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 based on them. Consider the estimator for the intensity for a given PWA solution, based on a sum over a Monte Carlo sample with N_gen phase space events generated and N reconstructed and passing all cuts. $I={\frac {1}{N_{gen}}}\sum _{i}^{N}{\sum _{\gamma ,\delta }{\rho _{\gamma \delta }\sum _{\alpha ,\beta }^{n}{u_{\alpha }u_{\beta }^{*}A_{\alpha }^{\gamma }(x_{i})A_{\beta }^{\delta *}(x_{i})}}}$

where the PWA sum is over n coherent amplitudes, and indices γ, δ represent the spins of the external particles (incoming photon, incoming and outgoing nucleon), with ρ representing their collective spin-density matrix. When amplitude analysis fits contain amplitudes with no free parameters, it is convenient to rearrange the summations above, to pre-compute the sum over the intensities of the events:

Failed to parse (syntax error): {\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}(x_i) A_\beta^{\delta *}(x_i) } \right] } } = \sum_{\alpha,\beta}^n{ u_\alpha u_\beta^* \left[ \sum_{\gamma,\delta}{\rho_{\gamma\delta} \frac{1}{N_{gen}}\sum_i^N{ A_\alpha^{\gamma}(x_i) A_\beta^{\delta *}(x_i) \right] } } } = \sum_{\alpha,\beta}^n{ u_\alpha u_\beta^* I_{\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:

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

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

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:

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