Positivity in Multi-Field EFTs

TL;DR

<3-5 sentence high-level summary> The paper tackles the limitation that leading forward positivity bounds from elastic two-state scattering are insufficient for EFTs with multiple low-energy modes. It reframes positivity as a dual cone problem on spectrahedra and shows that optimal bounds correspond to extremal rays, solvable via semidefinite programming (SDP). The authors demonstrate this approach across multiple EFT sectors—scalars, gauge bosons, fermions, and spin-2 fields—producing complete, tighter bounds and unveiling new constraints that elastic methods miss, often with greater computational efficiency. This methodology provides a practical, scalable tool for deriving fully robust positivity constraints in realistic multi-field EFTs, with direct applications to SMEFT and gravity-related EFTs.

Abstract

We discuss the general method for obtaining full positivity bounds on multi-field effective field theories (EFTs). While the leading order forward positivity bounds are commonly derived from the elastic scattering of two (superposed) external states, we show that for a generic EFT containing 3 or more low-energy modes, this approach only gives incomplete bounds. We then identify the allowed parameter space as the dual to a spectrahedron, constructed from crossing symmetries of the amplitude, and show that finding the optimal bounds for a given number of modes is equivalent to a geometric problem: finding the extremal rays of a spectrahedron. We show how this is done analytically for simple cases, and numerically formulated as semidefinite programming (SDP) problems for more complicated cases. We demonstrate this approach with a number of well-motivated examples in particle physics and cosmology, including EFTs of scalars, vectors, fermions and gravitons. In all these cases, we find that the SDP approach leads to results that either improve the previous ones or are completely new. We also find that the SDP approach is numerically much more efficient.

Figures (5)

• Figure 1: 3-dimensional slice of $\mathbf C^{2^4}$ (left) and $\mathbf Q^{2^4}$(right) for the bi-scalar toy example with $Z_2$ symmetry. The three axes in the left plot are taken to be $(x,y,z)= \left( 2\sqrt{6}\left( C_{1111}-C_{2222} \right), \sqrt{2}\left( 2C_{1111}-C_{1212}+2C_{2222} \right), \sqrt{3} C'_{1122} \right)$, normalized to $4C_{1111}+C_{1212}+4C_{2222}=1$. Those in the right plot are the same but with $C^{(}{}'{}^{)}_{ijkl}\to Q_{ijkl}$.
• Figure 2: Comparison of the elastic bounds ("Elastic") and the SDP bounds ("Exact"). The red dots ("Center") denote the set of coefficients $\vec{C}_0$ which saturates the non-elastic bound $\mathcal{Q}_{\rm ex}$ given in Eq. (\ref{['eq:U']}).
• Figure 3: Bounds on the flavor-conserving coefficient $|C_{1111}|$ as a function of $C_{1112}$, with other coefficients fixed at $\vec{C}_0$. We see that flavor-conserving signal of new physics is bounded by the flavor-violating ones from below, and our approach improves the elastic bounds.
• Figure 4: Improvement in constraining the dRGT parameters $c_3$ and $d_5$ (left) and the $Z_2$ bi-field spin-2 EFT parameters $d$ and $\lambda$ (right; see Ref. [66]). "Elastic" denotes the superposed elastic positivity bounds, while "Exact" bounds are obtained by SDP.
• Figure 5: Comparison between the elastic and full positivity bounds for $c$-$d$ cross sections in the bi-field cycle theory (\ref{['bispin2']}). Other parameters are chosen as $(\kappa_3,\kappa_4,\lambda)=(1.4,0.38,0.2)$ for the left figure and $(\kappa_3,\kappa_4,\lambda)=(1.1,0.38,0.2)$ for the right figure.