1,004
edits
Changes
Jump to navigation
Jump to search
Error propagation in Amplitude Analysis (view source)
Revision as of 00:50, 22 November 2011
, 00:50, 22 November 2011Created page with 'The following is a review of error propagation needed in amplitude analysis. Consider a Monte-Carlo (MC) integral over the intensities of N detected events out of N<sub>gen</su…'
The following is a review of error propagation needed in amplitude analysis.
Consider a Monte-Carlo (MC) integral over the intensities of N detected events out of N<sub>gen</sub> generated.
<math>
I=\frac{1}{N_{gen}}\sum_i^N{
\sum_{\gamma,\delta}{\rho_{\gamma\delta}
\left| \sum_\alpha^n{
u_\alpha A_\alpha^{\gamma \delta}(x_i)
}
\right|^2
}
}=
\frac{1}{N_{gen}}\sum_i^N{
\sum_{\gamma,\delta}{\rho_{\gamma\delta}
\sum_{\alpha,\beta}^n{
u_\alpha u_\beta^* A_\alpha^{\gamma \delta}(x_i) A_\beta^{\gamma \delta *}(x_i)
}
}
}
</math>
where we take ''n'' coherent amplitudes and allow incoherent sums indexed by
γ, δ to allow for applications like spin-density matrices (ρ).
When amplitude analysis fits contain amplitudes with not free parameters, it is convenient to
rearrange the summations above, to pre-compute the sum over the intensities of the events:
<math>
I=\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]
}
}
</math>
storing the term in square brackets, a matrix indexed by α,β, for contractions with varying free production parameters ''u'' in the course of a fit.
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
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:
<math>
\sigma_{MC}^2=
\sum_i^N{ \sigma_i^2 \left| \frac{1}{N_{gen}}
\sum_{\gamma,\delta}{\rho_{\gamma\delta}
\sum_{\alpha,\beta}^n{
u_\alpha u_\beta^* A_\alpha^{\gamma \delta}(x_i) A_\beta^{\gamma \delta *}(x_i)
}
}
\right|^2
}
</math>
::<math>
= \sum_{\gamma,\delta,\gamma',\delta'}{ \rho_{\gamma\delta}\rho_{\gamma'\delta'}
\sum_{\alpha,\beta,\alpha',\beta'}^n{
u_\alpha u_\beta^* u_{\alpha'} u_{\beta'}^*
\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)
}
}
}
</math>
Consider a Monte-Carlo (MC) integral over the intensities of N detected events out of N<sub>gen</sub> generated.
<math>
I=\frac{1}{N_{gen}}\sum_i^N{
\sum_{\gamma,\delta}{\rho_{\gamma\delta}
\left| \sum_\alpha^n{
u_\alpha A_\alpha^{\gamma \delta}(x_i)
}
\right|^2
}
}=
\frac{1}{N_{gen}}\sum_i^N{
\sum_{\gamma,\delta}{\rho_{\gamma\delta}
\sum_{\alpha,\beta}^n{
u_\alpha u_\beta^* A_\alpha^{\gamma \delta}(x_i) A_\beta^{\gamma \delta *}(x_i)
}
}
}
</math>
where we take ''n'' coherent amplitudes and allow incoherent sums indexed by
γ, δ to allow for applications like spin-density matrices (ρ).
When amplitude analysis fits contain amplitudes with not free parameters, it is convenient to
rearrange the summations above, to pre-compute the sum over the intensities of the events:
<math>
I=\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]
}
}
</math>
storing the term in square brackets, a matrix indexed by α,β, for contractions with varying free production parameters ''u'' in the course of a fit.
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
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:
<math>
\sigma_{MC}^2=
\sum_i^N{ \sigma_i^2 \left| \frac{1}{N_{gen}}
\sum_{\gamma,\delta}{\rho_{\gamma\delta}
\sum_{\alpha,\beta}^n{
u_\alpha u_\beta^* A_\alpha^{\gamma \delta}(x_i) A_\beta^{\gamma \delta *}(x_i)
}
}
\right|^2
}
</math>
::<math>
= \sum_{\gamma,\delta,\gamma',\delta'}{ \rho_{\gamma\delta}\rho_{\gamma'\delta'}
\sum_{\alpha,\beta,\alpha',\beta'}^n{
u_\alpha u_\beta^* u_{\alpha'} u_{\beta'}^*
\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)
}
}
}
</math>