---
title: "Collider Signals and Systematic Errors Interactive Lab | Sandbox Physics"
description: "When does a bump count as evidence for a new signal? A likelihood compares signal and background predictions with the observed counts."
canonical_url: "https://sandboxphysics.com/labs/collider-likelihood/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/collider-likelihood/"
---

Collider physics · counting evidence

# Collider Signals and Systematic Errors

**When does a bump count as evidence for a new signal?** A likelihood compares signal and background predictions with the observed counts. Fitting uncertain background parameters can change the local significance; repeated synthetic experiments reveal how often the stated intervals cover the input.

Fit a small excess in a falling synthetic mass spectrum. Let a background uncertainty move, compare repeated samples and test how an omitted background feature can imitate a signal.

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

Model assumptions

Synthetic counts with a fixed-mass asymptotic test. Open model notes for nuisance assumptions, interval limits and the toy EFT interpretation.

### How much signal can fit?

At each trial signal strength, the background parameter is fitted again. The dashed threshold gives an asymptotic 95% interval.

### Where the excess sits

The bars compare recorded counts with the fitted background. These residuals are descriptive; their height is not a discovery significance.

[From a collision to jets and inference: eight connected chapters](https://sandboxphysics.com/stories/collider-qcd/)

Start with four-momentum, work through clustering, response matrices and likelihoods, then check extrapolation against a public jet spectrum.

Physics tutorial

## How a spectrum becomes a statistical test

**Background** A collider spectrum contains an integer number of events in each mass bin. Even a perfectly understood background fluctuates from one acquisition to the next.

**Why it matters** A small excess may come from a signal, a fluctuation or an incomplete background model. A likelihood compares those explanations using the whole spectrum and any auxiliary measurement.

Start with the essentials

- **Focus question**: How much evidence does a bump provide when the background can move?
- **One-sentence intuition**: The background must be fitted alongside the signal, and a significance is meaningful only for the model and search that were actually tested.

### Core mathematical model

#### Count model and auxiliary measurement

$$
\begin{aligned}\nu_i&=\mu s_i+b_i(\theta)\\L&=\prod_i\operatorname{Pois}(n_i\mid\nu_i)\,\exp[-(a-\theta)^2/2]\end{aligned}
$$

The non-negative signal strength scales a fixed template. One nuisance parameter changes the background normalization and slope; a separately simulated Gaussian auxiliary measurement constrains that parameter.

#### Profile the nuisance parameter

$$
\lambda(\mu)=\frac{L(\mu,\hat{\hat\theta}(\mu))}{L(\hat\mu,\hat\theta)}
$$

At each trial signal strength, fit the background parameter again. This is why the profile curve can broaden when the background becomes uncertain.

#### Test for a positive signal at one mass

$$
\begin{aligned}q_0&=-2\log\lambda(0)\\Z_{\rm local}&\simeq\sqrt{q_0}\end{aligned}
$$

The fitted strength is constrained to be non-negative. A downward fluctuation stops at zero strength and gives zero discovery significance. The square-root conversion uses the asymptotic approximation.

### Common difficulties

#### A significance assumes a model

**Typical misconception** A large significance rules out every background explanation.

**Better mental model** It tests the chosen background family. The false-signal preset adds a broad feature that the fitted background cannot express. A narrow signal can absorb part of that missing feature and acquire a large local significance.

#### Searching many masses changes the question

**Typical misconception** The largest significance found while dragging the mass marker is already the significance of the whole search.

**Better mental model** Every mass is another opportunity for a fluctuation. The displayed number is local at the current mass; a global search needs repeated background-only searches or another justified trials correction.

#### An EFT shape needs a validity range

**Typical misconception** The fitted toy strength is a bound on an actual Wilson coefficient.

**Better mental model** The broad tail is a declared quadratic teaching template with a fixed reference scale. Only complete bins below the adjustable cutoff are fitted. A physical interpretation needs a specified operator basis, interference terms, order counting and data analysis.

### Run the experiment

1. 01

   #### Find the narrow peak

   Start with A narrow signal and drag the gold search-mass marker.

   **What to observe:** The injected peak stays at the same mass. A trial template farther away explains less of its excess.

2. 02

   #### Repeat the acquisition

   Choose Background alone, then press Repeat acquisition several times.

   **What to observe:** Poisson counts and the auxiliary measurement fluctuate. A positive fitted strength can occur even when the injected signal is zero.

3. 03

   #### Let the background move

   Increase Background uncertainty scale and inspect the profile curve.

   **What to observe:** A larger allowed background change can explain some of the excess and alter the fitted interval.

4. 04

   #### Fit an incomplete explanation

   Choose A false signal, then compare the residuals with the fitted signal strength.

   **What to observe:** The data were generated with no signal. An omitted background bump nevertheless gives the signal template something to fit.

5. 05

   #### Restrict a growing tail

   Choose A rising EFT tail and drag the gold validity cut.

   **What to observe:** Grey bins remain visible but are excluded from the likelihood. The inferred strength and interval use only the retained range.

Physical conclusion

A convincing signal survives alternative background descriptions and calibrated repeated-sample checks. A fixed-mass local statistic is one step in that argument.

## Related experiments and articles

Collider physics · Reading path

### Collider physics from events to inference

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

1. [Article **Decide whether an excess needs a new signal**](https://sandboxphysics.com/stories/collider-qcd/profile-likelihood-and-coverage/)
2. [Article **Fit an effective interaction without outrunning its expansion**](https://sandboxphysics.com/stories/collider-qcd/bounded-eft-inference/)
3. [Article **Recalculate a published benchmark and test a public jet spectrum**](https://sandboxphysics.com/stories/collider-qcd/reading-and-reproducing-a-paper/)
4. [Lab **Jet Clustering Microscope** Cluster a soft or collinear event](https://sandboxphysics.com/labs/jet-clustering/)

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)