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

This is the second one of a series four papers devoted to first calculation scalar and spinor QED four-photon amplitudes completely off-shell. We use worldline formalism which provides gauge-invariant decomposition for these as well compact integral representations. It also makes it straightforward integrate out any given photon leg in low-energy limit, present sequel we do this with two photons. For special case where unrestricted momenta are equal opposite information on contained constant-field vacuum polarisation tensors, check our results. Although finite, possible higher-loop building blocks evaluate all integrals dimensional regularisation. As an example, them construct two-loop tensors approximation, rederive from those $\beta$-function coefficients analyse their anatomy respect decomposition. application external-field problem, provide streamlined Delbr\"uck scattering limit. All calculations done parallel QED.