Scattering Analyticity, Crossing, and Rigorous Bounds

Locality and the spectrum condition give scattering amplitudes genuine but limited analyticity domains once stable external states and LSZ reduction are available. For massive two-particle scattering, crossing between physical channels is a theorem through a connected complex domain under specific stability and mass assumptions. Fixed-$t$ dispersion relations and Froissart–Martin growth bounds require further polynomial boundedness, unitarity, and a nonzero crossed-channel mass gap; none survives automatically in a massless forward limit.

Required background. Counterexamples, nonconverses, and hypothesis stress tests fixes the logical boundaries; tube domains and complex Lorentz covariance gives off-shell analyticity; Jost points and edge-of-the-wedge joins boundary values; and LSZ reduction and amputated distributions supplies on-shell amplitudes.

Helpful background. Analyticity and crossing of amplitudes gives the physical continuation; fixed-$t$ and partial-wave dispersion gives applications; and S-matrix and amplitude bootstrap uses the resulting constraints.

From local correlators to on-shell domains

For identical stable scalars of mass $m>0$, write

$$s=(p_1+p_2)^2,\qquad t=(p_1-p_3)^2,\qquad u=4m^2-s-t.$$

Assume a local positive-metric theory, stable isolated external shells, the LSZ regularity hypotheses, and no unlisted lighter poles. Wightman/retarded distributions are boundary values of holomorphic functions in primitive momentum domains. Edge-of-the-wedge continuation and amputation transfer part of this analyticity to the mass shell.

For fixed real $t$ in a nonzero interval around $0$ inside the Lehmann–Martin domain, the elastic amplitude is analytic in the complex $s$ plane away from the physical right-hand cut $s\geq4m^2$ and the crossed-channel left-hand cut $s\leq-t$, with bound-state poles added if the spectrum contains them. The interval in $t$ is bounded by crossed-channel singularities; it is not the entire $t$ plane.

Bros, Epstein, and Glaser prove on-shell crossing for two-in/two-out stable particles of strictly positive masses and connect crossed physical regions through a domain of analyticity in Bros, Epstein, and Glaser 1965, pp. 240–264. This does not establish unrestricted multiparticle crossing or crossing in a theory with massless external or exchanged particles.

Fixed-t dispersion

If the amplitude is polynomially bounded at fixed allowed $t$, choose the number of subtractions $N$ large enough that the contour at infinity vanishes. Cauchy’s theorem gives

$$\begin{aligned} A(s,t)={}&P_{N-1}(s,t) +\frac{s^N}{\pi}\int_{4m^2}^{\infty} \frac{\operatorname{Im}_s A(s'+i0,t)} {(s')^N(s'-s)}\,\mathrm ds'\\ &+\frac{u^N}{\pi}\int_{4m^2}^{\infty} \frac{\operatorname{Im}_u A(u'+i0,t)} {(u')^N(u'-u)}\,\mathrm du', \end{aligned}$$

with pole terms included separately and subtraction variables chosen consistently. Crossing for identical scalars relates the two absorptive parts in their common analytic continuation. Temperedness motivates finite-order growth, but the exact uniform bound, domain, and permitted $N$ must be proved for the amplitude under consideration.

An independent check is discontinuity: approaching the right cut from above and below, the second integral and subtraction polynomial remain analytic, while the first integral yields $\operatorname{Disc}_s A=2i\,\operatorname{Im}_sA$. This fixes the factor $1/\pi$.

What the Froissart–Martin bound needs

Partial-wave unitarity bounds each elastic partial wave. Analyticity in the scattering angle inside a Lehmann–Martin ellipse, with a nearest crossed-channel singularity at $t_0>0$, makes high angular momenta decrease exponentially. A global polynomial bound limits the ellipse maximum. Together they leave only $O(\sqrt{s}\log s)$ significant partial waves in four dimensions, leading schematically to

$$\sigma_{\mathrm{tot}}(s) \leq \frac{C}{t_0}\log^2\!\left(\frac{s}{s_0}\right) \quad (s\to\infty),$$

with the sharp coefficient depending on normalization and the precise lightest crossed threshold. Froissart first obtained the logarithmic form under a Mandelstam-representation assumption Froissart 1961, pp. 1053–1057; Martin enlarged the axiomatic domain using unitarity in Martin 1966, pp. 930–953.

The licensed conclusion is a high-energy upper bound for the stated massive, local, unitary setting. It is not a prediction that a given theory saturates the bound, and it is not obtained from crossing alone.

Massive-scalar application and massless failure

For massive scalar $2\to2$ scattering, choose $t$ within the proven fixed-$t$ interval, write the subtracted representation above, and use the positive $t$-channel gap plus partial-wave unitarity before invoking a Froissart–Martin bound. The physical amplitude technology and domain map are developed on causality, growth, and analytic domains.

Now take a massless forward limit with an unsubtracted $t$-channel pole $A(s,t)\sim g^2s/t$. The singularity reaches $t=0$, so there is no nonzero $t_0$ and no uniform ellipse around the forward direction. The forward optical-theorem expression itself is singular. Moreover, an unsubtracted perturbative formula has not established the global polynomial bound and contour falloff needed for the fixed-$t$ dispersion argument; the limit is nonuniform at $t=0$. Both the gap/domain input and the required growth input fail. Applying the massive bound is therefore invalid.

Exercises

Why does polynomial boundedness determine the need for subtractions but not by itself establish crossing?

Solution

A growth bound controls the contour at infinity and tells us how many powers must be removed for a convergent Cauchy representation. Crossing instead requires analytic continuation connecting distinct channel boundary values, obtained from locality, spectrum, LSZ stability, and edge-of-the-wedge arguments. A polynomially bounded function can lack that connected domain.

References

• Bros, Jacques, Henri Epstein, and Vladimir Glaser. 1965. “A Proof of the Crossing Property for Two-Particle Amplitudes in General Quantum Field Theory.” Communications in Mathematical Physics 1: 240–264. DOI.
• Froissart, Marcel. 1961. “Asymptotic Behavior and Subtractions in the Mandelstam Representation.” Physical Review 123: 1053–1057. DOI.
• Martin, André. 1966. “Extension of the Axiomatic Analyticity Domain of Scattering Amplitudes by Unitarity—I.” Il Nuovo Cimento A 42: 930–953. DOI.