Towards positive geometry of multi scalar field amplitudes. Accordiohedron and effective field theory
Mrunmay Jagadale, Alok Laddha
Abstract
A bstract The geometric structure of S-matrix encapsulated by the “Amplituhedron program” has begun to reveal itself even in non-supersymmetric quantum field theories. Starting with the seminal work of Arkani-Hamed, Bai, He and Yan [1] it is now understood that for a wide class of scalar quantum field theories, tree-level amplitudes are canonical forms associated to polytopes known as accordiohedra. Similarly the higher loop scalar integrands are canonical forms associated to so called type-D cluster polytopes for cubic interactions or recently discovered class of polytopes termed pseudo-accordiohedron for higher order scalar interactions. In this paper, we continue to probe the universality of these structures for a wider class of scalar quantum field theories. More in detail, we discover new realisations of the associahedron in planar kinematic space whose canonical forms generate (colour-ordered) tree-level S matrix of external massless particles with n − 4 massless poles and one massive pole at m 2 . The resulting amplitudes are associated to λ 1 $$ {\phi}_1^3 $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>ϕ</mml:mi> <mml:mn>1</mml:mn> <mml:mn>3</mml:mn> </mml:msubsup> </mml:math> + λ 2 $$ {\phi}_1^2 $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>ϕ</mml:mi> <mml:mn>1</mml:mn> <mml:mn>2</mml:mn> </mml:msubsup> </mml:math> ϕ 2 potential where ϕ 1 and ϕ 2 are massless and massive scalar fields with bi-adjoint colour indices respectively. We also show how in the “decoupling limit” (where m → ∞ , λ 2 → ∞ such that g : $$ \frac{\uplambda_2}{m} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mfrac> <mml:msub> <mml:mi>λ</mml:mi> <mml:mn>2</mml:mn> </mml:msub> <mml:mi>m</mml:mi> </mml:mfrac> </mml:math> = finite) these associahedra project onto a specific class of accordiohedron which are known to be positive geometries of amplitudes generated by λ $$ {\phi}_1^3 $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>ϕ</mml:mi> <mml:mn>1</mml:mn> <mml:mn>3</mml:mn> </mml:msubsup> </mml:math> + g $$ {\phi}_1^4 $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msubsup> <mml:mi>ϕ</mml:mi> <mml:mn>1</mml:mn> <mml:mn>4</mml:mn> </mml:msubsup> </mml:math> .