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Line 225:
Line 225: − + − − − − \langle \mathbf{q} \lambda_1 \lambda_2| U | J M \rangle = − \sum_L − \langle \Omega \lambda_1 \lambda_2| U | J M \rangle BW(q) + + + + + + + + + +
Amplitudes for the Exotic b1π Decay (view source)
Revision as of 23:02, 15 August 2011
, 23:02, 15 August 2011→Assembly of the full amplitude
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.
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.
:<math>
:<math>
a^J_{LS}(m_X) =
a^J_{LS}(m_X) = a^J_{LS} BW_L(m_X,\Gamma^0_X)
</math>
</math>
:<math>
b^1_{L1}(m_{b1}) = b^1_{L1} BW_L(m_{b1},\Gamma^0_{b1})
</math>
:<math>
c^1_{LJ}(m_{\omega}) = c^1_{LJ} BW_L(m_\omega,\Gamma^0_\omega)
</math>
:<math>
f^L_{L0}(m_{\omega}) = f^L_{L0} BW_L(m_\rho,\infty)
</math>
The factor $f$ is a constant and does not carry mass dependence because the mass of the pion is fixed.
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.
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.