1,004
edits
Changes
Jump to navigation
Jump to search
Line 189:
Line 189: − + − + + + + + + + + + + + + Line 207:
Line 218: + + + − + − + − +
Amplitudes for the Exotic b1π Decay (view source)
Revision as of 23:41, 13 August 2011
, 23:41, 13 August 2011→Assembly of the full amplitude
== Assembly of the full amplitude ==
== Assembly of the full amplitude ==
Putting together the amplitudes discussed above, we arrive at the complete amplitude of photo-production of resonance X and decay to the given final state.
Putting together the amplitudes discussed above, we arrive at the complete angular-dependent amplitude of photo-production of resonance X and decay to the given final state. The mass-dependent component of the amplitude is given by the Breit-Wigner form:
:<math>
:<math>
\sum_{L_X} A_{L_X \epsilon_\gamma \epsilon_R}^{J_X}=
BW_L(m)=\frac{m_0 \Gamma_L(m)}{m_0^2-m^2-im_0\Gamma_L(m)}
</math>
where,
:<math>
\Gamma_L(m)=\Gamma_0 \frac{m_0}{m} \frac{q}{q_0} \frac{F_L(q)}{F_L(q_0)}
</math>
where ''q'' is the breakup momentum of the daughter particles and ''q<sub>0</sub>'' is the same, evaluated at ''m<sub>0</sub>''.
The total expression becomes:
:<math>
\sum_{L_X \epsilon_R} A_{L_X \epsilon_\gamma \epsilon_R}^{J_X}=
\sum_{\lambda_{b_1},\lambda_\omega,\lambda_\rho}
\sum_{\lambda_{b_1},\lambda_\omega,\lambda_\rho}
\langle \Omega_X 0 \lambda_{b_1} | U_X |
\langle \Omega_X 0 \lambda_{b_1} | U_X |
\epsilon_\gamma e^{i\alpha}\left|1\; +1\right\rangle \right) ;
\epsilon_\gamma e^{i\alpha}\left|1\; +1\right\rangle \right) ;
J_R \lambda_R \epsilon_R ; t, s; \Omega_0 \rangle
J_R \lambda_R \epsilon_R ; t, s; \Omega_0 \rangle
</math>
::<math>
BW_{L_X}^X BW_{L_{b_1}}^{b_1} BW_{L_\omega}^\rho BW_{L_\rho}^\rho
</math>
</math>
Note that we leave the sum over <math>L_X</math> outside the amplitude of interest. This is convenient in partial wave analysis so that the hypothesized L-states can be listed and summed explicitly.
Note that we leave the sum over <math>L_X</math> outside the amplitude of interest. This is convenient in partial wave analysis so that the hypothesized L-states can be listed and summed explicitly. In mass-independent fits, the Breit-Wigner for the resonance is replaced with strength factors that are parameters of the fit.
The intensity would then require an incoherent summation of amplitudes with laboratory x and y polarization.
The intensity would then require an incoherent summation of amplitudes with laboratory x and y polarization.
:<math>I=
:<math>I=
\frac{1+f}{2}\left|\sum_{L_X} A_{L_X\;-1\;\epsilon_R}^{J_X}\right|^2 +
\frac{1+f}{2}\left|\sum_{L_X \epsilon_R} A_{L_X\;-1\;\epsilon_R}^{J_X}\right|^2 +
\frac{1-f}{2}\left|\sum_{L_X} A_{L_X\;+1\;\epsilon_R}^{J_X}\right|^2
\frac{1-f}{2}\left|\sum_{L_X \epsilon_R} A_{L_X\;+1\;\epsilon_R}^{J_X}\right|^2
</math>
</math>
where f is the polarization fraction varying from 1, 100% x-polarized, to 0, unpolarized.
where f is the polarization fraction varying from 1, 100% x-polarized, to 0, unpolarized.