Geometry of Effective Field Theory Positivity Cones

Abstract Positivity bounds are theoretical constraints on the Wilson coefficients of an effective field theory. These bounds emerge from the requirement that a given effective field theory must be the low-energy limit of a relativistic quantum theory that satisfies the fundamental principles of unitarity, locality, and causality. The task of deriving these bounds can be reformulated as the geometric problem of finding the extremal representation of a closed convex cone $$\mathcal C_W$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>C</mml:mi> <mml:mi>W</mml:mi> </mml:msub> </mml:math> . More precisely, in the presence of multiple particle flavors, the forward-limit positivity cone $$\mathcal C_W$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>C</mml:mi> <mml:mi>W</mml:mi> </mml:msub> </mml:math> consists of all positive semi-definite tensors in $$W =\left\{ S \in \textrm{Sym}^2 (\textrm{Sym}^2\, V^*)\oplus \textrm{Sym}^2 \left( {\Lambda }^2 V^*\right) :\right. \left. \tau S = S \right\} \subset \textrm{Sym}^2(V^*\otimes V^*)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>W</mml:mi> <mml:mo>=</mml:mo> <mml:mfenced> <mml:mi>S</mml:mi> <mml:mo>∈</mml:mo> <mml:msup> <mml:mtext>Sym</mml:mtext> <mml:mn>2</mml:mn> </mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:msup> <mml:mtext>Sym</mml:mtext> <mml:mn>2</mml:mn> </mml:msup> <mml:mspace/> <mml:msup> <mml:mi>V</mml:mi> <mml:mo>∗</mml:mo> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>⊕</mml:mo> <mml:msup> <mml:mtext>Sym</mml:mtext> <mml:mn>2</mml:mn> </mml:msup> <mml:mfenced> <mml:msup> <mml:mrow> <mml:mi>Λ</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> <mml:msup> <mml:mi>V</mml:mi> <mml:mo>∗</mml:mo> </mml:msup> </mml:mfenced> <mml:mo>:</mml:mo> </mml:mfenced> <mml:mfenced> <mml:mi>τ</mml:mi> <mml:mi>S</mml:mi> <mml:mo>=</mml:mo> <mml:mi>S</mml:mi> </mml:mfenced> <mml:mo>⊂</mml:mo> <mml:msup> <mml:mtext>Sym</mml:mtext> <mml:mn>2</mml:mn> </mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:msup> <mml:mi>V</mml:mi> <mml:mo>∗</mml:mo> </mml:msup> <mml:mo>⊗</mml:mo> <mml:msup> <mml:mi>V</mml:mi> <mml:mo>∗</mml:mo> </mml:msup> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> , where $$\tau $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>τ</mml:mi> </mml:math> denotes transposition in the second and fourth tensor factor and $$V\cong \mathbb {R}^n$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>V</mml:mi> <mml:mo>≅</mml:mo> <mml:msup> <mml:mrow> <mml:mi>R</mml:mi> </mml:mrow> <mml:mi>n</mml:mi> </mml:msup> </mml:mrow> </mml:math> , where n is the number of flavors. In this work, we solve this question up to three flavors, i.e., $$n=3$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>=</mml:mo> <mml:mn>3</mml:mn> </mml:mrow> </mml:math> , proving a full classification of all extremal elements in these cases. We furthermore study the implications of our findings, deriving the full positivity bounds for amplitudes with and without additional symmetries. In the cases with additional symmetries that we consider, we find that the so-called elastic bounds are sufficient to give rise to the full positivity bounds.