# Projectivity of Planar Zeros in Field and String Theory Amplitudes

## TL;DR

This paper investigates the projective properties of planar zeros in scattering amplitudes across various theories, revealing how gauge couplings, string effects, and soft limits influence their structure and invariance.

## Contribution

It uncovers the conditions under which planar zeros exhibit projective invariance and how string corrections modify these properties in different theories.

## Key findings

- Planar zeros in scalar theories lack projective invariance.
- Coupling scalars to gauge fields introduces homogeneous polynomial conditions for zeros.
- String corrections generally destroy the projective structure of planar zeros.

## Abstract

We study the projective properties of planar zeros of tree-level scattering amplitudes in various theories. Whereas for pure scalar field theories we find that the planar zeros of the five-point amplitude do not enjoy projective invariance, coupling scalars to gauge fields gives rise to tree-level amplitudes whose planar zeros are determined by homogeneous polynomials in the stereographic coordinates labelling the direction of flight of the outgoing particles. In the case of pure gauge theories, this projective structure is generically destroyed if string corrections are taken into account. Scattering amplitudes of two scalars with graviton emission vanish exactly in the planar limit, whereas planar graviton amplitudes are zero for helicity violating configurations. These results are corrected by string effects, computed using the single-valued projection, which render the planar amplitude nonzero. Finally, we discuss how the structure of planar zeros can be derived from the soft limit behavior of the scattering amplitudes.