Correlations between Polarisation States of W Particles in the Reaction e-e+ ->W-W+ at LEP2 Energies 189-209 GeV

In a study of the reaction e-e+ ->W-W+ with the DELPHI detector, the probabilities of the two W particles occurring in the joint polarisation states transverse-transverse (TT), longitudinal-transverse plus transverse-longitudinal (LT) and longitudinal-longitudinal (LL) have been determined using the final states WW ->l nu q qbar (l = e, mu). The two-particle joint polarisation probabilities, i.e. the spin density matrix elements rho_TT, rho_LT, rho_LL, are measured as functions of the W- production angle, theta_W-, at an average reaction energy of 198.2 GeV. Averaged over all cos(theta_W-), the following joint probabilities are obtained: rho_TT = (67 +/- 8)%, rho_LT = (30 +/- 8)%, rho_LL = (3 +/- 7)% . These results are in agreement with the Standard Model predictions of 63.0%, 28.9% and 8.1%, respectively. The related polarisation cross-sections sigma_TT, sigma_LT and sigma_LL are also presented.


Introduction
In the study of the reaction e − e + →W − W + at LEP2, the DELPHI Collaboration [1] and other LEP experiments [2,3] have established that on average ∼ 25% of W particles are longitudinally polarised, as predicted by the Standard Model. The present study measures the joint two-particle spin density matrix elements which give the probabilities that both W s are transversely polarised (W T W T ), both are longitudinally polarised (W L W L ) or that one W is transversely polarised while the associated W is longitudinal (W T W L +W L W T ). In what follows, these correlations will be referred to as T T , LL and LT , respectively. This is a more detailed test of the Standard Model prediction for the W polarisation than those previously published. Production of longitudinal W s is of particular interest because they are associated with the breaking of the electroweak symmetry. This study tests the theoretical prediction of the correlations and in particular, that the correlation LL is suppressed relative to LT .
The previously published measurements of the spin-dependent correlations between the W particles in the reaction e − e + →W − W + are by the OPAL collaboration [4] and by the L3 collaboration [5]. OPAL evaluated the cross-sections σ T T , σ LT and σ LL from their data at 189 GeV with a comparatively low statistics. Their results are in poor agreement with the Standard Model. L3 used the W W →lνqq and W W →qq qq events from their full LEP2 data set to establish the correlation between the decay planes of the two W s. The correlation was found to be consistent with the Standard Model prediction.
The analysis presented in this paper uses only the events in which one W decays into an electron plus a neutrino or a muon plus neutrino, while the other W decays into two hadron jets. These "semi-leptonic" events are kinematically well constrained and they offer the best available data for any detailed analysis of this reaction. The τ semi-leptonic events are excluded because the uncertainties in their identification cause problems in W W correlation measurements. The fully hadronic final state W W →qqqq is also excluded because of the uncertainties in jet reconstruction: the charges of the hadron jets cannot be well measured and the particles from the four jets tend to overlap in the space of the detector, resulting in uncertainties in associations between the W s and the measured jets.
A complete description of the polarisation states of the produced W particles is given in terms of the two-particle joint spin density matrix ρ λ 1 λ 1 ′ λ 2 λ 2 ′ , where λ 1 and λ 2 are the helicities of the W − and W + respectively. In terms of the W production helicity amplitudes, F (µ) λ 1 λ 2 , the spin density matrix elements are defined by The normalisation is such that the trace of the matrix is unity. The initial state helicity sum runs over µ = ±1/2 and the W particle helicities run over λ i , λ i ′ = ±1, 0.
The helicities of W particles can be determined from their centre-of-mass decay distribution asymmetries. The above definition of the ρ λ 1 λ 1 ′ λ 2 λ 2 ′ elements can be put [6] into the following form which is model independent and is directly applicable to experimental data corrected for backgrounds and detection efficiences: Here, θ W − is the angle of the W − production with respect to the e − beam, θ * 1 and θ * 2 are the polar decay angles of the W − and W + in their rest frames and φ * 1 , φ * 2 are the corresponding azimuthal decay angles, as shown in figure 5. The functions D λλ ′ are the theoretical decay distributions of the W particles in the helicity states specified by the λ indices. Precise definitions of the angles θ * 1,2 and φ * 1,2 and of the functions D λλ ′ relevant to the present analysis are given in section 3. It should be noted that the cross-section formula (1) is model-independent regarding the W W production process. The single-W particle spin density matrix ρ λλ ′ , derived from the W W spin density matrix ρ λ 1 λ 1 ′ λ 2 λ 2 ′ by summation over one of the indices (1,2), gives information about the polarisaton of one W regardless of the state of the other. All nine elements of the single-W particle spin density matrix can be determined using the data from semi-leptonic events. This was done in [1] and [2,3], where only the electron and muon decays of one W were used as the analyser of the W polarisation. The hadronic decays of W particles were not used because of the reasons outlined above and because the analysing power of polarisation in such decays is greatly reduced as the result of the severe practical difficulty to distinguish quark jets from anti-quark jets.
In the present study, it would ideally be desirable to measure the complete 9 × 9 matrix ρ λ 1 λ 1 ′ λ 2 λ 2 ′ . As the result of the limitations in the polarisation information from the hadronic W decays, only a small part of the joint W W spin density matrix can be measured. It is possible to measure 5 (out of 9) diagonal elements (ρ λ 1 λ 1 λ 2 λ 2 ) plus 9 complex off-diagonal elements. Instead of this incomplete set of individual matrix elements, the following three linear combinations of matrix elements are considered in this paper: The quantities ρ LL , ρ T T and ρ LT are composed of the diagonal elements of the full matrix and they can be interpreted as probabilities of the joint, i.e. correlated, polarisation states of the two W s. The elements ρ T T and ρ LT do not distinguish between the + and -transverse helicities, and also the polarisations of the states W − L W + T and W − T W + L are combined. This is a reduced set of information about the W W polarisations but it is nevertheless very useful. The elements ρ T T , ρ LT and ρ LL can be measured in semi-leptonic W W events because cancellations in the sums (2) imply that the incompleteness of the polarisation information in the hadronic W decays does not matter [6]. In section 3 it will be shown how ρ T T , ρ LT and ρ LL can be measured directly from the data without recourse to the individual spin density matrix elements in (2).

The Experiment, Treatment of Data and Simulation 2.1 The Experiment
The DELPHI detector is described in detail in [7,8] and its configuration during the LEP2 runs is given in [9]. The reference [9] gives a complete description of the selection of W W events in DELPHI. The present analysis uses the data taken at centre-of-mass energies between 189 and 209 GeV. The data are grouped into three sets at average energies of 189 GeV, 200 GeV and 206 GeV. The total integrated luminosity is 520 pb −1 , and the luminosity-weighted average energy of all data is 198.2 GeV. Jet reconstruction algorithms as well as electron and muon identification are also described in [9].

Data selection
The initial selection procedure for the channels µνqq and eνqq is based on the typical topology of those events. As already mentioned, events from the τ νqq channels are not included in this analysis and thus they are a part of the background. The starting values of the data cuts are those listed in [9]: The precise values of these cuts, in particular those on the event transverse energy and the minimum particle multiplicity in jets, were varied slightly for data taken at the three different average e − e + energies. Three-constraint kinematic fits were then performed to the reaction e − e + →W − W + on the selected data samples, requiring both W s to have the same mass consistent with 80.4 GeV/c 2 . Cuts on the χ 2 probability distribution were then applied, with the value of the cut (in the region 0.5% -1%) determined from the χ 2 distribution in each of the three data sets. The final sample, taken at all beam energies, consists of 800 electron plus 880 muon events. This sample is somewhat smaller than that reported in [9] because we require full functionality of all parts of DELPHI.
Particle momenta and angles obtained from kinematic fitting have been used in the analysis of this experiment.

Event Simulation
Simulation of events plays a crucial role in the experimental procedure to separate events corresponding to the reaction e − e + →W − W + , the "signal", from backgrounds. The signal is defined by the three CC03 Feynman diagrams shown in figure 1. These account only for a part of the four-fermion processes contributing to the data. In this experiment a customised version [10] of the WPHACT [11,12] generator program was used to simulate all the four-fermion processes. The DELPHI WPHACT program includes reweighting for the Double Pole Approximation (DPA) radiative corrections and the possibility to compute the matrix elements of different subsets of Feynman diagrams. The weights are the ratios of the squared matrix element for W W production only via the CC03 diagrams to that for production via the full set of four-fermion processes. It is thus possible to simulate CC03 events corresponding to production via the CC03 diagrams with or without inclusion of other four-fermion processes.
In addition to the four-fermion background, there is a significant two-fermion background, mostly fromqqγ final states. This background has been simulated using the KK2F generator [13].
The generators were interfaced to the PYTHIA [14][15][16] hadronisation program. Large simulated samples, of the order of 10 6 events, were produced by the programs listed above, interfaced to the DELPHI detector simulation program DELSIM [7,8].

Reconstruction of events
Event reconstruction efficiencies were determined using events simulated with the WPHACT Monte Carlo program adapted for DELPHI [10]. The efficiency is defined as the ratio of the number of reconstructed CC03 WPHACT Monte Carlo events to the number of generated events, including all four-fermion channels.
Efficiencies are determined as functions of cos θ W − , cos θ * 1 and cos θ * 2 , defined in section 1. The resulting reconstruction efficiency table is a 5×5×7 matrix corresponding to 7 bins of cos θ W − , as defined in section 4.2, and 5 equal size bins for each of cos θ * 1 and cos θ * 2 . The reconstruction efficiency matrices were determined separately for the electron and muon event samples at the three centre-of-mass energies. Typical values of the reconstruction efficiency, ǫ, as a function of cos θ W − are 0.65 to 0.74 when integrated over all cos θ * 1 and cos θ * 2 . The values of ǫ as a function of cos θ * 1 (or of cos θ * 2 ), averaged over cos θ W − , vary over the range 0.6 to 0.8.

Treatment of Backgrounds
The dominant backgrounds to the selection of eνjj and µνjj events (where j implies a hadronic jet) can be divided into two groups: (a) Events which result from problems in reconstruction or selection procedures. The dominant contribution to this class of events comes from two-fermion final states, in particular fromqqγ. Other contributions come from neutral current four-fermion final states which might be misidentified as eνjj or, more rarely, as µνjj. Misidentified charged current events τ νjj are also potentially a background in this experiment.
(b) The non-CC03 charged current four-fermion contributions to the global µνjj or eνjj final states. In contrast to the background (a), this background would exist even in a perfect detector and an ideal experimental event selection procedure. A number of the charged current four-fermion diagrams in this category can interfere with the CC03 amplitude 2 .
Following the treatment in [9], the backgrounds requiring special attention are: (i) the τ events, W W →τ νjj (belonging to group (a)); (ii) various four-fermion processes which for experimental or other reasons lead to the same final state as the reaction e − e + →W − W + (such backgrounds may arise from groups (a) and (b)); (iii) two-fermion events, mostlyqqγ interactions (group (a)). Discussions of each of these background sources follows in turn.
• The background from source (i) has been investigated by passing the simulated τ νjj events through the normal analysis chain and requiring eνqq or µνqq fits. The background from this source turns out to be negligible as the result of kinematic cuts and fitting.
• The background (ii) is due both to charged current and neutral current events.
The charged current processes consist of the three CC03 diagrams plus seven diagrams with s-channel exchange of Z 0 /γ, leading to the production of only one W . There are also ten charged current diagrams corresponding to t-channel processes which give rise to one W . The latter diagrams can give rise to backgrounds only in the electron final state eνjj.
The neutral current four-fermion states have two quarks and two leptons of the same flavour. If one of the electrons is not identified in the detector, the event may be classified as belonging to the channel W W → eνjj and may satisfy the criteria for an acceptable kinematic fit. Altogether, the four-fermion backgrounds affect the electron channel, eνqq, more than the muon channel µνqq.
It was found that the non-CC03 four-fermion background in the real events could be efficiently reduced to a level less than 4% by tuning the kinematic cuts and the χ 2 cuts, as described in section 2.2. The effectiveness of the removal of this class of background events can be demonstrated in the following way: Starting from a large sample of generated events, two data samples were produced making use of the event reweighting facilities in WPHACT. Sample A contained predominantly the CC03 events and sample B the non-CC03 four-fermion background. Each sample was processed through the experimental procedure described in section 2.2. The event ratio Π = B/A represents the proportion of the four-fermion background in the W W signal to be expected in the final sample of the real data. This quantity is of the order of 3% and is weakly dependent on cos θ W − . A plot of Π for WPHACT data at 200 GeV is shown in figure 2(a). The results for other run energies are similar.
A further test of the effectiveness of the method for dealing with the non-CC03 fourfermion background is to apply the analysis to a simulated data set where the expected result is known. For this purpose, the element ρ 00 of the single-W spin density matrix ρ λλ ′ is evaluated. This test was carried out for data generated at all three run energies, but here only the results from WPHACT at 200 GeV are shown. In figure 2(b), triangle symbols are used to plot the value of ρ 00 obtained from the WPHACT generated events using all the four-fermion diagrams, with no cuts except for the final kinematic χ 2 selection. Star symbols are used to plot the value of ρ 00 after the same generated events have been passed through the complete selection procedure described in section 2.2. The smooth solid curve is from an analytic calculation using only the CC03 diagrams. The conclusion drawn from this is that the procedure adopted for analysing the data removes essentially all the four-fermion background, leaving events which are attributable to the CC03 signal. 3 The residue of the non-W W four-fermion background in the real data at all run energies is estimated to be at the level of 3 ± 2%, the uncertainty is due to combining results from different energies.
• The background process (iii) is potentially very serious because the cross-section for qqγ production is about an order of magnitude larger than that for the W W signal. Although the topology of theqqγ events is quite different from that of the W W events, the reconstructed events of the background can resemble and fit the W W reaction. The problem of how to suppress this background has been investigated using the KK2F Monte Carlo generator [13]. It is found that this background shows a very characteristic kinematic signature in the distribution of the quantities cos Θ ll and cos Θ jj , where Θ ll is the angle between the momentum vectors of the two leptons (one charged and a neutrino) coming from one W and Θ jj is the angle between the jets from the accompanying W , with quantities defined in the laboratory frame (i.e. the reaction centre-of-mass). 4 The presence of theqqγ background shows up as accumulations of events at cos Θ ll ∼ −1 and cos Θ jj ∼ −1.
After the usual cuts and kinematic fitting, the real data samples show peaks in the cos Θ ll and cos Θ jj distributions indicating small but non-negligible contamination from theqqγ background. Below it will be shown that the contamination is of the order of 10% at 189 GeV. This contamination has to be evaluated accurately because it affects the angular distribution of W decays. The kinematics of the W decay in the laboratory frame of the reaction is such that the distributions of the quantities cos Θ ll and cos Θ jj have large discriminating power against theqqγ background. This enables the magnitude of the background to be determined and also provides a means of reducing the background by applying cuts on these distributions or by applying suitable event weights. Only the method of weights has been used in dealing with this background.  (b) Plots of the density matrix element ρ 00 evaluated from four-fermion events generated at 200 GeV with WPHACT before (triangle symbols) and after (star symbols) the event reconstruction procedure, including application of data cuts and kinematic fitting. The smooth curve is from an analytic calculation using CC03 diagrams. Figures 3(a) and 3(b) show, respectively, simulations of the cos Θ ll and cos Θ jj distributions at 189 GeV. The curves labelled W W were obtained from the WPHACT simulation of CC03, while the curves labelledqqγ are from the KK2F simulation of this background. At this stage of the analysis, the relative magnitudes of the W W andqqγ components are still unknown; therefore, the two curves are normalised to the same number of events. Figures 4(a) and 4(b) show the results of a least squares fit to the relative contributions from W W andqqγ, using the distributions from the simulated events, to the real data distributions. The fit to the 189 GeV data requires contributions of 90% from W W and  10 ± 2% fromqqγ, while the same analysis applied to the data at 200 GeV and 206 GeV require a 5 ± 2%qqγ background contribution at both energies.. This determination of the relative magnitude of theqqγ background with respect to the W W signal enables a combined WPHACT plus KK2F simulation of the real data. The combined simulation was subsequently used to derive weighting factors, p, defining the purity of the W W signal in each bin of cos θ W − , cos θ * 1 and cos θ * 2 , using the same binning as that used for the reconstruction efficiencies, ǫ. The application of the weight factors p is described in the following section.  3 Analysis W decays are well described by the V-A theory of the charged current weak interactions. The theory gives the functions D λλ ′ (θ * , φ * ) required for evaluating the spin density matrix elements ρ λ 1 λ 1 ′ λ 2 λ 2 ′ by applying equation (1) to the data. The W − production angle, θ W − , and the W −,+ decay angles θ * 1,2 , φ * 1,2 , which specify the direction of the final state fermion in the rest frame of the W − and of the final state anti-fermion in the rest frame of the W + , are defined in figure 5. As already pointed out, fermions can be distinguished from anti-fermions in the W leptonic decays but not so in the hadronic decays. However, some of the decay functions D λλ ′ (θ * , φ * ) are invariant under the transformation which rotates the vector momentum of a fermion in the W rest frame into the direction of its opposite anti-fermion vector. These functions will be called symmetric and designated by D (s) . Consequently, the symmetric D functions and the symmetric parts of the non-symmetric D functions can be used to analyse the polarisation of the W s decaying into the purely hadronic final states. The polarisation information obtained is thus incomplete but, nevertheless, it is useful and, in particular, the quantities ρ T T , ρ LT and ρ LL can be obtained from the data using only the symmetric decay distributions in both associated W s.
The theoretical formalism for extracting ρ T T , ρ LT and ρ LL from the data is based on equation (1) with two modifications: (1) is integrated over the full range of φ * 1 and φ * 2 . This removes the functions D λλ ′ having λ ′ = λ and eliminates all non-diagonal elements of the matrix ρ λ 1 λ ′ 1 λ 2 λ ′ 2 . The following three decay functions remain: (ii) Furthermore, only the symmetric parts, D Pursuing the above steps, the following modified form [6] of equation (1) is obtained: where ρ T T , ρ LL and ρ LT have been defined in (2)   The angular distribution of the W − production is not used explicitly in the data analysis for measuring the correlations ρ T T , ρ LT and ρ LL . However, it is appropriate to examine the shape of the distribution dN/d(cos θ W − ) as a test of the quality of the data. Figure 6 shows the angular distribution of the W − in the present data at all energies combined, compared to the WPHACT prediction. The agreement with WPHACT is satisfactory. The distribution shown also agrees well with that in the DELPHI publication on the W W production cross-section [9].
Before using the functions (3) to determine the intensities of different helicity states in the data, these functions must be transformed to a related set of functions with the property that each one is non-orthogonal to only one in the group of distribution functions (3) and is orthogonal to the other two. These functions are called projectors [17] and they can easily be worked out from (3). The projector functions needed to extract ρ T T , ρ LT and ρ LL are: These projectors are normalised to give the spin density matrices in the standard representation [6,17]. The quantities ρ LL , ρ T T and ρ LT are obtained from the data by evaluating the following sums: where summations are over all events i. N w = i w i is the sum of all event weights w i , and the event weights w are defined as where p is the W W purity factor, defined as in section 2.4, for each bin of (cos θ W − , cos θ * 1 , cos θ * 2 ), and ǫ is the event reconstruction efficiency in that bin, defined in section 2.3. In these formulae, index 1 refers to W − and index 2 refers to W + .
Monte Carlo studies have shown that the correlations ρ LT , ρ LL , ρ T T extracted from small samples of data (such as we have at each of the three energy points considered) are subject to large statistical fluctuations. These fluctuations are much larger than those encountered in the determination of the single-W spin density matrix elements. Because of that, all 1680 semi-leptonic electron and muon events have been taken as one sample for measuring ρ LT , ρ LL and ρ T T . The sum of weights of these events is 2844.
The fact that events which we analyse here as one sample come from a spread of centreof-mass energies presents no difficulty because the theoretical predictions which we want to test can be modified to take into account the spread. In particular, the Standard Model [6] predicts a negligible variation of ρ T T , ρ LT and ρ LL over the energy range of this experiment.

Systematic Effects and Errors
(a) Data cuts. Systematic effects resulting from residual backgrounds in real events have been estimated by processing the data several times with small variations in the cuts and, separately, with various χ 2 cuts. Variations in the resulting values of the spin density matrix elements ρ LT , ρ LL , ρ T T amount to 10% of their statistical error. This is interpreted as the magnitude of the systematic uncertainty and it is neglected.
(b) Hadron jet reconstruction. Problems in hadron jet reconstruction [9] can give rise to a shift in cos θ W − with the further consequence that the reconstruction efficiency is read from the wrong cell of the ǫ matrix. This migration and its effects have been examined using simulated events. The resulting uncertainty on θ W − is small: it varies from ±2 • at large angles to ±5 • at small angles, i.e. in the forward direction 5 . This is negligible by comparison with the sizes of the cos θ W − bins. The effect of this migration on the spin density matrix elements has been examined by moving the simulated events randomly by the above uncertainty in cos θ W − . The effect of this variation on the joint spin density matrix elements is 5% -8% of the statistical uncertainty and is therefore considered to be negligible.
Also, jet reconstruction problems can produce wrong momentum vectors of the W hadronic decay products. This has been investigated by processing the same events using different jet algorithms as described in reference [9]. No statistically significant effect was found when comparing the spin density matrix elements distributions obtained in these tests.
(c) Lepton charge determination. Tracks at small angle with respect to the e − and e + beams are susceptible to wrong charge determination. This problem is essentially eliminated [1] by the 20 • cut on the lepton polar angle (see section 2.2). The effect of this cut on the spin density matrix elements has been examined by simulation of events and was found to be negligible.
(d) Radiative corrections. The effect of the initial state radiation is essentially removed by the appropriate energy cut. The final state radiative corrections are implemented through the Double Pole Approximation in the WPHACT reweighting. The uncertainty due to the radiative corrections on the spin density matrix elements is negligible. This has been established by a comparison of the spin density matrix elements evaluated from events generated with WPHACT, including the corrections, with the same elements calculated analytically [6,17] without the radiative corrections. This can be understood because the spin density matrix elements are ratios of quantities which are similarly affected by the radiative effects.
(e) Use of a fixed W mass in kinematic fitting of events. Three-constraint kinematic fits of the reaction events are needed in order to separate the signal from a large background. Fixing the masses of both W s in the reaction to the same value, as stated in section 2.2.1, does not cause noticeable distortions of angular distributions and other quantities needed for the physics analysis. This has been checked by comparing the results obtained from the three-constraint fits with those from the oneconstraint fits of the same events. As a further test, somewhat different fixed values of the W mass were tried. Statistically insignificant differences in the results were found.

Presentation of Results
The measured values of ρ T T , ρ LT and ρ LL as functions of cos θ W − are shown in figure 7. Because the number of events in the negative hemisphere of W − production is much smaller than that in the positive hemisphere, the data have been divided into two bins in the negative hemisphere and five bins in the positive hemisphere. (The positive hemi-sphere is in the direction of the e − beam.) It is easy to derive from formulas (5) and (6) that the condition ρ LL + ρ T T + ρ LT = 1 (7) is valid on an event by event basis and is hence automatically satisfied by all data samples.
The curves shown in the plots are the Standard Model calculations based on the CC03 diagrams evaluated at 198.2 GeV using the expressions in [17]. The error bars shown in the plots are statistical, the systematic errors being negligible by comparison. Errors on all measured quantities are evaluated from the data as standard deviations. The distributions of the errors are approximately Gaussian.
Some events can contribute negative numbers to the sums shown in (6) while the final result is positive. However, if a particular correlation quantity ρ is very small, the measurement errors, which have a Gaussian distribution, can lead to an overall result which is negative. This happens in three out of the seven measured values of ρ LL presented here. They are very small negative quantities, consistent with zero within the measurement errors, i.e. | ρ LL |< δρ LL , where δρ LL is the measurement error. These negative values of ρ LL are included in figure 7. The condition (7) still holds in these cases as a result of the properties of the projector functions (5). Of course, the physical quantities ρ should satisfy the condition ρ ≥ 0. Several methods of adjusting the results to satisfy the positivity condition were tried, but not used in the end because they introduce biases that are more harmful than the small negative numbers among the results.
The data bins in figure 7 are too wide to show the possibly rapid variations of the measured quantities very near to cos θ W − = ±1. However, the exact values of these correlations at cos θ W − = ±1 follow from the conservation of angular momentum. Neglecting the electron mass, the vector and axial-vector interactions involved at the e − and e + vertices cause the initial e − and e + to interact only when their helicities are opposite. Thus, the initial system has total helicity ±1 and when the final state is collinear with the e − e + beams its total helicity must be the same. This means that at θ W − = 0 or π, ρ LT must be 1 and, at the same time, ρ LL = ρ T T = 0. These values have been indicated with diamond-shaped symbols in figure 7.
Because of the low statistics in this experiment, it is useful to examine the values of ρ T T , ρ LT and ρ LL averaged over all bins of cos θ W − and to compare them with the predictions of the Standard Model. The results are shown in Table 1.
ρ Measured fraction Standard Model ρ T T 67 ± 8% 63.0% ρ LT 30 ± 8% 28.9% ρ LL 3 ± 7% 8.1% Table 1: plus the analogous expressions for the states LT and LL. The first term on the right-hand side of (8) can be replaced by the data points from the angular distribution shown in figure 6, normalised to the total cross-section for e − e + →W − W + at the average energy of the experiment. The relevant measured cross-section is σ = (17.07 ± 0.57) pb at 198.2 GeV. It has been obtained by an interpolation of the DELPHI measurements [9] which cover the range 161 to 209 GeV. With this result and the measured ρ T T , ρ LT , ρ LL as functions of cos θ W − , one obtains the differential polarisation cross-sections shown in figure 8. Integration over cos θ W − yields the total polarisation cross-sections σ T T , σ LT and σ LL shown in Table 2.
These results are in good agreement with the Standard Model predictions. The polarisation fractions expressed in terms of the density matrix elements (Table 1) and those expressed in terms of the cross-section ratios ( Table 2) are different expressions of the two-particle polarisation correlations in the reaction. Measurement errors of the polarisations are themselves strongly correlated. The extent of all the correlations will be shown in figure 9(b).

Discussion and Conclusions
The measurements of the total cross-section for the reaction e − e + →W − W + by ALEPH, DELPHI, L3 and OPAL [20], have successfully tested the predicted gauge coupling cancellations in this reaction. The direct measurement of the longitudinal cross-section for the single (i.e. uncorrelated) W production [1][2][3] is an important test of the same cancellation because the cause of the potential divergence of the cross-section is in the longitudinal parts. The measurement of the joint W W helicity states presented in this paper is an advance towards a more complete test of the Standard Model in the context of the W W reaction. Our measurement of σ(e − e + →W L W L ) is consistent both with zero and with the small value predicted by the Standard Model. This confirms that the probability of longitudinal W production is predominantly in association with a transverse W . Figure 7 shows good agreement between the data and the Standard Model over the range of cos θ W − where there are sufficient data. An important aspect of these results is the interrelation between the three spin density correlations ρ T T , ρ LL and ρ LT determined in this analysis. These quantities can be displayed in one common plot due to the fact that they satisfy the condition (7). Thus, they can be plotted in a triangle plot, as shown in figure 9. Data can be plotted directly as indicated with arrows on the left side of figure 9(a) or by using Cartesian coordinates x and y, In figure 9(a) there are seven data points, each one corresponding to a different bin of cos θ W − . Error bars are also shown. The curve inside the triangle is the locus of points (x,y) calculated using CC03 diagrams. Each point on the curve corresponds to one particular value of cos θ W − . The point corresponding to cos θ W − = −1 is at the top   Table 1, presented in a triangle plot. The shaded area is the one standard deviation region around the average point (x,y). The star symbol is at the point predicted by the Standard Model using the CC03 diagrams.
It is clear that the TT correlation probability is large, the LT correlations are next in strength and the LL correlations are small. These are some of the most important features of the Standard Model predictions for the reaction e − e + →W − W + .
In spite of the limitations due to low statistics, these results show that the salient features of the Standard Model predictions for the W −W polarisation correlations are compatible with our data. This provides an additional test of the gauge theory relations between the SU(2) L ⊗ SU(1) Y couplings.