Custodial Decomposition of the \(LL\Delta\) Portal
Date: 17 October 2026 Branches: B03, B05, B06 and B11 Claim status: exact spin-one Clebsch–Gordan decomposition up to phase conventions; physical flavor coefficients remain open
1. Product basis
The bitriplet carries
Hypercharge gauges the right weight, and the complex triplet \(\Delta\) is the slice
Its components are
with \(Q=m_L+m_R\).
Under the diagonal custodial group,
2. Exact decomposition
With a consistent Clebsch–Gordan phase convention,
and
The squared coefficients sum to one in each charge sector. Independent rephasings can change displayed signs but not the representation weights or coupling magnitudes.
3. Lepton-pair couplings
The gauge-invariant portal
therefore contains the schematic component couplings
and
up to field-normalization and Lagrangian sign conventions.
The strongest model-independent statement is
4. Custodial breaking carried by the portal
The portal selects only \(m_R=+1\). It does not couple matter symmetrically to the full right triplet \(m_R=+1,0,-1\). Therefore it breaks the accidental custodial \(SO(3)_R\) unless the matter sector is enlarged into compatible right multiplets.
In the pure bitriplet model the custodial triplet is the Goldstone orbit. After adding a Higgs doublet, the eaten triplet and any physical custodial triplet are mixtures of doublet and bitriplet modes. The coefficients above describe the bitriplet projection before that additional mixing is diagonalized.
5. Neutrino and collider branches
The same matrix \(y_{ij}\) controls:
- the triplet contribution to Majorana neutrino masses;
- doubly charged dilepton vertices;
- singly charged lepton–neutrino vertices;
- neutral triplet–neutrino vertices.
This can become predictive only if the source theory relates \(y_{ij}\), the triplet vacuum and the scalar masses. Gauge representation theory alone does not.
6. Stop-everything and falsification tests
Positive: the Furey projector fixes a nontrivial relation between the symmetric neutrino matrix and the charged-fiveplet dilepton couplings.
Negative: \(y_{ij}\) is an arbitrary symmetric matrix unrelated to the chiral and flavor structures the programme aims to explain.
Current result: the exact custodial destinations of every \(LL\Delta\) charge component are known.
Unsafe shortcut: call the fiveplet fermiophobic after an allowed portal has been retained elsewhere in the same action.
7. Certificate
The custodial-decomposition certificate constructs normalized coupled-spin bases, projects the \(m_R=+1\) product states and verifies norm reconstruction in all three charge sectors.