Longitudinal top polarisation measurement and anomalous $Wtb$ coupling

Kinematical distributions of decay products of the top quark carry information on the polarisation of the top as well as on any possible new physics in the decay of the top quark. We construct observables in the form of asymmetries in the kinematical distributions to probe their effects. Charged-lepton angular distributions in the decay are insensitive to anomalous couplings to leading order. Hence these can be a robust probe of top polarisation. However, these are difficult to measure in the case of highly boosted top quarks as compared to energy distributions of decay products. These are then sensitive, in general, to both top polarisation and top anomalous couplings. We compare various asymmetries for their sensitivities to the longitudinal polarisation of the top quark as well as to possible new physics in the $Wtb$ vertex, paying special attention to the case of highly boosted top quarks. We perform a $\chi ^2$- analysis to determine the regions in the longitudinal polarisation of the top quark and the couplings of the $Wtb$ vertex constrained by different combinations of the asymmetries. Moreover, we find that use of observables sensitive to the longitudinal top polarisation can add to the sensitivity to which the $Wtb$ vertex can be probed.


I. INTRODUCTION
The top quark is the heaviest of all fundamental particles discovered so far in the Standard Model(SM). Since the mass of the top quark (m t = 173.5 GeV /c 2 ) [1] is very close to the electroweak symmetry breaking (EWSB) scale, effects of any New Physics (NP) associated with EWSB are likely to reveal themselves in the properties of the top quark. The LHC, during the course of its runs, is expected to determine several of the properties of the top quark [2]. A comparison of these with expectations from the SM will reveal NP, if present. In the search for NP, there are already some results from the LHC, which include those on the top-quark polarisation and anomalous couplings in the W tb vertex [3,4], which are relevant to the discussion in this paper.
New Physics may appear in the production of the top quark or its decay or both [5]. A model-independent way of probing NP in the top sector is provided by the effective-theory formalism where all gauge-invariant higher-dimensional operators suppressed by powers of the corresponding scale of NP are added to the SM Lagrangian ( [6][7][8][9]). This description is valid at scales much lower than the NP scale. A complete set of dimensionsix operators relevant to top production and decay can be found in [9].
The top quark, on account of its large mass, decays before it hadronises, thereby transferring its spin information to the decay products. The angular and energy distributions of the decay products carry information on the spin of the top quark [10].
The polarisation of top quarks produced in a hadron collider like the LHC depends upon the hard subprocesses that produce the top quarks. Since QCD and QED mainly responsible for top-pair production in the SM are vector interactions, there is no significant polarisation of the top quarks pair produced in the SM. While the singletop production, occurring via electro-weak interactions at much lower rates, does give rise to polarised top quarks, it is clear that observation of substantial polarisation in top-pair production will strongly indicate NP. Any nontrivial chiral structure in the top coupling induced by the NP can affect the polarisation of the produced top quarks [11][12][13].Hence measurement of the polarisation of the top quark can provide information on the chiral structure of the couplings involved in NP contributions to top quarkproduction [11][12][13][14][15].
New physics reflects itself in changes in total and differential cross sections for top production. Detailed study of angular distributions of the decay products of the top quark, which are also affected by top-quark polarisation, provides a useful handle for discrimination between different NP models. Moreover, when NP couplings are small and the deviations of the total cross section from theoretical predictions in the SM can be small, the kinematic distributions and final-state polarisations being sensitive to the interference between the SM contribution and NP contribution can lead to increased sensitivity. The top-quark polarisation can, in addition, give a handle on the chiral structure of the couplings in NP.
A number of interesting scenarios for the production of top quarks occur once extensions of the standard model are introduced. The most popular extensions include supersymmetry, theories with extra dimensions and theories with extended gauge groups, all of which introduce new particles, which would contribute to top quark production in various ways: through an on shell production of resonances or via virtual effects. As said before, a nontrivial chiral structure of the top-quark couplings induced by the NP will lead to a prediction for top-quark polarisation which depends on the values of the parameters of that particular extension of the SM being considered.
Some of the NP models predict new heavy resonances with masses at the TeV scale [16]. Such heavy resonances are produced effectively at rest in the parton centreof-mass (cm) frame. When these heavy resonances de-cay into top quarks, the resulting top quarks are highly boosted in the lab frame. The decay products of these highly boosted top quarks are collimated along the direction of motion of the parent top quark. In such a case observables based on the energy distributions of the decay products rather than their angular distributions are more suitable to probe the polarisation of the top quark [17]. For such highly boosted tops methods based on jet substructure have been proposed to extract information on the polarisation of the top which then can be used to get information on the production mechanism of top quarks [18]. Recently a new method for measuring the polarization of top when the top decays hadronically has been proposed [19]. This method, involving a weighted average in the top rest frame of the directions of two light-quark jets that come from the decay t → bjj, has been shown to perform better than methods based on other hadronic top spin analyzers.
Data from the LHC has placed stringent lower bounds on the masses of resonances [20]. If they do exist at higher masses, the observation in the invariant mass distribution would be difficult. On the other hand, NP production amplitude of the resonance giving rise to a top pair could have sizable interference with the SM amplitude. This could lead to observable top polarization provided the NP couplings have a nontrivial chiral structure. Top polarization can serve as a tool in testing these models [12].
Another example of significant top polarisation is in stop decayt in the minimal supersymmetric standard model (MSSM), wheret 1 is the lightest stop andχ 0 i , i = 1,...,4 stand for the four neutralinos, which can be used to study mixing in the sfermion sector as well as the neutralino-chargino sector [11,13,21]. In R-parity violating MSSM, top quarks pair produced via a t-channel exchange of a stau or stop or a top quark produced in association with a slepton, can have nonzero polarisation, whose measurement can be used to constrain the R-parity violating couplings [11,22,23]. There have been several NP explanations of the forward-backward asymmetry of the top quark observed at Tevatron (see, for example [24][25][26]), and top polarisation can be useful in discriminating among them [14,27].
Since the top quark mainly decays through the channel t → W b with a branching ratio of ≈ 100%, any new physics which appears through the W tb vertex can affect the measurement of polarisation of top quarks which is determined by the production process. In general, measures of top polarisation have a dependence both on the strength and the tensor structure of the W tb coupling associated with top decay. Measures of top polarisation which depend only on the energy integrated angular distributions are insensitive to the anomalous part of the decay vertex W tb [28][29][30]. Recently another measure of top-quark polarisation has been proposed in [31] which factors out the the effect of any possible anomalous W tb vertex from the polarization of the top quark. The factor which contains the information about the W tb vertex of the top decay can be extracted in a model-independent way from the angular distributions of the top quark's decay products [32]. Since the anomalous tbW coupling also affects the kinematic distributions of the decay products of the top quark, it too can be probed by studying these and such probes have been constructed [13,[32][33][34][35][36][37][38][39][40][41].
Since the NP can affect top polarisation as well as give rise to anomalous decay vertex, it is of interest to explore how well one can study simultaneously both the top polarisation and the anomalous W tb couplings and further see how probes of one are influenced by the other. We present in this note some observations on construction of various observables as a measure of top polarisation and how one can simultaneously probe top polarization and the anomalous W tb coupling, when neither of the two is known a priori.
Studies of spin effects in top physics have largely concentrated on spin correlations in top pair production, as these are nonzero even in the SM and the measurements are interesting, even if no NP effects exist. A comparison of experimental results with SM predictions can then be used to constrain the NP models. The results so obtained at the Tevatron and the LHC have so far shown consistency with the SM, though errors are large. These correlations are best measured using leptonic final states from both top and anti-top. It is conceivable that a single polarisation measurement on either the top or the anti-top which decays leptonically, allowing the other to decay hadronically, could add to the accuracy.
Moreover, attempts to measure single-top polarisation at the Tevatron and the LHC have so far been made by reconstructing the rest frame of the top quark. A method which does not require such full reconstruction of the top may be desirable. We have thus concentrated on the measures of the polarization of a single top in the laboratory frame.
We construct various kinematic observables (asymmetries), make a comparative study of their dependence on top polarisation and anomalous W tb vertex, and examine the possibility of simultaneously constraining the anomalous W tb vertex and the polarisation of the top quark. Our observables do not always require full reconstruction of the top momentum. We do not look at any specific top production mechanism, but simply consider the top quark to be produced in the lab frame with various momenta, paying special attention to highly boosted top quarks.
Our paper is divided into four sections. In section I we describe the structure of W tb vertex and constraints on various anomalous couplings. In section II we make a comparative study of the sensitivities of different asymmetries to the polarisation of the top and anomalous W tb couplings. In section III we use these asymmetries to constrain simultaneously the polarisation of the top quark and the W tb vertex. In section IV we give a summary and conclusions. 1

II. THE STRUCTURE OF W tb VERTEX
The W tb vertex in the SM has a V − A structure. Depending upon the NP the structure of W tb vertex can be modified from the V − A structure [42]. We follow a model-independent approach by writing down the most general W tb vertex [32]: where g is the SU (2) L gauge coupling constant, p t , p b are the four-momenta of the top and the bottom quarks respectively, and P L , P R are the left and right chiral projectors. In the SM, f 1L = 1 and f 1R = f 2L = f 2R = 0 at tree level. We take the CKM matrix element V tb ≈ 1.
Similarly, the vertextWb with anomalous couplings is given by [41]. Direct searches of NP in top decay at the Tevatron give constraints on the coefficients: |f 1R | 2 < 0.30, |f 2L | 2 < 0.05, |f 2R | 2 < 0.12 at 95 % C.L assuming f 1L = 1 [43]. Indirect constraints from the measurement of the branching ratio of b → sγ are stronger for f 1R , f 2L and weaker for f 2R :−0.15 ≤ Re(f 2R ) ≤ 0.57, −0.0007 ≤ f 1R ≤ 0.0025, −0.0013 ≤ f 2L ≤ 0.0004 [44] at 95% C.L. Direct search constraints are given also by the LHC: for f 1L = 1, f 1R = f 2L = 0 the CMS collaboration [4] obtained as a best fit value of the tensor part of the W tb coupling which in our notation reads as −f 2R = −0.070 ± 0.053 (stat) +0.073 −0.081 (syst) in a fit to measured W helicity fractions proposed by [10]. When CP is conserved, the constraints on the anomalous couplingsf 1L ,f 1R ,f 2L andf 2R are the same as those for In the analytical expressions for different kinematic distributions (see below) we have assumed that the 1 In this work all kinematic quantities in the rest frame of the top quark are denoted by a subscript '0'. All kinematic quantities which do not have subscript '0' are in the laboratory frame (lab frame), unless stated otherwise. The top rest frame is chosen such that the z-axis is along the top-quark direction of spin. The lab frame is obtained by a boost along the z-axis. Therefore in the lab frame, the direction of motion of top corresponds to the direction of top spin in the top rest frame. We use the word lepton to denote the charged anti-lepton¯ from t → b¯ ν.
anomalous couplings f 1L andf 1L are real valued while all other anomalous couplings are complex valued. However, in view of the strong constraints on the anomalous couplings, in our numerical work on W tb vertex, we set f 1L = 1, f 1R = f 2L = 0 and take f 2R as a real valued quantity varying in the range −0.2 to +0.2.

III. KINEMATIC DISTRIBUTIONS OF THE DECAY PRODUCTS OF THE TOP
Recall that a measurement of the polarisation of the top quark can only be done through the kinematic distributions of its decay products and these would also be affected by the nature of the W tb coupling.
We begin by looking at the details of the three-body decay of the top quark. The top quark decays into a b quark and a W boson which in turn decays into a charged anti-lepton (¯ ) and a neutrino ν . We assume that all the particles in the decay chain t → bW → b¯ ν are onshell (including the intermediate W boson). The angular distributions of the decay products are correlated to the polarisation P of the top quark. In the SM, in the rest frame of the top quark, the energy integrated distribution is given by where X = b, , W, ν , the quantity α X is called the spinanalysing power of the particle X and θ X is the angle between the direction of the momentum of the particle X and the top quark spin axis in the rest frame of the top quark. The spin-analyzing powers of the b quark, lepton and the neutrino in the SM at tree level are

A. Kinematics of the top decay
Before proceeding to the description of asymmetries, we give a brief description of the kinematics of the top quark decay: The conservation of energy and momentum and the on-shell condition of W give the following equations: and where p t , p b , p , p ν are the four-momenta of the particles. Solving these equations along with the on-shell condition of the particles fixes all but four variables in the rest frame of the top quark: energy of the lepton E ,0 , the polar and azimuthal angles of the lepton θ ,0 , φ ,0 and the azimuthal angle (α 0 ) of the b quark with respect to a coordinate system where the z azis is along the direction of the lepton momentum. We have energy of the b quark cosine of the angle between the b-quark momentum and the lepton momentum cos ζ = The energy E ,0 of the lepton is constrained to vary between m 2 W /2m t and m t /2.

B. Definition of asymmetry
Asymmetry in a kinematic variable X is defined (in a given frame) as where dΓ dX is the differential partial decay width of the top quark in the variable X and X c is a value of X between [x min , X max ] chosen as a reference point about which the asymmetry is evaluated.
The asymmetries vary in their sensitivity to the polarisation of the top quark and the anomalous coupling f 2R , and in their usefulness in a particular kinematic regime of top decay. We describe four such asymmetries in this section. We can classify them into broadly two categories: angular asymmetries and energy-based asymmetries. Examples of the former include A θ and of the latter include A x , A u , A z . When the top quarks are highly boosted, the decay products of the top are highly collimated. In this case measurement of angular distribution of visible decay products is possible but difficult [46]. Hence energy-based asymmetries are used to probe the top-quark polarisation [17] and W tb vertex.
The asymmetry is defined in terms of cos θ where θ is the angle between the momentum of the lepton from the W decay and the top quark direction of motion. The cos θ distribution is sensitive to the polarisation of the top quark (P ). In the rest frame of the top quark the distribution is given in the SM by This expression is valid even when the anomalous coupling f 2R is non zero provided it is small [28][29][30]. In the lab frame the cos θ distribution becomes [13] 1 Γ where t = cos θ is the cosine of the angle between the top quark direction of motion and the lepton momentum in the lab frame. In eq. 9, the factor X in the denominator of the right-hand side is given by and β is the boost required to go from the lab frame to the top-quark rest frame. When |f 2R | 1, keeping only terms which are first order in f 2R , we get which is completely independent of the anomalous coupling f 2R . In other words, the energy-integrated distribution 1/ΓdΓ/d cos θ is only very very weakly dependent on the anomalous coupling f 2R . Therefore the lepton angular asymmetry (A θ ) serves as a useful measure of polarisation of the top quark irrespective of NP effects at the decay vertex when they are small [30].
In the SM, the asymmetry about the point cos θ = 0 is given in the lab frame, using eq 7, by From this equation one can easily observe that the sensitivity of A θ to the polarisation of the top quarks decreases with increasing boost. This can be understood as follows: In the rest frame of the top quark, the lepton is preferentially emitted either in the forward direction or the backward direction depending upon the sign of the polarisation of the top quark (P ) (equation 8). But in the lab frame, at large values of boost, the lepton emission is strongly suppressed except in the direction cos θ = 1 due to kinematics which appears through the factor (1 − β 2 )/(1 − βt) 3 in eq 9 for all values of polarisation P . This means that the lepton angular distribution loses its sensitivity to P at large boosts as shown in eq 12.

D. The x asymmetry (Ax )
The variable x is defined as x = 2E /m t where E is the energy of the lepton from the top decay in a given frame. Unlike the θ ,0 distribution, the x ,0 distribution is not insensitive to f 2R in the top quark rest frame. The 1: The x distribution in the lab frame for different values of the polarisation of the top quark (P ) and the anomalous coupling (f 2R ).
analytical expression for the distribution (1/Γ)dΓ/dx ,0 in the top quark rest frame is given by where X is given in eq 10. In the case when |f 2R | 1 the distribution is not independent of f 2R as the factors that are linear in f 2R do not cancel each other from the denominator and the numerator of eq 13. The distribution 1/ΓdΓ/dx is plotted in fig 1 for different values of top polarisation P and anomalous coupling f 2R . The location of the peak of the distribution for a given top polarisation P depends upon the anomalous coupling f 2R as can be seen from the figure. It would be convenient if the value of x at the position of the peak is chosen as the reference point to evaluate the asymmetry A x . In view of the fact that this point varies with P as well as f 2R , for uniformity we take the value of x corresponding to the peak of the distribution for P = −1 and f 2R = 0 as a reference point of all values of P and f 2R .
The above equation also shows that the x ,0 distribution is independent of P in the rest frame of the top quark. Therefore the asymmetry A x ,0 has no sensitivity to the polarisation of the top quark (P ) in the top quark rest frame. But under a Lorentz transformation along the top quark direction of motion which takes the top quark rest frame to the lab frame, the energy of the lepton in the lab frame gets related to both the energy and the polar angle θ ,0 of the lepton measured in the top quark rest frame: Since the distribution in θ ,0 is correlated to the top quark polarisation (P ) (see eq 8) , the distribution in E (or x ) becomes dependent on P . Hence the asymmetry A x for β = 0 depends on the polarisation of the top quark (P ).
A variable similar to x has been proposed in [47]. It is defined as x B = 2E /E t and it is related to x by a boost dependent factor: x B = 1 − β 2 x /2. However, the asymmetry constructed out of x B -distribution is the same as the asymmetry A x at any given value of β.

E. The u Asymmetry (Au)
The variable u is defined as u = E /(E + E b ) where E and E b are the energies in the lab frame carried by the lepton and the b quark respectively [17]. The variable u can be written as where x ,0 = 2E ,0 /m t ,E ,0 and θ ,0 are the energy and the angle between the top quark direction of motion and the momentum of the lepton measured in the top quark rest frame respectively. cos θ b,0 is given by with cos ζ = 2−x ,0 (ξ+1) x ,0 (ξ−1) and 0 ≤ α 0 ≤ 2π,(1/ξ) ≤ x ,0 ≤ 1. u varies in the range (0, 1). The u distribution is given by where X is as defined in eq 10, and α 0,i ,(i = 1, 2) are the roots of the equation u = u(x ,0 , θ ,0 , α 0 ). Since u is invariant under the transformation α 0 → 2π − α 0 , we have two solutions: α 0,1 and 2π − α 0,1 with 0 ≤ α 0,1 ≤ π. The function J(α 0,i ) is given by where cos α 0,i = (1 + β cos θ ,0 cos ζ) β sin θ ,0 sin ζ This equation determines α 0,i which can be used to get the value of the distribution at u. The asymmetry A u is calculated the point u = u c = 0.5 using eq 7. We note that the u-distribution is independent of Im(f 2R ) to linear order: the integrand of eq 16 actually has an additional term that is proportional to P (1 − x )x sin α sin θ sin ζ. Since the u-distribution is obtained after summing over two values of α i.e α 0,1 and 2π − α 0,1 , this additional term does not give any contribution.
The u distribution for different value of P and f 2R is given in fig 2. F. The z asymmetry (Az) The variable z is defined as z = E b /E t where E b and E t are the energies in the lab frame carried by the b and t quarks respectively [17]. The variable z can be related to the variables defined in the rest frame of the top quark: where cos θ b,0 is as defined in eq 15. Since cos θ b,0 varies in the range [−1.0, 1.0], the variable z varies in the range [(1 − β)(ξ − 1)/2ξ, (1 + β)(ξ − 1)/2ξ]. The z distribution is given by The z distribution is plotted in fig 3 for different values of the top polarisation P and the anomalous coupling f 2R . One can easily observe that the effect of the anomalous coupling f 2R is to change the slope of the z distribution which will be explained below.
Since the distribution is linear in z, an analytical expression for the asymmetry can be easily found. We take as the reference point z c , the value of z which corresponds to cos θ b,0 = 0 i.e z c = (ξ − 1)/2ξ. To simplify the notaion let us define two functions of f 2R : . Then the distribution in z can be rewritten in terms of t b,0 = cos θ b,0 (see [13,40]): Now changing the variable from z to t b,0 we get the limits of the integration in 7 as t b,0,min = −1 and t b,0,max = 1.
Only the term linear in t b,0 survives in the numerator and the expression for A z is given by Therefore the asymmetry A z is directly proportional to the top-quark polarisation P and is independent of the boost factor β as V and X are independent of both P and β. Moreover the z distribution can be directly related to the angular distribution of the b quark in the top rest frame due to eq 19. In fact, substituting the relation 19 in 21, we get, where α is the spin-analysing power of the b quark in the presence of anomalous W tb coupling f 2R . It includes correction to the SM tree level value of α b . To second order in f 2R , α can be written as Substituting the values of m t = 173.5GeV/c 2 and m W = 80.385GeV/c 2 [1], we get the numerical value of α as Since the coefficiets of f 2R and f 2 2R are much greater than the constant term in the above equation and the alternate terms differ in their signs, cancellation between different terms can occur. Therefore the effect of the anomalous coupling f 2R on the z distribution is non-trivial. As an observable based on the ratio of the energy of the top quark decay products (in the lab frame), A z can be used along with A u to probe top-quark polarisation at large boosts.

G. CP violation in the top decay
Here we note that using the asymmetries constructed for the t andt decay, one can probe CP violation in the When the production process is CP-conserving, polarizations of the top (P ) and the anti-top (P ) are related:P = −P . In this limit, the difference in the u-asymmetries of the top and the anti-top decay is proportional to Re(f 2R ) − Re(f 2L ) to linear order in the anomalous couplings (see eq 16). The coefficient of proportionality is a function of top polarization (P ) and the boost β. Heref 1L is set to unity,f 1R andf 2R to zero. Similarly, the difference in z-asymmetries of the top and the anti-top is proportional to Re H. Sensitivity of the asymmetries to P and f2R The dependences of various asymmetries on the polarisation of the top quark are compared in fig 4. One can observe that for moderate values of boost β ∼ 0.5 all the four asymmetries are sensitive to the top polarisation while for large values of boost, A u and A z are more sensitive as compared to A x and A θ . For very small values of boosts (β ≈ 0), the angular asymmetry A θ has the highest sensitivity to the top-quark polarisation (P ) due the fact that the charged lepton has the maximal spin-analysing power. A z follows A θ in the sensitivity to P for β ≈ 0 as it is directly proportional to the spin-analysing power of the b quark (α). This is true as long as the value of the anomalous coupling f 2R does not reduce the value of |α|. From equation 24 or from the plots of A z in fig 5, one can easily see that the value of α monotonically increases with f 2R in the range [−0.2, 0.2]. Therefore as a measure of top quark polarisation, A z is better for negative values of f 2R than for positive values. In a detailed comparison, for β ≈ 0, A z turns out to be the second best in the sensitivity to P , surpassed only by A θ .
An additional result of the comparison is that the sensitivity of A u to P is higher than that of A x in general (see fig 4).
Regarding the sensitivity of the asymmetries to f 2R , an interesting interplay of the top-quark polarisation and the anomalous coupling f 2R can be seen from fig 5. For large boost values (β ∼ 1) A θ and A x have similar sensitivities to f 2R (for small values of f 2R ) which is clearly shown in fig 5. In the case of A z , a non-zero polarisation of the top quark (P ) is necessary to probe the anomalous coupling f 2R since the asymmetry is directly proportional to P (subsection III C). Moreover A z is independent of β as mentioned above (subsection III F). This makes A z a suitable probe of f 2R for all values of beta as long as P = 0. When P = 0, A u and A x can be used to measure f 2R although the sensitivity of A x to f 2R (for small f 2R ) is low at large values of boosts ( fig 5). The angular asymmetry is not suitable to measure f 2R (as long as f 2R is small) for any value of the boost as the spin analysing power α is insensitive to f 2R (subsection III C). From fig 5 one can say that for large boosts, A u can be used to measure f 2R irrespective of the top-quark polarisation P . Therefore A u is the only observable that can be used to measure f 2R at large boosts, for any production mechanism of the top quark.

IV. CONSTRAINING P AND f2R SIMULTANEOUSLY
When f 2R = 0 in the W tb vertex, the asymmetries considered above are affected by the anomalous coupling f 2R along with the polarisation P of the top. Therefore with the measurement of an asymmetry one constrains a region in the two-dimensional P -f 2R plane. In this section we compare the asymmetries based on the region each one constrains in the P -f 2R plane assuming a plausible set of values of asymmetries expected to be measured at the LHC. We also discuss combining these asymmetries in a χ 2 -based analysis.

A. Method of analysis
We assume that the statistical error associated with the measurement of an asymmetry A is given by where N is the number of top quarks in the sample of measurement. The number N is estimated from the expected number of top quarks produced at the LHC from heavy resonances with invariant masses of O(TeV) decaying into a top pair. Based on the estimated differential cross section for top-pair production calculated in QCD [48] for the LHC at √ s = 7 TeV in the invariant-mass window of 1.0 TeV and 1.2 TeV, we take the number of top quarks as N = 1.47 × 10 4 for an integrateed luminosity of 100 fb −1 . This number is obtained under the assumption that the top quark decays semileptonically t → b¯ ν with = e, µ and the anti-top decays hadronically. Theoretically an asymmetry is a function of P and f 2R and the factor β is close to unity as we consider only those top quarks which are highly boosted in the lab frame. In fact, the boost values of the top quarks produced in the above-mentioned invariant-mass window are in the range 0.94 to 0.96. As we intend to keep our analysis a qualitative one, we fix β to 0.9.
Suppose that an experimental measurement of A corresponds to a true value (P 0 , f 2R0 ) of the parameters P and f 2R . This measurement corresponds to an unknown point (P 0 , f 2R0 ) in the P -f 2R plane. We define a region of siginificance f around the point (P 0 , f 2R0 ) as the region where the value of the asymmetry A(P, f 2R ) is indistinguishable from the experimental value A exp to within f times the error in the measurement ∆A. In other words, Since our purpose in this paper is to demonstrate the use of asymmetries, we choose a value for P 0 and f 2R0 and evaluate ∆A exp from the formula for ∆A given above in eq 25 and A from the expressions of the corresponding distributions derived in the previous section. Similarly A(P, f 2R ) in eq 26 is calculated from the expressions of the corresponding distributions given in the previous section. The results are shown in fig 6. B. χ 2 analysis We combine three of the four asymmetries to make a χ 2 statistic. We assume that the asymmetries are measured independently and their errors are given according to eq 25. There are four ways in which three of the asymmetries A x , A θ , A u , A z can be combined. We discuss each of the combination. The χ 2 is defined by where i = x , θ , u, z. Since our purpose is to demonstrate the utility of combining asymmetries, we calculate A exp for a "true" value of P and f 2R i.e P 0 and f 2R0 and also ∆A exp using eq 25. We give contours of ∆χ 2 values 2.30 and 5.99 corresponding to 68.3% and 95% confidence level (C.L) (for 2 degrees of freedom) respectively. As in the previous section we set β = 0.9 and use the same number of events N . Fig 7 shows the ∆χ 2 contours for four different combinations of asymmetries for β = 0.9. Table I gives the upper bound obtained on P when f 2R = 0.0 for different combinations of asymmetries for β = 0.9. When the true value of P and f 2R are P 0 = 0, f 2R0 = 0 the combination of A θ ,A u and A z and A x , A u , A θ are better in constraining both P and f 2R than the other two combinations. Asymmetry Dependence on f 2R , P =0.5, β =0.9 x θ u z

FIG. 5:
Comparison of the f 2R dependence of various asymmetries for different values of P and f 2R for a boost factor β = 0.9. In each plot the u asymmetry is given in dot-dashed lines, the x asymmetry in solid lines, the θ asymmetry in dashed lines and the z asymmetry in dotted lines respectively.

C. Limits on f2R
In the table II and III we summarise the limits obtained on the anomalous coupling f 2R for two values of polarisation P = 0 and P = −1.0. The best 1σ limits on f 2R obtained in our analysis assuming the top polarisation to be zero is [−0.079, 0.069]. The sensitivity increases considerably if, for example, the expected polarization of the top would be −1.0. The corresponding limit on f 2R is: [−0.017, 0.006].
Note that we have analyzed the case of tt pair production using events with tt invariant masses in the range 1. tt events over the entire range of the invariant mass. Due to the lower statistics our limits on f 2R for the case of zero polarisation are weaker compared to the limits obtained in [32]. But they are compatible with the CMS measurement of −f 2R :−0.07 ± 0.053(stat) +0.073 −0.081 (syst) using 2.2 f b −1 of integrated luminosity at √ s = 7 TeV. Note that our results are compatible with the result obtained by the CMS experiment with 2.2 fb −1 luminosity (corresponding to a number of events N ∼ O(10 4 )) and hence smaller number of tt events than the ATLAS analysis, using the tt events with invariant-masses over the whole allowed range. This gives us confidence that the various limits indicated in this report are representative of what can be achieved in a real analysis.

V. SUMMARY
In this work we have taken up the study of observables constructed out of kinematical variables of top decay products for the purpose of measuring top polarization in the presence of anomalous W tb couplings as well as measuring the anomalous coupling f 2R itself. We concentrate on laboratory-frame variables which do not require the reconstruction of the top rest frame. An important consideration has been the degree of boost of the decaying top, since for many practical processes, as for example, a heavy resonance decaying into a top pair, the top quark is produced with large momentum in the lab frame.
We have considered four observables -asymmetries in the variables θ , u, x and z. They are compared for their sensitivities to the polarisation of the top quark and the anomalous coupling f 2R . We state the results of the comparison of asymmetries in two categories: 1. Asymmetries for the measurement the top-quark polarisation P , and 2. Asymmetries for the measurement of the anomalous coupling f 2R .
As for the first category of asymmetries for the measurement of the top-quark polarisation, for small values of boost from the top quark rest frame to the lab frame (β ≈ 0), A θ is the most sensitive observable. Next in sensitivity is A z as long as f 2R is small or negative. For large values of boosts (β ∼ 1), A u and A z can be used as they are much more sensitive to P compared to A x and A θ .
For the second category corresponding to the measurement of the anomalous coupling f 2R , for all values of β, The boost factor is set to β = 0.9. The "true" values of P and f 2R are P 0 = 0.0 and f 2R0 = 0.0 respectively.
A z can be used to measure f 2R as long as P = 0. For P = 0, A u , A z can be used to measure f 2R for any β.
The angular asymmetry A θ is not suitable as a measure of f 2R for any value of β as its sensitivity to f 2R is much more smaller than the sensitivities of A u and A z . Irrespective of the production mechanism of the top quark, A u can be used to measure f 2R at large values of the boost β.
In all cases, we determine the 1σ and 2σ limits that the measurement of asymmetries can put on the determination of the polarisation or f 2R with a chosen number of events. We also do an analysis of the use of combination of asymmetries for the simultaneous determination of the top polarisation as well as f 2R .