Polarization of a vector boson produced in decay of a heavy fermion in an arbitrary frame

We consider processes where an electroweak gauge boson ($W$ and $Z$) is produced in the decays of a heavy fermion. The polarization state of the produced gauge boson is given in terms of a vector polarization and a rank-2 tensor polarization. In the rest frame of the mother fermion, these are given directly by the dynamical parameters of the underlying theory. In a frame where the mother particle is moving, the polarization parameters of the gauge boson are dependent additionally upon kinematical factors. We show that these kinematical factors depend only on the magnitude of the velocity of the mother particle and derive analytical expressions for them. We apply the results to pair production of heavy fermions at the LHC with one of the fermions decaying to a gauge boson ($W$, $Z$) and a light Standard Model fermion. We construct estimators of laboratory frame values of polarization parameters of the produced gauge boson. These estimators can be used to estimate the laboratory frame values of polarization parameters of the produced gauge boson without a detailed simulation of the entire process. We validate our expressions with detailed Monte Carlo simulations in the context of beyond the Standard Model scenarios which have a vector-like top partner. We also indicate how to include finite width effects of the heavy fermion in some special cases.


I. INTRODUCTION
Polarization of unstable particles such as the top, the tau or the electroweak gauge bosons (W, Z) produced in high energy colliders is a powerful tool. It can be used to probe the production or decay of these particles. In particular, the top quark's polarization has been widely suggested as a probe of new physics [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20]. There exist works on the polarization of tau lepton [21] and the gauge bosons (W , Z) [22][23][24][25] as well. The polarization of a particle depends not only on the physics involved in its production but also on the kinematics of its production [26][27][28]. In this work, we investigate the effect of kinematics on the polarization state of a spin-1 particle, such as W or Z, in a class of processes where the boson is produced in the decays of a heavy fermion.
Consider the production of a gauge boson W at the LHC. It can be produced directly: e.g, pp → W + W − . It can as well be produced in decays of a heavy particle: e.g., pp → tt with t → bW andt decaying inclusively. The kinematics of W production in the second case is different from that in the first case. In the second case, there is a Lorentz boost from the rest frame of the top to the laboratory(lab) frame. This boost arises since the top quark, the parent particle of the W , moves with a non zero velocity in the lab frame. The effect of this boost on the value of W polarization measured in the lab frame in non trivial. To see this, we first note that the polarization of W is determined by the underlying theory in the rest frame of the top. Since the helicity states of W are not invariant under arbitrary Lorentz boosts(see, for example [29,30]), the polarization of W measured in the lab frame is, in general, different from its value in the top quark rest frame.
In this work, we generalize the above mentioned example to include all processes of the type where F denotes a heavy quark, V a vector boson (W , Z), and f a light quark 1 .. We assume that theF decays inclusively. As will be discussed below, the lab frame value of polarization of the particle is related to that in the rest frame of its mother particle by a kinematical factor. This opens up the possibility of predicting the lab frame polarization, given the theory. In an earlier work [28] we considered the polarization of a spin-1/2 particle (the top quark) in cases where the particle is produced in the decay of a heavy spin-1/2 or a spin-0 particle. This work may be considered as an extension of our previous work to the case of a spin-1 boson (such as W or Z) produced in the decay of a heavy quark. 1 The restriction to pair production of F is dictated by the requirement that the fermion F be unpolarized (see below). Since the pair production of these vector-like quarks is dominantly a QCD process, the produced vector-like quarks are unpolarized. This argument applies to the top quark pair production as well.
The gauge boson V produced in the decays of the heavy quark F further decays to SM fermions according to its polarization state. Since the vector bosons are spin-1 particles, the polarization state is described by a three component vector (P i (i = x, y, z)) and a five component tensor polarization (T xy , T yz , T zz , T xx − T yy , T xz ). In the rest frame of the heavy quark F , the polarization parameters of V are given by dynamical parameters such as couplings, mixing matrices and masses of the particles involved in the decay. On the other hand, the lab frame values of polarization parameters of V are simpler to be measured in experiments as they do not require the full reconstruction of the entire event.
To connect the experimentally measured values of polarization parameters with theory parameters (the dynamical parameters relevant in the decay of F ), one requires a method to predict the lab frame values of the polarization parameters from a theory. To achieve this, we propose estimators for lab frame polarization parameters of V [28].
These estimators (for the eight polarization parameters of V ) are of the following form: Here, σ F F is the cross section for the pair production of the parent quark F : pp → FF and β F is the velocity of F in the lab frame. P(β F ) can be interpreted as the polarization parameter of V in a single event. Then the above expression can be interpreted as a weighted average of a polarization parameter of V over the entire sample of events. The weighting factor dσ F F dβ F includes convolution over the parton distribution functions. Note that these estimators require only the magnitude of velocity of its mother particle. Hence, these estimators are simple to use in a Monte Carlo simulation of the relevant process. An advantage of the use of these estimators is that one can now predict the value of polarization parameters of V without any simulation of the decay of F , its mother particle. These estimators are derived under the assumption that the parent quark has a narrow width (Γ/m F 1). We shall later relax this assumption and consider the case of larger widths of the mother particle in Sec. V. We find that these estimators reproduce the polarization parameters of V to within a few percent in the narrow width case. In the finite width case, when the non resonant and off-shell contributions are small, we modify these estimators to include the Breit-Wigner shape of the propagator of F . We find that both the original and the modified estimators provide equally good approximations to the polarization parameters of V , in the finite width case. The masses of these vector-like quarks are constrained by the direct searches at the LHC [54,55].
The assumption is that they couple only to third generation SM quarks. In general, vector-like quarks can couple to the first two generations of SM fermions as well [33]. Depending upon the assumptions on the branching ratios, the lowest direct search bounds from the LHC read 790 GeV for the charge +2/3 quark, 730 GeV for the charge -1/3 quark [55]. masses. The constraints depend upon the representation of these vector-like quarks under the SM gauge group [32][33][34][35].
This paper is divided into six sections with the first section being the introduction to this work. Section II describes the expressions of polarization parameters of V in the rest frame of F .
Section III discusses two models of a vector-like quark T of charge +2/3. In this section, benchmark values of model parameters are provided and the corresponding Z polarization parameters are obtained for the decay T → Zt.
Section IV describes the formalism for the derivation of the polarization estimators. Section V completes the derivation of polarization estimators of V at the level of pp collisions and describes numerical validation of the polarization estimators. Section VI presents a summary of this work.

II. POLARIZATION PARAMETERS OF V
In this section, we give expressions for the polarization parameters of the vector boson V , in the rest frame of the parent particle F in the decay F → V f . The only non-vanishing polarization parameters are P z and T zz and they are defined by: where Γ i (i = +, 0, −) denote the partial decay widths corresponding to the decay of F into an unpolarized f and an electroweak gauge boson V with helicity i. Taking the vertex F f V vertex as γ µ (g L P L + g R P R ) where P L and P R are the left and right chiral projectors, we get, and m f denote the masses of the particles F , V and f , respectively. Note that T zz is non zero even when g L = g R .

III. MODELS
In this section we consider the models which have a vector-like quark (T ) of charge +2/3 mixing with the third generation quarks of the SM [32,34]. The models depend upon the representation of T under the SM gauge group. We consider the cases where T is an electroweak singlet, T forms an electroweak doublet with a vector-like quark B of charge -1/13. We refer to them as the Singlet Model and the Doublet Model, respectively. In both the models, we consider the decay: T → Zt where t is the top quark.

A. The Singlet Model
The Yukawa part of the Lagrangian reads The Yukawa coupling of the top is denoted as y t . We have not included the Yukawa part of the SM in the above equation. The primes refer to the fact that the Lagrangian is in gauge basis.
After the Higgs doublet acquires a vacuum expectation value (vev), Φ = (0, v/ √ 2) (the prime denotes transpose), T mixes with the SM fermions with the amount of mixing determined by λ.
Neglecting the top mixing with the first two generation fermions, we get two mass eigen states (t, T ) by performing a bi-diagonalization of the mass matrix. Taking the mass eigen values as the physical masses of the top and T , m t and m T , respectively, we have

B. The Doublet Model
The Yukawa part of the Lagrangian reads, where θ u L,R and θ d L,R denote mixing angles of the left chiral and right chiral parts of the t − T and b − B pairs, respectively and m b and m B denote the masses of the bottom quark and the B quark, respectively. Constraints from the appearance of tree level corrections to the Zbb vertex, contribution to oblique parameters and loop level contributions to Zbb vertex read sin θ d R < 0.06 and sin θ u R < 0.13 for m T > 750 GeV at 95% C.L [34]. The mass splitting between B and T is also severely constrained to lie within a few GeVs [34]. We assume that B is degenerate in mass with T : m B = m T . In this model, there are three free parameters: m T , θ d R and θ u R .

C. Benchmark points
In this subsection, we describe our choices of parameter values that are used throughout this work. In the case of the Singlet Model and the Doublet Model, we take the lower bound of the mass of T as 900 GeV, based on the current direct search constraints. We take θ L = 0.1 in the Singlet Model due to the constraints mentioned in the previous section We set m B = m T , θ u R = 0.09 and θ d R = 0.05 in the Doublet Model, to be consistent with the constraints. The values are tabulated in Table I.  parameter Singlet Doublet List of parameters and their values in the two models, the Singlet and the Doublet models.

D. Discussion
We now discuss the value of P z and T zz for specific models described above for the decay T → Zt, and t → bW in the SM. In the case of Singlet Model, the coupling ZtT which is responsible for the decay T → Zt, is purely left chiral: g L = (g/(2 cos θ W )) sin θ L cos θ L and g R = 0 where θ W denotes the weak mixing angle and g the SU (2) L gauge coupling. In the case of Doublet Model, the ZtT vertex is purely right chiral: For these two models, the expressions for P z and T zz can be obtained by the following replacements: t /m 2 T and substituting the values of g L and g R in Eq. 4. In the case of the SM top decay, the coupling is purely left chiral: g L = g/ √ 2 and g R = 0. The expression for P z and T zz of W can be obtained by the following replacements b /m 2 t and substituting the values of g L and g R in Eq. 4.

IV. FORMALISM
To obtain the expressions for the polarization estimators, we begin by looking at the parton level process of the form: where p 1 , p 2 denote partons. We shall assume that the produced V further decays to leptons: V → ¯ . The parton level amplitude can be written as: where the primes on amplitudes denote that the wave functions of the particles which are exchanged have not been included. The quantities P µν and P αβ denote the numerators of the propagators of V and F , respectively. The indices α and β (α, β = 1, · · · 4) denote the components of the spinors and Dirac matrices. For clarity, the denominators of the propagators have been explicitly taken out of the expressions for the amplitudes. We have are the helicities and momenta of the particles F and V , respectively, measured in the parton center of mass (PCM) frame. Substituting these relations in Eq. 10, and we write the amplitude as The phase space element of the parton level process can be written in terms of 2-body phase space elements as with where P in denotes the sum of initial state parton momenta and p 2 F , p 2 V denote the invariant masses of F and V , respectively (p 2 F > 0, p 2 V > 0). Squaring the amplitude iM, multiplying with the phase space element and flux factor, and averaging over initial state spin and color indices, summing over intermediate state spin and color indices, we get the parton level cross section for the process aŝ where f in avg denotes the spin and color averaging factor for the initial state and E i , v i (i = 1, 2) denote the energies and velocities of the initial state partons. In the above equation, ∆ BW (p 2 ) = ((p 2 − m 2 ) + m 2 Γ 2 ) −1 denote the Breit-Wigner factors arising from the propagators. When the width of an intermediate particle is much smaller than its width, ∆ BW (p 2 ) can be replaced by a Dirac delta function as follows 3 : Assuming that this approximation, the so-called Narrow-Width Approximation (NWA), holds for both F and V , using the factorized form of the phase space element, we write Eq. 16 aŝ where λ dσ FF λλ =σ FF , the cross section for the pair production of F , is the partial decay width for the decay V → ¯ and f V avg and f F avg denote the spin averaging factors included in the definitions of dΓ V and dΓ F . The Lorentz invariance of the phase space element factors allows evaluation of different parts of the squared amplitude in different frames. The pair production of F can be evaluated in the PCM frame, the decay of F in the rest frame of F and the decay of V in the rest frame of V . The rest frame of V can be reached from the PCM frame through the transformation: where β P CM The rest frame of F can be reached from the PCM frame by a Lorentz transformation whereβ,θ,φ define the velocity and the direction of motion of F in the PCM frame: This transformation transforms the momenta to the rest frame of F : Under this transformation, the helicities of F are unchanged as the Lorentz transformation is along its direction of motion. On the other hand, the helicity states of V transform in the following way: where R is a rotation matrix corresponding to the rotation In the above expressions, This implies that the expression for the parton-level cross section for the process becomes, where {λ} and {i} denote the set of helicity indices corresponding to F and V that appear in the above expression. In the above expression, the subscripts F and V on dΓ f F V /Γ F and dΓ V ¯ /Γ V indicate that they have to be evaluated in their respective frames. We assume that the production of the heavy fermion pair is through QCD interactions. Since QCD conserves parity, the produced fermion F is unpolarized. This allows the following simplification, after a partial integration over the FF phase space : 4 To see that the transformation in Eq. 18 defines a rotation, consider the application of the transformations on the momentum of V in its rest frame: 0)). The transformation takes the rest frame of V to itself. Hence, it is a rotation on the rest frame of V . This can also be checked explicitly using the expressions for the Lorentz transformations.
where f F avg = 2 and dσ FF /dΩ is the differential cross section for the production of FF pair, in the PCM frame. Substituting this in Eq. 26, we get, In the rest frame of V , the matrix dΓ V ¯ is given, after partial integration over phase space, by where dΩ V = sin θ dθ dφ and θ, φ define the direction of motion of in the rest frame of V . The matrix ρ V is a matrix with i ρ V ii = 1 and can be parameterized as follows: where s x = sin x, c x = cos x. In the above equation, g L , and g R denoting the left and right chiral couplings of the V ¯ vertex.
In the case where the polarization of V is measured in the rest frame of F obtained by a sequence of Lorentz transformations starting from the PCM frame, the helicity states of V do not undergo helicity rotation. In this case, where Γ F f V is the partial width for the decay F → f V , ρ F is a matrix with constant elements, and i ρ F ii = 1. In this case, the expression forσ becomes, where B.R denotes a branching ratio. From the above expression, differential angular distribution of in the rest frame of V can be obtained: This implies that the matrix ρ F can be regarded as the density matrix for the production of V in the rest frame of F . The matrix ρ F can be parameterized as follows [30]: In the above expression, P i and T ij (i = x, y, z) contain the information of the polarization of the vector boson V . T ij is a symmetric traceless tensor (T xx + T yy + T zz = 0) and P i is a vector.
Note that the angle ω is independent of the direction of motion of F in the PCM frame, i.e, independent ofθ andφ. We have the following expressions for χ: cos χ = cos φ cos ∆φ − sin ∆φ cosθ sin φ, sin χ = − sin ∆φ sinθ β2 + β 2 −β 2 β 2 sin 2 θ + 2ββ cos θ where ∆φ =φ − φ P CM , φ P CM is a function of β,β, θ, φ andθ. The terms with cos χ and/or sin χ drop out after integration over the azimuthal angle φ of the lepton, since these terms always appear with factors such as sin φ , cos 2φ , etc. Equation 36 becomes, The terms with cos ω and sin ω are independent of the direction of motion of F (θ,φ) in the PCM frame. This means that the integration (dσ FF /dΩ)dΩ can be performed independently to give a factor σ FF . The angle ω is also independent of the azimuthal angle (φ) of the vector boson V in the rest frame of F . Hence, a partial integration of dΩ = d cos θdφ can be performed independently. The simplified Eq. 39 reads, we rewrite the above equation as Comparing this expression with the expression for the azimuthal-averaged angular distribution of a decay product of a vector boson in its rest frame [24], where σ is the cross section for the production of V , we interpret the quantities P z (β) and T zz (β) as the polarization parameters of V , for a given parton level event, as seen by a direct boost from the parton center of mass frame to the rest frame of V . Note that such a simplification does not arise had the factors involving χ been kept in Eq. 39, since they are functions of all the angular variables in the problem and hence the integration would have become a multi-dimensional one.
In such a case, this method does not offer any advantage over a Monte Carlo simulation to extract the polarization parameters of V .
Before we complete the derivation of the expression of polarization estimators (see Sec. V), we discuss the expressions in Eq. 41. Performing the integrations over cos θ, we get, whereβ is the velocity of the parent particle in the PCM frame and β is the velocity of the vector boson V in the rest frame of the parent particle. Similarly, we get the expression for T zz (β):

A. Discussion
We study the expressions Eq. 44 and Eq. 45 for models of vector-like quarks given in Sec. III where such decays are possible. We also consider the case of top decays in the SM. One can evaluate the value of P z (β) and T zz (β) at any given value of the velocity of the heavy fermionβ using Eq. 44 and Eq. 45 and the expression for β: obtained in the rest frame of F . Figure 1 shows the polarization parameters P z and T zz in the rest frame of T (β = 0) as a function of m T , for the two models given in Table I. One can see that, for large values of m T , the value of P z tends to zero while that of T zz tends to a constant value. This can be understood from Eq. 4 (with ξ V → ξ Z , ξ f → ξ t ). In the limit of large m T , |P z | → 2ξ Z /(1 − ξ t ) → 0 and T zz → − 2/3 ≈ −0.82, due to the fixed masses of Z and t. Note that both T zz and P z are independent of the value of the mixing angles since one of the two couplings of ZtT , g L and g R , is always zero for the two models considered in this work. This leads to results of T zz (β) in the singlet model being identical to that of the doublet model, for any given value of β and m T , which can be seen on the right panel of Fig. 1. Figure 2 shows the values of polarization parameters P z (β) and T zz (β) forβ = 0.95, for the two models. Comparing Fig. 2 with Fig. 1, one can see that the value of T zz significantly differs in the two cases, for each model, when the mass of the heavy fermion T is low. Such a large difference is not observed in the case of P z . In other words, the effect of the boost of the rest frame of T relative to the PCM frame is more important in T zz than P z . However, such effects decrease when the mass of T becomes larger since the velocity β of Z in the rest frame of T approaches unity (for a fixedβ). In the limit β → 1, Eq. 44 and Eq. 45 give P z (β) → P z and T zz (β) → T zz , for a fixed  Table. I. In the figure on the right panel, the lines corresponding to the two models are identical and appear merged.  Table. I. In the figure on the right panel, the lines corresponding to the two models are identical and appear merged.
β. In other words, the values of P z (β) and T zz (β) tend to remain close to their values in the rest frame of F , for any value ofβ < β , thus reducing their sensitivity toβ (< β).
The dependence of the polarization parameters P z (β) and T zz (β) is shown in Fig. 3, for the two models and for m T = 1000 GeV. One can see that the effect of the boost of the rest frame of T relative to the PCM frame (β) is stronger at large values ofβ, as it should be.
For completeness, we show in Fig 4, the polarization parameters of W in the top decay, in the SM, as a function ofβ. In this case, due to the relatively lighter parent particle, the top, the dependence of P z (β) and T zz (β) onβ (the velocity of the top in the PCM frame) are stronger even for moderate values ofβ.  Table. I. In the figure on the right panel, the lines corresponding to the two models are identical and appear merged.

V. POLARIZATION ESTIMATORS
To obtain expressions for polarization parameters of V at the level of pp collisions, the expressions in Eq. 44 and Eq. 45 on P z (β) and T zz (β) need to be convoluted over the parton distribution functions (pdfs). This is required since the parton distribution functions and the cross section determine theβ distribution in the PCM frame. Defining the cross section for the process whereσ =σ FF B.R(F → f V )B.R(V → ), we get the expressions for the polarization parameters as The superscript NW refers to the fact that we have used the NWA for F (for V it is applicable).
This expression can also be written as an average over theβ distribution as it is the only variable in the problem (the c.m. energy √ŝ can be traded forβ throughβ = 1 − 4m 2 F /ŝ).
where 1/σdσ/dβ is the normalized velocity distribution of F in the PCM frame.
To covert the above expressions into the corresponding lab frame quantities, we replace thē β distribution with the normalized velocity distribution of F in the lab frame 1/σdσ/dβ lab and the polarization parameters by P z (β lab ) and T zz (β lab ) with β lab being the velocity of F in the lab frame. In this case, the rest frame of F is understood to be obtained by a direct boost of β lab from the lab frame. Hence, the final expressions for the polarization estimators of V produced in the decay of F , in the lab frame are This equation can be interpreted as the average of P z (β lab ) and T zz (β lab ) over all the events with a weighting factor (1/σ)dσ/dβ lab . We now present the numerical validation the expressions for the polarization estimators of V given Eq. (50). We generate events for the process pp → TT followed by the decay of T into Z and t with Z further decaying to ¯ , using MadGraph [59]. The events correspond to the two models in Table I For the models of T concerned, due to the strong constraints on the couplings, the width of T remains much smaller compared to its mass throughout the range of mass i.e. 900 GeV to 2000 GeV. This means that the NWA is a good approximation throughout the mass range considered and the expression in Eq. 44 and Eq. 45 can be expected to be valid. We also considered the case of W polarization in top decays both in the case of the singlet and the doublet Models and in the case of the SM 5 . For this purpose, we generated events for pp → tt and allowed the top to decay to W and b with W further decaying to¯ ν. This provides an additional verification of our method. We use the value of velocity (β lab ) of the heavy fermion (T or t) in the lab frame, from the generated sample and compute the quantities P z (β lab ) and T zz (β lab ) for each event and obtain an average over the entire event sample. This is equivalent to the use of Eq. 50. This method yields the values of P N W z and T N W zz , since the heavy fermion is assumed to be on-shell due to the NWA.
The value of polarization parameters can be directly extracted from the Monte Carlo event samples through the use of angular asymmetries of the lepton from the Z decay [24]. Consider the asymmetries A z and A zz defined by and where θ is the polar angle of the lepton (from the Z decay) in the rest frame of Z that is obtained by a direct boost from the lab frame. σ is the cross section for the production of V followed by V → ¯ and is the same as σ in Eq. 47. These asymmetries can be directly related to the values of polarization parameters of Z (see Eq. 43): performing the convolutions with parton distribution functions, we get where the M C superscript refers to the fact the polarization parameters are extracted from a Monte Carlo simulation. Note that in the case of W produced in the top decay, the lepton corresponds to a neutrino (W →¯ ν), which is unobservable. Hence, one need to use the asymmetries of¯ in place of the above mentioned asymmetries of . Since the neutrino and the anti-charged lepton have equal and opposite momenta , θ¯ = π − θ . This means that there is an additional factor of (−1) in the first expression of Eq. 53. Figure Table. I. GeV. The remaining parameters for the models are taken as in Table. I.
for four choices of pp center of mass energy √ S. One can see the excellent agreement between the two methods.
We have so far assumed that the width of the mother particle is small compared to its mass.
This justified our application of the Narrow Width Approximation by which we have taken the on-shell mass of the mother particle as its mass. Due to the strong constraints on the couplings of the vector-like quarks, in the two models that are considered here the width of T remains small (Γ/m < 0.02) compared to its mass throughout the mass range considered. This can be expected since the decays of T are of electroweak type. However, we indicate how to extend the applicability of our method to the case where the width of T is large (Γ/m ∼ 0.1). In this case, non-resonant  Table. I. The pp center of mass energy √ S is taken as 13 TeV.
contributions to the process in question can not be neglected, in general. For example, the vector boson may come from a t-channel heavy fermion exchange rather than coming from the decay of the heavy fermion. The kinematics of such a process is different from the decay process we are interested in. Moreover, there are additional spin-correlations between the production and the decay of the heavy fermion T when the fermion is off-shell [60][61][62]. These off-shell effects are not present in the narrow width case, since the application of NWA results in on-shell vector-like quark F . Hence, the extension of our method to the cases where T has a finite width is, in general, highly non-trivial. However, in the case where the non-resonant production and the additional spin-correlation effects are small, we can construct appropriately modified polarization estimators.
Since, in this case, the only additional effect is the smearing of mass of the parent particle, we take A comparison of the two sets of estimators is shown in Fig. 7, for the two models given in Table I with the width-to-mass ratio of 10% (Γ T /m T = 0. to within a few percent. This shows that the estimator derived in the narrow width case can also be used when the parent particle has a finite width. This is justified provided we neglect any off-shell spin and non-resonant contributions. The pp collision center of mass energy √ S is taken as 13 TeV. The remaining parameters for the models are taken as in Table. I.

VI. SUMMARY
In this work, we obtained expressions for the polarization parameters of a vector boson V (such as W, Z) produced in the decay of a heavy fermion such as a possible vector-like quark or the top quark, both in the rest frame of the mother particle and in a frame where the mother particle is moving. Based on these expressions we constructed simple estimators of the polarization parameters requiring only the velocity distribution of the mother particle, apart from the necessary couplings and masses involved in the decay of the mother particle. Since the vector boson has both a vector and a tensor polarization, we construct two non-trivial estimators one for each type of polarization, which survive in the azimuthal-averaged decay distribution of the vector boson. The estimators P N W , T N W zz are derived under assumption that the width of the mother fermion is small compared its mass, applying the Narrow Width Approximation for the mother particle. The advantage of this method is the possibility of a quick estimate of polarization parameters of the vector boson, in frames such as the lab frame. The polarization estimated by this method can be measured experimentally without any requirement to reconstruct any intermediate frame. We demonstrate the validity of this method in a set of models with a vector-like like quark and also in the case of the top decay in the SM.
We believe that this work will aid the study of the vector-like quark phenomenology. This is because the polarization information carried by the vector boson from the decays of vector-like quarks can be a probe of the coupling structure of the decay vertex. In addition to the case of vector-like quarks of narrow width, we also consider the cases where the vector-like quark has a finite width (width-to-mass ratio is taken to be 0.1) . We restrict ourselves to the cases where the non-resonant production of the same final states and additional spin correlation between the production and decay of the vector-like quark can be ignored. In this scenario, we propose two estimators P BW z , T BW zz similar to the previous estimators by introducing an additional convolution over the Breit-Wigner shape of the vector-like quark invariant mass distribution. We validate the modified estimators and compare them with the original estimators. We find that both the set of estimators provide equally good approximations to the polarization parameters of V , in the finite width case.