Reducing redundant operators and simplifying Lagrangians

The EFT Lagrangian produced by Match contains many operators that are not independent — they are related by integration-by-parts identities, Fierz relations, gauge-algebra identities, or the equations of motion. Matchete provides two complementary routines to reduce the Lagrangian to a minimal basis.

Note that the simplification routines have been mainly tested on dimension-six EFT operators, and might not fully simplify results at higher orders in the power counting.

Off-shell reduction

GreensSimplify applies all off-shell relations — integration-by-parts identities and algebraic identities from the gauge structure — to reduce the Lagrangian to a Green’s basis:

LEFToff = GreensSimplify[LEFT];

This removes all redundancies that hold independently of the equations of motion, i.e., those valid off-shell.

On-shell reduction

EOMSimplify additionally performs field redefinitions (equivalent to equations-of-motion substitutions at leading order) to further reduce the Lagrangian to a minimal on-shell basis:

LEFTon = EOMSimplify[LEFToff];

EOMSimplify internally applies all off-shell relations as well, so it is not necessary to call GreensSimplify separately beforehand.

The resulting basis is determined by Matchete’s internal reduction algorithm and does not, in general, coincide with a standard published basis such as the Warsaw basis. The choice of independent operators may also change between Matchete versions if the internal scoring is updated. To project onto a specific basis, use MapEffectiveCouplings.

Reduction identities

Both functions accept the ReductionIdentities option, which controls which algebraic identities are applied during the reduction:

  • dDimensional (default for GreensSimplify) — applies only identities that are strictly valid in $d$-dimensional spacetime. This is the appropriate choice for one-loop matching results obtained in dimensional regularisation.
  • EvanescenceFree (default for EOMSimplify) — additionally applies four-dimensional identities via an evanescent prescription, then performs a finite shift to remove the resulting evanescent operators. Requires a complete Lagrangian as input, not just a part of it.
  • Evanescent — same as EvanescenceFree, but keeps the evanescent operators in the output instead of removing them.
  • FourDimensional — uses strictly four-dimensional identities. Incompatible with one-loop matching results obtained in dimensional regularisation.

Inspecting the result

SelectOperatorClass extracts all operators of a given field content and derivative order from the result, making it straightforward to inspect specific pieces of the matching output:

(* Select all operators built from two Higgs doublets with 2 derivatives *)
ops = SelectOperatorClass[LEFTon, {H, Bar@H}, 2];
ops //NiceForm

The third argument counts covariant derivatives; field-strength tensors count as two. CollectOperators provides a flat grouped listing of all operators and their coefficients, and is useful for a complete overview. NiceForm renders either output in a readable mathematical notation.

For a full worked example see the Quickstart.