Measuring effective electroweak couplings in single top production at the LHC

We study the mechanism of single top production at the LHC in the framework of an effective electroweak Lagrangian, analyzing the sensitivity of different observables to the magnitude of the effective couplings that parametrize new physics beyond the Standard Model. The observables relevant to the distinction between left and right effective couplings involve in practice the measurement of the spin of the top and this can be achieved only indirectly by measuring the angular distribution of its decay products. We show that the presence of effective right-handed couplings implies that the top is not in a pure spin state. A unique spin basis is singled out which allows one to connect top decay products angular distribution with the polarized top differential cross section. We present a complete analytical expression of the differential polarized cross section of the relevant perturbative subprocess including general effective couplings. The mass of the bottom quark, which actually turns out to be more relevant than naively expected, is retained. Finally we analyze different aspects the total cross section relevant to the measurement of new physics through the effective couplings. The above analysis also applies to anti-top production in a straightforward way.


Introduction
The standard model of electroweak and strong interactions has been, to this day, tested to a remarkable degree of accuracy, particularly in what concerns the neutral current sector.
However it is clear that suffers from several theoretical drawbacks (naturalness, triviality,...) making it conceivable that it should be considered as an effective theory valid only at low energies ( 1 TeV ). With the current limit on the Higgs mass already placed at 113,5 GeV [1] and no clear evidence for the existence of an elementary scalar (despite much controversy regarding the results of the last days of LEP) it makes sense to envisage an alternative to the minimal Standard Model described by an effective theory without any physical light scalar fields. This in spite of the seemingly good agreement between experiment and radiative corrections computed in the framework of the minimal Standard Model (see [2] however). This effective theory should contain an infinite set of effective operators, of increasing dimensionality, compatible with the electroweak and strong symmetries SU (3) c × SU (2) L × U (1) Y and their coefficients would parametrize physics beyond the Standard Model. In this framework [3] one can describe the low energy physics of theories exhibiting the pattern of symmetry breaking SU (2) L × U (1) Y → U (1) em with full generality, in the understanding, that this approach is useful as long as those particles not explicitly included in the effective Lagrangian are much heavier than the scale of energies at which the effective Lagrangian is to be used.
In this work we plan to investigate the new features that physics beyond the standard model may introduce in the production of the top (or anti-top) quarks through W -gluon fusion at the LHC. When describing the appropriate effective vertex in the effective Lagrangian language, we will keep only the leading non-universal (i.e. not appearing in the standard model) effective operators in the low energy expansion. They correspond to the operators of dimension four, which were first classified by Appelquist et al. [4]. These operators realize the SU (2) L × U (1) Y symmetry non-linearly and are thus characteristic of strongly coupled theories and, strictly speaking, are absent in the minimal Standard Model and modifications thereof containing only light fields, such as supersymmetric extensions, where all degrees of freedom are weakly interacting and need to be included explicitly. When particularizing to interactions involving the W, Z bosons, the operators present in the effective electroweak Lagrangian induce effective vertices coupling the gauge boson to the matter fields (see e.g. [5]) Other possible effects are not physically observable, as we shall see in a moment. In practical terms, LHC will set bounds on these effective W vertices, and therefore our results are also relevant in a broader phenomenological context as a way to bound κ L and κ R , without any need to appeal to an underlying effective Lagrangian describing a specific model of symmetry breaking. Of course one then looses the power of an effective Lagrangian, namely a well defined set of counting rules and the ability to relate different processes.
Another point to note is that, even in the minimal At the LHC energy (14 TeV) the dominant mechanism of top production, with a cross section of 800 pb [6], is gluon-gluon fusion. This mechanism has nothing to do with the electroweak sector and thus is not the most adequate for our purposes (although is the one producing most of the tops and thus its consideration becomes necessary in order to study the top couplings through their decay, which will not be our main interest here, and also as a background to the process we shall be interested in).
Electroweak physics enters the game in single top production. (For a recent review see e.g. [7].) At LHC energies the (by far) dominant electroweak subprocess contributing to single top production is given by a gluon (g) coming from one proton and a light quark or anti-quark coming from the other (this process is also called t-channel production [8,9]). This process is depicted in Figs. 1 and 2, where light u-type quarks ord-type antiquarks are extracted from the proton, respectively. These quarks then radiate a W whose effective couplings is the object of our interest. The cross total section for this process at the LHC is estimated to be 250 pb [9], to be compared to 50 pb for the associated production with a W + boson and a b-quark extracted from the sea of the proton, and 10 pb corresponding to quark-quark fusion (s-channel production). For comparison, at the Tevatron (2 GeV) the cross section for W -gluon fusion is 2.5 pb, so the production of tops through this particular subprocess is copious at the LHC. Monte Carlo simulations including the analysis of the top decay products indicate that this process can be analyzed in detail at the LHC and traditionally has been regarded as the most important one for our purposes.
In a proton-proton collision a bottom-anti-top pair is also produced, through the subprocesses given in Figs. 3 and 4. At any rate qualitative results are very similar to those corresponding to top production, from where the cross sections can be easily derived doing the appropriate changes.
In the context of effective theories, the contribution from operators of dimension five to top production via longitudinal vector boson fusion was estimated some time ago in [11], although the study was by no means complete. It should be mentioned that t,t pair production through this mechanism is very much masked by the dominant mechanism of gluon-gluon fusion, while single top production, through W Z fusion, is expected to be much suppressed compared to the mechanism presented in this paper, the reason being that both vertices are electroweak in the process discussed in [11], and that operators of dimension five are expected to be suppressed, at least at moderate energies, by some large mass scale. The contribution from dimension four operators as such has not, to our knowledge, been considered before, although the potential for single top production for measuring the CKM matrix element K tb , has to some extent been analyzed in the past (see e.g. [9,10]).

Effective couplings and observables
Including family mixing and, possibly, CP violation, the complete set of dimension four effective operators which may contribute to the top effective couplings and are relevant for the present discussion is [4,12,13] and where L = 1−γ 5 2 , R = 1+γ 5 2 are the left and right projectors, the matrices M have family indices only, and the above set is written the non-physical 'weak' basis (that is, with matter fields transforming as irreducible representations of the gauge group) [13]. The unitary matrix U contains the three Goldstone bosons associated to the breakdown of the global symmetry SU (2) L × SU (2) R down to SU (2) V . The derivatives appearing in the effective operators are given by where f is a weak doublet of matter fields with family indices also. In addition, one has the following 'universal' terms which are present in the Standard Model. In Eq.(5) we allow for general couplings M L , ; in the Standard Model these couplings can be renormalized away via a change of basis, but in more general theories they leave traces in other operators not present in the Standard Model [13].
In [13] it was also shown that when we diagonalize the mass matrix present in Eq. (5) via a redefinition of the matter fields (f → f ) we change also the structure of operators (2,3). Taking that into account, the contribution to the different gauge boson-fermion-fermion vertices is as follows where τ u and τ d are the up and down projectors and f represents the matter fields in the physical, diagonal basis. It can be shown that once the all the renormalization (vertex, CKM elements, wave-function) countertems are taken into account [13] we obtain a L,R = 1, b L,R = Q; i.e. we have no contribution from the effective operators to the vertices of the gluon and photon. For the Z couplings we get instead where K is the CKM matrix, and the matrices N 's andM 's are redefined matrices according to the results of [13] (the exact relation of these matrices to the M 's of Eqs. (2,3) has no relevance for the present discussion). Finally for the charged couplings we have Since the set of operators (2-3) is the most general one allowed by general requirements of gauge invariance, locality and hermiticity; it is clear that radiative corrections, when expanded in powers of p 2 , can be incorporated into them. In fact, such an approach has proven to be very fruitful in the past. Once everything is included we are allowed to identify the couplings d L,R with κ CC LR . In this paper we shall be concerned with the bounds that the LHC experiments will be able to set on the couplings κ CC LR , more specifically on the entries tj of these matrices (those involving the top). In the rest of the article we do not consider mixing and we consider non-tree level and new physics contributions only on the tb effective couplings, therefore in the numerical simulations we have taken When we speak along the paper of the results for the Standard Model at tree level we mean g L = 1, and g R = 0. However, even though numerical results are presented considering only the tb entry (g L and g R ), since flavour indices and masses are kept all along in the analytical expressions (see Appendix), the appropriate changes to include other entries are immediate.
The effective couplings of the neutral sector (7) can be determined from the Z → ff vertex 1 [12], but at present not much is known from the 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 [14] |K tb | 2 |K td | 2 + |K ts | 2 + |K tb | 2 = 0.99 ± 0.29.
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 with a 5% accuracy [15]. The present work is a contribution to such an analysis in the case of the LHC experiments.
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 [16]. Due to a m t /m b enhancement of the chirality flipping contribution, a particular combination of mixing angles and κ CC R can be found. The authors of [17] reach the conclusion that |Re(κ CC R )| ≤ 0.4 × 10 −2 . However, considering κ CC 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 on the right-handed coupling is a very stringent one. It is pretty obvious that the LHC will not be able to compete with such a bound. Yet, the measurement will be a direct one, not through loop corrections. Equally important is that it will yield information on the td and ts elements too, by just replacing theb quark in Figs. 1,2 by ad or as respectively.
Now we shall proceed to analyze the bounds that single top production at the LHC can set on the effective couplings. This combined with the data from Z physics will allow an estimation of the six effective couplings (7)(8) in the matter sector of the effective electroweak Lagrangian. We will, in the present work limit ourselves to the consideration of the crosssections for production of polarized top quarks. We shall not consider at this stage the potential of measuring top decays angular distributions in order to establish relevant bounds on the effective electroweak couplings. This issue merits a more detailed analysis, including the possibility of detecting CP violation [18].

The cross section
In order to calculate the cross section σ of the process pp → tb we have used the CTEQ4 set of structure functions [19] to determine the probability of extracting a parton with a given fraction of momenta from the proton. Hence we write schematically where f q denote the parton distribution function (PDF) corresponding to the partonic quarks and antiquarks and f g indicate the PDF corresponding to the gluon. In Eq.  Typically the top quark decays weakly well before strong interactions become relevant, we can in principle measure its polarization state with virtually no contamination of strong interactions (see e.g. [20] for discussions this point and section 6). For this reason we have considered polarized cross sections and provide general formulas for the production of polarized tops or anti-tops. To this end one needs to introduce the spin projector as the polarization projector for a particle or anti-particle of momentum p 1 with spin in then direction. The calculation of the subprocesses cross sections have been performed for tops and anti-tops polarized in an arbitrary directionn. Later we have analyzed numerically different spin frames defined as follows • Lab helicity frame: the polarization vector is taken in the direction of the three momentum of the top or anti-top (right helicity) or in the opposite direction (left helicity).
• Lab spectator frame: the polarization vector is taken in the direction of the three momentum of the spectator quark jet or in the opposite direction. The spectator quark is the d-type quark in Fig.1 or theū-type quark in Fig.2.
• Rest spectator frame: like in the Lab spectator frame we choose the spectator jet to define the polarization of the top or anti-top. Here, however, we definen as ± the direction of the three momentum of the spectator quark in the top or anti-top rest frame (given by a pure boost transformation Λ of the lab frame). Then we have n r = (0,n) in that frame and n = Λ −1 n r back to the lab frame.
The calculation of the subprocess polarized cross-section we present is completely analytical from beginning to end and the results are given in the Appendix. Both the kinematics and the polarization vector of the top (or anti-top) are completely general. Since the calculation is of a certain complexity a number of checks have been done to ensure that no mistakes have been made. The integrated cross section agrees well with the results in [9] when the same cuts, scale, etc. are used. The mass of the top is obviously kept, but so is the bottom mass.
The latter in fact turns out to be more relevant than expected as we shall see in a moment.
As we have already discussed, the production of flavours other thanb in association with the top can be easily derived from our results.
In single top production a distinction is often made between 2 → 2 and 2 → 3 processes.
The latter corresponds, in fact, to the processes we have been discussing, the ones represented in Fig.1, in which a gluon from the sea splits into a bb pair. In the 2 → 2 process the b quark is assumed to be extracted from the sea of the proton, and both b andb are collinear.
Of course since the proton has no net b content, ab quark must be present somewhere in the final state and the distinction between the two processes is purely kinematical. As is well known, when calculating the total cross section for single top production a logarithmic mass singularity [9] appears in the total cross section due to the collinear regime where the b quark (and theb) quark have k T → 0. This kinematic singularity is actually regulated by the mass of the bottom; it appears to all orders in perturbation theory and a proper treatment of this singularity requires the use of the Altarelli-Parisi equations and its resumation into a b parton distribution function. It is thus indistinguishable of the so-called 2 → 2 process.
Clearly an appropriate cut in k T should allow us to retain the perturbative regime of the 2 → 3 process, while suppressing the 2 → 2 one.
Two approaches can be used at this point. One -advocated by Willenbrock and coworkers [9] is to focus on the low p T regime, to minimize the contribution of the t,t background. One actually is interested in processes where one does not see theb (resp. b) quark which is produced in association with the t quark (resp.t), and therefore sets an upper cut on the p T of theb. For this one has to take into account the 2 → 2 process and, in particular, one must pay attention not to double count the low p T region (for theb (or b) quark) of the 2 → 3 process, which is already included via a b PDF. The reason given in [9] is to separate this single top production process from the dominant gg →tt, which eventually also produces a b and ab. We view this strategy as somewhat risky since, first of all, the separation between the 2 → 3 and 2 → 2 is not a clear cut one and, on top of that, it is a region where the cross section is rapidly varying so the results do depend on the way the separation is done.
Furthermore it relies on assuming that we have a good knowledge of the b PDF at that scale; one will be forced to assume that it behaves largely like the s PDF at the end of the day. On top of that, that strategy does not completely avoid the background originated in tt production. That is, when in the decayingt → W −b →ūdb theb is missed along with theū-type anti-quark in which case the d-type quark is taken as the spectator or when thē b is missed along with the d-type quark in which case theū-type anti-quark is taken as the spectator.
On the other hand, measuring theb (or b for anti-top production) momenta will allow a better kinematic reconstruction of the individual processes. This should allow for a separation from the dominant mechanism of top production through gluon fusion. Setting a sufficiently high cut for the jet energy and a good jet separation should be sufficient to avoid contamination from t,t when one hadronic jet is missed. Finally, the spin structure of the top is completely different in both cases due to the chiral couplings in electroweak production. Therefore, according to this philosophy -stay perturbative-, we have implemented a lower cut of 30 GeV in the transversal momentum of theb (resp. b) in top (resp. anti-top) production.

A first look at the results
We shall now present the results of our analysis. To calculate the total event production corresponding to different observables we have used the integrating montecarlo program VE-GAS [22]. We present results after one year (defined as 10 7 seg.) run at full luminosity in one detector (100 fb −1 at LHC).
The total contribution to the electroweak vertices g L , g R has two sources: the effective operators parametrizing new physics, and the contribution from the universal radiative corrections. In the standard model, neglecting mixing, for example, we have a tree level contribution to thetW + µ b vertex given by − i √ 2 γ µ gK tb L. Radiative corrections (universal and M H dependent) modify g L and generate a non zero g R . These radiative corrections depend weakly on the energy of the process and thus in a first approximation we can take them as constant. Our purpose is to estimate the dependence of different LHC observables on these total effective couplings and how the experimental results can be used to set bounds on them. Assuming that the radiative corrections are known, this implies in turn a bound on the coefficients of the effective electroweak Lagrangian.
Let us start by discussing the experimental cuts. Due to geometrical detector constraints we cut off very low angles for the outgoing particles. The top, anti-bottom, and spectator quark have to come out with an angle of less than 10 degrees and more than 170 degrees.
These angular cuts correspond to a cut in pseudorapidity |η| < 2.44. In order to be able to detect the three jets corresponding to the outgoing particles we implements isolation cuts of 20 degrees between each other.
As already discussed we use a lower cut of 30 GeV in theb jet. This reduces the cross section to less than one third of its total value, since typically theb quark comes out in the same direction as the incoming muon 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 similar, but not identical to the ones used in [9] and [10]. To summarize detector geometry cuts : Numerically, the dominant contribution to the process comes from the diagram where a b quark is exchanged in the t channel, but a large amount of cancellation takes place with the crossed interference term with the diagram with a top quark in the t channel. The smallest contribution (but obviously non-negligible) corresponds to this last diagram. It is then easy to see, given the relative smallness of the b mass, why the process is so much forward.
Undoubtedly the largest theoretical uncertainty in the whole calculation is the choice of a scale for α s and the PDF's. We perform a leading order calculation in QCD and the scale dependence is large. We have made two different choices. We present some results with the scale p cut T used in α s 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. When we use these scales and compute, for instance, the total cross section above a cut of p T = 20 GeV in theb momentum, we get an excellent agreement with the calculations in [9]. Most of our results are however presented with a common scale 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. It remains to be seen which one is the correct choice.
From our Monte Carlo simulation for single top production at the LHC after 1 year of full luminosity and with the cuts given above we obtain the total number of events. This number depends on the value of the effective couplings and on the top polarization vector n given in the frames defined in section 3 . If we call N (g L , g R ,n, (f rame)) to this quantity, we obtain the following results N g L , g R ,n = ± p 1 | p 1 | , (lab) = g 2 L × (3.73 ∓ 1.31) × 10 5 + g 2 R × (3.54 ± .97) × 10 5 +g L g R × (−.237 ∓ .0283) × 10 5 , where we have omitted the √ N statistical errors and we have neglected possible CP phases (g L and g R real). One can observe from the simulations that the production of negative helicity (left) tops represents the 69% of the total single top production, this predominance of left tops in the tree level electroweak approximation is expected due to the suppression at We have also calculated single anti-top production. In Fig.8 we show the histograms corresponding to the production oft with the two possible helicities in the LAB frame. All the histograms correspond to the tree level electroweak approximation and clearly show that single anti-top production is approximately the 75% of single top production. The same pattern is observed in Fig. 9. This suppression is generated by the parton distribution functions corresponding to negatively charged quarks that are smaller than the ones corresponding to positively charged quarks. Because of that the conclusions for anti-top production are practically the same as the ones for top production taking into account this suppression and that, because of the transformations (27) (see Appendix), passing from top to anti-top is equivalent to changing the spin direction.
In Fig. 10 Table 1, where we indicate also the polarization vector chosen in each case. Of course those sensitivities (which, as said, are merely indicative) are calculated with the assumption that one could perfectly measure the top polarization in any of the above basis. As it is well known the top polarization is only measurable in an indirect way through the angular distribution of its decay products. In section 6 we outline the procedure to use our results to obtain a final angular distribution for the polarized top decay products (we believe that some confusion exists on this point).
Obtaining that angular distribution involves a convolution of the single top production cross section with the decay products angular distribution and because of that we expect the true sensitivity to become worse than the ones given in table 1. Obviously that true sensitivity must be independent of the spin basis one uses at an intermediate step (see section 6) so the whole discussion as to which is the best basis for the top polarization is academic. However it may still be useful to know that some basis are more sensitive to the effective couplings than others if one assumes (at least as a gedanken experiment) that the polarization of the top could be measured directly.
It is worth mentioning that the bottom mass, which appears in the cross section in crossed left-right terms, such as m b g L g R , plays a crucial role in the actual determination of g R . This is because from the |Re(κ CC R )| ≤ 0.4 × 10 −2 bound [17] we expect g L g R m b > g 2 R m t . Evidently for the ts or td couplings these terms are not expected to be so relevant.

The differential cross section for polarized tops
We define the matrix element of the subprocess of Figs.1, to 4 as M d + , Mū + , M u − , and Md − , respectively. With these definitions the differential cross section for polarized tops σ 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 c is a proportionality factor incorporating the polarization frame 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.. From Eq.
(13) we observe that A is an Hermitian matrix and therefore it is diagonalizable with real eigenvalues. Moreover, from the positivity of σ we immediately arrive at the constraints T rA ≥ 0, 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 (16) and (17) must be satisfied for any set of positive PDF's 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 (14) (that is Eq. (16)) 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 is only one solution (up to a global complex normalization factor α) for the pair (g R , g L ) to the equation σ = 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).
To illustrate these considerations let us give an example: in the unphysical situation where m t → 0 it can be shown that there exists two solutions to the saturated constraint (14), namely once we have found this result we plug it in the expression (18) 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 expect in general to end up with a top in a pure polarized state; although this is not impossible as it is shown the in the last example.
In the physical situation where m t = 0 (we use m t = 175.6 GeV. and m b = 5 GeV. in this paper) we have found that a spin basis with relatively high polarization is the one with the spinn taken in the direction of the spectator quark in the top rest frame. This is in accordance to the results in [10]. In general the degree of polarization ( N +n N +n +N −n ) depends not only on the spin frame but also on the particular cuts chosen. We have found that the lower cut for the transverse momentum of the bottom worsens the polarization degree but, in spite of that, from Eq.(12) we see that we have a 84% of polarization in the Standard Model (g L = 1, g R = 0) that is much bigger than the 69% obtained with the helicity frame. The above results follow the general trend of those presented by Mahlon and Parke [10], but in general, their degree of polarization is higher. We understand that this is due to the different cuts (in particular for the transversal momentum of the bottom) along with the different set of PDF's used in our simulations.

Measuring the top polarization from its decay products
A well know result in the tree level SM regarding the measure of the top polarization from its decay products is the formula that states the following: Given a top polarized in then direction in its rest frame, the lepton l + produced in the decay of the top via the process presents an angular distribution [23] σ l = α (1 + cos θ) , where α is a normalization factor and θ is the axial angle measured from the direction ofn.
What can we do when the top is in a mixed state with no 100% polarization in any direction?
The first naive answer would be: With any axisn in the top rest frame the top will have a polarization p + (with 0 ≤ p + ≤ 1) in that direction and a polarization p − = 1 − p + in the opposite direction so the angular distribution for the lepton is The problem with formula (22) is that the angular distribution for the lepton depends on the arbitrary chosen axisn and this cannot be correct. The correct answer can be obtained by noting the following facts: • Given an arbitrary chosen axisn in the rest frame and the associated spin basis to it {|+n , |−n } the top spin state in given by a 2 × 2 density matrix ρ which is in general not diagonal (b = 0) and whose coefficients depend on the rest of kinematical variables determining the differential cross section.
• From the calculation of the polarized cross section we only know the diagonal elements • Given ρ in any orthogonal basis determined (up to phases) byn we can change to another basis that diagonalizes ρ. Since the top is a spin 1/2 particle, this basis will correspond to another directionn d .
• Once we have ρ diagonalized then Eq.(22) is trivially correct with p ± = ρ ± and now θ is unambiguously measured from the direction ofn d .
From the above facts the first question that comes to our minds is if there exists a way to determinen d without knowing the off-diagonal matrix elements of ρ. The answer is yes. It is an easy exercise of elementary quantum mechanics that given a 2 × 2 Hermitian matrix ρ the eigenvector with largest (lowest) eigenvalue correspond to the unitary vector that maximizes (minimizes) the bilinear form v| ρ |v constrained to {|v , v|v = 1}. Since an arbitrary normalized |v can be written (up to phases) as |+n and in that case ρ + = p + then the correctn d entering in Eq. (22) is the one that maximizes the differential cross section |M | 2 n for each kinematical configuration. At the end, the correct angular distribution for the leptons is given by the cross section for polarized tops in this basis (n d ) convoluted with formula (21) (or improvements of it [21]).
The above analysis was carried out in the Standard Model (g R = 0) but it is correct also for g R = 0 using the complete formula for this case where h ≃ 0.566 [23]. Formula (24) deserves some comments: • First of all we remember that θ is the angle (in the top rest frame) between then that maximizes the difference (p + − p − ) and the three momentum of the lepton.
• Taking into account the above comment and that (p + − p − ) depends on g L and g R we see that also θ depends on g L and g R .
• From the computational point of view, formula (24) is not an explicit formula because involves a process of maximization for each kinematical configuration.
• In some works in the literature [10] formula (24) is presented for an arbitrary choice of the spin basis {|±n } in the top rest frame. This is incorrect because it does not take into account that, in general, the top spin density matrix is not diagonal.
• In a recent work [21] O (α s ) corrections are incorporated to the polarized top decay angular analysis. In this work the density matrix for the top spin is properly taken into account. To connect this work with ours we have to replace their polarization vector P by Pn d where the magnitude of the top polarization P is just the spin asymmetry in our language. This taken into account, the density matrix the authors of [21] quote is in the basis {|+n W + , |−n W + } wherê n W + is a normal vector in the direction of the three momentum of the W + (in the top rest frame).

Conclusions
We have done a complete calculation of the subprocess cross sections for polarized tops or anti-tops including the right effective coupling and bottom mass corrections. We have used a p T > 30 GeV. cut in the transverse momentum of the producedb quark and, accordingly we have retained only the so called 2 → 3 process, for the reasons described in the text.
Our analysis here is completely general. No approximation is made. We use the most general set of couplings and, since our approach is completely analytical, we can describe the contribution from other intermediate quarks in the t channel, mixing, etc. Masses and mixing angles are retained. On the contrary, the analysis has to be considered only preliminary from an experimental point of view. No detailed study of the backgrounds has been made, except for the dominant gg → tt process which has been considered to some extent (although again without quantitative evaluation).
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 statistical sensitivity to different values of this coupling is given in the text.
We present a variety of p T and angular distributions both for the t and theb quarks.
Obviously, the top decays shortly after production, but we have not made detailed simulations of this part. In fact, the interest of this decay is obvious: one can measure the spin of the top through the angular distribution of the leptons produced in this decay. In the Standard Model, single top production gives a high degree of polarization (84 % in the optimal basis, with the present set of cuts). This is a high degree of polarization, but well below the 90+ claimed by Mahlon and Parke in [10]. We understand this being due to the presence of the 30 GeV. cut. In fact, if we remove this cut completely we get a 91 % polarization. Still below the result of [10] but in rough agreement (note that we do not include the 2 → 2 process).
Inasmuch as they can be compared our results are in good agreement with those presented in [9] in what concerns the total cross-section. Two different choices for the strong scale µ 2 are presented.
In addition, it turns out that when g R = 0 the top can never be 100% polarized. In other words, it is in a mixed state. In this case we show that a unique spin basis is singled out which allows one to connect top decay products angular distribution with the polarized top differential cross section.
Finally it should be mentioned that a previous study for this process in the present context was performed by the present authors [24] using the effective W approximation [25], in which the W is treated as a parton of the proton. While this is certainly not an exact treatment, it is expected to be sufficiently good for our purposes. In the course of this work we have found, however, a number of interesting differences. A detailed comparison with this practical and widely used approach will be given elsewhere.

Acknowledgments
We would like to thank A. Dobado

A Subprocesses cross sections
In this appendix we present the analytical results obtained for the matrix elements corresponding to the processes of Figs.(1-4) as M d + , Mū + , M u − , and Md − , respectively. Defining we have the square modulus with and Finally, it can be shown that we can obtain the other matrix elements from the above expressions performing the following changes it is useful to note also that all matrix elements are symmetric under the change The subprocesses contributing to these histograms have been calculated at tree level in the electroweak theory. The cuts are described in the text. The degree of polarization in this spin basis and reference frame is only 69% . The QCD scale is taken to be µ 2 =ŝ = (q 1 + q 2 ) 2 .          to the couplings g L = 1, and g R = 0.1 has been omitted for the sake of clarity and it is located roughly in between the ones showed in the picture. Note that the sensitivity of this cross section to variations of the right coupling around its SM tree level value is comparable to the sensitivity to variations of the left coupling around its SM tree level value (see figure   16).