---
title: "S-matrix Bootstrap Interactive Lab | Sandbox Physics"
description: "How much does consistency tell us about a scattering amplitude?"
canonical_url: "https://sandboxphysics.com/labs/s-matrix-bootstrap/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/s-matrix-bootstrap/"
---

FR05-B · Which scattering functions survive?

# S-matrix Bootstrap

**How much does consistency tell us about a scattering amplitude?** Crossing relates two descriptions of the same process, the pole spectrum fixes which stable particles may appear, and unitarity limits the elastic probability. Together they leave fewer candidate scattering functions.

Drag a proposed amplitude through a coefficient plane. Watch crossing, the pole spectrum and probability conservation remove different parts of the plane.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

Model assumptions

This finite rational ansatz tests necessary conditions on an identical-particle elastic S-matrix element in one spatial dimension. The external mass is one and the prescribed bound-state mass is the square root of three. Analyticity includes the fixed physical-sheet pole spectrum and regular high-energy and threshold boundaries. Generic subunitary candidates lose elastic probability down to threshold. With a unique lightest external species and no other channel open at that threshold, a complete theory has a purely elastic interval; this ansatz does not enforce it or supply inelastic channels. The allowed interval is exact within this ansatz; it does not classify complete quantum field theories. Curves and exported values are calculated model data.

Finite ansatz · calculated curve

## Following the amplitude around the complex plane

Candidate Exact elastic CDD reference Selected rapidity

$\theta=1.00,\quad s/m^2=5.09$

The violet reference has unit magnitude at every real rapidity. A candidate outside the unit circle assigns too much probability to the elastic channel.

Paulos et al. · 1607.06110v2 · Figure 5a

## Two variables, one analytic scattering function

The reference uses the one-pole CDD solution at bound-state mass square root of three. The independent energy-domain formula below reproduces the analytic curves compared in Figure 5a; the paper’s numerical optimizer is outside this reproduction.

Real part Imaginary part Magnitude

$$
S(s)=\frac{\sqrt{s}\sqrt{4-s}+\sqrt3}{\sqrt{s}\sqrt{4-s}-\sqrt3}
$$

Lines use the energy-domain formula; dots independently use the rapidity CDD factor. Above threshold the physical upper rim uses the negative imaginary square root. Both calculations, their grid and discrepancies are saved in the model snapshot.

$\max|S_s-S_\theta|<10^{-12}$

[Source equations 9 and 14](https://arxiv.org/html/1607.06110v2#S2.SS1) [Read the course](https://sandboxphysics.com/stories/scattering-amplitudes/)

Physics tutorial

## S-matrix Bootstrap

**Background** Two identical particles approach, scatter and separate. In one spatial dimension the elastic outcome has a single complex coefficient: the S-matrix element. Its squared magnitude is the probability left in that channel.

**Why it matters** Writing a plausible complex function is easy. Requiring that it describe the same crossed process, contain only the specified stable-particle poles and conserve probability makes the choice much smaller.

Start with the essentials

- **Focus question**: How much does consistency tell us about a scattering amplitude?
- **One-sentence intuition**: A function can conserve probability on the real energy axis and still have forbidden poles inside the physical sheet.

### Core mathematical model

#### Putting the physical sheet inside a disk

$$
\begin{aligned}s&=4m^2\cosh^2(\theta/2),\quad m=1\\z&=\tanh\!\left(\frac{\theta-i\pi/2}{2}\right)\\\theta\mapsto i\pi-\theta&\quad\Longrightarrow\quad z\mapsto-z\end{aligned}
$$

Rapidity labels the collision energy. The physical sheet becomes the strip between zero and pi in imaginary rapidity; the change to z maps its interior into the unit disk. Crossing exchanges the two sides of that disk.

#### A fixed bound state and a finite candidate family

$$
\begin{aligned}C_\alpha(\theta)&=\frac{\sinh\theta+i\sin\alpha}{\sinh\theta-i\sin\alpha},\quad\alpha=\frac\pi3\\S_{a,b,r}(\theta)&=C_\alpha(\theta)\frac{r+iaz}{1-bz^2}\\m_1&=2\cos(\alpha/2)=\sqrt3\end{aligned}
$$

The CDD factor supplies the prescribed bound-state pole and its crossed partner. The rational multiplier changes the amplitude. Exact crossing sets a to zero. Keeping the prescribed pole spectrum and regular boundaries requires the magnitude of b to be smaller than one.

#### Testing probability conservation everywhere

$$
\begin{aligned}t&=\operatorname{sech}\theta\in[0,1]\\F(t)&=r^2+a^2-(1-b)^2+2rat-4bt^2\\|S(\theta)|\le1\quad&\Longleftrightarrow\quad\max_{0\le t\le1}F(t)\le0\\a=0,\ |b|<1\quad&\Longrightarrow\quad |b|\le1-r\end{aligned}
$$

For this ansatz unitarity reduces to the maximum of one quadratic polynomial. Checking both endpoints and its interior maximum tests the entire real rapidity axis, including the high-energy limit. The colored region uses this algebraic test; the plots sample the same function.

### Common difficulties

#### An allowed amplitude is only one consistency test

**Typical misconception** Every allowed point defines a complete quantum field theory.

**Better mental model** The test concerns one elastic matrix element and a finite ansatz. Other scattering channels, residue signs and the existence of a full theory require additional conditions.

#### A line really has zero width

**Typical misconception** The bright line is a narrow tolerance band.

**Better mental model** Crossing is an exact functional identity. The plot draws the zero-width set with a visible stroke; dragging near its center snaps the odd coefficient to zero.

### Run the experiment

1. 01

   #### Remove one requirement

   Predict what happens if you uncheck Crossing. Then drag the gold point sideways.

   **What to observe:** A two-dimensional region opens up. Restoring crossing leaves only the vertical line because the odd coefficient must vanish exactly.

2. 02

   #### Look away from the real axis

   Choose Hidden physical-sheet poles. Uncheck Fixed pole spectrum while keeping Unitarity enabled.

   **What to observe:** The candidate passes boundary probability conservation. Restoring the pole requirement rejects it: two additional denominator zeros sit inside the disk.

3. 03

   #### Reach the elastic solution

   Choose Elastic CDD solution. Lower Numerator scale and predict whether the allowed interval grows or shrinks.

   **What to observe:** At unit scale the allowed interval contracts to a point, and the complex trajectory lies on the unit circle. A smaller scale opens an interval inside the disk. Completing such a function with physical inelastic channels needs further conditions.

Physical conclusion

Check the complex domain as well as the physical energy axis. Each condition removes a different kind of false candidate.

## Related experiments and articles

Scattering amplitudes · Reading path

### From probability flow to scattering amplitudes and bootstrap

Articles on this path that use this experiment, followed by the other experiments they suggest.

1. [Article **Put two-dimensional scattering in a complex strip**](https://sandboxphysics.com/stories/scattering-amplitudes/analytic-strip-crossing/)
2. [Article **Shrink a family of S matrices and reproduce a published curve**](https://sandboxphysics.com/stories/scattering-amplitudes/finite-bootstrap-benchmarks/)

Quantum & fields · Starter path

## Start from a single measurement

Predict an outcome, then count what the measurements actually give. The path starts with one qubit and moves on to wave packets, bound states and two-photon interference.

[Start here → **Quantum State & Born Measurement** Predict a probability, turn the analyzer, then compare your prediction with the measured frequency over many shots. Observe · 15 min](https://sandboxphysics.com/labs/qubit-state/)

1. [1 Quantum State & Born Measurement](https://sandboxphysics.com/labs/qubit-state/)
2. [2 Stern–Gerlach Sequence](https://sandboxphysics.com/labs/stern-gerlach-sequence/)
3. [3 Wave Packet & Uncertainty](https://sandboxphysics.com/labs/wave-packet-uncertainty/)
4. [4 Quantum Harmonic Oscillator](https://sandboxphysics.com/labs/quantum-harmonic-oscillator/)
5. [5 Which Path & Quantum Eraser](https://sandboxphysics.com/labs/quantum-eraser/)
6. [6 Hong–Ou–Mandel Two-Photon Interference](https://sandboxphysics.com/labs/hong-ou-mandel/)

[View the full path →](https://sandboxphysics.com/topics/quantum/#starter-path)