Analyticity, dispersion, and positivity bounds

Consider identical stable scalars of mass m, the lightest species in a four-dimensional relativistic theory with its first continuum threshold at $4m^2$. Use a forward amplitude normalization with positive absorptive part, and assume crossing, analyticity, and high-energy boundedness sufficient for two subtractions. Set $\nu=s-2m^2$ at zero momentum transfer. Crossing exchanges s and u and makes the pole-subtracted amplitude $B(\nu)$ even. Its expansion point $\nu=0$ lies between the crossed physical cuts.

Subtract explicitly known light-particle poles before differentiating, and include their effects consistently in the coefficient being bounded. The remaining right-hand cut begins at the appropriate multiparticle threshold; in the simple elastic massive setup its location is $\nu_{\mathrm{th}}=2m^2$. A contour argument also needs control of the contribution at infinity. Analyticity alone does not make that arc vanish, and the number of subtractions depends on the growth assumption. These premises are part of the calculation rather than optional background.

The optical theorem relates the forward absorptive part to a sum of physical transition probabilities with positive kinematic factors. The dispersive kernel in the displayed moment is positive over the right-hand integration range. Their combination therefore constrains a low-energy derivative without requiring a microscopic calculation of every high-energy state. The statement is a weighted sum rule; it does not imply that each individual Feynman graph or each basis-dependent Wilson coefficient is separately positive.

Since $\nu$ has mass dimension two, $b_2$ has mass dimension minus four for a dimensionless amplitude. In an effective theory, the bounded combination can involve several operators, loop terms, and subtraction contributions. Matching and renormalization conventions must be included before translating the inequality into a claim about a named coefficient. A positive allowed coefficient remains only a necessary consistency condition: it does not establish all higher moments, nonforward constraints, or existence of a complete ultraviolet theory.

Massless exchange can put a singular pole at zero momentum transfer and remove the clean gapped forward setup. Gravity is a central example where infrared singularities, high-energy behavior, and subtraction choices require special care. Moving slightly away from forward scattering, subtracting calculable infrared contributions, or introducing a regulator may help in particular arguments, but each modification needs a justified limit. A massive-scalar theorem cannot simply be quoted unchanged after deleting its mass-gap assumption.

Research bounds also incorporate known low-energy absorptive contributions, higher derivatives, spin, internal symmetries, and several coupled channels. More information can sharpen the allowed coefficient region, while additional assumptions can narrow its applicability. An apparent violation should trigger checks of normalization, poles, loop accuracy, infrared treatment, and the effective-theory domain before claiming a failure of causality. Conversely, passing one positivity inequality supplies useful evidence but never substitutes for the full set of consistency requirements.