Error propagation in Amplitude Analysis

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_gen generated.

$I={\frac {1}{N_{gen}}}\sum _{i}^{N}{\sum _{\gamma ,\delta }{\rho _{\gamma \delta }\left|\sum _{\alpha }^{n}{u_{\alpha }A_{\alpha }^{\gamma \delta }(x_{i})}\right|^{2}}}={\frac {1}{N_{gen}}}\sum _{i}^{N}{\sum _{\gamma ,\delta }{\rho _{\gamma \delta }\sum _{\alpha ,\beta }^{n}{u_{\alpha }u_{\beta }^{*}A_{\alpha }^{\gamma \delta }(x_{i})A_{\beta }^{\gamma \delta *}(x_{i})}}}$

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:

$I=\sum _{\gamma ,\delta }{\rho _{\gamma \delta }\sum _{\alpha ,\beta }^{n}{u_{\alpha }u_{\beta }^{*}\left[{\frac {1}{N_{gen}}}\sum _{i}^{N}{A_{\alpha }^{\gamma \delta }(x_{i})A_{\beta }^{\gamma \delta *}(x_{i})}\right]}}=\sum _{\gamma ,\delta }{\rho _{\gamma \delta }\sum _{\alpha ,\beta }^{n}{u_{\alpha }u_{\beta }^{*}J_{\alpha \beta }^{\gamma \delta }}}$

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 in a process of independent events, having therefore a count uncertainty of σ_i=1. An integral over such events is then a weighted sum of such samples, having resulting in a contribution to the variance:

$\sigma _{MC}^{2}=\sum _{i}^{N}{\sigma _{i}^{2}\left|{\frac {1}{N_{gen}}}\sum _{\gamma ,\delta }{\rho _{\gamma \delta }\sum _{\alpha ,\beta }^{n}{u_{\alpha }u_{\beta }^{*}A_{\alpha }^{\gamma \delta }(x_{i})A_{\beta }^{\gamma \delta *}(x_{i})}}\right|^{2}}$

$=\sum _{\gamma ,\delta ,\gamma ',\delta '}{\rho _{\gamma \delta }\rho _{\gamma '\delta '}\sum _{\alpha ,\beta ,\alpha ',\beta '}^{n}{u_{\alpha }u_{\beta }^{*}u_{\alpha '}u_{\beta '}^{*}\left[{\frac {1}{N_{gen}^{2}}}\sum _{i}^{N}{A_{\alpha }^{\gamma \delta }(x_{i})A_{\beta }^{\gamma \delta *}(x_{i})A_{\alpha '}^{\gamma '\delta '}(x_{i})A_{\beta '}^{\gamma '\delta '*}(x_{i})}\right]}}$

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:

$\sigma _{fit}^{2}=\left|\sum _{k}^{n}{\sigma _{u_{k}}{\frac {\partial }{\partial u_{k}}}\left(\sum _{\gamma ,\delta }{\rho _{\gamma \delta }\sum _{\alpha ,\beta }^{n}{u_{\alpha }u_{\beta }^{*}J_{\alpha \beta }}}\right)}\right|^{2}$

$=\left(\sum _{k}^{n}{\sigma _{u_{k}}\sum _{\gamma ,\delta }{\rho _{\gamma \delta }\sum _{\alpha ,\beta }^{n}{\delta _{k\alpha }u_{\beta }^{*}J_{\alpha \beta }^{\gamma \delta }}}}\right)\left(\sum _{k'}^{n}{\sigma _{u_{k'}}^{*}\sum _{\gamma ',\delta '}{\rho _{\gamma '\delta '}\sum _{\alpha ',\beta '}^{n}{\delta _{k'\alpha '}u_{\beta '}J_{\alpha '\beta '}^{\gamma '\delta '*}}}}\right)$

$=\sum _{\gamma ,\delta ,\gamma ',\delta '}{\rho _{\gamma \delta }\rho _{\gamma '\delta '}\sum _{\alpha ,\beta ,\alpha ',\beta '}^{n}{\left(\sigma _{u_{\alpha }}\sigma _{u_{\alpha '}}^{*}\right)\left(u_{\beta }^{*}J_{\alpha \beta }^{\gamma \delta }\right)\left(u_{\beta '}J_{\alpha '\beta '}^{\gamma '\delta '*}\right)}}$

$=\sum _{\gamma ,\delta ,\gamma ',\delta '}{\rho _{\gamma \delta }\rho _{\gamma '\delta '}\sum _{\alpha ,\alpha '}^{n}{\left(\sigma _{u_{\alpha }}\sigma _{u_{\alpha '}}^{*}\right)G_{\alpha }^{\gamma \delta *}G_{\alpha '}^{\gamma \delta }}}$

The product of σ terms in the summation is represented by the error matrix derived from the fit. G was defined as

$G_{\alpha }^{\gamma \delta }=\sum _{\beta }{u_{\beta }J_{\alpha \beta }^{\gamma \delta }}$

The overall uncertainty in the integral I defined in the beginning comes out to:

$\sigma _{I}={\sqrt {\sigma _{MC}^{2}+\sigma _{fit}^{2}}}$