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Line 79: − + − + − + − + + + + + + + + + + + + + + + + + + + + + + + + − \right) + − \frac{\partial}{\partial u_l}\left(+ − \sum_{\gamma',\delta'}{\rho_{\gamma'\delta'}+ − + − + + + − \right) − } − + + + + + + + + + + + +
Error propagation in Amplitude Analysis (view source)
Revision as of 03:21, 22 November 2011
, 03:21, 22 November 2011no edit summary
= \sum_{\gamma,\delta}{\rho_{\gamma\delta}
= \sum_{\gamma,\delta}{\rho_{\gamma\delta}
\sum_{\alpha,\beta}^n{
\sum_{\alpha,\beta}^n{
u_\alpha u_\beta^* J_{\alpha \beta}
u_\alpha u_\beta^* J^{\gamma\delta}_{\alpha \beta}
}
}
}
}
When considering the uncertainty on the overall integral, both the errors on ''u'' parameters
When considering the uncertainty on the overall integral, both the errors on ''u'' parameters
and those from the finite MC set of events will contribute. A single detected event (i) can be viewed as one sample of a Poisson process, having therefore an uncertainty of
and those from the finite MC set of events will contribute. A single detected event (i) can be viewed as one sample in a process of independent events, having therefore a count uncertainty of
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 resulting in a contribution to the variance:
<math>
<math>
<math>
<math>
\sigma_{fit}^2=
\sigma_{fit}^2=
\sum_{k,l}^n{ \sigma_{u_k}\sigma_{u_l}
\left| \sum_k^n{ \sigma_{u_k}
\frac{\partial}{\partial u_k}\left(
\frac{\partial}{\partial u_k}\left(
\sum_{\gamma,\delta}{\rho_{\gamma\delta}
\sum_{\gamma,\delta}{ \rho_{\gamma\delta}
\sum_{\alpha,\beta}^n{
\sum_{\alpha,\beta}^n{u_\alpha u_\beta^* J_{\alpha\beta}}
u_\alpha u_\beta^* J_{\alpha\beta}
}
\right)
}
\right|^2
</math>
::<math>
=\left( \sum_k^n{ \sigma_{u_k}
\sum_{\gamma,\delta}{ \rho_{\gamma\delta}
\sum_{\alpha,\beta}^n{\delta_{k\alpha} u_\beta^* J^{\gamma\delta}_{\alpha\beta}}
}
}\right)
\left( \sum_{k'}^n{ \sigma^*_{u_{k'}}
\sum_{\gamma',\delta'}{ \rho_{\gamma'\delta'}
\sum_{\alpha',\beta'}^n{\delta_{k'\alpha'} u_{\beta'} J^{\gamma'\delta'*}_{\alpha'\beta'}}
}
}\right)
</math>
::<math>
= \sum_{\gamma,\delta,\gamma',\delta'}{
\rho_{\gamma\delta} \rho_{\gamma'\delta'}
\sum_{\alpha,\beta,\alpha',\beta'}^n{
\left(\sigma_{u_\alpha}\sigma^*_{u_{\alpha'}}\right)
\left(u_\beta^* J^{\gamma\delta}_{\alpha\beta}\right)
\left(u_{\beta'} J^{\gamma'\delta'*}_{\alpha'\beta'} \right)
}
}
}
}
</math>
::<math>
= \sum_{\gamma,\delta,\gamma',\delta'}{
\sum_{\alpha',\beta'}^n{
\rho_{\gamma\delta} \rho_{\gamma'\delta'}
u_{\alpha'} u_{\beta'}^* J_{\alpha'\beta'}
\sum_{\alpha,\alpha'}^n{
\left(\sigma_{u_\alpha}\sigma^*_{u_{\alpha'}}\right)
G^{\gamma\delta*}_\alpha G_{\alpha'}^{\gamma\delta}
}
}
}
}
</math>
</math>
The product of σ terms in the summation are the error matrix derived from the fit.
The product of σ terms in the summation is represented by the error matrix derived from the fit. ''G'' was defined as
<math>
G_\alpha^{\gamma\delta}=\sum_\beta{ u_\beta J_{\alpha\beta}^{\gamma\delta}}
</math>
The overall uncertainty in the integral ''I'' defined in the beginning comes out to:
<math>
\sigma_I=\sqrt{\sigma^2_{MC} + \sigma^2_{fit}}
</math>