The analytic structure of amplitudes on backgrounds from gauge invariance and the infra-red

Anton Ilderton, Alexander J. MacLeod
2020 Journal of High Energy Physics  
Gauge invariance and soft limits can be enough to determine the analytic structure of scattering amplitudes in certain theories. This prompts the question of how gauge invariance is connected to analytic structure in more general theories. Here we focus on QED in background plane waves. We show that imposing gauge invariance introduces new virtuality poles into internal momenta on which amplitudes factorise into a series of terms. Each term is gauge invariant, has a different analytic structure
more » ... analytic structure in external momenta, and exhibits a hard/soft factorisation. The introduced poles are dictated by infra-red behaviour, which allows us to extend our results to scalar Yukawa theory. The background is treated non-perturbatively throughout. Given that an arbitrary background (coupling to some set of fields in a theory) introduces an arbitrary amount of additional structure, it is not obvious if/how gauge invariance could (fully) determine properties of amplitudes in that background. We will find, though, that traces of the above results on gauge invariance and soft limits do persist. We consider QED with an additional background electromagnetic field. We will show, using tree-level amplitudes in the background, that imposing explicit gauge invariance uncovers a hidden analytic structure; gauge invariance demands a certain infra-red behaviour which introduces new poles in the internal momenta. These poles affect the analytic structure of the entire amplitude (not just the infra-red part); the amplitude factorises on the internal poles with the residues being individually gauge-invariant sub-amplitudes, each with distinct analytic structures in the external, scattered, momenta. The connection between gauge invariance of amplitudes and the infra-red allows us to extend our results to theories without gauge invariance. We will show for a simple scalar Yukawa theory that the infra-red structure of amplitudes leads to an almost identical factorisation of scattering amplitudes. Our chosen background is an electromagnetic (or later scalar) "sandwich" plane wave of finite extent. Here, the high degree of symmetry frequently allows exact solutions [19] [20] [21] [22] , and our results will be exact in the coupling to the background. The same background has been used to test the "double copy" conjecture (for a review see [23] ) beyond flat spacetimes [24, 25] . An outline of our results is as follows. Consider a tree-level four-point QED amplitude in an external field, where all external particles are fermions and hence there is an internal photon line. The corresponding amplitude is defined in position space, due to a nontrivial dependence of the background on position. For the case of plane waves, there is at each vertex a nontrivial dependence on a single spacetime coordinate x + := n·x for some lightlike vector n µ . As such only three momentum components are conserved at each vertex, and overall. Stripping off the δ-function conserving overall three-momentum, the amplitudes M for our processes may be written in the form
doi:10.1007/jhep04(2020)078 fatcat:l4sjsrofuvb2fgx74giedyzsw4