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Error propagation in Amplitude Analysis (view source)
Revision as of 01:42, 22 November 2011
, 01:42, 22 November 2011no edit summary
σ<sub>i</sub>=1. An integral over such events is then a weighted sum of such samples,
σ<sub>i</sub>=1. An integral over such events is then a weighted sum of such samples,
having therefore a contribution to the variance:
having therefore a contribution to the variance:
<math>
<math>
\sigma_{MC}^2=
\sigma_{MC}^2=
\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'}^*
\frac{1}{N_{gen}^2} \sum_i^N{
\left[ \frac{1}{N_{gen}^2} \sum_i^N{
A_\alpha^{\gamma \delta}(x_i) A_\beta^{\gamma \delta *}(x_i)
A_\alpha^{\gamma \delta}(x_i) A_\beta^{\gamma \delta *}(x_i)
A_{\alpha'}^{\gamma' \delta'}(x_i) A_{\beta'}^{\gamma' \delta' *}(x_i)
A_{\alpha'}^{\gamma' \delta'}(x_i) A_{\beta'}^{\gamma' \delta' *}(x_i)
}
}
\right]
}
}
}
}
</math>
The relevant piece to pre-compute over the event set for error calculation is shown in brackets.
Turning our attention now to the contribution to error on the production parameters ''u'':
<math>
\sigma_{fit}^2=
\sum_{k,l}^n{ \sigma_{u_k}\sigma_{u_l}
\frac{\partial}{\partial u_k}\left(
\sum_{\gamma,\delta}{\rho_{\gamma\delta}
\sum_{\alpha,\beta}^n{
u_\alpha u_\beta^*
\left[ \frac{1}{N_{gen}}\sum_i^N{
A_\alpha^{\gamma \delta}(x_i) A_\beta^{\gamma \delta *}(x_i)
}
\right]
}
}
\right)
\frac{\partial}{\partial u_l}
\sum_{\gamma,\delta}{\rho_{\gamma\delta}
\sum_{\alpha,\beta}^n{
u_\alpha u_\beta^*
\left[ \frac{1}{N_{gen}}\sum_i^N{
A_\alpha^{\gamma \delta}(x_i) A_\beta^{\gamma \delta *}(x_i)
}
\right]
}
}
\right)
}
</math>
</math>