Triple crossing positivity bounds for multi-field theories

TL;DR

This work extends positivity- bounds for EFTs from the forward limit to multi-field theories by developing triple-crossing, su-symmetric dispersion relations and incorporating $st$ null constraints via a convex-cone/spectrahedron framework. By formulating the bound extraction as a semidefinite program with a continuous UV scale $\ olinebreak[0]\mu$ and implementing it with SDPB, the authors obtain fully crossing, finite bounds on higher-order Wilson coefficients, including odd powers of $s$, in multi-field examples. They demonstrate the method concretely on bi-scalar theories with different discrete symmetries, showing that the allowed Wilson-coefficient space forms finite regions near the origin, thus constraining UV completions and SMEFT-like parameter spaces in a robust, model-independent way. The approach combines dispersive sum rules, null constraints, and partial-wave unitarity into a computationally tractable framework with clear geometric interpretations (cones and spectrahedra), offering a practical toolkit for exploring the parameter space of multi-field EFTs.

Abstract

We develop a formalism to extract triple crossing symmetric positivity bounds for effective field theories with multiple degrees of freedom, by making use of $su$ symmetric dispersion relations supplemented with positivity of the partial waves, $st$ null constraints and the generalized optical theorem. This generalizes the convex cone approach to constrain the $s^2$ coefficient space to higher orders. Optimal positive bounds can be extracted by semi-definite programs with a continuous decision variable, compared with linear programs for the case of a single field. As an example, we explicitly compute the positivity constraints on bi-scalar theories, and find all the Wilson coefficients can be constrained in a finite region, including the coefficients with odd powers of $s$, which are absent in the single scalar case.

Figures (2)

• Figure 1: Positivity bounds on $(\tilde{c}_{(2)}^{2,0}=4\tilde{c}^{2,0}_{1122},\tilde{c}_{(1)}^{2,1}=2\tilde{c}_{1111}^{2,1},\tilde{c}_{(2)}^{2,1}=4\tilde{c}^{2,1}_{1122})$ for bi-scalar theory with double $\mathbb{Z}_2$ symmetry in the $\tilde{c}^{m,n}_{(3)}=2\tilde{c}^{m,n}_{1221}=0$ subspace, agnostic about all higher order coefficients. $c_{ijkl}^{m,n}$ is the Wilson coefficient in front of the $(s+\frac{t}{2})^m t^n$ term in the $ij\to kl$ amplitude, and the tilded coefficients are defined as $\tilde{c}_{(a)}^{m,n}\equiv c_{(a)}^{m,n}/c_{(1)}^{2,0}$. We choose units such that the cutoff $\Lambda=1$. We see that the triple crossing positivity bounds form a closed, finite region in the parameter space, numerically of order $\mathcal{O}(1)$.
• Figure 2: Positivity bounds on $(\tilde{c}_{(3)}^{2,0}=2\tilde{c}_{1212}^{2,0},\tilde{c}_{(2)}^{2,1}=4\tilde{c}^{2,1}_{1122})$ and $(\tilde{c}_{(3)}^{2,0}=2\tilde{c}_{1212}^{2,0},\tilde{c}_{(3)}^{2,1}=2\tilde{c}^{2,1}_{1212})$ for double $\mathbb{Z}_2$ symmetric bi-scalar theory. The SDP includes 3 parameters, $c_{(1)}^{2,0}$, $c_{(3)}^{2,0}$ and $c_{(2)}^{2h,n}$, and is agnostic about all other coefficients. In both the left and right plot, the right hand side of the bound extends to infinity.