Variations on the Maiani-Testa approach and the inverse problem

We discuss a method to construct hadronic scattering and decay amplitudes from Euclidean correlators, by combining the approach of a regulated inverse Laplace transform with the work of Maiani and Testa. Revisiting the original result, we observe that the key observation, i.e. that only threshold scattering information can be extracted at large separations, can be understood by interpreting the correlator as a spectral function, $\rho(\omega)$, convoluted with the Euclidean kernel, $e^{- \omega t}$, which is sharply peaked at threshold. We therefore consider a modification in which a smooth step function, equal to one above a target energy, is inserted in the spectral decomposition. This can be achieved either through Backus-Gilbert-like methods or more directly using the variational approach. The result is a shifted resolution function, such that the large $t$ limit projects onto scattering or decay amplitudes above threshold. The utility of this method is highlighted through large $t$ expansions of both three- and four-point functions that include leading terms proportional to the real and imaginary parts (separately) of the target observable. This work also presents new results relevant for the un-modified correlator at threshold, including expressions for extracting the $N \pi$ scattering length from four-point functions and a new strategy to organize the large $t$ expansion that exhibits better convergence than the expansion in powers of $1/t$.