CGDecomposition
CGDecomposition[cg,decomposition]
specifies how a CG of the original symmetry group decomposes into CGs of the stability group.
Details and Options
- The decomposition should be a tensor with the rank equal to the number of indices in the original cg. 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, most conveniently, as rules for the non-vanishing elements following (as those generated by ArrayRules).
- All non-vanishing elements of the decomposition must be singlets under the stability group; in other words, the elements should be charge neutral and proportional to CGs of the stability group, with open indices to match the multi-dimensional representations of the appropriate element of the branching rule.
- Elements of the decomposition that corresponds to non-singlet elements of the branching rules of the original indices carry open indices in that direction themself. The open indices of the decomposition must have the same labels as the original label for Matchete to match-up the two.
- CGDecomposition 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.
- The normalization of the decomposition is checked against the normalization of the original cg tensor to verify consistency.
Examples
open allclose allBasic Examples (2)
Take the familiar case of EWSB: the symmetry-breaking pattern is specified by
and the fundamental representation decomposes as
The two-index anti-symmetric (Levi-Civita) tensor of the fundamental representation decomposes as
The elements at (1,1) and (2,2) have EM charges ±1 according to the branching rules of the fundamental representation and vanish accordingly (this also follows from anti-symmetry of εij.
Proceeding to the fundamental generator gen[SU2L@fund] of 
, it has one adjoint, a fundamental, and an anti-fundamental index. The branching rule for the adjoint index is
The branching rules have length 3 for the adjoint and length 2 for the (anti-)fundamental, so the CG decomposition of the generator should have size 3×2×2:
All elements of the branching rules are singlets, so all components of the decomposition are scalars.
Scope (2)
In the BSM symmetry breaking scenario 
(see e.g. https://arxiv.org/abs/1808.00942):
for the adjoint and fundamental representations of 
. The fundamental generator gen[SU4h@ fund] of 
has one adjoint, a fundamental, and an anti-fundamental index. The branching rules have length 4 (1 octet, 2 triplets, and one singlet of color) for the adjoint and length 2 (1 triplet and one singlet of color) for the (anti-)fundamental, so the resulting decomposition should have size 4×2×2.
The fundamental generator of 
decomposes as
By the branching rules for the adjoint and fundamental representations, we observe that the element of the decomposition at (1, 1, 1) should be a color octet along the direction of the original SU4h[adj] direction (with index A). Similarly it is a color (anti-)fundamental along the original SU4h[fund] and Bar@SU4h[fund], respectively (indices a and b). No indices appear in the elements of the decomposition in directions that corresponds to singlets of the branching rules.
Another example from the 4321 models, is the Levi-Civita tensor of 
. The fundamental index branches to a color triplet and a singlet, so the decomposition is a 2×2×2×2 tensor. Symmetries of the original CG (along with knowledge that the only singlet in the product of three 
triplets is the totally anti-symmetric combination) informs us that all components must be proportional to the 
Levi-Civita tensor. Normalization and anti-symmetry fixes the numerical factors; thus,