This paper shows a new way to compute hadron form factors at complex values of momentum directly from lattice data. Form factors are simple functions that describe how particles like pions respond to probes. The authors demonstrate that one can get those functions at complex momentum transfer by applying a bilateral Laplace transform — in practice a Fourier transform with a complex momentum — to position-space Euclidean correlators. Crucially, this avoids doing a separate analytic continuation of the data or solving a difficult inverse problem.

At a high level the method uses the fact that Euclidean correlators from lattice Quantum Chromodynamics (QCD) fall off exponentially in space and time. Multiplying those correlators by oscillating and growing complex phases and then summing (the complex Fourier or Laplace transform) converges inside a definite region because the correlator decay controls the growth. By isolating the long-time plateau that corresponds to the ground-state contribution, the transformed correlator gives the usual form factor analytically continued to complex momentum transfer.

Focusing on the pion form factor, the authors show the construction reaches an extended analytic domain. Concretely, it gives access to all complex values of Q^2 (the invariant momentum transfer) whose real part is larger than −4 m_π^2, where m_π is the pion mass. That includes real timelike Q^2 values below the first particle-production threshold. The authors point out possible uses such as improving determinations of the charge radius and providing denser input for fits that use conformal mapping of form-factor data.

The paper discusses how to make the idea practical in realistic finite-volume lattice calculations. They present several finite-volume estimators: direct truncated Fourier sums, fits of the correlator in position space with extrapolation, reconstruction of part of the spectrum to extend the correlator, and a novel idea of applying a spatial chemical potential implemented as a complex (imaginary) twist in boundary conditions. Each option trades off ease of use and systematic uncertainty.