Symbolic expressions for fully differential single top quark production cross section and decay width of polarized top quark in presence of anomalous Wtb couplings

Spin correlations in the t-channel single top quark production and its subsequent decay are investigated for the case of contributions involving anomalous Wtb couplings. We obtained analytical expressions for the differential width for the three-particle decay of a polarized t-quark in its rest frame and also expressions for the differential cross sections of the full process of production and decay of the t-quark ($2\to 4$) as a function of the energy of a charged lepton and two angles of orientation of the quantization axis of the t-quark spin. The expression is presented in the most general form for the case of real and imaginary vector and tensor anomalous Wtb couplings. It is shown that shapes of certain multidimensional kinematic distributions of final state particles are significantly different for the contributions proportional to different combinations of the anomalous couplings. The most noticeable differences appear in the shape of the surfaces of two-dimensional distributions, where one of the variables is the energy of a charged lepton and the other is one of the t-quark spin orientation angles. Observed properties are confirmed by two methods of computations either from the obtained symbolic expression for the differential cross sections of full process of production and decay of the t-quark or by means of the CompHEP program for the complete process involving the t-channel single top. In addition, using the obtained analytical expressions, we estimate the statistical accuracy of extracting values of the anomalous Wtb couplings for different levels of the expected integral luminosity at the LHC.


Introduction
With the Higgs boson discovery at the LHC the Standard Model (SM) is completed in a sense that all predicted particles are found and all the interactions between particles are fixed. However, most likely, the SM is a kind of effective field theory working (and working amazingly well) at the energy range determined by the electroweak (EW) energy scale. The top quark being the heaviest found fundamental particle with closest to the EW scale mass is a promising place to search for possible deviations from the SM(see, recent reviews on top quark [1][2][3][4][5][6]). In particular such deviations may be related to a presence of the top quark anomalous couplings which are usually parametrizes in terms of a number of the gauge invariant dimension 6 operators given in [7] and in the Warsaw basis [8] following the notations from [9]: uW = (q L3 σ µν τ I t R )φ W I µν , These operators lead to the effective Lagrangian [14] allowed by the Lorentz invariance parametrizing the anomalous terms in Wtb vertex where M W is the W-boson mass, P L,R = (1 ∓ γ 5 )/2 is the left(right)-handed projection operator, W − µν = ∂ µ W − ν − ∂ ν W − µ , g is the weak isospin gauge coupling, parameters f LV (T ) and f RV (T ) are the dimensionless coefficients that parametrize strengths of the left-vector (tensor) and the right-vector (tensor) structures in the Lagrangian.
The couplings in the Lagrangian (2) are related in the following way to constants C (3,33) φq , C (33) φud , C (33) dW , C (33) uW in front of the gauge-invariant dimension six operators [10][11][12][13] : If one assumes that naturally the constants in front of the operators are of the order of unity, the natural values for the anomalous couplings are of the order of v 2 Λ 2 and therefore rather small. This is confirmed by the recent most stringent experimental limits [15].
In the SM all fermions interact through the left-handed currents and all constants (3) are equal to zero, except f LV = V tb (CKM-matrix element). Note that the anomalous coupling parameters could be complex in the most general case.
In this paper we discuss a simple idea allowing to improve further the limits on anomalous couplings. The idea is based on the top quark spin correlation properties in the t-and s-channel single top production with its subsequent decay. Spin correlations in presence of anomalous couplings have been studied previously in [16][17][18][19][20][21][22][23][24][25][26][27]. The effect of decoupling of anomalous top decay vertices in angular distributions have been studied in [28][29][30]. T -odd correlations in polarized top quark decays have been studied in [31]. Decays of polarized top quarks in frame of SM at Next-to-Leading and Next-to-Next-to Leading Order in QCD have been studied in papers [32][33][34][35]    As well known in the SM, the positively charged lepton from the top quark decay in its rest frame tends to follow the top quark spin direction [16,17]. In the t-and s-channel single top production the direction of the top quark spin in the top rest frame is highly correlated with the d-quark momentum for the t-channel (outgoing light jet) and anti d-quark momentum (incoming parton corresponding to the beam axis) for the s-channel [18,19]. One can understand this very simply by considering the single top production as a decay back in time [22]. The direction of preferred spin configuration in the single top production in presence of anomalous couplings is changed insignificantly comparing to SM due to smallness of the anomalous couplings. So one can chose the direction related to mentioned d-quark momentum as a top spin quantization axis and to make sure that this will be a preferred spin direction of top quark in its rest frame similar to SM. After production, the highly polarized top quark decays (Fig. 1), and one can use properties of such a polarized decay.

Top decay
To investigate the effect of anomalous parameters on the decay of a polarized top quark, we will calculate the differential width of the 3-body t-quark decay. At first, using Feynman rules from Lagrangian (2), with a help of symbolic manipulation system FORM [37], we calculate the matrix element squared of the polarized t-quark decay into a charged lepton, d-quark and b-quark: where: s is the spin vector of the top quark.
For further analytic calculations we use coordinate system that shown in (Fig. 2). Here the angle θ is the angle between the charged lepton momentum and the direction of the top quark spin quantization axis, φ is the angle in the plane perpendicular to the lepton momenta counted from the decay plain formed by the top quark decay products, Therefore, in the top quark rest frame, we have the following parametrization for the direction of the quantization axis of top quark and for the 3-momentum of the positron and b-quark: s = (sin θ cos φ, sin θ sin φ, cos θ), p e + = |p e + | · (0, 0, 1), The cosine of the angle between b-quark and the top quark spin quantization axis is: cos θ bs = sin θ be ·sin θ·cos φ+cos θ be ·cos θ. One can also express the cos φ in terms of other angles: cos φ = (cos θ bs − cos θ be · cos θ)/(sin θ be · sin θ). It leads to: φ = arccos cos θ bs − cos θ be · cos θ sin θ be · sin θ The expression (6) is used to reconstruct the φ angle in numerical Monte-Carlo simulation. We substitute parametrization (5) to the matrix element squared (4). Using the 4-momentum conservation law in the t-quark rest system, one can express the neutrino momentum through the momentum of the b-quark and positron. Then we integrate the matrix element squared of the polarized t-quark decay (4) over the b-quark energy using the narrow width approximation for the W-boson decay and neglecting the b-quark mass in comparison to the top quark and W-boson masses. Symbolic computation gives the following expression for the fully differential partial decay width of the top quark in its rest frame: This expression was obtained for the first time in such a complete form for such a parameterization. The expression (7) contains of 8 terms corresponding to different possible quadratic combinations of the coupling products. Each term contains a multiplier (1 − ) or ( − r 2 ), as well as a polynomial multiplier being a function of kinematic variables , cos θ, sin θ, cosφ or sin φ. One should note that all the 8 terms are different functions of the variables resulting to different shapes of multidimensional distributions. Therefore such multidimensional distribution shapes (multidimensional surfaces) can be used to separate contributions from different anomalous coupling combinations.
Let us integrate the expression of the fully differential partial decay width of the top quark (7) over the angle φ from 0 to 2π and obtain the doubly differential partial width Since the first six terms in (7) are even functions of φ, their integral over the φ in the range from π to 2π is equal to the integral over the φ in the range from 0 to π. At the same time, the last two terms in (7) proportional to sin φ are odd functions of φ and their integral over the φ in the range from π to 2π equal to the integral over the φ in the range from 0 to π, taken with the opposite sign.
The angular dependence of the expression was simplified after integration, but the energydependent factors (1 − ) and ( − r 2 ) did not change and the differences between the various terms of the expression remained. The first six terms of the formula (8) agree with the expression previously obtained in [29]. The last two terms give zero contribution to the 2D distribution on the energy and cos θ. But we kept them in (8) as a factor (1-1) stressing that the term consists of two equal contributions with different signs at φ intervals from 0 to π and prom π to 2π. This fact allows one to extract corresponding anomalous couplings looking at differences (or asymmetry in φ) of the two distributions.
Also, integrating the expression (7) over cos θ from -1 to 1 one can obtain the doubly differential partial width Parts of polynomials containing the cos θ function disappear after integrating the expression (7) over cos θ. The function sin θ in the remaining parts of (7) is replaced by the factor π/2. However, in spite of these changes, the differences between the eight terms of the formula (9), proportional to anomalous couplings, remain. Now integrating the expression of the fully differential top quark partial decay width (7) over from r 2 to 1, we obtain the doubly differential partial width dΓ t→bνe + d cos θ · dφ as a function of the two orientation angles: (1 − r 2 ) · sin θ cos φ + 6r · (1 + cos θ) (1 − r 2 ) · sin θ cos φ + 6r · (1 + cos θ) The first term in the expression (10) is the well-known SM like contribution [16,17] corresponding to 100% spin correlation behaviour (1+cos θ). The interference between the left-vector term with real coupling and the right-tensor term with imaginary coupling corresponding to CP violating part coincides exactly with the expression presented in [23]. As one can see, integration (7) over eliminates the factors (1 − ) and ( − r 2 ) and makes the terms proportional to anomalous couplings in the expression (10) more similar to each other.
For the case of purely real anomalous couplings, this expression is consistent with the previously obtained [36].

Top production and decay
Now we use the expression for the differential decay width of the t-quark to derive a differential cross section for the complete process of the single top production with its subsequent decay.
The dominant process of single top production at the LHC collider is the t-channel process shown in the (Fig. 3). It is known that in the framework of the SM in a t-channel process ub → td, in its rest frame the t-quark is produced polarized in the direction of the d-quark. Therefore, we set the components of the spin vector of the t-quark along this direction in amplitudes of the production and decay and square the amplitude of the complete process ub → d, b, ν, e + . Going into the t-quark rest frame and writing explicit scalar products through the components of the 4-momentum and angles, summing over the spin components one gets an expression for the matrix element of the complete process ub → d, b, ν, e + .
To calculate the cross section of the complete process, we use the formula for t-channel anomalous production of the top-quark [38]. where: Integrating the matrix element of the complete process ub → d, b, ν, e + over all variables of the phase space, except ( , cos θ, φ), neglecting the b-quark mass in comparison to the top quark and W-boson masses, and using formulas (7) and (12), we get differential cross section of the polarized t-quark production and decay: where Γ t is the total decay width of the t-quark, taking into account the anomalous couplings and all decay modes, σ(ŝ) ub→td is the cross section of the unpolarized t-quark production (12) and σ R (ŝ) ub→td part of the cross section of the unpolarized t-quark production that proportional |f RV | 2 and |f RT | 2 , and dΓ t→bνe + d · d cos θ · dφ is the differential partial width of the polarized t-quark decay (7), θ and φ -orientation angles of the positron with respect to direction of the d-quark momentum.
The expression (13) was obtained for the first time. Integrating it over one of the variables, we obtain expressions for all possible double differential scattering cross sections: After integration over all variables, the cross section takes on a well-known simple form: where Br t→bνe + = Γ t→bνe + /Γ t .

Numerical illustration, real couplings
As the first numerical illustration, we draw normalized plots (Fig. 5) of differential partial width dΓ t→bνe + /(d · dcosθ) at φ interval from 0 to π for all eight terms of formula (8), which correspond to different combinations of anomalous couplings. As one can see, the shapes of the surfaces corresponding to the various terms of (8) are very different from each other. This set of 8 different surfaces can be used as a basis for multi-parametric fitting function to get limits on anomalous couplings. To verify this statement, we present the following numerical illustration for a few simple scenarios when we alternately set one of the anomalous couplings to a non-zero value. In each case we present surfaces corresponding to the formulas (8), (9), (10) for the differential t-quark width. Then we show surfaces for the same kinematic variables for differential cross sections as follows from the formulas (14), (15), (16) after integration with the parton distribution functions 3 at 14 TeV energy at the LHC collider. To validate the correctness of the results, we perform Monte-Carlo event generation for the 2 → 4 process for the dominating t-channel single top production with subsequent 3 body decay of t-quark ub → d, b, ν, l + (Fig. 3) using the CompHEP program [39] and show the same multidimensional distributions in the top rest frame taking all the angles with respect to the d-quark momentum (Fig. 4) and restore angle φ using an expression (6). Note that the anomalous couplings are included both in the top quark production and decay process. Values for anomalous couplings are taken according to the latest experimental upper limits [15]. To make deviations from the SM more pronounce we subtract the SM contribution. To correctly compare the shapes of different distributions, we normalized them to the value of the full integral. As follows from the formula (13) the leading terms up to second power on anomalous couplings have the form: where σ ub→td is the cross section of the unpolarized t-quark production (12) and (σ R ) ub→td is a part of this cross section that proportional |f RV | 2 and |f RT | 2 , (σ RT ) ub→td is a part that proportional V tb · Ref RT , (Γ RT ) t→bνe + is a part of top partial width that proportional V tb · Ref RT and V tb · Imf RT , (σ SM ) ub→td is the SM cross section of the unpolarized t-quark production, r Γ = Γt (Γ SM )t , Γ t is the total decay width of the t-quark, taking into account the anomalous couplings and all decay modes, (Γ SM ) t is the SM total decay width of the t-quark. The first term of the formula (18) reproduces shapes of the distributions as follows from the formula for the differential width (7), whereas the second, third and fourth terms give additional contributions.
As the first scenario, we consider the case when the left vector anomalous coupling (Ref LV − V tb ) is not equal to 0, and the remaining anomalous couplings are equal to 0. In this case, the last two terms of the expression (18) are zero, and the first two terms reproduce the same shapes of the surfaces as that for the Standard Model with 100% spin correlation behaviour (1 + cos θ), the maximum of the energy distribution at E e + = m t /4 ≈ 43 GeV and no dependence on φ angle. Figure (6) shows normalized distributions corresponding to this scenario. The upper figures show plots of the normalized double-differential t-quark decay partial width. The middle figures show plots of the normalized double-differential cross sections and the lower figures show plots of the normalized double-differential cross sections built from Monte Carlo events.
For the second scenario, Ref RV is set to be non-zero while the remaining anomalous couplings are taken to be zero. Unlike the previous one, in this scenario, all terms of the formula (18) have different behaviour with respect to kinematic variables and give contributions to twodimensional surfaces shown in (Fig. 7). The first term (18) depends on used variables , cos θ, and φ angle and reproduces the same shapes as those from the differential top width (upper plot in Fig. 7). The second term in (18) gives the shapes as for the SM slightly affecting the common shapes. The third term are zero. The fourth term does not depend of the φ angle but being proportional to the |f RV | 2 coupling changes the overall shapes significantly. Since this term is proportional to cos θ, its influence is most noticeable in cos θ distributions (Fig. 7  middle left and right).
For the third scenario, we set Ref LT not equal to 0, and the remaining anomalous couplings are equal to 0. In this case, the last two terms of the expression (18) are zero and there is no dependence on the angle φ in the first and second terms. The dependence of both terms on cosθ is the same as for SM, but the energy distributions are different. The second term of the formula (18) slightly deviates shapes of differential cross sections (Fig. 8 middle left and central) from the corresponding shapes for the differential width ( Fig. 8 upper left and central).
For the fourth scenario, we set Ref RT non-zero and positive, and the remaining anomalous couplings are equal to 0. For this scenario, the formula for the differential cross section (18) is the most complex of all the listed scenarios. The first part of formula (18) contains quadratic anomalous terms, as well as the leading linear interference term, which mainly determines the shape of the differential cross sections. The second, third and fourth terms of the formula (18) contain quadratic anomalous terms, which only slightly affect the shape of differential cross sections (Fig. 9 middle).
For the last scenario, Ref RT is set to be non-zero and negative, and the remaining anomalous couplings are equal to 0. The overall picture is almost similar to the previous case with the only difference that deviations from the prediction of the Standard Model have the opposite sign (Fig. 10).
One can see a good agreement of the plots obtained using the analytical formula (13) with the corresponding Monte-Carlo distributions for all listed scenarios. This comparison confirms the correctness of the analytical calculations performed. It should be noted that the presence of an additional subprocessdb → l + , ν l , b,ū does not have a noticeable effect on the shape of the distributions. It is clearly seen that in all the scenarios listed above, except for the second one, the shape of the differential cross section is determined mainly by the differential width of               and d(σ−σ SM ) d cos θ · dφ built from formulas (14), (15), (16). The lower figures show plots of the normalized double-differential cross sections built from Monte Carlo events.     and d(σ−σ SM ) d cos θ · dφ built from formulas (14), (15), (16). The lower figures show plots of the normalized double-differential cross sections built from Monte Carlo events.  Figure 10: Scenario Ref RT = -0.048. Plots of the normalized double-differential cross sections: the t-quark decay. But in the case of the scenario (Ref RV = 0), the second and fourth terms of the formula (18) significantly change the shape of the differential cross sections. As a main conclusion, it can be noted that the corresponding shapes of the surfaces in the coordinate space ( , cos θ) and ( , φ) are very different for various scenarios. The most spectacular differences can be observed in coordinate space ( , cos θ). This allows one to separate all scenarios from each other. At the same time, the plots in the space (cos θ, φ) are quite similar for listed scenarios (except for Ref RV = 0 case) and do not allow them to be uniquely identified.
In addition to the two-dimensional distributions, we draw plots showing the probability density d(σ−σ SM ) d · d cos θ · dφ in the 3D space ( , cos θ, φ) for the cases listed (Fig. 11). The size of each cell on the figures is proportional to the probability density. One can see that the areas of maximum density for different scenarios are located in different places of the 3D cube ( , cos θ, φ). Although the three-dimensional areas are almost the same for cases (Ref RT = 0.048) and (Ref RT = -0.048), the corresponding deviations from the Standard Model due to anomalous contributions for these scenarios have the opposite signs. Now we use the obtained analytical expressions to extract the values of anomalous coupling applying a fitting procedure. For completeness of MC simulation, we also included additional subprocessdb → l + , ν l , b,ū withd quark in the initial state. In the case of SM, the contribution to the rate from such events is about 13% while the impact on the distribution shapes are practically the same. We generate SM Monte-Carlo event samples for different values of the integral luminosity and collision energy of the LHC collider: 30 f b −1 at 13 TeV, 300 f b −1 and 3000 f b −1 at 14 TeV. Using analytical expressions corresponding to different scenarios and the method of maximum likelihood, we fit histograms built from these MC events. In this case, we are not interested in the absolute value of the couplings, but in the accuracy of their For fitting we use MINUIT algorithm [40] build into the ROOT package [41]. The results of two-parametric fitting of two-dimensional histogram dσ d · d cos θ are given in (Table 1). The values of the coupling Ref LV and one of the non-zero anomalous coupling of the above scenarios were used as fitting parameters.
Similar to the two-dimensional case, we fit the distribution of the SM events in 3D space ( , cos θ, φ). (Table 2) shows the results of this fitting. The values of coupling Ref LV and one of the non-zero anomalous coupling of the above scenarios were used as fitting parameters.
It can be seen that fitting in 3D space ( , cos θ, φ) made it possible to improve the accuracy of measuring the couplings f LV , f RV and f RT . But the accuracy of measuring the coupling f LT was slightly worse than the results of fitting in two-dimensional space ( , cos θ). Despite the fact that the 3D histogram contains more information than the 2D histogram, the fitting in case of 3D distribution is technically more complex. Here, much depends on the histogram binning and some other settings. By optimizing these parameters, it is possible to further improve the measurement accuracy of couplings. It can be seen that the accuracy of measuring of anomalous couplings is much higher (almost an order of magnitude) than the current available experimental accuracy [15] with the corresponding value of the integral luminosity of the LHC. Of course, our simulation corresponds to the ideal case when we can accurately restore the t-quark system and do not take into account the effects of the detector response. However, this illustration demonstrates the potential for increasing measurement accuracy using the proposed method.

Numerical illustration, imaginary couplings
In the previous examples, we considered cases when only φ-even terms of formula (13) are involved. To use the remaining terms, proportional to sin φ, we must consider a scenario with non-zero anomalous imaginary couplings. It should be noted that in the above simulation, we had in mind that the φ angle is restored by the formula (6). With this approach, events corresponding to the angles φ and (2π − φ) are counted as events with the same φ. Therefore, integration of the differential cross section (13) over the (6) from 0 to π is equivalent to integration over the angle φ from 0 to 2π. This made sense in the considered cases where the φ-even terms of formula (13) were involved. In this case, the analysis was simplified without loss of information. But not in the case of the scenarios when terms proportional to sin φ are involved. To reflect the contribution of such terms, it is necessary to carefully separate the events corresponding to the angle φ in the range from 0 to π, and the events corresponding to φ ranging from π to 2π by using triple-product T = (p e + × p b ) · s. If T > 0: φ ∈ (0, π). If T < 0: φ ∈ (π, 2π).
As the first numerical illustration with imaginary couplings, we draw plots of doubledifferential cross sections (Fig.12, up) and plots of probability density in 3D phase space (Fig.12, down) for the scenario when Imf RT are not equal to 0, and the remaining anomalous couplings are equal to 0. For a correct comparison with the case of real coupling, we set the value of imaginary coupling Imf RT =-0.048 similar to the corresponding real one. (Fig.12, up) shows dcosθ · dφ corresponding to φ ranging from 0 to 2π. It can be seen that deviation from the prediction of the Standard Model due to anomalous contributions (σ − σ SM ) on the intervals of the angle φ: (0, π) and (π, 2π) differ in sign for this scenario. You can also notice a slight difference in the absolute values of the deviations from the SM, corresponding to different intervals of φ (Fig.12, down). This is due to the fact that the linear anomalous term from first part of (18) is an odd function of φ and, when changing the interval of φ, it changes sign, while the other quadratic anomalous terms from (18) are even and do not change. Therefore, the total sum of even and odd terms will differ at different intervals. But since the linear term dominates, and the contribution of quadratic terms is substantially suppressed, this difference is small.
For another example, when (Imf LV and Imf RT = 0) the joint contribution of the quadratic terms becomes somewhat larger and the difference in the absolute values of the deviations at different intervals increases (Fig.13).
It is also interesting to consider the case (Ref RT and Imf RT = 0), when both linear terms of formula (7) are involved in the game, but one of them proportional to the Ref RT is even, and the other proportional to Imf RT is an odd function of φ. It is possible to choose a combination of couplings values in which even and odd components amplify each other in one interval φ: (0, π) and fully compensate each other in another interval φ: (π, 2π) as shown in the (Fig.14).
Finally, we consider a scenario (Ref LT and Imf RV = 0) where linear anomalous terms are absent and even and odd components are represented by quadratic terms only (Fig.15). It can be seen that in this case the even terms dominate, therefore the deviation from the SM does not change the sign on all intervals of the φ angle. However, the contribution of the odd components is manifested in the asymmetry of the deviations at different intervals of the φ angle.
As the main conclusion of this part, we note that various combinations of real and imaginary couplings belonging to the terms of formula (7) that are odd with respect to the φ angle can manifest themselves in the form of asymmetries of differential cross sections at different intervals of φ: (0, π) and (π, 2π). The shapes of such distributions differ significantly from the corresponding distributions for cases of pure real couplings, which will allow us to experimentally detect and identify imaginary couplings.
Similar to the considered case of real couplings, we apply the method of fitting the Monte-Carlo events with the obtained analytical formula, to estimate the accuracy of measuring imaginary anomalous couplings. Accuracy values for joint measurement of Ref LV and Imf RT couplings by fitting in 3D space ( , cos θ, φ) are given in (Table 3).

Conclusions
We obtained an analytical expression for the differential width for the three-particle decay of a polarized t-quark in its rest frame as a function of the energy of a charged lepton and two angles of orientation of the quantization axis of the t-quark spin. The expression is presented in the most general form for the case of real and imaginary vector and tensor anomalous Wtb couplings. The parts of this expression containing the contribution of the SM and its interference with the anomalous contributions are fully consistent with the published results. We have shown that the expression for the differential width of the t-quark can be divided into 8 kinematically different terms corresponding to possible combinations of anomalous parameters. These 8 terms have different dependencies on distribution variables and we suggest using them as basis functions when fitting experimental data and extracting anomalous parameter values. Also, these 8 analytic functions can be used as multidimensional variables for analysis based on the neural network method. This makes it possible to most effectively separate various anomalous contributions.
In addition, expressions for the differential width of the t-quark were obtained as functions of various combinations of two variables: the energy of the charged lepton and one of the t-quark spin orientation angles, as well as two spin orientation angles. Using the obtained analytical expressions, we constructed various two-dimensional plots corresponding to different anomalous scenarios The most noticeable differences appear in the shape of the surfaces of two-dimensional distributions, where one of the variables is the energy of a charged lepton, and the other is one of the t-quark spin orientation angles. At the same time, the dependence of the t-quark width only on two angles is less informative and does not effectively separate one anomalous component of the width from the other.
Also, we have shown that the formulas obtained for the differential width of the t-quark can be used to derive the differential cross section for the full process of production and decay of the t-quark (ub → d, b, ν, l + ). Where the optimal direction of the quantization axis of the t-quark in its rest frame is the direction of the d-quark from the t-quark production.
To verify the obtained analytical results, we performed a numerical simulation of the full tchannel processes of t-quark production and decay at the LHC collider for various scenarios with real and imaginary anomalous couplings. The values of the recent experimental upper limits were taken as the values of the anomalous parameters. Based on the Monte-Carlo generators created by CompHEP, various two-dimensional distributions with respect to the energy of the charged lepton and the orientation angles of the t-quark spin in its rest frame were constructed. Also, for the considered scenarios, using the obtained analytical expressions for the t-quark production and decay differential cross sections, the corresponding two-dimensional plots were constructed. Comparison of the shapes of Monte Carlo distributions with the plots obtained from the formulas showed their full agreement with each other and confirmed the correctness of the analytical calculations performed.
In addition, using the obtained analytical expressions, we estimated the accuracy of extracting the values of the anomalous Wtb couplings for different levels of the integral luminosity of the LHC collider using fitting methods. The predicted accuracy values are an order of magnitude higher than the current experimental accuracy for the same integral luminosity. Despite the ideal nature of our theoretical experiment, it shows the potential for improving the accuracy of measurements using the method we have proposed in addition to the experimental methods already used.

Acknowledgements
The work was supported by grant 16-12-10280 of Russian Science Foundation. The authors are grateful to L. Dudko and Y. Kurihara for useful discussions and critical remarks.