QCD Prediction and Uncertainty Ledger

A QCD uncertainty statement is part of the prediction, not a percentage attached afterward. It must identify the observable vector, central calculation, approximations, nuisance sources, correlations, validation tests, and provenance. The durable result of this page is a covariance-aware method that prevents shared scale, PDF, parameter, and fit variations from being counted twice.

Required background. Hadron-collider factorization and parton luminosities supplies the channel- and bin-level prediction to be propagated. Validation and theory uncertainties supplies the distinction between sensitivity studies, probability statements, and empirical coverage.

Helpful background. QCD radiation, jets, and event shapes supplies the multi-scale and matching variations that make correlations essential.

Define the prediction before its uncertainty

Let $y\in\mathbb R^B$ be the vector of predicted fiducial bins or derived observables. Freeze the central specification before generating variations:

$$\mathcal P_0= \left\{\mathcal M,\ \text{process},\ \text{orders},\ \text{schemes},\ \mu_i^0,\ n_f,\ \text{thresholds},\ \text{input release},\ \text{nonperturbative model},\ \text{code and numerical settings}\right\}.$$

Changing the jet radius, fiducial definition, heavy-flavor scheme, PDF release, or matching prescription can define a different prediction rather than a random fluctuation around $\mathcal P_0$. Classify that change before combining it with ordinary nuisance variations.

Each uncertainty source $a$ should have:

Field	Required content
physical origin	missing perturbative terms, PDF inference, parameter input, power correction, numerical integration, or experimental measurement
varied object	exact scales, replicas, eigenvectors, parameters, model terms, or random samples
response	signed bin vector $\Delta^{(a)}$ or a nonlinear map $y(\theta_a)$
interpretation	sensitivity range, probabilistic nuisance, replica ensemble, or discrete alternative
correlation scope	bins, channels, processes, energies, and other sources that share it
validation	expansion, closure, convergence, stability, or coverage test
provenance	source, version, date, configuration, and correction history

This classification is the main defense against double counting.

A covariance construction

For approximately linear nuisance parameters $\theta_a$ with unit variances and correlation matrix $\rho$, write

$$y(\theta)\simeq y_0+\sum_a\Delta^{(a)}\theta_a,$$

so the induced covariance is

$$C_{\mathrm{th}}= \sum_{a,b}\rho_{ab}\, \Delta^{(a)}\Delta^{(b)\mathsf T} +C_{\mathrm{rem}}.$$

$C_{\mathrm{rem}}$ contains components represented directly as covariance matrices. Independent quadrature is the special case $\rho_{ab}=\delta_{ab}$; it is not the default. Fully correlating every bin is equally unjustified unless one nuisance genuinely produces that response.

For a replica ensemble $y^{(r)}$,

$$\overline y=\frac1{N_{\mathrm{rep}}}\sum_r y^{(r)}, \qquad C_{\mathrm{rep}}= \frac1{N_{\mathrm{rep}}-1} \sum_r\left(y^{(r)}-\overline y\right) \left(y^{(r)}-\overline y\right)^{\mathsf T}.$$

Hessian eigenvector sets instead supply coordinated parameter displacements; propagate each displacement through all bins and channels before constructing the covariance. The Hessian method and its tolerance assumptions are explained by Pumplin et al. 2001, §§II–IV.

For asymmetric or nonlinear responses, retain nuisance samples, up/down response vectors, or a likelihood rather than forcing a symmetric Gaussian covariance. An envelope can be reported as a sensitivity range, but it becomes a probability interval only after a calibrated statistical model is supplied.

Component-by-component rules

Missing perturbative orders

Renormalization- and factorization-scale variation probes sensitivity to logarithmic terms generated by known renormalization-group structure. It is not a random draw from the unknown coefficient distribution. Record the central scales, allowed correlated combinations, excluded extreme ratios, and the binwise signed responses.

Use all available information:

• compare successive fixed orders at common inputs;
• vary $\mu_R$ and $\mu_F$ in a pattern consistent with factorization-scale cancellation;
• test alternative central scales when they represent equally natural power counting;
• for resummation, vary hard, jet/beam, soft, and rapidity profiles in ways that preserve their hierarchy; and
• expand the matched result and verify its fixed-order singular terms before treating resummation-scale changes as meaningful.

A probabilistic missing-order interval requires explicit assumptions about coefficient sizes and correlations. The Bayesian construction of Cacciari and Houdeau is one such model, not a universal reinterpretation of a scale envelope Cacciari and Houdeau 2011, §§2–4.

PDFs, αₛ, and fitted parameters

Evaluate the total prediction for every provided replica or eigenvector. Preserve correlations among flavors, $x$ regions, bins, and processes. If a PDF ensemble was fitted at several values of $\alpha_s$, use the release’s joint prescription or an explicitly reconstructed joint covariance. Do not add a separate $\alpha_s$ error to replicas that already sample that same uncertainty.

The same rule applies to masses, electroweak parameters, calibration inputs, and branching fractions. Propagate the joint input vector through running, matching, hard coefficients, phase space, and acceptance. A Jacobian $JC_pJ^{\mathsf T}$ is adequate only when the response is locally linear.

Thresholds, schemes, and resummation matching

Heavy-flavor matching-scale variations affect the coupling, masses, PDFs, and coefficients together. Varying each occurrence independently destroys the cancellation the matching was designed to enforce. A mass-scheme conversion similarly requires the coefficients and parameter covariance to move together.

Additive and multiplicative matching, profile freezing, and transition points can probe missing terms in a resummed calculation. Their common fixed-order content must first be aligned; otherwise their spread includes a known mismatch rather than an estimate of unknown orders.

Nonperturbative and factorization-limit effects

Hadronization, underlying event, power corrections, large-$b$ TMD input, fragmentation, and nuclear effects are different sources. Represent each by fitted nuisance parameters, ensembles, or controlled alternative models, preserving correlations with the data and parameters used to determine it. A Monte Carlo tune variation is not automatically a pure hadronization uncertainty.

When a leading-power theorem may fail—through Glauber exchange, an endpoint, a low scale, or a high-density boundary—state a domain restriction or a dedicated remainder. Ordinary scale variation does not estimate failure of the factorization hypothesis.

Numerical and experimental components

Monte Carlo integration provides a bin covariance because the same events, subtraction terms, or adaptation grids can feed several bins. Estimate and store that covariance, demonstrate convergence with increased statistics, and keep numerical noise small enough that it does not determine scale or PDF responses.

Experimental covariance belongs to the data model, not the theory prediction. Combine it only when forming a comparison. If theory inputs were inferred from overlapping data, a cross-covariance may be required; assuming independence should be stated, not silently imposed.

Preventing double counting

Use a source–response table before summing anything:

Apparent pair	Possible shared origin	Safe treatment
PDF and $\alpha_s$	coupling varied or fitted inside the PDF ensemble	use a joint release prescription or joint nuisance model
scale and threshold matching	the same logarithms respond to $\mu_F$ and heavy-flavor matching scales	vary coordinated theory ingredients; inspect overlap
hadronization and tune	the tune variation changes shower, underlying event, and fragmentation together	decompose with dedicated variations or retain one combined nuisance
resummation and fixed-order scale	matched singular terms occur in both components	use a matched variation scheme with common terms subtracted
PDF and experimental data	the compared dataset contributed to the PDF fit	account for dependence or qualify the comparison
integration noise and small theory response	common finite samples fluctuate across variations	use correlated sampling or a numerical covariance

If two labels map to the same underlying parameter, merge them. If one variation mixes several inseparable effects, keep it as one combined nuisance and name that limitation. Do not split it into independent components merely to make a longer table.

Validation of the final result

For a covariance $C$ and response matrix $D=(\Delta^{(1)},\ldots)$:

1. Symmetry and positivity: verify $C=C^{\mathsf T}$ and that negative eigenvalues are absent up to controlled numerical tolerance.
2. Reconstruction: check that $D\rho D^{\mathsf T}$ reproduces stored nuisance shifts and bin correlations.
3. Order stability: compare central values and uncertainty bands at successive perturbative orders without changing unrelated inputs.
4. Closure: inject known parameter or model shifts into synthetic data and confirm that the propagation recovers their effect.
5. Resampling stability: repeat replica, eigenvector, and Monte Carlo estimates with adequate samples or alternative partitions.
6. Boundary tests: expand resummation, approach threshold matches from both sides, and test the stated factorization domain.
7. Comparison residual: for data vector $d$, form $r=d-y$ and use the correctly combined covariance, including any cross-covariance, rather than adding published one-dimensional errors bin by bin.

Coverage is an empirical property of an ensemble of repeated problems or a calibrated probabilistic model. A band that contains one data set is neither validated nor invalidated by that fact alone.

The dated-evidence boundary

Numerical claims about the current strong coupling, a current PDF or fragmentation release, a collider combination, or a world summary are not timeless consequences of this method. A current claim must name the release or edition, dataset period, covariance or likelihood, theory setup, software version or checksum, correction history, and an explicit evidence date. Without those items, report the durable procedure and no current value, ranking, or status claim. Release-specific conclusions should be maintained in a dated research record rather than copied into an evergreen derivation.

Reproducible handoff

The final object is

$$\left\{y_0,\ \mathcal P_0,\ \{\theta_a,\Delta^{(a)},\rho_{ab}\},\ C_{\mathrm{th}},C_{\mathrm{num}},\ \text{domain},\ \text{validation results},\ \text{versions and dates}\right\}.$$

Keep experimental data and $C_{\mathrm{exp}}$ alongside—but distinct from—this object until a specified comparison is formed. This separation lets a later analysis update measurements without silently redefining the theory prediction.

Common pitfalls

Calling a scale envelope a confidence interval. It is a perturbative sensitivity study unless a calibrated probabilistic model says otherwise.

Adding named percentages in quadrature. Names do not establish independence. Trace each component to its underlying nuisance and preserve its signed bin response.

Dropping off-diagonal entries. A small uncertainty per bin can correspond to a large or small uncertainty in a shape, ratio, or fit depending on correlations.

Publishing central values without provenance. A result that cannot identify its inputs, code, settings, and evidence date cannot be reproduced or safely updated.

References

• Cacciari, Matteo, and Nicolas Houdeau. “Meaningful Characterisation of Perturbative Theoretical Uncertainties.” Journal of High Energy Physics 2011, no. 9 (2011): 039. DOI. Open PDF.
• Pumplin, Jon, Daniel Stump, Robert Brock, Daniel Casey, Joey Huston, Jon Kalk, H. L. Lai, and Wu-Ki Tung. “Uncertainties of Predictions from Parton Distribution Functions. II. The Hessian Method.” Physical Review D 65, no. 1 (2001): 014013. DOI. Open PDF.