Changes

\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>

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

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

\langle \mathbf{q}_\pi 0 0 | U_\rho | J_\rho , \lambda_\rho \rangle

\langle \mathbf{q}_\pi 0 0 | U_\rho | J_\rho , \lambda_\rho \rangle

</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, width and partial wave matrix elements 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.