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Building on the prior work in [1] we locate a family of positive geometriesin the kinematic space which are a specific class of convex realisations of theassociahedron. These realisations are obtained by scaling and translating thekinematic space associahedron discovered by by Arkani-Hamed, Bai, He and Yan(ABHY). We call the resulting polytopes, deformed realisations of theassociahedron. The deformed realisations shed new light on the CHY formula. Oneof the striking discoveries in [2] was the fact that the CHY scatteringequations generate diffeomorphism between the (compactified) CHY moduli spaceand the ABHY associahedron. As we argue, the deformed realisation of theassociahedron can also be interpreted as an diffeomorphic image of the CHYmoduli space under scattering equations that we call deformed scatteringequations. The canonical form in the kinematic space is thus once again thepush-forward of the Parke-Taylor form . A natural off-shoot of our analysis isthe universality of the Parke-Taylor form as a CHY Integrand for a class of(tree-level and planar) multi-scalar field amplitudes. These ideas help us inproving the existence of positive geometries for certain specific multi-scalarinteractions. We prove that in a field theory with a massless and a massivebi-adjoint scalar fields which interact via cubic interaction, the tree-levelS-matrix with massless external states and at most one massive propagator is aweighted sum over the canonical forms defined by certain deformed realisationsof the associahedron. Finally, we show that these ideas admit an extension toone-loop. In particular, the one loop S-matrix integrand with at most onemassive propagator is a weighted sum over canonical forms of a family ofdeformed realisations of the type-D cluster polytope, discovered in [3,4].

Towards Positive Geometries of Massive Scalar field theories