---
title: "Put two-dimensional scattering in a complex strip"
description: "Use rapidity to open the massive physical sheet, derive crossing and calculate the mass and residue of a fixed CDD pole."
canonical_url: "https://sandboxphysics.com/stories/scattering-amplitudes/analytic-strip-crossing/"
language: "en"
page_type: "story"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/stories/scattering-amplitudes/analytic-strip-crossing/"
last_modified: "2026-10-27"
---

[Scattering amplitudes](https://sandboxphysics.com/stories/scattering-amplitudes/)

L4 · Chapter 7

# Put two-dimensional scattering in a complex strip

Sandbox Physics · 2026-10-27

In one spatial dimension, elastic scattering of equal-mass particles has one independent kinematic variable. Rapidity opens its physical analytic sheet into a strip. Crossing, conservation of probability and bound-state poles can then be described in one complex plane.

## Before you begin

Amplitudes and probability conservation from Chapters 1–2, complex numbers, hyperbolic functions and simple poles. The gluon recursion is optional. This branch studies identical massive neutral particles and uses the corresponding scalar-element conventions.

[Review the preceding prerequisite](https://sandboxphysics.com/stories/scattering-amplitudes/partial-wave-unitarity/)

- Derive the threshold, crossing map and physical strip from rapidity.
- Calculate the bound-state mass, crossed pole and residue of the fixed CDD reference.

## One spatial dimension simplifies kinematics

Use one time and one space dimension with metric $(+,-)$, natural units $\hbar=c=1$ and particle mass $m$ as the energy unit. A single momentum is $p=m(\cosh\beta,\sinh\beta)$, since the squared hyperbolic cosine minus the squared hyperbolic sine equals one.

The incoming rapidity difference is $\theta=\beta_1-\beta_2$. In the center-of-mass frame the rapidities are opposite halves of this difference. The squared total energy therefore depends only on that one variable. Real rapidity describes physical scattering above the two-particle threshold.

$$
s=4m^2\cosh^2\!\frac\theta2=2m^2(1+\cosh\theta),\qquad s\geq4m^2\quad(\theta\in\mathbb R)
$$

The scalar elastic S-matrix element includes the identity contribution from free propagation. It differs from the earlier coupling-stripped gluon partial amplitude.

Source: M. F. Paulos · Patrick Dorey

## Crossing connects the two strip edges

Moving a particle line between final and initial states analytically maps $s$ to $4m^2-s$. For identical neutral particles, crossing equates these values. In rapidity the transformation is $\theta\mapsto i\pi-\theta$.

A complex square root requires a branch choice: approaching a cut from opposite sides can give different signs. The physical sheet between these cuts opens into $0<\operatorname{Im}\theta<\pi$, the physical strip. Real scattering lies on the lower edge; crossed scattering lies on the upper edge. Pure imaginary rapidity inside the strip describes real energy squared between thresholds, where stable-particle exchange may produce poles.

$$
S(\theta)=S(i\pi-\theta),\qquad S(s)=S(4m^2-s)
$$

Crossing relates analytic values. Mirroring a finite real-energy dataset does not determine the interior function.

$$
|S(\theta)|^2\leq1\qquad(\theta\in\mathbb R)
$$

One elastic element cannot carry more than the complete incoming probability. A purely elastic solution saturates this inequality.

Physical rapidity strip

$0<\operatorname{Im}\theta<\pi$

The lower edge represents real scattering. The paired poles use the fixed CDD reference; the figure is a schematic derived for this course. $\theta=i\pi/3,\;2i\pi/3\quad\Longleftrightarrow\quad s=3,\;1$

Source: M. F. Paulos

## Build a function unitary at every real rapidity

Choose real $0<\alpha<\pi/2$ and form the ratio below. On real rapidity its numerator and denominator are complex conjugates, giving unit modulus. The identity $\sinh(i\pi-\theta)=\sinh\theta$ also gives crossing. This is one CDD pole factor.

The denominator vanishes at $\theta=i\alpha$ and $i(\pi-\alpha)$. The first gives exchanged mass $m_1=2m\cos(\alpha/2)$; the second is its crossed partner. Boundary probability and interior spectrum appear in the same function.

$$
C_\alpha(\theta)=\frac{\sinh\theta+i\sin\alpha}{\sinh\theta-i\sin\alpha},\qquad C_\alpha(\theta)C_\alpha(-\theta)=1
$$

This is the positive single-factor choice of Paulos et al., equation (14). Its pole structure must agree with the assumed spectrum.

$$
s(i\alpha)=4m^2\cos^2\!\frac\alpha2=m_1^2,\qquad s(i(\pi-\alpha))=4m^2-m_1^2
$$

Crossing fixes the second pole location once the first has been chosen.

Source: M. F. Paulos

## Worked example with exchanged mass squared three

Set $m=1$ and $\alpha=\pi/3$. The poles lie at one and two thirds of the strip height, corresponding to $s=3$ and $s=1$. At the first pole the denominator derivative is $\cos\alpha=1/2$ and the numerator is $i\sqrt3$.

The rapidity residue is therefore $2i\sqrt3$. Residues change under a change of variable: multiply by $ds/d\theta=i\sqrt3$ to obtain the residue $-6$ in energy squared. Confusing these two numbers gives an incorrect coupling normalization.

$$
\operatorname*{Res}_{\theta=i\alpha}C_\alpha=2i\tan\alpha,\qquad\operatorname*{Res}_{s=3}S=\left(2i\sqrt3\right)\left(i\sqrt3\right)=-6
$$

The variable change uses the derivative at the pole. Masses are measured relative to the external-particle mass.

$$
S(s)\simeq-\frac{g_1^2}{2m_1\sqrt{4-m_1^2}}\frac{1}{s-m_1^2}\quad\Longrightarrow\quad g_1^2=12\sqrt3\qquad(m_1=\sqrt3)
$$

This dimensionless coupling follows precisely the paper’s equation (3). Other state or connected-amplitude normalizations require conversion.

Source: M. F. Paulos

## Necessary conditions leave further questions

Crossing, a fixed analytic domain and the elastic modulus exclude many candidate functions. Completing every survivor into a quantum field theory would also require other scattering channels, thresholds, residue signs and higher processes. The final chapter separates a finite analytic calculation from numerical sampling and from theory existence.

The single CDD reference describes the sine-Gordon lightest-breather scattering discussed in the source. We reproduce its analytic function and a specified parameter curve. Establishing the full model spectrum would require additional processes and information.

Source: M. F. Paulos

## Try the reasoning in an experiment

### [Follow one analytic function along the physical edge](https://sandboxphysics.com/labs/s-matrix-bootstrap/)

1. Choose the exact CDD reference and predict the modulus of its real-rapidity trajectory. Follow its motion in the complex plane.
2. Compare the crossed poles in this chapter’s static strip diagram and calculate their squared masses three and one. The experiment’s probe slider follows the real-rapidity boundary.
3. Change coefficients of the finite ansatz and identify the first necessary condition that fails.

## Exercises and explanations

1. What happens at $\alpha=\pi/2$? Can the simple-pole residue formula still be used directly?

   Read the explanation

   Both pole locations meet at $i\pi/2$, with $s=2m^2$. The denominator’s first derivative vanishes, so a simple-pole expansion fails and the function must be expanded again. Bounds for a fixed separated simple-pole spectrum cannot be continued to this degenerate point without analysis.

2. Does observing unit modulus or less at finitely many real rapidities establish the absence of extra poles inside the strip?

   Read the explanation

   No. Those samples constrain the modulus at their locations. Interior singularities require an analytic argument or another method controlling the whole domain. The next chapter makes the distinction visible in a finite family.

[Previous: Reconstruct a tree amplitude from its poles](https://sandboxphysics.com/stories/scattering-amplitudes/on-shell-recursion/) [Next: Shrink a family of S matrices and reproduce a published curve](https://sandboxphysics.com/stories/scattering-amplitudes/finite-bootstrap-benchmarks/)

## Sources and reproduced scope

Worked examples and schematic figures are authored for this course. Numerical values are calculated model references. The published comparison recalculates a specified analytic CDD curve; it does not copy experimental data, source artwork or the paper’s numerical optimization points. Source locations and reproduced scope are listed below.

1. [M. F. Paulos, J. Penedones, J. Toledo, B. C. van Rees and P. Vieira, The S-matrix Bootstrap II: Two Dimensional Amplitudes, arXiv:1607.06110v2; JHEP 11 (2017) 143](https://arxiv.org/html/1607.06110v2)

   Section 2 equations (1)–(3) fix crossing, elastic unitarity and coupling normalization. Equations (9), (13)–(14), Figures 5a and 6 give the sine-Gordon reference, CDD factor and physical strip. We recalculate the analytic Figure 5a reference at external mass one and exchanged mass square root of three, its pole residue and the complex value at energy squared six. We do not reproduce the dispersive optimization of section 2.1/Appendix A. The rational three-parameter family is our extension.

2. [Patrick Dorey, Exact S-matrices, arXiv:hep-th/9810026v1 (1998)](https://arxiv.org/pdf/hep-th/9810026v1)

   Section 3, printed pages 12–16, derives the rapidity map, physical strip, real analyticity, matrix unitarity and crossing. Section 4, pages 16–21, relates imaginary rapidity to bound-state masses and discusses sine-Gordon poles, including the limits of identifying every pole with a new particle. The course uses the scalar identical-particle specialization; its finite rational ansatz is separately defined.