Spectral densities from Euclidean lattice correlators via the Mellin transform

Spectral densities connect correlation functions computed in quantum field theory to observables measured in experiments. For strongly interacting theories, their nonperturbative determinations from lattice simulations are therefore of primary importance. They entail the inverse Laplace transform of correlation functions calculated in Euclidean time. By making use of the Mellin transform, we derive explicit analytic formulas to define spectral densities from the time dependence of correlation functions, both in the continuum and on the lattice. The generalization to smeared spectral densities turns out to be straightforward. The formulas obtained here within the context of lattice field theory can easily be applied or extended to other areas of research.