parameters. This model description is broken up into three parts: the strong sector, kinetic mixing, and π D decay. Using this setup, we explore the constraints on dark meson parameter space. Sec. 3 is devoted to constraints from single ρ D production, while we explore constraints from π D pair production in Sec. 4. We step through the details of the searches that provide constraints and provide insight into why other searches fail to. Finally, we present our conclusions in Sec. 5.

Phenomenological Description of Dark Mesons
The dark meson interactions will be described below using a phenomenological lagrangian. The core philosophy was formulated in "vector-like confinement" [88,89], and our discussion of resonant production of dark pions through a dark rho parallels theirs. The key distinction between our formulation and vector-like confinement is the presence of Higgs interactions among the dark fermions which breaks enough of the dark flavor symmetries to allow dark pions to decay. In the language of vector-like confinement, all species symmetries are broken by Higgs interactions in the dark sector (either Yukawa couplings or higher-dimensional interactions).

Dark Mesons in SU (2) Triplet Representations
The lagrangian can be written as L = L strong + L kinetic mixing + L decay . (2) The first contribution contains the meson sector of the theory as it arises from the stronglycoupled dark sector: It contains the kinetic terms of the vector (ρ D ) and pseudoscalar (π D ) mesons, mass terms, and the interactions among these mesons. As we indicated in the introduction, the mesons fill out representations of the SU (2) dark flavor symmetry, and the meson self-interactions respect the SU (2) dark flavor symmetry. Throughout all of these expressions, we have assumed that the dark sector contains (at least) one set of dark pions and (at least) one set of dark vector mesons in the triplet representation of the SU (2) dark flavor symmetry. Hence the a = 1, 2, 3 index attached to π a D and ρ a D . 2 We will only consider the phenomenological consequences of the lightest triplet dark vector meson (ρ a D ) and the lightest triplet dark pion (π a D ).
The coupling between the ρ D and π D is show in Eq. (5). This is the analogue of g ρππ in QCD. In the SU (2) R model, the full set of SU (2) R -symmetric interactions are present, though in practice only the ρ 0 D π + D π − D interaction is phenomenologically relevant since only 2 We use ρ 3 D and ρ 0 D interchangeably. ρ 0 D talks to SM fermions via kinetic mixing (see Sec. 2.2). The NDA estimate of the coupling strength is given by

Kinetic Mixing of ρ D with SM
The second term of Eq. (2) contains the kinetic mixing of the dark rhos and the electroweak gauge bosons: This provides the main "portal" from the Standard Model into the dark sector. There are two cases we detail below: F aµν identified with W aµν (the SU (2) L model), and F aµν identified with δ a0 B µν (the SU (2) R model).
In each of the models defined by Eq. (1), all or part of the SU (2) dark flavor symmetry is gauged. In SU (2) L model, the triplet of global SU (2) is identified as a triplet of the gauged electroweak SU (2) L group. In the SU (2) R model, the triplet of global SU (2) is identified as a triplet of the would-be gauged electroweak SU (2) R group, had the entire SU (2) R been gauged. Of course the entire SU (2) R is not gauged -just the U (1) B subgroup. After electroweak symmetry breaking, SU (2) L × U (1) B → U (1) em , the triplet of vector and pseudoscalar mesons of the SU (2) L and SU (2) R models have the same electric charges, Q = (+1, 0, −1).
In both models, we use naive dimensional analysis (NDA) to estimate the size of the kinetic mixing: strictly valid for a large number of colors N D of the confining dark gauge group. Diagonalizing the kinetic terms leads to a field redefinition of at leading order in . This leads to a ρ D interaction with the SM fermions with a coupling strength proportional to g 2 or g 2 , 6 where f i,j are left-handed SM fermions in the SU (2) L model, while f are any SM fermions with hypercharge Y f in the SU (2) R model.
The difference between the two models is mainly in the kinetic mixing. In the SU (2) L model, the entire triplet of ρ a D mixes with the triplet of W a bosons. In the SU (2) R model, only the neutral component of the triplet, ρ 0 D , mixes with the hypercharge gauge boson. Additionally, the kinetic mixing has one power of the gauge coupling: g in the SU (2) L model; g in the SU (2) R model. Here we emphasize that while the difference between g/g 2 may seem small or trivial, pp → ρ production is proportional to 3g 4 in the SU (2) L model (compared with g 4 in the SU (2) R model), and so this leads to a significant difference in the production rates of ρ D 's in the two models.
Neglecting mass differences among states within the triplets, the strong sector is thus described by three parameters: As our canonical example that we use throughout this paper, we have taken N D = 4 in the bulk of our results below. This choice was motivated by the Stealth Dark Matter model [75]; the phenomenology is broadly similar so long as the number of colors is not excessive. We quantify this in detail below.
Additionally, we will often replace one of the dark meson mass parameters for the ratio η = m π D /m ρ D . This ratio is important because it governs how the ρ D can decay. Specifically, if η < 0.5, ρ D can decay to a pair of dark pions, while if η > 0.5 the dark rhos must decay directly back to SM particles. As we will see, the latter case is strongly constrained by limits from Z , W searches. From now on, we will label our dark meson models by the type of kinetic mixing and the ratio of dark meson masses, i.e., Having specified N D , the production cross section for ρ D is completely determined for both models as shown in the left-side plot in Fig. 1. Figure 1 also shows the ρ D branching ratios for two different η values: as expected, if η < 0.5 (middle panel) then the interaction strength and form of the ρ D π D π D interaction make ρ D → π D π D . On the other hand, if the π D are too heavy (η > 0.5, right plot), the ρ D decay back through kinetic mixing and the branching ratios are simply determined by the SM color factors.
In focusing on the two models, we are ignoring scenarios where the SU faces constraints from Higgs coupling measurements and precision electroweak tests. In terms of ρ D phenomenology, having well defined SU (2) L × U (1) Y properties means that the ρ D → V π D (V = SM electroweak boson) decay modes are always small.
We would be remiss to not point out that the SU (2) R model invovling a dark U (1) vector boson mixing between the hypercharge is ubiquitious in the literature of simple dark sectors as "dark photons" (e.g., for a review [118]). While most of this literature focuses on (much) lighter dark photons, for simple dark photon models with a dark photon mass at or above the electroweak scale, we can map this toy model onto a special case of our strongly-coupled dark sector. The mapping utilizes the SU (2) R model with: η > 0.5 (so that the dark vector boson can decay only into SM states), m SM /v π small (so that single production of dark pions is negligible), and the number of dark colors N D chosen to obtain a kinetic mixing . Even with these parameter choices, our strongly-coupled dark sector obviously has differences from the simple toy models. One is that the kinetic mixing is at most one-loop suppressed. Another is that there is relationship between the smallness of the kinetic mixing, the number of dark colors, and the relative size of self-interactions of the dark mesons. While it would be interesting to map out this space more fully, this is beyond the scope of this paper.

Dark Pion Decay to SM
Finally, dark pion decay. This is the main subject of our companion paper [114]. There we show that strongly-coupled models with custodially-symmetric Higgs interactions among 8 the dark fermions leads to a low energy effective theory in which dark pions interact with the SM through: where ψ u,d are SM fermions. There are several important features of this Lagrangian. First, while we have used the language that the decay interactions 'break the flavor symmetry', this is slightly sloppy. Stated more correctly, we have married the SU (2) V symmetry of the dark pions to part of the O(4) symmetry group of the Higgs potential. Both the dark pions and the SM fields transform under the shared symmetry, so we can write down single pion interactions of the form π a O a where O a is some triplet of SM fields.
The overall scale of the operators is set by 1/v π for the fermions and ξ/v π for the gauge/Higgs bosons. The fact that the interactions do not further distinguish the fermions (i.e., one overall coupling for the first four terms) nor the gauge/Higgs interactions (one coupling for the last three terms) is due to the the dark sector's preservation of custodial symmetry. However, since custodial SU (2) is broken in the SM by differences of Yukawa couplings as well as hypercharge, there is a residual differentiation of the interactions by m u − m d as well as g = 0.
This form is convenient, since coupling π D to the SM fields requires breaking electroweak symmetry and hence the coupling strengths must be proportional to the mass of a SM field. The primary role of the 1/v π parameter is to set the total width of the π D . In this paper our main focus is on scenarios where the π D decay promptly. This sets a lower bound on m SM /v π , where m SM is the mass of the mass of the SM particle(s) in the dominant π D decay. Scenarios where π D is displaced or long-lived are also interesting to study. The main search methodologies are well-known from other displaced/long-lived searches (for a review, see e.g. [119]).
The remaining model-dependent parameter is the relative strength of the coupling to fermions versus the gauge/Higgs sector that we have parameterized by ξ. We will consider two possibilities for ξ: The scaling of the gaugephobic parameter with the electroweak scale and the dark pion mass scale deserves some discussion. The origin of this scaling is found from an analysis of the strongly-coupled effective theories that we have discussed in detail in Ref. [114]. In essence, 9 100 200 300 400 500 600 there are higher dimensional operators involving additional Higgs fields, suppressed by at least the scale of the dark pions, that can regenerate couplings to the gauge/Higgs sector even if they don't exist at leading order. As we show in Ref. [114], the Stealth Dark Matter model is gaugephobic with is a another dark pion that is at least slightly heavier than π D . Since the dark kaon scales with the parameters of the ultraviolet theory in exactly the same way as the dark pion, in our phenomenological study we take c ξ = λ h and do not distinguish between the dark pion and kaon masses.
In the limit that the dark pion mass scale is taken large, ξ → 0, and the dark pions can only decay back to fermions. However, when the dark pions are near to the electroweak scale, ξ can be "smallish" but, importantly, nonzero. This implies π D → ff dominate so long as there is no small coupling. For the specific case of π 0 D in the mass range m h + m Z < m π 0 D < 2m t , the decay π 0 D → Z + h dominates despite being gaugephobic. This is because the Zh mode is longitudinally enhanced, while the competing fermionic mode π 0 D → bb is suppressed by the small Yukawa coupling y b . For all other ranges of dark pion masses (both charged and neutral), π D → ff dominates. By contrast, in the gaugephilic case π D → W + h, Z + h dominate once they are kinematically open.
While the two choices in Eq. (13) may seem arbitrary at first, a large class of stronglycoupled models can be mapped into this categorization (see Ref. [114] for more details). Specifically, the Stealth Dark Matter model [75,76,87] and others similar to it are gaugephobic. By contrast, models of bosonic technicolor / induced symmetry breaking [14], as well as the triplet state in Georgi-Machacek models [120] have gaugephilic interactions.
In our taxonomy, the gaugephilic case only occurs for the SU (2) L model. This is not immediately obvious from our discussion thus far. Essentially the gauge/Higgs interactions  12) is permitted with order one ξ when π a D is in the same representation as W a µ , i.e., an SU (2) L triplet. The reader may then immediately wonder why the SU (2) R case does not have ξ = 0. At leading order it does, but at higher orders one finds gauge/Higgs interactions are generated albeit with a suppression typically of order m 2 h /m 2 π D . This is parametrically the suppression we find in the Stealth Dark Matter model [114], and is similar to what we find in generic 2-flavor custodially-symmetric models. More details can be found in Ref. [114].
Any given model may or may not permit arbitrary choices for v π and ξ; for instance, induced electroweak symmetry breaking requires v π fixed (up to order one coefficients) and ξ = 1 due to the requirements of proper electroweak symmetry breaking. However, as we detail in [114], there are models that span a wide range of (v π , ξ 1).
Given ξ, the branching fractions of the π D are fully specified as a function of the pion mass. As π D decay couplings are proportional to mass, they decay to the heaviest kinematically available SM particles. The branching ratios for the gaugephilic and gaugephobic scenarios are compared side by side in Fig 2 (charged π D ) and 3 (neutral π D ). 3 For the charged π D , the branching ratios in the two cases are similar at small masses. However, the unsuppressed gauge/Higgs couplings in the gaugephilic scenario imply π D → W + h quickly dominates once it is kinematically allowed (due to the kinematic enhancement of decays to longitudinal W ), while the π D → tb mode always dominates at heavy mass for the gaugephobic case. There is a similar pattern in the branching ratio of the neutral pions.
Again, when the pion is light, the decay modes between the two categories are similar and are dominated by the bb mode. This similarity persists after π D passes the Zh threshold. However, as π D is further increased past the tt threshold we can spot the difference, as the π D →tt branching ratio dominates at large π D masses in the gaugephobic case but stays subdominant to Zh in the gaugephilic case.

Constraints from single production
Having established the dark meson phenomenological Lagrangian and fleshed out the relevant parameters, we now move on to LHC production, sensitivities, and constraints.
The phenomenology of the dark meson sector that we pursue in this paper clearly bifurcates at η = 0.5 as evident from the branching fractions of the dark pions in Fig. 1. For η > 0.5, the ρ D is kinematically forbidden to decay to a pair of on-shell dark pions, and thus decays to SM fermions dominate. 4 The decays into SM fermions are determined solely by the gauge and color charges of the fermions, so the ρ D phenomenology is essentially independent of the details of how the pions interact with the SM.
When η < 0.5, ρ D → π D π D is open, and generally dominates so long as the number of dark colors, N D , is not large (we'll be more precise below). In this case, the most promising way to search for dark mesons is dark pion pair production. The largest contribution to dark pion pair production is resonant production pp → ρ D → π D π D through the dark rho, so long as it is not very heavy. Dark pions can also be pair-produced through Drell-Yan production, though this tends to give a smaller cross section due to the W or Z exchange being off-shell. We find that resonant production through ρ D dominates for η 0.2 for N D = 4.
The final states populated by dark pion pairs depends on how the dark pions decay, which in turn depends on whether we are in a gaugephilic or gaugephobic scenario. We have chosen 9 benchmarks spanning the phenomenology possibilities that we believe give a solid idea of the differing phenomenology, shown in Table 1. We provide the FeynRules [121] model files and corresponding UFO files on GitHub. 5 We used MadGraph5_aMC@NLO [122] to simulate the events. When studying constraints directly on the ρ D , we simulated pp → ρ D and then allowed for any decay mode. For the constraints on π D , we simulated pp → π D π D which then had resonant and Drell-Yan production. In all cases, showering and hadronization was performed by Pythia 8 [123] and Delphes 3 [124] was used for fast detector simulation. We used the default detector card because we recast both ATLAS and CMS results. Within Delphes, jets were calculated with Table 1: Benchmark models and parameters used in our study. Note that the gaugephilic case only occurs for the SU (2) L model, as discussed in Sec. 2 in the text.
FastJet [125] using the anti-k t algorithm [126] For each of the benchmark scenarios in Table 1, the mass of the π D was scanned with variable spacing in order to capture the different decay mode transitions. We take the lower limit of dark pion mass to be 100 GeV, coming from the bound on BSM charged particles from LEP II. At each mass point, 500k events were produced for pair production of dark pions (all allowable modes). This was done for both √ s = 8 TeV and √ s = 13 TeV collisions. The π D are decayed in the narrow width approximation using Pythia.
There is no dedicated search for dark mesons at the LHC. We therefore estimate the existing bounds by recasting a vast set of potentially constraining searches using Monte Carlo methods. We will present our results first, followed by a more detailed description of our recasting methods and a summary of why several searches which look promising at first glance fail to set strong bounds.

ρ D constraints
We first consider ρ D production and decay. The ρ D dark vector mesons kinetically mix with electroweak gauge bosons, shown in Eq. (7), giving direct couplings to SM fermions, shown in Eq. (10). In both the SU (2) L and SU (2) R models, there is a neutral ρ 0 D , better known as a new Z gauge boson. Via kinetic mixing, this ρ 0 D acquires a coupling to leptons.
The strongest constraints on generic Z gauge bosons (with masses near or above the electroweak scale) is from the absence of resonances in the the + − invariant mass spectrum The black line is the model-independent limit. To illustrate the impact of this bound on the model space, we have superimposed the predicted 2 × BR(ρ 0 D → + − ) for the SU (2) L model, varying the number of colors between 2 to 16. On the right, the 2-body decay ρ 0 D → π + d π − d is kinematically forbidden, leading to strong constraints: m ρ D > 1.5-2.5 TeV. On the left, the 2-body decay ρ 0 D → π + d π − d is open, and we see that when N D 4, there is no constraint from resonant ρ 0 D production and decay to dileptons. [127,128]. Using the ATLAS 13 TeV search with 36.1 fb −1 of integrated luminosity [127], we have recast the dilepton searches for the combined electron and muon channels into a limit on ρ D cross section times branching fraction to leptons. This is accomplished by simulating the production of ρ D and decaying them according to the branching ratios shown in Fig 1. After passing through a parton shower, hadronization, and detector simulation, we select events which contain same-flavor opposite-sign leptons within the ATLAS selection criteria. The combined efficiency (branching ratio times the detector efficiencies) multiplied by the cross section can then be compared against the exclusion limits provided by the ATLAS HEPData [129].
In Fig. 4, we illustrate the bounds that we have obtained by determining the largest coupling of the ρ 0 D to the SM for any choice of m ρ D within the range of interest in this paper. The coupling is completely determined by the model-independent quantity 2 × BR(ρ 0 D → + − ), that is shown as a black line in both panels of Fig. 4. Also superimposed on the panels are the predicted sizes of 2 × BR(ρ 0 D → + − ) for a given m π D /m ρ D and number of dark colors N D in the SU (2) L model. It is important to note that is the kinetic mixing parameter and not the detector efficiency. (Similar but weaker constraints are found in the SU (2) R model.) The right panel clearly shows that the neutral dark vector meson is strongly constrained by the dilepton data when m π D /m ρ D > 0.5.
The dependence on the number of dark colors is nontrivial: In the case η > 0.5, the one power of N D comes from 2 while in the branching fraction the N D dependence cancels. Contrast this with the case η < 0.5, where the branching fraction The left panel clearly shows that when ρ D → π D π D is both kinematically open (η < 0.5) and dominates (N D 4), there are virtually no LHC constraints on neutral dark vector meson production and decay. (The very narrow region near m ρ D ∼ 300 GeV is, as we will see, also constrained by other searches).
The bounds we have obtained from the ATLAS searches for dilepton resonances assumed the width of the new resonance is relatively narrow, Γ(Z )/M Z 0.03 [127]. In all of the cases with η = 0.55, where the ρ 0 D can only two-decay into SM states, the width is narrow, where we have evaluated the result for the two values η = 0.45, 0.25 used in our benchmarks for the paper. Despite the relative strong-coupling among mesons (g ρ D π D π D = 4π/ √ N D 1), the kinematic suppression of taking η = 0.45 suppresses the width of the ρ 0 D to a few percent, and so the ATLAS bounds are fully applicable. For η = 0.25, the width is now tens of percent that is large enough requiring a re-analysis of the dilepton data to set precise bounds on the ρ 0 D . For η = 0.25, N < 4, the ρ D width to mass ratio reaches ∼ tens of percent, so a simple recast of the ATLAS bounds is not completely precise. However, given that i.) the bounds on a wide resonance will be weaker than on a narrow resonance, and ii.) the narrow resonance bounds for N < 4 are already weak, we conclude that there is no bound on ρ 0 D for η = 0.25, N < 4.
There is one additional constraint on the kinetic mixing of ρ D with SM gauge bosons from LEP constraints on four-fermion effective operators [130]. Integrating out ρ 0 D results in four-fermion operators of the form where we have used the operator normalization of Ref. [130]. Matching the coefficient, The strongest constraints from the LEP data suggest Λ 20 TeV [130]. For the SU (2) R model, there is no constraint due to the smallness of g . For the SU (2) L model, the bound on m ρ 0 D varies from about 250-750 GeV for N D = 2-16. Given the order one uncertainties in the large N D estimate for the kinetic mixing, this bound is not any stronger than what we have already found from Fig. 4.

Constraints on the dark pion coupling to SM
Throughout the paper, we will generally work in the "vector-like" limit (See Ref. [114]) where m SM vπ is small and thus single production of π D is suppressed. This limit is automatically safe from constraints from electroweak precision observables as well as Higgs coupling measurements, and coincides with the demarcation of our model space into the two categories SU (2) L and SU (2) R . If, however, m SM vπ is not so small, single production of dark pions is possible and relevant to the phenomenology. In the model of bosonic technicolor / induced electroweak symmetry breaking, this sets the strongest constraints [14].
We can also characterize the parameter space of our effective theory by determining the constraints on 1/v π of Eq. (12). In Fig. 5, we consider several processes 6 where single dark pion production can set upper bounds on 1/v π . One process is top decay, t → π + Db . In this process the π + D must be somewhat lighter than the top quark, and thus π + D → τ + ν τ dominates for the charged pions, leading to an excess of τ 's in top decay. LHC analyses of top decay, however, are consistent with lepton universality [131,132]. For values of the pion mass slightly less than the top quark mass, the pion branching ratio to τ is similar to the SM branching ratio of the W to tau. Thus, in this region the branching ratio alone is not enough to constrain the coupling. Instead, we use the total width of the top quark [133][134][135] as a secondary constraint, and exclude any region where the BSM additions to the top decay change either the width or the tauonic branching ratio by more than two standard deviations away from the measured values. This constraint is shown in red in Fig. 5.
There are also many searches for the heavy Higgs particles of two-Higgs doublet models that can be recast into searches for single production of the charged or neutral dark pions. In Refs. [136,137], ATLAS searches for a charged Higgs produced association with tb. The two searches consist of one looking for H + → τ + ν τ while the other looks for H + → tb. The limits are presented in terms of σ(tbH + ) × BR, but unfortunately HEPData is not given. We therefore take the limits from plots in Refs. [136,137] and reinterpret them by replacing π + D for the charged Higgs boson. The upper bounds on 1/v π we obtain are shown in orange and blue in Fig. 5. Finally, in a similar approach, Ref. [138] performed searches for a heavy neutral Higgs boson produced in association with bb and decaying to τ + τ − . Upon recasting this search for neutral dark pions, we find somewhat weaker constraints -shown in green in Fig. 5 -compared with the bounds from charged dark pions.
Finally, while this is not a constraint on the parameter space per se, it is interesting to determine when 1/v π is small enough that the decays of the dark pions are no longer prompt in colliders. As a rough guide, we can use and estimate that if cτ = 1 mm, then the neutral pions would lead to displaced tracks, or the charged pions would lead to kinked (or disappearing) tracks when they decay. If cτ > 10 m, then the pions can escape the detectors before decaying, leading to missing energy or long-lived charged tracks. Search strategies for both of these types of signals are interesting but best explored through existing dedicated strategies for long-lived charged or neutral particles [139][140][141][142]. The smallness that 1/v π needs to be to lead to these long-lived signals is shown in Fig. 5.
There can also be a contribution to the S parameter as a result of the interactions in Eq. (12). However, in the ultraviolet strongly-coupled theories considered in Ref. [114], we find the contributions depend on the spectrum of the heavier mesons, and so there is no useful translation into bounds on 1/v π . Suffice to say that there are no bounds from the S parameter when the contributions to the dark fermion masses are mostly vector-like with only smaller contributions arising from electroweak symmetry breaking [6,143].
Clearly, there is a huge range in 1/v π -roughly values larger than 10 −7 and smaller than 10 −2 , with some slight variation depending on m π D -where dark pion decays are prompt but the rate for single dark pion production is too small to be detected. Our goal for the remainder of this paper is to explore how prompt LHC searches constrain paired dark pion production in this otherwise open region of parameter space.

Resonant Dark Pion Pair-Production at LHC
The rate for dark pion pair-production depends on the model -SU (2) L versus SU (2) R , and the NDA estimates for the kinetic mixing as well as the meson self-interactions. It does not depend on how the dark pions decay (gaugephilic versus gaugephobic) because the production rate is independent of 1/v π and ξ/v π from Eq. (12). However, the different decay modes require different search strategies. In Table 2, we have denoted different mass regions for each of the categories defined by which decay modes are dominant. The intermediate SM particles, which may subsequently decay, are listed for both the charged (π D ± π D 0 ) and the neutral (π D + π D − ) currents. Note that the symmetries do not allow for neutral currents of the type π D 0 π D 0 , so the SU (2) R model does not contain a resonantly enhanced charged current. Table 2 shows that there are many Standard Model particles in the final states, with possibly exotic combinations. We analyzed 13 searches (in addition to the ones already discussed), broken down into 6 searches at 8 TeV and 7 searches at 13 TeV. Surprisingly, we find that many of the searches are not sensitive to our benchmark models. The searches with sensitivity are further detailed here, while we save a discussion of the non sensitive searches for Sec. 4.4.
The results of our recasting are summarized in Fig. 6. This is the main result of our paper. The top line of each plot (colored in blue) shows the constraints on the model coming  from searches for resonant dilepton production. As discussed in the previous section, this depends only on if the ρ D can decay to leptons or not, and is independent of how the π D decay. The x axis for the plots is m π D , so the results are obtained from Fig. 4 by scaling the x axis by the ratio m π D /m ρ D .
The next two lines in the Fig. 6 display the best constraints we could find for 13 TeV searches. The first of these is a search for supersymmetry in final states with either same-sign leptons or three leptons. Recasted in terms of dark pions, it excludes m π D in the 200-300 GeV range for the gaugephilic and slightly worse for the gaugephobic categories when η = 0.45. This search does not work when η = 0.25 because for fixed m π D , smaller η implies a heavier ρ and therefore a smaller resonant contribution to pion pair production. The other 13 TeV search with moderate sensitivity is a supersymmetry search with final states of tau leptons. The bounds from this search limit the dark pion mass in all models with η < 0.5 that we examined to be 130 GeV, the mass above which π D + → τ + ν ceases to be the dominant decay mode.
The remaining lines in Fig. 6 come from the 8 TeV searches which have sensitivity to π D . Two are multilepton searches from ATLAS and CMS, which are general searches counting the numbers of events for many signal regions. These work well for the models at low masses, and are slightly better for the gaugephilic models. The other exclusion comes from a search for supersymmetry in states with same sign leptons. In particular, one of the signal regions trades the usual missing energy requirement for more b-jets, which works well for the gaugephobic models. Gaugephobic SU(2) 55 R Figure 6: Summary of the dark meson exclusions for the benchmark scenarios and values of the π D and ρ D masses. The scenarios are labeled by the type of kinetic mixing, the ratio of the dark pion to dark rho mass η = m π D /m ρ D , and the relative strength of the fermionic versus bosonic dark pion decay modes. All of the dark pions decay promptly. The top line indicates the bound on ρ 0 D inferred from recasting the latest dilepton bounds and interpreted in terms of m π D . The next five lines (in black) show the π D mass bound from the most constraining 8 and 13 TeV searches we could find. The union of the exclusions from all of the searches is shown in the last line.
Finally, the last line (shown in red) combines all of the previous constraints in the most naive method. The models where the ρ D cannot decay to π D are excluded to over m π D = 1100 GeV for SU (2) kinetic mixing and to 900 GeV for U (1) (SU (2) R model). If the mass ratio allows for decays to pions, the exclusion limits are drastically reduced. For m π D /m ρ D = 0.45, the gaugephilic limits are to around 425 GeV while the gaugephobic limits are at 500 GeV for SU (2) mixing. This corresponds to 13 TeV cross sections of 600 fb and 300 fb, respectively. It is surprising that processes with such distinct final states are still 20 allowed with these large of rates at the LHC. The SU (2) R model limits are m π D 130 GeV, with a cross section of a few pb. As the mass ratio is further extended, the decay products become more energetic, boosting some of the search efficiencies. However, the resulting decrease in the cross section from the heavier ρ D compensates for this and leads to reduced limits. All of the models with m π D /m ρ D = 0.25 have limits at or below m π D = 200 GeV, corresponding to a (13 TeV) cross section of around a pb.
The rates that are still allowed are much larger than one would expect, especially given the exotic combinations of final state particles. In the next subsections, we examine the constraining searches in more detail, looking at why the searches work and what the deficiencies are. The details we expose, combined with the information in Sec. 4.4, will help us identify important elements that future searches should incorporate in order to improve sensitivity to dark pion scenarios.

Searching for taus
Working from the bottom up of the dark pion mass range, O(100 − 150 GeV) dark pions in all of our benchmark models decay primarily as π D + → τ + ν τ . Therefore, we begin our survey of experimental searches with searches that explicitly look for taus.
ATLAS searches for supersymmetry in electroweak production of supersymmetric particles with final states with τ leptons using 14.8 fb −1 of √ s = 13 TeV data [144]. They interpret the search in terms of the leptons coming from the decays of charginos or neutralinos. As this search is aimed at a supersymmetric model with a neutralino also in the final state, they require a large amount of missing energy, which limits the sensitivity to our benchmarks. The general search strategy is:  Figure 7: Cut flow for the search for hadronically decaying taus, optimized for electroweak production of supersymmetric particles [144]. The efficiency is much larger for the η = 0.25 benchmarks than the η = 0.45 models because the larger ρ D mass leads to more energetic π D . This increase in efficiency is offset by the decrease in resonant production cross section.
The stransverse mass is defined as where the transverse momenta of the two taus are p T,τ 1(2) and q T is the transverse vector which minimizes the larger of the two transverse masses. The transverse mass is defined as, Figure 7 shows the efficiency of the signal as the various cuts are being made, and exemplifies the kinematic differences between models with different value of η. There is very little loss in efficiency from the b-and Z-vetos for masses less than 150 GeV. Additionally, the figure shows that at this stage, there is very little difference between the η values. However, there is a huge drop in efficiency when requiring large amounts of missing energy. This is not as dramatic in the η = 0.25 models, which produce more energetic π D because of the heavier ρ D .
The exclusions from this search are plotted in Fig. 8, where the y-axis is the cross section times search efficiency. The expected number of events in the signal region from standard model backgrounds was 5.9 ± 2.1, while only three events were actually observed. As fewer events were seen than expected, the observed limits of 0.32 fb is more stringent than the expected 0.  [144]. The dark mesons on the lighter side of our spectrum predominately decay to taus, and the cross sections are large. The SU (2) L type models are excluded if m π D 180 GeV while the SU (2) R models limits are around 130 GeV. This is the only search which limits the SU (2) R models where the ρ D can decay to π D π D .
mixing and η = 0.25 or η = 0.45 are excluded from this search if m π D 170 − 180 GeV. Surprisingly, this search also constrains the SU (2) models with η = 0.55 even though the π D are not produced through a resonant ρ D . These are only allowed if m π D > 160 GeV. 8 Additionally, the SU (2) R models [that kinetically mix through U (1) Y ] with η = 0.25, 0.45 are also constrained to be above m π D 130 GeV. As shown in the summary plot of Fig. 6, this is the only search we examined which had sensitivity to the SU (2) 25,45 R models.
The reason these limits are not stronger is because the branching ratio to taus is decreasing rapidly as the mass of the pions increases. This is compensated by an increase in the expected number of W s, Zs, and bs. The next sections examine searches which exploit these particles.

Generic multilepton searches
Examining Table 2, once m π D 150 GeV, pair produced dark pions decay to lots of bottom and top quarks, along with Z and W . It should be expected that searches utilizing bs and leptons could place strong constraints on the benchmark models. While we studied many model driven searches and found no limits (see Sec. 4.4), model-independent searches proved useful. Both ATLAS and CMS have a generic search at 8 TeV based on final states with multiple leptons. (Neither collaboration has repeated the analysis at 13 TeV).
The inclusive ATLAS search looks for 3 + leptons [145]. The basic search requirements are: 1 electron or muon for triggering purposes (p T > 26 GeV, |η| < 2.5), a second electron or muon with slightly looser requirements, and a third e/µ or hadronic τ . The events are broken into further sub-categories according to several kinematic variables, such as the b-jet multiplicity, or whether or not the event contains a same-flavor-opposite-sign lepton antilepton pair. The signal regions are not orthogonal, and they set bounds on the BSM cross section of roughly a few fb.
Applied to π D production, we find the most constraining signal regions are those containing a hadronic τ and that contain ≥ 1 b-jet or have low H T,L , defined as the scalar sum of the p T of the three leading leptons (or τ ) in the event. The limits depends strongly on the lepton and tau identification. In particular, the ATLAS study used only single-prong hadronic taus 9 in the analysis and a benchmark identification efficiency of 0.5. Compared to more recent τ reconstruction numbers [146] (which are in the default Delphes card), the ATLAS values are worse by a factor of ∼ 2. We artificially imposed the reduced tau reconstruction numbers for consistency.
The shape of the exclusion curves for two of the signal regions are shown in Fig. 9, and exemplify the difference between gaugephilic and gaugephobic models which were not observed in the ditau search discussed in Sec. 4.1. The shapes show that the exclusions closely follow the π D the branching ratios.
Out of the 144 signal regions defined in the ATLAS search, we find that 16 provide some level of constraint. These are summarized in Fig. 10. Picking the strongest limit from the signal regions, we find π D > 370 GeV in the gaugephilic, m π D /m ρ D = 0.45 case and π D > 330 GeV in the gaugephobic, m π D /m ρ D = 0.45 case. For m π D /m ρ D = 0.25 the bounds are looser, due to the fact that smaller m π D /m ρ D for fixed m π D implies a heavier ρ D , and therefore a smaller resonant contribution to the pp → π D π D cross section. The difference between the limits in the gaugephilic and gaugephobic can be traced to the presence of more Higgs bosons in the gaugephilic π D decays, since more Higgs bosons leads to more events with τ s or b-jets.
The CMS generic multilepton search is similar, but contains some important differences. 9 Also, there was no dedicated τ trigger in place for this analysis. It is based on 19.5 fb −1 of √ s = 8 TeV data [147] and looks for events with either three or four reconstructed leptons. In this case, the definition of leptons includes electrons with p T > 10 GeV and |η| < 2.4, muons with p T > 20 GeV and |η| < 2.4, or hadronically decaying taus with p T > 20 GeV and |η| < 2.3. To trigger, events must contain either an electron or muon with at least p T > 20 GeV and events are only allowed to have one hadronic tau.
The events are divided into 192 independent bins (96 for each of the three or four lepton cases). The bins are split based whether there are same-flavor-opposite-sign (OSSF) pairs of leptons, the invariant mass of existing OSSF pair, the presence of tagged b jets, the number of hadronic τ leptons, the amount of missing energy, and the scalar sum of accepted jet transverse momenta. When CMS combines their signal regions, they are able to set bounds on new physics on the order of σ × Br 100 fb.
While it would in principle be possible to combine signal regions within our study, CMS does not provide the correlation information. Therefore, we are forced to examine each bin individually. This is in contrast to the method used in the ATLAS search, which used overlapping signal regions, such that some of the regions were more inclusive. Because of this, we find that the exclusions on the benchmarks from the CMS search are not as strong at the ATLAS ones. They are summarized in Fig. 11 for the signal regions which provide a limit. While the limits are not as strong, we find that the pattern is similar to the ATLAS result, in that the gaugephilic modes have tighter constraints than the gaugephobic models. Gaugephobic SU(2) 45 L Figure 10: Out of the 144 signal regions defined in Ref. [145], 16 regions constrain some portion of the dark meson parameter space. The mass ranges which are colored are excluded. The gaugephilic models have larger branching ratios to Zh and W h than the gaugephobic models, which leads to greater search efficiency and larger bounds.
To date, there is no 13 TeV multi-lepton analysis. Given the success we see in the 8 TeV versions at catching models that fall through the cracks in dedicated searches (see Sec. 4.4), we encourage ATLAS and CMS to pursue similar model-independent, inclusive searches in the future.

Same sign lepton searches
The last type of search that we find has sensitivity to pair produced dark pions is also fairly generic. The main difference is that instead of looking for three or four leptons, they ). In addition, the effective mass is defined as the scalar sum of the transverse momentum of the leading two leptons, the selected jets, and the missing energy.
Three different signal regions are defined. The first signal region has a veto on b-jets, which severely restricts the efficiency for higher mass π D . For lower masses, there is not enough missing energy in the events to pass the cut of E miss > 100 GeV, and an effective mass > 700 GeV. There are no limits from this region as well, due to the large amount of missing energy required.
The third signal region takes an different approach. In addition to the two same sign leptons, at least three b jets and at least 4 jets overall are required as well. In order to be statistically independent of the other regions, this region looks for events with small amounts of missing energy or transverse mass. The dark pions have no intrinsic missing energy (other than leptonic W decays), but do produce a lot of b quarks, making this an ideal signal region.
In the gaugephobic model, the fraction of decays to W ± h (Z h) grows with increasing charged (neutral) π D mass, while dark pions in the gaugephobic case predominantly decays to tb (tt). The difference in branching fractions leads to a smaller average b-jet multiplicity in the gaugephilic case which results in a slightly lower efficiency and, as a consequence, weaker bounds.
To obtain the number of expected signal events, we multiply the cross section and luminosity by the efficiency derived from the analysis cuts. These are then compared to the limits set by ATLAS. In the signal region, 4 events were observed against an expected background of 3.1 ± 1.6. With this, models which would produce 7.0 expected signal events are excluded at the 95% CL. The left panel of Fig. 12 shows the results of this signal region with number of expected events for the different models are shown in the red, blue, and green lines. The regions where the expected events extends above the black line are excluded. The only benchmarks which are limited by this search are the SU (2) 45 L models. The gaugephilic version is excluded for 210 GeV m π D 420 GeV, while the gaugephobic model is ruled out if m π D is between 250 GeV and 500 GeV. These are the strongest limits that we obtained for all of the searches.
With the success of the 8 TeV analysis, there was hope that when the search was extended to 13 TeV, the limits would greatly improve. However, this is not that case. Using 36 fb −1 of √ s = 13 TeV collisions, ATLAS searched for supersymmetry in final states with two same-sign leptons or three leptons [149]. The basic requirements are nearly the same for the lepton reconstruction, however, the η cut is tightened to |η| < 2.0 for electrons and loosened to |η| < 2.5 for muons.
The signal regions are more complicated in the 13 TeV analysis. There are 19 nonexclusive signal regions defined in terms of the number of leptons required; the number of b-jets; the number of jets harder than 25, 40 or 50 GeV, regardless of flavor; the missing energy and effective mass, and the charge of the leptons.
Unlike the previous search at 8 TeV, the 13 TeV search does not have any signal regions which require at least three b-jets. Instead, to cut down on background, the signal regions either require more than 6 jets or large effective mass. This combination is aimed at TeV scale colored particles and does not bode well for searching for pair produced particles with masses in the hundreds of GeV.
The only one of the 19 regions that has sensitivity to heavy dark mesons is the one that does not have requirements on the number of jets, the effective mass, or the missing energy. Instead, it requires at least three leptons of the same-sign and one b-jet. In addition, it requires that no combination of same-sign leptons has an invariant mass around the Z pole (veto 81 < m e ± e ± < 101 GeV).
The limits from this region for the different models are shown in the right panel of Fig. 12. The efficiency is largest in the mass region where π D + → tb and π D 0 → Zh dominate. The π D 0 → Zh mode is suppressed in the gaugephobic models, hence the limits are not quite as strong as the gaugephilic case. From Fig. 12, we see that this search only excludes m π D ∼ 200-400 GeV for η = 0.45, while η = 0.25 models are not constrained at all. Thus, while we expected that updating the best 8 TeV search would yield impressive bounds, it was unable to extend the limits above the 500 GeV bound set at √ s = 8 TeV.
This result highlights a troubling trend. We found the strongest limits from the 8 TeV analysis, pushing the mass of the dark pion to 500 GeV for the most excluded model. However, that search was designed with supersymmetry in mind, and using a supersymmetric interpretation of the 8 TeV search excluded sparticles (stops, specifically) up to 1 TeV. In the supersymmetry interpretation, it makes sense to harden the cuts and focus on particles heavier than 1 TeV. As we have seen in this analysis, however, imposing harder cuts as done with the 13 TeV analyses is detrimental to the signals in our benchmark models, with the result that the older, 8 TeV analyses yielded the strongest constraints. In the next subsection, we discuss other searches which have been thwarted in a similar way.

Additional searches
According to Table 2 (or the branching ratios in Figs. 2 and 3), we expect pair produced dark pions to result in lots of third generation fermions or gauge/Higgs bosons. However, this is not a unique feature of heavy dark mesons. Many BSM scenarios involve new particles that couple predominantly to gauge/Higgs bosons and third generation fermions, and as a result there are numerous LHC searches (underway, or already done) looking for characteristics signals of, e.g. multiple b-jets, multiple τ s, multiple e/µ in association with b-jets or τ , etc. of this type of final states. Based on energy and luminosity alone, the expectation is that one of these 13 TeV searches should be the most constraining. Our results strikingly show this is not the case -we find only a few searches constrain dark pions, with the strongest searches coming from 8 TeV.
Our main result, Fig. 6 came from considering a wide array of BSM searches. While the details of the most successful five searches have been provided in the previous sections, we summarize the other, un-constraining searches in Table. 3. In addition to the search channel, we provide a short explanation of why dark pions were so inefficiently captured by the search strategy.
While there are varying reasons the searches in Table 3 are not sensitive, there are a few common themes: 1. Searches expect single production. This is especially true for scalars which decay to the Higgs and gauge bosons. To cut down backgrounds, events are vetoed if there are too many objects to be only V h.

Searches assume large E miss
T . The searches which allow for pair production assume that pair production comes from a sector preserving a Z 2 symmetry and that therefore result in an invisible/dark matter particle at the end of the decay chain. While dark pions in the parameter space we are interested are predominantly pair-produced, they only decay back to SM particles. 13 Looks for a bump on the falling soft-drop jet mass spectrum. Not enough information to recast the designed decorrelated tagger. Only sensitive to σ 10 3 pb.
CMS Vector-like T → t h [157] 13 Looking for t h resonance, only very heavy and needs QCD production. Table 3: Possible search strategies which seem like they should set bounds, but have limited-to-no sensitivity.
but are not produced at rates the ATLAS and CMS are sensitive to, especially given that there are no leading order QCD-mediated production modes.
4. Data is not presented in a way that is recast-friendly. For instance, the CMS pair produced leptoquark search actually has some minor sensitivity when only using the total number of events. The search then uses the shape of the scalar sum of the p T of the light lepton, the hadronic τ , and the two jets to set limits, but they do not provide a fit of the shape. Similarly, experiments trying to measure standard model processes (such as hh → bbτ + τ − ) may potentially be sensitive to some π D parameter space, but they rely on machine learning techniques which cannot be reproduced.
We encourage the experiments to continue to push the limits of the LHC searches using all of the techniques they have available. However, as it is not possible for them to test every theory model, it is important that the results be presented in such a way that they can be reproduced without insider knowledge.

Conclusions
• In this paper we have examined the phenomenology of dark pions -composite states with electroweak and Higgs interactions that may lurk at the electroweak scale. Dark pion -like states are a component of many BSM scenarios with new strong dynamics near the electroweak scale.
• In addition to electroweak interactions, dark pions are also resonantly produced via dark rhos that kinetically mix with SM gauge bosons and decay through interactions with SM fermions or into hV . The overall size of the single-pion to SM coupling and the relative strength of the fermionic versus V h decay modes encodes some information about the symmetry structure of the strong sector and is the subject of Ref. [114].
• Taken more abstractly, dark pions represent a type of new physics that is predominantly pair produced, is uncolored, and decays back to SM final states. This is a particularly tricky combination for the LHC, since the lack of strong interactions means the BSM cross sections are small and the fact that the final states are pure SM leaves few easy handles to separate signal from background.
• The phenomenology of the dark pions is governed largely by a few parameters; the relative strength of the dark pion decays to fermionic versus gauge bosons, the type of kinetic mixing [whether with SU (2) L or U (1) Y ], and the mass of π D relative to ρ D . Setting up nine benchmark models with different values for these key parameters, we explored the constraints on dark mesons from 8 and 13 TeV LHC searches.
• The only scenario where we find constraints in the TeV range is when the ρ 0 D is kinematically forbidden from decaying to dark pions and therefore decays with significant branching ratio into leptons, the SU (2) 55 L,R cases. For all other cases, ρ D → π D π D is kinematically accessible so the dilepton bounds are negligible and the best avenue is to look for signals of π D pairs. Depending on the type of kinetic mixing and the relative mass of the ρ D mesons, the bounds on m π D from π D pair production signals vary from slightly above the LEP II charged particle bound to ∼ 500 GeV. The strongest bounds come when the mass of ρ D is not too much heavier than 2 m π D , and kinetically mix with the SU (2) L , while the weakest bounds come when the kinetic mixing only involves U (1) Y . As the most extreme example of how light these particles can be while remaining undetected, consider the SU (2) 45 R model. There, dark pions as light as ∼ 130 GeV are still viable; perhaps more surprising, the vector ρ D in this scenario sits at ∼ 300 GeV!
• In our survey of LHC searches, we found the most useful features for bounding dark mesons to be signal regions with high multiplicity of leptons and/or b-jets without strong requirements on the energy (of the individual objects, or summed) or missing energy. As model-specific searches march towards higher masses in the 13 TeV era, this type of signal region has become rarer and rarer. For scenarios without a dedicated search, such as the dark meson explored here -or, more generally, for types of BSM physics that is pair produced with sub-QCD rates and does not bring a non-SM source of missing energy -the net result is that 13 TeV searches can be less sensitive than 8 TeV versions. Generic searches based on multiple leptons served as a catch-all for this type of "non-standard" BSM at 8 TeV, and we encourage ATLAS and CMS to repeat similar studies with 13 TeV.