FieldDecomposition
FieldDecomposition[field,decomposition]
specifies how a field of the original symmetry decomposes into mass eigenstates under the stability group.
Details and Options
- field is any Field charged under the original symmetry group or a singlet. It ignores any chiral projectors.
- The decomposition should be a tensor with the rank equal to the number of indices in the original field. The dimensionality of the tensor is such that the range of each index equals the number of representations in the branching rules dictated by the original representation according to RepresentationDecomposition.
- decomposition can be specified as a nested List, a SparseArray, or as rules for the non-vanishing elements following (as those generated by ArrayRules).
- Elements of the decomposition should have open indices and charges to match the corresponding elements of the branching rules from the indices of the broken-representation indices of the original field. The open indices of the decomposition must have the same labels as the original label for Matchete to match-up the two. All unbroken indices of field (e.g., Lorentz or flavor) must appear in all entries in the decomposition.
- FieldDecomposition can only be used after a symmetry-breaking pattern has been specified with SetSymmetryBreakingPattern and the branching rules for each representation of the indices of cg have been set with RepresentationDecomposition.
Examples
open allclose allBasic Examples (1)
Take the familiar case of EWSB: the symmetry-breaking pattern is specified by
and the fundamental representation and hyper charge decomposes as
The SM Higgs doublet H[i], decomposes into a two-dimensional vector (from the fundamental 
index):
The charged upper component is given to the charged would-be GB χ[]. The lower component is a mix of the VEV v[], the scalar Higgs boson [] and the neutral would-be GB χ0[].
Scope (1)
In the BSM symmetry breaking scenario 
(see e.g. https://arxiv.org/abs/1808.00942):
for the fundamental representations of 
. The diagonal-like breaking ensures that the fundamental of 
is identified with the fundamental of color after symmetry breaking:
The Ω3[a,b] field is charged in the fundamental of 
and the anti-fundamental of 
, so the decomposition of the field under symmetry breaking is a 2×1 tensor – not a 2-dimensional vector – due to the length of the two branching rules:
Here OG[A] and ϕG[A] are real scalars with color-octet indices; v3[] is the VEV; S1[], S2[], ϕA[], and Ap[] are real singlet scalars; and ϕT[a] and Tp[a] are color triplet scalars. The remaining symbols are various mixing angles. Note, the use of the dummy index summation over the adjoint A index in the decomposition and the inclusion of the gen[SU3c@fund] generator to properly embed the color octet fields.