A precise characterisation of the top quark electro-weak vertices at the ILC
M.S. Amjad, S. Bilokin, M. Boronat, P. Doublet, T. Frisson, I. Garcìa Garcìa, M. Perelló, R. Pöschl, F. Richard, E. Ros, J. Rouëné, P. Ruiz Femenia, M. Vos
IIFIC/15-15, LAL 15-111
A precise characterisation of the top quarkelectro-weak vertices at the ILC
M.S. Amjad a,1 , S. Bilokin a , M. Boronat b , P. Doublet a,3 , T. Frisson a,2 , I. García García b ,M. Perelló b , R. Pöschl a, ∗ , F. Richard a , E. Ros b , J. Rouëné a , P. Ruiz Femenia b,4 , M. Vos b a Laboratoire de l’Accélérateur Linéaire (LAL), Centre Scientifique d’Orsay, Université Paris-Sud XI,BP 34, Bâtiment 200, F-91898 Orsay CEDEX, France b IFIC, Universitat de Valencia CSIC, c/ Catedrático José Beltrán, 2 46980 Paterna, Spain.
Abstract
Top quark production in the process e + e − → t ¯ t at a future linear electron positron collider withpolarised beams is a powerful tool to determine indirectly the scale of new physics. The presented study,based on a detailed simulation of the ILD detector concept, assumes a centre-of-mass energy of √ s =500 GeV and a luminosity of L = 500 fb − equally shared between the incoming beam polarisations of P e − , P e + = ± . , ∓ . . Events are selected in which the top pair decays semi-leptonically and the crosssections and the forward-backward asymmetries are determined. Based on these results, the vector,axial vector and tensorial CP conserving couplings are extracted separately for the photon and the Z component. With the expected precision, a large number of models in which the top quark acts as amessenger to new physics can be distinguished with many standard deviations. This will dramaticallyimprove expectations from e.g. the LHC for electro-weak couplings of the top quark.
1. Introduction
The main goal of current and future machines at the energy frontier is to understand the nature ofelectro-weak symmetry breaking. This symmetry breaking can be generated by the existence of a newstrong sector, inspired by QCD, that may manifest itself at energies of around 1 TeV. In all realisationsof the new strong sector, as for example Randall-Sundrum models [1] or compositeness models [2], thestrength of the coupling to this new sector of the Standard Model fields are supposed to increase withtheir mass. For this and other reasons, the heavy top quark or t quark with a mass of approximately m t = 173 GeV [3] is expected to be a window to any new physics at the TeV energy scale. New physicscan modify the electro-weak t ¯ tX vertex described in the Standard Model by V ector and A xial vectorcouplings V and A to the vector bosons X = γ, Z . At the International Linear Collider, ILC [4],that will collide electron and positrons at a centre-of-mass energy of
500 GeV , t quark electro-weakcouplings can be measured at the % level.In contrast to the situation at hadron colliders, the leading-order pair production process e + e − → t ¯ t goes directly through the t ¯ tZ and t ¯ tγ vertices. There is no concurrent QCD production of t quarkpairs, which increases greatly the potential for a clean measurement. A parametrisation of the t ¯ tX vertex valid to all orders of perturbation theory may be written as ∗ : ∗ Corresponding author: [email protected] Now at COMSATS Institute of Information Technology, Islamabad, Pakistan. Now at CERN, 1211 Genève 23, Switzerland. Now at IUT d’Orsay (Université Paris-Sud), France. Now at Technische Universität München, 85748 Garching, Germany. ∗ A dependence on an additional term ( q + ¯ q ) µ · F can be neglected in the limit of a vanishing electron mass [5]. a r X i v : . [ h e p - e x ] S e p t ¯ tXµ ( k , q, ¯ q ) = ie (cid:26) γ µ (cid:0) F X V ( k ) + γ F X A ( k ) (cid:1) − σ µν m t ( q + ¯ q ) ν (cid:0) iF X V ( k ) + γ F X A ( k ) (cid:1)(cid:27) , (1)with e being the electrical charge of the electron, k = ( q + ¯ q ) being the squared four-momentum ofthe exchanged boson and q and ¯ q being the four-vectors of the t and ¯ t quark, respectively. Further, γ µ are the Dirac matrices leading to vector currents of fermions and γ is the Dirac matrix allowing tointroduce an axial-vector current into the theory. Finally, σ µν = i ( γ µ γ ν − γ ν γ µ ) allows for describingthe scattering of a particle with spin 1/2 and a given magnetic moment.Within the Standard Model the F have the following values at tree level: F γ,SM V = 23 , F γ,SM A = 0 , F Z,SM V = 14 s w c w (cid:18) − s w (cid:19) , F Z,SM A = − s w c w , (2)while all the F are zero. In Eq. 2 s w and c w are the sine and the cosine of the Weinberg angle θ W . Thescale dependence of the form factors is a consequence of higher order corrections. The corrections ofthe vector currents lead to the anomalous electro-magnetic and weak-magnetic moments representedby F X V that correct the gyromagnetic ratio g t of the t quark. Typical values for these corrections arein the range O (10 − − − ) [6]. Corrections to the axial-vector current result in the Form Factors F X A that are related to the dipole moment d Xt = ( e/ m t ) F X A (0) that in turn violates the combined C harge and P arity symmetry CP . Otherwise said, all couplings but F X A ( k ) conserve CP .The Form Factors F Z V,A are related to couplings of t quarks with left and right-handed helicity tothe Z : g ZL = F Z V − F Z A , g ZR = F Z V + F Z A (3)Trivially, the same equations apply correspondingly to the photon couplings g γL In this paper the precision of CP conserving form factors and couplings as introduced abovewill be derived by means of a full simulation study of the reaction e + e − → t ¯ t at a centre-of-massenergy of √ s = 500 GeV with 80% polarised electron beams and 30% polarised positron beams usingexperimentally well defined observables. Special emphasis will be put on the selection efficiency andthe polar angle of the final state t quarks. Both experimental quantities are suited to monitor carefullyexperimental systematics that may occur in the extraction of form factors and couplings. The resultspresented in the following are based on the studies described in detail in Refs. [7, 8].
2. Top quark production at the ILC
The tree level diagram for pair production of t quarks at the ILC is presented in Figure 1a. Thedecay of the top quarks proceeds predominantly through t → W ± b . The subsequent decays of the W ± bosons to a charged lepton and a neutrino or a quark-anti-quark pair lead to a six-fermion finalstate. The study presented in this article focuses on the ’lepton+jets’ final state l ± νb ¯ bq (cid:48) ¯ q representinga branching fraction of about 43.4% on all t ¯ t pair decays.Several other Standard Model processes give rise to the same final state. The most importantsource is single t quark production through the process e + e − → W W ∗ → W t ¯ b → l ± νb ¯ bq (cid:48) ¯ q . One ofthe diagrams contributing to this process is presented in Figure 1b. Another relevant source is the Z W + W − production. Due to the coupling of initial state electrons or positrons to W bosons bothsources contribute nearly exclusively in a configuration with left-handed polarised electron beams andright-handed polarised positron beams.In that case single t quark and Z W + W − boson production can yield a total production rate ofup to 10% of that of the pair production diagram of Fig. 1a. Experimentally, Z W + W − productioncan be distinguished rather efficiently from t ¯ t pair production, but a clean separation of final stateswith a single t quark seems impossible. A realistic experimental strategy must therefore consider the W + bW − ¯ b inclusively [9]. 2 γ ∗ /Z ∗ e − e + t ¯ t (a) t ¯ t pair production (cid:1) W + ∗ ν ∗ e e − e + W − ¯ bt (b) Single t quark production Figure 1:
Diagrams that contribute to the e + e − → lνb ¯ bq (cid:48) ¯ q production: (a) Tree level t ¯ t pair production,(b) single t quark production. In case of polarised beams Ref. [10] suggests to express the form factors introduced in Sec. 1 interms of the helicity of the incoming electrons, F Lij = − F γij + (cid:16) − + s w s w c w (cid:17)(cid:16) ss − m Z (cid:17) F Zij F Rij = − F γij + (cid:16) s w s w c w (cid:17)(cid:16) ss − m Z (cid:17) F Zij , (4)with i = 1 , and j = V, A and m Z being the mass of the Z boson. The tree level cross section for t ¯ t quark pair production for an electron beam polarisation I = L, R reads σ I = 2 A N c β (cid:104) (1 + 0 . γ − )( F I V ) + ( F I (cid:48) A ) + 3 F I V F I V + (1 + 0 . γ )( F I V ) (cid:105) , (5)where A = πα s with the running electromagnetic coupling α ( s ) and N c is the number of quarkcolours. Furthermore γ and β are the Lorentz factor and the velocity of the t quark, respectively. Theterm F I (cid:48) A = β F I A describes the reduced sensitivity to axial vector couplings near the t ¯ t productionthreshold. The cross sections at the Born level of the signal process e + e − → t ¯ t and the main StandardModel background processes at a centre-of-mass energy of
500 GeV are summarised in Table 1.Channel σ unpol. [fb] σ − , + [fb] σ + , − [fb] t ¯ t
572 1564 724 µ + µ −
456 969 854 (cid:80) q=u , d , s , c q ¯ q b ¯ b
372 1212 276 γZ W + W − Z Z
422 1106 582 Z W + W −
40 151 8.7 Z Z Z t for e + e − → e − ¯ ν e t ¯ b [11] 3.1 10.0 1.7Table 1: Unpolarised cross-sections and cross-sections at tree level for 100% beam polarisation forsignal and background processes.
The forward-backward asymmetry A tF B can be expressed as ( A tF B ) I = ∓ A N c β · F I (cid:48) A ( F I V + F I V ) σ I . (6)3he ’-’ sign applies in case of an initial left-handed polarised electron beam, i.e. I = L , and the ’+’applies correspondingly in case of an initial right-handed polarised electron beam, i.e. I = R . In theStandard Model the forward-backward asymmetry takes the values ( A tF B ) L = 0 . and ( A tF B ) R = 0 . at tree level.Neglecting CP violating form factors, the fraction of right-handed t quarks is given by the followingexpression: ( F R ) I = 0 . ∓
23 ( A tF B ) I (7)The ’-’ sign applies in case of an initial left-handed polarised electron beam, i.e. I = L , and the ’+’applies correspondingly in case of an initial right-handed polarised electron beam, i.e. I = R . Thevalues expected in the Standard Model at tree level are ( F R ) L = 0 . and ( F R ) R = 0 . .With the introduced observables the six CP conserving form factors defined for the Z and thephoton can in principle be extracted simultaneously. However, close to the t ¯ t threshold the observablesdepend always on the sum F V + F V . Therefore, a full disentangling of the form factors will beimprecise for energies below about . Hence, in the present study either the precision on theForm Factors F X V,A , or equivalently on the Couplings g XL,R , are determined simultaneously, while thetwo F V are kept at their Standard Model values or vice versa. Due to these considerations the studywill only make use of the cross section and A tF B since these are either the most precise observablein case of the cross section or the one that is most sensitive to axial couplings in case of A tF B . Itis however reminded that in [7] the fraction of right-handed t quarks is determined to a precision ofabout 2%. The extraction of form factors requires precise predictions of the inclusive top quark pair produc-tion rate and of several differential distributions. In this section the state-of-the-art calculations andestimate theoretical uncertainties are briefly reviewed.As discussed at the beginning of this section, the optimal experimental strategy should consider e + e − → W + bW − ¯ b inclusively, without attempting to distinguish single top and top quark pairproduction. However, today, sufficiently precise calculations are not available for the full process e + e − → W + bW − ¯ b . Therefore the discussion in this paper is based on the current state of the artcalculations for e + e − → t ¯ t , assuming that in the next decade theorists will rise to the challenge ofextending the calculations to e + e − → W + bW − ¯ b .The QCD corrections to e + e − → t ¯ t production are known up to N LO for the inclusive crosssection [12], and to N N LO for the forward backward asymmetry A tF B [13]. The perturbative seriesshows good convergence.In the kinematic region at around √ s = 500 GeV as relevant for this study the N LO correctionto the total cross-section is below 1 %. An estimate of the size of the next order - obtained from theconventional variation of the renormalisation scale by a factor two and one half - yields 0.3 %. It cantherefore be concluded that the uncertainty of today’s state-of-the-art calculations is at the per millevel.In a similar manner the QCD corrections to the prediction of differential distributions and quantitiessuch as the forward-backward asymmetry can be estimated. The size of the N LO correction to A tF B is estimated using the scale variation to be smaller than 1%, see also the discussion in e.g. [13].Electro-weak (EW) corrections to the same process have also been calculated. A full one-loopcalculation is presented in [14]. The correction to the total cross section is found to be approximately5%. The electro-weak correction to the forward-backward asymmetry is large, approximately 10% [14,15]. Recent studies [16] show that the corrections are notably different for different beam polarisations.They change for example the shape of the angular distribution in case of P e − , P e + = − , +1 beampolarisation while they only influence the normalisation in case of P e − , P e + = +1 , − .The above discussion refers to corrections to the process e + e − → t ¯ t . Further corrections of order Γ t /m t ∼
1% are expected to appear if the decay of the top quarks is included in the calculation.4t can be concluded that the state-of-the-art calculations of QCD corrections offer the precisionrequired for this study. Uncertainties are under relatively good control, with uncertainties to the crosssection of the order of a few per mil and order 1% on the forward-backward asymmetry. Electro-weak (one-loop) corrections are large. Further work is needed to estimate the size of the two-loopcorrection and, ultimately, to calculate this contribution. Currently these aspects are discussed withtheory groups.
3. Analysis of simulated events
The study has been carried out on a fully polarised sample albeit realistic values for the ILC are P e − , P e + = ± . , ∓ . . The cross section and therefore its uncertainty scales with the polarisation ina well defined way according to [17] σ P e − , P e + = 14 [(1 − P e − P e + )( σ − , + + σ + , − ) + ( P e − − P e + )( σ + , − − σ − , + )] . (8)The observable A tF B varies only very mildly with the beam polarisation. The realistic beam polar-isation will be correctly taken into account in the uncertainty of the results.Signal and background events corresponding to a luminosity of fb − at √ s = 500 GeV for each ofthe two polarisation configurations are generated with version 1.95 of the WHIZARD event generator [18,19] that provides lowest order calculations of the → fermions subprocess and simulates multiplephoton radiation from the initial state electron and positron in leading-logarithmic approximation. WHIZARD is interfaced to the
PYTHIA
Monte Carlo programme [20] for QCD and QED showering.The generated events were subject to a full simulation of the ILD Detector [4] and subsequent eventreconstruction using the version
ILD_o1_v05 of the ILD software.The decay of the t quarks proceeds predominantly through t → W ± b . The subsequent decays ofthe W ± bosons to a charged lepton and a neutrino or a quark-anti-quark pair lead to a six-fermionfinal state. The study presented in this article focuses on the semi-leptonic final state l ± νb ¯ bq (cid:48) ¯ q . Severalother Standard Model processes give rise to the same final state. The most important source is single t quark production. Another relevant source is the Z W + W − production. Experimentally, Z W + W − production can be distinguished rather efficiently from t quark pair production. The separation betweensingle t quark production and t ¯ t pair production is much more involved. Note however, that accordingto Table 1 single t quark production is strongly suppressed in case of P e − , P e + = +1 , − .The entire selection procedure including lepton and b jet identification, top quark reconstructionand suppression multi-peripheral γγ → hadrons background is explained in detail in [7, 8] and [21].The total selection efficiency of about 56% for semi-leptonic t ¯ t events includes events with a τ leptonin the final state. Background processes can be very efficiently removed down to a negligible level.A powerful tool is the b likeness or b -tag value that suppresses about 97% of the dominant W + W − background. Additional selection criteria comprise cuts on the t quark and W ± boson masses and ofthe invariant mass of the total hadronic final state.With the determined efficiencies a statistical uncertainty of the cross section e + e − → t ¯ t of 0.47%in case P e − , P e + = − . , +0 . and 0.63% in case P e − , P e + = +0 . , − . can be derived. A tF B The forward-backward asymmetry A tF B has the well known definition A tF B = N (cos θ top > − N (cos θ top < N (cos θ top >
0) + N (cos θ top < , (9)where N is the number of events in the two detector hemispheres w.r.t. the polar angle θ top of the t quark calculated from the decay products in the hadronic decay branch. The direction measurementdepends on the correct association of the b quarks to the jets of the hadronic W boson decays. Theanalysis is carried out separately for a left-handed polarised electron beam and for a right-handed5 top q cos( -1 -0.5 0 0.5 110002000300040005000 L+ e R- e R+ e L- eReconstructedGenerator - Whizard ) top q cos( -1 -0.5 0 0.5 110002000300040005000 L+ e R- e R+ e L- e c Reconstructed with cut on SM BackgroundGenerator - Whizard
Figure 2:
Left: Reconstructed forward-backward asymmetry compared with the prediction by the eventgenerator
WHIZARD [18] for two configurations of the beam polarisations. Right: The same but afterthe application of a on χ < for the beam polarisations P e − , P e + = − , +1 as explained in the text.Note, that in both figures no correction is applied for the beam polarisations P e − , P e + = +1 , − . Thefigure on the right hand side shows also the residual Standard Model background. polarised beam. In case of a right-handed electron beam the direction of the t quark can be preciselyreconstructed. In case of a left-handed electron beam the final state features two hard jets from the b quarks and soft jets from the hadronically decaying W boson. This configuration leads to migrationsin the polar angle distribution of the t quark as visible in the left part of Fig 2. This implicationmotivates to restrict the determination of A tF B in case of P e − , P e + = − , +1 to cleanly reconstructedevents. For this a test variable χ is defined that compares the measured values of the Lorentz factor γ of the top, the momentum of the b quark in the rest frame of the top and the angle cos θ bW betweenthe b quark and the W boson. The reconstructed polar angle distribution of the t quark is comparedwith the generated one for different cuts on χ . For a value of χ < an excellent agreement betweenthe generated and reconstructed polar angle distributions is obtained, see the right part of Fig. 2. Thetight selection however reduces the efficiency in case of left-handed initial electron beams from 55% to28%. With this the forward backward asymmetry can be determined to a statistical precision of betterthan 2%. The precise results corrected to the beam polarisations P e − , P e + = ± . , ∓ . are givenin Table 2 together with those for the cross section, see previous section. A more straightforward,albeit experimentally more challenging, way to control the migrations is to measure the charge of the b quarks that are issue of the t quark decay. References [22] and [23] describe the determination of the b quark charge using secondary tracks. The same value of A tF B is obtained at a comparable selectionefficiency [8]. This means that A tF B can be determined with two independent methods. P e − , P e + ( δσ/σ ) stat. [%] ( δA tF B /A tF B ) stat. [%] − . , +0 . +0 . , − . Statistical precisions expected for the cross sections and A tF B for different beam polarisations. Hard gluon radiation may alter the polar angle distribution of the final state t quarks. The WHIZARD version 1.95 used for the study generates hard gluons only via the interface to
PYTHIA that generatesthe parton shower. Therefore results presented before have been checked with a study on parton levelusing the most recent version 2.2.2 of
WHIZARD that correctly accounts for hard gluon radiation. Nosignificant difference has been observed. 6 . Discussion of systematic uncertainties
In the previous sections measurements of either cross sections or asymmetries have been presented.This section makes an attempt to identify and quantify systematic uncertainties, which may influencethe precision measurements. • Luminosity: The luminosity is a critical parameter for cross section measurements only. Theluminosity can be controlled to 0.1% [24]. • Polarisation: The polarisation is a critical parameter for all analyses. It enters directly thecross section measurements. The studies presented in [25] using W pair production lead to anuncertainty of 0.1% for the polarisation of the electron beam and to an uncertainty of 0.35% forthe polarisation of the positron beam. This translates into an uncertainty of 0.25% on the crosssection for P e − , P e + = − . , +0 . and 0.18% on the cross section for P e − , P e + = +0 . , − . . Theuncertainty on the polarisation can be neglected with respect to the statistical uncertainty for A tF B . • Beamstrahlung and beam energy spread: The mutual influence of the electromagnetic fields ofthe colliding bunches provokes radiation of photons known as
Beamstrahlung . This Beam-strahlung modulates the luminosity spectrum, i.e. moves particles from the nominal energy tosmaller energies. At the ILC for a centre-of-mass energy of
500 GeV about 60% of the particlesare expected to have 99% or more of the nominal energy [4]. The beam energy spread, i.e. theRMS of this main luminosity peak is 124 MeV for the electron beam and 70 MeV for the positronbeam [4]. Both effects play a role at the t ¯ t threshold [26] and can be neglected at energies wellabove this threshold. • Experimental uncertainties in top quark reconstruction: As discussed in Sec. 3.1 migrations haveto be taken into account for the measurement of A tF B , in particular for the polarisations P e − , P e + = − . , +0 . . These migrations are reduced by stringent requirements on the event selection usinga χ analysis. This in turn leads to a penalty in the efficiency. The success of the method dependsin addition on a very sharp measurement of the variables used for the χ analysis. It is expectedthat these ambiguities can be (partially) eliminated by an event-by-event determination of thecharge of the b quark from the t decay. As has been shown in Sec. 3, the effect will be very muchsuppressed in case of P e − , P e + = +0 . , − . beam polarisation. • Other experimental effects: There is a number of other experimental effects such as acceptance,uncertainties of the b tagging or the influence of passive detector material. The LEP1 experimentsquote a systematic uncertainty on R b of 0.2% a value which may serve as a guide line for values tobe expected at the ILC, which on the other hand will benefit from far superior detector resolutionand b tagging capabilities. • Theory: The uncertainties of today’s state-of-the-art calculations are discussed in Section 2.2.The uncertainties in the QCD corrections to the total cross section and A F B are of the same orderas the experimental uncertainties. Two-loop electro-weak calculations are required for a reliableestimation of the uncertainties due to electro-weak corrections. It is however intuitively clear thatthe latter will benefit from the insight of the different impact for different beam polarisations,see Sec. 2.2 and [16]. • Single-top production: Single top production at the LC in association with a W boson and bottomquark (through W W ∗ production) leads to the same final state as t quark pair production.Being largely suppressed in case of P e − , P e + = +0 . , − . beam polarisation, it forms a sizeablecontribution to the six-fermion final state in case of P e − , P e + = − . , +0 . beam polarisation. Itmust therefore be taken into account in a realistic experimental strategy. This is left for a futurestudy. 7 Beyond Standard Model Physics: Possible BSM effects may affect the various components ofthe background, in particular the t ¯ t induced background. This will therefore require a carefuliterative procedure with tuning of event generators. This procedure seems feasible without asignificant loss of accuracy.As a summary it can be concluded that the total systematic uncertainties will not exceed thestatistical uncertainties. This, however, requires an excellent control of a number of experimental andtheoretical quantities.
5. Precision of form factors and electro-weak couplings
The measured cross sections and A tF B lead for two polarisation configurations to a set of fourobservables. By means of Eqs. 5 and 6 the uncertainties on these observables are used to build upa system of linear equations to determine the variances of up to four variables † , The variances areequivalent to the square of the standard deviations of the variables under study. These variables canbe the form factors or alternatively directly the couplings. More explicitly, in this paper the followingquantities will be determined separately:1. The standard deviations of the Form Factors F γ V , F Z V , F Z A assuming no variation of the FormFactors F X j ;2. The standard deviations of the Form Factors F γ V , F Z V assuming no variation of the Form Factors F X j ;3. The standard deviations of the Couplings g γL , g γR , g ZL , g ZR .Note, that the Form Factor F Z A is fixed to be 0 in order to respect QED gauge invariance. On theother hand all four Couplings g XI are allowed to vary freely. The resulting standard deviations are listedin Table 3. Quantity F γ V F Z V F Z A F γ V F Z V g γL g γR g ZL g ZR SM Value at tree level 2/3 0.230 -0.595 0 0 2/3 2/3 0.824 -0.364Standard deviation 0.002 0.003 0.007 0.001 0.002 0.005 0.005 0.008 0.009Relative precision [%] 0.3 0.9 1.2 - - 0.8 0.8 1.0 2.5
Table 3:
Standard deviations and resulting relative precisions of form factors and couplings derivedfrom the statistical precisions on the observables cross section and A tF B as listed in Table 2. The complete covariance matrices are given in Appendix A. From there it can be told that e.g. g ZL and g ZR are strongly correlated while F Z V and F Z A are nearly uncorrelated. The expected high precisionat a linear e + e − collider allows for a profound discussion of effects by new physics. The findings canbe confronted with predictions in the framework of Randall-Sundrum models and/or compositenessmodels such as [2, 34, 36, 32, 31, 28, 29, 33] or Little Higgs models as e.g. [30]. All these modelsentail deviations from the Standard Model values of the t quark couplings to the Z boson that willbe measurable at the ILC as illustrated in Fig. 3. Therefore, the couplings of the t quark to the Z boson will discussed in a broader context in the following. Z Boson - Comparison with perspectives for LHC and FlavourPhysics
Electro-weak couplings can be measured at the LHC in associated ¯ ttγ and ¯ ttZ production. Acomprehensive compilation on the statistical precisions on the form factors that can be expected atthe end of the HL-LHC is given in [37] and [38] for an update on ¯ ttZ form factors. The values8.. (cid:45) δg ZR /g ZR -330% -20% -10% 10% 20% (cid:54) δg ZL /g ZL -20%-10%10%20% (cid:120) SM (cid:117) Light top partners [28] (cid:117)
Light top partnersAlternative 1 [29] (cid:117)
Light top partners Alternative 2 [29] (cid:117)
Little Higgs [30] (cid:117)
RS with Custodial SU(2) [31] (cid:117)
Composite Top [2] (cid:117)
5D Emergent [32] (cid:117)
4D Composite Higgs Models [33] (cid:117)
RS with Z-Z’ Mixing [34]
ILC Precision
Figure 3:
Predictions of several models that incorporate Randall-Sundrum (RS) models and/or com-positeness or Little Higgs models on the deviations of the left- and right-handed couplings of the t quark to the Z boson. The ellipse in the frame in the upper right corner indicates the precisionthat can be expected for the ILC running at a centre-of-mass energy of √ s = 500 GeV after havingaccumulated L = 500 fb − of integrated luminosity shared equally between the beam polarisations P e − , P e + = ± . , ∓ . . The original version of this figure can be found in [35]. V g F F F g F F U n c e r t a i n t y - - - -1 =500 GeV, L=500 fbsILC, Phys.Rev.D73 (2006) 034016Phys.Rev.D71 (2005) 054013 -1 =14 TeV, L=3000 fbsLHC, Figure 4:
Graphical comparison of statistical precisions on CP conserving form factors expected atthe LHC, taken from [37] and [38], and at the ILC. The LHC results assume an integrated luminosityof L = 3000 fb − at √ s = 14 TeV . The results for the ILC assume an integrated luminosity of L = 500 fb − at √ s = 500 GeV and a beam polarisation P e − = ± . , P e + = ∓ . . published there are compared with the results in the present study in Fig. 4. All but one form factorwill be measured at about a factor 10 better at the ILC for the scenario discussed in this paper thanit will be possible at the LHC. This exception is F Z A where [38] quotes a possible statistical precisionof δF Z A ≈ . . It should however be pointed out that the considerable precision expected for δF Z A benefits strongly from LEP/SLC bounds on the oblique parameters that e.g. render it unlikely that F Z A flips sign due to New Physics. The study presented by [38] is an analysis at leading order QCD.The analysis carried out in [39] suggests that higher-order effects in the theory may allow for animprovement of the LHC precision by up to 40%. Note at this point that the interference between the γ and the Z in case of e + e − → t ¯ t will allow for measuring flips of the signs of the form factors thatwill be unnoticed in associated ¯ ttZ at the LHC.While the prospects for the LHC discussed so far are based on analyses differential in given jetobservables of the final state, LHC experiments observe the process pp → ¯ ttZ [40, 41, 42, 43]. Theinterpretation of the results is however still limited by the small statistics available for the analyses.At the LHC electro-weak couplings are measured also in single t quark production. In the effectivefield theory approach, assuming SU (2) L × U (1) Y gauge symmetry for the operators, the relation δg tbWL g tbWL ≈ . δg ZL g ZL (10)can be established. Here g tbWL is the charged current coupling of the decay t → W b . The CMSCollaboration [44] reports a precision for the t - b transition probability V tb of about 4%. In the StandardModel V tb is identical to g tbWL . Hence, by means of Eq. 10 the precision of the coupling of left-handed t quarks to the Z boson can be derived to be of the order of 11%. Noting that σ ( pp → ¯ ttZ ) ∼ † For the Linear Algebra the software package
Eigen [27] version 3.2.2 has been used. g ZL ) + ( g ZR ) this allows in principle also for deriving ( g ZR ) , albeit with a poor precision given that ( g ZL ) (cid:29) ( g ZR ) . Loop corrections in heavy flavour physics as e.g. in the processes b → sγ , B → µ + µ − or K → µ + µ − , respectively, may also lead to competitive determinations of δg ZL [45]. However, again g ZR can only be constrained rather poorly.It follows that the ILC will allow for measurements superior to those that can be expected fromexisting experiments. This is particularly true for the determination of δg ZR . Given the fact that at the ILC in its current layout centre-of-mass energies of up to canbe reached and that the alternative project for a linear collider, CLIC [46], may even reach higherenergies, it is instructive to discuss the results presented in this paper with this possibility in mind.The selection and reconstruction of the decay topology of boosted t quarks is very different from thatof t quarks with moderate velocity. Therefore, the study must be extrapolated to high centre-of-massenergy with some care. Still, the following observations can be made: • Neglecting varying detector systematics and theory uncertainties with varying centre-of-mass en-ergy, and assuming the linear collider luminosity vs. centre-of-mass energy curve, the sensitivityto the form factors considered in this paper is greatest at approximately √ s = 400 −
700 GeV .At lower centre-of-mass energy, as e.g. studied in [47], the small velocity of the t quarks reducesthe potential of the A tF B measurement compromising thus the measurement of the axial vectorcouplings to the Z boson and by virtue of Eq. 3 the disentangling of left- and right-handedcouplings. On the other hand running at centre-of-mass energies close to the t ¯ t threshold offerssensitivity to virtual Higgs exchange [48, 49, 50]. In case the Higgs has a CP odd component thismay give rise to recognisable CP violating effects in the threshold region [51]. However, in thetransition region between the t ¯ t threshold and the continuum region starting at around
380 GeV the current QCD uncertainties are at least 10%. This is due to uncertainties on higher QCDorder corrections and on the correct matching procedure between the non-relativistic calculationsat the t ¯ t threshold and the relativistic continuum calculations [52]. • If an effect is seen at √ s = 500 GeV it will be crucial to know how it evolves with energy with adecent lever arm. If, for instance, the effect is due to mixing of the Z boson with a new Z (cid:48) bosonit will remain unchanged. If, however, a Z (cid:48) boson leads to a propagator term, the correspondingeffect will grow like s/M Z (cid:48) . In the case of Randall-Sundrum Models both effects are presentand therefore measurements at two energies are needed to extract M Z (cid:48) , see e.g. [35] for a deeperdiscussion. • The impact of high-scale new physics on the observables can increase strongly with centre-of-mass energy. Operators corresponding to the top quark dipole moments and four-fermioncontact interactions induce larger anomalous form factors at higher energy. For other anomalouscouplings, however, the impact is nearly independent of the centre-of-mass energy as is the casefor F Z V and F Z A .A full simulation study at different centre-of-mass energies is left for a future publication.
6. Summary and outlook
This article presents a comprehensive analysis fully simulated events of t ¯ t quark production at theInternational Linear Collider using the semi-leptonic decay channel. Results are given for a centre-of-mass energy of √ s = 500 GeV and an integrated luminosity of L = 500 fb − shared equally betweenthe beam polarisations P e − , P e + = ± . , ∓ . .Semi-leptonic events, including those with τ leptons in the final state can be selected with anefficiency of about 55%. The cross section of the semi-leptonic channel of t ¯ t quark production can11herefore be measured to a statistical precision of about 0.5%. The second observable is the forward-backward asymmetry A tF B . It was shown that in particular for predominantly left-handed polarisationof the initial electron beam the V − A structure leads to migrations, which distort the theoreticalexpected A tF B . These migrations can be remedied by tightening the selection criteria of the events oralternatively by measuring the charge of the b quark produced in the decay of the t quark. Taking intoaccount this correction the forward-backward asymmetry can be determined to a statistical precisionof better than 2% for both beam polarisations.The observables together with the unique feature of the ILC to provide polarised beams allow fora largely unbiased disentangling of the individual couplings of the t quark to the Z boson and thephoton. These couplings can be measured with high precision at the ILC and, when referring to theresults in [37, 38], considerably better than it will be possible at the LHC even with an integratedluminosity of L = 3000 fb − . The improving analyses of the LHC experiments will however be observedwith great interest.Beam polarisation is a critical asset for the high precision measurements of the electroweak t quark couplings. Experimental and theoretical effects manifest themselves differently for differentbeam polarisations. It seems to be that the configuration P e − , P e + = +0 . , − . is more benign inboth, experimental aspects due to the suppression of migrations in the polar angle spectrum of the finalstate t quark and theoretical aspects due to the somewhat simpler structure of higher order electroweakcorrections. It is intuitively clear that the described facts would greatly support the discovery of effectsdue to new physics.The precision as obtained in the present study for the ILC would allow for the verification ofa great number of models for physics beyond the Standard Model. Examples for these models areextra dimensions and compositeness. The results obtained here constitute therefore a perfect basis fordiscussions with theoretical groups. Note at this point that running scenarios for the ILC have beenproposed that would yield between 8 and 10 times more integrated luminosity [53] than it is assumedfor the present study. Moreover it can be expected that the event reconstruction will be improvedby e.g. the measurement of the b quark charge. It is therefore not statistics that will limit the finalaccuracy but most likely theory and experimental systematics.Hence, the study of systematic errors, only partially addressed in this study, will become veryimportant. Already from the achieved precision it is mandatory that systematics are controlled to the1% level or better in particular for the measurement of the cross section. This issue is addressed inongoing studies.The study presented in [16] based on generated events suggests that by exploiting the polarisationof the final state t quarks a simultaneous extraction of all ten form factors, see Eq. 1, to a precisionbelow the percent level is feasible. A detailed comparison between the advantages and drawbacks ofthe method applied there and the method presented in this paper is left for a future study. Acknowledgements
This work was supported within the ’Quarks and Leptons’ programme of the CNRS/IN2P3, France,and by the Spanish Grant Agreement "Detector developments and physics studies for future colliders,FPA2012-39055-C02-01". The results benefit from the enlightening discussions in the framework of theFrench-Japanese FJPPL/TYL ’virtual laboratory’ on top physics, particularly through comments byEmi Kou, Keisuke Fujii and François LeDiberder. We would like to thank Fabian Bach and MaximilanStahlhofen for the profound discussion on QCD uncertainties in the t ¯ t threshold region. A. Covariance matrices
For completeness the underlying covariance matrices of the results presented in Sec. 5 are given inthis appendix. 12
The covariance matrix resulting from the system of linear equations built for the Form Factors F γ V , F Z V , F Z A reads: var ( F γ V ) cov ( F γ V , F Z V ) cov ( F γ V , F Z A ) var ( F Z V ) cov ( F Z V , F Z A ) var ( F Z A ) = . − .
043 0 . .
791 0 . . × − . (11) • The covariance matrix resulting from the system of linear equations built for the Form Factors F γ V , F Z V reads: (cid:20) var ( F γ V ) cov ( F γ V , F Z V ) var ( F Z V ) (cid:21) = (cid:20) . − . . (cid:21) × − . (12) • The covariance matrix resulting from the system of linear equations built for the Couplings g γL , g γR , g ZL , g ZR reads: var ( g γL ) cov ( g γL , g γR ) cov ( g γL , g ZL ) cov ( g γL , g ZR ) var ( g γR ) cov ( g γR , g ZL ) cov ( g γR , g ZR ) var ( g ZL ) cov ( g ZL , g ZR ) var ( g ZR ) = . − . − . − . . − .
780 2 . . − . . × − . (13) ReferencesReferences [1] L. Randall and R. Sundrum, “A Large mass hierarchy from a small extra dimension”
Phys.Rev.Lett. (1999)3370–3373, arXiv:hep-ph/9905221 [hep-ph] .[2] A. Pomarol and J. Serra, “Top Quark Compositeness: Feasibility and Implications” Phys.Rev.
D78 (2008) 074026, arXiv:0806.3247 [hep-ph] .[3]
Particle Data Group , K. Olive et al. , “Review of Particle Physics”
Chin.Phys.
C38 (2014) 090001.[4] T. Behnke et al. , “ILC TDR and DBD”
ILC-Report-2013-040 . .[5] G. L. Kane, G. Ladinsky, and C. Yuan, “Using the Top Quark for Testing Standard Model Polarization and CPPredictions” Phys.Rev.
D45 (1992) 124–141.[6] L. Labun and J. Rafelski, “Top anomalous magnetic moment and the two photon decay of Higgs” arXiv:1209.1046 [hep-ph] .[7] M. Amjad, M. Boronat, T. Frisson, I. Garcia, R. Poschl, et al. , “A precise determination of top quark electro-weakcouplings at the ILC operating at √ s = 500 GeV” arXiv:1307.8102 [hep-ex] .[8] J. Rouëné, “Calorimètre électromagnétique silicium-tungstène hautement granulaire - Production du quark top àl’International Linear Collider”
LAL 14-154 (2014) .[9] J. Fuster, I. García, P. Gomis, M. Perelló, E. Ros, and M. Vos, “Study of single top production at high energyelectron positron colliders”
The European Physical Journal C (2015) no. 5, . http://dx.doi.org/10.1140/epjc/s10052-015-3453-2 .[10] C. R. Schmidt, “Top quark production and decay at next-to-leading order in e+ e- annihilation” Phys.Rev.
D54 (1996) 3250–3265, arXiv:hep-ph/9504434 [hep-ph] .
11] E. Boos, M. Dubinin, A. Pukhov, M. Sachwitz, and H. Schreiber, “Single top production in e+ e-, e- e-, gamma eand gamma gamma collisions”
Eur.Phys.J.
C21 (2001) 81–91, arXiv:hep-ph/0104279 [hep-ph] .[12] Y. Kiyo, A. Maier, P. Maierhofer, and P. Marquard, “Reconstruction of heavy quark current correlators atO(alpha(s)**3)”
Nucl.Phys.
B823 (2009) 269–287, arXiv:0907.2120 [hep-ph] .[13] W. Bernreuther, R. Bonciani, T. Gehrmann, R. Heinesch, T. Leineweber, et al. , “Two-Parton Contribution to theHeavy-Quark Forward-Backward Asymmetry in NNLO QCD”
Nucl.Phys.
B750 (2006) 83–107, arXiv:hep-ph/0604031 [hep-ph] .[14] J. Fleischer, A. Leike, T. Riemann, and A. Werthenbach, “Electroweak one loop corrections for e+ e- annihilationinto t anti-top including hard bremsstrahlung”
Eur.Phys.J.
C31 (2003) 37–56, arXiv:hep-ph/0302259 [hep-ph] .[15] P. Khiem, J. Fujimoto, T. Ishikawa, T. Kaneko, K. Kato, et al. , “Full O ( α ) electroweak radiative corrections to e + e − → t ¯ tγ with GRACE-Loop” arXiv:1211.1112 [hep-ph] .[16] P. Khiem, E. Kou, Y. Kurihara, and F. L. Diberder, “Probing New Physics using top quark polarization in the e + e − → t ¯ t process at future Linear Colliders” arXiv:1503.04247 [hep-ph] .[17] G. Moortgat-Pick, T. Abe, G. Alexander, B. Ananthanarayan, A. Babich, et al. , “The Role of polarized positronsand electrons in revealing fundamental interactions at the linear collider” Phys.Rept. (2008) 131–243, arXiv:hep-ph/0507011 [hep-ph] .[18] W. Kilian, T. Ohl, and J. Reuter, “WHIZARD: Simulating Multi-Particle Processes at LHC and ILC”
Eur.Phys.J.
C71 (2011) 1742, arXiv:0708.4233 [hep-ph] .[19] M. Moretti, T. Ohl, and J. Reuter, “O’Mega: An Optimizing matrix element generator” arXiv:hep-ph/0102195[hep-ph] .[20] T. Sjostrand, S. Mrenna, and P. Z. Skands, “PYTHIA 6.4 Physics and Manual”
JHEP (2006) 026, arXiv:hep-ph/0603175 [hep-ph] .[21] M. Boronat, I. Garcia, and M. Vos, “A new jet reconstruction algorithm for lepton colliders” arXiv:1404.4294[hep-ex] .[22] M. S. Amjad,
Forward-Backward asymmetry in top pair production at the ILC . Theses, Université Paris Sud -Paris XI, Feb., 2014. https://tel.archives-ouvertes.fr/tel-00949818 .[23] J. Rouëné,
A Highly Granular Silicon-Tungsten Electromagnetic Calorimeter and Top Quark Production at theInternational Linear Collider . Theses, Université Paris Sud - Paris XI, June, 2014. https://tel.archives-ouvertes.fr/tel-01062136 .[24] C. Rimbault, P. Bambade, K. Monig, and D. Schulte, “Impact of beam-beam effects on precision luminositymeasurements at the ILC”
JINST (2007) P09001.[25] A. Rosca, “Measurement of the beam polarisation at the ILC using the WW annihilation data” LC-REP-2013-009 . .[26] K. Seidel, F. Simon, M. Tesar, and S. Poss, “Top quark mass measurements at and above threshold at CLIC” arXiv:1303.3758 [hep-ex] (2013) , arXiv:1303.3758 [hep-ex] .[27] “Eigen homepage”. http://eigen.tuxfamily.org/index.php?title=Main_Page .[28] C. Grojean, O. Matsedonskyi, and G. Panico, “Light top partners and precision physics” arXiv:1306.4655[hep-ph] .[29] G. Panico, A. Wulzer, private communication, Possible deviations of couplings in framework described in [28].[30] C. Berger, M. Perelstein, and F. Petriello, “Top quark properties in little Higgs models” arXiv:hep-ph/0512053[hep-ph] .[31] M. S. Carena, E. Ponton, J. Santiago, and C. E. Wagner, “Light Kaluza Klein States in Randall-Sundrum Modelswith Custodial SU(2)” Nucl.Phys.
B759 (2006) 202–227, arXiv:hep-ph/0607106 [hep-ph] .[32] Y. Cui, T. Gherghetta, and J. Stokes, “Fermion Masses in Emergent Electroweak Symmetry Breaking”
JHEP (2010) 075, arXiv:1006.3322 [hep-ph] .[33] D. Barducci, S. De Curtis, S. Moretti, and G. M. Pruna, “Top pair production at a future e + e − machine in acomposite Higgs scenario” arXiv:1504.05407 [hep-ph] .
34] A. Djouadi, G. Moreau, and F. Richard, “Resolving the A(FB)**b puzzle in an extra dimensional model with anextended gauge structure”
Nucl.Phys.
B773 (2007) 43–64, arXiv:hep-ph/0610173 [hep-ph] .[35] F. Richard, “Present and future constraints on top EW couplings” arXiv:1403.2893 [hep-ph] .[36] Y. Hosotani and M. Mabe, “Higgs boson mass and electroweak-gravity hierarchy from dynamical gauge-Higgsunification in the warped spacetime”
Phys.Lett.
B615 (2005) 257–265, arXiv:hep-ph/0503020 [hep-ph] .[37] U. Baur, A. Juste, L. Orr, and D. Rainwater, “Probing electroweak top quark couplings at hadron colliders”
Phys.Rev.
D71 (2005) 054013, arXiv:hep-ph/0412021 [hep-ph] .[38] U. Baur, A. Juste, D. Rainwater, and L. Orr, “Improved measurement of ttZ couplings at the CERN LHC”
Phys.Rev.
D73 (2006) 034016, arXiv:hep-ph/0512262 [hep-ph] .[39] R. Röntsch and M. Schulze, “Probing top-Z dipole moments at the LHC and ILC” arXiv:1501.05939 [hep-ph] .[40]
CMS Collaboration , S. Chatrchyan et al. , “Measurement of associated production of vector bosons and topquark-antiquark pairs at sqrt(s) = 7 TeV”
Phys.Rev.Lett. (2013) 172002, arXiv:1303.3239 [hep-ex] .[41]
CMS , V. Khachatryan et al. , “Measurement of top quark-antiquark pair production in association with a W or Zboson in pp collisions at √ s = 8 TeV”
Eur.Phys.J.
C74 (2014) no. 9, 3060, arXiv:1406.7830 [hep-ex] .[42]
The ATLAS Collaboration , “Evidence for the associated production of a vector boson (W, Z) and top quarkpair in the dilepton and trilepton channels in pp collision data at √ s = 8 TeV collected by the ATLAS detector atthe LHC”.[43]
CMS Collaboration , “Observation of a Z and measurement of a W boson produced in association with a topquark pair in events with multiple leptons at CMS”. https://twiki.cern.ch/twiki/bin/view/CMSPublic/PhysicsResultsTOP14021 .[44]
CMS , V. Khachatryan et al. , “Measurement of the t-channel single-top-quark production cross section and of the | V tb | CKM matrix element in pp collisions at √ s = 8 TeV” JHEP (2014) 090, arXiv:1403.7366 [hep-ex] .[45] J. Brod, A. Greljo, E. Stamou, and P. Uttayarat, “Probing anomalous ttZ interactions with rare meson decays”
JHEP (2015) 141, arXiv:1408.0792 [hep-ph] .[46] P. Lebrun, L. Linssen, A. Lucaci-Timoce, D. Schulte, F. Simon, et al. , “The CLIC Programme: Towards a Stagede+e- Linear Collider Exploring the Terascale : CLIC Conceptual Design Report” arXiv:1209.2543[physics.ins-det] .[47] P. Janot, “Top-quark electroweak couplings at the FCC-ee”
JHEP (2015) 182, arXiv:1503.01325 [hep-ph] .[48] R. Harlander, M. Jezabek, and J. H. Kuhn, “Higgs effects in top quark pair production” Acta Phys.Polon.
B27 (1996) 1781–1788, arXiv:hep-ph/9506292 [hep-ph] .[49] D. Atwood, S. Bar-Shalom, G. Eilam, and A. Soni, “CP violation in top physics”
Phys.Rept. (2001) 1–222, arXiv:hep-ph/0006032 [hep-ph] .[50] T. Horiguchi, A. Ishikawa, T. Suehara, K. Fujii, Y. Sumino, et al. , “Study of top quark pair production nearthreshold at the ILC” arXiv:1310.0563 [hep-ex] .[51] W. Bernreuther, A. Brandenburg, and P. Overmann, “CP nonconservation in top quark production by(un)polarized e+ e- and gamma gamma collisions” arXiv:hep-ph/9602273 [hep-ph] .[52] F. Bach and M. Stahlhofen, “Private communication”.[53] T. Barklow, J. Brau, K. Fujii, J. Gao, J. List, N. Walker, and K. Yokoya, “ILC Operating Scenarios” arXiv:1506.07830 [hep-ex] ..