Todorov Superconnection Note

This note collects the Todorov-specific part of the Higgs-mass story that was previously making higgs.md too large.

Core Claim

Within the $u$-framework, the key octonionic input

is no longer postulated. After choosing

one gets

So the Todorov split is internal to the program.

Using the already-derived value

the mass formula becomes

This is about $1.5\%$ above the observed Higgs mass.

What Is Imported vs Derived

Derived inside the present framework

• the 1+3 octonionic split \(\mathbb{O} = \mathbb{C}_u \oplus \mathbb{C}^3_{u^\perp};\)
• the Weinberg-angle value \(\sin^2\theta_W = \frac{3}{8};\)
• the representation-theoretic singlet-plus-triplet reading of Todorov’s lepton/quark channels.

Imported from Todorov unless the remaining metric step is closed

• the exact odd-sector superconnection normalization;
• the resulting coupling relation \(\lambda = \frac{g^2\cos^2\theta_W}{2};\)
• hence \(m_H = 2\cos\theta_W\,m_W.\)

Formalized Normalization Statement

Let

be the positive quadratic form on the Higgs doublet sector induced by the odd part of the superconnection.

Assume:

1. Before selecting a specific $u$, $\mathcal{N}$ comes from a $G_2$-invariant positive Hermitian form on the relevant octonionic data.
2. After fixing $u$, the decomposition \(\mathbb{O} = \mathbb{C}_u \oplus \mathbb{C}^3_{u^\perp}\) is orthogonal for $\mathcal{N}$.
3. The same Higgs amplitude is copied into each colour channel.

Then there is a constant $c>0$ such that

so

This is the invariant octonionic version of Todorov’s lepton-vs-single-colour comparison.

Translation to Todorov’s \rho

Todorov does not package the normalization as a raw factor of 3. Instead he compares the leptonic contribution with one single-colour quark contribution inside the restricted projector formalism.

So the dictionary is:

• octonionic full quark sector: \(\mathcal{N}_Q = 3\mathcal{N}_L;\)
• Todorov single-colour comparison: \(\mathcal{N}_{Q,j} = \mathcal{N}_L.\)

In his basis this equality is encoded by

and substituting this into his mass formula gives

So the factor 1/2 is not contradicting the octonionic factor 3; it is the basis-dependent way the same equality appears after restricting to one quark channel.

Representation Side: Essentially Closed

With Todorov’s projector convention

the colour Fock space decomposes by occupation number:

Then:

• $\ell$ is the colour vacuum sector, so it is a singlet;
• each $q_j$ lies in the two-particle / one-hole sector, hence in a three-dimensional irreducible colour module, naturally $\bar{\mathbf 3}$ in these conventions.

Using Todorov’s state identifications

\(\ell aa^\dagger = \nu, \qquad \ell a^\dagger a = e,\) \(q_j aa^\dagger = u_j, \qquad q_j a^\dagger a = d_j,\)

and the fact that the weak oscillators preserve colour occupation number, one gets:

So:

• $(\nu,e)$ are colour singlets;
• $(u_1,u_2,u_3)$ and $(d_1,d_2,d_3)$ each furnish the same irreducible three-dimensional colour representation.

The representation part is therefore no longer the bottleneck.

What Todorov’s Explicit Formulas Already Force

The primary-source formulas sharpen the remaining metric problem a lot.

In the restricted projector formalism the odd Higgs operator has the form

with

\(\Phi_\ell=\ell(\phi_1A_1^\dagger-\overline{\phi}_1A_1+\phi_2A_2^\dagger-\overline{\phi}_2A_2),\) \(\Phi_q=\rho\,q\sum_{\alpha=1}^2(\phi_\alpha a_\alpha^\dagger-\overline{\phi}_\alpha a_\alpha), \qquad q=q_1+q_2+q_3.\)

Since

and the weak oscillators live in the separate $Cl_4$ factor, one may decompose

Then distinct-channel cross terms vanish formally:

So orthogonality of different quark channels is already built into the projector algebra.

For the diagonal terms, Todorov’s explicit formula gives

hence

for each $j$.

Therefore the quark block is automatically

The leptonic diagonal entry is the only genuinely nontrivial part left. Todorov computes

while a single quark channel contributes

Equality of the leptonic contribution with one coloured-quark contribution is exactly

So the source-level metric problem is no longer “show that an arbitrary $4\times 4$ channel matrix is scalar”. The formulas already collapse it to

and the open question is the remaining scalar equality

Sharpest Remaining Theorem Target

The whole normalization program has now been reduced to:

Target. Show that Todorov’s odd quadratic form on the channel basis

is not merely

but actually

Equivalently:

1. off-diagonal channel pairings vanish;
2. the three quark diagonal entries are equal;
3. the leptonic diagonal entry equals each quark one;
4. the common value is positive.

Items 1 and 2 are now essentially forced by the explicit projector algebra. The real remaining content is item 3.

Residual 1.5% Gap

The old idea that the 127.1 GeV -> 125.2 GeV discrepancy is explained by ordinary GUT-to-EW running is not credible.

Running the boundary condition

down from a high matching scale pushes the Higgs mass far too high, not gently down to the observed value.

So the better interpretation is:

• Todorov’s formula is a tree-level EW-scale relation inside the $sl(2

  1)$ framework;

• the 1.5% mismatch should be treated as a finite electroweak threshold / one-loop correction problem;
• there is also a genuine vacuum-stability tension if one insists on reading the formula as a high-scale UV boundary condition.

The live next calculation is therefore an explicit one-loop correction to

inside the superconnection framework.

Current Status