Extremal Effective Field Theories

TL;DR

This work develops a numerically rigorous S-matrix bootstrap for a massless real scalar EFT by combining twice-subtracted dispersion relations with null constraints and a semidefinite programming framework. It proves two-sided, order-one bounds on dimensionless EFT couplings and identifies extremal, analytically tractable amplitudes that realize kinks in the allowed region, notably at tilde g4 = 1/2. The analysis reveals that dimensional analysis scaling emerges from causality and unitarity, and the allowed space is largely captured by convex combinations of simple analytic models (spin-0 and stu-pole), with gravity and non-identical-scalar extensions left for future work. Overall, the paper quantifies how high-energy consistency conditions sharply constrain low-energy EFT parameter spaces and provides a practical pipeline for mapping these constraints numerically. The results have potential implications for EFT constructions in particle and nuclear physics and offer a blueprint for exploring extremal theories via S-matrix techniques.

Abstract

Effective field theories (EFT) parameterize the long-distance effects of short-distance dynamics whose details may or may not be known. It is known that EFT coefficients must obey certain positivity constraints if causality and unitarity are satisfied at all scales. We explore those constraints from the perspective of 2 to 2 scattering amplitudes of a light real scalar field, using semi-definite programming to carve out the space of allowed EFT coefficients for a given mass threshold M. We point out that all EFT parameters are bounded both below and above, effectively showing that dimensional analysis scaling is a consequence of causality. This includes the coefficients of four- and six-derivative interactions. We present simple extremal amplitudes which realize, or "rule in", kinks in coefficient space and whose convex hull span a large fraction of the allowed space.

Figures (12)

• Figure 1: The $2\to 2$ scattering process studied in this paper. For different choices of four-momenta, time can flow either horizontally or vertically (or diagonally).
• Figure 2: Analyticity in the upper-half-plane relates the $s$-channel amplitude and the complex conjugate (anti-time-ordered) $u$-channel amplitude. Note that the $s$- and $u$-channel cuts overlap in the physical region where $t<0$, which is not a problem since the crossing path avoids small $|s|$.
• Figure 3: Contour deformation which gives the sum rule \ref{['disp low high']} when low-energy loops are neglected: the integral over arcs at infinity vanishes, thus relating low-energy data and heavy cuts.
• Figure 4: Integration contour to be used when low-energy loops are included; the integral vanishes as it is equivalent to arcs at infinity. This relates high-energy cuts at $s>M^2$ and $u>M^2$ with EFT-computable data near the EFT cutoff $|s|\sim M^2$.
• Figure 5: The null constraint $m^8 n_4(m^2,J)$, which is a function of only $J$. It vanishes at $J=0$, is negative at $J=2$, but positive at $J=4,6,\dots$, and thus balances spin-two against higher-spin states. The sign change in various space-time dimensions (at $J_{\text{critical}} = \frac{1}{2}(3-d + \sqrt{d (d+4)+1})$) is always situated between $2<J<3$.