  Building on the seminal work of Arkani-Hamed, He, Salvatori and Thomas
(AHST), we explore the positive geometry encoding one loop scattering amplitude
for quartic scalar interactions. We define a new class of combinatorial
polytopes that we call pseudo-accordiohedra whose poset structures are
associated to singularities of the one loop integrand associated to scalar
quartic interactions. Pseudo-accordiohedra parametrize a family of projective
forms on the abstract kinematic space defined by AHST and restriction of these
forms to the type-D associahedra can be associated to one-loop integrands for
quartic interactions. The restriction (of the projective form) can also be
thought of as a canonical top form on certain geometric realisations of
pseudo-accordiohedra. Our work explores a large class of geometric realisations
of the type-D associahedra which include all the AHST realisations. These
realisations are based on the pseudo-triangulation model for type-D cluster
algebras discovered by Ceballos and Pilaud.
