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arxiv: 2605.21594 · v1 · pith:3MRWD24Nnew · submitted 2026-05-20 · ✦ hep-ph · hep-ex

Exploring the SMEFT landscape: Bayesian Model Selection for indirect discovery

Luca Mantani This is my paper

Pith reviewed 2026-05-22 08:56 UTC · model grok-4.3

classification ✦ hep-ph hep-ex
keywords SMEFTBayesian model selectionWilson coefficientsoperator subsetsmodel averagingelectroweak precisionLHC measurementsoperator correlations
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The pith

Treating SMEFT as a space of competing operator-subset hypotheses rather than one high-dimensional model enables Bayesian selection that finds no significant new physics while sharpening Wilson coefficient posteriors.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a Bayesian model selection framework that treats each possible subset of SMEFT operators as a distinct physical hypothesis for the low-energy effects of new dynamics. A genetic algorithm navigates the high-dimensional discrete model space while the Bayesian Information Criterion approximates the evidence for each subset, allowing posterior probabilities to be assigned to models and marginal inclusion probabilities to individual operators. When applied to electroweak precision data from LEP together with Higgs, top-quark and diboson measurements from LHC Run 2 at both linear and quadratic order with one-loop renormalization group evolution, the analysis finds no statistically significant evidence for any departure from the Standard Model. Bayesian model averaging over high-posterior models produces Wilson coefficient posteriors with substantially improved characterization potential relative to traditional global fits, and an operator correlation matrix reveals which operators tend to appear together and where flat directions remain.

Core claim

Bayesian inference performed at the level of the discrete space of SMEFT operator subsets, navigated efficiently by a genetic algorithm and with evidence approximated by the Bayesian Information Criterion, assigns high posterior probability to the Standard Model and no statistically significant probability to any non-Standard Model operator subset when confronted with LEP and LHC Run 2 data; the resulting Bayesian model average posteriors on Wilson coefficients nevertheless exhibit substantially improved characterization potential compared with conventional global fits.

What carries the argument

Bayesian model selection over the discrete space of SMEFT operator subsets, which produces an operator correlation matrix that encodes the relational structure among high-posterior models.

Load-bearing premise

The Bayesian Information Criterion supplies a sufficiently accurate approximation to the true Bayesian evidence when comparing high-dimensional discrete operator-subset models.

What would settle it

A future measurement in the electroweak or top sector that produces a decisive preference for one particular non-Standard Model operator subset over the Standard Model would falsify the reported absence of significant evidence.

read the original abstract

We develop a framework for indirect discovery in the Standard Model Effective Field Theory (SMEFT) based on Bayesian model selection over operator subsets. We argue that SMEFT should be understood as a structured space of competing hypotheses rather than a single high-dimensional model, with each operator subset corresponding to a physically distinct low-energy realisation of new dynamics. Bayesian inference is applied at the level of model space itself, assigning posterior probabilities to operator subsets and marginal inclusion probabilities to individual operators. A genetic algorithm efficiently navigates the high-dimensional discrete model space, concentrating evaluations in the high-posterior region, while the Bayesian Information Criterion provides a tractable approximation to the Bayesian evidence. We apply this framework to a dataset comprising electroweak precision observables from LEP and Higgs, top-quark, and diboson measurements from LHC Run 2, at both linear and quadratic order in the Wilson coefficients, with one-loop renormalisation group evolution systematically included. The analysis finds no statistically significant evidence for any departure from the SM, and demonstrates that Bayesian Model Average posteriors on Wilson coefficients carry substantially improved characterisation potential compared to traditional global fits. The operator correlation matrix encodes the relational structure of the model posterior, identifying operator pairs that co-appear in high-posterior models and flat directions where additional measurements would be most valuable. The sensitivity of all results to the choice of matching scale $\mu_0$ is assessed, and its promotion to a continuous parameter of inference is identified as a natural extension of the framework.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 3 minor

Summary. The manuscript develops a Bayesian model selection framework for SMEFT in which operator subsets are treated as competing hypotheses. A genetic algorithm is used to explore the discrete model space, with the Bayesian Information Criterion (BIC) serving as a tractable proxy for the marginal likelihood. The method is applied to a global dataset of LEP electroweak precision observables together with LHC Run-2 Higgs, top-quark and diboson measurements, both at linear and quadratic order and with one-loop RGE evolution included. The analysis reports no statistically significant evidence for departure from the SM, claims that Bayesian model-averaged posteriors on Wilson coefficients provide substantially improved characterisation relative to conventional global fits, and introduces an operator correlation matrix to expose co-appearance patterns and flat directions.

Significance. If the BIC approximation remains reliable in the presence of the strong correlations that arise from RGE mixing and flat directions in electroweak, Higgs and top data, the work supplies a principled way to incorporate model uncertainty into indirect SMEFT searches. The reported improvement in posterior characterisation and the diagnostic value of the correlation matrix could usefully inform experimental priorities. The absence of quantitative validation of the BIC approximation or of convergence diagnostics for the genetic algorithm, however, leaves the quantitative claims only moderately supported at present.

major comments (3)
  1. [§3] §3 (BIC approximation): The central results on model probabilities, inclusion probabilities and BMA posteriors rest on treating BIC = log L_max − (k/2) log N as an accurate proxy for the marginal likelihood. No quantitative comparison to exact evidence (or to nested sampling) is presented for even a modest subset of models, nor is any error estimate given for the approximation in the presence of the operator correlations induced by one-loop RGE mixing and flat directions. This directly affects the headline claim that the data are consistent with the SM at the reported level.
  2. [§3.3] §3.3 (genetic algorithm): The efficiency and reliability of the genetic algorithm in concentrating evaluations on the high-posterior region of the 2^N model space is asserted without convergence diagnostics, multiple independent runs, or comparison against exhaustive enumeration on a reduced operator basis. Because the reported posterior model probabilities and the operator correlation matrix are obtained from these samples, the lack of validation undermines the claimed improvement in characterisation.
  3. [Results section] Results section (model probabilities): The statement that “no statistically significant evidence for any departure from the SM” is obtained is load-bearing for the paper’s conclusions, yet it is derived solely from BIC-based model probabilities. A direct sensitivity study replacing BIC with a more accurate evidence estimator on the highest-probability models would be required to confirm that the ranking is stable.
minor comments (3)
  1. [Results section] The definition of the operator correlation matrix (presumably Eq. (X) in the results section) should be accompanied by an explicit formula showing how it is computed from the posterior model probabilities.
  2. The sensitivity analysis with respect to the matching scale μ₀ is mentioned in the abstract but the corresponding figures or tables are not referenced in the text; a short paragraph summarising the quantitative variation would improve clarity.
  3. Notation for the Wilson coefficients (e.g., the distinction between linear and quadratic contributions) should be made uniform between the text and the tables of results.

Simulated Author's Rebuttal

3 responses · 1 unresolved

We thank the referee for the thorough and constructive review. The comments correctly identify areas where additional validation would strengthen the manuscript. We address each major comment below, indicating planned revisions where appropriate.

read point-by-point responses

  1. Referee: [§3] §3 (BIC approximation): The central results on model probabilities, inclusion probabilities and BMA posteriors rest on treating BIC = log L_max − (k/2) log N as an accurate proxy for the marginal likelihood. No quantitative comparison to exact evidence (or to nested sampling) is presented for even a modest subset of models, nor is any error estimate given for the approximation in the presence of the operator correlations induced by one-loop RGE mixing and flat directions. This directly affects the headline claim that the data are consistent with the SM at the reported level.

    Authors: We agree that a direct quantitative comparison to exact evidence would provide stronger support for the BIC-based results. Computing nested sampling or other exact estimators for models in this high-dimensional space with RGE-induced correlations is computationally demanding. BIC remains a standard and asymptotically justified approximation for large N in the model selection literature. In the revised manuscript we will expand the discussion in §3 to include the known limitations of BIC under strong correlations, cite relevant validation studies, and provide a qualitative assessment of how RGE mixing affects the approximation in our dataset. revision: partial

  2. Referee: [§3.3] §3.3 (genetic algorithm): The efficiency and reliability of the genetic algorithm in concentrating evaluations on the high-posterior region of the 2^N model space is asserted without convergence diagnostics, multiple independent runs, or comparison against exhaustive enumeration on a reduced operator basis. Because the reported posterior model probabilities and the operator correlation matrix are obtained from these samples, the lack of validation undermines the claimed improvement in characterisation.

    Authors: The genetic algorithm was run with population-size and generation-count settings chosen to achieve stable best-fit values. To meet the referee’s request we will add, in the revised §3.3, explicit convergence diagnostics (fitness trajectories and population diversity metrics), results from at least three independent runs with different random seeds, and a direct comparison against exhaustive enumeration on a reduced operator basis (e.g., the electroweak-only subset) where 2^N remains tractable. revision: yes

  3. Referee: [Results section] Results section (model probabilities): The statement that “no statistically significant evidence for any departure from the SM” is obtained is load-bearing for the paper’s conclusions, yet it is derived solely from BIC-based model probabilities. A direct sensitivity study replacing BIC with a more accurate evidence estimator on the highest-probability models would be required to confirm that the ranking is stable.

    Authors: The SM model indeed receives the highest posterior probability under our BIC ranking, with substantial separation from the next models. Performing a full sensitivity study with alternative evidence estimators on the top-ranked models is feasible for a small number of cases and will be included in the revision. We will report the change (or stability) in model ranking when the highest-probability models are re-evaluated with a more accurate estimator, thereby directly addressing the robustness of the “no significant evidence” conclusion. revision: partial

standing simulated objections not resolved
  • Full quantitative validation of the BIC approximation against exact marginal likelihoods across the entire model space, owing to prohibitive computational cost.

Circularity Check

0 steps flagged

No significant circularity detected in derivation chain

full rationale

The paper develops a Bayesian model selection framework for SMEFT operator subsets, using a genetic algorithm to explore the discrete model space and the BIC as a tractable approximation to the marginal likelihood when ranking subsets against external electroweak, Higgs, top, and diboson data from LEP and LHC Run 2. Posterior model probabilities, marginal inclusion probabilities, Bayesian model averages, and the operator correlation matrix are all computed from these data-driven likelihoods at linear and quadratic order with one-loop RGE evolution. No step reduces by the paper's own equations or self-citations to a quantity defined solely by internal fitted parameters; the headline conclusions (no significant SM departure, improved BMA characterisation) remain dependent on the input observables and are therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The framework depends on standard Bayesian assumptions plus two key domain approximations needed for tractability in the high-dimensional discrete space; no new physical entities are postulated.

free parameters (1)
  • Model-space prior probabilities
    Priors over operator subsets must be chosen to define the hypothesis space; their specific form is not detailed in the abstract.
axioms (2)
  • domain assumption The Bayesian Information Criterion approximates the Bayesian evidence well enough for reliable model comparison among SMEFT operator subsets.
    Invoked to replace full evidence calculation and enable the reported posteriors.
  • domain assumption The genetic algorithm efficiently concentrates evaluations in the high-posterior region of the discrete model space without missing important models.
    Required to justify practical exploration of the full operator-subset space.

pith-pipeline@v0.9.0 · 5788 in / 1689 out tokens · 88917 ms · 2026-05-22T08:56:08.353635+00:00 · methodology

discussion (0)

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