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:

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

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

storing the term in square brackets, a matrix indexed by α,β, for contractions with varying free PWA parameters u in the course of a fit.

When considering the uncertainty on the overall integral, both the errors on the u parameters and those from the finite MC statistics will contribute. The part of the error on the intensity coming from the finite MC statistics is computed using the usual rules for error propagation.

A subtle point easy to miss at this stage is that the errors on the I_αβ are correlated because they are computed on the same MC sample. Therefore, while the I_αβ is rank 2 in the partial wave index, its covariance matrix is rank 4.

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

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

$-\left.{\frac {1}{N_{gen}^{3}}}\left(\sum _{i}^{N}{A_{\alpha }^{\gamma }(x_{i})A_{\beta }^{\delta *}(x_{i})}\right)\left(\sum _{i}^{N}{A_{\alpha '}^{\gamma '}(x_{i})A_{\beta '}^{\delta '*}(x_{i})}\right)\right]$

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 from the fit parameters u, it is convenient to write them explicitly in terms of their real and imaginary parts as u = a + ib.

$I=\sum _{\alpha \beta }^{n}{\left({\begin{array}{lr}a_{\alpha }&b_{\alpha }\end{array}}\right)\left({\begin{array}{lr}\Re (I_{\alpha \beta })&-\Im (I_{\alpha \beta })\\\Im (I_{\alpha \beta })&\Re (I_{\alpha \beta })\end{array}}\right)\left({\begin{array}{c}a_{\beta }\\b_{\beta }\end{array}}\right)}$

$=\sum _{\alpha \beta }^{2n}{a_{\alpha }a_{\beta }J_{\alpha \beta }}$

where the a coefficients stand for both the a and b terms introduced above, so that the sum over n complex parameters is expanded to a sum over 2n real ones. The matrix J represents the above 2nx2n matrix of the elements of I. The variance on the intensity sum is expressed in terms of the covariance matrix C_αβ among the fit parameters a_α as

$\sigma _{fit}^{2}=\sum _{\alpha ,\beta ,\alpha ',\beta '}^{2n}{\left(a_{\beta }a_{\alpha '}J_{\alpha \beta }J_{\alpha '\beta '}\right)C_{\alpha \beta '}}$

$=\sum _{\alpha ,\beta '}^{2n}{\left(\sum _{\beta }^{2n}{a_{\beta }J_{\alpha \beta }}\right)\left(\sum _{\alpha '}^{2n}{a_{\alpha '}J_{\alpha '\beta '}}\right)C_{\alpha \beta '}}$

$=\sum _{\alpha ,\alpha '}^{2n}{G_{\alpha }G_{\alpha '}C_{\alpha \alpha '}}$

where J and C are both symmetric matrices, and G is defined as

$G_{\alpha }=\sum _{\beta }^{2n}{a_{\beta }J_{\alpha \beta }}$

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

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