Looking for bSM physics using top-quark polarization and decay-lepton kinematic asymmetries
LLooking for bSM physics using top-quark polarization anddecay-lepton kinematic asymmetries
Rohini M. Godbole, Gaurav Mendiratta, and Saurabh Rindani Centre for High Energy Physics,Indian Institute of Science, Bangalore 560 012, India Theoretical Physics Division, Physical Research Laboratory,Navrangpura, Ahmedabad 380 009, India (Dated: October 9, 2015)
Abstract
We explore beyond Standard Model (bSM) physics signatures in the l + jets channel of tt pair production process at the Tevatron and the LHC. We study the effects of bSM physicsscenarios on the top quark polarization and on the kinematics of the decay leptons. To thisend, we construct asymmetries using the lepton energy and angular distributions. Further, wefind their correlations with the top polarization, net charge asymmetry and top forward back-ward asymmetry. We show that when used together, these observables can help discriminateeffectively between SM and different bSM scenarios which can lead to varying degrees of toppolarization at the Tevatron as well as the LHC. We use two types of coloured mediator mod-els to demonstrate the effectiveness of proposed observables, an s -channel axigluon and a u -channel diquark. a r X i v : . [ h e p - ph ] O c t ONTENTS
I. Introduction 3II. Status of experimental results 4III. Flavour non-universal axigluon model 7A. Constraints on the axigluon model 8IV. U-channel scalar exchange model 10A. Constraints on the coloured scalar model 11V. Constraints from dijet production at LHC 12VI. Polarization of the top quark and decay-lepton distributions 14VII. Correlations 19A. Correlations between charge and forward-backward asymmetries 20B. Correlations among lepton and top asymmetries 211. Asymmetry correlations for the Tevatron 212. Asymmetry correlations for the LHC 23VIII. Asymmetry correlations and top transverse polarization 25IX. Conclusions 27Acknowledgments 29Appendices 29A. AC at TeV LHC 29B. tt Production density matrices 301. axigluon density matrices 302. diquark density matrices 31References 322 . INTRODUCTION
Most experimental observations at particle accelerators fit the Standard Model (SM)very well. However, there are some major puzzles to be solved. One needs to havephysics beyond the standard model (bSM) to explain the presence of dark matter, ex-plain quantitatively the observed baryon asymmetry of the universe, and to explain thepuzzle of dark energy. Looking for signs of bSM, one finds that most of the terrestrialexperimental observations that are in tension with the SM results are in the propertiesof third generation fermions. For example B → τν [1] , h → µτ [2] and bb forwardbackward asymmetry (AFB) at LEP and Tevatron [3–5] show such deviations. One ofthese long standing puzzles is the top-quark AFB measured by the D0 and CDF de-tectors at the Tevatron collider in 2008 [6, 7]. These observations by two independentcollaborations were updated with full data from the Tevatron and were consistent witheach other and in tension with the SM calculations until 2015. Recent experimentalresults from D [8] and theoretical calculations [9] point towards the possibility that theanomalous nature of these observations may be a statistical phenomenon.Due to its large mass which is close to the electroweak scale and the implied con-nection with electroweak symmetry breaking, the top quark is an important laboratoryfor various bSM searches at colliders. In fact various proposals put forward to solvethe different theoretical problems of the SM often involve modifications in the top sec-tor. Various extensions to the SM have also been proposed inspired by the possiblyanomalous value of measured top-AFB at the Tevatron. These bSM proposals involveexplanation of the AFB in terms of processes involving a) s -channel resonances likethe axigluon, KK gluon, coloron [10–18] or b) t-channel exchange of particles with dif-ferent spin and SM charges like the Z (cid:48) , diquarks etc [14, 16, 17, 19–25]. Effectiveoperator approach has also been used in this context [26–28]. Measurements of otherrelated observables such as lepton angular asymmetries and t ¯ t invariant-mass depen-dence of the top quark AFB are also compatible with the hypothesis of a heavy bSMparticle, see for example [29, 30].In this study, we will focus on the lepton + jets final state ( pp / pp → tt → b ν lt ) of the tt pair production process. This channel has a larger cross-section as compared to thedilepton + jets channel, and it has a much smaller background compared to the all jets3hannel. For lighter quarks, hadronization smears the information available about theirspin and polarization. The mass of the top quark is large enough that it decays intoits daughter particles before strong interactions can initiate the hadronization process.Hence top-quark polarization leaves a memory in the kinematic distribution of the de-cay products and can be tracked [31, 32]. We study the correlations between variouskinematic asymmetries and polarization to distinguish between different sources ofthese asymmetries within an s -channel (axigluon) and a t -channel (diquark) extensionof the SM. For the Tevatron, the top pair production process is dominated by qq colli-sions and at the LHC it is dominated by gg collisions which means that new physicscan manifest at differently at the two colliders.A wide variety of observables have been studied in the literature to explore thetop sector as a bSM portal [11, 16, 23, 33–37]. A brief review of some of these ob-servables which have been experimentally measured and are relevant to this workis presented in section II. In section III,IV we describe the flavour non-universal ax-igluon and diquark models which we use as templates for our analysis. Constraintson these models from top pair production cross-section and forward backward asym-metry at Tevatron, charge asymmetry, top quark pair production, dijet and four jet pro-duction cross-sections at LHC are discussed in section V. In section VI we constructthe asymmetries which we use to explore the bSM models. In section VII we presentthe correlations between various asymmetries and discuss the role of top quark po-larization and kinematics in discerning the various regions of parameter space of thebSM models. We contrast our results for the axigluon and diquark models and theresulting conclusions can be generalized to other new physics scenarios. Our resultsare presented for the Tevatron √ s = and the LHC √ s = TeV , TeV. Wediscuss the effects of transverse polarization coming from the off-diagonal terms in thetop-quark density matrix in section VIII and then conclude in section IX.
II. STATUS OF EXPERIMENTAL RESULTS
We begin by summarizing some of the experimental results from the LHC and theTevatron concerning the top quark and compare them with the corresponding SM cal-culations from the literature. 4he measured t ¯ t production cross-section for the Tevatron at √ s = TeV is σ Tevatronpp → tt = ± pb [38] and that for the LHC at √ s = TeV is σ LHCpp → tt = ± pb [39, 40]. These agree with the calculated SM NNLO cross-sections σ Tevatronpp → tt = + − pb [41] for the Tevatron, and σ LHC TeVpp → tt = ± pb [42] forthe LHC, within σ . The uncertainties coming from the top-quark mass dependence of tt cross-section [43] have been included in the given LHC cross-sections. In the cal-culations in following sections, we use a common K factor for the bSM+SM to estimatethe NNLO total cross-section. For the Tevatron, the K factor is K Tevatron = + − [44]. The K factor for the LHC is calculated using the NNLO cross-section cited aboveand LO cross-section calculated using CTEQ6l parton distribution functions (pdf) withfactorization scale Q = m t . The errors in the K factors represent pdf uncertainties,scale dependence and statistical errors in the NNLO cross-section. For the LHC with √ s = TeV, K LHC = + − .The cross-sections impose a constraint on any new particle to have small couplingswith the top quark and/or have a sufficiently high mass. It is interesting to note that wecan still find a range of couplings of the bSM large enough to explain the measuredanomalous top-quark and lepton asymmetries reported at the Tevatron and remaincompatible with the measurements at the LHC. The AFB of the top quark in the tt centre-of-mass (CM) frame is defined as A Forward Backward = N F ( y t − y ¯ t > ) − N B ( y t − y ¯ t < ) N F ( y t − y ¯ t > ) + N B ( y t − y ¯ t < ) (1) = N ( cos θ t > ) − N ( cos θ t < ) N ( cos θ t > ) + N ( cos θ t < ) , (2)where y t , y ¯ t are respectively the rapidities of the t and the ¯ t and θ t and θ ¯ t are theirrespective polar angles measured with respect to the beam direction.The CDF measurement of the tt CM frame AFB with the full data set is A t ¯ tFB = ± [45]. The corresponding SM result is 0 at tree level in QCD. At NLO inQCD, the value predicted is + − (the errors only represent scale variation)which upon including NLO electroweak corrections becomes, + − [46]. Re-cently, AFB has been calculated at NNLO to be + − in pure QCD and ± including EW corrections [46] and including effective N LO QCD, A tt , SMFB = ± [9]. D0 has come out recently with a measurement A FB = ± [8] which agree with the theoretical results. However for the purpose of this study,5e use the CDF measurement which is still in tension with the SM and with the D0measurement.Since the LHC is a pp collider, its symmetric initial state makes the forward andbackward regions trivially symmetric. For the LHC, instead of top quark AFB, a chargeasymmetry (AC) is defined in the lab frame as, A C = N ( ∆ | y t | > ) − N ( ∆ | y t | < ) N ( ∆ | y t | > ) + N ( ∆ | y t | < ) (3)where ∆ | y t | = | y t | − | y ¯ t | . The AC at the LHC is much smaller than the AFB at theTevatron both in the case of the SM and of the bSM models aimed at explaining theTevatron’s anomalous AFB. The measured value of AC with CMS and ATLAS combi-nation is A c = ± [47]. The theoretical results for the SM values of the AC[48] (QED+EW+NLO QCD) are given in table I for different energies at the LHC. √ s (TeV) A C Table I: Charge asymmetry in the lab frame at the LHC, as defined in eqn (3).Measurements have also been made for a number of other observables including M t ¯ t , rapidity-dependent top AFB [45], lepton and di-lepton asymmetries [49, 50], someof which show a deviation from the standard model [35] of up-to 1-3 σ . Some CDFresults are shown in table II and D0 results [50] in table III. tt spin correlations have been measured using decay particle double distributionsin polar and azimuthal angles at the Tevatron [51, 52] and the LHC [53, 54]. The po-larization of the top quark, as defined in eqn (12), has also been observed at CMS,for the LHC 7 TeV run to be ± [54] compared to the corresponding SM pre-diction from MC@NLO [55] ± . The ATLAS collaboration also observed thepolarization at 7 TeV beam energy, assuming CP conserving tt production and decayprocess, to be ± [56], in agreement with the SM prediction.6 symmetry Experimental Value SM calculation A lFB ( or A θ l ) + − ± A M t ¯ t > GeVtt _ FB ± ± ± A M t ¯ t < GeVtt _ FB ± ± ± A l + l − FB ± + − ± Table II: CDF lepton and M tt dependent top level asymmetries [35, 45, 49] Asymmetry Experimental Value SM calculation A lFB ( or A θ l ) (extrapolated) ± ± A lFB ( | y l | < ) + − Table III: D0 lepton asymmetries [50]
III. FLAVOUR NON-UNIVERSAL AXIGLUON MODEL
An axigluon is a massive, coloured ( SU ( ) c adj), vector boson. Models of axigluonwhich have only axial couplings with the quarks have been suggested in the literature inmany GUT like theories as chiral extensions of the QCD [57, 58]. Contribution to AFBfor such a particle was studied even before the possible anomalous AFB was observedat the Tevatron in 2008 [10]. For this flavour universal, axially interacting massivegluon with coupling g s , the top quark AFB becomes negative for masses above m A ∼ GeV. Upon the observation of a positive AFB by Tevatron in 2008, this modelwas found to be incompatible in the mass parameter regions allowed by the di-jetconstraints from Tevatron. The AFB turns back positive if the assumption of universalityof the interaction of axigluon with the quark families is dropped [12]. In our studyhere, we have used a more general, flavour non-universal axigluon with axial vector +vector couplings [13]. This model is obtained by breaking a larger symmetry group of SU ( ) A × SU ( ) B to the QCD colour group SU ( ) C and a SU ( ) C (cid:48) . The axial-vectorcoupling of the axigluon to the first and second generation quarks is negative of thatfor the third generation and the vector couplings are the same for all three generations.7he couplings of the axigluon with quarks are described by the Lagrangian L = ¯ ψγ µ T a ( g V + g A γ ) ψ A a µ , (4)where T a are the Gell-Mann matrices. The couplings are parametrized by g V = − g s tan ( θ A ) , g A = g s sin ( θ A ) , for the third generation of quarks. The parameters in thismodel are θ A and m A . We vary the value of the coupling in the range θ A ∈ [ π ] whichcorresponds to varying the axial and vector couplings from a large value at small θ A to g V = g A = g s for θ A = π . A mass range of m A ∈ [
1, 3 ] TeV is scanned.The decay width of axigluon and the density matrices for top-pair production me-diated by an axigluon are given in Appendix B 1. For an s-channel resonance, theterms in the tt pair production amplitude which are proportional to the linear power of cos θ (where θ is the top-quark polar angle) contribute towards the AFB. The helicitydependent analysis of the top-quark decay distributions can give additional informa-tion about the bSM couplings. We will show in this study that this information can beaccessed at the experiments from correlations among top polarization, top-quark anddecay-particle asymmetries.We first discuss constraints coming from t ¯ t production cross section measurements,and top-quark level forward-backward and charge asymmetries measured at the Teva-tron and the LHC (as appropriate). A. Constraints on the axigluon model
We calculate the differential cross-section of the process ( pp ) p ¯ p → t ¯ t → l ν b ¯ t at theTevatron with √ s = and at the LHC with √ s = and √ s =
13 TeV forthe SM + bSM with CTEQ6l [59] parton distribution functions with factorization scalefixed at Q = m t = GeV, the top quark mass is taken to be m t = and α s ( m t ) = .The cross-section calculated for the Tevatron, LHC and the AC and AFB of the tt at those experiments in the axigluon model are shown in figure 1. We constrainthe model parameter space by limiting the predicted observables σ pp → tt , σ pp → tt , A FB and A C to within σ of the experimental values. As the values of θ A and m A growlarger, the couplings reduce, the mass of the mediating particle rises and the bSMcontributions to the observables reduce. At large values of θ A , the figures correspond8 A ( GeV ) θ A σ ( pb ) (a) Cross-section at the Tevatron with √ s = m A ( GeV ) - - - - θ A A F B t (b) AFB at the Tevatron with √ s = m A ( GeV )
10 15 20 25 30 35 40 45150200250300 θ A σ ( pb ) (c) Cross-section at the LHC with √ s = m A ( GeV ) - - - - - θ A A C (d) A C at the LHC with √ s = Figure 1: Observables at the top quark level at the Tevatron and the LHC as afunction of θ A for various values of masses for the axigluon. The experimentallymeasured values are marked in grey and the respective 2 σ errors in dotted blacklines. As the lines go from solid to dashed with larger gaps, the mass of the axigluonrises from 1 TeV to 2.7 TeV. .to an axigluon model with only an axial coupling with the top quark and no resultingtop polarization. For a lower mass range, constraints from the LHC allow only larger θ A and hence smaller coupling values, at the same time, interference with SM givesa constraint at the Tevatron which allows some region in the large coupling range aswell. A C gives a complimentary constraint and rules out large values of θ A (couplingsclose to g s ) for a smaller mass of the axigluon. The result is that for the low massesof the axigluon, a range of couplings corresponding to θ A ∼ ( ◦ − ◦ ) and masses9 A ∼ ( − ) GeV are allowed. Masses above these values are allowed foralmost all parameter space with the only constraints coming from the Tevatron cross-section.CMS results constrain the mass of an additional massive spin-1 colour octet of par-ticles (eg. Kaluza Klein-gluon) which couple to gluons and quarks to above
TeV,which excludes the parameter region favoured by the experimental results from the tt process mentioned above [60]. The constraints can be evaded if the assumption ofequal couplings of axigluons to light quarks and the top quark is relaxed. In this case,the values of coupling g V , g A we use can be split into g qV , g qA and g tV , g tA where, thecouplings with quarks would be constrained strongly from the axigluon direct produc-tion bounds. In the limit that the vector and axial couplings are equal or any one ofthe vector or axial coupling is small, our results can be recast into the modified modelby using g v / a = g qv / a g tv / a . A more generalized version of such an axigluon model hasalready been discussed in the literature [61] along with constraints on the model fromlepton and top quark asymmetries at the Tevatron and the LHC.The axigluon model can be constrained from B physics [62] results however, giventhe somewhat large hadronic uncertainties in some of the variables along with thepossibility of relaxing these constraints in various modified axigluon models and/orby constructing UV completions, for the purpose of this study, we do not take theseconstraints into account. IV. U-CHANNEL SCALAR EXCHANGE MODEL
In a second class of bSM models, AFB is explained due to contributions of a t -or u -channel exchange of new particles between the top quark-antiquark pair. Thecorresponding mediators do not show resonance behaviour and are elusive in thebump-hunting type analyses in tt pair production though they do contribute significantlyto the angular distributions. We consider here a scalar particle called diquark, which,similar to a squark with R parity violation, transforms as a triplet under SU ( ) c and hasa charge of − . The corresponding coupling is given by the lagrangian below, L = t c T a ( y s + y p γ ) u φ a + h . c . (5) t c = − i γ t ∗ − i ( t γ γ ) T (6)We assume a right-handed coupling of the scalar with the up type quarks with y = y s = y p . This ensures that flavour constraints and proton stability bounds are avoided.The density matrices for top pair production in the diquark model are given in AppendixB 2. All calculations in this work are performed at tree level. The NLO contributionsbecome important to study the effects on invariant mass distributions which we havenot included here. These calculations are under progress for both axigluon and diquarkmodels. A. Constraints on the coloured scalar model
As in the case of an s -channel resonance in the previous section, the constraintsare obtained from the measurements of the top pair production cross-section at theTevatron and the LHC (7 TeV), the AFB and AC. We explore a parameter space of m φ ∈ [ ] GeV and y ∈ [
0, 2 π ] chosen so as to explore all the values of thecoupling within the perturbative limit. As the value of the coupling rises, contribution ofthe bSM to all the observables becomes larger. For lighter diquarks, negative valuesof AFB and AC are predicted for large values of the coupling, though this mass rangeis ruled out by independent constraints from diquark pair production [63]. In figure 2we can notice from the top left panel that, as in the case of the axigluon, the Teva-tron cross-section provides the constraints in the parameter space of lower massesand couplings. In the next panel, the AFB measured at the Tevatron disallows lighterscalars and also constraints a part of coupling values for larger masses. The LHCcross-section constraints large coupling regions which give larger contribution and thecut-off coupling increases as mass of the scalar becomes heavier. The AC also allowslarger coupling parameter space for higher masses of the scalar.The constraints from pair production of the coloured scalar from gluon fusion at theLHC are weak (~300 GeV) as reinterpreted from corresponding constraints on squarks[64]. There are further constraints on lower mass scalars from atomic parity violation[65]. Constraints from uu → tt can be avoided by adding flavour symmetries (see forexample [66]). 11 ϕ ( GeV )
100 550 12001700 26000 π π π π π π y p ( = y s ) σ ( pb ) (a) Cross-section at the Tevatron with √ s = m ϕ ( GeV )
100 550 12001700 26000 π π π π π π - y p ( = y s ) A F B t (b) AFB at the Tevatron with √ s = m ϕ ( GeV )
100 550 12001700 26000 π π π π π π y p ( = y s ) σ ( pb ) (c) Cross-section at the LHC with √ s = m ϕ ( GeV )
100 550 12001700 26000 π π π π π π - - y p ( = y s ) A c (d) A C at the LHC with √ s = Figure 2: Observables at the top-quark level at the Tevatron and the LHC (7 TeV) asa function of the Yukawa coupling for various values of the diquark masses. Theexperimentally measured values are marked in grey and the respective 2 σ errors indotted black lines. The line spacing changes from solid to a dashed line with widerspaces as mass values rise from 100-2600 GeV. . V. CONSTRAINTS FROM DIJET PRODUCTION AT LHC
The coloured scalar and vector bSM models get constrained from searches for di-rect production of bSM particles and subsequent decay to di-jet and four jet final states( q ¯ q → A → j , gg → φφ † → j ). Earlier constraints on axigluon model were obtainedfrom searches of narrow resonances from dijet spectrum at 8 TeV LHC and were ex-12
000 1500 2000 2500 30001015202530354045 m A θ A (a) Constraints on axigluon parameter space m ϕ y (b) Constraints on diquark parameter space Figure 3: Allowed parameter space for axigluon and diquark models are depicted asGreen (lighter) coloured regions. Figure 3a shows the constraints from dijet searchesas the blue (darker) shaded area. In figure 3b the dotted line represents the boundfrom pair production of bSM particles at LHC.tended to the case of a wider width axigluon model with Γ A m A = where the axigluonhas only axial-vector couplings [67] (also see [68] for bSM particle off-shell effects indijet searches). We reinterpret these constraints to the case of the axigluon model inthis study which has both axial-vector and vector couplings with the quarks and findthe excluded parameter range where Γ A m A < . Figure 3a shows the parameter spaceallowed for the axigluon model. The following constraints are put on the model param-eters to obtain the allowed values : reinterpreted searches for bSM resonances in dijetproduction, t ¯ t cross-section and top charge asymmetry measurements at 7 TeV LHCand cross-section, top forward backward asymmetry measurements at the Tevatron asdiscussed in section III A. The coupling values corresponding to θ A > ◦ are ruled outfor axigluon masses up to ∼ TeVs as these narrow, resonant particles would havebeen detected in the dijet searches. The allowed values of couplings correspond to θ A ∼ ◦ − ◦ for the mass range between 1.5 TeV to 3 TeV. Note that the constraintsfrom dijet searches may be relaxed if the magnitude of coupling of axigluon is differentfor the third generation of quarks as compared to the first and the second generations.13he diquark mass is bound from below to m φ > ∼ GeV from pair production ofdiquarks via gluon fusion at the LHC [64]. The direct production bounds along withthe constraints obtained from top quark pair production cross-section and top charge,forward backward asymmetry measurements at LHC and Tevatron (see section IV)are shown in figure 3b. A narrow strip of parameter space is allowed when couplingsare large due to destructive interference effects. Besides this region, the rest of theallowed diquark parameter space follows the expected behaviour of small couplingvalues < for lower masses and for a diquark of mass 3 TeV, y s as large as 2 isallowed. VI. POLARIZATION OF THE TOP QUARK AND DECAY-LEPTON DISTRIBUTIONS
The decay kinematics of leptons embeds the information regarding top quark pro-duction dynamics, kinematics and polarization [20]. Different lepton observables em-bed these effects in different ways and so provide a number of probes which are allcorrelated with the top quark kinematics and polarization. For a detailed analysis of topquark decay see [31, 32]. In this section we discuss distributions of the lepton polar an-gle, azimuthal angle and energy in SM decay of top quark and construct asymmetriesbased on these distributions to probe top quark bSM interactions.A proper treatment of the decay distributions of the top quark requires the spindensity matrix formulation, which preserves correlations between the spin states inthe production and in the decay.The spin density matrix for t in the production of a tt pair with the spin of t summedover, can be expressed as ρ tt production (cid:0) λ t , λ (cid:48) t (cid:1) = ∑ λ ¯ t M production ( λ t , λ ¯ t ) M (cid:63) production (cid:0) λ (cid:48) t , λ ¯ t (cid:1) (7)The density matrix gets SM contributions ρ ggSM ( λ t , λ (cid:48) t ) and ρ q ¯ qSM ( λ t , λ (cid:48) t ) respectivelyfrom gluon-gluon and quark-anti-quark initial states, a contribution ρ bSM ( λ t , λ (cid:48) t ) fromthe bSM model, and a contribution ρ inter f erence ( λ t , λ (cid:48) t ) from the interference betweenthe SM amplitude and the bSM amplitude: ρ (cid:0) λ t , λ (cid:48) t (cid:1) = ρ ggSM (cid:0) λ t , λ (cid:48) t (cid:1) + ρ qqSM (cid:0) λ t , λ (cid:48) t (cid:1) + ρ bSM (cid:0) λ t , λ (cid:48) t (cid:1) + ρ Inter f erence (cid:0) λ t , λ (cid:48) t (cid:1) (8)14he spin density matrix for the decay of the top quark is given by Γ top decay (cid:0) λ t , λ (cid:48) t (cid:1) = M decay ( λ t ) M (cid:63) decay (cid:0) λ (cid:48) t (cid:1) , (9)with the spins of the decay products summed over.The squared amplitude for the combined process of production and decay is givenby |M| = πδ (cid:0) p t − m t (cid:1) Γ t m t ∑ λ t , λ (cid:48) t ρ (cid:0) λ t , λ (cid:48) t (cid:1) Γ (cid:0) λ t , λ (cid:48) t (cid:1) (10)This expression assumes a narrow-width approximation for the top quark. Top decayis assumed to progress through SM processes. In the rest frame of top quark, thedifferential decay distribution of the top quark is given by d Γ t Γ d cos ( θ ) = + A p k f cos θ (11)where θ is the angle between top-quark spin direction and the momentum of the decayproduct f . For N ( λ t ) number of top quarks with helicity λ t , polarization A p is definedas, A P = N ( λ t = +) − N ( λ t = − ) N ( λ t = +) + N ( λ t = − ) , (12)and the coefficient k f is called the top-spin analysing power of the decay particle f .For the case of leptons as the final state particles, the factor k f = at tree level in theSM. When the top quark is boosted in the direction of its spin quantization axis, eqn(11) gets modified tod Γ Boosted Γ d cos ( θ tl ) = (cid:0) − β (cid:1) ( + λ t cos θ tl − β ( cos θ tl + λ t )) ( − β cos θ tl ) , (13)where θ tl is defined as the angle between lepton and top quark momenta in theboosted frame. Lepton kinematic distributions for the tree-level SM differential cross-section for the process pp → tt → l + jets with √ s = TeV are presented in thefigure 4. The energy and azimuthal lepton distributions are uncorrelated with the po-larization in the rest frame of the top quark, though correlate with the polarization inthe boosted frame. Higher-order corrections to the production and decay processeshave been calculated for the SM and the distributions are found to be qualitativelyunchanged [69, 70]. Due to this reason, we expect that the effect of higher-order15 a) Lepton polar angle distribution in top quark restframe. (b) Lepton polar angle distribution in the lab frame.(c) Lepton Azimuthal angle distribution in the labframe (d) Lepton energy distribution in the lab frame
Figure 4: The tree-level lepton polar and azimuthal distributions for pp → tt → l + jets with √ s = . In the above plots, the average boost of the t ¯ t pairs is 0.34.corrections to the asymmetries constructed from decay-lepton distributions to be rela-tively small as the corrections partially cancel out within the difference and ratio takento derive the asymmetry.A lepton polar-angle asymmetry with respect to the top-quark direction can be de-fined by A tlFB = σ ( cos ( θ tl ) > ) − σ ( cos ( θ tl ) < ) σ ( cos ( θ tl ) > ) + σ ( cos ( θ tl ) < ) (14) A tlFB has been measured at both LHC and Tevatron in the lab and t ¯ t center of momen-16um frame, albeit with large statistical errors and different results from CDF and D0[49]. Integrating eqn (11) the top rest-frame lepton asymmetry can be related to thepolarization of the top quark, A P = A tl , t − restFB . (15)In QCD, A tl , restFB = A p = , though in the boosted frame, the lepton polar asymmetrywith respect to the top quark is large even in a tree-level SM calculation.In the lab frame where the top quarks and leptons are boosted and the cross-section convoluted with the pdf, the correlations between various angles and energiesbecome more complicated. The lepton polar angle with respect to the proton beamis a convenient observable which does not require top-quark rest frame or momentareconstruction. The lepton polar asymmetry A lFB in the lab frame is also 0 at treelevel in SM QCD. A lFB is identically 0 at the LHC due to the symmetric nature of theinitial state. This asymmetry according to our analysis correlates the best with the off-diagonal elements of the top quark density matrix for the l + jets process considered(see Section VIII) for both axigluon and diquark models. An analytic study of the leptonpolar angle and its correlation with top AFB and polarization has also been made byBerger et al. [71]. They relate the lepton and the top quark level polar asymmetriesand subsequently use this relation to distinguish between a sequential axigluon and aW’ type model [72].In the top-quark rest frame, other lepton kinematic variables : azimuthal angle andits energy have no dependence on the helicity of the top quark and hence the inte-grated asymmetries are uncorrelated with the polarization. It has been noted in theliterature that the lepton azimuthal distributions correlate with the polarization of thetop quark in a boosted frame [32, 73–75]. Sums and differences of azimuthal decayangle in top pair production process have also been used in the literature to study thepolarization and spin correlations of top quark in detail [76]. For a detailed analysis ofanalytic relation between polarization of a heavy particle and decay particle azimuthalasymmetry, see [77]. We reproduce the azimuthal distribution in the lab frame for theSM tt pair production process at the Tevatron in figure 4c. The azimuthal distributioncan be measured at both Tevatron and LHC and requires only partial reconstruction oftop quark rest frame. The azimuthal angle is defined by assuming that the top quark17ies in the x-z plane with proton (beam) direction as z-axis. From this distribution, anazimuthal asymmetry about a point φ can be defined as A l φ = σ ( π > φ l > φ ) − σ ( φ l < φ ) σ ( π > φ l > φ ) + σ ( φ l < φ ) (16)A natural choice for the value of φ would be the point of intersection of the distributionscorresponding to left and right helicity top quarks. For SM, this point is about φ = ◦ for both Tevatron and the LHC. The SM point would correspond to 0 polarization andwould maximize correlation with bSM contribution. We assume a value of φ a bit lowerat ◦ . Since the positive helicity top quark have larger differential cross-section inthis region, this choice enhances correlations of the lepton level asymmetry for largerpositive (or smaller negative) values of polarization. The standard model tree-levelresults for this asymmetry at the Tevatron and the LHC respectively are given in tableIV . In the lab frame, due to the boost and rotation from the direction of the top quark, A φ is sensitive to both the polarization and the parity breaking or t-channel structureof the top quark coupling. Another observable which can be constructed from thedecay-lepton kinematics is the lepton energy asymmetry about a chosen energy E : A lE l = σ ( E l > E ) − σ ( E l < E ) σ ( E l > E ) + σ ( E l < E ) (17)No reconstruction of the top-quark rest frame is needed to measure E l . Just like theazimuthal case, this asymmetry can be measured both at the LHC and the Tevatron.The lepton energy distribution is sensitive to the polarization of top quark [32], asshown in figure 4d. Similar asymmetries based on the energy of decay particles or theratios of these energies have been used in the literature to study bSM physics [74, 78–80]. We define the lepton energy asymmetry about a value of E = GeV, to actas a better discriminator between bSM and SM. Ideally, the point of intersection of thepositive and the negative top-polarization curves should form the best correlation withthe top polarization, though this point varies with the energy and the invariant mass ofthe initial state. Standard model values of asymmetries mentioned in this section aregiven in table IV. It would be interesting to use SM distributions at NLO to decide thereference points E , φ , but since in the end we construct asymmetries, we expect thatthe qualitative behaviour of our results would not change.In the recent past, polarization measurements have been made by collaborationsboth at the Tevatron and the LHC. The polarization at the Tevatron points towards a18 symmetry Q = m t Q = m t A tlFB A l φ − − A lE l (a) Tevatron √ s = Asymmetry Q = m t Q = m t A tlFB A l φ − − A lE l (b) LHC √ s = Asymmetry Q = m t Q = m t A tlFB A l φ − − A lE l (c) LHC √ s =
13 TeV
Table IV: Scale dependence of SM values of various asymmetries tree level.small positive value and that at the LHC to small negative values. This is consistentwith the small coupling and large mass regimes of both the models studied here.In the next section, we use correlations among top charge and forward-backwardasymmetries, decay lepton angular and energy asymmetries, and polarization to un-cover specific properties of bSM particles which can be inferred from the Tevatron andthe LHC data.
VII. CORRELATIONS
The parameter space of m A ∈ [ ] GeV and θ A ∈ [
10, 45 ] are explored forthe axigluon model and m φ ∈ [ ] GeV and y s ∈ [
0, 2 π ] for the coloured scalar.19he figures in the section VII B show parameter space allowed by the constraints men-tioned in sections III, IV. A. Correlations between charge and forward-backward asymmetries
The correlation between the AC at 7 TeV LHC and the AFB at the Tevatron havebeen used in the literature constrain various bSM models(see for example [14]). Theseconstraints are model dependent and the asymmetries are not in general tightly cor-related [81]. We show similar correlations in figure 5 where we plot A C vs A FB , usingthe relation A C / FB = A SM _ NLOC / FB + A bSMC / FB (18)This relation is valid as long as the bSM physics corrects the SM cross-section of the tt pair production process by a small amount. ++ ++ ++ ++++ ++ ++++++++++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++++++ ++ ++ ++ ++++++ ++ ++++++ ++ ++ ++ ++++++ ++++ ++ ++++ ++ ++ ++ ++ ++++++++ ++ ++ ++ ++++ ++ ++ ++ ++ ++ ++ ++ ++++ ++ ++ ++ ++ ++++++ ++++ ++ ++++ ++ ++ ++ ++ ++++++ ++ ++ ++ ++++++++ ++ ++ ++ ++++ ++ ++ ++ ++ ++++ ++ ++ ++ ++++ ++ ++ ++++++ ++ ++ ++ ++++++ ++++ ++ ++ ++ ++ ++ ++ ++ ++ ++++++++++ ++ ++ ++ ++ ++ ++++++ ++ ++ ++++ ++ ++++++ ++ ++ ++ ++ ++ ++ ++ ++ ++++ ++++ ++ ++ ++ ++ ++ ++ ++ ++++ ++ ++ ++ ++ ++ ++++++++ ++++ ++ ++ ++ ++ ++ ++++++ ++ ++ ++ ++ ++ ++++++ ++ ++ ++ ++ ++ ++ ++ ++++ ++ ++ ++ ++++++ ++ ++ ++ ++ ++ ++++ ++ ++ ++++++ ++ ++ ++++ ++ ++++ ++ ++ ++++++++ ++ ++ ++ ++ ++ ++ ++ ++ ++++++++++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++++++++++++++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++++++ ++ ++++ ++ ++++++++ ++ ++++++++ ++ ++++++++++++++++++++ ++ ++ ++ ++++++ ++ ++++++++++++++++++++++++++++++++++++++++++ ++ ++ ++ ++++++++++++++ ++ ++ ++++++ ++ ++ ++++++++ ++++ ++++++++++++++++++ ++ ++ ++ ++++ ++ ++++ ++ ++ ++ ++++ ++ ++ ++++ ++ ++++ ++ ++ ++ ++ ++++++++ ++ ++ ++++ ++ ++ ++++ ++++++ ++ ++ ++ ++++++ ++++ ++ ++ ++ ++ ++ ++ ++ ++++ ++++ ++++ ++ ++ ++ ++++ ++++++ ++ ++ ++++ ++ ++ ++++ ++ ++ ++ ++++++++ ++ ++ ++ ++++ ++ ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ ++++++++++++++++++++++++++++++ ++++++++++++++++++++++ ++++++++++ ++++++ ++++ - - A t FB A C SM (a) The red markers represent axigluon model withonly axial interactions( g V = g tA = g s ). The sizeof the plus marks represent a mass range from1000-3000 GeV ++++++++++++++++++++++++++++++++++++ ++ ++++++++ ++ ++++ ++ ++++++++++++++ ++ ++++++++++++++++++++++++++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++++ ++ ++ ++ ++ ++ ++++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++++ ++ ++++++ ++ ++ ++ ++++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++++ ++ ++ ++ ++ ++ ++ ++ ++ ++++ ++ ++ ++ ++ ++ ++ ++ ++ ++ ++++++++++++++ - - A t FB A C SM (b) diquark model with right handed couplings tou,t quarks. The size of the plus marks represent amass range from 100-3000 GeV Figure 5: Correlation between top-quark asymmetries A tFB vs A tC at the Tevatron andthe LHC ( √ s = ). The grey solid and dashed lines represent the observedvalues of the respective asymmetries and their σ errors.The pure axial axigluon which leads to unpolarized top quark is disfavoured as itdoes not have a parameter space where it can explain both A FB and A C experimentalvalues. In the diquark model, the coupling to right handed quarks is sampled from 020 .05 0.10 0.15 0.20 0.25 0.30 0.35 - - A FB A θ l - A p + SM - Diquark - Axigluon (a) Lepton polar asymmetry Tevatron √ s = TeV. A FB A θ t l - A p + SM - Diquark - Axigluon (b) Lepton polar asymmetry Tevatron √ s = TeV.
Figure 6: Correlations between top AFB and lab frame θ l , θ tl asymmetries at theTevatron √ s = to π where large mass or small couplings lead to a better agreement with SM NLOvalues of asymmetries. B. Correlations among lepton and top asymmetries
In this section, we study the correlations among top polarization, top asymmetriesand decay lepton asymmetries. We show that combined, they form sensitive dis-criminators between models with different dynamics. The top-quark and decay-leptonasymmetries are calculated at various points in the parameter space allowed by theexperimental constraints discussed in section III and IV. The expected polarizationof the top quark, for corresponding points in the parameter space, is represented incolour contrast form inside the graphs and clear trends for the polarization can be ob-served. In all the following figures, top-quark asymmetries represented on the x -axisare calculated as shown in eqn (18) and the lepton asymmetries shown on the y -axisare calculated including SM+bSM contributions at tree level.
1. Asymmetry correlations for the Tevatron
The correlations of the lepton-level asymmetries with the top AFB at the Tevatronare shown in figures 6a, 6b, 7a and 7b. For the case of axigluon as its mass is in-21 .05 0.10 0.15 0.20 0.25 0.30 0.35 - - - A FB A ϕ l - A p + SM - Diquark - Axigluon (a) Lepton Azimuthal asymmetry about φ for theTevatron √ s = TeV. A FB A E l - A p + SM - Diquark - Axigluon (b) Lepton energy asymmetry about energy E forthe Tevatron √ s = TeV.
Figure 7: Correlations between top AFB and the lepton energy and azimuthalasymmetries at the Tevatron √ s = creased, the polarization rises until m A ∼ GeV and then drops again for evenlarger masses. The diquark model predicts negative polarization for a significant por-tion of parameter space, turning positive only for large couplings. The large massregion for the diquark also favours large (negative) values of azimuthal asymmetry,smaller lepton polar asymmetry and larger lepton energy asymmetry. Lepton polarasymmetry correlation with top AFB shows large overlap between the two models.The observed value for the lab-frame lepton polar asymmetry in tables II,III points to-wards a positive polarization between A θ l = to . In this region (see figure 6a), alarge positive value for polarization is favoured for the diquark model and a large con-tribution to top AFB. The axigluon model is compatible with both the observed valueof A θ l and a small contribution towards longitudinal polarization for a significant partof its parameter space. Figure 6b shows the asymmetry in the lepton polar angle withrespect the top direction, A θ tl , which is equal to twice the top polarization, eqn (15),when calculated in top-quark rest frame. It receives contribution from bSM physics viathe boost of the parent top quark. In the lab frame, large deviations of A θ tl from theSM value correlate with large contribution to the top-quark polarization from the bSM.The values of asymmetries grow closer to the corresponding SM values with increasein mass and reduction in the bSM coupling strength. The azimuthal asymmetry andlepton polar asymmetry with respect to the top-momentum direction ( A θ tl ) and lepton22 - - - - - - A C A ϕ l - - - A p + SM - Diquark - Axigluon (a) Lepton Azimuthal asymmetry about φ = degrees for the LHC √ s = TeV - A C A θ t l - - - A p + SM - Diquark - Axigluon (b) Lepton polar asymmetry(w.r.t top quarkdirection) for the LHC √ s = TeV - - A C A E l - - - A p + SM - Diquark - Axigluon (c) Lepton Energy asymmetry about E = GeVfor the LHC √ s = TeV
Figure 8: Correlations between lepton and top kinematic asymmetries at the LHC-7TeV.energy asymmetry in figures 6b, 7a and 7b discriminate well between the s -channeland u -channel exchange models though the parameter spaces within the model areclumped together. When combined with polarization, all correlations enhance their dis-criminating power especially to distinguish between s -channel and u -channel modelsas they predict opposite signs of polarization for a large portion of parameter space.
2. Asymmetry correlations for the LHC
The lepton-level asymmetry correlations with tt charge asymmetry are shown infigures 8a-8c for the LHC TeV run and figures 9a-9c for the LHC TeV. The plots23 - - - - - A C A ϕ l - - - A p + SM - Diquark - Axigluon (a) Lepton Azimuthal asymmetry about φ = degrees for the LHC √ s = TeV - - - A C A θ t l - - - A p + SM - Diquark - Axigluon (b) Lepton polar asymmetry(w.r.t top quarkdirection) for the LHC √ s = TeV - - - - A C A E l - - - A p + SM - Diquark - Axigluon (c) Lepton Energy asymmetry about E = GeVfor the LHC √ s = TeV
Figure 9: Correlations between lepton and top kinematic asymmetries at the LHC-13TeV.are made for the region of the model parameter space constrained in sections III andIV. For the √ s = TeV calculation, we use m t = GeV and factorization scale Q = m t and α s = to remain consistent with the ATLAS and CMS reconstructionof A C . For √ s = TeV, , the mass of top quark is chosen at the updated central value m t = GeV and Q = m t with α s = and CTEQ6l pdf.A large portion of the parameter space predicts a negative polarization at 7 TeV LHCfor the axigluon model. The diquark model predicts a small negative polarization forsmall couplings with quarks. When heavier diquark models are considered, larger cou-plings are allowed leading to a large positive contribution to the polarization. Observedvalues of polarization from CMS and ATLAS are compatible with − < A p < ,24hich covers a large region of parameter space for both axigluon and diquark models.As in the case of the Tevatron, polarization is an important discriminant between mod-els for the LHC as well, especially when combined with decay-lepton asymmetries. Itis able to distinguish overlapping parameter space regions between the two models.This is more true when the couplings are small and the bSM effects are more difficult todetect as the s -channel and u -channel exchanges predict small polarization, but withopposite signs in this region. The energy asymmetry becomes smaller for the LHC at13 TeV due to the effect of the overall boost. The values of azimuthal and polar asym-metries do not change significantly for higher energy and so remain good observablesfor the study of top quark dynamics. VIII. ASYMMETRY CORRELATIONS AND TOP TRANSVERSE POLARIZATION
As remarked earlier, keeping full spin correlations between the production and de-cay of the top quark in a coherent manner requires the spin density matrix formalism.In this formalism, the top polarization, which played a significant role in the aboveanalysis, corresponds to the difference in the diagonal elements of the density matrix,as seen from eqn (12). The off-diagonal elements of the density matrix can also besignificant in practice, and they would contribute to the transverse polarization of thetop quark, corresponding to a spin quantization axis transverse to the momentum. InSM, these terms arise at loop level and have been studied in the literature along-withtransverse polarization and observables have been suggested to measure their con-tribution [82]. We examine in this section what role these off-diagonal matrix elementsand transverse top polarization play in the two models considered in this study.Following the formalism developed in [32], the spin density matrix integrated over asuitable final-state phase space can be written as σ ( λ , λ (cid:48) ) = σ tot P t ( λ , λ (cid:48) ) , where σ tot represents the unpolarized cross section. The matrix P t ( λ , λ (cid:48) ) can be written as P t ( λ , λ (cid:48) ) = + η η − i η η + i η − η . (19)Here η is the longitudinal polarization, η and η are polarizations along two transversedirections. The expressions for the η i in terms of the top-quark density matrix σ (cid:16) λ t , λ (cid:48) t (cid:17) η = ( σ (++) − σ ( −− )) σ tot (20) η = ( σ (+ − ) + σ ( − +)) σ tot (21) i η = ( σ (+ − ) − σ ( − +)) σ tot (22)Splitting the top density matrix as shown in eqn (10) under the narrow-width approx-imation, the helicity-dependent decay density matrix in the rest frame of top quarkseparates into a simple functions of the decay angle: d Γ ( λ , λ (cid:48) ) = c × A ( λ , λ (cid:48) ) d Ω l (23)where A = λ ↓ , λ (cid:48) → + − + + cos ( θ l ) sin ( θ l ) e i φ l − sin ( θ l ) e − i φ l − cos ( θ l ) . (24) Ω l is the solid angle in which the lepton is emitted and c is the integrated contributionof the rest of the decay kinematic variables. The resulting lepton angular distributionin the lab frame is , d σ d cos ( θ l ) d φ l = c σ tot ( + η cos ( θ l ) + η sin ( θ l ) cos ( φ l ) + η sin ( θ l ) sin ( φ l )) (25)The off-diagonal elements in the top-quark production density matrix do not contributeto the total cross-section due to an overall factor of sin ( θ l ) which integrates to 0. Theydo contribute instead to the kinematic distributions of the decay particle, although thiseffect is quite small for most observables.In this study, we find that the lepton polar angle asymmetry defined in the lab frameis sensitive to the off-diagonal terms in the top quark density matrix eqn (7). Thetransverse polarization originating from these off-diagonal terms contains further infor-mation about the dynamics of top-quark interaction. This relation has been pointed outbefore in the context of a wide-width colour octet bSM particle [83, 84].In figure 10 we study the contribution of off-diagonal terms to the lepton distributionsand present the distributions for a few sample masses of bSM particles.It can be seen that the contribution of the off-diagonal density matrix elements canbe significant, and is particularly important for the diquark model. These can in turn26 A ( GeV )
10 15 20 25 30 35 40 45 - - θ a A F B l (a) axigluon m ϕ ( GeV ) π / π / π - - - y p = s A F B l (b) diquark Figure 10: Contribution to Lepton asymmetry from the off-diagonal terms of topdensity matrix calculated in the lab frame for the Tevatron √ s = TeV for theaxigluon and diquark models. The red lines represent the asymmetry for diagonaldensity matrix and the black line represents the distribution for the case of totaldensity matrix. The darker lines represent allowed regions of parameter space.lead to significant transverse polarization of the top for appropriate range of parameterswhich could be measured experimentally.
IX. CONCLUSIONS
The forward backward asymmetry of the top quark in top-pair production processat the Tevatron collider was, for a long time, anomalously large and a persistent effectobserved independently by both D0 and CDF detectors. It has been demonstratedonly relatively recently that NNLO contributions give rise to an A tFB of the right orderof magnitude and seems to be in agreement with the values measured experimentally.Previously, many bSM models had been proposed with parity breaking interactions toexplain the observed A tFB . Many of these models predicted a charge asymmetry atthe LHC. Since LHC has a gluon dominated initial state as opposed to the Tevatronwhere q ¯ q was the primary initial state, the asymmetries predicted for LHC coming fromthe bSM couplings to the quarks get diluted. The data gives values for A C consistentwith the SM and so far there has been no evidence for the new particles predictedin the different bSM models. Under these circumstances there is a need to construct27easures which can distinguish between different sources of the A tFB : either SM orbSM.One such measure is provided by polarization of the top quark which has a non-zero value in the presence of a parity breaking interaction. Within SM, top polarizationis close to 0. Observables that correlate with top polarization can be used to distin-guish between various SM and bSM contributions. In continuation to a previous workwhere correlations between polarization and forward backward asymmetry were usedto constraint bSM [14], we have introduced the correlations between lepton polar, az-imuthal and energy asymmetries and top charge asymmetry and showed how theycan be used together with top longitudinal polarization to distinguish between SM andbSM.In the reference [85] the authors have constructed dilepton central charge and az-imuthal asymmetries and studied it along with top quark polarization, forward backwardasymmetry and t ¯ t spin correlations for benchmark models of G’ and W’. Subsequently,in reference [71] the authors have shown that the lepton polar asymmetry and top for-ward backward asymmetry and lepton charge asymmetry vs top charge asymmetrycorrelations can be useful in the study of W’ and G’ models. Our work adds multiplenew observables to the analysis of new physics in t ¯ t pair production which includesingle-lepton azimuthal angle, energy and polar angle (wrt top quark) in the lab framewhich show signatures from parity breaking in top interactions and help isolate andconstrain the interactions of bSM particles.We demonstrate the efficacy of the correlations between forward backward asym-metry and the lepton asymmetries at Tevatron and charge asymmetry and lepton an-gluar,energy asymmetries at LHC, by utilizing a representative s-channel model, ax-igluon and an u-channel model, diquark. Constraints on these models are obtainedbased on measured values of t ¯ t cross-section at Tevatron and LHC 7TeV, A tFB and A tC and resonance searches in dijet, four jet cross sections. The parameter spaceof axigluon allowed within 2- σ of the measured values of the stated observables in-cludes a lower bound on the mass of axigluon at 1.5 TeV with a corresponding cou-pling θ A > ◦ . The allowed mass of the diquark is bounded from below by 300 GeVand masses above are allowed with the coupling of the model bounded from above bya value of 0.2 for smaller masses which rises to y s < corresponding to m φ = TeV.28nother sliver of parameter space is allowed for larger couplings of the diquark due todestructive interference effects.For the first time we have presented the complete density matrix of the top quark,including the off-diagonal elements for top quark pair production process, in the ax-igluon and diquark models to aid further studies. We use these to show that the leptonpolar asymmetry in the lab frame shows a correlation with the transverse polarizationof the top quark for axigluon model and even more significantly for the diquark model.The lepton asymmetry usually considered in studies of top polarization is calculatedfrom lepton polar angle with respect to the top quark and it does not show this correla-tion with transverse polarization. The correlation of transverse polarization with leptonazimuthal or energy asymmetry is also very small.Finally, we extend our analysis to 13 TeV LHC where, even though the values of theasymmetries get diluted, the correlations between accurate measurements of chargeasymmetry and lepton asymmetries still separate out bSM and the SM in the 2 di-mensional space. Taken together, these correlations can indeed be used to improvesignificance of the constraints on bSM from LHC data even in the initial stages of lowluminosity.
ACKNOWLEDGMENTS
The authors are pleased to acknowledge conversations with Ritesh K. Singh,Pratishruti Saha and Arunprasath V. RMG wishes to acknowledge support from theDepartment of Science and Technology, India, under Grant No. SR/S2/JCB-64/2007.SDR acknowledges support from the Department of Science and Technology, India,under the Grant No. SR/SB/JCB-42/2009. The research of GM was supported byCSIR, India via SPM Grant No. 07/079(0095)/2011-EMR-I.
Appendix A: AC at TeV LHC
Since the AC calculated at NLO for 13TeV LHC was unavailable at the time ofsubmission of this work, we note that the available charge asymmetry values [48] forma smooth function of the beam energy and fit them to a polynomial to find the AC as29 function of the beam energy. We obtain a fit to a polynomial presented in eqn (A1)with goodness of fit parameter r = . Ac (cid:0) √ s (cid:1) = × − − × − √ s + × − s − × − s (A1)This gives a value of Ac(13 TeV)=0.0063. Appendix B: tt Production density matrices1. axigluon density matrices
With C θ = cos ( θ t ) , S θ = sin ( θ t ) , β = (cid:113) − m t ˆ s and β A = (cid:114) − m t m A . ρ ++ bSM = Γ A m A + ( m A − ˆ s ) ˆ s { ( g A + g V )( g At g V β + g At β − g V (cid:16) − + β (cid:17) )+ g A g V (cid:0) g V + g t A β (cid:1) C θ + (cid:16) g A + g V (cid:17) β (cid:16) g t A g V + g t A β + g V β (cid:17) C θ } (B1) ρ + − bSM = Γ A m A + ( m A − ˆ s ) ˆ s (cid:32) − g A g V m t S θ √ ˆ s − g t A g V (cid:0) g A + g V (cid:1) m t β C θ S θ √ ˆ s (cid:33) (B2) ρ − + bSM = ( ρ + − bSM ) (cid:63) (B3) ρ −− bSM = Γ A m A + ( m A − ˆ s ) ˆ s { ( g A + g V )( − g t A g V β + g t A β − g V (cid:16) − + β (cid:17) ) − g A g V ( g V − g A β ) C θ + (cid:16) g A + g V (cid:17) β (cid:16) − g t A g V + g t A β + g V β (cid:17) C θ } (B4) ρ ++ Inter f erence = g s ( Γ A m A + ( m A − ˆ s ) ) ˆ s (cid:16) − m A + ˆ s (cid:17) × (cid:16) g A (cid:0) g V + g t A β (cid:1) C θ + g V (cid:16) g v + g t A β − g V β + β (cid:0) g t A + g V β (cid:1) C θ (cid:17)(cid:17) (B5) ρ + − Inter f erence = g s ( Γ A m A + ( m A − ˆ s ) ) m t √ ˆ sS θ × (cid:16) g A (cid:16) g v (cid:16) m A − ˆ s (cid:17) + ig At Γ A m A β (cid:17) + g t A g V (cid:16) m A − ˆ s (cid:17) β C θ (cid:17) (B6) ρ − + Inter f erence = ( ρ + − Inter f erence ) (cid:63) (B7) ρ −− Inter f erence = g s ( Γ A m A + ( m A − ˆ s ) ) (cid:16) m A − ˆ s (cid:17) ˆ s (cid:16) g A (cid:0) g V − g t A β (cid:1) C θ + g V (cid:16) g t A β + g V (cid:16) β − (cid:17) + β (cid:0) g t A − g V β (cid:1) C θ (cid:17)(cid:17) (B8)To present the dependence on top boost and polar angle clearly the amplitudesquare is written in terms of the polar angle θ in tt center of momentum frame. Theoff-diagonal terms in the gluon initiated process are zero and the diagonal terms ingluon initiated process are not dependent on the top quark polarization therefore wehave omitted these here and they can be found in many references including [14].Decay width of axigluon at tree level is given by, Γ A = π m A { g A (cid:16) m A ( β A + ) − m t β A (cid:17) + g V (cid:16) m A ( β A + ) + m t β A (cid:17) } (B9)
2. diquark density matrices
The top-quark spin density matrix for the u -channel exchange is given below in tt center of momentum frame. The notation and SM only contributions remain the sameas for the case of the s -channel model. ρ ++ bSM = s (cid:0) y P + y S (cid:1) (cid:16) β ˆ sC θ − m t + m φ + ˆ s (cid:17) ×{ ˆ s (cid:16) y P y S (cid:16) β + β C θ + C θ (cid:17) + ( y P + y S ) (cid:16) β C θ + C θ + (cid:17)(cid:17) − C θ m t (cid:16) C θ (cid:16) y P + y S (cid:17) + y P y S (cid:17) } (B10) ρ + − bSM = − ˆ s m t y P y S ( β C θ + ) (cid:0) y P + y S (cid:1) S θ (cid:16) β ˆ sC θ − m t + m φ + ˆ s (cid:17) (B11) ρ − + bSM = ( ρ + − bSM ) (cid:63) (B12) ρ −− bSM = ˆ s (cid:0) y P + y S (cid:1) (cid:16) β ˆ sC θ − m t + m φ + ˆ s (cid:17) { ˆ s (cid:16) − y P y S (cid:16) β + β C θ + C θ (cid:17) + ( y P + y S ) (cid:16) β C θ + C θ + (cid:17)(cid:17) − C θ m t (cid:16) C θ (cid:16) y P + y S (cid:17) − y P y S (cid:17) } (B13) ρ ++ Inter f erence = g s (cid:16) β ˆ sC θ − m t + m φ + ˆ s (cid:17) × { (cid:16) C θ − (cid:17) m t (cid:16) y P + y S (cid:17) ˆ s (cid:16) y P y S (cid:16) β + β C θ + C θ (cid:17) + ( y P + y S ) (cid:16) β C θ + C θ + (cid:17)(cid:17) } (B14) ρ + − Inter f erence = g s √ ˆ sm t y P S θ y S ( β C θ + ) (cid:16) β ˆ sC θ − m t + m φ + ˆ s (cid:17) ρ − + Inter f erence = ( ρ + − Inter f erence ) (cid:63) (B15) ρ −− Inter f erence = g s (cid:16) β ˆ sC θ − m t + m φ + ˆ s (cid:17) × { (cid:16) C θ − (cid:17) m t (cid:16) y P + y S (cid:17) − ˆ s (cid:16) − y P y S (cid:16) β + β C θ + C θ (cid:17) + ( y P + y S ) (cid:16) β C θ + C θ + (cid:17)(cid:17) } (B16) [1] BaBar
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