Scattering amplitudes in YM and GR as minimal model brackets and their recursive characterization [article]

Andrea Nützi, Michael Reiterer
2018 arXiv   pre-print
Attached to both Yang-Mills and General Relativity about Minkowski spacetime are distinguished gauge independent objects known as the on-shell tree scattering amplitudes. We reinterpret and rigorously construct them as L_∞ minimal model brackets. This is based on formulating YM and GR as differential graded Lie algebras. Their minimal model brackets are then given by a sum of trivalent (cubic) Feynman tree graphs. The amplitudes are gauge independent when all internal lines are off-shell, not
more » ... rely up to L_∞ isomorphism, and we include a homological algebra proof of this fact. Using the homological perturbation lemma, we construct homotopies (propagators) that are optimal in bringing out the factorization of the residues of the amplitudes. Using a variant of Hartogs extension for singular varieties, we give a rigorous account of a recursive characterization of the amplitudes via their residues independent of their original definition in terms of Feynman graphs (this does neither involve so-called BCFW shifts nor conditions at infinity under such shifts). Roughly, the amplitude with N legs is the unique section of a sheaf on a variety of N complex momenta whose residues along a finite list of irreducible codimension one subvarieties (prime divisors) factor into amplitudes with less than N legs. The sheaf is a direct sum of rank one sheaves labeled by helicity signs. To emphasize that amplitudes are robust objects, we give a succinct list of properties that suffice for a dgLa so as to produce the YM and GR amplitudes respectively.
arXiv:1812.06454v1 fatcat:x5odu6vqz5cxbfxsnlpsgtlf5a