Selection rule
Check the supported SU(4) representation and Wilson-domain selection conditions, including the chiral-count test for a quark-generation embedding.
Control names are kept as they appear in the application. Start with a reference, change one input at a time, and retain the assumptions with your result.
What this tool is for
Use this before choosing a representation or reducing a vacuum-search domain. Boundary symmetries can relate some phase values, but the allowed reduction depends on the content. The embedding test asks a second question: whether the supported representation can contain the required chiral structure under the stated projection.
First steps
- Select the Dynkin labels (a, b, c), which identify the representation.
- Read the centre-charge condition and the permitted Wilson-line search domain.
- Inspect all three embedding conditions before using the displayed count.
- Compare a passing representation with one that fails a single condition.
Controls and inputs
| Control | Meaning | Try this |
|---|---|---|
| Dynkin labels a, b, c | Nonnegative integer labels of an SU(4) representation; changing them changes the representation, not a continuous coupling. | Start with a catalogue entry and change one label. |
| Test the rule on every representation | Runs the displayed finite catalogue comparison. | Read the tested range together with the verdict. |
| Shared SU(4) content | Connects the selection rule to the matter content used by the calculator. | Inspect the domain again after editing that content. |
Reading the results
| Result | Meaning |
|---|---|
| Permitted phase domain | The domain justified by the selected content's symmetry. An unjustified reduction can miss a vacuum. |
| Three embedding conditions | Centre-charge parity, a nonzero middle Dynkin label and the stated size condition must be read together. |
| Chiral count | The count for the stated projection when its hypotheses hold; it does not specify Yukawa couplings or measured quark masses. |
The count applies under the displayed projection and embedding conditions. Classical centre-charge facts and the series' chiral-count calculation have separate attribution. Passing these representation tests does not construct a complete quark sector.
Worked example
Check a proposed representation before calculating its vacuum
- Choose a catalogue representation.
- Record each condition and the permitted domain.
- Change one Dynkin label and inspect which condition changes.
- Follow the representation to Model calculator using a domain justified for that content.
What to expect: You can distinguish a representation-selection result from the later potential and mass calculations.
Formulas and conventions
N = (b + 1)(a + c + 1) / 2
The displayed count under the panel's three embedding conditions and chiral projection. The symbol here is a count, not the SU(4) gauge-group rank.
Scope and limitations
- The conditions use this SU(4) embedding and boundary projection; they are not a universal criterion for every GHU model.
- A finite catalogue check does not replace the stated proof assumptions.
- An allowed representation does not settle masses, interactions or anomaly completion.
If something is unclear
The count is unavailable or a condition fails.
Read the failed condition before interpreting the formula. Do not evaluate an out-of-scope count as an accepted embedding.
Terms used here
Sources and attribution
Current instrument guide. Archived editions retain their own scope.
Back to top