Variations on the Maiani-Testa approach and the inverse problem

Abstract

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 [1]. Revisiting the original result of ref. [1], 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, ρ(ω), convoluted with the Euclidean kernel, e−ωt, 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 Nπ 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.

Testing a new method for scattering in finite volume in the $$\phi ^4$$ theory

“…In this letter, we have investigated the BH method, proposed in Ref. [1], using a scalar theory on the lattice. We have indeed verified that it is a viable method to obtain the scattering length, and that it produces results that are compatible with Table 1 Values of a 0 , M 0 , M 1 and E 2 measured.…”

“…In Ref. [1], Bruno and Hansen derived a relation between the scattering length a 0 and the following combination of Euclidean four-point and two-point correlation functions at the two-particle threshold:…”

“…In Ref. [1], the authors propose a new strategy to extract scattering quantities. Henceforth, this will be referred to as the BH method.…”

Testing a new method for scattering in finite volume in the $ϕ^4$ theory

“…In this letter, we have investigated the BH method, proposed in Ref. [1], using a scalar theory on the lattice. We have indeed verified that it is a viable method to obtain the scattering length, and that it produces results that are compatible with those of the Lüscher method [14].…”

Scattering from generalised lattice $ϕ^4$ theory

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We investigate numerically different techniques to extract scattering amplitudes from the Euclidean Lattice 𝜙 4 theory with two fields, having different masses. We present an exploratory study of the recently proposed method by Bruno and Hansen for extracting the scattering length from a four-point function [1], and a study of the two-and three-particle quantization condition.

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