Matchete`

EOMSimplify

simplifies the input Lagrangian L to an on-shell basis by applying field-redefinitions. Kinetic terms of scalar and fermion fields are brought into canonical form.

Details and Options

The input Lagrangian is first treated using GreensSimplify. If the target basis is the on-shell one, GreensSimplify can thus be skipped after matching.
The following options can be given:

Verbose	True	whether or not the function reports on its progress. Can be True or False.
DummyCoefficients	False	determines whether the module replaces operator coefficients by temporary variables before applying the reductions. For Lagrangians where the matching coefficients consist of complicated loop functions, this can yield a better performance.
EFTOrder	All	specifies the targeted order in power-counting to which field redefinitions are applied. The standard option All determines the order from the input Lagrangian and keeps the output at the same order.
EffectiveCouplingSymbol	"C"	the standard symbol used for effective couplings which are generated for superleading terms.
ReductionIdentities	EvanescenceFree	specifies whether or not to use the subset of identities valid only in 4 dimensions (fierz, gamma reduction and Levi-Civita relations) in intermediate simplifications.

The option EFTOrder takes either an integer value greater or equal to 4, or the symbol All. In the latter case, EOMSimplify determines the order in power-counting from the input Lagrangian. When the option value is lower than the maximal order of the Lagrangian, redundant operators are only removed up to the specified order.
When the Lagrangian contains a term that is formally of lower order than dimension 4, it is replaced by an effective coupling with its power-counting defined such, that the term becomes dimension 4. These terms typically occur after matching when the light degrees of freedom do not have their masses protected by a symmetry (c.f. the Hierarchy Problem). Matchete then assumes the loop suppression to be equivalent to a power-suppression, trading a "superleading-power" loop function for a tree-level object that is of higher order in power-counting. The option EffectiveCouplingSymbol controls which letter is used for the generation of such couplings.
The allowed options for ReductionIdentities are

	EvanescenceFree	uses 4-dimensional identities and evaluates evanescent contributions
	Evanescent	uses 4-dimensional identities and keeps evanescent operators in the Lagrangian
	dDimensional	uses only reduction identities valid in dimensional regularization, which do not yield evanescent operators
	FourDimensional	uses 4-dimensional identities without including evanescent operators

Examples

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Basic Examples (2)

Defining a real scalar theory with one redundant operator at dimension 6

Calling EOMSimplify on the Lagrangian removes the redundant operator, the coefficients of the ϕ⁴ and ϕ⁶ operators have changed:

Adding a non-standard contribution to the kinetic term

The kinetic term is normalized by EOMSimplify, the field is rescaled and the new coefficient appears in all expressions where power-counting allows for it:

Scope (1)

Defining a Yukawa theory with a light scalar and two heavy fermions, and integrating out the latter at one-loop:

The resulting Lagrangian contains terms formally of dimension 2:

For the EFT power-counting to be valid, Matchete assumes the loop function to count as the necessary power-suppression and replaces it with an effective mass:

The mass term is replaced by a new effective coupling, and the term now counts as a tree-level dimension 4 object:

Options (5)

Verbose (1)

By default, the function notifies the user of the progress as well as replacements of effective couplings:

All output is muted:

DummyCoefficients (1)

Defining a Yukawa theory with a heavy fermion and matching at one-loop:

Replacing loop functions with intermediate effective couplings yields a speed-up:

1.187765`

0.923399`

EFTOrder (1)

Defining a Lagrangian with redundant operators at dimension 5 and 6

Only dimension-5 terms are reduced, redundant operators at dimension 6 are still present:

EffectiveCouplingSymbol (1)

A Lagrangian with a one-loop dimension-2 contribution to the scalar mass term:

The symbol for the effective mass can be specified:

ReductionIdentities (1)

Adding a four-fermion operator and a redundant operator

By default, contributions from evanescent operators are evaluated and added to the coefficients after simplifications:

If evanescent operators are kept explicit, the shift is not present, but there is an evanescent operator proportional to the coupling of the Fierz-transformed operator:

Possible Issues (2)

Lagrangians need kinetic terms for the occuring fields for field redefinitions to be successful:

If the kinetic term is present, the field redefinitions can be performed:

Excursions from canonical kinetic terms need to be perturbative, i.e. either loop- or power-suppressed, otherwise the input is returned unprocessed:

If the coefficient is defined to be of subleading power or comes with a loop-factor, the term can be removed: