Planar Radiation Zeros and Scattering Equations in Field Theory Amplitudes

Resumen

We have presented for the rst time a detailed description of planar radiation zeros as a novel mathematical structure giving rise to new insights on the internal behavior of a theory, such as the biadjoint scalar theory, the Yang-Mills theory or the Einstein-Hilbert gravity. The concept \radiation zero" makes reference to all the con gurations in phase space for which the full scattering amplitude of a given process vanishes. In our case, we have studied \planar zeros", meaning that our characterization applies to those processes where all particle momenta lie in the same spatial plane. Although being a rather naive concept, the obtained results are far from incidental. On one side, we have found that the conditions of emergence of gauge planar zeros in the maximally helicity violating sector live inside the projective space spanned by the stereographic coordinates labelling the direction of ight of the outgoing momenta. The existence of such a projective characterization implies that planar zeros are always realized inside the soft limit of any of the emitted particles, which might be of relevance for the infrared structure or the asymptotic symmetries of the theory. On a di erent side, we have found that gravitational amplitudes always vanish inside this planar limit for non-helicity conserving con gurations without imposing any further kinematic conditions. String 0-corrections of these behaviors have also been obtained. All the computations have been done in the context of the color-kinematics du- ality, used as a procedure to compute gravitational amplitudes from their gauge analogues; and the Cachazo-He-Yuan formalism, as a novel integral representation to write scattering amplitudes in contrast to the traditional Feynman diagram decomposition. In particular, the latter relies upon a rational map between the space of null D-dimensional momentum vectors and the moduli space of punctured Riemann spheres, given the name of scattering equations. Considered to be a challenging task, we have shown the advantages of using the Sudakov parametrization of particle momenta to simplify the computation of their exact solutions. In particular, we have shown that both punctures in the Riemann sphere and scattering amplitudes themselves adopt rather compact formulas when expressed in terms of Sudakov variables, suggesting the parametrization to be a natural candidate for an e cient description of scattering amplitudes inside the formalism.