Anomaly-free charge family

What is being checked

This note gives a conditional consistency test for a complete specified charge table. It concerns representations of observed four-dimensional chiral fields, not additional physical spatial dimensions, and does not derive those fields from the nested spatial sectors.

Use only left-handed Weyl fields, conjugating right-handed fields:

The mixed non-Abelian, mixed gravitational, and cubic Abelian conditions are

$$2q+u+d=0,\qquad 3q+l=0,$$ $$6q+3u+3d+2l+e+n=0, \qquad 6q^3+3u^3+3d^3+2l^3+e^3+n^3=0.$$

The cubic $SU(3)$ anomaly cancels between two fundamentals and two antifundamentals. There are four weak doublets counting colours, satisfying the usual even-doublet condition. These checks assume precisely this field content and chirality assignment.

A family of exact solutions

For one scalar doublet $H$ of charge $h$, invariance of the ordinary Yukawa terms $QHu^c$, $QH^\dagger d^c$, $LH^\dagger e^c$, and $LH\nu^c$ requires, respectively,

$$q+h+u=0,\quad q-h+d=0,\quad l-h+e=0,\quad l+h+n=0.$$

The weak-index contractions use the invariant alternating tensor where needed; these equations display the Abelian charge sums. Combining them with $l=-3q$ gives the family

$$(q,u,d,l,e,n)=(q,-q-h,-q+h,-3q,3q+h,3q-h).$$

Every displayed anomaly vanishes for arbitrary $q,h$. In particular, the cubic sum reduces to

$$6q^3+3(-q-h)^3+3(-q+h)^3+2(-3q)^3 +(3q+h)^3+(3q-h)^3=0.$$

In the field order $(Q,u^c,d^c,L,e^c,\nu^c)$, define the conventional vectors

$$Y=\left(\frac16,-\frac23,\frac13,-\frac12,1,0\right), \qquad B-L=\left(\frac13,-\frac13,-\frac13,-1,1,1\right).$$

Then the general displayed family is $X=aY+b(B-L)$, with $a=2h$, $b=3q-h$, by direct substitution in each entry. The two-parameter freedom and its relation to the neutrino charge are also described in Allanach, Davighi and Melville, §3.1. The calculation here is an explicit sufficient family, not a classification of every spectrum or of every solution without the Yukawa assumptions. For a compact global $U(1)$, compatible charge lattices impose further integrality conditions; rational examples already demonstrate nonuniqueness.

Imposing $n=0$ removes $b$ in this family. Choosing $h=1/2$ then yields

$$(q,u,d,l,e,n)=\left(\frac16,-\frac23,\frac13,-\frac12,1,0\right).$$

Conjugation restores the usual right-handed charges. Neutrino neutrality and the normalization are extra conditions, or must follow from an independently derived geometric charge map. They do not follow solely from having three colours. Conversely, dropping $\nu^c$ while calling the neutrino Dirac would omit a field from this anomaly calculation.

V2's existing determinant-trivial gauge carrier gives a particular charge assignment by a separate construction. The freedom here neither changes that assignment nor inserts a new gauge field into V2. It specifies what anomaly cancellation alone can establish, leaving the physical carrier, chirality, and occupied-mode bridge to be derived.

Applying a Geometrically Supplied Color Rank

The middle-Hodge application supplies a definite candidate color rank $k$ from spatial geometry. With one quark weak doublet, one lepton weak doublet, the corresponding right singlets, vector-like electric charges, a neutral neutrino and electron charge $-1$, write $Q_{\rm electric}=T_3+Y$. Then

$$Y_L=-\frac12,\qquad kY_Q+Y_L=0,\qquad q_u=\frac{k+1}{2k},\qquad q_d=\frac{1-k}{2k}.$$

The remaining local anomalies cancel for that specified field content. This derivation uses charge equality of left and right electromagnetic responses directly; it does not require adopting a Yukawa interaction as an IDWT mechanism.

The global weak anomaly further requires $k+1$ even for one, or an odd number of identical, generations of doublets and singlets without added cancellation. The middle-Hodge rank is odd exactly when its half-dimension label is a power of two. Consequently the non-Abelian full-channel construction passes at depths three/four among the maintained depths, and also at seven/eight and further controls. At rank three it gives the usual two quark charge fractions under the same lepton premises. The complete observed chiral spectrum is required before imposing this test on a physical field model.

Verification and related

interaction_operator_checks.py checks the polynomial identities symbolically, mixed $Y$/$B-L$ sums, the neutral-neutrino specialization, and nonstandard rational solutions.

• Interaction operator tools — summarizes this charge-consistency tool.
• Auxiliary gauge carrier — the existing global charge construction and its separate mode-spectrum obligations.