Error propagation in Amplitude Analysis

The following is a review of error propagation needed to compute the errors on the normalization integrals and the intensity sum that is formed based on them. Consider a Monte-Carlo (MC) integral over the intensities of N detected events out of N_gen generated.

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 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:

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 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^{\gamma\delta}_{\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 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:

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_{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:

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

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

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{ \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) } } }

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,\alpha'}^n{ \left(\sigma_{u_\alpha}\sigma^*_{u_{\alpha'}}\right) G^{\gamma\delta*}_\alpha 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

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 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:

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