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Amplitudes for the Exotic b1π Decay (view source)
Revision as of 16:22, 19 August 2011
, 16:22, 19 August 2011→Assembly of the full amplitude
\left(\frac{q_\pi dm_\rho}{16\pi^3}\right)
\left(\frac{q_\pi dm_\rho}{16\pi^3}\right)
</math>
</math>
where <math>p_i\,</math> [<math>p_f\,</math>] is the target [recoil] nucleon momentum in the center of mass frame of the overall reaction. The explicit kinematic factors from the initial-state flux and the density of final states for each of the decays are not factored into the T matrix so that we can make sure that it explicitly respects unitarity in each partial wave. In terms of the individual decay matrix elements introduced earlier, the T matrix element can be written as
where <math>p_i\,</math> [<math>p_f\,</math>] is the target [recoil] nucleon momentum in the center of mass frame of the overall reaction. In terms of the individual decay matrix elements introduced earlier, the T matrix element can be written as
:<math>
:<math>
T_{(f)(i)} =
T_{(f)(i)} =
T_{(\mathbf{q}_\pi \mathbf{q}_\rho \mathbf{q}_\omega \mathbf{q}_{b1} \mathbf{p}_f \lambda_f)
T_{(\mathbf{q}_\pi \mathbf{q}_\rho \mathbf{q}_\omega \mathbf{q}_{b1} \mathbf{p}_f \epsilon_f)
(\epsilon_\gamma \epsilon_R t \mathbf{p}_i \lambda_i)}=
(\mathbf{k}_\gamma \epsilon_\gamma \mathbf{p}_i \epsilon_i)}=
</math>
</math>
:::<math>
=\sum_{R,\lambda_R;\lambda_{b_1},\lambda_\omega,\lambda_\rho}
=\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
\langle \mathbf{q}_\pi 0 0; \mathbf{q}_\rho \lambda_\rho 0; \mathbf{q}_\omega \lambda_\omega 0; \mathbf{q}_{b1}\lambda_{b1} 0
</math>
</math>
:::::::::<math> \times
:::::::::<math> \times
\langle J_R \lambda_R \epsilon_R;s,t; \mathbf{p_f}, \lambda_f | W | \mathbf{p_i}, \lambda_i\rangle
\langle J_R \lambda_R \epsilon_R;s,t; \mathbf{p_f}, \epsilon_f | W | \mathbf{p_i}, \epsilon_i\rangle
</math>
</math>
To obtain the second line in the above equation, we factorized the T operator into two vertex factors U and W, and inserted between them a sum over a complete set of intermediate exchanges ''R''. Polarizations of all particles are represented by the reflectivity quantum numbers <math>\epsilon</math>. For the nucleon, the reflectivity is a complete description of its spin state. For reactions involving higher spin baryons, it would need to be supplemented by an additional <math>|m|</math> quantum number.
=== Proton states and individual decay amplitudes ===
=== Proton states and individual decay amplitudes ===
:<math>
:<math>
\frac{d^8\sigma}{d\Omega_{b1}\,d\Omega_\omega\,d\Omega_\rho\,d\Omega_\pi} \propto
\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'}
\sum_{\epsilon_\gamma \epsilon_\gamma' \epsilon_i \epsilon_f \epsilon_i' \epsilon_f'}
\rho_{\lambda_i \lambda_i'}
\rho_{\epsilon_\gamma \epsilon_\gamma'}
\rho_{\lambda_f \lambda_f'}
\rho_{\epsilon_i \epsilon_i'}
T_{(\mathbf{q}_\pi \mathbf{q}_\rho \mathbf{q}_\omega \mathbf{q}_{b1} \lambda_f)
\delta_{\epsilon_f \epsilon_f'}
(\epsilon_\gamma \epsilon_R t \lambda_i)}
T_{(\mathbf{q}_\pi \mathbf{q}_\rho \mathbf{q}_\omega \mathbf{q}_{b1} \mathbf{p}_f \epsilon_f)
T^*_{(\mathbf{q}_\pi \mathbf{q}_\rho \mathbf{q}_\omega \mathbf{q}_{b1} \lambda_f')
(\mathbf{k}_\gamma \epsilon_\gamma \mathbf{p}_i \epsilon_i)}
(\epsilon_\gamma \epsilon_R t \lambda_i')}
T^*_{(\mathbf{q}_\pi \mathbf{q}_\rho \mathbf{q}_\omega \mathbf{q}_{b1} \mathbf{p}_f \epsilon_f')
(\mathbf{k}_\gamma \epsilon_\gamma' \mathbf{p}_i \epsilon_i')}
</math>
</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
where density matrices <math>\rho</math> represent the initial state particles' spin states. The unpolarized target presents an initial state with both reflectivities 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>\rho_{\lambda_i \lambda_i'} = \frac{1}{2} \delta_{\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>
:<math>