Euclidean to Minkowski Bethe–Salpeter amplitude and observables

We propose a method to reconstruct the Bethe–Salpeter amplitude in Minkowski space given the Euclidean Bethe–Salpeter amplitude – or alternatively the light-front wave function – as input. The method is based on the numerical inversion of the Nakanishi integral representation and computing the corresponding weight function. This inversion procedure is, in general, rather unstable, and we propose several ways to considerably reduce the instabilities. In terms of the Nakanishi weight function, one can easily compute the BS amplitude, the LF wave function and the electromagnetic form factor. The latter ones are very stable in spite of residual instabilities in the weight function. This procedure allows both, to continue the Euclidean BS solution in the Minkowski space and to obtain a BS amplitude from a LF wave function.