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Line 267: − Since the polarization of the initial state photon differs from event to event, the weighted sum over polarization states is performed incoherently. + + + + + + + + + + + + + − + − + − + − + − + + + + + + + + + + + −
Amplitudes for the Exotic b1π Decay (view source)
Revision as of 21:20, 18 August 2011
, 21:20, 18 August 2011→Assembly of the full amplitude
=== Summing over polarizations ===
=== Summing over polarizations ===
The T Matrix, written in the photon reflectivity basis can be expanded in the photon's lab frame helicity basis
:<math>
T_{\epsilon_\gamma} = \frac{\sqrt{-\epsilon_\gamma}}{2}
\left( T_{-1}e^{-i\alpha} + \epsilon_\gamma T_{+1}e^{i\alpha} \right)
</math>
where,
:<math>
T_{\pm 1}=\sum_{R,\lambda_R;\lambda_{b_1},\lambda_\omega,\lambda_\rho}
\langle \mathbf{q}_\pi 0 0; \mathbf{q}_\rho \lambda_\rho 0; \mathbf{q}_\omega \lambda_\omega 0; \mathbf{q}_{b1}\lambda_{b1} 0
| U | (1\; \pm 1)_{\mathrm{lab}}; J_R \lambda_R \epsilon_R;s,t \rangle
</math>
Now, the average over the initial photon polarization states results in a cross section evaluated as follows:
:<math>
:<math>
\frac{d^8\sigma}{d\Omega_{b1}\,d\Omega_\omega\,d\Omega_\rho\,d\Omega_\pi} =
\frac{d^8\sigma}{d\Omega_{b1}\,d\Omega_\omega\,d\Omega_\rho\,d\Omega_\pi} \propto
\frac{1+g}{2}\frac{d^8\sigma_{(\epsilon_\gamma=-1)}}
\frac{1}{2}\left\{
{d\Omega_{b1}\,d\Omega_\omega\,d\Omega_\rho\,d\Omega_\pi} +
(1+g)\left| \frac{1}{\sqrt{2}}(T_{-1}e^{-i\alpha} + T_{+1}e^{i\alpha}) \right|^2
\frac{1-g}{2}\frac{d^8\sigma_{(\epsilon_\gamma=+1)}}
+
{d\Omega_{b1}\,d\Omega_\omega\,d\Omega_\rho\,d\Omega_\pi}
(1-g)\left| \frac{i}{\sqrt{2}}(T_{-1}e^{-i\alpha} - T_{+1}e^{i\alpha}) \right|^2 \right\}
</math>
:::<math>
=\frac{1}{2}\left[
|T_{-1}|^2 + |T_{+1}|^2 +
g(T_{+1}T_{-1}^*e^{2i\alpha} + T_{+1}^*T_{-1}e^{-2i\alpha}) \right]
</math>
:::<math>
=\frac{|T_{-1}|^2 + |T_{+1}|^2}{2} +
g\left[\mathrm{Re}(T_{+1}T_{-1}^*)\cos 2\alpha +
\mathrm{Im}(T_{+1}T_{-1}^*)\sin 2\alpha)\right]
</math>
</math>
where ''g'' is the polarization fraction ranging from 1 (100% x-polarized) to 0 (unpolarized.)
where ''g'' is the polarization fraction ranging from 1 (100% x-polarized) to 0 (unpolarized.)