Worldline quantum field theory and the eikonal expansion: classical limits and proof of eikonal exponentiation

The eikonal expansion of quantum field theory can be mapped to a first-quantized worldline formalism in the classical limit, called worldline quantum field theory. Keeping the equivalence in mind, we can show that the 2-body gravitational eikonal phase exponentiates at all orders of the eikonal expansion, showing that the S-matrix for gravitational 2-into-2 scattering of scalar fields factorizes exactly into classical and quantum pieces at all loop orders. The validity of this exponentiation is crucial to the eikonal method, and worldline formalism gives a clear framework to study and demonstrate this exponentiation. Additionally, in the conventional eikonal expansion method, quantum contributions will need to be accounted for and subtracted out of the amplitude at higher loops in order to extract the classical contribution. However, wordline quantum field theory allows one to specifically target and pick out classical contributions at all orders by considering a specific subset of diagrams. We shall also discuss the connection between classical limits of theories and eikonal exponentiation in scalar quantum electrodynamics, scalar Yukawa, and Yang-Mills theories.