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:

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

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_i=1. An integral over such events is then a weighted sum of such samples, having therefore a contribution to the variance:

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

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle = \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:

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \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^* J_{\alpha\beta} } } \right) \frac{\partial}{\partial u_l}\left( \sum_{\gamma',\delta'}{\rho_{\gamma'\delta'} \sum_{\alpha',\beta'}^n{ u_{\alpha'} u_{\beta'}^* J_{\alpha'\beta'} } } \right) } } The product of σ terms in the summation are the error matrix derived from the fit.