Collider Measurements, Fiducial Predictions, and Likelihood Provenance

A collider result is reusable only when the theory observable, detector response, observed and auxiliary data, likelihood or covariance, and immutable release identity form one traceable interface. A plot or unfolded central value alone is not that interface. The durable workflow is to define the fiducial measurement function, forward-fold predictions whenever possible, publish nuisance and correlation semantics, and bind every reused object to its exact version, corrections, and overlap information.

Required background. Standard Model pseudo-observables and unstable particles distinguishes pole, pseudo-observable, and fiducial layers. Validation and theory uncertainties supplies the approximation and correlation checks used below.

Helpful background. QCD prediction and uncertainty accounting supplies the scale, PDF, nonperturbative, and matching information that a collider prediction must pass forward.

The complete interface is shown below. A hard process reaches released data through event evolution, detector response, reconstruction, and fiducial selection, while pole definitions, theory uncertainties, nuisance semantics, and correlations enter the likelihood used for inference.

Collider inference is reproducible only when hard-process, detector, fiducial, pole, theory, data, and likelihood interfaces are versioned together. The schematic preserves these interfaces without claiming the internal algorithms of generators or detectors.

The measurement function defines the fiducial quantity

For stable particle-level final states, a fiducial cross section can be written schematically as

$$\sigma_F =\sum_n\int d\Phi_n\, \frac{d\sigma_n}{d\Phi_n}\,F_n(\Phi_n),$$

where $F_n$ assigns an event to the accepted region or to a histogram bin. A complete $F_n$ specifies particle lifetimes or status convention, jet algorithm and parameters, lepton/photon dressing, isolation, overlap removal, cuts, bin edges, normalization, and units. For a fixed-order QCD prediction, infrared safety requires the relevant soft and collinear limits to leave the bin assignment unchanged,

$$F_{n+1}(\Phi_{n+1})\longrightarrow F_n(\Phi_n),$$

after the unresolved momentum is mapped to the lower-multiplicity configuration. If this condition fails, real–virtual cancellation does not define the claimed perturbative observable without additional fragmentation or nonperturbative input.

Pole pseudo-observables and fiducial observables must remain distinct. A fiducial cross section uses stable final states and a measurement function. Extracting a pole mass, width, or residue from it adds a line-shape, radiation, background, interference, and acceptance model.

Forward folding and unfolding

Let $t$ label truth or fiducial bins and $r$ reconstructed categories. A prediction is connected to observed counts by

$$\nu_r(\boldsymbol\theta,\boldsymbol\eta) =\mathcal L\sum_t R_{rt}(\boldsymbol\eta)\, \sigma_t(\boldsymbol\theta,\boldsymbol\eta) +b_r(\boldsymbol\eta).$$

$R_{rt}$ is the response, including efficiency and migration under its declared convention; $b_r$ is the background expectation. If each column of $R$ is a probability distribution over selected reconstructed outcomes, its sum is the efficiency and lies between zero and one. Some software absorbs luminosity or bin-width factors into the response, so dimensions and normalization must be checked rather than assumed.

Forward folding evaluates this model in reconstructed space and retains the count distribution and response nuisances. Unfolding instead constructs a derived truth-space estimate, schematically

$$\widehat{\boldsymbol\sigma} =U(\mathbf n-\widehat{\mathbf b}),$$

where $U$ may depend on simulation, regularization, and a prior or reference spectrum. Its covariance is not merely the observed counting covariance transformed by a fixed inverse; response, background, and regularization choices contribute. An unfolded spectrum is valuable when these choices and the full covariance are released, but it is not automatically equivalent to the detector-level likelihood.

For any differentiable change of variables $\mathbf z=f(\mathbf y)$, the linear covariance rule is

$$C_z=J C_yJ^T, \qquad J_{ij}=\frac{\partial f_i}{\partial y_j}.$$

The ordering, units, and evaluation point of $J$ are part of the transformation. Applying a normalized-shape transformation creates a singular covariance because the bins obey a sum constraint; one must remove a redundant coordinate or use a generalized inverse in the constrained subspace.

A released likelihood is a statistical model at fixed data

Separate primary observations $\mathbf x$ from auxiliary observations $\mathbf a$. A statistical model has the form

$$p(\mathbf x,\mathbf a\mid \boldsymbol\theta,\boldsymbol\eta) =p(\mathbf x\mid\boldsymbol\theta,\boldsymbol\eta) p(\mathbf a\mid\boldsymbol\eta),$$

and the likelihood is this density evaluated at the observed data. A binned counting example is

$$L(\boldsymbol\theta,\boldsymbol\eta) =\prod_r\operatorname{Pois} \!\left(n_r\mid\nu_r(\boldsymbol\theta,\boldsymbol\eta)\right) \prod_k p_k(a_k\mid\boldsymbol\eta).$$

The second product often appears in software as “constraint terms.” In a frequentist construction these can encode auxiliary measurements, not Bayesian priors; the release must say which. A serialized model needs the observed counts, channel/bin structure, sample yields and interpolation, parameter domains, auxiliary data, nuisance correlations, and software/schema version. Publishing only a one-dimensional profile curve can lose the nuisance response needed for combination, while multiplying models that contain the same auxiliary measurement double counts information Cranmer et al. 2022, §§2.1–2.4 and Fig. 1, pp. 4–10.

HistFactory provides one concrete channel/sample/nuisance factorization for binned models, including interpolation and auxiliary constraints Cranmer et al. 2012, §§2–4, pp. 4–21. A release that uses it still has to state the exact workspace and software version; naming the framework does not identify the model.

The full collider-to-inference handoff

Stage	Object passed forward	Scientific check	Not claimed here
Hard process	Stable external states, perturbative order, scales, PDFs and parameter scheme	Gauge, infrared, dimension, and benchmark checks	A detector-level prediction
Shower and hadronization	Generator configuration, matching and nonperturbative model	Inclusive-rate preservation and matching-domain checks	First-principles detector response
Detector response	Calibration and migration model with nuisances	Closure and validation-domain checks	A universal response outside that domain
Reconstruction and selection	Dataset, object definitions, triggers, cuts, categories	Cut flow, event uniqueness, frozen selection	A pole observable without an extraction map
Fiducial or pole interpretation	Measurement function or pole expansion; acceptance and radiation map	Particle/reconstruction mapping and approximation tests	Model-independent extrapolation beyond the definition
Public release	Observed and auxiliary data; covariance or likelihood; exact identity	Reproduction of a documented benchmark	Permission to ignore correlations or corrections
Theory comparison	Calculation version and correlated uncertainty model	Independent benchmarks and limit checks	Validity outside the stated perturbative/EFT domain
Combination	Overlap rule, shared nuisances, frozen mask and test definition	Duplicate-input and covariance checks	A contemporary conclusion without dated inputs

This table specifies interfaces, not the internal algorithms of showering, detector simulation, or reconstruction.

Provenance fields that make a result reusable

Each machine-readable object should travel with a compact semantic record:

Field	Required content	Failure detected
Observable definition	Particle/object convention, cuts, bin edges, normalization and units; or pole/pseudo-observable convention	Comparing different quantities with the same label
Dataset identity	Experiment, collision system and energy, run/sample identity, integrated exposure where applicable	Reusing overlapping events as independent data
Repository identity	Record and table identifiers, DOI, exact HEPData or equivalent version, stable URL	Silently following a mutable latest version
Integrity	File name, format/schema version, checksum when supplied or locally recorded	Byte-level substitution or corrupted transfer
Covariance	Bin ordering, units, component meanings, normalization constraints, parameter dependence	Permuted or dimensionally inconsistent matrices
Likelihood	Observed and auxiliary data, nuisance definitions/domains, correlations, interpolation, software version	Hidden priors, duplicated constraints, nonreproducible profiling
Theory	Calculation code/version, inputs, scales, PDFs, nonperturbative corrections, accuracy	Comparing unlike central prescriptions
Overlap	Shared events, controls, auxiliary data, theory inputs, and a resolution rule	Double counting across releases
Validity mask	Predeclared bins or parameter domain and reason	Post-fit selection of favorable inputs
Lifecycle	Correction notice, withdrawal or supersession relation, exact object used downstream	Replacing the historical input without disclosure

HEPData’s record/table organization is designed to preserve numerical tables and their associated metadata, but authors and reusers still have to cite the exact record version and table rather than an unversioned landing page Maguire, Heinrich, and Watt 2017, pp. 1–6. Recommendations for reinterpretation likewise distinguish data tables, response information, simplified likelihoods, and full statistical models because they support different questions LHC Reinterpretation Forum 2020, §§2.2 and 3, pp. 8–25.

The method layer and numerical snapshot should be separate. Definitions, matrix-order rules, and reuse checks are durable. Central values, covariance files, workspace hashes, corrections, and any resulting contours are snapshot objects and must be dated and versioned together.

Independent checks and failure modes

Dimensions and bin normalization. Confirm whether each released number is a bin integral, density, normalized fraction, count, or cross section. Covariance entries carry the product of the corresponding units.

Response orientation. Inject a vector with one nonzero truth bin. The nonzero reconstructed pattern should be the documented column (or row); its sum should match the efficiency convention.

Likelihood reproduction. Evaluate the released model at a documented parameter point and reproduce its expected yields and likelihood ratio. Test nuisance values away from zero so interpolation and correlations are exercised.

Covariance geometry. Check symmetry and eigenvalues in the documented ordering. A normalized spectrum has a known null direction; an unexplained negative eigenvalue signals an invalid or mistyped matrix.

Duplicate inputs. Compare dataset identifiers, event selections, control regions, and auxiliary constraints before multiplying likelihoods. Statistical independence cannot be inferred from different paper titles.

Version mutation. Resolve every DOI or record version and calculate the expected checksum before fitting. If a correction is issued, preserve the old result’s exact input record and make a new inference with the corrected object.

Model validity. Freeze kinematic or EFT masks before examining fit residuals. A mask chosen after the outcome changes the procedure and requires its own calibration.

Common pitfalls

Digitizing a contour and calling it a likelihood. Contours encode a chosen threshold, profiling prescription, and plotting transformation, not the underlying statistical model. Use the published likelihood or limit the claim to what the contour actually supplies.

Treating an unfolded covariance as universal. Its response, prior, and regularization can depend on the reference model. Preserve those ingredients and test forward-folded alternatives where available.

Multiplying common constraint terms. Two channels may carry the same luminosity or calibration auxiliary measurement. Correlate one shared nuisance and include the auxiliary information once.

Using an unversioned repository page. A stable landing page can point to a corrected object later. Record the exact version, table, file, and checksum used.

Informal self-check

A release provides bin values, a covariance heat map, and a one-dimensional profile-likelihood plot. Which minimum objects are still required for an exact two-parameter reuse?

Solution

At minimum, obtain machine-readable bin values and ordered covariance or, preferably, the full statistical model; exact bin definitions and units; observed and auxiliary data; nuisance meanings, domains, and correlations; the response or acceptance needed by the new prediction; dataset-overlap information; parameter validity; and exact release/version identifiers with corrections. The heat map is not a numerical covariance, and a one-dimensional profile has discarded the second parameter and generally the nuisance response.

Handoffs

• Return to pseudo-observables and unstable particles when a pole or production–decay extraction is part of the published result.
• Use Higgs precision and coupling inference when response changes under coupling or tensor-structure variations.
• Use correlated Standard Model fits after identity, overlap, and nuisance semantics are complete.
• Check covariance, profiling, marginalization, and coverage on an exact synthetic fixture.

References

• Cranmer, Kyle, et al. HistFactory: A Tool for Creating Statistical Models for Use with RooFit and RooStats. CERN-OPEN-2012-016 (2012). DOI · Open PDF
• Cranmer, Kyle, et al. “Publishing Statistical Models: Getting the Most out of Particle Physics Experiments.” SciPost Physics 12 (2022) 037. DOI · Open PDF
• LHC Reinterpretation Forum. “Reinterpretation of LHC Results for New Physics: Status and Recommendations after Run 2.” SciPost Physics 9 (2020) 022. DOI · Open PDF
• Maguire, Eamonn, Lukas Heinrich, and Graeme Watt. “HEPData: A Repository for High Energy Physics Data.” Journal of Physics: Conference Series 898 (2017) 102006. DOI · Open PDF