Summation to Calculate Single W SDM Elements

In this appendix the summations of operators needed to calculate the elements of the single W Spin Density Matrix are listed. In each equation $\theta_{f_{1}}$ is the polar angle and $\phi_{f_{1}}$ is the azimuthal angle of the decay fermion from the ${\rm W}^{-}$ in the ${\rm W}^{-}$ rest frame. The elements are extracted in bins of $\cos\theta_{\rm W}$, the ${\rm W}^{-}$ production angle. $k$ is the bin of $\cos\theta_{\rm W}$ and $N_{k}$ is the number of events in that bin.

$$\rho^{W^{-}}_{++}(k) = \frac{1}{N_{k}}\sum_{i=1}^{N_{k}}\frac{1}{2}(5\cos^{2}\theta_{f_{1}} - 2\cos\theta_{f_{1}} - 1)$$

$$\rho^{W^{-}}_{--}(k) = \frac{1}{N_{k}}\sum_{i=1}^{N_{k}}\frac{1}{2}(5\cos^{2}\theta_{f_{1}} + 2\cos\theta_{f_{1}} - 1)$$

$$\rho^{W^{-}}_{00}(k) = \frac{1}{N_{k}}\sum_{i=1}^{N_{k}} 2-5\cos^{2}\theta_{f_{1}}$$

$$Re\left(\rho^{W^{-}}_{+-}(k)\right) = \frac{1}{N_{k}}\sum_{i=1}^{N_{k}} 2\cos2\phi_{f_{1}}$$

$$Im\left(\rho^{W^{-}}_{+-}(k)\right) = \frac{1}{N_{k}}\sum_{i=1}^{N_{k}} -2\sin2\phi_{f_{1}}$$

$$Re\left(\rho^{W^{-}}_{+0}(k)\right) = \frac{1}{N_{k}}\sum_{i=1}^{N_{k}}\frac{-8}{3\pi\sqrt{2}}(1-4\cos\theta_{f_{1}})\cos\phi_{f_{1}}$$

$$Im\left(\rho^{W^{-}}_{+0}(k)\right) = \frac{1}{N_{k}}\sum_{i=1}^{N_{k}}\frac{8}{3\pi\sqrt{2}}(1-4\cos\theta_{f_{1}})\sin\phi_{f_{1}}$$

$$Re\left(\rho^{W^{-}}_{-0}(k)\right) = \frac{1}{N_{k}}\sum_{i=1}^{N_{k}}\frac{-8}{3\pi\sqrt{2}}(1+4\cos\theta_{f_{1}})\cos\phi_{f_{1}}$$

$$Im\left(\rho^{W^{-}}_{-0}(k)\right) = \frac{1}{N_{k}}\sum_{i=1}^{N_{k}}\frac{-8}{3\pi\sqrt{2}}(1+4\cos\theta_{f_{1}})\sin\phi_{f_{1}}$$