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WCxf: an exchange format for Wilson coefficients beyond the Standard Model
Jason Aebischer, Ilaria Brivio, Alejandro Celis, Jared A. Evans, Yun Jiang, Jacky Kumar, Xuanyou Pan, Werner Porod, Janusz Rosiek, David Shih, Florian Staub, David M. Straub, Danny van Dyk, Avelino Vicente
TL;DR
Exchanging Wilson coefficients across EFT calculators, RG runners, and observable codes is hindered by incompatible bases, conventions, and file formats. The paper introduces WCxf, an unambiguous, extensible, and robust exchange format with tools and interfaces for SMEFT and WET. It demonstrates that the format facilitates interfacing public codes and supports future global likelihood analyses.
Problem
Exchanging numerical Wilson coefficients between EFT codes is difficult because software uses many operators, bases, normalizations, and programming-language-specific file formats.
Method
WCxf defines non-redundant operators and fixed normalizations within user-extensible EFT bases, using standard formats plus tools for translation, matching, validation, and code interfacing.
Results
WCxf was implemented for SMEFT and WET in several public codes, demonstrating facilitated interfacing for precision tests of the Standard Model.
Takeaways & Limitations
The format makes future global likelihoods combining flavour, electroweak, and Higgs observables more realistic and simplifies model-specific analyses.
Takeaways & Limitations
SMEFT basis rotations can depend on the scale-dependent vacuum expectation value, so the paper advocates an on-shell definition for affected terms.
Abstract
from arXiv · showhide
We define a data exchange format for numerical values of Wilson coefficients of local operators parameterising low-energy effects of physics beyond the Standard Model. The format facilitates interfacing model-specific Wilson coefficient calculators, renormalisation group (RG) runners, and observable calculators. It is designed to be unambiguous (defining a non-redundant set of operators with fixed normalisation in each basis), extensible (allowing the addition of new EFTs or bases by the user), and robust (being based on industry standard file formats with parsers implemented in many programming languages). We have implemented the format for the Standard Model EFT (SMEFT) and for the weak effective theory (WET) below the electroweak scale and have added interfaces to a number of public codes dealing with SMEFT or WET. We also provide command-line utilities and a Python module for convenient manipulation of WCxf files, including translation between different bases and matching from SMEFT to WET.
1. Introduction and Motivation
WCxf addresses the practical difficulty of exchanging Wilson coefficients across EFT codes by defining a common format that is unambiguous, extensible, and robust. It preserves basis flexibility while enabling translation and matching through shared tools.
- EFTs describe low-energy effects of heavy new particles through local operators, supporting model-independent analyses and separation of low-energy phenomenology from short-distance physics.
- Exchanging Wilson coefficients between public observable, RG, matching, SMEFT, and dark-matter codes is hindered by many operators, incompatible bases, normalization ambiguities, and inconsistent file formats.
- WCxf defines a common format that fixes non-redundant operators and normalization within each EFT basis, while allowing users to add EFTs or bases.
- Industry-standard file formats and parsers in many programming languages make WCxf robust across software ecosystems.
- WCxf does not impose one universal basis: public definition files support basis translation and EFT matching through a single shared tool.
- The format is implemented for SMEFT and WET, with command-line and Python tools for translation, matching, validation, and code interfaces.
2. Definition of the format
WCxf separates theory definitions, basis definitions, and exchanged coefficient data, using flexible machine-readable structures with optional metadata. Its sector organization and basis conventions support translation, RG evolution, and interoperable numerical exchange.
- 2. Definition of the format: WCxf defines EFT, basis, and Wilson coefficient files, separating immutable theory and basis definitions from numerical data exchanged between programs.
- 2. Definition of the format: The data structure uses key-value pairs and lists, with JSON and YAML recommended as parsable exchange formats; YAML adds readability, editing, and comments.
- 2. Definition of the format: WCxf data can also be exchanged directly as in-memory structures, such as a Python wcxf.WC instance or dictionary, without using files.
- 2.1. The EFT file: EFT files define named sectors whose fixed quantum numbers organize operators and allow basis translations and RG evolution to proceed sector by sector.
- 2.2. The basis file: Basis files list non-redundant operators by sector, use scalar Wilson coefficients without flavour indices, and may be partial when a problem needs only a subset.
- 2.2. The basis file: Basis conventions define coefficients in the effective Lagrangian, add Hermitian conjugates only for non-Hermitian operators, and place factored prefactors within the operator.
- 2.2. The basis file: Child bases can inherit omitted sectors from parent bases, while Wilson coefficient files specify EFT, basis, scale, and numerical coefficients with defined defaults and units.
- 2.4. Metadata: Metadata supports descriptions, software generators, and readable operator representations, with arbitrary top-level key-value pairs and per-operator fields such as tex.
3. Pre-defined EFTs and bases
WCxf predefines SMEFT and WET EFTs with complete, non-redundant operator bases, while accommodating basis choices and lower-energy EFT variants. Its SMEFT conventions fix a weak basis and mass-matrix structure, with explicit scale-dependent caveats.
- WCxf defines SMEFT and WET, including complete bases and additional WET bases tailored to specific codes.
- SMEFT: SMEFT includes dimension-6 baryon- and lepton-number-conserving operators and operators with ΔB = ΔL = 1; WCxf currently limits itself to dimension 6.
- SMEFT: The Warsaw basis is a complete non-redundant dimension-6 basis, while SMEFT basis construction must address the freedom of U(3)^5 fermion rotations.
- SMEFT: WCxf fixes a default weak basis through specified fermion mass-matrix forms, including diagonal Md and Me and CKM- and PMNS-like structures.
- SMEFT: The chosen diagonality is convenient for removing unphysical Yukawa parameters and translating to the electroweak-scale mass basis, but it is not invariant under SMEFT RG evolution.
- SMEFT: The neutrino mixing convention coincides with the standard PMNS matrix only for normally ordered neutrino masses; inverse ordering permutes the angles.
- SMEFT: The scale-dependent electroweak VEV can vanish at high scales, so WCxf advocates its on-shell definition for the O(v^2) mass-matrix terms.
- WET: WET contains leptons, all quarks except the top, the photon, and the gluon; below the b-quark scale, WCxf defines EFTs with fewer dynamical fields.
4. Python and command-line interface
The wcxf Python package and command-line interface centralize validation, basis translation, and matching for WCxf files. They support reusable translators and matchers, standard-format conversion, and both programmatic and command-line workflows.
- The wcxf package provides central tools for basis translation, EFT matching, and validation, including SMEFT Warsaw-to-WET JMS matching.
- The package requires Python 3.5 or above and NumPy, and includes public-repository EFT and basis files that are read at import time.
- The package represents EFT, basis, and Wilson coefficient files with classes that load YAML or JSON and validate their contents.
- Wilson coefficient objects expose numerical values as dictionaries, representing complex coefficients directly as complex numbers.
- Translators and matchers are registered as functions that map coefficient dictionaries between bases or EFTs, after which translate and match methods generate converted instances.
- The command-line interface supports JSON/YAML conversion, file validation, basis translation, and matching between EFTs.
5. Implementation in public codes
WCxf interfaces connect existing observable, matching, RG, and model-calculation codes to a common Wilson-coefficient representation. Implementations cover flavio, EOS, FlavorKit, and SPheno-related workflows, with code-specific bases and output conventions.
- flavio: flavio uses WCxf Wilson coefficients for WET observables and fits, evolving coefficients from their input scale to each process scale automatically.
- EOS: EOS computes flavour observables and infers SM and WET parameters, exposing both command-line clients and C++ and Python interfaces.
- EOS: The wcxf2eos script converts WCxf files into EOS parameter YAML files, while the EOS Python interface is recommended for examining more than a few points.
- FlavorKit: FlavorKit extends SARAH to calculate Wilson coefficients and numerical flavour observables across a wide range of models.
- FlavorKit: The FlavorKit basis is non-redundant but incomplete relative to the complete JMS basis and can be extended with new operators.
- FlavorKit: FlavorKit output configuration specifies the number of energy scales, EFT, basis, scale, and mappings from internally calculated coefficients to WCxf coefficients.
5.4. SPheno
SPheno exports calculated Wilson coefficients in WCxf JSON files at separate scales for quark- and lepton-flavour-violating operators. Its outputs use the FlavorKit basis and include only coefficients needed for SPheno’s flavour observables.
- SPheno exports calculated Wilson coefficients as WCxf JSON files when the corresponding Les Houches input flag is enabled.
- Quark flavour-violating coefficients are written at Q = 160 GeV, while lepton flavour-violating coefficients are written at Q = mZ.
- SPheno writes separate files for quark and lepton flavour violation: WC.SPheno_1.json and WC.SPheno_2.json, respectively.
- The output files use the FlavorKit basis, but contain only coefficients required for the flavour observables calculated in SPheno.
5.5. DsixTools
DsixTools supports WCxf input and output, with conversions through its native SLHA workflow and routines for SMEFT basis rotation. Converting from SLHA requires users to supply metadata absent from the SLHA file, including CP-violation status and coefficient scales.
- DsixTools supports WCxf Wilson-coefficient input and output from version 1.1.2.
- WCxf input can use JSON or YAML and must contain Warsaw-basis coefficients plus Options.dat and SMInput.dat.YAML input requires a Mathematica YAML importer.
- DsixTools translates WCxf input into SLHA format before continuing its usual processing workflow.The WCxf basis is a weak basis distinct from the fermion mass basis.
- SMEFT coefficients must be rotated to the WCXF basis before export, while standard CKM phase conventions are maintained.The resulting replacement rule can obtain individual coefficients, after which all coefficients can be exported.
- SLHA-to-WCxf conversion requires the high scale because DsixTools treats Wilson coefficients as dimensionless and rescales them by Λ^-2.The conversion relation is C_WCxf = Λ^-2 C_DsixTools.
- SLHA files omit CP-violation status, SCALE, and HIGHSCALE, so users must provide these three details when running the conversion routine.
5.6. wilson
wilson is a Python library that automatically matches and runs Wilson coefficients across SMEFT and WET regimes. It accepts ultraviolet-scale coefficients and can return them at lower scales through its WCxf-based interface.
- wilson provides complete 1-loop SMEFT running, full tree-level matching to WET, and complete 1-loop QCD and QED WET running.
- The library automatically performs running and matching from a user-specified UV scale to any lower scale, including below the electroweak scale.
- wilson uses the WCxf Python API, allowing initial Wilson coefficients to be returned as wcxf.WC instances.
- The WC class can also be initialized directly from a WCxf file.
5.7. FormFlavor
FormFlavor derives Wilson coefficients through modular analytic, numerical, and observable-calculation components and supports WCxf file exchange. Its WCxf output uses the FormFlavor basis and omits information about diagrammatic origins.
- FormFlavor is modularized into amplitude derivation, numerical Wilson-coefficient conversion, observable calculation, and model-specific components.
- Since version 1.2.0, FormFlavor can read and write WCxf files.
- FormFlavor writes FFWilson output to JSON in the FormFlavor basis at the scale where the coefficients were evaluated.
- WCxf JSON output loses information about which diagrams generated each contribution.
- JSON files can be read by FFObservables only when they are supplied in the FormFlavor basis.
5.8. SMEFTsim
SMEFTsim implements dimension-six Warsaw-basis operators in FeynRules and UFO models for symbolic calculations and numerical simulations. The WCxf package converts WCxf files into SMEFTsim parameter cards with configurable model, scheme, and cutoff settings.
- SMEFTsim implements the complete B- and L-conserving dimension-six Warsaw-basis operator set in FeynRules and UFO models.
- The package supports symbolic Mathematica calculations and numerical simulations with MadGraph5_aMC@NLO.
- SMEFTsim provides six models combining three flavour assumptions with two input-parameter schemes.
- wcxf2smeftsim converts JSON or YAML WCxf files into MadGraph-compatible param_card.dat files.
- The conversion script allows users to select the input scheme, model set, output filename, and cutoff scale Λ.
- Unspecified model-set, input-scheme, and cutoff-scale options default to A, alpha, and 1000, respectively.The default output filename is wcxf2smeftsim param card.dat.
5.9. SMEFT Feynman Rules
The SMEFT Feynman Rules package integrates WCxf values into its symbolic and numerical workflow, supporting import, model generation, and export to downstream formats. Its implementation accommodates SMEFT calculations in the Warsaw mass basis and repeated generation of outputs after coefficient updates.
- Flavour structure: The implementation includes Minimal Flavour Violation structures with flavour spurion insertions up to linear order in Yukawa combinations.The MFV setup assumes CKM CP violation and a U(3)5 flavour symmetry broken by Yukawa insertions.
- WCxf integration: SMEFT Feynman Rules can import and export Wilson coefficient values in WCxf format for use with downstream observable programs.The package produces analytical Lagrangians and interaction vertices, while numerical coefficients enter subsequent calculations.
- WCxf integration: The import routine reads WCxf JSON files and generates a FeynRules model file using the Warsaw mass basis.The imported coefficients are assumed to be defined in the Warsaw mass basis specified by the package.
- Output generation: The generated SMEFT model can produce Feynman rules in UFO format from numerically initialised Wilson coefficients.The workflow reads the physical SMEFT Lagrangian and generates UFO output, with adaptation to other FeynRules-supported formats possible.
- Output generation: The package separates analytical initialization from numerical coefficient assignment, allowing repeated execution after Wilson coefficient values change.Analytical physical SMEFT Lagrangian data are prepared first, then model files are used in subsequent notebooks.
6. Conclusions and outlook
The paper introduces WCxf as an extensible exchange format implemented across SMEFT, WET, and several public codes. Its central tooling supports interoperable precision analyses and can be extended by contributing new EFTs or bases to the public repository.
- Conclusions: WCxf provides a data exchange format for Wilson coefficients in SMEFT above and WET below the electroweak scale.Implementations in several public codes demonstrate interoperability for precision tests of the Standard Model.
- Conclusions: The format facilitates interfaces among codes and supports constructing global likelihoods spanning flavour, electroweak precision, and Higgs physics.The paper presents this as enabling future model-independent analyses of new physics.
- Outlook: New EFTs or bases can be added through submissions to the public repository, extending WCxf beyond its current definitions.Documentation for the bases and associated software is available on the project website.