Matchete`
Matchete`

GaugeFixLagrangian

GaugeFixLagrangian[lagrangian]

returns the gauge-fixed version of the ungauged lagrangian including ghost terms.

Details and Options

  • For broken-phase Lagrangians, GaugeFixLagrangian uses the definitions from the defined symmetry breaking pattern, gauge field decomposition and representation branching rules to derive the Rξ gauge fixing terms.
  • The unitary gauge is applicable only to broken-phase Lagrangians. The (massless) gauge fields associated with the stability group will be fixed as per the Rξ gauge.
  • The gauge parameters introduced by GaugeFixLagrangian are named ξGauge[V] for vector field V.
  • The following options can be given:
  • EFTOrderAutomaticdetermines the order in the EFT expansion used for the gauge-fixing terms.
    Gauge Rξsets the gauge of the resulting lagrangian.
    SubstituteMasses Automaticspecifies whether to substitute the mass terms of the heavy vectors and radial scalar modes, in terms of their canonical mass labels (also for the GB-vector kinetic mixing terms).
  • The supported values for the Gauge option is Rξ for the Rξ gauge and Unitary for the unitary gauge.
  • The SubstituteMasses option accepts Automatic, True, and False. By default, it will substitute in the heavy vector masses in the gauge-fixing terms, if this has been done in the input Lagrangian, i.e., in ImplementVacuumConditions. This behavior can be overwritten with the True/False options.

Examples

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Basic Examples  (1)

As an input, we take the true vacuum of the broken-phase SM:

The gauge-fixing and ghost terms are determined automatically with GaugeFixLagrangian (default Rξ gauge). Here we subtract off the original Lagrangian to show only the part that was generated:

Matchete employs the ξGauge[vectorField] symbol to denote the gauge parameters for the individual gauge fields.

Options  (2)

Gauge  (1)

As an input, we take the true vacuum of the broken-phase SM:

The Lagrangian in the unitary gauge is obtained with the GaugeUnitary option.

All would-be Goldstone bosons and ghosts of the massive vector fields vanish. The gauge fields of the stability(/unbroken/remnant) group have been fixed in the Rξ gauge.

SubstituteMasses  (1)

As an input, we take the true vacuum of the broken-phase SM:

The SubstituteMassesTrue option can be used to get the gauge-fixing terms, in terms of the canonical vector masses (those associated to the vectors in their definition with DefineField). In particular the associated ghost and would-be GBs of the massive vectors will now get the canonical mass labels. This works even if parametric mass expressions have been retained previously (e.g., with ImplementVacuumConditions):

(We have subtract off the original Lagrangian to show only the part that was generated).

Tech Notes
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  • SM Electroweak Symmetry Breaking