Complex $q$ parameters for helium $L=0,1,2$ autoionizing levels

We recently reported\footnote{B.A. deHarak, K. Bartschat, and N.L.S. Martin, {\em Phys. Rev. Lett.}, {\bf 100}, 063201 (2008).} out-of-plane \ee\ experiments on He autoionization. The data were presented as angular distributions of ejected electrons from the three autoionizing levels $^1S_0$, $^1D_2$, and $^1P_1$ and exhibited two well known features, the binary and recoil peaks. It was found that the recoil peak (relative to the binary peak) could be accurately reproduced by a second order distorted wave Born calculation using the $R$-matrix with pseudo-states approach, but not by the equivalent {\it first order} calculation, which underestimated the size of the recoil peak. It was also found that a plane wave Born approximation calculation could reproduce the results, but only if anomalously large values of Fano $q$-parameters were assumed. We will present an analysis of the first and second order calculations in terms of Fano $q$ parameters. We find that for the first order calculations the $q$ parameters are essentially real, but for the second order calculations they are complex, quantities. The $^1D_2$ parameters are particularly striking in this respect.