Abstract
We derive a recursion relation for loop-level scattering amplitudes of La- grangian field theories that generalises the tree-level Berends-Giele recursion relation in Yang-Mills theory. The origin of this recursion relation is the homological perturbation lemma, which allows us to compute scattering amplitudes from minimal models of quantum homotopy algebras in a recursive way. As an application of our techniques, we give an alternative proof of the relation between non-planar and planar colour-stripped scattering amplitudes.
Highlights
One can study the cohomology Hm 1(a) with respect to m1, and, for instance, Hm1 1(a) contains all free onshell fields. This cohomology extends to an A∞-algebra (a◦ := Hm 1(a), m◦i ) with m◦1 = 0, called the minimal model, which encodes the n-point tree-level scattering amplitudes, cf. [8]
We showed that full quantum scattering amplitudes of quantum field theories can be conveniently described in terms of minimal models of cyclic quantum A∞-algebras
This description allows for recursion relations for currents, which reproduce and generalise known recursion relations
Summary
Consider again four-dimensional Minkowski space R1,3 with metric η. Working with A∞-algebras amounts to working in the ‘color flow’ formalism or using double line Feynman diagrams. This implies that the fields take values in a matrix algebra and we have to extend the gauge algebra from u(n) to gl(n, C). To scalar field theory, 3On a technical note, the vector space a1 should again be decomposed into free (i.e. compactly supported on Cauchy surfaces) and interacting (i.e. Schwartz type) fields as before for scalar field theory. As before, scattering amplitudes are encoded in the corresponding minimal model and given by formulas of the form (2.8) and (2.19) with φ replaced by a To determine these from the homological perturbation lemma, we note that the relevant propagator h, which gives rise to H0 via (2.10b), acts as c.
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Free) 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a call
