Automation of a Matching On-Shell Calculator

Fuensanta Vilches; Javier L\'opez Miras

REVIEW 3 major objections 6 minor 53 references

Automation of a Matching On-Shell Calculator

T0 review · 3 major / 6 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read The paper introduces mosca, a package that automates the reduction of redundant effective-field-theory operator bases to physical bases by computing tree-level on-shell amplitudes in two equivalent Lagrangians and solving for the…

desk verdict A genuinely useful Mathematica package for automating on-shell basis reduction, with an overstated headline guarantee for RedBasis and a thin validation layer. read the letter →

arxiv 2505.21353 v1 pith:7SW7GFGJ submitted 2025-05-27 hep-ph hep-th

Javier López Miras , Fuensanta Vilches This is my paper

classification hep-phhep-th

keywords effectivefieldtheoryon-shellmatchingoperatorbasisreductionGreen'sWilsoncoefficientsredefinitionsrationalkinematicsFeynmandiagramisomorphisms

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper presents mosca, a symbolic-computation package that automates basis reduction in effective field theories by on-shell matching rather than by substituting equations of motion. Its central claim is that the function RedBasis[model1, model2, EFTOrder] takes two Lagrangians that describe the same physics in different operator bases and returns every Wilson coefficient of model2 expressed in terms of the Wilson coefficients of model1, up to a chosen EFT order. If this works generally, redundant Green's bases, which are convenient for computing but contain unphysical operators, could be reduced to compact physical bases exactly and without the tedious field redefinitions that are the correct alternative to equations-of-motion replacements beyond leading order. The package also provides tools for detecting isomorphic Feynman diagrams and for substituting exact rational on-shell kinematics, which together make the amplitude computations tractable and exact.

What carries the argument

The load-bearing object is the diagrammatic on-shell matching equation: for each process, the physical amplitude computed from model1 is set equal to the amplitude computed from model2 after replacing bare masses with physical masses and multiplying by the wavefunction factors required by LSZ reduction. The named functions that carry this out are PropagatorAttributes (two-point attributes), AmplitudeMatching and MassReduction (per-process coefficient solutions), SolvePerturbative (order-by-order solution in $1/\Lambda$), and RedBasis (process-list generation, iteration, and assembly). The exactness of the numerical step comes from rational kinematics produced by a momentum-twistor construction, with symbolic masses kept independent, and isomorphism detection works on graph topologies via canonical forms and double cosets $H g A_0$, so only permutation-inequivalent diagrams are computed.

What would settle it

Take a dimension-six Green's basis for the standard-model EFT with a known physical basis, run RedBasis to order six, and compare with the field-redefinition reduction computed independently; the central claim fails if any physical coefficient still contains Green's-basis Wilson coefficients after all processes are processed, or if any coefficient disagrees with the independent reduction. A smaller version of the same test is to rerun the paper's six-operator example with a redundant operator's Wilson coefficient shifted by a known field-redefinition-invariant amount and check that the returned redefinition shifts exactly as the equivalence principle requires.

Watch

Extended reading notes

Core claim

On the paper's own terms, the discovery is that the on-shell matching algorithm, originally developed for matching a UV model to its EFT, can be repurposed as a basis-reduction engine. Two Lagrangians are equivalent at the level of physics when a local field redefinition transforms one into the other, and such equivalence implies equal tree-level S-matrix elements. mosca exploits this by computing physical masses, wavefunction factors, and modified propagators from each model's two-point functions; generating connected amputated on-shell amplitudes for a list of processes in both models; evaluating those amplitudes at random rational kinematic points; and solving the resulting equations order by order in $1/\Lambda$. The paper demonstrates the full reduction of a six-operator model containing scalar, fermion, and vector fields, with all five Wilson coefficients of the physical Lagrangian determined in terms of the Green's-basis coefficients, and it presents the algorithm as the core of the RedBasis function.

Load-bearing premise

The load-bearing premise is that the finite list of processes mosca chooses, together with the random rational kinematic points it evaluates, is enough to determine every Wilson coefficient of model2 uniquely up to the target EFT order; the paper shows examples but does not prove this completeness, and it notes cases where a returned coefficient still mixes WCs from both models.

Editorial extensions

If this is right

RedBasis returns the Wilson-coefficient redefinition for any pair of same-theory Lagrangians related by a local field redefinition, up to the specified EFT order.
Basis reduction via on-shell matching is exact order by order in $1/\Lambda$, unlike naive equations-of-motion replacement, which the paper cites as only an approximation beyond linear order.
One amplitude can solve several coefficients at once, and the extra amplitude equations serve as built-in consistency checks.
Identifying isomorphic topologies before inserting fields reduces the number of amplitudes that must be computed, making some 76-process SMEFT calculations feasible where the full diagram set would exhaust memory.
The automatic computation of physical masses, wavefunction factors, and modified propagators is what makes LSZ-correct on-shell amplitudes possible in an arbitrary redundant basis.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A natural stress test would turn the completeness caveat into a theorem: for a basis with several redundant operators that only mix in high-multiplicity amplitudes, the process-ordering heuristic must provably generate enough independent equations; until then, users should check output for residual model2 coefficients.
The same code path could be pointed at genuine UV-to-EFT matching without changing the matching equations, since tree-level on-shell matching does not care whether the two Lagrangians differ by a basis choice or by integrated-out heavy fields.
The rational-kinematics generator is already decoupled from the amplitude machinery, so it could serve other exact numerical evaluations, such as helicity-summed cross-sections, outside basis reduction.
If the planned one-loop extension succeeds, physical-basis renormalization and finite matching with evanescent contributions would remove the need for Green's bases altogether; that is the paper's outlook, not a demonstrated result.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Read it. The short version: mosca is a real, usable Mathematica package for on-shell basis reduction in EFTs, and the implementation is more substantial than the title suggests. The methodological core is not new — it is the algorithm from Chala et al. [38] — but the paper turns that into working code and is honest about the lineage. The genuinely new pieces are the diagram-isomorphism detection via double cosets and the exact rational kinematic substitution; both are handled carefully and are useful beyond the main reduction workflow.

What it does well: the propagator treatment, including the nontrivial fermion case, is algebraically consistent in the worked examples; the package ships with models, example notebooks, and a timing comparison for SMEFT diagram generation that shows a real practical gain. That is reproducible evidence and should count.

Soft spots, in proportion. First, the headline claim for RedBasis overstates what it does. The function is defined as returning redefinitions of model2 WCs in terms of model1 WCs only, but Section 4 documents outputs like c1model2 -> c1model1 + rmodel1 c2model2, and the stored Reduction object explicitly allows mixed-model right-hand sides. The paper tells the careful reader to re-substitute, but the advertised guarantee is not what the program terminates on. This is not fatal, since the limitation is disclosed, but the abstract and Section 3.5 should be aligned with the documented behavior. Second, the completeness assumption — that the automatically generated process list determines every WC — is asserted, not proved. The examples are internally consistent, but they are toy models, and there is no independent cross-check against Matchete or an explicit field-redefinition calculation. A single nontrivial benchmark with a published basis would raise my confidence substantially. Third, the solved-WC count in the main example says 5/5 while the reduction contains seven redefinitions; that looks like a genuine documentation bug. Minor, but it should be corrected. The missing commit hash in the repository link is a small reproducibility annoyance.

Bottom line: this is a solid software manual with real code and a useful niche. It deserves a serious referee, and I would send it out. The revision should fix the RedBasis language, add a validation benchmark, and correct the count. My own skepticism is moderate, not deep.

Referee Report

3 major / 6 minor

Summary. The paper introduces mosca, a Mathematica package that implements a numerical-exact tree-level on-shell matching procedure for effective field theories, specialized in this initial release to basis transformations and to the reduction of redundant Green's bases to physical bases. The authors describe the main building blocks: computation of 2-point attributes (physical masses, wave-function factors, and modified propagators), amplitude generation with graph-isomorphism-based diagram classification, generation and substitution of rational on-shell kinematics with symbolic masses, and the main function RedBasis, which is advertised as returning Wilson-coefficient redefinitions of one model in terms of another. Worked examples include a scalar model and a scalar-fermion-vector model with redundant operators, together with timing benchmarks for diagram and amplitude construction.

Significance. If the advertised output contract can be made precise, this would be a practically useful and conceptually clean contribution: it automates a step that is otherwise algebraically tedious (removing redundant operators via field redefinitions), and it preserves exactness through rational kinematics. The paper ships the package, models, and example notebooks, and the isomorphism pre-classification is a concrete efficiency improvement. The core idea of using on-shell matching for basis reduction is sound, and the package is one of the first automated implementations of this approach. However, the documented behavior of RedBasis falls short of the stated guarantee unless the caveats in Section 4 are resolved, so the significance of the contribution currently depends on a fix rather than on the manuscript as written.

major comments (3)

[Section 3.5 / Section 4] The advertised output contract of RedBasis is not what the package is documented to deliver. Section 3.5 states that RedBasis "returns the redefinition of the WCs of model2 in terms of the WCs of model1 up to the dimension fixed by EFTOrder," but Section 4 explicitly says that an AmplitudeMatching output can have the form c1_model2 -> c1_model1 + r_model1 c2_model2, "revealing that the matching procedure was not able to disentangle c2 from c1," and that entries stored in Reduction can have RHSs depending on WCs of both models. The recommended cure is a repeated replacement of the reduction list over itself. Because RedBasis is the end-to-end function presented as the main result, the manuscript must either implement that post-processing automatically and check that no model2 WC remains on any RHS, or restate the weaker guarantee precisely; as written, the central claim is internally inconsistent.
[Section 3.5 / Section 4] The completeness of the reduction is asserted operationally, not established. The statement that RedBasis "does not stop until it either matches every WC in the model or computes every available process" is not a completeness criterion, because "matched" is not defined as disentangled, and no argument is given that the finite list of automatically generated processes with random rational kinematics spans the space of WC redefinitions up to the target EFT order. The package should expose the solved linear system (or an equivalent algebraic rank test) so that a user can verify that every model2 WC has been eliminated, and it should state what happens when the rank is insufficient. This is especially important because the paper itself documents the need for repeated substitution arising from later-solved WCs.
[Section 3.5, modelFull -> modelPhys example] The claimed reduction is only validated against the package's own amplitude computations. There is no independent check, for example against a direct field-redefinition calculation following [31,32] or against a known EOM reduction of the same Lagrangian. The message "Number of solved WCs: 5/5" demonstrates internal consistency of the solver, but it does not by itself establish that the mapping is the correct field-redefinition mapping. A comparison with an external calculation for at least one example would materially support the central claim that mosca correctly automates basis reduction.

minor comments (6)

[Section 1] The displayed spinor-helicity formulas for polarization vectors contain apparent notation errors (for example, `\lambda_\alpha e_{\dot\mu} \sigma_{\alpha\dot\alpha} \lambda_\beta \mu^\beta` mixes the spinor variables and the index structure is incomplete); these formulas should be retyped correctly because they underlie the kinematic substitution routines.
[Section 2.2] The installation instructions state that a manual edit of FeynArtsInterface.m (`>` replaced by `>=`) is required, while Patch2ptFunction is said to automate the 2-point modification; please clarify which steps remain manual and what happens if a model is generated without the patched interface.
[Section 3.1] The output in In[11] contains malformed expressions such as `inv\Lambda\Lambda\Lambda2` and the internal symbol `Prop$23001`; the notebook export should be cleaned so that displayed results use the same notation as the text.
[Section 3.3] In the double-coset discussion following Eq. (3.7), the index `k` in `a(k)_0` is introduced without definition, and the relation `gj = h gi a(k)_0` should be spelled out in words as well as in symbols.
[Section 3.4] The text says that massive vector kinematic configurations are not supported (vector mass entries must be zero), but the introduction promises independent symbolic masses for each particle; this limitation should be stated in the overview as well.
[Section 4 / Figure 2] The timing benchmark would be reproducible only with more environment information: Mathematica version, operating system, CPU, and available memory should be reported alongside the absolute timings.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the WC reductions are solved from amplitude equalities, not fitted or assumed; the only self-citation ([38]) is not load-bearing because the algorithm and its standard premise are restated in the paper.

full rationale

The derivation chain is self-contained. RedBasis computes tree-level on-shell amplitudes in both models and solves for the model2 Wilson coefficients from the matching equations (Sections 1 and 3.5, e.g. In[39]-In[45]), using SolvePerturbative to solve order by order (Section 3.6). The output reductions are therefore solutions of the equality of physical amplitudes, not quantities that were fitted to the claimed result or defined in terms of it. The citation to [38] for the statement that tree-level on-shell matching relates Lagrangians connected by a local field redefinition is a collaborative self-citation, but the paper itself summarizes the algorithm and the underlying equivalence-of-S-matrix principle, so the citation is not the sole load-bearing support. Section 4 documents a real limitation: an AmplitudeMatching output can be c1model2 -> c1model1 + rmodel1 c2model2, 'revealing that the matching procedure was not able to disentangle c2 from c1', and RedBasis terminates when processes are exhausted rather than when all RHSs are model1-only. This is an advertised-guarantee versus termination-condition gap (a completeness/correctness caveat), not a circular reduction: the solved coefficients still come from amplitude equalities, and the paper tells the user to resolve the remaining coupling by further processes or repeated substitutions. No pattern from the enumerated circularity classes is present, so the score is 1 rather than 0 only to acknowledge the minor self-citation.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No free parameters or invented entities; the package introduces only software objects. The main burden is on the completeness of the process selection and the correctness of the underlying field-redefinition equivalence.

assumptions (5)

domain assumption Two Lagrangians related by a local field redefinition produce identical on-shell amplitudes, so equating tree-level amplitudes determines the WC mapping.
Invoked throughout Section 1 and 3.5 as the foundation of RedBasis; standard EFT result from [31,32].
ad hoc to paper The automatically generated process list in RedBasis and the random rational kinematic configurations are sufficient to fully solve for all WCs up to the target EFT order.
Not proven; the paper relies on examples and warns in Section 4 that reductions can remain mixed between model1 and model2 WCs.
domain assumption The momentum twistor algorithm of [39] with the NumericalKinematics package produces rational, on-shell momenta with symbolic masses, and the spinor/polarization expressions are exact.
Used for kinematic substitution in Section 3.4; relies on external package, slightly modified.
domain assumption Landau gauge can be used for vector propagators without loss of generality for on-shell matching.
Section 3.1 sets xi=0 to simplify the vector propagator; assumes gauge invariance of on-shell amplitudes.
domain assumption FeynArts/FeynCalc correctly generate and evaluate the needed amplitudes for the user-supplied models.
The package builds on FeynArts/FeynCalc; correctness of the reduction depends on their correct operation.

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Cite this review

Pith. "Pith review of Automation of a Matching On-Shell Calculator." pith.science (2026). https://pith.science/paper/7SW7GFGJ

@misc{pith2026250521353,
  author       = {Pith},
  title        = {Pith review of: Automation of a Matching On-Shell Calculator},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7SW7GFGJ}},
  note         = {Machine review of arXiv:2505.21353}
}

abstract

We introduce $\texttt{mosca}$, a $\texttt{Mathematica}$ package designed to facilitate on-shell calculations in effective field theories (EFTs). This initial release focuses on the reduction of Green's bases to physical bases, as well as transformations between arbitrary operator bases. The core of the package is based on a diagrammatic on-shell matching procedure, grounded in the equivalence of physical observables derived from both redundant and non-redundant Lagrangians. $\texttt{mosca}$ offers a complete set of tools for performing basis transformations, diagram isomorphism detection, numerical substitution of kinematic configurations, and symbolic manipulation of algebraic expressions. Planned future developments include extension to one-loop computations, thus providing support for EFT renormalization directly in a physical basis and automated computation of one-loop finite matching, including contributions from evanescent operators. The package, along with example notebooks and documentation, is available at: https://gitlab.com/matchingonshell/mosca.

Figures

Figures reproduced from arXiv: 2505.21353 by the authors.

**Figure 1.** Schematic representation of mosca roadmap. Until now, the discussion of the on-shell matching procedure has focused on its traditional application in the construction of an EFT that reproduces the low-energy behavior of a more fundamental theory. However, in this initial version, mosca sets the usual matching of UV models aside and focuses on the matching of different Lagrangian representations of the same theory.… view at source ↗

**Figure 2.** Comparison of diagram and amplitude computation times for 76 different processes within the EW sector of dimension-6 SMEFT Green’s basis with two different methods: standard FeynArts techniques (A) and our isomorphisms-identification approach (B). The amplitude computation time in method (B) corresponds to the total time after applying a RecoverFullAmplitude[] command. deed, it could happen than we enocunter an out… view at source ↗

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Reviewed August 7, 2026 · model on record in the stance chip above.

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