On positive geometries of quartic interactions: one loop integrands from polytopes

A bstract Building on the seminal work of Arkani-Hamed, He, Salvatori and Thomas (AHST) [1] we explore positive geometry encoding one loop scattering amplitude for quartic scalar interactions. We define a new class combinatorial polytopes that call pseudo-accordiohedra whose poset structures are associated to singularities integrand Pseudo-accordiohedra parametrize family projective forms abstract kinematic space defined by AHST restriction these type-D associahedra can be one-loop integrands The (of form) also thought as canonical top form certain geometric realisations pseudo-accordiohedra. Our explores large which include all realisations. These based pseudo-triangulation model cluster algebras discovered Ceballos Pilaud [2].