Dispersion, Positivity, and UV Constraints
Dispersion relations connect low-energy amplitude coefficients to discontinuities at higher energies. Positivity emerges only when the contour, growth, crossing, pole subtraction, and physical-state sum make the relevant integral sign-definite. This chapter makes those conditions part of the bound.
Enter this chapter
Section titled “Enter this chapter”Begin with Subtracted Dispersion Relations to identify the analytic domain and subtraction freedom before applying positivity.
Topics
Section titled “Topics”- Subtracted Dispersion Relations — Determines which singularity data and subtraction constants reconstruct an amplitude.
- Forward Scattering Sum Rules — Relates low-energy moments to inclusive absorptive data.
- Fixed-t and Partial-Wave Dispersion — Retains angular information beyond the forward amplitude.
- Forward-Limit Positivity Bounds — Derives sign constraints in a gapped channel with a finite forward limit.
- Beyond-Forward Positivity — Adds angular derivatives and crossing relations under stronger hypotheses.
- Massless Exchange and Infrared Subtractions — Diagnoses forward poles, moving branch points, and infrared modifications.
- EFT Positivity and UV Consistency — Translates physical amplitude bounds into qualified EFT statements.
How the ideas fit together
Section titled “How the ideas fit together”A subtracted contour relation can express derivatives of a pole-subtracted forward amplitude as weighted absorptive integrals. Unitarity supplies the sign of the physical forward discontinuity, while analyticity and high-energy control justify the contour manipulation. If the channel has nonzero absorptive weight, suitable moments are strictly positive; a decoupled channel may saturate zero. See de Rham et al. 2017, equations (1)–(6).
Massless exchange can make the forward limit singular and requires a new argument. A bound also concerns physical amplitude combinations, not an arbitrary basis coefficient whose value can change under field redefinitions.
Review the chapter
Section titled “Review the chapter”Does satisfying all bounds checked in a finite calculation prove that an EFT has a conventional ultraviolet completion?
Solution
No. Those bounds are necessary consequences of the chosen completion assumptions, not a construction of a theory. Other channels, higher orders, untested inequalities, or additional consistency conditions can still obstruct a completion.
Where to go next
Section titled “Where to go next”Continue to EFT Positivity and UV Consistency for the precise translation between amplitude coefficients and effective interactions.