Study of W boson polarisations and Triple Gauge boson Couplings in the reaction e+e- ->W+W- at LEP 2

A determination of the single W Spin Density Matrix (SDM) elements in the reaction e+e- ->W+W- ->l nu q qbar (l=e/mu) is reported at centre-of-mass energies between 189 and 209 GeV. The data sample used corresponds to an integrated luminosity of 520 pb^{-1} taken by DELPHI between 1998 and 2000. The single W SDM elements, rho_{tau tau'}^{W+-} (tau,tau' = +/-1 or 0), are determined as a function of the W- production angle with respect to the e- beam direction and are obtained from measurements of the W decay products by the application of suitable projection operators, Lambda_{tau tau'}, which assume the V-A coupling of the W boson to fermions. The measured SDM elements are used to obtain the fraction of longitudinally polarised Ws, with the result: sigma_L/sigma_tot = 24.9 +/- 4.5(stat) +/- 2.2(syst) % at a mean energy of 198 GeV. The SDM elements are also used to determine the Triple Gauge Couplings Delta g_1^Z, Delta kappa_gamma, lambda_gamma and g_4^Z, kappa_Z and lambda_Z. For the CP-violating couplings the results of single parameter fits are: g_4^Z = -0.39 +0.19 -0.20 kappa_Z = -0.09 +0.08 -0.05 lambda_Z = -0.08 +/- 0.07 . The errors are a combination of statistical and systematic errors. All results are consistent with the Standard Model.


Introduction
This paper reports on a study of W boson polarisations and measurements of Triple Gauge Couplings (TGC's) in the reaction e + e − → W + W − , using data taken by the DELPHI experiment at LEP at centre-of-mass energies between 189 and 209 GeV. The amplitude of the reaction e + e − → W + W − results from t-channel neutrino and s-channel γ and Z exchange and is dominated by the lowest order, so-called CC03, diagrams (see figure 1). The s-channel diagrams contain trilinear γW + W − and ZW + W − gauge boson couplings whose possible deviations from the predictions of the Standard Model (anomalous TGC's), due to the effects of new physics, have been extensively discussed in the literature and are for instance described in references [1][2][3][4]. The decay angles of the charged lepton in the W − (W + ) rest frame are used to extract the single W CC03 Spin Density Matrix (SDM) elements as a function of the W − production angle with respect to the e − beam direction. The method of projection operators described in reference [4] is used. Measurements of the SDM elements in the reaction e + e − → W + W − have been reported by OPAL [5].
The diagonal SDM elements have been used to obtain the differential cross-sections for longitudinally polarised W bosons. The study of the longitudinal cross-section is particularly interesting as this degree of freedom of the W only arises in the Standard Model through the electroweak symmetry breaking mechanism. Measurements of the W polarisations at LEP have been reported previously by OPAL [5] and L3 [6]. The imaginary parts of the off-diagonal W + and W − SDM elements should vanish in the Standard Model and are particularly sensitive to CP-violation [7]. Previous studies of CPviolation in the reaction e + e − → W + W − have been performed by ALEPH [8], DELPHI [9] and OPAL [5].
Fits were performed to SDM elements measured as a function of the W − production angle with respect to the e − beam direction in order to extract CP-conserving and CPviolating charged Triple Gauge boson Couplings. In this paper the theoretical framework described in [1], based on the references given in [2], is used. The effective Lagrangian containing only the lowest dimension operators (up to dimension six; terms of higher dimensions should be negligible at LEP energies [1]) and describing the most general Lorentz invariant W W V vertex, with V = γ or Z, contains 14 terms with 14 corresponding couplings, g V 1 , κ V , λ V , g V 4 , g V 5 ,κ V ,λ V , representing the annihilation through the two virtual bosons ( γ and Z). Assuming SU(2) L × U(1) Y gauge invariance to be preserved, the following constraints between coupling constants are obtained [1,3]: with ∆κ Z = κ Z − 1, ∆κ γ = κ γ − 1, ∆g Z 1 = g Z 1 − 1 and θ W the weak mixing angle. Electromagnetic gauge invariance implies that g γ 1 = 1 and g γ 5 = 0 for on-shell photons (q 2 = 0) [1] . In the following the possible q 2 -dependence of all the TGC's will be assumed to be negligible and we set 1 g γ 1 = 1 and assume that the CP-violating coupling g γ 4 = 0 and that g γ 5 = g Z 5 = 0 at all q 2 . These last two coupling constants, although 1 The parameters g γ 1 , κγ and λγ are related to the charge Q W , the magnetic dipole moment µ W and the electric quadrupole moment q W of the W + with: CP-conserving, correspond to the only terms violating both C-and P-symmetry in the Lagrangian considered in this analysis. With these assumptions, the number of independent coupling parameters can be reduced to six, three of which correspond to CP-conserving interactions (∆g Z 1 , ∆κ γ and λ γ ), the remaining three being CP-violating (g Z 4 ,κ Z andλ Z ). In the Standard Model (SM) all these parameters are expected to be zero at tree level. Hence ∆g Z 1 and ∆κ γ explicitly parameterise possible anomalous deviations of the couplings g Z 1 and κ γ from their Standard Model values.
Triple Gauge Couplings have been measured by the four LEP experiments, ALEPH [8], DELPHI [10], L3 [11] and OPAL [12]. The most recent results from DELPHI on CPconserving TGC's [10] were derived from data taken at centre-of-mass energies ranging from 189 to 209 GeV. Hadronic as well as leptonic decay channels of the W bosons were considered using methods based on angular observables characterising both W production and decay. Measurements of CP-violating TGC's analogous to those described in this paper have been made by OPAL [13], while results from a different fit method have been published by ALEPH [8].
The selection of semi-leptonic e + e − → W + W − → lνqq(l = e, µ) events and the corrections for efficiency, resolution and purity are given in section 2. Section 3 discusses the determination of the single W SDM elements, the estimation of the fraction of longitudinally polarised Ws and the study of CP-violating effects on the imaginary elements. Section 4 is devoted to the estimation of the systematic errors on the SDM's. The TGC fits are described in section 5. A global summary is given in section 6.

Data sample and Monte Carlo simulation
For this analysis the data taken by DELPHI at centre-of-mass energies between 189 and 209 GeV were used. The DELPHI detector and its performance are described in reference [14]. The data consist of events of the type e + e − → W + W − → lνqq(l = e, µ). In order to take the energy dependence of the measurements into account, the data were grouped into three samples: 154 pb −1 taken in 1998 at 189 GeV, 218 pb −1 taken in 1999 at energies between 192 and 202 GeV (mean 198 GeV) and 149 pb −1 taken in 2000 at energies in the range 204 to 209 GeV (mean 206 GeV).
Events were selected in which one W decayed into a eν or µν pair while the other W decayed into a pair of quarks. These events are characterised by one isolated electron or muon, two hadronic jets and missing momentum coming from the neutrino. The major background comes from qqτ ν final states, from qq(γ) production and from neutral current four-fermion final states containing two quarks and two leptons.
After a loose preselection, an Iterative Discriminant Analysis (IDA) was used to make the final selection. This part of the event selection is identical to the procedure used to measure the WW production cross-sections [15]. Events were selected with a cut on the output of the IDA, chosen to optimise the product of efficiency and purity for each channel. Events were first passed to the qqµν selection; if they were not selected, they were passed to the qqeν selection; if they were still not selected, they were then finally passed to the qqτ ν selection for possible inclusion or rejection. In this analysis only the events tagged as qqeν or qqµν were retained.
A three-constraint kinematic fit was then applied in which the masses of the two W candidates were constrained to be equal to a reference mass (80.35 GeV/c 2 ). A cut was applied on the χ 2 probability of this fit at 0.005. Events for which the angle between the lepton track and the beam axis was less than 20 • were rejected to remove leptons with poor charge measurement.
The integrated luminosity used is 520 pb −1 , corresponding to data taking runs in which the subdetectors which were essential for this analysis were fully operational. A total of 1880 lνqq events was selected. The data were analysed separately for each of the three years. A breakdown of the collected statistics for different energies, as well as the mean energy for each sample, are given in table 1, with other details.
The signal refers to the WW-like CC03 diagrams leading to lνqq final states [4]. The efficiencies and purities were estimated by Monte Carlo methods with the WPHACT [16] program (charged and neutral current four-fermion events), and KK2F [17] (qq(γ) event generator) at energies of 188.6, 199.5 and 206.0 GeV. The hadronisation of quarks was simulated with the JETSET [18] package. To account for the full O(α) radiative corrections the generated charged current events were reweighted following the procedure described in [19]. The CC03 selection efficiency was around 70% while the purity was around 92%. Both were roughly energy independent as shown in table 1.
To obtain the SDM elements the selected events were corrected for the acceptance, the angular resolutions and the sample purity. The correction factors were obtained from samples of simulated events with sizes given in table 1.
The selection efficiency was calculated as a function of the W − production angle cos Θ W and the lepton decay angles cos θ * and φ * . The lepton decay angles are defined in the W rest frame as shown in figure 2. The efficiency is defined as the number of reconstructed events divided by the number of generated events in a given angular interval. Since the signal refers to the CC03 diagrams only, each event was reweighted by the ratio of the square of the matrix element for the CC03 diagrams only to the square of the matrix element for the full set of diagrams leading to qqeν and qqµν final states, including the full O(α) radiative corrections. The events were divided in 8 equal bins of cos Θ W , in 10 equal bins of cos θ * and in 10 equal bins of φ * . The corrections were computed in each of these three-dimensional bins. The average number of generated events in a bin was 80 and about 7% of the bins were populated by less than 10 events. Examples of the efficiency distributions at 199.5 GeV are shown in figure 3.
The typical resolution on the measured cos Θ W , after the 3C kinematic fit, was found to be 0.04, much smaller than the bin size of 0.25. For about 17% of the events the reconstructed cos Θ W deviates from the generated value by more than 0.125. Because of the definition of the selection efficiency as the convolution of efficiency and migration, correlations between neighbouring cos Θ W bins are expected after the correction procedure.
A study of simulated events shows that between 70% and 90% of the events are reconstructed in the correct bin, and that the remaining events are nearly all reconstructed in the directly neighbouring intervals. The typical resolution on the measured cosθ * was 0.05, while it was 0.08 radians on the measured φ * . This has to be compared to the bin widths of 0.2 and 0.628 radians respectively.
The purity with respect to CC03 e/µ production was calculated as a function of the three relevant angles with the same binning as used for the efficiencies. To estimate the signal contribution, the WW events were reweighted to obtain 'CC03 events' as explained above for the efficiency estimation. To estimate the background from τ νqq and fully hadronic WW final states the events were reweighted to account for full O(α) radiative corrections. The small contribution of non-CC03 semi-leptonic e/µ events was also accounted as background. The other background contributions come from qq(γ) and neutral current four-fermion final states. Examples of the purity distributions at 199.5 GeV are shown in figure 4. Effective purities can become slightly greater than 1 due to interference effects between CC03 and higher order diagrams affecting the CC03 reweighting procedure [19] The fully corrected production and decay angle distributions obtained from the data are shown in figure 5 for the three data-taking years. The cos Θ W and cos θ * distributions for W − and W + events, with the W decaying respectively in a negative or positive lepton, have been added together.

Single W Spin Density Matrix and W polarisation
For events of the type is the helicity of the electron (positron), τ − = ±1, 0 and τ + = ±1, 0 are the helicities of the W − and W + , respectively, the two-body spin density matrix (SDM) is defined as [1,3,4]: with cos Θ W the production angle of the W − with respect to the e − beam and F (λ) τ − τ + the amplitude for the production of a W − with helicity τ − and a W + with helicity τ + . If only W − decays are observed we have In an analogous way, one has: The differential cross-section for W + W − production with subsequent leptonic decay of the W − can be written as: are the angles of the lepton in the W − rest frame (see figure 2) and BR is the W − → ℓ −ν branching fraction. The coordinate system in which these angles are defined is that of ref. [3] and corresponds to the one shown in figure 2. This representation of the differential cross-section in terms of the spin density matrix is independent of the specific form of the helicity amplitudes, i.e. of the specific form of the W + W − production process. The empirical determination of the SDM elements thus amounts to a model-independent analysis of this process.
A set of projection operators Λ W − τ − τ ′ − can be found [4] which isolate the corresponding contributions when integrated over the full lepton spectrum: The SDM elements for W + production are obtained in a similar way.
For a CP-invariant interaction, such as in the standard SU(2) L × U(1) Y theory, the SDM elements of the produced W + and W − are related via [7]: The magnitude of any difference between the left-hand and right-hand sides of (6) constitutes a direct measure of the strength of a possible CP-violating interaction. At tree level, invariance under CPT transformations also implies the validity of relations (6) when applied to the real parts of the SDM, while for the imaginary parts, CPT invariance leads to the relation: Thus a violation of CP-invariance in WW production can best be investigated by looking for inequality of the imaginary parts of the SDM in (6), i.e. by testing the relations: Relations (7) and (8) result in the fact that the imaginary parts of the SDM should vanish.
Experimentally the SDM elements were obtained from the relation where N i is the number of selected events in a given cos Θ W bin. Each event was weighted with a correction factor w j dependent on (cos Θ W , cos θ * , φ * ) as explained in section 2, to account for detector acceptance, bin migration and sample purity. The event sample was divided into 8 equal bins of cos Θ W . As the W − production occurs mainly in the forward direction with respect to the e − beam, and the experimental statistics available are rather restricted, 75% of the cos Θ W bins in the backward region have less than 20 events when the cos Θ W values are sampled in eight equal bins. From WPHACT Monte Carlo studies of a large number (250) of data-sized samples simulated at energies of 189, 200 and 206 GeV, it appears that the number of events per bin should be at least about 20 to allow a reliable extraction of Triple Gauge Couplings from the data. In order to reach this goal, the SDM elements were redetermined in two equal-sized cos Θ W bins for W − bosons produced in the backward region. Figures 6, 7 and 8 show that the SDM elements computed for W + and W − separately are compatible with relation (6) imposed by CP-invariance. Only statistical errors are displayed as systematic effects are expected to be small compared to statistical fluctuations (see section 4) and are similar for W + and W − bosons. The measurements of the SDM elements are shown in figures 9, 10, and 11 for the three data samples taken in 1998, 1999 and 2000 separately. As the SDM elements computed for W + and W − separately are compatible, CP-invariance is assumed in these plots and both the W + and W − leptonic decays were used to compute the W − SDM elements, based on relation (6). The predictions from Standard Model signal events (about 50000 pb −1 at each energy simulated with WPHACT) are also shown together with the results from the analytical calculations used in the TGC fits (see section 5). The measured values agree with the SM expectation at all energies considered. Indeed, the χ 2 values for comparison with the analytical calculation, and taking into account the SDM elements in the 6 bins as shown in the figures 9 to 11, are respectively 45.3 (189 GeV), 43.5 (198 GeV) and 35.8 (206 GeV) for 48 degrees of freedom. In the calculation of the χ 2 the linear constraints on the diagonal elements were taken into account by removing the element ρ ++ , and the full covariance matrix based on the statistical and systematic errors as explained in section 5, was used. The corresponding χ 2 probabilities are 58.2%, 65.9% and 90.2% respectively.
In figures 9, 10 and 11 a comparison is made of the CC03 SDM elements calculated with WPHACT (open dots) and those obtained with the expressions from ref. [4] (full line), which do not include radiative corrections. It is seen that the two calculations agree well, which implies that the effect of radiative corrections is very small compared to the experimental errors.
The differential cross-section for the production of longitudinally polarised W bosons is In this formula dσ/d cos Θ W is the differential cross-section after correction for detector acceptance and sample purity. The differential cross-sections were determined for the three energies considered. Figure 12 shows the luminosity weighted average of the measured differential cross-sections, together with the Standard Model predictions from WPHACT. The two distributions are in good agreement.
Integration yields the fraction of longitudinally polarised W bosons:

Systematic errors on the SDM elements
The systematic uncertainties in the measurements of the SDM elements were calculated as described below. The list of systematic errors considered for ρ 00 is shown in table 2 as an example. The systematic errors on the differential cross-section and on the fraction of longitudinally polarised W bosons were estimated in the same way and are discussed at the end of this section.
1. Monte Carlo statistics. The detector corrections are binned in 8 bins in cos Θ W , 10 bins in cos θ * and 10 bins in φ * . Some bins have a low population of events which results in a large uncertainty in the correction factor. To estimate this effect on the SDM elements, the simulated data samples were divided in 9 subsamples of about 2600 pb −1 and detector corrections were computed for each subsample. The analysis was rerun on the data with each set of detector corrections and the differences of the new SDM elements with the SDM elements obtained with the standard corrections were computed. The standard deviation of the distributions of differences, corrected for the factor 9 difference in statistics between the subsamples and the full sample, was taken as the systematic error. 2. Signal and background cross-sections. The uncertainties on the signal and background cross-sections influence the purities. For the estimation of the systematic error arising from the uncertainty on the background cross-sections only the uncertainties on the qq(γ) and four-fermion neutral current cross-sections were taken into account, and were taken to be 5% [20]. The purities were recalculated with background cross-sections which were modified by plus and minus one standard deviation. The mean of the differences of the recomputed SDM elements and the standard elements was taken as systematic uncertainty. The uncertainty on the signal cross-section enters both in the denominator and the numerator and its effect is expected to be small. The purities were recalculated with signal cross-sections which were modified by plus and minus one standard deviation. The uncertainty on the signal cross-section was taken to be 0.5% , the theoretical error [20]. The mean of the differences of the recomputed SDM elements and the standard elements was taken as systematic uncertainty. These uncertainties are negligible at all energies considered. 3. Jet reconstruction, hadronisation modelling and migration of events between cos Θ W bins. The reconstruction of the hadronic jets influences the determination of the W production and decay angles and will hence lead to migration effects between bins in the cos Θ W distribution. On the other hand, the corrections for acceptance and purity are sensitive to the modelling of the hadronisation in the simulation. To estimate these effects, the differences between the SDM elements calculated with simulated events at generator level and at reconstruction level, using the HERWIG hadronisation modelling [21], were computed. The reconstructed SDM elements were obtained by reweighting the selected events with the standard detector corrections obtained from the JETSET hadronisation modelling. The absolute values of these differences were taken as systematic uncertainty. This uncertainty was estimated at 199.5 GeV and the same value was used for all 3 energies. A problem with the track reconstruction efficiency for low-momentum particles at low polar angles was corrected for as described in [22]. We have investigated the systematic error related to this correction and found that it was negligible. 4. Cut on lepton polar angle. In the analysis, events with a lepton close to the beam (polar angle below 20 • or above 160 • ) were rejected, and the standard detector corrections were calculated accordingly. To estimate the effect of the limited resolution in the reconstruction of the lepton angle, the analysis was redone with a cut at both 18 • and 22 • . The detector corrections were recalculated, one set for each cut, and the events were corrected with these new sets. The differences between the SDM elements obtained in the analysis with a cut at 22 • and the analysis with a cut at 18 • were rescaled to a difference corresponding to ±0.5 • . This is a conservative estimate compared to the estimated value of the resolution which is about 0.1 • , plus some tails. In addition, the SDM elements were recalculated with these new cuts, but corrected with the standard detector corrections, and the difference scaled down to ±0.5 • was also computed. This yields two estimates of the uncertainty related to the resolution on the lepton polar angle reconstruction and the modelling of this reconstruction in the simulation. The larger estimate was taken as systematic uncertainty.
5. Cut on the χ 2 probability of the 3C fit. The analysis was redone with two different cuts on the χ 2 probability, at 0.003 and at 0.007, in a region where the probability has a flat distribution. For each cut, detector corrections were recalculated and the data were corrected with these new sets of corrections. The mean difference between the elements obtained with each new set of corrections and the standard elements was taken as systematic uncertainty. 6. Radiative corrections and CC03 reweighting. The purities which enter in the detector corrections refer to CC03 events of the type e + e − → W + W − → lνqq(l = e, µ). The simulated event samples which were used to calculate these purities contain all four-fermion charged current processes. To obtain the signal angular distributions which are input to the purity calculations the events were reweighted with CC03 weights following the reweighting procedure explained in ref. [19]. The uncertainty on the calculation of the radiative corrections has only a small influence on the SDM elements (see section 3). The combined effect of the uncertainty from the CC03 reweighting and the radiative corrections was estimated by the difference between the analytical calculation of the SDM's used for the TGC fits (CC03 in the zero width approximation, no radiative corrections at all, see [4]) and the SDM elements calculated at generator level with samples of simulated signal events corresponding to about 50000 pb −1 (WPHACT MC). For the cases where the error on the Monte Carlo calculation was larger than this difference, this error was taken as systematic uncertainty. 7. Lepton charge determination. In the forward and backward regions of the detector the lepton charge is sometimes badly determined. To estimate this effect on the SDM elements, 10% of the events were artificially given a wrong charge and the elements were recalculated with standard detector corrections. From a study of two-lepton events [23] the fraction of leptons with a wrong charge assignment was estimated to be less than 1%. The uncertainty on the SDM elements from lepton charge determination was obtained from a rescaling by a factor 10 of the difference between the elements calculated with the 10% wrong charge data and the standard elements.
The systematic errors on the 9 SDM elements in a given bin at a given energy are fully correlated since the elements are determined from the same events. The systematic error from Monte Carlo statistics (1.) is uncorrelated between bins and energies. All other systematic errors are fully correlated between bins and energies. Therefore a luminosity weighted average of the values obtained at the three energies was used in the TGC fits, hence reducing the effects of statistical fluctuations. The systematic errors on the differential cross-sections and the fraction of longitudinally polarised W bosons were estimated with the same procedure as that used for the SDM elements. When computing the luminosity weighted average of these quantities all systematic errors were considered fully correlated between years, apart from the error from Monte Carlo statistics. The systematic error on the fraction f L is given in table 3.

Fits of Triple Gauge Couplings
Both CP-conserving and CP-violating TGC's are determined in this analysis, which is however particularly suited to the determination of CP-violating couplings, whose existence would be revealed by non-zero imaginary parts of the SDM's. To investigate the possible existence of the anomalous CP-violating TGC's g Z 4 ,κ Z ,λ Z in each of the three data samples defined in table 1, the experimental values of the single W SDM elements ρ W − τ τ ′ (s, cos Θ W ) and ρ W + τ τ ′ (s, cos Θ W ) determined in each of the cos Θ W bins considered in this analysis were fitted to theoretical expressions derived in Ref. [4]. For CP-invariant interactions the relationship (6) holds. This allows a combination of W − and W + elements in each cos Θ W bin. This procedure was applied in order to extract the CP-conserving couplings ∆g Z 1 , ∆κ γ and λ γ . In each of the cos Θ W bins the 9 SDM elements are correlated. The strongest correlations occur between ρ ++ , ρ −− and ρ 00 , whose sum is constrained to be one. The correlations were determined from the data and taken into account in the fit.
As the sum of the projection operators Λ ++ + Λ −− + Λ 00 = 1, it is seen from expression (9) that the sum of the experimentally determined diagonal SDM elements will always be exactly equal to one, whatever the sample used. The most straightforward way to take this constraint into account is to retain only two of the three diagonal elements in the fit, whose results are indeed totally insensitive to which of those elements is rejected. In the following, the element ρ ++ has been removed from the fits which are hence reduced to five real SDM elements per bin (ρ −− , ρ 00 , Re(ρ +− ), Re(ρ +0 ), Re(ρ −0 )) to determine the CP-conserving couplings, and to sets of 8 elements per bin (as above plus Im(ρ +− ), Im(ρ +0 ), Im(ρ −0 )) for the extraction of the CP-violating couplings.
A least squares fit was used in which the measured values of the SDM elements were compared to their theoretical predictions at the average centre-of-mass energies for each of the three data sets. The statistical covariance matrices were computed from the data. These were combined with the full systematic covariance matrix containing the systematic errors described in section 4. Table 4 shows the results of the one-parameter fits for the three data sets separately and for the combined fit to all data. The total (statistical and systematic) error matrices were used. In each χ 2 fit only one of the TGC's considered was varied, all other couplings being fixed at their SM value. The χ 2 curves of the fits are displayed in figure 14 for the CP-conserving couplings and in figure 15 for the CP-violating couplings. The minimum χ 2 values are displayed in table 4. The χ 2 probabilities of all fits to the full sample are acceptable, but are considerably lower for the CP-violating fits than for the CP-conserving fits. This is mainly due to the data at 189 GeV. The errors on the results of fits using only statistical errors on the SDM elements are given in the last column of table 4. It is seen that the results of the fits are dominated by the statistical errors. Using statistical errors only, the results of the Monte Carlo studies of 250 data-sized samples with SDM's computed at generation and at reconstruction level do not indicate any marked bias of the fitted values of the TGC's with respect to their SM input values. These Monte Carlo studies also revealed the existence of a double minimum in the fits of ∆κ γ which is confirmed by the data, as seen in figure 14. Such double minima can occur [1,24] as the helicity amplitudes are linear in the couplings.
In the fits to the data the average beam energies, displayed in table 1 for each of the data taking years, were used. However, as already mentioned in section 2, the beam energy of the data samples taken in 1999 varied from 192 to 202 GeV and from 204 to 209 GeV for the samples taken in the year 2000. The effect of these beam energy spreads on the errors on the fitted values of the TGC's was estimated by repeating the single parameter fits with beam energy values varying within the allowed energy ranges. The resulting shifts in the fitted values of the TGC parameters are very small and have been treated as systematic errors included in the full errors given in table 4. The maximum size of this systematic error is 0.02. Two-parameter fits of the TGC's at fixed central beam energy values were also performed, the results of which are shown in figures 16 and 17 for the full data set using the total (statistical and systematic) error matrix. The results are in reasonable agreement with the SM expectations. It is seen from figure 16 that the fit of ∆κ γ exhibits a second minimum which appears as an extension of the 95% probability contour. This second minimum also strongly affects the shape of the ∆χ 2 -plot at 189 GeV shown in figure 14.
Finally, three-parameter fits to the full data sample with full error matrices were also performed separately for the CP-conserving and CP-violating couplings respectively. The results are shown in table 5, in which the errors shown are the standard deviations of the marginal distributions of each of the parameters.
The results of the one, two and three-parameters fits are consistent with each other and agree with the Standard Model.

Summary
The data taken by the DELPHI experiment at centre-of-mass energies of 189, 192-202 and 204-209 GeV were used to select a sample of respectively 520, 838 and 522 events of the type e + e − → lνqq(l = e, µ). The decay angles of the leptonically decaying W bosons were used to calculate the single W − and W + spin density matrices, which are defined for CC03 events, and the average values assuming CP symmetry.
The SDM elements were used to determine the fractions of longitudinally polarised W bosons. For each of the three data samples the measured fraction of longitudinally polarised W bosons is in agreement with the SM prediction. For all data taken between 189 and 209 GeV an average value of σ L /σ tot = 24.9 ± 4.5(stat) ± 2.2(syst)% (13) is obtained at an average energy of 198 GeV, where 23.9 ± 0.2% is expected from the Standard Model. The SDM elements have been used to determine the CP-violating Triple Gauge Couplings. One-parameter fits to the full data sample yield: For the CP-conserving TGC's the values obtained in this analysis are less precise than those measured in the DELPHI analysis using optimal observables [10], but they confirm the good agreement of all the fitted couplings with the predictions of the Standard Model.            (6)), with statistical errors, measured with the data taken at 189 GeV, corrected for detector acceptance and sample purity as explained in the text.  (6)), with statistical errors, measured with the data taken at 198 GeV, corrected for detector acceptance and sample purity as explained in the text. Figure 8: Difference ∆ρ τ τ ′ = ρ W − τ τ ′ (s, cos Θ W ) − ρ W + −τ −τ ′ (s, cos Θ W ) (see equation (6)), with statistical errors, measured with the data taken at 206 GeV, corrected for detector acceptance and sample purity as explained in the text. Figure 9: Averages of W + and W − SDM elements, with statistical and total errors, measured with the data taken at 189 GeV (black dots), corrected for detector acceptance and sample purity as explained in the text. The full line shows the tree level SM prediction calculated with the analytical expression from ref. [4]. The open circles are the SM tree level predictions obtained with the WPHACT MC at generator level. Figure 10: Averages of W + and W − SDM elements, with statistical and total errors, measured with the data taken at an energy of 198 GeV (black dots), corrected for detector acceptance and sample purity as explained in the text. The full line shows the tree level SM prediction calculated with the analytical expression from ref. [4]. The open circles are the SM tree level predictions obtained with the WPHACT MC at generator level. Figure 11: Averages of W + and W − SDM elements, with statistical and total errors, measured with the data taken at an energy of 206 GeV (black dots), corrected for detector acceptance and sample purity as explained in the text. The full line shows the tree level SM prediction calculated with the analytical expression from ref. [4]. The open circles are the SM tree level predictions obtained with the WPHACT MC at generator level.