Study of top-quark polarization in single-top-quark production at the CERN LHC

This paper complements the study of single top-quark production at the CERN LHC aiming to estimate the sensitivity of different observables to the magnitude of the effective couplings. In a previous paper the dominantW-gluon fusion mechanism was considered, while here we extend the analysis to the subdominant ~10% with our set of experimental cuts ! -channel process. In order to distinguish left from right effective couplings it is required to consider polarized cross sections and/or include mb effects. The spin of the top quark is accessible only indirectly by measuring the angular distribution of its decay products. We show that the presence of effective right-handed couplings implies necessarily that the top quark is not in a pure spin state. We discuss to what extent quantum interference terms can be neglected in the measurement and therefore simply multiply production and decay probabilities classically. The coarsening involved in the measurement process makes this possible. We determine for each process the optimal spin basis where theoretical errors are minimized and, finally, discuss the sensitivity in the s channel to the effective right-handed coupling. The results presented here are all analytical and include mb corrections. They are derived within the narrow width approximation for the top quark.


I. INTRODUCTION
At present not a lot is known about the Wtb ¯effective coupling.This is perhaps best evidenced by the fact that the current experimental results for the ͑left-handed͒ K tb matrix element give ͓1͔ ͉K tb ͉ 2 ͉K td ͉ 2 ϩ͉K ts ͉ 2 ϩ͉K tb ͉ 2 ϭ0.99Ϯ0.29.͑1͒ In the standard model this matrix element is expected to be close to 1.It should be emphasized that these are the ''measured'' or ''effective'' values of the Cabibbo-Kobayashi-Maskawa ͑CKM͒ matrix elements, and that they do not necessarily correspond, even in the standard model, to the entries of a unitary matrix on account of the presence of radiative corrections.These deviations with respect to unitary are expected to be small-at the few percent level at most-unless new physics is present and makes an unexpectedly large contribution.At the Fermilab Tevatron the lefthanded couplings are expected to be eventually measured with a 5% accuracy ͓2͔.
As far as experimental bounds for the right handed effective couplings are concerned, the more stringent ones come at present from the measurements on the b→s␥ decay at CLEO ͓3͔.Because of an m t /m b enhancement of the chirality flipping contribution, a particular combination of mixing angles and effective right-handed couplings can be bound very precisely.The authors of ͓4͔ reach the conclusion that ͉Re(g R )͉р0.4ϫ10Ϫ2 .However, considering g R as a matrix in generation space, this bound only constrains the tb element.Other effective couplings involving the top remain virtually unrestricted from the data.The previous bound on the right-handed coupling is a very stringent one.It should be obvious that the CERN Large Hadron Collider ͑LHC͒ will not be able to compete with such a bound.Yet, the measurement will be a direct one, thus ruling out some contrived models where substantial cancellations might hypothetically avoid the b→s␥ constraint.For the value of the effective couplings in some specific models see e.g.͓5͔.
At LHC energies the mechanism underlying single topquark production, therefore allowing a direct test of the Wb ¯t effective couplings g L and g R , consists of several different processes ͑see e.g.͓6͔͒.The dominant process 1 is the socalled W-gluon fusion channel, or t-channel process.The electroweak subprocesses corresponding to this channel are depicted in Fig. 1, where light u-type quarks or d ¯-type antiquarks are extracted from the protons.Besides this dominant channel ͑250 pb at LHC ͓7͔͒ single top quarks are also produced through the process where the W ϩ boson interacts with a b quark extracted from the sea of the proton ͑50 pb͒ ͓7͔ and in the quark-quark fusion or s-channel process ͑10 pb͒ which is depicted in Fig. 2. The numbers quoted here *Email address: espriu@ecm.ub.es correspond to total cross sections.The separation between the subdominant processes and the dominant W-gluon fusion is purely kinematical ͓7,8͔.By placing a cut on the p T of the detected b ¯quark, the former process can be eliminated altogether.This also eliminates a sizeable fraction of the top quarks produced via the W-gluon fusion mechanism ͑about two thirds for the cuts we use͒.The cut on p T has the additional bonus of making the QCD corrections manageable.One is therefore left with those single top quarks coming from the W-gluon fusion mechanism (t-channel͒ and the subdominant s-channel process.The latter one is actually the main object of our interest in this article, although we will also have many comments to make on the t-channel process.
In a proton-proton collision a bottom-top-antiquark pair is also produced through analogous subprocesses.The analysis of such top antiquark production processes is similar to the top quark ones and the corresponding cross sections can be easily derived doing the appropriate changes.
In a previous paper ͓8͔ we have analyzed the sensitivity of different LHC observables to the magnitude of the charged current effective couplings considering only the dominant W-gluon fusion channel.In that work we did not consider the subsequent decay of the top quark in any detail.We did, however, a complete analytical calculation of the subprocess cross sections, for general left and right effective couplings and including bottom mass corrections.A p T Ͼ30 GeV cut in the transverse momentum of the produced b ¯quark was implemented in ͓8͔ and, accordingly, only the so-called 2 →3 process was retained, excluding top-quark production off a b quark from the proton Fermi sea.Given the ͑pre-sumed͒ smallness of the right handed couplings, the bottom mass plays a role which is more important than anticipated, as the mixed crossed g L g R term, which actually is the most sensitive one to g R , is accompanied by a b quark mass.The reader is encouraged to see ͓8͔, where a very detailed analysis is presented.
Typically the top quark decays weakly well before strong interactions become relevant, so we could in principle ''measure'' its polarization state with virtually no contamination of strong interactions ͑see e.g.͓9,10͔ for discussions of this point͒ and try to establish interesting observables based on this measurement.In fact it is not difficult to convince oneself that in order to disentangle left from right effective couplings, it is almost compulsory to be able to ''measure'' the polarization of the top quark.This will become apparent from the formulas presented in Sec.II.For this reason we have derived in this work and in ͓8͔ analytical expressions for the cross sections for the production of polarized top quarks or top antiquarks.To this end one introduces the spin projector as the polarization projector for a particle or anti-particle of momentum p 1 with spin in the n ˆdirection.The calculations of the subprocess cross sections have been performed in this work and in ͓8͔ for an arbitrary polarization vector n ˆ.Obviously, however, the top quark decays very shortly after production, so the only practical way one can measure the spin of the top quark is through its influence on the angular distribution of the leptons produced in the decay.It is tacitly assumed in most of the works published on this subject that the decaying top quark is in a pure spin state for all practical purposes; i.e. its polarization vector is pointing in a particular direction in space in a given reference frame.
In the tree-level standard model this is not quite true, but it is almost true.The tree level standard model corresponds in our notation to taking g L ϭ1 and g R ϭ0.Imposing the cut on p T we have mentioned, only two subprocesses contribute: W-gluon fusion and the s-channel process.The latter provides 100% polarized top quarks in a certain direction ͑to be discussed later͒.The situation in the t-channel process is a bit more complicated.The results from our previous analysis presented in ͓8͔ show that single top-quark production is highly, but not fully, polarized in this case too ͑84% in the optimal basis, with the present set of cuts͒.This is a high degree of polarization, but still well below the 90ϩ claimed by Mahlon and Parke in ͓10͔.We understand this as being due to the presence of a 30 GeV cut in p T .In fact, if we remove this cut completely we get 91% polarization, in rough agreement with ͓10͔ ͑note that we do not include the 2→2 or b-sea process͒.Inasmuch as they can be compared, our results for the tree-level standard model are in good agreement with those presented in ͓7͔ in what concerns the total cross section.These considerations are quite independent of the choice of the strong subtraction scale, which is by far the largest source of uncertainty.2Let us assume now for the sake of discussion that the polarization is indeed 100%.The top quark subsequently decays ͑say emitting a positively charged lepton͒.One can compute the angular probability distribution of the lepton with respect to the polarization direction in the standard model, multiply the two probabilities and compare the experimental result with the theoretical prediction.
In fact things are a lot more subtle.First of all, we have seen that even in the standard model, polarization is never 100%.Furthermore, it turns out that when g R 0, i.e. beyond the standard model, the top can never be 100% polarized ͑see the discussion in Sec.II and in ͓8͔͒, not even in principle.In other words, the top quark is necessarily in a quantum mixed state and is described by a density matrix.The entries of this density matrix depend on the momenta of the incoming and outgoing particles; that is to say, there is an entanglement between spin and momenta.
Of course this complication amounts to a small effect because g R is surely quite small, even in most models beyond the standard model, so in a first approximation the experimental consequences should be small.However, if our purpose is precisely to measure g R or at least to set a bound on it, it is clear that the effect needs to be taken into account.
The next step is to select the direction where one is to ''measure'' the spin of the top.By tracing the appropriate spin operator with the density matrix one would determine the expected probabilities of finding a top that ͑after the mea-sure͒ would point in the given direction of our choice.There is a privileged spin basis, namely the one where the density matrix is diagonal, where the calculation is greatly simplified since one need not compute the off-diagonal terms.This diagonalization process has to be done event by event and it selects a particular vector n ˆ͑event dependent͒.In sec.V we provide explicit formulas for this privileged direction.Elementary quantum mechanical considerations show that this is also the direction where the differential cross section is maximal ͑or minimal depending on the sign of the spin͒.Using this 3-vector as spin basis, for instance, one can multiply the probability of producing a top quark polarized in the positive ϩn ˆdirection with the corresponding decay angular probability distribution plus the probability of producing a top quark polarized in the negative Ϫn ˆdistribution times the corresponding decay angular probability distribution.The dependence on the effective left and right couplings g L and g R is obviously contained in the density matrix and also in the decay distributions.
Obviously, since the entries of the density matrix depend on the spin basis, the final physically observable result of the previous analysis will certainly depend on the spin basis too.How is this possible?In fact this is as it should be; we are multiplying probabilities and in fact we are neglecting the quantum interference terms because we are assuming that the polarization of the top quark is measured in the intermediate state before the top quark decays.Then there should be no surprise in the fact that the interaction between the top quark and the apparatus measuring its spin modifies the final physical results.
However, a proper measure of the top quark spin before it decays is impossible; the only way we learn about top-quark polarization is precisely from the final decay products.So, the previous procedure is conceptually incorrect. 3The final result has to be strictly independent of the intermediate spin basis one uses.Does this mean that the usual procedure -which is the one we just described-is totally flawed?In principle yes, however one expects that the coarsening involved in the measuring process washes some or all of the interference effects.Then perhaps the previous procedure where one assumes that the spin of the top quark is well defined and one proceeds as if it could be measured before it decays it could be approximately correct.But what then are the errors involved?Do they jeopardize the determination of some of the effective couplings, in particular the distinction between g L and g R ?These are some of the issues we would like to address in the present work.

II. THE DIFFERENTIAL CROSS SECTION FOR POLARIZED TOP-QUARK PRODUCTION
We shall discuss here the t-channel production for the sake of definiteness.This is the most involved process.We refer the reader to ͓8͔ for detailed expressions of the different amplitudes.We denote the matrix elements of the hard subprocess of Fig. 1 by M ϩ d .There will also be an M ϩ u ¯, corresponding to having instead a u ¯as spectator quark.We will also eventually define the matrix elements corresponding to the processes producing top antiquarks as M Ϫ u , and With these definitions the differential cross section for polarized top quarks d can be written schematically as where f u and f d ¯denote the parton distribution functions corresponding to extracting a u-type quark and a d ¯-type quark respectively and ␤ is a proportionality factor incorporating the kinematics and also the gluon distribution function.Us-ing our analytical results for the matrix elements given in the Appendix of ͓8͔ we obtain for the differential cross section

͑3͒
where and where a, b, c, a n , b n , c n and d n are independent of the effective couplings g R and g L and the subscripts n indicate linear dependence on the top-quark spin four-vector n.All these quantities depend only on masses and momenta.The c, c n and d n terms are proportional to the bottom-quark mass and are therefore absent if one neglects m b ͑this at first sight does not look unreasonable, given the energies involved͒.Inspection of the above differential cross section reveals that in the m b limit, the only way to tell left from right effective couplings is precisely by considering and measuring polarized cross sections ͑the terms in a n ,b n ) unless one is willing to rely strongly on the parton distribution functions. 4For these reasons, both polarization and m b terms are quite important.
We observe that A is a Hermitian matrix and therefore it is diagonalizable with real eigenvalues.Moreover, from the positivity of d we immediately arrive at the constraints det Aу0, ͑5͒ TrAу0, ͑6͒ that is Note that it is not possible to saturate both constraints for the same configuration because this would imply a vanishing A which in turn would imply relations such as which evidently do not hold.Moreover, since constraints ͑7͒ and ͑8͒ must be satisfied for any set of positive parton distribution functions we immediately obtain the bounds In order to have a 100% polarized top we need a spin fourvector n that saturates the constraint ͑5͒ ͓that is Eq.͑7͔͒ for each kinematical situation, that is we need A(n) to have a zero eigenvalue which is equivalent to have a unitary matrix C satisfying for some positive eigenvalue .In general such n need not exist and, should it exist, is in any case independent of the effective couplings g R and g L .Moreover, provided this n exists there is only one solution ͑up to a global complex normalization factor ␣) for the pair (g R ,g L ) to the equation dϭ0.This solution is just Note that if one of the effective couplings vanishes we can take the other constant and arbitrary.However if both effective couplings are nonvanishing we would have a quotient g R /g L that would depend in general on the kinematics.This is not possible so we can conclude that for a nonvanishing g R (g L is evidently nonvanishing͒ it is not possible to have a pure spin state ͑or else only for fine tuned g R a 100% polarization is possible͒.
Let us now give a very simple example to make the previous discussion more understandable: in the unphysical situation where m t →0 it can be shown that there exist two solutions to the saturated constraint ͑5͒, namely

͑10͒
Once we have found this result we plug it in the expression ͑9͒ and we find the solutions (0,g L ) with g L arbitrary for the ϩ sign and (g R ,0) with g R arbitrary for the Ϫ sign.That is, physically we have zero probability of producing a right handed top when we have only a left handed coupling and vice versa when we have only a right handed coupling.Note that in this case it is clear that having both effective couplings nonvanishing would imply the absence of 100% polarization in any spin basis.This can be understood in gen-eral remembering that the top-quark particle forms in general an entangled state with the other particles of the process.Since we are tracing over the unknown spin degrees of freedom and over the flavors of the spectator quark we do not end up with a top quark in a pure polarized state.

III. CROSS SECTIONS FOR TOP-QUARK PRODUCTION AND DECAY IN THE s CHANNEL
Let us now turn to the s-channel process.This is, as already mentioned, subdominant but non-negligible since it roughly amounts to 10% of all single top quarks produced after our set of cuts are imposed.It is also a lot cleaner from a theoretical point of view, as QCD corrections are small.As a by-product we shall derive the differential decay width, which is applicable to both the t-and s-channel processes.
Using the momenta conventions of Fig. 2 and averaging over colors and spins of the initial fermions and summing over colors and spins of the final fermions ͑remember that we have included a spin projector for the top quark͒ the squared amplitude for top-quark production is given by

͑11͒
where g ˜L and g ˜R are left and right couplings to light quarks and g L and g R are the effective couplings to the top-quarkbottom-quark system.In the numerical results we have taken g ˜Lϭ1, g ˜Rϭ0; i.e. we stick to the tree-level standard model values in the light sector, but is quite straightforward to include more general couplings.Notice that, exactly as for the t channel, the crossed g L g R terms vanish in the differential cross section in the m b →0 limit.Also for exactly the same reasons as in the t-channel analysis, modulo parton distribution function effects, the differential unpolarized production cross section would be proportional to ͉g L ͉ 2 ϩ͉g R ͉ 2 .
The differential cross section for producing polarized top-quarks is then where f "x ˜1 ,x ˜2 ,(q 1 ϩq 2 ) 2 ,⌳ QCD …dx ˜1dx ˜2 accounts for the quarks parton distribution functions.The total decay rate of the top-quark, on the other hand, with arbitrary left and right effective couplings is given by The squared amplitude corresponding to the decay rate in the channel depicted in Fig. 2 summing over the top-quark polarizations ͑with a spin projector inserted͒, averaging over its color and summing over colors and polarizations of decay products, is given by where ͉M n ͉ 2 (q 1 →k 2 ,q 2 →k 1 , p ˜2→Ϫp 2 ) is just expression ͑11͒ with the indicated changes in momenta.In the above expression g ˜L and g ˜R are the left and right couplings corresponding to the lepton-neutrino vertex.We have assumed g ˜Lϭ1, g ˜Rϭ0, but again this hypothesis can be relaxed.The decay rate differential distribution for this channel is given by Finally, using the narrow-width approximation, we have that the differential cross section d corresponding to Fig. 2 is given by

IV. THE ROLE OF SPIN IN THE NARROW-WIDTH APPROXIMATION
Within the narrow-width approximation we just discussed we decompose the process depicted in Fig. 2 in two consecutive processes: the top-quark production and its consecutive decay.In that setup we denote the single top-quark production amplitude as A p,Ϯn ˆ(p) and the top-quark decay amplitude as B p,Ϯn ˆ(p) .In the polar representation we write where p indicate external momenta and n ˆ(p) a given spin basis for the topquark.The differential cross section for the whole process is schematically given by

͑15͒
Since the axis with respect to which the spin basis is defined is completely arbitrary d is independent of this choice of basis.However within the narrow width approximation one never computes d following formula ͑14͒.The commonly used procedure ͓10,14͔ consists in computing the probability of producing a polarized top quark and then multiplying this probability by the probability of a given decay channel ͓see Eq. ͑13͔͒.This procedure is equivalent to the neglect of the interference term in formula ͑15͒ as indicated there.First of all, as discussed in the Introduction, if one neglects the interference term, the result depends on the spin basis, i.e. on the direction one chooses to measure the third component of the top-quark spin.This is of course acceptable if one really performs a physical measure of the spin ͓in the n(p ˆ) direction in this case͔ since the interaction with the apparatus modifies the state.A dependence on the spin frame is however unacceptable if the spin is not measured before the top quark decays.
Let us see whether this approximation can be justified nevertheless.Clearly, the integration over momenta enhances the positive-definite terms in front of the interference oscil-lating one.If in addition we make a choice for n ˆ(p) that diagonalizes the top-quark spin density matrix and thus maximizes ͉A p,ϩn ˆ(p) ͉ and minimizes ͉A p,Ϫn ˆ(p) ͉, then we expect the interference term to be negligible when compared to ͉͐A p,ϩn ˆ(p) ͉ 2 ͉B p,ϩn ˆ(p) ͉ 2 dp even for a small amount of phase space integration.In the s channel we will see in the next section that in the limit of g R →0 there exists a spin basis n ˆ(p) where ͉A p,Ϫn ˆ(p) ͉ is strictly zero.This basis is given by nϭ 1 From this it follows that for small g R if we use that basis the interference integrand is already negligible with respect to the dominant term ͉͐A p,ϩn ˆ(p) ͉ 2 ͉B p,ϩn ˆ(p) ͉ 2 dp.For g R 0 one can still find a basis that maximizes ͉A p,ϩn ˆ(p) ͉ ͑and minimizes ͉A p,Ϫn ˆ(p) ͉) and therefore diagonalizes the top-quark density matrix .In the next section we will show how to obtain such a basis that will be the one used in our numerical integration.In these simulations we have checked numeri-cally that this basis is the one that maximizes d and therefore, on the same grounds, the one that minimizes the interference term.The same considerations can be applied to the t-channel process.
Given that the observables are strictly independent of the choice of spin basis only if the interference term is included, we can easily assess the importance of the latter by checking to what extent a residual spin basis dependence is present.We have checked numerically this point by changing the definition of the spin basis n ˆ( p) and noting that our results are actually only weakly dependent on the choice of n ˆ(p) even for a small amount of coarsening.A 4% maximum variation in p T distributions was found between the optimal diagonal basis and another basis orthogonal to the beam axis ͑that is, almost orthogonal to all momenta͒.Moreover we have checked that if spin is ignored altogether ͑by considering unpolarized top-quark production͒ roughly the same amount of variation with respect to the diagonal basis is observed.Thus we conclude that even though the dependence on the choice of spin basis is not dramatic, its consideration is a must for a precise description using the narrowwidth approximation taking into account the presumed smallness of the effective coupling to be measured and how subtle the experimental distinction of left and right couplings turns out to be.

V. THE DIAGONAL BASIS
As stated in the previous section, in order to calculate the top-quark decay we have to find the basis where the polarized single top-quark production cross section is maximal.We can do this maximizing in the 4-dimensional space generated by the components of n constrained by where p 1 is the top-quark four-moment, that is where ʈn ជ ʈϭͱ(n 1 ) 2 ϩ(n 2 ) 2 ϩ(n 3 ) 2 , that is n i ϭʈn ជ ʈn ˆi with n the normalized spin three-vector.From above equations we obtain from which Eq. ͑2͒ follows immediately.Let us now find the polarization vector that maximizes and minimizes the differential cross section of single top-quark production.

A. The t channel
We will begin with the t channel that was analyzed in the previous section.Using Eq. ͑3͒ we define and using Lagrange multipliers 1 and 2 for constraints ͑16͒ we maximize and thus using Eqs.͑18͒ and ͑20͒ and therefore with the normalization factor 1 given by Eq. ͑19͒.Note that in the idealized case f u ϭ f d ¯ϭ f we obtain where ␣ is the normalization constant that does not depend on f or the effective couplings.In the SM (g R ϭ0) we obtain where ␣ is a normalizing factor.

B. The s channel
The s-channel differential cross section has the form where again ␤ is a proportionality incorporating the kinematics, and where f u,c and f d ¯,s ¯denote the parton distribution functions corresponding to extracting a u,c-type quark and a d ¯,s ¯-type quark respectively.Using again the decomposition ͑17͒ and proceeding analogously to the t-channel calculation we obtain

͑21͒
where ␣ is the normalizing factor that in this case ͑unlike in the t-channel result͒ does not depend on the parton distribution functions.From Eq. ͑11͒ we obtain hence replacing in Eq. ͑21͒ we arrive at which is the basis we use in our numerical simulations.If we neglect g R we obtain where we have included the normalization factor and since q 2 2 ϭ0 the above reduces to which is the result we have quoted in the previous section coinciding with ͓10͔.

VI. NUMERICAL ANALYSIS OF s-CHANNEL SINGLE TOP-QUARK PRODUCTION
Let us start this section by discussing the experimental cuts we have implemented.Due to geometrical detector constraints ͓15͔ we cut off very low angles for the outgoing particles.The charged particles in the final state have to come out with an angle in between 10 and 170 degrees to be detected.These angular cuts correspond to a cut in pseudorapidity ͉͉Ͻ2.44.In order to be able to separate the jets corresponding to the outgoing particles we implement isolation cuts of 20 degrees between each other.These are the appropriate cuts for general purpose experiments such as AT-LAS or CMS.
The set of cuts used in this work is compatible with the ones used in the t-channel.Since in the previous paper ͓8͔ top-quark decay was not considered, the equivalence is only approximate and a more detailed phenomenological analysis will be required in due course ͑it is actually quite straightforward with the help of the results presented here to redo the t-channel study, but this goes beyond the scope of this paper͒.The present analysis should however suffice in any case to identify the most promising observables and get a rough estimate of the precision that can be reached.
We use a lower cut of 20 GeV in the b ¯jet.5This completely eliminates top-quark production from a b quark from the proton sea and greatly reduces higher order QCD contributions.In the t channel reduces the cross section to about one third of its total value, since typically the b ¯quark comes out in the same direction as the incoming gluon and a large fraction of them do not pass the cut.Similarly, p T Ͼ20 GeV cuts are set for the top quark and spectator quark jets.These cuts guarantee the validity of perturbation theory and will serve to separate from the overwhelming background of low p T physics.These values come as a compromise to preserve a good signal, while suppressing unwanted contributions.They are very similar to the ones used in ͓7͔ and ͓10͔.
In order to calculate the cross section of the process pp→tb ¯we have used the CTEQ4 set of structure functions ͓17͔ to determine the probability of extracting a parton with a given fraction of momenta from the proton.To calculate the total event production corresponding to different observables we have used the integrating Monte Carlo program VEGAS ͓18͔.We present results after one year ͑defined as 10 7 seg.)run at full luminosity in one detector (100 fb Ϫ1 at LHC͒.
Since in order to be able to perform the effective Wtb coupling one definitely needs to tag the two b-type quarks, this value for the luminosity is surely too high.The b physics program at ATLAS ͓19͔, for instance, is planned to be done at one tenth of the total luminosity to avoid pile-up effects.This is even more so in a dedicated detector such as LHCb6 ͓16͔ where the appropriate figure is expected to be 2 fb Ϫ1 .ATLAS plans to do most of the b-physics runs before full luminosity is reached, for instance.We have nonetheless used the high luminosity figure since at this stage the experimental strategy is not totally settled yet.
The way we proceed is the following.We analyze the kinematics of each event including a b ¯that passes the experimental cuts and reconstruct the vector n ˆusing the analytic formulas presented in the previous sections.As the reader will remember, this provides us with a spin basis that minimizes the quantum interference terms.We then proceed to multiply the probabilities classically-just as if we pretend that the top-quark spin has been measured in the direction determined by n ˆand we determine the decay probability distribution.We retain only those final states that pass the remaining cuts.
In the same way and choosing arbitrary spin directions we are able to see how much the physical results depend on the interference term.We have found a 4% difference between the worst case ͑a spin direction perpendicular to almost all 3-momenta involved͒ and the optimal case ͑found analytically here͒.We have every reason to believe that, after the integration over momenta and the resulting coarsening, the interference term in this basis is all but negligible.The rest of the results presented in this section are all worked out in the optimal spin basis.
Let us first review the results obtained in the framework of the tree-level standard model.This corresponds to taking g L ϭ1 and g R ϭ0 in all our formulas.The results are summarized in Figs. 3, 4, and 5.The first of these figures shows how the final products of the process are predominantly emitted in the axis direction ͑albeit not so much as in the case of the t-channel production͒ and in the same direction.The plot shows the direction with respect to the beam of bottom quarks and bottom antiquarks.Recall that a 10 degree cut is implemented, as well as a separation cut of 20 degrees among all jets.Figure 4 shows the p T distribution for the b ¯, showing the 20 GeV cut on the p T of the b ¯enforced.Figure 5 shows the invariant mass of the lepton and bottom-quark system in the tree level standard model.Since we are working in the narrow width approximation, the distribution falls to zero just below the physical mass of the top FIG. 3. Distribution of the cosines of the polar angles of the bottom quark and bottom antiquark with respect to the beam line.The plot corresponds to single top-quark production at the LHC with top-quark decay included.The calculation was performed at tree level in the standard model.For the parton distribution functions we use 2 ϭs ˆϭ(q 1 ϩq 2 ) 2 .quark and this reflects the part of the total momentum of the top quark carried away by the undetected neutrino.Figures 6 and 7 actually show the p T distribution for the bottom quark and lepton, respectively, that are produced in the top-quark decay.As previously discussed, 20 GeV cuts on the respective p T are imposed.Even though some information is lost by the fact that the neutrino is not seen and therefore there is some amount of missing momentum, this does not seem to affect the sensitivity to the effective couplings too much.One could as well consider channels in which the W ϩ , pro-duced in the top-quark decay, decays hadronically.In hadronic decays of the top quark a full reconstruction of the top-quark mass would be feasible.
Let us now move beyond the standard model.Since changing the value of g L ͑while keeping g R ϭ0) amounts to a simple rescaling, we shall concentrate on the more interesting case of varying g R .As a rough order-of-magnitude estimate for the effective g R coupling we take ͉g R ͉ϭ5 ϫ10 Ϫ2 .This is still worse than the limit implied by b →s␥, but is the sort of sensitivity that LHC will be able to set.The effects are linear in g R , so it is easy to scale the results up or down.We have considered the possibility of g R FIG. 4. Bottom-antiquark transversal momentum distribution corresponding to single top-quark production at the LHC.The calculation has been performed at tree level in the SM (g L ϭ1,g R ϭ0).
FIG. 5. Distribution of the invariant mass of the lepton ͑electron or muon͒ plus bottom-quark system arising in top-quark decay from single top-quark production at the LHC.The calculation was performed at tree level in the standard model with 2 ϭs ˆϭ(q 1 ϩq 2 ) 2 .FIG. 6. Bottom-quark transversal momentum distribution corresponding to single top-quark production at the LHC.The calculation has been performed at tree level in the SM (g L ϭ1,g R ϭ0).FIG. 7. Lepton ͑electron or muon͒ transversal momentum distribution corresponding to single top-quark production at the LHC.The calculation has been performed at tree level in the SM (g L ϭ1,g R ϭ0).
having a phase and, accordingly, the experimental sensitivity to that phase.
We have found that the anti-lepton plus bottom-quark invariant mass distribution we just discussed in the previous paragraph is actually sensitive to g R .Figures 8 and 9 reflect this sensitivity with the second figure showing the statistical significance per bin.
We shall now show the dependence of the three p T distributions (b, b ¯and lepton͒ to the modulus and phase of the effective coupling g R .In all cases the value g L ϭ1 is taken.The sensitivity to departures from the tree level SM is shown in Figs. 10, 11 and 12.We also include the statistical significance per bin for the signal vs cos( tl ) in Fig. 13 and vs cos( tb ) in Fig. 14. cos( tl ) and cos( tb ) are the cosines of the angle between the best FIG. 8. Event production difference between nonvanishing g R coupling caculations and the tree level SM ones (g R ϭ0).Differences are plotted versus the invariant mass of the lepton ͑electron or muon͒ plus the bottom-quark system arising in top-quark decay from single top-quark production at the LHC.We have taken g R ϭϩ5ϫ10 Ϫ2 , ϩi5ϫ10 Ϫ2 , Ϫ5ϫ10 Ϫ2 and Ϫi5ϫ10 Ϫ2 in plots ͑a͒, ͑b͒, ͑c͒ and ͑d͒ respectively.With the present set of cuts, the total number of events in the standard model is 181 000.The total excess is 1200, roughly 1%.  9 we have taken g R ϭϩ5ϫ10 Ϫ2 , ϩi5ϫ10 Ϫ2 , Ϫ5ϫ10 Ϫ2 and Ϫi5ϫ10 Ϫ2 in plots ͑a͒, ͑b͒, ͑c͒ and ͑d͒ respectively.Note that here statistical significance has a strong dependence on the bottom-antiquark transversal momentum but is almost linear on Re(g R ) and almost insensible to the sign of Im(g R ).
FIG. 11.As in Fig. 9 we have taken g R ϭϩ5ϫ10 Ϫ2 , ϩi5 ϫ10 Ϫ2 , Ϫ5ϫ10 Ϫ2 and Ϫi5ϫ10 Ϫ2 in plots ͑a͒, ͑b͒, ͑c͒ and ͑d͒ respectively.Note that here statistical significance has a strong dependence on the bottom-quark transversal momentum and clearly favors positive values of Re(g R ) and again is insensible to the sign of Im(g R ).
reconstruction of top-quark momentum and the momenta of anti-lepton and bottom quark, respectively.In these figures we can clearly see that low angles correspond to bigger sensitivities.This is in qualitative accordance with Eq. ͑12͒ which, after inspection, tells us that anti-leptons are predominantly produced in the direction of the top-quark spin and therefore most of those produced predominantly in the top-quark direction come from a top quark mainly polarized in a positive helicity state.Thus the quantity of those antileptons is more sensitive to variations in g R .Even though this argument applies in the top-quark rest frame, the fact that most of the kinematics lies in the beam direction makes it valid at least for this kinematics.With the cuts considered here, the standard model prediction at tree level for the total number of events at LHC with one year full luminosity is 180 700.Using the values g L ϭ1, g R ϭϩ5ϫ10 Ϫ2 leads to an excess of 1220 events which corresponds to a 2.9 standard deviations signal.The g L ϭ1, g R ϭϪ5ϫ10 Ϫ2 model has a deficit of 480 events which corresponds to a 1.1 standard deviations signal.Finally the g L ϭ1, g R ϭϮi5ϫ10 Ϫ2 model has an excess of 367 events which corresponds to a 0.86 standard deviation.We see that there is a large dependence on the phase of g R .
It is perhaps interesting to remark that after considering the decay process, the sensitivity to g R is actually quite comparable to the one obtained in the t channel, where it was assumed that the polarized top quark was observable.From this point of view, not much information gets diluted through the process of top-quark decay.
The implementation of carefully selected cuts can slightly improve these statistical significances but since here we are interested in an order of magnitude estimate we will not enter into such analysis here.Moreover since backgrounds are bound to worsen the sensitivity the above results must be taken as order of magnitude estimates only.A more detailed analysis goes beyond the scope of this article.

VII. CONCLUSIONS
In this paper we have performed a full analysis of the sensitivity of single top-quark production in the s channel to FIG. 12. Statistical significance of the corresponding signals per bin with respect to lepton ͑electron or muon͒ transversal momentum.As in Fig. 9 we have taken g R ϭϩ5ϫ10 Ϫ2 , ϩi5ϫ10 Ϫ2 , Ϫ5ϫ10 Ϫ2 and Ϫi5ϫ10 Ϫ2 in plots ͑a͒, ͑b͒, ͑c͒ and ͑d͒ respectively.Note that again statistical significance has a strong dependence on the lepton transversal momentum and clearly favors positive values of Re(g R ).The sign of Im(g R ) cannot be distinguished.9 we have taken g R ϭϩ5ϫ10 Ϫ2 , ϩi5ϫ10 Ϫ2 , Ϫ5ϫ10 Ϫ2 and Ϫi5ϫ10 Ϫ2 in plots ͑a͒, ͑b͒, ͑c͒ and ͑d͒ respectively.Note that again statistical significance has a strong dependence on cos( tb ).
the presence of effective couplings in the effective electroweak theory.The analysis has been done in the context of the LHC experiments.We have implemented a set of cuts which is appropriate to general-purpose experiments such as ATLAS or CMS.The study complements the one presented in ͓8͔ that was devoted to the dominant t-channel process.
We have seen that the determination of the right effective coupling in such an experimental context is quite challenging.One has to include both polarization effects and m b corrections.Analytical formulas are presented.
Unlike in the discussion concerning the single top-quark production through the dominant t channel, top-quark decay has been considered.The only approximation involved is to consider the top quark as a real particle ͑narrow width ap-proximation͒.
We have paid careful attention to the issue of the topquark polarization.We have argued, first of all, why it is not unjustified to neglect the interference term and to proceed as if the top-quark spin were determined at an intermediate stage.We have provided a spin basis where the interference term is minimized.A similar analysis applies to the t-channel process.We present here an explicit basis for this case too.We get a sensitivity to g R in the same ballpark as the one obtained in the t channel ͑where decay was not considered͒.Finally we have obtained that observables most sensible to g R are those where anti-lepton and bottom-quark momenta are cut to be almost collinear.
Note added in proof.After submission of this work we became aware of the work by del Aguila and Aguilar-Saavedra ͑Ref.͓20͔͒ discussing somewhat similar issues in the context of top-pair production.We certainly agree with them in the fact that defining a properly normalized asymmetry is the way to eliminate systematic errors related to uncertainties in luminosity, parton distribution functions and so on.We stress that the results presented in the present paper are quite preliminary and that a more detailed analysis is required in due time.

FIG. 1 .
FIG. 1. Feynman diagrams contributing single top-quark production subprocess.In this case we have a d as spectator quark.

FIG. 9 .
FIG. 9. Plots corresponding to differences ͑a͒, ͑b͒ ͑c͒ and ͑d͒ of Fig. 8 divided by the square root of the event production per bin at LHC.The square of the quotient denominator can be obtained from Fig. 5 multiplying d/dm inv by the LHC 1-year full luminosity (100 fb Ϫ1 ) and by the width of each bin ͑4 GeV in Fig. 5͒.Taking the modulus of the above plots we obtain the statistical significance of the corresponding signals per bin.Note that statistical significance has a strong and nonlinear dependence both on the invariant mass and the right coupling g R .However purely imaginary couplings are almost insensible to their sign.

FIG. 13 .
FIG. 13.Statistical significance of the corresponding number of events per bin with respect to cos( tl )ϭp ជ l •(p ជ l ϩp ជ b )/͉p ជ l ͉͉p ជ l ϩp ជ b ͉ where p ជ l and p ជ b are respectively the tree momenta of the lepton ͑positron or antimuon͒ and bottom quark.The combination p ជ l ϩp ជ b is the best experimental reconstruction of the top-quark momentum provided the neutrino information is lost.As in Fig.9we have taken g R ϭ ϩ5ϫ10 Ϫ2 , ϩi5ϫ10 Ϫ2 , Ϫ5ϫ10 Ϫ2 and Ϫi5ϫ10 Ϫ2 in plots ͑a͒, ͑b͒, ͑c͒ and ͑d͒ respectively.Note that again statistical significance has a strong dependence on cos( tl ).

FIG. 14 .
FIG. 14. Statistical significance per bin with respect to cos( tb ) ϭp ជ b •(p ជ l ϩp ជ b )/͉p ជ l ͉͉p ជ l ϩp ជ b ͉ where p ជ l and p ជ b are respectively the tree momenta of the lepton ͑positron or antimuon͒ and bottom.The combination p ជ l ϩp ជ b is the best experimental reconstruction of the top-quark momentum provided the neutrino information is lost.As in Fig.9we have taken g R ϭϩ5ϫ10 Ϫ2 , ϩi5ϫ10 Ϫ2 , Ϫ5ϫ10 Ϫ2 and Ϫi5ϫ10 Ϫ2 in plots ͑a͒, ͑b͒, ͑c͒ and ͑d͒ respectively.Note that again statistical significance has a strong dependence on cos( tb ).