Abstract
We derive the first ever on-shell recursion relations applicable to effective field theories. Based solely on factorization and the soft behavior of amplitudes, these recursion relations employ a new rescaling momentum shift to construct all tree-level scattering amplitudes in the nonlinear sigma model, Dirac-Born-Infeld theory, and the Galileon. Our results prove that all theories with enhanced soft behavior are on-shell constructible.
Highlights
AnI ðzI ÞAnI ðzIÞ; ð3Þ establishing a recursion relation in terms of the lower-point amplitudes AnI and AnI where nI þ n I 1⁄4 n þ 2
Introduction.—The modern S-matrix program exploits physical criteria like Lorentz invariance and unitarity to construct scattering amplitudes directly and without the aid of a Lagrangian
Many S matrices are constructible via on-shell recursion, which elegantly encodes factorization as a physical input
Summary
AnI ðzIÞ; ð3Þ establishing a recursion relation in terms of the lower-point amplitudes AnI and AnI where nI þ n I 1⁄4 n þ 2. The above derivation fails when there is a nonzero residue at z 1⁄4 ∞ This boundary contribution is calculable in certain circumstances [12] and there exist any number of generalizations of BCFW recursion for which the amplitude vanishes at large z [3,4]. Since existing recursive technology already exploits the amplitudes’ singularities, a natural candidate for new physical information is the amplitudes’ zeros The former are dictated by factorization while the latter require special kinematics at which the amplitude vanishes. For n < D þ 1, a generic set of momenta pi are linearly independent, so the only solution to Eq (6) has all ai equal, corresponding to total momentum conservation Since this momentum shift rescales of all the momenta, it is not useful for recursion. To compute the amplitude we apply Cauchy’s theorem to a contour encircling all poles at finite z
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