Torsion Limits From $t\bar{t}$ Production at the LHC
Fernando Marroquim Fernando de Andrade Maria A. B. do Vale Andre Nepomuceno
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    []fmarroquim@ufsj.edu.br[]aline@ufsj.edu.brDepartamento de Ciências Naturais, Universidade Federal de São João del-Rei,
    
    São João del Rei, 36301-160, MG, Brazil[]asevedo@gmail.comDepartamento de Ciências da Natureza, Universidade Federal Fluminense
    
    Rua Recife, s/n, Rio das Ostras, RJ, 28890-000, Brazil
    Torsion models constitute a well known class of extended quantum gravity models. 
    In this work, one investigates the phenomenological consequences of a torsion field interacting  
    with top quarks at the LHC. A torsion field could appear as a new heavy state characterized by 
    its mass and couplings to fermions. This new state would form a resonance decaying into a top anti-top pair. 
    The latest ATLAS tt̅ production results from LHC 13 TeV data are used to set limits on torsion parameters. 
    The integrated luminosity needed to observe torsion resonance at the next LHC upgrades are also evaluated, 
    considering different values for the torsion mass and its couplings to Standard Model fermions. 
    Finally, prospects for torsion exclusion at the future LHC phases II and III are obtained using
    fast detector simulations. 
    
    Torsion Limits From tt̅ Production at the LHC
        A. A. Nepomuceno
        December 30, 2023
    =============================================
    
    
    
    § INTRODUCTION
    
    
    Since the start of the LHC era, its experiments have performed an impressive set of measurements 
    in different energy regimes that has strengthened our confidence in the Standard Model (SM).  
    The most significant result was the discovery of the Higgs boson by the ATLAS and CMS experiments <cit.>. 
    On the other hand, new physics has not yet appeared, and so far we have no indications from the data on what kind of 
    physics beyond SM lies in the high energy scales. Drastic experimental constraints have been put in many extensions of SM, 
    making some of them less appealing. As a result, the famous SM puzzles that motivated the pre-LHC model building 
    era are still unsolved.  
    
    The various SM extensions deal with one or more problems of the theory, but few of them try to 
    incorporate quantum gravity. In fact, it is generally accepted that a consistent quantum gravity theory does not exist. 
    In this scenario, the most realistic candidate to a universal theory would be the string theory, 
    which induces gravitational interactions in the low energy limit. However, there is no perspective in near future that 
    this theory could be verified experimentally. The alternative has been to apply effective approaches to the problem by 
    considering natural extension of General Relativity (GR), assuming they might come from a still unknown fundamental theory. 
    One of the most natural extensions of GR is the torsion gravity theory <cit.>. 
    
    There are different approaches to treat the torsion field, but for the purpose of this paper, 
    we consider the one where torsion is a fundamental propagating field, with a well-defined action and 
    characterized by a mass and couplings between torsion and fermions <cit.>. 
    As the torsion is taken as a dynamical field, it is incorporated into the SM along with the other vector fields. 
    The coupling between torsion and fermions can be, in principle, non-universal. This possibility is explored in this paper. 
    
    At the LHC, torsion signals could be observed through its decay into fermions. 
    The LHC theoretical reach to probe torsion was investigated in <cit.>. Limits on torsion parameters have 
    been derived using LEP and TEVATRON results <cit.>. The ATLAS experiment 
    has put limits on torsion mass and couplings using 7 TeV data in dilepton channel <cit.>. 
    
    The impact of a heavy torsion on top pair asymmetries was studied 
    in <cit.>.
    In most of these studies, the torsion-fermion couplings were considered universal.
    Limits on torsion coupling to a scalar field, when it is identified as a dark matter candidate, 
    were derived in <cit.>.
    
    In this paper, we use the latest ATLAS tt̅ production results to constrain torsion parameters 
    assuming representative values for the torsion-top coupling. Limits on torsion from this channel, using LHC 
    published data, are derived 
    here for the first time. The torsion discovery potential and exclusion at LHC Run II and III are also evaluated. 
    Torsion decaying into tt̅ at LHC was first investigated in <cit.>, 
    but the subsequent top decay and the final state reconstruction were not taken into account. 
    In the present study, we go a step further in understanding the actual collider 
    signature by considering measurable final states.
    
    This article is organized as follows. In section <ref> a very brief review of torsion model is given, 
    highlighting the features that are most relevant to the current analysis. Section <ref> describes the Monte Carlo 
    and detector simulation procedures. In section <ref>, experimental bounds on torsion mass and couplings are derived 
    from ATLAS published 13 TeV LHC collision data. 
    
    Section <ref> presents a fast detector simulation performed to obtain
    the LHC potential to observe torsion resonances decaying into tt̅. Prospects for torsion exclusion at next LHC 
    upgrades are also presented. Conclusions are drawn in Section <ref>.
    
    
    
    
    
    
    § THE TORSION INTERACTIONS
    
    
    The interaction between a Dirac field and torsion, assuming that the metric is flat, is described by the 
    following action <cit.> S_non-min^TS-matter =  i ∫ d^4x  ψ̅_(i) ( ^μ ∂_μ  
    + iη_i^μ^5 S_μ
    - im_i  ) ψ_(i) ,
    
    where ψ_(i) stands for each of the SM fermions, S_μ is a axial vector field and η_i 
    is the non-minimal interaction parameter for the corresponding spinor. 
    The spinor-torsion interaction enter the SM as interactions of fermions with the new axial vector S_μ, 
    characterized by new dimensionless parameter, the coupling constants η_i. 
    
    Unitarity and renormalization conditions in the effective low-energy quantum theory lead to the  torsion action of the form 
    
     S^TS-Free =  ∫ d^4x
    (-1/4S_μνS^μν + 1/4M_TS^2S_μS^μ) .
    
    where M_TS is the torsion mass and S_μν = ∂_μS_ν - ∂_νS_μ. 
    To preserve unitarity, it has been shown that the following relation must be satisfied for each fermion 
    of mass m_i<cit.>M_TS^2/η_i≫ m_i^2
    
    
    The torsion-fermion interactions are not necessary universal since the values of the couplings η_i may not be the same 
    for all fermions. The difference comes from the renormalization group equations for each η_i that depend on the 
    Yukawa coupling for the corresponding fermion. From simplified assumptions <cit.> it is possible to conclude that, at TeV scale, 
    the values of all η_i must be the same, except for the top quark. The torsion-top  coupling, denoted hereafter by η_t, 
    may be different because of the potentially stronger running between the Planck and TeV scales.  
    Hence, the free parameters of the theory include M_TS, 
    the torsion-top coupling η_t and the coupling between torsion and all other SM fermions, denoted by η_f. 
    In order to be as general as possible, using these assumptions, we explore in our analyses the parameter space regions where  
    η_t <  η_f,  η_t =  η_f and η_t >  η_f. 
    
    
    
    § MONTE CARLO AND DETECTOR SIMULATION
    
    The torsion effective model was implemented in calchep event generator <cit.> according to Eq. (<ref>). 
    The implementation was validated and tested for consistency and unitarity.
    calchep is used to calculate cross sections and to generate events in which a torsion resonance is produced in 
    proton-proton collisions and decays into a pair of top quarks. The simulation is done using the MSTW2008nlo 
    parton distribution function <cit.> and center-of-mass energy of 13 TeV. The generated events are processed by 
    pythia 8<cit.> for hadronization and decays. 
    pythia 8 is also used to generate SM tt̅ background events.
    For this study, the semileptonic decay of the top quark is selected. 
    A fast detector simulation is performed using delphes<cit.> with ATLAS detector configuration,
    but pile-up is not taking into account.
    The torsion masses were taken in the range 500 GeV to 4500 GeV. 
    The coupling η_f varied from 0.1 to 0.5 in steps of 0.1. 
    Three representative values η_t was chosen: 0.1, 0.5 and 1.0. 
    These coupling values were selected to produce significant variation 
    on σ(pp → TS → tt̅) without violating 
    the constraint imposed by Eq. (<ref>). 
    
    A torsion signal can be observed at the LHC as an increase in the number of tt̅ events 
    produced via s-channel torsion exchange. The Feynman diagram of the process is shown in Fig. <ref>.
    The LO cross-sections at 13 TeV, as function of torsion mass and η_f, are shown in 
    Figures <ref> and <ref> for η_t = 0.1 and η_t = 1, respectively.
    In the analysis that will follow, a K-factor of 1.3 is applied to the LO cross-sections to account for NLO effects. 
    
    
    
    As we can see in Fig.<ref>, for η_t = 0.1, 
    σ(pp → TS → tt̅) does not change significantly 
    with η_f. In this case, the torsion production increases with η_f, 
    but it is compensated by decrease of 
    Br (TS → tt̅). Changes in the cross-section 
    as a function of η_f are only important when η_t  > η_f, as illustrated in Fig. <ref>.
    For this reason, different values of η_f are considered in the next sections only 
    when η_t = 0.5 and η_t = 1.0.
    
    
    
    § OBSERVED EXCLUSION LIMITS
    
    
    The ATLAS experiment has searched for heavy particles decaying into tt̅ at center-of-mass energy of 13 TeV 
    with a data sample corresponding to an integrated luminosity of 3.2 fb^-1<cit.>. 
    The analysis selected events where the top and the 
    anti-top quarks decay to W boson and bottom quarks (t → W^+b, t̅→ W^-b̅ ), 
    one of the W's 
    decays to leptons and the other decays to quarks, 
    forming the lepton-plus-jets topology. The number of selected events from data and from different SM processes 
    estimated by the experiment, in the electron-plus jet channel (e + jets), is listed in Table <ref>. 
    As we can see, there is a deficit of events in data compared to total expected background, 
    but it is still compatible with the prediction within the uncertainty. 
    
    
    
    The results from Table <ref> and the theoretical cross-sections calculated in Sec. <ref> are used to set 
    limits on torsion mass and couplings. The ATLAS acceptance times efficiency in the e+jets channel is also used to 
    calculate the number of torsion signal events. Upper limits on the signal cross-section are obtained by applying a Bayesian technique implemented in the 
    mclimit program <cit.>. This approach assumes that the signal adds incoherently to the background. 
    The inputs for the calculations are the number of events observed in data, the expected number of torsion  
    events and the expected number of background events. 
    The limit on the cross-section is translated in the lower limit on the torsion mass for each combination of η_f and η_t. 
    Figures <ref> and <ref> show the 95% C.L. exclusion regions 
    on the M_TS×η_f plane for η_t = 0.5 and η_t = 1.0, respectively.
    The large difference between the observed and expected limits is due to the deficit of data events mentioned above. 
    
    
    
    In a scenario where torsion is strongly coupled to the top quark but the interaction with other fermions 
    is weak (η_f = 0.1), the current data excludes torsion with a mass between 1700 and 1800 GeV. 
    For the highest coupling values considered in this paper, the lower limit on torsion mass is pushed to ∼ 2700 GeV. 
    For η_t = 0.1, the observed limit is M_TS < 1200 GeV. The observed limits for η_t = 0.5 and η_t = 1.0 
    are summarized in Table <ref>.
    
    
    
    § DISCOVERY POTENTIAL AND LIMITS AT RUNS II AND III
    
    The aim of this section is to perform high mass tt̅ resonance reconstruction and investigate 
    the LHC potential to discover torsion at 13 TeV. A fast detector simulation using delphes is performed to determine the 
    efficiency for reconstructing the decaying tops from torsion and from SM processes. Jets are reconstructed using the anti-k_t 
    algorithm. The missing transverse momentum E^miss_T is calculated from calorimeter cell energies.
    
    The tt̅ system is reconstructed from the hadronic top (t → Wb → b qq̅) and from the semi-leptonic top 
    (t → Wb → beν_e). The events are required to have exactly one electron with p_T > 30 GeV 
    and pseudo-rapidity | η|< 2.5[η, without subscription, stands for pseudo-rapidity.].
    Their missing transverse momenta are required to be greater than 30 GeV. The three jets from the hadronic top quark decay can be so collimated 
    in the detector that they cannot be distinguished from each other and therefore are reconstructed as a single jet or two jets. 
    To select the high boosted top candidates, the following criteria are applied for events with at least two jets: 
    
    
      *  the leptonic-top jet is selected by requiring Δ R(jet_ℓ, e) < 1.5, where Δ R = √(Δη^2 + Δϕ^2) and jet_ℓ is the jet identified with the expected b-jet from the 
    	leptonic top quark. If more than one jet satisfies this condition, the one with highest p_T is chosen;
    	
      *  the hadronic-top jet candidate jet_h must be well separated from the lepton by an azimuthal angle distance of Δϕ (jet_h, e) > 2.3 rad. Additionally, the monojet jet_h must have p_T > 400 GeV and Δϕ (jet_h, jet_ℓ) > 1.5 rad.
    
     
    For the events that do not pass the above selection, three jets are required with one of them having a mass above 70 GeV. It is
    assumed that the highest mass jet contains the two merged jets from the W decay or one of W jets merged with the b-jet.
    The signal efficiency after these selections depends on M_TS and η_f, and it reaches the maximum value of ∼ 2%. This efficiency already includes the top and W decay branching ratios.
      
    The invariant mass m_tt̅ is reconstructed by adding the four-momentum of the selected semi-leptonic and hadronic top quarks. For the semi-leptonic 
    top quark, the longitudinal component of the neutrino momentum is calculated from E^miss_T and imposing an on-shell W mass constraint. 
    When two jets are selected, the monojet jet_h is assumed to be the reconstructed hadronic top quark.
    For the events where three
    jets are selected, the following χ^2 is used to determine which jet must be assigned to the semi-leptonic or hadronic tops <cit.>:  
    
    
        χ^2 = [ m_jj - m_t_h/σ_t_h]^2 + [ m_jℓν - m_t_ℓ/σ_t_ℓ]^2
    
    where m_t_hm_t_ℓ are the expected mean masses of hadronic and leptonic top quarks, and σ_t_h and σ_t_ℓ are their 
    respective standard deviations. These four parameters are determined from Monte Carlo simulation studies from <cit.>. 
    All jet combinations are tested, and the one with the lowest χ^2 value is chosen to calculate m_tt̅. Figure <ref> shows the 
    reconstructed m_tt̅ from the selected events for three torsion mass hypotheses and particular values of torsion-fermions couplings. 
    The broad distribution observed for M_TS = 3.0 TeV is due to the large resonance width.   
    
    
    
    The main background in this analysis is the SM tt̅ production. 
    The expected number of background events is estimated using the measured tt̅ cross-section at 13 TeV <cit.> and 
    the background event selection efficiency determined from simulation. 
    Other backgrounds include production of W and Z bosons associated with jets, single top production, multi-jet 
    and di-boson productions. 
    We have estimated the number of reconstructed events from these various backgrounds as 20% of the SM tt̅ reconstructed events. 
    Hence, the total number of background events N_b is
    
    
        N_b = 1.2 ×σ_tt̅^SM×ϵ_b ×ℒ
    
    where σ_tt̅^SM is the SM tt̅ cross-section, ϵ_b is the background selection efficiency and ℒ is the integrated luminosity. 
    
    In order to determine the LHC experimental sensitivity to probe torsion, the invariant mass m_tt̅ 
    is used as signal/background discriminant variable. The background is considerably suppressed by applying the cut m_tt̅ > 900 GeV.
    This selection criteria keeps more than 90% of the signal events for M_TS≥ 1500 GeV. The background invariant mass distribution, above 900 GeV, is modeled as an exponential using a large simulated MC sample, and a numerical PDF is used to model the signal shape. For a given integrated luminosity ℒ, a likelihood fit is performed to the signal-plus-background  invariant mass distribution in the m_tt̅ range of [900,5000] GeV.
    The number of signal and background events are the fitted parameters.
    Asymptotic formulae for likelihood-based tests <cit.> is used to calculate the P-value, the probability of a 
    background fluctuation being greater than or equal  to the excess observed in the simulated data.
    The value of ℒ is increased and the fitting procedure is repeated until P-value < 3 × 10^-7, 
    the probability required to claim a discovery.  This test is performed for various torsion mass and coupling hypotheses, 
    and for each one of them the minimal integrated luminosity needed to discover torsion is obtained. 
    The results are shown in Fig. <ref> for η_t = 0.5 and η_t = 1.0.
    
    
    
    From Fig. <ref> we can estimate that by the end of Run II, 
    torsion with mass ∼ 1500 GeV can be observed at LHC if η_t = 0.5 and η_f= 0.1. 
    The discovery reach of Run II (Run III) is ∼ 2500 GeV (∼ 3000 GeV) if 
    torsion is strongly coupled to quarks (η_t = 1.0, η_f = 0.5). 
    In the high-luminosity LHC scenario (ℒ = 3000 fb^-1), 
    torsion mass up to ∼ 4000 GeV can be probed. For η_t = 0.1, the maximum discovery reach at LHC is 
    M_TS∼ 1700 GeV.
    
    The invariant mass distribution is also used to calculate torsion expected limits at Runs II and III. 
    Expected upper limits on σ× Br(TS ⟶ tt̅) are calculated using simulated pseudo-experiments assuming that only 
    SM processes are present. 
    A Bayesian approach is used <cit.>, with a flat prior probability distribution for 
    σ Br. The most probable number of signal events, 
    and the corresponding confidence intervals, are determined from a likelihood function defined as the product of the 
    Poisson probabilities over all m_tt̅ mass bins in the search region, using the appropriate signal invariant mass distribution. 
    The limit on the number of events is converted into a limit 
    on σ Br. 95 % CL upper limit on σ Br for each pseudo-experiment is obtained, 
    and the median value is chosen to represent the expected limit. This calculation is performed for each combination of the 
    parameters (M_TS, η_f, η_t). The 68% and 95% error bands are also calculated. 
    By comparing the limits on σ Br with the torsion theoretical cross-sections, the lower limits on torsion 
    mass are determined. 
    The procedure result is illustrated in Fig. <ref> for η_t = 0.1 and ℒ = 300 fb^-1. 
    In this case, the lower bound M_TS < 1.52 TeV can be achieved. 
    
    
    
    Figures <ref> and <ref> show the expected exclusion region on M_TS×η_f plane for η_t = 0.5 
    and η_t = 1.0 considering ℒ = 100 fb^-1 and ℒ = 300 fb^-1, respectively. With 100 fb^-1 of data, 
    the torsion mass limits range from 1.96 TeV 
    to 3.10 TeV. With ℒ = 300 fb^-1, the limits can be extend to ∼ 3.5 TeV. 
    
    
    
    In Table <ref> we summarize the results obtained for η_t = 1.0 for the different integrated luminosities considered. 
    The second table column shows the limits obtained in the di-lepton channel by ATLAS Experiment at 7 TeV <cit.>. Since di-lepton is a cleaner signature
    with better efficiencies, the torsion limits obtained from tt̅ with the current data are still less restrictive. The tt̅ channel, 
    however, provides a unique way to test the torsion-top coupling and the torsion non-universality interactions.  
     
    
    
    § CONCLUSIONS
    
    
    Exclusion limits on torsion mass and couplings based on ATLAS tt̅ results from LHC 13 TeV data are derived. 
    Considering non-universal torsion-quarks couplings, torsion with masses between 1.2 TeV to 2.7 TeV 
    are excluded at 95% CL. The LHC potential to observe torsion decaying 
    into t t̅ is also investigated. Taking into account the reconstruction and selection efficiencies of 
    the decaying top quarks, 
    it is found that torsion with mass up to ∼ 3.0 TeV can be observed by the end of Run III. 
    The results also show that a torsion mass of 4.0 TeV 
    set the maximum M_TS value that can be probed at LHC from t t̅ production in the electron-plus-jets selection. 
    New data from Run II and Run III can extend the current 
    torsion mass limits to ∼ 3.5 TeV.  
    The results presented in this paper are complementary to torsion searches in di-lepton channel 
    and provide information on torsion-top coupling. 
    
    
    
    
    
    
    This work is supported by the National Council of Research (CNPq) under grants 308494/2015-6 and 402846/2016-8,
    and by Fundação de Amparo a Pesquisa do Estado de Minas Gerais (FAPEMIG), APQ 03179-15
