Positivity Constraints in Effective Field Theories

Effective field theories (EFTs) furnish a framework to describe low-energy phenomena without detailed knowledge of underlying high-energy dynamics. In such theories, interactions are organised as an expansion in energy or momentum, weighted by Wilson coefficients that encode the influence of heavier states. Positivity constraints arise by demanding fundamental principles—causality, unitarity and analyticity—upon scattering amplitudes. Analyticity ensures that amplitudes vary smoothly with complexified energy variables, unitarity guarantees conservation of probability, and causality enforces that signals do not propagate faster than light. Together these principles imply dispersion relations that relate the energy behaviour of amplitudes at low scales to weighted integrals over their imaginary parts at high scales. By examining forward and non-forward scattering of particles—such as photons, scalars or gravitons—one obtains inequalities on combinations of Wilson coefficients. These bounds serve as consistency checks on EFT constructions, ruling out pathological interactions that cannot admit any standard ultraviolet completion. The resulting positivity bounds have broad application: they sharpen predictions for new physics searches in collider experiments, constrain modifications of gravity at cosmological distances and inform the design of precision tests of the Standard Model. Recent theoretical advances have deepened the geometric understanding of these constraints, revealing that allowed EFT couplings inhabit a convex “positivity region” or “EFT-hedron”, whose facets correspond to physical unitarity and causality conditions.

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Positivity Constraints in Effective Field Theories publication trend

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Technical terms

Effective Field Theory (EFT): A low-energy approximation that organises interactions among light particles via a series of operators with increasing dimension, each multiplied by a Wilson coefficient.

Wilson Coefficient: A parameter in an EFT that measures the strength of a higher-dimension operator, encapsulating effects of heavy degrees of freedom.

Dispersion Relation: An integral representation of a scattering amplitude that relates its real part at low energies to an integral over its imaginary part (absorptive part) at higher energies, enforcing analyticity and unitarity.

Unitarity: The principle that total probability is conserved in quantum scattering, implying that the imaginary part of an amplitude is positive in the physical region.

Analyticity: The requirement that scattering amplitudes are analytic functions of complex energy variables, except for known singularities corresponding to physical thresholds.

Crossing Symmetry: The property that amplitudes for different scattering channels (s-, t-, u-channels) are related by analytic continuation, allowing the use of dispersion relations in non-forward kinematics.

References

1. Physics of the analytic S-matrix. Physics Reports (2024).
2. Gravitational Regge bounds. SciPost Physics (2024).
3. Causality constraints on corrections to Einstein gravity. Journal of High Energy Physics (2023).
4. The EFT-hedron. Journal of High Energy Physics (2021).
5. New positivity bounds from full crossing symmetry. Journal of High Energy Physics (2021).