Dispersion, Positivity, and UV Constraints

Analyticity turns a scattering amplitude into an integral transform of its singularities; unitarity can then make part of that spectral information positive. The resulting constraints are powerful but conditional. Every conclusion in this chapter keeps visible the analytic domain, the number of subtractions, the crossing sector, the treatment of poles and light cuts, and the assumed high-energy growth.

Enter this chapter

The clean benchmark is elastic scattering of identical massive scalars,

$$s+t+u=4m^2, \qquad v=s-2m^2+\frac{t}{2},$$

for which $s\leftrightarrow u$ sends $v\mapsto-v$. Under the stated fixed-$t$ analyticity hypotheses, the mass gap separates the right- and left-hand cuts; after known stable poles are removed and the center is checked for any remaining singularity, $v=t=0$ has an analytic neighborhood. It is the setting in which the logical chain is easiest to see:

$$\begin{gathered} \text{analyticity and growth} \Longrightarrow \text{subtracted dispersion relation}\\ \Longrightarrow \text{absorptive moments} \Longrightarrow \text{conditional inequalities}. \end{gathered}$$

The first implication is Cauchy’s theorem plus a bound strong enough to remove the large arc. The last additionally needs a sign-definite absorptive part in the chosen state and crossing sector. This is why a dispersion relation can remain valid when a positivity claim does not. The distinction is standard in forward-dispersion theory Weinberg 1995, § 10.8, pp. 462–469, while modern EFT bounds make the mass-gap and UV-completion assumptions explicit de Rham et al. 2017, pp. 1–4, PDF.

Choose a route

If you want to…	Start with…	What you will obtain
derive the integral representation	Subtracted Dispersion Relations	cuts, poles, subtraction data, and the large-arc criterion
connect low-energy coefficients to inclusive rates	Forward Scattering Sum Rules	crossing-even moments and crossing-odd sum rules
retain momentum transfer and angular information	Fixed-t and Partial-Wave Dispersion	fixed-$t$ kernels, partial-wave absorptive parts, and the allowed $t$ domain
prove the basic scalar sign constraint	Forward-Limit Positivity Bounds	positive even derivatives and moment inequalities
strengthen the result using $t$ derivatives and full crossing	Beyond-Forward Positivity	derivative combinations, improved bounds, and convex constraints
decide whether a massless exchange invalidates the argument	Massless Exchange and Infrared Subtractions	regulated limits, pole-subtraction tests, and gravity cautions
constrain an EFT coefficient responsibly	EFT Positivity and UV Consistency	an amplitude-level workflow with loop and truncation uncertainties

Assumptions that do different jobs

Input	Role	What fails without it
analyticity in a stated cut domain	permits contour deformation and Taylor expansion	extra singularities contribute, or the expansion point is unavailable
real analyticity	turns a discontinuity into $2i\operatorname{Im}A$	the two boundary values cannot be replaced by one imaginary part
polynomial boundedness or another explicit asymptotic estimate	fixes how many subtractions remove the large arc	an unknown arc term remains
crossing	rewrites the left cut using a physical crossed channel	the kernel is incomplete or not sign definite
unitarity	makes physical absorptive data nonnegative in suitable channels	a dispersion relation survives, but positivity need not
a mass gap and a finite forward limit	separates cuts and permits $t\to0$	massless poles or cuts can pinch the subtraction point
explicit pole and low-energy subtraction	prevents known singularities from masquerading as contact coefficients	the inferred Wilson-coefficient combination is wrong

These hypotheses are not interchangeable. In particular, the Froissart–Martin type growth used for the familiar twice-subtracted scalar relation depends on a gapped, local setting; it is not a universal theorem about gravitational amplitudes. Fixed-$t$ and full-crossing analyses must also remain inside their common analyticity domain Tolley, Wang, and Zhou 2021, § 2, pp. 4–6, PDF.

One representative forward moment displays the whole package:

$$\frac12B''(0) =\frac{2}{\pi}\int_{\nu_0}^{\infty} \frac{\operatorname{Im}A(2m^2+\nu+i0,0)}{\nu^3} \,\mathrm d\nu>0.$$

The equality needs the cut domain, explicit pole subtraction, crossing-even projection, and a twice-subtracted outer-arc bound. The strict sign additionally needs a positive-norm inclusive state sum with nonzero spectral weight and a finite gapped forward limit. If the selected state decouples, the conclusion weakens to nonnegativity; if a massless pole reaches $t=0$, this moment need not exist.

What the chapter does not claim

A satisfied positivity inequality is a necessary test under its assumptions, not a construction of a UV completion. A violated inequality excludes only the stated completion class after calculational and truncation uncertainties are controlled. The chapter does not perform phenomenological dispersive fits, build EFT operator bases, or claim a universal gravitational positivity theorem.

Begin with the derivation: Subtracted Dispersion Relations.

References

• de Rham, Claudia, Scott Melville, Andrew J. Tolley, and Shuang-Yong Zhou. “Positivity Bounds for Scalar Theories.” Physical Review D 96 (2017): 081702. DOI. Open PDF.
• Tolley, Andrew J., Zi-Yue Wang, and Shuang-Yong Zhou. “New Positivity Bounds from Full Crossing Symmetry.” Journal of High Energy Physics 05 (2021): 255. DOI. Open PDF.
• Weinberg, Steven. The Quantum Theory of Fields. Volume I: Foundations. Cambridge: Cambridge University Press, 1995. DOI.