Long-Lived, Colour-Triplet Scalars from Unnaturalness
UUMN-TH-3506/15
Long-Lived, Colour-Triplet Scalars fromUnnaturalness
James Barnard ∗ , Peter Cox † , Tony Gherghetta ‡ , and Andrew Spray § ARC Centre of Excellence for Particle Physics at the Terascale, School of Physics, The University ofMelbourne, Victoria 3010, Australia School of Physics and Astronomy, University of Minnesota, Minneapolis, Minnesota 55455, USA Center for Theoretical Physics of the Universe, Institute for Basic Science (IBS), Daejeon, 34051, Korea
Abstract
Long-lived, colour-triplet scalars are a generic prediction of unnatural, or split, com-posite Higgs models where the spontaneous global-symmetry breaking scale f (cid:38)
10 TeV and an unbroken SU (5) symmetry is preserved. Since the triplet scalars arepseudo Nambu-Goldstone bosons they are split from the much heavier composite-sector resonances and are the lightest exotic, coloured states. This makes them idealto search for at colliders. Due to discrete symmetries the triplet scalar decays viaa dimension-six term and given the large suppression scale f is often metastable.We show that existing searches for collider-stable R-hadrons from Run-I at the LHCforbid a triplet scalar mass below 845 GeV, whereas with 300 fb − at 13 TeV tripletscalar masses up to 1.4 TeV can be discovered. For shorter lifetimes displaced-vertexsearches provide a discovery reach of up to 1.8 TeV. In addition we present exclusionand discovery reaches of future hadron colliders as well as indirect limits that arisefrom modifications of the Higgs couplings. ∗ [email protected] † [email protected] ‡ [email protected] § [email protected] a r X i v : . [ h e p - ph ] M a r ontents The discovery of a light Higgs boson and the conspicuous absence of new states beyondthe Standard Model at Run-I of the Large Hadron Collider (LHC) suggests that the scaleof new physics may well be beyond that suggested by naturalness arguments. CompositeHiggs models (for a recent review see [1]), which are typically motivated as a possiblesolution to the hierarchy problem, have therefore come under increased scrutiny as lowerlimits on resonance masses strain the boundaries imposed by naturalness. This tension isfurther exacerbated by precision electroweak and flavour constraints, both of which prefera much larger value of the spontaneous global-symmetry breaking scale, f , than can bedirectly probed at the LHC.A simple solution that can satisfy even the most stringent constraints (typically dueto flavour) is to require that f (cid:38)
10 TeV. This leads to an unnatural, or split, compositeHiggs model [2] in which the Higgs mass-squared is tuned to the order of 10 − and theparticle spectrum splits into light pseudo Nambu-Goldstone bosons and heavy composite-sector resonances. Despite their unnaturalness these models still preserve gauge coupling2 (cid:61) T e V c Τ (cid:61) . mm c Τ (cid:61) m m T (cid:61) m S (cid:43) m t (cid:43) m b S t a b l e m T (cid:64) GeV (cid:68) m S (cid:64) G e V (cid:68) Figure 1: A schematic diagram of the possible types of decays as a function of the colour-triplet scalar mass m T and singlet scalar mass m S . The three shaded regions from leftto right correspond to decays that are collider stable, displaced and prompt, respectively.The dashed line represents the kinematic limit for the decay T → t c b c SS and the blacksolid line represents the limit when m T = 2 m S .unification due to the presence of a composite right-handed top quark and an unbroken SU (5) symmetry in the composite sector provided f (cid:46)
500 TeV. An immediate consequenceis that the low-energy spectrum always contains a colour-triplet, pseudo Nambu-Goldstoneboson; the colour-triplet partner of the composite Higgs doublet. In addition discretesymmetries, which arise from proton stability, furnish these models with a singlet scalardark matter candidate, S . In the minimal model, the same discrete symmetries imply thatthe colour-triplet scalar decays to quarks and a pair of singlet scalars via a dimension-sixterm in the low-energy, effective Lagrangian. Since this high-dimension term is suppressedby the large symmetry-breaking scale, f (cid:38)
10 TeV, the triplet-scalar is often metastable.Long-lived, colour-triplet scalars therefore provide a unique way to test unnaturalness incomposite Higgs models.Motivated by unnatural composite Higgs models we study the collider limits on long-lived, colour-triplet scalars and the prospects for detecting them at future colliders. The3olour-triplet will be pair-produced via QCD processes and has the same quantum numbersas a right-handed scalar bottom quark. If long-lived, a colour-triplet will hadronize toform an R-hadron and can be detected in various ways depending on its decay length. Therange of decay lengths as a function of the singlet mass m S and triplet mass m T is shownin Figure 1.First, if the colour-triplet scalar is collider stable (i.e. decaying outside the detector),charged R-hadrons will leave a track in the inner detector and possibly the muon chamber.R-hadron searches at the LHC can then be used to place mass limits on the colour-triplet.Current limits from LHC Run-I results forbid a collider-stable colour-triplet with a massbelow about 845 GeV. At Run-II similar searches will be performed and we show that with300 fb − of integrated luminosity triplet masses up to about 1.4 (1.5) TeV can be discovered(excluded) for lifetimes corresponding to cτ (cid:38)
10 m. The discovery reach is significantlyincreased at a 100 TeV proton collider where discovery of a colour-triplet scalar with amass up to 2-6 TeV, depending on its lifetime, will be possible, otherwise exclusion limitsranging from 2-7 TeV can be set. These limits depend only on the mass and width of thecolour-triplet, therefore the results we obtain are quite general and can be applied to anyother model predicting a similar, long-lived particle.A second possibility is that the colour-triplet scalar decays within the detector (at radialdistances greater than about 4 mm) and produces a displaced vertex (DV) in the innerdetector. The colour-triplet in the minimal model decays into a top quark, bottom quarkand two singlet scalars, so the collider signature is predominantly jets from the quarksand missing energy from the singlets. This signal has previously been used to search forlong-lived superparticles such as gluinos and squarks. While current results from displacedsearches do not constrain the colour-triplet mass, these searches will become increasinglyimportant at Run-II and beyond. With 300 fb − at √ s = 13 TeV we find that colour-triplet masses up to 1.8 (1.9) TeV can be discovered (excluded) for singlet masses below450 GeV. In the future a 100 TeV collider would significantly improve the discovery reach,up to colour-triplet masses in the range 3-10 TeV depending on the singlet mass.The final possibility is that the colour-triplet scalar decays promptly, dominantly pro-ducing jets and missing energy. These decays become relevant when the colour-triplet isheavier than about 4 TeV. For such heavy colour-triplets the production cross section atLHC energies is quite small and there will be too few events to detect them, even at thehigh-luminosity (HL) LHC. Instead, prompt decays could be searched for at a hypothet-ical 100 TeV proton collider. Using a similar search strategy to that used for gluinos weshow that a future collider is potentially able to exclude colour-triplet masses in the range4-7 TeV for singlet masses in the range 100-900 GeV.4ndirect limits on the colour-triplet scalar mass can be obtained by constraining mod-ifications to the Higgs couplings. Using the current LHC results we find that colour-triplet masses are mostly constrained by the Higgs coupling to gluons to be in the range m T (cid:38)
100 GeV. This limit will improve at the HL-LHC and ILC, although the most robustlimits are inferior to the bound obtained from requiring that the triplet be heavier thantwice the singlet scalar mass. The latter is constrained by direct detection experiments,with the current LUX bound giving m S (cid:38)
150 GeV and hence m T (cid:38)
300 GeV.Previous studies of long-lived particles have primarily focused on supersymmetric mod-els, motivated by the idea of split supersymmetry [3–6] or simplified toy models withR-parity violation [7–12]. Our work is the first analysis of models based on the compositeHiggs idea. It is a complete framework, incorporating gauge coupling unification, darkmatter and an explanation for the fermion mass hierarchies, that represents an alternativeto split supersymmetric models. Interestingly, unnatural (or split) composite Higgs modelslead to similar decay signals albeit with different properties of the decaying particle anddecay products. It will therefore be interesting to experimentally distinguish between thesetwo ideas at future colliders.The outline for the rest of this paper is as follows. In Section 2 we review the unnaturalcomposite Higgs model and derive the decay width and corresponding decay length for thecolour-triplet scalar. The limits from experimental searches are presented in Section 3. Wefirst discuss direct limits from R-hadron searches at the LHC and a future 100 TeV collider,followed by limits from displaced-vertex searches and limits from prompt decays. We thenend with indirect limits that are obtained by studying modifications of the Higgs couplings.We summarise our results in Section 4. Details of the four-body phase space integral aregiven in Appendix A and in Appendix B we compare the validity of our assumptions onthe displaced-vertex search with the full experimental analysis.
We begin by briefly reviewing the unnatural composite Higgs model. Further details can befound in Ref. [2]. The underlying strong dynamics responsible for producing a compositeHiggs has an SU (7) global symmetry group which is spontaneously broken to SU (6) × U (1) at a scale f (cid:38)
10 TeV. This scale of breaking is chosen to satisfy all precisionelectroweak and flavour constraints without requiring any further symmetry in the model.This contrasts with the usual composite Higgs models where f (cid:38)
750 GeV in order to5inimise the tuning in the Higgs potential as much as possible, but where extra symmetriesare needed to satisfy flavour and precision electroweak constraints.The coset space SU (7) /SU (6) × U (1) contains twelve Nambu-Goldstone bosons whicharrange themselves into a complex of SU (5) (containing the Higgs doublet, H , and acolour-triplet scalar, T ) and a complex singlet, S . Note that this is the smallest coset spacethat preserves an SU (5) symmetry and thus gauge coupling unification due to a compositetop quark. The coset space also contains enough symmetry to prevent proton decay andstabilise the dark matter candidate, S . In particular, the strong sector is forbidden frommediating proton decay as it respects baryon number, B . It follows that it preserves baryontriality, a Z symmetry defined as Z B = 3 B − n C mod 3 , (2.1)where n C is the number of fundamental colour ( SU (3) C ) indices. All SM fields are neutralunder this symmetry, while T has B ( T ) = B ( H ) = 0 and n C = 1. Since a stable T is trivially excluded, we must use baryon triality to stabilise S , achieved by arranging B ( S ) = . A similar Z symmetry was previously used to stabilise composite fermionicdark matter in Refs. [13, 14].The SU (7) global symmetry is explicitly broken by coupling elementary-sector fieldsto composite-sector operators. This partial compositeness generates the Higgs potentialwhereupon a tuning, at least of order 10 − , is needed to obtain a 125 GeV Higgs boson.It also gives rise to masses for the singlet and colour-triplet scalars. The pseudo Nambu-Goldstone bosons ( H, T, S ) are light ( (cid:46) f ) and split from the composite-sector resonanceswhich are much heavier ( (cid:29) f ). There are also extra elementary-sector states, some ofwhich are coloured, known as top companions. These are instrumental in decoupling themultiplet partners of the composite right-handed top quark and obtain a mass of order f . Thus the scalar triplet is the lightest, coloured exotic state predicted by the unnaturalcomposite Higgs model and will generally be the most promising state to search for atcolliders. Because T is charged under baryon triality ( Z B = +2) it must decay to S , which has Z B = +1. Since the composite sector additionally preserves baryon and lepton number(required to forbid too-large neutrino masses) then the minimal possible decay is T → u c d c SS , (2.2)6 l ¯ b ¯ tSS Figure 2: Leading Feynman diagram for colour-triplet scalar decay.where u c , d c are the SU (2) singlet quarks with Z B = 0. Further, it is clear that t c , b c willdominate other final states, as the third generation couples most strongly to the compositesector. We would expect this decay to correspond to a dimension-6 operator in the low-energy effective theory after integrating out the heavy composite resonances. However, nosuch operator is generated in our model due to accidental symmetries associated with thenecessary fermion representations. Instead, this decay is generated by the dimension-10operator L ⊃ Π f λ b c λ ν λ ∗ τ (cid:15) i j k ( b c ) i ( t c ) j ( T † ) k S l † /p l . (2.3)Here, Π ∼ O ( p / Λ ) is a form factor, Λ ≈ g ρ f is the approximate resonance mass, g ρ astrong sector coupling, and the λ ’s are spurions for the partial compositeness couplings ofthe SM fermions. This operator exploits the fact that the lepton doublet has two couplingsto the composite sector. It generates the decay of Eq. (2.2) after closing the leptons intoa loop and this turns out to be less suppressed than the six-body final state.Eq. (2.3) is only the leading contribution to the T decay. Integrating out the com-posite sector will generate additional operators at higher orders. Further contributionsto the decay (2.2) necessarily involve loops of elementary particles and are suppressed by λ / (16 π g ρ ), where λ is the appropriate elementary-composite spurion couplings. Otherdecay modes must involve at least two additional fermions, so are phase-space suppressedby m T / (8 π Λ ). It is therefore a good approximation to neglect alternative operators.The relevant Feynman diagram is shown in Figure 2. Neglecting the lepton mass thematrix element becomes i M = − i f λ b c λ ν λ ∗ τ (cid:15) i j k ¯ u ( p t ) P R u ( p b ) (cid:90) d p l (2 π ) ( − (cid:20) /p l p l /p l P L (cid:21) Π , (2.4)where i , j , k are colour indices, u, ¯ u are spinors and P L,R are projection operators. Theloop integral is cut off by the presence of composite resonances at the scale Λ. We cannotcompute this integral without knowledge of the physics at that scale, so we define (cid:90) d p l (2 π ) ( − (cid:20) /p l p l /p l P L (cid:21) Π = − (cid:90) d p l (2 π ) Π = − c T Λ (4 π ) , (2.5)7here c T is an order-one constant. The matrix element now takes a simple form13 (cid:88) |M| = (cid:18) c T π f (cid:19) | λ b c λ ν λ ∗ τ | p t · p b . (2.6)The calculation of the decay width is straightforward, though details regarding the four-body phase space integral are given in Appendix A. We define a dimensionless function, J , to capture the phase-space suppression from non-zero final state masses J ( m t , m S ) = 72 m T (cid:90) dQ dQ Q (cid:115) I (cid:18) Q m T , Q m T (cid:19) (cid:18) − m t Q (cid:19) (cid:115) − m S Q , (2.7)where the function I ( a, b ) is defined in Eq. (A.4). The limits on the integrals are given byEqs. (A.11) and (A.12). By construction, J (0 ,
0) = 1. The total width isΓ = ( c T ) π | λ b c λ ν λ ∗ τ | m T f J ( m t , m S ) . (2.8)Compared to the result in Ref. [2] the width in the zero-mass limit differs by a factor of5/16. Finally, making the replacements λ b c ∼ (cid:112) g ρ y b and λ ν ∼ λ τ ∼ (cid:112) g ρ y τ , where y b ( y τ ) are the bottom (tau) Yukawa couplings, we obtain the approximate expression forthe decay length cτ = 0 . (cid:18) c T (cid:19) (cid:18) g ρ (cid:19) (cid:18) m T (cid:19) (cid:18) f
10 TeV (cid:19) J ( m t , m S ) . (2.9)We see that for typical parameters in the unnatural composite Higgs model the decaylength is of order the millimetre scale. The decay length can be substantially larger byeither increasing the scale f , reducing the triplet mass, or having kinematic suppression m T ≈ m S + m t (i.e. J ( m t , m S ) ≈ Any composite Higgs model that unifies via an SU (5) gauge group will contain (at least) acolour-triplet pseudo Nambu-Goldstone boson like the one discussed here. Although otherunification patterns are possible, precision unification in composite Higgs models is onlyknown to occur via an SU (5) gauge group, and only when the right-handed top quarkis fully composite [15]. Unless a qualitatively different solution for precision unification isfound light, colour-triplet scalars can therefore be considered a generic feature of unnaturalcomposite Higgs models. 8hether the colour-triplet scalar is long-lived or not depends more on the details ofthe model. It will necessarily be charged under baryon triality, a symmetry that musthold at least approximately in order to prevent proton decay. This has a stabilising effecton the colour-triplet and means that it will preferentially decay via other exotic states.Furthermore, because the colour-triplet scalar is a pseudo Nambu-Goldstone boson theonly available states are other pseudo Nambu-Goldstone bosons. In itself this is not enoughto guarantee a long-lived state but, in the minimal model proposed in Ref. [2], includingthe SM matter content resulted in several additional, accidental symmetries that stabilisedthe scalar colour-triplet even more. Accidental symmetries like these are increasingly likelyto occur in more complicated models with larger initial symmetry groups so, while it isby no means certain, long-lived colour-triplet scalars seem likely to be a feature of mostunnatural composite Higgs models exhibiting precision gauge coupling unification. We next discuss experimental searches for colour-triplet scalars. We first present limitsfrom various direct searches that look for decays over a range of decay lengths. Afterwardswe discuss indirect limits on the colour-triplet mass that arise from the modification of theHiggs couplings.
ATLAS and CMS have published comprehensive R-hadron searches, including searchesfor charged R-hadrons escaping the detector [16, 17] and searches for R-hadrons gettingstopped by and then decaying within the detector [18, 19]. The former analyses give riseto the strongest bounds so we shall use them to derive constraints on unnatural compositeHiggs models, and also generalise them to estimate the R-hadron discovery and exclusionpotentials of future experiments. Since our results depend only on the mass and width ofthe colour-triplet scalar they can be applied to any model predicting a long-lived particleof a similar nature.The searches are characterised by low backgrounds, between zero and one event after20 fb − of 8 TeV collisions, and signal efficiencies around 10%. The ATLAS study inRef. [18] demonstrated that R-hadrons with more than 20 GeV of kinetic energy are notsignificantly slowed by the detector. We therefore take the following approach to deriveconstraints on unnatural composite Higgs models.9 Read in colour-triplet scalar production cross-sections from Ref. [20]. • Pair produce R-hadrons using the R-hadronisation routines in
PYTHIA 8.1 [21–23]. • Discard R-hadrons with less than 20 GeV of kinetic energy. • Record the mass, energy, and transverse momenta of all remaining R-hadrons. • Weight each event by a survival factor (the probability of both R-hadrons escapingthe detector). • Apply the reported signal acceptance-times-efficiency values.In several of these steps we exploit the fact that the colour-triplet has the same quantumnumbers as a (right-handed) sbottom, so various tools designed for SUSY searches can beeasily repurposed.Because the backgrounds are so low it is necessary to weight each event by a survivalfactor instead of allowing R-hadrons to decay directly in
PYTHIA . Prohibitively large num-bers of events are otherwise needed to investigate the discovery and exclusion potentialsof future experiments. The survival factor, p , for each R-hadron is given by p ( r decay > r detector ) = e − β T r detector Γ /γ (3.1)where β T is the R-hadron’s transverse speed and γ its overall Lorentz factor, both derivedfrom the mass, energy, and transverse momentum of the R-hadron. Γ is the colour-tripletwidth and we assume a value of r detector = 10 m for the detector radius throughout thisstudy.For the number of background events we assume that the existing values will simplyscale up with luminosity at future experiments. Taking a value from the ATLAS study inRef. [16] gives 0.27 events per 19.1 fb − . Similarly, we assume that the signal acceptance-times-efficiency will remain constant, the same study giving a value of 0.084.The results of this analysis are the discovery and exclusion contours shown in Figure 3.These are presented in the plane of the colour-triplet mass, m T , versus its lifetime, cτ .We find that the final LHC dataset will be able to discover long-lived, colour-triplets witha mass up to around 1.4 TeV, and exclude those with a mass up to around 1.5 TeV. A100 TeV collider would increase these values considerably, to 6 and 7 TeV respectively.10 .4 0.6 0.8 1.0 1.2 1.4 1.6 m T [TeV] -2 -1 c τ [ m ] LHC R-hadron searches σ discovery: 300 fb − @ 13 TeV95% exclusion: 300 fb − @ 13 TeV95% exclusion: 19.1 fb − @ 8 TeV m T [TeV] -3 -2 -1 c τ [ m ]
100 TeV R-hadron searches σ discovery: 3000 fb − @ 100 TeV95% exclusion: 3000 fb − @ 100 TeV Figure 3: Current status and future prospects for R-hadron searches as functions of colour-triplet scalar mass and lifetime.
Traditional heavy stable charged particle or R-hadron searches provide good sensitivitywhen the colour-triplet scalar is stable or has a long enough lifetime such that most of thedecays occur outside the detector. However for shorter lifetimes these types of searchesbegin to lose sensitivity as shown in Figure 3. Dedicated searches for displaced decaysare therefore essential in order to cover the entire parameter space of the model. Thereare now a variety of ATLAS [25–29] and CMS [30–33] searches specifically targeting dis-placed signals. However recasting limits from these searches is difficult without access tothe complete detector simulations used by the collaborations. Nevertheless several recentpapers [7–10, 12] have demonstrated that, with some reasonable assumptions, good agree-ment with the full experimental analyses can be achieved. The most relevant search for ourmodel is the ATLAS displaced-vertex search [29] and we shall take a similar approach tothat of Ref. [10], which also reinterpreted this search but in the context of supersymmetricmodels with R-parity violation.The ATLAS displaced-vertex search targets long-lived particles which decay withinthe inner detector, up to radial distances ∼
30 cm. The search looks for displaced verticescontaining at least five charged particle tracks in addition to the presence of a high- p T muon ATLAS has now also performed a search for metastable R-hadrons [24] which decay within the detectorat radial distances greater than 45 cm. This search is expected to have lower sensitivity than the displaced-vertex search we consider here except for a narrow range of lifetimes approaching the collider stable case.
11r electron, jets or missing energy ( /E T ). All channels are essentially background free withless than one event expected. We will focus on the DV+jets and DV+ /E T channels as theseare expected to give the highest sensitivity to our colour-triplet decay. The displaced vertexrequirements along with the final selection criteria in each of the channels are detailed inTable 1. Selection criteriadisplacedvertex ≥ p T > | d | > r DV <
300 mm, | z DV | <
300 mmand ≥ m DV >
10 GeV (assuming m ± π for individual tracks)material vetoDV+jets ≥ p T >
90 GeV) or ≥ p T >
65 GeV)or ≥ p T >
55 GeV) and | η | < . /E T /E T >
180 GeVTable 1: Displaced vertex requirements and final selection criteria for the ATLAS displaced-vertex search in the DV+jets and DV+ /E T channels.In replicating the experimental analysis we must also take into account the ATLAStracking and vertex reconstruction procedures in addition to the above selections. Thestandard ATLAS tracking algorithms have a low efficiency for reconstructing tracks withlarge impact parameters ( d , z ) arising from displaced vertices. Therefore additional offlineretracking is performed with looser requirements on d and z . In order to account for thiswe have included an additional | d | -dependent efficiency factor multiplying the standardprompt efficiencies in the DELPHES 3 [34] detector simulation.In simulating the ATLAS vertex reconstruction algorithm we adopt the same procedureas Ref. [10]. Firstly we consider only tracks with p T > | d | > < r <
300 mm and | z | <
300 mm. Vertices are then reconstructedby firstly combining all track pairs with origins separated by < (cid:126)p , of these tracks must also satisfy (cid:126)d · (cid:126)p/ | (cid:126)p | > −
20 mm, where thevertex position, (cid:126)d , with respect to the primary vertex (PV) is taken as the average positionof its constituent track origins. Any vertices separated by < /E T channel in order to accurately estimatethe missing energy. Firstly, we neglect any prompt tracks from R-hadrons that decay withinthe detector and which are anyway ignored when reconstructing displaced vertices. Wealso neglect the curvature of these R-hadron trajectories in the magnetic field, which willgenerally be small due to their large momenta. The decay products (excluding neutrinos)of R-hadrons decaying within the calorimeters are assumed to deposit all of their energy,although clearly this assumption is not expected to be valid for R-hadrons decaying nearthe outer edge. We neglect any energy deposits from the R-hadrons themselves which areexpected to be small. R-hadrons decaying within the muon spectrometers are unlikely tobe reconstructed as muons and are therefore assumed to contribute to /E T . Finally, chargedR-hadrons which escape the detector are assumed to be reconstructed as muons.Similarly to the R-hadron search, signal events were generated using the R-hadronisationroutines in PYTHIA although with additional matrix-element re-weighting to correctly cap-ture the kinematics of the 4-body decays of the triplet. The dominant (albeit very small)source of background for this search is due to low- m DV vertices which are crossed by anunrelated high- p T track. We assume that the current background expectations scale withincreased luminosity while the systematic uncertainties remain fixed. We also assume asystematic uncertainty of 20% on the signal efficiency. The 5 σ discovery reach and 95% CLsexclusion limits in the ( m T , m S ) plane are then shown in Figures 4 and 5. Limits werecomputed in the ROOSTATS [35] framework using the asymptotic formula for the profilelikelihood [36] and Gaussian constraints for the systematic uncertainties.We find that with the existing 8 TeV dataset this analysis does not have sufficientsensitivity to provide constraints on our colour-triplet scalar. This is due to the factthat for masses where the cross-section is sufficiently large the triplet is in most casesdecaying outside the detector and R-hadron searches provide the only constraints. Howeverdisplaced searches will become important to probe the full parameter space in Run-II andbeyond. In Figure 4 we see that with 300 fb − of integrated luminosity this search canpotentially discover our colour-triplet up to masses of 1.8 TeV and exclude it up to 1.9 TeV.Furthermore this search is clearly complementary to the R-hadron searches considered inthe previous section and the combination of both searches provides good coverage of the( m T , m S ) plane. For both searches the upper bound on the colour-triplet mass is cross-section limited and the reach is expected to improve with the increased dataset of theHL-LHC. Finally, we can also consider larger values of f (cid:38)
100 TeV, which increasesthe lifetime of the colour-triplet. In this case R-hadron searches will provide the onlyconstraints at the LHC.In Figure 5 we also consider the prospects for this search at a hypothetical √ s =13igure 4: Projections for the R-hadron and displaced-vertex searches at the LHC with300 fb − of integrated luminosity at √ s = 13 TeV as functions of the scalar mass m S andtriplet mass m T . The shaded regions can potentially be excluded at 95% CLs and thedashed lines denote the 5 σ discovery reach. The grey shaded region is excluded by currentR-hadron searches at √ s = 8 TeV.100 TeV collider. We have assumed the same experimental cuts as the current ATLASanalysis, which leads to signal efficiencies of up to ∼
70% for the highest colour-tripletmasses considered. Of course in practice the cuts are likely to be more stringent, driveneither by trigger considerations or background expectations derived from data. Althoughnote that the signal efficiency can reach 60% for some of the benchmark models consideredin the existing analysis, suggesting that our estimate is not unreasonable. Nevertheless wealso show results with the signal efficiency reduced by a factor of two in order to providea more conservative estimate of the discovery reach. Regardless, we find that the reachwould be significantly greater than at the LHC with potential discovery of the scalar tripletup to masses around 10 TeV. 14igure 5: Projections for a hypothetical √ s = 100 TeV collider with 3000 fb − of inte-grated luminosity as functions of the scalar mass m S and triplet mass m T . The shadedregions show the 5 σ discovery reach (95% CLs exclusion limit) for the R-hadron/displaced(prompt) searches. The dashed lines include an additional factor of two reduction in thesignal efficiency for DV searches to account for the impact of more stringent experimentalcuts. The left and right panels correspond to f = 10 and 100 TeV respectively. Standard searches for prompt decays of the colour-triplet are not expected to provide usefulconstraints at the LHC. This is simply due to the fact that for masses below about 4 TeV(assuming f = 10 TeV) most of the colour-triplet decays will be displaced, while for highermasses the LHC will not produce enough events even by the end of the planned HL-LHCupgrade. However future colliders may be able to probe this region of parameter spacewhere the colour-triplet lifetime is small enough to lead to prompt decays, less than about2 mm.We therefore investigate the potential limits from a hypothetical 100 TeV proton col-lider. Of course many assumptions have to be made about the future performance of sucha machine and we will use the Snowmass detector [37] implemented in DELPHES to modelthe detector performance. We also make use of the Snowmass background Monte-Carlo For larger values, f (cid:38)
100 TeV, prompt-decay searches will not be constraining even at a future √ s = 100 TeV collider and all limits will be from displaced-vertex and R-hadron searches. PYTHIA and we use the sameweighted event generation procedure as used for the background events in order to obtain asample suitable for studies with high integrated-luminosity. In our case the events are sep-arated in bins of p T to allow for straightforward implementation using PYTHIA and 50 000events are generated in each bin.The ATLAS experiment has recently performed a search for gluinos [39] which considersa similar final state to that which arises from the pair production of our colour-triplet. Wewill employ a similar search strategy for our 100 TeV analysis, however extracting thesignal for the colour-triplet case is significantly more challenging due to the reduced cross-section and, as we shall see, this leads to a relatively limited reach even at √ s = 100 TeV.We will focus on a search using the purely hadronic final state. Searches in the leptonicchannel were also considered but are expected to be less sensitive for higher triplet massesdue to the small cross-sections combined with a lower branching fraction. To begin wemake the following preselection cuts: • ≥ p T >
50 GeV, | η | < . • ≥ • leading jet p T >
150 GeV, • δφ jmin > . • /E T >
400 GeV, • m eff > • No isolated leptons ( p T >
20 GeV, | η | < . k T algorithm [40, 41] with R = 0 . δφ jmin is defined as the minimum azimuthal separationbetween /E T and each of the four leading jets with p T >
20 GeV and | η | < .
5. The cut onthis variable is designed to reduce the contribution to /E T from poorly reconstructed jets orneutrinos emitted in the direction of a jet. Combined with the cut on /E T this is expectedto reduce the QCD background to a negligible amount, although the QCD background hasnot been simulated as part of the background sample. Finally, m eff is defined as the scalarsum of /E T and all jets with p T >
50 GeV and | η | < .
5. We also neglect events where16he triplet decay vertex is displaced by more than 2 mm in the radial direction since theywould likely fail b-tagging track requirements [42, 43].After these preselection cuts the background still dominates over the signal in theselected sample by several orders of magnitude. The dominant background for this searchis t ¯ t + jets. While we expect our signal to exhibit a higher b-jet multiplicity and increased /E T compared to the background, the large t ¯ t cross-section means that the number ofbackground events can still easily exceed the signal expectation even in the tails of thebackground distributions. This can be clearly seen in Figure 6 where we have plotted thesignal and background distributions of /E T and m eff after applying the preselection cuts forthree benchmark signal points.Next we optimise the cuts on the number of b-jets ( N b ), /E T and m eff in order toobtain the optimal background rejection as a function of signal efficiency using the TMVA package [44] in
ROOT v5.34 . This was performed separately for each signal point in ascan over the ( m T , m S ) plane. However we find that the cuts yielding the maximumsignal significance do not vary significantly over the parameter ranges of interest. Wetherefore impose the following final cuts when deriving the exclusion limits: N b ≥ /E T > . m eff >
10 TeV. The background and signal yields for three benchmarkpoints after imposing the preselection and final cuts are shown in Table 2.Preselection Final selection( N b ≥ /E T > . m eff >
10 TeV) t ¯ t ( ∗ ) + jets 7 . × W/Z + jets 9 . × t ¯ t + W/Z . × . × . × m T = 4000 GeV m S = 200 GeV 1720 13 m T = 5975 GeV m S = 835 GeV 378 19 m T = 7020 GeV m S = 160 GeV 147 22Table 2: Background and signal event yields before and after the final selection for threebenchmark signal points. Additional cuts on the number of jets and leading jet p T were also considered but found not to providesignificant improvement in the background rejection. [GeV] T E E v en t s / G e V (*) ttW/Ztt+W/Z/hdibosonother )=(4000,200) S ,m T m ( )=(5975,835) S ,m T m ( )=(7020,160) S ,m T m ( [GeV] eff m E v en t s / G e V (*) ttW/Ztt+W/Z/hdibosonother )=(4000,200) S ,m T m ( )=(5975,835) S ,m T m ( )=(7020,160) S ,m T m ( Figure 6: The /E T (upper) and m eff (lower) distributions for the backgrounds and threebenchmark signal points after imposing the preselection cuts.We can now compute 95% CLs exclusion curves in the ( m T , m S ) plane. The followingsystematic uncertainties are assumed in computing the limits: background normalisation(20%), signal efficiency (15%), PDF (5%) and luminosity (2.8%). We also consider themore optimistic assumption of 10% and 5% systematics for the background normalisationand signal efficiency respectively . The final exclusion curves are shown in Figure 5. Wesee that for the lowest singlet masses we are able to potentially exclude triplet masses inthe range 4-7 TeV. This upper reach is consistent with previous studies of colour-triplets at √ s = 100 TeV colliders in the context of supersymmetric simplified models [45]. However With reduced systematic uncertainties the analysis does benefit from additional signal regions (e.g. /E T > . m eff > m T , m S ) point. m T , m S ) parameter space where weare able to achieve a 5 σ discovery potential. One might expect this to be attainable forlower masses, where the cross-section is larger, however the colour-triplet then becomeslong-lived and we must turn instead to displaced searches for the strongest limits. Onceagain this search is clearly complementary to the R-hadron and displaced-vertex searchesand all three search strategies will be essential in order to probe the entire ( m T , m S ) plane.Although we see from Figure 5 that there remains a narrow region between the promptand displaced regimes which may be challenging to explore.Finally, there are inevitably many assumptions which must be made in estimating thereach of future colliders. The analysis considered here relies heavily on b-tagging and thisis likely to provide the largest source of uncertainty. We have chosen to use the loose b-tagging point defined for the Snowmass detector in our analysis as this assumes a reasonablyconservative estimate on the mis-tag rate of 3%. Improvements in b-tagging at the LHChave demonstrated that this kind of performance is reasonable for both highly boostedjets [46] and in high pile-up environments [43, 47]. We have also neglected the effects ofpile-up in our analysis, however we do not expect this to have a significant effect beyondthe impact on b-tagging. The assumptions made about the systematic uncertainties alsohave a significant effect on the final exclusion limit. Most corrections to the SM Higgs properties in composite Higgs models scale like v /f ,and as such are unobservable given our lower bound f (cid:38)
10 TeV. There are two possibleexceptions to this rule: loop contributions from other Nambu-Goldstone bosons, and theHiggs coupling to Zγ . Loop corrections from the scalar triplet will scale as v /m T . We havealready considered m T (cid:28) f in the previous sections; in such cases the triplet contributionscould be substantially enhanced. Limits derived this way are also independent of the tripletdecay mode. Second, the hZγ coupling is unique in being loop-level in the SM yet allowedby the shift symmetry [48] (through the operator γ µν Z µ ∂ ν h ). This could potentially allow acontribution enhanced by strong sector couplings g ρ /g SM and large numerical multiplicities.The modifications to the Higgs coupling from loops of new particles are well-known(see [49] and references therein). We follow Ref. [50] in parameterising the shifts in the hγγ , hgg , and hZγ couplings in terms of effective scale factors κ i κ g,γ,Zγ = 1 + ∆ A g,γ,Zγ A SMg,γ,Zγ , (3.2)where ∆ A i ( A SMi ) is the new physics (SM) contribution to the loop. For the colour-triplet19
00 400 600 800 1000024681012 m T ( GeV ) λ H T
500 1000 1500 2000012345 m T ( GeV ) m χ / f Figure 7: The triplet mass m T regions excluded by the Higgs coupling to gluons as functionsof the Higgs-triplet quartic coupling, λ HT (left) and top companion coupling, λ χ = m χ /f (right). The shaded regions are excluded by the current LHC measurements and the solid(dashed) lines show the prospective exclusions from the HL-LHC and the ILC.scalar, T , the contributions to the photon and gluon decays are very similar∆ A g = λ HT v m T A ( τ h ) , A SMg ≈ . , (3.3)∆ A γ = λ HT v m T A ( τ h ) , A SMγ ≈ − , (3.4) A ( τ ) = − τ (cid:2) − τ f ( τ ) (cid:3) , f ( τ ) = arcsin (cid:112) /τ (if τ > , (3.5) τ h = 4 m T m h , v = 246 GeV , (3.6)and λ HT is the scalar quartic term in the potential L ⊃ λ HT H † H T † T . (3.7)The contribution to the Zγ decay is slightly more complex∆ A Zγ = λ HT v s W c W m T I ( τ h , τ Z ) , A SMZγ ≈ , (3.8)20here s W ( c W ) is the sine (cosine) of the Weinberg angle, and the loop function I ( a, b ) = ab a − b ) + a b a − b ) (cid:0) f ( a ) − f ( b ) (cid:1) + a b ( a − b ) (cid:0) g ( a ) − g ( b ) (cid:1) , (3.9) g ( τ ) = √ τ − (cid:112) /τ (if τ > , τ Z = 4 m T m Z . (3.10)The current bounds from ATLAS, assuming no corrections to the other Higgs couplings,are [51] κ γ = 1 . ± . , κ g = 1 . ± . , κ Zγ < . . (3.11)It is clear that T shifts κ g much more than κ γ , due to the relative size of the SM contri-butions. The shift to κ Zγ is also subdominant due to a cancellation in the loop function.We show the 95% exclusion contour in the mass quartic-coupling plane in the left panelof Figure 7. We also show projected limits from the HL-LHC and ILC from κ g , assuminga SM central value and uncertainties of 5% [52–54] and 1% [55, 56] respectively. Theseresults hold for the generic case where a colour-triplet scalar is the only new light colouredstate coupling to the Higgs.In the unnatural composite Higgs the quartic coupling (3.7) is calculable up to order-onecoefficients [2] λ HT = 116 π (cid:18) c χχ | λ χ | + 89 c tt | λ t | + 49 c bb | λ b | + 89 c b c b c | λ b c | − c χt | λ χ | | λ t | + 23 c tb | λ t | | λ b | − c bb c | λ b | | λ b c | (cid:19) . (3.12)The consequent exclusions in terms of m χ /f ≈ λ χ , where m χ is the mass of the topcompanions, are shown in the right panel of Figure 7, assuming λ t ≈ y t (where y t is thetop Yukawa coupling), λ b,b c ≈ (cid:112) g ρ y b , g ρ ≈
8, and all c i ≈
1. The contributions from thebottom Yukawa are negligible. There is a model-independent limit from the top Yukawa,which from the LHC is m T (cid:38)
100 GeV. For heavy or light top companions the limitsgets stronger, with the ILC able to exclude 2 TeV triplets for top companions with masses m χ = 5 f . In addition to the limits of the previous section these should be compared to thebound m T > m S required to avoid a stable colour-triplet. The current LUX bounds [57]enforce m S (cid:38)
150 GeV and hence m T (cid:38)
300 GeV, which is already superior to potentialHL-LHC bounds unless m χ (cid:38) f .Finally, we note that in the unnatural composite Higgs model the contributions tothe hZγ coupling from the strongly-interacting sector vanish at leading order. This is aconsequence of the unbroken SU (5) global symmetry, and thus applies to any composite21iggs model compatible with an SU (5) or SO (10) GUT. It is distinct from the parityargument discussed in Ref. [48], as that symmetry only exists in models with a custodial SU (2). The low-energy effective theory for the Nambu-Goldstone bosons in the absenceof explicit breaking of the global symmetry is given by the CCWZ expansion. Becausethis respects SU (5), the gauge fields can only appear in two forms: as part of the Nambu-Goldstone covariant derivatives, and in the SU (5) matrix form F µν = (cid:32) g s G aµν t aSU (3) − g (cid:48) B µν × ( gW iµν σ i + g (cid:48) B µν × ) (cid:33) , (3.13)where t aSU (3) are the Gell-Mann matrices and σ i are the Pauli matrices. The lower blockdiagonal term is the one which multiplies the Higgs field when F µν is contracted with theNambu-Goldstone field. In terms of mass-basis fields, we have F (2) µν ≡ (cid:0) gW iµν σ i + g (cid:48) B µν × (cid:1) ∼ (cid:32) γ µν W µν W † µν Z µν (cid:33) . (3.14)There are only three possible terms that can appear at dimension-6 involving the Higgsand gauge fields: H † F (2) µν F (2) µν H , ( D µ H ) † F (2) µν D ν H , (cid:15) µνρσ H † F (2) µν F (2) ρσ H . (3.15)In particular, a term like H † H Tr[ F (2) µν F (2) µν ] would break the shift symmetry. Expandingthese expressions in the unitary gauge, we see that none of them involve a coupling of theHiggs to the photon. At this order the hZγ coupling can then only be generated by thespurion couplings between the elementary and confining sectors and therefore will not beenhanced. In the unnatural, or split, composite Higgs model electroweak precision and flavour con-straints are simply eliminated by requiring that f (cid:38)
10 TeV. This causes a splitting ofthe particle spectrum as the pseudo Nambu-Goldstone bosons are much lighter than thecomposite-sector resonances. In order to preserve gauge-coupling unification the model hasa composite right-handed top quark and the strong sector must remain invariant under an SU (5) global symmetry. This means that the low-energy spectrum generically contains the SU (5) colour-triplet partner of the Higgs doublet, as well as a singlet scalar that plays therole of dark matter. In the minimal model residual symmetries related to proton and dark22atter stability cause the colour-triplet scalar to decay via a dimension-six term in theLagrangian and, since f (cid:38)
10 TeV, it can be metastable. Thus a long-lived colour-tripletscalar provides a distinctive experimental signal to test for unnaturalness.R-hadron searches can be used to place limits on the colour-triplet mass and the currentlower limit on a collider-stable ( cτ (cid:38)
10 m) colour-triplet from LHC Run-I results is around845 GeV. We have shown that with 300 fb − of integrated luminosity at √ s = 13 TeVthere is potential for a discovery up to a colour-triplet mass of 1.4 TeV or else it can beexcluded up to 1.5 TeV. These limits significantly increase at a 100 TeV collider where,depending on the lifetime, triplets with masses ranging from 2 to 6 TeV can be discovered.Note that our limits from R-hadron searches are actually quite general, depending only onthe mass and lifetime of the colour-triplet, and can be applied to any other model. If thetriplet decays in the inner detector (4 mm < r DV <
30 cm) then displaced-vertex searchescan be used to obtain limits. We find that the LHC can discover (exclude) colour-tripletmasses up to 1.8 (1.9) TeV for singlet masses below 450 GeV. At a 100 TeV collider thediscovery reach is extended up to colour-triplet masses in the range 3-10 TeV dependingon the singlet mass. There is also the possibility that the colour-triplet decays promptlywhen the mass (cid:38) m T (cid:38)
100 GeV. These limits can be improved upon at the HL-LHC or the ILC but remainweak compared to the direct detection limit of m S (cid:38)
150 GeV from LUX, which impliesthat m T (cid:38)
300 GeV assuming the singlet is the lightest stable particle.Finally it should be noted that long-lived colour-triplet scalars are a sign of unnatu-ralness in composite Higgs models in much the same way that long-lived gluinos signalunnaturalness in split supersymmetric models. In both cases the experimental signals arequite similar because the decays produce jets and missing energy. Nevertheless there aredifferences related to the spin of the decaying particle and the particle(s) carrying themissing energy, as well as the large difference in the production cross-section. Given thatcurrent LHC results suggest that the Higgs potential may be tuned, it would therefore beworthwhile to study how these two unnatural possibilities could be distinguished at futurecolliders. 23 cknowledgements
We thank C.-P. Yuan for useful discussions as well as Abi Soffer and Nimrod Taiblum forhelpful correspondence regarding Ref. [29]. This work was supported by the AustralianResearch Council. TG was supported by the Department of Energy grant DE-SC0011842.AS was also supported by IBS under the project code IBS-R018-D1. PC is grateful to theFine Theoretical Physics Institute for hospitality during the completion of this work.24
Four-Body Phase Space Integral
We present the calculation of the four-body phase space integral that is needed for obtainingthe decay width of the colour-triplet scalar. We follow the common approach for many-body phase space integrals, and rewrite them as several two-body integrals. Given that thecolour-triplet T decays to t c b c SS , where t ( b ) is the top (bottom) quark and S is a singletscalar, let Q = p t + p b and Q = p S + p S . Note that the squared matrix element (2.6)depends only on Q , and is independent of all other kinematic variables. The four-bodyphase space integral can be written (cid:90) d Π ( p T ; p t , p b , p S , p S ) = (cid:90) d (cid:101) Π ( p T ; Q , Q ) d Π ( Q ; p t , p b ) d Π ( Q ; p S , p S ) , (A.1)where d Π ( p a ; p , p ) = d p (2 π ) d p (2 π ) πθ ( p ) δ ( p − m ) 2 πθ ( p ) δ ( p − m ) × (2 π ) δ (4) ( p a − p − p ) , (A.2) d (cid:101) Π ( p a ; p , p ) = d p (2 π ) d p (2 π ) (2 π ) δ (4) ( p a − p − p ) . (A.3)We can then do the integrals over all momenta other than Q , trivially. Let us introducethe triangle function I ( a, b ) = 1 + a + b − a − b − ab . (A.4)Then the two-body phase space integral may be written (cid:90) d Π ( p a ; p , p ) = 18 π (cid:18) | (cid:126)p | p a (cid:19) COM = 18 π (cid:115) I (cid:18) m m a , m m a (cid:19) . (A.5)The first result is the well-known expression for the two-body phase space in the centre ofmass frame; the second result expresses this in Lorentz-invariant form. Since the integralis manifestly Lorentz-invariant this result holds in all frames. In the two specific cases werequire this simplifies further. Neglecting the bottom quark mass we have (cid:90) d Π ( Q ; p t , p b ) = 18 π (cid:18) − m t Q (cid:19) , (A.6) (cid:90) d Π ( Q ; p S , p S ) = 116 π (cid:115) − m S Q . (A.7)The additional factor of one-half in the latter equation is due to the presence of identicalfinal states. 25ext, we rewrite the integral over Q and Q . It is easy to see that, if p , are constrainedpositive, d (cid:101) Π ( p a ; p , p ) = dm π dm π d Π ( p a ; p , p ) . (A.8)This condition applies to Q , . Therefore we may write (cid:90) d (cid:101) Π ( p t ; Q , Q ) = (cid:90) dQ π dQ π π (cid:115) I (cid:18) Q m T , Q m T (cid:19) . (A.9)Putting all of this together, we have the final result (cid:90) d Π ( p T ; p t , p b , p S , p S ) = 12 π (cid:90) dQ dQ (cid:115) I (cid:18) Q m T , Q m T (cid:19) (cid:18) − m t Q (cid:19)(cid:115) − m S Q . (A.10)Finally we need the limits on the integral. It is straightforward to see that the absolutebounds on Q are m t < Q < ( m T − m S ) . (A.11)The lower bound occurs when the b quark is produced at rest, and the upper bound whenthe two S are at rest. For any given Q there is an upper bound on Q and so4 m S < Q < (cid:18) m T − (cid:113) Q (cid:19) . (A.12)The lower bound arises from when the two S are at rest, while the upper bound is obtainedwhen they are back-to-back. B Displaced-Vertex Search Validation
Given the challenges involved in recasting displaced searches and the various assumptionsthat must be made, it is important to check the validity of our implementation againstthe full experimental analysis. We have therefore also simulated events for one of thesignal processes considered in the ATLAS paper [29]. We have chosen the case of a long-lived gluino decaying to two top quarks and a 100 GeV neutralino since this most closelyresembles the final-state that is produced by the decay of our colour-triplet.In Figure 8 we compare the event-level efficiencies obtained from our analysis (datapoints) with the results reported by ATLAS (shaded regions) for both the DV+jets andDV+ /E T channels. Overall we find that our analysis gives reasonably good agreementwith the full experimental analysis, especially in the DV+jets channel. The discrepancies26n the DV+ /E T channel suggest that our assumptions regarding the reconstruction of thedecay products from displaced R-hadron decays leads to an underestimate of the missingenergy. The difference in signal efficiency is not expected to have a significant effect on theexclusion limits we derive, especially at higher center-of-mass energies where the expectedmissing energy from our signal can be significantly greater than the experimental cuts. (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230)(cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224)(cid:236) (cid:236) (cid:236) (cid:236) (cid:236) (cid:236) (cid:236) (cid:236) (cid:236) DV + jets1 10 100 10000.00.10.20.30.40.5 c Τ (cid:64) mm (cid:68) E v e n t (cid:45) l e v e l e ff i c i e n c y (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230) (cid:230)(cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224) (cid:224)(cid:236) (cid:236) (cid:236) (cid:236) (cid:236) (cid:236) (cid:236) (cid:236) (cid:236) DV (cid:43) E T Τ (cid:64) mm (cid:68) E v e n t (cid:45) l e v e l e ff i c i e n c y Figure 8: Comparison of the event-level efficiencies from our analysis (data points) andthe ATLAS analysis (shaded regions) for the case of a long-lived gluino decaying to tt ˜ χ .From top to bottom the curves correspond to gluino masses of 1400, 1000 and 600 GeV.The left and right panels are for the DV+jets and DV+ /E T channels respectively. References [1] G. Panico and A. Wulzer, arXiv:1506.01961 [hep-ph].[2] J. Barnard, T. Gherghetta, T. S. Ray, and A. Spray, JHEP , 067 (2015),[arXiv:1409.7391 [hep-ph]].[3] J. D. Wells, in ,arXiv:hep-ph/0306127 [hep-ph].[4] N. Arkani-Hamed and S. Dimopoulos, JHEP , 073 (2005), [arXiv:hep-th/0405159[hep-th]].[5] A. Arvanitaki, N. Craig, S. Dimopoulos, and G. Villadoro, JHEP , 126 (2013),[arXiv:1210.0555 [hep-ph]]. 276] N. Arkani-Hamed, A. Gupta, D. E. Kaplan, N. Weiner, and T. Zorawski,arXiv:1212.6971 [hep-ph].[7] Y. Cui and B. Shuve, JHEP , 049 (2015), [arXiv:1409.6729 [hep-ph]].[8] Z. Liu and B. Tweedie, JHEP , 042 (2015), [arXiv:1503.05923 [hep-ph]].[9] A. de la Puente and A. Szynkman, arXiv:1504.07293 [hep-ph].[10] C. Csaki, E. Kuflik, S. Lombardo, O. Slone, and T. Volansky, JHEP , 016 (2015),[arXiv:1505.00784 [hep-ph]].[11] N. Zwane, arXiv:1505.03479 [hep-ph].[12] N. Nagata, H. Otono, and S. Shirai, JHEP , 086 (2015), [arXiv:1506.08206[hep-ph]].[13] K. Agashe and G. Servant, Phys. Rev. Lett. , 231805 (2004),[arXiv:hep-ph/0403143 [hep-ph]].[14] K. Agashe and G. Servant, JCAP , 002 (2005), [arXiv:hep-ph/0411254[hep-ph]].[15] K. Agashe, R. Contino, and R. Sundrum, Phys.Rev.Lett. , 171804 (2005),[arXiv:hep-ph/0502222 [hep-ph]].[16] CMS, S. Chatrchyan et al. , JHEP , 122 (2013), [arXiv:1305.0491 [hep-ex]].[17] ATLAS, G. Aad et al. , JHEP , 068 (2015), [arXiv:1411.6795 [hep-ex]].[18] ATLAS, G. Aad et al. , Phys. Rev. D88 , 112003 (2013), [arXiv:1310.6584 [hep-ex]].[19] CMS, V. Khachatryan et al. , Eur. Phys. J.
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