Calculating the SDM Elements

The W-pair production process is characterised by a final state of four fermions. It was shown in equation 3.27 that the cross-section for the process ${\rm e}^{+}{\rm e}^{-} \rightarrow {\rm W}^{+}{\rm W}^{-} \rightarrow f_{1}\bar{f}_{2}f_{3}\bar{f}_{4}$ may be written in terms of five angles, the production angle of the ${\rm W}^{-}$ boson, the polar and azimuthal angles of the ${\rm W}^{-}$ decay fermion in the ${\rm W}^{-}$ rest frame, $\cos\theta_{f_{1}}$, $\phi_{f_{1}}$ and the polar and azimuthal angles of the ${\rm W}^{+}$ decay anti-fermion in the ${\rm W}^{+}$ rest frame, $\cos\theta_{\bar{f}_{4}}$, $\phi_{\bar{f}_{4}}$.

Monte Carlo generators can be used to generate pseudo-data events for the process ${\rm e}^{+}{\rm e}^{-} \rightarrow {\rm W}^{+}{\rm W}^{-} \rightarrow f_{1}\bar{f}_{2}f_{3}\bar{f}_{4}$. Figure 4.1 shows the Standard Model prediction of the distributions of the five angles in W-pair production and decay calculated from events generated by the EXCALIBUR Monte Carlo generator.

Figure 4.1: The angular distributions from Monte Carlo generated W-pair events. a) the ${\rm W}^{-}$ production angle. b) ${\rm W}^{+}$ production angle. c) & e) the polar and azimuthal angular distributions of the ${\rm W}^{-}$ decay fermion in the ${\rm W}^{-}$ rest frame. d) & f) the polar and azimuthal angular distributions of the ${\rm W}^{+}$ decay anti-fermion in the ${\rm W}^{+}$ rest frame.

The equation describing the 5-fold differential cross-section in terms of these angles contains the helicity amplitudes and the D-functions that described how the decay fermions couple to the W bosons through the standard V$-$A coupling. This coupling, and hence the angular distribution of the fermions depends on the helicity of the W bosons.

Measuring the angular distribution of the W decay products then gives an effective way of measuring the W bosons' helicities. The D-functions, given in equation 3.26, can be inverted, so that rather than giving the angular distribution for a certain helicity, it will give the helicities for a certain angular distribution. A set of so-called projection operators [41] can thus be formed from the D-functions. When these operators are applied to the angular distributions of the decay fermions, they effectively project out information about the helicities of the W bosons. The projection operators are given the form of $\Lambda^{W^{\pm}}_{\tau\tau^{\prime}}$, where the $\tau$ and $\tau^{\prime}$ relate to the interfering spins of individual W bosons, and the $W^{\pm}$ indicates that there is a different set of operators for the ${\rm W}^{+}$ boson and ${\rm W}^{-}$ boson. The full set of projection operators can be seen in equation 4.1, where $\Lambda_{\tau\tau^{\prime}} = \Lambda_{\tau^{\prime}\tau}^{\ast}$.

$$\Lambda^{W^{-}}_{\pm\pm} = \Lambda^{W^{+}}_{\mp\mp} = \frac{1}{2}(5\cos^{2}\theta_{f} \mp 2\cos\theta_{f} - 1)$$

$$\Lambda^{W^{-}}_{00} = \Lambda^{W^{+}}_{00} = 2-5\cos^{2}\theta_{f}$$

$$\Lambda^{W^{-}}_{+-} = \Lambda^{W^{+}}_{+-} = 2{\rm e}^{-2i\phi_{f}}$$ (4.1)

$$\Lambda^{W^{-}}_{\pm0} = -\left(\Lambda^{W^{+}}_{\mp0}\right)^{*} = \frac{-8}{3\pi\sqrt{2}}(1 \mp 4\cos\theta_{f}){\rm e}^{\mp i\phi_{f}}$$

The single W SDM elements that describe the helicity properties of one of the W bosons can now be calculated using these projection operators. The unnormalised single W density matrix elements can be extracted from the 3-fold angular distribution of the ${\rm W}^{-}$ decay fermion (or ${\rm W}^{+}$ decay anti-fermion), by integrating with the appropriate projection operators, for example:

$$\frac{d\sigma({\rm e}^{+}{\rm e}^{-} \rightarrow {\rm W}^{+}{\rm W}^{-})}{d\cos\theta_{\rm W}}\rho^{\rm W^{\pm}}_{\tau\tau^{\prime}}$$ (4.2)

$$= \frac{1}{{\rm Br}({\rm W}^{\pm} \rightarrow f\bar{f})}\int\frac... ...mbda^{W^{\pm}}_{\tau\tau^{\prime}}(\theta_{f},\phi_{f})d\cos\theta_{f}d\phi_{f}$$

Each projection operator projects out information about one of the W bosons in the W-pair. So, by integrating over combinations of the ${\rm W}^{-}$ and ${\rm W}^{+}$ projection operators, all 81 of the unnormalised two-particle joint SDMs can be calculated, equation 4.3.

$$\frac{d\sigma({\rm e}^{+}{\rm e}^{-} \rightarrow {\rm W}^{+}{\rm ... ...}{d\cos\theta_{\rm W}}\rho_{\tau_{1}\tau_{1}^{\prime}\tau_{2}\tau_{2}^{\prime}}$$ (4.3)

$$= \frac{1}{{\rm Br}({\rm W}^{-} \rightarrow f_{1}\bar{f}_{2}){\rm... ...W}d\cos\theta_{f_{1}}d\phi_{f_{1}}d\cos\theta_{\bar{f}_{4}}d\phi_{\bar{f}_{4}}}$$

$$\times \Lambda^{W^{-}}_{\tau_{1}{\tau^{\prime}}\!_{1}}(\theta_{f_... ...}})d\cos\theta_{f_{1}}d\phi_{f_{1}}d\cos\theta_{\bar{f}_{4}}d\phi_{\bar{f}_{4}}$$

If the full angular distributions of the decay fermion from the ${\rm W}^{-}$ and the decay anti-fermion from the ${\rm W}^{+}$ are known, all the SDM elements can be calculated. If the set of data are binned into bins of $\cos\theta_{\rm W}$, then experimentally equation 4.2 can be realised as a discrete summation over events, as in equation 4.4, where $k$ is the bin of $\cos\theta_{\rm W}$, and $N_{k}$ is the number of events in that bin.

$$\rho^{W^{\pm}}_{\tau\tau^{\prime}}(k) = \frac{1}{N_{k}}\sum_{i=1}^{N_{k}}\Lambda^{W^{\pm}}_{\tau\tau^{\prime}}(\cos\theta_{f},\phi_{f})_{i}$$ (4.4)

The summations needed for each of the single ${\rm W}^{-}$ SDM elements are shown appendix A.1. Performing these summations on the Monte Carlo data whose angular distributions are shown in figure 4.1 will give the single ${\rm W}^{-}$ SDM elements for this set of data. These SDM elements are shown in figure 4.2. Overlaid are the analytical predictions for the Standard Model calculated from equation 3.29.

Figure 4.2: The single ${\rm W}^{-}$ SDM elements calculated from a Standard Model EXCALIBUR Monte Carlo sample by application of the projection operators given in appendix A.1. Overlaid are the analytical predictions for the Standard Model.

When calculating these SDM elements CPT-invariance can be assumed, so that information from the ${\rm W}^{-}$ and ${\rm W}^{+}$ decay can be combined. CPT-invariance means $\rho^{W^{-}}_{\tau_{1}\tau_{2}} = (\rho^{W^{+}}_{-\tau_{1}-\tau_{2}})^{*}$. Therefore, the summation to calculate each single W$^-$ SDM element may now be written as a summation over the ${\rm W}^{-}\rightarrow f_{1}\bar{f}_{2}$ decays, plus a summation over the ${\rm W}^{+}\rightarrow f_{3}\bar{f}_{4}$ decays with the appropriate CPT transform applied to the projection operator.

The equation needed to calculate the single ${\rm W}^{-}$ SDM elements when both the ${\rm W}^{-}$ decay fermion and the ${\rm W}^{+}$ decay anti-fermion are measured in every event is shown in equation 4.5.

$$\rho^{W^{-}}_{\tau\tau^{\prime}}(k) = \frac{1}{2N_{k}}\left[\sum_... ...au^{\prime}}(\cos\theta_{\bar{f}_{4}},\phi_{\bar{f}_{4}})_{i}\right)^{*}\right]$$ (4.5)

If only one of the W bosons is measured in each event, the measurements from the ${\rm W}^{+}$ and ${\rm W}^{-}$ can still be combined to form just the single ${\rm W}^{-}$ SDM elements as shown in equation 4.6. In this equation $N^{W^{+}}_{k}$ are the number of events with the ${\rm W}^{+}$ $\rightarrow f\bar{f}$ decay measured in bin k of $\cos\theta_{\rm W}$, and $N^{W^{-}}_{k}$ are the number of events with the ${\rm W}^{-}$ $\rightarrow f\bar{f}$ decay measured in bin k of $\cos\theta_{\rm W}$. Thus $N^{W^{+}}_{k} + N^{W^{-}}_{k} = N_{k}$.

$$\rho^{W^{-}}_{\tau\tau^{\prime}}(k) = \frac{1}{N_{k}}\left[\sum_{... ...au^{\prime}}(\cos\theta_{\bar{f}_{4}},\phi_{\bar{f}_{4}})_{i}\right)^{*}\right]$$ (4.6)

For the two-particle joint SDM elements both the decay fermion from the ${\rm W}^{-}$ and the anti-fermion of the ${\rm W}^{+}$ in the W-pair event need to be measured. Experimentally, equation 4.3 can also be written in bins of $\cos\theta_{\rm W}$ as a discrete summation over events, as shown in equation 4.7.

$$\rho_{\tau_{1}\tau_{1}^{\prime}\tau_{2}\tau_{2}^{\prime}}(k) = \f... ...da_{\tau_{2}\tau_{2}^{\prime}}(\cos\theta_{\bar{f}_{4}},\phi_{\bar{f}_{4}})_{i}$$ (4.7)

The complete set of summations of operators giving all the two-particle joint SDM elements are given in appendix A.2. Taking, for example, the operators for the diagonal elements ( $\tau_{1} = {\tau^{\prime}}\!_{1},\tau_{2} = {\tau^{\prime}}\!_{2}$) of the two-particle joint SDM and performing the summations on the Monte Carlo data, the two-particle SDM elements shown in figure 4.3 are obtained. Overlaid are the analytical predictions for the Standard Model calculated from equation 3.29.

Figure 4.3: The diagonal elements of the two-particle joint W SDM calculated from a Standard Model EXCALIBUR Monte Carlo sample by application of the projection operators given in appendix A.2. Overlaid are the analytical predictions for the Standard Model.