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Deciphering the spin of new resonances in Higgsless models

We study the potential of the CERN Large Hadron Collider (LHC) to probe the spin of new massive vector boson resonances predicted by Higgsless models. We consider its production via weak boson fusion which relies only on the coupling between the new resonances and the weak gauge bosons. We show that the LHC will be able to unravel the spin of the particles associated with the partial restoration of unitarity in vector boson scattering for integrated luminosities of 150-560 fb^-1, depending on the new state mass and on the method used in the analyses.


I. INTRODUCTION
Despite the success of the Standard Model (SM) of particle physics in describing electroweak physics below ∼ 100 GeV in terms of a non-abelian gauge theory with spontaneously broken SU (2) L × U (1) Y gauge group, the gauge symmetry does not predict the precise mechanism of the electroweak symmetry breaking (EWSB). Indeed, up to this moment, there is no direct experimental signal of the mechanism of EWSB, being its search one of the main goals of the LHC.
The EWSB mechanism plays an important role in the high energy electroweak gauge boson scattering which violates partial wave unitarity or becomes strongly interacting at energies of the order of E ∼ 2 TeV, if there is no new state to cut off its growth [1,2]. In the context of the SM, as well as in its supersymmetric realization, electroweak symmetry is broken by the vacuum expectation value of some weakly coupled neutral scalar state(s), the Higgs boson(s), which will contribute to electroweak gauge boson scattering, preventing the unitarity violation of the process.
From the point of view of unitarity, all Higgsless models share the common feature that new weakly interacting spin-1 gauge bosons particles with the same quantum numbers as the SM gauge bosons appear and they are responsible for the partial restoration of unitarity in vector boson scattering and for rendering a theory weakly coupled to energies well above 2 TeV [25,26,27]. This property allows for an almost model independent search for the lightest charged resonance V ± 1 at the LHC through pp → V ± 1 W ∓ or via weak boson fusion pp → V ± 1 qq [28,29], as long as V ± 1 remains a narrow resonance. The LHC experiments will be able to unravel the existence of the charged state via these processes with modest integrated luminosities of 10-60 fb −1 . On the contrary, the corresponding search for the neutral vector resonance in gauge boson fusion is expected to be very difficult, since a generic feature of this class of models is the absence of coupling between the neutral resonance and ZZ pairs. Reconstructing the heavy neutral vector resonance decaying into W + W − requires at least one hadronic W decay, posing the challenge to dig it out from the large SM backgrounds.
Once a clear signal of the charged resonance is observed in the above channels, it is mandatory to study its spin to confirm that the new state is indeed a vector particle. In this work, our goal is to probe the V ± 1 spin via the study of weak boson fusion production of V ± 1 with its subsequent decay into leptons, i.e.
with ℓ and ℓ ′ = e or µ, considering final states where the W 's and Z's decay into different and same flavor charged leptons. To determine the spin of the state decaying into W ± Z we contrast the final state distributions arising from the production and decay of the vector charged state with the ones stemming from the decay of a scalar state; i.e. we work in the framework commonly used to analyze the spin of supersymmetric particles [30,31].
Here we show that it is possible to determine the spin of a new heavy resonance decaying into W ± Z at the LHC with 99% CL for luminosities of ∼ 150-560 fb −1 , depending on the particle mass and the method used in the analysis.

II. MODEL AND CALCULATION SETUP
The restoration of partial wave unitarity in Higgsless models is due to new Kaluza-Klein resonances V ± (i) and V W W ). In order to cancel the dangerous terms in the scattering W Z → W Z that depend on E 2 and E 4 , where E is the energy of the incoming W and Z in the center-of-mass system, the new vector state coupling constants must satisfy the following constraints: Eqs. (2) and (3) constrain the couplings of the lightest charged Kaluza-Klein state to W Z pairs, In our analysis we assume that this bound is saturated [32], which leads to the largest allowed value for g VWZ , and we evaluate the quartic coupling g WWZZ using Eq. (2). Moreover, we assume that the V ± 1 couplings to fermions are small and that the V ± 1 's mainly decay into W Z pairs. This hypothesis is, in fact, realized in some higgsless models [24].
Our study of the V ± 1 spin was carried out by comparing the kinematic distributions of its decay products with the predictions for the production of a spin-0 resonance. Since the signal for the new charged state is characterized by peak in the W Z invariant mass distribution, we use as template the kinematic distributions in a model which is the SM without a Higgs plus a scalar charged state, H ± , with an interaction H ± Z µ W ∓ µ . The coupling of the H ± Z µ W ∓ µ vertex is chosen such that the H ± production cross section is equal to the one for V ± 1 after all cuts. We also set the H ± width equal to the V ± 1 one. We performed a parton level study using the full tree level amplitude for the final state processes in order to keep track of spin correlations. The matrix elements were generated using the package MADGRAPH [33], where we included the higgsless (and template) model particles and interactions. We employed the CTEQ6L parton distribution functions [34] with the factorization scale µ F = (p 2 T j1 + p 2 T j2 )/2, where p T ji are the transverse momenta of the tagging jets. For the QCD backgrounds we chose the renormalization scale µ R = µ F . In order to have a crude simulation of the detector performance we smeared energies, but not directions, of all final state partons with a Gaussian error. For the jets, we assumed a resolution ∆E/ Furthermore, we considered the jet tagging efficiency to be 0.75 × 0.75 = 0.56, while the lepton detection efficiency is taken to be 0.9 3 = 0.73.

III. RESULTS
We analyzed the process which contains the contribution of the vector boson fusion production of new charged resonances decaying into leptons; see Eq. (1). This process possesses a significant irreducible background originating from electroweak and QCD W Zjj production. Moreover, the production of tt pair in association with a jet exhibits a large cross section after we demand the presence of two tagging jets [35] and can lead to trilepton events when both t's decay semileptonically and the decay of one of the b's leads to an isolated lepton 1 . Initially we imposed the following jet acceptance cuts designed to enhance events produced by vector boson fusion, cuts (5)-(6) cuts (5)-(7) cuts (5)-(8) cuts (5)  We also applied lepton acceptance and isolation cuts As we can see from Table I the SM background is still quite large after these cuts with the ttj production being the dominant contribution. In order to reduce this background we explore two features of the signal and backgrounds. First of all, in the ttj production the lepton coming from the b semi-leptonic decay is quite soft, therefore, it can be reduced by imposing an additional lepton transverse momentum cut: Moreover, two of the leptons in the signal come from a Z decay, consequently we also required that the events present a pair of same flavor opposite charge leptons (SFOC) with an invariant mass in a window around the Z mass. Thus we further demanded The presence of just one neutrino in the signal final state, Eq. (1), allows for full reconstruction of the neutrino momentum -up to a twofold ambiguity on its longitudinal component -by imposing the transverse momentum conservation and requiring that the invariant mass of the neutrino-ℓ ± pair, where ℓ is the charged lepton not identified as coming from the Z decay, is compatible with the W mass: Consequently, there are two distinct estimates for the W Z invariant mass which we label M rec,max for M V ± 1 = 700 GeV (10) The effect of these cuts on the W Z invariant mass spectrum can be seen in the right panel of Fig. 1 for M V ± 1 = 500 GeV and 700 GeV. As seen in the figure after these cuts, a good fraction of the peak signal events are retained.
The predicted cross sections for the signal and SM backgrounds after cuts (5)-(10) are listed in Table I. From these numbers we conclude that the above cuts lead to a good signal to background ratio of ≃ 2.4 (1.5) for M V ± 1 = 500 (700) GeV. Thus, a clear observation (5σ) of the new charged resonances V ± 1 with a 500 (700) GeV mass in the leptonic channel requires a modest integrated luminosity of 15 (66) fb −1 , which can be achieved in the low luminosity run of the LHC or in the early stages of the high luminosity run.
Similar sensitivity could be obtained by cutting, instead, on M rec,max W Z , though in general the cuts have to be chosen tighter and dependent on the M V ± 1 mass. This is so because the SM background is a decreasing function of the W Z mass, therefore when cutting on the maximum reconstructed W Z mass, the number of miss-reconstructed background events in the signal region tends to be larger.
After the new state coupled to W Z is discovered, it is important to probe its spin. The best way to accomplish that is to study angular correlations of the final state particles. In principle, useful information on the spin could be also extracted from the production cross section, however, at the LHC one measures the production cross section times the decay branching ratio, requiring additional information to disentangle these quantities. Here we em- ploy two methods to unravel the spin of the new charged state based exclusively on the kinematic distribution of the final state particles. In the first method, we contrast the kinematic distributions of the charged leptons produced in the decay of vector and scalar charged states, much in the spirit of the analysis carried out to study the spin of supersymmetric particles at the LHC [30,31]. A virtue of this method is that it does not rely on the reconstruction of the neutrino momentum (besides the invariant mass cut). In our second analysis, we used the reconstructed neutrino momentum to obtain the polar angle of the produced Z's in the W Z center-of-mass system.
In order to contrast the spin-0 and spin-1 resonances, we focused on the leptons whose momenta can be well determined. In previous studies [30], it has been shown that a convenient variable for such analysis is where ∆η ℓℓ is the rapidity difference between the same charge leptons. Notice that this quantity is invariant under longitudinal boosts. We plot in Fig. 2 the expected cos θ * ℓℓ distributions for the SM background and the production of scalar and vector resonances with mass 500 (700) GeV in the left (right) panel after cuts (5)-(10). In obtaining this figure, we imposed that the cross section for the production of spin-0 resonances is the same of the one for spin-1 states. We also display the SM background alone to show its impact on the distributions.
These figures clearly show that the cos θ * ℓℓ distribution for spin-1 and and spin-0 resonances are quite different and they can be used to quantify the required integrated luminosity needed to discriminate between them at a given CL. A simple χ 2 analysis of the distributions shown in Fig. 2 yields a 99% CL discrimination between spin-0 and spin-1 resonances of mass 500 (700) GeV for an integrated luminosity of 170 (215) fb −1 , considering only the statistical errors.
In order to eliminate possible normalization systematics in the angular distributions, we have also estimated the integrated luminosity needed to decipher the spin of the new charged state by constructing an angular asymmetry Considering only the statistical errors, this asymmetry allows a 99% CL distinction between spin-0 and spin-1 resonances of mass 500 (700) GeV for an integrated luminosity of 440 (560)  We also studied the resolving power of the reconstructed Z polar angle (θ W Z ) distribution evaluated in the W Z center-of-mass frame. We display in Fig. 3 the cos θ W Z distribution for spin-1 charged states after cuts (5)- (10). Since the reconstructed neutrino momentum has a twofold ambiguity, there is also a twofold ambiguity in the reconstructed Z polar angle in the W Z center-ofmass frame which lead to the two distributions shown in the figure. The dashed (dotted) lines correspond to the reconstructed Z polar angle distribution using the neutrino momentum that leads to the maximum (minimum) W Z invariant mass. As we can see, the two distributions differ appreciably for cos θ W Z close to zero. However, as shown in the figure, the average of the two distributions has a better behavior in the central region of the detector and resembles the true distribution. Consequently, we have considered the average of the two reconstructed distributions as discriminating observable. Fig. 4 depicts such averaged distributions for charged vector and scalar resonances, where we are included the SM background prediction for assessment of its impact on the spin determination. Clearly, the production of V ± 1 leads to more W Z pairs produced at small polar angles while the scalar resonance leads to more central events, as expected. As above, in order to quantify the discriminating power between the scalar and vector productions we constructed the asymmetry (13) We find that for the new state mass of 500 (700) GeV, it is necessary 400 (550) fb −1 to separate the two possibilities at 99% CL. With these choices of integrated luminosities, we have A W Z (scalar) = +0.057 ± 0.05 and A W Z (vector) = −0.125 ± 0.05, for M V ± 1 = 500 GeV, = 700 GeV, where we have again quoted only the statistical errors. Furthermore the use of a χ 2 analysis of the cos θ W Z distribution is able to reveal the spin of the new state at 99% CL for an integrated luminosity of 150 (220) fb −1 , for M V ± 1 = 500 (700) GeV.

IV. CONCLUSIONS
The observation of new charged vector resonances in Higgsless models decaying into W Z pairs can be carried out via weak boson production at the LHC and their subsequent decays into charged leptons [28,29]. Here we show how the LHC will be able to determine the spin of these new states using two different methodologies. In the first method, only the observed charged leptons are used to discriminate between spin-0 and spin-1 resonances using the variable defined in Eq. (11). In this case, an integrated luminosity of 170 (215) fb −1 is needed to establish the spin of the 500 (700) GeV resonance at 99% CL via a χ 2 analysis of the cos θ * ℓℓ distribution. On the other hand, the use of the asymmetry given by Eq. (12) requires 440 (560) fb −1 to determine the new resonance spin for a mass 500 (700) GeV. The second method is based on the two-folded reconstruction of the escaping neutrino momentum to obtain the W Z polar angle distribution in its center-of-mass frame. This procedure requires a good understanding and calibration of the hadronic calorimeters, therefore, being subject to larger systematic uncertainties. We determined that the later method can distinguish between spin-1 and spin-0 states at 99% CL for integrated luminosities of 150 (220) fb −1 for M V ± 1 = 500 (700) GeV, respectively, when we perform a χ 2 fit of the cos θ W Z distribution. If we use the asymmetry defined in Eq. (13) to perform the analysis, the integrated luminosities are 400 and 550 fb −1 , respectively. All results above only account for statistical errors and the inclusion of systematic uncertainties may render the efficiencies of the two methods rather similar.