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.

Fitted signal strength $0.896$

Local significance $6.46\,\sigma$

Asymptotic 95% interval $[0.60,1.22]$

Selected recorded events $1155$

A peak and its alternatives

How convincing is the excess?

Injected signal strength

\mu_{\rm true}

1

Relative exposure

\mathcal L/\mathcal L_0

1

Background uncertainty scale

\delta_b

0.15

Fixed search mass

m_0

1250\,\mathrm{GeV}

The test is local at the chosen mass. Searching many masses needs a trials correction.

Model, limits and recorded sample

Sixteen mass bins span 400–2000 GeV. The expected background falls exponentially; a Gaussian auxiliary measurement constrains a single parameter that changes its normalization and slope. Each acquisition saves integer Poisson counts and a fresh auxiliary measurement. The injected narrow peak always remains at 1250 GeV.

The local test allows only a positive signal; a downward fluctuation gives zero discovery significance. The 95% interval uses the large-sample likelihood threshold. Small samples and an omitted background shape require calibration with independent pseudoexperiments.

The EFT preset multiplies the reference background by the positive excess below. Its fixed reference scale is 2000 GeV. The fitted multiplier measures this toy shape only; whole bins above the chosen validity cut are omitted.

\begin{aligned}x_i&=\left(\frac{m_i}{2000\,\mathrm{GeV}}\right)^2\\\Delta N_i&=b_i(2x_i+x_i^2)\end{aligned}

Cowan et al. (2011)

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.

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 misconceptionA large significance rules out every background explanation.

Better mental modelIt 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 misconceptionThe largest significance found while dragging the mass marker is already the significance of the whole search.

Better mental modelEvery 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 misconceptionThe fitted toy strength is a bound on an actual Wilson coefficient.

Better mental modelThe 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

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.
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.
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.
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.
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.

Collider Signals and Systematic Errors

How much signal can fit?

Where the excess sits

How a spectrum becomes a statistical test

Core mathematical model

Count model and auxiliary measurement

Profile the nuisance parameter

Test for a positive signal at one mass

Common difficulties

A significance assumes a model

Searching many masses changes the question

An EFT shape needs a validity range

Run the experiment

Find the narrow peak

Repeat the acquisition

Let the background move

Fit an incomplete explanation

Restrict a growing tail

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