Matchete is a Mathematica package for the automated matching of UV theories onto effective field theories (EFTs) at tree level and one-loop order, using functional methods. This tech note walks through a complete example: the Standard Model (SM) extended by a heavy real scalar singlet ϕ. We integrate out ϕ at tree level and at one-loop order, simplify the resulting EFT Lagrangian to an on-shell basis, and finally express the result in terms of the Wilson coefficients of the Warsaw basis of the SMEFT. More details on the framework can be found in arXiv:2212.04510.

All of the following inputs are evaluated in a single Matchete session.

Matchete comes with a number of predefined models (see GetModels). We start by loading the SM with LoadModel, which defines the SM gauge groups, fields, and couplings, and returns the SM Lagrangian:

Next, we define the new heavy scalar with DefineField. The option Mass->{Heavy,M} specifies that ϕ is a heavy field with mass M, which is integrated out when matching. The option SelfConjugate->True makes ϕ a real field:

Fields and couplings are entered with the short-hand notation label[indices], where the indices are given in the order specified upon definition. Repeated indices are summed over. The kinetic and mass terms of ϕ are obtained with FreeLag.

The same model is also available as a predefined model and can be loaded with LoadModel["Singlet_Scalar_Extension"]. How to write your own model files is explained in Model Files.

Match integrates out all heavy fields. The option EFTOrder specifies the maximal mass dimension of the EFT operators and LoopOrder the order in the loop expansion (0 for tree level, 1 for tree level plus one loop, and {1} for the pure one-loop piece).

The matching result is not given in a minimal basis. GreensSimplify reduces it to a Green's basis using integration by parts and other operator identities, while EOMSimplify additionally removes redundant operators with field redefinitions, yielding an on-shell basis.

With LoopOrder->1, Match computes the tree-level and one-loop contributions. One-loop terms are multiplied by hbar, which stands for a loop factor , and depend on the renormalization scale through logarithms of (μbar2). Poles in ϵ, where 𝒹=4-2ϵ, are related to the difference of UV divergences between the UV model and the EFT and are removed by the appropriate counterterms; for the matching conditions in the scheme they can be dropped.

To keep the result compact, EOMSimplify may introduce effective couplings, e.g. for the shifted Higgs mass parameter. They can be shown with PrintEffectiveCouplings and replaced by their definitions with ReplaceEffectiveCouplings.

Finally, we express the matching result in terms of the Wilson coefficients of the SMEFT in the Warsaw basis. Matchete provides the SMEFT Lagrangian as a predefined model and MapEffectiveCouplings determines the coefficients of the target Lagrangian from the matching result.

Note that couplings appearing on both sides of the matching conditions, such as the Higgs quartic coupling λ, refer to the SMEFT on the left-hand side and to the UV theory on the right-hand side.

The matching results can be further processed, e.g. exported to the WCxf format with ExportWCxf for the running and phenomenological analysis with other tools, or used to derive Feynman rules with FeynmanRules and ExportUFO; see BSM to EFT: Universal Feynman Output. More details on the internal representation and the available functionality are given in the other tech notes and in the Matchete guide.