The QED four – photon amplitudes off-shell: Part 1

The present paper is the first in a series of four where we use the worldline formalism to obtain the QED four-photon amplitude completely off-shell. We present the result explicitly in terms of hypergeometric functions, and derivatives thereof, for both scalar and spinor QED. The formalism allows us to unify the scalar and spinor loop calculations, avoiding the usual breaking up of the amplitude into Feynman diagrams, and to achieve manifest transversality at the integrand level as well as UV finiteness term by term by an optimized version of the integration-by-parts procedure originally introduced by Bern and Kosower for gluon amplitudes. The full permutation symmetry is maintained throughout, and the amplitudes get projected naturally into the basis of five tensors introduced by Costantini et al. in 1971. Since in many applications of the “four-photon box” some of the photons can be taken in the low-energy limit, and the formalism makes it easy to integrate out any such leg, apart from the case of general kinematics (part 4) we also treat the special cases of one (part 3) or two (part 2) photons taken at low energy. In this first part of the series, we summarize the application of the worldline formalism to the N-photon amplitudes and its relation to Feynman diagrams, derive the optimized tensor-decomposed integrands of the four-photon amplitudes in scalar and spinor QED, and outline the computational strategy to be followed in parts 2 to 4. We also give an overview of the applications of the four-photon amplitudes, with an emphasis on processes that involve some off-shell photons. The case where all photons are taken at low energy (the “Euler-Heisenberg approximation”) is simple enough to be doable for arbitrary photon numbers, and we include it here for completeness.