EFT Inference, Power Counting, and Truncation

This map compares the choices that turn an effective Lagrangian into an inference: degrees of freedom and symmetry realization, operator basis, power counting, matching and running, likelihood, priors or profiling, truncation error, and ultraviolet interpretation. The target classes include low-energy precision observables, collider distributions, flavor and neutrino processes, nuclear amplitudes, and gravity EFT observables. The correct workflow is the one whose expansion parameter and data model remain controlled over the information-bearing region.

Evidence cutoff. 11 August 2026.

Required background. Power counting and predictive order defines what is retained; truncation errors and breakdown defines what must be tested.

Helpful background. EFT as a controlled expansion supplies the remainder logic, basis translation and reproducibility separates coordinates from observables, and correlated Standard Model fits supplies likelihood and covariance context.

The EFT inference chain and its alternatives

Choice	Main options	Required inputs	Material tradeoff
field content/symmetry	SMEFT, HEFT, low-energy EFT, process EFT	hierarchy, light spectrum, symmetry realization	stronger symmetry improves predictivity but can exclude the correct infrared realization
operator coordinates	nonredundant basis; amplitude or observable directions	equations of motion, flavor/global assumptions, scheme	convenient coordinates can hide flat directions or loop mixing
power counting	canonical dimension, chiral, velocity, loop, large-$N$, hybrid	estimated breakdown scale and coupling sizes	ranking depends on UV/kinematic assumptions
matching and running	top-down model matching; bottom-up coefficients; fixed or RG-evolved	scales, thresholds, perturbative order	omitting mixing or thresholds misaligns datasets at different scales
prediction order	linear interference; selected squares; complete next order	matrix elements and theory covariance	positivity/stability versus formal order consistency
statistical inference	profile likelihood; Bayesian posterior; information geometry	likelihood, priors, nuisance and covariance model	coverage, marginalization, and volume effects differ
UV interpretation	coefficient bounds, simplified classes, positivity, explicit matching	coupling and symmetry hypotheses	inverse map is nonunique and often many-to-one

Basis changes are field redefinitions plus coefficient transformations; observables agree only if matching, running, truncation, and priors are transformed consistently. The Warsaw basis supplies a complete dimension-six SMEFT coordinate set under its assumptions Grzadkowski et al. 2010, foundational. The broader matching, running, input-scheme, and observable interfaces are synthesized in Brivio and Trott 2019, orientation. A “one coefficient at a time” interval is a conditional slice, not a basis-invariant statement of global knowledge.

Truncation is part of the likelihood

Write an expansion for each bin or observable as

$$y_k=y_{k,0}+\sum_{n=1}^{N}c_{kn}Q_k^n+R_{k,N+1}, \qquad Q_k<1,$$

where the remainder is correlated across bins and processes when it shares coefficients or scales. Common treatments are: hard kinematic cuts; order-by-order envelopes; nuisance functions with naturalness scales; Gaussian-process or coefficient priors; and explicit inclusion of selected higher-order terms. None is assumption-free. Bayesian EFT uncertainty quantification makes coefficient-size and correlation assumptions visible and can be calibrated by order-by-order behavior Furnstahl, Phillips, and Wesolowski 2015, method.

The complete error model can include

$$\Sigma_{\rm total}=\Sigma_{\rm data}+\Sigma_{\rm SM\ theory} +\Sigma_{\rm EFT\ trunc}+\Sigma_{\rm emulation}+\Sigma_{\rm inputs},$$

with cross-covariances retained where known. Adding a diagonal “theory error” can falsely let many bins average down one shared missing term. Conversely, making truncation fully correlated can erase genuine kinematic information. Validate correlation length and coefficient-scale assumptions on known orders, injected pseudo-data, and held-out observables.

Internal checks and failure modes

Expansion control. Plot which bins drive the likelihood; scan maximum energy; compare successive orders; and test whether the inferred coefficient region keeps $Q$ and amplitude growth controlled. Dimension-six squares can stabilize positive event rates but are of the same $\Lambda^{-4}$ order as omitted dimension-eight interference. Label the hybrid rather than calling it complete.

Basis and scheme control. Reproduce the fit in a second basis with transformed priors and full covariance. Evolve coefficients to common scales and test matching-scale dependence. Input-scheme choices should change coordinates and truncation terms, not the all-order physics.

Inference control. Compare profiling and marginalization, publish nuisance pulls and weak directions, run simulation-based coverage or calibration tests, and leave out high-leverage datasets. A controlled SMEFT case study finds real profile–marginal differences in correlated weak directions Brivio et al. 2024, qualifying evidence. Public software such as SMEFiT supports basis and inference comparisons, but common theory tables and datasets create shared error lineage Giani, Magni, and Rojo 2023, benchmark/software.

Failure modes include fitting outside the breakdown scale; interpreting prior volume as data information; neglecting covariance between unfolded bins; double counting the same measurement in derived combinations; mixing perturbative orders; and translating coefficients to UV masses without coupling or loop assumptions.

Benchmarks and independent agreement

Useful benchmarks are a synthetic EFT with known higher orders; a basis rotation with identical predictions; a simple UV model matched at tree and loop level; injected-signal likelihood closure; a linear/quadratic/order-complete comparison; and leave-one-process-out prediction. Exercise each inference stage separately: compare matched Wilson coefficients at a fixed order and convention, evolve them against an independently integrated renormalization-group solution, and apply dispersion or positivity bounds only after checking their analyticity, unitarity, crossing, and subtraction hypotheses.

Two fits are independent only to the extent that their data, covariance construction, theory predictions, interpolation, and statistical machinery differ. Different priors applied to the same likelihood are a sensitivity analysis, not independent experimental evidence. Agreement between a coefficient fit and a UV-model fit is expected when one is algebraically derived from the other.

What EFT inference cannot establish

EFT can show that data constrain combinations of low-energy interactions under a declared expansion. It cannot uniquely reconstruct ultraviolet particle content, prove that an unconstrained coefficient vanishes, or turn a null fit into evidence that the Standard Model is complete. Positivity constraints do not apply without their analyticity, unitarity, locality, crossing, and infrared-subtraction hypotheses. A good global fit can still be uninformative along exact or approximate flat directions.

Decision aid

Situation	Defensible starting choice	Essential stress test
$E\ll\Lambda_b$ and one observable family	order-consistent linear EFT with calibrated remainder	energy-cut and next-order sensitivity
correlated multi-process data	global coefficient fit with full covariance and RG evolution	basis transform and leave-one-dataset-out
nonlinear Higgs dynamics plausible	compare SMEFT and HEFT rather than forcing one	power counting and observable coverage
UV model specified	explicit matching and running	recover bottom-up fit in the model subspace
weak directions dominate	report eigen/combinations, profile and marginal views	prior/parameterization sensitivity
tails drive sensitivity	include truncation as correlated nuisance or remove bins	verify posterior/interval lies in controlled region

Source selection and related pages

The finite search covered INSPIRE, arXiv, official collaboration records, journal/DOI pages, public fit frameworks, and targeted searches for basis, prior, and truncation sensitivity through 11 August 2026. Reassess when a new perturbative order or official covariance changes a fit, a calibrated remainder model shifts the information-bearing bins, or an inference benchmark reveals failed coverage.

See the EFT and Standard Model tests field guide, SMEFT validity in global fits, renormalization/EFT pathway, and the bounded W-mass and muon $g-2$ briefs.

References

• I. Brivio et al., “To Profile or to Marginalize — A SMEFT Case Study,” SciPost Physics 16 (2024) 035. DOI.
• I. Brivio and M. Trott, “The Standard Model as an Effective Field Theory,” Physics Reports 793 (2019) 1–98. DOI.
• R. J. Furnstahl, D. R. Phillips, and S. Wesolowski, “A Recipe for EFT Uncertainty Quantification in Nuclear Physics,” Journal of Physics G 42 (2015) 034028. DOI.
• T. Giani, G. Magni, and J. Rojo, “SMEFiT: a Flexible Toolbox for Global Interpretations of Particle Physics Data with Effective Field Theories,” European Physical Journal C 83 (2023) 393. arXiv.
• B. Grzadkowski, M. Iskrzyński, M. Misiak, and J. Rosiek, “Dimension-Six Terms in the Standard Model Lagrangian,” JHEP 10 (2010) 085. DOI.