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Amplitudes for the Exotic b1π Decay (view source)
Revision as of 23:43, 18 August 2011
, 23:43, 18 August 2011→Assembly of the full amplitude
:<math>
:<math>
T_{(f)(i)} =
T_{(f)(i)} =
T_{(\mathbf{q}_\pi \mathbf{q}_\rho \mathbf{q}_\omega \mathbf{q}_{b1})
T_{(\mathbf{q}_\pi \mathbf{q}_\rho \mathbf{q}_\omega \mathbf{q}_{b1} \lambda_f)
(\epsilon_\gamma \epsilon_R t)}=
(\epsilon_\gamma \epsilon_R t \lambda_i)}=
</math>
</math>
::::<math>
::::<math>
</math>
</math>
The aggregate decay matrix element can be further broken up into a product of individual decay amplitudes,
An average over the target proton initial state will be necessary to compute the cross section section. Also, because the polarization of the recoiling proton cannot be measured, a sum over the proton final states must be done. This can be represented as
:<math>
\frac{d^8\sigma}{d\Omega_{b1}\,d\Omega_\omega\,d\Omega_\rho\,d\Omega_\pi} \propto
\frac{1}{2}\sum_{\lambda_i \lambda_f \lambda_i' \lambda_f'}
\rho_{\lambda_i \lambda_i'}
\rho_{\lambda_f \lambda_f'}
T_{(\mathbf{q}_\pi \mathbf{q}_\rho \mathbf{q}_\omega \mathbf{q}_{b1} \lambda_f)
(\epsilon_\gamma \epsilon_R t \lambda_i)}
T^*_{(\mathbf{q}_\pi \mathbf{q}_\rho \mathbf{q}_\omega \mathbf{q}_{b1} \lambda_f')
(\epsilon_\gamma \epsilon_R t \lambda_i')}
</math>
where density matrices <math>\rho</math> represent weights of proton's states in the summations. The unpolarized target presents an initial state with both helicities equally likely, resulting in
<math>\rho_{\lambda_i \lambda_i'} \propto \rho_{\lambda_i \lambda_i'}</math>. The same property holds for <math>\rho_{\lambda_f \lambda_f'}</math> by definition of the summation over the final states. As a result, the term characterizing the target proton's transition with the emission of the Reggeon factorizes, allowing us to drop indices for the proton states in the T matrix:
:<math>
T_{(f)(i)} =
T_{(\mathbf{q}_\pi \mathbf{q}_\rho \mathbf{q}_\omega \mathbf{q}_{b1})
(\epsilon_\gamma \epsilon_R t)}
</math>
The remaining production and decay matrix elements can be further broken up into a product of individual decay amplitudes,
:<math>
:<math>
\langle \mathbf{q}_\pi 0 0; \mathbf{q}_\rho \lambda_\rho 0; \mathbf{q}_\omega \lambda_\omega 0; \mathbf{q}_{b1}\lambda_{b1} 0
\langle \mathbf{q}_\pi 0 0; \mathbf{q}_\rho \lambda_\rho 0; \mathbf{q}_\omega \lambda_\omega 0; \mathbf{q}_{b1}\lambda_{b1} 0
=== 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
The T Matrix, written in the photon reflectivity basis can be expanded in the photon's lab frame helicity basis. Temporarily omitting indices not pertaining to the photon:
:<math>
:<math>
T_{\epsilon_\gamma} = \frac{\sqrt{-\epsilon_\gamma}}{2}
T_{\epsilon_\gamma} = \frac{\sqrt{-\epsilon_\gamma}}{2}
:::<math>
:::<math>
=\frac{|T_{-1}|^2 + |T_{+1}|^2}{2} +
=\frac{|T_{-1}|^2 + |T_{+1}|^2}{2} +
g\left[\mathrm{Re}(T_{+1}T_{-1}^*)\cos 2\alpha +
g\,\mathrm{Re}(T_{+1}T_{-1}^*)e^{2i\alpha}
</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.)