Changes

</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.  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

:<math>

:<math>

T_{fi} = \sum_{R,\lambda_R\lambda_{b_1},\lambda_\omega,\lambda_\rho}

T_{(f)(i)} =

=\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

| U | \epsilon_\gamma; J_R \lambda_R \epsilon_R;s,t \rangle

| U | \epsilon_\gamma; J_R \lambda_R \epsilon_R;s,t \rangle

\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}, \lambda_f | W | \mathbf{p_i}, \lambda_i\rangle

</math>

</math>

The aggregate decay matrix element can be further broken up into a product of individual decay amplitudes,

The aggregate decay matrix element can be further broken up into a product of individual decay amplitudes,

:<math>

:<math>

</math>

</math>

Expressions for the angular dependence of the matrix elements of <math>U_X</math>, <math> U_{b1}</math>, <math> U_\omega</math>, and <math> U_\rho</math> have already been written down above, in terms of the unknown mass-dependent factors ''a'', ''b'', ''c'', and ''f''.  The mass dependence of the ''a'' factor can be written in terms of a standard relativistic Breit-Wigner resonance lineshape as follows, although often its mass dependence is determined empirically by binning in the mass of ''X'', and fitting each bin independently.  In a global mass-dependent fit, the central mass and width of ''X'' are free parameters in the fit.  The remaining factors ''b'', ''c'', and ''f'' are assigned standard resonance forms, with their central mass and width parameters fixed to agree with the established values for these resonances.

Expressions for the angular dependence of the matrix elements of <math>U_X</math>, <math> U_{b1}</math>, <math> U_\omega</math>, and <math> U_\rho</math> have already been written down above, in terms of the unknown mass-dependent factors ''a'', ''b'', ''c'', and ''f''.  The mass dependence of the ''a'' factor can be written in terms of a standard relativistic Breit-Wigner resonance lineshape as follows, although often its mass dependence is determined empirically by binning in the mass of ''X'', and fitting each bin independently.  In a global mass-dependent fit, the central mass and width of ''X'' are free parameters in the fit.  The remaining factors ''b'', ''c'', and ''f'' are assigned standard resonance forms, with their central mass, width and partial wave matrix elements fixed to agree with the established values for these resonances.

:<math>

:<math>

a^J_{LS}(m_X) = a^J_{LS} BW_L(m_X;m^0_X,\Gamma^0_X)

a^J_{LS}(m_X) = a^J_{LS} BW_L(m_X;m^0_X,\Gamma^0_X)

=== Summing over polarizations ===

=== Summing over polarizations ===

The intensity would then require an incoherent summation of amplitudes with laboratory x and y polarization, and over the unmeasured helicities of the incident and final-state nucleons.

 

:<math>I=

Since the polarization of the initial state photon differs from event to event, the weighted sum over polarization states is performed incoherently.  

\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 \epsilon_R} A_{L_X\;+1\;\epsilon_R}^{J_X}\right|^2

:<math>

\frac{d^8\sigma}{d\Omega_{b1}\,d\Omega_\omega\,d\Omega_\rho\,d\Omega_\pi} =

where f is the polarization fraction varying from 1, 100% x-polarized, to 0, unpolarized.

\frac{1+g}{2}\frac{d^8\sigma_{(\epsilon_\gamma=-1)}}

\sum_{L_X \epsilon_R} A_{L_X \epsilon_\gamma \epsilon_R}^{J_X}=

{d\Omega_{b1}\,d\Omega_\omega\,d\Omega_\rho\,d\Omega_\pi} +

\frac{1-g}{2}\frac{d^8\sigma_{(\epsilon_\gamma=+1)}}

BW_{L_X}^X BW_{L_{b_1}}^{b_1} BW_{L_\omega}^\omega BW_{L_\rho}^\rho e^{-kt}

{d\Omega_{b1}\,d\Omega_\omega\,d\Omega_\rho\,d\Omega_\pi}

</math>

</math>