Positivity in Multifield Effective Field Theories

We discuss the general method for obtaining full positivity bounds on multifield 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 three 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.

Figure 1

Three-dimensional slice of ${\mathbf{C}}^{2^4}$ (left) and ${\mathbf{Q}}^{2^4}$ (right) for the biscalar toy example with $Z_2$ symmetry. The three axes in the left plot are taken to be $(x,y,z)=(2\sqrt{6}(C_{1111}-C_{2222}),\sqrt{2}(2C_{1111}-C_{1212}+2C_{2222}),\sqrt{3}{C}_{1122}^{\prime })$, normalized to $4C_{1111}+C_{1212}+4C_{2222}=1$. Those in the right plot are the same but with $C_{ijkl}^{(\prime )}\to Q_{ijkl}$.

Figure 2

Comparison of the elastic bounds (“Elastic”) and the SDP bounds (“Exact”). The red dots (“Center”) denote the set of coefficients ${\overrightarrow{C}}_0$, which saturates the nonelastic bound ${\mathcal{Q}}_{\mathrm{ex}}$ given in Eq. (8).