MI-HET-759

Tumblers: A Novel Collider Signature for Long-Lived Particles

Keith R. Dienes,1, 2, ∗ Doojin Kim,3, † Tara Leininger,4, ‡ Brooks Thomas4, §

1Department of Physics, University of Arizona, Tucson, AZ 85721, USA2Department of Physics, University of Maryland, College Park, MD 20742, USA

3Mitchell Institute for Fundamental Physics and Astronomy,Department of Physics and Astronomy, Texas A&M University, College Station, TX 77843 USA

4Department of Physics, Lafayette College, Easton, PA 18042, USA

In this paper, we point out a novel signature of physics beyond the Standard Model which couldpotentially be observed both at the Large Hadron Collider (LHC) and at future colliders. Thissignature, which emerges naturally within many proposed extensions of the Standard Model, resultsfrom the multiple displaced vertices associated with the successive decays of unstable, long-livedparticles along the same decay chain. We call such a sequence of displaced vertices a “tumbler.” Weexamine the prospects for observing tumblers at the LHC and assess the extent to which tumblersignatures can be distinguished from other signatures of new physics which also involve multipledisplaced vertices within the same collider event. As part of this analysis, we also develop a procedurefor reconstructing the masses and lifetimes of the particles involved in the corresponding decaychains. We find that the prospects for discovering and distinguishing tumblers can be greatlyenhanced by exploiting precision timing information such as would be provided by the CMS timinglayer at the high-luminosity LHC. Our analysis therefore provides strong additional motivation forcontinued efforts to improve the timing capabilities of collider detectors at the LHC and beyond.

I. INTRODUCTION

Ever since the seminal work of Glashow, Weinberg, andSalam in the 1970s that gave birth to modern particlephysics, the Standard Model (SM) has reigned supreme.Although the discoveries of dark matter, dark energy,and even neutrino masses have indicated the need to ex-tend the SM into new domains, the core of the SM hasremained intact and continues to accurately describe allexisting collider data despite decades of intense experi-mental research. Indeed, unambiguous evidence for pos-sible SM extensions such as weak-scale supersymmetryor large extra dimensions has not yet been found.

There are, in principle, two possible reasons for thisstate of affairs. On the one hand, the energy scale asso-ciated with the new physics may be sufficiently high thatthis physics lies beyond the reach of current experiments.However, on the other hand, it is possible that the newphysics resides at energy scales which are potentially ac-cessible at current or imminent collider experiments, butthat this physics is manifested through collider signa-tures that have not yet received much attention withinthe community.

In this paper we point out a novel collider signaturewhich arises in a variety of scenarios for new physics.This signature rests on the possible existence of long-lived particles (LLPs). As discussed in Ref. [1], LLPscan arise in many proposed extensions of the StandardModel. These include models which attempt to addressthe gauge hierarchy problem, models which provide new

∗ dienes@arizona.edu† doojin.kim@tamu.edu‡ leiningt@lafayette.edu§ thomasbd@lafayette.edu

approaches to dark-matter physics, models which de-scribe different scenarios for baryogenesis and leptoge-nesis, and even non-minimal models of neutrino physics.Because of their relative long lifetimes, LLPs, once pro-duced, can propagate across macroscopic distances beforethey decay. This in turn gives rise to macroscopicallydisplaced vertices (DVs), with proper decay lengths cτranging from millimeters to hundreds of meters inside acollider detector. While searches for DVs are part of thestandard experimental program at colliders, the signa-ture on which we shall focus our attention involves thepresence of multiple displaced vertices which result fromthe successive decays of multiple unstable LLPs withinthe same decay chain. In such cases, the event unfoldsby “tumbling” down the steps of the decay chain, termi-nating only once a collider-stable particle is reached.

Given this decay topology, we shall refer to such a se-quence of DVs as a “tumbler.” In this work, we shallconsider the special case of tumblers in which each suchLLP decay yields a single, lighter LLP as well as oneor more SM particles which can be detected directly bya collider detector. The signatures of such tumblers arequite striking as they have very low SM backgrounds. Weshall examine the prospects for observing such tumblersat the LHC, and we shall assess the extent to which suchtumbler signatures can be distinguished from other sig-natures of new physics which also involve multiple DVswithin the same collider event. We shall also develop aprocedure for reconstructing the masses and lifetimes ofthe particles involved in the corresponding decay chains.

One important theme running through this work willbe the observation that the prospects for discovering anddistinguishing tumblers can be greatly enhanced by ex-ploiting precision timing information. Fortunately, thissort of information can be provided by a precision tim-ing layer of the sort that will be installed within the CMS

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detector during the forthcoming high-luminosity upgradeof the Large Hadron Collider (LHC) [2, 3]. As we shallsee, this timing information can significantly improve theprecision with which the masses and lifetimes of the par-ticles within a tumbler can be measured.

This paper is organized as follows. In Sect. II, wedescribe the basic properties of tumblers and discussthe role that timing information can play in charac-terizing them. In Sect. III, we introduce a concreteexample model which can give rise to tumblers. InSect. IV, we survey the parameter space of this modeland identify regions of this parameter space wherein theprospects for identifying tumblers are particularly aus-picious. In Sect. V, we investigate the extent to whichcurrent LHC data constrains this parameter space andassess the prospects for observing a significant number oftumbler events both before and after the high-luminosityLHC (HL-LHC) upgrade. In Sect. VI, we develop anevent-selection procedure which provides an efficient wayof distinguishing between events which involve tumblersand events which involve multiple DVs which were not infact produced by the successive decays of unstable par-ticles within the same decay chain. We also investigatethe degree to which the masses and lifetimes of the dark-sector states can be measured from tumbler events. InSect. VII, we conclude with a summary of our resultsand a discussion of the ways in which improvements inenergy and timing resolution could enhance our abilityto distinguish tumbler signatures at the HL-LHC or atfuture colliders.

II. TUMBLERS AT THE LHC

Macroscopically displaced vertices can result from thedecays of long-lived particles (LLPs) that decay on dis-tance scales O(1 mm) . cτ . O(100 m) inside a colliderdetector. Such vertices represent a striking potential sig-nal of new physics [1, 4]. While the DV signatures asso-ciated with the decays of even a single LLP species canyield a wealth of information about physics beyond theSM, the phenomenology associated with DVs can be farricher in extensions of the SM which involve multiple ofLLP species. One intriguing possibility arises in scenariosin which one species of LLP can decay into a final statewhich includes both SM particles and a lighter LLP of adifferent species. If this lighter LLP also decays withinthe detector, the result is a sequence of two or more DVswhich result from the successive decays of unstable parti-cles within the same decay chain. Like DVs themselves,such sequences of DVs — i.e., such “tumblers” — canarise naturally in many extensions of the SM. These in-clude models involving compressed supersymmetry [5],theories involving large numbers of additional degrees offreedom with a significant degree of disorder in their massmatrix [6], and scenarios involving non-minimal dark sec-tors [7].

An example of a tumbler is illustrated in Fig. 1. In

this example, an LLP χ2 is produced within a colliderdetector at the primary interaction vertex VP , along withone or more additional SM particles. This χ2 particletravels a measurable distance away from VP before itdecays into a pair of SM particles (which for concretenesswe take to be a quark q and an anti-quark q), along withanother, lighter LLP χ1 at the secondary vertex VS . Thisχ1 particle, in turn, travels a measurable distance awayfrom VS before it likewise decays at a tertiary vertexVT into a quark q′, an anti-quark q′, and another, evenlighter LLP χ0, which escapes the detector and manifestsitself as missing transverse energy /ET .

Fig. 1 illustrates the topology of a tumbler involvingonly two DVs, as appropriate for a decay chain involvingthree LLPs (χ2, χ1, and χ0). In some sense, this is theminimal possible tumbler, and this case will be the focusof this paper. However, there is nothing that requirestumblers to be limited to only two DVs or three LLPs,and indeed longer decay chains leading to more DVs arepossible. Indeed, in many SM extensions, entire ensem-bles of LLPs χn can arise. Such ensembles can then giverise to potentially long decay chains with many sequentialDVs. However, all such tumbler events share the samebasic event topology, with sequential decays proceedingin linear fashion down the decay chain.

How might such a tumbler be detected and distin-guished? Since the SM backgrounds for processes in-volving DVs are quite low, signals involving DVs provideparticularly striking indications of new physics. A vari-ety of LLP searches involving DVs have already been per-formed by the ATLAS and CMS Collaborations. More-over, the sensitivity of the ATLAS and CMS detectorsto DV signatures will be significantly enhanced duringthe forthcoming HL-LHC upgrade, in part as a result ofthe installation of additional apparatus within both ofthese detectors which provides precision timing informa-tion about the particles produced in a collider event. Inparticular, the upgraded ATLAS detector will include ahigh-granularity timing detector in front of each of theend-cap calorimeters in order to provide timing informa-tion for particles emitted in the forward direction [8].The upgraded CMS detector, by contrast, will includenot only a pair of timing detectors located in front of theend-caps, but also a thin cylindrical timing layer situatedbetween the tracker and the electromagnetic calorimeter(ECAL) which provides coverage within the barrel re-gion of the detector [2, 3]. This timing layer, which isincluded in the illustration in Fig. 1, will provide a tim-ing resolution of σt ≈ 30 ps — a vast improvement overthe timing resolution σt ≈ 150 ps currently afforded bythe ECAL itself [9]. Such a significant enhancement intiming precision will significantly improve the sensitivityof LLP searches at the HL-LHC. Indeed, not only caninformation from the timing layer be used to reduce SMbackgrounds for such searches [10, 11], but it can alsoaid in the reconstruction of the LLP masses [12, 13]. Inparticular, the momenta ~pq and ~pq of the hadronic jetsassociated with q and q, in conjunction with timing the

3

q qq'

q'

χ2

χ1

VP

VS

VT

p p

Timing Layer

Additional Particles

Tracker

χ0ECAL

HCAL

FIG. 1. Schematic of a tumbler event within a collider detector modeled after the CMS detector at the HL-LHC. In this event,a heavy LLP χ2 is initially produced at the primary vertex VP , along with some additional particles. The χ2 particle thentravels a measurable distance before decaying into a lighter LLP χ1, a quark q, and an anti-quark q at the secondary vertexVS . This χ1 particle then travels a measurable distance away from VS and subsequently decays into an even lighter LLP χ0,another quark q′, and another anti-quark q′ at the tertiary vertex VT . The χ0 particle manifests itself as missing energy /ET ,while the quarks and anti-quarks manifest themselves as hadronic jets. Information about when each jet interacts with thetiming layer, in conjunction with additional information about the momentum of the jet from the tracker and calorimeters, canbe used to reconstruct the locations and times at which VS and VT occurred.

information for these jets provided by either the timinglayer or the ECAL, can be used to identify both the timetS and spatial location ~xS of VS . Similarly, the momenta~pq′ and ~pq′ of the jets associated with q′ and q′, in con-junction with the corresponding timing information, canbe used to identify the time tT and spatial location ~xTof VT . Information about the momenta of the additionalSM particles produced at VP , in conjunction with thecorresponding timing information, can be used to iden-tify the time tP and spatial position ~xP of this vertex.

III. A CONCRETE EXAMPLE MODEL

In order to perform a more quantitative assessmentof the prospects for detecting tumbler signatures at theLHC and beyond, it is necessary to work within the con-text of a concrete model. Such a model can therefore alsoserve as an existence proof that tumblers may indeedarise at colliders such as the LHC, and yet be consis-tent with current experimental results. The model thatwe adopt for this purpose is drawn from a general classof non-minimal dark-sector scenarios in which there ex-ist multiple dark-sector states χn with similar quantumnumbers, all of which can interact with the fields of thevisible sector via a common mediator particle φ. Not onlydo these interactions provide a portal through which theχn can be produced, but they also render the heavier

χn unstable. Since the final states into which these χndecay in such scenarios generically involve both SM par-ticles and other, lighter dark-sector states χm, extendeddecay chains can develop.

In Ref. [7] we constructed such a model within thisclass and focused on a region of parameter space in whichthe χn particles involved in these decay chains had life-times leading to prompt decays rather than macroscopi-cally displaced vertices. We then discovered that such de-cay chains can lead to striking signatures involving largemultiplicities of produced SM states.

In this paper, by contrast, we shall focus on a dif-ferent region within the parameter space of this model,one in which the χn have lifetimes within the rangeO(1 mm) . cτn . O(100 m). As we shall explain fur-ther below, we thus obtain decay chains involving DVs —i.e., tumblers. Moreover, although our analysis in Ref. [7]considered arbitrary numbers of χn states within the as-sociated decay chains, we shall here restrict our attentionto cases with only three χn particles, with n = 0, 1, 2 la-beling these states in order of increasing mass.

More specifically, this model is defined as follows. Weshall take the χn to be Dirac fermions and to be singletsunder the SM gauge group. We take the masses mn of theχn to be free parameters, subject to the condition m2 >m1 > m0. The particle φ which mediates the interactionsbetween the χn and the fields of the SM in our model istaken to be a complex scalar which transforms as a triplet

4

both under the SM SU(3)c gauge group and under theapproximate U(3)u flavor symmetry of the right-handedup-type quarks. In order to alleviate issues involvingflavor-changing effects, we shall assume that the up-typequarks q ∈ {u, c, t} and the component fields φq withinφ share a common mass eigenbasis. Expressed in thiseigenbasis, the interaction Lagrangian which couples thedark and visible sectors then takes the form

Lint =∑q

2∑n=0

[cnqφ

†qχnPRq + h.c.

], (3.1)

where PR ≡ 12 (1 + γ5) is the usual right-handed projec-

tion operator and where cnq is a dimensionless couplingconstant which in principle depends both on the value ofthe index n for the dark-sector field and on the flavor ofthe quark. Such a coupling structure implies that eachof the φq couples only to a single quark flavor q.

For simplicity, we shall focus on the case in which themasses of the mediators φc and φt which couple to thecharm and top quarks are sufficiently large that thatthey greatly exceed the mass of the mediator φu (i.e.,mφc ,mφt � mφu) and also have no appreciable impacton the collider phenomenology of the model. From alow-energy perspective, this is equivalent to adopting acoupling structure in which cnc ≈ 0 and cnt ≈ 0 for all n,while the cn ≡ cnu are in general non-vanishing. More-over, we shall assume that the cn scale according to thepower-law relation

cn = c0

(mn

m0

)γ, (3.2)

where c0 is the coupling associated with the lightest en-semble constituent χ0 and where γ is a dimensionlessscaling exponent.

In summary, our model is characterized by six free pa-rameters. These are the masses mn of the three χn, theparameters c0 and γ which specify the couplings betweenthese fields and the mediator φu, and the mass mφu

ofthe mediator itself. For ease of notation, since we areassuming that φc and φt are sufficiently heavy that theyplay no role in the collider phenomenology of the model,we shall henceforth simply refer to φu and mφu as φ andmφ, respectively.

Our interest in this model is primarily due to the tum-bler signatures which result from successive decays of thedark-sector states. Indeed, the interaction Lagrangian inEq. (3.1) renders χ1 and χ2 unstable. We shall primarilybe interested in the regime within which the mediator isheavy, with mφ > mn for all n. Within this regime, theleading contribution to the decay width Γφ of the medi-ator arises from to two-body decay processes of the formφ → qχn. By contrast, the leading contribution to thedecay widths of each χ1 and χ2 arise from three-bodydecay processes of the form χn → qgχm with m < n,each of which involves an off-shell mediator. Thus, whena χ2 particle is produced at the primary interaction ver-tex VP , there is a non-vanishing probability that it will

χ2 χ0χ1 ϕ

q' q'

ϕ

q qVS VT

FIG. 2. Realization of the tumbler event topology shown inFig. 1 within the context of our model. In particular, withinour model, the secondary and tertiary vertices VS and VT inFig. 1 are each now effectively realized as a pair of three-pointvertices mediated by φ.

decay via the process χ2 → qgχ1, with χ1 in turn de-caying via the process χ1 → qgχ0. The resulting decaychain is illustrated in Fig. 2, where each black dot rep-resents an interaction vertex associated with one of theLagrangian terms in Eq. (3.1). Since the φ particles in-volved in the decay processes are both off shell, the redcircles indicated in the diagram, each of which encom-passes two such interaction vertices, represent localizedspacetime events. If the χ1 and χ2 particles are bothlong-lived and each travel a macroscopic distance beforethey decay, the result is a tumbler, with these spacetimeevents corresponding to the secondary and tertiary decayvertices VS and VT indicated in Fig. 1.

Although χ1 and χ2 are unstable, the lightest dark-sector state χ0 in our model is stabilized by an accidentalZ2 symmetry of the model under which φ and the χn areodd, whereas the fields of the SM are even. This sym-metry, if unbroken, would render this particle absolutelystable — and a potential dark-matter candidate [7]. Al-ternatively, this symmetry could be broken by additional,highly suppressed interactions which permit χ0 to decayinto final states involving SM particles alone. However,as long as χ0 is collider-stable — i.e., sufficiently long-lived that virtually every χ0 particle produced within acollider detector escapes the detector well before it decays— we shall not need to specify whether this Z2 symmetryis exact or approximate for the purposes of understand-ing the collider phenomenology of the model. In whatfollows, we shall therefore simply assume that χ0 is in-deed collider-stable and consequently manifests itself as/ET .

IV. SURVEYING THE PARAMETER SPACE

Our first step is to identify regions of the parameterspace of our model within which the prospects for observ-ing a tumbler signature, either at the LHC or at a futurehadron collider, are particularly auspicious. The eventrate for collider processes involving tumblers depends onseveral factors. These include the cross-sections for therelevant production processes; the lifetimes of χ1, χ2, andφ; and the probability that an on-shell φ or χ2 particleinitially produced via one of these production processeswill decay via an appropriate decay chain.

5

We begin by evaluating the total decay widths Γφ andΓn and the branching fractions BRφn and BRn` for de-cay processes of the form φ† → qχn and χn → qqχ`,respectively. In order to calculate these branching frac-tions, we must first evaluate the partial widths for allkinematically accessible decays of φ, χ2, and χ1. Thepartial width Γφn ≡ Γ(φ† → qχn) for the decay processin which an on-shell mediator decays into a quark and anensemble constituent χn is [7]

Γφn =c2n

16π

(m2φ −m2

n)2

m3φ

. (4.1)

Likewise, the partial width Γn` ≡ Γ(χn → qqχ`) takesthe form [7]

Γn` =3c2nc

2`

256π2

mφ

r3φn

[f

(1)φn` − f

(2)φn` ln(rn`)

+ f(3)φn` ln

(1− r2

φn

1− r2φnr

2n`

)], (4.2)

where rn` ≡ m`/mn, where rφn ≡ mn/mφ, and where

f(1)φn` ≡ 6r2

φn(1− r2n`)− 5r4

φn(1− r4n`)

+ 2r6φnr

2n`(1− r2

n`)

f(2)φn` ≡ 4r8

φnr4n`

f(3)φn` ≡ 6− 8r2

φn(1 + r2n`)− 2r8

φnr4n`

+ 2r4φn(1 + 4r2

n` + r4n`) . (4.3)

The branching fractions of interest are then given by

BRφn =ΓφnΓφ

, BRn` =Γn`Γn

, (4.4)

where the total widths of φ and χn are respectively givenby

Γφ =2∑

n=0

Γnφ , Γn =n−1∑`=0

Γn` . (4.5)

We observe from the partial-width expressions inEqs. (4.1) and (4.2) that Γφ ∝ c20, while Γn ∝ c40. Formn ∼ O(100 GeV) and mφ ∼ O(TeV), these expressionsalso imply that we must take c0 � 1 in order for χ1 andχ2 to be sufficiently long-lived that their decays give riseto DVs. Together, these two considerations imply thatΓφ � Γn within regions of parameter space which giverise to tumblers. As a result, within these regions of in-terest, any on-shell φ particle produced at the primaryinteraction vertex typically decays promptly into a quarkand one of the χn.

From the branching fractions in Eq. (4.4), we may inturn determine the probability that a particular decaychain will arise from the decay of a φ or χn particle. Wedenote the probability of a given decay chain Pa1a2...af ,where the sequence of ai ∈ {φ, 2, 1, 0} in the subscript

indicates the set of φ and χn particles produced alongthe decay chain. For example, Pφ20 represents the prob-ability that an on-shell φ particle, once produced, decaysdirectly to χ2, which subsequently decays directly to χ0.These decay-chain probabilities are simply the productsof the relevant branching fractions. Since χ1 decays viathe process χ1 → qqχ0 with branching fraction BR10 = 1,we have P10 = 1. There are two possible decay chainswhich can arise from the decay of a χ2 particle, given thatχ2 can decay either to χ0 directly, or to χ1 which thensubsequently decays to χ0. The respective decay-chainprobabilities are therefore P20 = BR20 and P210 = BR21.The probabilities associated with decay chains initiatedby the decays of φ and χ2 can be evaluated in a similarmanner.

We now consider the production processes throughwhich φ and χn particles can be produced at a hadroncollider. The accidental Z2 symmetry of our model en-sures that particles which are odd under this symmetrywill always be produced in pairs. The dominant scatter-ing processes which give rise to a signal in our toy modelare therefore pp → φ†φ, pp → φχn (and its Hermitian-conjugate process), and pp → χmχn. The Feynman di-agrams which provide the leading contributions to thecross-sections for these processes are shown in Ref. [7].

Since φ carries color charge, the dominant contributionto the cross-section σφφ for the process pp→ φ†φ comesfrom diagrams which involve strong interactions alone.By contrast, the diagrams which provide the dominantcontribution to the cross-section σφn for any process ofthe form pp → φχn each include one vertex which fol-lows from the interaction Lagrangian in Eq. (3.1). Like-wise, the diagrams which provide the dominant contri-bution to the cross-section σmn for any process of theform pp → χmχn each include two such vertices. Theseconsiderations imply that σφφ is independent of c0, whileσφn ∝ c20 and σmn ∝ c40. Thus, since c0 � 1 withinregions of parameter space which give rise to tumblers,pp → φ†φ typically dominates the production rate fortumbler events by several orders of magnitude withinthose regions.1 As a result, while the branching fractionsBRφn and BRn` depend on the values of γ, c0, m0, m1,and m2, the cross-section σφφ for the sole scattering pro-cess relevant for tumbler production at hadron collidersdepends essentially on mφ alone.

Since pp → φ†φ typically provides the dominant con-tribution to the tumbler event rate within our parameter-space region of interest, it is the decays of on-shell medi-ator particles which typically provide the dominant con-tribution to the tumbler-event rate. The sole decay chain

1 In unusual circumstances wherein BRφ2 is suppressed by phase-space considerations and φ decays do not tend to produce tum-blers, it is also possible that pp→ φχ2 dominates this event rate.However, since this possibility requires that the masses m2 andmφ be tuned such that they are nearly equal, we do not considerit further.

6

0 200 400 600 8000

100

200

300

400

500

600

700

Δm10 [Gev]

Δm21

[GeV

]

c0 = 0.001

γ = 0

m0 = 600 GeV

mϕ = 1750 GeV

★BM1

0 200 400 600 8000

100

200

300

400

500

600

700

Δm10 [Gev]Δm21

[GeV

]

c0 = 0.001

γ = 1

m0 = 600 GeV

mϕ = 1750 GeV

★BM2

★BM3

★BM4

log10(Pϕ210)

-5

-4

-3

-2

-1

FIG. 3. Contours within the (∆m10,∆m21) plane of the overall probability Pφ210 = BRφ2BR21 that an on-shell mediator φ willdecay via the three-step decay chain which yields a tumbler. The results shown in the left panel correspond to the parameterassignments mφ = 1750 GeV, m0 = 600 GeV, c0 = 0.001, and γ = 0. The results shown in the right panel correspond to thesame assignments for mφ, m0, and c0, but with γ = 1. Regions of parameter space shown in white are not of interest from atumbler perspective, either because one of the relevant decay processes is kinematically forbidden, because one or both of theproper decay lengths cτ1 and cτ2 of the unstable LLPs lies below 1 mm or above 10 m, or because Pφ210 < 10−6. The fourstars which appear in the panels of this figure indicate the parameter-space benchmarks defined in Table I.

through which an on-shell φ particle, once produced bythis process, can give rise to a tumbler is the chain inwhich this φ particle decays promptly to a χ2 particle,which then decays to a χ1 particle (which itself subse-quently decays to a χ0 particle with BR20 = 1). Thus,the decay-chain probability Pφ210 = BRφ2BR21 for thissequence of decays is a crucial figure of merit in assessingwhether or not a given choice of our model parameters islikely to lead to a significant number of tumbler eventsat a hadron collider.

In order to assess which regions of the parameter spaceof our model are the most promising for tumbler detec-tion, we search for points at which the following criteriaare satisfied. First, the proper decay distances cτ1 andcτ2 of the unstable LLPs must each lie within the range1 mm < cτn < 10 m. These conditions ensure not onlythat a χ1 or χ2 particle has a significant probability oftraveling an appreciable distance away from the locationat which it was produced before it decays, but also thatit has a significant probability of decaying before it leavesthe detector tracker. Second, we require that m2 < mφ

in order to ensure that the decay φ† → qχ2 is kinemati-cally allowed. Third, we require that Pφ210 exceed a cer-tain threshold. While this probability can be as high asPφ210 ∼ O(0.1) within the most auspicious regions of pa-rameter space, we adopt a far more modest requirementPφ210 & 10−6 in our survey in order that we may betterexplore how this decay-chain probability varies across theparameter space as a whole.

In Fig. 3, we plot contours of Pφ210 in (∆m10,∆m21)-space, where ∆m10 ≡ m1 −m0 and ∆m21 ≡ m2 −m1.Results are only shown for regions wherein all of the threecriteria discussed above are satisfied; other regions ap-pear in white. The results shown in the left panel cor-respond to the parameter assignments mφ = 1750 GeV,m0 = 600 GeV, c0 = 0.001, and γ = 0. The results shownin the right panel correspond to the same assignments formφ, m0, and c0, but with γ = 1.

Broadly speaking, within these regions, the largest val-ues of Pφ210 are obtained when ∆m10 is small and ∆m21

is large. Moreover, we see that tumbler decay-chain prob-abilities as large as Pφ210 ∼ O(0.1) can arise withinthis region for γ = 1, whereas probabilities as large asPφ210 ∼ O(0.01) can arise even for γ = 0. Within thewhite region on the left side of each panel, the availablephase space for the decay χ1 → qqχ0 is extremely small,and consequently cτ1 > 10 m. By contrast, within thewhite region in the upper right corner of each panel, m2

is quite large. As a result, either the partial width forthe decay χ2 → qqχ0 becomes so large that cτ2 < 1 mm,or else m2 > mφ and the three-step decay chain whichgives rise to tumblers is kinematically forbidden. Whilethe results shown in Fig. 3 by no means represent anexhaustive survey of the parameter space of our model,they serve to highlight those regions which could poten-tially yield a significant number of tumbler events at theLHC or at future colliders.

Guided by these results, then, we shall identify a set

7

BenchmarkInput Parameters Mass Splittings Proper Decay Lengths

c0 γm0 m1 m2 mφ ∆m10 ∆m21 cτ1 cτ2

(GeV) (GeV) (Gev) (GeV) (GeV) (GeV) (m) (m)BM1 0.001 0 600 800 1000 1750 200 200 2.42 8.33× 10−2

BM2 0.001 1 600 800 1000 1750 200 200 1.36 2.89× 10−2

BM3 0.001 1 600 800 1200 1750 200 400 1.36 2.14× 10−3

BM4 0.001 1 600 1000 1200 1750 400 200 3.15× 10−2 2.89× 10−3

TABLE I. Definitions of our parameter-space benchmarks BM1 – BM4. Values for the corresponding mass splittings ∆m10 ≡m1 −m0 and ∆m21 ≡ m2 −m1 and proper decay lengths cτ1 and cτ2 for the unstable LLPs are also provided.

0 200 400 600 800

-2

-1

0

1

2

Δm10 [Gev]

γ

c0 = 0.001

Δm21 = 200 GeV

m0 = 600 GeV

mϕ = 1750 GeV

★BM1

★BM2

★BM4

0 200 400 600 800

-2

-1

0

1

2

Δm10 [Gev]

γ

c0 = 0.001

Δm21 = 400 GeV

m0 = 600 GeV

mϕ = 1750 GeV★BM3

log10(Pϕ210)

-5

-4

-3

-2

-1

FIG. 4. Same as in Fig. 3, except that the contours of Pφ210 are shown within the (∆m10, γ) plane for ∆m21 = 200 GeV (leftpanel) and ∆m21 = 400 GeV (right panel).

of four benchmark points within these regions for furtherstudy. The parameter assignments which define thesebenchmark points are provided in Table I. Each pointis also labeled with a star in Fig. 3. These benchmarkpoints represent different combinations of the parametersγ, m1, and m2.

It is also interesting to consider how our results forPφ210 vary as a function of the choice of the scaling ex-ponent γ. In Fig. 4, we plot contours of Pφ210 withinthe (∆m10, γ) plane for ∆m21 = 200 GeV (left panel)and ∆m21 = 400 GeV (right panel). The values we haveadopted for mφ, m0, and c0 in both panels of the figureare the same as those adopted in Fig. 3. The locationsof our parameter-space benchmarks are once again indi-cated by the stars. We see that increasing γ with all otherparameters held fixed generally increases Pφ210. Indeed,increasing this scaling exponent increases the ratios c2/c1and c2/c0, and thereby increases the branching fractionBRφ2 for the decay φ→ qχ2 that initiates the three-stepdecay chain which gives rise to tumblers. By the sametoken, however, increasing γ also increases the total de-cay width of χ2. For sufficiently large γ, the lifetime of

this particle becomes such that cτ2 < 1 mm. This iswhat occurs in the white region in the upper right cor-ner of each panel. On the other hand, when γ < 0, thedecay φ → qχ0 dominates the width of φ. As a result,Pφ210 decreases rapidly with γ until it drops below thethreshold Pφ210 > 10−6, leading to the white region inthe lower right of the plot. As in Fig. 3, the white regionon the left side of each panel corresponds to the region inwhich the available phase space for the decay χ1 → qqχ0

is small and cτ1 > 10 m.

V. CONSTRAINTS AND EVENT RATES

In the previous section, we identified the parameter-space regions of our model which are particularly auspi-cious for producing tumblers. In this section, we focuson these parameter-space regions of interest and assesswhether a substantial population of tumbler events couldyet await us at the LHC, given that no significant excessin discovery channels involving multiple DVs has beenobserved to date.

8

One important consideration is that our model notonly gives rise to tumblers, but also yields contributionsto the event rates in several additional detection chan-nels for new physics. These channels include the mono-jet + /ET channel, the multi-jet + /ET channel, and variouschannels involving displaced hadronic jets. The resultsof new-physics searches which have been performed inthese channels by the ATLAS and CMS Collaborationsplace additional constraints on the parameter space ofour model. Thus, we begin our analysis with a summaryof the relevant constraints from these searches.

A. Displaced-Vertex Search Constraints

A variety of searches for signatures of new physicsinvolving displaced hadronic jets have been performedby both the CMS and ATLAS Collaborations. TheCMS Collaboration, for example, has recently performedone search for displaced jets with 137 fb−1 of inte-grated luminosity which incorporates timing informationfrom the ECAL [14], as well as another, similar searchwith 132 fb−1 of integrated luminosity in which ded-icated displaced-jet triggers and background-reductiontechniques were applied [15]. A CMS search for dis-placed jets emanating from a pair of DVs resulting fromthe decays of pair-produced of LLPs was also recentlyperformed with 140 fb−1 of integrated luminosity [16].The results of these searches collectively supersede thosefrom similar CMS searches for displaced jets performedat 36 fb−1 [17] and 38.5 fb−1 [18] of integrated luminosity.The extent to which machine-learning techniques couldbe used in order to further improve the reach of searchesinvolving displaced jets was investigated in Ref. [19].

The ATLAS Collaboration has likewise performed anumber of different searches for LLPs decaying into dis-placed jets. These include searches for events in whichthe decay which produces the jets occurs within thetracker [20], within the calorimeter [21], or in the muonchamber [22]. An ATLAS search has also been per-formed for multiple LLPs decaying to jets in the sameevent, where one LLP decays within the tracker and theother decays within the muon chamber [23]. All of thesesearches are performed with roughly 35 fb−1 of integratedluminosity, though the precise value of the integrated lu-minosity varies slightly among these searches. Owing pri-marily to the substantially lower integrated luminosity,these ATLAS searches are not as constraining as the CMSsearches. For this reason, we focus on the results of theCMS searches in what follows.

The results in Refs. [14–16] collectively constrain new-physics scenarios involving LLPs with lifetimes τχ in therange 10−4 m . cτχ . 10 m which decay into final statesinvolving hadronic jets. In particular, they impose anupper bound on the product σχχBR2

χj of the LLP pair-production cross-section and the square of the branchingfraction of the LLP into such final states. While theprecise numerical value of this upper bound depends on

the production and decay kinematics of the LLP and onτχ, the bound falls within the range 0.05 – 0.5 fb acrossalmost this entire range of τχ.

B. Multi-Jet Search Constraints

Searches performed by both the CMS and ATLASCollaborations also place constraints on beyond-the-Standard-Model (BSM) contributions to the event ratefor processes involving multiple hadronic jets and /ET .The searches most relevant for constraining the param-eter space of our model are those designed to uncoverevidence of heavy decaying particles — e.g., squarks andgluinos in supersymmetry. The leading CMS constraintsfrom multi-jet + /ET searches are those derived fromsearches [24, 25] performed with 137 fb−1 of integratedluminosity. These include searches involving standardtechniques developed in order to search for squarks andgluinos more generally, as well as searches which focuson specific scenarios for which the use of the MT2 vari-able is particularly advantageous in terms of discoverypotential. The results of these analyses supersede thoseof a prior CMS study [26] performed with 36 fb−1 of in-tegrated luminosity.

The leading ATLAS constraints on excesses in themulti-jet + /ET channel of the sort obtained in our modelare those derived from a search for squarks and gluinosperformed with 139 fb−1 of integrated luminosity [27].These results supersede those obtained from a prior AT-LAS study [28] performed with 36 fb−1 of integrated lu-minosity.

In each of these ATLAS or CMS analyses, 95%-C.L.exclusion limits on the product of the production cross-section σ, the signal acceptance A, and the detection effi-ciency ε are obtained for a variety of signal regions, whichare defined differently in the different studies. These lim-its are also interpreted in each case as constraints on theparameter space of a simplified supersymmetric model in-volving a single flavor of squark q which is pair-producedvia the process pp → q†q and subsequently decays di-rectly to a light quark and the lightest neutralino χ1. Allother sparticles are assumed to be extremely heavy inthis scenario, and therefore to play no role in the pair-production process. Since q and χ1 in this supersymmet-ric model have the same quantum numbers as φ and χ0

in our model, respectively, these bounds may be appliedto our model directly. The constraint contours withinthe (mq,mχ1

) plane obtained in Refs. [24, 25, 27] are allroughly commensurate and, roughly speaking, excludethe region of this plane wherein mq . 1250 GeV andmχ1

. 500 GeV.Given that the values of the parameters mφ and m0 for

all of our parameter-space benchmarks lie well outsidethe corresponding region in the (mφ,m0)-space, we maysafely assume that our benchmarks are consistent withthese constraints. Moreover, in many of these searches,events are vetoed in which a significant fraction of the

9

jets are produced at locations other than the primaryvertex.

C. Monojet Search Constraints

The most stringent bound on excesses of events in themonojet + /ET channel is that from an ATLAS study [29]performed with 139 fb−1 of integrated luminosity. Theresults of this study supersede those from a similar AT-LAS study [30] performed with 36 fb−1 of integrated lu-minosity. Similar searches have been performed by theCMS Collaboration, but with far lower integrated lumi-nosity.

The results in Ref. [29] are quoted in a model-independent way for several different signal regions cor-responding with different threshold values taken for the

magnitude |~p (rec)T | of the transverse momentum which

recoils against the jet. For each of these signal regions,a 95%-C.L. exclusion limit on the product of the pro-duction cross-section σ, signal acceptance A, and detec-tion efficiency ε is obtained. These limits range from

σ×A× ε < 736 fb for a threshold of |~p (rec)T | > 200 GeV

to σ ×A× ε < 0.3 fb for |~p (rec)T | > 1200 GeV.

In the context of our model, the region of parameterspace excluded by these constraints turns out to be quitesimilar to the region excluded by the multi-jet constraintsdiscussed above. Thus, we may safely assume that ourbenchmarks are consistent with these constraints. Insummary, then, it is clear that the dominant constraintson our model within our parameter-space region of in-terest are those from displaced-jet searches. We shalltherefore focus primarily on these constraints in whatfollows.

D. Effective Cross-Sections and Event Rates

In order to assess the impact of these experimentalconstraints on our model, we must evaluate the net con-tributions to the event rates for a number of differentdetection channels. In particular, we can identify fourrelevant channels, each of which is associated with a par-ticular set of collider processes:

• Tumbler class: processes which involve at least onetumbler. Processes in this class are the primaryfocus of this paper.

• DV class: processes which involve at least one DV,regardless of whether this DV is part of a tumbler.The event rates associated with processes in thisclass are constrained by the results of displaced-jetsearches.

• Multi-jet class: processes which do not give riseto any DVs, but instead yield a pair of prompt

hadronic jets and missing transverse energy. Pro-cesses in this class contribute to the event rate inthe multi-jet + /ET channel.

• Monojet class: processes which do not give riseto any DVs, but instead yield a single prompthadronic jet and missing transverse energy. Pro-cesses in this class contribute to the event rate inthe monojet + /ET channel.

We emphasize that these classes are not mutually exclu-sive. For example, all processes in the tumbler class nec-essarily include DVs and are therefore also part of the DVclass. We also emphasize that all processes within a par-ticular class are not completely equivalent. One exampleof this is that the contributions from some DV-class pro-cesses may not be as stringently constrained by existingDV searches as the contributions from other such pro-cesses as a consequence of differences in kinematics andthe event-selection criteria involved. Another example,as we shall discuss further in Sect. VI, is that tumbler-class processes in which one or more additional hard jetsare produced at the primary vertex are significantly moreuseful for reconstructing the masses and lifetimes of theχn. Nevertheless, as we shall see, this classification isuseful in categorizing the contributions from our modelto the event rates in different detection channels.

Contributions to the total event rate for each of thesefour classes of processes can in principle arise from a va-riety of different event topologies — i.e., different com-binations of production processes. In Table II, we list allpossible such event topologies which can arise from pair-production processes of the forms pp → φφ, pp → φχn,and pp → χmχn. The first column indicates the struc-ture of the longer decay chain in the event, while thesecond column indicates the structure of the shorter de-cay chain. An additional jet is produced by the decayof each mediator, while an additional pair of jets is pro-duced by the decay of each LLP. However, for clarity, wehave omitted mention of these particles in these columnsof the table. Moreover, since there is no heuristic dif-ference in terms of collider phenomenology between thedecay chains precipitated by the decays of φ and χn andthe decay chains precipitated by the decays of their anti-particles φ† and χn, we do not distinguish between par-ticle and anti-particle decay chains. The third columnof the table indicates whether the process gives rise toone or more tumblers at a collider. An entry of “T” inthis column indicates that the process gives rise to a sin-gle tumbler, while an entry of “2T” indicates that theprocess gives rise to two tumblers, one from each decaychain. Likewise, the fourth column indicates whether ornot the process gives rise to an isolated DV — i.e., aDV which is not part of a tumbler. An entry of “DV” inthis column indicates the presence of a single such vertex,while an entry of “2DV” indicates the presence of suchvertices. Finally, the fifth column indicates the presenceof one or more prompt jets in the event. An entry of “j”indicates the presence of one such jet, while an entry of

10

First Chain Second Chain TumblersDisplaced PromptVertices Jets

From pp→ φφ Productionφ→ χ2 → χ1 → χ0 φ→ χ2 → χ1 → χ0 2T 2jφ→ χ2 → χ1 → χ0 φ→ χ2 → χ0 T DV 2jφ→ χ2 → χ1 → χ0 φ→ χ1 → χ0 T DV 2jφ→ χ2 → χ1 → χ0 φ→ χ0 T 2j

φ→ χ2 → χ0 φ→ χ2 → χ0 2DV 2jφ→ χ2 → χ0 φ→ χ1 → χ0 2DV 2jφ→ χ2 → χ0 φ→ χ0 DV 2jφ→ χ1 → χ0 φ→ χ2 → χ0 2DV 2jφ→ χ1 → χ0 φ→ χ1 → χ0 DV 2j

φ→ χ0 φ→ χ0 2jFrom pp→ φχn Production

φ→ χ2 → χ1 → χ0 χ2 → χ1 → χ0 2T jφ→ χ2 → χ1 → χ0 χ2 → χ0 T DV jφ→ χ2 → χ1 → χ0 χ1 → χ0 T DV jφ→ χ2 → χ1 → χ0 χ0 T j

φ→ χ2 → χ0 χ2 → χ1 → χ0 T DV jφ→ χ2 → χ0 χ2 → χ0 2DV jφ→ χ2 → χ0 χ1 → χ0 2DV jφ→ χ2 → χ0 χ0 DV jφ→ χ1 → χ0 χ2 → χ1 → χ0 T DV jφ→ χ1 → χ0 χ2 → χ0 2DV jφ→ χ1 → χ0 χ1 → χ0 2DV jφ→ χ1 → χ0 χ0 DV j

φ→ χ0 χ2 → χ1 → χ0 T jφ→ χ0 χ2 → χ0 DV jφ→ χ0 χ1 → χ0 DV jφ→ χ0 χ0 j

From pp→ χmχn Productionχ2 → χ1 → χ0 χ2 → χ1 → χ0 2Tχ2 → χ1 → χ0 χ2 → χ0 T DVχ2 → χ1 → χ0 χ1 → χ0 T DVχ2 → χ1 → χ0 χ0 T

χ2 → χ0 χ2 → χ0 2DVχ2 → χ0 χ1 → χ0 2DVχ2 → χ0 χ0 DVχ1 → χ0 χ1 → χ0 2DVχ1 → χ0 χ0 DV

χ0 χ0

TABLE II. List of the possible event topologies which can arise within our model from pair-production process of the formpp → φφ, pp → φχn, and pp → χmχn. The entries in each column describe the corresponding properties of these topologies,with notation as described in the text.

“2j” indicates the presence of two such jets. We notethat since every decay chain which occurs in our modelterminates with χ0, every event which results from anyof the processes listed in this table also includes /ET .

For each of the four class of processes α itemized above,

we define an effective cross-section σ(α)eff which represents

the sum of the individual contributions from all combina-tions of production and decay processes listed in Table IIthat contribute to the overall event rate for processes

in that class. Each such individual contribution to σ(α)eff

is the product of the cross-section σa1a2 for the pair-production process pp → a1a2, where ai ∈ {φ, 2, 1, 0},and the two decay-chain probabilities Pa1,c1 and Pa2,c2associated with the decay chains on each side of the event.

The index ci appearing in these probabilities representsthe sequence of particles produced from the decay of thecorresponding initial particle ai and includes the null de-cay chain in the event that the initial particle is stable,in which case the corresponding decay-chain probabilityis unity. In other words, our effective cross-section is

σ(α)eff ≡

∑a1

∑a2

∑c1

∑c2

[σa1a2Pa1,c1Pa2,c2

]α, (5.1)

where the subscript α on the brackets enclosing the sum-mand indicates that only event topologies associated withthe corresponding class of processes are included in thesum. Indeed, it is the product of this effective cross-section and the integrated luminosity which yield the

11

0 200 400 600 8000

100

200

300

400

500

600

700

Δm10 [Gev]

Δm21

c0 = 0.001

γ = 0

m0 = 600 GeV

mϕ = 1750 GeV

★BM1

0 200 400 600 8000

100

200

300

400

500

600

700

Δm10 [Gev]

Δm21

c0 = 0.001

γ = 1

m0 = 600 GeV

mϕ = 1750 GeV

★BM2

★BM3

★BM4

log10(σeff(T)/fb)

-5.5

-5.0

-4.5

-4.0

-3.5

-3.0

-2.5

-2.0

-1.5

FIG. 5. Contours within the (∆m10,∆21) plane of the effective cross-section σ(T )eff defined in Eq. (5.1) for processes involving

at least one tumbler at the√s = 14 TeV HL-LHC. The results displayed in the left and right panels correspond to the

parameter assignments in corresponding panels of Fig. 3. As in Fig. 3, results are shown only within regions wherein alldecay processes involved in the production of a tumbler are kinematically allowed, where cτ1 and cτ2 both satisfy the criterion1 mm < cτn < 10 m, and where cτφ < 0.1 mm. The four stars indicate the locations of the parameter-space benchmarksdefined in Table I.

overall event count for the corresponding class of pro-cesses.

In Fig. 5, we show contours of the effective cross-section

σ(T )eff for tumbler-class processes in (∆m10,∆m21)-space.

Cross-sections for all of the individual production pro-cesses were computed using the MG5 aMC@NLO code pack-age [31] for a center-of-mass energy

√s = 14 TeV. The

results displayed in the left and right panels of the figurecorrespond to the parameter assignments in the corre-sponding panels of Fig. 3. As in Fig. 3, results are shownonly within regions wherein all decay processes involvedin the production of a tumbler are kinematically allowed,where the proper decay lengths cτ1 and cτ2 of the unsta-ble LLPs both satisfy the criterion 1 mm < cτn < 10 m,and where the proper decay length of the mediator satis-fies cτφ < 0.1 mm. However, no minimum threshold forPφ210 is imposed. The four stars once again indicate thelocations of the parameter-space benchmarks defined inTable I.

We observe that the contours of σ(T)eff displayed in Fig. 5

have roughly the same shape as the contours of Pφ210 dis-played in Fig. 3. This follows from the fact that pp→ φ†φvastly dominates the event rate within our parameter-space region of interest. As discussed in Sect. IV,the cross-section for this process depends essentially onmφ alone, and is therefore roughly uniform across the(∆m10,∆m21) plane shown in each panel. More impor-tantly, however, we also observe that an effective cross-

section of order σ(T)eff ∼ O(1 – 100 ab) for tumbler-class

processes can be achieved across a substantial region of

our parameters space — a region which includes the lo-cations of all four of our parameter-space benchmarks.Given the integrated luminosity Lint = 3000 fb−1 an-ticipated for the full HL-LHC run, cross-sections of thisorder are in principle expected to give rise to a significantnumber of tumbler events at the HL-LHC.

In Table III, we list the values of σ(T)eff obtained for each

of our four benchmarks, along with the the respective

effective cross-sections σ(DV)eff and σ

(Nj)eff for DV-class and

multi-jet-class processes. Also shown in the figure are thecorresponding total numbers of tumbler events expectedafter Run 2 of the LHC (Lint = 137 fb−1) and after thefull HL-LHC run (Lint = 3000 fb−1), accounting for thecontributions from both the CMS and ATLAS detectors.While σ

(T)eff varies significantly across the (∆m10,∆m21)

plane shown in the panels of Fig. 5, we find that σ(DV)eff

and σ(Nj)eff are far less sensitive to the values of ∆m10

and ∆m21 within these same regions. Indeed, we findthat both of these effective cross-sections remain roughlywithin a single order of magnitude across this same regionof (∆m10,∆m21)-space.

One of the primary messages of Table III is that the

effective cross-section σ(DV)eff for each of our parameter-

space benchmarks is σ(DV)eff . 0.06 fb−1. Such

cross-sections are consistent with the constraints fromdisplaced-jet searches discussed above. Thus, we con-clude that a significant number of both tumbler eventsand events involving DVs of any sort could potentiallystill be awaiting discovery at the LHC or at future collid-

12

Benchmarkσ

(α)eff (fb) Tumbler Events

Tumblers DV Multi-Jet + /ET LHC Run 2 (137 fb−1) HL-LHC (3000 fb−1)BM1 1.5× 10−3 5.3× 10−2 1.1× 10−2 0.4 9.2BM2 4.3× 10−3 6.1× 10−2 4.0× 10−3 1.1 25.6BM3 1.3× 10−2 6.0× 10−2 4.3× 10−3 3.7 76.1BM4 1.4× 10−3 6.1× 10−2 3.9× 10−3 0.4 8.1

TABLE III. The effective cross-sections σ(α)eff for tumbler-class, DV-class, and multi-jet-class processes for our parameter-space

benchmarks. Also shown are the total numbers of tumbler events expected after Run 2 of the LHC and after the full HL-LHCrun, including the contributions from both the CMS and ATLAS detectors.

ers, even though no significant excess in such events hasbeen observed to date. Although the above cross-sectionslie very close to the exclusion limits from displaced-jetsearches, we also note that there are regions of our pa-

rameter space wherein σ(DV)eff lies even further below the

bound from displaced-jet searches, tumblers still arise,and all additional constraints are satisfied.

Looking ahead, in order to assess what the results inTable III portend in terms of the prospects for identi-fying a signal of new physics within the context of ourmodel at the HL-LHC, we must take into account therelevant SM backgrounds. Fortunately, one of the ad-vantages of searching for signal processes which lead toDVs is that these backgrounds are typically extremelylow. The dominant contribution to the SM backgroundin searches for displaced jets at the LHC arises as aconsequence of multi-jet events in which poorly recon-structed tracks lead to the identification of spurious DVs.Most such events arise from purely strong-interactionprocesses. Since both τ1 and τ2 satisfy cτn > O(1 mm)for all of our benchmarks, the displacements of the DVsfrom χ1 and χ2 decay away from the primary vertex aretypically well above 0.1 µm. For displacements of thissize, events involving additional primary vertices frompile-up do not represent a significant background [16].Thus, a rough estimate of the background event rate atthe end of the full HL-LHC run can be obtained simplyby scaling the expected number of background events ob-tained from searches using Run 2 data after the applica-tion of all relevant cuts by the ratio of the correspondingintegrated luminosities.

The expected number of background events in anygiven displaced-jet search depends on the particularset of event-selection criteria employed, but the lead-ing searches discussed above yield 0.1 – 0.7 backgroundevents [15, 16] at an integrated luminosity of around137 fb−1. Thus, one would expect around 2.2 – 17.5background events at the end of the full HL-LHC run.We also note that this background estimate is actuallya conservative one, given that improvements in machine-learning approaches to LLP tagging have the potentialto further reduce SM backgrounds without a significantloss in the signal-event rate [19]. By contrast, the signalefficiency obtained for the same cuts is typically aroundεS ∼ 0.45 – 0.75 [15, 16]. Thus, given the results in Ta-ble III, we see that a significant excess of DV-class events

would be observed at the HL-LHC for all of our bench-marks. Moreover, for BM2 and BM3, this excess wouldinclude a substantial number of tumbler events. The ob-servation of such an excess would clearly prompt signif-icant additional investigation into how we might betterprobe the underlying physics responsible for this excess.It is toward this question that we now turn.

VI. DISTINGUISHING TUMBLERS VIA MASSRECONSTRUCTION

While we have shown that our model can give rise to asignificant number of tumbler events at the LHC, we havealso shown that it typically simultaneously gives rise toa far larger number of non-tumbler DV-class events — asubstantial fraction of which likewise involve more thanone DV. Indeed, any of the processes listed in Table IIin which each decay chain involves only a single χ1 orχ2 particle gives rise to a pair of DVs. At this stageof the analysis, such pairs of DVs are indistinguishablefrom tumblers. Thus, in this sense, our model not onlygives rise to tumblers but also simultaneously gives riseto a “background” of non-tumbler events, each involv-ing a pair of DVs which arise from decays within differ-ent chains. If a significant number of events involvingmultiple DVs is observed at the LHC either before or af-ter the high-luminosity upgrade, it will therefore becomeimperative to develop methods of assessing whether ornot a significant number of these events in fact involvetumblers. Indeed, without such methods of distinguish-ing between tumbler and non-tumbler events, one cannottruly claim to have detected a tumbler signature.

Fortunately the distinctive kinematics associated withtumbler decay chains provides a basis on which we maydiscriminate between tumbler and non-tumbler eventsat colliders. In this section, we develop a set of event-selection criteria which are capable of efficiently discrim-inating between tumbler and non-tumbler events. In theprocess, we shall also investigate the extent to which themasses and lifetimes of the χn can be reconstructed fromthe kinematic and timing information provided by a col-lider detector.

13

A. Mass Reconstruction

In order to distinguish between tumbler events andother events which involve multiple DVs, we employ anevent-selection procedure which makes use of the distinc-tive kinematic structure associated with tumbler decaychains. This procedure follows from the observation thatif two DVs in a given event arise from successive decaysalong the same decay chain, it is in principle possibleto reconstruct the masses of the χn involved in that de-cay chain. That such an event-by-event mass reconstruc-tion is possible for tumblers is itself noteworthy. Meth-ods for reconstructing the masses of unstable particles inmulti-step decay chains which terminate in invisible par-ticles typically rely on the identification of features suchas cusps [32–35], edges [35–49], or peaks [45, 50, 51] inthe distributions of kinematic variables — features whichemerge only in the aggregate, from a sizable populationof events. By contrast, when the vertices in the decaychain are macroscopically displaced from each other andfrom VP , as they are for a tumbler, additional informa-tion can be brought to bear in reconstructing the massesof the unstable particles.

The information we need in order to reconstruct themn for a tumbler includes the three-momenta of the fourdisplaced jets produced by the decays of χ1 and χ2, thethree-momenta of the additional jets produced at the pri-mary vertex, and the timing information supplied by theECAL or timing layer concerning the time at which thesejets exit the tracker. As discussed in Sect. II, these three-momenta, in conjunction with timing information, aresufficient to reconstruct the times tP , tS , and tT andspatial locations ~xP , ~xS , and ~xT of the primary, sec-ondary, and tertiary vertices. Taken together, these mea-surements are then sufficient to determine the velocities~β1 ≡ (~xT − ~xS)/(tT − tS) and ~β2 ≡ (~xS − ~xP )/(tS − tP )of χ1 and χ2, respectively.

Given these velocities, the mn can then be determinedin a straightforward manner. Approximating the quarksas massless and noting that the energy En and mo-mentum ~pn of each χn are given by En = γnmn and

~pn = γnmn~βn, we find that the equations which repre-

sent four-momentum conservation at VS may be writtenin the form

γ2m2 = γ1m1 + |~pq|+ |~pq|γ2m2

~β2 = γ1m1~β1 + ~pq + ~pq . (6.1)

Likewise, applying four-momentum conservation at VTyields

γ1m1 = γ0m0 + |~pq′ |+ |~pq′ |γ1m1

~β1 = ~p0 + ~pq′ + ~pq′ . (6.2)

Solving this system of equations for the three mn, we

obtain

m2 =

∣∣~pq + ~pq − ~β1

(|~pq|+ |~pq|

)∣∣γ2|~β1 − ~β2|

m1 =

∣∣~pq + ~pq − ~β2

(|~pq|+ |~pq|

)∣∣γ1|~β1 − ~β2|

m20 = m2

1 − 2γ1m1

[|~pq′ |+ |~pq′ | − ~β1 · (~pq′ + ~pq′)

]+2(|~pq′ ||~pq′ | − ~pq′ · ~pq′

). (6.3)

Were it possible to measure with arbitrary precisionboth the magnitude of the momentum of each jet in atumbler event and the time at which each jet exits thetracker, it would be possible to reconstruct the mn ex-actly from the relations in Eq. (6.3). In practice, ofcourse, our ability to reconstruct these masses is lim-ited by the precision with which the detector is capa-ble of measuring these quantities. Nevertheless, providedthat these uncertainties are sufficiently small, it is highlylikely that the mn values obtained when these recon-struction formulas are applied to the jets associated witha tumbler will satisfy certain basic self-consistency cri-teria. For example, these reconstructed mn values willbe real, positive, and properly ordered in the sense thatm2 > m1 > m0.

By contrast, when the mass-reconstruction formulas inEq. (6.3) are applied to the jets associated with a pair ofDVs in the same event which do not arise from succes-sive decays along the same decay chain, it is far less likelythat they will yield a set of masses for the χn which sat-isfy these criteria. This consideration suggest that thesemass-reconstruction formulas can be used in order to dis-tinguish tumbler events from the far larger “background”of non-tumbler events involving multiple DVs which alsoarises in our model — and indeed arises generically inscenarios wherein the LLPs involved in the tumbler de-cay chain have identical quantum numbers.

In order to assess the extent to which we are able todistinguish tumbler events from other events involvingmultiple DVs in this way, we perform a Monte-Carloanalysis. Our specific procedure is as follows. Usingthe MG5 aMC@NLO code package [31], and for each of ourparameter-space benchmarks, we generate 100,000 eventsfor the initial pair-production process pp → φ†φ at acenter-of-mass energy

√s = 14 TeV. This process over-

whelmingly dominates the event rate for both tumbler-class and all relevant DV-class processes. We have chosenthe number of events we include in our simulations to besufficiently large that, as we shall see, the mn can beadequately reconstructed when the timing uncertainty issmall. Nevertheless, we emphasize that this remains trueeven when the number of events in each sample is farsmaller.

We then simulate the kinematics of the subsequent de-cay chains using our own Monte-Carlo code. For eachjet we record not only the magnitude and direction ofits three-momentum vector, but also the time at whichthe jet exits the tracker. We work at the parton level

14

and do not consider the effects of initial-state or final-state radiation, parton-showering, or hadronization. Wedetermine the locations ~xS and ~xT of the secondary andtertiary vertices in each event from the momenta of thejets produced at these vertices using the parton-level ver-texing algorithm described in Appendix A. We likewisedetermine the location ~xP of the primary vertex from themomenta of the two jets produced by the prompt decaysof φ and φ† at this vertex.

Of course, this parton-level vertexing procedure doesnot incorporate any of the uncertainties involved in a fulltrack-based reconstruction of the locations of the primaryor displaced vertices in the event. Moreover, it does notaccount for the measurement uncertainties in the mo-menta of the jets. Thus, in order to account for theseuncertainties — which can be significant — when esti-mating the precision with which we might hope to mea-sure the values of the mn from tumbler data, we proceedas follows.

We account for the timing uncertainty by smearing thetime at which each jet exits the tracker using a Gaussiansmearing function with standard deviation σt. We like-wise account for the uncertainty in the magnitude of thejet momenta by smearing the magnitude of each momen-tum vector according to a Gaussian smearing functionwhose standard deviation σE(Ej) varies with the energyEj of the jet. Since the jet-energy resolution of a colliderdetector also depends on the pseudorapidity ηj of the jet,we adopt a conservative approach and model our σE(Ej)after the jet-energy resolution obtained in Ref. [52] forjets with 1.4 < ηj < 3.0 in the endcap region rather thanthe barrel region of the CMS detector.

The uncertainties ση and σφ in the pseudorapidity andazimuthal angle that characterize the direction of each jetwithin a given event affect the reconstructed values of themn in two ways. The first is directly through ~pq, ~pq, ~pq′ ,and ~pq′ themselves in Eq. (6.3). The second is indirectlythrough their effect on the reconstructed vertex positions~xP , ~xS , and ~xT , which in turn affects the reconstructed

LLP velocities ~β1 and ~β2. Since the CMS detector iscapable of measuring the directions of the momentumvectors of hadronic jets with excellent precision [52], thefirst effect turns out to be subleading in terms of its effecton the mn in comparison with the effect of jet-energysmearing. By contrast, the second effect can have a moresignificant impact on the mn. Indeed, ση and σφ can

dominate the uncertainty in ~β1 and ~β2 when σt is small.Our method for simulating the effect of these uncer-

tainties shall be the following. Since σE dominates theuncertainty in the mn that arises directly from the jetmomenta, we shall simply take ση = σφ = 0 in what fol-lows. However, in order to account for the effect of theseangular uncertainties and other uncertainties which en-ter into the track-based reconstruction of DVs at a realcollider detector, we also shift each of the three vertexpositions ~xP , ~xS , and ~xT that we obtain from our fittingprocedure by an independent random offset vector. Themagnitude of this offset vector is distributed according to

a single-sided Gaussian function with standard deviationσr, while its direction is distributed spherically uniformly.Since the estimated uncertainty in the vertex displace-ments for the CMS detector after the HL-LHC upgradeis roughly O(10 − 30 µm) [3], we take σr = 30 µm inwhat follows.

In order to extract a set of values for the mn from agiven sample of of events, as well as an estimate of theuncertainties in these values, we proceed as follows. Webegin by requiring that the decays of all unstable dark-sector particles in the event occur within the tracker re-gion of our hypothetical detector. Modeling this detec-tor after the CMS detector, we take this region to be acylinder of radius r = 1.161 m, centered at the interac-tion point z = 0 and extending longitudinally within therange −2.5 m < z < 2.5 m, whose axis of symmetry runsalong the beam. We note that events which satisfy thisrequirement necessarily involve a significant number ofenergetic jets — including two highly-energetic promptjets from the decays of φ and φ† — and typically also sig-nificant /ET . The overwhelming majority of such eventstherefore satisfy one or more of several Level-1 triggersappropriate for a detector in high-luminosity collider en-vironment [53].

We also require that the event contain at least twoDVs. We compute the time ti at which each such vertexVi occurred from the momentum and timing informationobtained for the pair of displaced jets produced at thatvertex. For each combination of DVs Vi and Vj in theevent which are appropriately time-ordered, in the sensethat ti < tj , we reconstruct a set of mn values usingEq. (6.3). We then check whether this set of mn val-

ues, taken together with the corresponding values of |~β1|,|~β2|, and the magnitude of the three-momentum vector~p0 obtained from Eq. (6.2), satisfy the following criteria,to which we shall henceforth refer as our reconstructioncriteria:

• m1 and m2 are real and positive

• m20 is real

• |~p0| is real and positive

• 0 < |~βn| < 1 for n = 1, 2

• m22 > m2

1 > m20.

For reasons to be discussed shortly, we shall not requirethat m2

0 > 0 at this stage of the analysis. If any appropri-ately time-ordered combination of DVs in the event yieldsa set of masses which satisfy these criteria, we retain theevent; if not, we reject it. If multiple combinations ofDVs within the same event satisfy all of these criteria,we take the set of mn for the combination which yieldsthe largest value of m2 to be the set of mn for the event.

In order to illustrate the effect of these cuts, we shallbegin by focusing on the reconstruction of m1. In Fig. 6,we show the distribution of reconstructed m1 values forthe set of events which survive these cuts for each of

15

0 500 1000 15000

50

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250

m1 [GeV]

Ne

BM1

σt = 30. ps

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50

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m1 [GeV]

Ne

BM1

σt = 5. ps

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50

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Ne

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σt = 0.01 ps

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Ne

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σt = 30. ps

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m1 [GeV]

Ne

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σt = 5. ps

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Ne

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σt = 0.01 ps

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m1 [GeV]

Ne

BM3

σt = 30. ps

0 500 1000 15000

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350

m1 [GeV]

Ne

BM3

σt = 5. ps

0 500 1000 15000

100

200

300

400

m1 [GeV]

Ne

BM3

σt = 0.01 ps

0 500 1000 15000

50

100

150

m1 [GeV]

Ne

BM4

σt = 30. ps

0 500 1000 15000

50

100

150

200

250

300

m1 [GeV]

Ne

BM4

σt = 5. ps

0 500 1000 15000

100

200

300

400

500

m1 [GeV]

Ne

BM4

σt = 1. ps

FIG. 6. The distribution of values of the mass m1 for the sample of Monte-Carlo events described in the text, as reconstructedfrom tumbler kinematics. The orange portion of each histogram bar represents the contribution from tumbler events, while theblue portion represents the contribution from events with multiple DVs which do not involve a tumbler. From top to bottom,the rows in the figure correspond to the parameter-space benchmarks BM1 – BM4 defined in Table I. The dashed black verticalline in each panel indicates the actual value of m1 for the corresponding benchmark. The results shown in the left, center, andright columns correspond respectively to the values σt = 30 ps, σt = 5 ps, and σt = 0.01 ps for the timing uncertainty of thedetector. Since the efficiency of the cuts depends on the benchmark and varies with σt, the scale of the vertical axis has beenvaried from panel to panel in order to facilitate comparison between the distributions.

our parameter-space benchmarks. The histogram in eachpanel of the figure is obtained by binning these m1 val-ues into bins of width ∆mn = 5 GeV. The blue por-tion of each histogram bar represents the contributionto that bin from non-tumbler processes, whereas the or-ange portion represents the contribution from processeswhich involve tumblers. From top to bottom, the rowsin the figure correspond to our parameter-space bench-marks BM1 – BM4. The dashed black vertical line ineach panel indicates the actual value of m1 for the cor-

responding benchmark. The results shown in the left,center, and right columns correspond respectively to thevalues σt = 30 ps, σt = 5 ps, and σt = 0.01 ps for thetiming uncertainty of the detector. The first of these σtvalues represents the timing uncertainty associated withthe barrel timing layer to be installed within the CMSdetector as part of the HL-LHC upgrade. The second isa value chose to reflect a moderate improvement in thistiming uncertainty, while the third is an extremely smallvalue representative of the regime in which jet-energy

16

and vertex-position smearing dominates the uncertaintyin the mass reconstruction. Since the efficiency of thecuts depends on the benchmark and varies with σt, thescale of the vertical axis has been varied from panel topanel in order to facilitate comparison between the dis-tributions.

First, we observe from Fig. 6 that the number of resid-ual non-tumbler events is still quite significant even afterthe imposition of these preliminary cuts. Moreover, weobserve that this distribution has a well-defined shapethat peaks at low values of m1 and falls off rapidly as m1

increases. By contrast, the m1 distribution for the tum-bler events exhibits a well-defined peak centered aroundthe actual value ofm1, as well as an additional populationof events with m1 values well below this peak. This addi-tional population of events arises in part due to smearingeffects and in part due to the combinatorial backgroundwhich arises from incorrect identifications of the verticesVS and VT in events which contain more than two DVs.The relative size of the peak in the m1 distribution forthe tumbler events at low σt is primarily controlled byPφ210. Indeed, we observe that this peak is more pro-nounced for BM3, which has by far the largest value ofPφ210, than for our other three benchmarks.

The presence of this peak in the m1 distribution isa unique and distinctive feature of tumbler events. Aswe shall see, similar peaks appear in the distributions ofm0 and m2 for tumbler events as well. An observationof these peaks, taken together, would constitute com-pelling evidence for tumblers. It is in this way, then, thatour mass-reconstruction procedure furnishes a methodthrough which tumblers can unambiguously be detected.

We also observe from Fig. 6 that as σt decreases, thepeak in the tumbler distribution becomes both narrowerand more pronounced for all of our benchmarks. Indeed,this is to be expected, since increasing σt renders thereconstructed values of tP , tS , and tT less reliable. How-ever, a greater reduction in timing uncertainty is requiredto resolve this peak for some of our benchmarks than forothers. For example, the peak obtained for BM4 remainseffectively washed out even for σt = 5 ps. We can makesense of these differences in sensitivity to σt by comparingthe lifetimes τ1 and τ2 quoted for each of our benchmarksin Table I to the value of σt itself. For BM1 and BM2,τ1 � τ2 ∼ O(100 ps), and thus the effect of the timinguncertainty on the times tS and tT reconstructed for theDVs in a tumbler event will be negligible for either ofthese benchmarks when σt � 100 ps. By contrast, forBM3 and BM4, τ2 ∼ O(10 ps), which implies that the ef-fect of the timing uncertainty on tS will only be negligiblewhen σt � 10 ps. Furthermore, for BM4, τ1 ∼ O(100 ps)is also quite small, and thus the timing uncertainty has anon-negligible impact on tT as well unless σt � 100 ps.As a result, the reconstructed value of m1 is more sensi-tive to the value of σt for BM4 than they are for BM3,and are more sensitive to this value for BM3 than theyare for BM1 and BM2.

In order to further suppress the contribution from non-

tumbler events, we shall impose one additional cut on thedata. In particular, in addition to the criteria describedabove, we shall also impose one additional reconstructioncriterion:

• m20 > 0.

We have separated out this particular criterion from theothers because it merits special attention. In particular,as we shall demonstrate, not only does requiring thatm2

0 > 0 induce a dramatic enhancement in the ratio oftumbler to non-tumbler events, but it also gives rise to anadditional feature in the distribution of reconstructed m1

values — a feature which reveals additional informationabout the mass spectrum of the χn, and in particularabout the mass splitting ∆m10.

In order to quantify the effect of the m20 > 0 criterion

on the ratio of of tumbler to non-tumbler events for eachof our four parameter-space benchmarks, in Fig. 7 we plotthe ratio of the number NT of tumbler events to the num-ber NNT of non-tumbler events obtained for each of ourparameter-space benchmarks after cuts as a function ofσt. The dash-dotted curves represent the NT/NNT ratiosobtained after the imposition of all of our event-selectioncriteria except the m2

0 > 0 criterion. By contrast, thesolid curves represent the NT/NNT ratios obtained afterthe m2

0 > 0 criterion is also imposed.It is evident from Fig. 7 that the imposition of the

m20 > 0 criterion has a significant impact on NT/NNT.

When σt is relatively large, as on the right side of this fig-ure, this enhancement factor is already significant for ourfirst three benchmarks, even up to σt = 30 pb. By con-trast, as σt decreases (towards the left side of this figure),this ratio is enhanced even further, ultimately reaching afactor of ∼ 10 for all of the benchmarks. The only excep-tion to this behavior arises for BM4. For BM4, the valueof σt has a proportionally greater effect on the times re-constructed for the DVs and the difference between τ1and τ2 is far smaller than for our other benchmarks. Asa result of these differences, the effect of smearing σtis more likely to result in a set of reconstructed masseswhich fail our reconstruction criteria for BM4 than it isfor our other three benchmarks.

We now turn to discuss the impact of the m20 > 0

criterion on the shapes of the mn distributions obtainedfrom our mass-reconstruction procedure, and in particu-lar on the shape of the m1 distribution — the distributionon which this criterion has the greatest impact. Indeed,since the presence of identifiable peaks in each of thethree reconstructed mn distributions is the characteristicfeature that distinguishes a population of tumbler eventsfrom a population of non-tumbler events, the shapes ofthese distributions are of crucial importance.

The shapes of these distributions also allow us to de-termine the masses of the LLPs involved in the tumbler.In order to assess the precision with which this can bedone, we need a method of estimating the width of thepeak in the corresponding mass distribution. We shalldo this in the following way. We begin by constructing

17

0.1 0.5 1 5 100.05

0.10

0.50

1

5

σt [ps]

NT/NNT

BM1

BM2

BM3

BM4

FIG. 7. The ratio of the number NT of tumbler events to thenumber NNT of non-tumbler events for each of our parameter-space benchmarks, shown as a function of the timing uncer-tainty σt. The dash-dotted curves in each panel representthe corresponding efficiencies obtained without imposing them2

0 > 0 criterion, whereas the solid curves represent the cor-responding efficiencies obtained with the m2

0 > 0 criterionincluded. The vertical arrows in each case therefore indicatethe improvements induced by imposing the m2

0 > 0 cut.

a template for the non-tumbler contribution to each mn

distribution after the application of our event-selectioncriteria. We construct each such template by perform-ing a smoothing procedure on the non-tumbler contribu-tion to the mn distribution obtained from an additionalsample of Monte-Carlo events — a smoothing procedurewherein we replace the number of events in each his-togram bin with the mean value of the event counts inall bins whose central mn values are within 25 GeV ofthe central mn value for that bin. We then subtract thistemplate from the correspondingmn distribution in orderto obtain the contribution to the mn distribution fromthe tumbler events alone. We then perform a fit of this“background-subtracted” mn distribution to the rescaledGaussian function

f(mn) =Nmn√2πσ2

mn

exp

[− (mn − 〈mn〉)2

2σ2mn

]. (6.4)

We take the values of 〈mn〉 and σmnas our best estimates

for mn and its uncertainty. While more sophisticatedmodeling of the shape of the mass peak would of courseimprove upon these results, this procedure provides areasonably reliable indicator of the extent to which onemight hope to extract a meaningful measurement of eachmn for a given set of model parameters at the LHC or atfuture colliders.

In Fig. 8, we display the m1 distributions obtained for

our benchmarks after the application of all of our event-selection criteria, including the m2

0 > 0 criterion. Thus,all differences between the m1 distribution shown in eachpanel of this figure and the distribution shown in thecorresponding panel of Fig. 6 are solely due to the effectof this criterion. The 〈m1〉 and σm1

values we obtainfrom our fitting procedure for the distribution shown ineach panel are also indicated.

We observe that the non-tumbler contribution to eachof the m1 distributions shown in Fig. 8 is significantly re-duced relative to the corresponding distribution in Fig. 6.However, somewhat surprisingly, we also see that each ofthese distributions now manifests a visible dip or troughat a particular reconstructed value of m1 well below thispeak. The origin of this dip can be understood as fol-lows. First, we see from Eq. (6.3) that events which failto satisfy the m2

0 > 0 criterion are events for which

m21 − 2m1E

∗jj +m2

jj ≤ 0 , (6.5)

where we have used the fact that the center-of-massenergy reconstructed for the q′q′ system is given by

E∗jj ≡ γ1[|~pq′ | + |~pq′ | − ~β1 · (~pq′ + ~pq′)] and the fact

that the invariant mass of this system is given by m2jj =

2(|~pq′ ||~pq′ | − ~pq′ · ~pq′) in order to write this conditionmore compactly. Thus, for any particular values of E∗jjand mjj , the range of reconstructed m1 values excludedby the m2

0 > 0 criterion is

E∗jj −√

(E∗jj)2 −m2

jj ≤ m1 ≤ E∗jj +√

(E∗jj)2 −m2

jj .

(6.6)We note that this range of excluded m1 values alwayscontains the point m1 = E∗jj .

We also observe that constraints which follow fromstandard three-body-decay kinematics restrict the truevalues of E∗jj and mjj to lie within the respective ranges

0 ≤ mjj ≤ m1 − m0 and (m21 − m2

0)/(2m1) ≤ E ≤m1 − m0. Of course, the reconstructed values of E∗jjand mjj will in general differ from these true valuesdue to timing, jet-energy, and vertex-position smearing,and can in principle lie outside these ranges. However,in the regime in which σt is negligible compared to τ1and τ2, we find that the vast majority of reconstructedvalues for E∗jj and mjj lie within or only slightly out-side these ranges. For all of our parameter-space bench-marks, we note that the range of kinematically-allowedE∗jj values is fairly narrow. For BM1 – BM3, this rangeis 175 GeV ≤ E∗jj ≤ 200 GeV; for BM4, this range is320 GeV ≤ E∗jj ≤ 400 GeV. As a result, when the re-constructed value of m1 for an event lies within this nar-row range of E∗jj values, Eq. (6.6) implies that the eventwill typically be excluded. Indeed, we observe a dramaticsuppression in each of the distributions shown in Fig. 8across the corresponding range of m1 values.

It is worth remarking that this dip in the m1 distribu-tion arises solely as a consequence of the decay kinemat-ics at the final vertex VT along the tumbler decay chain.Thus, the kinematic considerations which lead to the dip

18

0 500 1000 15000

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Ne

BM1

σt = 30. ps

⟨m1⟩ = 742.4 GeV

σm1 = 197.2 GeV

0 500 1000 15000

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Ne

BM1

σt = 5. ps

⟨m1⟩ = 788.3 GeV

σm1 = 85.1 GeV

0 500 1000 15000

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Ne

BM1

σt = 0.01 ps

⟨m1⟩ = 800.2 GeV

σm1 = 70.4 GeV

0 500 1000 15000

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Ne

BM2

σt = 30. ps

⟨m1⟩ = 692.1 GeV

σm1 = 247.8 GeV

0 500 1000 15000

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Ne

BM2

σt = 5. ps

⟨m1⟩ = 780.7 GeV

σm1 = 129.2 GeV

0 500 1000 15000

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Ne

BM2

σt = 0.01 ps

⟨m1⟩ = 801. GeV

σm1 = 81.9 GeV

0 500 1000 15000

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Ne

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σt = 30. ps

⟨m1⟩ = 626.9 GeV

σm1 = 289.9 GeV

0 500 1000 15000

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Ne

BM3

σt = 5. ps

⟨m1⟩ = 706.2 GeV

σm1 = 254.4 GeV

0 500 1000 15000

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Ne

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σt = 0.01 ps

⟨m1⟩ = 795.9 GeV

σm1 = 92. GeV

0 500 1000 15000

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Ne

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σt = 30. ps

⟨m1⟩ = 764.5 GeV

σm1 = 451.1 GeV

0 500 1000 15000

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Ne

BM4

σt = 5. ps

⟨m1⟩ = 870.7 GeV

σm1 = 245.6 GeV

0 500 1000 15000

20

40

60

80

m1 [GeV]

Ne

BM4

σt = 1. ps

⟨m1⟩ = 976.7 GeV

σm1 = 145.6 GeV

FIG. 8. Same as Fig. 6, but after the imposition of the additional m20 > 0 reconstruction criterion.

are insensitive to the full structure of that decay chain.We would therefore expect the contribution to the m1

distribution from non-tumbler events in which a χ1 par-ticle appears in either one or both of the decay chains toexhibit a similar dip. Indeed, we observe that a dip ap-pears in both tumbler and non-tumbler contributions tothe m1 distributions in Fig. 8. It is also worth remarkingthat the location and width of the dip provide additionalinformation about the mass spectrum of the χn. Indeed,we have seen that both E∗jj and mjj are bounded fromabove by ∆m10. Thus, in principle, correlations betweenthe properties of the dip and the locations of the tumblerpeaks in the mn distributions can be exploited to improvethe precision with which the mn can be measured.

We also observe from Fig. 8 that the width σm1of the

tumbler peak obtained for each benchmark depends quite

sensitively on σt. When σt is fairly large, as shown in theleft and center columns of this figure, timing uncertaintytends to dominate the widths of the peaks in the tum-bler distributions. By contrast, when σt is sufficientlysmall, as in the right column of this figure, the widths ofthese peaks are instead dominated by σE and σr. Thevalue of σt at which this transition occurs for each of ourbenchmarks depends once again on τ1 and τ2. Neverthe-less, it is clear that the identification of a tumbler peakin the m1 distribution will be extremely challenging witha timing resolution on the order of the σt = 30 ps thatthe CMS barrel timing layer will be able to provide atthe beginning of the upcoming HL-LHC run. However,it is also clear that a reduction in timing uncertainty byeven a factor of a few relative to this value would signifi-cantly enhance the capabilities of the HL-LHC or future

19

0 500 1000 15000

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m0 [GeV]

Ne

BM1

σt = 30. ps

⟨m0⟩ = 539.3 GeV

σm0 = 203.5 GeV

0 500 1000 15000

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m0 [GeV]

Ne

BM1

σt = 5. ps

⟨m0⟩ = 588.8 GeV

σm0 = 89.9 GeV

0 500 1000 15000

10

20

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Ne

BM1

σt = 0.01 ps

⟨m0⟩ = 599.9 GeV

σm0 = 75.5 GeV

0 500 1000 15000

20

40

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80

100

m0 [GeV]

Ne

BM2

σt = 30. ps

⟨m0⟩ = 474.9 GeV

σm0 = 276.3 GeV

0 500 1000 15000

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40

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m0 [GeV]

Ne

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σt = 5. ps

⟨m0⟩ = 580.8 GeV

σm0 = 134.5 GeV

0 500 1000 15000

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40

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m0 [GeV]

Ne

BM2

σt = 0.01 ps

⟨m0⟩ = 601.8 GeV

σm0 = 85.9 GeV

0 500 1000 15000

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40

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Ne

BM3

σt = 30. ps

⟨m0⟩ = 329.6 GeV

σm0 = 414.6 GeV

0 500 1000 15000

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40

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80

100

m0 [GeV]

Ne

BM3

σt = 5. ps

⟨m0⟩ = 490.5 GeV

σm0 = 288.7 GeV

0 500 1000 15000

50

100

150

200

250

300

m0 [GeV]

Ne

BM3

σt = 0.01 ps

⟨m0⟩ = 596.4 GeV

σm0 = 99.1 GeV

0 500 1000 15000

10

20

30

40

50

60

m0 [GeV]

Ne

BM4

σt = 30. ps

⟨m0⟩ = 45.2 GeV

σm0 = 535.8 GeV

0 500 1000 15000

20

40

60

80

m0 [GeV]

Ne

BM4

σt = 5. ps

⟨m0⟩ = 381.2 GeV

σm0 = 360.7 GeV

0 500 1000 15000

20

40

60

80

100

120

m0 [GeV]

Ne

BM4

σt = 1. ps

⟨m0⟩ = 577.6 GeV

σm0 = 159.1 GeV

FIG. 9. Same as Fig. 8, except that the distributions shown are for m0 rather than m1.

colliders — both in terms of distinguishing tumblers fromother signatures of new physics involving multiple DVsand in terms of extracting information about the massspectrum of the particles involved. Indeed, the resultsshown in Fig. 8 are an indication that we are on thedoorstep of being able to probe the underlying physicswhich gives rise to DVs at a much deeper level.

Thus far, we have focused on the reconstruction of themass m1. In Figs. 9 and 10, we show the correspond-ing distributions of reconstructed m0 and m2 values forour four benchmarks, respectively, after the applicationof our event-selection criteria, including the m2

0 > 0 cri-terion. As with the m1 distributions, there are no sig-nificant discernible peaks when σt is larger than O(1 –5 ps). This is true for all benchmarks. However, as σtdecreases, a discernible peak begins to appear in both the

m0 andm2 distributions, ultimately becoming higher andnarrower as σt drops. Moreover, for each benchmark,these peaks are centered around the true values of thecorresponding masses. However, unlike the distributionsshown in Fig. 8, the distributions in Figs. 9 and 10 donot exhibit a discernible dip at any particular value ofthe corresponding reconstructed mn.

Taken together, the results shown in Figs. 8 – 10 at-test that our mass-reconstruction procedure is quite effec-tive in discriminating between tumbler and non-tumblerevents, provided that the timing uncertainty is suffi-ciently small that the peaks in the mn distributions canbe resolved. On the one hand, it is clear from thesefigures that conclusively identifying tumblers at the HL-LHC with the σt ≈ 30 ps timing resolution the CMS tim-ing layer is anticipated to provide would prove challeng-

20

0 500 1000 15000

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10

15

20

25

m2 [GeV]

Ne

BM1

σt = 30. ps

⟨m2⟩ = 929.3 GeV

σm2 = 215.2 GeV

0 500 1000 15000

5

10

15

20

25

30

m2 [GeV]

Ne

BM1

σt = 5. ps

⟨m2⟩ = 982.2 GeV

σm2 = 102.9 GeV

0 500 1000 15000

5

10

15

20

25

30

35

m2 [GeV]

Ne

BM1

σt = 0.01 ps

⟨m2⟩ = 998.9 GeV

σm2 = 85.5 GeV

0 500 1000 15000

10

20

30

40

m2 [GeV]

Ne

BM2

σt = 30. ps

⟨m2⟩ = 880.7 GeV

σm2 = 269.3 GeV

0 500 1000 15000

10

20

30

40

50

60

70

m2 [GeV]

Ne

BM2

σt = 5. ps

⟨m2⟩ = 976.5 GeV

σm2 = 151.8 GeV

0 500 1000 15000

20

40

60

80

100

120

m2 [GeV]

Ne

BM2

σt = 0.01 ps

⟨m2⟩ = 1001.3 GeV

σm2 = 99.3 GeV

0 500 1000 15000

10

20

30

40

50

60

m2 [GeV]

Ne

BM3

σt = 30. ps

⟨m2⟩ = 1019.4 GeV

σm2 = 396.2 GeV

0 500 1000 15000

20

40

60

80

m2 [GeV]

Ne

BM3

σt = 5. ps

⟨m2⟩ = 1100.2 GeV

σm2 = 315.6 GeV

0 500 1000 15000

50

100

150

200

250

m2 [GeV]

Ne

BM3

σt = 0.01 ps

⟨m2⟩ = 1191.9 GeV

σm2 = 122.6 GeV

0 500 1000 15000

5

10

15

20

m2 [GeV]

Ne

BM4

σt = 30. ps

⟨m2⟩ = 1052.7 GeV

σm2 = 540.1 GeV

0 500 1000 15000

5

10

15

20

25

30

m2 [GeV]

Ne

BM4

σt = 5. ps

⟨m2⟩ = 1057.3 GeV

σm2 = 275.7 GeV

0 500 1000 15000

10

20

30

40

50

m2 [GeV]

Ne

BM4

σt = 1. ps

⟨m2⟩ = 1172.9 GeV

σm2 = 159.9 GeV

FIG. 10. Same as Fig. 8, except that the distributions shown are for m2 rather than m1.

ing indeed. On the other hand, it is also clear that a mod-erate reduction in timing uncertainty from σt ≈ 30 ps toσt ≈ 5 ps would have a dramatic effect on our abilityto probe the underlying structure of the decay chainsthat give rise to events involving multiple DVs. As wedemonstrated in Sect. V, a robust excess in the relevantdetection channels could yet be observed at the LHC. Ifsuch an excess is in fact observed, improvements in timingprecision, in conjunction with event-selection procedureslike the one we have developed here, will play a pivotalrole in determining whether or not this excess arises as aconsequence of successive decays within the same decaychain.

B. Lifetime Reconstruction

We now assess the degree to which we can likewisemeasure the respective lifetimes τ1 and τ2 of the unsta-ble LLPs involved in the tumbler decay chain. For anygiven tumbler, the proper-time intervals t1 and t2 be-tween the production and decay of each of these parti-cles are given by t1 = (tT − tS)/γ1 and t2 = (tS− tP )/γ2,

where γn ≡ (1 − |~βn|)−1/2 is the usual relativistic fac-tor. In order to estimate the proper lifetime τn of eachparticle from a given sample of events, we first selectevents which satisfy the same criteria we imposed in ourmass-reconstruction analysis. We then define Nn(t) torepresent the number of events in the sample for whichtn > t. We then perform a least-squares fit of the func-tion f(t) = Nn(0) exp(−t/τn) to the events in the sam-

21

10-3 10-2 10-1 1 1010-1

1

10

102

103

ct [m]

Ne

BM1

σt = 30. ps

cτ1 = 2.62 m

cτ2 = 8.71×10-2 m

10-3 10-2 10-1 1 1010-1

1

10

102

103

ct [m]

Ne

BM1

σt = 5. ps

cτ1 = 2.05 m

cτ2 = 7.94×10-2 m

10-3 10-2 10-1 1 1010-1

1

10

102

103

ct [m]

Ne

BM1

σt = 0.01 ps

cτ1 = 1.94 m

cτ2 = 8.1×10-2 m

10-3 10-2 10-1 1 1010-1

1

10

102

103

104

ct [m]

Ne

BM2

σt = 30. ps

cτ1 = 1.21 m

cτ2 = 4.02×10-2 m

10-3 10-2 10-1 1 1010-1

1

10

102

103

104

ct [m]

Ne

BM2

σt = 5. ps

cτ1 = 9.81×10-1 m

cτ2 = 3.05×10-2 m

10-3 10-2 10-1 1 1010-1

1

10

102

103

104

ct [m]

Ne

BM2

σt = 0.01 ps

cτ1 = 9.12×10-1 m

cτ2 = 2.68×10-2 m

10-3 10-2 10-1 1 1010-1

1

10

102

103

104

ct [m]

Ne

BM3

σt = 30. ps

cτ1 = 1.29 m

cτ2 = 1.1×10-2 m

10-3 10-2 10-1 1 1010-1

1

10

102

103

104

ct [m]

Ne

BM3

σt = 5. ps

cτ1 = 1.18 m

cτ2 = 3.98×10-3 m

10-3 10-2 10-1 1 1010-1

1

10

102

103

104

105

ct [m]

Ne

BM3

σt = 0.01 ps

cτ1 = 9.67×10-1 m

cτ2 = 2.22×10-3 m

10-3 10-2 10-1 1 1010-1

1

10

102

103

104

ct [m]

Ne

BM4

σt = 30. ps

cτ1 = 2.85×10-2 m

cτ2 = 1.02×10-2 m

10-3 10-2 10-1 1 1010-1

1

10

102

103

104

ct [m]

Ne

BM4

σt = 5. ps

cτ1 = 3.38×10-2 m

cτ2 = 3.97×10-3 m

10-3 10-2 10-1 1 1010-1

1

10

102

103

104

ct [m]

Ne

BM4

σt = 0.01 ps

cτ1 = 3.35×10-2 m

cτ2 = 2.69×10-3 m

FIG. 11. Distributions of the number of events N1(t) (orange histogram) and N2(t) (blue histogram) for which the correspondingLLP χ1 or χ2 has not yet decayed a proper time t after it was initially produced, displayed as a function of the correspondingproper decay distance ct. From top to bottom, the rows in the figure correspond to the parameter-space benchmarks BM1– BM4 defined in Table I. The results shown in the left, center, and right columns correspond respectively to the valuesσt = 30 ps, σt = 5 ps, and σt = 0.01 ps for the timing uncertainty of the detector. Exponential-decay curves constructed usingthe best-fit values of cτ1 (thick orange curve) and cτ2 (thick blue curve) are also shown in each panel. The dotted orange andblue vertical lines correspond to the best-fit values of cτ1 and cτ2, respectively, while the dashed black vertical lines indicatethe actual values of these proper decay lengths. The best-fit values of cτ1 and cτ2 are also quoted in the box in the lower leftcorner of each panel.

ple and interpret the value of τn as our estimate for theproper lifetime of χn. Since the goodness-of-fit statisticfor this non-linear fit is more sensitive to deviations inwhich t is small and Nn(t) is large, the resulting value ofτn is typically insensitive to the small, residual contribu-tion to Nn(t) at large t from non-tumbler events whichnevertheless survive our mass-reconstruction cuts.

In Fig. 11, we show the results of such a fit for the

parameter-space benchmarks defined in Table I. Theorange and blue histograms in each panel respectivelyrepresent the N1(t) and N2(t) distributions obtainedfor a Monte-Carlo data sample that once again consistsof 100,000 events before the imposition of our event-selection criteria. The thick orange and blue curvesrepresent the exponential-decay functions obtained forour best-fit values of cτ1 and cτ2, respectively. From

22

top to bottom, the rows in the figure correspond to ourparameter-space benchmarks BM1 – BM4. The resultsshown in the left, center, and right columns of Fig. 11once again correspond respectively to the timing uncer-tainties σt = 30 ps, σt = 5 ps, and σt = 0.01 ps.

The results shown in Fig. 11 demonstrate that for rela-tively large σt values, the accuracy with which these life-times can be measured differs among the different bench-marks. For example, the extent to which our fitting pro-cedure overestimates the value of τ2 for BM3 and BM4is significant, whereas this effect is less severe for BM1and BM2. This is once again primarily a reflection of thefact that τ2 is far shorter for BM3 and BM4 that it is forthese other benchmarks, and hence the effect of timinguncertainty on the results for BM3 and BM4 becomessignificant at a far lower value of σt. However, if σt isreduced to σt = 0.01 ps, we see that reliable measure-ments of both τ1 and τ2 can be made for all four of ourbenchmarks.

VII. CONCLUSIONS

In this paper, we have described a novel potential sig-nature of new physics at colliders. This signature involvesprocesses which we call tumblers — processes in whichmultiple successive decays of LLPs within the same decaychain give rise to multiple DVs within the same event.We have investigated the prospects for observing tum-blers at the LHC and at future colliders. Despite thestringent constraints that current LHC data impose onprocesses involving DVs, we have shown in the contextof a concrete model that a significant number of tum-bler events could yet be observed at the LHC. However,scenarios which give rise to a significant number of tum-bler events also often give rise to a significant number ofnon-tumbler events which also involve multiple DVs. Inorder to address this issue, we have developed an event-selection procedure which permits us to discriminate ef-ficiently between tumbler and non-tumbler events on thebasis of the distinctive kinematics associated with tum-bler decay chains. This procedure incorporates the tim-ing information provided by the collider detector regard-ing the SM particles produced by these decay chains. Asa result, the degree to which this procedure is capable ofdistinguishing tumbler from non-tumbler events dependscrucially on the timing resolution of the detector. Inter-estingly, we have shown that a modest enhancement intiming precision beyond the σ ≈ 30 ps timing resolutionthat will be provided by the CMS timing layer at theoutset of the forthcoming HL-LHC upgrade could have acrucial impact on the prospects for discerning tumblersamongst possible signals of new physics involving mul-tiple DVs. Moreover, via this same procedure, we haveshown that it is also possible to reconstruct the massesand lifetimes of these LLPs. Once again, the precisionto which these masses and lifetimes can be measured de-pends crucially on the timing uncertainty of the detector.

Several comments are in order. First, we have madea number of simplifications concerning the manner inwhich DVs are identified and reconstructed in our anal-ysis. In so doing, we have accounted for the relevant un-certainties in a manner sufficient to provide a reasonableestimate of the detector capabilities necessary in orderto detect a robust signature of tumblers. That said, aprecise, quantitative estimate of the discovery reach fortumblers at a particular detector would require a moredetailed, track-based analysis which incorporates infor-mation about the tracker geometry.

Second, in this paper, we have employed the mass-reconstruction procedure introduced in Sect. VI as ourprimary mechanism for distinguishing between tumblerand non-tumbler events. However, there may be more ef-ficient methods of distinguishing between these two typesof events. Various possibilities along these lines are underinvestigation [54].

Third, we have focused in this paper on the case inwhich the tumbler decay chains involve only three parti-cles: χ0, χ1, and χ2. Indeed, this is the minimum numberof χn needed in order to give rise to a tumbler. However,tumblers can also arise in more complicated scenarios inwhich the number N of χn particles is larger — perhapssubstantially so. It is therefore interesting to considerhow the tumbler phenomenology of the N = 3 model an-alyzed in this paper generalizes for larger values of N . Inkeeping with our established notation, we shall assumethat these additional χn, where n = 3, . . . , N − 1, areall heavier than χ2. We shall nevertheless continue toassume that mφ > mN−1. Several observations can thenimmediately be made.

One possibility is that the lifetimes τn of the addi-tional χn are sufficiently short that these particles decaypromptly. In this regime, tumbler events which arise asa consequence of pp→ φ†φ production will often includeadditional prompt jets which can be traced back to theprimary vertex. When the number of such jets is large,both triggering and the reconstruction of DVs from kine-matic information becomes more challenging. Further-more, when N becomes large, the contribution to the to-tal event rate from processes of the form pp → φχn andpp→ χmχn increases simply as a result of the multiplic-ity of the LLPs. For sufficiently largeN , the contribution

from these processes to the effective cross-section σ(T)eff for

tumbler events — and to the effective cross-sections forother classes of processes as well — can overwhelm thecontribution from pp→ φ†φ.

In cases in which the τn for one or more of the addi-tional states are within the DV regime, further complica-tions arise. The reconstruction of the mn and τn in thiscase becomes more challenging, since the tumblers them-selves can involve different sequences of χn, even for de-cay chains involving only two DVs. Moreover, tumblersinvolving more than two DV can also arise. Neverthe-less, the methods we have developed in Sect. VI can begeneralized in a straightforward manner. It is still thecase, for example, that the momentum and timing in-

23

formation for the jets produced by a tumbler involvingmore than two individual decay steps is sufficient to per-mit the reconstruction of the mn and τn of the LLPsinvolved in the corresponding decay chain. In particular,the reconstructed mn distributions corresponding to themaximal tumbler decay chain — i.e., the chain involv-ing the largest possible number of individual displaceddecay steps — will each exhibit a peak around the truevalue of mn. However, the reconstructed mass distri-butions of non-maximal such decay chains will manifesta more complicated peak structure as a result of differ-ent decay sequences involving the same number of steps.For example, a two-step decay sequence from χ3 to χ0

could proceed via χ3 → χ2 → χ0 or χ3 → χ1 → χ0.Following the procedures we have outlined in this paperfor such two-step decays, a reconstruction of the massof the intermediate state would then result in two peaks:one centered around m1 and one centered around m2. Ofcourse, given the limited spatial extent of the tracker andthe fact that lighter χn are typically longer-lived than theheavier χn, decay chains involving large numbers of stepsmay be difficult to resolve in this manner.

Fourth, one could also consider more complicated eventtopologies involving LLPs which are themselves producedat DVs. Indeed, tumblers are merely the simplest exam-ple of such an event topology. More complicated eventtopologies in which multiple LLPs are produced at thesame DV are also possible. Such possibilities would re-sult in a proliferation of decay chains, ultimately leadingto “showers” of LLPs within the collider environment.Of course, whether or not these showers are detectableas such depends on the lifetimes of the particles involved.

Fifth, in addition to considering changes in the topol-ogy of the decay chains, one might also consider changesin the properties of the individual decays themselves,such as their decay products. In this paper we have fo-cused on models in which each decay within the tum-bler produces two quarks, ultimately leading to two jets.However, it is also possible to consider models in whichonly a single quark is produced at each DV. In suchcases, the techniques we have employed in this paper forreconstructing DVs would not be appropriate. However,as discussed above, DVs can still be reconstructed via atrack-based analysis, even in such cases. Likewise, it ispossible to consider models in which the SM particles pro-duced by LLP decays include charged leptons as well asquarks and/or gluons. Methods for reconstructing DVslikewise exist for such cases.

Finally, in this paper, we have focused on the casewherein both LLPs involved in our (two-step) tumblerdecay chain decay within the collider tracker. One couldalso consider the case in which one or both of these LLPsdecays within the calorimeters or the muon chamber. In-deed, searches have been performed by the ATLAS Col-laboration [23] for events involving multiple displaced de-cays wherein one such decay occurs within the trackerand the other occurs within these outer layers of the de-tector. Moreover, one could also consider the case in

which the lighter LLP escapes the main detector entirelyand decays within an external detector designed specif-ically for the purpose of observing LLP decays, such asMATHUSLA [55] or FASER [56]. By incorporating in-formation from such dedicated LLP detectors, one wouldpotentially be able to extend an analysis of the sort wehave performed in this paper across a broader range ofLLP lifetimes. In fact, MATHUSLA may even be capa-ble of detecting evidence of a tower of LLPs, as discussedin Refs. [1, 57].

ACKNOWLEDGMENTS

We would like to thank Gabriel Facini and Zhen Liufor discussions. TL wishes to thank the EXCEL Schol-ars Program for Undergraduate Research at LafayetteCollege, which helped to facilitate this research. The re-search activities of KRD are supported in part by the De-partment of Energy under Grant DE-FG02-13ER41976(DE-SC0009913) and by the National Science Founda-tion through its employee IR/D program. The researchactivities of DK are supported in part by the Depart-ment of Energy under Grant DE-SC0010813. The re-search activities of TL and BT are supported in partby the National Science Foundation under Grant PHY-1720430. The opinions and conclusions expressed hereinare those of the authors, and do not represent any fund-ing agencies.

Appendix A: Vertexing Procedure

We consider a pair of SM particles A and B whichwe assume to have been produced at the same vertexwithin a collider detector. We refer to the lab-framethree-momenta of these particles as ~pA and ~pB , and werefer to the lab-frame coordinates at which they exit thetracker as ~rA and ~rB . The trajectories of these particleslie along two lines which are described parametrically by

~RA(a) ≡ ~rA + a~pA~RB(b) ≡ ~rB + b~pB . (A1)

A value of a or b identifies a particular location along thecorresponding line.

Given that A and B are produced at the same vertex,the lines in Eq. (A1) will intersect at the vertex location,assuming ~rA, ~rB , ~pA, and ~pB are all measured with in-finite precision. However, in an actual experiment, mea-surement uncertainties in these quantities will typicallyresult in the lines passing very close to each other, butnot actually intersecting. We can obtain a best estimatefor the intersection point by identifying the values of a

and b for which the vector ~D(a, b) ≡ ~RA(a) − ~RB(b) isperpendicular to both lines — i.e., for which

~D(a, b) · ~pA = 0

~D(a, b) · ~pB = 0 . (A2)

24

Solving the system of equations in Eq. (A2) for a andb, we find that

a =−~pA · (~rA −~rB)|~pB |2 + ~pB · (~rA −~rB)(~pA · ~pB)

|~pA|2|~pB |2 − (~pA · ~pB)2

b =~pB · (~rA −~rB)|~pA|2 − ~pA · (~rA −~rB)(~pA · ~pB)

|~pA|2|~pB |2 − (~pA · ~pB)2.

(A3)

Evaluating ~RA(a) and ~RB(b) at these values of a andb and taking the midpoint between them, we obtain anestimate for the location of the corresponding vertex.

We emphasize that this vertexing procedure not onlyprovides a way of pinpointing the location of a vertexfrom the measured momenta of a pair of particles pro-duced at that vertex, but can also be used in order toassess whether or not two particles in the event were infact produced at the same vertex. In cases in which the

two particles were in fact produced at the same vertex,

the magnitude of the vector ~D(a, b), when evaluated atthe values of a and b in Eq. (A3), will be extremely small.By contrast, if the particles were not in fact produced at

the same vertex, |~D(a, b)|, when evaluated at the corre-sponding values of a and b, typically will be far larger.

The processes that we have considered in this paperyield up to ten jets emanating from up to five displacedvertices when both decay chains are included. For thereasons discussed above, it is very unlikely that identify-ing unrelated pairs of jets as coming from the same vertexwill result in small minimum values of |D(a, b)|. Thus byconsidering different pairwise combinations of jets andevaluating their minimum values of |D(a, b)|, it shouldbe relatively straightforward to correctly identify thosethat emanate from the same vertex. We therefore expectthe combinatorial background from misidentifications ofjet pairs to be negligible.

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