Source-linked AI summary
Wilson: a Python package for the running and matching of Wilson coefficients above and below the electroweak scale
Jason Aebischer, Jacky Kumar, David M. Straub
TL;DR
Analyses of high-scale new physics require technically challenging evolution and matching of many dimension-six Wilson coefficients before low-energy predictions can be obtained. wilson automates these steps across SMEFT and WET and interfaces with other codes through WCxf. Its scope is currently one-loop RG evolution and tree-level matching, and an example finds simultaneous explanation of charged- and neutral-current anomalies disfavoured in a simplified scenario.
Problem
High-scale new-physics analyses require RG evolution, matching, and threshold treatment across many dimension-six Wilson coefficients, making the route to low-energy phenomenology technically challenging.
Method
wilson modularly automates SMEFT and WET running, SMEFT-to-WET matching, basis translation, and WCxf-based exchange with public codes.
Results
In a simplified scenario with Λ = 1 TeV, the example concludes that simultaneously explaining charged- and neutral-current anomalies is disfavoured.
Takeaways & Limitations
Starting from high-scale Wilson coefficients, wilson returns low-energy coefficients relevant for precision measurements and can directly connect to public prediction codes.
Takeaways & Limitations
wilson is currently limited to one-loop RG evolution in SMEFT and WET and tree-level matching.
Abstract
from arXiv · showhide
Wilson is a Python library for matching and running Wilson coefficients of higher-dimensional operators beyond the Standard Model. Provided with the numerical values of the Wilson coefficients at a high new physics scale, it automatically performs the renormalization group evolution within the Standard Model effective field theory (SMEFT), matching onto the weak effective theory (WET) at the electroweak scale, and QCD/QED renormalization group evolution below the electroweak scale down to hadronic scales relevant for low-energy precision tests. The matching and running encompasses the complete set of dimension-six operators in both SMEFT and WET. The program builds on the Wilson coefficient exchange format (WCxf) and can thus be easily combined with a number of existing public codes.
1 Introduction
The EFT treatment connects high-scale SMEFT coefficients to low-energy observables through RG evolution and matching. wilson automates the technically challenging steps between these scales and interoperates with public codes through WCxf.
- Motivation: The EFT approach describes high-scale new-physics effects through local SM-gauge-invariant interactions and supports calculations across electroweak and low-energy processes.Above the electroweak scale the theory is conventionally SMEFT, while below it the WET contains QED and QCD gauge interactions.
- Workflow: A phenomenological analysis typically computes SMEFT coefficients at Λ, evolves them to the electroweak scale, matches to WET, evolves again, and computes the low-energy process.Below the b-quark mass, the final evolution and matching steps are repeated for WET variants with reduced flavour content.
- Motivation: The full procedure is technically challenging because dimension-six theories contain a vast number of Wilson coefficients.wilson automates steps 2–5 and bridges high-scale coefficients to low-energy phenomenology implemented by other public codes such as flavio.
- Foundations: wilson builds on complete SMEFT and WET operator bases, one-loop RG results, tree-level SMEFT-to-WET matching, and the WCxf exchange format.Its components also draw on DsixTools, python-rundec, and other public packages.
- Workflow: From a user perspective, the match_run method performs basis translations, RG running, and matching to return low-scale WET coefficients.The workflow is represented schematically in figure 1.
2 Description
wilson combines parameter extraction, SMEFT running, electroweak matching, and WET evolution in modular subroutines. It uses WCxf conventions and implements complete tree-level matching with one-loop RG evolution in the stated operator sectors.
- Workflow: The package takes SMEFT coefficients in the Warsaw up basis at Λ and returns WET coefficients at a low scale in the flavio basis.Internally, WCxf Warsaw is used for SMEFT running and JMS for matching and WET running; match_run automates the procedure.
- SM parameter extraction: SMEFT running requires SM parameters whose electroweak-scale extraction receives dimension-six corrections and whose UV evolution depends on the Wilson coefficients.The implementation determines MS parameters at MZ, inverts SMEFT relations, and iteratively runs between MZ and Λ.
- RG evolution in SMEFT: The SMEFT RGEs are solved numerically after determining the input-scale SM parameters, following the implementation of DsixTools.The evolution integrates the RGE right-hand side numerically.
- Caveat: Numerical inputs and outputs in the WCxf non-redundant flavour basis can differ from redundant-basis conventions by symmetry factors.The authors direct readers to appendix A of for details.
- Matching and WET evolution: The complete tree-level SMEFT-to-WET matching is performed at the electroweak scale for the full non-redundant dimension-six operator sets.The WET evolution covers QCD and QED renormalization and is restricted to baryon- and lepton-number-conserving operators.
- Matching and WET evolution: WET evolution is simplified by rescaling dipole and three-gluon operators, yielding anomalous-dimension matrices containing numbers and fermion-mass ratios.These ratios are treated as approximately constant because they are RG invariant to O(αs), and evolution matrices are then constructed.
3 Installation
wilson requires Python 3.5 or later and runs on Linux, Mac OS, and Windows. Installation from the Python Package Index also installs WCxf when needed.
- Requirements: wilson supports Python version 3.5 or above on Linux, Mac OS, and Windows.These are the stated system requirements.
- Installation: The latest version can be installed from the Python Package Index without root privileges.The installation also automatically installs the wcxf package and command-line interface when unavailable.
- Installation: Installed versions can be upgraded when a new release becomes available.The package documentation provides the corresponding upgrade command.
4 Usage
The usage interface represents EFT parameter points with Wilson objects, which can be evolved, matched, translated, and exchanged with other codes. The package automates running across SMEFT and WET descriptions and supports configurable evolution and matching settings.
- Initializing Wilson objects: A Wilson object is initialized with coefficients at a specified scale, EFT, and basis, and can also be loaded from WCxf files.Unlike a fixed-scale wcxf.WC object, wilson.Wilson represents an evolvable EFT parameter-space point.
- Matching and running: The match_run method returns coefficient values in different EFTs, scales, and bases, covering SMEFT and WET running and threshold matching.The workflow includes matching to WET and reduced-flavour WET variants below the bottom- and charm-quark thresholds.
- Matching and running: Output scales may exceed input scales only within the same EFT; evolution across EFT thresholds cannot be reversed because matching is not bijective.In same-EFT cases, RG evolution proceeds from the low input scale to the high output scale.
- Options: Users can configure SMEFT accuracy, QCD/QED perturbative orders, and matching scales for the electroweak, bottom, and charm thresholds.The SMEFT evolution supports numerical integration or a faster leading-logarithmic approximation.
- Interfacing with other codes: WCxf support enables import and export with compatible programs, while flavio directly accepts Wilson instances and receives the translated low-energy coefficients automatically.For an SMEFT input, wilson performs SMEFT running, matching, WET running, and conversion to the flavio basis before observable calculation.
5 Example
The example applies wilson to a third-generation left-handed new-physics scenario proposed to address charged- and neutral-current B anomalies. Scanning its parameters reproduces earlier constraints and disfavors a simultaneous explanation in the simplified setup.
- Scenario: The example studies models coupling mainly to third-generation left-handed quarks and leptons, enhancing b →cτν while suppressing b →sµµ through small flavour-mixing angles.The scenario is motivated by left-handed semileptonic operators that fit low-energy flavour data.
- Scenario: The setup contains the [O(1,3)lq]3333 operators at Λ in a weak basis related to the mass basis by small mixing angles.The Warsaw-basis convention used has diagonal down-type-quark and charged-lepton masses.
- Implementation: The simplified initialization uses C1, C3, λq23, λℓ23, and Λ as free parameters before predictions are computed with flavio.The Wilson instance is passed to flavio to evaluate the relevant experimental constraints.
- Scan and result: The four parameters are scanned over λq23 ∈ [−0.05, 0], λℓ23 ∈ [−0.5, 0.5], C1,3 ∈ [−4, 0], with Λ = 1 TeV.These scan ranges reproduce the result of refs..
- Scan and result: The simplified scenario disfavors a simultaneous explanation of the charged- and neutral-current anomalies.This is the example’s stated conclusion from the reproduced constraints.
6 Summary
wilson automates the running, matching, and basis translation of Wilson coefficients from a high scale to low energies relevant for precision measurements. Its current implementation uses one-loop RG evolution and tree-level matching, while its general structure supports future higher-order extensions.
- wilson automatically evolves, matches, and translates Wilson coefficients from a high scale Λ to low energies relevant for precision measurements.It is built on WCxf and can link to public codes for computing low-energy observables.
- The package is currently limited to one-loop RG evolution in SMEFT and WET and tree-level matching.
- The code structure is general enough to support higher loop orders in running and loop-level matching in the future.Loop-level matching is only partially known at present.
- The package does not accept symbolic inputs.
- wilson has already been used in new-physics analyses of B anomalies and ε/′ε.