One-loop matching with Matchete

Matching is the process of integrating out heavy degrees of freedom from a UV theory to obtain the corresponding Effective Field Theory (EFT). Given a UV Lagrangian, Matchete computes the Wilson coefficients of the EFT up to one-loop order and a specified order in the EFT power expansion.

Matchete uses the manifestly covariant functional matching approach based on the Wilson line method. Currently Matchete supports integrating out both heavy scalars and heavy Dirac fermions. The extension to heavy vectors is under development.

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Running the matching

The central routine is Match:

LEFT = Match[LBSM,
  EFTOrder  -> 6,
  LoopOrder -> 1
];

EFTOrder sets the maximum operator dimension retained in the EFT (e.g., 6 for a dimension-six truncation). LoopOrder controls the loop expansion:

• 0 — tree level only
• {1} — one loop only (without the tree-level contribution)
• 1 — tree level plus one loop (default)

The output is a Matchete Lagrangian expression containing the EFT operators with their Wilson coefficients as functions of the UV parameters.

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What gets integrated out

All fields declared with Mass -> {Heavy, ...} in their DefineField call are integrated out. Fields declared with Mass -> {Light, ...} remain in the EFT. Should the mass terms in the Lagrangian not conform to the declared scalings (e.g., if a mass term with a different scaling is added manually), the scalings are adapted accordingly and the user is notified. The heavy–light split is determined entirely by the model definition.

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Next steps

The matched Lagrangian typically contains many redundant operators. These are removed in the operator reduction step. The simplified result can then be mapped onto a standard basis or exported for use with external tools.

For a full worked example see the Quickstart.