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Error propagation in Amplitude Analysis (view source)
Revision as of 18:30, 18 December 2011
, 18:30, 18 December 2011no edit summary
The following is a review of error propagation needed to compute the errors on the normalization integrals and the intensity sum that is based on them. Consider the estimator for the intensity for a given PWA solution, based on a sum over a Monte Carlo sample with N<sub>gen</sub> phase space events generated and N reconstructed and passing all cuts. <math>
The following is a review of error propagation needed to compute the errors on the normalization integrals and the intensity sum that is based on them. Consider the estimator for the intensity for a given PWA solution, based on a sum over a Monte Carlo sample with N<sub>gen</sub> phase space events generated and N reconstructed and passing all cuts.
<math>
I=\frac{1}{N_{gen}}\sum_i^N{
I=\frac{1}{N_{gen}}\sum_i^N{
\sum_{\gamma,\delta}{\rho_{\gamma\delta}
\sum_{\gamma,\delta}{\rho_{\gamma\delta}
\right|^2
\right|^2
}
}
-
\frac{1}{N_{gen}}
\left|
\sum_{\alpha,\beta}^n{
u_\alpha u_\beta^* I_{\alpha\beta}
}
\right|^2
</math>
</math>
:<math>
:<math>
= \sum_{\alpha,\beta,\alpha',\beta'}^n{
= \sum_{\alpha,\beta,\alpha',\beta'}^n{
u_\alpha u_\beta^* u_{\alpha'}^* u_{\beta'}
u_\alpha u_\beta^* u_{\alpha'}^* u_{\beta'}
\sum_{\gamma,\delta,\gamma',\delta'}{ \rho_{\gamma\delta}\rho_{\gamma'\delta'} \left[
\frac{1}{N_{gen}^2} \sum_i^N{
\frac{1}{N_{gen}^2} \sum_i^N{
A_\alpha^{\gamma}(x_i) A_\beta^{\delta *}(x_i)
A_\alpha^{\gamma}(x_i) A_\beta^{\delta *}(x_i)
A_{\alpha'}^{\gamma'*}(x_i) A_{\beta'}^{\delta'}(x_i)
A_{\alpha'}^{\gamma'*}(x_i) A_{\beta'}^{\delta'}(x_i)
}}
}
\right.
}
}
</math>
::::<math>
-
\left.
\frac{1}{N_{gen}^3} \left(
\sum_i^N{
A_\alpha^{\gamma}(x_i) A_\beta^{\delta *}(x_i)
}\right)
\left(\sum_i^N{
A_{\alpha'}^{\gamma'}(x_i) A_{\beta'}^{\delta' *}(x_i)
}\right)
\right]
\right]
</math>
</math>