  Recent years have seen the emergence of a new understanding of scattering
amplitudes in the simplest theory of colored scalar particles - the
Tr$(\phi^3)$ theory - based on combinatorial and geometric ideas in the
kinematic space of scattering data. In this paper we report a surprise: far
from the toy model it appears to be, the ''stringy'' Tr$(\phi^3)$ amplitudes
secretly contain the scattering amplitudes for pions, as well as
non-supersymmetric gluons, in any number of dimensions. The amplitudes for the
different theories are given by one and the same function, related by a simple
shift of the kinematics. This discovery was spurred by another fundamental
observation: the tree-level Tr$(\phi^3)$ field theory amplitudes have a hidden
pattern of zeros when a special set of non-planar Mandelstam invariants is set
to zero. Furthermore, near these zeros, the amplitudes simplify, by factoring
into a non-trivial product of smaller amplitudes. Remarkably the amplitudes for
pions and gluons are observed to also vanish in the same kinematical locus.
These properties further generalize to the ''stringy'' Tr$(\phi^3)$ amplitudes.
There is a unique shift of the kinematic data that preserves the zeros, and
this shift is precisely the one that unifies colored scalars, pions, and gluons
into a single object. We will focus in this paper on explaining the hidden
zeros and factorization properties and the connection between all the colored
theories, working for simplicity at tree-level. Subsequent works will describe
this new formulation for the Non-linear Sigma Model and non-supersymmetric
Yang-Mills theory, at all loop orders.
