POPxf: Polynomial Observable Format

• POPxf Format is a standardized machine-readable representation that encodes polynomial dependencies of observables on theory parameters with explicit metadata for robust EFT analyses.
• It employs a JSON structure to separate numerical coefficients from the underlying code, ensuring interoperability and consistent interpretation across workflows.
• The format supports linearized dimension-6 SMEFT predictions, facilitating NLO electroweak and QCD corrections for Higgs, gauge-boson, and related observables.

Searching arXiv for the cited POPxf papers to ground the article. arXiv query: (Bellafronte et al., 18 May 2026) POPxf SMEFT decays observables Higgs gauge boson decays POPxf, the Polynomial Observable Prediction Exchange Format, is a structured, machine-readable format for publishing and exchanging semi-analytical theoretical predictions written as polynomials in model parameters, or as functions of such polynomials. It was proposed as a standard way to encode observables, assumptions, and metadata for reuse in phenomenology, reinterpretation, and global fits, with particular emphasis on EFT applications (Brivio et al., 21 Nov 2025). In the dimension-6 SMEFT study of NLO electroweak and QCD corrections to Higgs and gauge-boson observables, POPxf is used as the common interface through which decays, electroweak precision observables, and Higgstrahlung predictions are packaged in JSON form for direct reuse in experimental and phenomenological analyses (Bellafronte et al., 18 May 2026).

1. Definition and rationale

POPxf is designed to standardize arbitrary polynomial dependences of observables on model parameters. Its motivating use case is the common situation in HEP phenomenology where predictions are available in semi-analytical form but are not published in a reusable machine-readable representation. The format is therefore intended to store not only the polynomial coefficients themselves, but also the metadata needed to interpret them consistently, including basis conventions, scales, and other theoretical assumptions (Brivio et al., 21 Nov 2025).

In the SMEFT application to Higgs and gauge-boson observables, this role is especially explicit. NLO SMEFT results are otherwise scattered across papers and involve many Wilson coefficients, particularly at NLO electroweak order where operators absent at tree level can enter through loops. In that setting, POPxf functions as a common interface between precision calculations and downstream fit implementations, rather than as an event-generation format or a full likelihood model (Bellafronte et al., 18 May 2026).

A central feature of the format is that it separates the numerical parameterization of observables from the code that produced it. This makes a prediction portable across workflows while keeping the underlying theoretical assumptions explicit. In the broader POPxf specification, the same design also accommodates observables that are direct polynomials and observables defined as functions of intermediate polynomials, such as ratios (Brivio et al., 21 Nov 2025).

2. Polynomial structure and SMEFT truncation

The general POPxf formalism encodes scalar polynomials in a real parameter vector. For a file $n$, a polynomial $P_k^{(n)}$ is written as

$$P_{k}^{(n)} = \vec p_{k}^{\ (n)} \cdot \vec V^{(n)}\,,$$

and for quadratic order

$$\vec V^{(n)} = \begin{pmatrix} 1 \ \vec C^{(n)} \ {\rm vech}( \vec C^{(n)} \otimes \vec C^{(n)}) \end{pmatrix}.$$

Accordingly,

$$P_{k}^{(n)} = a_{k}^{(n)} + \vec b_{k}^{\,(n)} \cdot \vec C^{(n)} + \vec c_{k}^{\ (n)} \cdot {\rm vech}( \vec C^{(n)} \otimes \vec C^{(n)})\,.$$

This is the generic POPxf object: a polynomial coefficient vector together with enough metadata to define the parameter basis and interpretation (Brivio et al., 21 Nov 2025).

The SMEFT decay-and-Higgstrahlung release specializes this generality very strongly. It starts from

$\mathcal{L}=\mathcal{L}_{SM}+\sum_{i,d}\frac{\hat C_i^{(d)}}{\Lambda^{d-4}}{\mathcal{O}_i^{(d)}\,,$

and expands observables as

$$O_\alpha=O_{\alpha,SM}+\sum_{i,j,d}\beta_{\alpha,i,j}\frac{\hat C_i^{(d)}} {(\Lambda^{d-4})(16 \pi^2)^j}\, .$$

In the actual POPxf files of that work, the stored numerical content is the tree-level SMEFT piece $\beta_{\alpha,i,0}$ and the one-loop NLO SMEFT piece $\beta_{\alpha,i,1}$ for dimension-6 operators. The published representation is therefore a linearized dimension-6 polynomial parameterization around the SM value, not the generic quadratic EFT expansion that POPxf can support (Bellafronte et al., 18 May 2026).

This truncation is explicit. Although POPxf allows arbitrary polynomial degree, the SMEFT release expands only up to linear $\mathcal O(1/\Lambda^2)$ order. Quadratic dimension-6 terms are not included, and neither $P_k^{(n)}$0 effects from double insertions nor dimension-8 contributions are part of the published files. In schematic form, the content is

$P_k^{(n)}$1

with higher-order EFT terms omitted (Bellafronte et al., 18 May 2026).

For Higgs observables, the paper also gives explicit normalized forms: $P_k^{(n)}$2

$P_k^{(n)}$3

with

$P_k^{(n)}$4

The numerical implementation fixes $P_k^{(n)}$5 TeV, so the coefficients in the files are to be read with that normalization (Bellafronte et al., 18 May 2026).

3. File model, metadata, and coefficient conventions

A POPxf prediction file is a single JSON object with exactly three required top-level fields:

• "$schema&quot;</code></li> <li><code>&quot;metadata&quot;</code></li> <li><code>&quot;data&quot;</code></li> </ul> <p>For version 1, the schema URI is $\beta_{\alpha,i,0}$$1 No additional top-level keys are permitted in the current schema (<a href="/papers/2511.17348" title="" rel="nofollow" data-turbo="false" class="assistant-link" x-data x-tooltip.raw="">Brivio et al., 21 Nov 2025</a>).</p> <p>The metadata layer defines how the numerical content is to be interpreted. In the general format, required metadata include <code>observable_names</code>, <code>parameters</code>, <code>basis</code>, and <code>scale</code>; <code>polynomial_names</code> and <code>observable_expressions</code> are additionally required in function-of-polynomials mode. Optional fields include <code>polynomial_degree</code>, <code>reproducibility</code>, and <code>misc</code> (<a href="/papers/2511.17348" title="" rel="nofollow" data-turbo="false" class="assistant-link" x-data x-tooltip.raw="">Brivio et al., 21 Nov 2025</a>). In the SMEFT NLO release, the metadata are process-dependent and encode items such as renormalization scales, input schemes, basis choice, flavor information, and settings such as fermion-mass renormalization scheme (<a href="/papers/2605.18679" title="" rel="nofollow" data-turbo="false" class="assistant-link" x-data x-tooltip.raw="">Bellafronte et al., 18 May 2026</a>).</p> <p>The basis field is particularly important. POPxf can point to a WCxf basis through</p> <ul> <li><code>eft</code></li> <li><code>basis</code></li> <li>optional <code>sectors</code></li> </ul> <p>or use a custom basis specification. This is the mechanism by which Wilson-coefficient conventions are made explicit (<a href="/papers/2511.17348" title="" rel="nofollow" data-turbo="false" class="assistant-link" x-data x-tooltip.raw="">Brivio et al., 21 Nov 2025</a>). In the SMEFT decay application, the underlying EFT is the dimension-6 SMEFT in the Warsaw basis, and the results are described as having arbitrary flavor structure with no built-in restriction such as$$P_k^{(n)}$6 (Bellafronte et al., 18 May 2026).

  The numerical data are indexed by monomial keys, represented as stringified Python-style tuples. For quadratic degree, examples are $\beta_{\alpha,i,0}$2 for constant, linear, and quadratic terms. For complex parameters, an additional R/I tag may be appended. Missing monomials are implicitly interpreted as zero coefficients (Brivio et al., 21 Nov 2025). In the specific SMEFT release, the effective content reduces to the SM reference value together with the linear LO and NLO coefficient sets, since quadratic terms are deliberately absent (Bellafronte et al., 18 May 2026).

  The POPxf results discussed for the SMEFT application are distributed as JSON files in a public GitLab repository. The paper does not reproduce the full schema inline, but it states that the files contain the numerical values of $P_k^{(n)}$7 and $P_k^{(n)}$8, along with sufficient metadata for reproducibility (Bellafronte et al., 18 May 2026).

  4. Observables encoded in the NLO SMEFT release

  The POPxf implementation in the SMEFT decay paper is broad in scope. It packages NLO QCD and electroweak dimension-6 SMEFT results for Higgs decays, gauge-boson decays, electroweak precision observables, and Higgstrahlung (Bellafronte et al., 18 May 2026).

  Sector	POPxf content
  Higgs decays	All 2-body and 4-body Higgs decays and branching ratios
  Differential Higgs decay	$P_k^{(n)}$9 inclusive with $P_{k}^{(n)} = \vec p_{k}^{\ (n)} \cdot \vec V^{(n)}\,,$0 GeV and $P_{k}^{(n)} = \vec p_{k}^{\ (n)} \cdot \vec V^{(n)}\,,$1
  Gauge-boson observables	All 2-body $P_{k}^{(n)} = \vec p_{k}^{\ (n)} \cdot \vec V^{(n)}\,,$2 and $P_{k}^{(n)} = \vec p_{k}^{\ (n)} \cdot \vec V^{(n)}\,,$3 decays and a standard EWPO set
  Higgstrahlung	$P_{k}^{(n)} = \vec p_{k}^{\ (n)} \cdot \vec V^{(n)}\,,$4 at $P_{k}^{(n)} = \vec p_{k}^{\ (n)} \cdot \vec V^{(n)}\,,$5 GeV

  For Higgs decays, the listed channels include

  $P_{k}^{(n)} = \vec p_{k}^{\ (n)} \cdot \vec V^{(n)}\,,$6

  $P_{k}^{(n)} = \vec p_{k}^{\ (n)} \cdot \vec V^{(n)}\,,$7

  and four-fermion final states

  $P_{k}^{(n)} = \vec p_{k}^{\ (n)} \cdot \vec V^{(n)}\,,$8

  with $P_{k}^{(n)} = \vec p_{k}^{\ (n)} \cdot \vec V^{(n)}\,,$9. The total Higgs width, including all dimension-6 contributions at NLO, is highlighted as a particularly useful output (Bellafronte et al., 18 May 2026).

  For differential $\vec V^{(n)} = \begin{pmatrix} 1 \ \vec C^{(n)} \ {\rm vech}( \vec C^{(n)} \otimes \vec C^{(n)}) \end{pmatrix}.$0 information, the files include inclusive $\vec V^{(n)} = \begin{pmatrix} 1 \ \vec C^{(n)} \ {\rm vech}( \vec C^{(n)} \otimes \vec C^{(n)}) \end{pmatrix}.$1 with a cut $\vec V^{(n)} = \begin{pmatrix} 1 \ \vec C^{(n)} \ {\rm vech}( \vec C^{(n)} \otimes \vec C^{(n)}) \end{pmatrix}.$2 GeV and the distributions

  $\vec V^{(n)} = \begin{pmatrix} 1 \ \vec C^{(n)} \ {\rm vech}( \vec C^{(n)} \otimes \vec C^{(n)}) \end{pmatrix}.$3

  for

  $\vec V^{(n)} = \begin{pmatrix} 1 \ \vec C^{(n)} \ {\rm vech}( \vec C^{(n)} \otimes \vec C^{(n)}) \end{pmatrix}.$4

  The paper defines $\vec V^{(n)} = \begin{pmatrix} 1 \ \vec C^{(n)} \ {\rm vech}( \vec C^{(n)} \otimes \vec C^{(n)}) \end{pmatrix}.$5 as the same-flavor opposite-sign pair whose invariant mass is closest to $\vec V^{(n)} = \begin{pmatrix} 1 \ \vec C^{(n)} \ {\rm vech}( \vec C^{(n)} \otimes \vec C^{(n)}) \end{pmatrix}.$6, and $\vec V^{(n)} = \begin{pmatrix} 1 \ \vec C^{(n)} \ {\rm vech}( \vec C^{(n)} \otimes \vec C^{(n)}) \end{pmatrix}.$7 as the opposite lepton pair (Bellafronte et al., 18 May 2026).

  For electroweak precision observables, the published set is

  $\vec V^{(n)} = \begin{pmatrix} 1 \ \vec C^{(n)} \ {\rm vech}( \vec C^{(n)} \otimes \vec C^{(n)}) \end{pmatrix}.$8

  In the $\vec V^{(n)} = \begin{pmatrix} 1 \ \vec C^{(n)} \ {\rm vech}( \vec C^{(n)} \otimes \vec C^{(n)}) \end{pmatrix}.$9 input scheme, $P_{k}^{(n)} = a_{k}^{(n)} + \vec b_{k}^{\,(n)} \cdot \vec C^{(n)} + \vec c_{k}^{\ (n)} \cdot {\rm vech}( \vec C^{(n)} \otimes \vec C^{(n)})\,.$0 replaces $P_{k}^{(n)} = a_{k}^{(n)} + \vec b_{k}^{\,(n)} \cdot \vec C^{(n)} + \vec c_{k}^{\ (n)} \cdot {\rm vech}( \vec C^{(n)} \otimes \vec C^{(n)})\,.$1 in that list (Bellafronte et al., 18 May 2026).

  For Higgstrahlung, the release provides total cross sections for

  $P_{k}^{(n)} = a_{k}^{(n)} + \vec b_{k}^{\,(n)} \cdot \vec C^{(n)} + \vec c_{k}^{\ (n)} \cdot {\rm vech}( \vec C^{(n)} \otimes \vec C^{(n)})\,.$2

  at

  $P_{k}^{(n)} = a_{k}^{(n)} + \vec b_{k}^{\,(n)} \cdot \vec C^{(n)} + \vec c_{k}^{\ (n)} \cdot {\rm vech}( \vec C^{(n)} \otimes \vec C^{(n)})\,.$3

  with both polarized and unpolarized results, and with full LO and NLO SMEFT effects including QCD and electroweak corrections (Bellafronte et al., 18 May 2026).

  5. Renormalization conventions and practical reconstruction

  The POPxf files in the SMEFT release are convention-dependent, and these conventions are part of their scientific content rather than incidental metadata. For Higgs decays, the renormalization scale is $P_{k}^{(n)} = a_{k}^{(n)} + \vec b_{k}^{\,(n)} \cdot \vec C^{(n)} + \vec c_{k}^{\ (n)} \cdot {\rm vech}( \vec C^{(n)} \otimes \vec C^{(n)})\,.$4; couplings and gauge-boson masses are renormalized in the on-shell scheme; and fermion masses are supplied in both on-shell and $P_{k}^{(n)} = a_{k}^{(n)} + \vec b_{k}^{\,(n)} \cdot \vec C^{(n)} + \vec c_{k}^{\ (n)} \cdot {\rm vech}( \vec C^{(n)} \otimes \vec C^{(n)})\,.$5 schemes as separate files. The quark-mass renormalization choice can significantly affect numerical coefficients, especially for $P_{k}^{(n)} = a_{k}^{(n)} + \vec b_{k}^{\,(n)} \cdot \vec C^{(n)} + \vec c_{k}^{\ (n)} \cdot {\rm vech}( \vec C^{(n)} \otimes \vec C^{(n)})\,.$6, and because $P_{k}^{(n)} = a_{k}^{(n)} + \vec b_{k}^{\,(n)} \cdot \vec C^{(n)} + \vec c_{k}^{\ (n)} \cdot {\rm vech}( \vec C^{(n)} \otimes \vec C^{(n)})\,.$7 dominates the total width this propagates into branching ratios generally (Bellafronte et al., 18 May 2026).

  For electroweak observables, separate files are provided in both

  $P_{k}^{(n)} = a_{k}^{(n)} + \vec b_{k}^{\,(n)} \cdot \vec C^{(n)} + \vec c_{k}^{\ (n)} \cdot {\rm vech}( \vec C^{(n)} \otimes \vec C^{(n)})\,.$8

  input schemes. The paper states that input-scheme dependence is sizable for some operators, so the choice changes the numerical POPxf coefficients (Bellafronte et al., 18 May 2026).

  For Higgstrahlung, the input parameters are $P_{k}^{(n)} = a_{k}^{(n)} + \vec b_{k}^{\,(n)} \cdot \vec C^{(n)} + \vec c_{k}^{\ (n)} \cdot {\rm vech}( \vec C^{(n)} \otimes \vec C^{(n)})\,.$9, and the renormalization scale is taken either as

  $\mathcal{L}=\mathcal{L}_{SM}+\sum_{i,d}\frac{\hat C_i^{(d)}}{\Lambda^{d-4}}{\mathcal{O}_i^{(d)}\,,$0

  Files are split accordingly, as well as by benchmark energy and polarization choice (Bellafronte et al., 18 May 2026).

  The practical reconstruction rule is straightforward. Each file supplies an SM reference value and a set of linear coefficients. One then inserts Wilson coefficients in the normalization

  $\mathcal{L}=\mathcal{L}_{SM}+\sum_{i,d}\frac{\hat C_i^{(d)}}{\Lambda^{d-4}}{\mathcal{O}_i^{(d)}\,,$1

  and evaluates

  $\mathcal{L}=\mathcal{L}_{SM}+\sum_{i,d}\frac{\hat C_i^{(d)}}{\Lambda^{d-4}}{\mathcal{O}_i^{(d)}\,,$2

  where $\mathcal{L}=\mathcal{L}_{SM}+\sum_{i,d}\frac{\hat C_i^{(d)}}{\Lambda^{d-4}}{\mathcal{O}_i^{(d)}\,,$3 denotes $\mathcal{L}=\mathcal{L}_{SM}+\sum_{i,d}\frac{\hat C_i^{(d)}}{\Lambda^{d-4}}{\mathcal{O}_i^{(d)}\,,$4, $\mathcal{L}=\mathcal{L}_{SM}+\sum_{i,d}\frac{\hat C_i^{(d)}}{\Lambda^{d-4}}{\mathcal{O}_i^{(d)}\,,$5, or the analogous process-dependent coefficient. For branching ratios, the dedicated $\mathcal{L}=\mathcal{L}_{SM}+\sum_{i,d}\frac{\hat C_i^{(d)}}{\Lambda^{d-4}}{\mathcal{O}_i^{(d)}\,,$6 coefficients may be used directly, or the result may be reconstructed from partial widths and total width if a consistency check is required (Bellafronte et al., 18 May 2026).

  The SM normalization is not fully uniform across all entries. For most Higgs rates, $\mathcal{L}=\mathcal{L}_{SM}+\sum_{i,d}\frac{\hat C_i^{(d)}}{\Lambda^{d-4}}{\mathcal{O}_i^{(d)}\,,$7 and $\mathcal{L}=\mathcal{L}_{SM}+\sum_{i,d}\frac{\hat C_i^{(d)}}{\Lambda^{d-4}}{\mathcal{O}_i^{(d)}\,,$8 are taken from the Higgs Cross Section Working Group “world’s best theory calculations,” whereas for the $\mathcal{L}=\mathcal{L}_{SM}+\sum_{i,d}\frac{\hat C_i^{(d)}}{\Lambda^{d-4}}{\mathcal{O}_i^{(d)}\,,$9 inclusive result with $O_\alpha=O_{\alpha,SM}+\sum_{i,j,d}\beta_{\alpha,i,j}\frac{\hat C_i^{(d)}} {(\Lambda^{d-4})(16 \pi^2)^j}\, .$0 GeV and for the $O_\alpha=O_{\alpha,SM}+\sum_{i,j,d}\beta_{\alpha,i,j}\frac{\hat C_i^{(d)}} {(\Lambda^{d-4})(16 \pi^2)^j}\, .$1 distributions the authors use their own NLO SM calculation (Bellafronte et al., 18 May 2026).

  6. Uncertainties, interoperability, and limitations

  In the general POPxf specification, uncertainties are attached to the observable coefficients rather than only to evaluated observable points. The format defines coefficient uncertainties $O_\alpha=O_{\alpha,SM}+\sum_{i,j,d}\beta_{\alpha,i,j}\frac{\hat C_i^{(d)}} {(\Lambda^{d-4})(16 \pi^2)^j}\, .$2, coefficient-level correlations $O_\alpha=O_{\alpha,SM}+\sum_{i,j,d}\beta_{\alpha,i,j}\frac{\hat C_i^{(d)}} {(\Lambda^{d-4})(16 \pi^2)^j}\, .$3, and corresponding covariance matrices. It also allows multiple uncertainty sources, such as MC statistics, scale, and PDFs, whose covariance contributions add linearly (Brivio et al., 21 Nov 2025). Correlations are stored separately from the main prediction file, either in JSON or HDF5, and parameter-dependent correlations use four-dimensional arrays indexed by observables and monomials (Brivio et al., 21 Nov 2025).

  This general design is complemented by explicit interoperability with WCxf through the basis.wcxf field, while remaining applicable to custom bases. POPxf is therefore best understood as a standard for observable predictions as functions of theory parameters, not as a replacement for Wilson-coefficient exchange formats or as a container for event samples (Brivio et al., 21 Nov 2025).

  The SMEFT implementation in the NLO decay paper carries several explicit limitations. The EFT expansion is linearized in dimension-6 coefficients, so no quadratic dimension-6 terms are included. Tree-level results are retained only to $O_\alpha=O_{\alpha,SM}+\sum_{i,j,d}\beta_{\alpha,i,j}\frac{\hat C_i^{(d)}} {(\Lambda^{d-4})(16 \pi^2)^j}\, .$4, and NLO calculations are accurate to $O_\alpha=O_{\alpha,SM}+\sum_{i,j,d}\beta_{\alpha,i,j}\frac{\hat C_i^{(d)}} {(\Lambda^{d-4})(16 \pi^2)^j}\, .$5, not to $O_\alpha=O_{\alpha,SM}+\sum_{i,j,d}\beta_{\alpha,i,j}\frac{\hat C_i^{(d)}} {(\Lambda^{d-4})(16 \pi^2)^j}\, .$6. The release provides NLO QCD and electroweak corrections for the listed processes, but not NNLO SMEFT, not full $O_\alpha=O_{\alpha,SM}+\sum_{i,j,d}\beta_{\alpha,i,j}\frac{\hat C_i^{(d)}} {(\Lambda^{d-4})(16 \pi^2)^j}\, .$7 one-loop effects, and not all process-specific additions such as QED corrections for Higgstrahlung, which may need to be added separately if desired (Bellafronte et al., 18 May 2026).

  For Higgs $O_\alpha=O_{\alpha,SM}+\sum_{i,j,d}\beta_{\alpha,i,j}\frac{\hat C_i^{(d)}} {(\Lambda^{d-4})(16 \pi^2)^j}\, .$8 decays, the treatment differs between perturbative orders: LO uses the full four-body final state, whereas NLO uses the narrow width approximation,

  $O_\alpha=O_{\alpha,SM}+\sum_{i,j,d}\beta_{\alpha,i,j}\frac{\hat C_i^{(d)}} {(\Lambda^{d-4})(16 \pi^2)^j}\, .$9

  The paper refers readers to the original Higgs-decay publication and repository documentation for details on the validity of that approximation (Bellafronte et al., 18 May 2026).

  The POPxf description in this SMEFT work does not spell out a CP-filtering convention. The text states “all dimension-6 operators” and does not advertise a CP-even-only restriction, but it does not provide a dedicated POPxf-level CP convention. This suggests that CP treatment is not encoded as a separate high-level simplification in the format description itself (Bellafronte et al., 18 May 2026).

  Finally, the practical validity of any POPxf-encoded SMEFT prediction remains subject to EFT convergence. The paper states that the truncation is meaningful only when Wilson-coefficient effects are small enough that neglected $\beta_{\alpha,i,0}$0 terms are subdominant, a caveat that is especially relevant at higher Higgstrahlung energies or in kinematic tails (Bellafronte et al., 18 May 2026).