The minimal stealth boson: models and benchmarks
IIFT-UAM/CSIC-19-71CFTP/19-017
The minimal stealth boson: models and benchmarks
J. A. Aguilar–Saavedra a,b, , F. R. Joaquim c, a Instituto de F´ısica Te´orica UAM-CSIC, Campus de Cantoblanco, E-28049 Madrid, Spain b Universidad de Granada, E-18071 Granada, Spain (on leave) c Departamento de F´ısica and CFTP, Instituto Superior T´ecnico, Universidade de Lisboa, Av.Rovisco Pais 1, 1049-001 Lisboa, Portugal
Abstract
Stealth bosons are relatively light boosted particles with a cascade decay S → A A → q ¯ qq ¯ q , reconstructed as a single fat jet. In this work, we establish minimalextensions of the Standard Model that allow for such processes. Namely, we con-sider models containing a new (leptophobic) neutral gauge boson Z (cid:48) and two scalarsinglets, plus extra matter required to cancel the U(1) (cid:48) anomalies. Our analysisshows that, depending on the model and benchmark scenario, the expected statis-tical significance of stealth boson signals (yet uncovered by current searches at theLarge Hadron Collider) is up to nine times larger than for the most sensitive of thestandard leptophobic Z (cid:48) signals such as dijets, t ¯ t pairs or dibosons. These resultsprovide strong motivation for model-independent searches that cover these complexsignals. New heavy resonances are easy to spot at the Large Hadron Collider (LHC) when theydecay into charged leptons, e.g. Z (cid:48) → e + e − /µ + µ − , W (cid:48) → eν/µν , but they are quite moredifficult to detect in hadronic final states, since the production of quarks and gluons byQCD interactions has a very large cross section. Still, heavy resonances decaying intoboosted hadronically-decaying W , Z or Higgs bosons, or top quarks, may be separatedfrom the background. In the last decade, great progress has been made in this directionwith the development of jet substructure techniques [1–13] and grooming algorithms [1,14–16]. These tools allow to distinguish jets originating from boosted hadronically-decayingbosons and top quarks from the Standard Model (SM) background, composed mainly byquark and gluon jets produced in QCD processes. In this way, searches for diboson [18–24], [email protected] fi[email protected] a r X i v : . [ h e p - ph ] O c t ¯ t [25] and t ¯ b [26, 27] resonances in purely hadronic channels have been performed, witha sensitivity that turns out to be competitive with channels involving leptons in the finalstate. Nevertheless, these searches are insensitive to heavy resonances decaying into morecomplex hadronic final states, giving rise to multi-pronged jets.One example of a multi-pronged jet signature is given by the ‘stealth bosons’ intro-duced in ref. [28], which are boosted particles S decaying hadronically into four collimatedquarks, via two (equal or different) intermediate particles A , A , namely S → A A → q ¯ qq ¯ q . (1)The particles A , in the above decay chain may be SM weak bosons W , Z , a Higgsboson, or new relatively light (pseudo-)scalars. When S is produced in the decay of amuch heavier parent resonance R , R → S + X , (2)(with X an additional particle) its experimental signature is a fat jet with four-prongedstructure. Jet substructure observables designed to distinguish two-pronged Z , W andHiggs decays from the QCD background, for example the so-called D [11] and τ [7, 10]variables respectively used by the ATLAS and CMS Collaborations, classify four-prongedjets as QCD-like. Therefore, should a new resonance involve one or more decay productsof this type, it would be very hard to identify it in current searches. On the other hand,generic searches that use a multivariate tool like an anti-QCD tagger [29] to pin downmulti-pronged jets from the QCD background are sensitive to this type of signals. Noticethat if S weakly couples to SM particles, for example if it is a neutral scalar, its directproduction cross section may be too small for this particle to be directly observed.The aim of this paper is to investigate the minimal SM extension in which stealthboson signals may appear, and contextualise the relevance of these signatures as a dis-covery channel for new leptophobic resonances, when compared to the usual decay modessearched for at the LHC, like dijets, dibosons or t ¯ t pairs. In section 2 we find, followinga bottom-up approach, that the minimal additional content that allows for the cascadedecays in (1) is a Z (cid:48) boson and two scalars that are singlets under the SM group butcharged under the extra U(1) (cid:48) (further details of the models are given in appendix A).An extension with one Z (cid:48) boson, a new scalar doublet and a scalar singlet, which is alsoattractively simple, does not serve our purposes, as briefly discussed in appendix B. Sec-tion 3 is devoted to the discussion of how benchmark scenarios for the scalar masses andmixings are tested, to ensure that the reconstructed model parameters lead to an absoluteminimum of the scalar potential. In section 5 and appendix C those benchmark scenariosare studied in detail performing fast simulations of the various Z (cid:48) signals in the decays2nto stealth bosons, dijets, and t ¯ t , as well as their SM backgrounds. We discuss our resultsin section 6. Following a minimalistic approach, we assume that the heavy resonance R in (2) is aneutral colour-singlet Z (cid:48) boson, so that the gauge symmetry of the SM is extended byan extra U(1) (cid:48) . We require the Z (cid:48) boson to be leptophobic, i.e. the left-handed leptondoublets (cid:96) L and right-handed singlets e R have zero hypercharge Y (cid:48) (cid:96) = Y (cid:48) e = 0 under thenew U(1) (cid:48) . Otherwise, the leptonic signals Z (cid:48) → e + e − and Z (cid:48) → µ + µ − would be easyto observe at the LHC. Gauge invariance of the Yukawa couplings with the SM Higgsdoublet Φ, L Y = − y u ¯ q L ˜Φ u R − y d ¯ q L Φ d R − y e ¯ (cid:96) L Φ e R + h.c. , (3)then requires that Y (cid:48) Φ = 0, and that the left-handed quark doublets q L and right-handedquark singlets u R , d R have the same hypercharge Y (cid:48) q = Y (cid:48) u = Y (cid:48) d ≡ z (for simplicity weomit generation indices and generically denote the Yukawa couplings by y u , y d , y e ). Thehypercharge assignments are collected in table 1, where z is unspecified.Cancellation of the anomalies associated to U(1) (cid:48) requires introducing extra matter,which we assume to be vector-like under the SM gauge group, to preserve SM anomalycancellation. Two simple choices for these extra degrees of freedom, which we will denoteas model 1 and 2, are: • Model 1:
One set of vector-like quarks, comprising a doublet ( T B ) with SMhypercharge Y = 1 / T , B of charge 2 / − /
3, respectively. • Model 2:
One set of vector-like leptons, with a doublet ( N E ) with SM hyper-charge Y = − / N , E with charges 0 and −
1, respectively.The hypercharge assignments for these fields is summarised in table 2. We note that model2, with z = 1 /
3, has been considered in previous literature [33], motivated by the searchfor an anomaly-free Z (cid:48) dark matter mediator (the dark matter particle corresponds to thesinglet N ) with weak constraints from direct detection experiments. A similar model, Alternatives in the context of left-right models can easily be worked out from the results in Ref. [30]. Inthat work we focused on the ‘resolved’ signatures where three or four well-separated bosons are producedfrom the cascade decay of a W (cid:48) boson into new scalars. When the masses of the intermediate particlesare lighter, their bosonic decay products are merged giving rise to signatures such as in (1). Cascadedecays can also be produced in a variety of other non-minimal scenarios, see for example refs. [31, 32]. Y (cid:48) Y Y (cid:48) ( u d ) L / z u R / zd R − / z ( ν e ) L − / e R − z a free parameter. Thecolumns labelled with Y collect the standard hypercharges with the normalisation Q = T + Y . Model 1 Y Y (cid:48)
Y Y (cid:48) ( T B ) L / − z/ T B ) R / z/ T L / z/ T R / − z/ B L − / z/ B R − / − z/ Y Y (cid:48)
Y Y (cid:48) ( N E ) L − / − z/ N E ) R − / z/ N L z/ N R − z/ E L − z/ E R − − z/ (cid:48) anomaly cancellation(top: vector-like quarks, bottom: vector-like leptons). The Y (cid:48) hypercharges are given interms of z .with a three-fold replication of the new lepton set, ( N i E i ) L , ( N i E i ) R , N jL , N jR , E jL , E jL, , with i = 1 , , j = 4 , ,
6, also preserves the cancellation of anomalies. In thiscase, the lepton hypercharges are 1 / Y = 7 / Y = − / O ∼
10 quark singlets. Notice that kinetic mixing would modifyour hypercharge assignments but, since both the U(1) (cid:48) coupling g Z (cid:48) and the hyperchargeparameter z are unspecified, it has no effect in our analysis and we do not consider it.The scalar sector of the SM must be extended in order to break the U(1) (cid:48) symmetryand generate the Z (cid:48) boson mass. The simplest possibility is to consider a neutral complexSU(2) L singlet χ with non-zero hypercharge Y (cid:48) χ under U(1) (cid:48) . Having the Z (cid:48) mass generatedby a higher SU(2) L multiplet is problematic, as its vacuum expectation value (VEV) wouldalso contribute to the weak boson masses. Notice that the heavy fermion masses can alsobe generated with the same singlet provided Y (cid:48) χ = 3 z (for model 1) or Y (cid:48) χ = 9 z (for4odel 2), and that the new fermions do not have Yukawa interactions with the SM ones.Moreover, in model 2, if the lightest new fermion is a neutral singlet, it may possibly bea dark matter particle [33], while in model 1 the lightest new quark would have exoticsignatures [34], not addressed here.Additional scalars, besides this singlet, are required to yield the cascade decays (1)and (2) (ref. [33] only considers one scalar singlet). As discussed in appendix B, addinga second scalar doublet is not a viable option. Thus, we instead consider a scalar sectorcomprising the SM doublet Φ and two complex singlets χ , χ with the same hypercharge.Further extensions of the scalar sector that allow interactions of the new fermions withthe SM ones are possible, but they are not required for our purposes. The most generalgauge-invariant scalar potential is V = V Z + V (cid:54) Z , with V Z = m Φ † Φ + m χ † χ + m χ † χ + λ † Φ) + λ χ † χ ) + λ χ † χ ) + λ ( χ † χ )( χ † χ )+ 12 (cid:104) λ ( χ † χ )( χ † χ ) + h.c. (cid:105) + λ † Φ)( χ † χ ) + λ † Φ)( χ † χ ) ,V (cid:54) Z = m χ † χ + 12 (cid:104) λ ( χ † χ )( χ † χ ) + λ ( χ † χ )( χ † χ ) + λ (Φ † Φ)( χ † χ ) (cid:105) + h.c. , (4)where V Z ( V (cid:54) Z ) contains the terms which are invariant under (break) a Z symmetryfor which χ → − χ and all remaining fields transform trivially. Among all the aboveparameters, m , m , m , λ − and λ , are real, while m , λ and λ − can be, ingeneral, complex. We write the neutral scalar in Φ = ( φ + φ ) T and the singlets χ , as φ = 1 √ ρ + v + iη ) , χ = 1 √ ρ + u + iη ) , χ = 1 √ ρ + iη + u e iϕ ) , (5)such that the VEVs are (cid:104) φ (cid:105) = v √ , (cid:104) χ (cid:105) = u √ , (cid:104) χ (cid:105) = u e iϕ √ . (6)Rephasing χ to χ (cid:48) = e − iϕ χ (which has real VEV (cid:104) χ (cid:48) (cid:105) = u / √ V canbe written in terms of χ (cid:48) as in (4) with the replacements m → m (cid:48) = m e iϕ , λ → λ (cid:48) = λ e i ϕ , λ − → λ (cid:48) − = λ − e iϕ , (7)while the remaining parameters stay invariant. Therefore, in the general complex case onecan always assume ϕ = 0 without loss of generality in order to simplify the expressions,with a possible non-vanishing phase absorbed by the above redefinition. On the otherhand, if the (unprimed) parameters in the potential are all real this cannot be done, anda non-zero phase ϕ could break CP spontaneously.5here are four minimisation conditions corresponding to the four parameters v , u , and ϕ ,0 = m + 12 v λ + 14 ( u λ + u λ ) + 12 u u Re( λ (cid:48) ) , u m + u Re( m (cid:48) ) + u u λ + u λ ) + 12 u u Re( λ (cid:48) ) + 34 u u Re( λ (cid:48) )+ 14 u Re( λ (cid:48) ) + 14 v u λ + 14 v u Re( λ (cid:48) ) , u m + u Re( m (cid:48) ) + u u λ + u λ ) + 12 u u Re( λ (cid:48) ) + 34 u u Re( λ (cid:48) )+ 14 u Re( λ (cid:48) ) + 14 v u λ + 14 v u Re( λ (cid:48) ) , u u (cid:26) Im( m (cid:48) ) + 12 u u Im( λ (cid:48) ) + 14 (cid:2) u Im( λ (cid:48) ) + u Im( λ (cid:48) ) + v Im( λ (cid:48) ) (cid:3)(cid:27) . (8)Since we will be interested in those vacuum configurations with u , (cid:54) = 0, we adopt thecommon definitions: u = (cid:113) u + u , tan β = u u . (9)The minimisation conditions are used to express m , m , m and Im( m (cid:48) ) as functionsof the remaining potential parameters and VEVs. The two would-be Goldstone bosonsare G = η and G = cos β η + sin β η . The orthogonal state A = − sin β η + cos β η is CP-odd, being a mass eigenstate if the parameters in the scalar potential are real and ϕ = 0. In the basis H (cid:48) i = ( ρ ρ ρ A ), the squared mass matrix, denoted as M ij , haselements (with M ij = M ji ) M = v λ ,M = 12 uv [ Re( λ (cid:48) ) sin β + λ cos β ] ,M = 12 uv [ Re( λ (cid:48) ) cos β + λ sin β ] ,M = − uv Im( λ (cid:48) ) ,M = − (cid:20) Re( m (cid:48) ) + 14 v Re( λ (cid:48) ) + 14 u Re( λ (cid:48) ) sin β (cid:21) tan β + u (cid:20) λ cos β + 38 Re( λ (cid:48) ) sin(2 β ) (cid:21) ,M = Re( m (cid:48) ) + 14 v Re( λ (cid:48) ) + 12 u { sin(2 β )[ λ + Re( λ (cid:48) ) ]+ 32 [ Re( λ (cid:48) ) cos β + Re( λ (cid:48) ) sin β ] (cid:27) ,M = − u [ Im( λ (cid:48) ) cos β + Im( λ (cid:48) ) sin β ] , = − (cid:20) Re( m (cid:48) ) + 14 v Re( λ (cid:48) ) + 14 u Re( λ (cid:48) ) cos β (cid:21) cot β + u (cid:20) λ sin β + 38 Re( λ (cid:48) ) sin(2 β ) (cid:21) ,M = − u [ Im( λ (cid:48) ) cos β + Im( λ (cid:48) ) sin β ] ,M = − β ) (cid:20) Re( m (cid:48) ) + 14 v Re( λ (cid:48) ) (cid:21) − u (cid:26) Re( λ (cid:48) ) −
14 [ Re( λ (cid:48) ) cot β + Re( λ (cid:48) ) tan β ] (cid:27) . (10)We remark again that the minimum conditions and the expressions for M ij are validboth for a general potential with complex parameters, in which case one can just dropthe primes and assume ϕ = 0, or for a potential with real parameters, in which casethe primed parameters are defined by (7). We also note in passing that, should we havechosen χ and χ with different hypercharges, m and λ − would vanish, in which case M i = 0 and A would be massless.The scalar interactions with the Z (cid:48) boson field originate from the term L = ig Z (cid:48) Y (cid:48) χ (cid:16) χ ∗ ←→ ∂ µ χ + χ ∗ ←→ ∂ µ χ (cid:17) B (cid:48) µ . (11)Since for the scalar doublet Y (cid:48) Φ = 0, there is no Z − Z (cid:48) mixing and B (cid:48) µ ≡ Z (cid:48) µ is a masseigenstate, with mass M Z (cid:48) = ( g Z (cid:48) Y (cid:48) χ ) u . (12)We express the scalar weak eigenstates as H (cid:48) i = O ij H j , where H i are the mass eigenstateswith mass M H i , and O is a 4 × Z (cid:48) H i H j ( i < j ) couplingsare then L Z (cid:48) H i H j = g Z (cid:48) Y (cid:48) χ R ij H i ←→ ∂ µ H j Z (cid:48) µ , (13)with mixing factors R ij = cos β [ O i O j − O j O i ] − sin β [ O i O j − O j O i ] . (14)Notice that R ij are anti-symmetric and therefore R ii = 0, reflecting the fact that Z (cid:48) → H i H i is forbidden. Also, it can be shown that (cid:80) i 1, the scalars will dominantly decayto lighter scalars, if kinematically allowed. Otherwise, they will decay into W + W − , ZZ or fermion pairs, like a Higgs boson with a mass M H i (decays such as H i → H j Z areabsent). For lighter masses the dominant mode will be H i → b ¯ b . The number of parameters in the scalar potential (4) is large enough to reproduce anypattern of scalar masses and mixing. Thus, we will focus on setting benchmark scenariosrepresentative of the signals we are interested in. Within this approach, and with the goalof reducing the number of parameters, we will consider a simpler version of the modelwith λ − = 0 in the scalar potential. This corresponds to having a softly broken Z symmetry under which χ → − χ , i.e. the only term remaining in V (cid:54) Z = 0 is the bilinear m soft-breaking term. We are then left with twelve real parameters in the scalar massmatrix: Re( m ), λ − , , , Re( λ ), Im( λ ), β , v and u (determined by the Z (cid:48) mass througheq. (12)), of which only ten are independent due to the relations M = 0 , tan β = M M . (22)These ten parameters match the four scalar masses and six independent parameters ofthe (real) 4 × M H i through M = (cid:88) i =1 O i O i M H i = 0 , (23)while the second one determines tan β . Taking O ij , v , M Z (cid:48) , λ and three of the masses M H i as inputs, the remaining parameters in the potential can be determined asRe( m ) = ( u λ sin β − M ) tan β ,λ = M v , λ = Re( m ) tan β + M u , λ = − Re( λ ) − M − Re( m ) u , Re( λ ) = M u − Re( m ) u sin(2 β ) , Im( λ ) = − M u sin β , λ = 2 M v u cos β , λ = 2 M v u sin β , (24)where M ij are expressed in terms of M H i and O ij using M ij = (cid:88) k =1 O ik O jk M H k . (25)The 4 × × O = (cid:98) O (cid:98) O (cid:98) O (cid:98) O (cid:98) O (cid:98) O , (26)being (cid:98) O kl a rotation in the ( k, l ) plane by an angle θ kl , whose ( i, j ) matrix element canbe written as( (cid:98) O kl ) ij = δ ij + ( δ ik δ jk + δ il δ jl )(cos θ kl − 1) + ( δ ik δ jl − δ il δ jk ) sin θ kl . (27)The constraints on the couplings of the SM 125 GeV Higgs boson ( H ≡ H in our models)obtained by the ATLAS Collaboration [35] imply O + O + O ≤ . 05 at 95% confidencelevel (CL), in which case O j and O j are small for j (cid:54) = 1. This, in turn, implies that themixing angles θ , θ and θ are small.The minimisation of the scalar potential and the viability analysis of a given vacuumconfiguration with v, u (cid:54) = 0 proceeds as follows. Setting the mass of the SM Higgs bosonto M H ≡ M H = 125 GeV, for a given set of input parameters O ij , v , M Z (cid:48) , λ and M H , The strength parameter µ = 1 . +0 . − . corresponds to O in our notation, from which limits on theother matrix elements can be obtained using unitarity and approximating the Gaussian distribution by asymmetric one. Ignoring the fact that O ≤ O + O + O ≤ . 05 at 95% CL. By restricting ourselves to the physical region O ≤ 1, the 95% CL limit is relaxed to0.11. In all our benchmark scenarios the matrix elements O j and O j ( j (cid:54) = 1) are more than two ordersof magnitude below these bounds. 10e first determine M H and tan β using eqs. (22). Afterwards, the parameters in (24) arecomputed using also eq. (25). At this point, for the chosen set of inputs, the potential iscompletely defined. It now remains to check whether stability and perturbative unitaritycriteria are fulfilled, and ensure that our VEV corresponds to the global minimum of thepotential. For the stability analysis we follow the method of refs. [36] and [37] based onrequiring copositivity of the quartic coupling matrix Λ. Parameterising the field bilinearsas | Φ | ≡ h , | χ , | ≡ h , and χ ∗ χ ≡ ρh h e iϕ (with | ρ | ∈ [0 , ϕ = 0, wehave Λ = λ λ λ λ λ ρ Re( λ ) + 2 λ λ ρ Re( λ ) + 2 λ λ , (28)defined in the basis ( h , h , h ). The stability of the potential is ensured if the abovematrix is copositive, i.e., if the following conditions hold [36]:Λ ii ≥ , Λ (cid:48) ij = Λ ij + (cid:112) Λ ii Λ jj ≥ i < j = 1 , , , (cid:112) Λ Λ Λ + Λ (cid:112) Λ + Λ (cid:112) Λ + +Λ (cid:112) Λ + (cid:112) (cid:48) Λ (cid:48) Λ (cid:48) ≥ . (29)In the above relations ρ = 0 , λ ) > λ ) < 0, respec-tively. Since in the cases we are interested in the quartic couplings λ i are typically verysmall (due to the fact that M H i /u (cid:28) S -matrix unitarity for elastic scatteringof two scalar boson states.It now remains to check whether our minimum is the global minimum of the potential.For that, we must compare the value of the potential at our minimum, V = − (cid:2) λ u + λ u + 2 λ u u + 2 u u Re( λ ) + v ( λ v + λ u + λ u ) (cid:3) , (30)with the values at any other minima obeying the minimisation conditions (8). The moststraightforward alternative solutions correspond to vacua with vanishing VEVs. Namely,we have u = u = v = 0 → V = 0 u = u = 0 , v = − m λ → V = − m λ , (31)where V i corresponds to the value of the potential at the corresponding set of VEVs.Notice that these two solutions must be discarded as being the global minimum of thepotential since they imply no spontaneous symmetry breaking of the SM and/or the U(1) (cid:48) ϕ (cid:54) = 0 and v, u , (cid:54) = 0 may exist. Sincefor these cases the analytical treatment is quite involved, we use a numerical routine tospot those solutions. For all alternative minima we check positivity of the scalar massesand if V i < V . At the end, only those sets of input parameters which lead to a stablepotential and to a global minimum are considered viable in the parameter space scanperformed in the next section. We are interested in scenarios with Z (cid:48) and H i masses close to those used for the anti-QCDtagger in ref. [29]. We remark that this assumption is done only for the sake of simplicity,and with the purpose of using the performance for signals and backgrounds obtained inprevious work without the need of training neural networks for new taggers. Thus, werestrict our study to benchmarks where the Z (cid:48) boson decays into two stealth bosons, thatis for example Z (cid:48) → H H , H , → H H , (32)with H subsequently decaying into quark pairs. Scenarios with Z (cid:48) → H , H , H , → H H are also interesting and lead to signals that are quite elusive as well, since a lightboosted H → q ¯ q produces two-pronged jets that closely resemble one-pronged QCD jets.However, their analysis requires the development of new taggers, which is out of the scopeof this work.In the following we will identify three representative scenarios. In scenario 1 withrelatively light scalars the decay pattern is quite simple, with dominant decays H , → H H . For heavier scalars H , , their decays into W W , ZZ , t ¯ t and H H are possible,besides H H , if the latter is kinematically open. For illustration, we set scenario 2 wheredecays H , → H H dominate, and scenario 3 where H , → W W, ZZ dominate. Noticethat these are extreme cases and, in general, for H , one could have similar branchingratios for W W/ZZ and H H final states. One of the virtues of a generic tagger is thatit is sensitive to all of them at once. For simplicity, the extra fermions (quarks in model1 and leptons in model 2) are assumed heavy enough not to be produced in the decays ofthe Z (cid:48) boson. We choose M Z (cid:48) = 2 . M H (cid:39) M H (cid:39) 80 GeV, M H (cid:39) 30 GeV, as in one of thebenchmark points used in the tagger labelled as std1000 in ref. [29]. The coupling is12igure 1: Z (cid:48) branching ratio to different scalar pairs in scenario 1 of model 1. We scanover the input parameters θ ij and λ keeping only those points which lead to a viableminimum of the potential. The mixing angles θ , θ and θ are unrestricted, exceptin the bottom right panel where θ and θ are taken small (see the text). The U(1) (cid:48) coupling is such that g Z (cid:48) z = 0 . g Z (cid:48) z = 0 . We perform a scan of the allowed parameter space by varying θ , θ and θ with a flat distribution (keeping the other mixing angles small as requiredby constraints on the couplings of the SM Higgs) and compute the Z (cid:48) branching ratio toscalars. The results for model 1 are presented in figure 1. The branching ratio for quarkpairs (not summed over flavours) is included for comparison. For model 2 the branchingratios for scalars trivially scale by a factor of 4 . 8, and the branching ratios to quark pairsby 0 . Z (cid:48) → H H can be dominant in wide regions of the The results for decay branching ratios are quite independent of the actual value used for the coupling,which determines u for fixed Z (cid:48) mass and sets the scale for the λ − couplings. θ , vary in the interval [0 , π ] while in the right panel | θ , | ≤ . 01. For completenesswe include the branching ratios of H , which do not depend on the mixing angles.parameter space, as long as θ and θ are not close to π/ 2. In particular, if these twoangles are small, the mixing matrix is approximately O (cid:39) ε ε ε ε ε ε ε ε cos θ sin θ ε ε − sin θ cos θ , (33)with ε ij (cid:46) . 01. Neglecting these small parameters, the Z (cid:48) couplings to the mass eigen-states are L = g Z (cid:48) Y (cid:48) χ (cid:104) − sin β sin θ H ←→ ∂ µ H + sin β cos θ H ←→ ∂ µ H − cos βH ←→ ∂ µ H (cid:105) B (cid:48) µ , (34)with sin β (cid:28) cos β , from which it can be clearly seen that the dominant Z (cid:48) decay toscalars is Z (cid:48) → H H . This is seen in the bottom right panel of figure 1, where we restrict | θ , | ≤ . θ , θ and θ free), Br( H , → H H ) (cid:39) λ and λ , respectively, which are not suppressed. The small mixingwith ρ allows the decay of H into quarks, but has little effect on the decay of the heavierparticles. The same holds when | θ , | ≤ . 01, as shown in the right panel of the same The decay of H does not produce displaced vertices even with this small mixing. For example, for M H = 30 GeV and O = 0 . 01, the decay length is of 1.5 nm. H decays into different final states in scenario 2 ofmodel 1, resulting from a scan of allowed points in parameter space. Right: the same asin the left panel but for scenario 3 of model 1.figure. The masses are set to M Z (cid:48) = 3 . M H (cid:39) M H (cid:39) 400 GeV and M H (cid:39) 80 GeV asin the tagger benchmark labelled as std1500 in ref. [29]. The U(1) (cid:48) coupling is set to g Z (cid:48) z = 0 . 2. For the Z (cid:48) branching ratios the results obtained from the parameter spacescan are the same as in scenario 1, and are omitted for brevity. This is so because thescalars are much lighter than the Z (cid:48) boson, and kinematical effects are unimportant. Thedecay Z (cid:48) → H H can dominate the scalar decays of the Z (cid:48) boson, in particular when θ and θ are small.The results for the decay of the heaviest scalar H are shown in figure 3 (left panel).For H , which has nearly the same mass, the outcome is similar. In most of the parameterspace Br( H , → H H ) (cid:39) 1, and the same happens for model 2. The masses are set to M Z (cid:48) = 3 . M H (cid:39) M H (cid:39) 400 GeV as in the tagger becnhmarklabelled as std1500 in ref. [29] but, in contrast with scenario 2, M H (cid:39) 300 GeV in orderto forbid the decay H , → H H . The coupling is set to g Z (cid:48) z = 0 . 2. The results of theparameter space scan are the same as in scenarios 1 and 2 for the Z (cid:48) branching ratios15nd, thus, they are not presented. Regarding H decays, the results are shown in theright panel of figure 3 (for H , with nearly the same mass, the outcome is similar). Bothscalars decay into pairs of SM particles with branching ratios that are nearly independentof the mixing angles. The partial widths are determined by the matrix element O i andthe small triple couplings λ i ( i = 3 , The potential relevance of stealth boson signals as a discovery channel is assessed in thissection by a comparative study of the sensitivity of three searches, • Z (cid:48) → jj . • Z (cid:48) → t ¯ t . • A generic search, using the efficiencies for signals and background previously ob-tained for the anti-QCD tagger.In addition, for scenario 1 we investigate whether the decay Z (cid:48) → H H is visible in adiboson resonance search. The various processes in our analysis are generated using Mad-Graph5 [38], followed by hadronisation and parton showering with Pythia 8 [39] anddetector simulation using Delphes 3.4 [40] using the configuration for the CMS detector.The reconstruction of jets and their substructure analysis is done using FastJet [41].For the signal processes the relevant Lagrangian is implemented in Feynrules [42] andinterfaced to MadGraph5 using the universal Feynrules output [43]. As backgroundprocesses we consider QCD dijet production pp → jj , with j being a light jet, pp → b ¯ b ,and t ¯ t production. In order to populate with sufficient Monte Carlo statistics the entiremass and transverse momentum range under consideration, we split the samples in 100GeV slices in the transverse momentum of the leading jet (or top quark), from 300 GeVto 2.2 TeV and above, generating 2 × events for jj , 10 events for b ¯ b and 10 eventsfor t ¯ t in each slice. The different samples are then recombined with weights proportionalto the cross sections. Signal samples for Z (cid:48) → jj (including b ¯ b ), Z (cid:48) → t ¯ t and Z (cid:48) → H H have 10 events each, except for Z (cid:48) → t ¯ t in scenarios 2 and 3 and Z (cid:48) → H H in scenario3, which have 2 × events.The dijet and t ¯ t analyses are common to the two signal benchmark scenarios studied.We do not recast any specific experimental search but we choose event selections similarto the ones commonly adopted by the ATLAS and CMS Collaborations:16 Dijet resonance analysis: jets are reconstructed with the anti- k T algorithm [44] usinga radius R = 0 . 8, and groomed using Soft Drop [16] with the parameters z cut = 0 . β = 0. The use of large-radius jets is motivated by the possible presence of hardradiation accompanying the energetic decay products of a heavy resonance, and thegrooming is implemented in order to clean the jets from pile-up and initial stateradiation (see for example ref. [45]). The leading and subleading jets are requiredto have pseudo-rapidity | η | ≤ . p T ≥ 500 GeV, whiletheir pseudo-rapidity difference must satisfy | ∆ η | ≤ . • t ¯ t resonance analysis: we use large-radius jets with R = 0 . | η | ≤ . | ∆ η | ≤ . p T ≥ b tagging, a collection of ‘track jets’ of radius R = 0 . 2, reconstructed withtracks only, is used. A large- R jet is considered as b -tagged if a b -tagged track jet(using the 70% efficiency working point) within ∆ R = 0 . b -tagged, have a (groomed) mass 100 ≤ m J ≤ 220 GeV, and a value of the(ungroomed) subjettiness variable [7] τ ≤ . In this scenario we have M Z (cid:48) = 2 . Z (cid:48) → H H , with M H (cid:39) M H (cid:39) 80 GeV. For the signal coupling we choose g Z (cid:48) z = 0 . 15, yielding a total Z (cid:48) production cross section of 142.6 fb. The total Z (cid:48) width is Γ = 26 . . Z (cid:48) width, compared to the experimental resolution,justifies using the same signal samples for both models. We set the mixing factor R in eq. (14) to unity for simplicity — as seen in the previous section for the numericalexamples provided, R is often very close to one. Therefore, we obtain for the branchingratios Br( Z (cid:48) → H H ) = 0 . 11 (model 1) , Br( Z (cid:48) → H H ) = 0 . 53 (model 2) , (35)and Br( H , → H H ) (cid:39) 1. ( Z (cid:48) decays to other scalar pairs are suppressed.) We select M H = 30 GeV as one of the benchmark points studied in ref. [29]. With M H = 15GeV, the tagger efficiency is quite close but the acceptance of the stealth boson signal in Larger couplings are compatible with current bounds from dijet, diboson and t ¯ t resonance searches.We prefer to select for our benchmarks in this section small values of the couplings, still yielding asignificance around 5 σ for the generic searches. t ¯ t diboson Z (cid:48) (model 1) 0.71 fb 27 fb 0.40 fb 0.073 fb Z (cid:48) (model 2) 3.4 fb 31 fb 0.21 fb 0.057 fb jj b ¯ b t ¯ t — — 78 fb —Table 3: Signal and background cross sections for the Z (cid:48) signal in scenario 1 and mainSM backgrounds (in rows) under the four different event selections for generic, dijet, t ¯ t and diboson resonance searches, in columns.usual diboson resonance searches is slightly larger. That scenario is examined in detail inappendix C. The branching ratios for the decay of the lighter scalar into quark pairs areBr( H → b ¯ b ) = 0 . , Br( H → c ¯ c ) = 0 . . (36)It is expected that the tagger performance for b ¯ bb ¯ b , b ¯ bc ¯ c and c ¯ cc ¯ c multi-pronged jets issimilar, so we include both channels. With four scalars H from the Z (cid:48) cascade decay, thebranching ratio factor is Br( H → b ¯ b, c ¯ c ) = 0 . ≤ m J ≤ 100 GeV, and • For the generic search we apply the tagger performance efficiency factors obtained inref. [29] of 0 . 01 for QCD jets and 0.31 for the signal jets ( M H , = 80 GeV, M H = 30GeV). • For the diboson search we require τ ≤ . ≤ m J ≤ W and Z jets by the CMS Collaboration) but we preferto keep the same window as in the generic search for better comparison.The presence of the heavy Z (cid:48) resonance can be detected as a bump in the dijet or t ¯ t invariant mass distribution. We present in figure 4 these distributions for the generic (toppanels), t ¯ t (middle panels) and dijet (bottom panels) analyses. For model 1 we use anintegrated luminosity of L = 15 fb − , while for model 2, for which the signal is muchlarger, we take L = 2 fb − . The signal and background cross sections for the differentevent selections are collected in table 3. 18 500 2000 2500 3000 3500 4000 m JJ (GeV) -4 -3 -2 -1 E v en t s / G e V backgroundbackground + signal Z ′ → H H , H → H H Model 1 , M H = 30 GeVL = 15 fb -1 m JJ (GeV) -5 -4 -3 -2 -1 E v en t s / G e V backgroundbackground + signal Z ′ → H H , H → H H Model 2 , M H = 30 GeVL = 2 fb -1 m tt (GeV) -2 -1 E v en t s / G e V backgroundsignal Z ′ → ttModel 1 , M H = 30 GeVL = 15 fb -1 m tt (GeV) -3 -2 -1 E v en t s / G e V backgroundsignal Z ′ → ttModel 2 , M H = 30 GeVL = 2 fb -1 m jj (GeV) E v en t s / G e V backgroundsignal Z ′ → jjModel 1 , M H = 30 GeVL = 15 fb -1 m jj (GeV) -2 -1 E v en t s / G e V backgroundsignal Z ′ → jjModel 2 , M H = 30 GeVL = 2 fb -1 Figure 4: Invariant mass distribution for the Z (cid:48) signals in scenario 1 and their back-grounds, in the stealth boson (top), t ¯ t (middle) and dijet (bottom) analyses, for model 1(left) and model 2 (right). 19lthough the resonance is relatively narrow, the detector resolution effects yield awider distribution, especially for the decays into scalars which produce four-pronged jets.As it has previously been shown [17, 28], standard grooming algorithms are not adequatefor multi-pronged jets, shifting jet masses and momenta from their original values. Theeffect can clearly be seen in the signal profile for the dijet analysis, which is much widerin model 2, where more than half of the dijet events are actually Z (cid:48) → H H .The expected significance of the Z (cid:48) signal in the different searches is computed byusing the Monte Carlo predictions for signal plus background as pseudo-data, performinglikelihood tests for the presence of narrow resonances over the expected background, usingthe CL s method [46] with the asymptotic approximation of ref. [47], and computing the p -value corresponding to each hypothesis for the resonance mass. The probability densityfunctions of the potential narrow resonance signals are Gaussians with centre M (i.e. theresonance mass probed) and standard deviation of 0 . M . The likelihood function is L ( µ ) = (cid:89) i e − ( b i + µs i ) ( b i + µs i ) n i n i ! , (37)where i runs over the different bins with numbers of events n i , b i is the predicted number ofbackground events and s i the predicted number of signal events in each bin, and µ a scalefactor. We do not include any systematic uncertainty in the form of nuisance parameters.For each mass hypothesis the value µ b that maximises the likelihood function (37) iscalculated, and local p -value is computed as p = 1 − Φ( (cid:112) L ( µ b ) − L (0)]) , (38)with Φ( x ) = 12 (cid:20) (cid:18) x √ (cid:19)(cid:21) . (39)The results, assuming luminosities of 15 fb − (model 1) and 2 fb − (model 2) are presentedin figure 5. As it can be seen, stealth boson signals in the generic search are by far moresignificant than standard dijet signals. In terms of standard deviations, the significancein generic searches is 4 (8) times larger for model 1 (model 2). As we have noted at thebeginning of this section, the actual values depend on the product g Z (cid:48) z , which is a freeparameter. Several additional comments and clarifications are in order. • In our generic search, sensitive to stealth boson signals, we have focused on a jet masswindow 40 ≤ m J ≤ 100 GeV, adequate for the benchmark point of the anti-QCDtagger considered. In order to cover all masses for the new scalars, experimen-tal searches should explore bidimensional phase space, also using varying jet masswindows (see for example ref. [48]). 20 000 1500 2000 2500 3000 3500 4000M (GeV)10 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 Lo c a l p σ σ σ σ σ σ σ Z ′ → jj, ttZ ′ → H H , H → H H diboson search Model 1, M H = 30 GeV L = 15 fb -1 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 Lo c a l p σ σ σ σ σ σ σ Z ′ → jj, ttZ ′ → H H , H → H H diboson search Model 2, M H = 30 GeVL = 2 fb -1 Figure 5: Expected local p -value for the Z (cid:48) signal in the various searches, for scenario 1of model 1 (left) and model 2 (right). • The use of b tagging in the generic search would significantly improve the significancefor stealth boson signals. Requiring one b tag in either jet enhances the ratio S/ √ B by a factor of 2, and requiring two b tags by 3 . 4, where S stands for signal and B for background cross sections. We have chosen not to make use of b taggingin our analysis because the signals are already quite conspicuous, especially formodel 2, and the background is already small. In this regard, our results are quiteconservative. The use of b tagging would be very useful for large luminosities, tofurther reduce the background keeping the same signal efficiency for the anti-QCDjet tagger. • We have not considered systematic uncertainties in our estimation of the significanceof the different signals. These uncertainties will be more important in the channelswhere the background is larger, that is, jj and t ¯ t . In the generic search, where theexpected background lies between 0 . − . • As aforementioned, existing grooming algorithms are not designed nor optimisedfor multi-pronged jets and may shift the mass peaks. (Several other groomingalgorithms and parameters were explored in ref. [17] with similar results.) This isclearly observed in figure 5, where the maximum significance is for the stealth bosonsignal is near 2 TeV while the Z (cid:48) mass is 2.2 TeV. We have not attempted any massrecalibration because this small shift does not affect our results and conclusions. • The relative (in)significance of the Z (cid:48) signal in dijet, t ¯ t and diboson searches dependson the model and the mass of the lightest scalar, as the signal for the dijet anddiboson event selections receive contributions from various Z (cid:48) decay modes.21 For all the final states considered, the global significances of the deviations — whichdepend on the mass range studied in each experimental search — are smaller thanthe local significances in figure 5. In any case, for the point addressed in this section,namely to show that the sensitivity of a generic search is a factor of ten (in termsof significance) or more than for current searches, local significances suffice. • For lighter H , the four-pronged stealth boson jets have a more two-pronged struc-ture, and the acceptance in diboson searches is slightly larger (see appendix C).Actually, for M H = 30 GeV most of the signal that passes the diboson event selec-tion is Z (cid:48) → jj and Z (cid:48) → t ¯ t , not Z (cid:48) → H H .For completeness, let us comment about the direct production of the new scalars H − ,which can be produced in the same processes (gluon-gluon fusion, associated productionwith a W/Z boson, etc.) as the SM Higgs H . The cross sections are the ones that wouldcorrespond to a SM Higgs of the same mass M H − , multiplied by the small factor O i . Inthe benchmark scenarios considered, with mixing angles | θ j | ≤ . 01 ( j = 2 , , O i (cid:46) − ,leading to unobservable signals.The most stringent constraints on the lightest scalar H result from the Higgs bosonsearches at LEP experiments [49]. For a mass of 30 GeV, the LEP constraints imply O i ≤ . 02, two orders of magnitude above the maximum used in our benchmark. For H , with M H , = 80 GeV there are no searches targeting the production and decay pp → H , → H H , with subsequent decay of H . Still, one can use similar analysesperformed for Higgs decays into light pseudoscalars, H → aa , to obtain an estimate ofthe sensitivity. In gluon-gluon fusion, a search for H → aa → b ¯ bµ + µ − by the ATLASCollaboration [50] obtains an upper limit on cross section times branching ratio of σ ( H ) × Br( b ¯ bµ + µ − ) ≤ . m a = 30 GeV. Using the cross sections from ref. [51] for a 80GeV SM-like Higgs, we find that in our benchmark σ ( H , ) × Br( b ¯ bµ + µ − ) (cid:46) . W H/ZH associated production, a search for H → aa → b ¯ bb ¯ b [52] sets the upper limit σ ( H ) × Br( b ¯ bb ¯ b ) ≤ . m a = 30 GeV.In our benchmark, σ ( H , ) × Br( b ¯ bµ + µ − ) (cid:46) . 67 fb, three orders of magnitude smaller.At the Tevatron, the CDF Collaboration performed a search for pair production of newparticles Y , each decaying into two jets, p ¯ p → Y Y → jjjj . The mass range explored M Y ≥ 50 GeV does not cover M H = 30 GeV, but for illustration we can take the limitfor M Y = 50 GeV, namely σ ( Y Y → jjjj ) ≤ 200 pb. In our benchmarks, the predictionis σ ( H , ) × Br( jjjj ) (cid:46) . t ¯ tZ (cid:48) (model 1) 0.079 fb 8.1 fb 0.080 fb Z (cid:48) (model 2) 0.37 fb 9.3 fb 0.043 fb jj b ¯ b t ¯ t — — 78 fbTable 4: Signal and background cross sections for the Z (cid:48) signal in scenario 2 and mainSM backgrounds (in rows) under the three different event selections for generic, dijet, and t ¯ t resonance searches. The event selection for dijet and t ¯ t is the same as in scenario 1,and the quoted background numbers are the same as in table 3. This scenario is similar to scenario 1 but with heavier masses, M Z (cid:48) = 3 . M H (cid:39) M H (cid:39) 400 GeV, and M H (cid:39) 80 GeV. For the signal coupling we choose g Z (cid:48) z = 0 . Z (cid:48) production cross section of 20.1 fb. The total Z (cid:48) width is Γ = 70 . . R = 1 ineq. (14), leading to the branching ratiosBr( Z (cid:48) → H H ) = 0 . 10 (model 1) , Br( Z (cid:48) → H H ) = 0 . 51 (model 2) . (40)Again, Br( H , → H H ) (cid:39) 1, and for the decay of H into quark pairsBr( H → b ¯ b ) = 0 . , Br( H → c ¯ c ) = 0 . . (41)The combined branching ratio factor for the four H decays into quark pairs is Br( H → b ¯ b, c ¯ c ) = 0 . m J ≥ 250 GeV. We apply the taggerperformance efficiency factors obtained in ref. [29] of 0 . 01 for QCD jets and 0.33 for thesignal jets ( M H , = 400 GeV, M H = 80 GeV).The dijet / t ¯ t invariant mass distributions are presented in figure 6 for the generic (top), t ¯ t (middle) and dijet (bottom) analyses. Given that the cross sections for Z (cid:48) productionare smaller to those of scenario 1, we present our results for integrated luminosities L = 150fb − for model 1, and L = 20 fb − for model 2. The signal and background cross sectionsfor the different event selections are collected in table 4.23 000 2500 3000 3500 4000 4500 m JJ (GeV) -4 -3 -2 -1 E v en t s / G e V backgroundbackground + signal Z ′ → H H , H → H H Model 1L = 150 fb -1 m JJ (GeV) -4 -3 -2 -1 E v en t s / G e V backgroundbackground + signal Z ′ → H H , H → H H Model 2L = 20 fb -1 m tt (GeV) -2 -1 E v en t s / G e V backgroundsignal Z ′ → ttModel 1L = 150 fb -1 m tt (GeV) -3 -2 -1 E v en t s / G e V backgroundsignal Z ′ → ttModel 2L = 20 fb -1 m jj (GeV) -2 -1 E v en t s / G e V backgroundsignal Z ′ → jjModel 1L = 150 fb -1 m jj (GeV) -2 -1 E v en t s / G e V backgroundsignal Z ′ → jjModel 2L = 20 fb -1 Figure 6: Invariant mass distribution for the Z (cid:48) signals in scenario 2 and their back-grounds, in the generic (top), t ¯ t (middle) and dijet (bottom) analyses, for model 1 (left)and model 2 (right). 24 500 2000 2500 3000 3500 4000 4500 5000M (GeV)10 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 Lo c a l p σ σ σ σ σ σ σ Z ′ → jjZ ′ → H H , H → H H Model 1 L = 150 fb -1 Z ′ → tt 1500 2000 2500 3000 3500 4000 4500 5000M (GeV)10 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 Lo c a l p σ σ σ σ σ σ σ Z ′ → jjZ ′ → H H , H → H H Model 2 L = 20 fb -1 Z ′ → tt Figure 7: Expected local p -value for the Z (cid:48) signal in the various searches, for scenario 2of model 1 (left) and model 2 (right).The expected significance of the Z (cid:48) signal in the different searches is presented in fig-ure 7, assuming luminosities of 150 fb − in model 1, and 20 fb − in model 2. The differencebetween stealth boson modes and standard t ¯ t and dijet decays is quite pronounced. Thesignificance of the signals in the generic search (expressed in terms of standard deviations)is 3 and 6 times larger than in dijets, for model 1 and model 2, respectively. Still, weremind the reader that we have not taken advantage of b tagging, which would improvethe signal significance by a factor of 2 − H with decay into SM particles (withits decay branching ratios corresponding to a SM Higgs with a mass of 80 GeV) can beconstrained from Higgs boson searches. At LEP, the non-observation of a signal constrains O i ≤ . 04 for M H = 80 GeV [49], two orders of magnitude above the bound O i (cid:46) − in our benchmark. At the Tevatron, searches by the D0 and CDF Collaborations [54, 55]only cover masses above 90 GeV, and also are less sensitive.For the heaviest scalars the production and decay chain is pp → H , → H H , withsubsequent decay of H . A search for pair produced resonances decaying into quark pairsby the CMS Collaboration [56] is sensitive to (but no optimised for) this process. Thelimits corresponding to stop masses m ˜ t = 80 GeV are σ (˜ t ˜ t ∗ ) ≤ 400 pb, assuming 100%branching ratio for the R -parity violating decay ˜ t → b ¯ q . In our benchmark, the crosssection times branching ratio for final states with b quarks is σ ( H , ) × Br( b ¯ bb ¯ b ) (cid:46) . M H , = 400 GeV. This is six orders of magnitude below the above experimentallimit. 25eneric dijet t ¯ tZ (cid:48) (model 1) 0.062 fb 7.8 fb 0.080 fb Z (cid:48) (model 2) 0.30 fb 7.6 fb 0.043 fb jj b ¯ b t ¯ t — — 78 fbTable 5: Signal and background cross sections for the Z (cid:48) signal in scenario 3 and mainSM backgrounds (in rows) under the three different event selections for generic, dijet, and t ¯ t resonance searches. The event selection is the same as in scenario 2, and the quotedbackgrounds are the same as in table 4. The signal in the t ¯ t selection is also the same. Here we take M Z (cid:48) = 3 . M H (cid:39) M H (cid:39) 400 GeV, as in scenario 2, and we keepthe same signal coupling g Z (cid:48) z = 0 . 2. Therefore, the Z (cid:48) production cross section and widthare the same, σ ( Z (cid:48) ) = 20 . . . R to unity, hence the branching ratios are the same as in scenario 2,Br( Z (cid:48) → H H ) = 0 . 10 (model 1) , Br( Z (cid:48) → H H ) = 0 . 51 (model 2) . (42)The differences with respect to scenario 2 stem from the fact that H is now heavier,namely M H (cid:39) 300 GeV. This forbids the decays H , → H H . We therefore focus on H decays into gauge boson pairs, taking (see figure 3)Br( H , → W W ) = 0 . , Br( H , → ZZ ) = 0 . . (43)The event selection for the three analyses is the same as in scenario 2 (for the dijet and t ¯ t searches it is also the same as in scenario 1). The dijet invariant mass distributions arepresented in figure 8 for the generic (top panel) and dijet (bottom panel) analyses, forintegrated luminosities L = 150 fb − (model 1) and L = 20 fb − (model 2). The resultsfor the t ¯ t analysis are the same as in scenario 2, and were already shown in figure 6.The signal and background cross sections for the different event selections are collectedin table 5.The expected significance of the Z (cid:48) signal in the different searches is presented infigure 9, assuming luminosities of 150 fb − in model 1 and 20 fb − in model 2. Thesignificance of the signals in the generic search (expressed in terms of standard deviations)26 000 2500 3000 3500 4000 4500 m JJ (GeV) -4 -3 -2 -1 E v en t s / G e V backgroundbackground + signal Z ′ → H H , H → WW, ZZModel 1L = 150 fb -1 m JJ (GeV) -4 -3 -2 -1 E v en t s / G e V backgroundbackground + signal Z ′ → H H , H → WW, ZZModel 2L = 20 fb -1 m jj (GeV) -2 -1 E v en t s / G e V backgroundsignal Z ′ → jjModel 1L = 150 fb -1 m jj (GeV) -2 -1 E v en t s / G e V backgroundsignal Z ′ → jjModel 2L = 20 fb -1 Figure 8: Invariant mass distribution for the Z (cid:48) signals in scenario 3 and their back-grounds, in the generic (top) and dijet (bottom) analyses, for model 1 (left) and model 2(right). -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 Lo c a l p σ σ σ σ σ σ σ Z ′ → jjZ ′ → H H , H → WW, ZZModel 1 L = 150 fb -1 Z ′ → tt 1500 2000 2500 3000 3500 4000 4500 5000M (GeV)10 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 Lo c a l p σ σ σ σ σ σ σ Z ′ → jjZ ′ → H H , H → WW, ZZModel 2 L = 20 fb -1 Z ′ → tt Figure 9: Expected local p -value for the Z (cid:48) signal in the various searches, for scenario 3of model 1 (left) and model 2 (right).is 2.5 and 9 times larger than in dijets, for model 1 and model 2, respectively. Regarding27he results for the generic analysis, it is worth remarking a few points. • The requirement of jet masses m J ≥ 250 GeV in the generic analysis filters thehadronic decays of both gauge bosons, which have branching ratio of 0.45 for W W and 0.49 for ZZ , reducing the signal with respect to scenario 2. This explains whythe expected sensitivity in the generic analysis is slightly worse, despite the largerefficiency of the tagger for jets corresponding to H , → W W (0.49 for the signalfor a background rejection of 10 ) than for jets with H , → H H in scenario 2. • The decays H , → HH have a branching ratio of 0.13, and the generic searchwould also be sensitive to jets containing two SM Higgs bosons. Because there is nobenchmark working point of the tagger available, we have not included these signalcontributions in our analysis. • The tagger has an efficiency of 0.21 for jets containing two top quarks, resultingfrom H , → t ¯ t . For simplicity we have not included this signal contribution to thegeneric search, given the small branching ratio Br( H , → t ¯ t ) = 0 . V V pairs set a limit of approximately σ × Br( V V ) ≤ 350 fb for a scalar mass of 400 GeV. Inthis benchmark we have σ ( H , ) × Br( V V ) ≤ . 26 fb, more than two orders of magnitudebelow the limit. The lightest new scalar H with M H = 300 GeV is not covered by thisanalysis, which only considers masses above 400 GeV. Still, σ ( H ) × Br( V V ) ≤ . 66 fb iswell below potential constraints at this mass. The search for elusive new physics signals yielding various types of multi-pronged jetsrequires a model-independent approach, with the use of novel tools like the anti-QCDtagger [29] or non-supervised learning methods [58–60]. In order to contextualise therelevance of these signals as discovery channel for new (leptophobic) resonances, it iscrucial to provide examples of consistent models that may produce them. We have doneso for stealth bosons, boosted particles with a cascade decay giving a four-pronged fatjet. We have worked out the minimal implementation, adding to the SM a leptophobic Z (cid:48) boson, two complex scalar singlets and extra matter, either new vector-like quarks28model 1) or new vector-like leptons (model 2), to cancel anomalies. In these modelsone can compare the potential significance of stealth boson signals, still unexplored atthe LHC, with the standard signals (dijets, top pairs and dibosons) already searchedfor. Depending on the model and benchmark scenario considered, the significance of theformer may be up to 9 times larger than the most sensitive of the latter. Therefore, it isclear that stealth boson signals might well be hidden in LHC data, yet invisible to currentsearches. Besides, direct production of the new light scalars is suppressed by the squareof the small mixing, and signals are too small to be observed.In the two models considered in this work the branching ratios of Z (cid:48) decays into scalarsare sizeable (around 10% in model 1 and 50% in model 2). Moreover, cascade decays ofthe new scalars are likely to happen, provided one of the following conditions are fulfilled: • There is a hierarchy among the masses of the new scalars, so that the decays of oneinto others are possible. These decays are not suppressed by mixing with the SMscalar doublet, and will therefore be dominant, as in our scenario 1. • The scalars are heavy enough to decay into W + W − (and possibly ZZ , HH and t ¯ t ).If the decay into other new scalars is kinematically allowed, it will be the dominantchannel, as in our scenario 2. Otherwise, decays into pairs of SM bosons will bedominant, as in our scenario 3.Therefore, as it has been shown with a scan on parameter space, it is natural to havestealth bosons as decay products of the Z (cid:48) . For simplicity, we have restricted our detailedsimulations to Z (cid:48) decays into a pair of stealth bosons giving two four-pronged jets. Still,those processes giving one stealth boson (four-pronged jet) and one scalar that subse-quently decays into quarks (two-pronged jet) are also possible and interesting. A genericsearch would be sensitive to all these possibilities at once, and this is one of the mainvirtues of the anti-QCD tagger.In conclusion, we stress that despite the fact that stealth bosons are rather stealth forcurrent LHC searches, they would be quite conspicuous in a generic search. Moreover,these signals may well appear in decays of heavy Z (cid:48) resonances. These facts alreadyprovide a strong motivation for model-independent searches. Acknowledgements F.R.J. thanks the AHEP group at IFIC/CSIC (Valencia) and the Theoretical Physics De-partment of the University of Valencia for the warm hospitality and for financial support29uring the final stage of this work. J.A.A.S. thanks A. Casas and C. Mu˜noz for usefuldiscussions. The work of F.R.J. is supported by Funda¸c˜ao para a Ciˆencia e a Tecnolo-gia (FCT, Portugal) through the projects CFTP-FCT Unit 777 (UID/FIS/00777/2013),CERN/FIS-PAR/0004/2017 and PTDC/FIS-PAR/29436/2017, which are partly fundedthrough POCTI (FEDER), COMPETE, QREN and EU. The work of J.A.A.S. is sup-ported by Spanish Agencia Estatal de Investigaci´on through the grant ‘IFT Centro deExcelencia Severo Ochoa SEV-2016-0597’ and by MINECO project FPA 2013-47836-C3-2-P (including ERDF). A Triple scalar couplings In the weak interaction basis H (cid:48) i = ( ρ ρ ρ A ) the trilinear scalar interactions can beexpressed in the condensed form: L H = − (cid:88) u C pqr H (cid:48) p H (cid:48) q H (cid:48) r , (44)with the sum over p ≤ q ≤ r running from from 1 to 4. The C ijk coefficients are explicitlygiven by: C = v u λ ,C = 14 [ λ cos β + Re( λ (cid:48) ) sin β ] ,C = 14 [ Re( λ (cid:48) ) cos β + λ sin β ] ,C = − 14 Im( λ (cid:48) ) ,C = v u λ ,C = v u Re( λ (cid:48) ) ,C = − v u Im( λ (cid:48) ) cos β ,C = v u λ ,C = − v u Im( λ (cid:48) ) sin β ,C = v u [ λ cos β − Re( λ (cid:48) ) sin(2 β ) + λ sin β ] ,C = 14 [ 2 λ cos β + Re( λ (cid:48) ) sin β ] ,C = 14 { λ (cid:48) ) cos β + 2 [ λ + Re( λ (cid:48) ) ] sin β } , = − { Im( λ (cid:48) ) sin(2 β ) + Im( λ (cid:48) )[2 + cos(2 β ) ] } ,C = 14 { λ (cid:48) ) sin β + 2 [ λ + Re( λ (cid:48) ) ] cos β } ,C = − { λ (cid:48) ) + Im( λ (cid:48) + λ (cid:48) ) sin 2 β } ,C = 14 (cid:8) λ − Re( λ (cid:48) ) cos β ] + Re( λ (cid:48) − λ (cid:48) ) cos β sin β +2 [ λ − λ (cid:48) ) ] sin β cos β + Re( λ (cid:48) ) sin β (cid:9) ,C = 14 [ 2 λ sin β + Re( λ (cid:48) ) cos β ] ,C = 14 [ − Im( λ (cid:48) ) sin(2 β ) + Im( λ (cid:48) )( − β ) ] ,C = 14 { Re( λ (cid:48) ) cos β + 2[ λ − λ (cid:48) )] cos β sin β + Re( λ (cid:48) − λ (cid:48) ) cos β sin β + 2[ λ − Re( λ (cid:48) )] sin β } ,C = 14 [ Im( λ (cid:48) ) sin(2 β ) − Im( λ (cid:48) ) sin β − Im( λ (cid:48) ) cos β ] . (45)In the H a mass eigenstate basis, with H (cid:48) i = O ia H a , L H = − (cid:88) u C pqr O pa O qb O rc H a H b H c , (46)where the sums over a, b, c and p ≤ q ≤ r run from 1 to 4. From this, one can readthe interactions H i H j H k which, when the indices i, j, k are different, can be written as − uλ ijk H i H j H k , with λ ijk = (cid:88) p ≤ q ≤ r, ( s ) C pqr O ps O qs O rs . (47)The sum above runs over all permutations( s , s , s ) = { ( i, j, k ) , ( i, k, j ) , ( j, k, i ) , ( j, i, k ) , ( k, i, j ) , ( k, j, i ) } . (48)When two of the indices i, j, k are equal, the sum (48) contains each of the three inde-pendent permutations twice, thus introducing a double counting. When the three indicesare equal, i = j = k , this sum counts six times the single term H i H i H i present in thesum (46). One can take this fact into account by introducing a symmetry factor S ijk ,which is one if the three indices are different, two if two of the indices are equal, and sixif i = j = k . With this convention, the interaction is (no sum over indices) − u λ ijk S ijk H i H j H k , (49)keeping the definition (47) for λ ijk and all permutations (48), even repeated ones. Whenderiving the Feynman rule for the three-scalar interaction, one has to multiply by asymmetry factor for the presence of identical particles, which is precisely S ijk . Therefore,the Feynman rule for the vertex is simply − iuλ ijk .31 Model with two scalar doublets and one singlet An attractive SM extension which apparently could lead to stealth boson decays wouldbe that with and extra scalar doublet and a scalar singlet. However, this model does notproduce any of the desired processes Z (cid:48) → H i Z ,Z (cid:48) → H i H j . (50)For illustration and completeness we summarise why. We label the two existent scalardoublets as Φ = ( φ +1 φ ) T and Φ = ( φ +2 φ ) T , and the singlet as χ . The scalar potentialcompatible with the SU(2) L × U(1) Y symmetry is V = m Φ † Φ + m Φ † Φ + m χ † χ − (cid:104) m Φ † Φ + h.c. (cid:105) + λ † Φ ) + λ † Φ ) + λ (Φ † Φ )(Φ † Φ ) + λ (Φ † Φ )(Φ † Φ )+ 12 (cid:104) λ (Φ † Φ )(Φ † Φ ) + λ (Φ † Φ )(Φ † Φ ) + λ (Φ † Φ )(Φ † Φ ) + h.c. (cid:105) + λ χ † χ ) + λ † Φ )( χ † χ ) + λ † Φ )( χ † χ )+ 12 (cid:104) λ (Φ † Φ ) + h.c. (cid:105) ( χ † χ ) . (51)Among the parameters above, m , m , m , λ − and λ − are real, while m , λ − and λ can be complex. Writing the neutral scalar fields in the usual way: φ = 1 √ ρ + v + iη ) , φ = 1 √ ρ + v + iη ) , χ = 1 √ ρ + u + iη ) , (52)the would-be Goldstone bosons associated to the breaking of the U(1) (cid:48) and electroweaksymmetries are η , and a combination of η and η (as in the usual two-Higgs doubletmodel), respectively. Therefore, we have three scalars and a pseudo-scalar, which can inprinciple mix.At least one of the two doublets must have vanishing hypercharge Y (cid:48) , as requiredby the existence of Yukawa terms (3). We choose it to be Φ . If the other doubletΦ has hypercharge Y (cid:48) Φ (cid:54) = 0, then invariance under U(1) (cid:48) requires m = 0, λ − = 0, λ = 0. After applying the potential minimisation conditions it is found that the physicalpseudoscalar is massless, which is unacceptable. Besides this obvious drawback, we notethat the vacuum expectation value (cid:104) φ (cid:105) = v / √ Z − Z (cid:48) mixing [34], whichis constrained to be very small. Because the Z − Z (cid:48) coupling to the scalars in Φ isproportional to v , the width for Z (cid:48) → H i Z , which would be characteristic for this model,32 500 2000 2500 3000 3500 4000 m tt (GeV) -3 -2 -1 E v en t s / G e V backgroundsignal Diboson searchModel 2 , M H = 30 GeVL = 2 fb -1 m tt (GeV) -3 -2 -1 E v en t s / G e V backgroundsignal Diboson searchModel 2 , M H = 15 GeVL = 2 fb -1 Figure 10: Invariant mass distribution for the Z (cid:48) signals and their backgrounds in thediboson analysis, for and alternative scenario 1 of model 2 with M H = 15 GeV.is also very small. If both doublets have Y (cid:48) = 0, neither of the decays in (50) is present,the former because of the vanishing doublet hypercharges, and the latter because the only Z (cid:48) coupling to scalars is Y (cid:48) χ Z (cid:48) µ η ←→ ∂ µ ρ and η is not physical. C Alternative scenario 1 We present here results for an alternative scenario 1 for model 2, with M H = 15 GeV, inwhich the substructure of the four-pronged fat jets resulting from H , → H H → q ¯ qq ¯ q resembles more a two-pronged structure because of the lighter H . For this mass, we haveBr( H → b ¯ b ) = 0 . , Br( H → c ¯ c ) = 0 . . (53)so that Br( H → b ¯ b, c ¯ c ) = 0 . M H = 30 GeV (left panel ) and M H = 15GeV (right panel). Besides the large differences in the cross section, due to the largeracceptance for Z (cid:48) → H H , in the latter case we observe a resonant signal structure thatis not present in the former. The p -value for the different Z (cid:48) signals is given in figure 11.Notice that, despite the fact that a possible signal would be more visible in the dibosonresonance searches than in t ¯ t and dijet final states, it is by far surpassed by the signalthat would be visible in a generic search using the anti-QCD tagger.33 000 1500 2000 2500 3000 3500 4000M (GeV)10 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 Lo c a l p σ σ σ σ σ σ σ Z ′ → jj, ttZ ′ → H H , H → H H diboson search Model 2, M H = 15 GeVL = 2 fb -1 Figure 11: Local p -value for the Z (cid:48) signal in the different searches for and alternativescenario 1 of model 2 with M H = 15 GeV. References [1] J. M. Butterworth, A. R. Davison, M. Rubin and G. P. Salam, Jet substructureas a new Higgs search channel at the LHC, Phys. Rev. Lett. (2008) 242001[arXiv:0802.2470 [hep-ph]].[2] J. Thaler and L. T. Wang, Strategies to Identify Boosted Tops, JHEP (2008)092 [arXiv:0806.0023 [hep-ph]].[3] D. E. Kaplan, K. Rehermann, M. D. Schwartz and B. 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