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Line 231: − <math>\rho_{\lambda_i \lambda_i'} = \frac{1}{2} \delta_{\lambda_i \lambda_i'}</math>. + + + + + − 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, − + − +
Amplitudes for the Exotic b1π Decay (view source)
Revision as of 17:06, 19 August 2011
, 17:06, 19 August 2011→Assembly of the full amplitude
</math>
</math>
:::<math>
:::<math>
=\sum_{R,\lambda_R;\lambda_{b_1},\lambda_\omega,\lambda_\rho}
=\sum_{R,\lambda_R,\epsilon_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
| U | \epsilon_\gamma; J_R \lambda_R \epsilon_R;s,t \rangle
| UV | \epsilon_\gamma; J_R \lambda_R \epsilon_R;s,t \rangle
</math>
</math>
:::::::::<math> \times
:::::::::<math> \times
</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.
To obtain the second line in the above equation, we factorized the T operator into two vertex factors ''UV'' and ''W'', and inserted between them a sum over a complete set of intermediate exchanges ''R''. The upper vertex operator has been written as ''UV'' in anticipation of its further factorization into the primary resonance production operator ''V'' and its decay operator ''U''. 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.
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
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'} = \frac{1}{2} \delta_{\lambda_i \lambda_i'}</math>.
In analogy to the reflectivity conservation relation shown above for ''V'' vertex, there is a similar relation for the ''W'' vertex:
:<math>\epsilon_R=\epsilon_i \epsilon_f</math>
Identification of <math>\epsilon_i</math> with <math>\epsilon_i'</math> and <math>\epsilon_f</math> with <math>\epsilon_f'</math> implies that only terms with <math>\epsilon_R=\epsilon_R'</math> survive in the sum over exchange quantum numbers.
:<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
| U | \epsilon_\gamma; J_R \lambda_R \epsilon_R; s,t \rangle
| UV | \epsilon_\gamma; J_R \lambda_R \epsilon_R; s,t \rangle
</math>
</math>
::<math>=\sum_{\lambda_R,\lambda_{b_1},\lambda_\omega,\lambda_\rho,m_X}
::<math>=\sum_{R,\lambda_R,\epsilon_R,\lambda_{b_1},\lambda_\omega,\lambda_\rho,X,M_X,\epsilon_X}
\langle \mathbf{q}_{b1} \lambda_{b_1} 0| U_X | J_X M_X \epsilon_X\rangle
\langle \mathbf{q}_{b1} \lambda_{b_1} 0| U_X | J_X M_X \epsilon_X\rangle
\langle J_X M_X \epsilon_X | V |
\langle J_X M_X \epsilon_X | V |