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Line 181: − + − :<math> − 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^2_L(q)}{F^2_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>''. − − Line 240:
Line 230: − + + + + + + + + + + − 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. An exponential dependence of ''t'' is inserted with a coefficient that can be deduced from fits in separate ''t'' bins.+
Amplitudes for the Exotic b1π Decay (view source)
Revision as of 23:09, 15 August 2011
, 23:09, 15 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 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:
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 final expression for the measured cross section becomes:
The final expression for the measured cross section becomes:
:<math>
:<math>
\frac{d^8\sigma}{d\Omega_{b1}\,d\Omega_\omega\,d\Omega_\rho\,d\Omega_\pi} =
\frac{d^8\sigma}{d\Omega_{b1}\,d\Omega_\omega\,d\Omega_\rho\,d\Omega_\pi} =
</math>
</math>
The factor $f$ is a constant and does not carry mass dependence because the mass of the pion is fixed.
The di-pion system in the decay of the <math>\omega</math> is non-resonant, so its width is entered as infinity. The mass-dependent terms in the expressions above are given by the explicitly unitary Breit-Wigner form:
:<math>
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^2_L(q)}{F^2_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>''.
An exponential dependence of ''t'' is inserted with a coefficient that can be deduced from fits in separate ''t'' bins.
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.