Positivity Bounds without Boosts

TL;DR

The paper derives the first positivity bounds for EFTs without Lorentz boosts by introducing a Breit-parametrised boosted amplitude $\tilde{\mathcal{A}}_s(s,t,M,\omega_t,\gamma)$ that preserves analyticity and crossing, enabling a dispersion relation and a suite of inequalities on derivatives of the EFT amplitude. This yields an infinite ladder of forward-limit bounds $\tilde{\mathcal{A}}_s^{(2N)}|_{t=0,\omega_t=0}\ge0$ for all $N\ge1$, with locality ensuring the high-energy contribution vanishes. When applied to concrete models (a simple scalar EFT, a superfluid, and the inflationary EFT), the bounds constrain combinations of Wilson coefficients (e.g. $f=\Lambda^4 \tilde{\mathcal{A}}_s^{(2)}|_{\omega_t=0,t=0}$ with $f(\gamma)$ given) and relate UV-scale requirements to observational limits on primordial non-Gaussianity, implying that in inflation new physics must appear near the Hubble scale for much of the Planck-allowed region. Overall, the work broadens positivity bounds to systems with broken boosts, offering a robust framework for constraining UV completions in cosmology and condensed matter and setting the stage for extensions to higher spins and wavefunction-based approaches in cosmology.

Abstract

We derive the first positivity bounds for low-energy Effective Field Theories (EFTs) that are not invariant under Lorentz boosts. "Positivity bounds" are the low-energy manifestation of certain fundamental properties in the UV -- to date they have been used to constrain a wide variety of EFTs, however since all of the existing bounds require Lorentz invariance they are not directly applicable when this symmetry is broken, such as for most cosmological and condensed matter systems. From the UV axioms of unitarity, causality and locality, we derive an infinite family of bounds which (derivatives of) the $2\to2$ EFT scattering amplitude must satisfy even when Lorentz boosts are broken (either spontaneously or explicitly). We apply these bounds to the leading-order EFT of both a superfluid and the scalar fluctuations produced during inflation, comparing in the latter case with the current observational constraints on primordial non-Gaussianity.

Figures (1)

• Figure 1: Assuming that the UV completion is unitary, causal and local, our positivity bound \ref{['eqn:pos_3_superfluid']} sets a scale $\omega_{\rm NP}$ beyond which inflation can no longer be single-field and weakly coupled. The red shaded regions correspond to $\omega_{\rm NP} < 5H, \, 10H, \, 20H$ and $\Lambda = 58.89 H$, and the grey contours show the $68\%, 95\%$ and $99.7\%$ confidence intervals from the Planck 2018 observations of the equilateral and orthogonal bispectrum $(f_{\rm NL}^{\rm eq}$ and $f_{\rm NL}^{\rm or}$), which are determined by the Wilson coefficients ($c_s$ and $\alpha_1$) appearing in the EFT of Inflation \ref{['eqn:EFT_action']}. For the majority of the $68\%$ confidence interval, the leading-order EFT requires the inclusion of new UV physics beyond the scalar fluctuations $\pi$ (or a non-perturbative treatment of $\pi$ loops) at an energies below $10H$.