Matchete`
Matchete`

GreensSimplify

GreensSimplify[expr]

simplifies (part of) a Lagrangian to a Green's (off-shell) basis.

Details and Options

  • The operator basis used by GreensSimplify is determined from a heuristic scoring and will not typically agree with literature basis choices. The output is expressed in terms of the basis operators and GreensSimplify will not attempt to reduce an expression to the minimal number of operators. As a result simple expressions may look more complicated after application of the simplification function.
  • In addition to simplifying operators to a set of basis operators, GreensSimplify also collects and simplify the coefficients in front of coefficients.
  • The following options can be given:
  • ReductionIdentities dDimensionaldetermines how to treat identies related to Dirac and Lorentz algebra that are valid only in four dimensions.
  • The option values for ReductionIdentities are
  • dDimensionalapplies only the identities that are strictly valid in -dimensional space time.
    EvanescenceFreereduces operators to a physical (four-dimensional) basis using the evanescent prescription and then performs a finite evanescent shift to remove resulting evanescent operators. This option requires that the imput is a full valid Lagrangian and not just a part.
    Evanescentreduces operators to a physical (four-dimensional) basis using the evanescent prescription. The resulting evanescent operators are kept in the output.
    FourDimensionalsimplifies with four-dimensional identities. This is incompatible with other -dimensional calculations, such as one-loop matching.

Examples

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Basic Examples  (1)

In a real-scalar toy model, GreensSimplify performs integration by parts relations (a real scalar is defined with DefineField)

Options  (1)

ReductionIdentities  (1)

When dealing with operators involving -dimensional Dirac algebra such as (the SM definitions are loaded with LoadModel)

it is often insufficient to simplify using strictly -dimensional identities. For instance,

In such cases the ReductionIdentitiesEvanescent option can be used to reduce the expression to a physical operator basis:

Here the physical basis has explicit symmetry in the r↔t indices. This option often leaves remnant evanescent operators (E1, E2 above), which may or may not be relevant in further calculations. When simplifying an entire Lagrangian, the user will typically want to use the option ReductionIdentitiesEvanescenceFree to remove the evanescent operators in favor of a finite shift in the physical operator coefficients.

To obtain a strictly four-dimensional reduction (incompatible with -dimensional matching and running), the option value FourDimensional can be used:

Tech Notes
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Evanescent Prescription