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

The QED four-photon amplitude has been well-studied by many authors, and on-shell is treated in many textbooks. However, a calculation with all four photons off-shell is presently still lacking, despite of the fact that this amplitude appears off-shell as a subprocess in many different contexts, in vacuum as well as with some photons connecting to external fields. The present paper is the first in a series of four where we use the worldline formalism to obtain this amplitude 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, to avoid the usual breaking up of the amplitude into three inequivalent Feynman diagrams, and to achieve manifest transversality as well as UV finiteness at the integrand level 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 relationto Feynman diagrams, derive the optimized tensor-decomposed integrands of the four-photon amplitudes in scalar and spinor QED, and outline the computational strategy tobe 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 naturally involve some off-shell photons, either because external fields are involved or we use the amplitude as a building block for higher-order process. 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