  Inspired by the closed contour of momentum conservation in an interaction, we
introduce an integrable one-dimensional theory that underlies some integrable
models such as the Kadomtsev-Petviashvili (KP)-hierarchy and the amplituhedron.
In this regard, by defining the action and partition function of the presented
theory, while introducing a perturbation, we obtain its scattering matrix
(S-matrix) with Grassmannian structure. This Grassmannian corresponds to a
chord diagram, which specifies the closed contour of the one-dimensional
theory. Then, we extract a solution of the KP-hierarchy using the S-matrix of
theory. Furthermore, we indicate that the volume of phase-space of the
one-dimensional manifold of the theory is equal to a corresponding Grassmannian
integral. Actually, without any use of supersymmetry, we obtain a sort of
general structure in comparison with the conventional Grassmannian integral and
the resulted amplituhedron that is closely related to the Yang-Mills scattering
amplitudes in four dimensions. The proposed theory is capable to express both
the tree- and loop-levels amplituhedron (without employing hidden particles)
and scattering amplitude in four dimensions in the twistor space as particular
cases.
