Non-perturbative approach to high-energy physics I know that main numerical approach to modeling high-energy physics events are Monte-Carlo event generators. But they are using perturbative description of collision and decay processes of particles.
On other hand we have Lattice QCD - a fully non-perturbative approach to quantum chromodynamics at low energies. It was very successful to investigating confinement and calculating masses of hadrons.
But my question is following - can we apply lattice techniques to simulate high-energy events from the first principles?
E.g Proton-proton collision.
 A: In the current day calculations, Lattice-QCD usually refers to calculations in Euclidean time, after wick rotation. The reason for this is that it transforms the usual complex measure (that has lots of oscillations), to a well-defined probability distribution:
$$\text{exp}\left(i\int d^4 x F_{\mu \nu}F^{\mu \nu}\right) \mapsto  \text{exp}\left(-\int d^4 x F^E_{\mu \nu}F^E_{\mu \nu}\right)  $$
where I write $F^E$ just to remind us these are in the Euclidean picture, the $\mu,\nu$ indices are Euclidean signature now. This is done because the sign problem prevents efficient simulation of complex measures, and a completely positive measure has many effective Monte Carlo methods to simulate it such as HMC.
Because of this, some quantities such as masses of hadrons or matrix elements are relatively straight-forward to extract from Lattice-QCD calculations, but some quantities such as those arising from real-time scattering are more tricky. It's still possible through methods such as Luscher's finite-volume formalism, it's a fairly 'indirect' extraction of physical quantities though (somehow, the energy states of multi-particle states in finite volume are related to the infinite volume scattering matrix elements). In practice, people do investigations of meson-scattering on the lattice, I believe a hot topic right now is three-body interactions as well.
Ignoring practicality however, in principle there's nothing stopping us from doing real-time lattice-QCD calculations, it's just really computationally expensive. People work on using Machine Learning or Quantum Computers to try and speed up this kind of calculation.
