Source-linked AI summary
A Proof of Concept for Matchete: An Automated Tool for Matching Effective Theories
Javier Fuentes-Martín, Matthias König, Julie Pagès, Anders Eller Thomsen, Felix Wilsch
TL;DR
Determining how new-physics models affect low-energy observables requires matching UV models to EFTs at relevant mass thresholds, a longstanding challenge for varied models and target EFTs. Matchete uses functional methods for automated matching and simplifies the resulting EFT Lagrangian. Its current version handles specified classes of heavy fields and can reduce outputs toward a useful EFT form, though its identity set is incomplete.
Problem
Matching BSM models to EFTs is a longstanding challenge because UV models and target EFTs vary, including intermediate-scale EFTs and extensions of the SMEFT.
Source: §1 · pp. 2–3Method
Matchete applies functional methods to match UV Lagrangians directly to EFT Lagrangians, then uses identities and field redefinitions to simplify the output.
Source: §1 · p. 3, §2.3 · p. 7Results
Matchete can integrate out heavy scalars and fermions at one loop and heavy vectors at tree level, with EFT dimension limited by computing time.
Source: §4 · p. 19Takeaways & Limitations
The simplification routines can reduce Lagrangians toward a basis-like form, including when Matchete is used without performing matching.
Source: §4 · p. 19The current approach requires hard-coding identity types and omits Fierz identities, which can leave LEFT in a non-minimal operator basis without invalidating the result.
Source: §2.3.1 · p. 8
Abstract
from arXiv · showhide
Studying the impact of new-physics models on low-energy observables necessitates matching to effective field theories at the relevant mass thresholds. We introduce the first public version of Matchete, a computer tool for matching weakly-coupled models at one-loop order. It uses functional methods to directly compute all matching contributions in a manifestly gauge-covariant manner, while simplification methods eliminate redundant operators from the output. We sketch the workings of the program and provide examples of how to match simple Standard Model extensions. The package, documentation, and example notebooks are publicly available at https://gitlab.com/matchete/matchete.
1 Introduction
With new physics still undiscovered, determining its low-energy effects requires matching UV models to EFTs at relevant thresholds, a longstanding challenge made harder by diverse UV models and target EFTs. Matchete is introduced as a public, proof-of-concept tool for automated one-loop matching and EFT simplification.
1 Introduction
- The unknown nature of new physics calls for broadly general methods, while realistic models may require intermediate-scale EFTs or extensions of the SMEFT.Sequential matching at relevant thresholds and renormalization-group running connect high-scale new physics to low-energy observables.Source: §1 · pp. 2–3
- Matching BSM models to their EFTs has been a longstanding challenge because both the UV model and target EFT can vary.Functional methods compute the EFT Lagrangian directly without matching individual amplitudes or requiring prior knowledge of the EFT structure or symmetries.Source: §1 · pp. 2–3
- Matchete is a public proof-of-concept Mathematica package for matching weakly coupled UV models to EFTs at relevant thresholds.It uses functional methods without requiring a target EFT basis and automates simplification toward an on-shell basis.Source: §1 · p. 3
- The paper introduces the package’s organization and use, then demonstrates it through examples including simple Standard Model extensions.The authors characterize the paper as an introduction, not a comprehensive user guide, and point readers to the distribution’s documentation notebook for detailed instructions.Source: §1 · pp. 3–4
2 Organization of the Package
Matchete takes a UV Lagrangian through tree- and one-loop matching, then simplifies the EFT output toward a useful basis. Its workflow supports generic gauge and symmetry groups, with matching performed under a common-heavy-mass assumption unless thresholds are handled sequentially.
2.1 Model setup and internal representations
- Users specify groups and representations before fields and couplings, then write the Lagrangian using objects that carry the properties needed for algebraic manipulations.Fields include spin, mass, flavor and gauge representations, and conjugation properties; couplings include flavor indices and mass dimensions.Source: §2.1 · p. 5, §2.1.2 · pp. 5–6
2.1.1 Symmetry groups
- Matchete constructs Clebsch–Gordan tensors for generic simple Lie groups using a linear-algebra algorithm, then evaluates contractions numerically and projects them onto a CG basis.Users may initialize additional representations and CG coefficients through built-in routines.Source: §2.1.1 · p. 5
2.2 Matching step
- The matching assumes heavy masses are of the same order; otherwise, thresholds require sequential matching with RG running.This common scale sets the heavy scale used to organize the EFT expansion.Source: §2.2 · p. 6
- Tree-level matching solves the heavy-field equations of motion iteratively in the mass expansion and uses the solution to construct the tree-level EFT.The heavy-field action separates a kinetic operator from the interacting part, and higher-order terms are truncated for performance.Source: §2.2.1 · p. 6
- Functional matching treats all particle spins and mixed heavy–light loop states together, evaluates traces manifestly gauge-covariantly, and can target contributions by the fields in the loops.Hard loop momenta are expanded around q ∼ Λ, with the series truncated at the relevant light-mass dimension; one-loop terms beyond dimension six are possible but time-intensive.Source: §2.2 · p. 6, §2.2.2 · p. 7
2.3 Simplifications
- The simplification pipeline uses exact identities to reach a Green’s basis, then field redefinitions to remove further redundant operators from the EFT Lagrangian.Exact identities act linearly on individual operators or the full Lagrangian; field redefinitions act non-linearly on the Lagrangian as a whole.Source: §2.3 · p. 7, §2.3.1 · pp. 7–8
- The current identity set must be hard-coded and omits Fierz identities, so simplification can leave a non-minimal LEFT operator basis without invalidating the result.The omitted Fierz identities require handling evanescent contributions; additional identity types can be added modularly.Source: §2.3.1 · p. 8
2.4 Conventions
- Computations use dimensional regularization with d = 4 − 2ϵ and MS; the initial release uses naive dimensional regularization with anticommuting γ5.Ambiguities cancel when EFT computations follow the matching computation’s Dirac-trace reading point.Source: §2.4 · p. 11
3 Using Matchete
This section presents Matchete workflows for defining models, matching them to EFTs, and simplifying the resulting operators, with examples from a toy U(1) model and two Standard Model extensions. It describes the tool’s tree-level and one-loop matching capabilities, validation against prior calculations, and limitations in computational reach and basis reduction.
3.1 Vector-like fermion toy-model
- On-shell simplification of the toy-model EFT removes redundant operators and leaves canonically normalized matter kinetic terms, with a nontrivial gauge kinetic factor.The hard-scale scalar mass correction is represented by an automatically introduced effective coupling.Source: §3.1 · pp. 13–14
3.2 Real singlet scalar BSM extension
- The full one-loop dimension-six singlet-scalar matching was validated against Ref. [82], although agreement requires the stated μ condition because its reported log terms otherwise include contributions matching cannot generate.The log terms were also partially cross-checked against Ref., with agreement for non-logarithmic contributions.Source: §3.2 · pp. 14–15
- Tree-level singlet matching yields Higgs-sector contributions including QH, but Matchete’s derivative product rule prevents direct output of QH□ and requires basis conversions or simplification.The output also includes terms exchangeable for QH□ and terms removable by field redefinitions or the Higgs equation of motion.Source: §3.2 · pp. 15–16
3.3 Vector-like lepton BSM extension
- The vector-like lepton matching result agrees fully with the diagrammatic matchmakereft result, providing validation for both implementations.Matchete’s CovariantLoop routine can isolate individual loop topologies; its fermion-hexagon contribution to (H†H)3 coincides with the six-NP-Yukawa term in Ref..Source: §3.3 · pp. 17–18
- Automated reduction to the Warsaw basis is not implemented, and field redefinitions must be applied to the full EFT because they can redistribute contributions across operators.The example applies EOMSimplify to the full EFT Lagrangian; additional identities needed for Warsaw-basis conversion remain manual.Source: §3.3 · p. 18
3.4 Further examples and applications
- Matchete automates tree-level matching for arbitrary particle content and one-loop matching for heavy-scalar/fermion models, to arbitrary heavy-mass order, but excludes heavy vectors.Computing time and memory grow rapidly with mass dimension, leaving room for improvement.Source: §3.4 · p. 18
- Illustrative applications span heavy scalars, heavy fermions, and mixed scalar–fermion models.The listed recent literature results are non-exhaustive examples of phenomenologically interesting applications.Source: §3.4 · p. 19
- Its non-exhaustive simplifier reduces redundant operators toward a mostly basis-form EFT without basis-specific hard-coded identities, preserving on-/off-shell equivalence when redundancies remain.The routines are particularly useful for higher-dimensional operators, whose reduction identities are harder to implement as terms proliferate.Source: §3.4 · p. 19
4 Conclusions and Future Prospects
Matchete v0.1.0 automates matching across a broad range of UV models and simplifies outputs toward an EFT basis, while future plans target additional capabilities and known limitations.
Current capabilities
- Matchete matches heavy scalars and fermions at one loop and heavy vectors at tree level, with no EFT-dimension limit beyond computing time.It requires no additional input for group theory or EFT bases, and its simplification routines can reduce Lagrangians even without matching.Source: §4 · p. 19
Operator simplification
- Redundant operators remain because flavor-index handling is not robust and Fierz and other four-dimensional identities require dedicated treatment of evanescent operators.Strictly d-dimensional matching currently prevents simplification to a four-dimensional basis; planned physical projection would address this.Source: §4 · pp. 19, 21
Standard-format output
- Future basis reduction could yield Wilson coefficients in established bases and enable export to WCxf for interfacing with phenomenology tools.The roadmap also identifies interfaces such as UFO as desirable.Source: §4 · p. 21
Heavy vectors
- The primary limitation is the treatment of heavy vectors, whose broken-phase interactions involve vectors, ghosts, and Goldstone bosons.The planned response is to develop (semi-)automated methods for determining the broken-phase Lagrangian.Source: §4 · p. 21
Renormalization-group running
- Matchete could determine RG functions for intermediate-scale EFTs and simplify sequential matching scenarios by computing EFT counterterms.This functionality is proposed for future implementation.Source: §4 · p. 21
Future prospects
- The roadmap is provisional, and automated tools such as Matchete may shift BSM workflows from mechanical tasks toward physics questions.The authors describe functional methods as a natural formulation of matching and RG running, while noting the proof of concept has limitations.Source: §4 · p. 21
A Defining the SM in Matchete
Matchete lets users define the Standard Model’s gauge groups, fields, couplings, and Lagrangian, then check its consistency before matching. Its model-loading routine also lets BSM examples reuse the SM definition.
Model input
- A BSM model can reuse the SM definition through LoadModel["SM"], rather than repeating the input.The paper illustrates this routine in later examples.Source: A Defining the SM in Matchete · p. 22
Gauge groups and fields
- The Standard Model example first uses DefineGaugeGroup to define SU(3)c × SU(2)L × U(1)Y; this automatically defines field-strength tensors G, W, and B and couplings gs, gL, and gY.Source: A Defining the SM in Matchete · p. 22
Higgs input
- The SM input defines the Higgs as massless and includes its tachyonic mass parameter manually as an interaction.The parameter is counted as light-mass dimension one in EFT power counting.Source: A Defining the SM in Matchete · p. 23
Validation and scope
- CheckLagrangian checks properties including Hermiticity, gauge invariance, and gauge anomalies, and Match runs only after the Lagrangian passes its checks.For extended gauge symmetries, the BSM Lagrangian must be supplied in the broken phase; this is not presently supported because of the complexity of ghosts, heavy vectors, and gauge fixing.Source: A Defining the SM in Matchete · p. 24