---
title: "Gluon Scattering Amplitudes Interactive Lab | Sandbox Physics"
description: "Why can changing a polarization reference move every diagram and leave their sum fixed?"
canonical_url: "https://sandboxphysics.com/labs/gluon-amplitudes/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/gluon-amplitudes/"
---

FR05-A · How three diagrams become one expression

# Gluon Scattering Amplitudes

**Why can changing a polarization reference move every diagram and leave their sum fixed?** A polarization reference changes how each diagram describes the same transverse gluon. The exchange diagrams and contact term cancel the reference dependence. Their sum agrees with the compact helicity expression.

Drag an outgoing beam and compare three Feynman contributions with one helicity formula. Follow a fifth gluon toward soft or collinear emission and see factorization emerge.

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

Model assumptions

The model evaluates coupling-stripped, color-ordered, massless tree amplitudes with two negative helicities. The momentum arrows show directions, with lengths rescaled for visibility. Soft and collinear comparisons use an explicitly mapped on-shell four-point state. Loop corrections, color sums, parton showers and detector events are outside this calculation.

Analytic tree model

## The three arrows add to the same answer

First exchange$s_{12}$

$5.83+0.875i$

Second exchange$s_{23}$

$-5.98+1.09i$

Contact graph$\text{contact}$

$0.147-0.478i$

First exchange Second exchange Contact graph Sum / helicity answer

Complex arrows are placed head to tail. The white endpoint is the compact helicity result. Changing the reference moves the intermediate endpoints.

Numerical details

$\sum_i k_i^\mu=0,\qquad k_i^2=0$

The four-point condition is the sum of the magnitudes of the three graph contributions divided by the magnitude of their sum. In factorization modes it is the inverse opening-angle scale of the adjacent spinor bracket. Both are local estimates, not rigorous forward-error bounds.

[Dixon, A brief introduction to modern amplitude methods (2013), sections 3–5](https://arxiv.org/abs/1310.5353v1)

Physics tutorial

## Gluon Scattering Amplitudes

**Background** A gluon has two transverse polarization states. Helicity tells whether its spin points along or opposite its momentum. Fixing those states can turn a long tensor calculation into a short complex number.

**Why it matters** Individual Feynman graphs depend on a freely chosen polarization reference. Adding every graph removes that dependence. This gives a stringent test before trusting a compact formula.

Start with the essentials

- **Focus question**: Why can changing a polarization reference move every diagram and leave their sum fixed?
- **One-sentence intuition**: The simple helicity answer contains the cancellation that the separate Feynman graphs perform explicitly.

### Core mathematical model

#### Four-point amplitude with two negative helicities

$$
\begin{aligned}A_4(1^-,2^-,3^+,4^+)&\\&=i\frac{\langle12\rangle^4}{\langle12\rangle\langle23\rangle\langle34\rangle\langle41\rangle}\end{aligned}
$$

Angle brackets are antisymmetric products of two-component momentum spinors. Their product with the reversed square bracket gives the pair invariant. All momenta are outgoing; incoming energies are negative. Couplings and the color trace are stripped.

#### Replacing a polarization by its momentum

$$
A_4(\ldots,\varepsilon_j\mapsto k_j,\ldots)=0
$$

The three diagram contributions are contracted again after the replacement, for each of the four legs. The displayed maximum is normalized by the larger of one and the compact amplitude magnitude.

#### Adding a soft positive-helicity gluon

$$
\begin{aligned}A_5(1,2,3,5^+,4)&\\\underset{E_5\to0}{\longrightarrow}\;&\frac{\langle34\rangle}{\langle35\rangle\langle54\rangle}A_4(1,2,3,4)\end{aligned}
$$

The soft factor depends on the two neighbors in the color order. The finite-energy comparison maps the hard pair to back-to-back unit-energy momenta, so its relative residual tends to zero as the emission softens.

### Common difficulties

#### A graph is one part of the calculation

**Typical misconception** Each graph should already be independent of the polarization reference.

**Better mental model** The reference cancels in the sum required by gauge symmetry. Dropping the contact graph generally spoils that cancellation.

#### A large amplitude needs careful arithmetic

**Typical misconception** A small factorization residual guarantees numerical stability at the pole.

**Better mental model** The residual tests the approximation at the selected point. Small spinor brackets or cancellations can amplify roundoff separately; the condition readout helps reveal this.

### Run the experiment

1. 01

   #### Change the reference

   Choose Three graphs, one answer. Open Polarization reference and move its direction through a full turn. Watch the three arrows in the data row.

   **What to observe:** The separate arrows move; their sum stays on the compact result. The Ward residual remains near floating-point roundoff.

2. 02

   #### Make one gluon soft

   Choose A gluon loses its energy. Predict the trend, then reduce the emitted energy by a factor of ten at a time.

   **What to observe:** The amplitude grows, yet the relative difference from the soft approximation falls. The log plot shows this convergence across the full selected scan.

3. 03

   #### Bring two directions together

   Choose Two outgoing beams merge. Keep the emitted energy fixed and reduce the opening angle. Save a snapshot with the full scan.

   **What to observe:** The adjacent pair invariant tends to zero. The collinear splitting approximation improves, while the spinor-bracket condition estimate grows.

Physical conclusion

Check the full amplitude with gauge invariance and factorization, then use the compact expression to understand why the answer is simple.

## 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 **Why split a momentum into two spinors?**](https://sandboxphysics.com/stories/scattering-amplitudes/helicity-and-color/)
2. [Article **Why four-gluon scattering reduces to one line**](https://sandboxphysics.com/stories/scattering-amplitudes/four-gluon-ward/)
3. [Article **Why a soft gluon forgets most of the collision**](https://sandboxphysics.com/stories/scattering-amplitudes/soft-collinear-factorization/)
4. [Article **Reconstruct a tree amplitude from its poles**](https://sandboxphysics.com/stories/scattering-amplitudes/on-shell-recursion/)

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)