A study of top polarization in single-top production at the LHC

This paper complements the study of single top production at the LHC aiming to estimate the sensitivity of different observables to the magnitude of the effective couplings. In a previous paper the dominant $W$-gluon fusion mechanism was considered, while here we extend the analysis to the subdominant (10% with our set of experimental cuts) s-channel process. In order to distinguish left from right effective couplings it is required to consider polarized cross-sections and/or include $m_b$ effects. The spin of the top 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 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 clasically. 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 $m_b$ corrections. They are derived within the narrow width approximation for the top.


Introduction
At present not a lot is known about the W tb effective coupling. This is perhaps best evidenced by the fact that the current experimental results for the (left-handed) K tb matrix element give [1] |K tb | 2 |K td | 2 + |K ts | 2 + |K tb | 2 = 0.99 ± 0.29. (1) 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 CKM 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 per cent level at most-unless new physics is present and makes an unexpectedly large contribution. At the Tevatron the left-handed couplings are expected to be eventually measured with a 5% accuracy [2].
As far as experimental bounds for the right handed effective couplings is concerned, the more stringent ones come at present from the measurements on the b → sγ decay at CLEO [3].
Due to a 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 [4] reach the conclusion that |Re(g R )| ≤ 0.4 × 10 −2 . However, considering g R as a matrix in generation space, this bound only constraints the tb element. Other effective couplings involving the top remain virtually unrestricted from the data. The previous bound (about two thirds for the cuts we use). The cut on p T has the additional bonus of making the QCD corrections manageable. One is therefore left with those single tops coming from the W -gluon fusion mechanism (t-channel) and the subdominant s-channel process. The later 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-anti-top pair is also produced through analogous subprocesses. The analysis of such anti-top production processes is similar to the top ones and the corresponding cross sections can be easily derived doing the appropriate changes.
In a previous paper [8] we have analyzed the sensitivity of different LHC observables to the magnitude of the charged current effective couplings considering only the dominant Wgluon fusion channel. In that work we did not consider the subsequent decay of the top 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 > 30 GeV cut in the transverse momentum of the producedb quark was implemented in [8] and, accordingly, only the so-called 2 → 3 process was retained, excluding top production off a b-quark from the proton Fermi sea. Given the (presumed) 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 [8], 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. [9,10] for discussions this point) 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. This will become apparent from the formulae presented in section 2. For this reason we have derived in this work and in [8] analytical expressions for the cross sections for the production of polarized tops or anti-tops. To this end one introduces the spin projector 1 + γ 5 n 2 , with n µ = 1 as the polarization projector for a particle or anti-particle of momentum p 1 with spin in then direction. The calculation of the subprocess cross sections have been performed in this work and in [8] for an arbitrary polarization vectorn.
Obviously, however, the top decays very shortly after production, so the only practical way one can measure the spin of the top 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 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 = 1 and g R = 0. Imposing the a cut on p T we have mentioned, only two subprocesses contribute; W -gluon fusion and the s-channel process. The later provides 100% polarized tops in a certain direction (to be discussed latter). The situation in the t-channel process is a bit more complicated. The results from our previous analysis presented in [8] show that single top production is highly, but not fully, polarized in this case too (84 % in the optimal basis, with the present set of cuts). This is a high degree of polarization, but still well below the 90+ claimed by Mahlon and Parke in [10]. We understand this 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 [10] (note that we do not include the 2 → 2 or b-sea process). Inasmuch as they can be compared our results for the tree-level Standard Model are in good agreement with those presented in [7] 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 1 .
Let us assume now for the sake of discussion that the polarization is indeed 100% . The top subsequently decays (say emitting a positively charged lepton). 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 section 2 and in [8]), not even in principle. In other words, the top is necessarily in a quantum mixed state and is described by a density matrix. The entries of this density matrix depend on the 1 Since we perform a leading order calculation in QCD, the scale dependence is large. We have made two different choices: (a) µ = p cut T is used as scale in αs and the gluon PDF, while the virtuality of the W boson is used as scale for the PDF of the light quarks in the proton. This gives an excellent agreement with the calculations in [7]. (b) µ 2 =ŝ,ŝ being the center-of-mass energy squared of the qg subprocess. The total cross section above the cut is then roughly speaking two thirds of the previous one, but no substantial change in the distributions takes place. This is the typical error for LO calculations in the present kinematical regime.
The total cross section has been known to NLO for some time [11], while NLO results for the differential cross section have become available just recently [12] 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 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. As already emphasized, to be able to tell left from right effective couplings one absolutely needs to consider the spin of the top.
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 measure) 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 needs not compute the off-diagonal terms. This diagonalization process has to be done event by event and it selects a particular vectorn (event dependent). In section 5 we provide explicit formulae 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 spin). Using this 3-vector as spin basis, for instance, one can multiply the probability of producing a top polarized in the positive +n direction with the corresponding decay angular probability distribution plus the probability of producing a top polarized in the negative −n distribution times the 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 is measured in the intermediate state before the top quark decays.
Then there should be no surprise in that the interaction between the top and the apparatus measuring its spin modifies the final physical results.
However, a proper measure of the top spin before it decays is impossible; the only way we learn about top polarization is precisely from the final decay products. So, the previous procedure it is conceptually incorrect 2 . The 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 is well defined and one proceeds as if it could be measured before it decays it could be approximately correct. But what are then 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.

The differential cross section for polarized top 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 [8] for detailed expressions of the different amplitudes.
We denote the matrix elements of the hard subprocess of Fig. 1 by M d + . There will also be a Mū + , corresponding to having instead aū as spectator quark. We will also eventually define the matrix elements corresponding to the processes producing anti-tops as M u − , and Md − . With these definitions the differential cross section for polarized tops dσ can be written schematically as where f u and fd denote the parton distribution functions corresponding to extracting a utype quark and ad-type quark respectively and β is a proportionality factor incorporating the kinematics and also the gluon distribution function. Using our analytical results for the matrix elements given in the appendix of [8] we obtain for the differential cross-section 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 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 mass and are therefore absent if one neglects m b (this at first sight does not look unreasonable, given the energies involved). Inspection of the above differential crosssection 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 3 . For these reasons, both polarization and m b terms are quite important.
We observe that A is an Hermitian matrix and therefore it is diagonalizable with real eigenvalues. Moreover, from the positivity of dσ we immediately arrive at the constraints that is and 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 (7) and (8) must be satisfied for any set of positive parton distribution functions we immediately obtain the bounds ab + a n b n − 1 4 In order to have a 100% polarized top we need a spin four-vector n that saturates the constraint (5) (that is Eq.(7)) 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 λ. 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 3 The statement is exact if one uses the so-called effective W approximation, which is not terribly accurate for the present case and certainly not recommended [13], but widely used in LHC physics is only one solution (up to a global complex normalization factor α) for the pair (g R , g L ) to the equation dσ = 0, 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 non-vanishing 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 non-vanishing g R ( g L is evidently non-vanishing) it is not possible to have a pure spin state (or, else, only for fine tuned g R a 100% polarization is possible).
Let us now give a very simple example to make the previous discussion more understandable: in the un-physical situation where m t → 0 it can be shown that there exists two solutions to the saturated constraint (5), namely once we have found this result we plug it in the expression (9) and we find the solutions (0, g L ) with g L arbitrary for the + sign and (g R , 0) with g R arbitrary for the − sign. That is, physically we have zero probability of producing a right handed top when we have only a left handed coupling and viceversa when we have only a right handed coupling. Note that in this case it is clear that having both effective couplings non-vanishing would imply the absence of 100 % polarization in any spin basis. This can be understood in general remembering that the top 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 in a pure polarized state.

Cross-sections for top 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 a 10% of all single tops 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) the squared amplitude for top production is given by whereg L , andg R are left and right couplings to light quarks and g L and g R are the effective couplings to the top-bottom system. In the numerical results we have takeng L = 1,g R = 0; 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 functions effects, the differential unpolarized production cross section would be proportional to |g L | 2 + |g R | 2 .
The differential cross section for producing polarized tops is then where f x 1 ,x 2 , (q 1 + q 2 ) 2 , Λ QCD dx 1 dx 2 accounts for the quarks parton distribution functions.
The total decay rate of the top, 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 polarizations (with a spin projector inserted), 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 → −p 2 ) is just expression (11) with the indicated changes in momenta. In the above expressiong L , andg R are the left and right couplings corresponding to the lepton-neutrino vertex. We have assumedg L = 1,g R = 0, 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 dσ corresponding to Fig. 2 is given by 4 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 production and its consecutive decay. In that set up we denote the single top production amplitude as A p,±n(p) and the top decay amplitude as B p,±n(p) . In the polar representation we write A p,±n(p) = A p,±n(p) e iϕ ± (p) , where p indicate external momenta andn (p) a given spin basis for the top. The differential cross section for the whole process is schematically given by Hence dσ = A p,+n(p) 2 B p,+n(p) 2 dp + A p,−n(p) 2 B p,−n(p) 2 dp +2 A p,+n(p) B p,+n(p) A p,−n(p) B p,−n(p) × cos (ϕ + (p) − ϕ − (p) + ω + (p) − ω − (p)) dp ≃ A p,+n(p) 2 B p,+n(p) 2 dp + A p,−n(p) 2 B p,−n(p) 2 dp.
Since the axis with respect to which the spin basis is defined is completely arbitrary dσ is independent on this choice of basis. However within the narrow width approximation one never computes dσ following formula (14). The commonly used procedure [10,14] consists in computing the probability of producing a polarized top and then multiplying this probability by the probability of a given decay channel (see Eq. (13)). This procedure is equivalent to the neglection of the interference term in formula (15)  A dependence on the spin frame is however unacceptable if the spin is not measured before the top decays.
Let us see whether this approximation can justified nevertheless. Clearly, the integration over momenta enhances the positive-definite terms in front of the interference oscillating one.
If in addition we make a choice forn (p) that diagonalizes the top spin density matrix and thus maximizes A p,+n(p) and minimizes A p,−n(p) , then we expect the interference term to be negligible when compared to A p,+n(p) 2 B p,+n(p) 2 dp even for small amount of phase space integration. In the s-channel we will see in the next section that the limit of g R → 0 there exists a spin basisn (p) where A p,−n(p) is strictly zero. This basis is given by 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,+n(p) 2 B p,+n(p) 2 dp. For g R = 0 one can still find a basis that maximizes A p,+n(p) (and minimizes A p,−n(p) ) and therefore diagonalizes the top density matrix ρ. 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 numerically that this basis is the one that maximizes dσ 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 basisn (p) and noting that our results are actually only weakly dependent on the choice ofn (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 momenta).
Moreover we have checked that if spin is ignored altogether (by considering unpolarized top production) 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 narrow-width 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.

The diagonal basis
As stated in the previous section in order to calculate the top decay we have to find the basis where the polarized single top production cross section is maximal. The can do this maximizing in the 4-dimensional space generated by the components of n constrained by n · p 1 = 0, where p 1 is the top four-moment, that is where n = (n 1 ) 2 + (n 2 ) 2 + (n 3 ) 2 , that is n i = n n i withn the normalized spin threevector. From above equations we obtain , from which Eq. (2) follows immediately. Let us now find the polarization vector that maximizes and minimizes the differential cross section of single top production.

The t-channel
We will begin with the t-channel the was analyzed in the previous chapter. Using Eq. (3) we define a n = n · a, b n = n · b, and using Lagrange multipliers λ 1 and λ 2 for constraints (16) we maximize σ + λ 1 n 2 + 1 + λ 2 n · p 1 , obtaining the equations and thus using Eqs. (18) and (20) and therefore with the normalization factor λ 1 given by Eq. (19). Note that in the idealized case f u = fd = f we obtain where α is the normalization constant that does not depend on f or the effective couplings.
In the SM (g R = 0) we obtain where α is a normalizing factor.

The s-channel
The s-channel differential cross section has the form where again β is a proportionality incorporating the kinematics, and where f u,c and fd ,s denote the parton distribution functions corresponding to extracting a u, c-type quarks and a d ,s-type quarks respectively. Using again the decomposition (17) and proceeding analogously to the t-channel calculation we obtain where α is the normalizing factor that in this case (unlike in the t-channel result) does not depend on the parton distribution functions. From Eq. (11) we obtain hence replacing in Eq. (21) 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 = 0 the above reduces to which is the result we have quoted in the previous section coinciding with [10] 6 Numerical analysis of s-channel single top production Let us start this section by discussing the experimental cuts we have implemented. Due to geometrical detector constraints [15] 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 |η| < 2.44.
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 ATLAS or CMS.
The set of cuts used in this work are compatible with the ones used in the t-channel. Since in the previous paper [8] top 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 paper). The present analysis should however suffice in any case to identify the most promising observables and get a rough estimate of the precision that it can be reached.
We use a lower cut of 20 GeV in theb jet 4 . This completely eliminates top 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 thē 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 > 20 GeV cuts are set for the top 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 [7] and [10].
In order to calculate the cross section σ of the process pp → tb we have used the CTEQ4 set of structure functions [17] 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 [18]. We present results after one year (defined as 10 7 seg.) run at full luminosity in one detector (100 fb −1 at LHC).
Since in order to be able to perform the effective W tb 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 [19], for instance, it 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 LHCb 5 [16] where the appropriate figure is expected to be 2 fb −1 . ATLAS plans to do most of the b-physics runs before full luminosity is reached, for instance. We have nonetheless used the high luminosity 4 In the previous paper [8] the value used was 30GeV. We have decided to use this lower value here to have a larger total cross-section without compromising the theoretical accuracy 5 This type of analysis is anyway not well suited for such a detector. The rapidity for LHCb is in the range 1.6 < η < 4.9 and the angular separation cut between jets imposed here is totally unfeasible. Furthermore, jet reconstruction is not possible. Clearly the implementation of this type of physics to this detector requires a lot more ingenuity.
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 ā b that passes the experimental cuts and reconstruct the vectorn using the analytic formulae 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 spin has been measured in the direction determined byn and 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 involved) and the optimal case (found analytically here). We have every reason to believe that, after the integration over momenta and the resulting coarsening, the interference term is in this basis 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.  for the bottom and lepton, respectively, that are produced in the top 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 + , produced in the top decay, decays hadronically. In hadronic decays of the top a full reconstruction of the top mass would be feasible.
Let us now move beyond the Standard Model. Since changing the value of g L (while keeping g R = 0) 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 | = 5 × 10 −2 . This is still worse than the limit implied by b → sγ, 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 up or down the results. We have consider the possibility of g R having a phase and, accordingly, the experimental sensitivity to that phase.
We have found that the anti-lepton plus bottom invariant mass distribution we just discussed in the previous paragraph is actually sensitive to g R . Figs  We shall now show the dependence of the three p T distributions (b,b and lepton) to the modulus and phase of the effective coupling g R . In all cases the value g L = 1 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 (θ tl ) in  Fig. (5)). 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 non-linear dependence both on the invariant mass and the right coupling g R. However purely imaginary couplings are almost insensible to their sign. almost insensible to the sign of Im (g R ).  and vs cos (θ tb ) in Fig. (14). cos (θ tl ) and cos (θ tb ) are the cosines of the angle between the best reconstruction of top momentum and the momenta of anti-lepton and bottom, respectively.
In these figures we can clearly see that low angles corresponds to bigger sensitivities. This is in qualitative accordance with Eq. (12)  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 was observable. From this point of view, not much information gets diluted through the process of top decay.
The implementation of careful 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.

Conclusions
In this paper we have performed a full analysis of the sensitivity of single top production in the s-channel to 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 [8] 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.
Unlike in the discussion concerning the single top production through the dominant tchannel, top decay has been considered. The only approximation involved is to consider the top as a real particle (narrow width approximation).
We have paid careful attention to the issue of the top polarization. We have argued, first of all, why it is not unjustified to neglect the interference term and to proceed as if the top spin was 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 and 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 considered). Finally we have obtained that observables most sensible to g R are those where anti-lepton and bottom momenta are cut to be almost collinear.