New observables for testing Bell inequalities in W boson pair production

We show that testing Bell inequalities in W ± pair systems by measuring their angular correlation suffers from the ambiguity in kinetical reconstruction of the di-lepton decay mode. We further propose a new set of Bell observables based on the measurement of the linear polarization of the W bosons that can be used in the semi-leptonic decay mode of W ± pair, and we analyze the prospects of testing the violation of Bell inequalities at e + e − colliders.

While the violation of Bell inequalities has already been confirmed in qubit systems, tests of Bell inequalities in massive vector boson pair systems, the only fundamental qutrit systems in our nature, are still pending.It is fascinating to check the violation of Bell inequalities in entangled "quNit" systems with N ⩾ 3 since the results for large N are shown to be more resistant to noise with a suitable choice of the observables [28][29][30][31].Although the W ± pair system is shown to be theoretically more promising than Z pair to test the 3-dimensional Bell inequalities [23,24], a feasible experimental approach to test the Bell inequalities in W ± system is yet to be proposed.In previous studies [21][22][23][24], it was a common practice to use the di-lepton decay mode of W ± pair to test the entanglement, because the complete density matrix ρW W of W ± system can be reconstructed in this channel then all entanglement criteria can be calculated from ρW W directly.But it was also pointed out that a two-fold ambiguity in the kinetic reconstruction is unavoidable in the di-lepton decay mode of W ± .In this work, we show that this ambiguity in the di-lepton decay mode can lead to a fake signal of the violation of Bell inequalities.Therefore, it is necessary to search for an alternative approach to test the Bell inequalities in the W ± system.
For W ± pairs produced at electron-positron colliders, an event-by-event kinetical reconstruction of W ± pair can be performed without any ambiguity in their semileptonic decay mode.Though the semi-leptonic decay mode is a major channel to measure W ± spin correlations at the LEP [32][33][34][35], this decay mode was not used to test entanglement before due to the loss of angular momentum information of the W boson as it is hard to identify the flavor of light jets.Therefore, the conventional approach to test Bell inequalities, which relies on measuring angular momentum correlations between W + and W − , cannot be applied in the semileptonic decay mode.Although only partial information on the density matrix ρW W can be reconstructed in the semi-leptonic decay mode of the W ± pairs, we succeed in finding a new observable to test the Bell inequalities.More specifically, we construct new Bell observables based on the linear polarization of W bosons, which does not require tagging the flavor of the decay products of W ± pairs.We show that these new Bell observables can be correctly measured from the semileptonic decay of W ± pairs, providing a feasible way to test Bell inequalities in W ± pair production.

II. METHOD
We begin by introducing the theoretical framework of testing Bell inequalities in W ± pairs.Ignoring the interactions between the W ± bosons, the W ± pair system can be described by the tensor product Hilbert space H = H A ⊗ H B of the state Hilbert space H A of W + and the state Hilbert space H B of W − .Fixing the momentum of the W ± boson, the subspace H A/B is 3-dimensional representation space of the rotation group SU (2).Considering some measurements Âi and Bi carried out in these 3-dimensional spin spaces of H A and H B , their outcomes A i and B i have three possible values in {−1, 0, 1}, where the index i = 1, 2 is used to denote different measurements on the same system.The optimal [36] generalized Bell inequality for 3-dimensional systems, also referred as Collins-Gisin-Linden-Massar-Popescu (CGLMP) inequality [31], states that the upper limit of the following expression, is 2 for any local theory, i.e., I 3 ≤ 2. Here, P (A i = B j + k) denotes the probability that the measurement outcomes A i and B j differ by k modulo 3.
For a non-local theory, the inequality I 3 ≤ 2 no longer holds, and the upper limit of I 3 is 4 instead.In other words, as long as there exists a set of measurements such that the corresponding CGLMP inequality is violated, i.e., max Â1, Â2, B1, B2 the non-locality of the system is confirmed.
A direct way to evaluate I 3 is to project the density matrix ρW W to the eigenstates of the operators Âi and Bi , e.g., the first term in Eq. ( 1) is where Π|ψ⟩ ≡ |ψ⟩ ⟨ψ| is the projection operator.At lepton colliders, ρW W could be theoretically calculated with the transition amplitudes M W W of the e + e − → W + W − process in the electroweak standard model (SM) to where M W W is a 9×4 matrix in spin space, and ρee is the 4 × 4 spin density matrix of the initial state e + e − which is Î4 /4 for unpolarized beam.Here, Îd is the identity operator in d-dimensional Hilbert space.Unfortunately, the spin state of the W bosons could not be directly measured at colliders.Therefore, we next introduce how to obtain ρ W W from the decay products of W ± pairs.As a preliminary, we start with the spin density matrix of one W boson, which could be generally parameterized as where Ŝi is the i-th component of the 3-dimensional angular momentum operator, Ŝ{ij} ≡ Ŝi Ŝj + Ŝj Ŝi , and the coefficients q ij is symmetric traceless.Note that the two sets of operators S i and S {ij} are orthogonal, i.e., Tr(S i S {jk} ) = 0.1 The parametrization separates the information of angular momentum and linear polarization of the W -boson explicitly.On the one hand, the expectation value of the angular momentum of the W -boson along direction ⃗ a yields Tr( ⃗ S • ⃗ a ρW ) = 2 ⃗ d • ⃗ a, which only depends on d i .On the other hand, a (partly) linear polarized W -boson has zero angular momentum with d i = 0, and its polarization information only depends on q ij .With the polarization information of each term of ρW in mind, we continue to reconstruct the density matrix of a W boson from its decay products.In its rest frame, ignoring the tiny mass of the final state fermion and anti-fermion, the W boson always decays into a negative helicity fermion and a positive helicity antifermion since the weak interaction only couples to lefthanded fermions, and we denote the normalized direction of outgoing anti-fermion in the rest frame of the W boson as ⃗ n, which is just the direction of the (experimentally measured) total angular momentum.In additional to n i , we define a symmetric and traceless tensor of rank-2 (the quadrupole) to describe the high-order information on the distribution of decay products.The probability of finding an antifermion in an infinitesimal solid angle dΩ of direction ⃗ n(θ, ϕ) from the W -boson decay products is [21] p(⃗ n; ρW ) = 3 4π Tr ρW Π⃗ n , where the projection operator Π⃗ n selects the positive helicity anti-fermion in the direction ⃗ n.The explicit expression of p(⃗ n; ρ W ) is shown in Appendix B. By integrating the probability with the kinetic observables n i and q ij , it is found that the parameters d i and q ij in Eq. ( 5) are directly determined by the averages of these kinetic observables, which are defined as Therefore, the parameters d i 's, which are related to the angular momentum of the W boson, are determined by ⟨n i ⟩, the dipole distributions of the anti-fermion, and require distinguishing fermion from anti-fermion (or flavor tagging).The parameters q ij 's, which are related to the linear polarization of the W boson, are determined by ⟨q ij ⟩, the quadrupole distributions of the decay products, and do not need flavor tagging.Likewise, the density matrix of W ± pair can be reconstructed from the distribution of their decay products.The density matrix ρW W is parameterized as (see Ref. [37] for a similar formulation) where Ŝ+ i ( Ŝ− i ) and Ŝ+ {ij} ( Ŝ− {ij} ) is the Ŝi and Ŝ{ij} operator defined in the rest frame of the W + (W − ) boson, respectively, and the repeated indices are summed as in Eq. ( 5).
We use ⃗ n ± to denote the normalized directions of two outgoing anti-fermions decayed from W ± in the rest frame of W ± , respectively.The quadrupole kinetic observables q ± ij ≡ n ± i n ± j − 1 3 δ ij are defined similarly.Again, all the parameters in ρW W can be reconstructed from the average of the observables n ± i , q ± ij and their correlations.With a detailed calculation in Appendix B, we enumerate the kinetic observables needed to obtain each term of ρW W as follows: The first two lines of Eq. ( 11) are determined by the decay products distributions of each W boson itself, The terms in the third line of Eq. ( 11) are determined by the correlations between the dipole or quadrupole distributions of the decay products of W + and W − .
The terms in the fourth line of Eq. ( 11) are determined by the correlations between the dipole distribution of the decay products of one W boson and the quadrupole distribution of the decay products of the other.
With Eqs. ( 12)-( 17), we are ready to obtain the complete density matrix ρW W of the system and test the Bell inequalities.Besides, it is worth emphasizing that tagging the flavor of the decay product W + or W − is necessary to fix the overall sign of n + i or n − i , but not necessary to obtain q ± ij .

III. NEUTRINO RECONSTRUCTION IN DI-LEPTON DECAY MODE
As a usual practice, the Bell inequalities in W ± system are tested by measuring the angular momentum correlations of the two W bosons.In that case, the operators in Eq. ( 2) are chosen as angular momentum operators and the Bell observable I 3 is defined as where Ŝ⃗ n ≡ ⃗ S • ⃗ n, and (⃗ a i , ⃗ b i ) are a set of directions in the rest frames of W ± respectively, and the maximum of I (S) 3 is obtained by scanning all possible directions ⃗ a i and ⃗ b i to measure the angular momentum.
To measure the angular momentum of each Wboson, the Ŝi dependent terms of ρW W such as C d ij Ŝi ⊗ Ŝj must be correctly obtained.Since these terms are reconstructed from the kinetic observable n ± i , distinguishing fermion from anti-fermion in both W boson decay processes is necessary.In the hadronic decay mode of W boson, it is shown that the jet substructures such as jet charge can help to distinguish light quark flavor, but the tagging efficiency is still very low [38].Therefore, only di-lepton decay mode, W + (→ ℓ + ν ℓ )W − (→ ℓ − νℓ ), is considered in previous studies to calculate the criteria of entanglement [21][22][23][24].
However, in the di-lepton decay mode of W ± , there are two undetectable neutrinos and the momenta of W ± cannot be obtained directly.To reconstruct the rest frame of W ± and obtain n ± i and q ± ij , the neutrino momenta must be solved from two observed leptons using on-shell conditions, but the solution suffers from twofold discrete ambiguity [39] even if we ignore W boson width and experimental uncertainties.In other words, the false solutions behave like irreducible backgrounds that are comparable with signals.As a result, attempting to measure the theoretical value of I (S) 3 calculated in previous studies are subject to experimental difficulties in kinetical reconstruction.
To illustrate the impact of the twofold ambiguity, we use the unpolarized scattering process e + e − → W + W − with √ s = 240 GeV as an example.We perform a parton level simulation using MadGraph5 aMC@NLO [40] with full spin correlations and Breit-Wigner effects included.From the momenta of the detected leptons, we obtain two degenerate solutions of the neutrino momentum that satisfying the kinetic conditions [39].We choose the solution with a larger or smaller transverse momentum (ν 1 and ν 2 in Fig. 1, respectively) to reconstruct the rest frame of W ± and then reconstruct calculated with true neutrino momentum (solid line) or solved neutrino momentum (dashed lines) at √ s = 240 GeV electron-positron collider.Here, θ is the scattering angle between W + and incoming e + beam, ν1 or ν2 denotes the neutrino solution with larger or smaller transverse momentum respectively.the density matrix from Eqs. ( 12)- (17).When averaging the kinetic observables in Eqs. ( 12)-( 17), we choose to work in the beam basis [16,41], where ẑ is along the incoming e + beam direction, x ∝ ẑ × ⃗ p W + is the normal direction of the scattering plan, and ŷ = ẑ × x.For comparison, we also include the results calculated with the knowledge of the true momentum of each neutrino, as shown in Fig. 1.For a better illustration, the results in Fig. 1 is reconstructed from the parton-level momenta of leptons directly without any selection cuts, so that the two fold ambiguity makes the only difference between the reconstructed results and the true results.It is found that the twofold ambiguity is destructive for testing Bell inequalities with I (S) 3 , as the observed value of I (S) 3 can be much larger than its theoretical value and may even exceed the physical upper limit, indicating a fake signal of entanglement.Considering momentum smearing effect and kinetic cuts further obscure the test of Bell inequalities.
Therefore, it is shown that the experimentally observed I (S) 3 cannot directly represent the entanglements between the W ± pair.In addition, other entanglement criteria that can only be measured at full leptonic decay channel of W ± pair, such as the concurrence and partial trace, also suffer from the two-fold solutions of neutrino momentum.The ambiguity of neutrinos also exists in the more studied t t case, where some reconstruction techniques such as unfolding [18,42,43] and parameter fitting [44,45] are commonly used. 2 Similarly, to test In the semi-leptonic decay modes of W ± pair produced at lepton colliders, all momenta can be determined without any ambiguity.
Despite the convenience in kinetical reconstruction in the semi-leptonic decay modes, a complete density matrix ρ W W cannot be reconstructed in these modes, because the angular momentum of the W -boson decaying to hadrons cannot be measured without jet flavor tagging.Consequently, the Bell observable I (S) 3 is not valid in these decay channels.
However, the linear polarization of the W -boson decaying to hadrons can still be measured correctly, because the linear polarization of a W -boson is determined from the quadrupole distribution ⟨q ij ⟩ of its decay products, which does not depend on the overall sign of ⃗ n.To construct a Bell observable that can be measured in the semi-leptonic decay mode of W ± , we choose operator Ŝ{xy} ≡ { Ŝx , Ŝy } to measure the linear polarization of the W -boson decaying to hadrons.Note that the eigenstates |S {xy} = ±1⟩ are purely linear polarized states with different polarization directions on the xy-plane, and the expectation value of Ŝ{xy} , E( Ŝ{xy} ), is directly determined by the quadrupole distribution of the decay products with E( Ŝ{xy} ) = 10 ⟨q xy ⟩, as shown in Fig. 2. We first consider the decay channel W + (→ ℓ + ν ℓ )W − (→ jj), where the lepton ℓ is electron or muon.In this channel, both the angular momentum of W + and the linear polarization of W − can be determined correctly.Therefore, we choose to measure the correlation between the angular momentum of W + and the linear polarization of W − to test the Bell inequalities in this channel, and the new Bell observable is defined as see, e.g., Ref. [45,46].where (x i , y i ) are the coordinates in the rest frame of W − , and ⃗ a i are the directions in the rest frame of W + .The observable is reconstructed in following steps.First, measure the distribution of W ± decay products and obtain the parameters of the density matrix using Eqs.( 12)-( 17).The W − decays hadronically and only the quadruple distribution of it is needed.Second, construct the density matrix ρW W in Eq. ( 11) with the parameters obtained in the previous step.Third, calculate the probabilities P (S ⃗ a = S {xy} ) and P (S ⃗ a = S {xy} ± 1) by projecting the density matrix to the eigenstates |S ⃗ a = A⟩ ⊗ |S {xy} = B⟩ , (A, B = −1, 0, 1) 3 , and then construct I (S,L) 3 according to Eq. (1).Note that the coefficients related to the angular momentum of W − , namely d − i , C d ij and C qd ij,k are independent of the projection, which is the reason why they are not needed in the first step.
We perform a Monte-Carlo simulation of e + e − → W + (→ ℓ + ν ℓ )W − (→ jj) processes with √ s = 240 GeV.The parton level events are generated by MadGraph5 aMC@NLO [40] and then passed to Pythia8 [47] for showering and hadronization.The showered events are passed to Fastjet [48] for jet clustering with Durham algorithm, and the clustering is taken to be stopped when it reaches 2 exclusive jets.We require the energy of lepton and jets to be larger than 15 GeV and 5 GeV respectively, and the invariant mass of the two jets satisfy |m jj − m W | < 20 GeV.In addition, we require the angle between the lepton missing vector, θ ℓpmiss , to satisfy cos θ ℓpmiss < 0.2 [49], so that the background from W → τ ν are negligible.As shown in Fig. 3, we find that the showering and selection cuts slightly dilute the signal of entanglements, but the observed I (S,L) 3 is still in good consistency with the parton level results, making I observable to test Bell inequalities in W ± pair system.The statistical significance of observing the violation of the Bell inequalities can be calculated with the standard χ 2 statistical test, where the sum runs over the bins with I 3 > 2, and the statistical uncertainty δ i are calculated from the standard error of mean in Eqs. ( 12)- (17).Using the observable I (S,L) 3 , the Bell inequality violation can be tested at 3σ significance at 240 GeV e + e − collider with 150 fb −1 luminosity.
Likewise, another semi-leptonic decay mode, W + (→ jj)W − (→ ℓ − νℓ ), can also be used to test the Bell inequalities.In this decay mode, we choose to measure the linear polarization of the W + and the angular momentum of W − , and the Bell inequalities I 3 ≤ 2 are tested by another observable, Combining the two semi-leptonic decay modes of W ± pair produced at 240 GeV e + e − collider, one can verify the violation of the Bell inequality at 5 σ significance with 180 fb −1 integrated luminosity.

V. CONCLUSION
The commonly used criteria of entanglement rely on the di-lepton decay mode of W ± , because the di-lepton decay mode is the only decay mode that can be used to reconstruct the complete density matrix.However, we show that due to the irreducible ambiguity of neutrino momentum solutions in the di-lepton decay mode, testing entanglement in the di-lepton decay mode of W ± pair may yield fake signals.
We provide a more realistic approach to test Bell inequalities in W ± pair systems using a new set of Bell observables based on measuring the linear polarization of W bosons.Our observables depend on only part of the density matrix that can be correctly measured in the semi-leptonic decay mode of W ± .With these new Bell observables, it is found that the violation of Bell inequalities in W ± pair produced at 240 GeV electropositron colliders can be tested at 5σ significance with an integrated luminosity of 180 fb −1 .

= Tr(4 Ŝ2
The second term contributes To estimate the first term, we notice that which immediately gives Tr( Ŝ2 eigenstates of Ŝ3 , the matrix representation of this basis is It is worth to emphasize that A 3 itself is not a real associative algebra under the matrix product, because the product of self-adjoint operators could not be selfadjoint operators.It is also a common practice to expand ρ W with eight Gell-Mann matrices, as in Refs.[21,23,26].The matrix representation of Ŝi and Ŝ{ij} in the basis of the eigenstates of Ŝz , and their relation with the Gell-Mann matrices λ a (a = 1, Appendix B: Density Matrix Reconstruction

One W -boson
Because the moving direction of the anti-fermion in the W -rest frame is just the measured spin direction of the W boson, we have which is not only a projective operator but also a density matrix of a pure state.So it could be represented by Along any direction ⃗ v, the spin expectation value of this state is 2⃗ v • d(n).Because it is the spin eigenstate along ⃗ n whose eigenvalue is 1, we have For a density matrix of a pure state (a projective operator), by definition we have The square of the density matrix With the normalization condition, we have Tr( Π2 Next we check the relations Tr( Π2 n Ŝi ) = Tr( Πn Ŝi ) = n i .Notice that the inner product Tr(A † B) is invariant under the transformation Î3 → Î3 , Ŝ{ij} → Ŝ{ij} , Ŝi → − Ŝi (i, j = 1, 2, 3), it is easy to see that Tr(( Ŝ{ij} Ŝ{kℓ} + Ŝ{kℓ} Ŝ{ij} ) Ŝm ) = 0, so Finally, we check that Tr( Π2 To estimate the last trace, we notice that the inner product is invariant under the "parity" transformation: Î3 → Î3 , Ŝ{ij} → (−1) Na Ŝ{ij} , Ŝi → (−1) Na Ŝi for specific a, where N a is the times the number a appears in the indices.So to have a non-vanished Tr[( Ŝ{ab} Ŝ{cd} + Ŝ{cd} Ŝ{ab} ) Ŝ{ij} ], each one of 1, 2, 3 must appear even times in the indices.With this result and the symmetric structure, one could verify that and when j = i The traceless solution of Eq. (B7), Eq. (B8) and Eq.(B11) is where Ŝn ≡ 3 i=1 n i Ŝi .The probability of finding an anti-fermion in an infinitesimal solid angle dΩ of direction ⃗ n = (sin θ cos φ, sin θ sin φ, cos θ) from the W -boson decay products is + d 1 sin θ cos φ + d 2 sin θ sin φ + d 3 cos θ + q 12 sin 2 θ sin 2φ + q 31 sin 2θ cos φ + q 23 sin 2θ sin φ + q 11 sin 2 θ cos 2 φ + q 22 sin 2 θ sin 2 φ + q 33 cos 2 θ .(B15) The normalization constant N is calculated with   Eqs.(B17)-(B25) can be summarized as It is found that the parameters d i and q ij , which are related to the circular polarization (spin eigenstates) and linear polarization of the W boson, are determined by the dipole and quadrupole distributions of the antifermion respectively.To see this fact clearly and quickly, we notice that the basis of the matrix representation of the SO(3) group we used is the eigenstates of the rotation transformation around the 3rd axis.However, in the vector representation (under the basis of the linear polarization states |j⟩ , j = 1, 2, 3), So (only keep the relative phase between the normalized state vectors) the eigenvectors of Ŝ3 are which immediately gives It is easy to see that the non-zero phase factor δ reflects the difference between the phase of the left-hand and right-hand circular polarization eigenstates, which could be removed by a rotation around the 3rd axis.Generally, the density matrix operator of the linear polarization state along the direction ⃗ n is which does not depend on the sign of n.It is easy to check that the components of the direction of the linear polarization could be written as Tr( Πnn Ŝ{ii} ).(B30)

W -boson pair
Likewise, the density matrix of W ± pair system can be reconstructed from the distribution of their decay products.In their rest frame of W ± respectively, we use ⃗ n ± to denote the normalized directions of two outgoing anti-fermions decayed from W ± .The probability of finding a pair of anti-fermions along directions ⃗ n ± is where N = 4π/3 is normalization constant as in Eq. (B15).As the density matrix of W ± pair system is the direct product of the two subsystems, the parameters can be obtained by calculating the averages of the kinetic observables n ± i , q ± ij and their combinations similarly.The average of an observable X is calculated by where dΩ ± denotes infinitesimal solid angles to find the anti-fermion direction ⃗ n ± (θ ± , ϕ ± ) from the W ± -boson decay products in the rest frame of W ± , respectively.
FIG. 1.The maximum value of I (S) 3

|S xy = 1 FIG. 2 .
FIG. 2. Distributions of the decay products of W bosons in different eigenstates of S {xy} , viewed from the z-direction.The color stands for the density of distribution.The decay products of the W boson in the state |S {xy} = ±1⟩ have positive or negative quadrupole distribution respectively.

) which gives N = 4π/ 3 .4π π 0 d 1 sin 3 θ dθ 2π 0 cos 2 φ dφ = d 1 , (B17) ⟨n 2 ⟩⟨n 3 ⟩
The averages of these kinetic observables n i (the ith component of the normalized 3-vector of the moving direction of the anti-fermion in the rest frame of the W -boson, or equivalently the ith component of the spin of the W -boson) and q ij (the correlations between the spin components) give the information of the density matrix ⟨n 1 ⟩ = sin θ cos φ p(⃗ n; ρ W ) dΩ = 3 = sin θ sin φ p(⃗ n; ρ W = cos θ p(⃗ n; ρ W