---
title: 'POPxf: Polynomial Observable Format'
url: https://www.emergentmind.com/topics/popxf-format
type: topic
---

# POPxf: Polynomial Observable Format

Searching arXiv for the cited POPxf papers to ground the article.
arXiv query: 2605.18679 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 [2511.17348]. 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 [2605.18679].

## 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 [2511.17348].

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 [2605.18679].

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 [2511.17348].

## 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 [2511.17348].

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 [2605.18679].

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 \(\mathcal O(1/\Lambda^4)\) effects from double insertions nor dimension-8 contributions are part of the published files. In schematic form, the content is
\[
O_\alpha = O_{\alpha,SM}
+ \sum_i \beta_{\alpha,i,0}\,\hat C_i^{(6)}\,\Lambda^{-2}
+ \sum_i \beta_{\alpha,i,1}\,\hat C_i^{(6)}\,\Lambda^{-2}(16\pi^2)^{-1},
\]
with higher-order EFT terms omitted [2605.18679].

For Higgs observables, the paper also gives explicit normalized forms:
\[
\frac{\Gamma(H\rightarrow X)^{(0,1)}}{\Gamma(H\rightarrow X)_{\mathrm{SM}}}
=
1+\sum_i \delta_i^{(0,1)}\,C_i\,,
\]
\[
\frac{BR(H\rightarrow X)^{(0,1)}}{BR(H\rightarrow X)_{\mathrm{SM}}}
=
1+\sum_i \Delta_i^{(0,1)}\,C_i\,,
\]
with
\[
C_i\equiv \frac{\hat C_i^{(6)}}{\Lambda^2}\,.
\]
The numerical implementation fixes \(\Lambda=1\) TeV, so the coefficients in the files are to be read with that normalization [2605.18679].

## 3. File model, metadata, and coefficient conventions

A POPxf prediction file is a single JSON object with exactly three required top-level fields:
- `"$schema"`
- `"metadata"`
- `"data"`

For version 1, the schema URI is
```json
"$schema": "https://json.schemastore.org/popxf-1.0.json"
```
No additional top-level keys are permitted in the current schema [2511.17348].

The metadata layer defines how the numerical content is to be interpreted. In the general format, required metadata include `observable_names`, `parameters`, `basis`, and `scale`; `polynomial_names` and `observable_expressions` are additionally required in function-of-polynomials mode. Optional fields include `polynomial_degree`, `reproducibility`, and `misc` [2511.17348]. 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 [2605.18679].

The basis field is particularly important. POPxf can point to a WCxf basis through
- `eft`
- `basis`
- optional `sectors`

or use a custom basis specification. This is the mechanism by which Wilson-coefficient conventions are made explicit [2511.17348]. 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 \(U(3)^5\) [2605.18679].

The numerical data are indexed by **monomial keys**, represented as stringified Python-style tuples. For quadratic degree, examples are
```json
"('', '')"
"('', 'C1')"
"('C1', 'C2')"
```
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 [2511.17348]. 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 [2605.18679].

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 \(\beta_{\alpha,i,0}\) and \(\beta_{\alpha,i,1}\), along with sufficient metadata for reproducibility [2605.18679].

## 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 [2605.18679].

| Sector | POPxf content |
|---|---|
| Higgs decays | All 2-body and 4-body Higgs decays and branching ratios |
| Differential Higgs decay | \(H\to 4\ell\) inclusive with \(M_{Z^*}>12\) GeV and \(d\Gamma/dM_{Z^*}\) |
| Gauge-boson observables | All 2-body \(Z\) and \(W\) decays and a standard EWPO set |
| Higgstrahlung | \(e^+e^-\to ZH\) at \(\sqrt{s}=240,365,500\) GeV |

For Higgs decays, the listed channels include
\[
H\rightarrow \gamma\gamma,\qquad H\rightarrow \gamma Z,\qquad H\rightarrow gg,
\]
\[
H\to f\bar f,\qquad f\in[b,\tau,\mu,c,s],
\]
and four-fermion final states
\[
H\rightarrow (f_{g_1}\overline{f}_{g_1})(f_{g_2}\overline{f}_{g_2}),
\qquad
f\in[\ell,\nu_\ell,u,d],
\]
with \(g_k=1,2,3\). The total Higgs width, including all dimension-6 contributions at NLO, is highlighted as a particularly useful output [2605.18679].

For differential \(H\to4\ell\) information, the files include inclusive \(H\to4\ell\) with a cut \(M_{Z^*}>12\) GeV and the distributions
\[
\frac{d\Gamma}{dM_{Z^*}}
\]
for
\[
H\to e^+e^-e^+e^-,
\qquad
H\to e^+e^-\mu^+\mu^-,
\qquad
H\to \mu^+\mu^-\mu^+\mu^-.
\]
The paper defines \(M_Z^{\ell\ell}\) as the same-flavor opposite-sign pair whose invariant mass is closest to \(M_Z\), and \(M_{Z^*}\) as the opposite lepton pair [2605.18679].

For electroweak precision observables, the published set is
\[
\alpha,\Gamma_W(\textrm{total}),\Gamma_Z(\textrm{total}),R_e,R_\mu,R_\tau,R_c,\sigma(\textrm{had}),A_\mu,A_\tau,A_s,A_c,A_b,A_{FB}^e,A_{FB}^\tau,A_{FB}^s,A_{FB}^c,A_{FB}^b.
\]
In the \(\{M_W,M_Z,G_F\}\) input scheme, \(M_W\) replaces \(\alpha\) in that list [2605.18679].

For Higgstrahlung, the release provides total cross sections for
\[
e^+e^-\rightarrow ZH
\]
at
\[
\sqrt{s}=240,\ 365,\ 500\ \mathrm{GeV},
\]
with both polarized and unpolarized results, and with full LO and NLO SMEFT effects including QCD and electroweak corrections [2605.18679].

## 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 \(\mu=m_H\); couplings and gauge-boson masses are renormalized in the on-shell scheme; and fermion masses are supplied in both on-shell and \(\overline{\rm MS}\) schemes as separate files. The quark-mass renormalization choice can significantly affect numerical coefficients, especially for \(H\to q\bar q\), and because \(H\to b\bar b\) dominates the total width this propagates into branching ratios generally [2605.18679].

For electroweak observables, separate files are provided in both
\[
\{\alpha(0),M_Z,G_F\}
\qquad\text{and}\qquad
\{M_W,M_Z,G_F\}
\]
input schemes. The paper states that input-scheme dependence is sizable for some operators, so the choice changes the numerical POPxf coefficients [2605.18679].

For Higgstrahlung, the input parameters are \(M_W,M_Z,G_F\), and the renormalization scale is taken either as
\[
\mu=1~\mathrm{TeV}
\qquad\text{or}\qquad
\mu=\sqrt{s}.
\]
Files are split accordingly, as well as by benchmark energy and polarization choice [2605.18679].

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
\[
C_i=\hat C_i^{(6)}/\Lambda^2,
\qquad \Lambda=1~\mathrm{TeV},
\]
and evaluates
\[
O = O_{\rm SM}\left(1+\sum_i k_i C_i\right),
\]
where \(k_i\) denotes \(\delta_i\), \(\Delta_i\), or the analogous process-dependent coefficient. For branching ratios, the dedicated \(BR\) coefficients may be used directly, or the result may be reconstructed from partial widths and total width if a consistency check is required [2605.18679].

The SM normalization is not fully uniform across all entries. For most Higgs rates, \(\Gamma_{\rm SM}\) and \(BR_{\rm SM}\) are taken from the Higgs Cross Section Working Group “world’s best theory calculations,” whereas for the \(H\to4\ell\) inclusive result with \(M_{Z^*}>12\) GeV and for the \(d\Gamma/dM_{Z^*}\) distributions the authors use their own NLO SM calculation [2605.18679].

## 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 \(\vec \sigma_m^{\ (n)}\), coefficient-level correlations \(\rho_{m m^\prime}^{(n n^\prime)}\), and corresponding covariance matrices. It also allows multiple uncertainty sources, such as MC statistics, scale, and PDFs, whose covariance contributions add linearly [2511.17348]. 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 [2511.17348].

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 [2511.17348].

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 \(\mathcal O(1/\Lambda^2)\), and NLO calculations are accurate to \(\mathcal O(1/(16\pi^2\Lambda^2))\), not to \(\mathcal O(1/(16\pi^2\Lambda^4))\). The release provides NLO QCD and electroweak corrections for the listed processes, but not NNLO SMEFT, not full \(\Lambda^{-4}\) one-loop effects, and not all process-specific additions such as QED corrections for Higgstrahlung, which may need to be added separately if desired [2605.18679].

For Higgs \(4f\) decays, the treatment differs between perturbative orders: LO uses the full four-body final state, whereas NLO uses the narrow width approximation,
\[
H \to f_{g_1}\bar f_{g_1}V,\qquad V\to f_{g_2}\bar f_{g_2}.
\]
The paper refers readers to the original Higgs-decay publication and repository documentation for details on the validity of that approximation [2605.18679].

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 [2605.18679].

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 \(\Lambda^{-4}\) terms are subdominant, a caveat that is especially relevant at higher Higgstrahlung energies or in kinematic tails [2605.18679].

Source: https://www.emergentmind.com/topics/popxf-format