Collider physics · recovering a mass spectrum

Detector Response and Unfolding

Can two nearby peaks survive a blurred measurement? A response matrix connects the original spectrum to the detector record. Unfolding estimates the original distribution, but finite resolution and a mismatched response can amplify noise or bias the result.

Follow saved simulated events from true mass to reconstructed mass. Change the detector resolution, then unfold the recorded spectrum and compare it with the known simulated truth.

Recorded events $1234$

Unfolded event yield $1514$

Error against simulated truth $14.2\%$

Refolded Poisson deviance $17.97$

From events to a spectrum

Blur, record, recover

Mass resolution

\sigma_m

6\,\mathrm{GeV}

True peak separation

\Delta m

28\,\mathrm{GeV}

Unfolding iterations

k

8

Detection efficiency

\epsilon

0.82

The calibration and acquisition use independent events.

Model and saved events

A one-dimensional Gaussian detector has constant detection efficiency. The response is fixed; calibration uncertainty is not propagated. Unfolding starts from a flat positive cause-bin prior, and its iteration count controls regularization. The animation repeats 70 recorded events. A true mass and its reconstructed mass share the same event ID. Faint paths fade when the event was lost. The spectrum uses all 1500 generated events; calibration uses 700 events in each true bin.

A good refolded fit can still hide a biased unfolded answer. The gold curve is an estimate; cyan is the separately labelled simulated truth.

D’Agostini (2010)

Model assumptions

Synthetic mass records with a Gaussian detector response. Open the model notes for calibration, prior and uncertainty assumptions.

The mass spectrum

Simulated truthRecorded countsUnfolded estimate

Bars show square-root count errors on the recorded bins; unfolded uncertainties are not shown.

Where each true bin is recorded

$R_{ji}=P(j\mid i)$

Brighter cells mean a larger probability. Each column sums to its acceptance; lost events stay outside the matrix.

Start with the essentials

Focus question: What can a blurred mass spectrum tell us about two nearby structures?
One-sentence intuition: Unfolding reallocates recorded counts using a calibrated response; its apparent sharpness depends on the information in the data and on regularization.

Core mathematical model

From true bins to recorded bins

\nu_j=\sum_i R_{ji}t_i,\qquad \epsilon_i=\sum_j R_{ji}

The column index labels true mass and the row labels reconstructed mass. A column can sum to less than one because detection and the displayed mass window lose events.

One iterative update

\begin{aligned}\nu_j^{(k)}&=\sum_\ell R_{j\ell}t_\ell^{(k)}\\t_i^{(k+1)}&=\frac{t_i^{(k)}}{\epsilon_i}\sum_j R_{ji}\frac{n_j}{\nu_j^{(k)}}\end{aligned}

Start with a positive flat cause-bin spectrum. Each update allocates observed counts among their possible causes and corrects for efficiency. Stopping early smooths the answer; more iterations can amplify noise.

Compare the estimate with simulated truth

e=\frac{\sqrt{\sum_i(\hat t_i-t_i)^2}}{\sqrt{\sum_i t_i^2}}

This relative error uses the true histogram saved by the simulator. A real measurement cannot compute it from recorded counts alone.

Common difficulties

A good refolded spectrum can conceal a wrong answer

Typical misconceptionIf the unfolded result reproduces the measured spectrum after folding, it must be the true distribution.

Better mental modelSeveral sharply different true spectra can blur into similar recorded counts. A response offset missing from calibration can also move the inferred peaks. Independent truth shapes and repeated simulated acquisitions test that sensitivity.

Iteration count changes the estimator

Typical misconceptionEnough iterations will always reveal every narrow peak.

Better mental modelFinite resolution discards information. Iterations trade smoothing bias for noise, and the saved sample has finite statistics. A confidence interval must account for this bias and for uncertainty in the response.

Run the experiment

01
Follow one event through the measurement
Watch the cyan-to-coral paths, then drag the gold resolution handle.
What to observe: Paths spread sideways as resolution worsens. Lost events fade; they do not enter the recorded mass histogram.
02
Merge the peaks
Choose Peaks merge and compare recorded counts with simulated truth.
What to observe: Two narrow truth structures produce one broad measured hill. The response matrix spreads across several reconstructed bins.
03
Change the regularization
Increase Unfolding iterations while keeping the acquisition fixed.
What to observe: The gold estimate can develop sharper structure. The recorded coral points do not change because the same saved events are being analysed.
04
Miss a calibration shift
Choose Calibration misses a shift and compare truth error with refolded Poisson deviance.
What to observe: The fitted recorded spectrum can look plausible while the recovered masses move away from the known simulated truth.

Detector Response and Unfolding

The mass spectrum

Where each true bin is recorded

Recovering a spectrum after the detector blurs it

Core mathematical model

From true bins to recorded bins

One iterative update

Compare the estimate with simulated truth

Common difficulties

A good refolded spectrum can conceal a wrong answer

Iteration count changes the estimator

Run the experiment

Follow one event through the measurement

Merge the peaks

Change the regularization

Miss a calibration shift

Related experiments and articles

Collider physics from events to inference