ToBrokenPhase
ToBrokenPhase[expr]
takes a Lagrangian (term) to the broken phase as dictated by the symmetry breaking definitions.
Details and Options
- The use of ToBrokenPhase is predicated on the user having already loaded the kernel with the relevant symmetry breaking information with SetSymmetryBreakingPattern, RepresentationDecomposition, FieldDecomposition, and CGDecomposition.
- ToBrokenPhase can be applied to individual Lagrangian terms or subsets of a complete Lagrangian.
- ToBrokenPhase automatically applies the replacement rules specified with SetSSBReplacements.
- The following options can be given to ToBrokenPhase:
-
EFTOrder Automatic truncates the broken-phase expansion at the given order in the EFT expansion. GoldstoneBoson True determines if Goldstone bosons should be kept in the broken phase expansion or set to vanish immediately.
Examples
open allclose allBasic Examples (1)
The symmetric-phase SM along with all symmetry breaking details can be loaded with the broken SM model file (here keeping only the bosonic sector for simplicity):
To take the symmetric-phase Lagrangian to the broken phase, one then simply calls ToBrokenPhase:
Options (2)
EFTOrder (1)
The substitutions of fields are truncated in the EFT expansion for performance reasons and to not include terms of higher order than is justified by the validity of the symmetric phase theory. For illustration we will load the symmetry breaking pattern supplied by the dimension-6 SMEFT include higher order corrections to the Higgs field:
As an example let's look at the breaking of the Higgs-bilinear. The default EFTOrder Automatic option, will identify the symmetric phase bilinear as EFT order 2, and truncated the broken phase accordingly:
We may manually overwrite the default contraction and, e.g., keep the expansion up to order 4:
Note that the counting of the EFT order will count the order of the couplings. A light-field only renormalizable Lagrangian is therefore expected to have EFT order 4.
GoldstoneBoson (1)
Goldstone bosons can be omitted from the field replacements when going to the broken phase (the user may want to do this for performance reasons in extreme cases). This directly takes the broken phase into the unitary gauge. Here with the Higgs bilinear in the SM:
without GBs in the broken phase, but keeping the radial mode (the physical HiggBoson).