Higgs, Furey Matching, and the Weinberg Angle

The Quaternionic Slice and the Higgs

When $u \in \mathrm{Im}(\mathbb{O})$ is selected, it determines not just the $G_2 \to SU(3)$ breaking but also a preferred family of quaternionic subalgebras $\mathbb{H} \subset \mathbb{O}$ containing $u$. This is the quaternion algebra generated by $u$ together with any two mutually anticommuting imaginary octonions orthogonal to $u$. Such a slice always exists, but it is only determined by $u$ up to an $SU(2)$ rotation.

This is precisely the representation space of an $SU(2)$ doublet — which is exactly what the Higgs field is.

The structural cascade:

u selected in Im(O)          [octonionic complex structure selected]
    ↓
quaternionic slice H ⊂ O     [quaternionic complex structure selected next]
    ↓
tri(H): quaternionic triality [threefold symmetry of H]
    ↓
triality triple (Ψ₊, Ψ₋, V)  [two fermion generations + scalar sector]
    ↓
V contains Higgs doublet      [scalar component, SU(2)_L doublet, Lorentz scalar]
    ↓
breaking SU(2)_R → residual Standard Model Higgs
    ↓
U(1)_EM

What Furey & Hughes (2022) Establish

The key paper is Furey & Hughes, Division algebraic symmetry breaking, arXiv:2210.10126, Phys. Lett. B (2022). This paper demonstrates:

1. A cascade of breaking symmetries driven by sequential complex structures — first from the octonions, then from the quaternions, then from the complex numbers — producing: $Spin(10) \to \text{Pati-Salam} \to \text{Left-Right symmetric} \to \text{SM} + B-L$, both pre- and post-Higgs. This cascade mechanism is structurally identical to the one this chain proposes.

2. The first explicit demonstration of left-right symmetric Higgs representations from quaternionic triality tri$(\mathbb{H})$. The Higgs is the scalar component $V$ of the triality triple $(\Psi_+, \Psi_-, V)$, where $\Psi_+$ and $\Psi_-$ give two generations of fermions and $V$ provides the third generation plus the Higgs representations. Upon breaking $SU(2)_R$, this Higgs reduces to the familiar Standard Model Higgs doublet.

The Higgs in this construction is a $2\mathbb{C}$-dimensional Lorentz scalar that transforms as a doublet under $\mathfrak{su}(2)_L$ and remains invariant under $\mathfrak{su}(2)_R$ and spatial rotations. It is not inserted by hand — it emerges from the triality structure of the quaternions.

Mapping to this framework

The Idempotent Matching

Furey’s idempotent $qq^\dagger$

In Furey’s construction, the algebra is Cl(6) — the Clifford algebra of 6 real dimensions — realised via the complex octonions $\mathbb{C} \otimes \mathbb{O}$. The Witt decomposition introduces ladder operators:

where $e_1, \ldots, e_6$ are six of the seven imaginary octonionic basis elements. The primitive idempotent is:

This satisfies $P^2 = P$ and generates the minimal left ideal carrying one generation of Standard Model fermions. Particles are elements of $\text{Cl}(6) \cdot P$.

The seventh imaginary octonion $e_7$ is left out — it is the distinguished direction, singled out precisely because it does not enter the ladder operator construction.

The matching condition

The matching condition is:

$u$ selects which octonionic direction is not in the Witt decomposition. The remaining six directions $u^\perp \cap \mathrm{Im}(\mathbb{O})$ form the six-dimensional space that Cl(6) is built on. $P = qq^\dagger$ is the idempotent associated to the Witt decomposition of $u^\perp$.

The matching is not $P_u = qq^\dagger$ directly (note: $P_u = \frac{1}{2}(1+\hat{u})$ is not idempotent in the raw octonion algebra since $\hat{u}^2 = -1$ gives $P_u^2 = \hat{u}/2 \neq P_u$). The correct statement is a derivation:

The idempotent $qq^\dagger$ is the $u$-selection idempotent, one step removed.

What this means for the Higgs

The full picture:

The chain is algebraically consistent. The Higgs is not just structurally compatible with the $u$-selection — it is the scalar component of the triality of the quaternionic subalgebra of $u^\perp$.

Still open: The explicit identification of which three directions in $u^\perp$ form $\mathbb{H}$ — there is an $SU(2)$ worth of choices, corresponding to the residual freedom after $u$ is fixed. Whether physics selects a preferred one (the vev direction) requires the dynamical argument.

Summary statement

The $u$-selection determines $u^\perp$, which is the six-dimensional space on which Furey’s Cl(6) is built. Furey’s primitive idempotent $qq^\dagger$ is the Witt decomposition idempotent of $u^\perp$. The Higgs doublet is then matched structurally to the scalar component of the quaternionic triality of some $\mathbb{H} \subset u^\perp$. The idempotent matching is therefore not an equality $P_u = qq^\dagger$ but a derivation: $u$ determines $u^\perp$ which determines Cl(6) which determines $qq^\dagger$. What remains open is the electroweak choice inside the residual $SU(2)$ family of quaternionic slices, together with the dynamical/normalization steps needed to turn the match into a closed Higgs derivation.

Weinberg Angle: Full Derivation from $u$-Selection

The Core Result

Derived from the $u$-selection with no free parameters.

Step 1: $u$-Selection Forces the 3+2 Split

Selecting $u \in \mathrm{Im}(\mathbb{O})$ leaves $u^\perp$ as a 6-dimensional real space. Under the residual $SU(3)$ stabilizer of $u$, this space splits as:

Three complex directions form the colour triplet $\mathbf{3}$ of $SU(3)$. Two complex directions form the quaternionic slice $\mathbb{H} \cong \mathbb{C}^2$, which carries the $SU(2)_L$ isospin doublet. This 3+2 split is forced by the $u$-selection, not chosen separately.

Step 2: The Hypercharge Generator is Forced

In $\text{Cl}_6$ built on $u^\perp$ (following Furey/Todorov), the weak hypercharge operator is:

where $b_j$ are colour ladder operators (from $\mathbb{C}^3$) and $a_\alpha$ are isospin ladder operators (from $\mathbb{C}^2$). The coefficients $1/3$ and $1/2$ are not inputs — forced by the dimension count: three colour directions get weight $1/3$, two isospin directions get weight $1/2$. This is the only hypercharge assignment consistent with the octonionic structure.

Step 3: Trace Ratio Over One Generation Gives $\sin^2\theta_W$

The Weinberg angle satisfies $\sin^2\theta_W = g’^2/(g^2 + g’^2)$, equivalently:

where traces are taken over one complete generation with generators normalised so that $\text{Tr}(T_a T_b) = \frac{1}{2}\delta_{ab}$.

The first generation (left-handed states: lepton doublet $(\nu_L, e_L)$, three quark doublets $(u_L^i, d_L^i)$, plus right-handed singlets) with hypercharge assignments from $Y = \frac{1}{3}\sum_{\text{colour}} - \frac{1}{2}\sum_{\text{isospin}}$ gives:

This is the same ratio as in standard $SU(5)$ unification, for the same reason: the 3+2 structure of colour vs isospin. In $SU(5)$ it comes from the group-theoretic embedding $SU(3) \times SU(2) \subset SU(5)$; here it comes from the $u$-selection forcing $u^\perp = \mathbb{C}^3 \oplus \mathbb{C}^2$. Therefore:

Honest Assessment

What is derived: The tree-level / high-energy value $\sin^2\theta_W = 3/8$, forced by the 3+2 split of $u^\perp$.

What matches experiment: The measured value at $M_Z \approx 91$ GeV is $\sin^2\theta_W \approx 0.231$. Standard one-loop RG running from a GUT scale $\sim 10^{15}$ GeV takes $3/8 \to 0.231$. This running is not specific to this framework.

What is not yet derived: The GUT scale itself — the energy at which $3/8$ applies — is not determined by the octonionic geometry. Without knowing this scale, the connection to the measured value requires importing the standard RG machinery.

Is this the same as SU(5)? The prediction $\sin^2\theta_W = 3/8$ is identical to $SU(5)$’s tree-level prediction. The mechanism is different — $SU(5)$ gets it from a group-theoretic embedding; this framework gets it from the 3+2 octonionic split forced by $u$. The number is the same; the same RG running is needed to reach experiment.

What this framework adds beyond Todorov: The derivation of the 3+2 split from the exceptional chain, rather than taking it as input. In Todorov’s construction the split $\mathbb{O} = \mathbb{C} \oplus \mathbb{C}^3$ is a starting postulate. Here it is derived: $u$ is the $\mathbb{C}$ factor (the lepton direction), $u^\perp \cap \text{Im}(\mathbb{O})$ splits as $\mathbb{C}^3 \oplus \mathbb{C}^2$ (colour + isospin) is forced.

Weinberg Angle at Low Energy: RG Running and the GUT Scale

The Problem

The tree-level prediction $\sin^2\theta_W = 3/8$ holds at the scale $M_X$ where the $u$-selection operates. Connecting this to the measured value $\sin^2\theta_W(M_Z) = 0.231$ requires knowing $M_X$. This section investigates what constrains $M_X$ from the octonionic geometry and what must still be imported.

Step 1: Standard One-Loop Running

Define $L \equiv \ln(M_X/M_Z)$. The one-loop SM running equations are:

with $\beta$-function coefficients (convention: $b_i > 0$ for asymptotic freedom):

Note: $\alpha_1^{-1}$ decreases with $\mu$ (no asymptotic freedom); $\alpha_2^{-1}, \alpha_3^{-1}$ increase.

Step 2: The GUT Scale Implied by $\sin^2\theta_W = 3/8$

The condition $\sin^2\theta_W = 3/8$ is equivalent to $\alpha_1(M_X) = \alpha_2(M_X)$ in $SU(5)$ normalisation. Setting $\alpha_1^{-1}(M_X) = \alpha_2^{-1}(M_X)$:

This is the scale at which the $u$-framework’s tree-level condition applies. It is determined entirely from the measured $\alpha_1(M_Z)$ and $\alpha_2(M_Z)$, with no additional geometric input.

Step 3: Full Unification Check — Three Couplings at $M_X$

At $M_X \approx 10^{13}$ GeV ($L = 25.4$), the strong coupling is:

The two electroweak couplings unify: $\alpha_1^{-1}(M_X) = \alpha_2^{-1}(M_X) \approx 42.4$.

But $\alpha_3^{-1}(M_X) = 36.7 \neq 42.4$. The strong coupling does not meet the electroweak couplings at $M_X$:

The three couplings do not unify. This is the same known failure as non-SUSY $SU(5)$: partial unification of $\alpha_1$ and $\alpha_2$ at $\sim 10^{13}$ GeV, but $\alpha_3$ does not meet them. (Full unification in $SU(5)$ requires SUSY, which shifts $M_X \to 2\times10^{16}$ GeV and makes all three couplings approximately meet.)

Step 4: What Is Needed for a Geometric $M_X$

The one-loop RG analysis has two unknowns at $M_X$: the matching scale $M_X$ and the GUT coupling $\alpha_\text{GUT}$. The condition $\alpha_1 = \alpha_2$ at $M_X$ gives one equation relating $M_X$ to measured low-energy quantities — determining $M_X = 10^{13}$ GeV by importing the measured $\alpha_1(M_Z)$ and $\alpha_2(M_Z)$.

A geometric determination of $M_X$ — without importing measured coupling values — requires a second condition from the octonionic structure. Candidates:

1. A condition on $\alpha_3/\alpha_2$ at $M_X$ from the $G_2 \to SU(3)$ reduction: the $G_2$ root structure assigns different Killing form normalizations to the colour and electroweak generators, potentially fixing $\alpha_3(M_X)/\alpha_2(M_X)$.
2. The coupling $\alpha_\text{GUT}$ from the $E_8$ embedding: if the $E_8$ gauge coupling is determined by some UV condition (e.g., the $E_8$ root system normalisation), and the SM coupling is $\alpha_\text{SM} = j \cdot \alpha_{E_8}$ for a calculable embedding index $j$, then $\alpha_\text{GUT}$ follows from $\alpha_{E_8}$.

Step 5: The $G_2$ Root Ratio — What Scale Does It Correspond To?

depending on which roots correspond to $SU(3)$ vs $SU(2)$.

Explicit SM running check: Setting $\alpha_3^{-1}(M_{G_2})/\alpha_2^{-1}(M_{G_2}) = 1/3$ (i.e., $\alpha_3 = 3\alpha_2$):

This is the electroweak scale, not the GUT scale. The $G_2$ coupling ratio condition $\alpha_3/\alpha_2 = 3$ is satisfied at $\mu \approx 3m_Z$, not at $\mu \approx 10^{13}$ GeV.

Verification at $\mu = 3m_Z$: Taking $L = \ln 3 = 1.099$: \(\alpha_3^{-1}(3m_Z) = 8.47 + 1.224 = 9.69 \qquad \alpha_2^{-1}(3m_Z) = 29.6 - 0.553 = 29.05\) \(\frac{\alpha_3}{\alpha_2} = \frac{29.05}{9.69} = 2.998 \approx 3 \quad \checkmark\)

The SM ratio $\alpha_3/\alpha_2$ passes through exactly 3 at $\mu = 3m_Z = 273$ GeV.

Implication: The $G_2$ Killing form ratio is an EW-scale condition, not a GUT-scale condition. The two physical scales are:

The $G_2$ root ratio marks the EW symmetry-breaking scale, not the GUT scale. This is suggestive: the $u$-selection that breaks $G_2 \to SU(3)$ in the chain operates at the EW scale, and the Killing form ratio is its EW-scale signature.

Step 6: What Remains Missing

The GUT scale $M_X \approx 10^{13}$ GeV is currently determined by importing measured couplings $\alpha_1(M_Z)$ and $\alpha_2(M_Z)$, not from the geometry. To derive it geometrically requires:

• The $E_8$ coupling $\alpha_{E_8}$, which requires a UV completion (string theory or an intrinsic $E_8$ normalisation condition), or
• A condition on $\alpha_3(M_X)/\alpha_2(M_X)$ from the octonionic structure. The $G_2$ root ratio does not supply this (it applies at the EW scale). The embedding index chain from $E_8 \to E_6 \to G_2 \to SU(3)$ may, but the index computation has not been done.

The non-unification of $\alpha_3$ at $M_X$: the $u$-framework gives the same partial unification as non-SUSY $SU(5)$, with $\alpha_3(M_X)/\alpha_2(M_X) \approx 1.15$. Full unification would require either SUSY particle content (changing the $\beta$-functions) or a specific embedding index from the $E_8$ chain that accounts for the 15% discrepancy.

The $G_2$ root ratio at the EW scale is a new numerical observation — $\alpha_3/\alpha_2 = 3$ exactly at $\mu = 3m_Z$ — whose meaning for the framework is not yet clear. It may indicate that the $G_2$ Killing form normalisation is an EW condition rather than a GUT condition, but the connection to the $u$-selection dynamics has not been made precise.

Status

Embedding Index Chain: $SU(3)_C, SU(2)_L \hookrightarrow E_8$

Setup. The Dynkin embedding index $j_{H\hookrightarrow G}$ measures how the $H$-coupling relates to the $G$-coupling: $\alpha_H = \alpha_G / j$. It is computed via any representation $\mathbf{R}$ of $G$:

where $I_2(\mathbf{r}) = \dim(\mathbf{r})\cdot C_2(\mathbf{r})/\dim(G)$ and the normalisation convention is $I_2(\mathbf{N}_\text{fund}) = 1/2$ for $SU(N)$. Equivalently, $I_2(\mathbf{adj}_G) = h^\vee(G)$ (the dual Coxeter number).

Dual Coxeter numbers used: $h^\vee(SU(N)) = N$, $h^\vee(G_2) = 4$, $h^\vee(F_4) = 9$, $h^\vee(E_6) = 12$, $h^\vee(SO(10)) = 8$, $h^\vee(E_8) = 30$.

Step A: $SU(3)_C \hookrightarrow G_2$.

The $G_2$ fundamental $\mathbf{7}$ decomposes under $SU(3)$ as: \(\mathbf{7}_{G_2}\big|_{SU(3)} = \mathbf{3} + \bar{\mathbf{3}} + \mathbf{1}\)

Indices: $I_2(\mathbf{3}) = I_2(\bar{\mathbf{3}}) = 1/2$, $I_2(\mathbf{1}) = 0$. \(I_2(\mathbf{7}|_{SU(3)}) = \tfrac{1}{2} + \tfrac{1}{2} = 1\)

Index of $G_2$ fundamental: $I_2(\mathbf{7}{G_2}) = 7 \times C_2(\mathbf{7}) / \dim(G_2) = 7 \times 2 / 14 = 1$. ($C_2(\mathbf{7}{G_2}) = 2$ is the standard result.)

Step B: $G_2 \hookrightarrow E_8$.

The maximal subgroup decomposition $E_8 \supset G_2 \times F_4$ gives: \(\mathbf{248}_{E_8}\big|_{G_2\times F_4} = (\mathbf{14},\mathbf{1}) + (\mathbf{1},\mathbf{52}) + (\mathbf{7},\mathbf{26})\)

(Dimension check: $14 + 52 + 7\times26 = 14 + 52 + 182 = 248$ ✓)

Restricting to $G_2$ alone: \(I_2(\mathbf{248}|_{G_2}) = I_2(\mathbf{14}) + I_2(\mathbf{52})\cdot 0 + 26\cdot I_2(\mathbf{7}) = h^\vee(G_2) + 0 + 26\times 1 = 4 + 26 = 30\)

Reference index: $I_2(\mathbf{248}_{E_8}) = h^\vee(E_8) = 30$.

Consequence: $j_{SU(3)C \hookrightarrow E_8} = j{SU(3)C\hookrightarrow G_2} \times j{G_2\hookrightarrow E_8} = 1$.

Step C: $E_6 \hookrightarrow E_8$.

The maximal subgroup decomposition $E_8 \supset E_6 \times SU(3)$ gives: \(\mathbf{248}_{E_8}\big|_{E_6\times SU(3)} = (\mathbf{78},\mathbf{1}) + (\mathbf{1},\mathbf{8}) + (\mathbf{27},\mathbf{3}) + (\overline{\mathbf{27}},\bar{\mathbf{3}})\)

(Dimension check: $78 + 8 + 81 + 81 = 248$ ✓)

Restricting to $E_6$: \(I_2(\mathbf{248}|_{E_6}) = I_2(\mathbf{78}_{E_6}) + 0 + 3\cdot I_2(\mathbf{27}) + 3\cdot I_2(\overline{\mathbf{27}}) = 12 + 3\cdot 3 + 3\cdot 3 = 12 + 18 = 30\)

where $I_2(\mathbf{27}{E_6}) = 27\cdot C_2/78$, and using $C_2(\mathbf{27}{E_6}) = 26/3$: $I_2(\mathbf{27}) = 27\times(26/3)/78 = 3$.

Step D: $SO(10) \hookrightarrow E_6$.

The $E_6$ fundamental $\mathbf{27}$ decomposes under $SO(10)\times U(1)$ as: \(\mathbf{27}_{E_6}\big|_{SO(10)} = \mathbf{16} + \mathbf{10} + \mathbf{1}\)

For $I_2(\mathbf{16}{SO(10)})$: decompose further under $SU(5)$, where $\mathbf{16}|{SU(5)} = \mathbf{10} + \bar{\mathbf{5}} + \mathbf{1}$. Using $I_2(\mathbf{10}_{SU(5)}) = 3/2$ (second antisymmetric) and $I_2(\bar{\mathbf{5}}) = 1/2$: \(I_2(\mathbf{16}_{SO(10)}) = \tfrac{3}{2} + \tfrac{1}{2} = 2\)

Step E: $SU(5) \hookrightarrow SO(10)$.

Step F: $SU(3)_C, SU(2)_L \hookrightarrow SU(5)$.

Step G: $U(1)_Y \hookrightarrow SU(5)$.

The hypercharge generator in the $\mathbf{5}$ of $SU(5)$ is $Y = \mathrm{diag}(-\tfrac{1}{3},-\tfrac{1}{3},-\tfrac{1}{3},+\tfrac{1}{2},+\tfrac{1}{2})$ (the down-type quark and lepton doublet charges):

Compared to $I_2(\mathbf{5}_{SU(5)}) = 1/2$:

This is the GUT normalisation factor. In $SU(5)$ normalisation $\alpha_1^{SU(5)} = (5/3)\alpha_Y$, so the $SU(5)$-normalised $U(1)$ has $j = 1$.

Summary: Full Embedding Index Table

Total chain indices: \(j_{SU(3)_C \hookrightarrow E_8} = 1 \quad j_{SU(2)_L \hookrightarrow E_8} = 1 \quad j_{U(1)_Y^{SU(5)} \hookrightarrow E_8} = 1\)

Physical Consequence: Complete Unification Predicted

All SM non-abelian couplings are embedded in $E_8$ with total index $j = 1$. Therefore at the matching scale $M_{E_8}$:

and $\alpha_Y = (3/5)\alpha_{E_8}$. The Weinberg angle at $M_{E_8}$:

This is exact and automatic. The prediction $\sin^2\theta_W = 3/8$ from the 3+2 octonionic split (derived earlier in this file) is now confirmed to be the direct consequence of the $E_8$ embedding structure: all SM gauge groups sit in $E_8$ with index 1, which forces their couplings to be equal at $M_{E_8}$, which forces $\sin^2\theta_W = 3/8$. The two derivations agree.

The Remaining Gap: $\alpha_{E_8}$ and the Absolute Scale

The embedding indices do not determine $\alpha_{E_8}$ at $M_{E_8}$. The absolute scale requires knowing the value of the common coupling at unification, which comes from where the three SM running couplings actually meet.

For non-SUSY SM: The three couplings do not exactly unify at any single scale (the standard GUT problem). The best-fit meeting point is $M_{E_8} \approx 10^{13\text{-}14}$ GeV with $\alpha_{E_8} \approx 1/42$ — but the three lines are not concurrent; the strong coupling misses by $\sim15\%$.

For SUSY SM (or with additional matter at intermediate scales): The three couplings meet precisely at $M_{E_8} \approx 2\times10^{16}$ GeV with $\alpha_{E_8} \approx 1/25$.

What is now established from the $E_8$ structure:

1. The $u$-framework’s $\sin^2\theta_W = 3/8$ is the direct consequence of all SM embedding indices in $E_8$ being 1. The two independent derivations (3+2 split, embedding indices) are consistent — they are two aspects of the same algebraic fact.
2. Complete unification (not merely electroweak unification) is predicted: $\alpha_3 = \alpha_2 = \alpha_1$ at $M_{E_8}$.
3. The value $\alpha_{E_8}$ and the scale $M_{E_8}$ are not geometrically fixed; they are determined together by where the SM running couplings meet.

Updated status table:

$m_H/m_W$ from $u$-Selection via $sl(2|1)$ Superconnection

The detailed Todorov/superconnection notebook has been split out to higgs-todorov.md.

At the high level, the chain now reads:

1. choosing $u \in \mathrm{Im}(\mathbb{O})$ forces \(\mathbb{O} = \mathbb{C}_u \oplus \mathbb{C}^3_{u^\perp};\)

2. this supplies the octonionic 1+3 input used in Todorov’s $sl(2

   1)$ superconnection;

3. with Todorov’s tree-level relation one gets \(m_H = 2\cos\theta_W\,m_W;\)
4. using the already-derived value \(\sin^2\theta_W = \frac{3}{8}, \qquad \cos^2\theta_W = \frac{5}{8},\) this becomes \(m_H = \sqrt{\frac{5}{2}}\,m_W \approx 127.1 \text{ GeV}.\)

The normalization issue is also much cleaner than before. The representation part is essentially closed, and Todorov’s explicit projector formulas already force the quark-sector block to have the form

So the live gap is no longer a generic $4\times 4$ channel-matrix problem from scratch. It has collapsed to the remaining scalar equality

between the leptonic weight and the single-colour quark weight in the channel form

The extracted note contains:

• the detailed derivation chain;
• the sharpened normalization target;
• the explicit source-formula reduction using Todorov’s projector basis;
• the revised reading of the 1.5% residual as an EW-threshold issue rather than successful GUT-to-EW RG running.

Status

Reading note: the status table below should be read in light of the revised Gap 1 discussion above. In particular, the normalization step is now best treated as a formalized conditional proposition rather than a closed theorem internal to the superconnection framework.