Forward Scattering Sum Rules

In a gapped forward amplitude, crossing combines the two cuts and the optical theorem turns the right-cut discontinuity into inclusive production data. Even low-energy Taylor coefficients then become inverse-energy moments of a physical absorptive part; crossing-odd combinations instead give generally signed sum rules unless extra structure supplies a sign.

Required background. Subtracted dispersion relations supplies the contour formula and subtraction count. The optical theorem and cut interpretation supplies the inclusive state sum and its normalization.

The crossing-even forward moment expansion

Consider identical massive scalars and first remove all stable one-particle poles below the two-particle threshold. At $t=0$ define

$$\nu=s-2m^2, \qquad B(\nu)=B(-\nu), \qquad \nu_0=2m^2,$$

so the cuts begin at $\nu=\pm\nu_0$. Assume real analyticity, no other first-sheet singularities, and $B(\nu)/\nu^2\to0$ on the large complex arc. The twice-subtracted relation about $\nu=0$ is

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

The factor of two is the combined right- and left-cut contribution. Its sign can be checked by expanding both kernels $1/(\nu'-\nu)$ and $1/(\nu'+\nu)$ before combining them. This forward crossing construction and its subtraction dependence are developed in Weinberg 1995, § 10.8, pp. 465–469.

For $|\nu|<\nu_0$, expand the denominator geometrically:

$$\begin{aligned} B(\nu)&=B(0)+\sum_{n=1}^{\infty}c_{2n}\nu^{2n},\\ c_{2n}&=\frac{2}{\pi} \int_{\nu_0}^{\infty} \frac{\operatorname{Im}A(2m^2+\nu'+i0,0)} {\nu'^{2n+1}}\,\mathrm d\nu'. \end{aligned}$$

Thus $c_{2n}=B^{(2n)}(0)/(2n)!$ is an inverse-energy moment. The first undetermined subtraction constant is $B(0)$; crossing has removed the linear one. More subtractions would leave more Taylor data outside the integral.

Inclusive data and convergence

With the common relativistic normalization in four dimensions, the forward optical theorem reads

$$\operatorname{Im}A(s+i0,0) =\sqrt{s(s-4m^2)}\,\sigma_{\mathrm{tot}}(s)\ge0, \qquad s\ge4m^2.$$

Other amplitude normalizations change the positive kinematic factor, not the sign. The moment therefore sums every on-shell final state accessible from the chosen initial state. It is not just the elastic cross section and is not a tree-level identity. Adams and collaborators use exactly this inclusive input to turn a forward contour integral into a positive coefficient Adams et al. 2006, § 4, pp. 14–19, PDF.

Convergence is checked twice:

• Near threshold, phase space and any threshold singularity must make the weighted integral locally integrable.
• At infinity, the power $\nu'^{-(2n+1)}$ must beat the absorptive growth. The same growth estimate must also justify discarding the original large arc.

A convergent moment does not retroactively justify the contour step: an amplitude could have acceptable real-axis data but uncontrolled growth in other complex directions.

The contour figure can now be read as a sum rule. The right and left lips become the same inclusive spectrum only after crossing; the small pole circles must be removed before a Taylor coefficient is identified.

Schematic forward contour, not to scale. Crossing maps the left cut to a physical crossed-channel cut, the optical theorem supplies its inclusive absorptive data, and two subtractions leave $B(0)$ while determining the even coefficients $c_{2n}$ for $n\ge1$. The dashed arc still requires the stated high-energy bound.

Contour contribution	Forward sum-rule meaning
right cut	inclusive $s$-channel production
left cut	crossed-channel production, with crossing matrix or parity retained
pole circles	known stable exchange terms, removed explicitly
subtraction point	low-energy constants not fixed at the chosen subtraction order

Crossing-odd sum rules

For particles with charge, flavor, or spin, crossing acts on a vector of amplitudes. In a crossing eigenchannel write

$$F_\eta(-\nu)=\eta F_\eta(\nu), \qquad \eta=\pm1.$$

$F_+$ has the even moment expansion above. For $F_-$, the ratio $F_-(\nu)/\nu$ is even, but its right-cut absorptive part is a crossing-weighted combination of physical processes. In ordinary particle–antiparticle examples the resulting integrand contains a difference of inclusive cross sections. Unitarity makes each cross section nonnegative, not their difference. Consequently an odd sum rule can be predictive without being a positivity bound.

If the odd amplitude decreases unusually fast, an otherwise free odd subtraction constant may obey a superconvergent sum rule. That statement requires the stronger falloff explicitly; it must not be inferred from crossing alone.

Moment checks

Write $\rho(\nu')=\operatorname{Im}A(2m^2+\nu'+i0,0)$. Because $\rho\ge0$,

$$c_{2n}>0$$

whenever the chosen channel has nonzero absorptive weight. Cauchy–Schwarz supplies a stronger internal check,

$$c_{2n+2}^{2}\le c_{2n}c_{2n+4},$$

and support on $\nu'\ge\nu_0$ gives

$$0<c_{2n+2}\le\frac{c_{2n}}{\nu_0^2}.$$

Indeed, with

$$\mathrm d\mu_n(\nu') =\frac{2}{\pi} \frac{\rho(\nu')}{\nu'^{2n+1}}\,\mathrm d\nu',$$

the three entries are $\int1\,\mathrm d\mu_n$, $\int\nu'^{-2}\mathrm d\mu_n$, and $\int\nu'^{-4}\mathrm d\mu_n$. Cauchy–Schwarz gives the first inequality, while $\nu'^{-2}\le\nu_0^{-2}$ gives the second. They are consequences of the positive measure and its support, not additional assumptions. The systematic moment formulation and its finite Hankel constraints are given in Bellazzini et al. 2021, § I.A–B, pp. 2–5, PDF.

Exercises

Put all spectral weight at one narrow scale $M^2=2m^2+\nu_M$. Show that $c_{2n}\propto\nu_M^{-(2n+1)}$ and that the Cauchy–Schwarz inequality is saturated.

Check

For $\rho(\nu')=w\,\delta(\nu'-\nu_M)$ with $w>0$,

$$c_{2n}=\frac{2w}{\pi\nu_M^{2n+1}},$$

so $c_{2n+2}^2=c_{2n}c_{2n+4}$. A mixture of distinct scales generally moves the sequence into the interior of the moment region.

Continue to the sign theorem: Forward-Limit Positivity Bounds. For nonzero momentum transfer: Fixed-t and Partial-Wave Dispersion.

References

• Adams, Allan, Nima Arkani-Hamed, Sergei Dubovsky, Alberto Nicolis, and Riccardo Rattazzi. “Causality, Analyticity and an IR Obstruction to UV Completion.” Journal of High Energy Physics 10 (2006): 014. DOI. Open PDF.
• Bellazzini, Brando, Joan Elias Miró, Riccardo Rattazzi, Marc Riembau, and Francesco Riva. “Positive Moments for Scattering Amplitudes.” Physical Review D 104 (2021): 036006. DOI. Open PDF.
• Weinberg, Steven. The Quantum Theory of Fields. Volume I: Foundations. Cambridge: Cambridge University Press, 1995. DOI.