Nuclear Predictions, Uncertainties, and Evidence Across Methods

A defensible nuclear prediction reports a joint distribution—or a clearly delimited nonprobabilistic bound or diagnostic—for the observable, with correlations retained across EFT truncation, low-energy constants, regulators, numerical solution, currents, radiative inputs, lattice matching, and experiment. Method comparison is meaningful only after observable definitions and calibration data are aligned. This page gives the durable procedure; it makes no claim about which interaction, calculation, or numerical value is currently preferred.

Required background. Electroweak Currents in Few-Body Systems supplies current-operator matching and shared force–current coefficients. Validation and Theory Uncertainties supplies the separation of calibration, prediction, and validation.

Helpful background. Three-Body Renormalization and Universality supplies an example in which an additional datum is required by renormalization rather than optional model tuning.

The prediction and its identity

Begin with the random or bounded object actually being predicted, not with a list of error percentages. A sufficient identity record includes:

Field	What must be fixed
Observable	Operator definition, normalization, units, kinematic point or binning, target state, and electromagnetic/weak convention
Degrees of freedom	Pionless, chiral, lattice, or other representation; included channels and breakdown assumptions
Approximation	EFT order, promoted iterations, many-body/current rank, radiative order, and numerical tolerances
Calibration	Data points, likelihood, priors, nuisance parameters, regulator, and fitted parameter combinations
Prediction	Held-out data or new kinematics, posterior/predictive construction, and every covariance component
Validation	Prespecified residual, coverage, or consistency test and its acceptance interpretation
Reproducibility	Interaction and code version, input snapshot date, corrections or withdrawals, and machine-readable covariance

This identity prevents three common category errors: comparing differently normalized quantities, calling a potential an observable, and treating agreement with fitted data as validation. It also makes shared inputs visible before uncertainties are combined.

EFT truncation as a correlated remainder

For one kinematic variable $x$, write a declared expansion

$$y_k(x)=y_{\rm ref}(x) \sum_{n\in\mathcal I_k}c_n(x)Q(x)^n, \qquad 0\le Q(x)<1,$$

where $\mathcal I_k$ contains the orders allowed by the counting. The omitted remainder is

$$\delta_k(x)=y_{\rm ref}(x) \sum_{n\notin\mathcal I_k,\,n>k}c_n(x)Q(x)^n.$$

Neither $Q$ nor “natural coefficients” is self-defining. State the low scale, breakdown scale $\Lambda_b$, reference size, missing orders, and whether zeros are symmetry enforced. Successive calculations give diagnostics

$$\widehat c_n(x) =\frac{y_n(x)-y_{n^-}(x)}{y_{\rm ref}(x)Q(x)^n},$$

where $n^-$ is the preceding calculated order. Coefficients that grow systematically, sharp kinematic failures, or $Q\ge1$ challenge the expansion assumptions; they should not be hidden by widening an interval after seeing the validation data.

A probability distribution for $\delta_k$ requires an explicit coefficient model. For example, if the omitted $c_n(x)$ are modeled as correlated zero-mean variables with scale $\bar c$, kernel $r(x,x')$, and independent permitted orders, then

$$\begin{aligned} C_{\rm trunc}(x,x')={}&y_{\rm ref}(x)y_{\rm ref}(x')r(x,x')\bar c^2\\ &\times\sum_{n\notin\mathcal I_k,\,n>k} [Q(x)Q(x')]^n. \end{aligned}$$

The kernel, coefficient distribution, and hyperparameter training are hypotheses to validate. The Bayesian construction and its dependence on the EFT series are set out in Furnstahl et al. 2015, §§ II–IV; correlated kinematic models and diagnostics are developed in Melendez et al. 2019, §§ II–IV.

Cutoff variation is different. After refitting the prescribed calibration data at every cutoff, residual variation tests whether counterterms and numerical implementation remove regulator dependence to the expected order. The selected cutoffs are not random samples from nature, so their envelope is not automatically a credible interval. It can inform a discrepancy model only after that model and its coverage have been stated and checked.

Building the joint covariance

Let $y_i$ denote observables or bins and let $s$ index uncertainty sources. The complete covariance is

$$C_{\rm tot}=\sum_s C_s +\sum_{s<t}\left(C_{st}+C_{st}^{T}\right).$$

For shared parameters $\theta$, the leading propagated component is

$$C_{\rm param}=J_\theta C_\theta J_\theta^T, \qquad (J_\theta)_{ia}=\frac{\partial y_i}{\partial\theta_a}.$$

Sampling the full posterior is preferable when the response is nonlinear or non-Gaussian. Relevant sources and their correlation mechanisms include:

Source	Correlation that must be considered	Independent check
EFT truncation	Across energy, angle, observable, nucleus, and neighboring orders through common coefficients	Withheld-order and coverage tests
Strong LECs	Same posterior parameters enter forces, bound states, and scattering	Recompute Jacobians or joint samples after every refit
Current operators	Shared one-body inputs and contact LECs couple multiple reactions and can correlate with force LECs	Change current order and regulator consistently with the Hamiltonian
Regulator artifacts	Common cutoff and omitted counterterms produce structured shifts	Refit over a justified cutoff window and test predicted scaling
Numerical solution	Basis truncation, mesh, Monte Carlo, solver, and finite-volume errors can share configurations	Independent convergence ladder or replicated calculation
Lattice inputs	Ensembles, scale setting, renormalization factors, finite-volume and continuum fits are shared	Preserve ensemble-level covariance and continuum model identity
Experiment and radiative input	Normalization, efficiency, luminosity, external constants, and unfolding can be common	Use the released covariance and nuisance model, not diagonal errors

To see why off-diagonal terms matter, consider two dimensionless synthetic predictions. A shared parameter with sensitivity vector $g=(1,2)$ and standard deviation $0.10$ gives

$$C_{\rm LEC}=0.01 \begin{pmatrix}1&2\\2&4\end{pmatrix}.$$

Let the truncation standard deviations be $(0.20,0.30)$ with correlation $0.5$, and let independent numerical variances be $(0.0025,0.0016)$. Then

$$C_{\rm tot}= \begin{pmatrix}0.0525&0.0500\\0.0500&0.1316\end{pmatrix}.$$

For the difference $d=y_2-y_1$, the correct variance is

$$\operatorname{Var}(d)=(-1,1)C_{\rm tot}(-1,1)^T=0.0841.$$

Discarding covariance would give $0.1841$. This fixture illustrates covariance algebra only; it is not evidence about a physical nuclear observable.

Calibration, validation, and method comparison

Partition data by role before fitting:

1. Calibration data determine LECs, discrepancy hyperparameters, nuisance parameters, or regulator-dependent contacts.
2. Validation data remain held out while those choices are made and test predictions under a prespecified residual.
3. Application data may be absent entirely; the validated model then predicts a new system or kinematic region with an explicit extrapolation warning.

For validation residual $r=y_{\rm val}-\mu_{\rm pred}$, whiten with the full predictive-plus-experimental covariance,

$$z=(C_{\rm pred}+C_{\rm exp})^{-1/2}r, \qquad D^2=z^Tz.$$

Inspect $z$ versus kinematics as well as the scalar $D^2$: a smooth bias can be hidden by a global statistic. Coverage is assessed over genuinely repeated or exchangeable tests; one point inside a nominal interval is neither proof nor a calibrated coverage frequency. Bayesian truncation calibration for nucleon–nucleon observables provides worked examples of these distinctions Melendez, Wesolowski, and Furnstahl 2017, §§ II–IV.

When methods $A$ and $B$ predict the same vector, compare $\Delta=y_A-y_B$ with

$$C_\Delta=C_A+C_B-C_{AB}-C_{AB}^T.$$

The cross-covariance $C_{AB}$ is nonzero when methods share experimental calibration, single-nucleon inputs, EFT coefficients, gauge ensembles, or radiative constants. Agreement after both methods were tuned to the same datum is a consistency check at that datum, not independent confirmation. Before interpreting tension, align kinematics, units, observable definitions, correction conventions, and the data cutoff.

Evidence boundary and update procedure

The equations above are durable. Numerical interaction rankings, fitted LECs, experimental averages, lattice continuum results, code capabilities, and claimed anomalies are mutable evidence. Any such statement needs a dated evidence record containing the source release, version or ensemble identity, covariance, corrections or withdrawals, data cutoff, and the exact observable definition. No current ranking or mutable numerical claim is asserted on this page in the absence of that record.

When an evidence snapshot is available, update it without rewriting the method:

• preserve the earlier version and state what changed;
• verify corrections and withdrawals at the primary source;
• recompute comparisons with the full covariance and shared-data map;
• keep calibration and held-out sets fixed unless the change is declared as a new analysis;
• label extrapolations beyond the validated $Q$, mass, volume, or kinematic domain.

Use the following exits for the next technical object:

Need	Continue to
Continuum extrapolation with correlated scale setting and quark-mass tuning	Lines of Constant Physics and Continuum Extrapolation
Complete lattice covariance and uncertainty provenance	Complete Lattice Error Budgets
Convert finite-volume levels into amplitudes and pole information	Scattering Amplitudes and Resonance Poles from Finite-Volume Spectra
Convert finite-volume current matrix elements	Finite-Volume Matrix Elements and One-to-Two Transitions
Build and validate a correlated EFT truncation model	EFT Inference, Power Counting, and Truncation

Failure modes

Double use of data. A datum used to fit a force, current, or discrepancy model cannot also be counted as held-out evidence. Record every path by which it enters, including external averages and lattice-informed priors.

Regulator spread as a probability. A cutoff scan is a renormalization diagnostic. Do not attach a confidence level unless a statistical model maps the scan to a validated predictive distribution.

Omitted current covariance. A shared axial or electromagnetic contact can move several processes coherently and may be correlated with strong LECs. Independent per-observable errors generally misstate ratios and differences.

Unsupported heavy extrapolation. A small posterior interval does not repair $Q\ge1$, a new threshold, an omitted channel, or a nucleus far outside the calibrated domain. Report the extrapolation and expand the theory or evidence set before treating it as controlled.

References

• Furnstahl, R. J., N. Klco, D. R. Phillips, and S. Wesolowski. “Quantifying Truncation Errors in Effective Field Theory.” Physical Review C 92 (2015): 024005. DOI.
• Melendez, J. A., R. J. Furnstahl, D. R. Phillips, M. T. Pratola, and S. Wesolowski. “Quantifying Correlated Truncation Errors in Effective Field Theory.” Physical Review C 100 (2019): 044001. DOI.
• Melendez, J. A., S. Wesolowski, and R. J. Furnstahl. “Bayesian Truncation Errors in Chiral Effective Field Theory: Nucleon–Nucleon Observables.” Physical Review C 96 (2017): 024003. DOI.