  In this note, we introduce and study a new class of "half integrands" in
Cachazo-He-Yuan (CHY) formula, which naturally generalize the so-called
Parke-Taylor factors; these are dubbed Cayley functions as each of them
corresponds to a labelled tree graph. The CHY formula with a Cayley function
squared gives a sum of Feynman diagrams, and we represent it by a combinatoric
polytope whose vertices correspond to Feynman diagrams. We provide a simple
graphic rule to derive the polytope from a labelled tree graph, and classify
such polytopes ranging from the associahedron to the permutohedron.
Furthermore, we study the linear space of such half integrands and find (1) a
nice formula reducing any Cayley function to a sum of Parke-Taylor factors in
the Kleiss-Kuijf basis (2) a set of Cayley functions as a new basis of the
space; each element has the remarkable property that its CHY formula with a
given Parke-Taylor factor gives either a single Feynman diagram or zero. We
also briefly discuss applications of Cayley functions and the new basis in
certain disk integrals of superstring theory.
